Matrix Finite Element Methods in Dynamics

D

Mass Matrix Diagonalization

D–1

Appendix D: MASS MATRIX DIAGONALIZATION

TABLE OF CONTENTS Page

§D.1 §D.2 §D.3 §D.4

HRZ Lumping . . . . . . . . . Lobatto Mass Lumping . . . . . . Nonconforming Velocity Shape Functions Congruential Mass Transformation .

D–2

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . .

D–3 D–4 D–6 D–6

§D.1

HRZ LUMPING

The construction of the consistent mass matrix (CMM) is fully defined by the choice of kinetic energy functional and shape functions. No significant procedural deviation is possible, other than possibly using reduced integration to obtain a singular matrix. On the other hand, the construction of a diagonally lumped mass matrix (DLMM) is not so clear cut, except for simple elements in which the lumping is uniquely defined by conservation and symmetry considerations. A consequence of this ambiguity is that various methods have been proposed in the literature, ranging from heuristic through algorithmic. A good discussion of mass diagonalization schemes starting from the CMM can be found in the textbook by Cook et al. [148]. Its use in explicit DTI is well covered in [74]. This Appendix gives a quick overview of proven methods, as well as a promising but as yet untried one. §D.1. HRZ Lumping This scheme is acronymed after the authors of [363]. It produces a DLMM given the CMM. Let m e denote the total element mass. The procedure is as follows. 1.

For each coordinate direction, select the DOFs that contribute to motion in that direction. From this set, separate translational DOF and rotational DOF subsets.

2.

Add up the CMM diagonal entries pertaining to the translational DOF subset only. Call the sum S.

3.

Apportion m e to DLMM entries of both subsets on dividing the CMM diagonal entries by S.

4.

Repeat for all coordinate directions.

The see HRZ in action, consider the three-node prismatic bar with CMM given by (17.2). Only one direction (x) is involved and all DOFs are translational. Excluding the factor ρ A/30, which does not affect the results, the diagonal entries are 4, 4 and 16, which add up to S = 24. Apportion the total element mass ρ A to nodes with weights 4/S = 1/6, 4/S = 1/6 and 16/S = 2/3. The result is the DLMM (17.3). Next consider the 2-node Bernoulli-Euler plane beam element. Again only one direction (y) is involved but now there are translational and rotational freedoms. Excluding the factor ρ A/420, the diagonal entries of the CMM (?), are 156, 42 , 156 and 42 . Add the translational DOF entries: S = 156+156 = 312. Apportion the element mass ρ A to the four DOFs with weights 156/312 = 1/2, 42 /312 = 2 /78, 156/312 = 1/2 and 42 /312 = 2 /78. The result is the DLMM (18.4) with α = 1/78. The procedure is heuristic but widely used on account of three advantages: easy to explain and implement, applicable to any element as long as a CMM is available, and retaining nonnegativity. The last attribute is particularly important: it means that the DLMM is physically admissible, precluding numerical instability headaches. As a general assessment, it gives reasonable results if the element has only translational freedoms. If there are rotational freedoms the results can be poor compared to customized templates.

D–3


Table D.1.

One-Dimensional Lobatto Integration Rules

Points

Abscissas ξi ∈ [−1, 1]

Weights wi

2 3 4

−ξ1 = 1 = ξ2 −ξ1 = 1 = ξ3 , ξ2 = 0 √ −ξ1 = 1 = ξ4 , −ξ2 = 1/ 5 = ξ3

w1 = w2 = 1 w1 = w3 = 13 , w2 = 43 w1 = w4 = 16 , w2 = w3 =

5 6

Common names: Trapezoidal rule and Simpson’s rule for p = √ 2, 3, respectively. In the p = 4 rule, interior points are not thirdpoints, since 1/ 5 ≈ 0.447213596 = 13 . Lobatto rules with 5 ≤ p ≤ 10, rarely important in FEM work, are tabulated in [2, Table 25.6]. Table D.2. Points

Abscissas ξi ∈ [−1, 1]

2 3 4 5

One-Dimensional Newton-Cotes Integration Rules Weights wi

Same as 2-point Lobatto; see Table D.1 Same as 3-point Lobatto; see Table D.1 −ξ1 = 1 = ξ4 , −ξ2 = 1/3 = ξ3 −ξ1 = 1 = ξ5 , −ξ2 = 1/2 = ξ4 , ξ3 = 0

w1 = w4 = 14 , w2 = w3 = 34 7 w1 = w5 = 90 , w2 = w3 = 32 , w3 = 90

12 90

Common names for p = 4, 5: Simpson’s 3/8 rule and Boole’s rule, respectively. Additional NC formulas may be found in [2, Table 25.4]. For p > 5 they have negative weights.

§D.2. Lobatto Mass Lumping A DLMM with n eF diagonal entries m i is formally equivalent to a numerical integration formula with n eF points for the element kinetic energy: Te =

n e

F

i=1

m i Ti ,

where

Ti = 12 u˙ i2

(D.1)

Assume the element is one-dimensional (1D), possesses only translational DOF, and that its geometry is described by the natural coordinate ξ that varies from −1 through 1 at the end nodes. Then (D.1) can be placed in correspondence with the so-called Lobatto quadrature in numerical analysis. (Also called Radau quadrature by some authors, e.g. [130]; however the handbook [2, p. 888] says that Lobatto and Radau rules are slightly different.) A Lobatto rule is a 1D Gaussian quadrature formula in which the endpoints of the interval ξ ∈ [−1, 1] are sample points. If the formula has p ≥ 2 abscissas, only p − 2 of those are free. Abscissas are symmetric about the origin ξ = 0 and all weights are positive. The general form is

1 −1

f (ξ ) dξ = w1 f (−1) + w p f (1) +

p−1

wi f (ξi ).

(D.2)

i=2

The rules for p = 2, 3, 4 are collected in Table D.1. Comparing (D.1) with (D.2) clearly indicates that if the nodes of a 1D element are placed at the Lobatto abscissas, the diagonal masses m i are simply the weights wi . This correspondence was first observed in [285], and further explored in [463,464]. For the type of elements noted, the equivalence works well for p = 2, 3. For p = 4 a minor difficulty arises: the interior Lobatto points are not at the thirdpoints, as can be seen in D–4

§D.2 (a)

(b)

LOBATTO MASS LUMPING

(c)

Figure D.1. A pair of degree-2, 3-point Gauss quadrature rules for the six-node plane stress triangle with constant metric: (a) node configuration; (b) the 3-interior point rule; (c) the 3-midpoint rule, which is a Lobatto rule for this node configuration. All weights are 1/3. Lines within triangle mark triangular natural coordinates (a.k.a. barycentric coordinates) of constant value, to illustrate constant metric.

(a)

(b)

(c)

Figure D.2. As in Figure D.1, but triangle has now curved sides and variable metric.

Table D.1. If the element nodes are collocated there, one must switch to the “Simpson 3/8 rule”, which is a Newton-Cotes formula listed in Table D.2. and adjust diagonal masses accordingly. As a generalization to multiple dimensions, for conciseness we call FEM Lobatto quadrature one in which the DOF-endowed element nodes are sample points of an integration rule. (Sample points at other than nodal locations are precluded.) If so, the equivalence with (D.1) still holds. But one quickly runs into difficulties: Negative Masses. If one insists in higher order accuracy, weights of 2D and 3D Lobatto rules are not necessarily positive, a feature noted in [203]. The subject is studied in detail in [285]. This shortcoming can be alleviated, however, by accepting lower accuracy, or sticking to product rules in suitable geometries. For example, applying a product 1D Lobatto rule over each side of a triangle or quadrilateral. Of course a more flexible alternative is provided by templates, because these allow the stiffness to be concurrently adjusted. Rotational Freedoms. If the element has rotational DOF, Lobatto rules do not exist. Any attempt to extend (D.2) to node rotations inevitably leads to translation-rotation coupling. Varying Properties. If the element is nonhomogeneous or has varying properties (for instance, a tapered bar element, or a plate of variable thickness) the construction of accurate Lobatto rules runs into additional difficulties, for the problem effectively becomes the construction of a quadrature formula with non-unity kernel. As a general assessment, Lobatto mass lumping is useful when the diagonalization problem happens to fit a Gaussian quadrature rule with element nodes as sample points and nonnegative weights. Formulas of that type were developed for multidimensional domains of simple geometry during the 1950s and 60s. They are can be found in handbooks such as [706,707], along with many other rules. D–5


As noted above, an obvious hindrance is the emergence of negative weights as the rule degree gets higher. This feature excludes those from contention except under extreme caution, whereas zero weights are less deadly. Rules useful for FEM work are compiled in [244], as well as Appendix I of [253], for seven element geometries. The six-node plane stress triangle, shown in Figure D.1(a), illuminates obstacles typically encountered in multiple space dimensions. The total element mass is m e = ρ A h, in which A denotes the plane area and h the plate thickness, assumed uniform. There are two 3-point Gauss quadrature rules of degree 2 for a constant metric triangle, shown in Figure D.1(b,c), which is extracted from [255]. The midpoint rule, illustrated in Figure D.1(c), is also a Lobatto rule for this element, but the 3-interior-point rule pictured in Figure D.1(b) is not. Using the midpoint rule to build the DLMM results in three masses of m e /3 collocated at the midpoints, while all corner masses vanish [203]. The HRZ scheme leads to the same result. This DLMM has rank 6 and rank deficiency 6. To attain full rank one must take some mass from the midpoints and move it to the corners: not a well defined process. An heuristic way out would be to apply the Simpson rule line lumping along the three edges. This results in m e /9 at corners and 2m e /9 at midpoints but the degree drops to 1. To retain accuracy, a simultaneous change of the stiffness matrix could be tried within the template framework. For a curved-side six-node triangle with variable metric, a case illustrated in Figure D.2, node and sample points remain at the same location in terms of natural coordinates, but local Jacobian determinants enter the formula. §D.3. Nonconforming Velocity Shape Functions This is a variational technique based on assuming velocity shape functions (VSF) that differ from the usual displacement shape functions (DSF). To produce a diagonal mass matrix, the VSF must satisfy additional “mass orthogonality” conditions that effectively decouples each VSF with respect to all others in the kinetic energy integral. This can be practically realized by making each VSF vanish at all points of a Gauss integration rule except one. Which rule? That appropriate to the correct integration of the kinetic energy over the element. Rather than explaining the technique further, the interested reader may want to study the examples provided in Appendix V. §D.4. Congruential Mass Transformation A congruential mass transformation, or CMT, is a general framework than can be applied to transform a given source mass matrix into a target one. In particular all model reduction techniques mentioned in §H.3. Here it is specialized to the following case of importance in diagonalization: (i)

The source mass matrix M S is nondiagonal and positive definite (PD); for example a CMM.

(ii) The target mass matrix MT is diagonal and nonnegative (that is, zero diagonal entries are permitted) Both matrices have order n D O F . The congruential transformation that converts source to target is MT = HT M S H. D–6

(D.3)

§D.4

CONGRUENTIAL MASS TRANSFORMATION

If H is nonsingular, the inverse mapping is M S = GT MT G, in which G = H−1 . We will say that M S and MT are congruentially linked through H. Even if M S and MT are both given and parameter-free, there are generally many H matrices that satisfy (D.3). In fact the number of solutions typically grows exponentially with n D O F . One particular form, however, is unique under conditions (i)-(ii). Perform the Cholesky factorization M S = L S LTS = L S U S , where L S is lower triangular and U S = LTS is upper triangular. If M S is PD, 1/2 this factorization is unique and both L S and U S are nonsingular [797]. Let MT be the principal square root of MT , obtained by taking the positive square root of each diagonal entry. By inspection 1/2

H = U S MT ,

HT = MT L−1 S . 1/2

(D.4)

This will be called the Cholesky form of H, and identified by subscript ‘CF’ if necessary. Since the inverse of a nonsingular upper triangular matrix is also lower triangular, and scaling by the 1/2 diagonal matrix MT does not alter that configuration, HT and H are lower and upper triangular, respectively. As an example, the CMM and DLMM of the two-node prismatic bar given in (?) are linked by the Cholesky form √ 1 3 −1 1.22474 −0.707107 = . (D.5) HC F = √ 2 0 1.41421 2 0 For the CMM and DLMM of the three-node prismatic bar studied in ?, the Cholesky form is √ √ √ 1.180303 0.288675 −0.816497 5/2 1/(2√ 3) −√2/3 (D.6) HC F = 0 1.154701 −0.816497 . −√ 2/3 = 0 2/ 3 3/2 0 0 1.224745 0 0 The Cholesky form of (D.3) is unique and easy to obtain, but does not link naturally to the algebraic Riccati equation mentioned below. For that purpose finding a symmetric H is more convenient. Those will be identified by subscript ’Sy’ if necessary. Symmetric forms are not unique; in fact typically one generally finds 2n D O F different solutions. It is rather easy, however, to extract a principal solution. For the two-node prismatic bar, the transformation (D.3) from CMM to DLMM with symmetric H has 22 = 4 solutions. The only one with positive eigenvalues is √ √ 1.366025 −0.366025 1 1 + √3 1 − √ 3 H Sy = 2 = . (D.7) −0.366025 1.366025 1− 3 1+ 3 For the 3-node pristamic bar, one gets 23 = 8 solutions. The only one with all eigenvalues positive is (only given numerically, as its analytical expression is complicated): 1.2051889 0.2051889 −0.1472036 (D.8) H Sy = 0.2051889 1.2051889 −0.1472036 . −0.1472036 −0.1472036 1.1518024 The determination of H in (D.3) is related to the quadratic matrix equation XT AX = B, where M S → A and MT → B are data and H → X the unknown. If H → X is symmetric so XT = X, the equation X AX = B is a specialization of the algebraic Riccati equation extensively studied in optimal control systems [7,433]. Hopefully this interdisciplinary resource could be eventually be applied to devise robust mass diagonalization schemes using a matrix function library [359]. But as of now, templates remain the most practical method to find optimal diagonalizations. D–7

H

A Short History of Mass Matrices

H–1

Appendix H: A SHORT HISTORY OF MASS MATRICES


§H.1 §H.2 §H.3 §H.4 §H.5 §H.6 §H.7 §H.8 §H.9 §H.10

Pre-FEM Work . . . . . . Consistent Mass Matrices Appear Dynamic Model Reduction . . Selective Mass Scaling . . . . Singular Mass Matrices . . . Frequency Dependent Matrices . Templates . . . . . . . . Multidimensional Elements . . Connection To Molecular Physics Conclusions and Future Work .

. . . . . . . . . . . . . .

H–2

. . . . .

. . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . . . . .

H–3 H–3 H–4 H–4 H–5 H–5 H–6 H–6 H–7 H–7

§H.2

CONSISTENT MASS MATRICES APPEAR

This Appendix summarizes previous developments on various topics addressed by this paper. §H.1. Pre-FEM Work The first appearance of a structural mass matrix in a journal article occurs in two papers by Duncan and Collar [190,191], which appeared in the mid 1930s. The authors were members of the world famous aeroelasticity team at the National Physics Laboratory in Teddington (UK), led by Frazier. As narrated in [238] those two papers represent the birth of Matrix Structural Analysis (MSA). Befitting the overdesigned aircraft structures of the time, the focus is on dynamics and vibrations rather than on statics. In [190] the mass matrix is called “inertia matrix” and denoted by [m]. The first example [190, p. 869] displays the 3 × 3 diagonal mass of a triple pendulum. In the 1938 book [282], which collects that early work plus intermediate papers, the notation changes to A. Diagonally lumped mass matrices (DLMM) were strongly preferred in early publications. In fact they dominate all pre-1963 work. Three reasons may be aduced for the preference: •

The three-century astronomical heritage of Newtonian Mechanics. For unperturbed orbit calculations, celestial bodies were idealized as point masses regardless of actual size.

•

Computational simplicity in vibration analysis and explicit direct time integartion (DTI).

•

Direct lumping gives an obvious way to account for nonstructural masses in simple discrete models of the spring-dashpot-point-mass variety. For example, in a multistory building “stick model” wherein each floor is treated as one DOF in lateral sway under earthquake or wind action, it is natural to take the entire mass of the floor (including furniture, insulation, etc.) and assign it to that freedom.

Nondiagonal (but not consistent) masses pop up ocassionally in pre-1960 aircraft matrix analysis — e.g. wing oscillations in [282,§10.11] — as a result of measurements. As such they necessarily account for nonstructural masses due to fuel, avionics equipment, etc. §H.2. Consistent Mass Matrices Appear The formulation of the consistent mass matrix (CMM) in structural mechanics by Archer [34,35] was a major advance. Most of the CMM derived in Chapters 17–19 appear in those papers. The underlying idea, however, is older. In fact, Irons and Ahmad observe [397] that consistent masses had been used in acoustics for over two decades before Archer’s papers; see e.g., the textbook [808]. The CMM at the system (master) level follow directly from the Lagrange dynamic equations established in the late XVIII Century [430]. If T is the kinetic energy of a FEM-discretized structural system occupying volume V , and u˙ i (xi ) the velocity field defined by the nodal velocities ˙ the master CMM is simply the Hessian of T with respect to nodal velocities: collected in u, ∂2T 1 ˙ . (H.1) ρ u˙ i u˙ i d V, u i = u i (u), M= T =2 ∂ u˙ ∂ u˙ V ˙ as happens in linear structural dynamics. Two key This matrix is constant if T is quadratic in u, decisions had to be reached, however, before this idea was applicable to FEM. Localization: (H.1) is applied element by element, and the master M assembled through the standard steps of the Direct Stiffness Method (DSM). H–3


Consistent Interpolation: the interpolation of velocities and displacements is made identical. That is, velocity shape functions (VSF) and displacement shape functions (DSF) coincide. These in turn had to wait until three major ingredients became slowly established during the 1960s: (i) the DSM of Turner [759,761], (ii) the concept of shape functions progressively evolving in early FEM publications [197,203,393,394,484], and (iii) the FEM connection to Rayleigh-Ritz. The last one was critical. It was established in Melosh’s thesis work [484,485]. The link to dynamics was closed with Archer’s contributions, and CMM became a staple of FEM. But only a loose staple. Problems persisted: (a) Nonstructural masses are not naturally handled by CMM. In vehicle systems such as ships or aircraft, the structural mass is only a fraction (10 to 20%) of the total. (b) It is inefficient in explicit DTI, since M is never diagonal. (c) It may not give the best results compared to other alternatives. Why? If K results from a conforming displacement interpolation, pairing it with the CMM is a form of the conventional Rayleigh-Ritz, and thus guaranteed to provide upper bounds on natural frequencies. This is not necessarily a good thing. In practice it is observed that errors increase rapidly as one moves up the frequency spectrum. If the response is strongly influenced by intermediate and high frequencies, as in contact-impact and wave propagation dynamics, the CMM may give poor results. (d) For elements derived outside the assumed-displacement framework, the displacement shape functions may be either unknown or altogether missing. Problem (a) can be addressed by “rigid mass elements” accounting for inertia (and possibly gravity or centrifugal forces) but no stiffness. Nodes of these elements are linked to structural (elastic) nodes by multifreedom kinematic constraints. This is more of an implementation issue than a research topic, although numerical difficulties typical of multibody dynamics may crop up. Problems (b,c,d) can be attacked by parametrization. MacNeal was the first to observe [20,452,461] that averaging the DLMM and CMM of Bar2 produced better results than using either alone. This idea was further studied by Belytschko and Mullen [71] using Fourier analysis; they also studied the CMM and SLMM of Bar3 but not their parametrizations. Krieg and Key [422] had emphasized that in transient analysis by DTI the introduction of a time discretization operator brings new compensation phenomena, and consequently the time integrator and the mass matrix should not be chosen separately. The template approach addresses this problem by allowing and encouraging customizing of the mass and stiffness to the problem at hand. §H.3. Dynamic Model Reduction Concurrently with advances in variational mass lumping leading to the CMM, the 1960s and early 1970 were a fertile time for the development of reduced order dynamic models based on Component Mode Synthesis (CMS), a name coined by Hurty [386,387]. These were motivated by the high cost of dynamic and vibration analysis in the computers of the time, and blended well with the emerging use of substructuring methods in aerospace engineering, summarized in [596]. Seminal publications of the period include [157,454,649], which originated the widely used CraigBampton and MacNeal-Rubin CMS methods. There is abundant literature since. Good textbooklevel descriptions are provided in [158,305]. H–4

§H.6

FREQUENCY DEPENDENT MATRICES

§H.4. Selective Mass Scaling The Selective Mass Scaling (SMS) method, proposed in the mid 2000s [522,523], has attracted attention for rapid-transient simulations involving contact-impact, such as vehicle crash or explosions. Those are typically treated by explicit direct time integration (DTI), in which ephemeral high frequencies produced by transient shocks may require extremely small timesteps for stability, as well as producing significant spurious noise (pollution). In this approach, a diagonally lumped mass matrix (DLMM) is augmented by a scaled version of the stiffness matrix. The underlyting idea is to knock down the high frequency of the “mesh modes,” as quantitatively shown in ? The derivation of the SMS variant for Bar3 in ? shows that for the three-node bar, SMS can be presented as a subset of the general mass template. Nevertheless, it deserves consideration on its own because of two attractive features: •

It involves only one free parameter, which may be adjusted during the response simulation process. This makes it especially suitable for multidimensional elements.

•

It does not depends on knowledge of element shape functions. In fact it may be used without knowing the source of the FEM matrices, an attractive feature for some commercial codes.

These are counteracted, however, by two disadvantages: •

Loss of low-frequency accuracy in the acoustic branch. Consequently SMS is not recommended for conventional structural dynamics and vibration analysis.

•

Adding the stiffness term necessarily makes M nondiagonal, complicating explicit DTI. If the stiffness contribution is relatively small, however, DLMM diagonal dominance might be retained, which permits the use of iterative schemes [547].

The method has been further explored in several recent publications, some of which focus on the use of singular mass matrices as well as connection with parametrized variational principles (PVP) [413,627,740,741]. The variational connection outlined in Appendix V, in which VSF and DSF are independent, has the potential to link to that recent line of research as well as earlier work cited there. §H.5. Singular Mass Matrices This approach to RHFP has been primarily developed with an applied mathematics flavor and with multibody dynamics as focus [834,762,763,764,765]. The key idea is to get the optical branch (or branches) out of the way at low frequencies to increase the acoustoptical gap. It can be readily subsumed under templates through spectral parametrization, as worked out for the Bar3 element in §23.1.11. While the BSSM instance constructed there shows promise in meeting both LFCF and RHFP customization goals, more numerical experimentation will be required to substantiate that promise.

§H.6. Frequency Dependent Matrices Making mass and stiffness frequency dependent (FD) was proposed by Przemieniecki [596], who expanded both Me (ω) and Ke (ω) as Taylor series in ω2 . An indirect derivation scheme, which preceeds [596], consists of starting from dynamic transfer matrices and convert them to mass and H–5


stiffness by partial inversion. The procedure is described in detail in [572]. It is restricted to 1D elements. The idea was subsequently pursued by other authors. Pilkey [581,582] derived such matrices by using exact solutions of the unforced EOM as shape functions. For the two-node bar, those matrices are not instances of the FDMS template presented in ? because the baseline mass matrix for zero frequency is the CMM. Since such exact solutions are available only for a limited number of 1D models, the approach is hardly extendible beyond prismatic bars and beams. More general 1D elements have been handled by numerical ODE integration [438,838] on the way to transfer matrices. The approach can be generalized to the template context by making free parameters frequency dependent, as illustrated in §22.1.12 for Bar2. As noted there, this extension might be of interest for problems dominated by a driving frequency, such as some electronic and optical devices. For more general use, keeping the parameters frequency independent is far more practical. In multiple dimensions it merges with boundary integral and spectral element methods in elastodynamics. These are specialized topics beyond the scope of this historical review; for recent work and references, see [438,683,686,806]. §H.7. Templates The template approach originally evolved in the late 1980s and early 1990s to construct highperformance stiffness matrices [228,229]. Its roots can be traced back to the Free Formulation of Bergan [80,82,86,87,216], in which the stiffness matrix was decomposed into basic and higher order components. A historical account is provided in a tutorial chapter [243]. The general concept of template as parametrized forms of FEM matrix equations is discussed in [235,241,246]. Mass templates in the form presented here were first described in [236,237] for a Bernoulli-Euler plane beam analyzed with Fourier methods. The study addressed mass-stiffness (MS) pairs. The idea was extended to other elements in [248]. The Bar3 stiffness template (23.4) was first stated in [228], in which the only free parameter has a slightly different definition. The Timoshenko beam model first appeared in [735]; see also [738]. The symbolic derivation scheme used for the Timoshenko beam EOM in (24.14) is due to Flaggs [263]; see also [551,553,554]. The optimal static stiffness matrix (24.22) for the Ti-beam element appeared originally in [758]. A detailed derivation may be found in [596,§5.6]. It is an instance of a template given in [245]. Two powerful customization techniques used regularly for templates are Fourier analysis and modified differential equations (MoDE). Fourier analysis are limited to separable systems but are straightforward to apply, requiring only undergraduate mathematics. (As tutorials for applied Fourier analysis, Hamming’s textbooks [335,334] are highly recommended.) MoDE methods, first published in correct form in 1974 [783] are less restrictive but more demanding on two fronts: mathematical ability and support of a computer algebra system (CAS). Processing power limitations presently restrict MoDE to two-dimensional elements and regular meshes. In the present exposition, only Fourier methods are used since those are likely to be more familiar to potential readers. The selection of a priori constraint criteria for template free parameters is not yet on firm ground. For example: is conservation of angular momentum useful in mass templates? The answer seem to depend on element type and complexity. H–6

§H.10

CONCLUSIONS AND FUTURE WORK

§H.8. Multidimensional Elements . Stiffness templates for 2D and 3D structural elements have been considered since a modest beginning in the 1980s, as Free Formulation plate elements that incorporated a scaling parameter for the higher order stiffness [87,216]. A significant number has been developed since; see references in §H.7. The development of mass matrix templates for multidimensional elements has lagged because of four complications: •

Mesh directionality effects that require angular averaging

•

Additional dependence on elastic material properties

•

Multiple plane wave types (pressure and shear waves in the case of an isotropic material)

•

The free parameter explosion as matrices get larger

The first study of this nature for a nontrivial 2D element has appeared as a 2012 thesis [320], which dealt with triangular membrane elements with and without corner node drilling freedoms. An example extracted from this thesis is presented in §25.2. §H.9. Connection To Molecular Physics DLMM results for regular lattices of structural elements have counterparts in a very different area: molecular physics. More precisely, the wave mechanics of crystalline solids created in the XX Century by particle mechanicians; e.g., [101,601,832]. In crystal models, lattice nodes are occupied by molecules interacting with adjacent ones. Thus the “element dimension” acquires the physical meaning of molecular gap. Both acoustic and optical branches have physical significance. In such models masses are always lumped at molecule locations, and atoms vibrate as harmonic oscillators in the potential well of the force fields of their neighbors. Dispersion curves govern energy transmission. In a linear atomic chain, the dimensionless wavenumber range κ ∈ [−π, π], or κ ∈ [0, 2π], is called the first Brillouin zone [108,402]. The happy connection of mass templates to periodic materials may be of interest as FEM and related discretization methods are extended into multiscale applications of crystal, micro- and nano-mechanics, and phononics [45,186]. §H.10. Conclusions and Future Work It is clear from the material presented in Chapters 21ff that mass matrix customization by templates can be effective in structural dynamics. The examples of §22.1.9, §23.1.14 and §24.2.8 illustrates two typical advantages: •

Orders of magnitude improvements in frequency accuracy can be achieved for the same computational effort.

•

The space discretization need not be changed at all. Only the template free parameters need to be adjusted by supplying the appropiate signature.

These should be attractive to engineers for practical FEM computations. The last one is particularly important, since redoing a structural dynamics model not amenable to mesh generation may take a significant portion of a design and analysis process. Would availability of customized templates eliminate the need for h and p adaptivity? Certainly not. Elements have performance limits, so such refinement schemes cannot be ruled out. It should H–7


be noted, however, that mesh adaptivity is less effective in dynamics, particularly in problems with rapid transients and shocks. Irregular meshes and high order elements are notorious sources of HF pollution, and adaptivity can make things worse by exacerbating nonphysical dispersion. As regards future work, the most ambitious plan is extension to multiple space dimensions. Additional challenges emerge there: •

Directionality. This means that the dynamic accuracy of the FEM model, as compared to the continuum, depends on the direction of plane wave propagation. This is a phenomenon missing in 1D. Integration averaging over propagation angles necessarily appears as another component of the Fourier analysis.

•

Material property dependence. Optimal free parameters become dependent on additional elastic material properties missing from the 1D treatment. For example if the material is isotropic Poisson’s ratio appears. Actually this is not unique to mass templates, but affects stiffness templates as well.

•

Multiple plane wave types. In isotropic 2D and 3D continua, one needs to consider pressure (P-waves) and shear (S-waves). Plainly this impacts customization. If the medium is nonisotropic, more complicated wave types may have to be considered.

•

Parameter explosion. This can be expected to hinder symbolic calculations. At first sight it seems inevitable given the rapidly increasing size of the element matrices. Growth could be controlled, however, by making use of a priori reduction techniques such as those mentioned in §21.3.1, §21.3.2, and §21.3.4.

The first three additional ingredients are illustrated for a simple 2D element (three-node linear triangle) in §25.2. The number of parameters is kept to one therein by assuming a linear combination of the CMM and DLMM. Less ambitious research thrusts may focus on extending the results presented here for 1D elements. For example: •

Assess the performance of different template variant construction approaches to reduce highfrequency pollution caused in direct time integration. Three were described for the Bar3 element: singular mass, selective mass scaling and constant optical branch. As of this writing, no comparison with numerical experiments is available.

•

Find out whether the impressive gains in accuracy observed in LFF-customized templates survive in irregular meshes and/or heterogeneous element mixtures.

•

Unfinished business remains for the Timoshenko beam. First, the unexpected deterioration in vibration accuracy as the coefficients Φ and Ψ increase is presently unexplained. Could the deterioration be arrested using MS template pairs? Prior experience with the BE-beam, referenced in §H.7, shows only modest improvements, but such continuum model is comparatively well behaved. Second, the relative performnace of the various template instances listed in Tables 24.1 and 24.2 for direct time integration (DTI) remains to be assessed.

H–8

I

Implementation of Bar3 Template in Mathematica

I–1

Appendix I: IMPLEMENTATION OF BAR3 TEMPLATE IN MATHEMATICA


§I.1

§I.2 §I.3

§I.4

Auxiliary Modules . . . . . . . . . . . . . . . . . . . §I.1.1 Name To Signature Mapper . . . . . . . . . . . . §I.1.2 Mass Template Variant Parameter Mapper . . . . . . . . Element Level Modules . . . . . . . . . . . . . . . . . §I.2.1 Element Mass and Stiffness Modules . . . . . . . . . Assembly Level Modules . . . . . . . . . . . . . . . . §I.3.1 Lattice Master Mass and Stiffness Modules . . . . . . . §I.3.2 Lattice Patch Modules . . . . . . . . . . . . . . Dispersion Analysis and Display Modules . . . . . . . . . . . §I.4.1 Characteristic Equation Module . . . . . . . . . . . §I.4.2 Dispersion Branches And Taylor Series . . . . . . . . . §I.4.3 Dispersion Diagram Plotting . . . . . . . . . . . .

I–2

I–3 I–3 I–4 I–5 I–5 I–6 I–7 I–8 I–8 I–8 I–9 I–11

§I.1

AUXILIARY MODULES

This Appendix presents the computer implementation of the mass-stiffness template pair for the three-node bar element, abbreviated to Bar3. It is written in the Mathematica language. Although the element is admittedly simple it is not trivial. In fact the implementation illustrates the use of template variants to simplify customization. Why Mathematica?. As observed in the Introduction, use of a CAS is essential for template development because analytical derivations soon exceed human endurance. Once the development phase is completed, a production version in a compiled language can be easily produced. But the CAS version should not be discarded.

Bar3CharFreq Forms the characteristic equation for plane wave propagation over a regular Bar3 patch, and returns characteristic frequencies

inlined from

Bar3Dispersion Given a template, finds its characteristic frequencies (using inlined results from Bar3CharFreq) and their Taylor series about κ=0

Bar3DispersionPlot Given a template instance, plot its DDD and/or DGVD over a specified κ range. Several display options available

Bar3MassPatch Extracts the 2 x 5 patch equation mass matrix from a 2-element lattice patch for subsequent Fourier analysis

Bar3MassLattice Assembles the master mass matrix of a regular, homogeneous,Bar3 FEM-discretized lattice with arbitrary # of elements

Bar3StiffLattice Assembles the master stiffness matrix of a regular, homogeneous, Bar3 FEM-discretized lattice with arbitrary # of elements

Bar3StiffPatch Extracts the 2 x 5 patch equation stiffness matrix from a 2-element lattice patch for subsequent Fourier analysis

Bar3TempSignature Given a template instance mnemonic name (such as CMM for Consistent Mass Matrix), returns template variant id and signature

Bar3ElemMassTemp Returns mass matrix of Bar3 element as template instance. Six template variants implemented: GEN through COBB

Bar3ElemStiffTemp Returns stiffness matrix of Bar3 element as template instance. Same form for all template variants

Bar3MassVarParMap Given a target mass template variant, return a rule to replace the general template parameters by those of the variant

Figure I.1. Organization of Bar3 template analysis modules presented in this Appendix.

The hierarchical organization of the modules presented in this Appendix is shown in Figure I.1. The bottom-up description that follows starts from the lowest level of that chart, going up and traversing against the arrows. §I.1. Auxiliary Modules The two outside modules at the lowest level of the chart of Figure I.1 provide auxiliary services to modules at all levels. §I.1.1. Name To Signature Mapper Auxiliary module Bar3TempSignature, listed at the top of Figure I.2, maps an abbreviated template instance name to its full signature definition. It is invoked by tsign=Bar3TempSignature[name]

(I.1)

The only argument is name: a character string of 3 or 4 letters that abbreviates a template instance. Examples: "CMM" for the consistent mass matrix or "SLMM" for Simpson-lumped mass matrix. I–3


Table I.1. Bar3 Template Signature List Specification Template variant

Ref. eqn.

# of Signature format, identified as pars tsign in Mathematica code

Mass conservation constraint∗

General, µi pars

(23.2)

5

{ "GEN",{ β },{ µ1 , µ2 , µ3 , µ4 } } 2µ1 +µ2 +2µ3 +4µ4 = 0

General, χi pars

(23.7)

4

{ "GEX",{ β },{ χ1 , χ2 , χ3 } }

preimposed

Lumped

(23.26)

3

{ "LUM",{ β },{ µ L1 , µ L2 } }

2µ L1 +µ L2 = 0

Spectral

(23.34)

3

{ "SPE",{ β },{ µ S1 , µ S2 } }

preimposed

Selective Mass Scaling

(23.44)

4

{ "SMS",{ β },{ µ L1 , µ L2 , µ K } }

2µ L1 +µ L2 = 0

1

{ "COBA",{ 1 },{ ν A } } or { "COBB",{ 1 },{ ν B } }

preimposed

Constant Optical (23.47,23.49) Branch† ∗

When doing symbolic work, the mass conservation constraint is not always preimposed in some template variants, as that may complicate intermediate expressions † For this variant, two families: COBA and COBB, are implemented. Cf. §23.1.13

Names currently implemented can be gathered by examining the code. Some of these are also listed in Table 23.1. The function returns tsign as template signature. This is a list of the form { tvar,{ kpars },{ mpars } }

(I.2)

Here tvar is a character string that identifies template variants, kpars a list of stiffness parameters, and mpars a list of mass parameters. Configuration details for this data structure are given in Table I.1. If name is not recognized, a warning message is printed and "CMM" is assumed. Example: Bar3TempSignature["SLMM"] returns { "LUM",{ 1 },{ 0,0 } } as function value. §I.1.2. Mass Template Variant Parameter Mapper Auxiliary module Bar3MasVarParMap, listed at the bottom of Figure I.2, returns a replacement rule that maps the four µ parameters of the Bar3 general mass template (23.2) to those of a variant. The latter is called the target form. The rule is used to specialize results such as dispersion equations; cf. the link drawn in Figure I.1. It is invoked by rule=Bar3MasVarParMap[gmpars,tsign]

(I.3)

The arguments are: gmpars

A list of symbols used for the free parameters of of the Bar3 mass template (23.2). Normally the parameters are labeled { µ1 ,µ2 ,µ3 ,µ4 }. Those symbols will appear in the left side of the replacement rule.

tsign

Signature of the target form.

Items in gmpars must be individual symbols, while those in tsign may be symbolic or numeric (see examples below). Note that β in tsign is used if the target pertains to the SMS variant. I–4

§I.2

ELEMENT LEVEL MODULES

Bar3TempSignature[name_]:=Module[{a1,a2,β,µL1,µS1,µS2,µK, modname="Bar3TempSignature"}, If [name=="CMM", Return[{"GEN",{1},{0,0,0,0}}]]; If [name=="SLMM", Return[{"LUM",{1},{0,0}}]]; If [name=="BLCD", Return[{"GEN",{1},{2,8,2,-4}/3}]]; If [name=="BLFM", a1=375^(1/4); a2=Sqrt[15]; Return[{"GEN",{1},{91-12*a1-7*a2,64-16*a2, 61-12*a1-a2,-92+12*a1+8*a2}/6}]]; If [name=="BLFD", β=3/(4*(Sqrt[3]-1)); µL1=5*(2-Sqrt[3]); Return[{"LUM",{β},{µL1,-2*µL1}}]]; If [name=="BSSM", β=(5+Sqrt[10])/12; µS1=3*(5+Sqrt[10])/2; Return[{"SPE",{β},{µS1,0}}]]; If [name=="SMS1", µK=1/24; β=1/(1-12*µK); Return[{"SMS",{β},{0,0,µK}}]]; If [name=="SMS2", µK=1/2; β=1; Return[{"SMS",{β},{0,0,µK}}]]; If [name=="SMS3", µK=2; β=1; Return[{"SMS",{β},{0,0,µK}}]]; If [name=="COB0", Return[{"COBA",{1},{-5/3}}]]; If [name=="COB1", Return[{"COBA",{1},{0}}]]; If [name=="COB2", Return[{"COBB",{1},{0}}]]; Print[modname,": illegal template name ",name," CMM assumed"]; Return[{"GEN",{0},{0,0,0,0}}]]; Bar3MassVarParMap[gmpars_,tsign_]:=Module[{tvar,β,mpars,µ1,µ2,µ3,µ4, ν1,ν2,ν3,ν4,χ1,χ2,χ3,µL1,µL2,µS1,µS2,µK,νA,νB,s,d,r,kw,rep={}}, {µ1,µ2,µ3,µ4}=gmpars; {tvar,{β},mpars}=tsign; kw=ToUpperCase[tvar]; If [kw=="GEN", {ν1,ν2,ν3,ν4}=mpars; rep={µ1->ν1,µ2->ν2,µ3->ν3,µ4->ν4}]; If [kw=="GEX", {χ1,χ2,χ3}=mpars; r=Sqrt[30]*Sqrt[χ1-χ3]-2*χ1; s=χ1+χ2; d=χ1-χ2; rep={µ1->s-4,µ2->14-4*χ1-4*r,µ3->d+1,µ4->r-2}]; If [kw=="LUM", {µL1,µL2}=mpars; rep={µ1->µL1+1,µ2->µL2+4,µ3->1,µ4->-2}]; If [kw=="SPE", {µS1,µS2}=mpars; rep={µ1->(µS1+µS2-2)/3,µ2->(4*µS2-38)/3, µ3->(13-µS1+µS2)/3,µ4->(4-2*µS2)/3}]; If [kw=="SMS", {µL1,µL2,µK}=mpars; rep={µ1->1+µL1+10*(4*β+3)*µK, µ2->4+µL2+160*β*µK,µ3->1+10*(4*β-3)*µK,µ4->-2-80*β*µK}]; If [kw=="COBA",{νA}=mpars; rep={µ1->11-5*νA/2,µ2->-2*(3+5*νA), µ3->6-5*νA/2,µ4->-7+5*νA}]; If [kw=="COBB",{νB}=mpars; rep={µ1->8/3-5*νB/2-5*νB^2/72,µ2->32/3, µ3->8/3-5*νB/6+(5*νB^2)/72,µ4->(5*νB-16)/3}]; Return[rep]];

Figure I.2. Two auxiliary modules. Bar3TempSignature maps a template name to its signature. Bar3MassVarParMap returns a replacement rule that maps the general mass template to a target variant.

The function returns rule

Replacement rule. If the target is not recognized, the empty list { } is returned.

Example 1. Let gmpars={ µ1 ,µ2 ,µ3 ,µ4 } and tsign={ "LUM",{ β },{ µ L1 ,µ L2 } }. The module call Bar3MasVarParMap[tsign,gmpars] returns { µ1 ->µ L1 +1,µ2 ->µ L2 +4,µ3 ->1,µ4 ->-2 }, a rule that maps the general template (23.2) to the lumped mass variant as per (23.25). Example 2. Same gmpars as above but now tsign={ "LUM",{ β },{ 0,0 } }. The call returns { µ1 ->1,µ2 ->4,µ3 ->1,µ4 ->-2 }, a replacement rule that produces the SLMM instance. §I.2. Element Level Modules This section describes element modules that return mass and stiffness templates. §I.2.1. Element Mass and Stiffness Modules The Bar3 element mass and stiffness template modules are called Bar3ElemMassTemp and Bar3ElemStiffTemp, respectively. They are listed in Figure I.3. The call sequences are Me=Bar3ElemMassTemp[Le,rho,A,tsign,numer] I–5

(I.4)

Appendix I: IMPLEMENTATION OF BAR3 TEMPLATE IN MATHEMATICA Bar3MassTemp[Le_,ρ_,A_,tsign_,numer_]:=Module[ {me=ρ*A*Le,µ1,µ2,µ3,µ4,µL1,µL2,µS1,µS2,µK,νA,νB,MKe,χ1,χ2,χ3, m1,m2,m3,tvar,β,mpars,kw,varOK,MeZ,Me,modname="Bar3MassTemp"}, {tvar,{β},mpars}=tsign; varOK=False; kw=ToUpperCase[tvar]; MeZ=Table[0,{3},{3}]; If [kw=="GEN", varOK=True; {µ1,µ2,µ3,µ4}=mpars; Me=me*{{4+µ1,-1+µ3,2+µ4},{-1+µ3,4+µ1,2+µ4}, {2+µ4,2+µ4,16+µ2}}/30]; If [kw=="GEX", varOK=True; {χ1,χ2,χ3}=mpars; m1=χ1+χ2; m2=χ1-χ2; m3=Sqrt[30]*Sqrt[χ1-χ3]-2*χ1; Me=me*{{m1,m2,m3},{m2,m1,m3},{m3,m3,30-4*χ1-4*m3}}/30]; If [kw=="LUM", varOK=True; {µL1,µL2}=mpars; Me=me*{{5+µL1,0,0},{0,5+µL1,0},{0,0,20+µL2}}/30]; If [kw=="SPE", varOK=True; {µS1,µS2}=mpars; Me=me*{{10+µS1+µS2, 10-µS1+µS2, 10-2*µS2}, {10-µS1+µS2, 10+µS1+µS2, 10-2*µS2}, {10-2*µS2, 10-2*µS2, 10+4*µS2}}/90]; If [kw=="SMS", varOK=True; {µL1,µL2,µK}=mpars; MKe=µK*me*Bar3StiffTemp[1,1,1,tsign,numer]; Me=me*{{5+µL1,0,0},{0,5+µL1,0},{0,0,20+µL2}}/30+MKe]; If [kw=="COBA", varOK=True; {νA}=mpars; Me=me*{{ 6-νA, 2-νA,-2+2*νA}, { 2-νA, 6-νA,-2+2*νA}, {-2+2*νA,-2+2*νA, 4-4*νA}}/12]; If [kw=="COBB", varOK=True; {νB}=mpars; Me=me*{{96-36*νB-νB^2, 24-12*νB+νB^2, -48+24*νB}, {24-12*νB+νB^2, 96-36*νB-νB^2, -48+24*νB}, {-48+24*νB, -48+24*νB, 384}}/432]; If [!varOK, Print[modname,": bad template var ",tvar, ", zero matrix returned"]; Return[MeZ]]; If [!numer, Me=Simplify[Me]]; Return[Me]]; Bar3StiffTemp[Le_,Em_,A_,tsign_,numer_]:=Module[ {ke=Em*A/Le,tvar,β,mpars,Keb,Keh,Ke}, {tvar,{β},mpars}=tsign; Keb=ke*{{1,-1,0},{-1,1,0},{0,0,0}}; Keh=(4/3)*β*ke*{{1,1,-2},{1,1,-2},{-2,-2,4}}; Ke=Keb+Keh; If[!numer, Ke=Simplify[Ke]]; Return[Ke]];

Figure I.3. Bar3 element mass and stiffness template modules.

Ke=Bar3ElemStiffTemp[Le,Em,A,tsign,numer]

(I.5)

The arguments are: Le

Element length

Em,A,rho Elastic modulus, cross section area, and mass density, respectively, of bar tsign

Template signature. See Table I.1 for configuration details.

numer

Logical flag. If True, process in floating point. If False, process symbolically.

As function values the modules return Me

3 × 3 element mass matrix

Ke

3 × 3 element stiffness matrix

I–6

§I.3

ASSEMBLY LEVEL MODULES

Bar3MassTempLattice[numele_,Le_,ρ_,A_,tdef_,numer_]:=Module[ {tsign=tdef,e,i,j,ii,jj,eft,Me,M,numnod=2*numele+1}, If [Head[tdef]==String, tsign=Bar3TempSignature[tdef]]; M=Table[0,{numnod},{numnod}]; Me=Bar3MassTemp[Le,ρ,A,tsign,numer]; For [e=1,e<=numele,e++, eft={1,3,2}+(e-1)*{2,2,2}; For [i=1,i<=3,i++, ii=eft[[i]]; For [j=1,j<=3,j++, jj=eft[[j]]; M[[ii,jj]]+=Me[[i,j]] ]; ]; ]; If [!numer, M=Simplify[M]]; Return[M] ]; Bar3StiffTempLattice[numele_,Le_,Em_,A_,tdef_,numer_]:=Module[ {tsign=tdef,e,i,j,ii,jj,eft,Ke,K,numnod=2*numele+1}, If [Head[tdef]==String, tsign=Bar3TempSignature[tdef]]; K=Table[0,{numnod},{numnod}]; Ke=Bar3StiffTemp[Le,Em,A,tsign,numer]; For [e=1,e<=numele,e++, eft={1,3,2}+(e-1)*{2,2,2}; For [i=1,i<=3,i++, ii=eft[[i]]; For [j=1,j<=3,j++, jj=eft[[j]]; K[[ii,jj]]+=Ke[[i,j]] ]; ]; ]; If [!numer, K=Simplify[K]]; Return[K] ];

Figure I.4. Bar3 master mass and stiffness assembler modules. Bar3StiffTempPatch[Le_,Em_,A_,tdef_,numer_]:=Module[ {e,i,j,ii,jj,K}, K=Bar3StiffTempLattice[2,Le,Em,A,tdef,numer]; Return[{K[[2]],K[[3]]}] ]; Bar3MassTempPatch[Le_,ρ_,A_,tdef_, numer_]:=Module[ {e,i,j,ii,jj,M}, M=Bar3MassTempLattice[2,Le,ρ,A,tdef,numer]; Return[{M[[2]],M[[3]]}] ];

Figure I.5. Bar3 mass and stiffness patch extraction modules.

