A mixed boundary integral — Finite element approach to plate vibration problems

MECHANICS

RESEARCH

COMMUNICATIONS

0093-6413/80/030141-10502.00/0

7(3),141-150, 1980. Printed in the USA. Copyright (c) Pergamon Press Ltd.

Vol.

A MIXED BOUNDARY INTEGRAL - FINITE ELEMENT APPROACH TO PLATE VIBRATION PROBLEMS

G. B~zine Laboratoire de Mgcanique des Solides E.R.A. 2|8 - Universit@ de Poitiers 40, Avenue du Recteur Pineau, 86022 Poitiers, France.

(Received 4 February 1980; accepted for print 12 April 1980)

Introduction

~Cnile for static plate problems the boundary integral method is now well known, it has not been used as much as might be expected for vibration problems, due to the complexity of the fundamental solution to the plate vibration differential equation. Indeed a treatment of this problem by boundary integral equation method has already been proposed in the works of Vivoli and Filippi [I, 2] through the research of a fundamental solution of the differential equation (A2v - %2v = 0). This formulation involves Hankel functions and its numerical computation is rather awkward. An easier approach is desirable,

and it is the purpose of this paper.

We propose to represent the inertia loading due to the vibrations by concentrated l o a d s a p p l i e d at each node of a mesh used to discretize the plate domain. By using the integral representation of the s t a d c deflection and that of its normal derivative along the boundary, and simultaneously the representation of the deflection inside the domain, it is possible to eliminate the boundary unknowns, so as to obtain equations which only inw~ive deflections and inertia forces inside the domain. The investigation of sinuso[dal vibrations (free vibrations) reduces to computing the eigenvalues and eigenvectors of some matrix. Problems pertaining to clamped, simply supported or cantilever square plates are treated to illustrate the potentialities of this method.

Integral formul.ation of a plate vibration .p.roblem

Let S be the domain occupied by a rectangular plate and let F be its boundary. According to Kirchhoff's

theory of thin plate vibrations,

deflection w is governed by the differential equation : 2 AAw + ph ~ w _ O in S D 2 ~t

141

the transverse

(I)

142

G. B E Z I N E

where D denotes the flexural rigidity h

is the thickness of the plate

0 is the mass density of plate material 32 32 A denotes the Laplacian -- + -(~x 2 ~y 2) Since in the following we confine our study to free harmonic vibrations,

the

deflection w must also satisfy some homogeneous boundary conditions on the edge of the plate. When the plate is undergoing sinusoldal vibrations with circular frequency %, the inertia force per unit mass is : 32w and the differential equation becomes AAw - ~

~2w

(2)

~2w = 0

(3)

:

At a given instant, the inertia loading (+ oh %2w) produces the same deflection as a static load per unit area of magnitude p(p = ph %2w) would, which entitles us to use the fundamental solution of the static problem. The formulation of such problems is now well established the following equations

[3], [4], and it leads to

:

B w(P) = JrS v(P,Q) ~

%2w(Q) dSQ - ~! I F [Kn(V(P,Q)) w(Q) - Mn(V(P,Q))

3--~w(Q) + T~n3V(P,Q) Mn(W(Q)) - v(P,Q) Kn(W(Q))]. dSQ - ~

lw(Ai) Mnt(V(P,Ai) ) - Mnt(W(Ai) ) v(P,Ai) ~

N ~ i=~ (4)

with B = !

if

P e S

(5)

B = ~

if

P K F

(6)

and 1 ?w (p) = 2 ~n p

3v

IS Tn--np

%2w(Q) dSQ

1

- ~ IF

[~ ---n-n (v(P,Q)) w(Q) p

~Mn (v(P,Q)) ~w (Q) + ~2 v (p,Q) Mn~W~p,~, ~ ~ _ ~v / /o~l ~n ~ ~n ~ ~ Kn.W.~..~ P P P N ~M dSQ - ~~ E ~w(A i) ~ nt ( v ( P , A i ) ) - Mnt(W(Ai)) ~~v (P,Ai) ~ i~ p ~

PLATE VIBRATION PROBLEMS

143

with P K r

(7)

where Q i s

a point

on t h e b o u n d a r y

dSQ (or dSQ) denotes

the integration

to the coordinates

element over S (or F) with respect

of Q

n is the outward unit normal n

F

at the point Q of the boundary

F

is the outward unit normal at the point P of the boundary

F

P K (u) is the Kirchhoff n flection field u M (u) is the normal

transverse

shear force associated with

flexure moment

the de-

associated with the deflection

field

n u

Mnt(U ) is the torsional moment v(P,Q)

for the deflection

field u

solution of Aflv = 0 which exhibits a singula1 2 rity at P, such that v(P,Q) = ~-~ r Log r where r = II~II

~'~A.

is the fundamental

is the jump of the function which may occur at corners Aol of cur-

Ivilinear

abscissa

s.

defined by

I~

A. = (')st -

")

i ~ i

Along the boundary,

t h e known q u a n t i t i e s

K (w) a n d M (w) on a f r e e n

S~ i

i

are

edge

n

w and M (w) on a s i m p l y

supported

n

edge

~w w and ~ n on a clamped edge.

