Cyclic Symmetric Modal Analysis

Full model of bladed disk assembly

Harish N. Dave

Lalit Kr. Mishra

2

Technical Report on

Cyclic Symmetric Modal Analysis

By

Harish N. Dave

Lalit Kr. Mishra

Wi th val val uab uabll e Co Contr ntr i butions f r om

A ns il Ge or e B ar at B N Biju Kumar Girish Kulkarni

Raghavendra S S Sourabh Sawant Suresh S

© 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

2

Technical Report on

Cyclic Symmetric Modal Analysis

By

Harish N. Dave

Lalit Kr. Mishra

Wi th val val uab uabll e Co Contr ntr i butions f r om

A ns il Ge or e B ar at B N Biju Kumar Girish Kulkarni

Raghavendra S S Sourabh Sawant Suresh S


3

Contents Chapter – 1 1 Introduction 1.1 Concepts of symmetry 1.2 General Concepts of Modal Analysis

…4 …7

Chapter – 2 2 Cyclic Symmetric Modal Analysis 2.0 Introduction 2.1 Cyclic Symmetric Modal Analysis 2.1.1 Eigenvalue Problem 2.1.2 Concept of Nodal Diameters

…9

...10 … 12 … 14

Chapter – 3 3 Results and Discussion 3.0 Introduction 3.1 Sources of Excitation in a Steam Turbine Assembly 3.2 Definition of Real Resonance 3.3 Results Interpretation 3.3.1 Why do we get DOUBLETS? 3.4 Interference Diagram 3.4.1 Plotting an Interference Diagram 3.4.2 Interference Diagram beyond NB/2 Limit 3.4.3 Studying an Interference Diagram 3.4.4 Interference Diagram vs. Campbell Diagram 3.5 Some Thoughts on Fatigue Life Estimation

… 19 … 19 … 22 … 22 … 24 … 27 … 28 … 29 … 30 … 32 … 34

Conclusions

… 35

Scope of future study References

...36 … 37


4

Chapter – 1 Introduction 1.0 Introduction: Many engineering problems have identical substructures in the circumferential direction. It is the field of mechanical engineering where one finds a variety of such applications. To highlight a few – turbomachinery blades and discs, centrifugal fans and impellers, rotary pumps, gear wheels, pressure vessels, milling cutters, reamers, gear 0 hobbers etc. These systems can be analyzed in full 360 but at a cost that is prohibitive. 0 Moreover, it also becomes cumbersome to analyze the full 360 FE model. It is good to exploit the cyclically symmetric nature of the bladed assembly and reduce the core, time and labor involved in analyzing these systems. Using the cyclic symmetric principle both static and modal analysis can be done. These problems can be broadly classified under three heads, 1. Static or steady state analysis of cyclic symmetric objects subjected to cyclic symmetric loading. 2. Static or steady state analysis of cyclic symmetric objects subjected to general loading (asymmetric loading). 3. Eigenvalue problems (modal analysis). Any structure that is circumferentially symmetric can be analyzed with the concept of cyclic symmetry. According to the principle of cyclic symmetry, it is enough to model only one sector and behavior of the other sectors can be accurately predicted from the modeled sector. These results in enormous reduction in computer core and labor, which means reduced cost and a economical solution to the problem. Further by utilizing this principle, the analyst can have the privilege of using more number of elements in single sector model. This report is an effort in making this concept of cyclic symmetry more clear, to the reader and to explain cyclic symmetric modal analysis in particular. In first chapter, the concept of symmetry will be discussed with its certain advantages in general. Later, some of the concepts of modal analysis will also be discussed. In second chapter a focus on cyclic symmetric modal analysis of a bladed disc assembly of a steam turbine is presented. Next mathematical formulation to the Eigenvalue problem for cyclic symmetric structures will be carried out. The concept of nodal diameters and their physical interpretation will also be discussed at a great length in this chapter. Third chapter discusses the means of interpreting the results obtained from cyclic symmetric modal analysis. Results interpreted are those of a bladed disc assembly of a steam turbine. Interference diagram is most important tool used by engine ers to evaluate the risk of resonance. This will be discussed in detail and compared with conventionally used Campbell diagram. Finally the conclusions of this study on cyclic symmetric modal analysis will be presented. © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

5

1.1 Concepts of symmetry Symmetry is a law of nature. Any symmetric properties in an analysis should be exploited to the full if at all possible, as it can drastically reduce solution times and hence facilitates more effective use of resources. In general, it is important that loads and boundary conditions are symmetric in nature to allow for symmetry to be used in the model. There are four major ways that symmetry can be exploited in the model, which are discussed below.

Type of Symmetries: 1) Planar or Reflective Symmetry: -

This is the most common type of symmetry found in finite element models. Reflective symmetry is a condition where the same pattern is mirrored in a plane. The image shown here indicates that the model is doubly symmetric, one plane in the horizontal direction and another in the vertical direction. As this is a 2D model the symmetric planes are comprised of lines. Planar symmetry can occur in 3D also, the plane of symmetry would be defined by a surface. It is important to apply appropriate symmetric boundary conditions. In the given case, the vertical constraint is to prevent any horizontal movement, while the horizontal constraint would be to prevent any vertical movement.

1 Full Model

Model utilizing reflective symmetry

2) Repetitive Symmetry: -

Repetitive symmetry is a symmetry condition that is seen to repeat along the length of a body. A typical example of this would be circular cooling fins equally spaced along the axis of heat exchanger tubing. An example of this type of symmetry is given in figure below.


2. Repetitive Symmetry

6 3) Axisymmetry or Rotational Symmetry: -

If a shape can be defined by rotating a cross-section about an axis, then it is said to be axisymmetric. If the loads and boundary conditions are also axisymmetric in nature, then an axisymmetric analysis may be carried out. Axisymmetric elements are 2D planar in nature, and are used to model a revolved 3D part in 2D space. No special boundary conditions are applied to exploit axisymmetry. An example showing axisymmetry is given in figure below.

3. Full Model

Model utilizing Axi-Symmetry

4) Cyclic Symmetry: -

Cyclic symmetry is the geometric repetition in the form of cyclic sectors. The structure is composed of a series of identical sectors that are arranged circumferentially to form a ring. In order to utilize cyclic symmetry, appropriate cyclic symmetric boundary conditions must be applied. Cyclic symmetry is further discussed and explained in detail in this report. An example of the symmetry is shown in figure below.

