Modal methods for the analysis of cyclic symmetric structures

Compurrrs

& Swurrurc.~ Vol. 50. No. I, pp. 67-77. 1994 ( 1994 Ekvier Science Ltd Pnnted in Great Bntain. All rights reserved 0045-7949194 16.00 + 0.00

MODAL METHODS FOR THE ANALYSIS OF CYCLIC SYMMETRIC STRUCTURES P. BALASUBRAMANIAN,~ J. G. JAGADEESH,? H. K. SuHAst and V. RAMAMURTI~$ TGraphics, Design and Modelling Group, National Informatics Centre, New Delhi- I 10 003, India fDepartment of Applied Mechanics, Machine Dynamics Laboratory, Indian Institute of Technology, Madras-600 036, India (Received

30 July 1992)

Abstract-In this paper, the modal displacement and modal acceleration methods have been extended to the cyclic symmetry analysis in the semi-complex domain. The study of the influence of forcing frequency on the relative accuracy of the two modal methods is also made. It is shown that only few modes are sufficient to get accurate stresses in the case of modal acceleration method.

[C,l

NOTATION real force vector on jth substructure real displacement of kth DOF of jth substructure error norm which varies from 0.0 to I.0 error vector which represents the difference between the actual load vector and the load vector approximated by the mode shapes for the pth harmonic force vector of pth Fourier harmonic corresponding to the constrained equilibrium equations imaginary term: J( - 1.O) force term of rth mode of pth harmonic modal force vector of pth harmonic complex displacement of k th DOF of p th harmonic displacement, velocity and acceleration vector of pth Fourier harmonic corresponding to the constrained equilibrium equations number of modes considered for the modal analysis which is less than or equal to J displacement, velocity and acceleration terms of rth mode of pth harmonic displacement velocity and acceleration vectors in modal coordinates displacement, velocity and acceleration of the whole structure displacement, velocity and acceleration of pth Fourier harmonic for the entire structure eigenvector matrix of p th harmonic rth mode shape of pth harmonic force vector of the whole structure damping matrix of the whole structure damping matrix of the constrained system unconstrained damping matrix of the basic substructure

r”;l IA(x) J

[Kl [&I [&I Dfl M,l Wpl N

Re(x)

ITI

F-1” %

BP’ 1; p P W/Jr

bpl

modal damping matrix of pth harmonic Young’s modulus identity matrix of size J imaginary part of complex term x total number of independent DOF in the constrained equilibrium equations stiffness matrix of the whole structure unconstrained stiffness matrix of the basic structure stiffness matrix of the constrained system mass matrix of the whole structure unconstrained mass matrix of the basic substructure mass matrix of the constrained system total number of substructures in the whole structure real part of the complete term x transformation matrix Hermitian transpose of [T] damping ratio corresponding to rth mode of p th harmonic frequency ratio of rth mode of pth harmonic Poisson’s ratio forcing frequency wave propagation constant = 27tn/N mass density of the material rth natural frequency of pth harmonic real diagonal matrix of pth harmonic in which diagonal values represents eigenvalues.

INTRODUCTION

It is well known that all structures have an infinite number of degrees of freedom when subjected to dynamic loading. One has to choose an appropriate mathematical model in order to reduce the infinite number of degrees of freedom system. At the same time the significant physical behaviour of the system should be preserved. An economical analysis should minimize the computer storage and the computer execution time. Keeping the above facts in mind, one has to see whether there is any repetition of geometry.

QVisiting Research Professor from the Department of Mechanical Engineering, Concordia University, Montreal, H3G lM8, Canada. 67

68

P.

