1290 THEORY OF VIBRATION/Fundamentals
THEORY OF VIBRATION Fundamentals B Yang, University of Southern California, Los Angeles, CA, USA Copyright # 2001 Academic Press doi:10.1006/rwvb.2001.0112
Vibrations are the oscillating motion of an object or a dynamic system about an equilibrium position. This oscillating motion can be either periodic, or nonperiodic and transient. Examples of vibrations are diverse, including the back-and-forth motion of an engine piston, the oscillations of a violin string, the fluctuations of a highway bridge with passing vehicles, and the rolling of a ship under the influence of ocean waves. For vibrations to occur, two elements are essential: a mass or inertia element that stores kinetic energy; and a spring or elastic element that stores potential energy. The spring element provides a restoring force that continually pulls the mass toward its equilibrium position, and thus causes the oscillations of the mass. During vibrations, potential energy and kinetic energy are converted to each other as they are stored in and released from the two elements, respectively. Vibrations of a system are initiated when energy is imparted to it. A free vibration occurs when external excitations are absent, but kinetic energy or potential energy is present initially in the system. Free vibrations of many systems are harmonic motion with periods or natural frequencies that are inherently dependent upon inertia and elastic properties. A forced vibration occurs with the application of external forces to the system. Forced vibrations can be periodic, nonperiodic, or random. Under a harmonic (sinusoidal) excitation, the response of a system becomes increasingly large if the excitation frequency is close to one of the natural frequencies of the system. This phenomenon is called resonance. Control of resonant vibrations is one important issue in design of structures and machines. For any real vibrating system, there always exists energy dissipation in motion, which is termed damping. Damping occurs as a result of friction among moving objects, or interactions of moving objects with their surrounding environments (e.g., rough
surface, air, fluids, and electromagnetic fields). Damping converts the mechanical energy in a vibrating system to other forms of energy such as heat and acoustic energy. This energy conversion process in general is irreversible. The presence of damping causes the amplitude of free vibrations to delay, and leads to the reduction in amplitude of forced vibrations. In addition, damping can be purposely introduced to suppress resonant vibrations. Inertia, elasticity, and damping are inherent properties of vibrating systems. Vibrations can degrade the performance and reliability of structures and machines, cause property damage, and in worst cases result in loss of human lives. Hence, vibration control is essential for proper operation of many systems. On the other hand, vibration phenomena are useful in certain devices such as musical instruments, shakers, and cardiac pacemakers. In either way, it is essential to model and analyze the behaviors of vibrating systems. Any study of vibrations inevitably turns to dynamics, as vibration involves motion (displacement, velocity, and acceleration), forces, and inertia. Kinematics, the study of geometry of motion without reference to forces and inertia, is part of dynamics. In this article, the fundamentals of vibration theory are presented under the following headings: classification of vibrating systems; equations of motion; free vibration; forced vibration; and modal analysis. For the sake of simplicity, linear lumped systems will be mainly considered.
Classification of Vibrating Systems There are several ways to classify vibrating systems, depending on different perspectives. In view of the distributions of inertia and elastic parameters, there are distributed parameter systems such as structures composed of flexible beams and plates, and lumped parameter systems such as systems of lumped masses, and rigid bodies. Vibrations of distributed parameter systems are described by partial differential equations, while vibrations of lumped parameters are described by ordinary differential equations. Because exact solutions of partial differential equations are difficult to obtain, distributed parameter systems are often approximated by lumped parameter models, through a procedure called discretization. In engineering practice, discretization via the finite element method is widely adopted.
THEORY OF VIBRATION/Fundamentals 1291
From the viewpoint of energy conversion, vibrating systems are divided into two kinds: (1) conservative systems which have no damping, and no energy exchange with the surrounding environments; and (2) nonconservative systems which exchange energy with the surrounding environments via damping, gyroscopic, and circulatory forces, magnetoelectromechanical interactions, or other means. One important feature of conservative systems is the existence of normal modes of vibration, which are composed of real natural frequencies (eigenvalues) and real orthogonal mode shapes (eigenvectors). The concept of normal modes has an important position in vibration theory, and lays a foundation for the development of a useful vibration analysis technique called modal analysis. Vibrating systems can also be categorized as linear systems whose motion is described by linear differential equations; and nonlinear systems whose motion is governed by nonlinear differential equations, due to nonlinear material and/or geometric properties of the systems. For linear systems, many well-developed methods of vibration analysis are available, as the superposition principle is valid. In the study of small oscillations, a nonlinear system can be treated as a linear one by proper linearization. However, nonlinearities must be considered when large oscillations are of concern. Physically, nonlinear systems have certain unique characteristics that linear systems lack, such as subharmonic and superharmonic resonances, parametrically excited vibrations, limit cycles, self-excited oscillations, and chaotic motion. Analysis of nonlinear vibrations is much more difficult, and still needs further development. Vibrations become random when there are uncertainties in motion. The cause of uncertain motion may be the unpredictability of excitations, or the lack of accurate information about the physical properties of the system. Buildings under earthquake excitations and large space structures with uncertain elastic and damping parameters are two examples. Random vibrations are usually studied based on the theory of stochastic processes. Thus, vibrating systems are also classified as deterministic systems and stochastic systems.
of degrees of freedom is the number of independent coordinates needed to describe motion completely. Systems with One Degree of Freedom
The simplest vibrating system is shown in Figure 1, where x
t is the horizontal displacement of the mass, measured from the equilibrium position at which the spring is unstretched, and f
t is an external force. The mass does not move vertically, due to the constraint of the frictionless surface. The system has one DOF as the coordinate x
t completely describes the motion. According to Newton's second law, the motion of the spring±mass system is governed by: m x
t f
t ÿ fs
t
where x d2 x=dt2 is the acceleration of the mass, and fs is the spring force which always pulls the mass towards the equilibrium position. For small oscillations, the spring force can be assumed to be proportional to the mass displacement; that is, fs
t kx where k is a spring coefficient or stiffness. It follows that the motion of the spring±mass system is described by the linear differential equation: m x
t kx
t f
t
2
For large oscillations, however, the spring force may become a nonlinear function of the displacement, say fNL
x, which results in a nonlinear equation of motion: m x
t fNL
x f
t
3
Eqns [2] and [3] are generic of 1-DOF vibrating systems. For instance, the angular displacement of the shaft±disk system in Figure 2A is governed by the differential equation: Jy
t Ky
t M0
t
4
where M0 is an external torque; and the swaying motion of a simple pendulum in Figure 2B is described by:
Equations of Motion Mathematically, the motion of a vibrating system is described by time-dependent coordinates or generalized displacements. These coordinates are governed by differential equations of motion, which are derived from basic laws of physics, like Newton's laws. Vibrating systems can be classified by the number of degrees of freedom (DOF) of motion. The number
1
Figure 1 A spring±mass system.
1292 THEORY OF VIBRATION/Fundamentals
Figure 2 (A) Shaft±disk system: J, polar moment of inertia of the disk; K, torsional stiffness of the shaft. (B) Simple pendulum: g, gravitational acceleration.
y
t g sin y
t 1 f
t l ml
5
Systems with Multiple Degrees of Freedom
Many vibrating systems either have more than one mass element, or have a single mass that is in multidimensional motion. These systems are called multiDOF systems because the complete description of their motion requires more than one coordinate. The motion of multi-DOF systems is governed by a set of simultaneous differential equations. As two examples, the equations of motion of the two-mass system in Figure 3A are of the matrix form:
m1
0
0 m2 f1
t
x1
t x2
t
k1 k2
ÿk2
ÿk2
k2
x1
t
x2
t
f2
t 6
while the spring-supported rigid body in Figure 3B, whose motion involves both vertical translation y
t and rotation y
t, is described by:
m
0
y
t y
t
0 I y
t a
k2 ÿ k1 f
t k1 k2 y
t a
k2 ÿ k1 a2
k1 k2 t
t 7 In general, the motion of an n-DOF linear vibrating system can be described by the matrix differential equation: M x
t Kx
t f
t
8
where M and K are n-by-n mass and stiffness matrices, respectively, x
t is a vector of n coordinates or
Figure 3 Two-DOF systems.
generalized displacements, and f
t is a vector of external forces. Eqn [8] can be derived by either Newton's laws or energy methods. Distributed Parameter Systems
Eqn [8] represents lumped parameter models of vibrating systems because their mass and spring elements are concentrated at a finite number of points. For a distributed parameter system, like a flexible structure, its mass is distributed over the entire body which it occupies in space, and so are its elastic elements. Distributed parameter systems are also called infinite-dimensional systems because complete description of motion requires an infinite number of coordinates (degrees of freedom). The motion of a distributed parameter system is governed by partial differential equations like: M w;tt
x; t Kw
x; t f
x; t
9
where M and K are mass and stiffness differential operators, respectively, x represents a point in space, w
x; t is the generalized displacement, w;tt @ 2 w=@t2 , and f
x; t is the distribution of external forces applied to the distributed system. One example is the tensioned uniform string shown in Figure 4A. The transverse displacement w of the string, under a transverse load f
x; t, is governed by the partial differential eqn [9] with the mass and stiffness operators given by: M r;
K ÿT @ 2 =@x2
10
where r and T are the linear density (mass per unit length) and tension of the string, respectively. Eqn [9] represents a distributed model of vibrating systems. Because exact solutions of partial differential
THEORY OF VIBRATION/Fundamentals 1293
respectively. The displacement is assumed as x
t Aelt , which, according to eqn [11], results in the characteristic equation: ml2 k 0
12
The roots or eigenvalues of the characteristic equap tion are l1 ion , and l2 ÿion ; i ÿ1. The parameter on is given by: r k on m
Figure 4 A string in transverse vibration: (A) distributed parameter model; (B) lumped parameter model.
equations are only available for a few simple systems, distributed vibrating systems are often approximated by lumped-parameter models. In such approximations, a vibrating continuum is treated as a finite number of elastically interconnected lumped masses (see Figure 4B, for instance). By certain mathematical algorithms such as the Rayleigh±Ritz method and the finite element method, the partial differential equation of motion is reduced to a set of ordinary differential equations like eqn [8]. This process is called discretization. Besides equations of motion, a description of the vibrating systems also needs to assign initial conditions, and boundary conditions (for distributed parameter systems). A fundamental issue in vibration theory is to solve initial-boundary value problems associated with differential equations.
Free Vibration Free vibration occurs without externally applied forces; it arises when kinetic energy or potential energy is present initially in the vibrating system. The energy input is due to initial displacements and velocities, which are also called initial disturbances. Harmonic Motion
Free vibrations of an undamped 1-DOF system are described by the differential equation: m x
t kx
t 0
11
_ v0 , with the initial conditions x
0 x0 and x
0 where x0 and v0 are initial displacement and velocity,
13
which is called the natural circular frequency of the vibrating system, of which the units are radians per second. The natural frequency depends on inertia and elastic properties of the system. The free vibration is of the form: x
t A sin on t B cos on t
14
which, by the initial conditions, is determined as: v0 sin on t on Am sin
on t f
x
t x0 cos on t
15
where the amplitude and phase angle are: s v2 Am x20 02 ; on
on x0 f tan v0 ÿ1
16
The displacement in eqn [15] is sinusoidal, and hence is called simple harmonic motion. The period of harmonic motion (the time needed to complete a cycle of motion) is: T
2p 1 o n fn
17
where fn on =
2p is the natural frequency of the vibrating system, in hertz (Hz). Figure 5 shows that the harmonic motion can be viewed as the projection of a rotating vector OP of amplitude Am and rotation speed on on the vertical axis OX. Natural frequency, amplitude, and the phase angle are three major parameters of harmonic motion. Modes of Vibration
The free vibration of an undamped multi-DOF system is described by:
1294 THEORY OF VIBRATION/Fundamentals Effects of Viscous Damping
Among the available damping models, viscous damping is most commonly used. Figure 6 shows a viscously damped 1-DOF system, where viscous damping is indicated by a dashpot or damper. The damping force is proportional to the velocity of the mass, but _ opposite to the motion of the mass, i.e., fc
t cx
t, ÿ1 where c is the damping coefficient, in kg s . The equation of motion of the damped system is:
Figure 5 Simple harmonic motion.
_ kx
t f
t m x
t cx
t M x
t Kx
t 0 x
0 x0
and
18
x_
0 v0
19
where the vectors x0 and v0 contain initial displacements and velocities, respectively. Extending the concept of harmonic motion, assume x
t ueiot . According to this, eqn [18] leads to the eigenvalue problem: 2
o Mu Ku o21 ;
o22
20 o2n ,
There are n eigenvalues, ... roots of the characteristic equation:
which are the
ÿ det ÿo2 M K 0
21
The ok are called the natural frequencies of the system. Corresponding to ok , the eigenvector uk describes a specific distribution of displacements, and is called mode shape. The pair
ok ; uk defines the kth mode of vibration of the system by: xk
t
Ak sin ok t Bk cos ok tuk
22
For free vibration analysis, the solution of eqn [18], is represented by a linear combination of all modes: x
t
n X l1 n X
xl
t
The characteristic equation of the damped system, obtained by assuming x
t Aelt and f
t 0 in eqn [24], is: ml2 cl k 0
25
The roots of the characteristic equation are: q l1;2 ÿxon i 1 ÿ x2 on ;
i
p ÿ1
26
p where on k=m is called the undamped natural frequency, and x C=
2mon the damping ratio. When 0 < x < 1, the roots are complex, and the free response is: x
t eÿxon t
A sin od t B cos od t
27
p where od 1 ÿ x2 on is called the damped natural frequency, and A and B are determined by the initial conditions. This motion, which is oscillatory with decaying amplitude, is called underdamped vibration. In Figure 7 the dotted curves indicate the decay in the amplitude of free vibration, which is controlled by the damping ratio x. This type of response is commonly seen in many vibrating systems. When 4 1, the characteristic roots are both real and negative, and the system response is called overdamped vibration. In this case, the amplitude of free vibrations decays, without oscillatory motion:
23
Al sin ol t Bl cos ol tul
l1
where Al and Bl are determined by the initial conditions. The concept of modes of vibration is extremely important in the theory of vibration, and will be explained further later on.
24
Figure 6 A 1-DOF system with viscous damping.
THEORY OF VIBRATION/Fundamentals 1295
Forced Vibration The motion of a vibrating system that results from externally applied forces is called forced vibration. A forced vibration often consists of two parts: transient response, which is the motion that disappears after a period of time; and steady-state response, which is the motion that remains after the transient response has disappeared. Harmonic Excitation
Consider the motion of a damped 1-DOF system subject to a harmonically varying force: Figure 7 Free vibration of an underdamped system.
ÿ x
t eÿxon t Aeÿbt Bebt ; q b o n x2 ÿ 1 When = 1, there are two repeated l1 = l2 = 7on, and the system response is: x
t eÿon t
A Bt
_ kx
t F0 sin ot m x
t cx
t
28
roots,
29
which decays without oscillation. This motion is called critically damped vibration as it is the case which separates oscillation (0 5 5 1) from nonoscillation ( 4 1). Besides decay in vibration amplitude, the effects of damping are also seen from decay in mechanical energy. The rate of change of total mechanical energy (sum of kinetic energy and potential energy) is: d ÿ1 mx_ 2 12 kx2 ÿcx_ 2 2 dt
30
which is negative most of the time. (It is only zero at those discrete times when the displacement peaks.) Thus, the damping renders the mechanical energy decreasing continuously. If damping is absent
c 0, the mechanical energy is conserved as its rate of change is zero all the time. For a multi-DOF system with viscous damping, its motion can be described by the matrix differential equation: _ Kx
t f
t M x
t Cx
t
31
where C is the damping matrix consisting of the coefficients of dampers. The concepts of free vibrations and modes of vibration presented previously can be extended here.
32
where o is the excitation frequency or forcing frequency. The steady-state response xss
t of the system is assumed to be sinusoidal with the same frequency, i.e.: xss
t X0 sin
ot f
33
Substitute eqn [33] into eqn [32], to find the amplitude and phase angle: F0 1 q ; k 2
1 ÿ r2 4x2 r2 2xr f tanÿ1 1 ÿ r2
X0
34
where r is the frequency ratio r o=on , and on and x are the undamped natural frequency and damping ratio introduced in eqn [26]. As shown in Figure 8 and eqn [34], the amplitude of the steady-state response depends on the excitation frequency and damping ratio. For light damping, say x 0:1, the amplitude approaches a maximum value as the excitation is near the undamped natural frequency on . Moreover, the smaller the damping ratio, the higher the peak. The steady-state response xss
t is a particular solution. The total solution of eqn [32] is the sum of a particular solution and the solution of the corresponding homogeneous equation. For the underdamped case
0 < x < 1 this becomes: x
t eÿxon t
A sin od t B cos od t X0 sin
ot f
35
where A and B are determined by the initial conditions. The first term in eqn [35] is the transient response because it approaches zero for large values of t.
1296 THEORY OF VIBRATION/Fundamentals
The response of multi-DOF systems subject to harmonic excitations can be similarly determined. To this end, consider: _ Kx
t F0 sin ot M x
t Cx
t
37
where the complex vector X0 by eqn [36] is determined as: ÿ ÿ1 X0 ÿo2 M ioC K F0
x
t
36
By complex analysis, the excitation is the imaginary p part of the exponential form F0 eiot ; i ÿ1. Thus, the steady-state response can be written as: ÿ x
t Im X0 eiot
under the harmonic excitation F0 sin on t, according to the theory of differential equations, is:
38
The amplitudes and phase angles of the displacement parameters can be evaluated from eqns [37] and [38]. As in Figure 8, the amplitude can be plotted versus the excitation frequency. The difference is that the amplitude±frequency curves have multiple peaks, because there are n undamped natural frequencies from the characteristic eqn [21]. Resonance
Without damping ( = 0), the amplitude given in eqn [34] becomes infinite as the excitation frequency approaches the undamped natural frequency; see also Figure 8. This phenomenon of unbounded response is called resonance and o = on is called the resonance condition. The expression given in eqn [33], however, is unable to describe the gradual growth of the amplitude. The resonant response
of
steady-state
response
versus
39
A plot of the resonant response given in Figure 9 shows the unbounded growth of the vibration amplitude. Physically, the amplitude cannot keep growing, as the spring would be already broken. The phenomenon of resonance is seen in general vibrating systems. Control of resonant vibrations is critical in many engineering applications. Three commonly used techniques for resonance control, among others, are: (1) passive or active damping, which reduces resonance peaks (see Figure 8); (2) frequency tuning, which shifts the natural frequencies away from the excitation frequencies; and (3) dynamic vibration absorption, which confines the vibration energy within an added vibrating subsystem, called vibration absorber. Periodic Excitation
Response to arbitrary periodic excitations can be determined by extending the approach given in eqn [33]. In Figure 10, a general periodic forcing function F
t repeats itself in a fixed time period T (the period), such that F
t T F
t for all values of t. A sinusoidal function is a periodic function, but the reverse may not be true. A periodic forcing function can be expanded in an infinite series of sinusoidal functions, called the Fourier series:
F
t
Figure 8 Amplitude frequency ratio.
F0 t sin on t 2mon
1 a0 X
an cos not bn sin not 2 n1
40
Figure 9 Resonant vibration of an undamped 1-DOF system.
THEORY OF VIBRATION/Fundamentals 1297
as shock and impact loads. The time history of an impact or shock load in general is difficult to measure. However, its temporal effect can be quantified. For instance, consider a damped 1-DOF system subject to an impulsive force with the time history shown in Figure 11, where the duration " is arbitrarily small, i.e., " = 0+. By the impulse-momentum principle deduced from Newton's second law: Ze
Figure 10 A periodic forcing function.
I0
F
tdt 45
0
where o 2p=T, and the Fourier coefficients an and bn are calculated by the formulas: 2 an T
2 bn T
ZT F
t cos not dt;
n 0; 1; 2 . . .
41
F
t sin not dt;
n 1; 2; 3 . . .
42
0
ZT 0
Following the superposition principle, the response to a periodic excitation is the sum of the contributions of all individual harmonic components. For instance, a particular solution of eqn [24] to a periodic force can be written as: xp
t X0
1 X n1
Xn sin
not fn
It can be shown that X0 a0 =
2k, Xn sin
not fn is the solution of:
43 and
_ kx
t an cos not bn sin not m x
t cx
t 44 which can be obtained by following eqns [33] and [34]. The total response of the system is the sum of the particular solution and the homogeneous solution accounting for the initial conditions. The harmonic analysis given above is also applicable to multi-DOF systems. It should be pointed out that, under periodic excitations, resonance occurs if one of the natural frequencies is identical to no for any integer n.
_ ÿ mx_
0 mx
e mx_
0 ÿ mv0
where I0 is the impulse of the force. This indicates that the impulsive force causes a jump in the initial _ _ ÿ x
0 I0 =m. Note that velocity, Dx_ x
0 F
t 0 for t > 0 . Thus, the impulse-excited motion is equivalent to the free vibration problem: _ kx
t 0; m x
t cx
t x
0 x0 ;
x_
0 v0 I0 =m
46 47
For an underdamped system (0 5 5 1) initially at rest i.e., x0 and v0 , its impulse response is: x
t
I0 ÿxon t e sin od t mod
48
The above analysis is also applicable to multi-DOF systems. Response to Arbitrary Forces
There are many solution methods for the response to arbitrary external forces, among which are convolution integral, Laplace and Fourier transforms, and modal analysis. In addition, numerical methods, such
Impulse Response
A common source of vibrations is the sudden application of large-magnitude, short-duration forces, such
t > 0
Figure 11 An impulsive force, e 0 .