§I.3. Assembly Level Modules This section covers modules that work at the assembly (master) level. These are the midlevel four pictured in Figure I.1. §I.3.1. Lattice Master Mass and Stiffness Modules Modules Bar3ElemMassTempLattice and Bar3StiffTempLattice, listed in Figure I.4, assemble the master mass and stiffness matrices, respectively, of a prismatic homogeneous bar member discretized as a regular lattice with a given number of elements. Since all elements are identical, only one call to the appropriate element-level module is made. The returning matrix is reused in the merge loop. The call sequences are similar: M=Bar3MassTempLattice[numele,Le,rho,A,tdef,numer] K=Bar3StiffTempLattice[numele,Le,Em,A,tdef,numer] I–7

(I.6) (I.7)


The arguments are: numele

Number of elements in lattice. The number of freedoms is numdof=2*numele+1

Le

Element length. Total member length will be Le*numele

Em,A,rho Elastic modulus, cross section area, and mass density, respectively, of bar tdef

Template definition argument. Two possibilities: If a list, tdef is taken to be the template signature tsign configured as shown in Table I.1, and thus passed directly to the element module If a character string (for example: "CMM"), tdef is interpreted as a template instance abbreviation and Bar3TempSignature called as per (I.3) to build tsign, which is then passed to the element modules

numer

Logical flag; see §I.2.1

As function values the modules return M

Master mass matrix of order ndof×ndof

K

Master stiffness matrix of order ndof×ndof

§I.3.2. Lattice Patch Modules Modules Bar3ElemMassPatch and Bar3StiffPatch, listed in Figure I.5, return the assembled mass and stiffness matrices, respectively, of a patch of two identical Bar3 elements. This is done by calling Bar3MassTempLattice and Bar3StiffTempLattice with numele=2 and returning only the second and third equations. The call sequences are similar: M=Bar3MassTempPatch[Le,rho,A,tdef,numer] K=Bar3StiffTemppatch[Le,Em,A,tdef,numer]

(I.8) (I.9)

The arguments are identical to those for the lattice master mass and stiffness modules, respectively, described in §I.3.1, except that numele is not supplied. As function values the modules return Mp

Patch mass equations as a coefficient matrix of order 2 × 5; see (?)–(?)

Kp

Patch stiffness equations as a coefficient matrix of order 2 × 5; see (?)–(?)

§I.4. Dispersion Analysis and Display Modules This section describe modules that produce and display dispersion diagrams. Those are the top three shown in Figure I.1. §I.4.1. Characteristic Equation Module Module Bar3CharFreq, listed in Figure I.6, forms the characteristic equation of a plane wave propagating over a regular Bar3 lattice patch and solves it for the two characteristic frequencies. The calling sequence is { detCm, 2aco, 2opt }=Bar3CharFreq[wavars,tdef,numer] I–8

(I.10)

§I.4

DISPERSION ANALYSIS AND DISPLAY MODULES

Bar3CharFreq[wavars_,tdef_,numer_]:=Module[ {κ,ζ ,Ω,Ω2,τ ,Bc,Bm,Kp,Mp,up,fm,fc,mfac,cfac,Cm,detCm, Ωsol,Ω21,Ω22,Ω2aco,Ω2opt}, {κ,ζ ,Ω,Ω2,τ ,Bc,Bm}=wavars; PlaneWave[k_,ω_,B_,x_,t_]:=B*Exp[I*(k*x-ω*t)]; Mp=Bar3MassTempPatch [1,1,1,tdef,numer]; Kp=Bar3StiffTempPatch[1,1,1,tdef,numer]; up={PlaneWave[κ,Ω,Bc,-1,τ ],PlaneWave[κ,Ω,Bm,-1/2,τ ], PlaneWave[κ,Ω,Bc, 0,τ ],PlaneWave[κ,Ω,Bm, 1/2,τ ], PlaneWave[κ,Ω,Bc, 1,τ ]}; {fm,fc}=Simplify[ExpToTrig[(Kp-Ω2*Mp).up]]; mfac=(Cos[κ/2+Ω*τ ]-I*Sin[κ/2+Ω*τ ]); cfac=(Cos[Ω*τ ]-I*Sin[Ω*τ ]); {fm,fc}=Simplify[{fm/mfac,fc/cfac}]; Cm={{Coefficient[fm,Bm],Coefficient[fm,Bc]}, {Coefficient[fc,Bm],Coefficient[fc,Bc]}}; detCm=Simplify[Det[Cm]]; Ωsol=Simplify[Solve[detCm==0,Ω2]]; Ω2aco=Ω21=Simplify[Ω2/.Ωsol[[1]]]; Ω2opt=Ω22=Simplify[Ω2/.Ωsol[[2]]]; {Ω210,Ω220}=Simplify[Limit[{Ω21,Ω22},κ->0]]; If [Ω220==0, Ω2aco=Ω22; Ω2opt=Ω21]; Return[{detCm,Ω2aco,Ω2opt}]];

Figure I.6. Bar3 characteristic equation module.

The arguments are: wavars

A list of symbols representing plane wave dispersion analysis variables, configured as the list { κ,ζ , , 2,τ ,Bc,Bm }, in which κ

Dimensionless wavenumber κ = k

ζ

Dimensionless space coordinate ζ = x/ ∈ [−1, 1] over patch.

Dimensionless characteristic frequency ω /c0

2

Dimensionless characteristic squared frequency

τ

Dimensionless time τ = tc0 /

Bc,Bm Corner and midpoint wave component amplitudes, respectively These must be individual symbols. No numbers or expressions should be in this list, because they are internally used as variables. For instance, entering * or ^2 for 2 will cause errors. tdef

Template definition argument; see §I.3.1

numer

Logical flag; see §I.2.1

The module returns the list { detCm, 2aco, 2opt }, in which detCm

Determinant of characteristic matrix Cm as a function of κ and 2

2aco

Dimensionless characteristic squared frequency a2 of acoustic branch (AB), expressed as function of κ

2opt

Dimensionless characteristic squared frequency a2 of optical branch (OB), expressed as function of κ

The last two expressions: 2aco and 2opt, collectively define the dimensionless dispersion diagram (DDD) for the template specified by tdef. I–9

Appendix I: IMPLEMENTATION OF BAR3 TEMPLATE IN MATHEMATICA Bar3Dispersion[κv_,tdef_,{ma_,mo_},slevel_]:=Module[ {κ=κv,kw,kw3,tsign=tdef,tvar,β,mpars,µ1,µ2,µ3,µ4,µL1,µL2, µS1,µS2,µK,νA,νB,P,Q,R,c1,c2,c3,c4,c5,c6,c7,c8,c9, assume,parmap,rep,Ω2aco,Ω2opt,Ω2acos=Null,Ω2opts=Null}, If [Head[tdef]==String, tsign=Bar3TempSignature[tdef]]; {tvar,{β},mpars}=tsign; kw=ToUpperCase[tvar]; kw3=StringTake[kw,3]; parmap=MemberQ[{"GEX","LUM","SPE","SMS"},kw3]; assume=κ>=0&&β>=0; If [kw3=="COB", Ω2aco=12*(1-Cos[κ])/(5+Cos[κ]); If [kw=="COBA", νA=mpars[[1]]; Ω2opt=16/(1-νA)]; If [kw=="COBB", νB=mpars[[1]]; Ω2opt=432/(36-12*νB-νB^2)]; If [ma>=0, Ω2acos=Series[Ω2aco,{κ,0,ma}]]; If [mo>=0, Ω2opts=Series[Ω2opt,{κ,0,mo}]]; Return[{Ω2aco,Ω2opt,Ω2acos,Ω2opts}]]; If [kw=="GEN", {µ1,µ2,µ3,µ4}=mpars]; c1=4*β*(40+4*µ1+µ2+4*µ4); c2=16+µ2; c3=µ4*(4+µ4); c4=4*(5+µ3+µ4); c5=60+4*µ2+µ1*c2; c6=16*µ3-c3; c7=4*β*(µ2+c4); c8=c6+µ2*(µ3-1)-20; c9=c7-3*c2; {c1,c2,c3,c4,c5,c6,c7,c8,c9}=Simplify[{c1,c2,c3,c4,c5,c6,c7,c8,c9}]; P=c1+3*c2+c9*Cos[κ]; R=c5-4*µ4-µ4^2+c8*Cos[κ]; Q=192*β*(Cos[κ]-1)*(c5-c3+c8*Cos[κ])+(c1+3*c2+c9*Cos[κ])^2; If [parmap, rep=Bar3MassVarParMap[{µ1,µ2,µ3,µ4},tsign]; {P,Q,R}={P,Q,R}/.rep]; If [slevel==1, {P,Q,R}=Simplify[{P,Q,R},assume]]; If [slevel>1, {P,Q,R}=FullSimplify[{P,Q,R},assume]]; Ω2aco=5*(P-Sqrt[Q])/R; Ω2opt=5*(P+Sqrt[Q])/R; If [slevel==1, {Ω2aco,Ω2opt}=Simplify[{Ω2aco,Ω2opt},assume]]; If [slevel>1, {Ω2aco,Ω2opt}=FullSimplify[{Ω2aco,Ω2opt},assume]]; If [ma>=0, Ω2acos=Series[Ω2aco,{κ,0,ma}]]; If [mo>=0, Ω2opts=Series[Ω2opt,{κ,0,mo}]]; If [slevel==1,{Ω2acos,Ω2opts}=Simplify[{Ω2acos,Ω2opts},assume]]; If [slevel>1, {Ω2acos,Ω2opts}=FullSimplify[{Ω2acos,Ω2opts},assume]]; Return[{Ω2aco,Ω2opt,Ω2acos,Ω2opts}]];

Figure I.7. Bar3 dispersion module that returns dimensionless characteristic squared frequencies as function of dimensionless wavenumber, and their Taylor series up to given order about κ = 0

§I.4.2. Dispersion Branches And Taylor Series Given a Bar3 template (or instance) definition, module Bar3Dispersion, listed in Figure I.7, returns the dimensionless characteristic squared frequencies a2 and 2o of the AB and OB as function of the dimensionless wavenumber κ. This module was built by inlining symbolic results produced by Bar3CharFreq with the goal of speeding up direct retrieval of those expressions. In addition, this module can compute and return their Taylor series expansions about κ = 0 up to specified orders. The calling sequence is { 2aco, 2opt, 2acos, 2opts }= Bar3Dispersion[κ,tdef,{ ma,mo },slevel] (I.11) The arguments are: κ

Dimensionless wavenumber

tdef


ma

If ma ≥ 0, return acoustic branch Taylor series (ABTS): a2 expanded in κ about κ = 0, up to and including order ma. If a negative integer, return Null.

mo

If mo ≥ 0, return optical branch Taylor series (OBTS): 2o expanded in κ about κ = 0, up to and including order mo. If a negative integer, return Null.

slevel

Simplification level for output results: an integer 0, 1 or 2. I–10

§I.4

DISPERSION ANALYSIS AND DISPLAY MODULES

Bar3DispersionPlot[κ_,tdef_,plotwhat_,κrange_,DVrange_, imgsiz_,title_]:=Module[{κv,tsign=tdef,tvar,kpars,mpars, cat,Ω2aco,Ω2opt,Ω2acos,Ω2opts,κmin,κmax,Drange,Vrange, style,pD=False,pV=False,dfun=$DisplayFunction}, If [Head[tdef]==String, tsign=Bar3TempSignature[tdef]]; {tvar,kpars,mpars}=tsign; cat=tvar<>" "<>ToString[N[kpars]]<>ToString[N[mpars]]; If [plotwhat=="D"||plotwhat=="DV", pD=True]; If [plotwhat=="V"||plotwhat=="DV", pV=True]; {κmin,κmax}=κrange; {Drange,Vrange}=DVrange; style={{AbsoluteThickness[1.80],RGBColor[1,0,0]}, {AbsoluteThickness[1.80],RGBColor[0,0,1]}, {AbsoluteThickness[1.80],RGBColor[0,0,0]}}; κv=κ; {Ω2aco,Ω2opt,Ω2acos,Ω2opts}=Bar3Dispersion[κv,tsign,{-1,-1},1]; {Ω2aco,Ω2opt}={Ω2aco,Ω2opt}/.κv->κ; If [title!=" ", Print[title]]; If [$VersionNumber>=6.0, dfun=Print]; If [pD, Plot[{Sqrt[Ω2aco],Sqrt[Ω2opt],κ},{κ,κmin,κmax}, PlotStyle->style, Frame->True, PlotRange->Drange, ImageSize->imgsiz, DisplayFunction->dfun, PlotLabel->"DDD for "<>cat]]; caco=D[Sqrt[Ω2aco],κ]; copt=D[Sqrt[Ω2opt],κ]; If [pV, Plot[{caco,copt,1},{κ,κmin,κmax}, PlotStyle->style, Frame->True, PlotRange->Vrange, ImageSize->imgsiz, DisplayFunction->dfun, PlotLabel->"DGVD for "<>cat]]; Return[]];

Figure I.8. Bar3 dispersion diagram plotting module.

0: (or negative): no simplifications 1: ordinary simplification using the Simplify function 2: more exhaustive simplification using the FullSimplify function. Note: this level should be used with caution. Reason: full simplification may return unexpected weird results with terms involving Abs, or conditional expressions. The module returns the list { 2aco, 2opt, 2acos, 2opts }, in which

2aco

AB dimensionless characteristic squared frequency a2 expressed as function of κ

2opt

OB dimensionless characteristic squared frequency 2o expressed as function of κ

2acos

ABTS about κ = 0 up to and including order ma. If ma < 0, Null is returned.

2ocos

OBTS about κ = 0 up to and inclusing order ma. If mo < 0, Null is returned.

§I.4.3. Dispersion Diagram Plotting Given a Bar3 template instance (that is, with all-numeric signature) module Bar3DispersionPlot, listed in Figure I.8, can plot its DDD, which includes the acoustic and optical branches returned by Bar3Dispersion. It may also plot its DGVD, which is the ratio γc = c/c0 of the FEM plane wave speed c to that of the continuum wave speed c0 . The module has been used to produce all Bar3 dispersion plots of this paper. The call sequence is Bar3DispersionPlot[κ,tdef,plotwhat,κrange,DVrange,imgsiz,title] I–11

(I.12)


The arguments are: κ

Dimensionless wavenumber

tdef


plotwhat A character string of the form "D", "V", or "DV". If the letter D appears, plot the DDD (the so-called “D-plot”). If the letter V appears, plot the DGVD (the so-called “V-plot”). If none of those strings is given, no plot is produced. κrange

A 2-item list { κmin,κmax } that specifies the (horizontal) plot range for κ. It is used for both DDD and DGVD plots. Usual range is { 0,2 Pi }.

DVrange

A list of the form { Drange,Vrange }. Drange is in turn a two-item list: { min, max } that specifies the DDD plot range for . Vrange has a similar configuration: { γ cmin,γ cmax } and specifies the DGVD range for γc = c/c0 . Both lists must be supplied even if only one plot is requested. Common specifications are { 0,8 } for Drange and { -2,2 } for Vrange.

imgsiz

Width of plot in points. Normally set to 300 to 400.

title

An optional character string to be printed before the plot. If " " no title appears.

The module does not return a value. Its output is the plot image object written to the Mathematica default display function. (Its name changed in Version 6.0 from $DisplayFunction to Print.)

I–12

V

Mass Templates in a Variational Framework

V–1

Appendix V: MASS TEMPLATES IN A VARIATIONAL FRAMEWORK


§V.1 §V.2 §V.3

Variationally Derived Bar2 Mass Template . . . . . . . . . . Variationally Derived Bar3 Mass Template . . . . . . . . . A Comment on the Variational Formulations of Elastodynamics . .

V–2

V–3 V–4 V–6

§V.1

VARIATIONALLY DERIVED BAR2 MASS TEMPLATE

A question that may be interest to FEM theoreticians: can any mass template be produced by a conventional variational framework? By “conventional” is meant based on shape functions injected in the kinetic energy. More precisely: velocities are interpolated over the element from nodal velocities using velocity shape functions (VSF), and the element kinetic energy T e evaluated by integration. The mass matrix follows as the Hessian of T e with respect to nodal velocities, as per (16.5). In short, a variationally derived mass matrix (VDMM). For practical template construction and customization, the variational interpretation is superfluous, since templates can be expediently postulated and algebraically customized. The reformulation may be worthwhile, however, for mathematical investigations, as well as linkage to work conducted by other researchers. Presently it is unknown whether the template-to-VDMM connection for arbitrary elements can be established. It has been only investigated for the two simplest bar elements: Bar2 and Bar3. In both cases, the general template was considered. The findings may be summarized as follows: (1) VSF that reproduce the general template as a VDMM can be found. They are not unique. (2) For any template instance that deviates from the CMM, the VSF do not coincide with the displacement shape functions (DSF) used in the derivation of the element stiffness. (3) VSF that deviate from the DSF are noninterpolatory and nonconforming with respect to nodal velocities computed from the displacements by time differentiation. They do not necessarily satisfy the unit-sum condition (also called partition of unity in the literature). A uniform velocity field, however, must produce the exact kinetic energy. Two simple elements are analyzed below. §V.1. Variationally Derived Bar2 Mass Template We investigate whether the general one-parameter Bar2 mass template (22.3) can be produced as a VDMM. The velocity field derived from the axial displacement u e (x, t) is u˙ e (x, t) = d e (x, t)/dt. Evaluation at the nodes yields the nodal velocities u˙ 1 and u˙ 2 , collected in u˙ e = [ u˙ 1 u˙ 2 ]T . Let N1 (ξ ) = (1 − ξ )/2 and N2 (ξ ) = (1 + ξ )/2 denote the well known displacement shape functions (DS)F of Bar2, ξ being the usual iso-P natural coordinate. The element velocity interpolation is taken to be v e (ξ ) = u˙ 1 Nv1 (ξ ) + u˙ 2 Nv2 (ξ ),

(V.1)

in which the velocity shape functions (VSF) Nv1 and Nv2 are linked to the DSF through the linear map Nv1 (ξ ) = (1 + 12 δ1 ) N1 (ξ ) + 12 δ2 N2 (ξ ),

Nv2 (ξ ) = 12 δ2 N2 (ξ ) + (1 + 12 δ1 ) N1 (ξ ).

(V.2)

In (V.2), δ1 and δ2 are functions of the template parameter (but not of ξ ), representing the deviations of the VSF from the DSF. Note that prismatic bar symmetry is built-in: Nv1 (ξ ) = Nv2 (−ξ ). The 1 2 associated kinetic energy T e is ρ A (/2) −1 v e (ξ ) dξ , which can be evaluated either analytically or through 2-point Gauss integration. Taking its Hessian with respect to u˙ e gives a mass matrix denoted by Meδ below. As for the Bar2 template, it is preferable to use the alternative form Meχ of (22.4) rather than Meµ of (22.3) because solutions are simpler. Summarizing, the two matrices to V–3


(a) 1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv1(ξ)

−1

(b)

VSF: Bar2 mass template instance with µ=0 (CMM)

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv2(ξ)

−0.5

0

0.5

ξ

1

Nv1(ξ)

−1

(c)

VSF: Bar2 mass template instance with µ=1/2 (BLFM)

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv2(ξ)

−0.5

0

0.5

ξ

Nv1(ξ)

Nv2(ξ)

Abcissas of 2-Point Gauss Rule

−1

1

VSF: Bar2 mass template instance with µ=1 (DLMM)

−0.5

0

0.5

ξ

1

Figure V.1. Velocity shape functions (VSF) that produce the general Bar2 mass template (22.2) in a variational framework, for three instances: (a) µ=0 (CMM); (b) µ = 1/2 (BLFM) and (c) µ = 1 (DLMM).

be matched are Meδ

=

1 24

ρ A

ψ11 ψ21

ψ12 , ψ22

Meχ

=

1 12

3+χ ρA 3−χ

ψ11 = ψ22 = 2(4 + 4δ1 + 2δ2 + δ12 + δ1 δ2 + δ22 ), ψ12 = ψ21 = 4 + 4δ1 + 8δ2 + χ = 1 + 2µ,

δ12

+ 4δ1 δ2 +

3−χ , 3+χ (V.3)

δ22 ,

µ = 12 (χ − 1),

in which (22.4) is reproduced for convenience.

√ √ On equating Meδ = Meχ we get four solutions: {δ1 = −3− χ, δ2 = −1+ χ}, {δ1 = √ √ √ √ √ √ −1− χ, δ2 = 1+ χ}, {δ1 = −3+ χ, δ2 = −1− χ}, and {δ1 = −1 + χ, δ2 = 1− χ }. Only the last one reduces the VSF to DSF when µ = 0 or χ = 1. Inserting it into (V.2) and simplifying yields √ √ Nv1 (ξ ) = 12 (1 − ξ χ) = 12 (1 − ξ 1 + 2µ), Nv2 (ξ ) = 12 (1 + ξ χ) = 12 (1 + ξ 1 + 2µ). (V.4) These VSF satisfy the conservation condition Nv1 + Nv2 = 1 for any µ and ξ . They are plotted in Figure V.1 for three instances: µ = 0 (CMM) µ = 12 (BLFM), and 1 (DLMM). If µ = 0, the VSF depart from the DSF, and are plainly nonconforming. √ display a distinguishing geometric feature: each VSF The two VSF for µ = 1, namely (1±ξ 3)/2,√ vanishes at one of the sample points ξ = ±1/ 3 of the 2-point Gauss rule; see Figure V.1(c). This 1 effectively orthogonalizes them in the sense that the kinetic energy cross integral −1 Nv1 Nv2 dξ is zero. The result is the diagonal mass matrix MeL of (22.1). Comparing the results (V.4) with the ansatz (V.2), plainly the latter was too elaborate. Little harm is done, however, for this simple element. For more complicated ones, such as the Bar3 studied next, a recursive adjustment is recommended using an interactive CAS. §V.2. Variationally Derived Bar3 Mass Template Next we find whether the general mass template for Bar3 can be derived variationally. The well known displacement shape functions are N1 (ξ ) = ξ(ξ − 1)/2, N2 (ξ ) = ξ(ξ + 1)/2, and N3 (ξ ) = V–4

§V.2

VARIATIONALLY DERIVED BAR3 MASS TEMPLATE

1 − ξ 2 . The velocity interpolation is assumed to be v e (ξ ) = u˙ 1 Nv1 (ξ ) + u˙ 2 Nv2 (ξ ) + u˙ 3 Nv3 (ξ ),

(V.5)

in which Nv1 (ξ ) = N1 (ξ )− 12 ξ(δ1 −δ2 ξ ), Nv2 (ξ ) = N2 (ξ )+ 12 ξ(δ1 +δ2 ξ ), Nv3 (ξ ) = N3 (ξ )+(δ3+δ4 ξ 2 . (V.6) Here δ1 through δ4 are functions of the template parameters to be determined. The VSF ansatz (V.6) was obtained after some simplifying initial computations. Note that prismatic bar symmetry is preimposed: Nv1 (ξ ) = Nv2 (−ξ ) and Nv3 (ξ ) = Nv3 (−ξ ). The associated kinetic energy T e can be evaluated either analytically or from 3-point Gauss integration. Taking its Hessian with respect to u˙ e gives the mass matrix

ψ11 ψ12 ψ13 ψ21 ψ22 ψ23 , ψ31 ψ32 ψ33 = 2(2 + 5δ1 + 5δ12 + 3δ2 + 3δ22 ),

ρ A Meδ = 30 ψ11 = ψ22

ψ12 = ψ21 = −1 − 10δ1 − 10δ12 + 6δ2 + 6δ22 , ψ13 = ψ23 = ψ31 = ψ32 = (1 + 2δ2 )(2 + 5δ3 + 3δ4 ),

(V.7)

ψ33 = 2(8 + 15δ32 + 4δ4 + 3δ42 + 10δ3 (2 + δ4 )). It is convenient to match Meδ to the 3-parameter, χ-form of mass matrix template (23.7) instead of against (23.2). Matching entries gives 8 solutions, of which the one that yields δ1 = δ2 = δ3 = δ4 = 0 for the CMM (χ1 = 5/2, χ2 = 3/2, χ3 = 2/3) is picked: δ1 = φ1 − 1/2, δ2 = φ2 − 1/2, δ3 = 3φ3 /2 − 1, δ4 = 1 − 2φ2 + φ3 − 5φ4 /2, (V.8) √ √ √ √ in which φ1 = χ2 /10, φ2 = χ1 /6, φ3 = χ3 /χ1 , and φ4 = 5(1 − χ3 /χ1 ). Except for the CMM, these VSF do not verify the strong (pointwise) unit sum condition Nv1 + Nv2 + Nv3 = 1 for each ξ , but do satisfy the more lenient element mass conservation constraint

1 2

1

−1

(Nv1 + Nv2 + Nv3 )2 dξ = 1.

(V.9)

In terms of the δi , (V.9) is 12δ22 + 15δ32 + 10δ3 (3 + δ4 ) + δ4 (10 + 3δ4 ) + 4δ2 (5 + 5δ3 + 3δ4 ) = 0. The VSF produced by (V.8) are plotted in Figure V.2 for nine Bar3 mass instances, as labeled therein. Except for the CMM they depart from the DSF, and are nonconforming. Some mass matrix properties can be discerned visually: •

For the diagonally lumped instances SLMM√and BLFD shown in Figure V.2(b,e), two VSF vanish at each of the sample points ξ ∈ {0, ± 3/5} of the 3-point Gauss rule. Those points are marked in the Figure. This feature effectively energy-orthogonalizes the VSF in the sense of 1 kinetic energy, since all cross integrals −1 Nvi Nv j dξ for i = j vanish. As a result, diagonal mass matrices are produced. V–5


(a)

VSF: Bar3 mass template instance CMM

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv1(ξ) Nv2(ξ)

(d)

−0.5

0.5

ξ

1

VSF: Bar3 mass template instance BLFM

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv1(ξ) Nv2(ξ)

−0.5

0

0.5

ξ

Nv1(ξ)

Nv2(ξ)

−0.5

0

0.5

ξ

1

−0.5

0

0.5

ξ

−0.5

0

0.5

ξ

−0.5

Nv2(ξ)

0

0.5

Nv1(ξ) Nv2(ξ)

ξ

1

0

0.5

ξ

1

Nv2(ξ)

Nv1(ξ)

Nv3(ξ) −1

(i)

−0.5

VSF: Bar3 mass template instance BSSM

−0.5

0

0.5

ξ

1

VSF: Bar3 mass template instance COB1

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv3(ξ)

−1

(f)

1

VSF: Bar3 mass template instance COB0

Nv1(ξ)

Nv3(ξ)

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv1(ξ) Nv2(ξ)


VSF: Bar3 mass template instance LCDM

−1

1

Nv3(ξ)

−1

(h)


VSF: Bar3 mass template instance BLFD

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv3(ξ)

−1

(e)

1

VSF: Bar3 mass template instance SMS2

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv1(ξ) Nv2(ξ)

−1

(c) 1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv3(ξ)

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv3(ξ)

−1

(g)

0

VSF: Bar3 mass template instance SLMM

1.5 1.25 1 0.75 0.5 0.25 0 −0.25

Nv3(ξ)

−1

(b)

Nv1(ξ)

Nv2(ξ)

Nv3(ξ)

−1

−0.5

0

0.5

ξ

1

Figure V.2. Velocity shape functions (VSF) that produce the general Bar3 mass template (23.7) in a variational framework, for the nine labeled instances.

•

The VSF for the singular mass instance BSSM shown in Figure V.2(f), clearly displays linear dependence among the VSF.

Aside from those special cases, it is difficult to draw general conclusions from a glance at Figure V.2 as to performance. For example, why does the VSF in (d) provides the best low frequency matching? Shapes for say, (a) through (e) look quite similar (once you’ve seen one parabola ...). The obvious conclusion: Fourier analysis is a much sharper tool in dynamics. §V.3. A Comment on the Variational Formulations of Elastodynamics The use of VSF that differ from DSF dates back to the early days of FEM. It was done, for example, in [203] for the HCT plate bending element, following suggestions by R. W. Clough. (The consistent mass of that tricubic macroelement was quite complicated for hand derivations in 1966.) The idea can be incorporated into the well-known stationary-action variational principle (VP) of elastodynamics, called Hamilton-Kirchhoff by Gurtin [321, p. 225], by weakening the temporal kinematic link. V–6

§V.3

A COMMENT ON THE VARIATIONAL FORMULATIONS OF ELASTODYNAMICS

That minor generalization of the primal VP of elastodynamics should not be confused with the use of dual (also called complementary or reciprocal) forms. Research in that subject took off with Toupin’s formulation [748] of a dual form of Hamilton’s principle for a system of mass particles with interaction impulses as unknown variables. For corrections and evolution into continua see [196,718] and references therein. FEM applications to vibrations and dynamics emerged during the early 1970s; see e.g., [277,300,717], but have stagnated since. Reason: impulse DOF are foreign to the DSM, which dominates general purpose codes.

V–7

1

Overview of Dynamical Systems

1–1

Chapter 1: OVERVIEW OF DYNAMICAL SYSTEMS


§1.1. §1.2. §1.3. §1.4. §1.5. §1.6.

Scope Dynamics versus Statics and Quasi-Statics Overcoming Mustiness Terminology and Notation Systems Theory Terminology Open Systems and Hierarchical Decomposition

1–2

1–3 1–3 1–4 1–5 1–6 1–8

§1.2

DYNAMICS VERSUS STATICS AND QUASI-STATICS

§1.1. Scope This is a book about dynamics. Nowadays this term has acquired several meanings. It is used here in the traditional sense: “The study of the relationship between motion and the forces affecting motion” This is in fact meaning (1a) in the American Heritage Dictionary of the English Language. As such, it pertains to the science of Mechanics.1 . The corresponding adjectives are dynamic2 and its equivalent dynamical.3 But even the traditional definition is far too broad in two respects. First, Mechanics embodies a wide range of scales that span from cosmological through atomic and sub-atomic. Second, the study can focus on three aspects: theoretical, applied and computational. Our focus is restricted to a particular subset: •

Classical Mechanics, which obeys Newton’s laws. This allows the use of continuum (field) models as well as certain “lumped” idealizations (point masses) that can be derived directly from such laws.

•

Computational Mechanics, which relies on model-based simulation on digital computers. Of the various discretization methods, our focus will be on the Finite Element Method (FEM).

The main applications of this subset are to Solid and Structural Mechanics. Although on first sight this appears to be a bit limited in scope, many modeling and computational tools covered here are application independent in the sense discussed in the IFEM book [?, Chapter 7]. Hence the inclusion of qualifiers such as “Structural Mechanics” in the book title would be too confining. §1.2. Dynamics versus Statics and Quasi-Statics Dynamic models in Classical Mechanics possess a common feature: the appearance of inertial effects modeled by Newton’s second law, which states that inertia forces are proportional to mass times accelerations. Since accelerations are time derivatives of displacements, which characterize the motion, the formulation inevitably leads to differential equations in time, whose solutions exhibit time dependence. The converse is not true: time dependence does not necessarily require a dynamic model, as discussed below. At the other extreme lies statics. This is the equilibrium mechanics of stationary bodies. The corresponding adjective is static, which means motionless, at rest, quiescent. Quantities associated with stationary bodies do not vary with time, whence modeling is greatly simplified. 1

The term dynamics has nowadays acquired a generalized meaning beyond Mechanics, as illustrated by definition (2) of that Dictionary: “The physical and moral forces that produce motion and changes in any field or system.”

2

Etymology: French dynamique, from Greek dunamikos: powerful, from dunamis, power, from dunashai, to be able. Its use in terms of active, energetic, vigorous, forceful, and the like, is comparatively recent: 1856 (from Emerson).

3

The variant Dynamical tends to be often used in a more abstract sense. For example, MathWorld defines dynamical system as “a means of describing how a state evolves into another over the course of time.” This is followed by “Technically, a dynamical system is a smooth action of the reals or integers into another object (usually a manifold).” This gobbledygook brings to mind one tongue-in-cheek comment about Lamb’s Hydrodynamics: one can read the whole book without realizing that water is wet.

1–3


In between dynamics and statics lies the world of quasi-static scenarios, in which quantities vary with time but do so slowly that inertial and damping effects can be ignored.4 For example one may imagine situations such as a roof progressively burdened by falling snow before collapse, the gradual filling of a dam over a decade, or the construction of a tunnel. Or foundation settlements: think of the Pisa tower before leaning was stopped.5 By contrast dynamic analysis is appropriate when the variation of displacements with time is so rapid that inertial effects cannot be ignored. There are numerous practical examples: earthquakes, rocket launches, vehicle crashes, explosive forming, air blasts, underground explosions, rotating machinery, airplane flutter, dancing robots. The structural accelerations, which are second derivatives with respect to time, must be kept in the governing equations. Damping effects, which are usually associated with velocities (the first temporal derivatives of displacements), may be also part of a dynamic model. Passive damping effects are often neglected, however, since they tend to take energy out of a system and thus reduce the response amplitude. Hence ignoring such kind of damping may lead to conservative designs. Developments in the AFEM [?], IFEM [?], MFEMS [?], and NFEM [?] books pertain to statics and quasi-statics. In this book, inertial effects will be always included, whereas damping effects are occasionally considered. Remark 1.1. Quasi-static behavior should not be confused with steady-state. The latter describes the response to certain kinds of forced excitation (usually periodic) once effects of initial conditions disappear over the course of time. The opposite of steady-state is transient, as in “transient response.” Those two qualifiers pertain only to dynamic systems.

§1.3. Overcoming Mustiness Face it: Classical Mechanics smells musty. Its heyday was attained during the Victorian and Edwardian eras: the world of “Upstairs, Downstairs,” just before relativity theory emerged. Reading Euler, Lagrange or Hamilton one can imagine horse carriages, powdered wigs, feather pens and blotting paper. But despite cowebs, it is not an obsolete subject. Far from it. But it needs to be spruced up with modern language and tools. Three “pick-me-ups” are used in this book. Systems Nomenclature. General System Theory (GST) emerged as a discipline by 1968 [775]. Certainly not musty: GST is still a vigorous topic that has brought about a fresh and holistic approach to old questions. Lipstick on a pig? Perhaps, but terms like “open system” and “system environment” are shiny lipstick that connect well with related topics such as control. Linear Algebra. Matrices are not exactly spring chickens. They were invented by Cayley and Sylvester in the mid XIX century. For over a century they were concealed behind suffocating tracts on determinants such as [501]. Linear and matrix algebra come to the expository forefront, however, once digital computers appeared in the early 1950s, and the baby boomer generation was 4

Quasi-statics includes statics as limit when the motion (or the time) is frozen. It does not include dynamics as the opposite limit because inertial and damping effects cannot be recovered from quasi-static models. Thus the name is apt since there is no such a thing as quasi-dynamics.

5

The quasi-static assumption can be done during design if dynamic effects can be accounted for through appropiate safety factors. For many types of structures (e.g., buildings, bridges, offshore towers) these are specified in building codes. This saves analysis time when dynamic effects are inherently nondeterministic, as in traffic, winds or water wave effects.

1–4

§1.4 TERMINOLOGY AND NOTATION Table 1.1. Some Terms Frequently Used in Dynamics Term

Definition

Body Mass Position Point Particle

Any bounded aggregate of matter. A measure of the resistance of a body to acceleration. A location in space. A geometric object devoid of any properties except position. A body whose spatial extent and internal configuration are irrelevant in a specific context. A finite mass assigned to a position. Its spatial extent and internal configuration are irrelevant. A punctiform particle with a finite mass. (Abbreviation: PMP)

Point-Mass Point-Mass Particle

Space and time are not defined as they are considered primitives.

trained in their use. Linear Algebra is not only the natural language of numerical computation but that of the Finite Element Method. And it blends smoothly with the vector notation that is now the bread and butter of college Physics. Finite Elements. The Finite Element Method (FEM), as used today, came out in 1956 [758] as an offspring of three interlaced developments: matrix structural analysis, energy methods, and digital computers [238]. Although FEM has reached middle age, it retains sufficient vigor to freshen up dynamics. There is a surprising formal equivalence between certain widely-used dynamic FEM models and point-mass particle dynamics. This “one-to-one mapping” is discussed in Chapter 2. Suffices to say that decorating portions of the old musty stuff with the lipstick of modern tools makes the pig tolerably attractive. §1.4. Terminology and Notation Some definitions of terms used frequently in this book are collected in Table 1.1 for further reference. Of those the definitions of mass6 and particle7 are the ones that have changed most over history. Although particle and point-mass are often viewed as synonymous, they are not. The term particle is more versatile, especially in molecular and atomic physics. Thus saying “particle dynamics” means little until one specifies what kind of particles and effects we are talking about. On the other hand, 6

From Latin massa, a large irregular lump of something. Newton spoke of “quantity of matter” and defined mass indirectly via proportions. The related term inertia is less specific, as it informally conveys inertness, inactivity, or sluggishness. In Mechanics inertia is the property of matter by which it retains its state of rest, or its velocity along a straight line, unless acted upon by an external force. This is actually Newton’s First Law of Motion; cf. §3.2.1.

7

From Latin particula, a small part. As noted by Truesdell in a critical essay on the axiomatics of classical mechanics [754, p. 512], the intended meaning has varied from author to author over the centuries. The roughly equivalent term corpusculum, which means “small body”, is used by Newton and his contemporaries. In modern Physics, “particle” generally implies portions of matter considered at molecular or atomic scales.

1–5


Table 1.2. Terminology from General Systems Theory Term

Definition

GST System GST State

An entity that maintains its existence through the interaction of its parts. The relevant properties, values or characteristics of a system component, or of the entire system. A change in the state of the system, or part of a system. A system event that initiates other events. A set of entities and their relevant properties that are not part of the system, but such that a change on any of which can produce a change in the system. A system that is influenced by entities outside the system. A self-contained system that is not affected by entities outside the system. A system whose state changes over time. It may be open or closed.

GST Event GST Behavior GST Environment GST Open System GST Closed System GST Dynamic System

By “maintains its existence” the definition of ‘GST System emphasizes self-regulation processes.

“point-mass particle dynamics” or “PMP dynamics” is precise: it means Newtonian mechanics applied to particles of finite mass. Its essentials have not changed over the past 4 centuries. As implied by the book title, matrix notation will be heavily used. Both matrices and vectors are identified by bold symbols. These symbols are lower case for vectors and upper case for matrices. Occasionally an arrow will be placed above a vector symbol to emphasize that it is a field. Real time will be always denoted by t. Derivatives with respect to t are abbreviated by superposed dots. For example, if u(t) is a scalar motion, the associated velocity and acceleration are compactly written d2 u du(t) . (1.1) , u¨ ≡ u˙ ≡ dt dt 2 Sometimes we will denote velocity by v and acceleration by a when appropriate. Note that partial differentiation with respect to t is not necessary when working with individual point-mass particles. §1.5. Systems Theory Terminology The name General Systems Theory (GST), was introduced by von Bertalanffy in the late 1960 [775].8 Some GST definitions are collected in Table 1.2, taken from [4,659]. The contrast with those in Table 1.1 is obvious. The GST definitions are nebulous: no images are conveyed. This is a consequence of ambitious goals: GST was originally intended to apply conjointly to areas as diverse as physical, climatological, biological, and even cultural systems, with initial emphasis on self-regulatory closed systems.9 This kind of generality has a silver lining: it illuminates features that connect different disciplines as long as they are not too discordant. For example: structural dynamics and control — or, to make 8

GST development was helped by the success of operations research and control theory for servomechanisms during World War II and its aftermath.

9

A contemporaneous but shorter-lived overreach: catastrophe theory, hampered by ridiculous claims from its founders, is mercifully gone. GST is still around, but has wisely evolved toward concrete applications.

1–6

§1.5

SYSTEMS THEORY TERMINOLOGY

Table 1.3. Systems-Related Terminology for Physical Systems Term

Definition

System Configuration State State Variables Dynamic System Response Event Behavior Environment Open System Closed System Interaction

A functionally related set of components regarded as a physical entity. The relative disposition or arrangement of system components. The condition of the system as regards its form, structure or constitution. A set of variables that uniquely characterizes the system state. A system whose state changes over time. It may be open or closed. The value of the state variables as a function of time. A change in the state variables produced by an agent. A pattern of events. A set of entities that do not belong to the system, but can influence its behavior. A system that is influenced by entities outside the system (its environment). A system that is not affected by entities outside the system. The mutual effect of a system component, or group of such components, on other components. The action agents through which effects are transmitted between system components, or between environment entities and system components. Forces that act between system components. Forces that act between environment entities and system components. The set of all actions that can influence the state of a system, or component. The external forces acting on the system, or component, as function of time. The set of all quantities that characterize the state of a system, or component. The response of the system, or system component. A system, or system component, which may be viewed solely as a transformer from input to output without knowledge of its internal workings. A system, or system component, which is described in sufficient level of detail to fully derive the transformation from input to output. A system, or system component, which is described at a level of detail intermediate between that of a white box and a black box.

Forces Internal Forces External Forces Input (general) Input (restricted) Output (general) Output (restricted) Black Box White Box Grey Box

Several terms are adaptations to physical systems of corresponding ones in Table 1.2.

it a bit more general: multiphysics. Table 1.3 takes up several useful concepts suggested by GST, and redefines them concretely for the dynamics of physical systems.10 Several changes, as well as a panoply of new terms, may be noticed. As regards system, the GST phrase “maintains its existence” is gone: a physical system, or a component thereof, may cease to exist through catastrophic events such as collision. The definition of state variables, missing in Table 1.2, is important, since those will be the primary unknowns to be carried along and solved for to get the system response. The notion of force, also missing there, is indirectly introduced as an agent or carrier of change.11 The distinction between the system proper and its environment is an important contribution of 10

Meaning systems considered in the physical sciences: mechanics, thermomechanics, optics, electromagnetics, and so on, at all observable scales.

11

The question of what is force is omitted since, as Joda says, it has no obvious answer.

1–7


;;;;; ;;;;; ;;;;;;;;;; (a)

(b)

DYNAMIC SYSTEM

Input: Forces

DYNAMIC SYSTEM

Input: Forces

Output: Motion

ENVIRONMENT

Output: Motion

ENVIRONMENT

Figure 1.1. Open dynamic system that exchanges input and output with its environment: (a) black box view; (b) grey box view. Double arrow symbols in the latter stand for component interaction effects.

GST. It allows a clear distinction between closed systems, in which the dynamic model includes everything, and open systems, in which the environment is viewed as separate: environment entities influence the system response but not the other way around. The distinction is crucial in practical applications since it allows engineers and scientists to focus on important interactions while neglecting unimportant feedback. The open vs. closed separation also offers an unambiguous way of classifying forces into external and internal. Many expositions still struggle with this particular decomposition. Note that definitions of input and output span two levels: general and specific. In a general sense everything coming into the system from outside (its environment) is input, while everything going out is output. The specific sense restricts these to external forces and state variable response, respectively. This duality is necessary to accomodate the unsettled use of such terms in the literature. Finally the definition of white, grey and black boxes responds primarily to te use of block diagrams to show system schematics. The distinction is important in verification and validation testing. §1.6. Open Systems and Hierarchical Decomposition All dynamical systems considered in this book are open. The system proper is distinguished from its environment. An input-output relationship is established, as schematically shown in Figure 1.1. In this figure, (a) depicts the system as a black box, which conceals internal details. On the other hand, (b) gives some system details and thus may qualify as a grey box. This view of open system can be continued hierarchically. Suppose the dynamic system is decomposed into first-level components. Each of these may be considered as an open subsystem, with the rest of the system as its environment. A first-level component may in turn be decomposed into second-level system components, which may also be viewed as open sub-subsystems, with the rest of the subsystem as environment. And so on. This process is continued into as many levels as necessary until further decomposition is deemed unnecessary. This “divide and conquer”, hierarchical multistage decomposition is natural for many engineering systems. Obviously it is not restricted to dynamics. It allows a problem to be decomposed into subproblems, subproblems into sub-subproblems, etc. Often the lowest component level is populated with black boxes, meaning that only their input-output transformer relationships need to be defined, while internal details of the transformer are of no relevance. The Finite Element Method provides a systematic way to do multilevel decomposition using the concept of superelements. For static problems this technique is described in Chapter 10 of [?]. 1–8

2

.

PMP & LMDFE Systems: Kinematics

2–1

2–2

Chapter 2: PMP & LMDFE SYSTEMS: KINEMATICS


§2.1.

Equivalence of Dynamical Systems §2.1.1. PMP-LMDFE Equivalence . . . . . . . §2.1.2. Generalization: More General Interaction Forces §2.1.3. Generalization: Continuum Models . . . . §2.2. Configurational Properties §2.2.1. Particle Coordinates . . . . . . . . . . §2.2.2. Velocities and Accelerations from Coordinates §2.2.3. Initial, Current and Final Configurations . . . §2.2.4. Initial Conditions . . . . . . . . . . §2.3. Motion Properties §2.3.1. Displacements . . . . . . . . . . . §2.3.2. Velocities and Accelerations From Displacements §2.3.3. Kinematic Constraints . . . . . . . . . §2.3.4. Response and Trajectories . . . . . . . §2.3.5. Possible and Actual Displacements . . . . . §2.3.6. Virtual Displacements . . . . . . . . §2.4. Mass Distribution Properties §2.4.1. Center of mass . . . . . . . . . . . §2.4.2. Linear Moment . . . . . . . . . . . §2.4.3. Quadratic Mass Moment . . . . . . . . §2.4.4. Mass Moment of Inertia . . . . . . . . §2.4.5. Mass Product of Inertia . . . . . . . . §2.4.6. Inertia Tensor . . . . . . . . . . . §2.4.7. Inertia Tensor Transformations . . . . . .

2–2

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . . . . .

. . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . . . . . .

. . . . . . .

2–3 2–3 2–5 2–6 2–7 2–7 2–7 2–7 2–8 2–8 2–8 2–9 2–9 2–10 2–10 2–11 2–11 2–12 2–12 2–12 2–12 2–13 2–13 2–14

2–3

§2.1

EQUIVALENCE OF DYNAMICAL SYSTEMS

§2.1. Equivalence of Dynamical Systems Classical dynamics reached maturity in the last half of the XIX Century. Over its 300+ year development, this discipline has been primarily concerned with point-mass particle or PMP systems.1 Such systems are composed by interacting particles of finite mass that evolve in time as per the action of external and internal forces. But the primary concern of this book are dynamic models constructed through the Finite Element method (FEM). Why should we worry about PMP systems? §2.1.1. PMP-LMDFE Equivalence The surprising answer is

Point-Mass Particle (PMP) systems with short-range interaction forces and Lumped-Mass-Discretized Finite Element (LMDFE) dynamic models are computationally equivalent. The system classes referred to above may be easier to grasp by looking at Figure 2.1. Instances are pictured as two-dimensional models for easier visualization. The model pictured in Figure 2.1(a) is a classical PMP system: a set of point-mass particles interacting with their neighbors (particle self interactions are not pictured to reduce clutter).2 The model shown in Figure 2.1(b) is a typical FEM mesh that discretizes a circular domain with 3-node triangles, in which the distributed mass is lumped at the nodes. This model only has translational freedoms, which are the node displacements. The presence of rotational freedoms is discussed in the next subsection. What does “computational equivalence” mean? That these seemingly disparate models produce semidiscrete equations of motion (EOM) in one-to-one correspondence. For example, the matrix EOM for linear, undamped dynamic models from either source can be written M u¨ + K u = f,

(2.1)

in which M, and K are the mass and stiffness matrices, respectively, u is a vector of displacements from a reference position, and f the corresponding vector of known forces. Although the EOM (2.1) for PMP and LMDFE models are constructed in completely different ways, once that is done the systems have exactly the same form. Table 2.1 shows an item-by-item correspondence. Note that there is none for one component of the LMDFE system: elements. This means that element-by-element operations, which for those models are the basis for forming and updating the semidiscrete EOM, have no PMP counterpart. But if abstraction is made on how such operations are carried out (or are viewed as a “black box”) the equivalence holds. 1

Recall that a PMP was defined in Chapter 1 as: “a particle that possesses only two essential attributes: position in space xi and finite mass m i . All other properties, such as spatial extent and internal structure, are irrelevant.”

2

Node-to-node FEM interactions forces are not necessarily node colinear (that is, act along the line that join the two nodes) if elements model a flexible medium. They are depicted that way in Figure 2.1 for visual simplicity.

2–3


(a)

2–4

(b)

Figure 2.1. Two dynamical models that are computationally equivalent in a formal sense: (a) a PMP system with short-range interaction forces, with particle self interactions not drawn to reduce clutter; (b) a Lumped-Mass-Discretized Finite Element (LMDFE) model built of 3-node triangles.

Why is the equivalence important? Knowledge reuse. Scientists from Newton through Poincar´e developed the theory and practice of PMP systems over three centuries. The fruits of that labor should not be allowed to go through waste just becase FEM and computers changed the way we do dynamic simulations. It is true that many of the classical expositions look cluttered and obscure in modern eyes. For example, Euler, Lagrange and Hamilton published their masterpieces way before matrix and vectors were invented, and lacked present knowledge of numerical methods and computation3 But their discoveries and results endure. There is no point in reinventing the wheel and letting all that fine work go to waste. What we can do is to present relevant excerpts of their results in a clearer and more compact notation, and reinterpret them as necessary in light of current modeling tools and computational methods. Remark 2.1. Note that the formal equivalence holds only between two spatially discrete models. It completely bypasses the sticky question of continuum model equivalence, which is briefly discussed in §2.1.3. Remark 2.2. Both models in 2.1 have only translational degrees of freedom. An immediate generaliztion is to allow for more general freedoms, such as, for instance, nodal rotations in FEM. On the PMP side, this equivalence is easily covered by the dynamic of rotating bodies, which in turn is a special case of the general description of Lagrangian dynamical systems by generalized coordinates.