BALASUBRAMANIAN

Often one finds problems involving design with repeating substructures in the circumferential direction. These are known as circumferentially periodic or cyclic symmetric structures. Some of the examples are cooling towers, gear, flywheel, railwheel, antenna, turbine bladed disk, fan impeller, etc. It is sufficient to model the repeating substructure in order to compare the significant physical behaviour of the entire system without losing any accuracy [ 1,2]. Because the discretized model of the repeating substructure may have several hundreds or even thousands of degrees of freedom, it is genera1 practice to reduce the equations of motion to a very small number before the dynamic response is computed. The dynamic response may be expressed adequately in terms of the undamped free vibration mode shapes of lower order since the lower modes predominate and higher order ones gradually diminishing in the order of their contribution. The above scheme, the mode superposition method, is further classified as the modal displacement method (MDM) and the modal acceleration method (MAM). In this paper the modal acceleration method which employs a pseudo-static solution method [5] is extended to a circumferentially periodic structure. Study of load frequency on relative accuracy of modal displacement and modal acceleration methods is also made.

et al.

Lx,) =

(4)

The velocity {5} and the acceleration {I?,,} can be expressed similarly. The above constraints may be used to reduce the order of eqn (1) to J for the specified harmonics as W,l{L’,~ + [CPl{+PI+

KJ{r,I = &I.

(5)

Here

(64

[&I = mH[~oI[~l and

(6b)

Similarly

(74 ANALYSIS

OF CYCLIC

SYMMETRIC

STRUCTURES

131

and

The matrix form of the dynamic equilibrium equations of the full structure at a given time is given by It is sufficient to solve eqn (5) for [M](a) + [C](1) + [K](X) = ,4(t).

(1) p=o,1,2.3..

The order of the equation is NJ. The arbitrary force acting on the full system at a given time can be expanded in vector form for the substructure using finite Fourier harmonics as [4] if,) = (lIN),$,

{a,)exp{i(j - 1)~).

(2)

Equation (1) is rearranged sequentially in the same way as the degrees of freedom of each substructure. Then the equation of motion of the entire structure for the p th Fourier harmonics at the given time may be expressed as

integer (N/2).

(8)

The complex displacement vector {r,,} is used to get real displacements on full structure. The displacement of k th degree of freedom of the jth substructure is calculated as follows [4] for N odd CN I),2 &

=

rOk

+

2

c

p-1

fRetrj,k

bos{

+

Irntr,k

(j

-

l)p

bin{(_i

)

-

I )p

)I

(9a)

and for N even

Pfl{xp) + [Cl{xp) + [Kl{xp)=

j$*

(3)

d,, = rok+ (- I)(‘+ “ro:,,,

I -6 e-ItN - “PI Since all substructures are identical and the forces acting on each substructure differ from the next substructure by the same phase constant (~0, the dynamic response of each substructure must be related in the same way, i.e.

(N,Z) I

+

c ]Be(r,,)cos{(j p= I

- 1)~)

+ Pm@,, )sin{(j - 1)~ )I.

C’b)

The above is performed for all the given time steps.

69

Analysis of cyclic symmetric structures MODE SlJPERPOSlTlON METHOD

ing yields a real submatrix in the first portion of [K,,]. This portion is independent of harmonics and hence it needs to be factorized only once [6]. This saves a lot of computational effort while analysing for various harmonics. The following orthogonality conditions are satisfied by the eigenvectors of eqn (10)

At first eqn (5) can be transformed into a set of uncoupled equations through the normal modes obtained by solving the undamped free vibration equations of the substructure of pth harmonics

b/J”[~,lbpl= b:l

U la)

~~,lHPfplbJ= [Gl.

(1lb)

Because of complex constraints, [K,,] is a Hermitian

matrix. If [MO]is a consistent mass matrix, then [M,,] is also Hermitian. On the other hand if [M,] is a diagonal matrix, then [M,,] is real diagonal. Since [K,,] is a Hermitian matrix, the eigenvalues of eqn (10) are real and eigenvectors are complex. If there are more internal nodes in the basic repeating substructure, then [K,] may be rearranged according to internal nodes, nodes adjacent to boundary nodes and boundary nodes. This partition-

and

The response of vector (rP} is transformed into a modal coordinate response vector {sP} by the transformation

1= [Xpl{~,I.