1298 THEORY OF VIBRATION/Fundamentals
as finite difference methods and numerical integration, are often used for complex vibrating systems. For demonstrative purposes, a Laplace transform± Green's function approach is briefly discussed here. Consider the motion of a damped multi-DOF system subject to arbitrary external and initial disturbances:
First, consider undamped systems modeled by eqn [8]. Under the condition of symmetric mass and stiffness matrices, and the condition of distinct eigenvalues, the eigenvectors defined in eqn [20] share the orthogonality relations:
_ Kx
t f
t M x
t Cx
t
Eigenvectors satisfying the above relations are usually called orthogonal or normal modes. The eigenvectors can be scaled or normalized such that:
49 x
0 x0 ; x_
0 v0 Laplace transformation of eqn [49] with respect to t yields: n o ^
s H
s Mv0 Cx0 sMx0 ^ f
s x
50
^
s and ^ where x f
s are the Laplace transforms of x
t and f
t, respectively, s is the complex Laplace transform parameter, and H
s
s2 M sC Kÿ1 is called the transfer function of the system. Inverse Laplace transform of eqn [50] gives the total response: x
t G
t
Mv0 Cx0 Zt d G
t ÿ tf
t dt G
tMv0 dt
51
0
where G
t, the inverse Laplace transform of the transfer function H
s, is called the impulse response function or Green's function of the system. Physically, the first two terms of eqn [51] represent vibration due to initial disturbances; the last term represents vibration due to external forces. Modal Analysis
Modal analysis is a useful technique of analysis and solution for general vibrating systems. Its use depends on the existence of certain orthogonality relations among system eigenvectors (mode shapes). Orthogonality relations are used to decouple the original equations of vibration into a set of independent differential equations. The solution of those decoupled equations yields the response in a series of system eigenvectors. Solution by this technique is usually called modal expansion or eigenfunction expansion. The basic concept of modal analysis is detailed as follows.
uTk Mul 0;
uTk Kul 0 for
k 6 l
uTk Muk 1
52
53
The orthogonality relations provide a convenient way of determining a dynamic response to arbitrary excitations. As an example, let the solution be expressed by a series of system eigenvectors: x
t
n X
qk
tfuk g
54
k1
where the unknown time-dependent coefficients qk
t are called modal coordinates. Eqn [54] represents a real transformation from physical coordinates to modal coordinates. Substitution of the expression [54] into eqn [8], and use of the orthogonality relations [52] and normalization condition [53], leads to n independent second-order differential equations: qk
t o2k qk
t uTk f
t;
k 1; 2; . . . n
55
which can be easily solved by many methods. With the determined qk
t, a closed-form modal expansion of the total response is given by eqn [54]. Furthermore, having solved eqn [55], the Green's function in eqn [51] is found to be: G
t
n X 1 sin ok t uT k uk o k k1
56
Next, consider the damped systems described by eqn [49]. A system is called proportionally damped if the condition: KMÿ1 C CMÿ1 K
57
holds. Under this condition, the eigenvalues are complex, but the eigenvectors are real. More importantly, the eigenvectors are the same as those of the corresponding undamped system. Consequently, the eigenvectors enjoy the orthogonality relations given in eqn
THEORY OF VIBRATION/Superposition 1299
[52]. In addition, they also satisfy: uTk Cul 0
for k 6 l
58
It follows that the modal expansion, eqn [54], can be directly used to decouple eqn [49] into: qk
t 2xk ok q_ k
t o2k qk
t uTk f
t
59
for k 1; 2 . . . n, where xk is called the modal damping ratio. The response of the damped system can be obtained in closed form. If the condition, eqn [57], is not met, the eigenvalues and eigenvectors are both complex. The system is then called nonproportionally damped. Besides nonproportional damping, complex modes are also caused by other effects such as gyroscopic forces induced by mass transport or Coriolis acceleration, and circulatory forces. These complex modes of vibration in general are not orthogonal in the sense of eqns [52] and [58]. As a result, the conventional modal expansion fails to decouple the equations of motion. In this case, complex modal analysis can be adopted. In a complex modal analysis, the original equations of vibration are cast in to a first-order statespace form with the state-space vector z
t containing displacements and velocities: z
t
x
t _ x
t
60
In this form, the orthogonality relations among statespace eigenvectors can be established. Consequently, the state equation is decoupled into a set of independent first-order differential equations, and the modal expansion of the state-space vector is obtained.
Nomenclature fs g I0 k K M M0 r T v0 x0 x r xk
spring force gravitational acceleration impulse of the force spring coefficient or stiffness stiffness differential operator mass differential operator external torque frequency ratio tension of the string initial velocity initial displacement acceleration of the mass linear density modal damping ratio
See also: Absorbers, active; Absorbers, vibration; Chaos; Commercial software; Computation for transient and impact dynamics; Eigenvalue analysis; Forced response; Krylov-Lanczos methods; Modal analysis, experimental, Basic principles; Mode of vibration; Nonlinear normal modes; Nonlinear systems, overview; Resonance and antiresonance; Shock isolation systems; Testing, nonlinear systems; Theory of vibration, Duhamel's Principle and convolution; Theory of vibration, Energy methods; Theory of vibration, Equations of motion; Theory of vibration, Impulse response function; Theory of vibration, Substructuring; Theory of vibration, Superposition; Theory of vibration, Variational methods; Viscous damping.
Further Reading Bishop RED (1979) Mechanics of Vibration. Cambridge, UK: Cambridge University Press. Dahlquist G and BjoÈrck AÊ Numerical Methods, Prentice Hall, Inc., Englewood Cliffs, New Jersey. Goldstein H (1980) Classical Mechanics, 3rd edn. Massachusetts: Addison Wesley. Harris CM (ed.) (1988) Shock and Vibration Handbook, 3rd edn. New York: McGraw-Hill. Horn RA and Johnson CR (1985) Matrix Analysis. Cambridge, UK: Cambridge University Press. Hughes TJR (1987) The Finite Element Method. Englewood Cliffs, NJ: Prentice Hall. Huseyin K (1978) Vibration and Stability of Multiple Parameter Systems. Alphen aan den Rijn, The Netherlands: Sijthoff & Noordhof. Inman DJ (1994) Engineering Vibration. Englewood Cliffs, NJ: Prentice Hall. Meirovitch L (1967) Analytical Methods in Vibrations. New York: Macmillan. Nayfeh AH and Mook DT (1979) Nonlinear Oscillations. New York: Wiley-Interscience. Newland DE (1975) An Introduction to Random Vibrations and Spectral Analysis. London: Longman. Rao SS (1995) Mechanical Vibrations, 3rd edn. Massachusetts: Addison Wesley. Rayleigh JWS (1945) The Theory of Sound. New York: Dover. Sun CT and Lu YP (1995) Vibration Damping of Structural Elements. Englewood Cliffs, NJ: Prentice Hall. Weaver, W Jr, Timoshenko SP and Young, DH (1990) Vibration Problems in Engineering. John Wiley.
Superposition M G Prasad, Stevens Institute of Technology, Hoboken, NJ, USA Copyright # 2001 Academic Press doi:10.1006/rwvb.2001.0109
1300 THEORY OF VIBRATION/Superposition
Introduction Linear system models are very important in vibration analysis. Linear models enable the designer to obtain a basic understanding of the component interactions in a system. Although nonlinearity exists in real systems, linear modeling and analysis are essential in most cases as they yield simpler mathematical formulation. The principle of superposition plays an extremely important role in mechanical vibrations and dynamic analysis of linear systems. It is known generally that all physical systems are nonlinear. However, assumptions and approximations are made such that the mathematical model and the resulting equations are linear. This is done for important reasons: the solution of linear problem is feasible and provides a good insight into the system behavior. Also, the results obtained using the linear approximation are often sufficient for many engineering applications. The importance of the principle of linear superposition to vibration analysis is seen by observing some of the following ramifications of the nonvalidity of the superposition to nonlinear systems. 1. Two linearly independent solutions cannot be linearly combined to obtain the solution of a nonlinear second-order differential equation. 2. The general solution to a nonlinear system equation cannot be obtained by a summation of the free vibration and forced vibration responses. This is because of the interaction between the free and forced vibration responses. 3. The development of a convolution integral approach based on the impulse response is not valid for nonlinear systems. 4. Superposition of modes will not be valid for nonlinear systems. This is due to mode coupling. 5. The mathematical techniques, such as Fourier series and Laplace transform, cannot be used to obtain responses of nonlinear systems under combination of excitations. Thus, it is seen that the superposition is an extremely important part of linear vibration analysis. It enables
Figure 1 A single-degree-of-freedom system model.
a solution of linear system models to a variety of excitations such as periodic functions, transients, modal excitations, etc. In the following section, the applications of the principle of linear superposition to wave addition, system response, convolution integral, and modal superposition are presented. It is to be noted that, although the principle of superposition is quite simple in its statement and application, it has a profound impact on all types of linear systems modeling and analysis.
Linearity and Superposition Figure 1 shows a linear lumped system model of a single-degree-of-freedom system with mass m, spring stiffness k and viscous damping coefficient c. Figure 2 shows a black-box model of the single-degree-of-freedom system with a harmonic input, f
t, and solution, x
t. The system equation is given by: m
d2 x dt2 c
dx=dt kx f
t
1
The principle of superposition says that the harmonic motions can be combined linearly to obtain the total motion. In reference to system analysis, the total response of a linear system can be obtained by the linear combination or addition of the individual responses of the system for the corresponding individual excitations. In other words, for the linear blackbox model of Figure 2, if x1
t and x2
t are the harmonic solutions, for the harmonic inputs are f1
t and f2
t respectively, then the total solution, as shown in Figure 3, is given by: x
t x1
t x2
t
2
It is seen that, whenever linearity holds, the powerful principle of superposition can be applied effectively. Thus, superposition can also be described as that, whenever many waves pass through the same given region, their effects are simply additive. The total solution is obtained by superposition.
THEORY OF VIBRATION/Superposition 1301
Applications of Superposition Addition of Simple Harmonic Waves Figure 2 An input±output model of a system.
In many situations, there is a need to combine the effects of individual vibrations to obtain the superposition effect. Generally, the presence of one vibration does not alter the medium significantly such that the characteristics of the other vibrations are disturbed. Thus the total vibration is obtained by a linear superposition of individual vibrations. Given that the two displacements have same angular frequency: x1
t A1 ej
otf1
and
x2
t A2 ej
otf2
The linear superposition from eqn [2] gives: Figure 3 Principle of superposition.
Aej
otf
A1 ejf1 A2 ejf2 ejot
To illustrate the importance of the principle of superposition, let the system shown in Figure 3 have a nonlinear spring in place of a linear spring. Let the nonlinear spring term be k
x ex2 , where e is the nonlinearity factor. The system equation is given by: .
m
d2 x dt2 c
dx=dt k
x ex2 f
t
3
Using phasor representation, the combined effect is shown in Figure 4. The real displacement is given by: x
t A cos
ot f where: A2 A2x A2y
and
f tanÿ1
Ay =Ax
Ax A1 cos f1 A2 cos f2 Ay A1 sin f1 A2 sin f2
By introducing the solutions x1 and x2 we get: . m
d2 x1 dt2 c
dx1 =dt k
x1 ex21 f1
t 4 . m
d2 x2 dt2 c
dx2 =dt k
x2 ex22 f2
t 5 By adding eqns [4] and (5) and then comparing them with eqn [1] we can see that the following inequality exists due to nonlinearity:
x1 x2 2 6
x21 x22
6
Thus solutions x1 and x2 for forcing functions f1 and f2 respectively, cannot be added to obtain the total solutions.
7
Figure 4 Superposition of two waves of same frequency.
8
1302 THEORY OF VIBRATION/Superposition
Thus, the linear superposition of two simple harmonic vibrations of the same frequency yields another simple harmonic vibrations of the same frequency. Further, the superposition can be extended to the case where there are n number of simple harmonic vibrations of the same frequency: 2 A4
X n
!2 An cos fn
!2 31=2 X An sin fn 5 n
9 f tan
ÿ1
X n
An sin fn
, X n
! An cos fn
10
When the frequencies of the waves are not identical, then the linear superposition yields a combined solution which is dependent on the ratio of the two frequencies. If the ratio is close to unity, then the well-known beating phenomenon occurs. If the ratio is very large, then a nonperiodic solution results.
periodic forcing function in terms of harmonic components. Then by the application of the superposition principle the total response is obtained using the individual responses for the corresponding component of the periodic forcing function. Superposition Integral
Another important application of the superposition principle is in the case of a linear system under arbitrary excitation. When an undamped linear system is excited by a unit impulse function, then the unit impulse response is given by: h
t
1=
mon sin
on t
Any arbitrary excitation can be considered to be a series of impulse excitations, as shown in Figure 5. Using an elapsed time variable,
t ÿ x, the unit impulse response at time t x is given by h
t ÿ x. Then the superposition principle can be used to combine these contributions for the varying amplitudes as per the given function. The response is given by:
System Response
The linear equation of motion of a single-degree-offreedom system as shown in Figure 1 is given by: . m
d2 x dt2 c
dx=dt kx f
t
11
The total response of the system eqn [6] is given by combining the solutions of the homogeneous part with f
t 0 and of the nonhomogeneous part with f
t as a given excitation function. Using the principle of superposition, the total response of the system is given by: x
t xc
t xp
t
12
where xc
t is the complementary solution or free response and xp
t is the particular solution or the forced response. It is to be noted that the free response xc
t depends on the initial conditions whereas the forced response is dependent on the given particular forcing function f
t. In addition it is noted that, for given different forcing functions, f
t, the corresponding particular solutions, xp
t, can be obtained and the total response is given by: x
t xc
t xp1
t xp2
t xp3
t
Zt f
x h
t ÿ x dx
x
t
15
0
Eqn [15] is the convolution integral and is also referred to as the superposition integral. Modal Superposition
In dealing with the linear vibrations of a continuous system, the superposition principle is very useful in evaluating the response of the system. The general solution of a wave equation is obtained by the superposition of various normal modes. A property of the normal modes is that they are linearly independent of each other. As an illustration in the case of a transverse vibration of a string, the wave equation is given by (Figure 6):
13
In the case of f
t being a periodic forcing function, the Fourier series approach is used to express the
14
Figure 5 An arbitrary excitation.
THEORY OF VIBRATION/Superposition 1303
c2
d 2 y d x2
d 2 y dt2
16
Then, for a prescribed boundary condition such as both ends fixed, the transverse displacements, y, at both ends are zero. Using the superposition principle, the total solution is given by: y
x; t
X m
y
x; t
X
ym
x; t
sin
mpx=l
m
Cn cos
mcpt=L Dn sin
mcpt=L
17
18
where L is the length of the string, m is the mode number, c is the wave speed, and Cn and Dn are the constants dependent on the initial conditions. Thus
Figure 6 Three normal modes of a vibrating string.
the principle of superposition has resulted in the total solution y
x; t. Figure 6 shows the first three modes and their superposition, which represents the net vibration of the string. This modal superposition is based on the summation of harmonic motions that is also seen in Fourier series representation of periodic functions. To illustrate the additive effects in superposition, the following example of a square wave is presented. A square wave, f
t, is given by: f
t
1 ÿ1
0 < t < T=2 T=2 < t < T
19
where T is the period of the wave. It can easily be shown that this wave is represented using the Fourier series as:
1304 THEORY OF VIBRATION/Duhamel's Principle and Convolution
f
t
4 p
1 1 sin ot sin 3ot sin 5ot 3 5 20
The superposition of these motions can be seen in Figure 7. The linear addition effect of the superposition is clearly seen as the fundamental wave transforms into a square wave. Thus, the principle of superposition can be used in both ways. This means that sinusoidal waves can be added to obtain the total
solution and a periodic wave can also be represented as a summation of sinusoidal waves.
Conclusion The importance of principle of superposition is seen through its many applications in linear system analysis. An understanding of the application of the principle of superposition is essential in vibration analysis of linear systems. Although most systems are nonlinear in nature, the linearization of the systems gives an insight into the system performance. Such linear analysis of systems in the presence of complex forcing functions is possible mainly due to the principle of linear superposition.
Nomenclature c e f
t L m T
wave speed nonlinearity factor periodic forcing function length of string mode number period of wave
See also: Linear algebra; Linear damping matrix methods; Theory of vibration, Fundamentals.
Further Reading Dimarogonas AD and Haddad S (1992) Vibrations for Engineers. New Jersey, USA: Prentice-Hall. Findeisen D (2000) System Dynamics and Mechanical Vibrations: An Introduction. Berlin, Germany: SpringerVerlag. Hall DE (1993) Basic Acoustics. Florida, USA: Krieger Publishing. Kelly SG (2000) Fundamentals of Mechanical Vibrations. New York, USA: McGraw-Hill. Kinsler LE, Frey AR, Coppens AB and Sanders JV (2000) Fundamentals of Acoustics. New York, USA: John Wiley. Meirovitch L (2001) Fundamentals of Vibrations. New York, USA: McGraw-Hill. Rao SS (1995) Mechanical Vibrations. Massachusetts, USA: Addison-Wesley. Thomson WT and Dahleh MD (1998) Theory of Vibration with Applications. New Jersey, USA: Prentice-Hall.
Duhamel's Principle and Convolution G Rosenhouse, Technion - Israel Institute of Technology, Haifa, Israel Copyright # 2001 Academic Press
Figure 7 Summation of first three sinusoidal waves.
doi:10.1006/rwvb.2001.0113
THEORY OF VIBRATION/Duhamel's Principle and Convolution 1305
Duhamel's principle or convolution expresses the solution for a problem as a superposition of a set of solutions of the homogeneous wave equation and certain initial conditions. In problems that involve mechanical vibrations a force±time function that excites a system is replaced by a sequence of pulses that are represented by initial conditions at each segment of time. Now, decreasing the intervals into infinitesimal ones converts the sums into integrals, which are a special case of Duhamel's integral. This procedure allows us to obtain the aforementioned set of solutions. Finally, the total solution is obtained by superposition at any time of interest. Duhamel's principle is useful for either deterministic or random vibrations. For example, it is frequently applied in seismic analysis of structures.
About Duhamel
system of coordinates. Other equivalent coordinates systems can also be used. A very important version of Duhamel's principle is defined as convolution. The convolution of two functions f and g is defined by f g and is given as: Z1 f g
f
tg
t ÿ t dt
2
ÿ1
If the integrals involved with the convolution exist, then it is a linear operation that satisfies:
f h g f g h g f
g h f g f h
3
The convolution is also commutative:
Jean-Marie Duhamel (1797±1872) studied at the Ecole Polytechnique. He was considered to be a remarkable teacher of mathematics. He was a professor there since 1830 for 39 years, and was elected to the AcadeÂmie des Sciences in Paris in 1840. He was director of studies at the Ecole Polytechnique for the period 1848±51. In 1851 he took the chair in analysis and also became a professor at the Faculty des Science, Paris. Duhamel worked on partial differential equations and applied his methods to acoustics, physics of harmonic overtones, rational mechanics, and heat. His acoustical studies involved vibrating strings and vibration of air in cylindrical and conical pipes. He developed a heat theory that is mathematically similar to the work of Fresnel in optics. Duhamel based his theory on the transmission of heat in crystal structures on the works of Fourier and Poisson. Duhamel's principle in partial differential equations was a consequence of his research on the distribution of heat in a solid with a variable boundary temperature.
Duhamel's Theorem: Definitions
1
0
Derivation of the Convolution Theorem from Fourier Transforms The Fourier transforms of f
t; g
t and h
t are, respectively, F
o; G
o, and H
o, with o as the radian frequency of the signals. We assume the relation H
o F
oG
o, and the aim is to find the time domain relation between the signals f
t; g
t, and h
t. By definition, the use of the Fourier transform yields: 1 h
t 2p 1 2p
1 2p
Z1 ÿ1 Z1
ÿ1 Z1
ÿ
H
oeiot do
ÿ
F
oG
oeÿiot do 2
F
o 4
ÿ1
Z1
5 3
g
teiot dt5eiot do
ÿ1
Now, by changing the order of integration we have: Z1 h
t ÿ1
where
x; y; z represents the radius vector of the point of interest in the Cartesian orthogonal spatial
4
There are other forms of Duhamel's principle that suit the solution of problems of mechanical and electrical vibrations.
The solution v
x; y; z; t of the boundary value problem with variable source and surface conditions is given in terms of the solution f
x; y; z; t; t0 of the boundary value problem with constant source and surface conditions by the formula: 2 t 3 Z @ v
x; y; z; t 4 f
x; y; z; t ÿ t0 ; t0 5 dt @t
f ggf
2 1 g
t4 2p
Z1
3 f
oeio
tÿt do5 dt 6
ÿ1
Which, in turn, results in the convolution theorem:
1306 THEORY OF VIBRATION/Duhamel's Principle and Convolution
Z1 h
t
The Engineering Approach
g
tf
t ÿ t du ÿ1 Z1
7 f
tg
t ÿ t dt f
t g
t
ÿ1
Because of its symmetry, a frequency domain convolution can be obtained. Using eqn [7] for the time domain, we obtain:
H
o
1 2p
1 2p
Z1 ÿ1 Z1
2
Z1
f
t4
3 G
OeiOt dO5eÿiot dt
ÿ1
2
G
O4
ÿ1
Z1
3
8
f
tei
oÿOt dt5 dO
ÿ1
or: 1 H
o 2p 1 2p
Z1 G
OF
o ÿ O dO ÿ1 Z1
9 F
OG
o ÿ O dO
ÿ1
The response of a mechanical system to a step function excitation or to a unit impulse can serve as a tool in the analysis of the response of a linear system to an excitation by an arbitrary function of time. This method is based on the principle of superposition that can be applied to any linear system that can be simulated by a set of linear differential equations. Figure 1A provides a simple illustration where a mass±spring system, with m as the mass and k as the spring constant, is vibrating in the x-direction due to an arbitrary force, f
t. If the coefficients of the differential equations of the system are constant, it is assumed that a unit step function or a unit impulse applied at the time t, as shown in Figures 1B and 1C, respectively, yields a response x
t as a function of the elapsed time t ÿ t only. However, if the coefficients are functions of time this assumption is, in general, not satisfied. The applied force f
t at the time t can be split along the time axis into a sequence of step functions Df , at each time step Dt, as illustrated in Figure 1B, by application of steps at 0, Dt; 2Dt; . . . . This procedure yields a finite difference approximation. A unit step function that begins at t results in the displacement F
t ÿ t which depends on the properties of the system, and is defined as the indicial admittance of the system. The total displacement x
t becomes:
Thus: 1 F
o G
o H
o 2p
10
Since O is generally complex, H
o is called the complex convolution theorem.