3

In fact, Lagrange boasts in his famous book [227] about not having a single figure or diagram, and there are no numerical examples. (Contrast with Euler and the Bernoullis, who favored a balanced mix of theories and applications.) This lack of practical exercitation led Lagrange and others hands-off purists into various errors, some of which propagated for several generations. Most errors were fixed, however, over the course of time.

2–4

2–5

§2.1

EQUIVALENCE OF DYNAMICAL SYSTEMS

Table 2.1. Correspondence Between PMP and LMDFE Dynamical System Models LMDFE system

PMP system

Node Lumped node mass Stiffness and damping forces Applied node forces Node position Node displacements Node velocities Node accelerations Element

Particle Point mass Interaction forces Applied particle forces Particle position Particle displacements Particle velocities Particle accelerations No counterpart

§2.1.2. Generalization: More General Interaction Forces Each of the dynamic models illustrated in Figure 2.1 obeys certain restrictions: •

The PMP system in Figure 2.1(a) assumes short range interaction forces. More precisely, each particle interacts only with its neighbors as well as itself. What is behind this restriction? In the LMDFE model, the node-by-node interaction extends only over one element because of the local support feature of FEM.

•

The LMDFE system in Figure 2.1(b) assumes that the mass of the discretized domain is lumped at the nodes. This restriction is dictated by the definition of point-mass particle.

Can these restrictions be removed while mantaining computational equivalence? The answer is yes, but “equivalence” must be understood in a less formal sense. To make the discussion concrete, reference to the linear matrix EOM (2.1) will be convenient. Long range interaction. In PMP systems, allowing each particle interact with all others, as well as with itself, poses no problem. The effect on the EOM (2.1) is that the stiffness matrix K becomes fully populated. In conventional FEM models, K is sparse on account of the local support of element interpolation. There is no conceptual difficulty, however, in imagining LMDFE models based on a global interpolation (across elements) that couples all nodal freedoms. Although such models would be computationally inefficient, there are no laws that preclude them. Generalized mass discretization. A lumped-mass FEM discretization scheme produces a diagonal mass matrix M in (2.1). But more general mass discretizations, abbreviated by GMDFE, are in common use in FEM, and lead to a nondiagonal M. An example is consistent mass lumping. Accounting for this more encompassing approach has only one side effect on the EOM (2.1): the mass matrix M becomes nondiagonal. Such a change is practically important only as regard computational efficiency of certain procedures, for example explicit time integration. Corresponding PMP systems are less common because they would involve particle-to-particle interaction forces that depend on accelerations.4 4

Some controversy exists in this regard. In his well known book on Analytical Dynamics, Pars [298] states that accelerationdependent forces would violate the tenets of Newtonian dynamics. This claim has been shown to be fallacious in a recent paper [419].

2–5

2–6

Chapter 2: PMP & LMDFE SYSTEMS: KINEMATICS No generally known equivalence Limit continuum models

Source continuum models

Finite element discretization

Continuification

Point-Mass Particle (PMP) systems with long-range interaction

Informally equivalent

Generalized mass discretization

Generalized interaction range Point-Mass Particle (PMP) systems with short-range interaction

Generally-mass-discretized finite element (GMDFE) models

Formally equivalent Models illustrated in Figure 2.1

Lumped-mass-discretized finite element (LMDFE) models

Figure 2.2. Flowchart showing two generalizations levels beyond the equivalent dynamic models of Figure 2.1.

In summary, lifting both restrictions has only side effects on the configuration of the matrices K and M in the EOM (2.1). We can say that informal equivalence persists, since those changes have consequences on computational efficiency but not on methodology. See Figure 2.2 for a diagramatic representation of those extensions. §2.1.3. Generalization: Continuum Models A more challenging extension is to continuum models, governed by field equations. This can be done in two conceptually different ways: •

Passing to the continuous limit in a PMP system.

•

Going back to the source continuum model from which a FEM discretization was derived.

Although some forms of continuum-discrete equivalence can be proven between specific systems, the general case is still the matter of much debate and controversy in the physical sciences, as recently reviewed in [232]. Thus the “outer loop” of Figure 2.2 is not yet closed. Since that kind of correspondence is beyond the scope of this book, nothing more will be said about the topic. The following sections summarize well known kinematic properties of PMP systems. The description largely follow standard expositions of Classical Dynamics. Some deviations are made, however, when there is explicit need to link up with the finite element models that are the main target of the book. In compliance with this objective, matrix and vector notation will be emphasized.

2–6

2–7

§2.2

CONFIGURATIONAL PROPERTIES

§2.2. Configurational Properties The properties described here pertain to the composition of the system and its time evolution. The PMP system consists of N particles of mass m i in 3D space, in which i denotes the particle index. As regards terminology, for a equivalent FEM model, use the correspondences annotated in Table 2.1; for example “particle” → “node,” “point mass” → “lumped node mass,” etc. §2.2.1. Particle Coordinates The position coordinates of the i th particle are taken with respect to a fixed Rectangular Cartesian Coordinate (RCC) reference frame O x yz with origin at O and axes {x, y, z}. The coordinates of the i th point mass are {xi , yi , z i }.5 The coordinates are grouped into the 3-vector xi = [ xi

yi

z i ]T .

(2.2)

All particle coordinates are grouped into a 3 × N matrix form as X = [ x1

x2

. . . xN ] =

x1 y1 z1

x2 y2 z2

... ... ...

xN yN zN

.

(2.3)

For some derivations it is useful to have all particle coordinates arranged as a 3N -vector configured as (2.4) x = [ x1 y1 z 1 x2 y2 z 2 . . . x N y N z N ]T . In a dynamical system, particle positions evolve in time, so xi = xi (t), etc. For all particles one writes X = X(t) or x = x(t). §2.2.2. Velocities and Accelerations from Coordinates Particle velocities and accelerations are obtained by taking derivatives of the position coordinates with respect to time, viz., d 2 x(t) dx(t) . (2.5) , x¨ (t) = x˙ (t) = dt dt 2 These can be alternatively expressed in terms of particle displacements taken with respect to a fixed reference configuration, as described later in §2.3.2. §2.2.3. Initial, Current and Final Configurations Dynamic system simulations start at an initial time t0 , which is usually taken to be t0 = 0, and end at a final time t f . A generic time t considered within the interval [t0 , t f ] is called the current time. Positions occupied by the particles at initial, current and final times are called initial, current and final coordinates, respectively. The particle arrangements at t0 , t and t f are collectively called initial, current and final configuration, respectively. Configurations will be labeled by calligraphic letters: C0 , Ct and C f , respectively, to 5

Note that if the axes were labeled {x1 , x2 , x3 }, an additional subscript would be needed, cluttering formulas.

2–7

2–8


avoid confusion with symbols such as C0 , which express function continuity order.6 The subscript t in Ct may be dropped for brevity. It is often convenient for brevity to exhibit the time as a subscript of the position coordinate symbols (2.3) or (2.4); for example X0 = X(0),

Xt = X(t),

X f = X(t f ),

x0 = x(0),

xt = x(t),

x f = x(t f ).

(2.6)

§2.2.4. Initial Conditions For a dynamic system the initial conditions specified at C0 include Initial Position. The positions x0 = x(t0 ) of the particles at are given. Initial Velocities. The velocities x˙ 0 = x(t0 ) of the particles are given. The former may be substituted by initial displacements, while the latter may be substituted by initial momenta. If the motion is subject to kinematic constraints, as discussed in §2.3.3, the initial conditions are assumed to satisfy those restrictions. §2.3. Motion Properties This section introduces terminology associated with specific aspects of the kinematics of motion. Displacements are given a key role comparable to that of particle coordinates. This balanced viewpoint is motivated by finite element applications; see Remark 2.1 at the end of §2.3.2. §2.3.1. Displacements Displacements measure change in particle positions. The vector symbol for displacements is u, with its O x yz components denoted by u x , u y and u z . Suppose that the i th particle moves from xi (t1 ) to x(t2 ). The corresponding displacement vector is ui (t2 , t1 ) = xi (t2 ) − xi (t1 ) =

u xi (t2 ) − u xi (t1 ) u yi (t2 ) − u yi (t1 ) . u zi (t2 ) − u zi (t1 )

(2.7)

If t1 = t0 , the initial time, and t2 = t, the current time, the notation is simplified by dropping the arguments: ui = ui (t) = ui (t, t0 ) = xi (t) − xi (t0 ). (2.8) This ui is called the total displacement of the particle, or simply its displacement unless a distinction need to be made. For the set of all particles a 3 × N displacement matrix arranged like X in (2.3) is rarely used. More convenient is to have all displacements arranged as a 3N -vector configured as u = [ u x1 6

u y1

u z1

u x1

u y1

u z1

. . . ux N

uyN

u z N ]T ,

(2.9)

State and configuration should not be confused. The former term is quantitative: it refers to values of the state variables, which is a set that fully describes the system and its response to inputs. The latter term is qualitative: it refers to the spatial arrangement of the particles that make up the system. Emphasis is on form visualization or, informally, how the system is set up.

2–8

2–9

§2.3 MOTION PROPERTIES

in direct correspondence to the position coordinate vector displayed in (2.4). The omission of arguments in (2.9) implies change from the initial configuration C0 to to the current one C. For other time intervals, arguments may be used as necessary for clarity. Thus u = u(t) = u(t, t0 ) = x(t) − x(t0 ) = xt − x0 ,

u(t2 , t1 ) = x(t2 ) − x(t1 ).

(2.10)

For equivalent FEM models, u denotes the node displacement vector; cf. Table 2.1. In this case the entries in u are ordered node-by-node. §2.3.2. Velocities and Accelerations From Displacements As described in §2.2.2, particle velocities and accelerations result on taking time derivatives of the position coordinates; cf. (2.5). Since the initial configuration is fixed, they may be also obtained by taking time derivatives of total displacements: ˙ u(t) =

d(x(t)−x(t0 )) du(t) = = x˙ (t), dt dt

¨ u(t) =

d 2 u(t) d 2 (x(t)−x(t0 )) = = x¨ (t). dt 2 dt 2

(2.11)

Again for brevity time arguments may be dropped, or replaced by appropriate subscripts. For ˙ 0 ), etc. example, x˙ = x˙ t = x˙ (t), u˙ 0 = u(t Remark 2.3. In many textbooks and monographs on Classical Mechanics, total displacements are conspicuous for their absence. One may look in vain for symbols such as (2.7) through (2.10). More likely is to find xi for incremental displacements and δxi for virtual displacements (or xi and δxi if the exposition uses vectors). In sharp contrast, node displacements are the key state variables in the Direct Stiffness Method (DSM) of FEM, and play a fundamental role therein. Since FEM dynamic models are our ultimate objective, both displacements and position coordinates will be used in a balanced fashion.

§2.3.3. Kinematic Constraints Kinematic constraints are restrictions on the motion of a dynamic system. Although those restrictions are studied in greater detail in Chapter 5, we introduce here basic definitions required to discuss the distinction between possible and virtual displacements in §2.3.6. Two possible constraint forms are of interest. Holonomic constraints involve particle coordinates and possibly time: ci (x, t) = 0,

i = 1, . . . m h ,

(2.12)

A holonomic constraint that does not depend on time: ci (x) = 0, is called rheonomic. It is called scleronomic otherwise. Nonholonomic constraints are those that involve particle coordinates, particle velocities, and possibly time, and that cannot be integrated into a holonomic form. In this subsection we will consider only the subclass that is linear in velocities: di (x, t) x˙ + gi (x, t) = 0,

i = 1, . . . m n ,

(2.13)

The total number of constraints, m = m h + m n , must be less than the total number of degrees of freedom n DOF of a N -particle system, which is n DOF = 3N in 3D and n DOF = 2N in 2D. (Note that either m h or m n may be zero.) Differentiating (2.12) with respect to time yields ∂ci (x, t) ∂ci (x, t) x˙ + , ∂x ∂t 2–9

i = 1, . . . m h ,

(2.14)

2–10


The nonholonomic constraints (2.13) and the differentiated holonomic constraints (2.14) can be combined in the compact matrix form A x˙ = g, (2.15) in which A = A(x, t) is a m × n DOF matrix and g = g(x, t) is a n DOF -vector. Multiplying both sides by dt gives the differential form A dx = g dt.

(2.16)

A x˙ = 0.

(2.17)

The homogeneous version of (2.15) is Form (2.17) results if all holonomic constraints are rheonomic (that is, do not depend on time), and the velocity-independent terms gi of all nonholonomic constraints vanish. Note that the compact matrix forms (2.15) through (2.17) only apply if the nonholonomic constraints (2.13) are linear in the velocities, or if all constraints are holonomic, that is, m n = 0. A matrix form that accomodates nonlinear-in-velocities nonholonomic constraints is worked out later. Example 2.1. Suppose that a single particle moving in 3D space and referenced to the O x yz frame is subjected

to the two constraints x1 + y1 + z 1 =

1 2

B t 2,

x1 x˙1 + y1 y˙1 + z 1 z˙ 1 = 3 C,

(2.18)

in which x1 , y1 and z 1 denote the coordinates of the particle while B and C are coefficients that restore the proper physical dimensions. Time differentiation of the first (holonomic) constraint gives x˙1 + y˙1 + z˙ 1 = B t. Combining with the second (nonholonomic) constraints yields

1 x1

1 y1

1 z1

x˙1 y˙1 z˙ 1

=

Bt , 3C

(2.19)

which befits the matrix form (2.15). Here A is a 2 × 3 matrix.

§2.3.4. Response and Trajectories The vector function x(t) for t0 ≤ t ≤ t f that satisfies the equations of motion, initial conditions and kinematic constraints is called the response history, or simply response, of the PMP system. ˙ ¨ By analogy the associated functions u(t) = x(t) − x(0), u(t) = x˙ (t), and u(t) = x¨ (t), are called the displacement, velocity and acceleration response, respectively, of the system. If the focus is on an individual particle, say the i th one, the locus of its position history xi (t) is called the trajectory or orbit of the particle. By extension, x(t) is sometimes called the system trajectory, especially in control system applications. §2.3.5. Possible and Actual Displacements The system is assumed to be in configuration Ct at current time t. We now take a deeper look ˙ at motions in that neighborhood. The particle displacements at Ct are ut = u(t) and u˙ t = u(t), respectively. These are assumed to satisfy the EOM as well as any kinematic constraints such as ˙ is called the actual velocity at Ct . (2.12) and (2.13). Under these assumptions, u˙ t = u(t) 2–10

2–11

§2.4

MASS DISTRIBUTION PROPERTIES

Consider the displacement change between the current configuration at t and a possible one taken at an infinitesimal time increment t + dt: du P = x(t+dt) − x(t),

or

du P = u(t+dt) − u(t).

(2.20)

By possible configuration is meant one that satisfies the kinematic constraints grouped in (2.15), at t + dt. If so du P is called a possible displacement increment or simply possible displacement.7 Notice the omission of the time arguments in du P for brevity. If a possible displacement satisfies the equations of motion at Ct , in addition to the constraints, it is called a actual displacement increment or simply actual displacement. If the actual velocity u˙ t (t) at Ct is unique, the actual displacement du = u˙ t dt is also unique. §2.3.6. Virtual Displacements The vector of virtual displacements, denoted by δu, plays a central role in analytical mechanics. If there are no restrictions on the motion, any set of displacements relative to that at Ct qualifies. In this case, δu coalesces with the possible displacements defined by (2.20), whence δu ≡ du P . The definition becomes more subtle when there are kinematic constraints, which are collectively described by (2.15). Several equivalent definitions are possible; three of which are listed next. Algebraic Definition. Consider the homogeneous version (2.17) of (2.15). Any n DOF -vector that satisfies this homogeneous equation is a virtual displacement vector. More formally: multiply both sides of A x˙ = 0 by dt, and call δu = x˙ dt. Thus A δu = 0.

(2.21)

Slightly different track favored by some authors: take the differential version (2.16), set g = 0 and replace dx by δu. Definition From Possible Displacements. Consider two possible (and distinct) displacements du1P and du2P taken fromthe same configurationCt . Their difference is a virtual displacement: δ u = du2P − du1P .

(2.22)

Equivalence with the previous definition is readily established; see page 133 of [390]. Variational Calculus Definition. In the variational framework of PMP dynamics, either x or u is considered the state variable vector, whereas time t is the independent variable. Then δx or δu is the conventional state variation taken at “frozen” time t. Remark 2.4. Readers should be warned that many expositions have conceptual glitches as regards virtual

displacements. Some define it as “a displacement compatible with the constraints”, which is wrong unless (2.15) is homogeneous to begin with. The correct assertion is “a displacement compatible with the homogenized constraints.” Even more common is the statement that virtual displacements are infinitesimal. That is nonsense. By the algebraic definition A δu = 0. Therefore A (c δu) = 0 where c is an arbitrary factor, whence the virtual displacement order-of-magnitude is irrelevant: microns, kilometers or light-years make no difference. To exemplify how the aforemetioned glitches appear even in reputable expositions, consider the following statement in [347, p. 50]: “A virtual displacement is an arbitrary, instantaneous, infinitesimal change of the position of the system compatible with the conditions of constraint.” Score: two out of four. 7

Other names for du P in the literature are real displacements, admissible displacements and feasible displacements. Some expositions replace the differential symbol d by the increment symbol ; this makes no essential difference.

2–11

2–12


§2.4. Mass Distribution Properties This section collects properties associated with the distribution of masses in PMP systems and equivalent LMDFE models. As noted in Remark 2.5, these can be extended without difficulty to continuum mass distributions. §2.4.1. Center of mass The center of mass of a PMP system is the point located at i m i xi m x 1 1 i i = m i xi = (2.23) xC = i m i yi , m tot i m tot i m z i mi i i i in which i ranges from 1 through N , and m tot = i m i is the total mass. For a continuum body, see Remark 2.5. §2.4.2. Linear Moment The linear mass moment of an N -particle PMP system with respect to a point Q of coordinates x Q = [ x Q y Q z Q ]T is the 3-vector i m i xi Q pQ = m i (xi −x Q ) = i m i yi Q , (2.24) i i m i zi Q in which i ranges from 1 through N , xi Q = xi − x Q , yi Q = yi − y Q , and z i Q = z i − z Q . If point Q is the center of mass C, the three components of p Q → pC vanish. For a continuum body, see Remark 2.5. §2.4.3. Quadratic Mass Moment The quadratic mass moment of an N -particle PMP system with respect to a point Q of coordinates x Q = [ x Q y Q z Q ]T is the 3 × 3 symmetric matrix   m x x m x y i i i Q i Q i Q i Q i i i m i xi Q z i Q  , (2.25) PQ = m i (xi −x Q ) (xi −x Q )T =  i m i yi Q xi Q i m i yi Q yi Q i m i yi Q z i Q i i m i z i Q xi Q i m i z i Q yi Q i m i zi Q zi Q in which i ranges from 1 through N , xi Q = xi − x Q , yi Q = yi − y Q , and z i Q = z i − z Q . Taking Q to be the center of mass minimizes P Q in some norms. For a continuum body, see Remark 2.5. Remark 2.5. For a continuum body of density ρ and volume , the foregoing definitions apply if summa-

tions are replaced by integrations over the volume. More precisely, (2.23), (2.24), and (2.25) are replaced, respectively, by

ρ x dV , xC =

ρ dV

  ρ (x−x Q ) d V

p Q =  ρ (y−y Q ) d V  , ρ (z−z Q ) d V

  ρ (x−x ) (x−x ) d V ρ (x−x Q ) (y−y Q ) d V ρ (x−x Q ) (z−z Q ) d V Q Q

P Q =  ρ (y−y Q ) (x−x Q ) d V ρ (y−y Q ) (y−y Q ) d V ρ (y−y Q ) (z−z Q ) d V  .

ρ (z−z Q ) (x−x Q ) d V

ρ (z−z Q ) (y−y Q ) d V

2–12

ρ (z−z Q ) (z−z Q ) d V

(2.26)

2–13

§2.4

(a) dai m i

MASS DISTRIBUTION PROPERTIES

(b)

(c)

plane p dpi

z

plane q d mi qi

xi x

axis a

O yi

mi

zi

y

Figure 2.3. Moments and product of inertia of an individual particle of mass m i : (a) mass moment of inertia with respect to a line L at distance di ; (b) mass product of inertia with respect to two orthogonal planes at distances di1 and di2 , respectively; (c) mass moments and products of inertia with respect to a RCC frame Ox yz.

§2.4.4. Mass Moment of Inertia The mass moment of inertia,8 is closely related to the quadratic mass moment defined in §2.4.3 but is taken with respect to an axis instead of a point. It is a measure of the resistance of a massendowed object to rotation. It plays roughly the same role in rotational dynamics as mass does in translational dynamics. It connects related quantities such as angular momentum and angular velocity, as well as torque and angular acceleration. There is an associated quantity called the mass product of inertia, which completes the formation of the moment of inertia tensor, and appears in transformation equations. In the sequel the “mass” qualifier is often implied and may be omitted for brevity; for example moment of inertia written instead of the more precise mass moment of inertia. The mass moment of inertia of a particle of mass m i with respect to a given axis a is Ia = m i dai2 , where dai is the distance from the mass point to a (the perpendicular distance from a to the mass point). See Figure 2.3(a). For a PMP system of N particles the moment of inertia is obtained by summation: m i dai2 , i = 1, . . . N . (2.27) Ia = i

§2.4.5. Mass Product of Inertia Connected with moments of inertia is a quantity called the mass product of inertia. This is not very useful by itself, but completes the formation of the moment of inertia tensor described in the next subsection, and appears in its transformation equations. The mass product of inertia of a particle of mass m i with respect to a pair of given perpendicular planes p and q is m i d pi dqi , in which d pi and dqi are signed distances from the mass point to the planes. See Figure 2.3(b). For a PMP system of N particles, the product of inertia is obtained by summation: m i d pi dqi , i = 1, . . . N . (2.28) I pq = i

8

Also called rotational inertia, polar mass moment of inertia and angular mass

2–13

2–14


§2.4.6. Inertia Tensor To study properties when rotation axes are not known in advance, it is convenient to introduce the system mass inertia tensor with respect to a RCC frame Ox yz. This tensor is fully defined by a system mass inertia matrix. This is a symmetric 3 × 3 matrix formed by grouping mass moments and products of inertia as Ix x −Ix y −Ix z (2.29) I O x yz = −I yx I yy −I yz −I yx −Izy −I yz in which Ix x =

m i (yi2 + z i2 ),

i

Ix y = I yx =

I yy =

m i (z i2 + xi2 ),

i

m i xi yi ,

I yz = Izy =

i

Izz =

m i (xi2 + yi2 ),

i

m i yi z i ,

Izx = Ix z =

i

m i z i xi .

(2.30)

i

Here xi . yi and z i are coordinates of the i th particle of mass m i ; see Figure 2.3(c). §2.4.7. Inertia Tensor Transformations Consider an axis a passing through the RCC frame origin O, having direction cosines ax , a y and az with respect to x, y and z, respectively. Define the unit-vector t = [ ax a y az ]T . Then the moment of inertia with respect to a is given by Ia = tT I O x yz t.

(2.31)

Next consider a totated RCC system {a, b, c} with origin at O Construct the transformation matrix ax bx cx T = ay by cy az bz cz whose columns store the direction cosides of {a, b, c}, respectively, with respect to {x, y, z}. The transformed inertia tensor is I Oabc = TT I O x yz T. The associated eigenvalue problem is Ix yz c j = I j c j . The eigenvalues I j , j = 1, 2, 3, are the principal moments of inertia. Since Ix yz is symmetric real, the three eigenvalues are positive. It can be shown that Ix yz is also nonnegative, whence the eigenvalues are nonnegative. They are typically ordered as (2.32) I1 ≥ I2 ≥ I3 ≥ 0. The associated eigenvectors form the principal inertia directions. (TBC)

2–14

2

.

PMP & LMDFE Systems: Kinetics

2–1

2–2

Chapter 2: PMP & LMDFE SYSTEMS: KINETICS


§2.1. §2.2.

§2.3.

§2.4.

Introduction Newton’ Laws of Motion §2.2.1. First Law . . . . . . . . . . . . §2.2.2. Second Law . . . . . . . . . . §2.2.3. Third Law . . . . . . . . . . . Forces §2.3.1. Concept of Force . . . . . . . . . §2.3.2. Force Decomposition . . . . . . . . §2.3.3. Action-Reaction (AR) Force Decomposition §2.3.4. Internal-External Force Decomposition . . Particle Forces §2.4.1. External and Internal Forces . . . . . §2.4.2. Resultant Force and Moment . . . . .

2–2

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . .

. . . . . . . . . . . . . .

2–3 2–3 2–3 2–3 2–4 2–4 2–5 2–5 2–6 2–6 2–7 2–7 2–7

2–3

§2.2

NEWTON’ LAWS OF MOTION

§2.1. Introduction This Chapter continues the study of point-mass particle (PMP) systems by going over their kinetics. This is the study of the effect of forces upon the motion. Modeling such effects using Newton’s laws results in the equations of motion (EOM) of the system. The concept of work is briefly introduced in this Chapter, since it is necessary to support some criterion for force decomposition. The concept and applications of work and energy principles are elaborated in following Chapters. §2.2. Newton’ Laws of Motion The laws of motion enuntiated by Newton in his Principia [269] will be stated in axiomatic form, as derivable from observation. §2.2.1. First Law Original Statement. Lex I: Corpus omne perseverare in status suo quiescendivei movendi uniformiter in directum, nisi quatenus a viribus impressis cogito statum illum mutare. Literal English Translation. An object at rest will remain at rest unless acted upon by an external and unbalanced force. An object in motion will remain in motion unless acted upon by an external and unbalanced force. Modern Version. A body remains in a state of rest or of uniform rectilinear motion unless compelled to change its state by acting forces. This is essentially Galileo’s law of inertia, enunciated by him long before Newton: “A body moving on a level surface will continue in the same direction at a constant speed unless disturbed.” The concept was further developed by Descartes, whereas the term “inertia” was introduced by Kepler. However Newton recognized the fundamental important of the concept and stated it as the First Law of Motion. In a definition preceding the statement of this law, Newton introduces the body “quantity of motion” as “the measure of the same, arising from the velocity and the quantity of matter conjunctly” [347, p. 3]. By “quantity of matter” Newton meant the mass of the body. The product of mass and velocity is what we now call momentum. Since the mass is a scalar whereas velocity is a vector, the momentum is also a vector: ˙ p = m v = m u. (2.1) Of course vectors have not been invented at Newton’s time.1 Using the modern notation (2.1) we can restate the First Law succintly as: if no forces act on the body, the momentum p is conserved: p = C,

(2.2)

in which C is independent of time. 1

Vector analysis was simultaneously created by the American mathematician J. W. Gibbs and the English electrical engineer O. Heaviside in the late XIX Century; cf. [76]. The notation of Gibbs is that primarily in use today. Vector analysis and matrix algebra were not linked until the appearance of digital computers in the 1950w. This synergistic marriage was delayed by the unseemly preoccupation of algebrists with determinants for over a century.

2–3


2–4

§2.2.2. Second Law Original Statement. Lex II: Mutationem motus proportionalem vi motrici impressate, et fieri secumdum lineam rectam qua vis illa imprimatur. Literal English Translation. The alteration of motion is ever proportional to the motive force impressed, and is made in the right line in which the force is impressed. Modern Version. The change in motion is proportional to the acting force, and takes place in the direction of the straight line along which the force acts. Using again the modern concept of momentum (2.1), Newton’s “change in motion” can be briefly stated as: the change with time of the momentum: p˙ = dp/dt is proportional to the acting force on the body, denoted by f. Thus the second law can be written p˙ = f.

(2.3)

This may be labeled as the law of momentum. Evidently if f = 0, p is constant in time, thus recovering the conservation law (2.2). If the mass m is constant in time, (2.3) reduces to the more familiar “mass times acceleration” expression m v˙ = m u¨ = f.

(2.4)

This form is called Newton’s law of acceleration in the literature. Although more restrictive, (2.4) has a wide application range, since time-varying mass problems are rare in Classical Mechanics.2 §2.2.3. Third Law Original Statement. Lex III. Actionem contrariam semper et æqualem esse reactionem: sive corporus duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi. Literal English Translation. All forces occur in pairs, and these two forces are equal in magnitude and of opposite directions. Modern Version. For every action there is an equal and opposite reaction. This is the principle of action and reaction. It says that forces are always paired in nature. Example from [347, p. 6]: the falling stone attracts the Earth just as strongly as the Earth attracts the stone. This law makes it possible to move from the mechanics of single mass points to that of systems, whether at rest (statics) or moving (dynamics). §2.3. Forces The term force appears in each of the three laws introduced above. In [384, p. 527], Truesdell cites Hamel as writing “in the concept of force lies the chief difficulty in the whole of mechanics.” These difficulties motivated Hertz [190] to construct a Mechanics without the force concept. His attempt, however, was not successful within the framework of Classical Mechanics. So what is force? The dictionary is no help. In fact, definitions pertinent to mechanical forces given by the American Heritage Dictionary are: 2

They are more important in relativistic mechanics, because particle mass depends on its velocity relative to light speed.

2–4

2–5

§2.3

1.

Capacity to do work or cause physical change; strength; power.

2.

Power made operative against resistance; exertion.

3.

The use of such power or exertion.

FORCES

These three statements confuse power and force, a serious mistake (as defined later, power is work done in unit time). The “capacity to do work” is energy, which has dimensions of force times motion. Hence the foregoing definitions are technically useless. §2.3.1. Concept of Force As suggested by Sommerfeld [347], in Classical Mechanics force can be concretely defined as the right hand side of the law of momentum (2.3). It is true that the left side still contains the mass, which so far has only been defined qualitately as “quantity of matter.” But some properties of forces can now be stated. ˙ Consequently force is First, since the momentum defined in (2.1) is a vector, so is its time rate p. a vector: it has direction and magnitude. Second, the rule of composition of forces appears as a corollary to the laws of motion: two forces applied to the same mass point compound to act like the diagonal of the parallelogram formed by their vectors. If the forces are denoted by f1 and f2 , that parallelogram law states that the law of momentum (2.3) is simply p˙ = f1 + f2 = f.

(2.5)

Thus forces add as vectors to provide a resultant force. The above equation expresses succintly the principle of superposition of forces.3 §2.3.2. Force Decomposition The three Newtonian laws of motion, plus the principle (2.5), are sufficient to develop all of Classical Mechanics for PMP systems. Because of the importance of the force superposition principle, some authors elevate its status to that of a Fourth Law of Motion. Without taking sides in this issue, the crucial role of (2.5) lies in the possibility of decomposing forces. Force decomposition is crucial to the analysis of general PMP systems. The total force acting on each particle is in general a composition of effects that originate from different sources. For example, some forces may depend on position, some on displacements, others on velocities, and others on accelerations. Some forces may be prescribed as data, while others, called reactions, depend on kinematic constraints4 and are unknown until the response is obtained. In summary, force decomposition is essential to the construction of the correct equations of motion, their boundary and initial conditions, as well as their solution. It turns out that for complex dynamic systems there are several possible decompositions. Obvious question: which one should be used? Quick generic answer: whatever it works. More precisely, a decomposition that allows solvable equations of motion (EOM) to be formulated and solved in a 3

This is a precise restatement in vector notation of Galileo’s fuzzier Principle of Superposition: If a body is subjected to two physical influences that are independent of each other, it responds to each without modifying its response to the other. Galileo used this principle to study the motion of projectiles by superposing vertical and horizontal motions.

4

Kinematic constraints were introduced in §2.3.3.

2–5


2–6

systematic way. The qualifier systematic is important. Often ad-hoc decompositions allow a case by case investigation of specific problems, but are not extendible beyond those. Decompositions based on physical intuition do exist. A common one in older textbooks is: contact forces (e.g. collision) versus action-at-a-distance forces (e.g. gravity). The distinction is easy to understand and explain but largely useless aside of some special problems. Several proven top-level force decompositions are introduced in the following sections. §2.3.3. Action-Reaction (AR) Force Decomposition A useful top-level force decomposition givide forces into action and reaction:  Applied forces   Action forces Interaction forces Forces (2.6)   Reaction forces E-reactions C-reactions Reaction forces are those associated with kinematic constraints. They are the forces required to exactly enforce the constraint if the kinematic devices on which it depends are removed. All remaining forces are action forces. If the dynamic system is unconstrained, all forces are of action type. Reaction forces are subclassified into E-reactions and C-reactions. E-reactions are associated with the environment of the dynamic systemwhile C-reactions are associated with constraints internal to the system They are also called external reactions and internal reactions, respectively. Reaction forces are in subclassified into applied forces and interaction forces. Applied forces are those specified directly on a particle from physical effects external to the dynamic system. For example: gravity, fluid pressure, electromagnetic fields. Interaction forces are particle-to-particle effects, which typically depend on relative displacements, velocities or accelerations of the linked particles. Why is this decomposition useful? It helps to select unknowns. If all or part of the reaction forces are carried along in the EOM, the result is the Lagrange multiplier method, which is more fully described in the next Chapter. (TBC)

2–6

4

.

PMP Dynamical Systems: Matrix EOM Formulation

4–1

4–2

Chapter 4: PMP DYNAMICAL SYSTEMS: MATRIX EOM FORMULATION


§4.1. §4.2.

§4.3.

§4.4. §4.

Matrix Equations of Motion §4.1.1. Approaches to EOM Formulation . . . . EOM Derivation by Force Equilibrium §4.2.1. PMP Dynamic System Example . . . . . §4.2.2. Equilibrium Equations . . . . . . . §4.2.3. Matrix Form . . . . . . . . . . . §4.2.4. Matrix Symmetry Check . . . . . . . §4.2.5. Center of Mass Motions Check . . . . . §4.2.6. PMP Equations of Motion for Numerical Data EOM Derivation by the Finite Element Method §4.3.1. Element-Level Calculations . . . . . . §4.3.2. Assembly of Master Mass Matrix . . . . §4.3.3. The FEM Equations of Motion . . . . . Comparison Between Formulation Approaches Notes and Bibliography . . . . . . . . . . . . . . .

4–2

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

4–3 4–3 4–3 4–3 4–5 4–6 4–6 4–7 4–7 4–8 4–8 4–9 4–9 4–10 4–11

4–3

§4.2

EOM DERIVATION BY FORCE EQUILIBRIUM

§4.1. Matrix Equations of Motion This Chapter describe how to construct the dynamic equations of motion (EOM) in matrix form. Several simplifying assumptions are made for clarity of exposition: • The dynamical model is a point-mass particle (PMP) system, with point masses linked by collinear interaction forces. • Motions from the reference configuration are small in the sense that the initial geometry can be used throughout • Interaction forces depend linearly on displacements and may be idealized as extenssional springs. Damping effects are ignored although they are briefly covered in other Chapters. The last two assumptions ensure that the EOM are linear in displacements and accelerations. The first assumption (dynamic model is PMP) is partially lifted when considering the equivalence between two ways of constructing the matrix EOM: by equilibrium or via FEM, as outlined next. §4.1.1. Approaches to EOM Formulation Sections §4.2 and §4.3 cover two approaches to the formulation of matrix EOM: Equilibrium Method. Writing down force balance expressions for each point-mass particle via Free Body Diagrams (FBD) that account for dynamic effects modeled as forces. Finite Element Method. Assembling element-level EOM provided by a FEM discretization. The two methods are quite different in philosophy as well as procedure. The equilibrium method reflects the direct application of Newton’s laws to free bodies isolated from the rest of the system by appropriate external and internal forces. Technically those are known as Free Body Diagrams or FBD. By contrast, FEM equations are generally derived in a variational framework, such as the Principle of Virtual Work, Lagrange’s equations, or Hamilton’s principle. Nonetheless, for the PMP systems considered in this Chapter, the two methods produce exactly the same matrix EOM. Why then bother to go over both? The answer is that each has distinct advantages as well as shortcomings. Those are contrasted in §4.4. §4.2. EOM Derivation by Force Equilibrium The force equilibrium approach to deriving matrix EOM will be illustrated in the two-dimensional PMP system pictured in Figure 4.1. A reader familiar with IFEM may note its similarity with the example truss used in [106, Chapters 2–3]. That is not accidental: the stiffness matrix equations derived there will be reused for part of the EOM derivation via FEM carried out in §4.3. §4.2.1. PMP Dynamic System Example The two-dimensional PMP system of Figure 4.1(a) is referred to a fixed RCC frame {x, y}, with origin as shown in the Figure. The geometry is defined by the reference position of the three point masses, labeled 1,2 and 3. 4–3


fy3 , uy3 m3 = 4

(a)

3(10,10) 45

k(3) = 20 (3)

y 1(0,0) 45o m1 = 5 x

(b)

(1) k(1) = 10

(2)

3

o

4–4

fx3 , ux3

45o

k(2) = 5 (3) 2(10,0)

fx1 , ux1

(2)

(1)

45o

fx2 , ux2 2 fy2 , uy2

1 fy1 , uy1

m2 = 3

External forces: fx3ext=2H(t), fy3ext=H(t), others zero. Here H(t) is the Heaviside unit-step function

Figure 4.1. Example PMP system to illustrate derivation of matrix EOM by force equilibrium: (a) reference geometry and physical properties; (b) kinematic DOF and associate forces.

Point masses are connected by linearly elastic extensional springs, which are labelled as (1), (2) and (3) for convenience.1 The springs possess extensional stiffnesses k (1) , k (2) and k (3) with values indicated as in the Figure. The system only moves on the {x, y} plane. Consequently it has 6 kinematic degrees of freedom (DOF), which are taken to be the point mass displacements in the coordinate directions. These DOF are collected in a system displacement 6-vector with a mass-bymass arrangement: (4.1) u = [ u x1 u y1 u x2 u y2 u x3 u y3 ]T . The velocity and acceleration 6-vectors are configured accordingly as u˙ = [ u˙ x1

u˙ y1

u˙ x2

u˙ y2

u˙ x3

u˙ y3 ]T ,

u¨ = [ u¨ x1

u¨ y1

u¨ x2

u¨ y2

u¨ x3

u¨ y3 ]T .

f x3

f y3 ]T

(4.2)

The 6-vector of generic forces associated to (4.1) is f = [ f x1

f y1

f x2

f y2

(4.3)

Unlike displacements, several kinds of forces may be considered: external, internal, interaction, constraint (also called reaction), inertial, effective, residual, and so on. When there is need to distinguish force type, the generic symbols of (4.3) are adorned with appropriate 3-letter superscripts. For example, ext ext ext ext ext ext T f ext = [ f x1 f y1 f x2 f y2 f x3 f y3 ] , (4.4) int int int int int int T f int = [ f x1 f y1 f x2 f y2 f x3 f y3 ] , denote vectors of external and internal forces, respectively, for the example system. To distinguish “internal” from “interaction”, the latter is identified with superscript “iac”, as in f iac . For consistency with NFEM [107], residual forces may be denoted by r instead of f r es to reduce clutter. 1

This follows the element-identification convention of IFEM [106, Chapter 2], since in §4.3 the springs of Figure 4.1 are shown to be equivalent to bar elements in a truss FEM model. Enclosing parentheses distinguish these from point-mass labels, while avoiding confusion with exponents when placed as superscripts of spring properties.

4–4

4–5

§4.2


fy3ext

(a)

3 f s(3)

(3) f s = k(3) (ux3 cos

45 o + uy3 sin o

m3

(b)

(2)

1 m1

f s(1)

ext

fs

f s(3) f s(1)

(1) f s = k(1)(ux2 −ux1 )

..

m 3 u y3

f s(2)= k(2)(uy3 −uy2 )

− ux2 cos 45 − uy2 sin 45 )

2 m2

..

m 1 u x1

fx3ext f s(2)

f s(3)

f s(2)

45o o

3

.. m 3u x3

ext fx1

fy1 f (3) s f s(1)

1 m u..y1 1

..

m 2 u x2

f s(1) ext

fy2

f s(2) ext fx2

2 ..

m 2 u y2 Figure 4.2. Auxiliary diagrams in equilibrium EOM derivation for PMP system of Figure 4.1: (a) springinteraction forces in terms of displacements; (b) point-mass FBD diagrams. External, spring-interaction and inertial forces are pictured in black, blue and red, respectively, for visualization clarity.

Derivations are carried out first for a floating PMP system, which has no motion constraints. In Chapter 5 the EOM will be modified to account for kinematic constraints of various types. §4.2.2. Equilibrium Equations Begin by disconnecting the point-masses. Replace the springs by interaction forces f s(1) , f s(2) and f s(3) . These are depicted in Figure 4.2(a), in which their expressions in terms of point-mass displacements is listed. The + sense of the interaction forces is governed by an easy-to-remember convention: assume that each spring is in tension. To complete the Free Body Diagrams (FBD) shown in Figure 4.2(b), two more force sets are added: ext ext • The external forces f x1 through f y3 . ine ine • The inertial forces2 f x1 = −m 1 u¨ x1 through f y3 = −m 3 u¨ y3 . Their displayed sense is dictated by a simple rule: the pertinent acceleration is assumed positive. Since masses are positive, these forces point in the opposite sense of a + acceleration. See Figure 4.2(b).

The three force sets: external, interaction and inertial, are pictured in different colors in Figure 4.2(b) for easier visualization. Next, force equilibrium conditions for each degree of freedom (DOF) are written down. Interaction and inertia forces are expressed in terms of displacements and accelerations. For example, force balance of the motion of mass 1 along x gives ext ine + f s(1) + f s(3) cos 45◦ + f x1 f x1 = f x1 ext = f x1 + k (1) (u x2 −u x1 ) + k (3) u x3 cos 45◦ + u y3 sin 45◦ (4.5) − u x1 cos 45◦ − u y1 sin 45◦ cos 45◦ − m 1 u¨ x1 ext = f x1 + k (1) (u x2 −u x1 ) + 12 k (3) u x3 +u y3 −u x1 −u y1 − m 1 u¨ x1 = 0. 2

Also known as D’Alembert forces, pseudo forces, or effective forces in the literature.

4–5


4–6

At this point it is convenient to revert the sign of the LHS of (4.5) so as to make the inertia term −m 1 u¨ x1 positive. While doing this, common displacements are collected, and the external force term passed to the RHS. The rearranged expression is ext m 1 u¨ x1 + (k (1) + 12 k (3) ) u x1 − k (1) u x2 + 12 k (3) u y1 − 12 k (3) u x3 − 12 k (3) u y3 = f x1 .

(4.6)

Following exactly the same pattern, five more balance equations are obtained. The end result is six equilibrium equations, one for each DOF. §4.2.3. Matrix Form Once the equilibrium equations for all DOF are obtained and appropriate arranged as discussed above, several steps are carried out to put them in matrix form: • The external forces, which are givens, are kept in the RHS and arranged as a vector fext , configured as per the first of (4.4). • LHS terms that depend on accelerations are organized as a diagonal master mass matrix M postmultiplied by the acceleration vector u¨ configured as in the second of (4.2). • LHS terms that depend on displacements are collected and organized as a master stiffness matrix K postmultiplied by the displacement vector u configured as in (4.1). The resulting matrix EOM have the compact form M u¨ + K u = f ext . For the example system the mass and stiffness matrices are   m1 0 0 0 0 0 0 0 0   0 m1 0   0 0  0 m2 0  0 M=  = diag [ m1, m1, m2, m2, m3, m3 ]. 0  0 0 m2 0  0   0 0 0 0 m3 0 0 0 0 0 0 m3  (1) 1 (3)  1 (3) k + 2k k −k (1) 0 − 12 k (3) − 12 k (3) 2 1 (3) 1 (3)  k k 0 0 − 12 k (3) − 12 k (3)    2 2 (1) (1)  −k  0 k 0 0 0   K= (2) (2)  0 0 0 k 0 −k   1 1 1 1 (3) (3) (3) (3)  − k  − k 0 0 k k 2 2 2 2 1 (3) 1 (3) 1 (3) 1 (3) (2) (2) −2k −2k 0 −k k k + 2k 2

(4.7)

(4.8)

(4.9)

If the RHS of (4.7) consists entirely of external (applied, given) forces, we will often drop the superscript, and write M u¨ + K u = f, being tacitly understood that f contains only given forces. 4–6

(4.10)

4–7

§4.2


§4.2.4. Matrix Symmetry Check Observe that both M and K in (4.8) and (4.9) are symmetric. For M that is not a surprise, since real diagonal matrices are necessarily symmetric. Can that be expected for K? Yes: it is not an accident. It previews the equivalence of this approach with the Finite Element Method, as covered in §4.3. Since FEM equations are automatically symmetric when derived in a variational framework, equivalence implies symmetry. Should K come out unsymmetric, two fixes can be applied: • Reverse sign of individual equations (which become matrix rows) as necessary so that diagonal entries of K and M are positive. If they are not of the same sign, look for derivation errors. • If diagonal signs check out but unsymmetry persists, try row scaling by appropriate factors. §4.2.5. Center of Mass Motions Check After verifying symmetry, reduction to center-of-mass motion provides another useful EOM check. Assume that x and y translations are rigid by setting u x1 = u x2 = u x3 = u x and u y1 = u y2 = u y3 = u y for all t. Hence u¨ x1 = u¨ x2 = u¨ x3 = u¨ x and u¨ y1 = u¨ y2 = u¨ y3 = u¨ y . Replace these into vectors u and u¨ of (4.10). Add equations 1, 3, and 5, factoring u x and u¨ x , and add equations 2, 4, and 6, factoring u y and u¨ y . The result is M u¨ x = Fx ,

M u¨ y = F y ,

ext ext ext + f x2 + f x3 in which M = m 1 + m 2 + m 3 is the total mass of the system, whereas Fx = f x1 ext ext ext and F y = f y1 + f y2 + f y3 are the resultants of the x and y external forces, respectively. These are the well known equations for the translational motion of the center of mass, which is located at xC = 10(m 2 + m 3 )/M and yC = 10m 3 /M, as may be easily verified. A similar check can be performed for an infinitesimal rotation θ about the center of mass: the result is IC θ¨ = TC , where IC is the mass moment about C and TC the torque of the external forces about C.

In these reductions the contribution of the stiffness term K u cancels out. This results from the rigid-body property of the stiffness matrix: K u R = 0 if u R is a rigid body motion. A direct verification can be done by extracting the eigenvalues of K and checking that its rank is 3 [106, Chapter 20], but symbolic eigenvalue verification becomes unwieldy for larger systems. §4.2.6. PMP Equations of Motion for Numerical Data Numerical values for point masses and spring constants given in Figure 4.1(a) are: m 1 = 5, m 2 = 3, m 3 = 4, k (1) = 10, k (2) = 5, and k (3) = 20. For the external forces, Figure 4.1(b) states ext ext ext ext ext ext = f y1 = f x2 = f y2 = 0, f x3 = 2H (t), and f y3 = H (t), in which H (t) is the that f x1 Heaviside unit-step function. Replacing into the foregoing matrix expression yields 

5 0  0  0  0 0

0 5 0 0 0 0

0 0 3 0 0 0

0 0 0 3 0 0

0 0 0 0 4 0

   0 u¨ x1 20 0   u¨ y1   10    0   u¨ x2   −10  + 0   u¨ y2   0    u¨ x3 0 −10 u¨ y3 4 −10

10 −10 0 10 0 0 0 10 0 0 0 5 −10 0 0 −10 0 −5 4–7

−10 −10 0 0 10 10

    −10 u x1 0 −10   u y1   0      0   u x2   0   =  −5   u y2   0      u x3 10 2H (t) u y3 15 H (t) (4.11)


fy3 , uy3 3 (3)

(3)

= 2 2,

(3)

y fx1 , ux1 1 fy1 , uy1

45o

External forces: fx3ext=2H(t), fy3ext=H(t), others zero. Here H(t) is the Heaviside unit-step function

(2)

E (2) = 50, A(2) = 1, (2) L = 10, ρ(2) = 1/5

fx2 , ux2

x (1) (1)

E (1) = (1)

fx3 , ux3

45o

E = 100, A (3) 2, ρ(3) = 3/20 L = 10

4–8

50, A = 2, L = 10, ρ(1) = 1/5

2 fy2 , uy2

Figure 4.3. The three-member example truss of IFEM [106, Chapters 2-3] reproduced for convenience. The dynamic model is (for now) “floating”: it has no supports and thus may move freely on the {x, y} plane. It is subject to the external forces indicated in the box.