FINITE ELEMENT MODEL OF ONE REPEATINGSUBSTRUCTURE; DYNAMIC LOADS ON EACH SUBSTRUCTURE ; DAMPING COEFFICIENTS

ASSEMBLY OF STIFFNESS AND MASS MATRICES ; PARTITIONING OF MATRICES _--A

. GENERATION OF FOURIER HARMONICS COMPONENTS FOR INITIAL DISPLACEMENT AND VELOCITY : LOADS FOR ALL TIME STEPS

MODAL DISPLACEMENT METHOD

MODAL ACCELERATION METHOD

Fig. I(a).

(124

P. BALASUBRAMANIAN et al.

70

h4ODACDISPLACEMENT METHOD

FOURIER HARMONICS INDEX LOOP

__L___._

__~_._

I-

- ._.- --. __-_

-

CONSTRAINING DEGREES OF FREEDOM OF BOUNDARY & AXIAL NODES _-__..I -_ EXTRACTION OF SPECIFIED NU~R OF EIGENPAIRS FROM UNDATED FREE VIERATIONS OF CONSTRAINED SYSTEM

GENERATION OF MODAL LOADS FROM FOURIER COMP.OF DYN.LOADS

-.--. :--- ’ SOLWNG FOR MODAL RESPONSE

L__-_--.

CALCULATION OF FOURIER HARMONIC RESPONSE i ,..._ _-. __

Fig.

I(b).

Similarly

$r + mp”qi,)4w f

(W,i,)Spr = qp.

(15a)

(12W where (12c) Considering proportional damping and substituting eqn (12) in eqn (5) yields

fu, 1 = bp1”i.f ai.’

(14)

Since the matrices [II], [C,,] and [wf] are diagonal, eqn (13) can be expressed into J number of uncoupled equations in single degrees of freedom as

r=1,2,3

MODAL

DISPLACEMENT

. . . . . f.

METHOD

(1%)

(MDM)

it is well known that when the system is excited, the system has to respond in one or more of its natural modes of vibration. Since the lower order predominates, it is customary to truncate higher order modes. In other words, by selecting s number of modes, the contribution to the response is restricted to the first s modes only. That is

r=l,2,3

Here s + J.

,._.,.

s.

(16)

71

Analysis of cyclic symmetric structures

Though the terms a,,, and CO,,,in eqn (15) are real, q,,, is complex and hence the terms s,,,, S,,, and $,. This leads to the following uncoupled real equations

Im(s,, 1= Wq,, )(4,/oj,

(18b)

)cos 04

where

(18~) W.$,) + (2q,,a,,,YW,,,) + (oji)Re(s,,,) = Retq,,,) (17a)

and

and

dp,

=

((1- Pj,)+ (2ap,Bp, tanet)) ((1- Sj,)’ + (2ap,&,)2}

fm(&) + (2+a,,)Im($,)

+ (w,Z,)Im(s,,) = Im(q,,). (17b)

If the loading is assumed to be varying load (harmonic), {A(t)fcos Or, then the solutions for eqn (17) are Re(+) = Re(q,,,)($/c+$)cos

or

(18a)

U8d)

For higher modes of the underdamped system, the value of &,, becomes negligible and hence the value of d,,, tends to unity. That is the response becomes static for higher modes. Further decrease in response for higher modes is also observed from the term w:, in the denominator of eqn (18). If the contributions of some of the harmonics are neglected, then eqn (5) may be considered only for the

MODAL ACCELWATION METHOD FOURIER HARMONICS INDEX LOOP

CONSTRAINING DEGREES OF FREEDOM OF BOUNDARY 8 AXIAL NODES

I EXTRACTKIN OF SPECIFIED NO.OF EIGENPAIRS AND COMPUTATION OF PSEUDO-STATIC RESPONSE I-

GENERATION OF MODAL STIFFNESS. MASS& DAMPING PARAMETERS

I READING INITIAL DISPLACEMENT AND VELOCITY CONDITIONS -

I I I

’

CALCULATION OF FOURIER HARMONIC RESPONSE AND INCLUSION OF PSEUDO-STATIC RESPONSE

Fig. I(c).