Figure 1 Schemes illustrating Duhamel's principle.
x
t
NX
tt
Df
nDtF
t ÿ nDt
t0
If we write Df
Df =DtDt, and nDt t, then:
11
THEORY OF VIBRATION/Duhamel's Principle and Convolution 1307 t1 X Df F
t ÿ tDt Dt t0
x
t
12
x
t lim fx
tg Dt!0
0
d
t ÿ t2
kDx
t ÿ t 0
17
and the last boundary conditions. Solving eqn [17] and substituting the initial conditions to find the solution constants yields:
and, in the limit: Z1
d2 Dx
t ÿ t
m
Df F
t ÿ t dt Dt
13
Eqn [13] has the form of Duhamel's integral that we have met before. By integrating by parts of eqn [13] another version of Duhamel's integral can be obtained:
f
tDt sin
o0
t ÿ t o0 Dx
t ÿ t mo0
r k m 18
Following eqn [16], the whole displacement of the specific problem, at t becomes: Zt
Zt x
t
14
0 0
F
t ÿ t h
t ÿ t: h
t ÿ t is the impulse response function (the response at t to a unit impulse at t). This result leads to another form of solution, which collects the contributions of discrete impulses f
tDt, that exist in the interval t 0 to t t as shown in Figure 1C, on the displacement x
t:
x
t
tt X
f
th
t ÿ tDt
0
1 mo0
19
Zt f
t sin
o0
t ÿ t dt 0
The solution of the contribution Dx
t ÿ t for the step function that begins at t is: Df
t 1 ÿ cos
o0
t ÿ t o0 Dx
t ÿ t k
15
t0
r k m 20
and the whole displacement at t becomes:
In the limit Dt!0, Duhamel's integral or the convolution is obtained: Zt f
th
t ÿ t dt
x
t
f
th
t ÿ t dt
x
t f
tF0
t ÿ t dt
16
0
An important feature of this approach is that the solution can be expressed in terms of the initial conditions of each step or contribution. If from Figures 1A and 1C we take the impulse at t and calculate its contribution at the time t, then from the multiplication of the impulsive force f
t by the increment of time Dt, we obtain the impulse f
tDt. The impulse equals the resulting momentum, m dx=dt. The initial conditions for the single impulse at t are: x
t 0, and dx=dt v
t. Hence, v
t f
tDt=m. We remain with the homogeneous equation of motion of the mass±spring for the incremental contribution Dx of the impulse at t to the displacement at t:
1 x
t k
Zt 0
df
t 1 ÿ cos
o0
t ÿ t dt dt
21
Obviously eqns [19] and [21] should lead to the same result. Further development of this formulation includes the addition of damping, etc. These extensions enable us to solve more complicated problems. In the case of a damped single-degree-of-freedom (SDOF) system, a dashpot is used where a viscous damping coefficient c, is added in parallel to the spring. The equation of motion for an impulse at t becomes: d2 Dx
t ÿ t
dDx
t ÿ t o20 Dx
t ÿ t 2b d
t ÿ t d
t ÿ t2 r f
td
t ÿ t c k b ; o0 m 2m m
22
and the initial conditions for the single impulse at t are: x
t 0, and dx=dt v
t. Hence,
1308 THEORY OF VIBRATION/Energy Methods
v
t f
tDt=m. Hence, for a damped SDOF system, Duhamel's integral becomes: Zt f
th
t ÿ t dt
x
t
See also: Earthquake excitation and response of buildings; Forced response; Theory of vibration, Energy methods; Theory of vibration, Equations of motion; Theory of vibration, Fundamentals; Theory of vibration, Superposition.
0
Zt 1 f
t exp ÿb
t ÿ t sin
o
t ÿ t dt mo 0 qÿ qÿ 1 ÿ z2 ; z b=o0 o o20 ÿ b2 o0
Further Reading Karman T, Biot MA (1940 Mathematical Methods in Engineering, pp. 403±405. New York: McGraw Hill.
23 Eqn [16] has some other variants. For example, since all values of f
t at t < 0 are zero, and the integral from t ÿ1 to t t is zero, the limits of the integral can be extended to this range. Moreover, since the response function h
t ÿ t for t < t is zero (these impulses do not exist so far), the time limits can be extended to the interval
ÿ11 without changing the result of eqn [16]: f
th
t ÿ t dt f h
t2R
24
ÿ1
The Discrete Time Convolution Whenever the right-hand side of eqn [17] exists, the discrete time convolution of the discrete time signals becomes:
ff hg
t
1 X
f mhn ÿ m
n2Z
25
mÿ1
Hence, a discrete time convolution is an operator acting on a pair of discrete time signals f and h, resulting in a discrete time signal f h, the value of which at the time n is ff hgn. The circular convolution of a finite duration discrete time signal, fxn; yn; 0 n N ÿ 1g, is: ff hgn
N ÿ1 X
S S Rao, University of Miami, Coral Gables, FL, USA Copyright # 2001 Academic Press doi:10.1006/rwvb.2001.0115
Introduction
Z1 x
t
Energy Methods
f mh
n ÿ m mod N
0nNÿ1
m0
26 This is an operator acting on a pair of N-dimensional vectors f and h, which yields an N-dimensional vector f h, and its nth component is ff hgn.
Energy methods can be used to find the solution of solid mechanics problems without having to solve the governing differential equations. The foundations of energy methods can be traced to the concepts of work, potential energy, and kinetic energy that were introduced by Huygens, Leibniz, Bernoulli, and Lagrange. The energy principles of mechanics are sufficiently general to allow Newton's second law to be deduced from them. The principle of virtual work, first presented by Jean Bernoulli, leads to the principle of minimum potential energy. The principle of complementary energy can be derived from the principle of virtual work by means of Legendre transformation. The principle of minimum complementary energy can be considered as the generalization of Castigliano's theorem. The principle of virtual work, coupled with the concept of inertia force, as presented by D'Alembert, can be used to derive the Hamilton's principle. Lagrange equations of motion can be obtained directly from the Hamilton's principle. Newton's second law of motion, Lagrange equations, or Hamilton's principle can be used to derive the equations governing the dynamics of rigid bodies as well as vibrating systems. For the solution of equations of motion of a vibrating system, exact methods of solving ordinary and partial differential equations can be used. These methods can only be used for simple systems with simple boundary and initial conditions. Useful approximate solution methods, based on variational principles and energy methods, were devised by Rayleigh, Ritz, Galerkin, and
Plate 51 Shape Memory Alloys. Memory metal. Close-up view of a wire of memory metal being 'educated' by being bent around a spindle. Memory metals are alloys of nickel, titanium and aluminium or copper. Once manufactured in a certain shape the metal may be deformed with little effort, but will regain its shape upon gentle heating. The heat makes the metal plastic, allowing it to return to the structural arrangement in which it was made (its minimum energy configuration). Repeated bending of the metal, each time to the same position, alters the original structure permanently and 'educates' it to a new shape. (With permission from Science Photo Library).
Plate 52 (left) Vibro-Impact Systems. Regions of stability and existence of impact motions for the periodic harmonic and rectangular excitations.
Plate 53 (below) Vibro-Impact Systems. Regions of periodic and chaotic impact motions near the resonance of the linear oscillator.
Plate 54 (above) Vibro-Impact Systems. Regions of chaotic impact motions and classification of their boundaries (four types of way into and from chaos and one type of exit from chaos). Plate 55 (right) Vibro-Impact Systems. Regions of periodic and chaotic impact motions in enlarged rectangular subregion of Plate 54.
THEORY OF VIBRATION/Energy Methods 1309
others. Approximate solutions, based on numerical analysis, such as finite difference and finite element methods have also been developed. This article deals with the approximate methods based on energy principles. These methods include the Rayleigh's method and the Rayleigh±Ritz method.
Rayleigh's Method for Discrete Systems The Rayleigh's method is useful to find an approximate value of the fundamental natural frequency of a vibrating system. The method makes use of energy expression and does not require either the differential equation of motion of the system or its solution. Rayleigh's method can be stated as follows: The frequency of vibration of a conservative system vibrating about an equilibrium position has a stationary value in the neighborhood of a natural mode. This stationary value, in fact, is a minimum value in the neighborhood of the fundamental natural mode. Although Rayleigh's method is more useful for continuous than for discrete systems, the method is described for both in this article. To derive an expression for the approximate value of the first natural frequency of a multi-degree-offreedom (discrete) system according to Rayleigh's method, let the displacement xi
t of mass mi be of the form: xi
t Xi sin ot
1
where Xi is the amplitude and o is the frequency. The displacements of all the n masses of the system can be expressed as a vector as: sin ot
t X x
2
where: 9 8 x1
t > > > > > > < x2
t =
t x . > > > .. > > > ; : xn
t
and
9 8 X1 > > > = < X2 > X .. > > > ; : . > Xn
The kinetic energy of the system can be written as: 1 _
tT Mx _
t T
t x 2 and the potential (strain) energy in the form:
3
1 U
t x
tT Kx
t 2
4
_
t
d=dx
x
t is the vector of velocities where x of the masses: 9 8 x_ 1
t > > > > > = < x_ 2
t > _x
t .. > . > > > > > ; : x_ n
t M is the mass matrix and K is the stiffness matrix of the system. Using eqn [2], the kinetic and potential energies can be expressed as: 1 T
t XT MXo2 cos 2 ot 2
5
1 U
t XT KX sin 2 ot 2
6
When the system is conservative, the maximum kinetic energy is equal to the maximum potential energy: Tmax Umax
7
Using eqns [5] and [6], eqn [7] can be rewritten as: 1 T 1 X MXo2 XT KX 2 2
8
which yields the frequency of vibration as: R o2
XT KX XT MX
9
The expression on the right-hand side of eqn [9] is known as Rayleigh's quotient or Rayleigh's function. Eqn [9] can be used to find an approximate value of the fundamental natural frequency of a system. The procedure involves selecting an arbitrary trial vector X to represent the fundamental natural mode. The substitution of the trial vector into eqn [9] yields an approximate value of the fundamental natural frequency. Because of the stationary property of the Rayleigh's quotient, a very good estimate of the fundamental natural frequency can be found even when the trial vector deviates greatly from the exact mode (eigenvector). Naturally, a better value of the fundamental natural frequency can be obtained when the trial vector resembles the first eigenvector more closely.
1310 THEORY OF VIBRATION/Energy Methods Stationarity of Rayleigh's quotient
The Rayleigh's quotient is stationary when the vector X is in the neighborhood of an eigenvector X
r . To prove this, let the arbitrary vector X be expressed as a linear combination of the normal modes of the system, X
i , as: X a1 X
1 a 2 X
2
10
where a1 , a2 . . . are constants. Using eqn [10], the numerator and denominator of the Rayleigh's quotient can be expressed as: XT KX a21 X
1T KX
1 a22 X
2T KX
2 11
XT MX a21 X
1T MX
1 a22 X
2T MX
2 12
iT
j
Note that the mixed terms of the form ai aj X KX and ai aj X
iT MX
j , are zero due to the orthogonality property of eigenvectors. The eigenvectors also satisfy the relation:
a2r o2r a2r R
X
a2r a2r
13
i1;2;... i6r
ai =ar 2 o2i
ai =ar 2
17
ÿ R
X o2r 1 O e2
18
with O
e2 denoting a function of e of the second order. Eqn [18] implies that if the arbitrary vector X differs from the true eigenvector X
r by a small quantity of the first order, R
X differs from the eigenvalue o2r by a small quantity of the second order. This shows that the Rayleigh's quotient is stationary in the neighborhood of an eigenvector. Minimum Value of R (X)
The stationary value can be shown to be a minimum in the neighborhood of the fundamental mode, X
1 . To investigate this property, rewrite eqn [17] with r 1: P
2 2 i2;3;...
ai =a1 oi P 1 i2;3;...
ai =a1 2 X ÿ o21 o2i ÿ o21 e2i
o21
19
i2;3;...
Using eqns [11]±[13], eqn [9] can be written as:
In view of the fact that o2i < o2l for i 2; 3, . . . , eqn [19] leads to:
R
X o2
P
i1;2;... i6r
Since jai =ar j ei 1 where ei , is a small number for all i 6 r, eqn [17] yields:
R
X X
iT KX
i o2i X
iT MX
i
P
a21 o21 X
1T MX
1 a22 o22 X
2T MX
2 a21 X
1T MX
1 a22 X
2T MX
2
14 If the eigenvectors are M-orthogonalized; X
iT MX
j 1
a21 o21 a22 o22 a21 a22
20
which indicates that Rayleigh's quotient is never lower than the first eigenvalue, o2l . Application ± Free Vibration of a Rod Carrying an End Mass
15
and eqn [14] reduces to: R
X o2
R
X o21
16
If X differs from the eigenvector X
r by a small amount, the constant ar , will be much larger than all other constants ai
i 6 r. Hence eqn [16] can be expressed as:
The natural frequency of longitudinal vibration of a rod carrying an end mass (Figure 1) is considered. The axial displacement of the cross-section of the rod (under a static load) is assumed to vary linearly along the length of the rod, as shown in Figure 1B. The amplitude of vibration is also assumed to vary linearly, as shown in Figure 1B. If x0 denotes the amplitude of the rod at z L, the amplitude of vibration
x of the cross-section at z can be expressed as: x
x0 z L
a
THEORY OF VIBRATION/Energy Methods 1311
Figure 1 A rod carrying an end mass. (A) System; (B) variation of axial displacement.
When the natural frequency of vibration is on , the maximum velocity of the cross-section is given by: x0 z on xon L
b
The maximum kinetic energy of an element of the rod of length dz can be expressed as: dT
x zo 2 1 0 n
r dz L 2
c
where r is the mass of rod per unit length and r dz is the mass of the element of length dz. The maximum kinetic energy of the system (rod and the end mass) can be determined as:
Tmax
1 1 M
x0 o n 2 2 2 1
ZL 2 x0 z r o2n dz L 0
m0 2 2 M x0 on 3 2
d
where m0 rL is the mass of the rod. The maximum strain energy of the system can be expressed as: 1 Umax kx20 2
e
where k
AE=L is the axial stiffness, A = crosssectional area, and E = Young's modulus of the rod. Equating Tmax and Umax , we find that the natural frequency of vibration of the system is:
1=2 k 1=2 k on meq M
m0 =3 1=2 AE L
M
m0 =3
f
where meq M
m0 =3 is the equivalent mass of the system. Eqn [f] shows that one-third of the mass of the rod is to be added to the attached mass to find the equivalent mass of the system. Application ± Natural Frequency of Vibration of a Three-Degree-of-Freedom System
The natural frequency of vibration of the three degree-of-freedom spring-mass system shown in Figure 2 is considered. The strain energy of the system can be expressed as: 1 1 1 U k1 x21 k2
x2 ÿ x1 2 k3
x3 ÿ x2 2 2 2 2 kÿ 2 3x1 5x22 3x23 ÿ 4x1 x2 ÿ 6x2 x3 2
a
The kinetic energy of the system is given by: 1 1 1 T m1 x_ 21 m2 x_ 22 m3 x_ 23 2 2 2 mÿ 2 2 2 x_ 2x_ 2 3x_ 3 2 1
b
For free vibration, harmonic motion at frequency on , is assumed as:
1312 THEORY OF VIBRATION/Energy Methods
Umax
kÿ 2 3X1 5X22 3X23 ÿ 4X1 X2 ÿ 6X2 X3 2 g
Tmax
mo2n ÿ 2 X1 2X22 3X23 2
h
By assuming the mode shape as: 8 9 8 9 < X1 = < 1 = X 2 X : 2; : ; 1 X3 3
i
where X1 is the amplitude of vibration of the first mass, the maximum potential and kinetic energies can be expressed as: Umax 3kX21 ;
Tmax 18 mX21 o2n
j
By setting Tmax Umax , the natural frequency of vibration can be found as: o2n
Figure 2 A three-degree-of-freedom spring±mass system.
k k 0:166667 6m m
k
The exact value of the natural frequency is given by: xi
t Xi cos on t;
i 1; 2; 3
c
so that: x_ i ÿon Xi sin on t;
i 1; 2; 3
d
Substituting eqns [c] and [d] into eqns [a] and [b] yields: kÿ U 3X21 5X22 3X23 ÿ 4X1 X2 ÿ 6X2 X3 cos2 on t 2 e
T
mo2n ÿ 2 X1 2X22 3X23 sin2 on t 2
f
Thus, the maximum potential energy (equal to maximum strain energy in the absence of external forces) and maximum kinetic energy are given by:
o2n 0:113992
k m
l
Rayleigh's Method for Continuous Systems The eigenvalue problem of a continuous (distributed) system can be expressed, in general terms, as: L
f lM
f
21
where L
f and M
f are linear homogeneous differential operators of orders p and q, respectively, with p < q and p and q even numbers, l is the eigenvalue and f is the field variable such as displacement. Eqn [21] is to be satisfied over the domain of the system (D) and f
D denotes the distribution of f over D. If lr represents an eigenvalue and fr the corresponding eigenfunction is: L
fr lr M
fr
22
The multiplication of eqn [22] by fr and integration over the domain D gives:
THEORY OF VIBRATION/Energy Methods 1313
R
f f L
fr dD N R lr D r r D
fr D fr M
fr dD
23
ZL
0 EAf0r
x fr
x
dx
ÿo2r
0
, can be shown to be where the numerator, N
f r proportional to the strain energy of mode r and the can be shown to be proportional denominator, D
f r and D
f to the kinetic energy of mode r. Both N
f r r are functions of the eigenfunction fr , and are positive for a positive definite system. Eqn [23] is called the Rayleigh's quotient. The explicit expressions for the Rayleigh's quotient are given below for a few simple cases.
ZL
rA
xf2r
x dx
0
28 Integrating the left-hand side integral of eqn [28] by parts results in:
EAf0r
xfr
x L0
ZL ÿ
EA f0r
x dx
0
ÿo2r
Longitudinal Vibration of a Thin Bar
ZL
29
rAfr
x2 dx
0
The equation of motion governing the longitudinal vibration of a thin bar is given by:
Eqn [29] can be rewritten as: @ @u @2u EA rA 2 @x @x @t
24
where E = Young's modulus, A
x = area of crosssection, r = density, and u
x; t = longitudinal displacement. For harmonic motion: u
x; t f
x sin ot
R
o2r
RL
0
2 EA f0r
x dx ÿ EAfr
xf0r
x L0 RL 2 0 rAfr
x dx 30
For simple boundary conditions such as free ends 0
fr 0 and fixed ends
fr 0, eqn [30] reduces to:
25 R
o2r
and eqn [24] becomes:
or: EAf0
x ÿrAo2 f
x
26
where a prime denotes differentiation with respect to x. For rth mode, eqn [26] gives:
0 EAf0r
x ÿrAo2r fr
x
2 0 dx 0 EA fr
x RL 2 0 rAfr
x dx
31
The expressions on the right-hand side of eqns [30] and [31] denote Rayleigh's quotients. Note that eqn [31] can also be derived from potential and kinetic energy expressions. The potential or strain energy of a bar is given by:
d df EA ÿrAo2 f dx dx
0
RL
27
Multiplying eqn [27] by fr
x and integrating over the length of the bar
L leads to:
1 U
t 2
ZL 0
@u
x; t 2 EA dx @x
32
and the kinetic energy by:
T
t
1 2
ZL 0
@u
x; t 2 rA dx @t
33
Using u
x; t f
xsin ot, eqns [32] and [33] can be expressed as:
1314 THEORY OF VIBRATION/Energy Methods
1 U
t sin2 ot 2
ZL 0
1 T
t o2 cos2 ot 2
df
x 2 EA dx dx ZL
rAf
x2 dx
34
35
0
Equation Umax to Tmax results in eqn [31].
Umax
Tmax
Consider the longitudinal vibration of a linearly tapered fixed-free rod shown in Figure 3. The area of cross-section of the rod is given by:
1 2
ZL
2 EA
x f0r
x dx
a Tmax
b
0
where fr
x is the deflection (mode) shape that satisfies the fixed boundary condition. Let fr
x be assumed as: fr
x sin
px 2L
c
so that: p px cos f0r
x 2L 2L and:
Figure 3 A tapered fixed-free rod.
o2n 2
ZL
rA
xf2r
x dx
f
0
where r is the density of the rod. Using eqn [c], we obtain:
where A0 denotes the area at the root
x 0. The maximum strain energy of the rod can be expressed as:
Umax
e
The maximum kinetic energy of the rod, during harmonic vibration at frequency on , can be determined as:
Application ± Longitudinal Vibration of a Tapered Rod
x A
x A0 1 ÿ L
ZL
x p 2 2 px dx EA0 1 ÿ cos L 2L 2L 0 EA0 p2 1 8L 4 1 2
ZL
x px dx A0 1 ÿ sin2 L 2L 0 rA0 Lo2n 4 1ÿ 2 8 p ro2n 2
g
By setting Tmax Vmax , we obtain: rA0 Lo2n 4 EA0 p2 1ÿ 2 1 8 8L 4 p
h
from which: o2n
Ep2 4 p2 E 5:830356 2 2 2 rL 4rL p ÿ 4
i
This yields the natural frequency of vibration as: d
s E on 2:414613 rL2
j
THEORY OF VIBRATION/Energy Methods 1315 Transverse Vibration of a String
The equation governing the transverse vibration of a string is given by: @ @w @2w T
x m
x @x @x @t2
36
where T
x is the tension in the string, m
x is the mass per unit length of string and w
x; t is the RL R o2r
0
R
R L 0 2 T fr
x dx R0L 2 0 mfr
x dx
R
Torsional Vibration of a Shaft
The torsional vibration of a shaft is governed by the equation of motion: @ @y
x; t @ 2 y
x; t GJ J0 @x @x @t2
38
where GJ
x is the torsional stiffness, G is the shear modulus, J
x is the polar moment of inertia of the cross-section (for circular sections), J0
x is the mass polar moment of inertia is the rJ for shafts with uniform cross-section, and y
x; t is the angular displacement is f
x sin ot for harmonic motion with frequency o. The Rayleigh's quotient corresponding to eqn [38] is given by: 0 2 dx 2 0 GJ
x fr
x R or R L 2 0 J0 fr
x dx where L is the length of the shaft. Transverse Vibration of a Thin Beam
The equation of motion governing the bending vibration of thin beams is given by:
o2r
RL
2 00 0 EI
x fr
x RL 2 0 rA
xfr
x
dx
42
dx
The expression on the right-hand sides of eqns [41] and [42] are called Rayleigh's quotients. Application ± Transverse Vibration of a Beam with Central Mass
Consider a uniform beam of mass m per unit length and bending stiffness EI, fixed at both ends and carrying a central mass, as shown in Figure 4. The transverse deflection of the beam during free vibration is assumed to be similar to the static deflection shape of a uniform fixed-fixed beam subject to a concentrated load at the middle: fr
x
c ÿ 3Lx2 ÿ 4x3 ; 48EI
0x
L 2
a
where c is a constant, denoting the magnitude of the load applied at the middle of the beam. The maximum strain energy of the beam can be found as:
RL
39
41
where L is the length of the beam. For common boundary conditions such as fixed end
fr 0, f0r 0, pinned end
fr 0; f00r 0 and free end
f00r 0;
EI
xf00r
x0 0, eqn [41] reduces to:
37
where L is the length of the string, w
x; t is the f
x sin ot and rth mode is assumed.
