Finite element analysis of non-linear vibration of a circular plate with a concentric rigid mass

Journal ofSound and Vibration (1989) 131(2), 215-227

FINITE ELEMENT ANALYSIS OF NON-LINEAR OF A CIRCULAR PLATE WITHA CONCENTRIC RIGID MASS C.

L. D.

VIBRATION

HUANG

Kansas State University, Manhattan, Kansas 66506, U.S.A. AND

S.

T.

HUANG

Zheng Zhou Institute of Technology, Henan, People’s Republic of China (Received 24 May 1988, and in revised form 27 September 1988)

A finite element analysis of non-linear vibration of an elastic circular plate with a concentric rigid mass is examined for different cases of boundary conditions. The non-linear partial differential equations and the associated boundary conditions are derived from an energy method, by using the Hamiltonian principle. The applications of the finite element method to the dynamic problem rely on the use of a variational principle to derive the necessary element properties or equations. The assembled equations for the plate are

formed by summing each element equation obtained in consideration of a single element. Then, the boundary conditions are imposed on the vector of nodal field variables, so that the appropriate boundary conditions are satisfied. The assembled equations form an eigenvalue problem and are solved for the eigenvalues as well as the unknown field variables. The relations between the fundamental frequencies and the amplitudes of non-linear vibrations of the circular isotropic elastic plate with a concentric rigid mass are obtained. The results agree well with the solutions obtained by the Kantorovich averaging method. Also, the results yield the upper bounds of the solution to the stated problem.

1.

INTRODUCTION

The transverse oscillations of a thin circular plate carrying a concentric rigid mass are found in many engineering applications, ranging from telephone receiver diaphragm oscillations to structures supporting vibratory machinery and to modeling printed circuit boards. Since vibrations may be disastrous, reliable predictions of their nature are of great importance. The amplitude of vibrations may be of sufficient magnitude to result in malfunction of delicate components. When the amplitude of vibration is of the same order of magnitude as the thickness of the plate, classical linear plate theory must be extended to include the membrane effects of the middle plane. Considering this membrane effect results in a set of two basic equations, well known as the dynamical von Karman equations, which are non-linear and coupled. Due to the complexity of the governing equations, the only present means of solution is by numerical approximate methods. Employing various approximate numerical methods, several authors have examined the vibratory characteristics of circular plates of constant thickness [ 1,2]. However, limited attention has been given to the large amplitude vibration of supported plates with a rigidly attached concentric rigid mass. 215 0022~460X/89/110215+

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@ 1989 Academic F’ressLimited

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C.

L. D. HUANG

AND

S. T. HUANG

Handleman and Cohen [3] studied the effects of adding a concentric rigid mass to a clamped circular plate. Small amplitude vibration response curves were obtained for various mass and radii ratios by employing a minimum principle. Extending the problem, Laura and Gutierrez [4] solved the linear problem of variable thickness circular plates carrying a concentric rigid mass by means of the Ritz method. Defining a thickness ratio and a flexibility parameter, they obtained frequencies for various boundary conditions. The solutions for the axisymmetric non-linear vibrations of a thin isotropic circular plate carrying a concentric rigid mass have been obtained by a Kantorovich averaging method [ 51. The present investigation is concerned with the axisymmetric non-linear vibrations of an isotropic circular plate with a concentric rigid mass by the method of finite elements. A non-linear eigenvalue problem in finite element equations is formed. Numerical solutions are obtained for the stated problem. Free vibrations of the plate-mass system are investigated for various mass and radius ratios, and the corresponding fundamental frequencies are presented. The same result as in the prior work [5] is obtained for the cases in the neighborhood of the linear oscillation. For large amplitude, the solutions obtained by the method of finite elements yield the upper bound solutions. Solutions obtained previously by a Kantorovich averaging method [5] are also given in the figures for comparison.