This is a linear system of six second-order ODEs in time. The problem specification is closed by providing 12 initial conditions: six displacements and six velocities at a start time, which is often taken to be t = 0. For example, the rest condition at t = 0 is specified by u x1 (0) = u y1 (0) = . . . , u y3 (0) = 0,

u˙ x1 (0) = u˙ y1 (0) = . . . , u˙ y3 (0) = 0.

(4.12)

A rest initial condition would be appropriate for a structure hit by a transient event at t = 0, e.g., earthquake, explosion or impact. §4.3. EOM Derivation by the Finite Element Method The FEM model for the example truss of IFEM [106, Chapters 2–3] is reproduced in Figure 4.3. One additional property is now activated: the mass density of the three elements (truss members). This is denoted by ρ e . §4.3.1. Element-Level Calculations As typical of FEM, computations proceed element by element. In this case two element-level matrices are generated: stiffness and mass. The element stiffness matrices are exactly those derived in IFEM [106, Chapters 2–3]. The reader is referred there for details. The mass matrices are produced by a nodal lumping scheme that proceeds as follows. The total mass of element e is m e = ρ e Ae L e , in which L e and Ae are the length and cross section area, respectively. Assign one half of this to each end node, in both the x and y directions. As a result the so-called lumped mass matrix of the truss (bar) element is 1 e  m 0 0 0 2 1 e  0 m 0 0  2  = 1 ρ e A e L e I4 , (4.13) Me =  1 e  0 0 m 0  2 2

0

0

0 4–8

1 e m 2

4–9

§4.3

EOM DERIVATION BY THE FINITE ELEMENT METHOD

(3)

m3 =3

(a)

3

m3 =1+3=4

(2)

m3 =1

(b) (3)

(2)

(3)

m1 =3 1

(2)

(1) (1)

m1 =2

(1)

m2 =2

2

m2 =1 m1 =3+2=5

m2 =2+1=3

Figure 4.4. Mass lumping procedure for the example truss of Figure ?: (a) element-level lumping; (b) addition at nodes to form master mass matrix.

in which I4 denotes the identity matrix of order 4. What happens to Me if the axes are rotated by an angle ϕ to be {x, ˜ y˜ }? It becomes e = (Te )T Me Te = ρ e Ae L e (Te )T I4 Te = ρ e Ae L e (Te )T Te , M where Te is a 4×4 transformation matrix very similar to that used in [106, §2.8]. But (Te )T Te = I4 e = Me for any ϕ. It follows that Me is invariant with respect to because Te is orthogonal, and M rotation of axes. Consequence: the machinery for deriving FEM equations in a local system and converting to global coordinates, as done for the truss element stiffness matrix, can be bypassed for the mass matrix. §4.3.2. Assembly of Master Mass Matrix The assembly of the master mass matrix follows exactly the same set of rules used for the master stiffness matrix. Since Me is diagonal, so will be the master mass matrix. A simple hand calculation proceed by inspecting which elements contribute to a given node. For the example truss: Node 1

gets half of the mass of elements (1) and (3): m 1 = 12 m (1) + 12 m (3) .

Node 2


Node 3


These values are assigned to both x and y directions. Consequently the master mass matrix is M=

1 2

diag [ m (1) +m (3) , m (1) +m (3) , m (1) +m (2) , m (1) +m (2) , m (2) +m (3) , m (2) +m (3) ]

= diag [ m 1 , m 1 , m 2 , m 2 , m 3 , m 3 ] = diag [ 5, 5, 3, 3, 4, 4 ].

(4.14)

This two-step construction of the master mass matrix is schematized in Figure 4.4. The node masses are m 1 = 5, m 2 = 3 and m 3 = 4. Remark 4.1. In later Chapters we will see that the lumped mass matrix is not the only choice; in fact there is an

infinite number of possible element mass matrices. Those other than the lumped one are no longer diagonal. Consequently the assembly process becomes more involved, but can be carried out by the same rules followed for the master stiffness matrix.

4–9


4–10

§4.3.3. The FEM Equations of Motion Combining the master mass matrix derived above with the master stiffness derived in IFEM [106, Chapter 3] and the external forces specified in Figure 4.3, we obtain         20 10 −10 0 −10 −10 u x1 0 u¨ x1 5 0 0 0 0 0 0 0 −10 −10   u y1   0   0 5 0 0 0 0   u¨ y1   10 10         10 0 0 0   u x2   0   0 0 3 0 0 0   u¨ x2   −10 0 + =         0 0 5 0 −5   u y2   0   0 0 0 3 0 0   u¨ y2   0         u¨ x3 u x3 −10 −10 0 0 10 10 2H (t) 0 0 0 0 4 0 u¨ y3 u y3 −10 −10 0 −5 10 15 H (t) 0 0 0 0 0 4 (4.15) Comparing(4.15) with (4.11) shows these matrix EOM to be identical. Is this a fluke? No. It is a consequence of two modeling choices. The mass matrices are the same because of the lumping choice in FEM: conservation conditions forces diagonal matrix entries to agree. If another choice had been made, the FEM mass matrix would be nondiagonal and clearly different from that of the PMP model. Stiffness matrices coalesce because we have set the spring constants of the PMP model of Figure 4.1 to be (4.16) k e = E e Ae /L e , e = (1), (2), (3), where E e , Ae and L e are member properties of the FEM truss model of Figure 4.3. As shown in Mechanics of Materials textbooks, as well as IFEM [106, Chapter 2], (4.16) provides the equivalent axial stiffness of a truss (bar) element. Given that the same EOM are obtained, downstream tasks such as • Application of constraint conditions • Modal analysis • Direct time integration • Internal force recovery need not distinguish between the two derivation methods. Those topics are covered in the following Chapters. §4.4. Comparison Between Formulation Approaches Having gone through the two EOM formulation methods, it is appropriate to summarize their relative advantages and shortcomings. The key advantages of the force equilibrium method are simplicity and physical transparency. These emanate from the use of just one modeling tool: Newton’s laws, as well as basic linear algebra. This makes it the method of choice in undergraduate instruction. Students in those courses have been exposed to those laws, as well as trained in static FBD and basic linear algebra through introductory lower-division courses, but have not yet encountered the more advanced mathematical machinery used in FEM.

4–10

4–11

§4. Notes and Bibliography

For modeling dynamic systems in practical projects in science or engineering, the equilibrium method is recommended in the following scenarios: • Preliminary design. The nature of the method encourages the use of highly simplified dynamic models. For example, an experienced engineer designing a car suspension would likely “lump” the car and the wheels as link-connected point masses, thus bypassing the overkill ingrained in using FEM models too early. Likewise, seismic design of a multistory building often starts with a point-mass “stick” model. Keeping-it-simple can significantly reduce design flowthrough, while avoiding costly modifications. • Particle model fits problem best. There are some scientific applications where PMP models are proper and natural. The classical example is Astronomy, which is precisely where Newtonian mechanics emerged as “system of the world.” The key advantage is the ability to discard ab initio internal particle structural details, which may not be known anyway. This enforces retention of key effects, such as inertia and gravitational interaction in that application. A FEM model of the Solar System at astronomical scales (or the trajectory of a space vehicle) would be not only absurd but a total waste of time. The key advantages of FEM are generality and automation. Once system details need to be included, the equilibrium force method rapidly loses its attractive simplicity. FEM is naturally adapted to distributed systems modeled by field (continuum) theories within a variational framework. Its computer implementation favors abstraction and modularity, reducing formulation errors while concealing internal processing details. Summarizing: in system design that involves dynamics, the two approaches form a natural hierarchy. The PMP approach may capture gross effects during preliminary design, while FEM can zero in details in subsequent stages. At the other extreme: verification and validation of existing (in situ) systems, FEM is typically the preferred choice. Notes and Bibliography (To be completed)

4–11

5

.

PMP Dynamical Systems: Kinematic Constraints

5–1

Chapter 5: PMP DYNAMICAL SYSTEMS: KINEMATIC CONSTRAINTS

5–2


§5.1. §5.2.

§5.3.

§5.4. §5.5.

§5.6.

§5.7. §5.8. §5.

Overview Analytically-Related Constraint Classification §5.2.1. Algebraic, Differential or Integral . . . . . . . §5.2.2. Stationary or Time Dependent . . . . . . . §5.2.3. Equality or Inequality . . . . . . . . . . Implementation-Related Constraint Classification §5.3.1. Single or Multiple Freedoms . . . . . . . . §5.3.2. Homogeneous or Nonhomogeneous . . . . . . §5.3.3. Linear or Nonlinear . . . . . . . . . . . Unified Treatment By Constraint Forces Holonomic SFC Examples §5.5.1. Scleronomic, Homogeneous SFC . . . . . . . §5.5.2. Rheonomic, Nonhomogeneous SFC . . . . . §5.5.3. Pseudo-Nonholonomic SFC . . . . . . . . Holonomic, Homogeneous MFC Examples §5.6.1. Scleronomic, Homogeneous MFC by Master-Slave §5.6.2. Rheonomic, Nonhomogeneous MFC by Master-Slave Penalty Function Augmentation Methods for MFC Lagrange Multiplier Adjunction Methods for MFC Notes and Bibliography . . . . . . . . . . . . . . . . . .

5–2

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . .

5–3 5–3 5–4 5–4 5–5 5–5 5–5 5–5 5–7 5–7 5–8 5–8 5–9 5–10 5–10 5–10 5–11 5–12 5–12 5–12

5–3

§5.2

ANALYTICALLY-RELATED CONSTRAINT CLASSIFICATION

§5.1. Overview Kinematic constraints are restrictions on the motion of a dynamic system.1 If there are no constraints, the system is said to be unconstrained or unrestrained. If an unconstrained model represents a structure or solid body, it is also called free-free or floating.2 The PMP example system treated in the previous Chapter is of this type. Familiar examples: an airplane or bird in flight, a planet, satellite or rocket in orbit. Constrained systems are more common, however, especially in Civil and Mechanical Engineering. For functional reasons, such as stability, control, safety, or operational usability, motions may be restricted by devices such as building foundations, bridge supports, member joints, failsafe stops, sliding guides, etc. This Chapter explains how to impose some practically important constraint types into a PMP dynamical system, once the unconstrained semidiscrete EOM are set up by one of the methods covered in the previous Chapter. Although the example is admittedly simplistic, most techniques can be extended to the more complex systems treated in later Chapters. The introduction of kinematic constraints into a dynamical system comprises three stages: • Idealization: stating physical restrictions on the motion as mathematical expressions. • Implementation: modifying unconstrained EOM so that the idealized constraints are verified. • Solution: accounting for constraints in the solution procedure if and as necessary. The first step results in a mathematical statement that may include displacements and/or velocities (involvement of accelerations is relatively rare), as well as possibly time and associated forces. In simple cases the idealization may be obvious from inspection. For instance, suppose that point mass 2 of the PMP example system of the previous Chapter, illustrated in Figure 4.1, can only move along x. Then evidently u y2 = 0. Such local or single-DOF constraint forms are often called spatial boundary conditions in the literature.3 More complicated scenarios lead to constraints that link multiple DOF. These are discussed in [106, Chapters 8–9] for the static case. But the spectrum of kinematic constraints in dynamics is far richer, because they may display time dependence and/or involve temporal state derivatives such as velocities. Accordingly, constraint implementation techniques span a wider range, and may bring about additional modeling as well as computational difficulties. To prevent such troubles it helps to beware of the different types of constraint that may occur in practice. Hence a categorization is in order. For convenient reference, the classifications outlined in §5.2 and §5.3 are summarized in Tables 5.1 and 5.2, respectively. 1

The appropriate definition for constraint, taken from the American Heritage Dictionary, is “something that restricts, limits or regulates.” Synonyms: coercion, control, limitation, prevention, regulation, restriction, restraint, suppression. The qualifier “kinematic” introduces the flavor of physical motion. The term “constraint” is relatively recent; other names found in the older literature are: side conditions and auxiliary conditions.

2

Qualifier free-free has the connotation of “exempt from external authority, interference or restrictions,” whereas qualifier floating is used in the sense of “having little or no attachment to a particular place.”

3

In Mathematics, the term “boundary condition” technically applies to situations where the boundary of a region can be readily identified, as in boundary-value problems. But occasionally the meaning is informally extended to encompass constraints of local type, whether pertaining to a boundary or not.

5–3


5–4

§5.2. Analytically-Related Constraint Classification In Classical Mechanics4 constraints were primarily classified according to mathematical features relevant to the use of analytical solution techniques. These include: (i) algebraic, differential or integral form; (ii) time dependence or independence, and (iii) equality or inequality. The following definitions assume that the idealized expression is of equality type since inequality constraints are beyond the scope of the book; see §5.2.3 for more details. §5.2.1. Algebraic, Differential or Integral An equality kinematic constraint is holonomic5 if it can be expressed in the algebraic form f k (u, t) = 0,

(5.1)

in which k denotes a constraint index, u is the vector of state variables (in PMP systems, point-mass displacements). Holonomic constraints are often associated with conservative systems. An equality kinematic constraint is nonholonomic if it is expressed as a differential or algebraic˙ differential form that includes state variable velocities collected in vector u: ˙ t) = 0, f k (u, u,

(5.2)

and is not reducible to (5.1) by either integration, or differentiation followed by elimination. [Constraints originally stated as (5.2) but reducible to (5.1) are sometimes called pseudo-holonomic or semiholonomic.] These are often associated with nonconservative systems (those exhibiting dissipative effects such as friction), as well as rolling contact or robotic motions. An equality constraint is isoparametric if it is expressed as a definite integral in time; for example

t2

˙ t) dt = C, f k (u, u,

(5.3)

t1

where C is a constant. Such conditions are important in contexts that involve conservation of some quantity such as path length or energy, but will not be treated here. An obvious generalization of (5.2) is an equality differential form that includes accelerations: ˙ u, ¨ t) = 0, and is not reducible to either (5.1) or (5.2). But as previously noted such forms f k (u, u, are comparatively rare.6 4

“Classical” refers to a period extending roughly from Newton to the advent of digital computers in the early 1950s. For major advances in the treatment of kinematic constraints during this period, see Notes and Bibliography.

5

The distinction between holonomic and nonholonomic constraints was first emphasized by Hertz in 1894 [381], and further studied by Ferrers [375], Routh [414] and other scientists of the time. For a historical account of these developments, see [390].

6

One reason for their rarity is that selected accelerations can often be eliminated using the original EOM.

5–4

5–5

§5.3

IMPLEMENTATION-RELATED CONSTRAINT CLASSIFICATION

§5.2.2. Stationary or Time Dependent Here are two more Victorian-era adjectives for impressing your Facebook friends. The holonomic constraint (5.1) is called scleronomic if it is stationary (that is, time independent), thus reducing to f k (u) = 0. It is rheonomic otherwise.7 Such a distinction is not relevant for the nonholonomic form (5.2), which depends at least implicitly on time because it contains temporal derivatives. This classification is vacuous for the static case, in which only the scleronomic holonomic form f k (u) = 0 occurs since both time and velocities are absent. §5.2.3. Equality or Inequality Note the qualifier equality used for the definitions in §5.2.1. If the equals sign in (5.1) or (5.2) is replaced by > or ≥, the expression becomes an inequality constraint. In Mechanics these occur primarily in contact and impact problems as well as optimization. Those forms are beyond the scope of this book, and will not be considered further. Computational handling of inequality constraints normally demands application-specific tools, such as search algorithms for contact-impact problems and sequential quadratic programming (SQP) methods in optimization. Those highly specialized topics are covered in advanced textbooks and monographs. §5.3. Implementation-Related Constraint Classification The classification discussed in §5.2 was important in the pre-computer era, where long and laborious dynamic calculations were carried out by hand.8 As computational dynamics evolved in the computer era, certain constraint features previously ignored or neglected were found to have significant impact on the computer implementation as well as in organizing time-stepping calculations. These are collected in Table 5.2 and summarized in the following subsections. §5.3.1. Single or Multiple Freedoms If the constraint involves just one state variable, it is called Single Freedom Constraint or SFC. if it links more than one, it is a Multiple Freedom Constraint, or MFC. The distinction was introduced for static FEM analysis in IFEM [106, Chapters 8–9], and is also important in dynamics. Suppose that for the example PMP system we specify a holonomic SFC such as u y2 = 0, or u y2 = 2 + 12 t 2 . The accelerations u¨ y2 are computed, and together with u y2 replaced into the matrix EOM. Known data is transferred to the RHS and the modified EOM partitioned to exhibit the remaining, unconstrained DOF. This reduction technique is illustrated with examples in §5.5. How about u y2 = 4u˙ y2 ? Some preprocessing is required. It is integrated to yield u y2 = exp(t/4) + C y2 , with C y2 determined by initial conditions. Then one proceeds as in the previous case. If integration is difficult or impossible, treat as a MFC. For a holonomic MFC, the three implementation methods described in IFEM [106, Chapter 89], and collected in Table 5.3, are still applicable with some extensions illustrated later. For a nonholonomic MFC, only the Lagrange multiplier method, used with extreme care, offers hope. 7

Roots come from Greek sclero: hard or solid, and rheo: flowing.

8

For example, the mathematician E. W. Brown (1866–1938) spent most of his professional life doing accurate calculations of the Moon orbit, reaching 1500 perturbation terms. A laptop can reproduce his 1908 tables in a few seconds.

5–5


5–6

Table 5.1. Constraint Classification Related to Analytical Issues Type

Definition

Comments

Holonomic Nonholonomic Isoparametric

Algebraic form (5.1) Differential form (5.2) Integral form (5.3)

If not reducible to (5.1) Not treated here

Scleronomic Rheonomic

Holonomic form (5.1), with no t dependence Holonomic form (5.1), with explicit t dependence

Equality Inequality

= separates LHS and RHS ≥ or > separates LHS and RHS

Not treated here

Table 5.2. Constraint Classification Related to Implementation Issues Type

Description

Comments

Single freedom (SFC) Multiple freedom (MFC)

One DOF in (5.1) or (5.2) Multiple DOF in (5.1) or (5.2)

Tougher to implement

Homogeneous Nonhomogeneous

Homogeneous polynomial in state variables Not meeting previous definition

Linear Nonlinear

State variables appear linearly State variables appear nonlinearly

No RHS modification

Not treated here

Table 5.3. Constraint Implementation Methods Method

Variants

Procedure

Elimination

Direct Master-slave

Substitute, then partition Congruential transformation on stiffness & mass

Penalty function

Standard (one pass) Augmented Lagrangian

Inject penalty elements of fixed weights Inject penalty elements but iterate on weights

Lagrange multiplier

Standard Regularized

Constraint forces are made system unknowns Similar to Augmented Lagrangian

§5.3.2. Homogeneous or Nonhomogeneous A scleronomic holonomic constraint is homogeneous if it is expressible as a homogeneous polynomial in the state variables. Otherwise it is nonhomogeneous. For the PMP example system, u y2 = 0 is homogeneous, and so is (u x2 − u x1 )n = 0 for any integer exponent n > 0. On the other hand u x2 = t, u x2 − u x1 = 2, and u 2x2 = 1 + u 2x1 , are nonhomogeneous. Rheonomic (time dependent) holonomic constraints, as well as nonholonomic ones, are always considered nonhomogeneous. 5–6

5–7

§5.4

UNIFIED TREATMENT BY CONSTRAINT FORCES

;;;;;; ;;;;;; ;;;;;;

f con

Figure 5.1. Remove the bridge (log) and replace its effect on the elephant by constraint forces fcon on the legs. The elephant stays happy: nothing happens. This illustrates Newton’s Third Law of action and reaction, whence constraint forces are often called reaction forces by engineers.

The significance of this feature in computer implementation is that, if the constraint is nonhomogeneous, applied force terms must be modified and converted to effective forces if elimination or penalty function methods are used. No such modifications are needed in the homogeneous case. §5.3.3. Linear or Nonlinear A holonomic constraint is linear if any displacements present in the constraint appear linearly. Otherwise the constraint is nonlinear. The dependence on time may be arbitrary. For instance u x1 − 2u y1 = t 2 is linear but (u x2 − u x1 )2 + (u y2 − u y1 )2 = 0 is not. Likewise, a nonholonomic constraint is linear if any displacements and velocities present in the constraint appear linearly. The dependence on time may be arbitrary. The presence of nonlinear constraints affects computer implementation if the time marching solution procedure iterates at each step, as is often the case with implicit time integrators. If so, such constraints must be appropriately linearized. Nonlinear constraints are not treated in this book. For convenient reference, Table 5.3 collects constraint implementation methods. Their basic features are described in IFEM [106, Chapters 3, 8 and 9]. Modifications required for the dynamic case can be learned from the examples that follow. §5.4. Unified Treatment By Constraint Forces Looking at the Tables 5.1 through 5.3 may discourage the reader. There are 12 constraint classes listed, along with 3 implementation methods with 2 variants each. But there is no cause for alarm. Only a few of the simplest constraints are those overwhelmingly found in practical engineering and physics problems. In previous chapters the unifying role of forces in unconstrained dynamic systems has been emphasized. Their role is equally strong once constraints are introduced. The basic rule is Any kinematic constraint may be replaced by a system of forces

(5.4)

See cartoon in Figure 5.1. Unsurpringly, these forces are called constraint forces. Since this recipe is nothing more than Newton’s Third Law of action and reaction under another name, they are often 5–7


fy3 , uy3 3

5–8

fx3 , ux3

45o

(3)

(2)

fy1 , uy1

;;

45o

(1)

2 fy2 , uy2 fx2 , ux2

;;

fx1 , ux1 1

Figure 5.2. Support conditions for Example worked out in §5.5.1.

called reaction forces by engineers, particularly when they happen at a foundation or support. From a mathematical standpoint, constraint forces can always be identified with Lagrange multipliers, which explains the power and generality of this methodology. The examples that follow will illustrate the rule (5.4) in action. §5.5. Holonomic SFC Examples We begin our set of examples with the simplest kinematic constraints. These are holonomic, single freedom constraints (SFC). Three variations on this theme are considered in this Section. §5.5.1. Scleronomic, Homogeneous SFC The floating example truss studied in Chapter 4 (see Figure 4.3) is now supported as shown in Figure 5.2. Node 1 is fixed whereas node 2 can only moved along x. Thus displacements u x1 , u y1 , and u y2 vanish and so do the associated accelerations u¨ x1 , u¨ y1 , and u¨ y2 . We can state the constraints as u x1 = u y1 = u y2 = 0, u¨ x1 = u¨ y1 = u¨ y2 = 0. (5.5) In the classification covered previously, these are scleronomic (time-independent), holonomic, and homogeneous SFC. To implement, insert (5.5) into the unsconstrained EOM (4.15) derived in the previous Chapter to get 

0 0  3  0  0 0

0 0 0 0 4 0

  0 −10 0

 0  u¨ x2  0  10  u¨ x3 +  0  0  u¨ y3  0 0 4 0

−10 −10 0 0 10 −10

    −10 0 −10 

 0     u x2   0   0    u x3 =  + −5   0    u y3    10 2H (t) 15 H (t)

con  f x1 con f y1   0  ext +f con . (5.6) con  = f f y2   0 0

in which fcon collects unknown constraint (support reaction) forces. System (5.6) is obtained from the matrix EOM (4.15) by striking out columns 1, 2 and 4 of M and K. (Those can be deleted 5–8

5–9

§5.5

HOLONOMIC SFC EXAMPLES

because they multiply zero entries.) Now (5.6) partitions naturally into two subsets. Keeping rows 3, 5 and 6 produces the reduced system

10 0 0 u x2 0 3 0 0 u¨ x2 (5.7) u¨ x3 + 0 u x3 = 2H (t) 10 10 0 4 0 u¨ y3 u y3 0 −10 15 H (t) 0 0 4 which has no constraint forces. This system provides three linear ODE for three unknowns, and may be solved once the initial conditions are specified. The other subset consists of rows 1, 2 and 4. Because accelerations drop out (note the 3 null rows in the reduced M), we get an algebraic system: con

f x1 −10 −10 −10 u x2 con = (5.8) f con = f y1 u x3 , 0 −10 −10 con f y2 u y3 0 0 −5 This is called the reaction recovery system because the constraint (reaction) forces may be computed, if so desired, once u x2 , u x3 and u y3 are obtained by solving (5.7) in time. Note that (5.8) is time independent, and merely reflects static force equilibrium. §5.5.2. Rheonomic, Nonhomogeneous SFC The truss support conditions of the foregoing example are modified by making u x1 , u y1 , and u y2 specified functions of time, which are denoted by gx1 , g y1 , and g y2 , respectively: u x1 = gx1 (t), u y1 = g y1 (t), u y2 = g y2 (t),

u¨ x1 = g¨ x1 (t), u¨ y1 = g¨ y1 (t), u¨ y2 = g¨ y2 (t). (5.9)

These are rheonomic (time-dependent), holonomic, and nonhomogeneous SFC. As before insert (5.14) into (4.15). But now contributions from columns 1, 2 and 4 do not vanish, and are transferred to the RHS before those columns are removed:     −10 −10 −10 0 0 0 0 0 0

 0 −10 −10 

  u x2   u¨ x2 0 0   10 3 0 0 (5.10)  u x3 = f ext + f,tra + f con .   u¨ x3 +  0 0 −5 0 0 0     u ¨ u     y3 y3 0 10 10 0 4 0 0 −10 15 0 0 4 in which



f ext

 0  0     0  = ,  0    2H (t) H (t)



f tra

 −5g¨ x1 − 20gx1 − 10g y1  −5g¨ y1 − 10gx1 − 10g y1    10gx1   = , −3g¨ y2 − 5g y2     10gx1 + 10g y1 10gx1 + 10g y1 + 5g y2

 f con

   =  

con  f x1 con f y1   0  con  , f y2   0 0

denote external (applied), transfer and constraint forces, respectively. Only f con carries unknowns. Keeping rows 3, 5 and 6 provides the reduced EOM system

10 0 0 u x2 10gx1 3 0 0 u¨ x2 (5.11) 10 10 u¨ x3 + 0 u x3 = 2H (t) + 10gx1 + 10g y1 0 4 0 u¨ y3 u y3 H (t) + 10gx1 + 10g y1 + 5g y2 0 −10 15 0 0 4 5–9


5–10

This RHS is a known function of time; so (5.11) can be directly solved once the initial conditions are given. The other subset, which consists of rows 1, 2 and 4, gives the reaction recovery system con

f x1 −10 −10 −10 u x2 5g¨ x1 + 20gx1 + 10g y1 con = (5.12) f y1 u x3 + 5g¨ y1 + 10gx1 + 10g y1 . 0 −10 −10 con f y2 u y3 3g¨ y2 + 5g y2 0 0 −5 This can be used to recover the constraint (support reaction) forces once u x2 , u x3 and u y3 are obtained by solving (?) in time. §5.5.3. Pseudo-Nonholonomic SFC The truss support conditions are specified in terms of time-dependent velocities, and time differentated to get accelerations: u˙ x1 = 1 + t,

u˙ y1 = 2t,

u y2 = −3t 2 ,

u¨ x1 = 1,

u¨ y1 = 2,

u¨ y2 = −6t.

(5.13)

These look like nonholonomic constraints, but actually can be integrated in time to get displacements: u x1 = t + 12 t 2 + C x1 , u x1 = t 2 + C y1 , u y2 = −t 3 + C y2 . (5.14) The constants of integration can be determined from the initial conditions. For example suppose that at t = 0 we have u x1 (0) = u y1 (0) = u y2 (0) = 0. Then C x1 = C y1 = C y2 = 0, and we are back to the rheonomic, holonomic SFC case treated in §5.5.2. §5.6. Holonomic, Homogeneous MFC Examples The examples in this Section pertain to a more complicated case: a set of holonomic homogeneous constraints, at least one of which involves multiple freedoms (MFC). The appearance of MFC calls for more powerful implementation methodologies that straightforward elimination. §5.6.1. Scleronomic, Homogeneous MFC by Master-Slave The example truss is subject to three MFC: u y1 = −u x1 ,

u y2 = u x2 ,

u y3 = −u x3 .

(5.15)

Interpretation for infinitesimal displacements: the truss rotates about the midpoint of element (3) halfway between nodes 1 and 3. Differentiate (5.23) twice in time to establish acceleration constraints: u¨ y1 = −u¨ x1 , u¨ y2 = u¨ x2 , u¨ y3 = −u¨ x3 . (5.16) To apply the master-slave elimination method, pick 3 slave displacement DOF, say the y components u y1 , u y2 and u y3 , which will be eliminated.9 The x components: u x1 , u x2 and u x3 are the master DOF. Write down the DOF transformation         u x1 u¨ x1 1 0 0 1 0 0  u y1   −1 0 0   u¨ y1   −1 0 0 

     u x1    u¨ x1  u x2   0 1 0   u¨ x2   0 1 0  (5.17)   =  u x2 , =  u¨ x2 .  u y2   0 1 0   u¨ y2   0 1 0  u u ¨         x3 x3 u x3 u¨ x3 0 0 1 0 0 1 u y3 u¨ y3 0 0 −1 0 0 −1 9

For details of this method in the static case, see IFEM [106, Chapter 8].

5–10

5–11

§5.6

HOLONOMIC, HOMOGENEOUS MFC EXAMPLES

In compact form ˆ u = T u,

u¨ = T uˆ¨

(5.18)

ˆ˙ which may be used for initial conditions as noted below. Substitute Likewise for velocities: uˆ˙ = T u, u and u¨ in the unconstrained matrix EOM M u¨ + Ku = f and premultiply both sides by TT to get the transformed EOM (5.19) TT M T uˆ¨ + TT K T u = TT f. or ˆ uˆ¨ + K ˆ u = ˆf. M

(5.20)

in which ˆ = TT M T = M

8 0 0 10 10 0 0 ˆ = TT K T = −10 15 5 , ˆf = TT f = . (5.21) 0 6 0 , K 0 0 0 10 0 5 5 2H (t)

ˆ has rank 2 and nullity 1. The initial conditions u(0) = u0 and The transformed stiffness matrix M ˆ˙ ˆ ˙ = uˆ 0 = T u0 and u(0) = vˆ 0 = T v0 . Once the reduced matrix u(0) = v0 transform as per u(0) ˆ EOM (5.26) is solved in time for uˆ = u(t), the slave displacements, velocities and accelerations may be recovered from the constraints (5.23); or, equivalently, from the master/slave transformation equations. §5.6.2. Rheonomic, Nonhomogeneous MFC by Master-Slave The example truss is subject again to three MFC but now they are nonhomogeneous and rheonomic: u y1 = −u x1 + g1 (t),

u y2 = u x2 + g2 (t),

u y3 = −u x3 + g3 (t).

(5.22)

Here gi (t), i = 1, 2, 3 are arbitrary but twice differentiable functions of time, which may reduce to constants. If all three functions vanish (5.23) coalesces with the homogeneous case (5.23). Differentiating (5.23) twice in time gives the acceleration constraints: u¨ y1 = −u¨ x1 + g¨ 1 (t),

u¨ y2 = u¨ x2 + g¨ 2 (t),

u¨ y3 = −u¨ x3 + g¨ 3 (t).

(5.23)

As in the previous example, the slave displacements are u y1 , u y2 and u y3 , while u x1 , u x2 and u x3 are the master DOF. The master-slave transformations may be written in the compact matrix form u = T uˆ + g,

u¨ = T uˆ¨ + g¨ ,

(5.24)

in which T, u and uˆ are the same as in (5.17), and g = [ g1

g2

g3 ]T ,

g¨ = [ g¨ 1

g¨ 2

g¨ 3 ]T ,

(5.25)

The transformed matrix EOM are TT M T uˆ¨ + TT K T u = TT (f − Kg)

(5.26)

ˆ uˆ¨ + K ˆ u = ˆf. M

(5.27)

or

5–11


ˆ anf K ˆ are the same as above, but ˆf is different: in which M

4g¨ 1 ˆf = . −5g2 + 5g3 − 3g¨ 2 H (t) − 5g2 + 5g3 + 5g¨ 3

5–12

(5.28)

Examples on nonholonomic MFC will not be given here, since they are non treatable by master-slave methods unless they integrate to holonomic constraints. §5.7. Penalty Function Augmentation Methods for MFC General recommendation: be careful in using penalty function methods to treat dynamic constraints. To justify this statement one needs to resort to variational techniques, which are the topic of next Chapter. §5.8. Lagrange Multiplier Adjunction Methods for MFC These is the most powerful technique available for the treatment of constraints. But explanation of use is deferred until next Chapter, which introduces variational forms. Notes and Bibliography Constraints were not important in early days of particle dynamics since the main application was astronomical: the movement of celestial bodies. Few kinematic constraints appear there except collitions. (Curiously the same thing appears at the other end of the scale: atomic physics and quantum mechanics.) Collitions were studied in the XVIII Century, still using particle models. Euler treated isoparametric constraints using the calculus of variations that he invented. The first investigator that studied constraints in some generality was Lagrange [389], who developed his method of multipliers as a powerful technique that has been heavily used to date. In his Principles of Mechanics book, Hertz was the first to emphasize a clear dictinction between holonomic and nonholonomic constraints. The importance of the latter had increased during the XIX Century as the Industrial Revolution focused attention on rolling motions. But the treatment of those constraints opened up a controversy that persists to the current day, complete with incorrect and misleading statements in even the most reputed textbooks and monographs. This unfortunate saga is well documented in the Hertz biography by L¨utzen [390, Chapter 23]. Within the FEM context, development in this topic have been minimal and tyipcally application dependent.

5–12

9

Dynamics & Vibration Overview

9–1

Chapter 9: DYNAMICS & VIBRATION OVERVIEW


Chapter 9: DYNAMICS & VIBRATION OVERVIEW §9.1 Introduction . . . . . . . . . . . . . . . . . . . . . §9.2 Semidiscrete Equations of Motion . . . . . . . . . . . . . §9.2.1 Vibrations as Equilibrium Disturbance . . . . . . . . . §9.2.2 Undamped Free Vibrations . . . . . . . . . . . . §9.2.3 The Vibration Eigenproblem . . . . . . . . . . . . §9.2.4 Eigensystem Properties . . . . . . . . . . . . . §9.3 Solving the Vibration Eigenproblem . . . . . . . . . . . . §9.3.1 Determinant Roots . . . . . . . . . . . . . . . §9.3.2 Reduction to the Standard Eigenproblem . . . . . . . . §9.3.3 Unsymmetric Reduction . . . . . . . . . . . . . §9.3.4 Symmetry Preserving Reduction . . . . . . . . . . . §9. Notes and Bibliography . . . . . . . . . . . . . . . . . . . . . . §9. References . . . . . . . . . . . . . . . . . . . . . . §9. Exercises . . . . . . . . . . . . . . . . . . . . . .

9–2

9–3 9–3 9–4 9–5 9–6 9–6 9–7 9–7 9–7 9–7 9–8 9–8 9–9 9–10

§9.2

SEMIDISCRETE EQUATIONS OF MOTION

§9.1. Introduction Developments in the IFEM-AFEM-NFEM book sequence pertain to static analysis, in which results are independent of time. This kind of analysis applies also to quasi-static scenarios, in which the state varies with time but does so slowly that inertial and damping effects can be ignored. For example one may imagine situations such as a roof progressively burdened by falling snow before collapse, the filling of a dam, or the construction of a tunnel. Or foundation settlements: think of the Pisa tower before leaning stopped. The quasi-static assumption is commonly used in design even for loads that vary in a faster time scale. For example, vehicles travelling over a bridge or wind effects on buildings.1 By contrast dynamic analysis is appropriate when the variation of displacements with time is so rapid that inertial effects cannot be ignored. There are numerous practical examples: earthquakes, rocket launches, vehicle crashes, explosive forming, air blasts, underground explosions, rotating machinery, airplane flutter. The structural accelerations, which are second derivatives with respect to time, must be kept in the governing equations. Damping effects, which are associated with velocities (the first temporal derivatives of displacements), may be also included. However, passive damping effects are often neglected as they tend to take energy out of a system and thus reduce the response amplitude. Dynamic analysis may be performed in the time domain or the frequency domain. The latter is restricted in scope in that it applies to linear structural models, or to linearized fluctuations about an equilibrium state. The frequency domain embodies naturally the analysis of free vibrations, which is the focus of the present Chapter. Remark 9.1. Mathematically, a dynamical system consists of a phase space together with an evolution law. (J.C. Yoccoz). A major goal of the theory is to understand the long term behaviour of the system.

§9.2. Semidiscrete Equations of Motion The essence of structural analysis is mastering forces. In the development of FEM, this was understood by the pioneers of the first generation, as narrated in §1.7.1. With the victory of the Direct Stiffness Method (DSM) by 1970, displacements came to the foreground as primary computational variables because they scale well into complicated systems. To understand dynamic analysis, that dual role must be kept in mind. Displacements become even more important as computational variables. After all, velocities and accelerations are temporal derivatives of displacements. There is no easy way to do the job with forces only, since dynamics is about motion. On the other hand, the fundamental governing equations of structural dynamics are force balance statements. They are elaborate versions of Newtonian mechanics. This Newtonian viewpoint is illustrated in Table 9.1 for several modeling scenarios that span statics, dynamics and vibrations. For notational simplicity it is assumed that the structure has been discretized in space, for example by the FEM. The right column shows the vector form of the governing equations as force balance statements. The table defines nomenclature. 1

The quasi-static assumption can be done during design if dynamic effects can be accounted for through appropiate safety factors. For many types of structures (e.g., buildings, bridges, offshore towers) these are specified in building codes. This saves time when dynamic effects are inherently nondeterministic, as in traffic, winds or wave effects.

9–3


Table 9.1. Discrete Structural Mechanics Expressed as Force Balance Statements Case Problem type I

General nonlinear dynamics

Governing force balance equations ˙ u, ¨ t) = f(u, u, u, p(u, ˙ t) exter nal

internal

II

General nonlinear statics

p(u) = f(u) exter nal

internal

III

Flexible structure nonlinear dynamics

˙ u, ¨ t) + pd (u, u, ˙ t) + pe (u, t) = f(u, t) p (u, u,

i

damping

elastic

iner tial

IV Flexible structure linear dynamics Linear elastostatics

exter nal

damping

elastic

static equilibrium

damping

elastic

periodic

¨ + C u(t) ˙ + K u(t) = 0 u(t) M iner tial

IX Undamped free vibrations

exter nal

¨ + C u(t) ˙ + K u(t) = f p (t) u(t) M iner tial

VIII Damped free vibrations

elastic

¨ ˙ = f(u) M(u) d(t) + C(u) d(t) + K(u) d(t) + p(u) iner tial

VII Damped forced vibrations

damping

K u = f elastic

VI Dynamic perturbations

exter nal

¨ + C u(t) ˙ + K u(t) = f(t) u(t) M iner tial

V

damping

elastic

¨ + K u(t) = 0 u(t) M iner tial

elastic

Symbol u is array of total displacement DOFs; d in case VI is a linearized perturbation of u. Symbol t denotes time. Superposed dots abbreviate time derivatives: u˙ = du/dt, u¨ = d 2 u/dt 2 , etc. The history u = u(t) is called the response of the system. This term is extendible to nonlinear statics. Initial force effects f I may be accommodated in forced cases by taking f = f I when u = 0.

When the model is time dependent, the relations shown in the right column of Table 9.1 are called semidiscrete equations of motion. The qualifier “semidiscrete” says that the time dimension has not been discretized: t is still a continuous variable. This legalizes the use of time differentiation, abbreviated by superposed dots, to bring in velocities and accelerations. This table may be scanned “top down” by starting with the most general case I: nonlinear structural dynamics, branching down to more restricted but specific forms. Along the way one finds in case V an old friend: the DSM master equations K u = f for linear elastostatics, treated in previous Chapters. The last case IX: undamped free vibrations, is that treated in this and next two Chapters. Some brief comments are made as regards damped and forced vibrations. §9.2.1. Vibrations as Equilibrium Disturbance An elastic structure is placed in motion through some short-term disturbance, for example an impulse. Remove the disturbance. If wave propagation effects are ignored and the structure remains 9–4

§9.2

SEMIDISCRETE EQUATIONS OF MOTION

elastic, it will keep on oscillating in a combination of time-periodic patterns called vibration modes. Associated with each vibration mode is a characteristic time called vibration period. The inverse of a period, normalized by appropriate scaling factors, is called a vibration frequency. The structure is said to be vibrating, or more precisely undergoing free vibrations. In the absence of damping mechanisms an elastic structure will vibrate forever. The presence of even minute amounts of viscous damping, however, will cause a gradual decrease in the amplitude of the oscillations. These will eventually cease.2 If the disturbances are sufficiently small to warrant linearization, this scenario fits case VI of Table 9.1, therein labeled “dynamic perturbations.” Its main application is the investigation of dynamic stability of equilibrium configurations. If the perturbation d(t) is unbounded under some initial conditions, that equilibrium configuration3 is said to be dynamically unstable. The analysis of case VI does not belong to an introductory course because it requires advanced mathematical tools. Moreover it often involve nondeterministic (stochastic) effects. Cases VII through IX are more tractable in an introductory course. In these, fluctuations are linearized about an undeformed and unstressed state defined by u = 0. Thus d (the perturbed displacement) becomes simply u (the total displacement). Matrices M, C and K are called the mass, damping and stiffness matrices, respectively. These matrices are independent of u since they are evaluated at the undeformed state u = 0. Two scenarios are of interest in practice: 1.

Forced Vibrations. The system is subjected to a time dependent force f(t). The response u(t) ¨ + C u(t) ˙ + K u(t) = f(t) of case IV. is determined from the linear dynamics equation: M u(t) Of particular interest in resonance studies in when f(t) is periodic in time, which is case VII.

2.

Free Vibrations. The external force is zero for t > 0. The response u(t) is determined from initial conditions. If damping is viscous and light, the undamped model gives conservative answers and is much easier to handle numerically. Consequently the model of case IX is that generally adopted during design studies.

§9.2.2. Undamped Free Vibrations From the foregoing discussion it follows that case IX: undamped free vibrations is of paramount importance in design. The governing equation is ¨ + K u(t) = 0. M u(t)

(9.1)

This expresses a force balance4 in the following sense: in the absence of external loads the internal ¨ The only ingredient beyond the elastic forces K u balance the negative of the inertial forces Mu. by now familiar K is the mass matrix M. The size of these matrices will be denoted by n f , the number of degrees of freedom upon application of support conditions. 2

Mathematically a damped oscillation also goes on forever. Eventually, however, the motion amplitude reaches a molecular scale level at which a macroscopic idealization does not apply. At such point the oscillations in the physical structure can be considered to have ceased.

3

Usually obtained through a nonlinear static analysis. This kind of study, called dynamic stability analysis, is covered under Nonlinear Finite Element Methods.

4

Where is f = ma? To pass to internal forces change the sign of f : f int + ma = ku + ma = 0. Replace by matrices and vectors and you have (9.1).

9–5


Equation (9.1) is linear and homogeneous. Its general solution is a linear combination of exponentials. Under matrix definiteness conditions discussed later the exponentials can be expressed as a combination of trigonometric functions: sines and cosines of argument ωt. A compact √ representation of such functions is obtained by using the exponential form e jωt , where j = −1: u(t) = vi e jωi t . (9.2) i

Here ωi is the i th circular frequency, expressed in radians per second, and vi = 0 the corresponding vibration mode shape, which is independent of t. §9.2.3. The Vibration Eigenproblem Replacing u(t) = v e jωt in (9.1) segregates the time dependence to the exponential: (−ω2 M + K) v e jωt = 0. Since e jωt is not identically zero, it can be dropped leaving the algebraic condition: (−ω2 M + K) v = 0.

(9.3)

Because v cannot be the null vector, this equation is an algebraic eigenvalue problem in ω2 . The eigenvalues λi = ωi2 are the roots of the characteristic polynomial be indexed by i: det(K − ωi2 M) = 0.

(9.4)

Dropping the index i this eigenproblem is usually written as K v = ω2 Mv.

(9.5)

If M and K satisfy some mild conditions, solutions of (9.5) are denoted by ωi and vi . This are called the vibration frequencies or eigenfrequencies, and the it vibration modes or eigenmodes, respectively. The set of all ωi is called the frequency spectrum or simply spectrum. §9.2.4. Eigensystem Properties Both stiffness K and mass M are symmetric matrices. In addition M is nonnegative. Nothing more can be assumed in general. For example, if K incorporates Lagrangian multipliers from the treatment of a MFC, as explained in Chapter 10, it will be indefinite. If M is positive definite, the following properties hold. 1.

There are n f squared vibration frequencies ωi2 , which are roots of the characteristic polynomial (9.4). These are not necessarily distinct. A root of (9.6) that appears m times is said to have multiplicity m.5

2.

All roots ωi2 of (9.6) are real. The corresponding eigenmodes vi have real entries. 2 If K is nonnegative, ωi ≥ 0 and the frequencies ωi = + ωi2 are also real and nonnegative. √ Furthermore, if K is positive definite, all ωi2 > 0 and consequently + ωi > 0.

3.

If M is nonnegative, care must be exercised; this case is discussed in an Exercise. If M is indefinite (which should never happen in structures) all of the foregoing properties are lost. 5

For example, a free-free (fully unsupported) structure has n R zero frequencies, where n R is the number of rigid body modes.

9–6

§9.3

SOLVING THE VIBRATION EIGENPROBLEM

Example 9.1. This illustrates the weird things that can happen if M is indefinite. Consider

K=

α 1

1 , 2

M=

0 1

1 , 1+β

(9.6)

where α and β vary from 1 to −1. Then M−1 K =

1 1 2α − 1 α

The eigenvalues are −2 + α + αβ ±

(1 − β) . (−1 + α + αβ)

(9.7)

α[4 − 4β + α(1 + β)2 ] . (9.8) 4α − 2 These are complex if the radicand is negative. But that is not all. If α → 0 one eigenvalue goes to ∞. If α = 0, A = M−1 K is a 2 × 2 Jordan block and one eigenvector is lost. 2 = ω1,2

§9.3. Solving the Vibration Eigenproblem In what follows we often denote λi = ωi2 to agree more closely with the conventional notation for the algebraic eigenproblem. §9.3.1. Determinant Roots Mathematically the ωi2 are the roots of the characteristic equation (9.4). The simple minded approach is to expand the determinant to get the characteristic polynomial P(ωi2 ) and get their roots: det(K − ωi2 M) = P(ωi2 ) = 0.

(9.9)

This approach is deprecated by numerical analysts. It seems as welcome as anthrax. Indeed for numerical floating point computations of large systems it risks numerical overflow; moreover the roots of the characteristic polynomial can be very ill-conditioned with respect to coefficients. For small systems and using either exact or symbolic computation there is nothing wrong with this if the roots can be expressed exactly in terms of the coefficients, as in the above example. §9.3.2. Reduction to the Standard Eigenproblem The standard algebraic eigenproblem has the form Ax = λx.

(9.10)

Most library routines included in packages such as Matlab and Mathematica are designed to solve this eigenproblem. If A is symmetric the eigenvalues λi are real; moreover there exist a complete system of eigenvectors xi . If these are normalied to length one: ||xi ||2 = 1 they satisfy the orthonormality conditions 1 if i = j T xi x j = δi j = , xiT Ax j = λi , (9.11) 0 if i = j where δi j is the Kronecker delta. If the xi are collected as columns of a matrix X, the foregoing conditions can be expressed as XT X = I and XT KX = Λ = diagλi . 9–7


§9.3.3. Unsymmetric Reduction If M is nonsingular, a simple way to reduce Kv = ω2 Mv to standard form is to premultiply both sides by M−1 whence M−1 Kv = ω2 v

⇒ Ax = λx,

with

A = M−1 K,

λ = ω2 ,

x = v.