Additional Considerations


4. Full Model consisting of 75 sectors

One sector modeled using cyclic symmetry principle

7 Additional Considerations:

It is important to note that taking advantages of symmetric properties may have a detrimental effect in frequency (modal) or eigenvalue buckling problems. The reason for this is that the symmetric model will not be able to predict non-symmetric mode shapes. Therefore, if a symmetric analysis is carried out, only the mode shapes that also have the same symmetric properties will be resolved. In cases where only minor details disrupt a structure's symmetry, one can often ignore them or treat them as being symmetric, in order to gain the benefits of using a smaller analysis model.

1.2 General Concepts of Modal Analysis Modal analysis is a vibration analysis. Objective of any modal analysis is to determine the natural frequencies and its associated mode shapes. Any physical system possesses its own natural frequency. Further any physical system has 3 unique properties i.e. stiffness, mass and damping. Damping can be expressed as a function of both stiffness and mass. Hence the stiffness and mass, govern the behavior of the system subjected to any loading.

Natural Frequency and mode shapes:

Natural frequency is the frequency at which stiffness force becomes equal and opposite of inertia force and the structure will vibrate in the absence of any external loading. At natural frequency, damping of the system governs the amplitude of response, and since material damping is usually very small, the amplitude of response becomes very high. When excitation frequency is approximately equal to natural U frequency, the energy build up is maximum and stresses induced in the system are very large and the system may fail due to excessive vibrations. Mode shape is the vibration pattern of the system excited at its natural frequency. Mathematically mode shape is nothing but the eigenvector. Each natural frequency has its own unique mode shape. Mode shape does not represent the absolute displacement of the system but only represents the relative displacement.

Frequency Ratio  /  n


8 Undamped Free Vibration Problem:

Consider an undamped free vibration problem, Fundamental equation of motion:

m x  kx  0 Where,

k is stiffness of the system m is the mass of the system x is the displacement about the mean position

The solution of the above can be assumed to be,

x  u cos( t )

Where u is the stationary amplitude

Thus we can formulate the eigenvalue problem as under,

 n

ku   n2 mu

u

is the natural frequency is the eigenvector or mode shape

The figure represents the simplest idealization of any physical system, whose Note: governing parameters are stiffness and mass. i.e. it should not be thought as merely a spring connected with a mass.

For a single DOF system, the natural frequency is given by

 n 

k m

Thus higher the mass lower is its natural frequency. Stiffer the structure, higher its natural frequency. On similar lines, the eigenvalue problem of a multi DOF system system can be formulated as,

 K u   mu 2 n

Where,

 K  is the stiffness matrix m  is the mass matrix u is the eigenvector or mode shape  n

is the natural frequency

The eigenvalue problem as formulated above can be solved by any of the following techniques,  Simultaneous iteration scheme  Subspace iteration scheme  Lanczos algorithm  Power dynamics scheme © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

9 

Reduced Householder’s Method

Chapter – 2 Cyclic Symmetric Modal Analysis 2.0 Introduction Turbomachinery bladed disk assembly is the most ideal example of cyclic symmetric modal analysis. In this chapter cyclic symmetric modal analysis of bladed disk assembly of a steam turbine rotor will be discussed in detail. A rotating turbine blade is the component, which converts the energy of the flowing fluid into mechanical energy. The reliability of these blades is very important for the successful operation of a turbine. Researchers have ascertained the major cause of failure to fatigue of metal. Usually the turbine blades are fabricated by single crystal solidification techniques to avoid grain boundaries and hence regions of fatigue crack initiation. Fluctuating forces cause fatigue failure. Turbine blades experience fluctuating forces when they pass through non-uniform fluid flow from stationary vanes (nozzles). Thus the basic design consideration is to minimize the dynamic stresses produced by these fluctuating forces. Based on the theory of vibration of mechanical structures, the dynamic behavior of the bladed disc assembly can be predicted. Combining the dynamic behavior with the nature of the fluctuating forces helps a design engineer in evaluating a blade design. Dynamic stresses reach a maximum at resonance. Thus a complete risk evaluation due to resonance is of utmost importance. The turbine rotor disk with the blades mounted evenly around its circumference possesses rotational periodicity in geometry. It can be analyzed with the help of cyclic symmetric principle.

Shroud

Vane

Bucket

Wheel

Fig-1 A typical industrial Steam Turbine Rotor One sector analyzed using UNCONTROLLED WHENconcept PRINTEDof cyclic symmetry © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. having 22 blades

10

Modeling Considerations:

FE Model is generated for one sector as shown in the above figure. A typical turbomachinery rotor disk assembly consists of a wheel, bucket (or dovetail), vane or blade and shroud (or cover plate). These various parts are shown clearly in the above figure. First we need to generate the meshed model and then generate a full FE Model, which is to be used in the subsequent modal analysis. Modeling procedure will not be discussed here as it changes from situation to situation. But however some modeling and meshing considerations are highlighted here, 1. The sole objective of a cyclic symmetric modal analysis is to extract natural frequencies; thus the mesh need not be too fine, unless there is a special need to maintain a finer mesh. 2. It is not required to capture all the minute geometric details, like very small fillets. 3. A flow through mesh between bucket and vane is very difficult to realize, thus constraint equations need to be applied between vane bottom and bucket top. In the same way, it needs to be done between vane top and shroud bottom. 4. Depending on the modeling requirements, couplings need to be generated at force transmitting points between bucket and wheel. 5. Displacement constraints must be applied at the appropriate locations. In the model shown it was applied at the bore region of the wheel. 6. It should be taken care that the displacement constraint, couplings, constraint equation should not contradict each other i.e. they should not appear on the same node. 7. Cyclic symmetric B.C`s needs to be defined along the two extreme faces along the hoop direction. Use CECYC or CPCYC command in ANSYS to define the same. Thus a FE Model can be built taking into account the above points. Now we proceed to carry out a cyclic symmetric modal analysis.