-

P. BALASUBRAMANIAN ef al

12

TIME STEPPING LOOP b

I

____.~~

RECOMBINATION OF FOURIER RESPONSES TO GET GLOBALRESPONSEONEACHSUBSTRUCTURE

___ -.___

I

CALCULATION OF INTERNAL FORCES MOMENTS AND STRESSES ON ALL ELEMENTS

,

Fig. l(d). Fig. l(a)
Flow chart of modal methods used in cyclic symmetry analysis.

selected range of harmonics. Hence the modal eqns (15) or (17) are solved for the specified range of harmonics. Then eqn (12) is used to compute the response in physical coordinates for all the harmonics selected. Finally the response of each substructure is computed by using eqn (9). The calculated response of each substructure may be used to find internal response and stresses. The above procedure may be repeated for each time step.

If {f,) is orthogonal to any eigenvector {x,,, } then the eigenvector is not at all excited. Based on the spatial distribution of harmonic load, the value of modal force q,,,varies. If any qp,value is relatively low, then the contribution of the mode need not be included when the highest frequency content of the load is far less than the natural frequency of the mode [9]. The modal forces are defined in eqn (14) as

{qpl =

b,lY&~.

(19)

APPROXIMATE LOAD ERROR NORM

The number of modes to be included depends on the spatial distribution and frequency content of {f,}.

This gives an indication how much each mode contributes in the modal analysis and hence it is called as modal participation factor.

YOUNG'S MODULUS 0.21E+OS POISSON'SRATIO

0.3

Y

Fig. 2. Annular

subjected

to in-plane

vibration

Analysis Table

of cyclic symmetric

I. In-plane

vibration

results of annular

plate

32

I

-0.361 x IO-’ (0.0)

4

5

3

5

2

5

I

5

-0.361 x (0.0) -0.325 x (- 10.0) -0.316 x (- 12.5) -0.685 x (-81.0)

-0.176 x (0.0) -0.206 x (17.0) -0.20 x (17.0) -0.342 x (-80.6)

No. of harmonics

per harmonic

SESTRA

73

In-plane element stress in x-direction in element I, x 10-l .. _~ -0.172 x IO-’ (0.0)

No. of eigenvalues Description

structures

Cyclic modal displacement

Cyclic modal acceleration

5

IO-” Io-x IO-’ IO-”

5

-0.361 x 10mH (0.0) -0.361 x lO-8

5

(0.0) -0.361 x IO--*

5 0

Radial displacement at outer edge

5

(0.0) -0.361 x l0-R (0.0) -0.361 x IO-* (0.0)

If eqn

(19) is premultiplied

on

both

sides

-0.176 x (0.0) -0.176 x (0.0) -0.176 x (0.0) -0.176 x (0.0) -0.176 x (0.0)

IO-’ IO-’ 10-l IO-’ IO-’ 10-l IO-’ IO-’ IO-’

by

Pf,JI~,J then [~,l[xpl~q,~= w/J[.u,l[~plH~f,I. Using the mass-orthonormal tors, ein (20) reduces to

property

(22)

(20)

of eigenvec-

In the modal displacement method only the lowest few natural frequencies are included. Hence the error in the representation of harmonic load is

(21) (23) i.e.

/SIMPLY

YOUNG’S

POISSON’S

MASS

MODULUS

RATIO

DENSITY

0.21E09

0.3

0.785E-05

h

CENTER

OF PLAT Fig. 3. Square

plate subjected

to transverse

vibration

SUPPORTED

P. BALASUBRAMANIANet al.

74

Table 2. Flexural Harmonic NO.

_~__.______.

vibration

Eigen No.