(9.12)

The fastest way to form A is by solving MA = K for A. One nice feature of (9.12) is that the eigenvectors need not be backtransformed, as happens in symmetry-preserving methods. As in the case of the characteristic polynomial, this is deprecated by numerical analysts, also not so vehemently. Their objection is that A is not generally symmetric even if K and M are. So Ax = λx has to be submitted to an unsymmetric eigensolver. Thus risks contaminating the spectrum with complex numbers. Plus, it is slower. The writer’s experience is that (9.12) works perfectly fine for small systems. If tiny imaginary components appear, they are set to zero and life goes on. §9.3.4. Symmetry Preserving Reduction It is possible to retain symmetry by proceeding as follows. Decompose the mass matrix as M = LLT

(9.13)

This is the Cholesky decomposition, which can be carried out to completion if M is positive definite. Then (9.14) A = L−1 KL−T . The demonstraion is in one of the Exercises. The symmetric eigenproblem can be handled by standard library routines, which give back all the eigenvalues and eigenvectors. The square root of the eigenvalues give the vibration frequencies and the vibration modes are recovered from the relation Lvi = xi , which can be handled by standard library routines. Notes and Bibliography The literature on dynamics and vibrations of structures is quite large. It is sufficient to cite here titles that incorporate modern analysis methods: Clough and Penzien [142], Geradin and Rixen [305], Meirovich [479,?] and Wilson [804]. Several books in matrix methods and FEM books contain at least an introductory treatment of dynamics. Citable textbooks include Bathe [54], Cook, Malkus and Plesha [148], Hughes [385]. Despite their age, Przemieniecki [596] remains a useful source of mass matrices, while Pestel and Leckie [572] contains a catalog of transfer matrices (an early-1960 method suitable for small computers but restricted to 1D models). As regards books on linear algebra matrix theory and matrix calculus see the Bibliography cited in Appendix A. The most elegant coverage is that of Strang [699]. Two comprehensive references on matrix computations in general are Golub and VanLoan [311] and Stewart [693]. The former is more up to date as regard recent literature. Bellman [?] contains more advanced material. Stewart and Sun [694] cover the sensitivity analysis of standard and generalized eigenproblems. There are comprehensive books that treat the algebraic eigenproblem. Wilkinson’s masterpiece [796] is dated in several subjects, particularly the generalized eigenproblem and the treatment of large eigenproblems. But it is still unsurpassed as the “bible” of backward error analysis. More up to date in methods is Parlett [542], which is however restricted to the symmetric eigenproblem.

9–8

§9.

Notes and Bibliography

As regards source code for matrix computations, the Handbook compilation of Algol 60 procedures by Wilkinson and Reisch [797] is elegant and still useful as template for other languages. Half of the handbook deals with eigenvalue problems. By contrast, the description of Fortran EISPACK code [291] suffers from the inherent ugliness and unreadability of Fortran IV. And of course there is Numerical Recipes in various flavors. To borrow from the immortal words of Winston Churchill, “never have so few wasted the time of so many.” References Referenced items moved to Appendix R.

9–9


Homework Exercises for Chapter 9 - Dynamics & Vibration Overview EXERCISE 9.1 [A:15]. A 3-element model of a bar in 1D gives



2 1 M= 0 0

1 4 1 0

0 1 4 1



0 0 , 1 2





1 −1 0 0  −1 2 −1 0  K= . 0 −1 2 −1  0 0 −1 1

(E9.1)

Solve the vibration eigenproblem and show the natural frequencies and associated vibration modes. Normalize the latter so that VT MV = I (“mass normalized eigenvectors”). EXERCISE 9.2 [A:25]. In (E9.1) replace the (4,4) mass entry by 2−α and the (4,4) stiffness entry by 1−α/2.

Using Matlab or Mathematica, solve the eigenproblem for α varing from 0to4 in 0.5 increments. Discuss what happens to the frequencies and vibration modes as α goes to 2 and beyond. Explain. EXERCISE 9.3 [D:20]. Eigenvectors can be scaled by arbitrary nonzero factors. Discuss 4 ways in which the eigenvectors vi of Kvi = ωi2 Mvi can be normalized, and what assumptions are necessary in each case.

9–10

16

Mass Matrix Construction Overview

16–1

Chapter 16: MASS MATRIX CONSTRUCTION OVERVIEW


§16.1 §16.2 §16.3 §16.4

§16.5 §16.6 §16.7

Introduction . . . . . . . . . . . . . . . . . . . . . 16–3 Mass Matrix Construction Steps . . . . . . . . . . . . . 16–3 Mass Matrix Construction Methods . . . . . . . . . . . . . 16–3 Element Mass Matrix Construction . . . . . . . . . . . . 16–6 §16.4.1 Direct Mass Lumping . . . . . . . . . . . . . . 16–6 §16.4.2 Variational Mass Lumping . . . . . . . . . . . . 16–7 Mass Matrix Properties . . . . . . . . . . . . . . . . . 16–7 Rank and Numerical Integration . . . . . . . . . . . . . 16–8 Globalization . . . . . . . . . . . . . . . . . . . . . 16–9 §16.7.1 Directional Invariance . . . . . . . . . . . . . . 16–9 §16.7.2 Failure of Repetition Rule . . . . . . . . . . . . . 16–10

16–2

§16.3

MASS MATRIX CONSTRUCTION METHODS

§16.1. Introduction This Part focuses on the construction of the mass matrix for FEM dynamic analysis of structures. Chapters 16 through 20 present and illustrate the two standard methods used to accomplish that goal: direct mass lumping and variational mass lumping. Those methods gave been used for decades, are well understood by now, and are implemented in production FEM programs. Chapters 21ff deal with a more general and advanced technique: templates. This approach produces parametrized mass matrices that include the standard ones as special instances. This newer scheme is still undergoing development. For the convenience of the reader, Table 16.1 collects acronyms often used in this Part. §16.2. Mass Matrix Construction Steps In Part II the master (system) mass matrix of a structural FEM model emerged as the discrete operator that converts nodal accelerations to inertial nodal forces: ¨ f I = M u.

(16.1)

This relation expresses Newton’s second law for a discrete dynamic system with masses constant in time.1 In the framework of the Direct Stiffness Method (DSM), the construction of M is done through three steps: Step 1: Localization: Form the mass matrix of each element, using a local frame if convenient. Step 2: Globalization: Transform element mass matrices to global coordinates if necessary. Step 3: Assembly: Merge the globalized element mass matrices to form M. In practice these operations are carried out concurrently within an element-by-element loop. On loop exit, the master mass matrix is complete. Readers familiar with the Direct Stiffness Method (DSM) for FEM static analysis [255] may notice that the formation of M through the preceding steps largely parallels that of the master stiffness matrix K. In particular, merging of element mass matrices into the master mass matrix follows exactly the same techniques. Consequence: assemblers for K and M, before application of boundary conditions, can be made identical — except for obvious indexing shortcuts in the case of diagonal mass matrices. This procedural uniformity is one of the strengths of DSM. A notable difference with the stiffness matrix is the possibility of using a diagonally lumped mass matrix (DLMM) based on the direct mass lumping scheme described below. A master DLMM can be stored as a vector. If all entries are nonzero, it is easily inverted in place, since the inverse of a diagonal matrix is diagonal. Plainly using a DLMM entails significant advantages in computations that involve M−1 ; for example explicit time integration [74,766] as well as symmetric eigenproblem solution [542]. Those benefits are counteracted by some negative features discussed later. 1

For the relatively rare cases in which the mass varies with time, the law must be used in the original form stated by Newton: the time derivative of momentum equals the inertial force.

16–3


Table §16.1. Acronyms Used in Part III Acronym

Stands for

AB ABTS BLCD BLFD BLFM CMM CMS CMT COB COF DDD DGVD DIMM DLMM DML DSF DTI DCF DWN FFB FPMM HF LCD LF LFF LLMM MOF NCT OB OBTS SDAV SFB SLMM SMS TML VDMM VML VSF

Acoustic branch in DDD: has physical meaning in continuum models AB Taylor series in DWN κ, centered at κ = 0 Best linear combination (LFF sense) of the CMM and a selected DLMM Best possible DLMM (LFF sense); acronym also applies to MS pair with this mass Best possible FPMM (LFF sense); acronym also applies to MS pair with this mass Consistent mass matrix: a special VDMM in which VSM and DSF coalesce Component Mode Synthesis: model reduction framework for structural dynamics Congruential (also spelled congruent) mass transformation Constant optical branch: OB frequency is independent of wavenumber Cutoff frequency: OB frequency at zero wavenumber (lowest one if multiple OB) Dimensionless dispersion diagram: DCF vs. DWN κ Dimensionless group velocity diagram: γc = c/c0 vs. DWN κ Directionally invariant mass matrix: repeats with respect to any RCC frame Diagonally lumped mass matrix; qualifier “diagonally” is often omitted Direct mass lumping Displacement shape functions to interpolate displacements over element Direct time integration of EOM Dimensionless circular frequency, always denoted by Dimensionless wavenumber, always denoted by κ Flexural frequency branch in Bernoulli-Euler or Timoshenko beam models Fully populated mass matrix (at element level); includes CMM as special case High frequency: short wavelength, small DWN, typically κ > 1 Mass matrix obtained as linear combination of the CMM and a selected DLMM Low frequency: long wavelength, small DWN, typically κ < 1 Low frequency fitting of AB to that of continuum Lobatto lumped mass matrix: a DLMM based on a Lobatto quadrature rule Maximum overall frequency: largest frequency in DDD over Brillouin zone Non-continuum term: a term in the ABTS that is not present in the continuum Optical branch (or branches) in DDD: no physical meaning in continuum models OB Taylor series in DWN κ, centered at κ = 0 Structural dynamics and vibration applications: low frequency range important Shear frequency branch in the Timoshenko beam model Simpson lumped mass matrix: a LLMM based on Simpson’s 3-pt quadrature rule Selective mass scaling: modifying a mass matrix by adding a scaled stiffness Template mass lumping Variational derived mass matrix: Hessian of discretized kinetic energy Variational mass lumping Velocity shape functions to interpolate velocities and produce a VDMM

16–4

§16.3

MASS MATRIX CONSTRUCTION METHODS

§16.3. Mass Matrix Construction Methods Structural elements based on continuum models have distributed mass characterized by the material density. Associated inertia forces are body forces: forces per unit volume triggered by motioninduced accelerations. The FEM discretization transforms those distributed forces to nodal (point) forces f I , which are linked to nodal accelerations by M as per (16.1). The discretization process is generically called mass lumping or simply lumping.2 A mass matrix that satisfied certain behavioral constraints, such as nonnegativity and mass conservation, will be called admissible. As noted in §16.2 the construction of M starts at the element level. Three methods to construct a element mass matrix can be distinguished: Direct Mass Lumping (DML). The total mass of the element is distributed to the nodes so that a diagonally lumped mass matrix (DLMM) is produced. The physical interpretation of this particular matrix configuration is that no inertial interaction occurs between the lumped masses. Variational Mass Lumping (VML). The kinetic energy of the element is expressed in terms of the degrees of freedom (DOF) using an interpolation scheme for the nodal velocities. The Hessian of the kinetic energy taken with respect to the DOF produces a variational mass matrix, which is generally non-diagonal. If the element stiffness is constructed using displacement shape functions, and these are used to interpolate velocities, the so-called consistent mass matrix or CMM is obtained. Template Mass Lumping (TML). This approach aims to produce an element mass matrix Me that has free parameters. Their range is restricted so that Me satisfies admissibility conditions. This is called a mass matrix template. Setting parameters to specific numeric values produces a template instance or simply instance. If the template happens to include all admissible mass matrices, it is called a general template. The union of the first two methods: direct and variational mass lumping, will be collectively called the standard approach to the construction of mass matrices. Since both CMM and DLMM configurations are by construction admissible, they are simply instances of a general template. Why then bother to make a distinction? Practical reasons: •

Both direct and variational lumping have been used in structural FEM for a fairly long time.3 They are straightforward to explain and implement, and are well understood after decades of experience. Furthermore, they produce specific matrices, avoiding parameters.

•

The template approach is more recent and there is less experience with it. The presence of free parameters makes derivations far more involved and usually require the help of a computer algebra system (CAS) to arrive at useful results in reasonable time. Most of the derivations to date concern one-dimensional elements. Extension to multidimensional elements poses challenges outlined later.

2

In English the verb “to lump” conveys the following meanings: combine, put together, group, bunch, aggregate, aglomerate, unite, pool, merge, collect, throw together, consider together. Its use for mass discretization comes from the direct lumping process through which an object of finite extent is idealized as a point mass. The idea can be traced back to centuries of orbital computations: celestial objects such as planets were idealized as point masses.

3

DLMM since the 1930s and CMM since the 1960s, as narrated in Appendix H.

16–5


(a)

Element total mass is m = ρA 1

m1 =

me

2 m2 =

(b) x ux1

me m1

ux2 m2

massless connector Figure 16.1. Direct mass lumping for two-node prismatic bar element: (a) lumping element mass to end nodes; (b) endowing the element with 2 translational degrees of freedom.

Because of the newness, dimensionality restrictions and computational demands imposed by templates, the derivation of mass matrices is restricted to the time-tested standard approach in this and following chapters. §16.4. Element Mass Matrix Construction The master mass matrix is built up from element contributions, and we start at that level. The construction of the mass matrix of individual elements with distributed mass density can be carried out through any of the three methods outlined above.4 By now both direct and variational mass lumping enjoy extensive coverage in the structural dynamics literature at the textbook level; see e.g., [142,158,305,596,685], and references therein. They are implemented in all general purpose FEM codes. §16.4.1. Direct Mass Lumping This is the simplest procedure. The total mass of element e is directly apportioned to nodal freedoms, ignoring any cross coupling. The goal is to build a diagonally lumped mass matrix or DLMM, denoted here by MeL . As the simplest example, consider a 2-node prismatic bar element with length , cross section area A, and mass density ρ, which can only move in the axial direction x, as shown in Figure 16.1(a). We often denote this element as Bar2 in the sequel. The total element mass is m e = ρ A. This is divided into two equal parts and assigned to each end node. The element is endowed with the two freedoms shown in Figure 16.1(b). Thus 1 0 e 1 = 12 m e I2 , (16.2) M L = 2 ρ A 0 1 in which m e = ρ A is the element mass and I2 denotes the 2 × 2 identity matrix. As sketched in Figure 16.1, we have effectively replaced the continuum bar with a dumbbell: two masses separated by a massless connector. This process conserves the translational kinetic energy or, equivalently, the linear momentum. To check this property for the bar example, take the constant x-velocity vector u˙ e = v [ 1 1 ]T . The kinetic energy of the element is T e = 12 (u˙ e )T MeL u˙ e = 12 ρ A v 2 = 12 m e v 2 . Thus the linear 4

Beyond the element level, methods to produce the master mass matrix coalesce.

16–6

§16.5

MASS MATRIX PROPERTIES

momentum p e = ∂ T e /∂v = m e v is preserved. When applied to simple elements that can rotate, however, the direct lumping process generally doe not preserve angular momentum. Historical motivations for direct lumping are noted in §H.1. Most crucial, it covers naturally the case where concentrated (point) masses are natural part of model building. For example, in aircraft engineering it is common to idealize nonstructural masses (fuel, cargo, engines, etc.) as concentrated at given locations. (Such point masses in general have rotational freedoms; rotational inertia lumping is then part of the process.) §16.4.2. Variational Mass Lumping The second standard procedure is based on a variational formulation. This is done by taking the kinetic energy as part of the governing functional. The kinetic energy of an element of mass density ρ that occupies the domain e and moves with velocity field ve is T = e

1 2

e

ρ(ve )T ve de .

(16.3)

Following the conventional FEM philosophy, the element velocity field is interpolated using shape functions: ve = Nv u˙ e , in which u˙ e are node DOF velocities and Nv a shape function matrix. (For 1D elements, Nv is a row vector.). Inserting into (16.3) and taking the node velocities out of the integral yields def e 1 e T T = 2 (u˙ ) ρ(Nv )T Nv d u˙ e = 12 (u˙ e )T Me u˙ e , (16.4) e

whence the element mass matrix follows as the Hessian of T e : ∂2T e M = e e = ∂ u˙ ∂ u˙

e

e

ρ (Nev )T Nv d.

(16.5)

If the same shape functions used in the derivation of the stiffness matrix are chosen, that is, Nev = Ne , (16.5) is called the consistent mass matrix or CMM. It is denoted here by MCe . A better name for (16.5) would be stiffness-consistent mass matrix. The shorter sobricket has the unfortunate implication that other choices are “inconsistent,” which is far from the truth. In fact, the consistent mass is not necessarily the best performer, a topic elaborated in Chapters that deal with templates. The shorter name is, however, by now ingrained in the FEM literature. For the Bar2 element moving along x, pictured in Figure 16.1(a), the well known stiffness shape functions are N1 = 1 − (x−x1 )/ = (1 − ξ )/2 and N2 = (x−x2 )/ = (1 + ξ )/2, in which ξ = 2(x − x1 )/ − 1 is the isoparametric natural coordinate that varies from −1 at node 1 to +1 at node 2. With d x = 12 dξ , the consistent mass is easily obtained as 1 = ρ A (N ) N d x = A . [ 1 − ξ 1 + ξ ] dξ = m 2 0 −1 (16.6) It can be verified that this mass matrix preserves linear momentum along x. If allowed to move in the x y plane, as considered in §16.7, it also preserves angular momentum about z.

MCe

e T

e

1 ρ 4

+1

1−ξ 1+ξ

16–7

1 6

e

2 1


§16.5. Mass Matrix Properties Mass matrices must comply with conditions that can be used for verification and debugging at the element level. They are: matrix symmetry, physical symmetries, conservation and positivity. Matrix Symmetry. This means (Me )T = Me , which is easy to check. For a variationally derived mass matrix this follows directly from the definition (16.5), whereas for a DLMM is automatic. Physical Symmetries. Also called geometric or fabrication symmetries. They are dictated by the physical configuration. For example, the CMM or DLMM of the prismatic Bar2 element must be symmetric about the antidiagonal: M11 = M22 . To see this, flip the end nodes: the element remains the same and so does the mass matrix.5 Conservation. At a minimum, total element mass must be preserved (we are talking about classical mechanics here; in relativistic mechanics mass and energy can be exchanged). This is easily verified by applying a uniform translational velocity and checking that linear momentum is conserved. Higher order conditions, such as conservation of angular momentum, are optional and not necessarily desirable. Positivity. For any nonzero velocity field defined by the node values u˙ e = 0, (u˙ e )T Me u˙ e ≥ 0. That is, Me must be nonnegative. Unlike the previous three conditions, this constraint is nonlinear in the mass matrix entries. It can be checked in two ways: through the eigenvalues of Me , or the sequence of principal minors. The second technique is more practical if the entries of Me are symbolic. A stricter form of the last condition requires that Me be positive definite: (u˙ e )T Me u˙ e > 0 for any u˙ e = 0. This is physically reassuring because one half of that form is the kinetic energy associated with the velocity field defined by u˙ e . In a continuum T can vanish only for zero velocities (a rest state). But allowing T e = 0 for some nonzero u˙ e makes life easier in some situations; e.g., elements with rotational or multiplier freedoms, or in the rapid-transient applications noted in §H.4. The u˙ e for which T e = 0 collectively form the null space of Me . Because of the conservation requirement, a rigid velocity field (that is, the time derivative u˙ eR of a rigid body mode ueR ) cannot be in the mass matrix null space, as it would imply zero total mass. This scenario is dual to that of the element stiffness matrix. For the latter, Ke ueR = 0 because a rigid body motion produces no strain energy. Thus ueR must be in the null space of the stiffness matrix. §16.6. Rank and Numerical Integration Suppose the element has a total of n eF freedoms. A mass matrix Me is called rank sufficient or of e = n eF . Because of the positivity requirement, a rank-sufficient mass matrix full rank if its rank is r M must be positive definite. Such matrices are preferred from a numerical stability standpoint. e e e < n eF the mass is called rank deficient by d M = n eF − r M . Equivalently Me is If Me has rank r M e times singular. For a numerical matrix the rank is easily computed by taking its eigenvalues dM and looking at how many of them are zero. The null space can be extracted by functions such as NullSpace in Mathematica without the need of computing eigenvalues.

The computation of Me by the variational formulation (16.5) is often done using Gauss numerical quadrature. Each Gauss points adds n D to the rank, where n D is the row dimension of the shape 5

The antisymmetry property would not generally hold if the element is not prismatic.

16–8

§16.7

GLOBALIZATION

function matrix Ne , up to a maximum of n eF . For most elements n D is the same as element spatial dimensionality; that is, n D = 1, 2 and 3 for 1, 2 and 3 dimensions, respectively. This property can be used to pick the minimum Gauss integration rule that makes Me positive definite. §16.7. Globalization Like their stiffness counterparts, mass matrices are often developed in a local or element frame. Should globalization be necessary before merge, a congruent transformation is applied: ¯ Te . Me = (Te )T M e

(16.7)

¯ is the element mass referred to a local frame x¯i (a.k.a. element frame), whereas Te is Here M the local-to-global displacement transformation matrix. The recipe (16.7) follows readily from the Principle of Virtual Work, or equivalently the invariance of the first variation of the element kinetic energy: e ¯ e δ u¯˙e = (u˙ e )T (Te )T M ¯ e Te δ u˙ e = (u˙¯ e )T Me δ u˙ e = δT e . (16.8) δ T¯ e = (u˙¯ )T M e

Matrix Te is in principle the same used for the stiffness globalization. Some procedural differences, however, must be noted. For stiffness matrices Te is often rectangular if the local stiffness has lower dimensionality. For example, two-node bar, shaft and spar elements have 2 × 2 local stiffnesses. Globalization to 2D and 3D involves application of 2 × 4 and 2 × 6 transformation matrices, respectively. This works fine because the local element has zero stiffness in some directions, and those zero rows and columns may be omitted at the local level. In contrast to stiffnesses, translational masses never vanish. One way to realize this is to think of an element moving in a translational rigid motion u R with acceleration u¨ R . According to Newton’s second law, f R = m e u¨ R , where m e is the element translational mass. Regardless of direction, this inertia force cannot vanish. Conclusion: all translational masses must be retained in the local mass matrix. A two-node prismatic bar, moving in the {x, y} plane as in Figure 16.2, furnishes a simple illustration. With the element freedoms arranged as ue = [ u x1 u x2 u y1 u y2 ]T , the local mass matrix constructed by variationally consistent and diagonalized lumping are, respectively, 

2 e 1 ¯ C = 1 me  M  6 0 0

1 2 0 0

0 0 2 1

 0 0 , 1 2



1 e 0 ¯ L = 1 me  M  2 0 0

0 1 0 0

0 0 1 0

 0 0 = 0 1

1 2

m e I4 ,

(16.9)

in which m e = ρ A is the total element mass. For 3D, repeat the diagonal block once more. §16.7.1. Directional Invariance For the case illustratedin in Figure 16.2 the local-to-global freedom transformation u¯ e = Te ue is   c u¯ x1  u¯ x2   0 =  u¯ y1 −s u¯ y2 0 

0 c 0 −s

s 0 c 0

  0 u x1 s   u x2   , u y1 0 u y2 c

in which

16–9

c = cos ϕ,

s = sin ϕ.

(16.10)

Chapter 16: MASS MATRIX CONSTRUCTION OVERVIEW _

uy2

uy2

_

ux2

Element RCC frame _

_

y _

y

uy1

uy1

ϕ

_

x Global RCC frame

m1

ux2

m2

x

ux1 ux1

massless connector

Figure 16.2. Bar2 element with diagonally lumped mass moving in 2D.

¯ Ce and Now apply (16.7) to either MCe or MeL of (16.9) using (16.10). The result is MCe = M ¯ eL : no change. We say that these mass matrices repeat. Verification for the DLMM is MeL = M ¯ eL Te = 1 m e (Te )T I4 Te = 1 m e (Te )T Te = 1 m e I4 . For easy because Te is orthogonal: (Te )T M 2 2 2 the CMM, however, repetition is not obvious. It can be shown to hold by expressing MCe and Te in 2 × 2 partitioned form ˜ e M ¯ MC = 0

0 cI2 e , T = ˜ −sI2 M

sI2 , cI2

˜ = with M

1 6

m

e

2 1

1 . 2

(16.11)

Carrying out the congruent transformation in block form gives ˜ ˜ (cs − cs)M ˜ M (c2 + s 2 )M = = (T ) MC T = 2 2 ˜ ˜ (c + s )M 0 (cs − cs)M

MCe

e T

e

0 ¯e ˜ = MC . M

(16.12)

A mass matrix that repeats upon transformation to any global frame is called a directionally invariant ˜ are irrelevant to the result (16.12). mass matrix, or DIMM. Note that the contents and order of M Hence the following generalization follows. If upon rearranging the element DOF so that they are grouped node by node: ¯ e has a repeating block diagonal form, and (i) M (ii) Te is configured as the block form shown above, then local and global matrices will coalesce. For (ii) to hold, it is sufficient that all nodal DOF be translational and be referred to the same coordinate system. The same conclusion is easily extended to 3D, and to any arrangement of the element freedoms. This repetition rule can be summarized as follows: A local mass matrix is DIMM if all element DOFs are translational and all of them are referred to the same global RCC system.

(16.13)

This property should be taken advantage of to skip superfluous local-to-global transformations. That operation may cost more than forming the local mass matrix. If the rule fails on actual computation, something (mass matrix or transformation) is wrong and must be fixed. 16–10

§16.7

GLOBALIZATION

§16.7.2. Failure of Repetition Rule ¯ e is not a DIMM. This occurs under the following The repetition rule can be expected to fail if M scenarios: 1.

The element has non-translational freedoms; for example node rotations, or displacement derivatives. (Occasionally the rule may work, but that should not be taken for granted.)

2.

The mass blocks are different in content and/or size. This occurs if different continuum models are used in different local directions. Examples are furnished by beam-column elements, shell elements, and elements with curved sides or faces.

3.

Nodes are referred to different coordinate frames in the global system. This can happen if certain nodes are referred to special frames to facilitate the application of boundary conditions.

16–11

17

Standard Mass Matrices for Bar Elements

17–1

Chapter 17: STANDARD MASS MATRICES FOR BAR ELEMENTS


§17.1

§17.2

The Three-Node Bar . . . . . . . . . . . §17.1.1 The Three-Node Bar With Equidistant Nodes §17.1.2 The Three-Node Bar With Offset Midnode . §17.1.3 The Three-Node Bar With Variable Area . . The Four-Node Bar . . . . . . . . . . .

17–2

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . .

17–3 17–3 17–4 17–4 17–4

§17.1

(a)

(b) Midnode 3 @ center

ρ,E,A = const 1 ξ = −1

(e) 3

Center

ρ,E,A = const (e) 1

2 x ξ=1

ξ=0

THE THREE-NODE BAR

Midnode 3 offset from center by α

α

2 x 3

ξ = −1

ξ=0

ξ=1

Isoparametric natural coordinate


= Le

= Le

Figure 17.1. Three-node prismatic bar element with (a) centered midnode; (b) offset midnode.

To further illustrate the use of the standard mass lumping methods, this Chapter derives the consistent mass matrix (CMM) and diagonally lumped mass matrix (DLMM) for two bar elements: Bar3 (3 nodes), and Bar4 (4 nodes).1 Since all examples pertain to the element level, overbars to distinguigh local and global frames are omitted for brevity. §17.1. The Three-Node Bar The three-node bar element, henceforth abbreviated to Bar3, is shown in Figure 17.1(a). It is prismatic with length , area A, and uniform mass density ρ, Midnode 3 is at the center. The DOFs are arranged ue = [ u x1 u x2 u x3 ]T . The three shape functions, collected in the shape function matrix, are Ne = [ N1 (ξ ) N2 (ξ ) N3 (ξ ) ] = [ ξ(1 − ξ )/2

ξ(1 + ξ )/2

1 − ξ2 ] .

(17.1)

in which ξ is the isoparametric natural coordinate pictured in Figure 17.1(a). §17.1.1. The Three-Node Bar With Equidistant Nodes Midnode 3 is at the center. The consistent mass follows as

MCe M M

1

me e T e = ρA (N ) N J dξ = 30 −1

4 −1 2

−1 4 2

2 2 , 16

(17.2)

in which the Jacobian J = d x/dξ = /2, and m e = ρ A . To produce a DLMM, the total mass of the element is divided into three parts: αρ A, αρ A, and (1 − 2α)ρ A, which are assigned to nodes 1, 2 and 3, respectively. See Figure 17.2. As discussed below the standard choice is α = 1/6. Consequently 2/3 of the total mass goes to the midpoint, and what is left to the corners, giving MeS L M M = 16 m e 1

1 0 0

0 1 0

0 0 . 4

Matrices for the Bar2 (2 nodes) element were derived in the previous Chapter.

17–3

(17.3)

Chapter 17: STANDARD MASS MATRICES FOR BAR ELEMENTS

Element total mass is me = ρA 3 1

(a)

m1 =

2 x

m2 =

me

m3 = massless connector

(b) ux1

me m1

ux2

ux3 m3

m2

me

Figure 17.2. Diagonal mass lumping for 3-node bar element (a) lumping as per Simpson’s integration rule; (b) assigning element freedoms.

The 1:1:4 allocation happens to be Simpson’s rule for integration, whence the label SLMM. This meshes in with the interpretation of diagonal mass lumping as a Lobatto integration rule, a topic discussed in §D.2. Both (17.2) and (17.3) are DIMM, and may be used as 3 × 3 building for blocks to expand the element to 2D or 3D space. The repetition rule (16.13) holds. §17.1.2. The Three-Node Bar With Offset Midnode To be done §17.1.3. The Three-Node Bar With Variable Area To be done §17.2. The Four-Node Bar To be done

17–4

18

Standard Mass Matrices For Plane Beam Elements

18–1

Chapter 18: STANDARD MASS MATRICES FOR PLANE BEAM ELEMENTS


§18.1 §18.2

The Plane BE Beam . . . . . . . . . . . . . . . . . . The Plane Timoshenko Beam . . . . . . . . . . . . . .

18–2

18–3 18–4

§18.1

THE PLANE BE BEAM

§18.1. The Plane BE Beam The two-node plane BE-beam element has length , cross section area A and uniform mass density ρ. Only the translational inertia due to the lateral motion of the beam is considered in the kinetic ˙ x) ¯ 2 d x¯ of the element, whereas its rotational inertia is ignored. The freedoms energy T = 12 0 ρ v( are arranged as ue = [ v1 θ1 v2 θ2 ]T . The natural coordinate ξ varies from ξ = −1 at node 1 (x = 0) to ξ = +1 at node 2 (x = ), whence d x/dξ = 12 and dξ/d x = 2/. The well known cubic shape functions in terms of ξ are collected in the shape function matrix Ne = [ 14 (1 − ξ )2 (2 + ξ )

1 (1 8

− ξ )2 (1 + ξ )

1 (1 4

+ ξ )2 (2 − ξ )

− 18 (1 + ξ )2 (1 − ξ ) ] (18.1)

The CMM obtained by analytical integration is MCe M M = ρ A



1

−1

156 22 e m 22 42  J (Ne )T Ne dξ =  54 13 420 −13 −32

in which the Jacobian J = d x/dξ = /2 and m e = ρ integration rules of 1, 2 and 3 points are    16 4 16 −4 86 13 22 2 2 4 −   4  13 22 5 C1   , C2  16 4 16 −4 22 5 86 2 2 −4 − −4 −5 −2 −13

54 13 156 −22

 −13 −32  . −22 42

(18.2)

A . The mass matrices obtained with Gauss   −5 444 62 2 −   62 112  , C3  −13 156 38 2 2 −38 −92

 −38 −92  , −62 112 (18.3) in which C1 = m e /64, C2 = m e /216 and C3 = m e /1200. Their eigenvalue analysis shows that all three are singular, with rank 1, 2 and 3, respectively. The result for 4 and more points agrees with (18.2), which has full rank. The main purpose of this example is to illustrate the rank property stated in §16.6: each Gauss point adds one to the rank up to 4, since the problem is one-dimensional. 156 38 444 −62

The matrix (18.2) conserves linear and angular momentum. So do the reduced-integration mass matrices (18.3) if the number of Gauss points is 2 or greater. To get a diagonally lumped mass matrix is trickier. Obviously the translational nodal masses must be the same as that of a bar: 12 ρ A. See Figure 18.1. But there is no easy road on rotational masses. To accommodate these variations, it is convenient to leave the latter parametrized as follows 1  0 0 0 2 0  0 α2 0 ¯ eL = m e  (18.4) M   , α ≥ 0. 1 0 0 0 2 0 0 0 α2 Here α is a nonnegative parameter, typically between 0 and 1/100. The choice of α has been argued in the FEM literature over several decades, but the whole discussion is largely futile. Matching the angular momentum of the beam element gyrating about its midpoint gives α = −1/24. This violates the positivity condition. It follows that the best possible α — as opposed to possible best — is zero. This choice gives, however, a singular mass matrix. This is undesirable in scenarios where a mass-inverse appears. 18–3

Chapter 18: STANDARD MASS MATRICES FOR PLANE BEAM ELEMENTS

Total mass ρA αρA

xαρA

3

ρA

ρA

1

3

2

Figure 18.1. Direct mass lumping for two-node plane BE beam element.

This result can be readily understood physically. The m e /2 translational end node masses grossly overestimate (in fact, by a factor of 3) the angular momentum of the element. Hence adding any rotational lumped mass only makes things worse. §18.2. The Plane Timoshenko Beam The Timoshenko beam (Ti-beam) incorporates two refinements over the Bernoulli-Euler (BE) model: 1.

For both statics and dynamics: plane sections remain plane but not necessarily normal to the deflected midsurface. See Figure 24.4 for the kinematics. This assumption allows the averaged shear distortion to be included in both strain and kinetic energies.

2.

In dynamics: the rotary inertia is included in the kinetic energy.

This model is more important for dynamics and vibration than BE, and indispensable for rapid transient and wave propagation analysis. More specifically, the BE beam has infinite phase velocity, because the EOM is parabolic, and thus becomes useless for high-fidelity wave propagation. According to the second assumption, the kinetic energy of the Ti-beam element is given by T =

1 2

˙ 2 d x. ρ A v(x) ˙ 2 + ρ I R θ(x)

(18.5)

0

Here I R is the second moment of inertia to be used in the computation of the rotary inertia and θ = v + γ is the cross-section rotation angle shown in Figure 24.4; γ = V /(G As ) being the section-averaged shear distortion. The element DOF are ordered ue = [ v1 θ1 v1 θ2 ]T . The lateral displacement interpolation is e e v(ξ ) = v1 Nv1 (ξ ) + v1 Nve 1 (ξ ) + v2 Nv2 (ξ ) + v2 Nve 2 (ξ ),

ξ=

2x − 1,

(18.6)

in which cubic interpolation functions are used. A complication over BE is that the rotational freedoms are θ1 and θ2 but the interpolation (18.6) is in terms of the neutral surface end slopes: v1 = (dv/d x)1 = θ1 − γ and v2 = (dv/d x)2 = θ2 − γ . From a kinmatic analysis we can derive the relation   v1 −2 − 1+ 2 1 v1  θ1  = (18.7)  , 1+ v2 v2 1 + − − 2 2 θ2 18–4

§18.2

THE PLANE TIMOSHENKO BEAM

in which the dimensionless parameter = 12E I /(G As 2 ) characterizes the ratio of bending and shear rigidities. The end slopes of (18.7) are replaced into (18.6), the interpolation for θ obtained, and v and θ inserted into the kinetic energy (18.5). After lengthy algebra the CMM emerges as the sum of two contributions: MCe M M = MCe T + MCe R =  13 7  11 11 1 9 13 3 1 + + 13 2 ( 210 + 120 + 24 2 ) + 3 + 16 2 −( 420 + 40 + 24 2 ) 35 10 70 10 1 1 1 13 3 1 1 1 1  ( 105 + 60 + 120 2 )2 ( 420 + 40 + 24 2 ) −( 140 + 60 + 120 2 )2    CT   7 1 11 11 1 13 2 2  + + 3 ( 210 + 120 + 24 )  35 10 symmetric  6   + CR  

5

1 ( 10 − 12 )

− 65

2 ( 15 + 16 + 13 2 )2

1 (− 10 + 12 ) 6 5

1 1 1 ( 105 + 60 + 120 2 )2  1 ( 10 − 12 ) 1 −( 30 + 16 − 16 2 )2   . 1 1  (− 10 + 2 ) 2 + 16 + 13 2 )2 ( 15

symmetric

(18.8) in which C T = ρ A /(1 + ) = m /(1 + ) and C R = ρ I R /((1 + ) ). Matrices MC T and MC R account for translational and rotary inertia, respectively. Caveat: the I in = 12E I /(G As 2 ) is the second moment of inertia that enters in the elastic flexural elastic rigidity. If the beam is homogeneous I R = I , but that is not necessarily the case if, as sometimes happens, the beam has nonstructural attachments that contribute rotary inertia. 2

e

2

2

The scale factor of MCe R can be further transformed to facilitate parametric studies by introducing r R2 = I R /A as cross-section gyration radius and = r R / as element slenderness ratio. Then C R = ρ I R /((1 + )2 ) = ρ A 2 /(1 + )2 = m e 2 /(1 + )2 . If = 0 and = 0, MCe R vanishes and MCe T in (18.8) reduces to (18.2). A DLMM can be obtained through the HRZ scheme explained in §D.1. The optimal lumped mass is derived in §24.2.5 via templates.

18–5

19

Standard Mass Matrices for Plane Stress Elements

19–1

Chapter 19: STANDARD MASS MATRICES FOR PLANE STRESS ELEMENTS


§19.1 §19.2

The Plane Stress Linear Triangle . . . . . . . . . . . . . . The Plane Stress Bilinear Quadrilateral . . . . . . . . . . .

19–2

19–3 19–3

§19.2

THE PLANE STRESS BILINEAR QUADRILATERAL

Here we pass to two dimensional elements in a plane stress state, also called membrane elements in the literature. §19.1. The Plane Stress Linear Triangle We consider the three-node linear displacement triangle to model a plate in plane stress. The element will be identified as Trig3 in the sequel. Its formulation using triangular natural coordinates ζi is available online [255]. For the following Me derivations, the plate is assumed to have constant mass density ρ, area A, and uniform thickness h. The motion is restricted to the {x, y} plane. The six DOFs are arranged as ue = [ u x1 u y1 u x2 u y2 u x3 u y3 ]T . The CMM is obtained using the well known displacement shape functions (DSF), which are simply the triangular coordinates ζi . Accordingly the shape function matrix is

ζ N = 1 0 e

0 ζ1

ζ2 0

0 ζ2

ζ3 0

0 . ζ3

(19.1)

Expanding (Ne )T Ne gives a 6 × 6 matrix quadratic in the triangular coordinates. This can be area integrated with formulas exemplified by e ζ12 d = A/3, e ζ1 ζ2 d = A/6, etc. The result is   2 ζ1 ζ1 0 ζ1 ζ2 0 ζ1 ζ3 0 0 ζ1 ζ1 0 ζ1 ζ2 0 ζ1 ζ3  0    me  1  ζ2 ζ1 0 ζ2 ζ2 0 ζ2 ζ3 0  = ρh    d = ζ2 ζ1 0 ζ2 ζ2 0 ζ2 ζ3  12  0 e  0    ζ3 ζ1 0 ζ3 ζ2 0 ζ3 ζ3 0 1 0 ζ3 ζ1 0 ζ3 ζ2 0 ζ3 ζ3 0 

MCe M M

0 2 0 1 0 1

1 0 2 0 1 0

0 1 0 2 0 1

1 0 1 0 2 0

 0 1  0  (19.2) 1  0 2

in which m e = ρ Ah. This computation may be done by numerical integration, using Gauss rules. Since the order of Me is 6, and each Gauss point adds two (the number of space dimensions) to the rank, a rule with 3 or more points is required to reach full rank, as can be verified by simple numerical experiments. The diagonally lumped mass matrix is constructed by taking the total element mass element, which is ρ Ah, dividing it by 3 and assigning those to the corner nodes. See Figure 19.1. This process produces a diagonal matrix: MeDL M M =

ρ Ah me diag [ 1 1 1 1 1 1 ] = I6 . 3 3

(19.3)

If this element is used in three dimensions (for example as membrane component of a shell element), it is necessary to insert the normal-to-the-plate z mass components in either (19.2) or (19.3). According to the invariance rule (16.13) the globalization process is trivial because MCe or MeL becomes RBD on grouping the element DOFs by component. Plainly the local element mass matrix repeats in the global frame. 19–3

Chapter 19: STANDARD MASS MATRICES FOR PLANE STRESS ELEMENTS

ρAh

Total mass ρAh one third goes to each node

y x

ρAh

ρAh

massless wireframe

Figure 19.1. Diagonal mass lumping for the Trig3 element in plane stress.

§19.2. The Plane Stress Bilinear Quadrilateral We finally consider the 4-node, 8 DOF bilinear quadrilateral modeling a plate in plane stress. The element is identified as Quad4 in the sequel. It is assumed homogeneous with density ρ and constant thickness h. It moves in the x, y plane. The nodal displacement vector is ue = [ u x1 u y1 u x2 . . . u y4 ]. The shape functions and appropriate Gauss quadrature rules are described in [255]. The integration is carried out numerically using a p× p Gauss product rule, with p variable. Testing the mass matrix module on a rectangular element of dimensions a and b in the x and y directions, respectively, returns the following CMMs for the 1×1 and 2×2 Gauss rules:     1 0 1 0 1 0 1 0 4 0 2 0 1 0 2 0 0 1 0 1 0 1 0 1 0 4 0 2 0 1 0 2     1 0 1 0 1 0 1 0 2 0 4 0 2 0 1 0     0 1 0 1 0 1 0 1 0 2 0 4 0 2 0 1 MCe M M1 = C1   , MCe M M2 = C2  , 1 0 1 0 1 0 1 0 1 0 2 0 4 0 2 0     0 1 0 1 0 1 0 1 0 1 0 2 0 4 0 2     1 0 1 0 1 0 1 0 2 0 1 0 2 0 4 0 0 1 0 1 0 1 0 1 0 2 0 1 0 2 0 4 (19.4) in which C1 = ρabh/32 = m e /32 and C2 = ρabh/72 = m e /72. The mass given by 1-point integration has rank 2 and 6 zero eigenvalues, and thus it is rank-deficient by 6. The mass given by the 2×2 rule is rank-sufficient and positive definite. Either matrix repeats on globalization. Using rules with 3 or more points returns the same matrix. The DLMM is obtained by assigning one fourth of the total element mass m e = ρabh to each freedom. For a quadrilateral of general geometry, use of 2 × 2 Gauss quadrature rule is recommended, as it provides full mass matrix ran

19–4

21

Mass Matrix Templates: General Description

21–1

Chapter 21: MASS MATRIX TEMPLATES: GENERAL DESCRIPTION


§21.1 §21.2 §21.3

Templates: A Tool for Mass Matrix Customization . . . . . . . 21–3 Is Customization Worth The Trouble? . . . . . . . . . . . 21–3 Mass Parametrization Techniques . . . . . . . . . . . . . 21–4 §21.3.1 Matrix-Weighted Parametrization . . . . . . . . . . 21–6 §21.3.2 Spectral Parametrization . . . . . . . . . . . . . . 21–7 §21.3.3 Entry-Weighted Parametrization . . . . . . . . . . . 21–8 §21.3.4 Multilevel Parametrization . . . . . . . . . . . . . 21–8 §21.3.5 Selective Mass Scaling . . . . . . . . . . . . . . 21–8 §21.3.6 Singular Mass Matrices . . . . . . . . . . . . . . 21–9 §21.3.7 Constant Optical Branch Variant . . . . . . . . . . 21–9 §21.3.8 Mass-Stiffness Template Pairs . . . . . . . . . . . . 21–9 §21.3.9 Frequency Dependent Templates . . . . . . . . . . 21–10

21–2

§21.2

IS CUSTOMIZATION WORTH THE TROUBLE?

§21.1. Templates: A Tool for Mass Matrix Customization The present Chapter provides a general description of template mass lumping. This is a general approach through which customized mass matrices can be constructed for specific structural elements.The qualifier “customized” is defined more precisely later. The standard procedures for constructing FEM mass matrices are well known. They are presented in Chapter 16–20. They lead to consistent and diagonally-lumped forms, respectively. Conventional forms of those models are denoted by MC and M L , respectively, with additional subscripts or superscripts as necessary or convenient. Abbreviations CMM and DLMM, respectively, are also used. Collectively those two models take care of many engineering applications in structural dynamics. Occasionally, however, they fall short. The gap can be filled with a more general approach that relies on templates. These are algebraic forms that carry free parameters. The set of parameters is called the template signature. When given numerical values, the signature uniquely characterizes a mass matrix instance. Templates are described in this and the next 5 chapters. The template approach has the virtue of generating a set of mass matrices that satisfy certain a priori constraints; for example symmetry, nonnegativity, invariance and linear momentum conservation. A mass matrix that satisfies those will be called admissible. In particular, the diagonally-lumped and consistent mass matrices should be obtained as instances. Thus those standard models are not excluded. Availability of free parameters, however, allows the mass matrix to be customized to special requirements. Several customization scenarios are listed in Table 21.1, along with their acronyms. The last one: reduction of directional anisotropy in wave propagation, is not applicable to one-dimensional elements and therefore not treated in this paper. The versatility of application will be evident from the examples. It will be also seen that optimizing templates for one scenario generally does not help with others, and in fact may make things worse. Thus, ability to adapt the mass matrix to particular needs as well as problem regions is an important virtue. Note that mesh and freedom configuration need not be modified in any way; only template signatures are adjusted. An attractive feature of templates for FEM programming is that each “custom mass matrix” need not be coded and tested individually. It is sufficient to implement the template as a single elementlevel module, with free parameters as arguments. (Alternatively, useful instances may be identified by predefined mnemonic character strings, and converted to numerical signatures internally.) The signature is adjusted according to goals and needs. In particular the same module should be able to produce the conventional DLMM and CMM models as instances. This can provide valuable crosschecking with other programs while doing benchmarks. In problems characterized by rapid transients, such as contact-impact and fragmentation, templates allow a flexible customization: reduced high-frequency pollution in elements in or near shock regions while maintaining low-frequency continuum fit away from such regions. In these scenarios, signatures may evolve in time. §21.2. Is Customization Worth The Trouble? The ability to customize a mass matrix is not free of development costs. The presence of free parameters makes template derivations considerably more complicated than those based on the 21–3


Table 21.1 Acronym LFCF AMC RHFP MSTS RDAW

Template Customization Scenarios

Customization Low-frequency continuum fit: matching acoustic branch (AB) to continuum model Angular momentum (= rotary inertia) conservation: useful for transverse motions. Reduced high-frequency pollution (spurious noise) in direct time integration (DTI) Maximum stable time step in conditionally stable direct time integration (DTI) Reduced directional anisotropy in wave propagation (not relevant to 1D meshes)

two standard procedures described in Chapter 16. Reason: everything must be carried along symbolically: geometry, material and fabrication properties, in addition to the free parameters. Consequence: hand computations rapidly become unfeasible, even for fairly simple 1D elements. Help from a computer algebra system (CAS) is needed to get timely results. A key issue is: when is this additional work justified? Two specific cases may be mentioned. One is high fidelity systems. Dynamic analysis covers a wide range of applications. There is a subclass that calls for a level of simulation accuracy beyond that customary in engineering analysis. Examples are deployment of precision space structures, resonance analysis of machinery or equipment, adaptive active control systems, medical imaging, phononics (wave guidance at molecular level), vehicle signature detection, radiation loss in layered circuits, and molecular- and crystal-level simulations in micro- and nano-mechanics. In static structural analysis an error of 20% or 30% in peak stresses is not cause for alarm — such discrepancies are usually covered adequately by safety factors. But a similar error in frequency analysis or impedance response of a high fidelity system can be disastrous. Achieving acceptable precision with a fine mesh, however, can be expensive. Model adaptivity comes to the rescue in statics. This approach is less effective in dynamics, however, on account of the time dimension and the fact that irregular meshes are prone to develop numerical pollution. Customized elements may provide a practical solution: achieving adequate accuracy with a coarse regular mesh. Another possibility is that the stiffness matrix comes from a method that avoids displacement shape functions. For example, assumed-stress or strain elements. [Or, it could simply be an array of numbers provided by a black-box program, with no documentation explaining its source.] If this happens the concept of consistent mass matrix, in which velocity shape functions (VSF) are taken to coincide with displacement shape functions (DSF), loses the comfortable variational meaning outlined in §16.4.2. An expedient way out is to choose an element with similar geometry and freedom configuration derived with DSF and take those as VSF. But which element to pick? If time allows, constructing and customizing a template avoids uncritically rolling the dice. §21.3. Mass Parametrization Techniques There are several ways to parametrize mass matrices. Techniques found effective in practice are summarized below. Most of them are illustrated in the worked out examples of ensuing sections. It is often advantageous to have several template expressions for the same element configuration. For example, to study the subset of diagonally lumped mass matrices (DLMM) it may be convenient to streamline the general form to one that produces only such matrices. Likewise for singular mass matrices. In that case we speak of template variants. These may overlap totally or partially: the 21–4

§21.3 Table 21.2 Acronym AB ABTS BLCD BLFD BLFM CMM CMS CMT COB COF DDD DGVD DIMM DOF DLMM DSF DSM DTI DCF DWN EOM FEM FFB FPMM HF LCD LF LFF LLMM MOF MSA NCT NND PD PVP OB OBTS RCC SDAV SF SFB SLMM SMS VDMM VP VSF

MASS PARAMETRIZATION TECHNIQUES

Acronyms Used in Paper

Stands for Acoustic branch in DDD: has physical meaning in continuum models AB Taylor series in DWN κ, centered at κ = 0 Best linear combination (LFF sense) of the CMM and a selected DLMM Best possible DLMM (LFF sense); acronym also applies to MS pair with this mass Best possible FPMM (LFF sense); acronym also applies to MS pair with this mass Consistent mass matrix: a special VDMM in which VSM and DSF coalesce Component Mode Synthesis: model reduction framework for structural dynamics Congruential (also spelled congruent) mass transformation Constant optical branch: OB frequency is independent of wavenumber Cutoff frequency: OB frequency at zero wavenumber (lowest one if multiple OB) Dimensionless dispersion diagram: DCF vs. DWN κ Dimensionless group velocity diagram: γc = c/c0 vs. DWN κ Directionally invariant mass matrix: repeats with respect to any RCC frame Degree(s) of freedom Diagonally lumped mass matrix; qualifier “diagonally” is often omitted Displacement shape functions to interpolate displacements over element Direct Stiffness Method: the most widely used FEM implementation Direct time integration of EOM Dimensionless circular frequency, always denoted by Dimensionless wavenumber, always denoted by κ Equations of motion Finite Element Method Flexural frequency branch in Bernoulli-Euler or Timoshenko beam models Fully populated mass matrix (at element level); includes CMM as special case High frequency: short wavelength, small DWN, typically κ > 1 Mass matrix obtained as linear combination of the CMM and a selected DLMM Low frequency: long wavelength, small DWN, typically κ < 1 Low frequency fitting of AB to that of continuum Lobatto lumped mass matrix: a DLMM based on a Lobatto quadrature rule Maximum overall frequency: largest frequency in DDD over Brillouin zone Matrix Structural Analysis: invented by Duncan and Frazier at NPL (1934) Non-continuum term: a term in the ABTS that is not present in the continuum Nonnegative definite; a qualifier reserved for symmetric real matrices Positive definite; a qualifier reserved for symmetric real matrices Parametrized variational principle Optical branch (or branches) in DDD: no physical meaning in continuum models OB Taylor series in DWN κ, centered at κ = 0 Rectangular Cartesian Coordinate: qualifier to frame, system, axes, etc. Structural dynamics and vibration applications: low frequency range important Shape function Shear frequency branch in the Timoshenko beam model Simpson lumped mass matrix: a LLMM based on Simpson’s 3-pt quadrature rule Selective mass scaling: modifying a mass matrix by adding a scaled stiffness Variational derived mass matrix: Hessian of discretized kinetic energy Variational principle Velocity shape functions to interpolate velocities and produce a VDMM

21–5


Table 21.3

Template Related Nomenclature

Term or abbreviation Meaning Template Signature Instance Subset Family Variant Admissible MS template FD template FDM template FDS template FDMS template EW template ML template MW template SP template

An algebraic expression for a FEM matrix that contains free parameters. So far used to construct stiffness and mass matrices of linear FEM models The set of free parameters that uniquely defines a template Matrix (or matrices) obtained by setting the signature to numeric values Generic term for template specialization: includes families and variants A template subset in which some free parameters are linked by constraints A template subset that introduces free parameters from scratch (the “subset“ may be the original template if reparametrized) Qualifier applied to instances that satisfy predefined conditions such as positiveness, element mass conservation, and fabrication symmetries Mass-stiffness pair template: both M and K have free parameters Frequency-dependent template: free parameters may depend on frequency Frequency-dependent mass template Frequency-dependent stiffness template Frequency-dependent mass-stiffness template Entry weighted parametrization of a template; see §21.3.3 Multilevel parametrization of a template; see §21.3.4 Matrix weighted parametrization of a template; see §21.3.1 Spectral parametrization of a template; see §21.3.2

DLMM variant is plainly a subset of the general mass template. The key difference between a template subset and a variant is that the latter redefines free parameters from scratch. For the reader’s convenience, acronyms often used in this paper are listed in Table 21.2. A set of definitions and abbreviations pertaining to templates are collected in Table 21.3. Notational conventions for mathematical expressions that appear in this paper are summarized in Table 21.4. Specific conventions used for free template parameters are given in Table 21.5. §21.3.1. Matrix-Weighted Parametrization A matrix-weighted (MW) mass template for element e is a linear combination of (k + 1) component mass matrices, k ≥ 1 of which are weighted by parameters µi , (i = 1, . . . k): def

Me = Me0 + µ1 Me1 + . . . µk Mek .