40
where E is Young's modulus, I
x is the area moment of inertia of cross-section, r is the density, A
X is the area of cross-section and w
x; t is the transverse displacement. When harmonic motion is assumed, w
x; t f
x sin ot, eqn [40] leads to (for rth mode):
n oL 2 0 EI
x f00r
x dx EI
xf00r
x fr
x ÿ EI
xf00
xf0r
x 0 RL 2 rA
x f
x dx r 0
transverse displacement. The Rayleigh's quotient corresponding to eqn [36] is given by: o2r
@2 @ 2 w
x; t @ 2 w
x; t EI
x ÿrA
x @x2 @x2 @t2
Umax
!2 ZL=2 EI d2 fr 2 dx dx2 2 0
ZL=2 EI 0
h c i2 c2 L3
L ÿ 4x dx 384EI 8EI
b
The maximum kinetic energy of the beam and the central mass M0 can be found as:
1316 THEORY OF VIBRATION/Energy Methods
Figure 4 A fixed±fixed beam with a central mass.
m Z M0 2
fr 2max o2n ffr on g2 dx 2 2 L=2
Tmax
( 2 @2w @2w D 2 @x2 @y A c " 2 2 #) @2w @2w @ w ÿ 2
1 ÿ n ÿ dx dy @x2 @y2 @x @y
1 U 2
0
ZL=2 h
i2 c ÿ 3Lx2 ÿ 4x3 dx 48EI 0 M0 2 2 L on fr x 2 2 c 2 13 M o2 c 2 L2 0 n L7 mo2n 2 48EI 1120 48EI 16 mo2n
c
ZZ
1 T 2
L3
192EI ÿ13 35 m0 M0
A
d
the Rayleigh's quotient can be expressed as:
N
ZZ
( D
2 fr;xx fr;yy
A 2
@ w @ w @ w m@ w ÿ2 2 2 4 0 4 @x @x @y @y D @t2
Eh3 12
1 ÿ n2
b
with E is Young's modulus, n is Poisson's ratio, h is the thickness of the plate and m is the mass of the plate per unit area. The strain and kinetic energies of the plate are given by:
h
ÿ 2
1 ÿ n f2r;xy ÿ fr;xx fr;yy
a
where w w
x; y; t is the transverse deflection, D is the flexural rigidity given by: D
f
and D are given by: where N
The equation governing the vibration of a thin rectangular plate is given by: 4
e
N R o2r D
Transverse Vibration of a Thin Plate
4
d
w
x; y; t fr
x sin or t
where m0 denotes the mass of the beam.
4
2 @w m dx dy @t
where A is the area of the plate. By assuming harmonic motion in rth mode as:
Using the relation Tmax Umax , we obtain: o2n
ZZ
D
ZZ
) i
g dx dy
m f2r dx dy
h
A
where fr;xx
@ 2 fr ; @x2
fr;yy
@ 2 fr ; @y2
fr;xy
@ 2 fr @x @y
A simpler expression can be derived for the approximate natural frequency of transverse vibration of
THEORY OF VIBRATION/Energy Methods 1317
plates. By considering the plate to be composed of several lumped masses with associated effective spring constants, the natural frequency of vibration in rth mode can be expressed as: o2r
RR g A a
x; yfr
x; y dx dy RR 2 A a
x; yfr
x; y dx dy
i
where a
x; y is the weight of the plate per unit area, g is the gravitational acceleration and fr
x; y is the deflection (mode) shape of the plate during vibration. Eqn [i] is known as the modified Rayleigh's quotient or Morley's formula.
o2r
( Ra Ra ) g 0 0 sin
px=a sin
py=a dx dy R R c 0a 0a sin 2
px=a sin 2
py=a dx dy
16g 2 cp
and hence the frequency of vibration is given by: 19:722 or a2
fr
x; y c sin
px py sin a a
a
r D m
19:74 o1 a2
r D m
c 0:00416
p0 a4 mga4 0:00416 D D
b
with p0 mg is the distributed load (self weight). Substitution of eqn [a] into eqn [i] of the section on Transverse vibration of a thin plate yields:
e
Minimum Value of Rayleigh's Quotient
The Rayleigh's quotient gives an approximate value of the fundamental natural frequency that is higher than the exact value. To show this, let an arbitrary
x, be given by a linear combinaeigenfunction, f tion of the exact eigenfunctions, fi
x, as:
x a1 f
x a2 f
x f 1 2
where:
d
The exact natural frequency of the plate in the fundamental mode is given by:
Application ± Transverse Vibration of a Square Plate
Let the dimensions of the plate be a a (Figure 5) and the deflection shape be assumed to be the static deflection shape of the plate under uniformly distributed load as:
c
1 X i1
ai fi
x 43
where a1 , a2 . . . are constants. For specificness, consider the Rayleigh's quotient corresponding to the transverse vibration of a thin beam, eqn [42], using eqn [43] for fr
x: ÿ 00 2
x dx EI
x f ÿ 2 dx 0 rA
x f
x ÿP1 2 RL 00 dx i1 ai f
x 0 EI
x RL ÿ P1 2
x dx rA
x ai f RL
R R0L
44
i1
0
The orthogonality of eigenfunctions leads to the following relation (from eqn [42]): P1 2 i1 ki ai R P1 2 i1 mi ai
45
where ki and mi denote modal stiffness and modal mass, respectively, in mode i: Figure 5 A simply supported square plate. SS = simple support.
Z ki
0
L
ÿ 2 EI
x f00i
x dx
46
1318 THEORY OF VIBRATION/Energy Methods
ZL mi
rA
x
fi
x2 dx
Pn ai fi
x N i1 R Pn D i1 ai fi
x
47
53
0
Considering R as a function of the unknown constants a1 ; a2 . . . an , the conditions for the stationariness of R can be stated as:
with: a2i ki a2i o2i mi
48
@R 0; @ai
Thus eqn [45] can be rewritten as: P1 2 2 i1 ai oi mi R P 1 2 i1 ai mi
or: 49
since o1 < o2 <. . ., eqn [49] implies that: R
o21 a21 m1 o21
o2 =o1 2 a22 m2 a21 m1 a22 m2
o2 a2 m1 o21 a22 m2 o21 1 12 a1 m1 a22 m2
ÿ ÿ D @ N @ai ÿ N @ D @ai 0; 2 D
i 1; 2; . . . ; n 54
which can be rearranged as: 50
Eqn [50] indicates that the Rayleigh's quotient attains a minimum at the fundamental frequency and any assumed mode shape (eigenfunction) yields an approximate natural frequency that is higher than the exact fundamental frequency.
Rayleigh±Ritz Method The Rayleigh quotient is defined as:
f N R D
f
i 1; 2; . . . ; n
51
where f is the eigenfunction. A trial (or approximate) solution f
x is chosen for f
x as:
ÿ @ai @N N R ÿ ; @ D @ai D
i 1; 2; . . . ; n
55
Eqn [55] yields a set of n equations of the type: @N @D li ; @ai @ai
i 1; 2; . . . ; n
56
since the Rayleigh's quotient gives the (approximate) value of the eigenvalue, li . Eqn [56] represents an algebraic eigenvalue problem which can be solved using any of the standard techniques. The correspondence between eqn [56] and the algebraic eigenvalue problem can be shown by expressing the numerator and denominator of the Rayleigh's quotient of eqn [53], more generally, as: h i Z h i N f
x f
xL f
x dV
57
V
x f
n X
ai fi
x
52
i1
where ai are constants, fi
x are trial functions satisfying only the essential boundary conditions and n is the number of terms considered in f
x. The functions fi
x are called admissible functions and may not satisfy the natural or free boundary conditions. Basically, by assuming the solution as indicated by eqn [52], the continuous (or infinitely many-degreesof-freedom) system is approximated by an n-degreesof-freedom system. When eqn [52] is substituted into eqn [51], one obtains:
h i Z h i D f
x f
x M f
x dV
58
V
where V is the volume of the body and L[ ] and M[ ] are the differential operators used in the definition of the eigenvalue problem of the system: Lw lMw
59
For example, the eigenvalue problem corresponding to the longitudinal vibration of a rod is given by:
THEORY OF VIBRATION/Energy Methods 1319
d dw
x EA
x lm
xw
x dx dx
60
k ÿ l ma 0
68
8 9 a1 > > > > > > > > a > 2> > = < > a > > > > > > > > > > > ; : > an
69
where:
where m
x is the mass per unit length of the rod. In this case, the expressions of L[ ] and M[ ] can be identified (from eqn [60]) as: d dw EA
x 61 Lw dx dx Mw rA
xw
x
62
When n terms are used in the assumed solution, eqn [52], eqns [61] and [62] can be used to define the stiffness and mass coefficients as follows: Z kij
fi L fj dV;
i; j 1; 2; . . . ; n
63
fi M fj dV;
i; j 1; 2; . . . ; n
64
V
Z mij
k kij and m mij . Equation [67] or [68] defines an algebraic eigenvalue problem. Application ± Longitudinal Vibration of a Tapered Rod
The natural frequencies of longitudinal vibration of the tapered rod shown in Figure 3 is considered using a three-term solution as: f
x
ai fi
x
a
f1
x sin
px 2L
b
f2
x sin
3px 2L
c
f3
x sin
5px 2L
d
i1
V
Thus the numerator and denominator of eqn [53] can be expressed as: N
"
Z X n
ai f i L
i1
V
n X n X
n X n X
n X
Z
aj fj dV
fi L fj dV
65
V
ai aj kij
i1 j1
D
Z X n
" ai f i M
i1
V
n X n Z X i1 j1
n X n X
with:
#
j1
ai aj
i1 j1
n X
where each of the functions fi
x satisfies the fixed boundary condition of the problem. Since the area of cross-section of the rod varies as:
# aj fj dV
j1
fi M fj dV
66
V
x A
x A0 1 ÿ L
ai aj mij
(
Using eqns [65] and [66], eqn [56] can be written, in equivalent form, as:
kij ÿ lmij aj 0;
e
the operators L and M can be expressed, from eqns [61] and [62], as:
i1 j1
n ÿ X
3 X
i 1; 2; . . . ; n
67
L EA0
) 1 d x d2 ÿ 1ÿ L dx L dx2
x M rA0 1 ÿ L
f
g
j1
In matrix form, eqn [67] becomes:
Substituting eqns [a] to [g] into eqns [63] and [64], we obtain the matrices k and m as:
1320 THEORY OF VIBRATION/Energy Methods
2
k11 6 k 4 k21
k12 k22
3 k13 7 k23 5
k31
k32
k33
2
p2 1 6 4 6 6 EA0 6 6 3 4L 6 6 6 4 5 9
3
9p2 1 4 15
5 9
3
7 7 7 7 7 15 7 7 2 7 25p 5 1 4 h
2
m11 6 m 4 m21
m12 m22
3 m13 7 m23 5
m31
m32
m33
2 6 6 rA0 L 6 6 6 4 6 6 4
4 1ÿ 2 p 4 p2 4 ÿ 2 9p
Principle of Minimum Total Potential Energy
4 p2
3 4 7 9p2 7 7 4 7 7 7 p2 7 5 4 1ÿ 25p2
When s is expressed as s
e and e as Bu where B is an appropriate differential operator matrix, then a strain energy function, U, exists such that:
i
and the virtual work principle will be equivalent to the minimum total potential energy principle. The total potential energy, (pp ), is given by:
j
pp U
e W
u
ÿ
4 1ÿ 2 9p 4 p2
The solution of the eigenvalue problem: ka l ma
k
l2 o22 30:4878
E rL2
l
l3 o23 75:0751
E rL2
m
Virtual Work Principle
The virtual work principle can be stated as:
V
d eT s dV ÿ
V
dV ÿ duT B
St
duT t dS 0 70
T
Z
u B dV ÿ
Wÿ V
Energy Methods in Finite Element Analysis
Z
d eT s dV
71
V
Z
E l1 o21 5:7837 2 rL
Z
Z dU
72
with:
gives the eigenvalues of the rod as:
Z
is the vector of where s is the vector of stresses, B body forces, t is the vector of prescribed forces or tractions on the surface St of the body, V is the volume of the body, de is the vector of virtual strains, and du is the vector of virtual displacements that are compatible and consistent with the specified boundary conditions. The virtual work principle can be described as follows. Let an elastic body subject to and the forces B t be in equilibrium. Let the body be given virtual displacements du, corresponding to the and forces B t, that are geometrically compatible with the constraints. If de is the virtual strain vector due to the virtual displacements, then the stress vector must satisfy eqn [70]. Here the forces B, t and s are statically compatible and du and de are geometrically compatible.
uT t dS
73
St
and hence eqn [70] is equivalent to dpp 0. In words, the principle of minimum total potential energy can be stated as follows. Of all the admissible displacement fields that are consistent with the prescribed boundary conditions, the one that minimizes the potential energy also satisfies the conditions of equilibrium. Thus, by minimizing the potential energy, we can satisfy the equilibrium equations of the body. Principle of Minimum Total Complementary Energy
Let ds be the virtual stress vector satisfying equilibrium with both the body forces and tractions on St set to zero. By equating the sum of internal and external virtual work to zero, we have:
THEORY OF VIBRATION/Energy Methods 1321
Z
dsT e dV ÿ
V
Z
Z
dS 0 dtT u
Su
Z
denotes the vector of prescribed displacewhere u ments on the boundary Su and dt [E] ds is the vector of virtual tractions with [E] defining the boundary tractions in terms of stresses. Eqn [74] assures that the boundary displacements on Su and strains are compatible. Then a complementary strain energy function, Uc , exists such that: Z dUc
ÿ Z
ÿ
lT1
de ÿ Bdu dV
80 Z
dS ÿ dlT2
u ÿ u
Su
lT2 du dS 0
Su
Using the relation:
75
V
V
V
Z
d sT e dV
dlT1
e ÿ Bu dV
dphw dpp ÿ
74
lT1 Bdu dV ÿ
V
Z
duT BT l1 dV
Z
duT El1 dS
s
V
81
By defining the total complementary energy, pc , as: Z pc Uc
s ÿ
dS Uc Wc tT u
76
Su
Eqn [74] is equivalent to dpc 0, which is called the principle of minimum total complementary energy. In words, the principle of minimum total complementary energy can be stated as follows. Of all the admissible stress states that satisfy the equations of equilibrium with the given applied loads, the one that minimizes the total complementary energy satisfies the prescribed boundary conditions of displacement.
where [E] is an operator that gives boundary tractions in terms of stresses and using the variation of pp given in eqn [72], eqn [80] can be rewritten as: Z V
77
0 on Su uÿu
78
in defining the potential energy pp in eqn [72]. By relaxing these constraints, we can define a new function, phw , as: phw pp ÿ V
Z ÿ Bu dV ÿ
lT2
u
dS ÿu
Su
duT
El1 ÿ t dS Z
T
Z
dlT1
e ÿ Bu dV
V
Su
dS 0 dlT2
u ÿ u
Su
82 As eqn [82] is true for any variation, we find that the various terms yield the relations: l1 s;
0; BT l1 B
El1 l2 ;
e Bu ;
El1 t uu
83
Noting that the Lagrange multipliers are given by t, the variational principle can be l1 s, and l2 stated as: dphw 0
84
with:
79 where l1 and l2 are Lagrange multipliers defined in V and Su , respectively. By varying each quantity in eqn [79] independently, we can write:
ÿ dV du BT l1 B
du
El1 ÿ l2 dS ÿ
and the displacements at the boundary are assumed to satisfy the prescribed values as:
lT1
e
Sl
Z
The strains are assumed to be related to the displacements as: e Bu in V
Z V
Z
ÿ
Hu±Washizu Stationary Principle
Z
deT
s
e ÿ l1 dV ÿ
Z phw pp
e; u ÿ V
sT
e ÿ Bu dV ÿ
Z
dS tT
u ÿ u
Su
85
1322 THEORY OF VIBRATION/Energy Methods
This is known as the Hu±Washizu stationary principle. Finite Element Equations
The finite element equations can be derived using any of the energy principles. For example, the use of the principle of minimum total potential energy results in the following finite element equations: KQ P
86
where K is the stiffness matrix, Q is the nodal displacement vector and P is the nodal load vector of the body or assemblage of finite elements given by: n X
K
K
e
ÿ _ k drk 0 ; dWI Fk ÿ B
93 where dWI is the virtual work done on mass particle k, drk is a kinematically admissible virtual displacement (with time held fixed), Fk is a force acting on _ k is the D'Alembert's inertial force (B is mass k, ÿB the linear momentum), and N is the number of mass particles. Hamilton's Principle
The integral of D'Alembert's virtual work expression over an arbitrary period of time is: Zt2
87
K
Z V
P
n X e1
BT DB dV
88
e
e
e
e
Pi Pt Pb
Pc
ÿ _ k drk dt 0 Fk ÿ B
94
t1
e1
e
k 1; 2; . . . ; N
When the virtual displacement vanishes at times t1 and t2 and the applied forces possess a potential energy function V, eqn [94] reduces to: Zt2
89
d
T ÿ V dt dpH 0
95
t1
Here K
e is the element stiffness matrix of element e, n is the total number of finite elements in the body, D
e
e
e is the elasticity matrix, Pi , Pt , Pb are the load vectors of element e due to initial strains, tractions, body forces, respectively, Pc is the vector of concentrated loads at the nodes of the body, and N is the matrix of shape functions used for the variation of displacement within the element e. The element load vectors are given by: Z
e
Pi
BT De0 dV
90
NT t dS
91
dV NT B
92
V
e
e
Z
Pt
e St
e
Z
Pb V
e
where: Zt2 pH
96
L T ÿ V
97
and T is the complementary kinetic energy. In Newtonian mechanics, T will be same as T, the kinetic energy of the system. Eqn [95] is known as Hamilton's principle and states that `among all kinematically possible motions in the interval t1 to t2 , the actual one is characterized by the stationary condition of the functional pH '. The stationary conditions of pH yield the equations of motion of the system. Complementary Hamilton's Principle
The complementary form of Hamilton's principle can be stated as: Zt2
Variational Formulations in Dynamics The variational formulations are useful to describe motion of a system over an arbitrary period of time. D'Alembert's principle states that:
L dt t1
t1
where:
d
T ÿ V dt dpT 0
98
THEORY OF VIBRATION/Energy Methods 1323
Zt2 pT
L dt
Zt2 D
99
t1
100
and T is the kinetic energy and V is the complementary potential energy of the system. In words, the complementary form of Hamilton's principle can be stated as `amongst all possible equilibrating motions in the interval t1 to t2 , the actual one is characterized by the stationary condition of the functional pT '. The stationary conditions of pT yield the conditions of compatibility.
In words, the principle of least action states that `amongst all possible motions of conservative systems between any two prescribed configurations, the actual motion will be such as to render the functional in eqn [104] an extremum'. Using the principle of conservation of energy, T V E constant, the principle of least action can also be expressed as: Zt2 105
t1
where:
The Lagrange's equations can be stated as: i 1; 2; . . . ; N
i 1; 2; . . . ; N
102
Complementary Lagrange's Equations
The complementary form of Lagrange's equations is given by: i 1; 2; . . . ; N
L T ÿ V
101
where L T ÿ V is the Lagrangian, qi is the generalized displacement and q_ i is the generalized velocity. In addition to the forces that possess a potential, where generalized forces Qi (that are not derivable from a potential function) act on the system, then the Lagrange's equations are given by:
d @L @L Si ; ÿ dt @ s_i @si
L E dt 0
D
Lagrange's Equations
d @L @L Qi ; ÿ dt @ q_ i @qi
104
t1
L T ÿ V
d @L @L 0; ÿ dt @ q_ i @qi
T T dt 0
103
is the complementary where L T ÿ V Lagrangian, si is the generalized impulse, and Si is the generalized velocity. Principle of Least Action
Although the interval of time t1 to t2 is arbitrarily in Hamilton's principle, the limits t1 and t2 are fixed. Thus time is held fixed during the variation of the coordinates qi . The principle of least action can be used when t1 and t2 are not fixed:
Nomenclature A B D D E EI g G GJ(x) h I J(x) L Mo N P Q S t T T(x) u u
x; t U
t V w(x,t) Xi X n l
cross-sectional area vector of body forces domain; flexural rigidity elasticity matrix Young's modulus bending stiffness gravitational acceleration shear modulus torsional stiffness thickness inertia polar moment of inertia length mass of the beam matrix of shape functions nodal load vector nodal displacement vector surface vector of tractions kinetic energy tension in the string vector of displacement longitudinal displacement potential (strain) energy volume transverse displacement amplitude arbitrary vector Poisson's ratio eigenvalue
106
1324 THEORY OF VIBRATION/Equations of Motion
f r s
eigenfunction mass of rod; density vector of stresses
See also: Discrete elements; Eigenvalue analysis; Finite difference methods; Finite element methods; Theory of vibration, Variational methods.