2.1 Cyclic Symmetric Modal Analysis This method takes advantage of harmonic variation of modal displacement around the circumference. Referring to figure 2 the cyclic symmetric object shown has N = 8 repeating sectors. Only 1 sector is modeled in FE Analysis. The forces acting on the whole object can be expanded in Fourier series as below,

F k  f 1  f 2ei ( k 1)  ...... f N 1ei ( k 1)( N 2 )  f N ei ( k 1)( N 1)

... (2.1)

th

Here F k is the force on the k sector, f 1 , f 2 ,...... f N are the Fourier coefficients and the angle   2 / N is the angle between any two repeated sectors. Thus if we have a force vector we can formulate a matrix equation in between the forces acting and the Fourier coefficients and subsequently solve it to obtain the Fourier coefficients. We may thus interpret the force on the structure to act in N harmonics, with the complete behavior of the structure being a superposition of its responses in all the N harmonics. © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

11

All calculations in cyclic symmetric analysis consist of both real and imaginary parts. For harmonic n  0 , the uniform force of f 1 is considered as acting in all the sectors. Thus displacement and stresses in all the sectors would be identical for the harmonic n  0 . For the harmonic n  1 , a force of f 2 acts in the first sector and

f 2 e i acts on the second sector, f 2 e i 2 acts in the third and so on. The displacement also follows the same rule. Similarly for st the harmonic n  2 , force f 3 acts in the 1 2 nd 4 sector f 3 e i  in the 2 sector, f 3 e i  in the

rd

3 sector and so on. Thus all the harmonics can be easily analyzed for the behavior of the structure, since the displacement and force in one sector determine the displacement and forces in the remaining sectors for each harmonic.

Fig-2 Cyclic symmetric object with N = 8 repeating sectors

To summarize:

1. If the forces in the repeating sectors are related as in above equation then the corresponding displacements in the substructures are also related. 2. The structure behavior can then be obtained from the analysis of a single repeating sector, stipulating the above variation (in the form e i ) as a precondition. Thus, if a generalized external force acting on a cyclic symmetric structure (e.g. load carried be a pair of meshing gears) is represented as a sum Fourier harmonics, then for each such harmonic, analysis of a single repeated sector is possible. It is worthwhile to mention here that cyclic symmetric analysis, in fact is an extension of the standard practice for axisymmetric structures under arbitrary external loading. The forces and corresponding displacements are expressed in terms of Fourier series and analysis is carried out for each harmonic individually. The total response is obtained as a sum of individual harmonic responses. In general for a problem having N repeated structures, for values of   2 p / N and   2 ( N  p ) / N the imaginary parts of the displacements values will be equal and opposite i.e. they will be complex conjucate. Hence cumulative sum of displacements will always be real. This is the reason why stress values are always real in a cyclic symmetric static analysis. For a complete problem, it is enough to compute only (N/2) + 1 harmonics when N is even and (N+1/2 harmonics when N is odd. © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

12

2.1.1 Eigenvalue Problem: In this section eigenvalue problem for cyclic symmetric structures will be formulated. General definition of eigenvalue problem and various solvers was discussed st i th repeated in 1 chapter. For a cyclic symmetric model, the eigenvector ui  if the sector will be connected to the (i  1) th by the relationship

ui 1  ei ui  

2 

N

… (2.2)

p

Where, N – Number of repeated structures, m – number of harmonics to be computed and p = 0, 1, 2……..m. The nodes in a one sector modeled using cyclic symmetric principle can be classified as: Adj acent nodes Slave Nodes

I nteri or Nodes

Fig – 3Concept of Master, Slave, Adjacent and Interior nodes.

M aster Nodes

Master / Slave boundary nodes: Slave boundary nodes are the nodes that also occur on the next consecutive sector. They are common to i th and (i  1) th sector. Hence the slave boundary nodes are connected to the master boundary nodes by a complex constraint, namely © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

13 i

Adjacent nodes: These are the nodes.

e

2  N

p

where p is the harmonic.

nodes that are adjacent to the slave boundary

Interior Nodes: These are the nodes that are neither master/slave nor adjacent nodes.

Thus it is clear that master and slave boundary nodes have a complex relationship in between them.

The Eigenvalue problem of cyclic symmetric modal analysis in matrix form can be arrange in the following form,

  B1   A1   A T  B   A  2 2  1 T   A2   B3   A3   T    B4  A 3   ... ... ... ... ... ... ...  ... ... ... ... ... ...  ...   Bn1   i  An 1 T  An e 

  2

=  n

 A e  i

n

... ...

 An1   Bn 

          

 z 1   z   2  z 3     z 4     ...   ...     ...    z n  

 M 1    M 2   M 3     M 4   ... ... ... ...... ... .... ...        M n 1   M n  

 z 1   z   2  z 3     z 4     ...   ...     ...    z n  

2 Where  z i are the eigenvectors,  M i  is the mass matrix,  n is the eigenvalue and the

stiffness matrix is given as,

  B1   A1   A T  B   A  2 2  1 T   A2   B3   A3   T    B4  A 3  Interior/Adjacent  ... Nodes ... ... ... ... ... ...  ... ... ... ... ... ...  ... © 2012, Quality Engineering and Software Technologies, Inc.  All rights reserved. UNCONTROLLED WHEN PRINTED  Bn 1   i Slave Nodes  An 1 T  An e  M aster N odes

 A e  i

n

... ...

 An 1   Bn 

          

14

Summarizing the above form of stiffness matrix:

1. The interaction stiffness terms of master and slave nodes are complex in nature. 2. The stiffness matrix has both real and imaginary parts. Comparatively, real part occupies more core than imaginary part. Imaginary stiffness terms occur only in the interaction terms between master and slave nodes. 3. Mathematically, the stiffness matrix in a cyclic symmetric modal analysis is Hermitian in nature, because real parts are symmetric and imaginary parts are skew-symmetric. Hence it is an eigenvalue problem of a Hermitian matrix. If the transpose of a complex matrix is its conjucate, then a matrix is said to be Hermitian. Its eigenvalues are real and eigenvectors are imaginary . 4. Cyclic symmetric modal analysis gives rise to a complex eigenvalue problem. However, this eigenvalue problem becomes real for   0 and   . 5. The harmonic, at which it is excited, has a considerable influence on the stiffness of the structure. Further if consistent mass is used then the mass matrix is also complex. But for a lumped mass formulation it remains real. The Eigenvector and natural frequency can be obtained by solving this complex Eigenvalue problem. Normally, Lanczos algorithm is used. The Eigenvector thus obtained are that of a single sector. Eigenvector of other sectors can be determined from the mathematical relation given by equation 2.2 and hence the mode shape of the entire disk can be easily obtained. Thus both natural frequency and mode shape are also dependent on the harmonic and this harmonic is popularly referred to as Nodal Diameter in literature.