Natural frequency. rad/sec

2

Table 3. Square

__ ~~~ _

X X x x x x x x x x

104 10s IO” 105 103 10’ los 105 IO5 IO’

0.7157 0.3775 0.2773 0.2773 0.181 I 0.1256 0.1256 0. I 190 0.1179 0.1179

2 3 4 5 6 7 8 9 10

0.7702 0.1992 0.2605 0.3797 0.4410 0.5624 0.6143 0.6757 0.7968 0.8956

X x x x x x X x x x

104 IOf 10’ 105 IO’ 10’ 105 10’ IO’ 10f

0.9467 0.9139 0.7704 0.6811 0.6704 0.5094 0.4797 0.3771 0.3763 0.3403

1 2 3 4 5 6 I 8 9 10

0.1229 0.1537 0.3054 0.3970 0.4829 0.5136 0.6052 0.7557 0.7721 0.8644

x x X x x x x X X x

10’ 10s 105 los 10s IO’ IO5 los 105 iOf

0.9908 0.9908 0.941 I 0.941 I 0.8729 0.8729 0.8166 0.8166 0.6566 0.6566

2 3 4 5 6 7 8 9 10

plate results

Description

No. of eigenvalues per harmonic

Cyclic-MDM

10 8 5 3 2

Cyclic-MAM

plate Load error norm

0.3086 0.1536 0.2746 0.3054 0.3970 0.5136 cl6OS2 0.7421 0.7556 0.7729

1

1

____.___

.._____

I

0

results for a square

IO 8 5 3 2

for forcing

frequency

of 0 = 14,000 rad/sec

Transverse displacement (W) at centre of plate -.-

0.1377 X (-0.2) 0.1373 x (-0.1) 0.1360 x (-1.0) 0.1335 x (-2.8) 0.1293 x (-5.9)

10-a 10-e lO-4 lo-4 1O‘4

0.1374 X 10-4 W.0) 0.1374 x 1o-4 (0.0) 0.1374 x 10-e (-0.1) 0.1370 X IO-4 (-0.3) 0.1359 x 10-J (-1.1)

Average nodal moment at oentre of plate, X 10-’ -~~--~---GGx 11.9) 0.214 x (0.9) 0.207 x (-2.4) 0.198 x (-6.6) 0.188 x (-11.3) 0.212 x W) 0.212 X (0.0) 0.212 x (0.0) 0.211 x (-0.5) 0.208 x (- 1.9)

IO’ 10: IO’ 10’ IO’ 10’ 10’ IO’ IO’ 10’

Analysis of cyclic symmetric structures

75

Table 4. Square plate results for forcing frequency of 9 = 18,OOrad/sec

Description

No. of eigenvalues per harmonic

Cyclic-MDM

10 8 5 3 2

Cyclic-MAM

The error norm for the harmonic load is defined as (24) The error norm values is ranging from 1.0 (when no eigenvector is selected) to 0.0 (when all eigenvectors are included).

Transverse displacement ( W) at centre of plate

Average nodal moment at centre of plate x 10-j

-0.7120 x IO-’ (-0.1) -0.7138 x IO-’ (0.2) -0.7270 x 1O-5 (2.0) -0.7544 x 10-5 (5.9) -0.8095 x IO-’ (13.6)

-0.518 (-4.9) -0.617 (-1.9) -0.685 (8.9) -0.783 (24.5) -0.916 (45.6)

-0.7126 x IO-* (0.0) -0.7127 x IO-’ (0.0) -0.7142 x IO-’ (0.2) -0.1799 x 1o-5 (1.0) -0.7435 x 1o-5 (4.3)

-0.629 (0.0) -0.630 (0.2) -0.638 (1.4) -0.658 (4.6) -0.715 (13.7)

Several versions of MAM have been published [3, 71.One of the most efficient methods for the numerical evaluation of modal equations has been described by Anagnostopoulos [8]. This method is extended to the complex modal eqn (15). Equation (I 3) is rearranged as

Therefore MODAL ACCELERATION METHOD (MAM)

As discussed in the previous section, the term d,,, becomes unity and response becomes static for higher modes. On the other hand the modal contribution to the spring force is equal to the product of the modal displacement response and the modal stiffness (wi,); this leads to a more significant contribution from higher modes. Hence the static effect of higher modes are to be included with the response of lower modes in order to calculate sufficiently accurate forces/ stresses. This method is known as modal acceleration or static correction method. In general, if only a few lower order mode shapes are used in MDM for the dynamic response analysis of linear structures, the accuracy obtained for the displacement is quite satisfactory. On the other hand, more mode shapes are needed to compute the internal forces and stresses to the same accuracy. Frequently the convergence rate of higher modes in the undamped free vibration analysis is slow. The error introduced by higher order mode truncation can be minimized or controlled by MAM [3]. In MAM the effects of truncated modes are considered by their static effects only and only a fewer modes are needed from free vibration analysis because of better convergence properties of MAM.