Further Reading James ML, Smith GM, Wolford JC, Whaley PW (1989) Vibration of Mechanical and Structural Systems. New York: Harper & Row. Langhaar HL (1962) Energy Methods in Applied Mechanics. New York: John Wiley. Meirovitch L (1967) Analytical Methods in Vibrations. New York: Macmillan. Rao SS (1995) Mechanical Vibrations, 3rd edn. Reading, MA: Addison-Wesley. Rao SS (1999) The Finite Element Method in Engineering. 3rd edn. Boston: Butterworth-Heinemann. Szilard R (1974) Theory and Analysis of Plates: Classical and Numerical Methods. Englewood Cliffs, NJ: PrenticeHall. Tabarrok B, Rimrott FPJ (1994) Variational Methods and Complementary Formulations in Dynamics. Dordrecht, The Netherlands: Kluwer Academic Publishers. Tedesco JW, McDougal WG, Ross CA (1999) Structural Dynamics: Theory and Applications. Menlo Park, CA: Addison-Wesley. Zienkiewicz OC (1977) The Finite Element Method, 3rd edn. London: McGraw-Hill.
Equations of Motion J Wickert, Carnegie Mellon University, Pittsburgh, PA, USA Copyright # 2001 Academic Press doi:10.1006/rwvb.2001.0116
Introduction Early procedural steps taken in the mathematical analysis of a mechanical vibration problem include: (1) modeling, in which the essentials of a physical system are extracted to produce an idealized model amenable to solution; (2) choosing appropriate coordinates which uniquely describe the system's configuration; and (3) obtaining one or more equations which describe vibration of the modeled system.
The resulting equation(s) of motion (EOM) are ordinary or partial differential equations that govern the manner in which displacements of the system evolve with time. They are used extensively in subsequent analysis and computation in order to predict transient response, steady-state forced vibration, natural frequencies, and mode shapes. EOM are derived through the application of physical laws and such methods of dynamics as Newton's laws of motion, Lagrange's equation, and Hamilton's principle. For discrete systems, the EOM take the form of ordinary differential equations, while partial differential equations in one or more independent spatial coordinates govern continuous or distributed-parameter models. The mass, stiffness, and damping properties of the model can be constant or time-dependent, and they appear in the EOM as parameters or coefficients of the coordinates and their derivatives. The external forces or torques which are applied to the system in order to excite vibration render the EOM inhomogeneous. This article presents a brief exposition of EOM, their derivation, mathematical structure, and properties for the illustrative cases of discrete systems having one or more degrees-of-freedom and prototypical models of continuous systems.
Single-degree-of-freedom Models In the simplest case, a mechanical system can be modeled using methods of lumped parameter analysis whereby motion is described fully by only one timedependent coordinate. Such a system is termed a 1-degree-of-freedom (DOF) vibration model. Although the transient or steady-state vibration of the system could, in fact, be rather complicated, only one translational or rotational coordinate is presumed necessary to specify within acceptable bounds of accuracy the configuration of the system at any instant. When more than one coordinate becomes necessary, the discrete system is said to have multiple degrees-of-freedom. Figure 1A illustrates the prototypical vibration model having lumped mass m, stiffness k, and damping c elements, in addition to the concentrated external force F
t. The parameters m, c, and k are generally interpreted as being effective values that represent combinations of other, multiple, interconnected components. An example in that regard is several springs that are connected in series and/or parallel connections so as to form a single element of equivalent stiffness k. In order to obtain the equation of motion, the free-body diagram of the inertia element is drawn as indicated in Figure 1B, and Newton's second law of motion, f ma, for a particle is applied, wherein f
THEORY OF VIBRATION/Equations of Motion 1325
in terms of the parameters: r k on m
Figure 1 (A) Single-degree-of-freedom vibration model for a particle in translation. (B) Free-body diagram of the inertia element.
represents the resultant force vector, and a is the particle's absolute acceleration. With displacement measured by the coordinate x
t, the particle's velo_ and city and acceleration are denoted v dx=dt x, a d2 x=dt2 x, respectively. The dot-superscript notation is used conventionally in vibration engineering to denote derivatives taken with respect to time. With the positive sign convention directed rightward in Figure 1, application of the second law provides 7kx ÿ cx_ F m x. In its standard form, the EOM is written with all terms involving displacement, velocity, and acceleration on one side, with one or more forcing terms grouped on the other. The final form of the equation of motion becomes: m x cx_ kx F
t
and
c z p 2 mk
which are the undamped, circular, natural frequency (in units radians per second) and the dimensionless damping ratio. An analogous procedure is applied to rotational systems, the only distinction being that the principle of angular momentum balance is invoked instead of the second law. The precise form taken by the balance law differs depending on whether it is resolved about a point O fixed in space, the mass center G of a rigid body, or an arbitrary point P; either method, when properly applied, will result in a valid equation of motion. When rotational vibration y
t occurs about a fixed point, for instance, M0 J0 y where M0 is the resultant moment, J0 is the mass moment of inertia, and y denotes the angular acceleration. For illustration, in Figure 2A, the rigid-body pendulum has centroid located distance L from the support, and oscillation occurs under the restoring influence of gravity g alone. Reactions Ox and Oy produce no moment about the support, and the equation of motion becomes JO y mgL sin y 0. Due to the explicit presence of the transcendental function, the equation of motion as derived is nonlinear and therefore is valid for arbitrarily large angles of rotation, including multiple clockwise or counterclockwise circulations of the pendulum about O. For the more common circumstance of `small' amplitude motion, the equation of motion is linearized about its stable, trivial, equilibrium point by using the Taylor series expansion sin y ~ y 7 y3/6 + . . ., and subsequently retaining only the first-order term. The desired linear EOM then becomes J0 y mgLy 0 in the standard form of eqn [1]. The rigid body undergoes free vibration
1
which is a second-order, linear, inhomogeneous, ordinary differential equation having constant coefficients. The solution is found subject to the initial conditions on displacement x
0 x0 and velocity _ x
0 v0 as evaluated at t 0. By convention, the EOM is also written: x 2zon x_ o2n x
F
t m
2
3
Figure 2 Rotational oscillation of a rigid-body pendulum.
1326 THEORY OF VIBRATION/Equations of Motion
since its equation of motion is homogeneous. In a straightforward application of some practical utility, with both the mass m and offset distance L known, a measurement of the natural frequency can be used to determine the rigid body's mass moment of inertia.
Two or More Degrees-of-freedom Models For a discrete system that requires more than one coordinate to describe fully its configuration, a system of ordinary differential equations, one written for each coordinate or inertia element, arises. With N such coordinates, the system's overall mass, damping, and stiffness characteristics are embodied by matrices of dimension N N. Likewise, the translational and/ or rotational displacements of individual inertia elements are collected to form an N-dimensional vector, as is also the case for any externally imposed forces or moments that may be present. In the most general case, the system's matrices will be full, with off-diagonal elements describing the degree of coupling, if any, that exists between the various coordinates. When the mass matrix is not diagonal, for instance, the system is said to be inertially coupled; analogous terminology is applied should off-diagonal elements be present in the damping or stiffness matrices. Such coupling and interaction among the multiple equations of motion is precisely what complicates vibration analysis relative to that seen for a 1-DOF model. By proper choice of coordinates, however, coupling can be reduced, transformed, and even eliminated altogether when modal coordinates are invoked. The latter opportunity motivates the study of normal modes, and provides the emphasis given to them throughout vibration engineering. In the lumped parameter system of Figure 3A, two masses vibrate with responses x1
t and x2
t under action of the impressed forces F1
t and F2
t and the indicated stiffness and damping elements. To establish the equations of motion, the second law is applied to m1 and m2 individually using the free-body diagrams of Figure 3B, providing:
Figure 3 (A) Two-degrees-of-freedom vibration model. (B) Free-body diagrams of the inertia elements.
m1
0
0 m2 k1 k2 ÿk2
x1 c1 c2 ÿc2 x_ 1 x2 x_ 2 ÿc2 c2 ÿk2 x1 F1
t x2 F2
t k2
6
where coupling is present in both damping and stiffness. More generally, for an N-degrees-of-freedom system subjected to viscous damping forces, the equations of motion will take the form: M x Cx_ Kx F
t
7
where M, C, and K are the mass, stiffness, and damping matrices, and x
t and F
t are the state and excitation vectors. Solutions are found subject to _ v0 . the initial conditions x
0 x0 and x
0 The precise nature of coupling that arises in the system matrices changes depending on the coordinates that are chosen to describe vibration. In Figure 4, a rigid bar is supported by springs having differing stiffness k1 and k2 , and the centroid G is offset from the bar's geometric center. In the first choice of coordinates, translation x of the mass center and rotation y are used to describe small-amplitude motion, and the equations of motion are:
m1 x1
c1 c2 x_ 1 ÿ c2 x_ 2
k1 k2 x1 ÿ k2 x2 F1
t
4
m2 x2 ÿ c2 x_ 1 c2 x_ 2 ÿ k2 x1 k2 x2 F2
t
5
The EOM are then conveniently written in the matrix vector form:
m 0 x y 0 JG k1 k2 k2 L2 ÿ k1 L1
8 x 0 2 2 k1 L1 k2 L2 y 0
k2 L2 ÿ k1 L1
where m and JG are the bar's mass and mass moment of inertia about G, respectively. There is no inertial coupling, and the EOM couple statically in stiffness
THEORY OF VIBRATION/Equations of Motion 1327
Figure 4 Rigid body with motion described alternatively by
x; y or by
x1 ; x2 .
through the off-diagonal elements of K. With the special parameter values satisfying k1 L1 ÿ k2 L2 0, translational and pitch motions as described by x and y fully decouple; vibration in that case can be analyzed as if two uncoupled single DOF systems were being treated in parallel. Alternatively, when displacements x1 and x2 of the bar's two ends are chosen as coordinates, the equations of motion take the form:
1
"
mL22 JG
mL1 L2 ÿ JG
mL21 JG
L1 L2 2 mL1 L2 ÿ JG x1 0 k1 0 x2 0 0 k2
#
x1 x2
[9] and the system suffers only inertial coupling. Modal analysis, in short, involves a coordinate transformation which simultaneously decouples both system matrices. In physical modeling and deriving equations of motion, numerical values for the mass and stiffness parameters can typically be obtained through straightforward analyses or experiments, and they are therefore generally known to within an acceptable level of accuracy. Further, within the context of a conservative mechanical system vibrating about a position of stable equilibrium, substantial theoretical justification exists for the mass and stiffness matrices being symmetric and positive definite, a result which underlies the calculation of normal modes, natural frequencies, and orthogonality among the modes. However, to the degree that a wide variety of energy dissipation mechanisms exists, including internal material damping, friction, sound radiation, and vibration transmission at joints and connections to other structures, damping models, and numerical values for viscous damping coefficients in particular,
are less well-developed. Beyond issues of mathematical convenience and certain restricted cases of laminar fluid flow through a slot or an orifice, there is somewhat less justification for the introduction of the viscous damping matrix C in an equation of motion. In certain practical applications, while damping in its various embodiments is recognized as being an important attribute of the system at hand, it will generally not be easily modeled in terms of damping coefficients. In such circumstances and when the effects of damping are judged to be `light', damping is conventionally ignored insofar as deriving the equations of motion is concerned, so that the EOM are written in terms of M and K only. Vibration analysis proceeds with the calculation of natural frequencies and normal modes, on to decoupling of the EOM through modal analysis, at which point dissipation is incorporated within the context of modal damping. In short, an effective damping ratio for each mode, rather than for each physical coordinate as was present in the original model, is introduced with a numerical value determined through appropriate measurements. The EOM for multiple-degrees-of-freedom systems can also be derived by using the Lagrangian approach of analytical dyamics, which is viewed often as a preferred technique when complicating factors of geometry, kinematics, or modeling are present. Written in terms of the scalar kinetic T and potential U energy functions, Lagrange's equation for each DOF k 1; 2; . . . N is: d @L @L Qk
t ÿ dt @ q_ k @qk
10
in terms of the Lagrangian function L T ÿ U, the generalized coordinates, qk
t, and the generalized forces Qk
t.
Continuous System Models Although discrete models of mechanical systems are often readily established and can be useful intuitive and predictive tools, in some applications and particularly at moderate or higher frequencies, they are inherently limited by the level of approximation made by lumping mass, stiffness, and damping properties into a relatively small number of elements. To the extent that in all real systems, physical properties are not concentrated at discrete points, the class of continuous or distributed-parameter vibration models provides more accurate structural representations. The higher level of model fidelity, however, is realized
1328 THEORY OF VIBRATION/Equations of Motion
at the expense of EOM having multiple independent variables, both spatial and temporal, and which are commensurately more complicated than their discrete counterparts. The EOM of continuous systems are partial differential equations that govern motion within an elastic body's interior, and they are solved subject to initial conditions in time as well as boundary conditions in space. A seemingly direct, but primitive, method for deriving the EOM of a continuous system involves approximating it by one that contains a finite number of discrete particles, and then examining the form taken by the EOM as the number of particles becomes large, namely, as the continuous limit is approached. For instance, should more discrete mass, stiffness, and damping elements be added to the configuration of Figure 3, the serial chain would behave in a manner representative of a rod undergoing longitudinal vibration. For that reason, a continuous system can be viewed conceptually as one having infinite degreesof-freedom, whereby the set of N functions which describe the discrete system's vibration is replaced by a single continuous function as evaluated over a continuum of positions and instants. While indeed pedagogically useful, such a viewpoint suffers from lack of generality, and so it is not used commonly in practice. The more systematic technique by which to obtain the EOM and boundary conditions is to isolate an infinitesimal element from the body, apply an appropriate constitutive law so as to relate restoring forces or moments to translational or rotational displacements, and invoke Newton's second law or the angular momentum balance principle to a free-body diagram of the element. Specifically, the approach for continuous system vibration models comprises the following major steps: . Choose coordinates to describe motion of the system and, in particular, an infinitesimal element isolated from it. . Determine the restoring force/moment-displacement, or stress-strain, relation using an appropriate model applied from strength of materials or elasticity theories. . Apply force or moment balance principles over the interior of the body to obtain the equation of motion. . Apply force or moment balance principles over the structure's periphery to obtain the boundary conditions. In a manner analogous to the utility of Lagrange's equation in analyzing discrete systems, more sophisticated approaches are available for deriving the EOM of continuous systems, including treatments
of three-dimensional vibration, continuum mechanics, and elastic-wave propagation. For structures having coupling between motions, complicated kinematics, or certain constitutive properties, application of the extended Hamilton's principle: Zt2
dT ÿ dU dWNC dt 0
11
t1
from the field of analytical dynamics can be a preferred starting point in deriving the EOM and boundary conditions. Here T and U are expressions for the kinetic and potential energies, d denotes variation, and dWNC is the nonconservative virtual work. Rod, String, and Shaft Vibration
A simple model of continuous system vibration, and one which enjoys significant practical application, describes longitudinal motion of a rod wherein material particles oscillate in a direction aligned with the rod's axis. The resulting deformation includes local tension and compression, which can be viewed within the context of propagation and superposition of tension and compression waves. Figure 5A illustrates the geometry in which the rod is fixed to a rigid support at its end x 0, free at the other end x L, and excited by the distributed force-per-unit-length f
x; t directed along the rod's axis. Motion is measured by u
x; t, a function of longitudinal position and time. In the strength of materials theory for uniaxial tension and compression, the rod deforms with axial stiffness EA
x, where E and A denote local values of the elastic modulus and cross-sectional area, and the product is not necessarily constant. With reference to the free-body diagram of Figure 5B, the internal elastic force within the rod at any point is EAu; x , where the comma-subscript notation is used
Figure 5 (A) Longitudinal vibration of an elastic rod. (B) Freebody diagram of an infinitesimal element isolated from the rod's interior.
THEORY OF VIBRATION/Equations of Motion 1329
conventionally to indicate a partial derivative, which in this case is taken with respect to x. Application of Newton's second law to the element provides:
rAdxu;tt ÿEAu;x EAu;x
EAu;x ;x dx f dx 12
where r is the material's volumetric density, and the internal force within the rod has already been expanded in a Taylor series about x 0. The equation of motion becomes: rAu;tt ÿ
EAu;x ;x f
x; t
13
which is solved subject to initial conditions on displacement u and velocity u; t at time t 0, as well as boundary conditions that are applied to u and/or u; x at x 0 and L. Various boundary conditions are possible and admissible depending on the manner in which the rod is supported. One condition must be applied at each end of the rod. In the presence of a built-in support at either end, for instance, the displacement at that point vanishes. Should an end of the rod be free, the tension or compression internal to the rod would likewise vanish at that point, providing the condition EAu; x 0. In cases involving interactions at the boundary between the rod and other components modeled as ideal discrete elements, the appropriate expression for the boundary condition is derived by constructing the free-body diagram of an isolated boundary element, as suggested by the examples given in Figure 6. When no external forces act on the rod, and the density, cross-sectional area, and elastic modulus do not vary along its length, the EOM assumes the form:
u;tt ÿ c2 u;xx 0
14
p of the standard-wave equation, where c E=r is the speed at which tension and compression waves propagate. The wave equation enjoys a rich history in the theory of vibration, and it describes a variety of physical phenomena beyond longitudinal rod vibration. With appropriate substitutions, the wave equation also describes the transverse vibration of a pretensioned flexible string, and the torsional vibration of an elastic shaft ± analogies which are outlined further in Figure 7. Beam Vibration
A beam is defined as a structural member that is long relative to its cross-sectional dimensions, and that executes small amplitude motion v in the direction transverse to its axis, as illustrated in Figure 8A. The most elementary and widely used treatment of beam deformation is attributed to Euler and Bernoulli, and it specifies that the beam's rotary inertia and deformation in shear are each negligible. Those assumptions are relaxed in the Timoshenko beam model, which offers higher-order accuracy at the expense of additional analytical complexity. In the free-body diagram of Figure 8B for an element that has been isolated from the beam, the internal shear V and bending moment M are expanded in Taylor series about x 0, and their gradients across the element dx produce imbalances in force or moment necessary to generate the requisite acceleration and vibration. With EI denoting the beam's flexural rigidity, subject to the standard restrictions implicit in the linear moment curvature M EIv; xx and shear V ÿ
EIv; xx ; x relationships invoked from the strength of materials theory, the equation of motion for transverse bending vibration of the beam becomes:
Figure 6 Examples of boundary conditions applied to the longitudinal vibration of an elastic rod, as evaluated on the right-hand end of the rod.
1330 THEORY OF VIBRATION/Equations of Motion
Figure 7 Analogies between the longitudinal vibration of a rod, transverse vibration of a string, and torsional vibration of a shaft, for cases in which the models' properties are lengthwise constant. In the string model, P denotes the pre-tension. In the shaft model, JG is the torsional rigidity, and I is the polar moment of inertia per unit-length of the shaft.
Figure 8 (A) Transverse bending vibration of a beam. (B) Free-body diagram of an infinitesimal interior element.
rAv;tt
EIv;xx ;xx f
x; t
15
where f
x; t is the transverse force-per-unit-length. The EOM is a linear partial differential equation and, because it is fourth-order, it is necessary to specify two conditions at each point on the beam's boundary. Specifically, either kinematic conditions on displacement or slope, or force conditions on shear or moment, can be imposed in accordance with specific restrictions on the combinations of conditions that are allowable. For a simply supported beam, the boundary conditions become v 0 and EIv; xx 0.
Conversely, the set of conditions v; x 0 and EIv; xx 0 at the same point cannot be physically realized and is therefore inadmissible. For a clampedfree beam, the kinematic conditions v v; x 0 are specified at x 0, with vanishing shear and moment at x L through EIv; xx
EIv; xx ; x 0. Those conditions and others are briefly outlined in Figure 9. Membrane and Plate Vibration
Membranes and plates are two-dimensional structures that vibrate in the direction transverse to their equilibrium plane. A membrane has negligible
THEORY OF VIBRATION/Equations of Motion 1331
Figure 9 Examples of boundary conditions for lateral bending vibration of a beam, as evaluated on its right-hand end.
bending rigidity and derives restoring stiffness from its initial pretension; a plate, on the other hand, develops curvature-related bending stress as it deforms. In that sense, the more elementary string and beam models are interpreted as being one-dimensional realizations of membranes and plates, respectively. For a membrane, the equation of motion in rectangular coordinates
x; y is: ÿ rhv;tt ÿ Nx v;xx Ny v;yy f
x; t
16
where h is the material's thickness, Nx and Ny are the initial tensions-per-unit-length resolved along the coordinate directions, and f
x; t denotes the transverse force applied per unit of the membrane's surface area. When the tension field is isotropic, Nx Ny N0 , the EOM takes on its standard form: rhv;tt ÿ N0 r2 v f
x; t
18
or polar coordinates
r; y: r2
@2 1 @ 1 @2 @r2 r @r r2 @y2
rhv;tt Dr4 v f
x; t
19
The most straightforward boundary conditions that can be considered are the constrained v 0 and free v; n 0 cases, the latter of which requires vanishing slope in the direction normal to the
20
in terms of the biharmonic operator r4 = (r2)2. The plate's stiffness parameter, analogous to the familiar flexural rigidity from the strength of materials theory for beam deformation, is defined:
17
in terms of the Laplacian operator in either Cartesian coordinates: @2 @2 r2 2 2 @x @y
membrane's boundary, namely, a directional derivative. Along a free edge of constant x, for instance, the condition becomes v; x 0. Other conditions which account for coupling between the membrane and discrete inertia, stiffness, or damping elements on the boundary are treated by application of appropriate balance laws to infinitesimal elements of the membrane isolated in a free-body diagram. In the case of a plate's bending vibration, the EOM is:
D
Eh3 12
1 ÿ v2
21
in terms of the plate's thickness h, elastic modulus E, and Poisson ratio v. Because the equation of motion is fourth-order, two conditions must be specified at each point on the plate's boundary. Those conditions are expressed in terms of requirements on the displacement, slope, bending moment, or equivalent shear force along the plate's edges.