(21.1)

Here Me0 is the baseline mass matrix. This should be an admissible mass matrix on its own if µ1 = . . . µk = 0. The simplest instance of (21.1) is a linear combination of the consistent mass matrix (CMM) and a diagonally-lumped mass matrix (DLMM): def

Me = (1 − µ)MCe + µMeL .

(21.2)

This can be reformatted as (21.1) by writing Me = MCe +µ(MeL −MCe ) = Me0 +µ Me1 . Here k = 1, the baseline is Me0 ≡ MCe , µ ≡ µ1 and Me1 is the “mass deviator” MeL − MCe . The specialization (21.2) is often abbreviated to “linear combination of consistent and diagonally lumped masses,” with acronym LCD; cf. Table 21.2. The rationale behind (21.2) is that the CMM typically overestimates 21–6

§21.3 Table 21.4


General Notational Conventions For Mathematical Expressions

Letter symbol∗

Used for

Examples

UC bold LC bold US roman SS LC roman SS LC roman DS UC roman Greek letters Superposed dot Prime

Matrices Vectors Scalar coefficients or functions Subscripted variants of scalar coefficients Vector entries conforming with vector symbol Matrix entries conforming with matrix symbol Dimensionless quantities except as noted below† Temporal differentiation 1D spatial differentiation, usually with respect to x

K, M u, u˙ ¯ u(x, t) a, b, Q, cˆ1 , cˆ2 u i : entries of u K i j : entries of K θ, ψ, u¨ ≡ d 2 u(t)/dt 2 v (x) ≡ dv(x)/d x

∗

UC:uppercase; LC:lowercase; US:unsubcripted; SS:single subscripted; DS:double subscripted † Exemption made for well established symbols; e.g. ω: frequency or ρ: mass density Table 21.5 Symbol∗ αi βi µi νi , χi ∗

Notational Conventions For Template Parameters

Used for Free parameters in basic stiffness matrix template (not used in this paper) Free parameters in higher order stiffness matrix template Original free parameters in mass template. Additional letter subscripts may be appended as appropriate to distinguish template families or variants Alternative notations for mass template parameters. Often derived from the original µi to streamline expressions, or to identify families or variants

The subscript index is suppressed if only one parameter appears; e.g. β, µ.

natural frequencies while a DLMM usually underestimates them. Thus a linear combination has a good chance of improving low-frequency accuracy for some µ ∈ [0, 1]. A MW mass template represents a tradeoff. It cuts down on the number of free parameters. Such a reduction is essential for 2D and 3D elements. It makes it easier to satisfy conservation and nonnegativity conditions through appropriate choice of the Mie . On the minus side it generally spans only a subspace of admissible mass matrices. §21.3.2. Spectral Parametrization A spectrally parametrized (SP) mass template has the form def

Me = HT Dµ H,

Dµ = diag [ c0 µ0 c1 µ1 . . . ck µk ] .

(21.3)

in which H is a generally full matrix. Parameters µ0 . . . µk appear as entries of the diagonal matrix Dµ . Scaling coefficients ci may be introduced for convenience so the µi are dimensionless. Often the values of µ0 and/or µ1 are preset from conservation conditions. Configuration (21.3) occurs naturally when Me is constructed first in generalized coordinates, followed by congruential transformation to physical coordinates via H. If the generalized mass is derived using mass-orthogonal functions (for example, Legendre polynomials in 1D elements), the unparametrized generalized mass D = diag [ c0 c1 . . . ck ] is diagonal. Parametrization is effected by scaling its entries. As noted, some entries may be left fixed to satisfy a priori constraints. 21–7


Expanding (21.3) and collecting matrices that multiply each µi leads to a matrix weighted combination form (21.1) in which each Mie is a rank-one matrix. The analogy with the spectral representation theorem of symmetric matrices is obvious. But in practice it is usually better to work directly with the congruent representation (21.3). As remarked later in §21.3.6, SP is especially convenient for constructing singular mass matrices under customization scenario RHFP of Table 21.1. §21.3.3. Entry-Weighted Parametrization An entry-weighted (EW) mass template applies free parameters directly to each entry of the mass matrix, except for a priori constraints on symmetry, invariance and conservation. As an example, for a one-dimensional (1D) element with three translational DOF we may start from   µ11 µ12 µ13 def (21.4) Me = m e  µ12 µ22 µ23  , µ13 µ23 µ33 in which m e is the total element mass, and the sum of all row sums is one. EW is often applied to entries of a “deviator matrix” that measures the change from a baseline matrix such as MC . For example, see the three-node bar template (23.2). Because of its generality, EP can be expected to lead to optimal customized instances. But it is restricted to simple (usually 1D) elements because the number of parameters grows quadratically in the matrix size, whereas for the foregoing two schemes either it grows linearly, or stays constant. §21.3.4. Multilevel Parametrization A hierarchical combination of parametrization schemes can be used to advantage if the kinetic energy can be naturally decomposed from physical considerations. For example, the Timoshenko beam element covered in §24.2 uses a two-matrix-split template combined by a weighted form similar to (21.2) as top level. (The energy split is between translational and rotational inertia, respectively.) The two components are constructed by spectral and entry-weighted parametrization, respectively. Such combinations fall under the scope of multilevel (ML) parametrization. §21.3.5. Selective Mass Scaling Selective Mass Scaling, or SMS, is a method proposed recently (references given in §H.4) in which the mass matrix is modified by a scaled version of the stiffness matrix. Thus M becomes MK = M +

µK K. ωr2e f

(21.5)

Here µ K ≥ 0 is a dimensionless scaling factor whereas ωr2e f is a “reference” frequency used to homogenize physical dimensions. The modification (21.5) may be done at the element or system level. The objective is to “filter down” high frequencies in explicit DTI for applications such as contact-impact; e.g., vehicle crash simulation. Filtering aims to reduce spurious noise as well as increasing the stable timestep. It thus follows under customization scenarios RHFP and MSTS of Table 21.1. The basic idea can be explained as follows. Let ωi and vi denote the natural frequencies 21–8

§21.3


and associated orthonormalized eigenvectors, respectively, whereas ωˆ i and vˆ i are their counterparts for the modified eigenproblem (M K + ωˆ i2 K) vˆ i = 0. By inspection the eigenvectors are preserved: vˆ i = vi . Taking the Rayleigh quotient shows that the modified frequencies are ωˆ i2 =

ωi2 , Ri

in which

Ri = 1 + µ K

ωi2 . ωr2e f

(21.6)

Choosing µ K > 0 cuts down each frequency by Ri > 0. For low frequencies the modification is negligible if µ K and ωr2e f are appropriately selected so that Ri ≈ 1. For nonphysical high frequencies (mesh modes) the reduction can be significant In fact note that if ωi2 >> µ K /ωr2e f , ωˆ i2 2 cannot exceed the fixed bound ωmax = ωr2e f /µ K . The downside is that low frequency accuracy may suffer significantly, as illustrated later. Although SMS may be presented as a variant of the MW parametrization technique of §21.3.1, it deserved to be considered on its own for the reasons stated in §H.4. §21.3.6. Singular Mass Matrices A thread linked to SMS but independently developed is that of singular mass matrices. This has been primarily advocated for multibody dynamics, as well as dynamical systems leading to differential-difference EOM that occur in active control with time lags. References are provided in §H.5. The objective is roughly similar to SMS: reduce high frequency noise pollution triggered by rapid transients and/or time lags. But now this is done by raising the optical branch (or branches) so as to widen the acoustoptical gap pictured described in §23.1.1 and illustrated in Figure 23.2. Noisy frequencies that fall in the gap decay exponentially. There are several ways to produce such matrices. Under the template framework, the use of spectral parametrization (SP) is particularly convenient, as observed in §21.3.2. Other approaches include reduced numerical integration or injection of a convenient null space using mass matrix projection. §21.3.7. Constant Optical Branch Variant Instead of rising the optical branch (or branches) by making Me singular, one may try to make the OB frequency independent. Templates that accomplish that feat are tagged as having a Constant Optical Branch, or COB for short. They form subsets collectively identified as the COB variant. The group velocity pertaining to a COB vanishes, so associated waveforms with that particular frequency do not propagate. COB templates were discovered during the course of this work, and are briefly studied in §23.1.13 for the three-node bar element. §21.3.8. Mass-Stiffness Template Pairs The concept of template was first developed for element stiffness matrices, as a natural generalization of its decomposition into basic and higher order parts. A brief historical account is provided in §H.7. Normally the stiffness template is optimized by imposing superconvergence conditions dealing with higher order patch tests while element aspect ratios are kept arbitrary. That optimal instance, if found, is kept fixed while a mass matrix template is subsequently investigated. Maximum customization for dynamics can be expected if both stiffness and mass matrix templates can be simultanously adjusted. This is known as a mass-stiffness (MS) template. These may 21–9


be of interest when improving dynamic behavior is paramount. Presently there is relatively little experience with this more ambitious approach. A note of caution: highly optimized MS templates may be abnormally sensitive to geometric or material perturbations away from a regular mesh. §21.3.9. Frequency Dependent Templates One final generalization should be mentioned: allowing free parameters to be function of the frequency. If this is done for the mass matrix, we speak of a frequency dependent mass (FDM) template. If this is done for both the mass and stiffness matrices, we call the combination a frequency dependent mass-stiffness (FDMS) template. Both cases are illustrated in §22.1.11–§22.1.13 for the two-node bar element. Although this ultimate complication is largely a curiosity, it might be occasionally useful in problems that profit from transformation to the frequency domain. For example: a linear dynamic system driven by a harmonic excitation of slowly varying frequency, if only the long term (steady-state) response is considered. Such systems may arise in parametric stability and active control.

21–10

22

Mass Templates for Bar2 Elements

22–1

Chapter 22: MASS TEMPLATES FOR BAR2 ELEMENTS


§22.1

The Two-Node Bar Element . . . . . . . . . . §22.1.1 Bar2 Entry Weighted Template . . . . . . . §22.1.2 Bar2 One Parameter Mass Template . . . . . §22.1.3 Bar2 Alternative Parametrization . . . . . . §22.1.4 Bar2 Angular Momentum Conservation . . . . §22.1.5 Bar2 Fourier Analysis . . . . . . . . . . §22.1.6 Bar2 Dispersion Diagrams . . . . . . . . §22.1.7 Best µ By Low Frequency Fitting . . . . . . §22.1.8 Folding Frequency . . . . . . . . . . §22.1.9 Bar2 Test: Vibrations of a Fixed-Free Bar Member §22.1.10 Other Customization Options . . . . . . . §22.1.11 Bar2 Frequency Dependent Mass . . . . . . §22.1.12 Bar2 Frequency Dependent Mass-Stiffness Pair . §22.1.13 Bar2 Frequency Dependent Mass Instances . . .

22–2

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

22–3 22–3 22–3 22–4 22–4 22–4 22–8 22–9 22–9 22–9 22–11 22–11 22–12 22–13

§22.1

THE TWO-NODE BAR ELEMENT

§22.1. The Two-Node Bar Element The template approach is best grasped through an example that involves the simplest nontrivial structural finite element: a two-node prismatic bar of mass density ρ, area A and length , that can only move along the longitudinal axis x. See Figure 22.1(a). This element is often acronymed Bar2 for brevity’s sake. The well known consistent and diagonally-lumped mass matrix forms are me 2 1 me 1 0 e e , ML = . (22.1) MC = 6 1 2 2 0 1 in which m e = ρ A is the total element mass. These are derived in §16.4. §22.1.1. Bar2 Entry Weighted Template The most general mass matrix form for Bar2 is the entry-weighted template e e M11 M12 µ11 µ12 µ11 µ12 e e = ρA =m . M = e e M21 M22 µ21 µ22 µ21 µ22

(22.2)

The first form is merely a list of entries. Next the element mass m e = ρ A is factored out. The emerging parameters µ11 through µ22 are numbers, which illustrates a general rule: template free parameters should be dimensionless. This simplifies analysis and implementation. To cut down on parameters one looks at configuration constraints. The most obvious ones are: Matrix symmetry: Me = (Me )T . For the expression (22.2) this requires µ21 = µ12 . Physical symmetry: For a prismatic bar, Me in (22.2) must exhibit antidiagonal symmetry: µ22 = µ11 . Conservation of total translational mass: same as conservation of linear momentum or of kinetic energy. Apply the uniform velocity field u˙ = v to the bar. The associated nodal velocity vector is u˙ e = ve = v [ 1 1 ]T . The kinetic energy is T e = 12 (ve )T Me ve = 12 m e v 2 (µ11 + µ12 + ν2 + µ22 ). This must equal 12 m e v 2 , whence µ11 + µ12 + µ21 + µ22 = 2(µ11 + µ12 ) = 1. Nonnegativity: Me should not be indefinite. [This is not an absolute must, and it is actually relaxed in some elements discussed later.] Whether checked by computing eigenvalues or principal minors, this constraint is nonlinear and of inequality type. Consequently it is not often applied ab initio, unless the element is quite simple, as in this case, or can be stated through simple expressions. §22.1.2. Bar2 One Parameter Mass Template On applying the symmetry and conservation rules three parameters of (22.2) are eliminated. The remaining one, called µ, is taken for convenience to be µ11 = µ22 = (2 + µ)/6 and µ12 = µ21 = (1 − µ)/6. This rearrangement gives 2+µ 1−µ e 1 = (1 − µ)MCe + µMeL . (22.3) Mµ = 6 ρ A 1−µ 2+µ Expression (22.3) shows that the general Bar2 mass template can be recast as a linear combination of the CMM and DLMM instances listed in (22.1). Summarizing, we end up with a one-parameter, matrix-weighted (MW) template that befits the LCD form (21.2). If µ = 0 and µ = 1, (22.3) 22–3


(a) 1

(e)

Total mass ρA

(b)

2 x

x

e = L

ρA

ρA

massless connector Figure 22.1. The two-node prismatic bar element: (a) element configuration; (b) direct mass lumping to end nodes.

reduces to MCe and MeL , respectively. This illustrates another requirement: the CMM and DLMM forms must be instances of the mass template. Finally we can apply the nonnegativity constraint. For the two principal minors of Meµ to be nonnegative, 2 + µ ≥ 0 and (2 + µ)2 − (1 − µ)2 = 3 + 6µ ≥ 0. Both are satisfied if µ ≥ − 12 . Unlike the others, this constraint is of inequality type, and only limits the range of µ. The remaining task is to select the parameter. This is done by introducing an optimality criterion that fits the problem at hand. This is where customization comes in. Even for this simple case the answer is not unique. Thus the statement “the best mass matrix for Bar2 is so-and-so” has to be qualified. Two specific optimization criteria are considered in §22.1.4 and §22.1.5. §22.1.3. Bar2 Alternative Parametrization An alternative template expression that is useful in some investigations, such as those undertaken in Appendix V, is obtained by reparametrizing via χ = 1 + 2µ, the inverse of which is µ = (χ − 1)/2. The resulting form is 3+χ 3−χ e 1 . (22.4) Mχ = 12 ρ A 3−χ 3+χ This is called the “χ form” of the general Bar2 mass template. Observing that its determinant is ρ A χ , Meχ is seen to be singular if χ = 0, and nonnegative if χ ≥ 0. §22.1.4. Bar2 Angular Momentum Conservation This criterion can only be applied in multiple dimensions, since angular rotations do not exist in 1D. Accordingly we allow the bar to move in the {x, y} plane by expanding its nodal DOF to ue = [ u x1 u y1 u x2 u y2 ]T , whence (22.3) becomes a 4 × 4 matrix   2+µ 0 1−µ 0 2+µ 0 1 − µ (22.5) Meµ = 16 ρ A  0 1−µ 0 2+µ 0 0 1−µ 0 2+µ Apply a uniform angular velocity θ˙ about the midpoint. The associated node velocity vector at θ = 0 is u˙ e = 12 θ˙ [ 0 −1 0 1 ]T . The discrete and continuum energies are /2 2 e e e 3 e 1 e T 1 1 ρ A θ˙ x d x = 24 ρ A3 . (22.6) Tµ = 2 (u˙ ) Mµ u˙ = 24 ρ A (1 + 2µ), T = −/2

Matching = T gives µ = 0. So according to this criterion the optimal mass matrix is the consistent one (CMM). Note that if µ = 1, Tµe = 3T e , whence the DLMM overestimates the element rotational (rotary) inertia by a factor or 3. Tµe

e

22–4

§22.1

CONTINUUM BAR

infinite continuum bar with ρ, E and A constant along x

(a)


c0 u0(x,t) x Plane axial wave with phase velocity c0 and wavelength λ0

λ0

(axial displacement drawn transversally to bar for visualization conveniency)

(b)

x

d

Over lattice, phase velocity and wavelength change to c and λ, respectively

FEM-DISCRETIZED BAR

FEM discretization as infinite lattice

(c)

(d) c

uj (t)

xj xj −d

x

xj +d j

x

j

xj

d

λ

j−1

j

j+1

2-element, 3-node patch extracted at generic node j

Figure 22.2. Propagation of a harmonic plane wave over an infinite, prismatic, elastic bar: (a) propagation over a continuum bar; (b) FEM discretization as infinite regular lattice; (c) propagation of plane wave over Bar2-discretized lattice; (d) extraction of a typical two-element patch. For visualization convenience, the wave-profile axial displacement u(x, t) is plotted normal to the bar.

§22.1.5. Bar2 Fourier Analysis For longitudinal motions, a more useful customization criterion is to improve accuracy in the long wavelength, low-frequency limit; this is labeled LFCF in Table 21.1. This is carried out by a well known tool: Fourier analysis. Physical interpretation: probe the fidelity with which planes waves are propagated over a FEM-discretized regular lattice, when compared to the propagation over a continuum bar. The essentials are illustrated in Figure 22.2. The top half depicts the continuum bar whereas the bottom half shows stages of the Fourier analysis of its FEM-discretized counterpart. Symbols used for the analysis of plane wave propagation are collected in Table 22.1 for the reader’s convenience. [The same notation is reused in later Sections.] Corresponding nomenclature for the FEM-discretized two-node bar lattice is collected in Table 22.2. The continuum-versus-lattice notational rule is: corresponding quantities use the same symbol but the zero subscript is suppressed in the lattice. For example, the continuum wavelength λ0 becomes the lattice wavelength λ. Plane wave propagation over a regular spring-mass lattice is governed by the semidiscrete linear equation of motion (EOM): M u¨ + K u = 0, (22.7) 22–5


Table 22.1 Quantity ∗ ρ, E, A ¨ () , () ρ u¨ 0 = E u 0 u 0 (x, t) B0 λ0 k0 κ0 ω0 f0 T0 0 c0 ∗

Nomenclature for Harmonic Plane Wave Propagation over Continuum Bar Meaning (physical dimension in brackets) Mass density, elastic modulus, and cross section area of bar Abbreviations for derivatives with respect to space x and time t, respectively 2 2 2 Bar wave equation. Frequency domain forms: −ω 0 u = 0.

0 u = c0 u and u + k√ Plane wave function u 0 = B0 exp i(k0 x − ω0 t) [length], in which i = −1 Wave amplitude [length] Wavelength [length] Wavenumber k0 = 2π/λ0 [1/length] Dimensionless wavenumber κ0 = k0 λ0 Circular (a.k.a. angular) frequency ω0 = k0 c0 = 2π f 0 = 2πc0 /λ0 [radians/time] Cyclic frequency f 0 = ω0 /(2π) [cycles/time: Hz if time in seconds] Period T0 = 1/ f 0 = 2π/ω0 = λ0 /c0 [time] Dimensionless circular frequency 0 = ω0 T0 √ = ω0 λ0 /c0 Group wave velocity c0 = ω0 /k0 = λ0 /T0 = E/ρ [length/time]. Often abbreviated to wavespeed. Physically, c0 is the longitudinal speed of sound.

Unsubscripted counterpart symbols, such as k or c, pertain to a discrete FEM lattice; cf. Table 22.2 Table 22.2 Quantity ∗

u(x, t) u Mu¨ + Ku = 0 B λ k κ Neλ ω f T c γc

Nomenclature for Harmonic Plane Wave Propagation over Bar2 Lattice Meaning (physical dimension in brackets) Plane wave function (22.8) [length] Node displacement vector, constructed by evaluating u(x, t) at nodes [length] Semidiscrete lattice wave equation (22.7). K and M are infinite Toeplitz matrices Wave amplitude [length] Bar element length [length] Wavelength λ = 2π/k = 2π/κ [length] Wavenumber k = 2π/λ = κ/ [1/length] Dimensionless wavenumber κ = k = 2π /λ Number of elements per wavelength: λ/: same as signal sampling rate Circular (a.k.a. angular) frequency ω = c0 / [radians/time] Cyclic frequency f = ω/(2π) [cycles/time: Hz if time in seconds] Period T = 1/ f = 2π/ω = λ/c [time] Dimensionless circular frequency = ω /c0 Group wave velocity over lattice: c = ∂ω/∂k = c0 (∂/∂κ) [length/time] Wavespeed ratio c/c0 = ∂/∂κ from discrete to continuum

∗

Quantities unchanged from continuum to lattice, such as E,√are not repeated in this Table. Note that the definition of uses the continuum wavespeed c0 = E/ρ; not the discrete wavespeed c.

in which M and K are infinite, tridiagonal Toeplitz matrices. This EOM can be solved by Fourier methods. Figure 22.2(b) displays two characteristic lengths: λ and . The element length-towavelength ratio is called ϒ = /λ. The floor function of its inverse: Neλ = λ/ is the number of elements per wavelengths. Those ratios characterize the fineness of the discretization, as illustrated in Figure 22.2(b). Within constraints noted later the lattice can propagate real, travelling, harmonic plane waves of wavelength λ and grpup velocity c, as depicted in Figure 22.2(b,c). The wavenumber is k = 2π/λ 22–6

§22.1


(a) Wavelength λ = , dimensionless wavenumber κ = 0. Sampling rate = elements per wavelength N eλ = c>0

x

(b) Wavelength λ =8 , dimensionless wavenumber κ = π/4. Sampling rate = elements per wavelength N eλ = 8 c>0

x

λ (c) Wavelength λ =2 , dimensionless wavenumber κ = π, Sampling rate = elements per wavelength Neλ = 2

x λ

c=0 (folding wavenumber)

(d) Wavelength λ = , dimensionless wavenumber κ = 2π, Sampling rate = elements per wavelength Neλ = 1

x

c<0

λ

Figure 22.3. Selected plane waves of various wavelengths, illustrating the physical meaning of the dimensionless wavenumber (DWN) κ = k = 2π /λ. The number of elements per wavelength is Neλ = λ/ = 2π/κ, in which . denotes the floor function. (This is equivalent to the spatial sampling rate of filter technology.) The case λ = 2 pictured in (c) pertains to the folding or Nyquist frequency, at which κ = π , Neλ = 2, and the group velocity c vanishes.

and the circular frequency ω = 2π/T = 2π c/λ = k c. The range of wavelengths that the lattice may transport is illustrated in Figure 22.3. To study plane wave solutions it is sufficient to extract a two-element patch, a process depicted in Figure 22.2(d). A harmonic plane wave of amplitude B is described by the function √

(22.8) u(x, t) = B exp [ j (kx − ω t)] = B exp j κ x − c0 t ], j = −1. Here the dimensionless wavenumber κ and dimensionless circular frequency √ were introduced as κ = k = 2π/λ = 2πχ and = ω /c0 , respectively, in which c0 = E/ρ is the elastic bar group velocity, which for the continuum is the same as the phase velocity. (In physical acoustics c0 is the sound speed of the material.) Using the well-known Bar2 static stiffness matrix and the mass template (22.3) gives the patch equations ρ A 6

2+µ 1−µ 0

1−µ 4 + 2µ 1−µ

0 1−µ 2+µ

u¨ j−1 u¨ j u¨ j+1

EA +

1 −1 0

−1 2 −1

0 −1 1

u j−1 uj u j+1

= 0.

(22.9)

From this one takes the middle (node j) equation, which repeats in the infinite lattice: ρ A [1 − µ 6

4 + 2µ

u¨ j−1 1 − µ ] u¨ j u¨ j+1

EA + [ −1

22–7

2

u j−1 −1 ] u j u j+1

= 0.

(22.10)


(b)

6 5 4

CMM: µ = 0

2 Continuum Bar

1.5

Continuum Bar

BLFM: µ = 1/2

3

Wavespeed ratio γc = c/c0

Dimensionless frequency Ω=ω /c0

(a)

1 0.5 BLFM: µ = 1/2

0

DLMM: µ = 1

−0.5

2 1

DLMM: µ = 1 0

5 1 4 2 3 Dimensionless wavenumber κ = k

CMM: µ = 0

−1

−1.5

6

0

5 1 4 2 3 Dimensionless wavenumber κ = k

6

Figure 22.4. Results from Fourier analysis of Bar2 infinite regular lattice for three choices of µ, plus continuum: (a) dimensionless dispersion diagram (DDD); (b) dimensionless group velocity diagram (DGVD).

Evaluate the wave motion (22.8) at x = x j−1 = x j − , x = x j and x = x j+1 = x j + while keeping t continuous. Substitution into (22.10) gives the wave propagation condition

c0 t c0 t ρ A c02 2 2 6 − (2 + µ) − 6 − (1 − µ) cos κ cos − i sin B = 0. (22.11) 3 If this is to be zero for any t and B, the expression in brackets, called the characteristic equation, must vanish. Solving gives the dimensionless frequency versus wavenumber relation 2 = Its inverse is

6(1 − cos κ) 1 − 2µ 4 = κ2 + κ + C6 κ 6 + . . . 2 + µ + (1 − µ) cos κ 12

(22.12)

6 − (2 + µ)2 1 − 2µ 3 9 − 20µ + 20µ2 5 κ = arccos + + ... = − 6 + (1 − µ)2 24 1920

(22.13)

Transforming (22.12) to physical wavenumber k = κ/ and circular frequency ω = c0 / gives 2 6c0 1 − cos(k) 1−2µ 2 2 2 2 2 4 4 (22.14) = c0 k 1 + k + C6 k + . . . ω = 2 2 + µ + (1−µ) cos(k) 12 in which C6 = (1 − 10µ + 10µ2 )/360. §22.1.6. Bar2 Dispersion Diagrams An equation that links frequency and wavenumber: = (κ) as in (22.12), or ω = ω(k), as in (22.14), is a dispersion relation. A plot of the dispersion relation with k and ω along horizontal and vertical axes, respectively, is called a dispersion diagram. When this is done in terms of dimensionless wavenumber κ and dimensionless frequency , the plot is called a dimensionless dispersion diagram, or DDD. Such diagrams exhibit a 2π period: (κ) = (κ + 2π n) for integer 22–8

§22.1


n. Thus it is enough to plot (π) over either [−π, π ] or [0, 2π ], a range called a Brillouin zone. All DDD in this paper use the [0, 2π] range choice. Why is = 0 at κ = 2π? The wavelenth λ = pictured in Figure 22.3(d) has the same value at all nodes for each time t. This nodal sampling cannot be distinguished from the case λ = ∞ (that is, κ = 0) shown in Figure 22.3(a). They must share the same frequency, which is zero; associated plane waves propagate with the same speed but in opposite directions. Similar arguments can be made to justify the dispersion curve symmetry about wavenumber κ = π, as well as the 2π periodicity. §22.1.7. Best µ By Low Frequency Fitting An oscillatory dynamical system is nondispersive if ω is linear in k, in which case c = ω/k is constant and the wavespeed (the group velocity) is the same √ for all frequencies. The physical dispersion relation for the continuum bar is c0 = ω0 /k0 = E/ρ. Hence all waves propagate with the same speed in this model. Group and phase velocities coalesce. The FEM-discretized lattice group velocity is c = ∂ω/∂k = c0 (∂/∂κ), which differs from c0 except at ω = κ = 0. The Bar2 discrete model is dispersive for any fixed µ, since from (22.14) we get 1 1 − 2µ 2 1 − 20µ + 20µ2 4 6(1 − cos κ) c ∂ = =1+ κ + κ + ... = γc = c0 ∂κ κ 2 + µ + (1 − µ) cos κ 24 1920 (22.15) Plainly the best fit to the continuum for small wavenumbers κ = k<<1 is obtained by taking µ = 1/2, which makes the second term of the series (22.12) or (22.15) vanish. So for LFCF customization the best mass matrix is the average of the lumped and consistent ones: ρ A 5 1 e e e e 1 1 . (22.16) M B L F M = Mµ 1 = 2 MC + 2 M L = µ= 12 1 5 2 This instance is labeled BLFM, for best low-frequency match. Figure 22.4(a) plots the dimensionless dispersion relation (22.12) for the CMM (µ = 0), DLMM (µ = 1) and BLFM (µ = 12 ) mass matrices, along with the continuum-bar relation 0 = κ0 . The superior small-κ fit provided by the BLFM is evident. §22.1.8. Folding Frequency The maximum lattice frequency occurs at the folding wavenumber κ = k = π or λ = 2, which is waveform (c) in Figure 22.3. The sampling rate Neλ is then 2 values per element. This is called the folding or Nyquist frequency, and is denoted as 12 . (22.17) a2 f = 1 + 2µ (The “a” in the subscript stands √ √ for acoustic branch; this notation is explained in §23.1.1). This varies from a f = 12 = 2 3 for the CMM through a f = 2 for the DLMM. Frequencies higher than a f cannot be propagated over the lattice. As shown in Figure 22.4(b), the lattice wavespeed vanishes at the folding wavenumber κ = π , and is negative over the range (π, 2π]. Waveforms in that rage move with negative speed: c < 0. As discussed in §22.1.6, the waveform with = λ, or κ = 2π , cannot be distinguished from a rigid motion such as that pictured in Figure 22.3(a), and the lattice frequency falls to zero. 22–9


ρ=E=A =1 throughout

;;

ρ=E=A =1 throughout

;; (a)

(b)

L=π/2

L=π/2

(a) LFCF Bar2 Template Instances, Fix-Free Mode 1, Exact Frequency ω 1 = 1

6

BLFM 4

DLMM 2 0

CMM 1

2

4

8

Number of elements Ne

16

8

(b) LFCF Bar2 Template Instances, Fix-Free Mode 2, Exact Frequency ω 2 = 3

6 4

BLFM

2

DLMM CMM

0

4

2

16

8

Correct digits in computed frequency

8



Figure 22.5. Fixed-free homogeneous prismatic elastic bar member used in vibration test for Bar2 and Bar3 template instances. Both pictured discretizations display 4 elements. (a): member modeled with Bar2 elements; results reported in Table 22.3 and Figure 22.6. (b): member modeled with Bar3 elements; results reported in Figure 23.6.

8

(c)

6

LFCF Bar2 Template Instances, Fix-Free Mode 3, Exact Frequency ω 3 = 5

4

BLFM 2

DLMM 0

4


CMM

8

16


Figure 22.6. Performance of selected Bar2 template instances in predicting the first three natural frequencies ωi , i = 1, 2, 3 of the fixed-free prismatic homogeneous bar shown in Figure 22.5(a). This is a graphical, log-log representation of the results of Table 22.3. Horizontal axis shows number of elements while vertical axis displays correct digits of computed frequency. See text for details of what is shown along each axis.

§22.1.9. Bar2 Test: Vibrations of a Fixed-Free Bar Member Natural frequency predictions of three Bar2 template instances are compared for predicting natural frequencies of longitudinal vibrations of the fixed-free elastic bar member pictured in Figure 22.5. The member is prismatic, with constant E = 1, A = 1, and ρ = 1. The total member length is taken as L = π/2 for convenience. With those numerical properties the continuum eigenfrequencies are (2i − 1)π E = 2i − 1, i = 1, 2, 3, . . . (22.18) ω0i = 2L ρ The member is divided into Ne identical elements, with Ne = 1, 2, . . . 16. Figure 22.5(a) pictures the case Ne = 4. Three template instances are compared: CMM (µ = 0), DLMM (µ = 1) and BLFM (µ = 1/2). Numerical results obtained for the first three frequencies are collected in Table 23.1. The O(κ 4 ) convergence of BLFM is obvious. For example, 4 elements give ω2 correct to 4 digits while both CMM and DLMM, which converge as O(κ 2 ), give only 2. As expected, CMM overestimates the continuum frequencies while DLMM underestimates them. The results of Table 22.3 are graphically reformatted in Figure 22.6, as accuracy versus elements log-log plots. The horizontal axis shows number of elements Ne in log2 scale. The vertical axis displays correct digits of computed frequency, computed as d = − log10 |ωi | ,

in which 22–10

ωi = ωi − ω0i .

(22.19).

§22.1


Table 3. Bar2 Instance Results for Vibrations of a Fixed-Free Bar Member Instance

Ne

ω1

ω2

ω3

CMM

1 2 4 8 16

1.102658 1.025859 1.006437 1.001607 1.000402

∗ 3.583726 3.174947 3.043539 3.010855

∗ ∗ 5.767394 5.202396 5.050339

DLMM

1 2 4 8 16

0.900316 0.974495 0.993587 0.998394 0.999598

∗ 2.352640 2.829496 2.956815 2.989169

∗ ∗ 4.234640 4.801608 4.949951

BLFM

1 2 4 8 16

0.986247 0.999188 0.999950 0.999997 1.000000

∗ 2.781352 2.987344 2.999237 2.999953

∗ ∗ 4.827222 4.989971 4.999389

* frequency not provided by discrete FEM model

Here ωi is the frequency error of computed values with respect to continuum frequencies ω0i = 2 i − 1, given by (22.18). The plots clearly show at a glance that, for the same Ne , BLFM roughly doubles the number of correct digits provided by the other two instances. It also illustrates that CMM and DLMM give the same error magnitude (within plot accuracy) although of different signs. Thus log-log plots such as those in Figure 22.6 are unable to show whether the convergence is from above or below, because of the taking of absolute values in (22.19). That visualization deficiency should be kept in mind should error signs be important.

§22.1.10. Other Customization Options The last three customization options listed in Table 21.1 are not relevant to this element. RHFP is unnecessary because the dispersion diagram does not have an optical branch. MSTS is pointless because the DLMM in (22.1) is unique. Finally, RDAW does not apply to 1D elements. §22.1.11. Bar2 Frequency Dependent Mass As noted in §21.3.9, it is occasionally useful to make the mass and/or stiffness matrix frequency dependent. The goal is to exactly match the continuum dispersion relation = κ for all frequencies, or at least a finite range that includes = κ = 0. Such an exact fit allows for coarser discretizations. The cost paid is that matrix entries become trigonometric functions of frequency. Both the EOM and associated eigenproblems become trascendental. Unless the frequency is specified beforehand (for example, in pure harmonic excitation) an iterative process is unavoidable. Therefore “exactness” gains might be illusory: the dog chases its own tail. 22–11


Early publications that follow this approach are cited in §H.6. For reasons indicated there, those formulations are not necessarily instances of the general template derived in §22.1.12. The simplest way to introduce frequency dependency is to allow the mass template parameter µ in (22.3) to be frequency dependent, while the stiffness matrix is held fixed. To find the expression of µ, set κ → in the characteristic equation extracted from (22.11):

(22.20) 6 − (2 + µω )2 − 6 − (1 − µω )2 cos = 0, in which µ has been renamed µω . Solving for it gives µω = 1 +

6 3 1 2 4 6 − = − − − − ... 2 1− cos 2 40 1008 28800

(22.21)

Since κ = for the continuum, µω = 1 +

6 3 1 κ2 κ4 κ6 − = − − − − ... κ2 1− cos κ 2 40 1008 28800

(22.22)

As → 0 or κ → 0 both (22.21) and (22.22) approach 0/0. The indeterminacy is removed by the Taylor expansions given above, which show that the limit is µω → 12 , as may be expected. As or κ grows, µω decreases so the template gradually favors the CMM more. Two interesting values should be noted. If κ = 3.38742306673364, µω = 0, which makes the CMM frequency exact; this occurs at the intersection of the continuum and CMM dispersion curves in Figure 22.4(a). If κ = κlim = 4.05751567622863, µω = − 12 , which makes Me singular. If κ > κlim , Me becomes indefinite. It follows that the match (22.21) or (22.22) is practically limited to the DWN range 0 ≤ κ < 4. §22.1.12. Bar2 Frequency Dependent Mass-Stiffness Pair The most general FDMS template for Bar2 has 8 free parameters. These are chosen as deviations from the optimal frequency-independent matrices: ω ω ω 5 1 µ11 −µω12 1 −1 β11 −β12 e e + , K = CK + , M = CM ω ω −µω21 µω22 −β21 β22 1 5 −1 1 (22.23) in which C M = ρ A/12 and C K = E A/. All parameters may be frequency dependent. For brevity that dependency will not be explicitly shown unless necessary. If all µiωj vanish, Me reduces to (22.16), which is BLFM optimal. If all βiωj vanish, Ke reduces to the well known stiffness of a 2-node prismatic bar. Thus in the zero-frequency (static) limit all parameters must vanish, which provides useful checks. To cut down on parameters, we impose diagonal and antidiagonal ω ω ω ω = β12 , and β22 = β11 . In addition symmetry conditions a priori: µω21 = µω12 , µω22 = µω11 , β21 ω ω ω = β21 = β11 avoids singularities in the static limit, as noted later. Thus (22.23) reduces setting β12 to ω ω ω 5 1 µ11 −µω12 1 −1 β11 −β11 e e + , K = CK + . M = CM ω ω −µω12 µ11 −β11 β11 1 5 −1 1 (22.24) 22–12

§22.1


Table 4. General FDMS Template For Bar2 Free parameters∗ Linkage equation (top line); Taylor series† at ω = κ = 0 (bottom line) µω11 , µω12

ω ω ω 2 2 β11 = (κ (5 +ωµ11 ) + (12 + κ2 (1 − µ312 )) cos κ − 12)/(12(1 − cos κ)) ω ω β11 κ→0 = [ µ11 0 − µ12 0 ] κ + O(κ )

ω β11 , µω12

ω ω µω11 = (12 + 12β11 − 5κ 2 − (12 + 12β11 + κ 2 (1 − µω12 )) cos κ)/κ 2 ω ω ω 2 3 µ11 κ→0 = [ µ11 0 − µ12 0 ] κ + O(κ )

ω β11 , µω11

ω ω µω12 = (12 + 12β11 + κ2 − − κ 2 (5 + µω11 )) sec κ)/κ 2 (122 + 12β11 ω ω ω µ12 κ→0 = [ µ11 0 − µ12 0 ] κ + O(κ 3 )

Template parameters are generally functions of κ or ; e.g., µω11 = µω11 (κ) = µω11 (), etc. Parameter arguments are usually omitted to reduce clutter unless necessary. † In the bottom-line series, β ω , µω and µω denote parameter values at κ = = 0. 11 0 11 0 12 0 ∗

ω which has 3 free parameters: µω11 , µω12 and β11 . These matrices are nonnegative if

4 + µω11 + µω12 ≥ 0,

6 + µω11 − µω12 ≥ 0,

ω 1 + β11 ≥ 0.

(22.25)

Imposing the plane wave motion (22.8) on a two-element patch, extracting the middle node equation and dropping extraneous factors yields the complex characteristic equation

12(1+β11 ) − (5+µω11 ) 2 − 12 + 12β11 + (1−µω12 ) 2 cos κ exp( jκ) = 0.

(22.26)

Since the complex exponential never vanishes, it may be dropped and (22.26) reduces to the real equation

ω ω ) − (5+µω11 ) 2 − 12 + 12β11 + (1−µω12 ) 2 cos κ = 0. 12 (1+β11

(22.27)

To match the continuum, is replaced by κ, whence ω ω ) − (5+µω11 ) κ 2 − (12 + 12β11 + (1−µω12 ) κ 2 ) cos κ = 0. f cm = 12 (1+β11

(22.28)

This establishes a linear constraint among the 3 parameters. Consider these as functions of κ: ω ω = β11 (κ), µω11 = µω11 (κ), and µω12 = µω12 (κ). Expanding in Taylor series about κ = 0 yields β11

f cm = 6

ω β11 0

−

µω11 0

+

µω12 0

ω ω ω ∂β11 ∂µ ∂µ 11 12 − + κ 3 + . . . = 0, (22.29) κ + 6 ∂κ 0 ∂κ 0 ∂κ 0

2

This shows that = κ = 0 the continuum equation is identically satisfied. If in the static limit ω ω ω −2 , however, a term in κ appears in (22.29); this is the reason for presetting β12 = β11 . β12 |0 = β11 0 Further developments depend on which parameter pair is kept. Table 22.4 lists three possibilities: ω ω , µω11 ), (β11 , µω12 ), and (µω11 , µω12 ). (β11 22–13


§22.1.13. Bar2 Frequency Dependent Mass Instances ω = 0 in (22.24). Taking Some relatively simple FDM instances can be obtained by setting β11 ω ω µ12 = µ11 and solving for the latter gives

µω11 = µω12 = 1 +

κ2 κ4 κ6 12 6 = − − − − ... − κ2 1 − cos κ 20 540 14400

(22.30)

The resulting Me is indefinite if κ > 4.05752. This is the equivalent of the FDM instance considered in §22.1.11. The difference between (22.30) and (22.22) lies in the choice of baseline matrix for ω = 0 yields null free parameters. On the other hand, setting µω12 = 0 along with β11 µω11 =

−12 + 5κ 2 + (12 + κ 2 ) cos κ κ4 11κ 6 = − − − ..., κ2 40 14400

(22.31)

This correction is smaller than (22.30) if κ < π/2. The resulting Me is indefinite if κ > 4.46192.

22–14

23

Mass Templates for Bar3 Elements

23–1



§23.1

The Three-Node Bar Element . . . . . . . . . . §23.1.1 Dispersion Diagram Terminology . . . . . . §23.1.2 Bar3 General Mass-Stiffness Template . . . . §23.1.3 Bar3 Alternative Mass Template . . . . . . §23.1.4 Bar3 Patch Equations . . . . . . . . . . §23.1.5 Bar3 Fourier Analysis . . . . . . . . . . §23.1.6 Bar3 Standard Template Instances . . . . . §23.1.7 Bar3 Low-Frequency Fitting . . . . . . . . §23.1.8 Bar3 Lumped Mass Template Variant . . . . §23.1.9 Bar3 MSTS Customization . . . . . . . . §23.1.10 Reducing High Frequency Pollution . . . . . §23.1.11 Bar3 Spectral Mass Variant . . . . . . . . §23.1.12 Bar3 Selective Mass Scaling Variant . . . . . §23.1.13 Bar3 Constant Optical Branch Variant . . . . . §23.1.14 Bar3 Test: Vibrations of a Fixed-Free Bar Member

23–2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

23–3 23–3 23–3 23–4 23–6 23–6 23–7 23–8 23–10 23–11 23–12 23–13 23–14 23–15 23–19

§23.1

THE THREE-NODE BAR ELEMENT

§23.1. The Three-Node Bar Element The three-node bar element configuration is shown in Figure 23.1(a). The element is prismatic with length = L e , uniform cross section area A and mass density ρ. Midnode 3 is at the center. The element DOFs are arranged as ue = [ u 1 u 2 u 3 ]T . The element name is often abbreviated to Bar3 in the sequel. We will consider only frequency independent templates here. Despite its simplicity, the Bar3 template is sufficiently feature rich so it may be used to illustrate most of the customization scenarios listed in Table 21.1. Two reasons: it has multiple dispersion branches, and the stiffness has a free parameter. But additional terminology on dispersion diagrams has to be introduced first. Readers familiar with the topic should skip to §23.1.2. §23.1.1. Dispersion Diagram Terminology The characteristic equation of the Bar3 element, derived in §23.1.5 below, gives two positive real frequencies for each plane wave wavelength. The dimensionless forms are identified by a and o , ordered so a ≤ o . The functions a (κ) and o (κ), in which κ is the dimensionless wavenumber, are called acoustic and optical branches, respectively, of the DDD. This terminology originated in crystal physics, in which both branches have physical meaning in modeling molecular level oscillations. [In crystallography, acoustic waves are lower frequency waves caused by sonic-like disturbances, in which adjacent molecules move in the same direction. Optical waves are higher frequency oscillations caused by interaction with light or electromagnetics, in which adjacent molecules move in opposite directions. Textbook references are provided in ?.] In FEM discretization work, only the acoustic branch has physical meaning because for small κ (that is, long wavelengths) it approaches the continuum bar relation = κ, as plainly illustrated by the a2 series in §22.1.5. On the other hand, the optical branch is spurious. It is caused by the discretization and pertains to higher frequency lattice oscillations, also known as “mesh modes.’ Figure 23.2 displays nomenclature used for a two-branch dispersion diagram, such as that exhibited by the Bar3 element. As noted, the acoustic branch is the long-wavelength counterpart of the continuum model, for which = κ; thus a |κ→0 = 0. On the other hand, the optical branch has a nonzero frequency oc = o |κ→0 called the cutoff frequency (COF) that cannot vanish, although it may go to infinity under certain conditions; such as a singular mass matrix. Also of interest are the values of a and o at the folding frequency κ f = π ; these are denoted a f and o f , respectively. The lowest and highest values of o are called max and min o o , respectively, whereas the largest max a is called a . For the plots drawn in Figure 23.2 (note disclaimer on the right): amax = a f ,

min = o f , o

max = oc . o

(23.1)

Often, but not always, min and amax occur at κ = π, the folding (Nyquist) wavenumber at which o > amax , the frequency range min > > amax is group velocities vanish. In any case, if min o o called the acoustoptical frequency gap. Frequencies within the gap are said to pertain to portion I of the stopping band or stopband, a term derived from filter technology. Frequencies > max o pertain to portion II of the stopping band. A frequency that falls within a stopping band cannot propagate as plane wave over the FEM lattice, since there the characteristic equation has complex roots with negative real parts. This causes exponential attenuation so any periodic disturbance with that frequency will die out. 23–3


(a)

(b)

ρ,E,A = const 1

(e)

3 = Le

ξ = −1

j−2

Midnode 3 @ center

2 x

j

j−1

j+1

j+2

Two-element patch

xj −

ξ=1


x j

xj

xj +

Figure 23.1. Three-node prismatic bar element: (a) configuration; (b) extraction of two-element patch from a regular lattice.