-~~,l~~~l~~p~~~~~l~~p). + {$,)I. Substituting

(26)

eqn (11) and eqn (14) in eqn (26) gives

{rpJ = W,l-‘If,) - [x,lbjl-‘(Km,)

+ &‘,),. (27)

Rearranging

eqn (I 3)

Hence eqn (27) can be expressed as

The first term in the above equation is the pseudo static response which accounts for the higher modes truncated. The second term is the dynamic effect of the participating dynamic modes. The number of natural frequency modes used in eqn (29) may be

P. BALASUBRAMANIAN er ul

76

restricted as

to s or even less. Hence eqn (29) is modified

pseudo-static correction required result.

alone is enough

to get the

E.uumplu 2

The pseudo static response in eqn (30) can be computed by static solution and the dynamic response (second term) by modifying MDM. The flow chart is given in Fig. 1. If the magnitude of load on the full structure and not the spatial distribution varies with time then the pseudo static correction can be done with a very reduced computational effort as it is calculated at each time by applying only the corresponding scaling factor for each harmonic. EXAMPLES

Two examples representing the in-plane and llexural vibration problems have been considered. The units used in the examples are in kg, N and mm. The results have been obtained in CYBER 180/830 machine using double precision real and single precision complex arithmatics available in CYBER machine. The load is considered to be cosine varying.

Example I In order to validate the theory, a 45” segment of an annular disk with an in-plane concentrated radial edge load as shown in Fig. 2 is considered. The inner edge of the disk is completely fixed and constant strain triangle [lo] element is used. Here the results of the 45” sector obtained using cyclic symmetry theory is compared with the results obtained by the analysis of full structure by the conventional super element/substructure method using SESAM software developed by Veritas Sesam Systems, Norway [ 1 I]. The 45’ sector shown in Fig. 2, analysed by using cyclic symmetric theory has four elements and six nodes; whereas the full structure analysed by SESAM software has 32 elements and 24 nodes. After applying boundary conditions and complex constraints, the size of the eigenproblem is four in cyclic symmetry analysis; the complex constraints are applied five times and 20 natural frequencies are extracted. The whole structure model used in SESAM considers 32 natural frequencies and some of them have multiplicity of two. All the natural frequencies obtained from the present analysis agree with SESAM results. natural frequency values vary The from 76.78 x IO’ rad/sec to 438.90 x IO’ radjscc. The results are presented in Table I. It can be observed that dynamic response results obtained from MDM and MAM are in good agreement with those obtained from SESAM software. Since /j,,, values are ranging from 0.13 x IO ’ to 0.23 x 10 ‘. the value of d,,, approaches unity and hence the modal response given by eqn (16) becomes static for all natural frequencies. Because of this effect. the