2. BASIC DIFFERENTIAL EQUATIONS Consider a flat circular plate having an outer radius a, constant thickness h, and an attached concentric rigid mass, M,. The radius of the rigid mass is b and equals the inner radius of the plate. Let the origin of polar co-ordinates (r, 0, z) be located at the center of the middle plane of the plate, as shown in Figure 1. The plate material is assumed to be elastic, homogeneous, and isotropic. By Hamilton’s principle, it can be shown that the non-dimensional equations of motion for a finite amplitude axisymmetric vibration of a circular plate with a concentric rigid mass as shown in Figure 1 are

where subscripts denote partial derivatives, except MC, which in this paper denotes the total mass of the rigid core. The dimensionless variables X=w/a, y = MC/ rb*ph 3

5=rla, U= u/h,

R=b/a, T = t(D/pha4)1’2

have been used to transform the governing equations of motion and the boundary conditions into the non-dimensional forms. The symbols u and w are the radial and transverse displacements of the middle plane, respectively. D is the flexural rigidity, p is the density of the plate material, v denotes the Poisson ratio and f is time.

CONCENTRIC

MASS-LOADED

h

CIRCULAR

217

PLATE

I

Fixed edge (Ff

Figure 1. Plate-mass

The associated boundary Figure 1 are as follows:

conditions in non-dimensional

form with plates shown in

(b) immovable hinge at 6 = 1

(a) movable hinge at ,$= 1 x =o,

systems.

x =o,

x,,+--x,=0, ;

x,,+-x,=0, ;

u,+ju+; 0f x:=0;

u=o; (d) connected with rigid mass 5 = R

(c) fixed edge at 5 = 1

x,=0,

x=0,

u =o,

x,=0,

u = 0; -;R’X_=O.

3.

FINITE

ELEMENT

(3)

ANALYSIS

An exact solution to the problem defined by equations (l), (2), and (3) is at present unknown. Thus, the analysis of this problem is accomplished by numerical approaches. Approximate solutions to large amplitude vibration problems are commonly obtained by

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C. L. D. HUANG AND S. T. HUANG

using functional space methods to eliminate the space co-ordinate with an assumed mode shape function [l]. Thus, the non-linear partial differential equations are reduced to a set of non-linear ordinary differential equations, with time T as an independent variable. On the other hand, a harmonic time function can be assumed [5] and then a Kantorovich averaging method employed to reduce the non-linear partial differential equations to a set of ordinary differential equations, with 5 as an independent variable. However, in this investigation, the problem is studied by the method of finite elements. Most applications of the finite element method to dynamic problems rely on the use of a variational principle to derive the element properties or equations. The energy approach is used in this paper. Since the plate problems considered herein as shown in Figure 1 are axisymmetric, the ring elements are chosen. The normalized region of the plate [I?, l] is divided into n finite elements of a width 2S, where S = (1 - R)/2n. A generic element is shown in Figure 2. Each element has three nodes (two exterior nodes and one interior node), and each nodal point has three degrees of freedom (Xj, Xj~, Uj)“‘, presenting the transverse deflection X, slope X,, and the in-plane displacement U at the jth nodal point of the ith element. The values of displacements and slopes within the ith element are expressed as {X(Q where { W"'}

T)} = LA+‘] { W”‘},

{ U”‘(& 7)) = [L”‘] { V”‘} ,

(da, b)

and {V”‘} are nodal variable vectors, defined as

Also [NC"] and [I,“‘] are row vectors of shape functions which play a most important role in all finite element analysis. Hermitian functions are used for interpolation, and then the shape functions are (see the Appendix),

l-4~-s 1: 1

2 4

I4

x

{$:;I=

;

-$ ;

2

-a

1

0

0

x5\

a

0

0

x4

01 0

01 0

x2 ’ x3 x1

-20

_;

5 4

.k;;!_ii-Y;

1

j+(i).

Their derivations are shown in the Appendices.

Figure 2. A generic finite element.