2.1.2 Concept of Nodal Diameters: In previous sections, the Eigenvalue problem for cyclic symmetric structures was explained. Nodal diameters play vital role in cyclic symmetric modal analysis and helps to understand the deflection pattern of the bladed disk assemblies, thus become an essential feature of the cyclic symmetry analysis. The influence of ND on stiffness of the structure is evident. It should be noted here that the word nodal used here is in vibration sense and not in finite element sense. Node here means a point of zero displacement, a virtual hinge. Nodal diameter can be understood by the following,  Nodal diameter defines the number of complete sine waves that pass through the

circumference. If we plot the displacement of the tip of the blade with angular position, it follows a sinusoidal characteristic which is designated as a diametrical pattern  Nodal diameter indicates number of zero displacement planes that pass around the

circumference.  Mathematically, nodal diameter represents the harmonic content. © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

15  Nodal diameter is the number of distinct harmonics that are required to completely

define the system. The amplitude of vibration varies harmonically around the disk and ND is nothing but the no. of distinct harmonics that are present in the system.  Nodal diameter will govern the cyclic symmetric B.C`s that are to be applied while

carrying out cyclic symmetric modal analysis.  Physically, Nodal diameter is nothing but the disk effect. It simulates the disk mode

shape.  Nodal diameter has a direct influence on natural frequency.

The following figure illustrates the harmonic variation of amplitude along the circumference. At ND = 0 there exists no zero displacement planes. At ND = 1 +ve indicates a positive displacement and – ve indicates a negative displacement and hence one zero displacement plane exists. Similarly at ND = 2, we have 2 zero displacement planes. The variation of amplitude along circumferential position is also depicted below. Similarly nth ND will have N complete sine waves passing around its circumference and N zero displacement lines in the disc. In a cyclic symmetric structure, the number of harmonics depends on the number of repeating sectors. If there are N repeating sectors in the structure, then the number of distinct harmonics i.e. (nodal diameters) can be calculated as,

(a) (N/2) + 1 nodal diameters,

if “N” is eve n

(b) (N+1)/2 nodal diameters,

if “N” is odd.

It should be noted that this also includes the zeroth ND, and hence maximum count of ND will be one less than the above quantity.


16

0

Let us workout a simple example of a 360 disc constrained at its bore and 0 conduct a cyclic symmetric modal analysis and study the nodal diameter pattern. One 60 model was analyzed in ANSYS and we have ND = 0,1,2,3. The nodal diameter patterns are as shown.

Full 360 disk having 6  60 sectors 0

0

0

One 60 sector with constraint at bore.


17

Nodal Diameter 0

The disc mode shapes are shown in the given figure. In ND = 0 disc mode, all the sectors move in the same direction, either up or down. They all are in phase i.e. there is no relative motion between the sectors. The displacement pattern is similar in all the sectors. All the sectors move together in this ND. In other words, no interaction exists between different sectors of the disc. The entire disc behaves like a cantilever beam constrained at its bore. No sine waves pass through the circumference. All the sectors are in phase. ND 0

Nodal Diameter 1

The disc mode shapes are shown in the given figure. In ND = 1 disc mode, the sectors 0

belonging to the 180 moves in one direction 0

and the other 180 moves in another direction i.e. in a flexure mode, half of the disc moves up while

the

other

half

moves

down. The

displacement pattern of the disc is as shown. One zero displacement plane occurs at this ND i.e.1-sine wave pass through the circumference. ND 1


18

Nodal Diameter 2

The disc mode shapes are shown in the given figure. In ND = 2 disc mode, first 0

0

90 moves in one direction, second 90 0

moves in another, third 90 moves in same direction as first and the last one moves in same direction as second.

Two zero

displacement plane occurs at this ND. The two planes are perpendicular to each other. 2sine waves pass through the circumference.

ND 2

Nodal Diameter 3 The disc mode shapes are shown in given figure. In ND = 3 i.e. at the max. ND all the sectors move in directions opposite to their

neighboring

sectors.

Three

zero

displacement plane occurs at this ND. 3-sine waves pass through the circumference. All the sectors are out of phase.


ND 3

19

Thus the main conclusions from the above study on nodal diameters can be summarized as, 1. For the ND = 0 and ND = NDMAX nodal diameters, the sectors of the bladed disk are in phase and out of phase respectively i.e. these two ND’s are unique. 2. The mode shapes shown in the above figures were flexure modes. Similarly we can have tangential, axial or a complex modes for each nodal diameter. 3. ND is the parameter that depicts the interaction of each sector in the whole disc. In bladed disc assemblies, it depicts the interaction in between different blades. In addition to nodal diameter there also exists a nodal circle mode in disc vibrations. Nodal circle mode is similar to nodal diameter mode except that, in nodal circle mode a zero displacement circle exists in contrast to that of a zero displacement line in nodal diameter modes. Depending on the nature there also can exist combination modes of nodal diameter and nodal circle. We limit our scope of discussion to nodal diameter modes only. Thus the principle of cyclic symmetry can be successfully applied to modal analysis. The next chapter is dedicated to results that we get from a cyclic symmetric modal analysis and tools to analyze the results of a cyclic symmetric modal analysis namely – Interference Diagram and Campbell Diagram .

Chapter – 3 Results and Discussion 3.0 Introduction: Evaluation of the turbine bladed disc assembly design is the sole purpose of any FE Analysis. Untimely shutdown disturbs the economics of both the user and the manufacturer. And hence there is an urge for reliable turbine operation. High Cycle Fatigue (HCF) plays a significant role in causing turbine blade failures. HCF failures can result from resonant vibratory stresses sustained over a relatively shorter time. During operation, the periodic fluctuation in the steam forces occurs at frequencies corresponding to running speed and its harmonics and thus causes the disc to vibrate. The amplitude of these vibrations depends on the proximity of the forcing frequencies to the natural frequency. Large amplitude vibration can occur when the forcing frequency © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

20 approaches or becomes resonant with a natural frequency of the bladed disk. Vibratory stresses at or near resonant condition aggravate HCF d amage. Steam turbine manufacturers typically design and manufacture bladed disks with adequate margins between the forcing frequencies and the fundamental natural frequencies to avoid resonance. However it is not possible to completely avoid resonance under normal operating conditions of a steam turbine. This is because of manufacturing variations, assembly variations, routine wear, varying operating speeds or any other factor that modifies the mass or stiffness of the structure. A practical case of resonance in operating conditions was due to solid build up at the root of blade because of corrosion, which shifted the natural frequency, and the excitation frequency coincided with natural frequency and produced resonance. Thus blade design must be reviewed continuously for possible interference of natural frequency with excitation frequency.