Special-interest Topics Other topics which are related to the derivation, structure, and properties of the equations of motion for discrete and continuous systems, but which are
1332 THEORY OF VIBRATION/Substructuring
not discussed in this section for reasons of brevity, include: . Gyroscopic systems where small-amplitude vibration occurs in the presence of a steady superposed motion. Applications include the vibration of rotating disks and shafts, and such translating media as power transmission belts and sheet metal webs. . Nonconservative forces, including the followertype loading of the classical Beck or PfluÈger problem in the field of dynamic stability, which render the stiffness matrix nonsymmetric. . Parametric excitation which is present when a parameter in a vibration model, such as rotation speed or tension, is time-dependent and, in particular, periodic. . Nonlinear systems wherein motion is of sufficiently large amplitude that nonlinear terms must be retained in the equations of motion, complicating their analysis and solution. . Discretization in which a continuous system model is systematically reduced to an equivalent, potentially high-DOF, discrete model. Local or global techniques such as finite element, Galerkin, and Ritz comprise a powerful class of methods available for developing accurate approximate solutions to the EOM. . Linear operators for mass, damping, and stiffness effects which are a mathematical formalism facilitating the analysis of continuous system vibration, particularly with respect to modal analysis.
Nomenclature A E F(t) G T U x(t) d y(t) y r r
cross-sectional area elastic modulus excitation vector centroid kinetic energy potential energy state vector velocity variation rotational vibration angular acceleration density Laplacian operator
See also: Beams; Discrete elements; Linear damping matrix methods; Membranes; Modal analysis, experimental, Basic principles; Nonlinear systems, overview; Parametric excitation; Plates; Theory of vibration, Duhamel's Principle and convolution; Theory of vibration, Energy methods; Theory of vibration, Fundamentals; Theory of vibration, Impulse response function;
Theory of vibration, Substructuring; Theory of vibration, Superposition; Theory of vibration, Variational methods.
Further Reading Bishop RED and Johnson DC (1979) The Mechanics of Vibration. Cambridge: Cambridge University Press. Greenwodd DT (1988) Principles of Dynamics, 2nd edn. New Jersey: Prentice-Hall. Leissa A (1993) Vibration of Plates. New York: Acoustical Society of America. Pars LA (1979) A Treatise on Analytical Dynamics. Woodbridge, CN: Ox Bow Press. Thomson WT and Dahleh MD (1998) Theory of Vibration with Applications. New Jersey: Prentice-Hall. Timoshenko SP and Woinowsky-Krieger S (1970) Theory of Plates and Shells. Singapore: McGraw-Hill.
Substructuring M Sunar, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Copyright # 2001 Academic Press doi:10.1006/rwvb.2001.0117
Substructuring is usually carried out for a vibrating structure, which is large in order. The substructuring can result in substantial savings in computations. The goal in substructuring is to reduce the order of the structure without changing many vibrational characteristics of the remainder. The truncation can be done on those coordinates whose effects are relatively insignificant in vibrations of the structure. Furthermore, if various substructures are analyzed by different people, assembling them by the use of substructuring methods may be very beneficial to predict the complete structural behavior. Among many substructuring methods, three basic ones are considered here: the Rayleigh±Ritz and Guyan reduction methods, and component mode synthesis.
The Rayleigh±Ritz Method The Rayleigh±Ritz method is used to estimate eigenvalues and eigenvectors (eigenfunctions) of a discrete (continuous) vibrating structure on its reduced-order model. The difficulty with this method lies in the selection of Ritz basis vectors. The rate of convergence of the eigenproblem solution largely depends on the choice of these basis vectors. Guyan reduction
THEORY OF VIBRATION/Substructuring 1333
method and component mode synthesis discussed in the following sections are basically Rayleigh±Ritz methods with certain Ritz basis vectors (see Basic principles).
Guyan Reduction Method Static condensation method can be used to reduce the order of the vibrating structure for approximating its eigenvalues and eigenvectors. The structure is usually discretized using the finite element procedure, which may yield a mass matrix with some zero diagonal elements due to the usage of lumped mass matrix. The method is applied to the structure to reduce its order on the degrees-of-freedom (DOF) corresponding to these zero diagonal elements. Consider the eigenproblem for the whole (fullorder) structure: Kf lMf
1
Assume that some lumping is done on M and the problem is written in the partitioned form as: KAA KBA
KAB KBB
fA fB
l
0 0
0 MBB
fA fB
8
General Method
M x Kx f
3
9
Consider any lth substructure of the structure (Figure 1) whose EOM are written in the partitioned form as:
MAA MBA
MAB MBB
l
A x B x
l
KAA KBA
KAB KBB
l
xA xB
l
fA fB
l
10
where the subscript A denotes the DOF to be condensed (slave DOF) and B represents the DOF to be retained (master DOF). The above equation can be expressed as:
KAA ÿ MAA o2 KBA ÿ MBA o2
KAB ÿ MAB o2 KBB ÿ MBB o2
l
uA uB
f A fB l l 11
2
The following relation can be written from eqn [2]: KAA fA KAB fB 0
0 IBB
The finite element model of the structure yields undamped equations of motion (EOM) of the form:
Static Condensation
F
Eqn [11] is due to the fact that if a force in the form of: f
r X
fj eioj t
12
j1
and hence: fA ÿKÿ1 AA KAB fB
4
is applied, then the substructural response can be written as:
Substituting fA in eqn [2] yields: KB fB lMBB fB
5
KB KBB ÿ KBA Kÿ1 AA KAB
6
where:
It can be shown that the static condensation method is a Rayleigh±Ritz method with Ritz basis vectors given as: C Kÿ1 F where F is defined as:
7 Figure 1 A typical lth substructure.
1334 THEORY OF VIBRATION/Substructuring
u
r X
uj eioj t
Bl Kl xBl f l Ml x
13
j1
where:
The assumption of no applied force at the slave DOF
f A 0 yields:
Ml TTABl Ml TABl Kl TTABl Kl TABl
KAA ÿ MAA o2 l uAl KAB ÿ MAB o2 l uBl 0
f l
14 which can be used to condense the slave DOF as:
uAl ÿ KAA ÿ MAA o
2 ÿ1 l
KAB ÿ MAB o2 l uBl 15
uAl
B K xB f M x
2 2 I MAA Kÿ1 AA o l KAB ÿ MAB o l uBl
17
For the approximation in eqn [17] to be justified, coefficients of o2 must be small. In other words, entries of MAA and MAB must be much smaller than those of KAA and KAB . This means low mass and high stiffness for the areas of slave DOF. Eqn [17] can also be justified if o is taken to be nearly zero. This implies that the frequencies of the applied force are small compared to the natural frequencies of the structure. From eqn [17], slave nodal displacements are also condensed as: xAl ÿKÿ1 AAl KABl xBl
18
The nodal displacement vector is now written as:
xA TABl xBl xl xB l
TABl
ÿKÿ1 AA KAB IBB
l
l1 n X
f
n X l1
Ml ;
K
f l
n X l1
24 Kl
The summations in eqn [24] must be comformable. The EOM of the reduced-order structure given by eqn [23] are solved for xB : Thereafter, xAl for each substructure is recovered from eqn [18].
Component Mode Synthesis In the constraint mode method of the component mode synthesis, vibrational mode shapes and general nodal displacements at the boundary are included for the lth substructure. The vibrational mode shapes are normal modes of the substructure with totally constrained boundary DOF. On the other hand, the boundary displacements occur due to successive generalized unit displacements at boundary DOF, while all other boundary DOF are totally constrained. The component mode synthesis relates the internal displacements to the vibrational mode shapes and boundary displacements for the lth substructure shown in Figure 1. This relation is given by: xAl fl xMl ÿ Kÿ1 AAl KABl xBl
25
The generalized displacement vector xl for the lth substructure can be expressed as:
20
Substituting eqn [19] in eqn [10] and premultiplying the resulting equation by TTABl yield:
xBl ;
l1
19
The transformation matrix TABl in the above equation is given by:
n X
M
16
uAl ÿKÿ1 AAl KABl uBl
23
where: xB
which is further approximated by:
22
TTABl f l
The EOM for the whole structure can be written as:
Equation (15) can be approximated as: ÿKÿ1 AAl
21
xl where:
x xA Tl M Tl xl xB l xB l
26
THEORY OF VIBRATION/Impulse Response Function 1335
Tl
f 0
ÿKÿ1 AA KAB IBB
27
l
The generalized displacement vector xl in eqn [26] couples modal coordinates with boundary displacements. The original EOM for the lth substructure are assumed as: l Kl xl f l Ml x
28
The order of this equation is reduced by substituting eqn [27] and premultiplying the resulting equation by TTl . The reduced-order EOM are now expressed as: l Kl xl f l Ml x
xA xB xM l
number of substrictures in structure magnitude of jth component of structural response nodal vector of internal generalized displacements nodal vector of generalized displacements at boundary vector of modal coordinates eigenvector eigenvalue
See also: Basic principles; Theory of vibration, Energy methods; Theory of vibration, Equations of motion; Theory of vibration, Fundamentals; Theory of vibration, Variational methods.
29
where:
Further Reading Ml TTl Ml Tl Kl TTl Kl Tl f l Tl f l
30
The reduced-order EOM for the whole structure are given by: K x f M x
2
.. .
6 6f 6 lÿ1 6 0 6 F6 6 fl 6 0 6 6 .. 4 . fn
.. . 0
Ilÿ1;l 0 0 .. . 0
Bathe K±J (1982) Finite Element Procedures in Engineering Analysis. New Jersey, USA: Prentice-Hall. Craig RR (1981) Structural Dynamics. New York, USA: Wiley. Meirovitch L (1980) Computational Methods in Structural Dynamics. Maryland, USA: Sijthoff and Noordhoff. Weaver W Jr, Johnston PR (1987) Structural Dynamics by Finite Elements. New Jersey, USA: Prentice-Hall.
31
where M ; K and f are found via comformable summations as in eqn [24]. Eqn [31] for the reduced-order structure can be solved for x . Thereafter, xA for each substructure is computed from eqn [25]. The component mode synthesis is also a Rayleigh± Ritz method. The Ritz basis vectors are found as in eqn [7], where F now contains eigenvectors due to the inclusion of vibrational mode shapes. F is given by: 3 7 7 7 7 7 7 7 7 7 7 5
Nomenclature fj I
n uj
magnitude of jth force component identity matrix
32
Impulse Response Function R K Kapania, Virginia Polytechnic Institute & State University, Blacksburg, VA, USA Copyright # 2001 Academic Press doi:10.1006/rwvb.2001.0118
Response of structures to short-duration loads (i.e., the time duration of the external force is much shorter than the time period), is of great interest to a designer. Especially important is the limiting case of these loads, the loads whose time duration approaches zero but their total impulse, area under the force± time curve, stays finite. This happens because the response of a structure to such a load plays an important role both in describing the system and in acting as a building block towards finding the response of the system to any arbitrary excitation. Here we will describe the methods to determine the impulse response function, defined as the response of a single-degree-of-freedom spring-mass-damper system with zero initial displacement and velocity to a unit impulse.
1336 THEORY OF VIBRATION/Impulse Response Function
The governing equation of a single-degree-of-freedom system of mass m, stiffness k, damping c, and subjected to an external force f
t is given as: m x cx_ kx f
t
1
where x denotes the displacement from the static equilibrium, and a dot denotes the derivative with respect to t. Mathematically, the impulse applied at a given time t t is described using Dirac's delta function, and is represented as d
t ÿ t. Dirac's delta function belongs to a class of functions called generalized functions. To understand this function, consider a uniform force f
t acting at t t over duration Dt such that the area under the curve, termed impulse I and given as I f Dt, remains unity. Now, as the value of the load duration Dt is reduced, the value of the force f will increase so that the product f Dt remains unity (Figure 1). As Dt approaches zero, the force f approaches infinity, but the area under the force±time curve remains unity. This function, with value approaching infinity and acting over a vanishingly small time (the independent variable) is called Dirac's delta function. This function, denoted by d
t, has the following important properties: Z1 d
t ÿ t dt 1
2
ÿ1
Z1 g
td
t ÿ t dt g
t
3
ÿ1
Here g
t is an arbitrary function of time. Consider the single degree of freedom system of eqn [1], and subjected to an impulse of magnitude I at t 0. The governing equation then becomes: m x cx_ kx Id
t
The response of the system to a unit impulse, termed the impulse response function, is represented as h
t. It is the solution of the above equation with _ vimp 1=m and initial conditions x
0 x
0 0, i.e. both initial velocity and initial displacement are zero. In terms of h
t, the above equation becomes: 1 h 2zon h_ o2n h d
t m h
0 0; and h_
0 0
6
A number of methods can be used to obtain h
t. Here, we present two classical methods: first, the Laplace transformation method; and second the Fourier series and integral method. Both methods are extensively used in determining the response of a system to an arbitrary excitation.
Laplace Transformation Method The Laplace transform of a function g
t, written as G
s is defined as:
4
Dividing both sides of the above equation by the mass m, we obtain: x 2zon x_ o2n x vimp d
t
Figure 1 Dirac's delta function. The area under the load±time duration remains unity.
5
p where on
k=m is the natural frequency of the system in radiansp per second, z c=cc is the damping ratio with cc 2
km; vimp I=m. Note that cc is called critical damping. Physically vimp is the change in the velocity of a mass m when subjected to an impulse Id
t.
Z1 G
s Lg
t
eÿst g
t dt
7
0
The function g
t is defined for t < 0 and s is a complex variable. It is assumed that the function g
t is such that the above integral exists. The Laplace transform is a linear operator, i.e.: Lag
t br
t aLg
t bLr
t aG
s bR
s
8
THEORY OF VIBRATION/Impulse Response Function 1337
Here a and b are constants, and g and r both are functions of t. Other properties of importance to us here are: Lg_
t sG
s ÿ g
0 Lg
t s2 G
s ÿ sg
0 ÿ g_
0
9
Knowing the Laplace transform of a function, the original function is obtained by taking the inverse Laplace transform: 1 g
t L G
s 2pi ÿ1
gi1 Z
G
s est ds
10
gÿi1
p Here i
ÿ1. Often tables of Laplace transform pairs, widely available in many books on engineering or operational mathematics, are used to obtain the Laplace transform of a function of interest. Tables are also used to obtain the inverse transform. Determining the inverse Laplace transform of a function, not given in the tables of Laplace transform pairs, may be a rather difficult task. Laplace transform of some functions that are important in the theory of vibrations are given in Table 1. The impulse response function h
t can now be obtained by taking the Laplace transform of both sides of eqn [6]: s2 H
s ÿ sh
0 ÿ h_
0 2zon
sH
s ÿ h
0 o2n H
s
1 11 m
_ Substituting h
0 0, and h
0 0, and combining all the coefficients of H
s, we obtain: ÿ2 1 s 2zon s o2n H
s m
12
Note that the function H
s is called the transfer function for the given system and it represents the ratio of the output of the system to the input to the system in the Laplace domain. It is given as: 1 H
s ÿ 2 m s 2zon s o2n
13
The impulse response function h
t is obtained by taking the inverse Laplace transform of H
s, i.e. h
t Lÿ1 H
s. We can obtain the inverse transform of H
s by using partial fractions. To that end, we need to factor the denominator, i.e. represent the
denominator in the form (s ÿ s1 )(s ÿ s2 ), where s1 and s2 , the two roots of the quadratic equation, s2 2zon s w2n 0, are called the poles of the transfer function. The poles, s1 , and s2 are given as: s1;2 ÿzon on
qÿ z2 ÿ 1
14
Thus, the nature of the poles depends upon the value of the nondimensional damping parameter z. The two roots would be complex and distinct if z < 1 (underdamped case), equal and real negative if z 1 (critically damped), and distinct and real negative if z < 1 (overdamped). These three cases lead to three completely distinct type of systems as is shown in the following. Underdamped case (z < 1): The two, complex, poles are: s1;2 ÿzon ion
qÿ
1 ÿ z2 ÿzon iod
15
p where od on
1 ÿ z2 is the so-called damped natural frequency. The transfer function in terms of the partial fractions is given as: 1 1 A B ÿ m s ÿ s1 s ÿ s2 m s2 2zon s o2n
16
Table 1 Laplace transform of some functions that are important in the theory of vibrations Ld
tÿt eÿts Lu
t L t n
1 s
n! sn1
L eat
L sin ot
o s2 o2
L cosh ot
n 0; 1; 2; . . .
1
s ÿ a 3
L sinh ot
s s2 ÿ o2
o s 2 ÿ o2
L eat cos ot
sÿa
s ÿ a2 o2
2 t Z 1 L4 g
tdt5 G
s s
L eat sin ot
Leat g
t G
sÿa
L eat cosh ot
0
L cos ot
s s2 o2
o
s ÿ a2 o2
L eat sinh ot
sÿa
s ÿ a2 ÿo2 o
s ÿ a2 ÿo2
1338 THEORY OF VIBRATION/Impulse Response Function
Multiplying both sides by the denominator in the above equation, we obtain: A
s ÿ s2 B
s ÿ s1 1
17
Substituting s s1 and s s2 , respectively, we obtain: A ÿB
1 1 i ÿ
s1 ÿ s2 2iod 2od
18
p where od on
z2 ÿ 1. Note that the impulse response function for the critical and overdamped case are nonoscillatory in nature. This can be seen in Figure 2B. In both Figures 2A and 2B, the plot of nondimensional displacement mon h
t=I is given as a function of nondimensional time on t. Observe that as the value of the damping is increased from the critical value, the value of the nondimensional peak amplitude decreases but the time it takes for the response to die down increases.
The impulse response function, for the underdamped case, is thus given as: h
t Lÿ1 H
s; t > 0 ÿi 1 1 ÿ1 L ÿ 2mod s ÿ s1 s ÿ s2 Recalling Lÿ1 1=
s ÿ a eat , and s1; 2 ÿzon iod , we obtain: ÿi es1 t ÿ es2 t 2mod ÿi ÿzon t iod t e e ÿ eÿiod t 2mod 1 ÿzon t e sin od t; t > 0 mod
h
t
19
Here we have made use of the Euler formula eiy cos y i sin y. Note that the impulse function for the underdamped case, because of the two poles being complex conjugate for this case, is oscillatory in nature. This is shown in Figure 2A. Critically damped case (z 1): Both the roots are equal s1; 2 ÿon . The impulse response function is given as: " h
t Lÿ1
1 m
s on 2
#
teÿon t ; m
t>0
20
Overdamped case (z < 1): Both the roots are distinct and are given as: s1; 2 ÿzon p on
z2 ÿ 1 ÿzon od . The impulse response function becomes: 1 h
t L m
s ÿ s1
s ÿ s2 1 ÿzon t e sin hod t ; t > 0 mod ÿ1
21
Figure 2 Nondimensional response of a single-degree-of-freedom spring-mass-damper system of mass m, and natural frequency on, to an impulse I applied at t 0 for different values of nondimensional damping
z. (A) underdamped systems
z < 1; (B) critically
z 1 and overdamped systems
z > 1. Note that the impulse response function h
t is simply x
t=I.
THEORY OF VIBRATION/Impulse Response Function 1339
Overdamped case
Response to Initial Conditions In the preceding section, we obtained the response of a dynamic system to a unit impulse by treating it as a special forcing function applied at t 0. Since the effect of applying a unit impulse is to impart a sudden velocity vimp 1=m at t 0, the impulse response function can also be obtained by studying the response of the system under an initial velocity _ x
0 1=m but keeping both the force f
t and the initial displacement x0 equal to zero. In this section, we present the response of a second-order system under the influence of nonzero initial conditions: _ v0 . The impulse response x
0 x0 , and x
0 function can then be obtained by substituting x0 0, and v0 1=m. The response under these initial conditions, in the absence of f
t, can be obtained by taking the Laplace transform of both sides of eqn [1]: s2 X
s ÿ sx
0 ÿ x_
0 2zon
sX
s ÿ x
0 o2n X
s 0
sx0 v0 2zon x0 s2 2zon s o2n
26 For the special case of x0 0 and v0 1=m, all the three equations reduce to the respective expressions for the impulse response function given in eqns [19]± [21], respectively. Most of the systems of practical interest are underdamped systems where the nondimensional damping factor z is of the order of 0.02±0.05. The response due to the initial conditions (as was also seen for the case of the impulse response function in Figures 2A and 2B) are oscillatory in nature only for the underdamped case.