O xo

(5b)

CONCENTRIC

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CIRCULAR

If the whole domain of the plate contains n elements, the representation variable over the whole plate is given by X(5, r) = i [N$“] { W:=‘}= N’W,

219

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of the field

U(& T) = i [L!“]{ Vje’}= LTV,

i=l

(6a, b)

i=l

where WT

(lx2(2n+1))

= (Xl, X(,)6, . . . , X(Zn+l),X(2n+l)O

represents the unknown nodal values of the transverse deflections and slope, and where VT

= (Q,

(lx(zn+l))

u2, - * - 9 q2n+ld

represents the unknown nodal values of the in-plane displacements. Consider the strain-displacement relations for an elastic media with large deformations (von K&man’s plate theory). The strain vector {E} is expressed as

{E} = (4X1)

_i;

+

1,1

=(f~}+{~~}=(E.)+(E.)(

(7)

0 -w*lr Ill where {ei} and {E”} are linear and non-linear strain vectors, respectively. dimensional forms of the strain parts E;, EL and E; are i

and

9

E;=

The non-

(8)

For an isotropic elastic material, the elastic constant matrix is

,

(9)

where

and z is the distance from the middle plane of the plate along the direction of thickness. To establish the global stiffness and mass matrices, the potential energy and the kinetic energy are determined. By using equations (6)-(9), the total strain energy for the plate is obtained as

n=+

J

E~CE d V

V

={t~v~‘TC~fdV}+{~Iv~E’TCE.f+E”TCF’+a’~TCEl]dV}=~‘+~~,

(10)

in which the first term is a quadratic function representing the linear part of the strain energy, and the second term is the non-linear part of the strain energy. After substitution of equations (8) and (9) into equation (lo), and integration through the thickness of the plate h, the linear and non-linear parts of strain energy can be written as E~~C,E~ ds+-

s

D 2

a2

Is

EbTC,&bds,

[&bTc,&;+ Ep"=C,E;+E;~C&']

ds.

(114 (lib)

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C. L. D. HUANG

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S. T. HUANG

After substitution of equations (6a) and (6b) into equation (S), the strain parts appearing in equation (11) can be written as E’ =-

h LT

v

’ a 1 L’/t I ’

&=

-i{

$}w,

“:={y}N;W

(12)

Thus, their first variations are

Now, the first variation of the total strain energy iV7 is

.

(14)

By use of the expressions (12) and (13), equation (14) can be written as

where Kp =

LcL;+;

LL;+;

LcLT++

LLT

f

Ni.N'K++

1

ds,

N,N;

K2” = I, (X:/2)

1

ds,

N,N;

ds.

The total kinetic energy of the plate can be determined as

(16) where and

NNT ds + yrR2( NNT)

M= IS

By applying the Hamilton principle, the Euler-Lagrange be obtained and written in the matrix form

(:

LIM){

i}+(12?

Dib

. c=R

equations for the plate can

ll;~ci~,){;}=o’ h2

2

or in a reduced form as K,V+;

K:‘W=O,

M++KbW+6;KfTV+12

0 ;

2

K;W=O.

(180)

CONCENTRIC

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PLATE

221

Equations (18) are the motion equations describing the non-linear free vibrations of an isotropic circular plate with a concentric rigid mass. There are a number of ways to impose the boundary conditions on equations (18). However, it is most convenient to introduce the known nodal variables in a way that leaves the original dimensions of equations unchanged and avoids major restructuring of equations. The boundary cases are as follows: (a) hinged movable case,

VT=(0, u2, 4,. . ., u2,, u*“+l), WT=

(X(l),

(b) hinged-immovable

0, X(2),

X(2)5,.

. * 9 X(2”),

X(2n&

0, X(2n+1)5k

(194

case,

VT=(0, u2, u,,-**, U,“, Oh WT=

(X(l),

0,

X(2,,

X(2)5,

* * * 9

X(Zn)*,

03 X(2n+d

(19b)

(c) clamped case VT= wT=

WC,,,

(0,

u2,

0, X(2),

us,

* * -,

X(2)&

u2n, (0,

* *. 3 X(2”)&

090).