§23.1.2. Bar3 General Mass-Stiffness Template We begin by introducing a general template for the mass-stiffness (MS) pair. The mass template is given four parameters: 4 + µ1 −1 + µ3 2 + µ4 µ1 µ3 µ4 e e m m (23.2) Meµ = −1 + µ3 4 + µ1 2 + µ4 = MCe M M + µ3 µ1 µ4 . 30 30 µ µ µ 2+µ 2+µ 16 + µ 4

4

2

4

4

2

is the consistent mass matrix (CMM), obtained for µ1 = µ2 = in which m = ρ A . Here µ3 = µ4 = 0, which is derived in §17.1. The template (23.2) incorporates matrix, geometric and fabrication symmetries ab initio. It includes all DLMM by setting µ3 = 1 and µ4 = −2. Because of its practical importance, however, that “lumped mass” subset is studied in §23.1.8 using a two-parameter template variant. e

MCe M M

For (23.2) to be nonnegative definite (NND), three inequality constraints have to be satisfied. Those are more elegantly expressed in terms of the alternative “χ -form” derived in §23.1.3. Conservation of total element mass ρ A (invariance of linear momentum) imposes the following homogeneous constraint: (23.3) 2µ1 + µ2 + 2µ3 + 4µ4 = 0. This constraint is not always preimposed as it may complicate intermediate expressions, but it is eventually applied at some point. Conservation of angular momentum in 2D or 3D requires µ1 = µ3 , as verified by the CMM. This is ignored, however, as it hinders customization. As regards the stiffness matrix, the following one-parameter template is used 1 1 −2 1 −1 0 e β 4 k Ke = Keb + Keh = k e −1 1 0 + 1 1 −2 . 3 −2 −2 4 0 0 0

(23.4)

in which k e = E A/. Here Keb and Keh denote the basic and higher-order stiffness matrices, respectively. This decomposition was introduced by Bergan and coworkers in the 1980s for the development of the Free Formulation; references are provided in §H.7. The higher order stiffness is scaled by the free parameter β ≥ 0. Setting β = 1 produces the well known stiffness of the quadratic (isoparametric) displacement model, whereas β = 0 reduces Ke to the Bar2 stiffness considered in §22.1.2. 23–4

§23.1 κ=0


κ=2π

Brillouin zone

Dimensionless frequency Ω

(branches have period 2π)

stopping band II

Ωoc

Ωoc optical branch Ωo= Ωo(κ)

cutoff frequency

Ωof

max

folding optical frequency

Ωof

acoustoptical gap: same as stopping band I

Ωaf

folding acoustic frequency

Ωaf

acoustic branch Ωa= Ωa(κ) folding 0

wavenumber

π

For the diagram as drawn

Dimensionless wavenumber κ

Ωa = Ω af Ωomax = Ω oc min Ω o = Ω of

These do not generally hold in more complex elements, multiple space dimensions or irregular lattices

2π

Figure 23.2. Nomenclature for a two-branch dispersion diagram typical of 1D structural elements such as Bar3. The stopping band is the union of I and II. (Symmetry and monotonicity about the folding wavenumber κ f = π is typical of prismatic, simple 1D elements in regular lattices; else those features are typically lost. In addition, multiple optical branches will appear for characteristic equations with more than two roots for each κ.)

§23.1.3. Bar3 Alternative Mass Template An alternative configuration of the general Bar3 mass template is obtained by changing the four µi free parameters to three: χ1 , χ2 , and χ3 , through the replacement rule µ1 = χ1 + χ2 − 4, µ2 = 14 + 4χ1 − 4χ13 , µ3 = χ1 − χ2 + 1, µ4 = χ13 − 2χ1 − 2. (23.5) √ √ in which χ13 = 30 χ1 −χ3 . The mass conservation condition (23.3) is identically satisfied by (23.5), which is why the number of free parameters can be cut by one. Conversely, if the µi are given and do satisfy (23.3), the χi can be computed from χ1 = χ3 =

1 (3 + µ1 + µ3 ), 2 1 4µ1 (40 + µ2 480

χ2 = 12 (5 + µ1 − µ3 ),

− 2µ3 ) + 40 (8 + µ2 + 4µ3 ) − 4µ21 − (µ2 − 2µ3 )2 .

(23.6)

(If µ1 = µ2 = µ3 = 0, this gives χ1 = 3/2, χ2 = 5/2 and χ3 = 2/3 for the CMM.) On inserting (23.5) into (23.2) the so-called “χ-form” of the Bar3 mass template emerges:   χ1 + χ2 χ1 − χ2 −2χ1 + χ13 e m  (23.7) Meχ = χ1 − χ2 χ1 + χ2 −2χ1 + χ13  30 −2χ1 + χ13 −2χ1 + χ13 30 + 4χ1 − 4χ13 An attractive feature of (23.7) is that mass matrix admissibility can be readily correlated to parameter values. Specifically, Meχ is positive definite (PD) if and only if χ1 , χ2 and χ3 are positive. This can be proven from the following properties: λ1 (Meχ ) =

m e χ2 , 15

det(Meχ ) =

m e χ2 χ3 , 225

det(Meχ 2×2 ) =

m e χ1 χ2 , 225

(23.8)

The first equality gives one eigenvalue of Meχ (the other two have more complicated expressions), whence PD mandates χ2 > 0. The last equalities give the determinants of Meχ and of its 2 × 2 23–5


upper principal minor, respectively. Accordingly, PD requires also χ3 > 0 and χ1 > 0. For Meχ to be nonnegative definite (NND) simply change > to ≥. Those conditions can be harked back to the µi of Meµ using (23.6), but the expressions are noticeably messier. The second equality of (23.8) gives another nice feature: Meχ becomes singular if and only if χ2 = 0, or χ3 = 0, or both. The main advantage of Meµ over Meχ is the linear dependence of entries on the µi . This simplifies patch analysis as well as reparametrization for several template variants studied later. §23.1.4. Bar3 Patch Equations To assess wave propagation and dispersion performance of the MS template defined by (23.2) and (23.4), we carry out the Fourier analysis of the infinite bar lattice shown in Figure 17.1(b). Extract a typical two node patch as illustrated. The patch has five nodes: three endpoints and two midpoints, which are assigned global numbers j−2, j−1, . . . j+2. The unforced semidiscrete dynamical equations of the patch are M p u¨ P + K p u P = 0, in which 

4 + µ1 2 + µ4 16 + µ2 2 + µ  4 me  Mp =  −1 + µ3 2 + µ4 30  0 0 0 0  3 + 4 β −8 β 16 β  E A  −8 β Kp =  −3 + 4 β −8 β 3  0 0 0 0 u P = [ u j−2 u j−1 u j u j+1

 −1 + µ3 0 0 2 + µ4 0 0   2(4 + µ1 ) 2 + µ4 −1 + µ3  ,  2 + µ4 16 + µ2 2 + µ4 −1 + µ3 2 + µ4 4 + µ1  −3 + 4 β 0 0 −8 β 0 0   6 + 8 β −8 β −3 + 4 β  ,  −8 β 16 β −8 β −3 + 4 β −8 β 3 + 4 β u j+2 ]T .

(23.9)

Note that the element mass conservation constraint (23.3) is not preimposed as it would complicate intermediate expressions. It is enforced later. Keep the second and third equations, namely those for nodes j − 1 and j. This selection picks the equations for a typical corner and midpoint node. Accordingly, the patch equations are p u P = 0. p u¨ P + K M

(23.10)

p result on deleting rows 1,4,5 of M p and K p , respectively. p and K The 2 × 5 matrices M §23.1.5. Bar3 Fourier Analysis We study the propagation of harmonic plane waves of wavelength λ, wavenumber k = 2π/λ, and circular frequency ω over the lattice of Figure 17.1(b). For convenience they are separated into corner and midpoint waves: √

u c (x, t) = Bc exp j (kx − ωt) , u m (x, t) = Bm exp j (kx − ωt) , j = −1. (23.11) Wave u c (x, t) propagates only over corner nodes and vanishes at midpoints, whereas u m (x, t) propagates only over midpoints and vanishes at corner nodes. Both have the same wavenumber and frequency but different amplitudes and phases. [The wave pair (23.11) can be combined to 23–6

§23.1


form a single waveform that propagates over all nodes. The combination has two components that propagate at the same speed but in opposite directions. This is useful when studying boundary conditions or transitions in finite lattices, but unecessary for a periodic infinite lattice.] √ As in §22.1.5, we will work with the dimensionless frequency = ω /c0 with c0 = E/ρ, and the dimensionless wavenumber κ = k. Inserting (23.11) into (23.10), passing to dimensionless variables, removing scale factors, and requiring that solutions exist for any t yields the characteristic equation 1 A (160β−(16+µ2 )2 ) −A1 A3 (80β+(2+µ4 )2 ) Bc 2 3 = 0. Bm −A1 (80β+(2+µ4 )2 ) 30+40β−(4+µ1 )2 +A1 (−30+40β+(1−µ3 )2 ) (23.12) in which A1 = cos κ, A2 = cos(κ/2) and A3 = cos(κ/2) − j sin(κ/2). For nontrivial solutions the determinant of the characteristic matrix must vanish, which provides a quadratic equation in 2 . For each wavenumber κ, solving the equation gives two squared frequencies. Their expressions, found by Mathematica, are √ √ P− Q P+ Q 2 2 , o = 5 , (23.13) a = 5 R R in which coefficients P, Q, and R are given by P = c1 + 3 c2 + c9 cos κ,

R = c5 − 4 µ4 − µ24 + c8 cos κ,

Q = 192 β (cos κ − 1) (c5 − c3 + c8 cos κ) + (c1 + 3 c2 + c9 cos κ)2 , c1 = 4 β (40 + 4 µ1 + µ2 + 4 µ4 ), c2 = 16 + µ2 , c3 = µ4 (4 + µ4 ), c4 = 4 (5 + µ3 + µ4 ), c5 = 60 + 4 µ2 + µ1 c2 , c6 = 16 µ3 − c3 , c7 = 4 β (µ2 + c4 ), c8 = c6 + µ2 (µ3 − 1) − 20, c9 = c7 − 3 c2 .

(23.14)

Subscripts a and o stand for acoustic and optical branches, respectively, a terminology explained in §23.1.1. If Me is positive definite and the conservation condition (23.3) holds, the branch frequencies (23.13) have small κ (low frequency, long wavelength) Taylor series of the generic form a2 = κ 2 +

C4 κ 4 C6 κ 6 C8 κ 8 + + + ..., 4! 6! 8!

2o = D0 +

D2 κ 2 D4 κ 4 + + ..., 2! 4!

(23.15)

Coefficients Cn and Dn were obtained through the Mathematica built-in Series function up to n = 10 and are displayed for some interesting instances below. §23.1.6. Bar3 Standard Template Instances We start by considering two instances available in the FEM literature since the mid 1960s. The CMM instance MCe is obtained for µ1 = µ2 = µ3 = µ4 = 0. Using β = 1 for Ke we get P = 208 + 32 cos κ, Q = 128 (237 + 224 cos κ − 11 cos(2κ)), and R = 20 (3 − cos κ). The squared frequencies have the small-κ expansions a2 = κ 2 +

κ6 11 κ 8 − + ..., 720 151200

2o = 60 − 20 κ 2 + 23–7

19 κ 4 + .... 3

(23.16)

Chapter 23: MASS TEMPLATES FOR BAR3 ELEMENTS (b) 2

7

1.5

6 Optical

5 4

Continuum

3

1

Continuum

0.5 0

−0.5

Acoustical

2

Acoustical

−1 Optical

−1.5

1 0

Velocity ratio γc =c/c0


(a) 8

1

0

2

3

4

5


6

5

4

3

6

(d)

7 Continuum

6 5 Optical

3

1.5 Continuum

1 0.5 0

−0.5

Acoustical

2

Acoustical

Optical

−1

−1.5

1 0

2


2



(c) 8

4

1

0

0

1

2

3

4

5


6

1

0

2

4

3

5


6

Figure 23.3. DDD and DGVD plots for well known Bar3 template instances treated in §23.1.6. (a,b): diagrams for CMM instance; (c,d): diagrams for SLMM (Simpson DLMM) instance. Acoustic and optical branches shown in red and blue, respectively. Continuum case = κ and γc = 1 shown in black.

The SLMM (Simpson lumped diagonal mass matrix) instance derived in (17.3) of §17.1 results if µ1 = 1, µ2 = 4, µ3 = 1, and µ4 = −2. Using β = 1 in (23.4) gives P = 220 + 20 cos κ, Q = 200 (147 + 140 + cos(2κ)), and R = 100. The squared frequencies have the small-κ expansions a2 = κ 2 −

κ8 κ6 − + ..., 1440 48384

2o = 24 − 2κ 2 +

κ4 + .... 12

(23.17)

For small κ, SLMM fits the continuum better than CMM. Dispersion diagrams for the foregoing instances are plotted in Figures 23.3(a,c). Corresponding group velocity diagrams are shown in Figures 23.3(b,d). As in the case of the two-node bar, the consistent mass overestimates the continuum frequency = κ for 0 ≤ κ ≤ π, whereas the lumped mass underestimates it. §23.1.7. Bar3 Low-Frequency Fitting Inspection of the coefficient of κ 6 in (23.16) and (23.17) suggests combining one third of MCe with two thirds of MeL to cancel it. Setting µ1 = 2/3,

µ2 = 8/3,

µ3 = −2/3,

in (23.4) gives MCe L

ρ A = 13 MCe + 23 MeL = 90 23–8

µ4 = 4/3, 14 −1 2

−1 14 2

β = 1,

2 2 . 56

(23.18)

(23.19)

§23.1


For this instance, labeled BLCD, P = 24 (9 + cos κ), Q = 32 (927 + 884 cos κ − 11 cos 2κ), and R = 20 (13 − cos κ)/3. It has the small κ expansions a2 = κ 2 −

κ8 + ..., 37800

2o = 30 −

15 κ 2 11 κ 4 + + .... 4 32

(23.20)

Dispersion and group velocity diagrams are shown in Figure ?(a,b). Despite the O(κ 8 ) accuracy achieved in the acoustic branch of BLCD, it is shown next that this instance is not optimal. Considering next the general MS template (23.2)–(23.4), let us find the MS pair for which the acoustic branch a best matches the continuum = κ forsmall κ. Given the expansion of a2 in (23.15), the goal is to make as many coefficients beyond κ 2 vanish as possible, and to minimize the magnitude of the first surviving one. The analysis was actually performed using the χ -form (23.7) of the general mass template. Four free parameters are available: χ1 , χ2 , χ3 , and β. Only a procedural summary and final results are given. It is possible to make C4 = C6 = 0 without difficulty, which permits elimination of χ1 and χ2 . But all solutions of C8 (χ3 , β) = 0 are imaginary, so the term in κ 8 cannot be cancelled. Extremization to χ3 and β gives only one constraint: 160β 2 − 120βχ3 + 9χ32 = 0, from which of C8 with respect √ χ3 = 4 (5 ∓ 15β)/3. Both signs give the same C8 . Taking β = 1 for convenience, the − sign in χ3 , and working back we get √ √ √ 3 85 4 4 (23.21) −2 ∓ 2 375, χ2 = 5 − 12 15, χ3 = 5 − 15 , β = 1. χ1 = 6 5 3 The − sign for χ1 gives better conditioned mass matrices and still the same C8 , so we pick that one. The numeric values to 16 places are χ1 = 2.7835604012611213, χ2 = 3.0635083268962915, √ and χ3 = 1.50268887172344. The resulting minimum of |C8 | is C8best = 64/3 − 28 3/5 = −0.355373. This is about 3 times smaller than the C8 = −8!/37800 = −16/15 from (23.19). Converting to the µi parameters via (23.5) yields 91−12 a1 −7 a2 32−8 a2 61−12 a1 −a2 −46+6 a1 +4 a2 , µ2 = , µ3 = , µ4 = , (23.22) 6 3 6 3 √ √ in which a1 = 31/4 53/4 = 4 375 and a2 = 15. Numerical values to 16 places are µ1 = 1.8470687281574132, µ2 = 0.3387110767802213, µ3 = 0.7200520743648302, and µ4 = −1.3682381704561768. The resulting mass matrix, labeled BLFM (for best low frequency match), given to 16 places, is 0.1949022909385804 −0.0093315975211724 0.0210587276514608 MeB L F M = m e −0.0093315975211724 0.1949022909385804 0.0210587276514608 . 0.0210587276514608 0.0210587276514608 0.5446237025593408 (23.23) e = ρ A . Its eigenvalues (to 6 places) are positive: 0.547077, 0.204234, and 0.183117 times m √ Hence For this instance P = 8 (35 − 2 15 − √ this mass matrix is admissible √ and well scaled. √ √ (5 − 2 15)√ cos κ), Q = 1920 (15 + 4 15 + (15 − 4 15) cos κ) cos(κ/2)2 , and R = 20 (53 − 10 15 + (7 − 2 15) cos κ)/3. It has the small-κ expansions √ √ 5(29 − 8 1127 − 291 30 15 15)) 2 a2 = κ 2 + κ 8 + . . . 2o = κ − . . . (23.24) √ √ − √ 3024(5 − 15)3 5 − 15 32(5 − 15)3 µ1 =

23–9


7 Continuum

6 5 4



(a) 8

Optical

3

Continuum

1 0.5 0

−0.5

Acoustical

2

Acoustical

−1

Optical

−1.5

1 0

1.5

1

0

2

3

4

5


6

7 Continuum Optical

3 Acoustical

6

1.5 Continuum

1 0.5 0

Acoustical

−1

Optical

1

0

2

3

4

5


6

1

0

(e)

2

3

4

5


6

(f)

8

2

7 6



5

−1.5

1

Continuum

5

Optical

4 3

1.5 Continuum

1 0.5 0

−0.5

Acoustical

2

Acoustical

Optical

−1

−1.5

1 0

4

−0.5

2 0

3

(d)

6 4

2


2



(c) 8

5

1

0

0

1

2

3

4

5


6

0

1

2

3

4

5


6

Figure 23.4. DDD and DGVD plots for RHFP instances derived in §23.1.7 and §23.1.8. (a,b): diagrams for the BLCD instance (23.19). (c,d): diagrams for the BLFM instance (23.23). (d,e): diagrams for the BLFD instance (23.29). Acoustic and optical branches shown in red and blue, respectively. Continuum case = κ and γc = c/c0 = 1 shown in black.

Dispersion and group velocity diagrams are shown in Figure 23.4(c,d). Note that √ values at the√folding (Nyquist) wavenumber κ = π are identical: a2 (π) = 2o (π) = 12(10 − 15)/(23 − 4 15) = 9.792694126734647, which is amazingly close to the continuum value of π 2 = 9.86960440108935. (In fact, the AB for κ < π and the continuum are indistinguishable at plot resolution.) There is no acoustoptical gap; instead we observe a bifurcation point. §23.1.8. Bar3 Lumped Mass Template Variant Although diagonally lumped mass matrices (DLMM) plainly form a subset of the general template (23.2), their practical importance justifies the use of a more compact two-parameter form. This is done by taking µ1 = µ L1 + 1, µ2 = µ L2 + 4, µ3 = 1, µ4 = −2. (23.25) 23–10

§23.1


Replacing into (23.2) produces the lumped mass template variant 0 = MeS L M M 0 . µ L2 (23.26) The baseline mass matrix is the DLMM (17.3) produced by Simpson’s 3-point integration rule, and now labeled SLMM. The stiffness matrix template is still (23.4). Parameters µ L2 and β can be eliminated in favor of µ L1 through ρ A MeL = 30

5 + µ L1 0 0

0 5 + µ L1 0

0 0 20 + µ L2

2µ L1 + µ L2 = 0,

β=

ρ A + 30

µ L1 0 0

0 µ L1 0

(10 − µ L1 )2 . 20 (5 − µ L1 )

(23.27)

The first constraint expresses element mass conservation while the second one enforces C4 = 0 and makes the AB agree with the continuum through κ 4 in the expansion (23.15). (This agreement is considered essential as otherwise there would be no advantage in using this element instead of Bar2.) As only one parameter remains, customization is straightforward. The admissible range in µ L1 for PD mass is −5 < µ L1 < 10, but if µ L1 > 5, β < 0 and Ke becomes indefinite. On applying (23.27), the first NZ term in the ABTS beyond κ 2 is C6 κ 6 /6!, in which C6 = (5 − 3µ L1 + µ2L1 )/(µ L1 − 10). Trying to attain O(κ 6 ) accuracy by setting C6 = 0 is futile since the√µ L1 roots are complex conjugate. √ Solving ∂C6/∂µ L1 = 0 gives two real solutions: µ L1 = 5(2 ± 3). Of these only the one with − 3 keeps Me admissible. Replacing into (23.27) gives the signature µ L1 = 5(2 −

√ 3),

µ L2 = −2µ L1 = −10(2 −

√ 3),

β=

3 . √ 4( 3 − 1)

(23.28)

Numerical values to 16 places are µ L1 = 1.339745962155614, µ L2 = −2.679491924311228, and β = 1.024519052838329. The κ 6 term in the AB is about 36% smaller than that of SLMM: ≈ −κ 6 /2246 versus −κ 6 /1440. The template instance, labeled BLFD, is MeB L F D KeB L F D

√ 0 ρ A 5 3 − 5 √ 0 = 0 √ , 0 5 3−5 30 0 0 20 − 10 3 1 1 4E A E A 1 −1 0 = −1 1 0 + √ 1 1 ( 3 − 1) 0 0 0 −2 −2

−2 −2 . 4

(23.29)

The Taylor series of the dispersion branches are a2

√ 10 3 − 10 6 κ − ..., =κ − 720 2

2o

√

3 κ2 − . . . = √ − √ 2 ( 3 − 1) 4 3−6 12

(23.30)

The DDD and DGVD are shown in Figure 23.4(e,f). As in the case of the√BLFM, pictured in (c,d) √ 2 2 of that figure, the branches intersect at κ = π, where a f = o f = (6−2 3)/(2− 3) ≈ 9.4641. 23–11


§23.1.9. Bar3 MSTS Customization Since DLMM are often used in explicit DTI, it is of some interest to find whether the stable time stepsize can be maximized while still satisfying O(κ 4 ) accuracy. This goal pertains to the MSTS customization of Table 21.1. Let max be the maximum of a and o over the Brillouin zone κ ∈ [0, 2π ]. To maximize the stable time step, one minimizes max over free parameters, while trying to keep both mass and stiffness admissible. This procedure can be streamlined by assuming a DDD configured as in Figure 23.2, whence only frequency values at κ = 0 and κ = π need to be considered. Since a |κ=0 = 0, the search only involves oc , a f and o f , or (for convenience) their squares. For the DLMM template variant (23.26) under the accuracy constraints (23.27) the process boils down to solving the max-min problem in one variable:

2 60 60(10 − µ L1 ) 4(10 − µ L1 ) 2 2 , , . (23.31) min max oc , a f , o f = min max µ L1 µ L1 µ L1 − 5 5 + µ L1 25 − µ2L1 A simple plot shows that the cutoff frequency dominates for admissible µ L1 ∈ (−5, 5), so it is sufficient to minimize with respect to 2oc . This again leads to the solution (23.25). Consequently the BLFD instance also maximizes the explicit DTI time step. The reward, however, is marginal with respect to SLMM: only about a 3.5% gain. To get a more significant improvement, it is necessary to keep β free, and accept that O(κ 4 ) accuracy is lost. It may be verified that the largest possible stable timestep is produced by the signature µ L1 = 5/2,

µ L2 = −2µ L1 = −5,

β = 3/8.

(23.32)

which apportions nodal masses as 1:1:2, while substantially modifying the stiffness matrix. Setting (23.32) gives an instance with a constant optical branch (COB) 2o = 8. Its stable stepsize is 1.673 times that of BLFD. But its LF performance is exactly the same as that of the lumped-mass Bar2, which does not have an OB. So it is largely a curiosity. §23.1.10. Reducing High Frequency Pollution The presence of the optical branch (OB) does not affect vibration calculations in structural dynamics. One simply ignores those eigenfrequencies as nonphysical. However, the OB may become a nuisance in direct time integration (DTI) for problems that involve discontinuities, such as pulse propagation, or contact-impact, because it may feed spurious noise. To alleviate this problem three approaches may be tried at the template level: 1.

Singular Mass Matrix. If Me is made singular with an appropriate null eigenvector, the optical branch is raised. In fact it becomes infinite at κ = 0. The net effect is that the acoustoptical gap is increased at low wavenumbers. This heps to filter out frequencies that fall in the gap, since they will decay exponentially. One drawback of singularity is that explicit DTI is excluded, even if Me is diagonal.

2.

Selective Mass Scaling. A scaled stiffness matrix is added to the mass. As discussed in §21.3.5, eigenvectors are unchanged but higher natural frequencies are effectively reduced. The effect is similar to that of adding stiffness proportional damping, but without altering vibration modes. It may be done at the element or assembly (master) level. In the study of §23.1.12 it is done at the element level. 23–12

§23.1 3.


Constant Optical Branch. A constant optical branch (COB) is one independent of wavenumber. It stays at a constant frequency o = oc over the entire Brillouin zone, and has zero group velocity since ∂oc /∂κ = 0. To be effective in cutting noise pollution, oc ≥ a f , in which a f is the folding acoustic frequency. If that holds, the stopping band above a f is effectively maximized. (Even if a mesh frequency hits oc exactly, it will not propagate since its group velocity vanishes.) A COB template is one that possesses that property (for each OB should there be more than one).

The three foregoing approaches are studied below for the Bar3 element. §23.1.11. Bar3 Spectral Mass Variant Making Me singular is not sufficient; it is important to have the correct null eigenvector. To achieve that it is convenient to use the Spectral Parametrization (SP) outlined in §21.3.2. Select three generalized coordinates: g0 , g1 and g2 as amplitudes of three physically transparent eigenmotions: g0

Amplitude of rigid body motion: v0 = [ 1

g1

Amplitude of acoustic bar motion: : v1 = [ −1

1

g2

Amplitude of optical bar motion: : v2 = [ 1

−2 ]T .

1 ]T .

1 1

0 ]T .

√ √ 3, 2 and Those three vectors are mutually orthogonal. To make them orthonormal, divide by √ 6, respectively. Stacking the orthonormalized vectors as columns, the linkage between physical and generalized coordinates can be expressed as √ √ √ 1/√3 −1/√ 2 1/√6 g0 ue = [ u 1 u 2 u 3 ] = 1/√3 1/ 2 (23.33) 1/ √6 = g1 = HT g. 0 −2/ 6 g2 1/ 3 The inverse relation is g = H ue because H is orthogonal by construction, and thus H−1 = HT . As mass matrix in generalized coordinates we stipulate the 3 × 3 diagonal matrix Dµ of entries m e µ S0 /3, m e µ S1 /45, and m e µ S2 /15, in which m e = ρ A , and the scaling factors were chosen for convenience in cleaning up downstream expressions. Element mass conservation will require µ S0 = 1, so the first entry is simply m e /3. Transforming to physical coordinates yields the spectral mass template variant 10 + µ S1 + µ S2 10 − µ S1 + µ S2 10 − 2 µ S2 15 0 0 e e m m MeS = HT 0 µ S1 0 H= 10 − µ S1 + µ S2 10 + µ S1 + µ S2 10 − 2 µ S2 . 45 90 10 − 2 µ S2 10 − 2 µ S2 10 + 4 µ S2 0 0 3µ S2 (23.34) The variant (23.34) is a subset of the general template (23.2) that results by taking µ1 = 13 (µ S1 +µ S2 −2), µ2 = 13 (4 µ S2 −38), µ3 = 13 (13−µ S1 +µ S2 ), µ4 = 13 (4−2 µ S2 ). (23.35) e e e By construction, the eigenvalues of (23.34) are m /3, m µ S1 /45 and m µ S2 /15, whence the nonnegativity condition is fulfilled if µ S1 and µ S2 are nonnegative. To make MeS singular, set µ S2 = 0, which produces 10 + µ S1 10 − µ S1 10 e m (23.36) MeS = HT Dµ H = 10 − µ S1 10 + µ S1 10 . 90 10 10 10 23–13


Solving C4 = 0 and C6 = 0 yields two solutions for β and µ S1 , of which we pick that with larger µ S1 (to get a better Me eigenvalue). This gives √ √ 5 + 10 3(5 + 10) = 12.24341649025257, β = = 0.6801898050140316, µ S1 = 2 12 (23.37) in addition to µ S2 = 0. Inserting into (23.39) gives the instance labeled BSSM for Best Singular Spectral Mass. The mass matrix, with numerical values given to 6 places, is M11 M12 M13 0.247149 −0.024927 0.111111 e m MeB SS M = 0.247149 0.111111 . (23.38) M12 M22 M23 ≈ m e −0.024927 180 M M M 0.111111 0.111111 0.111111 13

23

√

33

√ in which M11 = M22 = 35+ 10, M12 = 5−3 10, and M13 = M23 = M33 = 20. The associated stiffness matrix, with numerical values given to 6 places, is K 11 K 12 K 13 1.906920 −0.093080 −1.813839 e k KeB SS M = 1.906920 −1.813839 . (23.39) K 12 K 22 K 23 ≈ k e −0.093080 9 K K 23 K 33 −1.813839 −1.813839 3.627679 13 √ √ √ in which k e = E A/, √ K 11 = K 22 = 14 + 10, K 12 = −4 + 10, K 13 = K 23 = −10 − 2 10 and K 33 = 20 + 4 10. Dispersion and group velocity diagrams are shown in Figure 23.5(a,b). The Taylor series of the dispersion branches are a2 = κ 2 −

κ8 − ..., 6048

2o = 240 κ −2 +

5κ 4 + ... 126

(23.40)

The O(κ 6 ) AB accuracy of this element is comparable to that of BLCD and BLFM, but its OB gets out of the way. Is this the template instance for all seasons? Only future experimentation in direct time integration (DTI) will tell. §23.1.12. Bar3 Selective Mass Scaling Variant In the Selective Mass Scaling (SMS) approach outlined in §21.3.5, the mass matrix is modified by adding a scaled version of the stiffness matrix: MeK = Meu + c K Ke .

(23.41)

Here Me is an unmodified mass matrix, and c K a scaling coefficient with appropriate physical dimensions. Both Me and Ke may generally be template forms. Since Me and Ke have different physical dimensions, it is convenient to change the raw expression (23.41) to MeK = Meu + µ K s e Ke ,

(23.42)

in which s e is a scaling coefficient with dimension of mass-over-stiffness (equivalently, 1/s e has dimensions of squared physical frequency) while µ K is a dimensionless free parameter. For the Bar3 element we take s e = (ρ A )/(E A/) = ρ 2 /E. This can be maneuvered to the following equivalent form, which is convenient for implementation: . MeK = Meu + µ K m e K e

23–14

(23.43)

§23.1


e is a dimensionless stiffness matrix Here m e = ρ A is (as usual) the element mass, whereas K u obtained by setting E = 1, A = 1 and = 1. To reduce the overall number of parameters, we pick Meu to be the diagonally-lumped template subset (23.26); this agrees with the common use of SMS e . Hence in explicit DTI. The general stiffness template (23.4) with unit E, A and is used for K MeK = m e

1 30

5 + µ L1 0 0

0 5 + µ L1 0

0 0 20 + µ L2

4β µ K + 3

1 1 −2

1 1 −2

−2 −2 4

.

(23.44)

Mass conservation is enforced if µ L2 = −2µ L1 . Inserting this in (23.44) we have three free parameters: µ L1 , µ K and β. This MeK with µ L2 = −2µ L1 is a particular case of the general mass template if µ1 = 1 + µ L1 + 10 (4β + 3)µ K , µ2 = 4 − 2µ L1 + 160 βµ K , (23.45) µ3 = 1 + 10 (4β − 3)µ K , µ4 = −2 − 80 βµ K . Unlike previous variants, now β appears in the mass template. The linkage (23.45) becomes linear if β is preset, for example to 1, and nonlinear otherwise. Further experimentation with the SMS template variant (23.44) was confined to µ L1 = µ L2 = 0, which takes SLMM as original mass matrix. That leaves out two free parameters: µ K and β. Suppose µ K is chosen. Then O(κ 4 ) AB accuracy can be maintained by taking β=

1 . 1 − 12µ K

(23.46)

If µ K > 1/12, β < 0 and Ke becomes indefinite. But setting 0 ≤ µ K ≤ 1/12 hardly change the higher frequencies. For that one needs a much larger µ K ; say µ K = O(1). If so, adjusting Ke as per (23.46) is precluded: the cure is worst than the disease. One may as well set β = 1. The high frequencies are cut down, but LF accuracy is seriously lost. This tradeoff is vividly displayed in the vibration benchmarks reported in §23.1.14. Three instances labeled SMS1, SMS2 and SMS3, are tested there. Their signatures are {µ K = 1/24, β = 2}, {µ K = 1/2, β = 1}, and {µ K = 2, β = 1}, respectively. Dispersion and group velocity diagrams for SMS2 are shown in Figure 23.5(c,d). The poor LF fit is obvious. §23.1.13. Bar3 Constant Optical Branch Variant The investigation of the general Bar3 template (23.2)–(23.4) for COB instances was done under two preset conditions: C4 = 0, which enforces order O(κ 4 ) accuracy in the acoustic branch (AB), and β = 1 in the stiffness template (23.4). Several one-parameter families satisfying these conditions were found. The two that produced simpler mass matrices were retained, reparametrized, and labeled COBA and COBB. Associated mass matrices are subscripted OA and OB, respectively. The COBA family is defined by MCe O B A

me = 12

6 − νA 2 − νA −2 + 2ν A

2 − νA 6 − νA −2 + 2ν A

23–15

−2 + 2ν A −2 + 2ν A 4 − 4ν A

.

(23.47)


7 6



(a) 8

Continuum Optical

5 4 3

Continuum

1 0.5 0

−0.5

2

Acoustical

Acoustical

−1 Optical

−1.5

1 0

1.5

1

0

2

5

4

3


6

7

1.5

6

Continuum

5 4 3 1 0

Optical

6

Continuum

0.5 0 Optical

1

2

5

4

3


Acoustical

−1

6

1

0

(e)

2

3

5

4


6

(f) 2

7 6



5

4

−1.5

Acoustical

8

Continuum

5 4 3

Optical

1 1

Continuum

1 0.5 0

Optical

2

3

4

Acoustical

−1

−1.5

Acoustical 0

1.5

−0.5

2 0

3

1

−0.5

0

2


(d) 2



(c) 8

2

1

0

5


6

0

1

2

3

5

4


6

Figure 23.5. DDD and DGVD plots for three RHFP instances derived in §23.1.11–§23.1.13. (a,b) Diagrams for the BSSM instance (23.19). (c,d) Diagrams for the SMS2 instance (23.23). (e,f) Diagrams for the COB0 instance; first of (23.53). Acoustic and optical branches shown in red and blue, respectively. Continuum case = κ and γc = c/c0 = 1 shown in black.

in which m e = ρ A . The determinant is (1 − ν A )/18. MC O B A is PD if ν A < 1. Parameter ν A is linked to those of the general template (23.2) by µ1 = 11 −

5ν A , 2

µ2 = −2(3 + 5ν A ),

µ3 = 6 −

5ν A , 2

µ4 = −7 + 5ν A .

(23.48)

The COBB family is defined by MCe O B B

me = 432

96 − 36ν B − ν B2 24 − 12ν B + ν B2 −48 + 24ν B

24 − 12ν B + ν B2 96 − 36ν B − ν B2 −48 + 24ν B

−48 + 24ν B −48 + 24ν B 384

.

(23.49)

√ The determinant is (36 − 12ν B − ν B2 )2 /34992. MC O B B is positive definite (PD) if −6( 2 + 1) < 23–16

§23.1


Table 1. Bar3 Instances Compared In Fixed-Free Bar Vibrations Tests

Variant ref. eqn.

Full or diag Me

Signature

First NCT in ABTS †

Cutoff & folding freq‡ oc a f o f

CMM

(23.2)

F

β=1, µ1 =µ2 =µ3 =µ4 =0

+κ 6 /720

7.746 3.162 3.464

SLMM

(23.26)

D

β=1, µ L1 =µ L2 =0

−κ 6 /1440

4.899 2.828 3.464

BLCD

(23.2)

F

See (23.18)

−κ 8 /37800

5.477 2.928 3.464

BLFM

(23.2)

F

See (23.22)

−κ 8 /113458

5.159 3.129 3.129

BLFD

(23.26)

D

See (23.28)

−κ 6 /2246

4.732 3.076 3.076

BSSM

(23.34)

F

See (23.37)

−κ 8 /6048

SMS1

(23.44)

F

β=2, µ K =1/24

−κ 6 /640

4.000 2.828 3.098

SMS2

(23.44)

F

β=1, µ K =1/2

−κ 4 /2

1.359 1.265 1.309

SMS3

(23.44)

F

β=1, µ K = 2

−2κ 4

0.700 0.686 0.692

COB0

(23.47)

F

β=1, ν A = −5/3

−κ 6 /240

2.449 2.449 2.449

Instance name

∞

2.711 5.714

† NCT: non-continuum term, ABTS: AB Taylor series of 2 wrt κ, centered at κ = 0 a ‡ : at κ = 0; : at κ = π; : at κ = π. oc o af a of o

√ ν B < 6( 2 − 1). Parameter ν B is linked to those of the general template (23.2) by µ1 =

8 5ν B 5ν 2 − − B, 3 2 72

µ2 =

32 , 3

µ3 =

8 5ν B 5ν 2 − + B, 3 6 72

µ4 =

−16 + 5ν B . 3

(23.50)

These two families are taken to collectively define the Bar3 template variant identified as COB. They coalesce only for ν A = −5/3 and ν B = −6, which produces an instance discussed below. An interesting result is that the acoustic branch is identical for all COB instances: a2 =

12(1 − cos κ) κ6 κ8 = κ2 − − − ... 5 + cos κ 240 6048

(23.51)

whereas the constant OB value is family and parameter dependent: 2oc A =

16 , 1 − νA

2ocB =

432 . 36 − 12ν B − ν B2

(23.52)

It follows that the only role played by ν A and ν B is to adjust the “OB height” along the vertical DDD axis. As noted in §23.1.10, it should equal or exceed the folding acoustic frequency a2 f = 6, which is the same for all COB instances on account of (23.51). This requires ν A ≥ −5/3 and ν B ≥ −6. As ν A → 1 and ν B → 12 the OB moves to ∞ and the mass matrices assume different 23–17


BLCD

BLFD

BLFM 6 4

CMM

SLMM

2 0 1

2

4

8

16

8

BLFM 6 4

SLMM

2

1

2

SMS1

4

COB0 SMS2 SMS3

0 1

2

4

8


16



BSSM

2

16

8


8

BLFM

6

10

2

CMM SLMM

0 2

4

(f)

6

SMS1

4

BSSM COB0

2

SMS2

0

SMS3 1

16

8


RHFP Bar3 Template Instances, Fix-Free Mode 3, Exact Frequency ω 2 = 3

8

BLFD

BLCD

4

(e)


6

4

10


(d) 8

CMM

0

Number of elements Ne 10

BLFD

BLCD


8

10

(c)


2

4

8


16


10

(b)




(a)

10


8 6 4

SMS1

BSSM

2

SMS2 0 2

4

8

COB0 SMS3

16


Figure 23.6. Performance of ten Bar3 template instances in predicting the first three natural frequencies ωi , i = 1, 2, 3 of the fixed-free prismatic homogeneous bar shown in Figure 22.5(b). See text in §22.1.9 for a detailed description of the log-log plots.

limits. For COBA, MCe O B A ν A →1 is the optimal Bar2 matrix (22.16), which is PD. On the other hand the limit MCe O B B ν B →12 falls in the indefinite range. Three noteworthy instances of the mass matrices produced by these two families are 23 11 −16 3 1 −1 e e m m MCe O B0 = 11 23 −16 , MCe O B1 = 1 3 −1 , 36 −16 −16 6 −1 −1 32 2 (23.53) 4 1 −2 e m MCe O B2 = 1 4 −2 . 18 −2 −2 16 MCe O B0 is the unique mass matrix for which 2oc = a2 = 6; that is, the COB passes through the folding (Nyquist) frequency. It emerges by setting either ν A = −5/3 in (23.47) or ν B = −6 in (23.49). MCe O B1 which gives 2oc A = 16, is the simplest mass matrix that produces a COB. It is obtained by setting ν A = 0 in (23.47). Finally MC O B2 , which yields 2ocB = 12, was the first COB instance discovered, as noted in §H.7. It is obtained by setting ν B = 0 in (23.49). Dispersion and group velocity diagrams for COB0 are shown in Figure 23.5(e,f). The DDD for COB1 and COB2 would possess an identical AB branch but the flat OB would appear higher, whereas the DGVD would be identical. Those two diagrams are omitted to save space.

23–18

§23.1


§23.1.14. Bar3 Test: Vibrations of a Fixed-Free Bar Member The natural frequency benchmark test presented in §22.1.9 for three Bar2 discretizations is repeated for the ten Bar3 template instances listed in Table 23.1. The fixed-free bar member is pictured in Figure 22.5(b). It is prismatic, with constant E = 1, A = 1 ρ = 1. The total member length is L = π/2. With those numerical properties, the continuum eigenfrequencies ω0i are given by (22.18). The member is divided into Ne identical elements, with Ne = 1, 2, . . . 16. To reduce cluttering the instances in Table 23.1 are divided into two groups of five each. Results are presented in number of correct digits versus number of elements for the first three frequencies, exactly as described for the Bar2 test in §22.1.9. Group 1 include CMM and SLMM as well as instances constructed with optimal LFF customization in mind: BLCD, BLFM and BLFD. Results are displayed in Figure 23.6(a,b,c). Group 2 includes instances derived with RHFP in mind: BSSM, SMSx (x = 1, 2, 3) and COB0. Results are displayed in Figure 23.6(d,e,f). BLFM is the clear winner in the first group, with BLCD close behind, while the others, with only O(κ 4 ) AB accuracy, lag appreciably. In the second group, BSSM is the clear winner, with performance comparable to BLFM and BLCD of the first group. SMS1 and COB0 are way behind, while SMS2 and SMS3 are highly inaccurate. (As observed in §23.1.12, SMS1 would hardly effect any HF reduction, so its reasonable LF accuracy is misleading.)

23–19

24

Mass Templates for Plane Beam Elements

24–1

Chapter 24: MASS TEMPLATES FOR PLANE BEAM ELEMENTS


§24.1

§24.2

The Bernoulli-Euler Plane Beam Element . . . . . . . . . . §24.1.1 The BE Beam Mass Template . . . . . . . . . . . §24.1.2 BE Beam Template Fourier Analysis . . . . . . . . . . The Timoshenko Plane Beam Element . . . . . . . . . . . §24.2.1 Ti-Beam Continuum Elastodynamic Analysis . . . . . . . §24.2.2 Ti-Beam Element . . . . . . . . . . . . . . . §24.2.3 The Ti-Beam Mass Template . . . . . . . . . . . . §24.2.4 Ti-Beam Full Mass Parametrization . . . . . . . . . §24.2.5 Ti-Beam Block-Diagonal Mass Parametrization . . . . . . §24.2.6 Ti-Beam Fourier Analysis . . . . . . . . . . . . . §24.2.7 Ti-Beam Selected Template Instances . . . . . . . . . §24.2.8 Ti-Beam Vibration Analysis Example . . . . . . . . .

24–2

24–3 24–3 24–3 24–6 24–6 24–9 24–9 24–10 24–11 24–12 24–13 24–14

§24.1

THE BERNOULLI-EULER PLANE BEAM ELEMENT

§24.1. The Bernoulli-Euler Plane Beam Element This Section and the next one study templates for two-node plane beam elements constructed from the Bernoulli-Euler (BE) and Timoshenko models, respectively. To keep the material relatively compact, two restrictions are observed: •

Only mass matrix templates are developed.

•

The only customization is low frequency continuum fit (LFCF)

To enforce the first one, the optimal stiffness matrix for statics (“optimal” means that it satisfies the homogeneous static equilibrium equations over the element) is chosen and kept fixed. Simultaneous adjustment of the mass and stiffness templates to form MS pairs is relegated to future research. Prior experience in this regard, cited in §H.7, suggest that the improvement is marginal. §24.1.1. The BE Beam Mass Template The Bernoulli-Euler (BE) beam model is a special case of the Timoshenko model treated in §24.2. Nevertheless it is useful to build its mass template separately, since results provide a valuable cross check with the more complicated Timoshenko beam. The well known CMM of this element is derived in §18.1, to which the reader is referred for notation; the derivation assumes a prismatic two-node element with four nodal DOF with the standard cubic shape functions. This matrix is augmented to produce the following entry-weighted template:  13 Meµ

  =m  

35

+ µ11

11 ( 210 + µ12 ) 1 + µ22 )2 ( 105

e

9 70 13 ( 420 13 35

+ µ13 + µ23 ) + µ11

13 −( 420 + µ14 )



1 −( 140 + µ24 )2    11 −( 210 + µ12 ) 

(24.1)

1 + µ22 )2 ( 105

symm

in which m e = ρ A . The parameters in (24.1) are µi j , in which i j identifies the mass matrix entry. The template (24.1) accounts for matrix symmetry and some physical symmetries. Three more conditions can be imposed right away: µ14 = µ23 ,

µ13 = −µ11 ,

2µ12 = µ11 + 2µ22 + 2µ23 − 2µ24 .

(24.2)

The first comes from prismatic fabrication, and the others from conservation of total translational mass and angular momentum, respectively. Four free parameters remain: {µ11 , µ22 , µ23 , µ24 }. For the stiffness matrix we take the well known one for a plane prismatic homogeneous BE beam element   12 6 −12 6 EI  42 −6 22  (24.3) Ke = 3   12 −6 symm 42 in which I = Izz is the second moment of inertial of the cross section with the respect to z, which is chosen to go along the neutral axis. If the FEM model contains only prismatic beams, this Ke is nodally exact, and consequently statically optimal. (It can also be derived from the equilibrium equations.) This stiffness is kept fixed throughout the Fourier analysis. 24–3


§24.1.2. BE Beam Template Fourier Analysis The Fourier analysis procedure should be by now familiar to the reader. An infinite lattice of identical beam elements of length is set up. This will look like Figure 22.2(b–d), except that the member is now a plane beam. Plane waves of wavenumber k and frequency ω propagating over the lattice are represented by √ (24.4) v(x, t) = Bv exp j (kx − ωt , θ(x, t) = Bθ exp j (kx − ωt , j = −1. At a typical lattice node j there are two DOF: v j and θ j . Two patch equations are extracted, and converted to dimensioneless form on defining κ = k and = ωc0 /, in which c0 = E I /(ρ A4 ) is a reference phase velocity. The condition for wave propagation gives the characteristic matrix equation

Cvv Cvθ (24.5) = Cvv Cθ θ − Cvθ Cθ v = 0, det Cθ v Cθ θ in which Cvv = 840−2(13+35µ11 ) 2 − (840+(9−70µ11 ) 2 ) cos κ /35, −Cθ v = Cvθ = j 2520 + (13+420µ23 ) 2 sin κ/210, Cθ θ = 1680 − 4(1+105µ22 ) 2 + (840 + 3(1+140µ24 ) 2 ) cos κ /210.