Fourier Series and Transform 22
The Laplace transform, X
s, of the desired response becomes
X
s
v0 zon x0 x
t eÿzon t x0 cos od t sinh o t d od
23
The response x
t is obtained using inverse Laplace transform by keeping in mind that three distinct cases (underdamped, critically damped, and overdamped) arise, depending on the value of the nondimensional damping parameter z. The response for the three cases, obtained using inverse Laplace transform (see Table 1), is given as Underdamped case
v0 zon x0 sin od t x
t eÿzon t x0 cos od t od 24
For many practical cases, we are interested in finding the response of the system to external excitation, such as buildings subjected to wind and earthquake loads, an aircraft wing subjected to gusts, an automobile on uneven pavements and so on. The impulse response function derived above can be used to obtain the response of any system to any external excitation, say f
t. The response can be written as a sum of a complementary solution and a particular solution: x
t xc
t xp
t
The complementary part xc is the solution of the governing equation with the right-hand side equal to zero, the so-called homogeneous equation, as was done in the previous section on determining the response of the solution to initial conditions. The complementary part with arbitrary constants, due to the presence of eÿzon t term, is transitory in nature and approaches zero as t increases. While determining the response due to loads that are of longer duration, the complementary solution is often ignored and emphasis is placed only on the particular solution xp
t. This part of the solution, which does not have any arbitrary constants, can be obtained using the convolution (Duhamel's) integral:
Critically damped case x
t eÿon t x0
v0 on x0 t
27
Zt 25
xp
t
f h
t
f
th
t ÿ t dt 0
28
1340 THEORY OF VIBRATION/Impulse Response Function
Performing the above integral, at times, may be quite a difficult task, especially when the forcing function f
t is not a simple function. Numerical and transform methods, Laplace and Fourier, are used in such cases. Using transform methods we can transform the above integral into an algebraic product which is easy to compute. Then, the solution in the time domain is obtained by performing the inverse transform. In previous years, the Laplace transform was the transform of choice for many engineers. However, its use is limited to only those functions whose Laplace transform is easy to obtain and also the inverse transform of the product [H(s)F(s)] is easily available. Moreover, determining the Laplace transform of a function and determining the response as function in time by performing inverse Laplace transformation does not lend itself easily to the tremendous power of a modern digital computer. It should, however, be mentioned that the availability of symbolic manipulators like Mathematica and MAPLE has considerably improved our ability to obtain inverse Laplace transform. (Caution: an effective way to compute inverse Laplace transforms of long or complex expressions in Mathematica is to first produce the partial fraction expansion in the Laplace domain and then to take the inverse of each resulting term separately. Also, it has been our experience that if inverse Laplace transforms of long expressions are taken without first performing the partial fraction expansion, one may get incorrect results.) It is because of its ability in using the power of modern computers that the Fourier transform has gained a widespread popularity for determining the system response as well as in determining the system model using experimental methods. Moreover, since the Fourier transform works in the frequency domain, it provides an additional advantage: the system parameters for many systems of practical interest can be described easier in the frequency domain than in the time domain. To understand the use of Fourier transform in vibration theory it is important to understand the use of Fourier sine and cosine series in expressing periodic, not necessarily harmonic, functions, and to understand an extension of these series for aperiodic functions, called Fourier integrals. The Fourier series for a function f
t of period 2T can be written as: 1 1 X a0 X ak cos ok t bk sin ok t; 2 k1 k1 2p k ok 2T
f
t
29
In general, one only needs few terms in the series to achieve a good accuracy with respect to the given function. Here ok , called harmonics, are the discrete frequencies, and the Fourier coeficients ak and bk in eqn [29] are given by: 1 ak T 1 bk T
ZT f
t cos ok t dt
k 0; 1; 2 . . .
ÿT ZT
30 f
t sin ok t dt
k 1; 2; 3 . . .
ÿT
Note that the coefficients ak vanish for odd functions f
t ÿf
ÿ t and the coefficients bk vanish for even functions f
t f
ÿ t. The Fourier series converges to f
t for all t if f
t is continuous for
0 t 2T, and converges to f
t f
tÿ =2 if there is a discontinuity at t. It can be shown that the coefficients obtained by eqn [30] yield the best approximation for a given number of terms used in the Fourier expansion. That is, for a given number of terms used in the Fourier series, the square of the error between the function and its Fourier representation is a minimum if the coefficients given by eqn [30] are used in eqn [29]. This implies that the accuracy of the Fourier series can only be improved by adding additional terms in the Fourier series. In vibration theory, especially in experimental structural dynamics, a complex representation is often used. It can be shown that for a periodic function f
t, the Fourier representation in terms of complex exponential harmonic function, eiok t can be written as: f
t
X 1 k1 f
ok eiok t Do 2p kÿ1
31
where, Do, the difference between two consecutive discrete frequencies ok and ok 1 is given as: Do ok1 ÿ ok
2p 2T
32
In eqn [31], f
ok is given as: f
ok
ZT
f
t eÿiok t dt
33
ÿT
Note that in the complex representation, the integer k varies from ÿ1 to 1. The negative frequencies are thus also included. But this representation is convenient in extending the notion of Fourier series to Fourier transform as described below.
THEORY OF VIBRATION/Impulse Response Function 1341
The Fourier series described above can be used to represent an aperiodic function, since such a function can be considered as a periodic function with the period 2T approaching 1. For this case, Do!0. Thus, instead of a discrete representation in the frequency domain, one gets a continuous variation over o. The Fourier series, in eqn [31], becomes an integral, called the Fourier integral, as given below: 1 f
t 2p
Z1
f
oeiot do
f
o
The transfer function in the frequency domain H
o is the Fourier transform of the impulse response function. For the underdamped case the result obtained by, using eqns [34] and [19], is given as: Z1 H
o
In the above equation, f
o is given as: f
teÿiot dt
ÿ1
Functions f
t and f
o are considered to form a transform pair, the Fourier transform pair. Note that, in some references on Fourier transform, the factor 2p, seen here in the denominator of eqn [34] may be placed in the denominator of eqn [35] instead p and still in some other sources it may be placed as
2p in the denominator of both eqns [34] and [35]. Also note that, unlike the case here, some references define eqn [34] as the Fourier transform and eqn [35] as the inverse Fourier transform whereas in other references the names of the two equations are reversed. As is the case for the Laplace transform, the Fourier transform, denoted as F:
t, is a linear operation: Fag
t br
t aFg
t bFr
t aG
o bR
o
36
Also, as is the case for Laplace transform, the Fourier transform of a convolution integral in the time domain becomes an algebraic product of the respective transforms in the frequency domain. That is: 2 t 3 Z F
g h
t F4 g
th
t ÿ t dt5 0
Fg
tFh
t G
oH
o
h
teÿiot dt
ÿ1
35
37
where G
o is the Fourier transform of g
t and H
o is the Fourier transform of h
t as defined by eqn [35]. To obtain
gh
t from G
oH
o, one needs to apply the inverse Fourier transform given by eqn [34]. That is:
38
ÿ1
34
ÿ1
Z1
g h
t Fÿ1 G
oH
o Z1 1 G
oH
oeiot do 2p
1 mod
Z1
39 eÿzon t sin od teÿiot dt
0
Here we have used the fact that h
t 0 for t 0. The integral on the right-hand side of this equation can be easily obtained as: 1 ÿ m o2n 2izoon ÿ o2 "ÿ # 1 1 ÿ b2 ÿ i
2zb k ÿ1 ÿ b2 2
2zb2
H
o
40
where b
o=on . It is interesting to note that the transfer function in the frequency domain is the same as the steady-state (i.e. the transient part of the response containing eÿzon t term has died down) response of the system to a harmonic exponential excitation of unit amplitude and excitation frequency o (i.e. f
t eiot in eqn [1]) and is called the frequency response function (FRF). The frequency of the steady-state response will be the same as that of the excitation. The complex nature of H
o implies that there exists a phase difference (f) between the harmonic excitation and the resulting harmonic steadystate response. This steady-state response, xss
o, is given as: xss
o H
oeiot jH
ojei
otÿf " #1=2 1 1 ei
otÿf k ÿ1 ÿ b2 2
2zb2
41
Here f represents the phase difference between the harmonic excitation and the resulting steady-state response. This phase angle is given as: ÿ f tanÿ1
2zb 1 ÿ b2
1342 THEORY OF VIBRATION/Impulse Response Function
Also, since 1=k represents the response of the system under a unit static force, the term inside the square brackets in eqn [40] can be considered as the magnification of the static response if the unit load is instead applied in a harmonic manner with frequency o. The term, inside the square brackets, is called the magnification factor. In some references, the transfer function (or the FRF) given above is denoted as H
io. The impulse response function can be obtained by taking the inverse Fourier transform of H
o as shown in the following: h
t Fÿ1 H
o Z1 42 1 1 ÿ eiot do 2p m o2n 2izoon ÿ o2
z o and the semicircular arc (denoted as S) of radius r with r ! 1; as: Ic
1 2pm
1 2pm
Z1 ÿ1
Z S
ÿ
eiot do o2n 2i zoon ÿ o2
eizt ÿ dz 2 on 2i zzon ÿ z2
The second integral on the right-hand side of the above integral approaches zero as the integrand contains z2 term in the denominator. Comparing eqns [44] and [45], we obtain the impulse response function to be given as:
ÿ1
The integral on the right of the preceding equation can be performed by using Cauchy's relation, namely: I C
f
z dz 2pi f
z0
z ÿ z0
Here C is any simple closed path around z0 which is traversed in the anticlockwise direction and f
z is analytic within C. To apply this relation to the integral in eqn [42], consider the following integral, obtained by replacing o with z, a complex variable, in eqn [42]: 1 Ic 2pm
I C
ÿ
eizt dz o2n 2i zzon ÿ z2
43
Consider the path C to be the upper half plane, i.e., a semicircle of radius r as r ! 1 (Figure 3). Realizing that the denominator in the integrand in eqn [43] when equated to zero has two complex roots: z1; 2 izon od , the integral becomes: Ic ÿ
1 2pm
I C
eizt dz 2od
z ÿ i zon ÿ od
I 1 eizt dz 2od
z ÿ i zon od 2pm C h i 2pi ÿei
izon od t ei
izon ÿod t 4pmod 1 ÿzon t e sin od t mod
h
t
1 ÿzon t e sin od t mod
46
The transfer function in the frequency domain (or the FRF) H
o and the impulse response function h
t thus form a Fourier transform pair. In determining the impulse response function, the inverse Fourier transform was performed using an analytical approach that used the so-called Cauchy relation (or contour integral) to perform the improper integral. Such an approach is limited only to simple functions for which such integrals can be easily obtained either using Cauchy's relation or from the tables of Fourier and inverse Fourier transform (similar to the case for the Laplace transform). However, the Fourier transform, using its discretized version, can be performed numerically for all sorts of functions using the fast Fourier transform (FFT).
Fast Fourier Transform Let fn represent the value of a function at a discrete point tn nDt where Dt 2T=N, N being the total number of discrete points in the time period 2T or the time-history duration. Similarly fm represents the Fourier coefficient corresponding to the mth discrete frequency om given as om 2pm=
2T. Values of fn and fm are given as:
44
The integral in eqn [43] can also be written, by splitting the contour C in two parts: the real axis
45
fn
ÿ1 X 1N fm exp i
2pmn=N N m0
n 0; 1; 2; 3 . . . ; N ÿ 1 Nÿ1 X fn exp i
2pmn=N fm n0
m 0; 1; 2; 3 . . . ; N ÿ 1
47
THEORY OF VIBRATION/Impulse Response Function 1343
The preceding set of equations constitutes the discrete Fourier transform (DFT) pair and can easily be computed. However, if the computations are done using these equations in a direct manner, a transform will take N 2 complex multiplications. A fast method to evaluate the DFT was proposed by Cooley and Tukey. This algorithm called the fast Fourier transform (FFT), needs only O
Nlog2 N complex multiplications if N 2g , where g is an integer. This results in a substantial reduction in computational time. For example, for N 1024 210 , the direct evaluation to calculate the Fourier transform will take 10242 1 048 576 complex multiplications. On the other hand, one requires only O
10 240 multiplications if FFT is employed, around 1% of the effort required otherwise. To understand the logic behind FFT, let us consider any one of the two equations in eqn [47], say the second one. It can be rewritten as: fm
Nÿ1 X
fn exp ÿi
2pmn=N
n0
Nÿ1 X
fn W mn
n0
48
m 0; 1; 2; 3 . . . ; N ÿ 1 Here W expÿi
2p=N. The FFT is based on the fact that W 0 W N W 2N . . . 1, and W N m W m . As a result, for a given value of g
N 2g , the DFT can be performed in g stages. This is easily understood for N 4
i:e: g 2. For this case, eqn [48], becomes: 8 9 2 W0 > f > > = 6 0 < 0 > f1 6W 2 > 4 W 0 > f > ; :> W0 f3
W0 W1 W2 W3
W0 W2 W4 W6
3 W0 W3 7 7 W6 5 W9
8 9 f0 > > > = < > f1 f > > > ; : 2> f3
N 4 case, we calculate the DFT, once again using N complex multiplications, as: f0 g0 W 0 g1 f2 W 2 g1 g0 f1 g2 W 1 g3 f3 W 3 g3 g2 Note that the values of f are scrambled. The same approach can be easily implemented for other values of g. For example, for N 8
g 3 we will require three stages and each stage will have eight complex multiplications. The computer programs capable of performing FFT are readily available. For example, the commercially available packages Mathematica and MATLAB both contain commands for obtaining the FFT.
Nomenclature C f I v X(s) d(t) z
closed path force impulse velocity Laplace transform Dirac's delta function phase angle damping factor
See also: Critical damping; Transform methods.
49
Obviously a straightforward matrix-vector multiplication will take 42 complex multiplications. If, however, we make use of the fact that W 0 W 4 1 and W 6 W 2 , etc., we can reduce the number of multiplications to eight by performing the above multiplication in two
g stages. In the first stage, using N complex multiplications, we calculate an intermediatory vector g whose components are given as: g0 f 0 W 0 f 2 g1 f 1 W 0 f 3 g2 W 2 f2 f0 g3 W 2 f3 f1 In the second (gth or last) stage, also the last stage for
Further Reading Clough RW, Penzien, J (1993) Dynamics of Structures, 2nd edn. New York: McGraw Hill. Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 19(90): 297±301. Craig RR (1981) Structural Dynamics, An Introduction to Computer Methods. New York: Wiley. Gerald CF, Wheatley PO (1989) Applied Numerical Analysis, 4th edn. Reading, MA: Addison Wesley. Inman DJ (1996) Engineering Vibration. Englewood Cliffs: Prentice Hall. Kelly SG (1993) Fundamentals of Mechanical Vibrations. New York: McGraw Hill. Kreyszig E (1993) Advanced Engineering Mathematics, 7th edn. New York: Wiley. Meirovitch L (1997) Principles and Techniques of Vibrations. Upper Saddle River: Prentice Hall. Rao, SS (1995) Mechanical Vibrations, 3rd edn. Reading, MA: Addison Wesley.
1344 THEORY OF VlBRATlONNariational Methods
Variational Methods S S Rao, University of Miami, Coral Gables, FL, USA Copyright
0 2001 Academic
variational principles of solid mechanics and the application of Hamilton’s principle in the formulation of the equations of motion of a variety of discrete and continuous vibrating systems.
Press
doi:lO.1006/rwvb.2001.0119
Introduction Vibration problems can be formulated either according to vectorial dynamics, based on the concepts of force and momentum, or to variational dynamics, based on the concepts of kinetic energy and work done by the forces. In vectorial dynamics, Newton’s laws are used directly to derive the equations of motion. In variational dynamics, the conditions of extremization of a functional are used to derive the equations of motion of a system. Whereas Newton is considered by most people to be the originator of vectorial dynamics, Leibniz is considered by some and John Bernoulli and/or d’Alembert by others as the originator of variational dynamics. In solid mechanics, the principles of minimum potential energy, minimum complementary energy, and stationary Reissner energy can be used to formulate statics problems while the Hamilton’s principle can be used t o formulate dynamics problems. All these principles represent variational approaches that can also be used to formulate the finite element equations. Hamilton’s principle can be considered to be rooted in the concept that nature will insure efficiency as a system moves freely from one configuration to another. Of all possible paths a system might traverse, that chosen by nature will always be one that minimizes a quantity that can be called action. In Hamilton’s principle, the action is the integral of the Lagrangian from time t 1 to t 2 , where these two times are arbitrary. When calculus of variations is used, the necessary conditions for the minimum of this action result in Lagrange’s equations. The variational approaches have the following advantages:
Calculus of Variations The calculus of variations deals with the determination of extrema (maxima and minima) or stationary values of functionals. The basic problem in variational calculus is to find the function $(x) which makes the integral functional:
stationary. Here, x is the independent variable and 4x= d@/dx and as the condition for the stationaryness of I , the variation of I is set equal to zero so that:
The mathematical manipulation of eqn [2] leads to the governing differential equation of the system, known as Euler or Euler-Lagrange equation, as well as the boundary conditions. The rules of variation used in the manipulation of eqn [2] are summarized in Table 1. The Euler-Lagrange equation and the boundary conditions corresponding to eqn [2] can be expressed as:
[41
1. The forces that do no work, such as forces of constraint on masses, need not be considered. 2. The accelerations of masses need not be considered; only velocities are needed. 3. The mathematical operations are to be performed on scalars, not on vectors, in deriving the equations of motion. This article presents the basic ideas of calculus of variations followed by the description of different
When the function F is defined as F(x, 4, 4x, 4xx) where 4xx= d24/dx2, the Euler-Lagrange equation and the boundary conditions are given by:
dF
84
d (d F ) I d2 ( O F ) dx2 d4xx
dx
= 0,
x1 < x
THEORY OF VIBRATlONNariationaI Methods 1345 Table 1 6(41
-
Rules of variation
42) =
641
If the integrand involves derivatives of higher order than the second order as:
+ 842
/
x2
I
=
F ( x . 4,,4,(1)
(2)
. . . . . 4;'))
dx
[lo]
x1
where 4:) = d'4,/dx', i = 1,2...n, then the corresponding Euler-Lagrange equations are given by:
i = 1 , 2 , . . . ,n
x1 < x < x 2 ,
If the functional I is defined in terms of several independent variables and one dependent variable as:
6
j! tl
6
p
f(4) dt =
where $x = 84/ax: = 84/ d y and 4z = 84 / d z , the Euler-Lagrange equation can be obtained as:
1
$ J ~
6f(4) dt
tl
Some variational problems, known as isoperimetric problems, involve auxiliary constraints:
f(0) d0 = f(4) d4
Extremize I
=
7
XI
F(x, 4 34x)dx
1141
subject to the condition that $ ( x ) also satisfy the equation:
/
x2
When several dependent variables are involved as:
J =
G ( x ,4, q5x) dx = known constant
1151
x1
Using the technique of Lagrange multipliers, the Euler-Lagrange equation corresponding to the constrained variational problem can be expressed as:
, where ( + i ) x = d4i/dx and (q5i)xx = d 2 4 i / d ~ 2the Euler-Lagrange equation can be expressed as:
191 xl
i = 1 . 2.....n
aFs
a4
("">
- = 0:
dx 84%
XI
< x < x2
1161
where F, = F + i G and l is the Lagrange multiplier whose value is chosen so that eqn [15] is satisfied. When potential energy, U , is used as the functional, I , eqn [2] yields the principle of minimum potential
1346 THEORY OF VlBRATlONNariational Methods
energy. The principles of minimum potential energy, minimum complementary energy, and Reissner energy are used to formulate the problems in solid mechanics, that is, to derive the governing differential equations and the boundary conditions. When the Lagrangian function, L, is used as the functional, I , eqn [2] yields the governing equations of motion and the boundary conditions of the system. The corresponding variational principle, in this case, is called Hamilton’s principle. Hamilton’s principle is an integral principle that considers the entire motion of the system between two instants of time tl and t 2 . This permits the treatment of a dynamics problem in terms of a scalar integral. A specific advantage of this formulation is that it is invariant with respect to the coordinate system used. All the variational methods of static and dynamic problems are discussed in this article.
1201
U=DE
where D is the elasticity matrix, the strain energy can be expressed (in the absence of initial strains) as:
If initial strains are present, with the initial strain vector EO, the strain energy of the body is given by: li
///
1
=2
E ~ Dd E V-
///
E~DE dVO
[22]
V
V
The work done by the external forces is given by:
Variational Methods in Solid Mechanics Principle of Minimum Potential Energy
where SI is the surface of the body on which surface forces (tractions) are prescribed. By denoting the known body force vector, 6, the prescribed surface U=li-W, ~ 7 1 force (traction) vector, 6 and the displacement vector, u, as: where li is the strain energy and W , is the work done on the body by the external forces. The principle of minimum potential energy can be stated as follows: ‘Of all possible displacement states ( u ,v, and w ) a body can assume which satisfy compatibility and specified kinematic or displacement boundary conditions, the state Equation [23] can be rewritten as: which satisfies the equilibrium equations makes the potential energy assume a minimum value’. If the potential energy is expressed in terms of the displacement components u, v, and w , the principle of minimum potential energy gives, at the equilibrium state: Using eqns [22] and [24] the potential energy of the body can be expressed as: The potential energy of an elastic body, U , is defined as:
It is important to note that the variation is taken with respect to the displacements in eqn [18] while the forces and stresses are assumed to be constant. The strain energy of a linear elastic body is given by:
U ( u ,v,w ) = A/// V
V li
=
2
dV V
where E is the strain vector, u is the stress vector, V is the volume of the body and the superscript T denotes the transpose. By expressing the stress-strain relations as:
E ~ D (E 2 ~ dV ~ )
2
17 c1
s1
As stated above, the displacement field u(x, y , z ) that minimizes U and satisfies all the boundary conditions is the one that satisfies the equilibrium equations. If the principle of minimum potential energy is used, we minimize the functional U and the resulting equations denote the equilibrium equations while the compatibility conditions are identically satisfied.
THEORY OF VlBRATlONNariationalMethods 1347
.={ i}
Principle of Minimum Complementary Energy
The complementary energy of an elastic body (U,) is defined as: U , = complementary strain energy in terms of stresses ( n )- work done by the applied loads during stress changes (W,) The principle of minimum complementary energy can be stated as follows: 'Of all possible stress states which satisfy the equilibrium equations and the stress boundary conditions, the state which satisfies the compatibility conditions will make the complementary energy assume a minimum value'. By expressing the complementary energy U , in terms of the stresses oil, the principle of minimum complementary energy gives, for compatibility:
Eqns [28] and (30) can be used to express the complementary energy of the body as: uc(oxx,~ y y , .. . ,a,)
+
aT(Ca
=
EO) dV
V
-
//
dii dSz
s2
If the principle of minimum complementary energy is used, we minimize the functional U, and the resulting equations denote the compatibility equations while the equilibrium equations are identically satisfied. Principle of Stationary Reissner Energy
It is to be noted that variation is taken with respect to stress components in eqn [26] while the displacements are assumed to be constant. The complementary strain energy of a linear elastic body can be expressed as:
The complementary strain energy can be written, in the presence of known initial strains E O , as:
In the principle of minimum potential energy, we expressed the potential energy, U , in terms of displacements and permitted variations of u , v , and w . Similarly, in the case of the principle of minimum complementary energy, we expressed the complementary energy, U,, in terms of stresses and permitted variations of gxxroyy . . . ozx.In the present case, we express Reissner energy, U,, in terms of both displacements and stresses and permit variations with respect to both displacements and stresses. The Reissner energy for a linearly elastic body is defined as: U, = JJJ { internal stresses) x (strains expressed in v! terms of displacements) - complementary strain energy in terms of stresses] dV - work done by applied forces
///{ + (g+ g)] [oxx
ay + . . .
gyy d V
V
where the strain-stress relations are assumed to be of the form: E=Ca
ozx
-E}
dV
~ 9 1
The work done by the applied loads during stress change, also known as the complementary work, is given by:
/I/ V
where SI is the part of the surface of the body on which the values of displacements are prescribed as:
// s1
1
[aT& -Z 1 a T C a- @Tu dV
uT@dS1 -
// s2
(u - U)T@ dS2
~321
1348 THEORY OF VlBRATlONNariationalMethods
The variation of U,is set equal to zero by considering variations in both displacements and stresses:
The first term on the right-hand side of eqn [33] gives the stress-displacement relations while the second term gives the equilibrium equations and the boundary conditions. The principle of stationary Reissner energy can be stated in words as follows: ‘Of all possible stress and displacement states the body can have, the particular set which makes the Reissner energy stationary gives the correct stress-displacement and equilibrium equations along with the boundary conditions’.