(I9c)

Because of the boundary conditions, the coefficient matrices of W and V must be properly modified so that the original dimensions of coefficient matrices remain unchanged. The procedure for modification is given in the standard textbooks on finite element methods.

4. COMPUTATIONAL

PROCEDURES

To determine the responses of non-linear vibration of the circular plate, or to obtain the solutions of equations (18) and (19), the procedure listed below has been followed. (1) The values of the assembled mass matrix M, the linear bending stiffness matrix Kb, and of the linear in-plane stiffness matrix KP are determined first. (2) For vibrations of infinitesimal amplitude, equation (18b) reduces to a linear motion equation, namely MW+ Kb W =O. Then, the fundamental natural frequency w and the associated eigenvector W can be calculated numerically. It should be noted that the eigenvector W contains not only the transverse nodal deflection Xi but also the angle of rotation Xci)c. To obtain a unique solution for the mode shape, an arbitrary small deflection A, is imposed at the first nodal point, i.e., X, = A. Thus, the deflection mode of the plate W=(A,X,,X, ,..., X,,, 0) is obtained. In order to obtain a better approximate slope function interpolated in a generic finite element, the set of calculated values XCijE is disregarded from W and replaced by a new set of values X&, which are obtained by using the technique of second order interpolation (see the Appendix). The new set of slopes at each ith nodal point is calculated by X&e = 2&i + b, where a = (Xi+1 -2Xi

+ Xi_*)/2S2,

b=[(Xi-Xi-1)/S]-a[&+6-11.

(3) By using the function Xc,, the values of non-linear stiffness matrices KY and Kq can be calculated.

222

C. L. D. HUANG AND S. T. HUANG

(4) Upon using the appropriate boundary condition for V given in equation (19) and equation (18a), the in-plane nodal displacement values V are determined as v= -(a/h)(K,*)-‘K;‘W, where Kf is the modified matrix of Kp after imposing boundary conditions on K (5) After introducing the V vector into equation (18b), the motion equation yields the eigenvalue problem Mti+(K,+AK)W=O,

where AK = 12(4h)7zc;-f(~;‘)~(~,*)-~(~;)I.

The fundamental eigenvalue with the amplitude ratio X, = ( W,/h) = A and its eigenvector W are calculated by use of an iteration scheme. (6) By increasing X, to nA (n = 2,3, . . .) and repeating the previous steps, the relation between the fundamental frequency and the amplitude is determined.

5. RESULTS OF NUMERICAL

COMPUTATION

The results obtained by the method of finite elements are plotted in Figures 3, 4, and 5, as shown in solid lines. The plate-mass system behaves as the hard-spring system of the non-linear Duffing equation.

4

6

8

10

12 w

14

16

18

20

Figure 3. Non-linear free frequency responses; y = 1.5. M, movable; IM, immovable.

Same problems have been studied by using a ICantorovich averaging method [5]. For comparison, these solutions are also presented in Figures 3,4, and 5 as dotted lines. For infinitesimal amplitude vibrations, solutions obtained by both methods agree identically. For non-linear cases, the frequencies obtained by the finite element method are higher than those obtained by the Kantorovich averaging method. The reason is that the eigenvalues obtained by a finite element method, in general, yield the upper bounds of the exact solution, just as the Ritz method yields the upper bound solutions for eigenvalue problems. In fact, from the mathematical point of view, the finite element variational method is equivalent to the Ritz method.

CONCENTRIC

MASS-LOADED

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223

PLATE

1.6

Figure 4. Non-linear free frequency responses; y = 2.0. Key as Figure 3.

1.6 -

1.2 T

t r 0.8 -

0.4 -

-4

6

8

10

12 w

14

16

18

3

Figure 5. Non-linear free frequency responses; y = 3.0. Key as Figure 3.