The effect of the excitation frequency is considered next. One quadrant of a simply supported square plate is modelled with a DKT element [ 121 as shown in Fig. 3. The mesh contains 128 elements and 81 nodes. The application of complex constraints yields the matrix of size 170. Since the structure contains four quadrants/substructures, the first three harmonics, namely 0, I, 2 are considered. The loading is considered to be acting on the shaded region of the first quadrant. The first ten natural frequencies and the corresponding load distribution error norms are shown in Table 2. It can be observed from Table 2 that some of the frequencies are not contributing to the modal analysis; for example, 7th and 10th natural frequencies of fundamental harmonic. The results for the forcing frequency of 14,000 rad/sec is presented in Table 3. Two test cases were run for I.5 and IO eigenvalues and the values of transverse deflections and average nodal moment at the centre of the plate were found to be 0. I37 x IO ‘. 0.213 x IO’ and 0.137 x 10m4. 0.212 x lo’, respectively for the MAM method. In view of the negligible difference between the results for the cases with I5 and 10 eigenpairs using MAM method, the results have been tabulated for IO eigenpairs only. The modal acceleration result which includes IO eigenpairs for the analysis is taken as the reference and the percentage errors (shown within the brackets) are calculated for the displacement and the average nodal moment at the centre of the plate. Since the forcing frequency lies between first and second natural frequency, it is sufficient to include the dynamic effects of first two modes and static correction for higher modes when MAM is opted. When static correction is not included in MDM the error is higher than that of MAM. Further it is observed that the error in average nodal moment is higher than the error in nodal displacement. This can also be verified from eqn (16); the square of the frequency ((I,$) is appearing in the denominator and hence the eflect of higher modes are negligible. On the other hand. the modal spring force is equivalent to the product of modal displacement (.s,,,) and the modal stiffness (co,?,,). Hence the effect of higher modes arc more significant for calculating forces. moments and stresses. Table 4 shows the results for the forcing frequency of 18.000 rdd/sec which lies in between second and third natural frequencies of fundamental and second harmonics. It is seen that the effect of static correction is mom predominant. CONCLCSION

The cyclic symmetry is the best tool available for the dynamic analysis of structures which have rota-

Analysis of cyclic symmetric structures tionally periodic symmetry. The calculation of eigenpairs for the modal analysis is reduced by considering only the constrained equations of small size. The node renumbering [6] yields further saving of computational efforts. Though few lower modes are enough to obtain sufficiently accurate dynamic displacements, the force/moment/stress calculations require a higher number of modes in MDM. Whereas the effects of higher modes are included by the use of static correction in MAM and hence fewer modes are sufficient for the calculation of forces/moments/stresses.

77

3. R. R. Craig, Jr, Structural Dynamics; An Introduction to ComDufer Merhod.7. John Wilev. New York (1981). 4. A. J: Fricker and S. Potter, Transient forced‘vibration of rotationally periodic structures. Inr. J. Numer. Meth. Engng 17, 957-974 (1981).

5. V. Ramamurti and M. Ananda Rao, Dynamic analysis of spur gear teeth. Comput. Swucf. 29, 831-843 (1988).

6. P. Balasubramanian, H. K. Suhas and V. Ramamurti, Skyline solver for the static analysis of cyclic symmetric structures. Compuf. S/ruct. 38, 259-268 (1991). 7. N. R. Maddox, On the number of modes necessary for accurate response and resulting forces in dynamic analysis. J. appl. Mech., ASME 42, 516-511 (1975).

authors gratefully acknowledge the help and encouragement given by Dr B. K. Gairola, Head of Graphics, Design and Modelling Group, National Informatics Centre for carrying out the above study. Acknowledgement-The

REFERENCES I.

D. L. Thomas, Standing waves in rotationally periodic structures. J. Sound Vibr. 31, 288-290 (1974). 2. D. L. Thomas, Dynamics of rotationally periodic structures. Inf. J. Numer. Meth. Engng 14, 81-102 (1979).

8. S. A. Anagnostopoulos, Wave and earthquake response of offshore structures; evaluation of modal solutions. J. Srrucf. Dia., ASCE 108, 21752191 (1982). 9. E. L. Wilson, M. W. Yuan and J. M. Dickens, Dynamic analysis of direct superposition of Ritz vectors. Earthquake Engng Srruct. Dyn. 10, 813-821 (1982). IO. 0. C. Zienkiewicz, The Finite Element Method. Tata McGraw-Hill, New Delhi (1979). Il. SESAM-SESTRA User’s Manual. Veritas Sesam Systems, AS., Novik, Norway (1988). 12. J. L. Batoz, K. J. Bathe and L. W. Ho, Study of three node triangular plate bending elements. Inr. J. Numer. Me/h. Engng 15, 1771-1812 (1980).