3.1 Sources of Excitation in a Steam Turbine Assembly: The root cause of vibration in a steam turbine is due to flow induced vibrations . In a steam turbine, most common sources of excitation are nozzle-passing frequency (NPF), running speed and harmonics of both NPF and running speed. The main source of turbine blade vibration problems arising in turbomachines is the fact that the distribution of the steam load on the blade is not constant with respect to the rotational angle but varies during one rotation of blade. This angular dependence of blade load is caused by various circumstances. The main reason for that is the fact that the rotor blades run behind a row of still standing stator blades, which are fixed in the casing of the machine. This causes a circumferential distortion in the flow, which may be due to inlet or exhaust ducts. The pressure variation in the fluid flow causes the forces to vary circumferentially. These forces depend on the relative position of blading with respect to nozzles and any other interruptions like struts, diaphragms etc. Another reason for non-uniform blade load in one revolution can occur by partial admission e.g. the loading of the rotor blades by fluid forces in one or more sectors around the circumference while other sectors remain unloaded. In both the cases, the load on the rotating blade is periodic and can be described in the form of Fourier series as, 

F (t ) 

 F cos( t   ) n

n

… (3.1)

n

n 0

Where F (t ) is the forcing function, F n are the Fourier coefficients which define the forcing function F (t ) ,  n is the phase angle, t is the time,  n are the excitation frequencies given by,

 n  n  

(or)

 n  n  NN  

n = 1, 2, 3 ………….

Where  is the running speed of the turbine, NN is the number of nozzles (i.e. the number of periodics of the blade load during one revolution). Here it should be noted that © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

21 excitation frequency arises either from running speed harmonics or nozzle passing frequency. Nozzle passing frequency excitations are caused by steam flowing through a set of nozzles or diaphragms, which are used to turn and direct steam flow onto the blades. Because of their design, nozzles have flow interruptions at regular or cyclic intervals, which cause the force imparted on the blades to be cyclic. These excitations are caused by nozzle vane trailing edge wakes as the flow leaves the vanes. A blade passes through nozzle vane wakes a number of times equal to the number of nozzle vanes in one revolution. An example of this wake formation zone is shown in the given figure below. Due to formation of wakes in the airfoil region of nozzles, velocity of steam remains no more constant and a variation in pressure takes place. Net effect of this variation in velocity and pressure is to produce unbalance forces.

WAKES (REGIONS OF REDUCED VELOCITY)

SHOCKS (LINES OF VELOCITY AND DENSITY CHANGE)

P2 > P1 V2 < V1

P2

V1


V2

P1

22

Fig: Wake formation in Stator vanes Running speed harmonics occur due to interruptions in the fluid flow path. Harmonics of running speed are multiples of rotor operating speed. For example, the fifth harmonic of running speed would be a force that occurs five times for every revolution of the wheel. For example, a turbine rotor running at a speed of 3600 RPM (60 cycles/sec or Hz) would have running speed harmonics occurring at 120Hz, 180Hz, 240Hz and so on.

3.2 Definition of Real Resonance: A good design of turbomachinery blading consists of following steps, 1. The investigation of natural frequency and the mode shapes of the rotor blades 2. The determination of amplitude, frequency, and shape of aerodynamic forces acting on the rotor blades. 3. The determination of damping values and generating appropriate damping models. 4. Determination of dynamic stresses. 5. Life estimation based on cumulative damage fatigue theories. Thus determination of true resonance condition is critical for turbomachinery design. A turbine bladed disc can get into a state or resonance when the energy build up is maximum. There are two simultaneous conditions for the energy build up per cycle of vibration to be maximum. These conditions are:  The frequency of exciting force equals the natural frequency of the structure,  The exciting force profile has the same shape, as the associated mode shape

of vibration i.e. the harmonic number must represent the nodal diameter. Both of these must exist together for a real resonance to occur. If both the

conditions are satisfied, then work done by the system will be maximum and hence energy build up will be maximal, thus causing maximum damage to the system.

3.3 Results Interpretation: Results from cyclic symmetric modal analysis with objective of investigation of natural frequency and mode shape pattern are discussed in this section. Basically it should be noted that natural frequency would vary with nodal diameter. Consider a case of 44 blades around the circumference and thus we have nodal diameter range from © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

23 0,1,2…22. Natural frequencies were extracted for first 12 modes using Lanczos algorithm in ANSYS. A typical list of natural frequencies thus obtained from a cyclic symmetric modal analysis is given below:

Natural Frequencies at each Nodal Diameter

Mode # 1 Mode # 2 Mode # 3 Mode # 4 Mode # 5 Mode # 6 Mode # 7 Mode # 8 Mode # 9 Mode # 10 Mode # 11 Mode # 12

ND=0

ND=1

ND=2

ND=3

ND=4

ND=20

ND=21

ND=22

1018.551

1051.936

1102.742

1169.668

…..

3156.285

3198.778

3212.413

1296.730

1051.936

1102.742

…..

…..

3156.285

3198.778

3548.476

2918.921

1390.389

…..

…..

…..

3535.594

3544.492

4504.928

3549.565

2406.908

…..

…..

…..

3535.594

3544.492

5405.417

3081.082

3272.481

…..

…..

…..

4499.048

4503.265

5955.164

2982.150

3081.082

3272.481

…..

…..

…..

4499.048

4503.265

6352.461

3861.894

4225.162

4260.962

4304.176

…..

…..

…..

5400.065

5404.021

7297.392

5998.684

3861.894

4225.162

4260.962

4304.176

…..

…..

…..

5400.065

5404.021

7521.234

6625.923

4300.977

4540.467

4766.320

4886.864

…..

…..

…..

5981.648

5962.567

8351.090

7116.876

4300.977

4540.467

4766.320

4886.864

…..

…..

…..

5981.648

5962.567

8723.034

8135.735

4857.238

4985.760

5294.683

5491.265

…..

…..

…..

6371.552

6357.133

9671.878

8213.375

4857.238

4985.760

5294.683

5491.265

…..

…..

…..

6371.552

6357.133

10201.245

…..

…..

1264.132

…..

…..

1169.668

1264.132

…..

1657.615

2020.396

2406.908

1390.389

1657.615

2020.396

4288.775

2939.808

2982.150

4838.825

2939.808

5669.021

…..

ND = 0

NDMAX

The modes as indicated in first column are the mode shapes of the individual buckets/sectors and nodal diameter is the one that indicates the vibration mode of disc.