When a rigid mass occupying a finite area is concentrically added to a thin circular plate, the effect upon the dynamic responses, even for linear cases, is not immediately apparent. Insertion of a rigid mass produces an increment in kinetic energy which must match with a change in potential energy due to the additional stiffening effect at the inner boundary. Besides, the stiffness of an annular plate depends on the radius ratio. Thus, the behavior of the plate-mass system depends upon not only the specified rigid mass ratio but also the radius ratios of the system under consideration. An examination of the linear vibration characteristics is found to facilitate the study not only of the linear cases but also of the non-linear vibration of plate-mass systems. Accordingly, the linear problem has also been solved by the method of finite elements for various mass and radius ratios. The results agree identically with previous findings [5]. The dimensionless linear frequencies are presented as functions of radius ratio in Figure 6. Results are valid for both immovable and movable edge constraints, because the effects of in-plane forces in linear cases are negligible. For mass ratios of less than 2.0, the stiffening produced by insertion of a rigid core of any radius will increase the natural frequency of the system. Increasing the radius ratio simultaneously increases the stiffness and the frequency of the plate-mass system. For mass ratios greater than 2.0, there exists a critical radius ratio for which the frequency

C. L. D. HUANG

224

1.51 00

I 0.1

I 0.2

AND

, 0.3

S. T. HUANG

1 0.4

I 0.5

I 0.6

0.7

R Figure 6. Linear frequency of hinged plate-mass systems; 1W,/hl + 0.

decreases for radius ratios less than the critical radius ratio. Hence, the effect of additional mass is seen to have more influence than the increase in stiffness on the frequency. The frequency increases as the radius ratio is increased beyond the critical value. Here, the influence of the stiffening effect becomes dominant over any addition of mass. For the non-linear vibration cases, Figures 3, 4, and 5 illustrate how the effects of the concentric rigid mass and radius ratios, and the edge constraints, influence the response of the plate-mass system. Responses of the plate-mass system are the same as those of the hard-spring Duffing system.

6. CONCLUSIONS Utilizing the method of variational calculus facilitated the derivation of the governing differential equations of motion, geometric and the natural boundary conditions. By the principle of variation, the global governing non-linear equations for the transverse and in-plane displacements were derived in terms of nodal point variables. Results for the responses of a plate-mass system were obtained by using a computational algorithm. This method can easily be extended to investigate forced vibrations and stress distributions of a plate-mass system. The non-linear behavior of the plate-mass system under consideration is found to be dependent upon the specified mass, radius ratios, and the amplitude ratios W,/h. Agreement was obtained with previous results [5] for all cases. With the edge constraints considered, characteristics exhibited by the responses of the plate are similar to those of a hard-spring Duffing system. The findings of this investigation yield some essential information on the behavior of plate-mass systems. With the edge constraints analyzed being mathematical idealizations, Figures 3, 4, and 5 are valuable to the designer in guiding estimates of the frequency response of a thin circular plate carrying a concentric rigid mass. REFERENCES 1.

J. L.

NOWINSKI

ASME,

325-334.

1962 Proceedings of the Fourth U.S. National Congress of Applied Mechanics, Non-linear transverse vibrations of circular elastic plates built-in at the

boundary. 2. A. W. LEISSA 1969 NASA SP-160. Vibrations of Plates. See pp. 7-19.

CONCENTRIC MASS-LOADED CIRCULAR PLATE

225

3. G. HANDELMAN and H. COHEN 1957 Proceedings of 9th International Congress of Applied Mechanics. 7, 509-516. On the effects of the additions of mass to vibration systems. 4. P. A. A. LAURA and R. H. GUTIERREZ 1981 Journal of Sound and Vibration 79, 311-315. Transverse vibrations of annular plates of variable thickness with rigid mass on inside. 5. C. L. D. HUANG and H. S. WALKER, JR. 1988 Journal of Sound and Vibration 126, 9-17. Non-linear vibration of a hinged circular plate with a concentric rigid mass.