(24.6)

The condition (24.5) gives a quadratic equation in 2 that provides two dispersion solutions: acoustic branch (AB) a2 (κ) and optical branch (OB) 2o (κ). The AB represent genuine flexural modes, whereas the OB is a spurious byproduct of the FEM discretization. The small-κ (low freqnecy, long wavelength) expansions of these roots are

a2 = κ 4 + C6 κ 6 + C8 κ 8 + C10 κ 10 + C12 κ 12 + . . . ,

2o = D0 + D2 κ 2 + D4 κ 4 + . . . , (24.7) 2 2 in which C6 = −µ11 − 2µ22 − 4µ23 + 2µ24 , C8 = 1/720 + µ11 + 4µ22 + 2µ23 /3 + 16µ22 µ23 + 16µ223 + µ11 (1/12 + 4µ22 + 8µ23 − 4µ24 ) − µ24 − 8µ22 µ24 − 16µ23 µ24 + 4µ224 , etc.; and D0 = 2520/(1 + 420µ22 − 420µ24 ), etc. Mathematica calculated these series up to C14 and D4 . The continuum dispersion curve is 2 = κ 4 , which automatically matches a2 as κ → 0. Thus four free parameters offer the opportunity to match coefficients of four powers: {κ 6 , κ 8 , κ 10 , κ 12 }. But it will be seen that the last match is unfeasible if Me is to stay nonnegative. We settle for a scheme that agrees up to κ 10 . Setting C6 = C8 = C10 = 0 while keeping µ22 free yields two sets of solutions, of which the most useful one is µ11 = 4µ22 − 67/540 − (4/27) 38/35 − 108µ22 , (24.8) µ23 = 43/1080 − 2µ22 + 95/14 − 675µ22 /54, µ24 = 19/1080 − µ22 + 19/70 − 27µ22 /27. The positivity behavior of Meµ as µ22 is varied is shown in Figure 24.1(a). M(e) is indefinite for √ µ22 < µmin 22 = (27 − 4 35)/5040 = 0.0006618414419844316. At the other extreme the solutions of (24.8) become complex if µ22 > µmax 22 = 19/1890 = 0.010052910052910053. 24–4

§24.1

1

0

THE BERNOULLI-EULER PLANE BEAM ELEMENT

d 4 = det(Me)

0 −5

d1 µmax 22 =

d2 µmin 22

−1

0.01005291 −10

−0.02

µ b22 =0.00281659 =−0.0283026

C12 106

a =0.00412698 µ 22

= 0.00066184 −15

d3 −0.03

µ z22

−0.01

0

µ22

−0.03

0.01

−0.02

−0.01

0

µ22

0.01

Figure 24.1. Behavior of Meµ as function of µ22 with other parameters given by (24.8): (a) determinants dk of max 12 in ABTS principal minors of order k of Meµ , showing legal positivity range {µmin 22 , µ22 }; (b) coefficient C 12 of κ series.

Figure √ 24.1(b) √ plots C12 (µ22 ) = (−111545 − 3008ψ + z15120(525 + 4ψ)µ22 )/685843200, with ψ = 70 19 − 1890µ22 . This has one real root µ22 = −0.02830257472322391, but that max gives an indefinite mass matrix. For µ22 in the legal range [µmin 22 , µ22 ], C 12 is minimized for √ µb22 = (25 105 − 171)/30240 = 0.0028165928951385567, which substituted gives the optimal mass matrix:   a11 1788 a13 −732 me  a22 2 732 a24 2  MeL F F O pt =   a33 1788 30240 symm a44 2 (24.9)   0.389589 0.059127 0.110410 −0.024206 0.0123402 0.024206 −0.0055482   = me  . 0.389589 −0.059127 0.0123402 √ √ √ in which a11 = a√ 2724 + 60 105, a22 = a44 = 117 + 25 105 and 33 = 12396 − 60 105, a13 =√ a24 = −219 + 5 105. For this set, C12 = (25 105 − 441)/91445760 = −2.021 10−6 . Another interesting value is µ22 = 13/3150 = 0.004126984126984127, which substituted in (24.8) yields rational values for the other parameters: µ11 = −µ13 = 23/2100, µ12 = −µ14 = −µ23 = 23/4200, µ24 = 23/4200 and µ24 = −17/12600. Substitution into (24.1) gives   4818 729 1482 −321 me  1722 321 −732  MeB L F M =   4818 −729 12600 symm 1722 (24.10)   0.382381 0.057857 0.117619 −0.025476 0.0136512 0.025476 −0.0057942   = me  . 0.382381 −0.057857 symm 0.0136512 For this matrix, C12 = −41/18144000 = −2.26 10−6 . Its magnitude is only about 10% higher than for the truly LFF optimal (24.9). Since its entries are simpler, (24.10) is adopted as BLFM 24–5


matrix for the BE element, and used as a baseline for the Timoshenko beam element investigated in §24.2. §24.2. The Timoshenko Plane Beam Element This last example is far more elaborate than the previous ones. The goal is to construct a mass template for the prismatic, plane beam Timoshenko model, a name often abbreviated to Ti-beam. It includes the BE model as special case; consequently results can be crosschecked with those of §24.1. One interesting feature of this model is that the continuum dispersion diagram has two branches, both of which are physical. The “acoustical-like branch”, which has zero frequency at zero wavenumber, corresponds to lowerfrequency bending oscillations in which the beam displaces transversely. The “optical-like branch”, which exhibits a nonzero cutoff frequency, corresponds to higher frequency shear oscillations. Because of these interpretations, they are called the flexural frequency branch (FFB) and the shear frequency branch (SFB), respectively. They are identified by subscripts f and s, respectively, for the continuum model, while a and o are reused for the FEM discretization. Both branches are dispersive, meaning that group velocity depends on wavenumber. Those velocities tend to finite values, except in the BE limit. The upshoot of these complications is that LFCF customization, which was so clear-cut with Bar3, becomes ambiguous: do we want to fit the FFB or the SSB? For thin beams, (as well as in the BE limit, in which case the SSB moves to ∞) the FFB is dominant. But as the beam becomes progressively thicker (as measured by a slenderness coefficient introduced in §24.2.1) the situation is less clear: for an extremely thick beam the shear oscillations may well dominate. (Of course in that case the Timoshenko model is questionable.) The continuum model is first studied in some detail, since frequency expansion formulas applicable to template customization by characteristic root fitting are not available in the literature. §24.2.1. Ti-Beam Continuum Elastodynamic Analysis Consider a structural beam member modeled as a shear-flexible Timoshenko plane beam (Ti-beam), as illustrated in Figure 24.2. This figure provides the notation used below. Section properties {ρ, E, A, As , I, I R } are constant along x. The beam is transversally loaded by line load q(x, t) (not shown in figure), with dimension of force per length. The primary kinematic variables are the transverse deflection v(x, t) and the total cross-section rotation θ(x, t) = v (x, t) + γ (x, t), where γ = V /(G As ) is the mean shear rotation. The kinetic and potential energies in terms of those variables are

L

L 2 2 2 2 1 1 1 ˙ E I (v ) + G A (θ −v − qv d x. ρ A v˙ + ρ I R θ d x, [v, θ] = T [v, θ ] = 2 s 2 2 0

0

(24.11) where superposed dots denote time derivatives. The equations of motion (EOM) follow on forming the Euler equations from the Lagrangian L = T − : δL = 0 → G As (θ −v )+ρ Av¨ = q, δv

δL = 0 → E I θ +G As (v −θ)−ρ I R θ¨ = 0. (24.12) δθ 24–6

§24.2

Section-averaged shear rotation Deformed cross y, v section

Normal to deformed longitudinal axis

γ

v(x)

THE TIMOSHENKO PLANE BEAM ELEMENT

slope v' = dv/dx

A positive transverse shear force V = GAs γ produces a CCW rotation (+γ) of the beam cross section

y

ρ, E, G, A, As , I and IR constant along beam

M x

z

x, u

+γ

V

+V

Positive bending moment and transverse shear conventions

L

Figure 24.2. Plane beam member modeled as Ti-beam, illustrating notation followed in §24.2.1. Transverse load q(x) not shown to reduce clutter. Infinitesimal deflections and deformations grossly exaggerated for visibility.

An expedient way to eliminate θ is to rewrite the coupled equations (24.12) in transform space:

ρ As 2 − G As p 2 v˜ q˜ G As p = , (24.13) E I p 2 − G As − ρ I R s 2 θ˜ 0 G As p ˜ q} in which { p, s, v, ˜ θ, ˜ denote transforms of {d/d x, d/dt, v, θ, q}, respectively (Fourier in x and Laplace in t). Eliminating θ˜ and returning to the physical domain yields ρ AE I ρ 2 AI R .... ρ IR E I q + q. ¨ (24.14) v¨ + v =q− E I v + ρ Av¨ − ρ I R + G As G As G As G As (This derivation does not preset I ≡ I R , as usually done in textbooks.) For the unforced case q = 0, (24.14) has plane wave solutions v = B exp i (k0 x − ω0 t) . The propagation condition yields a characteristic equation relating k0 and ω0 . To render it dimensionless, introduce a reference phase velocity c02 = E I /(ρ AL 4 ) so that k0 = ω0 /c0 = 2π/λ0 , a dimensionless frequency = ω0 L/c0 and a dimensionless wavenumber κ = k0 L. As dimensionless measures of relative bending-to-shear rigidities and rotary inertia take 0 = 12E I /(G As L 2 ),

r R2 = I R /A,

0 = r R /L .

(24.15)

The resulting dimensionless characteristic equation is 1 κ 4 − 2 − ( 12 0 + 02 ) κ 2 2 +

1 12 0

02 4 = 0.

(24.16)

This is quadratic in 2 . Its solution yields two kinds of squared-frequencies, which will be denoted by 2f and 2s because they are associated with flexural and shear modes, respectively. Their expressions are listed below along with their small-κ (long wavelength) Taylor series: √ P− Q 2 1 1

f = 6 = κ 4 − ( 12 0 + 02 ) κ 6 + ( 144 20 + 14 0 02 + 04 ) κ 8 0 02 (24.17) 1 − ( 1728 30 +

1 2 2 24 0 0

+ 12 0 04 + 06 ) κ 10 + . . . = A4 κ 4 + A6 κ 6 + A8 κ 8 + . . . 24–7


(a)

40

Shear branches of Timoshenko model

250

200 Cutoff ν=0 frequencies 150 ν = 1/2 100

(b) Shear branches of Timoshenko model

35

Bernoulli-Euler model ν=0

Dimensionless speed Ω/κ


300

30

25

ν = 1/2

20

ν = 1/2

15

Bernoulli-Euler model

10

Flexural branches of Timoshenko model

50 0

ν=0

5 15 10 20 Dimensionless wavenumber κ

25

5

ν=0

ν = 1/2

0

Flexural branches of Timoshenko model 25 5 15 10 20 Dimensionless wavenumber κ

Figure 24.3. Spectral behavior of continuum Ti-beam model for a narrow b × h rectangular cross section. (a): dispersion curves (κ) for = h/ = 1/4 and two Poisson’s ratios; Timoshenko flexural and shear branches in red and blue, respectively; Bernoulli-Euler curve = κ 2 in black. (b) Wavespeed /κ.

√ P+ Q 12 12 1 1 =6 = + + 2 κ 2 − κ 4 + ( 12 0 + 02 )κ 6 + . . . = B0 + B2 κ 2 + . . . 2 2 0 0 0 0 0 0 (24.18) 1 0 ) and Q = P 2 − 13 κ 4 0 02 . The dispersion relation 2f (κ) defines in which P = 1 + κ 2 (02 + 12 the flexural frequency branch (FFB) whereas 2s (κ) defines the shear frequency branch (SFB). If 0 → 0 and 0 → 0, which reduces the Ti- model to the BE one, (24.16) collapses to 2 = κ 4 or (in principal value) = κ 2 . This surviving branch pertains to flexural motions whereas the shear branch disappears; more precisely, 2s (κ) → ∞.

2s

It is easily shown that the radicand Q in the exact expressions is strictly positive for any {0 > 0, 0 > 0, κ ≥ 0}. Thus for any such triple, 2f and 2s are real, finite and distinct with 2f (κ) <

2s (κ). Further { 2f , 2s } increase indefinitely as κ → ∞. Following the dispersion-diagram nomenclature introduced in Figure 23.2, the value s at κ = 0 is called the cutoff frequency. To see what branches look like, consider a beam of narrow rectangular cross section of width b and height h, fabricated of isotropic material with Poisson’s ratio ν. Accordingly E/G = 2(1 + ν) and As /A ≈ 5/6. [Actually a more refined As /A ratio would be 10(1 + ν)/(12 + 11ν), but that makes little difference in the results.] We have A = bh, I = I R = bh 3 /12, r R2 = I R /A = h 2 /12, 1 2 h /L 2 and 0 = 12E I /(G As L 2 ) = 12(1 + ν)h 2 /(5L 2 ). Since 0 /12 = 02 = r R2 /L 2 = 12 2 12(1 + ν) 0 /5, the first-order effect of shear on 2f , as measured by the κ 6 term in (24.17), is 2.4 to 3.6 times that from rotary inertia, depending on ν. Replacing into (24.17) and (24.18) yields 2 2 2 2 60 + κ 2 (17 + 12ν)2 − 240κ 4 (1 + ν)4 (17 + 12ν) ∓ 60 + κ

f = 2(1 + ν)4

2s  4 (24.19) 1 1  κ − 60 (17 + 12ν)2 κ 6 + 3600 (349 + 468ν + 144ν 2 )4 κ 8 + . . . = 60 + (17 + 12ν)2 κ 2 − (1 + ν)4 κ 4 + . . .  (1 + ν)4 in which = h/L. Dispersion curves (κ) for = h/L = / and ν = {0, 12 } are plotted 24–8

§24.2

−γ 2

Deformed cross section

−γ1 v'1 =[dv/dx]1

θ2

γ

y, v θ1


v'2 =[dv/dx]2

v1

v2

v(x)

1

2

x, u

Figure 24.4. Two-node element for Timoshenko plane beam, illustrating kinematics.

in Figure 24.3(a). Phase velocities /κ are shown in Figure 24.3(b). The figure also shows the flexural branch of the BE model. The phase velocities of the Timoshenko model tend to finite values in the shortwave, high-frequency limit κ → ∞, which is physically correct. The BE model is physically wrong in that limit because it predicts an infinite propagation speed. §24.2.2. Ti-Beam Element The shear-flexible plane beam member of Figure 24.2 is discretized by two-node elements. An individual element of this type is shown in Figure (24.4), which illustrates its kinematics. The element has four nodal freedoms arranged as ue = [ v1 θ1 v2 θ2 ]T

(24.20)

Here θ1 = v1 + γ1 and θ2 = v2 + γ2 are the total cross section rotations evaluated at the end nodes. The dimensionless properties (24.15) that characterize relative shear rigidity and rotary inertia are redefined using the element length: = 12E I /(G As 2 ),

r R2 = I R /A,

= r R /.

(24.21)

If the beam member is divided into Ne elements of equal length, = L/Ne whence = 0 Ne2 and = 0 Ne . Thus even if 0 and 0 are small with respect to one, they can grow without bound as the mesh is refined. For example if 0 = 1/4 and 02 = 1/100, which are typical values for a moderately thick beam, and we take Ne = 32, then ≈ 250 and 2 ≈ 10. Those are no longer small numbers, a fact that will impact performance as Ne increases. The stiffness matrix to be paired with the mass template is taken to be that of the equilibrium element:   12 6 −12 6 EI  6 2 (4 + ) −6 2 (2 − )  (24.22) Ke = 3 .  −6 12 −6 (1 + ) −12 6 2 (2 − ) −6 2 (4 + ) This is known to be optimal in static analysis for a prismatic beam member. It will be kept fixed in the ensuing derivations. It reduces to the stiffness matrix (24.3) of the BE model if = 0. §24.2.3. The Ti-Beam Mass Template FEM derivations usually split the 4 × 4 mass matrix of this element into Me = Mev + Meθ , where Mev and Meθ come from the translational inertia and rotary inertia terms, respectively, of the kinetic 24–9


energy functional T [v, θ ] of (24.11). The most general mass template would result from applying a entry-weighted parametrization of those two matrices. This would require a set of 20 parameters (10 in each matrix), reducible to 9 through 11 on account of invariance and conservation conditions. Attacking the problem this way, however, leads to unwieldy algebraic equations even with the help of a computer algebra system, while concealing the underlying physics. A divide and conquer approach works better. This is briefly outlined next and covered in more detail in the next subsections. (I) Express Me as the one-parameter matrix-weighted form Me = (1 − µ0 ) MeF + µ0 MeD . Here MeF is full and includes the CMM as instance, whereas MeD is 2 × 2 block diagonal and includes the DLMM as instance. This is plainly a generalization of the LC linear combination (21.2). (II) Decompose the foregoing mass components as MeF = MeF T + M F R and MeD = MeDT + MeD R , where T and R subscripts identify their source in the kinetic energy functional: T if coming from the translational inertia term 12 ρ A v˙ 2 and R from the rotary inertia term 12 ρ I R θ˙ 2 . (III) Both components of MeF are expressed as parametrized spectral forms, whereas those of MeD are expressed as entry-weighted. The main reasons for choosing spectral forms for the full matrix are reduction of parameters and physical transparency. No such concerns apply to MeD . The analysis follows a “bottom up” sequence, in order (III)-(II)-(I). This has the advantage that if a satisfactory custom mass matrix for a target application emerges during (III), stages (II) and (I) need not be carried out, and that matrix directly used by setting the remaining parameters to zero. §24.2.4. Ti-Beam Full Mass Parametrization As noted above, one starts with full-matrix spectral forms. Let ξ denote the natural iso-P coordinate that varies from −1 at node 1 to +1 at node 2. Two element transverse displacement expansions in generalized coordinates are introduced: vT (ξ ) = L 1 (ξ ) cT 1 + L 2 (ξ ) cT 2 + L 3 (ξ ) cT 3 + L 4 (ξ ) cT 4 = LT cT , L 4 (ξ ) c R4 = L R c R , v R (ξ ) = L 1 (ξ ) c R1 + L 2 (ξ ) c R2 + L 3 (ξ ) c R3 + L 1 (ξ ) = 1, L 2 (ξ ) = ξ, L 3 (ξ ) = 12 (3ξ 2 − 1), L 4 (ξ ) = 12 (5ξ 3 − 3ξ ), L 4 (ξ ) = 1 5ξ 3 − (5 + 10)ξ = L 4 (ξ ) − (1 + 5)ξ.

(24.23)

2

The vT and v R expansions are used for the translational and rotational parts of the kinetic energy, respectively. The interpolation function set {L i } used for vT is formed by the first four Legendre L 4 to polynomials over ξ = [−1, 1]. The set used for v R is the same except that L 4 is adjusted to produce a diagonal rotational mass matrix. All amplitudes cT i and c Ri have dimension of length. Unlike the usual Hermite cubic shape functions, the polynomials in (24.23) have a direct physical interpretation. L 1 : translational rigid mode; L 2 : rotational rigid mode; L 3 : pure-bending mode L 4 : bending-with-shear mode antisymmetric about ξ = 0. symmetric about ξ = 0; L 4 and With the abbreviation (.) ≡ d(.)/d x = (2/)d(.)/dξ , the associated cross section rotations are compactly expressed as θT = vT + γT = LT cT + γT ,

θ R = v R + γ R = LR c R + γ R ,

(24.24)

in which the mean shear distortions are constant over the element: γT =

2 10 vT = cT 4 , 12

γR =

24–10

2 10 vR = c R4 . 12

(24.25)

§24.2


The kinetic energy of the element in generalized coordinates is

T = e

1 2

0

ρ A v˙ T2

+ ρ I R θ˙R2 d x = 4

1

−1

ρ A v˙ T2 + ρ I R θ˙R2 dξ = 12 c˙ TT DT c˙ T + 12 c˙ TR D R c˙ R . (24.26)

Both generalized mass matrices turn out to be diagonal as expected: DT = m e diag [ 1

1 3

1 5

1 7

],

D R = 4m e 2 diag [ 0 1 3 5 ] ,

in which as usual m e = ρ A . To convert DT and D R to physical coordinates (24.20), vT , v R , θT and θ R are evaluated at the nodes by setting ξ = ±1. This establishes the transformations ue = GT cT −1 and ue = G R c R . Inverting: cT = HT ue and c R = H R ue with HT = G−1 T and H R = G R . A symbolic calculation yields for HT :   30(1 + ) 5(1 + ) 30(1 + ) −5(1 + ) 1 −3 36 + 30 −3  −36 − 30  HT = (24.27)  . 0 −5(1 + ) 0 5(1 + ) 60(1 + ) 6 3 −6 3 Matrix H R differs only in the second row:  30(1 + ) 5(1 + ) 1 15  −30 HR =  0 −5(1 + ) 60(1 + ) 6 3

30(1 + ) 30 0 −6

 −5(1 + ) 15  . 5(1 + ) 3

(24.28)

The difference comes from adjusting L 4 to L 4 in (24.23). To map this into a spectral template, inject six free parameters in the generalized masses while moving 4 2 inside D Rµ : DT µ = m e diag [ 1

1 1 1 µ µ µ 3 T1 5 T2 7 T3

],

D Rµ = m e diag [ 0 µ R1 3µ R2 5µ R3 ] .

(24.29)

The transformation matrices (24.27) and (24.28) can be reused without change to produce MeF = HTT DT µ HT + HTR D Rµ H R . If µT 1 = µT 2 = µT 3 = 1 and µ R1 = µ R2 = µ R3 = 4 2 one obtains the well known consistent mass matrix (CMM) as a valuable check. The configuration (24.29) already accounts for linear momentum conservation, which is why the upper diagonal entries are not parametrized. Enforcing also angular momentum conservation requires µT 1 = 1 and µ R1 = 4 2 , whence the template is reduced to four parameters:     1 0 0 0 0 0 0 0 2  0 13 0 0  0 0  e T  0 4  + m H H MeF = m e HTT    H R . (24.30) T R  0 0 1 µT 2  0 0 0 0 3µ R2 5 1 0 0 0 5µ R3 µ 0 0 0 7 T3 Since both HT and H R are nonsingular, choosing all parameters in (24.30) to be nonnegative guarantees that MeF is nonnegative. This property eliminates lengthy a posteriori checks. Setting µT 2 = µT 3 = µ R2 = µ R3 = 0 and = 0 yields the correct mass matrix for a rigid beam, including rotary inertia. This simple result highlights the physical transparency of spectral forms. 24–11


§24.2.5. Ti-Beam Block-Diagonal Mass Parametrization Template (24.30) has a flaw: it does not include the DLMM. To remedy the omission, a block diagonal form, with four free parameters: νT 1 , νT 2 , ν R1 , and ν R2 is separately constructed:  1    νT 1 0 0 0 0 0 ν R1 2 2 0 0  0 0   ν νT 2 2 e  ν R1 ν R2 MeD = m e  T 1 (24.31) +m  . 1 −ν 0 0 0 0 0 −ν T1 R1 2 0 0 −νT 1 νT 2 2 0 0 −ν R1 ν R2 2 Four parameters can be merged into two by adding the foregoing matrices:  1  ν1 0 0 2 0 0   ν ν2 2 MeD = m e  1 . 1 −ν1 0 0 2 0 0 −ν1 ν2 2

(24.32)

in which ν1 = νT 1 + ν R1 and ν2 = νT 2 + ν R2 . Sometimes it is convenient to use the split form (24.31), for example in lattices with varying beam properties or lengths, a topic not considered there. Otherwise (24.32) suffices. If ν1 = 0, MeD is diagonal. However for computational purposes a block diagonal form is just as good and provides additional customization power. Terms in the (1,1) and (3,3) positions must be as shown to satisfy linear momentum conservation. If angular momentum conservation is imposed a priori it is necessary to set ν2 = 12 2 , and only one parameter: ν1 , remains. The general template is obtained as a linear combination of MeF and MeD : Me = (1 − µ0 )MeF + µ0 MeD

(24.33)

In summary, there is a total of 7 parameters to play with: 4 in MeF , 2 in MeD , plus µ0 . This is less that the 9-to-11 count that would result from a full entry-weighted parametrization, so not all possible mass matrices for this element are included by (24.33). §24.2.6. Ti-Beam Fourier Analysis An infinite lattice of identical Ti-beam elements of length is set up in th usual manner. As in §24.1, plane waves of wavenumber k and frequency ω propagating over the lattice are represented by (24.34) v(x, t) = Bv exp i(kx − ωt , θ(x, t) = Bθ exp i(kx − ωt . At each typical lattice node j there are two freedoms: v j and θ j . Two patch equations are extracted, and converted to dimensionless form on defining κ = k and = ω c/, in which c = E I /(ρ A4 ) is a reference phase velocity. (These should not be confused with c0 .) The condition for wave propagation gives the characteristic matrix equation

Cvv Cvθ det (24.35) = Cvv Cθ θ − Cvθ Cθ v = 0, Cθ v Cθ θ in which the coefficients are complicated functions computed by Mathematica and omitted for brevity. Solving the equation provides two equations: a2 and 2o , where a and o denote acoustic 24–12

§24.2


and optical branch, respectively. These are expanded in powers of κ for matching to the continuum. For the full mass matrix one obtains

a2 = κ 4 + C6 κ 6 + C8 κ 8 + C10 κ 10 + . . . ,

2o = D0 + D2 κ 2 + . . .

(24.36)

Coefficients of terms up to κ 12 were computed by Mathematica. Those relevant for parameter selection are C6 = −/12 − 2 , C8 = 2 − 15µ R2 − µT 2 + 5(1 + ) + 60(1 + 3) 2 + 720 4 /720, C10 = − 44 + 35µT 2 − 3µT 3 − 282 + 525µ R2 (1 + ) − 105µ R3 (1 + )+ 1575µ R2 (1 + ) − (3µT 3 − 35µT 2 (4 + 3) + 35(17 + 5(3 + )))+ (−2940 + 12600µ R2 (1 + ) + 420(2µT 2 (1 + ) − 5(7 + 6(2 + )))) 2 − 25200(2 + (7 + 6)) 4 − 302400(1 + ) 6 / 302400(1 + ) , D0 = 25200(1 + )/ 7 + 105µ R2 + 3µT 3 + 21002 2 , D2 = 2100(1 + )(−56 − 35µT 2 + 3µT 3 − 63 + 3µT 3 + 105µ R3 (1 + )− 525µ R2 (1 + )2 − 35µT 2 (2 + )) + 2100(1 + )(3360 + 63002 + 2 21003 ) 2 + 529200002 (1 + ) 4 / 7 + 105µ R2 + 3µT 3 + 21002 2 . (24.37) For the block-diagonal template (24.32):

a2 = κ 4 + F6 κ 6 + F8 κ 8 + F10 κ 10 + . . . ,

2o = G 0 + G 2 κ 2 + . . .

(24.38)

2880ν2 − 5 + 360ν2 − 1 − 5 + 52 F6 = −24ν2 − , F8 = 720 6 24ν2 + − 2 , G2 = . G0 = ν2 (1 + ) 2ν2 (1 + )

(24.39)

in which

The expansions for the 7-parameter template (24.33) are considerably more involved than the above ones, and are omitted for brevity. §24.2.7. Ti-Beam Selected Template Instances Seven useful instances of the foregoing templates are identified and described in Table 24.1. Table 24.2 lists the template signatures that produce those instances. These tables include two well known mass matrices (CMM and DLMM) re-expressed in the template context, and five new ones. The latter were primarily obtained by matching series such as (24.37) and (24.38) to the continuum ones (24.17) and (24.18), up to a certain number of terms as indicated in Table 24.2. For the spectral template it is possible to match the flexure branch up to O(κ 10 ). Trying to match O(κ 12 ) leads to complex solutions. For the diagonal template the choice is more restrictive. It is only possible to match flexure up to O(κ 6 ), which leads to the instance called FLMM. Trying to go further gives imaginary solutions. For the 7-parameter template (24.33) it is again possible to 24–13


Table 1. Ti-Beam Selected Template Instances Instance name

Description

Comments

CMM

Consistent mass matrix derived in §18.2. Matches FFB up to O(κ 6 ).

A popular choice. Fairly inaccurate, however, as beam gets thicker. Grossly overestimates intermediate frequencies.

FBMS

FFB matched to O(κ 10 ) with spectral (Legendre) template (24.30).

Converges faster than CMM. Performance degrades as beam gets thicker, however, and becomes inferior to CDLA.

SBM0

SFB matched to O(κ 0 ) while flexure fitted to O(κ 10 )

Custom application: to roughly match SBF and cutoff frequency as mesh is refined. Warning: indefinite for certain ranges of and : use with caution.

SBM2

SBF matched to O(κ 2 ) while flexure fitted to O(κ 8 )

Custom application: to finely match SFB and cutoff frequency as mesh is refined. Warning: indefinite for wide ranges of and : use with extreme caution.

FLMM

Diagonally lumped mass matrix with rotational mass picked to match FFB to O(κ 6 ).

Obvious choice for explicit dynamics. Accuracy degrades significantly, however, as beam gets thicker. Underestimates frequencies. Becomes singular in the BE limit.

CDLA

Average of CMM and FLMM. Matches FFB to O(κ 8 ).

Robust all-around choice. Less accurate than FBMS and FBMG for thin beams, but becomes top performer as beam gets thicker. Easily constructed if CMM and FLMM available in code.

FBMG

FFB matched to O(κ 10 ) with 7parameter template (24.33).

Known to be the globally optimal positivedefinite choice for matching flexure in the BE limit. Accuracy, however, is only marginally better than FBMS. As in the case of the latter, performance degrades as beam gets thicker.

match up to O(κ 10 ) but no further. The instance that exhibits least truncation error while retaining positivity is FBMG. This is globally optimal for the BE limit = = 0, but the results are only slightly better for the reasons discussed below. Matching both flexure and shear branches leads to instances SBM0 and SBM2, which have the disadvantages noted in Table 24.1. The exact dispersion curves of these instances are shown in Figure 24.5 for = 48/125 and 2 = 1/75, which pertains to a thick beam. On examining Figure ?(c) it is obvious that trying to match the shear branch is quite difficult; the fit only works well over a tiny range near κ = 0. §24.2.8. Ti-Beam Vibration Analysis Example The vibration analysis performance of the seven Ti-beam template instances listed in Tables 24.1 24–14

§24.2


Table 2. Signatures of Selected Ti-Beam Template Instances

Instance name

Templ. form

CMM

(24.30)

1

FBMS

(24.30)

2

26/3 4 2 +/3

SMB0

(24.30)

2

SMB2

(24.30)

2

FLMM

(24.32)

CDLA

(24.33)

1

1

4 2

FBMG

(24.33)

c3

c4

c5

µT 2 µT 3

Template signature µ R2 µ R3 ν1

ν2

µ0

Fit to continuum freqs.

2f (flexural) 2s (shear)

4 2

up to κ 6

none

c1

up to κ 10

none

−7/3 4 2 +/3 20 2

up to κ 10

up to κ 0

−7/3

up to κ 8

up to κ 2

up to κ 6

none

1

4 2

c2

20 2 0

1 2 2

4 2

0

1 2 2

1/2

up to κ 8

none

c6

1/12

1 2 2

c7

up to κ 10

none

c1 = 253 + 120 2 + 2 (45 − 300 2 ) + 3(7 − 20 2 + 1200 4 ) / 15(1 + ) ,

c2 = − 19 + 102 (90 2 − 1) − 30(1 − 26 2 + 120 4 ) / 75(1 + )2 , √ √ √ c3 = (9 + 105)/10, c4 = (61 105 − 483)/18, c5 = ( 105 − 1)/30,

c6 = − 48 + 7272 + 8403 + 22128 2 + 19848 2 − 100802 2 − 113040 4 √ +120960 4 + 5 105 (48 + 872 + 403 ) − 24(6 + 21 + 202 ) 2 √ √ +720(3 + 8) 4 / 60(21 + 105)(1 + ) , c7 = (3 − 5 5/21)/8.

and 24.2 is evaluated on a simply supported (SS) prismatic plane beam. The beam has length L and is divided into Ne identical elements. The cross section is rectangular with width b and height h. The material is isotropic with Poisson’s ratio ν = 0. Three different height-to-span ratios h/L that characterize a thin, moderately thick and thick beam, respectively, are considered. Results for the three configurations are collected in Figures 24.6, 24.7 and 24.8, respectively, for the first three vibration frequencies. All calculations are rendered dimensionless using appropriate scaling. The accuracy of the computed frequencies is depicted using log-log plots of dimensionless natural frequency error versus Ne . The error is displayed as d = log10 (| comp − exact |, which gives at a glance the number of correct digits d, versus log2 Ne for Ne = 1 through 32. If the LF error is approximately controlled by a truncation term of the form ∝ κ m , the log-log plot should be roughly a straight line of slope ∝ m, inasmuch as κ = k = k L/Ne . (Note that the accuracy curves for CMM and FLMM are virtually on top of each other for Ne ≥ 4, although errors have opposite signs; that is why their average CDLA does much better.) The results for the Bernoulli-Euler (BE) model, shown in Figure 24.6, agree perfectly with the truncation error in the 2f branch as listed in Table 24.1. For example, the top performers FBMG and FBMS gain digits twice as fast as CMM, DLMM and SBM2, since the formers match 2f to O(κ 10 ) whereas the latter do that only to O(κ 6 ). Instances CDLA and SMB0, which agree 24–15

Chapter 24: MASS TEMPLATES FOR PLANE BEAM ELEMENTS 60

Ωcutoff

50



60

Ω s continuum

40

Ωo CMM

30

Ωf continuum

Ωa CMM

10

Ωa 2DLMM3 5 4 Dimensionless wavenumber κ= k

1

6

Ωcutoff

Ωa SBM0 5 1 4 2 3 Dimensionless wavenumber κ= k

0 60

50

Ωcutoff

40

6

Ωs continuum

50

Ωs continuum

Ωf continuum

Ωa SBM2



Ωo SBM0

10

60

40

Ωo CDLA

30

Ωf continuum

Ωo FBMG

20

Ωo FBMS 0

Ωa FBMS

Ω CDLA 5 4 2 a 3 Dimensionless wavenumber κ= k

1

Ωf continuum

30

20 10

Ωo SBM2

30 20

0

Ωs continuum

40

Ωo DLMM

20

Ωcutoff

50

Ωa FBMG

10

6

0

1

2

3

4

5

Dimensionless wavenumber κ= k

6

Figure 24.5. DDD of selected Ti-beam mass template instances for = 48/125 and 2 = 1/75.

through O(κ 8 ), come in between. The highly complicated FBMG is only slightly better than the much simpler FBMS. The case for their high accuracy should be emphasized. For example, four FBMS elements give 1 to six figures: 9.86960281 . . . versus π 2 = 9.86960440 . . ., whereas CMM gives less than three: 9.87216716 . . .. The “accuracy ceiling” of about 11 digits for FBMS and FBMG observable for Ne > 16 is due to the eigensolver working in double precision (≈ 16 digits). Rerunning with higher (quad) floating point precision, the plots continues marching up as straight lines before leveling off at approximately 25 digits. On passing to the Timoshenko model, the well ordered BE world of Figure 24.6 unravels. The culprits are and . These figure prominently in the branch series and grow without bound as Ne increases, as discussed in §24.2.2. Figure 24.7 collects results for a moderately thick beam with h/L = 1/8, which corresponds to 0 = 3/80 and 02 = 1/768. The BE top performers, FBMS and FBMG, gradually slow down and are caught by CDLA by Ne = 32. All other instances trail, with the standard ones: CMM and FLMM, becoming the worst performers. Note that for Ne = 32, CMM and FLMM provide only 1 digit of accuracy in 3 , although there are 32/1.5 ≈ 21 elements per wavelength. Figure 24.8 collects results for a thick beam with h/L = 2/5, corresponding to 0 = 24/625 and 02 = 1/75. The trends of Figure 24.7 are exacerbated, with FBMS and FBMG running out of steam by Ne = 4 and CDLA clearly emerging as best for Ne ≥ 8. Again CMM and FLMM trail badly. The reason for the performance degradation of FBMS and FBMG as the Ti-beam gets thicker is unclear as of this writing. Eigensolver accuracy is not responsible since rerunning the cases of Figures 24.7 and 24.8 in quad precision did not change the plots. A numerical study of the 2f truncation error shows that FBMS and FBMG fit the continuum branch better than CDLA even for 24–16

§24.2 12

10

FBMS

FBMG

CDLA 8

SBM0 6

4

SBM2 2

CMM FLMM 1

2

4

10

8

FBMG CDLA

6

SBM0 4

SBM2

2

FLMM CMM

32

16

8

12

Digits of accuracy in third vibration frequency

10

Digits of accuracy in second vibration frequency

12

Digits of accuracy in first vibration frequency


1

2

4

16

8

8

FBMG 6

4

SBM0

2

SBM2 FLMM CMM

32

1

Number of elements along span


FBMS CDLA

2

4

8

32

16


Figure 24.6. Accuracy of first 3 natural vibration frequencies of SS prismatic beam using mass matrices of instances listed in Tables 24.1 and 24.2. Bernoulli-Euler model with 0 = 0 = 0. Exact (12-decimal) frequencies

1 = π 2 = 9.869604401089, 2 = 4π 2 = 39.478417604357 and 3 = 9π 2 = 88.826439609804. Cutoff frequency +∞.

10

10

FBMG

8

FBMS CDLA

6

SBM0 4

SBM2 FLMM

2

CMM 1

12

2

4

8

16

10

8

6

FBMS FBMG CDLA SBM2

4

2

CMM

32

1



12



12

2

4

SBM0 FLMM 16

8

8

6

CDLA FBMS FBMG

4

SBM2 SBM0

2

FLMM CMM

32

1


2

4

8

32

16


Figure 24.7. Accuracy of first 3 natural vibration frequencies of SS prismatic beam using mass matrices of instances listed in Tables 24.1 and 24.2. Timoshenko model with 0 = 3/80 = 0.0375 and 02 = 1/768 = 0.00130, pertaining to a rectangular x-section with h/L = 1/8 and ν = 0. Exact (12-decimal) frequencies 1 = 9.662562122511,

2 = 36.507937703548 and 3 = 75.894968024537. Cutoff frequency cut = 12/(0 02 ) = 495.741868314549.

10

10

8

CDLA 6

FBMS FBMG

4

SBM0 SBM2

2

FLMM CMM 1

12

2

4

8

16


32

10

8

6

CDLA 4

FBMS FBMG SBM0 SBM2

2

FLMM CMM 1


12



12

2

4

8

16


32

8

6

4

CDLA FBMS FBMG SBM0 SBM2

2

FLMM CMM

1

2

4

8

16

32


Figure 24.8. Accuracy of first 3 natural vibration frequencies of SS prismatic beam using mass matrices of instances listed in Tables 24.1 and 24.2. Timoshenko model with 0 = 24/625 = 0.384 and 02 = 1/75 = 0.0133, pertaining to a rectangular x-section with h/L = 2/5 and ν = 0. Exact (12-decimal) frequencies 1 = 8.287891683498,

2 = 24.837128591729 and 3 = 43.182948411234. Cutoff frequency cut = 12/(0 02 ) = 48.412291827593.

24–17


very thick beams. Possible contamination of vibration mode shapes with the shear branch was not investigated.

24–18

25

Mass Templates for Plane Stress Elements

25–1

Chapter 25: MASS TEMPLATES FOR PLANE STRESS ELEMENTS


§25.1 §25.2 §25.3 §25.4

Longitudinal Wave Propagation in a Continuum Thin Plate . . . . Trig3 LCD Template . . . . . . . . . . . . . . . . . . Trig3 Template Fourier Analysis . . . . . . . . . . . . . . Can Plane Waves Actually Propagate Over a 2D FEM Lattice? . .

25–2

25–3 25–3 25–4 25–6

§25.2

TRIG3 LCD TEMPLATE

This chapter presents an example of the construction and LFF custiomization of a mass template for the simplest two-dimensional element, namely the three-node plane stress (membrane) linear triangle, identified as Trig3 in the sequel. The example is intended to illustrate three additional features that must be considered in multiple dimensions; in particular mesh directionality, multiple plane wave types, and (for an isotropic material) Poisson’s ratio. It is the only multidimensional template presented in this paper, since the subject has barely lifted off the ground. The results are taken from the a recent thesis [320] in which intermdiate calculation details (omitted here for brevity) may be found. The thesis also works out a more complicated version of this element type that includes corner drilling freedoms. That development is too complex to be briefly covered here. §25.1. Longitudinal Wave Propagation in a Continuum Thin Plate Wave propagation in continuum models of elastic solids is a classical subject of elastodynamics that is well covered in several textbooks and monographs; e.g., [833,835,315,836,837]. Here we consider longitudinal waves propagating in an isotropic, homogeneous, elastic flat thin plate. Thickness effects are ignored. The continuum model has elastic modulus E, Poisson’s ratio ν, shear modulus G = E/(2(1 + ν)), mass density ρ, and uniform thickness h. Two plane wave types are possible in that model: Pressure waves or P-waves, in which material points harmonically oscillate in the direction of the wave propagation. Also known as longitudinal, dilatational and “push” waves. A P-wave ˆ with Eˆ = E/(1 − ν 2 ). propagates at speed c P0 , in which c2P0 = E/ρ, Shear waves or S-waves, in which material points harmonically oscillate in the direction normal to wave propagation. Also known as transverse, rotational and “shake” waves. An S-wave propagates at speed c S0 , in which c2S0 = G/ρ. The dispersion equations are ω2P0 = c2P0 k 2 ,

ω2S0 = c2S0 k 2 ,

(25.1)

in which k is the wavenumber while ω P0 and ω S0 denote the characteristic frequencies for the P- and S-wave, respectively. In dimensionless variables κ = k a, 2P0 = ω2P a 2 /c2P0 , and 2S0 = ω2S a 2 /c2S0 , these reduce to 2P0 = κ 2 , 2S0 = κ 2 . (25.2) Note that both plane wave types are nondispersive (i.e., speed is independent of frequency) and can propagate along any in-plane direction, as befits isotropy. That is not the case in a FEM lattice of Trig3 elements, as studied next. §25.2. Trig3 LCD Template The Trig3 mass template is assumed to be of the LCD form (21.2): it linearly combines the CMM and DLMM given in (19.2) and (19.3), respectively, through a weight parameter:   2 0 1 0 1 0 0 2 0 1 0 1   e e m (1 − µ) 1 0 2 0 1 0 m µ I6 (25.3) + Meµ = (1 − µ)MCe M M + µ MeDL L M =   0 1 0 2 0 1 12 3   1 0 1 0 2 0 0 1 0 1 0 2 25–3


y ym (a)

Normal to wave propagation direction

Mesh symmetry axes

(b)

ym Wave propagation direction

xm φ

xm

4 Square cells of side dimension a

(c)

x

5

3 1

6 7

2

six-triangle, seven-node patch

Figure 25.1. Mesh for Fourier analysis of Trig3 LCD mass template: (a) infinite regular lattice of square cells; (b) rotated lattice to account for wave propagation directionality, forming angle φ with respect to wave propagation direction (x-axis); (c) six-triangle, seven-node patch.

This is paired with the well known Trig3 stiffness Ke , which is kept fixed. To study the performance of (25.3) an infinite 2D regular lattice such as pictured in Figure 25.1(a) was chosen. The triangles form square cells of side a. To account for the effect of wave directionality in the FEM lattice, the direction of propagation of the P- and S-waves is assumed to be always along the x axis. The lattice is rotated by an angle φ with respect to x, as illustrated in Figure 25.1(b). Thus the mesh symmetry axes {xm , ym } for an angle φ with the propagation axis {x, y}. The advantage of this choice, as opposed to rotating the propagation direction, is that the wave functions are kept simple. The kinematic expressions for the P- and S-waves are u x = Bx exp(k x − ω P t), u y = 0, S-wave: ux = 0, u y = B y exp(k x − ω S t). P-wave:

(25.4)

in which Bx and B y are nonzero amplitudes. Note that the wavenumber k is kept the same in both expressions. All calculations summarized below were carried out symbolically with Mathematica. §25.3. Trig3 Template Fourier Analysis From the 2D infinite lattice pictured in Figure 25.1(b), extract a (repeating) six-triangle, seven-node patch highlighted in (c) of that Figure. Assemble the 14 × 14 patch mass matrix M p from (25.3), p and the and the 14 × 14 patch stiffness matrix K p . From these extract the 2 × 14 mass matrix M p for the patch center node. The dynamic force residual equations at that 2 × 14 stiffness matrix K node are p ) uB , p − ω2 M (25.5) r = (K in which r = [ r x r y ]T is the residual vector measuring force equilibrium in the x and y directions, and u p is a 14-vector containing wave displacements of the seven patch nodes. Two u P : u p P and 25–4

φ−averaged optimal µ

§25.3

TRIG3 TEMPLATE FOURIER ANALYSIS

1.0 0.8

best µ for P-wave

0.6 0.4

best µ for S-wave

0.2 0.0 0.0

0.1

0.2

0.3

Poisson's ratio ν

0.4

0.5

Figure 25.2. Best ψ-averaged values of free parameter µ as function of Poisson’s ratio ν, for two plane wave types (P and S).

u pS are constructed by evaluating the P- and S- plane waves, respectively at the patch nodes, and inserted in (25.5). For the P-wave, u y = 0 and setting r x = 0 we solve for ω2 , renamed ω2P . For the S-wave, u x = 0 and setting r y = 0 we solve for ω2 , renamed ω2S . Thse are then Taylor series expanded at k = 0 in powers of k, up to k 4 . Passing to dimensionless variables: κ = k a, 2P = ω2P a 2 /c2P0 , and 2S = ω2S a 2 /c2S0 , the result for the P-wave is 2P = κ 2 −

2C1 + 2(1 − 3ν) cos(4φ) + C2 sin(2φ) − 3(1 + ν) sin(6φ) 4 κ + ... 384

(25.6)

in which C1 = 32µ + 3ν − 17 and C2 = 32µ + 9ν − 23. For the S-wave: 2S = κ 2 −

2D1 + 2(5 + ν) cos(4φ) + D2 sin(2φ) + 3(1 + ν) sin(6φ) 4 κ + ... 192(1 − ν)

(25.7)

in which D1 = 7ν +16µ(1−ν)−13 and D2 = 7ν +16µ(1−ν)−25. Comparing to the continuum expressions (25.2), which may be rewritten we observe a O(κ 4 ) error than depends on φ and ν. To eliminate the effect of φ we integrate that error term over φ ∈ [−π/2, pi/2] and divide by π to get an average error. Setting that to zero and solving for µ, we get two “averaged-optimal” values for the free parameter, one for each wave type, denoted by µ P and µ S : √ √ √ √ −45 + 40 3 − (45 − 24 3)ν 45 − 16 3 + (45 + 32 3)ν µ P (ν) = , µ S (ν) = , (25.8) 48 24(1 − ν) These are plotted as functions of Poisson’s ratio in Figure 25.2. As can be observed, the dependence on ν is failrly mild. For the S − wave it is approximately 0.5 whereas for the S-wave it varies from 0.72 to 1.00. Which one to pick? Assuming the given ν is uniform, select either µ P or µ S according to the wave type expected to dominate the solution. This may require some interaction. If S-waves dominate µ = 1/2 is recommended for any Poisson’s ratio.

25–5


§25.4. Can Plane Waves Actually Propagate Over a 2D FEM Lattice? The answer to the title question is: only for selected direction angles. To check that for the P-wave, pick the frequency ω2P obtained by solving r x = 0, and insert that solution into r y . It will be observed that for φ ∈ [−90◦ , 90◦ ], r y = 0 is only satisfied exactly at φ = ±90◦ , φ = ±45◦ , and φ = 0◦ . For those angles the assumption of plane wave propagation is exact. Othwerwise r y = 0 means that the plane wave is distorted by the FEM discretization even if the lattice is regular. A similar result is obtained for the S-wave on replacing ω2S into r x . In practice the distortion is not of great concern since the goal of this analysis is to find reasonable values for the free parameter, rather than finding a solution to the dynamic problem. Those are two distict objectives.

25–6

Matrix Finite Element Methods in Dynamics

Recommend Documents