Observations: 1. Mode of disc vibration can be completely defined if and only if both ND and the individual bucket (or) sector mode shape of one sector are specified i.e. 1F (3) st indicates 1 Flexure mode with 3 nodal diameters. If a combination of nodal st circle modes and nodal diameter modes occur, then we specify it as 1F (2C3D) 1 nd flexure mode occurring in conjunction with a combination of 2 nodal circle rd mode and 3 nodal diameter mode. 2. It is clearly evident from the above results that natural frequency depends to a considerable extent on nodal diameter. 3. Doublets in natural frequencies are observed for all nodal diameters except ND = 0 and ND = NB/2. 4. As the number of nodal diameter increases, the disk assembly becomes more and more stiffer in nature. 5. It may also happen that a natural frequency value can appear in more than one nodal diameters, that means a particular natural frequency can excite many nodal diameters. © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

24

3.3.1 Why do we get DOUBLETS? In general the mode shape of the bladed disc assembly can be given as, 

X ( M ) 

 A

L

sin( L   l )

… (3.2)

l 0

Where L  l  n  M , l  0,1,2,3.......... . , n is the number of buckets and if assembly is packeted then it is the number of packets and  l is the phase lag between the force and response, which occurs due to damping. The basic reason for repetition of natural frequencies at all ND’s except at ND = 0 and ND = NB/2 or (NB-1)/2 depending on whether the no. of buckets are even or odd respectively , is that they represent orthogonal mode shapes. Consider the case of 40 bladed disk assembly, thus we have no. buckets n  40 , no. of nodal diameters (ND) from 0 to 20 and phase angle  = 0 . Except at ND = 0 and ND = 20 we expect doublets. Now let us expand the above equation for the following four cases by substituting the various values of l, n and M.

Case 1: - ND = 0

Case 3: - ND = 15

L  0 L  40  0  40 L  80  0  80

L  15 L  40  15  25,55 L  80  15  65,95

Case 2: - ND = 20

Case 4: - ND =17

L  20 L  40  20  20,60 L  80  20  60,100

L  17 L  40  17  23,57 L  80  17  63,97

The values of L represent the harmonic content in the mode shape of the disc. 0 The 360 model of the turbine rotor consists of 40 blades. Thus 0 to 40 blades represents 0 0 0 to 360 in a cyclic manner. For understanding purpose one can assume the angular © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

25 0

position  of any nodal line can be calculated as … [()ang. = (360 / no. of buckets n) 0 0 *{harmonic no. L})] starting from 0 to 360 clockwise direction if it has been assumed 0 0 angular position at the top of the disk. This angular position will be the starting point of the sinusoidal wave to that deflection pattern of bladed disk will follow. For better understanding consider case 3 mentioned above in the table, this nodal diameter has th th mode shapes at 15 and 25 harmonic, means angular position of the nodal lines i.e. 0 0 sinusoidal waves will start either at 135 or 225 respectively. Similarly considering the th rd 0 0 case 4 we have only 17 and 23 harmonic, representing 153 and 207 respectively 0 within first 360 . Thus for the case, 3 and 4 we have only two harmonics that 0 encompasses within full 360 and other higher harmonic nodal lines will start exactly at any of the two angular positions. Consider case 2, this nodal diameter has only one starting angular position of the sinusoidal wave for each and every harmonics and which is unique. For each harmonics for case 2, there will be 20 sinusoidal waves and same 0 number of nodal lines, which will be at 18 interval. Finally considering case 1 it has th th 0 0 mode shapes occurring at 0 and 40 harmonics thus it represents 0 and 360 degrees 0 0 respectively. Mathematically we can say 360 in nothing but 0 (as the function is harmonic in nature cos(0)  cos(360) and sin(0)  sin(360) ). For the zeroth nodal diameter there is no sine wave which means no nodal line to be shown along the circumference. Again for the zeroth nodal diameter all the system is moving all together i.e. there is no relative motion among the components. Thus in the cases 1 and 2 we have 0 only one harmonic that encompasses within full 360 . The above study reveals an interesting observation that, except at ND = 0 and 0 ND = NB/2 we have two different harmonic contents within 360 . This leads to the natural modes to occur in spatially orthogonal pairs. Orthogonal mode shapes are the ones in which nodal radii of one mode in the pair become antinodes in the other and vice-versa.


26 The above figure represents an ND = 3 nodal diametric mode showing nodes and antinodes. Solid lines show one mode and dotted the other of the two orthogonal mode shapes i.e. nodes of solid lines are the antinodes of dotted line and vice-versa. These two orthogonal mode shapes have same frequency as they represent nothing but  and 360   angular position. The basic nature of function assumed in cyclic symmetric analysis is harmonic in nature i.e. cos , sin( ) and from mathematics we know that cos( )  cos(360   ) and sin   sin( 360   ) . Thus there is no change in mode

shape as such, except an orientation change. Since mode shape remain unaffected, the natural frequencies also occur in pairs for a pair of orthogonal mode shapes. This is true only if the disc is perfectly cyclic symmetric.

Following observations can be made from the study :  The reason for repetition of frequencies is because of occurrence of orthogonal

mode shapes at all ND’s except at ND = 0 and ND = NB/2.  The mode designation does not change in a pair of orthogonal mode shapes i.e.  The only change in a pair of orthogonal mode shape is the orientation i.e. the

location of nodal line is indeterminate and depends usually on the position of applied excitation. th

 Excitation at any ND can have harmonics i.e. it is not necessary that only 15

harmonic excites the system at ND = 15 and the system can get excited at ND = 15 even at 25, 55, 65, 95 …… harmonics. The next obvious question is - how are orthogonal mode shapes implemented in ANSYS?

In ANSYS 5.6 for cyclic symmetric modal analysis we use two macros CYCGEN and CYCSOL. For generating cyclic symmetric B. C`s ANSYS offers CECYC and CPCYC commands. The main purpose of CYCGEN is to generate a duplicate sector. This duplicate sector is generated to take care of orthogonal mode shape and leads to occurrence of natural frequencies in pairs. Thus duplicate sector and the original sector put together ensure an orthogonal mode shape pair. CYCGEN macro has to be executed before CYCSOL. The main purpose of CYCSOL macro is to generate cyclic symmetric B.C corresponding to the nodal diameter and solve it. To generate cyclic symmetric B.C, CYCSOL uses CECYC command. Thus we need to give in the nodal diameter number or number of buckets as an input to CYCSOL macro. But we also know that at ND = 0 and ND = NB/2 we do not have an occurrence of orthogonal mode shapes and hence natural frequencies do not occur in pairs. ANSYS ensures that at ND = 0 and ND = NB/2 DOF’s of duplicate sector as generated from CYCGEN is not included in the FE Analysis. This is taken care in CYCSOL macro by constraining all DOF of the duplicate sector, for ND = 0 and ND = NB/2 respectively. At Ansys 10 and above, the command CYCGEN is replaced by CYCLIC command. This command also instructs Ansys to automatically identify the sector angle © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

27 and creates the Constraint equations between the sector end faces. The CYCOPT command should be used in conjunction to the CYCLIC command to specify the number of nodal diameters to be extracted, tolerance of variation in co-ordinates of nodes on the end faces etc. The CYCSOL is no more used in the present version and a simple SOLVE command suffices. However the total number of modes to be extracted & the mode extraction algorithm is to be specified using the MODOPT command as in earlier versions. The reason for repetition of natural frequencies is explained and we now proceed to interpret the results from a cyclic symmetric modal analysis using an Interference diagram.