APPENDIX

A.l.

SHAPE FUNCTIONS N$” AND Lie’

Consider

t generic element with domain 0 = [b, [i+l]. The element 0 is transformed

linearly to 0 = [ 17= -1, 77= l] by 77= (,$-ei+l)/S. (l),

(2), and (3). The following

polynomial

The nodal points are desginated

functions

as

are chosen as the interpolation

functions:

X(~)=A$+B~4+C”r73+D~2+E~+F, u=H7q+Ir]+J.

X,,(q)=5A~4+4B~3+3C~2+2D~+E,

The coefficients (A, B, . . . , I, .I) can be expressed in terms of the nodal-point ( w”qT Therefore,

=

(X, , X(,jq, . . . ,x3,

the transverse deflection

X(3,?)

X”’

and

( VCeQT= (U,,

and the in-plane displacement

u,,

variables

U3).

U”’ are written

as x”’

= NT( W”‘)

= (T, Y1)‘( WC”),

Uce) = LT( V”‘)

= ( T* Yz)‘( V”‘),

where

A.2. LINEAR STIFFNESS MATRICES Kze’ AND Kb(e’ With the given shape functions N and L, the linear stiffness matrices Kb(=’ and K:” can be calculated by integration. (a) For the linear bending stiffness K$“,

=

T

1

2-6’‘T=1,

C.

226

I -800 7

L. D. HUANG

AND

HUANG

-485iil 144 5 S3 s2 1

-288 _480 57 6+1 s3 s2 1

&+I s3

S. T.

24%

,,btl 4S S3

000

0

8 -T S

0

0

0

0

0

0

,&l

Sym

S3

\

0

\ 40 7 S

3

2o S

0 24

0

sz

0 12 1 S

0

+2?rv

0

0

8 1 s

O

0

0

4 i S

0

0

0

-5

0 Sym

0. 25A,

20A, 16A,

+21r/s2

15A6 12A5 9A,

lOAs 8A4 6A, 4Az

Sym

5A, 4A, 3A2 2A,

0 0 0 0

A,

0 0

Here k-l Ao=

c

M&+z/5i),

(i)k-i($L)i+(_l)*

(-$+

c, =k(k-l)...(k-i+l) k

(b) For linear

(%)*jnf]:_‘,

i=O

in-plane

i!

stiffness

3

k=1,2

,...,

i,

Kb(<),

where ’ &

,Sw

&

A2

A2

A,

.

A0

(c) For the non-linear stiffness matrices KY”’ and K$e), the analytical slope function X, must be reconstructed for the purpose of integration. Thus, the second-order polynomial function is used as the interpolation function in a generic element X = a[‘+ b.$ + c,

CONCENTRIC

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where the coefficients can be expressed in terms of the nodal variables (Xi, Xi+*, Xi+*). Hence the slope function in the element is XF) = 2~5 + b, where a=(1/2S2)(Xi+z-2X,+1+X,),

b = (llS)(Xi+l

-Xi) -c(5i+l+

5;).

With the Xf), the first non-linear stiffness matrix K rce) is obtained by integration,

where SaS+- 16 -C, 5 s

yaS+-

%2

\o

0

Here

$S+-- 8 C, 3s 2G

0

0

J$aS fC, $C, YaS

faS $C,

$C, ZaS

2G

YaS

!aS

2C,

(

Cl = 2&+, + Mi+,

!C,

S

YaS

+Z,

+~PV

yc2

2C,

4C, 3 gzs+-

0

2G

S

00

0

C2 = 4a&+l+ b,

C, = 2a&+I + b.

The second non-linear stiffness matrix K,““’ is

where

Here C, = 12a2&+, + 4ab,

C2 = 4a25:+,, +4abtf+, + b2&+, , Cd= 12a2[f+,, + 8ab&+l+ b2.

C, = 4a2,

0