Hamilton’s principle can be stated in words as follows: Of all possible time histories of displacement states which satisfy the compatibility equations and the constraints or the kinematic boundary conditions and which also satisfy the conditions at initial and final times (tl and tz), the history corresponding to the actual solution makes the Lagrangian functional a minimum. Hamilton’s principle can thus be expressed as:
6
1
Ldt=O
1371
tl
Generalized Hamilton’s Principle The generalized Hamilton’s principle can be stated as follows:
Hamilton’s Principle
The variational principle that can be used for dynamic problems is called the Hamilton’s principle. According to this principle, the variation of the functional is taken with respect to time. The functional, similar to U,U,,and U,,used in Hamilton’s principle is called the Lagrangian (L) and is defined as follows: L=T-U = kinetic energy - potential energy
tl
= d]
(T-
7c
+ W) dt = 0
[38]
tl
where T is the kinetic energy, 7c is the strain energy, W is the work done by external forces, and L = T - 7c + W is called the generalized Lagrangian function. The expressions of T , 71 and W are given by:
[341 1391
The kinetic energy of the body, T , can be expressed as:
where p is the density of the material and li is the vector of velocity components at any point in the body:
u =
6 1 L dt
{ :} w
where uj is the ith component of displacement, hi is the ith component of velocity and p is the density:
1401 where CT,~are the components of stress vector, yfl are the expressions of strain in terms of the derivatives of u,,E,] are the components of the strain vector, and 7c0 is the the initial strain energy function:
Thus, the Lagrangian can be written as:
V
1361 where qhi is the ith component of body force, Q j is the ith component of surface tractions, a bar over a
THEORY OF VIBRATIONNariational Methods 1349
symbol denotes a prescribed quantity, S2 is part of the surface of the body on which displacements are prescribed and ST is part of the surface of the body on which surface tractions are prescribed. Note that Cartesian tensor notations and summation convention for repeated indices is used in eqns [39]-[41]. In addition, eqn [38] has the same form as in the ordinary Hamilton’s principle, but n and W are now different. When variations of the displacements, strains, and stresses are taken independently, we obtain:
/ /// { tz
dt
V
tl
1 /// dt
[471
+
] // dt
tl
( 3 +a)] 61, de,, a%,
[utl
,n
(yij - cii)60ii dV
V
tl
where yrl = yrl(emn.w,,) for a general nonlinear problem ( m ,n = 1, 2. 3 ) with:
= - (urn,, - U n m )
with u,, = du, /ax,. x1 = x , x2 = y . x3 = z , v, = cos(vn) = direction cosine and 61, = Kronecker delta. Using eqns [42]-[45], eqn [38] can be rewritten as:
-
Since yIi are functions of the first derivatives of u;, we can obtain:
1 2
wn,
(ui - iii)6@i dS2 dt = 0
SZ
Since the variations 6ul, dol1, and 6ct1 are arbitrary throughout the volume of the body (V), 6u, is arbitrary on S1 and is arbitrary on S2, their coefficients in the five integrands in eqn [48] must be zero independently. This gives the stress equations of motion, traction boundary conditions, stress-strain relations, strain-displacement relations, and displacement boundary conditions. These equations constitute the complete set of equations (and are valid for large elastic deflections also). The generalized Hamilton’s principle can be stated in words as follows: The displacements, strains and stresses, which, over the time interval tl to t 2 , satisfy the stress equations of motion and the stress-strain-displacement relations throughout the body, the traction boundary conditions on S1 and the displacement boundary conditions on S2, are determined by setting the variation of the time integral of the generalized Lagrangian function equal to zero, provided that the variations of the displacements, strains and stresses are taken independently and simultaneously, and the variations of displacements vanish at tl and t 2 throughout the body
1350 THEORY OF VlBRATlONNariational Methods
and the variations of displacements and tractions satisfy the prescribed boundary conditions.
Applications
- Continuous Systems
where E is the Young's modulus of the bar. The strain energy associated with the element dx (of volume A dx) can be expressed as:
Equation of Motion of a Uniform Bar under Axial Load
The derivation of the equation governing the longitudinal motion of a uniform bar is considered to illustrate the application of Hamilton's principle. Strain energy, U To derive a general expression for the strain energy of the bar, consider the bar to be of variable cross-section under an axial load as shown in Figure 1. If u(x,t) denotes the axial displacement of the cross-section at x, an element of length dx will assume a value of dx + (au / dx)dx due to axial deformation under the axial load F (Figure 1B). The elongation of the element dx is given by 6(dx) = ( a u / d x ) d x and hence the strain at point x can be expressed as: &=-
The stress induced Hooke's law, as:
d U = ( i m ) A = I1E A ( g )
2
where A = A(x) denotes the cross-section area of the bar at x. Thus the strain energy of the bar can be expressed as: U=
1
d U d ~ = ~ ] E Adu(x; ( x )t) [ ~ ] ~[a] d x 2 0
x=o
where L is the length of the bar. Kinetic energy, T The kinetic energy of the bar with variable cross-section can be expressed as:
du(x,t) 6(dx) dx ax ~
(0)can
be expressed, using
Z A
where m(x) = pA(x) is the mass per unit length of the bar and p is the density of the bar. By using eqns [a] and [b], the Hamilton's principle can be used to obtain:
tl
=d/ tl
t2
[ i /L m [ $ ] 2 d x - i / ELA [ g ] 2 d x ] d t 0
0
=o IC1
Z
By carrying out the variation operation, the various terms of eqn [c] can be rewritten, noting that 6 and d/dt as well as 6 and d/dx are commutative, as:
tl
Figure 1AB Longitudinal vibration of a bar. (A) Before deformation; (B) after deformation.
0
assuming that u is prescribed at tl and t2 so that 6u = 0 at tl and t2. Similarly:
THEORY OF VlBRATlONNariational Methods 1351
C
/ 0 0 0
a
I I I
a’
I
I
+
- -- -- -- -
I
I
I
Y au
U + -dX
ax
Figure 1C An element of bar.
6 1 [$5A(g)2 0
tl
1
1& L
t2
=
Assuming that 6u = 0 at x = 0 and x = L, and 6u is arbitrary in 0 < x < L, eqn [fl requires that:
dx] d t
b A zdu 6ulo
L -
d2U
zf3X( E A g ) - m s = O ,
( t A E ) B u dx] dt
O
[g]
and x = L
[h]
0
tl
and: Thus eqn [c] becomes: ( E A E ) ~ u = O at x = O
[{lL
[ & ( E A g ) -m$]6udx
[f] - ( E A ~ ) ~ u d~ t ~= ~0 }
Thus the Hamilton’s principle yields eqn [g], which denotes the equation of motion of the bar, and eqn [h], which indicates the boundary conditions.
1352 THEORY OF VIBRATlONNariationaI Methods
The boundary conditions require that either EA(du / a x ) = 0 (stress = 0) or 6u = 0 ( u = specified) at x = 0 and x = L. where + ( x , t ) = angular rotation due to bending and y(x, t ) = angular rotation due to shear deformation. The bending moment (M)and shear force ( V ) at x are given by:
Transverse Vibration of a Timoshenko Beam
Consider an element of a thick beam in bending as shown in Figure 2B. For a thick beam, when the effects of shear deformation and rotatory (or rotary) inertia are considered, the theory is known as the Timoshenko beam theory. The deflection of the beam at any point x is caused by bending as well as shear deformation. The slope of the deflected center line of the beam, aw / ax, can be expressed as:
V ( Xt,) = kA(x)GP(x,t )
[CI
where E is Young’s modulus, I ( x ) is the moment of inertia of the cross-section, A(x) is the area of cross-
f(x,t). dx
M+dM
(B) Figure 2 A nonuniform thick beam in bending. (A) Deformed configuration; (B) free-body diagram of an element of beam.
THEORY OF VlBRATlONNariational Methods 1353
section of the beam and k is a constant whose value depends on the shape of the cross-section of the beam.
6U = 6
/f
EI
(g)
0
tl
composed of energies due to bending and shear deformation:
=
/f
kGAy2 dx d t
0
tl
L
t2
Strain energy, U The strain energy of the beam is
dx d t + 6
/
/
( E l g ) 6 4 dx dt
[Eig6+1 -; 0
tl
+kGA
--4 (Z )
6wI;
L
U ( t ) = -1 j M-84 d x + -1/ V j d x 2 ax 2 0
-f&kGA(g-q5)]6wdx 0
0
L
0
L
L
t2
kGA(x)
6T=6/
0
d~+i//(:)~
[;/WZ(%)~
0
tl
L
Kinetic energy, T The kinetic energy of the beam due to translational and rotational motions can be expressed as:
L
12
=-/ 1 [/fi
[/;(m$)O-dt]
0
dx
tl
at
-
0
l.
dx] dt
0
(/2)6 at :
dt] dx
fl
Using eqns [fl, [g], and [h], the Hamilton’s principle can be stated as:
0 T
where m(x) = pA(x) is the mass per unit length of the beam, p is density, A(x) is the area of cross-section of the beam and J(x) = pI(x) is the mass moment of inertia of the beam per unit length.
L
-
f (x. t)bw(x, t) dx
6W(t) = J 0
[KGA - - 4
f& +f +
If1
Noting that the order of integrations with respect to t and x can be interchanged and the operators 6 and d / dx or 6 and d / dt are commutative, the variations of U and T can be written as:
/&
( E I g ) 6 $ dx
11
6wii
(E (E + / }
[kGA
0
kGA
L
+
0 -
Virtual work of nonconservative forces, 6 W The virtual work of the applied distributed force, f (x, t), is given by:
(EIg)d+l;
-
- 4)
4)] 6w dx
64dx
0
I1
-
f 6w dx
0
\
m
u
e + f}6w dx + at2
1{ [& 0
(EI
g)
dt
1354 THEORY OF VlBRATlONNariational Methods
Assuming that the virtual displacements, 6w and 64, to be zero, at x = 0 and x = L, and arbitrary in 0 < x < L, eqn [i] leads to:
zdx{ k G A ( g -
+)}
-
d2W
m=+
f
= 0,o
Section 1-1
(A)
O
Eqns [j] and [k] denote the equations of motion for the transverse and shear motions of the beam, respectively, and eqns [l] and [m] represent the boundary 111. requires that either conditions. Eqn . El(dqb/dz)= O or 6+=0 at x = O and x = L. Similarly, eqn [m] requires that either k G A [ ( d w / d x ) - + ] = O or 6 w = O at x = O and x = L. Eqns [l] and [m] can be seen to be satisfied for the following common boundary conditions: 0
0
0
Clamped (fixed) end: w = transverse deflection = 0; = bending slope = 0 Hinged (pinned) end: w = transverse deflection = 0, M = EI(d+ / ax) = bending moment = 0 Free end: M = E l ( d & / d s )=bending moment = 0, V = k G A [ ( d w/ ax) - +] = shear force = 0
+
Coupled Twist-bending Vibrations of Circular Rings
Figure 3 shows an element of a ring along with the sign convention used for the forces and moments. The slope of the transverse deflection curve can be written as:
1 dv R de = x + p
__
1.
where R is the radius of the undeformed center line of the ring, v is the transverse deflection, 8 is the angular coordinate, x is the slope of the deflection curve when the shearing force is neglected, and j3 is the angle of shear at the neutral axis in the same cross-section. The transverse shearing force, F , is given by: F = kAGP
[bl
where k is a numerical factor used to account for the variation of /3 through the cross-section, and is a constant for any given cross-section, A is the area of cross-section and G is the shear modulus. The moment-displacement relations, with a consideration for the shear deformation effect, can be expressed as:
THEORY OF VIBRATlONNariational Methods 1355
4( ):
M--
E+-
where M is the bending moment about a radial axis, M t is the torsional moment about the tangential axis, E is the Young’s modulus, I is the moment of inertia of the cross-section about a radial axis, C is the torsional stiffness, and R is the angular deformation of the cross-section of the ring. The strain and kinetic energies of a ring segment can be written as:
k AG + -p2 2 9
T
=
J’ 0
- pv - qR
Hamilton’s Principle for Discrete Systems
:{ (g)2+i(E)l+j A dt
A
( @ dt ) 2 ] ~
dd [f]
where 4 is the angle subtended by the ring segment at the center, p is the distributed transverse load acting through the shear center, q is the distributed twisting moment acting about the shear center, m is the mass per unit length, and J is the polar moment of inertia of the cross-section of the ring. When the expressions for U and T are used, the Hamilton’s principle results in the following equations:
+Rp---
asd [iiEI (.
-
+-C ( x + R
E)]
-
Consider a system of N mass particles. Let a particle of mass mi be acted upon by a system of external and constraint forces with resultants Fi and fi, respectively. If the system is in equilibrium, the resultant force is zero and the work done over a virtual displacement, 6ri, must be zero, that is: SW, = (F, + f l ) 6rl = 0
kAGR
(kg
-
ct)
[@I
The constraint forces, usually, do not perform any work since the displacements do not have any components in the direction of the constraint forces. Thus, for the system of all N particles, the total virtual work can be expressed as:
yAR d2v =o g at2
N
N
i= 1
i=l
D’Alembert’s principle states that a particle of mass mi subject to an unbalanced force Fi can be considered to be in equilibrium with the inertia force, miYi, so that:
Z) +--=o 7;;;
yJR d2R + Rq ---= g dt2
Eqns [g], [h], and [i] denote the equations of motion corresponding to the variables v , a, and R and eqns [j], [k], and [l] represent the boundary conditions which are satisfied for any combination of the pinned, free, and fixed end conditions.
where it.i is the acceleration of the particle. Thus, for N particles, the virtual work expression, in conjunction with d’alembert’s principle, can be stated as:
0
i=
[j 1
with:
1
1356 THEORY OF VlBRATlONNariational Methods N i=l
denoting the virtual work done by the applied forces. The time derivative of ti.6ri yields:
which can be rewritten as:
By multiplying eqn [55]by m, and summing over all the masses gives: N
mi?, . 6ri =
i= 1
N i=l
=
N i=l
d dt
mi - (i-i .6ri) - 6
N 1 -mi(i-i . t i ) 2 i= 1
Figure 4 True and varied paths of the system.
d dt
mi - ( t i . 6ri) - 6T
]
/ 6 ( T + W) dt = tl
where T denotes the kinetic energy of the system. Using eqns [53] and [56], eqn [52] can be expressed as:
(i-,
$mi
dt
tl
N =
1 t2
mi
i=l
d
1591
N (ti
. 6ri) dt =
m,Ei . 6r,lrz i=l
tl
fl
Since 6ri = 0 at tl and t 2 (eqn [ 5 8 ] ) eqn , [59] reduces to: Let the system of mass particles be defined in terms of n generalized coordinates. Then the configuration of the system at any instant is defined by the values of the generalized coordinates which define a point in the ndimensional configuration space. When the configuration of the system changes with time, the various configurations define points which yield a path, which can be called the true path of the system in the configuration space. At any specific instant, if a small variation in the position of the ith mass, h i , with no associated change in time (6t = 0), is considered, we obtain another path, known as the varied path of the system. The true and varied paths of the system are shown in Figure 4. If the true and varied paths are assumed to coincide at two instants tl and t2, we have: 6rj = 0 at t = tl
and t = t2
[58]
The multiplication of eqn [57]by dt and integration from tl to t2 yields:
6
1
(T+W)dt=O
[601
tl
In the particular case where the forces are conservative, the virtual work is equal to the negative of the so variation in the potential energy of the system (6U) that:
6W = -6U
1611
and hence eqn [60] can be rewritten as:
6
1
Ldt=O
1621
tl
where L is called the Lagrangian and is defined as:
THEORY OF VIBRATIONNariational Methods 1357
L=T-U
-631
Eqn [62] denotes the mathematical statement of Hamilton’s principle which can be stated in words as follows: Of all possible paths that the system can assume between the instants of time tl and t 2 , the path that makes the value of the integral Jt: L dt stationary represents the true or actual path provided that the initial and final configurations of the system are prescribed.
C
The stationary value is actually a minimum. Eqn [62] can be considered as the generalized Hamilton’s principle where W denotes the virtual work due to conservative and nonconservative forces. Hamilton’s principle is often written as:
( T - U )d t +
1
SW,, dt = O
[64]
where U is the potential energy, which includes both strain energy and the potential of any conservative external forces, and 6 W,, is the virtual work done by the nonconservative forces acting on the system. As can be seen from eqns [60] and [62], Hamilton’s principle reduces the problems of dynamics to the study of a scalar integral that does not depend on the coordinates used. Note that Hamilton’s principle yields merely the equations of motion of the system but not the solution of the dynamics problem.
I
1
Figure5 A single-degree-of-freedom system,
SW,,
= F(t)Sx - C X S X
The application of ~
- Discrete Systems
1 2
= -mx
.2
[CI
~principle i yields: l ~
~
~
t2
-
U ) dt
J
+ / SW,,
dt
=0
J
tl
The derivation of the differential equation of motion of a single-degree-of-freedom system, shown in Figure 5, is considered t o illustrate the application of Hamilton’s principle to discrete systems. The kinetic and strain energies of the system are given by:
T
/ (T
6
Single-degree-of-freedom System -
+X
F(t)
t2
Applications
I
4
[d]
tl
with T , U , and SW,, given by eqns [a], [b] and [c], respectively. The first term of eqn [d] can be rewritten
~~.
aa.
[a] The integration of the first term on the right-hand side of eqn [el by parts results in:
1 U =-kX2 2
t2
(T - U ) dt
6 J
The virtual work corresponding to the applied force F(t) and the damping (nonconservative) force cx is given by:
[fl
tl
= mi6xI:
-
.b”
( m x 6 x + kxdx) dt
~
1358 THEORY OF VIBRATlONNariational Methods
Assuming that x is prescribed at tl and t2, eqn [f] gives:
9
(mx + kx
-
F(t)
+ci)&
dt
=0
[81
tl
Since 6x is arbitrary, eqn [g] leads to the desired equation of motion of the system: m2
+ cx + kx = F(t)
[hl
By using integration by parts, eqn [d] can be rewritten as:
6
I.? T dt =
tl
(-m1x16xl
-
m2x26x2) dt
[f]
tl
Tivo-degree-of-freedom System
Next, the two-degree-of-freedom system, shown in Figure 6 , is considered. The kinetic and strain energies of the system can be expressed as:
Hamilton’s principle can be expressed as:
1 1 .2 T = -mix; + -m2x2 2 2 which, in view of eqns [c]-[f], becomes: 1 U = 411 x ; +-k2(x2 2 2
-XI)
2
[bl
The virtual work associated with the applied forces and the damping (nonconservative) forces can be written as:
The variations of the time integrals of T and U are given by: 1 +-m2X:) 2
Figure 6 A two-degree-of-freedom system.
dt
Assuming that x1 and x2 are independent coordinates, eqn [h] yields the equations of motion of the system as:
Next Page THEORY OF VlBRATlONNariational Methods 1359
Nomenclature area of cross-section of a bar torsional stiffness damping constant matrix relating strain and stress vectors elasticity matrix Young’s modulus external force function ith external force ith constraint force function; shear modulus area moment of inertia of a bar; integral function integral function; mass moment of inertia of a rod per unit length shear coefficient stiffness Lagrangian; length of a bar bending moment torsional moment mass for unit length of a bar; mass ith mass number of masses number of dependent variables distributed loads radius of a ring ith radial vector ith acceleration vector surface on which tractions are prescribed surface on which displacements are prescribed kinetic energy time potential energy complementary energy Reissner energy displacements along x,y,zaxes vector of displacement components ith displacement component ith prescribed displacement component volume of a body (or domain of a system); shear force work done by external forces work done by the applied loads during stress changes work done by nonconservative forces Cartesian coordinates initial and final values of x slopes variation operator ith virtual displacement vector Lagrange multiplier
7t
strain energy density stress components stress vector function of x;dependent variable; angular rotation first derivative of 4 with respect to x second derivative of 4 with respect to x ith component of body force body force per unit volume along x,y,z axes vector of prescribed body forces surface traction along x,y,zaxes ith component of surface traction prescribed surface forces (tractions) vector of prescribed surface forces (tractions) angular coordinate Kronecker delta angular rotation due to shear deformation strain component in terms of derivatives of u, strain component strain vector initial strain vector angular deformation transpose of ( )
See also: Finite element methods: Theory of vibration, Equations of motion; Theory of vibration, Fundamentals; Theory of vibration, Impulse response function.
Further Reading Meirovitch L (1967) Analytical Methods in Vibrations. New York: Macmillan. Moiseiwitsch BL (1966) Variational Principles. London: Interscience John Wiley. Rao SS (1971) Effects of transverse shear and rotatory inertia on the coupled twist-bending vibrations of circular rings. Journal of Sound and Vibration 16: 551566. Rao SS (1989) The Finite Element Method in Engineering, 2nd edn. Oxford: Pergamon Press. Rao SS (1995) Mechanical Vibrations, 3rd edn. Reading, MA: Addison-Wesley. Schechter RS (1967) The Variational Method in Engineering. New York: McGraw-Hill. Shabana AA (1997) Vibration of Dzscrete and Continuous Systems, 2nd edn. New York: Springer. Tabarrok B, Rimrott FPJ (1994) Variational Methods and Complementary Formulations in Dynamics. Dordrecht: Kluwer Academic Publishers.