3.4 Interference Diagram: In cyclic symmetric structures, natural frequency also depends on nodal diameter apart from rotating speed, material properties, geometry and applied constraints. For a given geometry, material property and applied constraints natural frequency varies with rotating speed and nodal diameter. Thus we really need a three dimensional surface ( n , ND,  ) to conduct a full-fledged investigation on risk due to resonance.

Engineers do not always prefer a three-dimensional surface. This is compounded by the fact that, it is difficult to visualize a 3D plot and it is also a time consuming © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

28 process to create a 3D plot of ( n , ND,  ) . So we need to simplify a 3D plot into 2D plot for better and quicker analysis. Thus we can either create a ( n , ND) plot or a

( n ,  ) plot. Plot of ( n , ND) is nothing but Interference Diagram and plot of ( n ,  ) is a Campbell Diagram.

3.4.1 Plotting an Interference Diagram: The interference diagram is a plot of nodal diameter along its absisca and natural frequency along its ordinate. Since interference diagram is a tool for risk evaluation arising due to resonance, we also need to express all sources of excitations in the diagram. A typical interference diagram is as shown below:

3.4.2 Interference Diagram beyond NB/2 Limit: © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

29 In chapter 2 we defined nodal diameters as the number of distinct harmonics that are required to define the complete system. Again consider the case of a rotor with 44 blades around the circumference and let us plot interference diagram beyond NB/2 limit:

Full Range Interference Diagram

Critical Nodal Diameters

NPF Lines

Running Speed Harmonics Line

7000

6000

5000

y c n e u 4000 q e r F l a r u 3000 t a N

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

2000

1000

0 0

4

8

12

16

20

24

28

32

36

40

44

48

52

56

60

64

68

72

76

80

84

88

Harmonic Number Harmonic no(NDMAX) after which natural frequency values periodically repeats.


No of Blades = 44

30 Points to be noted:

1. ND = NB/2 for even no of blades and ND = (NB-1)/2 fully defines the system. Behavior at all harmonics higher than ND = NB/2 are just a replica of behavior defined until ND = NB/2. This is possible due to the beauty of harmonic functions, which was the basic assumption in the case of cyclic symmetric structures. 2. If we map the running speed harmonic line in 0 to NB/2 range then it would be enough to completely interpret the risk due to resonance and hence carry out a full vibration analysis. The plot thus obtained after mapping is nothing but Interference Diagram.

3. It should also be noted that interference diagram is drawn keeping the speed as constant. In the present case running speed was assumed to be 60Hz.

3.4.3 Studying an Interference Diagram: This section is devoted to physical understanding of interference diagram and carrying out a risk evaluation due to resonance. The following important points are to be noted while studying an interference diagram:  Resonance will occur at all points where running speed harmonics and/or NPF

line cut the natural frequency line. But however the level of damage caused may be different.  At any nodal diameter, the harmonics that can excite the system are given by the

equation (3.2) i.e. it is not possible for any harmonic to excite any ND. There exists a relationship between the nodal diameter to be excited and the possible harmonics that can excite the system at that nodal diameter.  Experimentally or by experience the max harmonic NPF is decided. In the present

case it was assumed till 2  NPF .

 There exists no general rule for change in natural frequency with respect to nodal

diameters. In the present case we observe an increase in natural frequency with increase in nodal diameter. This may be due to the fact that the system becomes stiffer at higher modes and nodal diameter is nothing but disc mode. Some researchers have observed that shrouded assembly is more sensitive to nodal diameters than unshrouded assemblies. But strictly speaking there is no defined pattern of natural frequency variation with respect to nodal diameters.  Among all possible nodal diameters it is possible to single out a few which will

cause maximal damage to the system i.e. pump in max energy into the system. These nodal diameters are referred to as critical nodal diameters.


31



Why are critical nodal diameters really critical?

First what are critical nodal diameters? They are nothing but the nodal diameter at which both NPF, its harmonics have same forcing frequency as that of running speed and its harmonics i.e. they are the nodal diameters corresponding to the points of intersection of running speed harmonics line and NPF line. Thus the no. of critical nodal diameters that can occur are usually equal to the no of harmonics of NPF that are expected to excite the system i.e. maximum no. of harmonics of the NPF for which the risk evaluation due to resonance is to be carried out. In present case study upto 2  NPF was carried out and hence we had only 2 critical nodal diameters. Sometimes both the intersections can occur at same nodal diameter.

The main edge interference diagram possesses over other tools like Campbell diagram are due to the critical nodal diameters. Critical nodal diameters represent the interference of two waves i.e. running speed harmonic wave and NPF harmonic wave. From elementary physics, interference occurs when two waves with same frequency are in phase, the waves add up and produce a net wave, which has amplitude equal to the summation of amplitudes of both the waves. Thus in the present case at critical nodal diameters, the amplitude of force due to running speed harmonic and NPF harmonic add up and inflict maximum damage to the system, if the forcing frequency equals natural frequency. It should be taken care while designing that; the natural frequency of any mode should not cut the point of interference. In actual practice resonance condition prevails in a certain frequency zone. Thus we also need to evaluate the % of frequency margin that exists between natural frequency and forcing frequency at critical nodal diameters and compare it with design criteria. The name “Interference Diagram” itself arises from the occurrence of “interference” at critical nodal diameters. Necessary conditions, for real resonance was stated earlier. The number of nodal diameter must coincide with the number of harmonic. This is taken care in an interference diagram by the concept of interference at critical nodal diameter and hence interference diagram also helps to design the turbine rotor against the possibility of occurrence of real resonance. Thus capabilities of interference diagram are studied and an appreciation of its suitability towards better turbine blade design is presented.


32

3.4.4 Interference Diagram Vs. Campbell Diagram: In this section we make a comparative study of an interference diagram with a Campbell diagram. Campbell Diagram is a ( n ,  ) plot with natural frequency,  n along its ordinate and running speed, along its absisca. This diagram predicts where the blade’s natural frequencies coincide with the exciting frequencies (in above diagram we have plotted running speed harmonics). This condition of frequency coincidence will be termed a PROBABLE RESONANCE. Steady speed

Operating range of speed A further look at these coincident points with the knowledge of mode shape(s) and the shape(s) of exciting forces allows the POSSIBLE RESONANCE points to be identified. Figure below is a representation of such an analysis where the solid circles indicate the few POSSIBLE RESONANCES among the many PROBABLE RESONANCES of the blade frequency and force frequency intersections.

From these PROBABLE RESONANCES we need to focus our energies in avoiding REAL RESONANCE. In interference diagram it occurs at the point of intersection (or within the frequency margin) of NPF line, running speed harmonic line and natural frequency line i.e. at critical nodal diameter. Condition of REAL RESONANCE in both Interference diagram and Campbell diagram is illustrated below:


33


34

Interference Diagram showing Real Resonance

Campbell Diagram showing Real Resonance

Thus interference diagram has an edge over Campbell diagram that it readily indicates the condition of real resonance than a Campbell diagram, which on the first count shows all probable resonance conditions, which may or may not occur in actual practice. Thus a Campbell diagram can possibly misguide a designer by showing resonance’s that actually is not at all occurring. So care is needed in using a Campbell diagram while an interference diagram is foolproof.

3.5 Some Thoughts on Fatigue Life Estimation: Fatigue lifing follows a dynamic stress analysis. Blade fatigue is a multitechnological problem, which is undergoing continuous investigation by researchers. Blade fatigue failure depends on steady stress, vibratory stresses, fatigue properties of the blade material, loading history, material imperfections and environmental conditions under which the blade is operating. Precise evaluation of excitation forces in terms of magnitude and frequencies is really crucial for successful prediction of dynamic stresses. A detailed study of both steady and unsteady aerodynamics is thus essential. Turbomachinery blading is designed to have high steady stress levels. HCF occurs because of high mean stress and low amplitude vibratory loading. HCF is initiated by a formation of small, often microscopic crack. These fatigue cracks usually initiate at a zone of high stress at some metallurgical or structural discontinuity and it such critical conditions of blade operation sustain, the crack may grow and lead to a failure. It is customary design practice to use a modified Goodman diagram for fatigue studies. Various cumulative damage theories help in predicting life of turbine. Miner’s cumulative damage rule is one such theory. Researchers have presented various advanced theories and the subject is still under constant technological advancement. S-N curve is extensively used to predict the endurance limit of the material. In summary, HCF of turbomachinery blading is a significant design problem because fatigue failures can result from resonant vibratory stresses sustained over a relatively short duration of time. Blade life and hence turbine life are dependent on vibratory stress levels for a given steady operating conditions. And thus these vibratory © 2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

35 stresses must be accurately predicted. Further discussion on Fatigue of turbomachinery blading is out of scope of the present study.

Conclusions The basic objective of this technical report was to present various Concepts of Cyclic Symmetric Modal Analysis. In the process of fulfilling it, we discussed the following aspects:

 Various types of symmetry, and its merits were discussed. A very b asic discussion

on modal analysis was presented.  Principle of cyclic symmetry was outlined with emphasis on cyclic symmetric

modal analysis.  Eigenvalue problem for cyclic symmetric problems was presented.  Concept of nodal diameter was discussed at a considerable length.  Evaluation of resonance condition for a typical steam turbine rotor was discussed.

Various sources of nuisances in a steam turbine were outlined.  Condition of Real Resonance was presented.  Some aspects of cyclic symmetric modal analysis in ANSYS were discussed.

Reason for pairing of natural frequencies was explained.  Interference diagram was discussed and its use in risk evaluation due to resonance

was presented.  A comparison of Interference Diagram and Campbell Diagram was presented.  Finally some thoughts on fatigue life estimation were outlined.


36

Scope of future study Some of the related areas of future studies are details of effects such as Spin Softening, Gyroscopic effects, Temperature effects and Damping. The analysis with orthotropic and anisotropic materials is another area of interest to be exploited. Further more the study can be extended to Harmonic and Transient Dynamic analysis. All these studies can also be carried out for Gas turbines and other Turbomachinery components, both cyclically symmetric and partially symmetric. The Elastoplastic nature of components and its effect on the vibration behavior can also be studied.


37

References 1.

Arslan, M. A., “Cyclic Symmetry Analysis Procedure”, Training

Document QuEST-LLC (USA). 2. ANSYS 5.6 Structural Analysis Guide, Chapter 3: Modal Analysis. Balaji, K., Sriram, K., and Ramamurti, V., Development of User Friendly Program for Static and Dynamic Analysis of Impellers, Yet to be published. Dello, J. D., Frequency Evaluation of Steam Turbine Blade Disk, Technical Publication, Dresser Rand, Inc. Ewins, D. J., Stanbridge, A. B., and Martarelli, M., Rotating Disc Vibration Analysis with Circular Scanning LDV, Imperial College, London. Fleeter, S., and Zhou, Chenn., Fatigue Life Prediction of Turbomachine Blading, Technical Report, Purdue University. 3. Imregun, M., Structural and Aeroelastic Vibration Analysis of Bladed Systems, Ph.D., Thesis, Imperial College London, 1983. Irretier, H., Free and Forced Vibration of Turbine Blades, Institute of Mechanics, University of Kassel, Germany. Longtin, R. S., Crash Course in Modal Analysis, Training Document, QuEST-LLC (USA). th

Ramamurti, V., Computer Aided Mechanical Design and Analysis, 4 edition, Tata - McGraw – Hill, 2000. st

Rao, J.S., Turbomachine Blade Vibration, 1 edition, New Age International, 1991. Roemer, M. J., and Orsagh, R. F., Examination of Successful Modal Analysis Techniques Used for Bladed Disk Assemblies, Technical Publication, Impact Technologies, LLC (USA). Singh, M. P., Matthews, T., and Ramsey, C., Fatigue Damage of Steam Turbine Blade Caused by Frequency Shift Due to Solid Buildup – A Case Study, Technical Publication, Dresser Rand, Inc.


Cyclic Symmetric Modal Analysis

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