Furthermore, ~w :

the quantity

Mnt(W) a t

a point

Q can be expressed

in terms

of

~

~n

Mnt(W)

Matrix

= - D(I - ~) ~

[~-~]

formulation

Using the approach we proposed the vibration

for static problems[5]

problem can be achieved

- by discretizing

the boundary

a matrix

formulation

:

F into m segments,

the values of deflec-

tion w, its normal derivative

~__w bending moment M (w) and transverse ~n' n force K (w) being defined at the middle of each segment. n - hy dividing

the domain S into n finite elements,

(4), ( 6 )

the homogeneous

boundary

conditions,

shear

the value of the de-

flection w being defined at the center point of each element. into account

of

Thus taking

discretization

of equations

and (7) leads to :

[Gr~ {~ _ ~2 [Jrl {Ws~ = {0}

(~)

144

G. B E Z I N E

w!~ere [GF] is a 2m by 2m matrix whose coefficients

result from the curvili-

near integrals of (4) and (7) [JF] is a 2m by n matrix whose coefficients

result from the surface

integrals of (4) and (7) {I}

is the vector whose 2m components are the 2m boundary unknowns ~w (among the m values of w, ~n' Mn and Kn )

{w } is the vector whose components are the deflections at the n nodal s points inside the domain S. In the same way equation (4) together with (5) written for each nodal point P inside the domain S gives

:

{Ws } = -"[Gs] { I } where

%2fJ s]., {Ws }

(9)

[Gs] is a n by 2m rectangular matrix [Js] is a n by n matrix {~}

is the vector whose n components are the values taken by the deflection at the n nodal points inside the domain.

It is possible to eliminate equation

the unknown vector {I} in (9), by inverting

(8) in the form :
where

-I

] [JF] {w s}

]O)

[G~ ]] is the inverse matrix of [GF]

By substituting

(|O) into (9) we obtain

:

{ws} = zX(EGs] EG~I] Ejr] - Ejs]) {ws}

I~)

Equation (II) can be recast in the form :

([%] [GT'] [Jr] from which the eigenvalues

Dsl)

and eigenvectors

squared circular frequencies

= (j)

,2)

can be computed° These are the

and the corresponding modes of vibrations

of

the plate respectively.

Numerical results

Three distinct problems for square plate vibrations are considered in this paper : - Clamped plate - Simply supported plate - Plate with three free edges and the fourth edge clamped.

PLATE VIBRATION

.in e.~ch case Poisson'~ dimensionless The boundary

PROBLEMS

145

~-atio is 0,3 and frequency values are obtained

in

form. is partitioned

into 48 elements

(figure

l).

The nodal points are at the middle

of each segment.

The

unknown quantities (two among ~w w, ~-~n' Mn(W) or K (w) are n assumed to be constant over the element.

The interior

main is divided gure

into

l) or 64 square

do-

16 (fifinite

elements with the deflection w supposed

to be constant.

~. . . . . . . . . .

over each element. a) Clamped plate We compare

I. Integral

equation mesh

our results with

approximations Young

Fig.

obtained by

[6] through Ritz's me-

thod. We give in table

I the boundary

per cent of Young's values

integral

frequencies

for the first harmonic

which are within

|

and within 5 per cent for

the sixth with a mesh of 64 elements. b) Simply supported

plate

For these edge conditions The frequencies

obtained

we know an exact solution by our mixed boundary

integral-finite

are plotted and compared with the exact frequencies again,

that for the first frequency

given by Timoshenko

the error is less

than

| per cent,

and

the plate domain.

that our method gives an upper limit for frequencies.

c) Cantilever

plate

Table 3 shows the results finite element results

element method

in table 2. We see, once

close to 4 per cent for the fifth using a 8 x 8 mesh inside We remark

[7].

results

obtained by our method and we use as a reference

[8], Ritz's method frequencies

[8]. Error is calculated with respect

on our frequency

results

[~

to Young's

and experimental results.

The error

is less than 3 per cent. We see the good agreement

146

G. B E Z I N E

with finite element results. We remark that convergence is faster for the mixed boundary integral-finite element method than for finite element method and moreover we see that for the same computation time, our mixed method gives a slightly better accuracy and allows a finer discretization, which permits to plot the modal shapes with a greater precision. These modal shapes are given in figure 2, 3, 4, 5 and 6, and they are in good agreement with those already published

[6], [8].

Conclusion

The transverse vibration of rectangular plates has been investigated using a mixed boundary integral-finite element method where inertia forces are considered as a static loading, which led us to treat these problems with the static fundamental solution. This method offers the advantage of a simple computation and results obtained were seen to be in good agreement with those given by Ritz's method or finite elements. Moreover for a same computation time, the accuracy of results obtained by our method or finite elements is the same. Finally it should be pointed out that so far all our results give an upper bound for the true frequencies,

in contrast to finite elements. This method

gives a systematic procedure to solve any vibration problem, once the boundary integral solution of the corresponding static problem is known. Thus it can readily be extended to elastodynamics problems, torsion problems,

....

Acknowledgements

This research was partially supported by the "Centre National de la Recherche Scientifique".

References

I. J. Vivoli "Vibrations de plaques et potentiels de couches", Th~se de Doctorat ~s Sciences Physiques. Universit~ de Provence. Aix - Marseille Io 1972. 2. J. Vivoli and P. Filippi "Eigenfrequencies of thin plates and layer potentials", J. Acoust. Soc. Am. Vol. 55, n ° 3, March 1974. 3. G. Bezine and D. Gamby, "A new integral equation formulation for plate bending problems", Recent Advances in Boundary Element Methods, |978. Editor C.A. Brebbia. Pentech Press London. 4. G. Bgzine "Boundary integral formulation for plate flexure with arbitrary boundary conditions", Mech. Res. Comm. Vol. 5 (4), 1978, pp. 197-206.

PLATE VIBRATION PROBLEMS

147

5. G. B~zine "Mixed boundary integral-Finite element method for plate flexure with conditions inside the domain", Submitted for publication. 6. D. Young "Vibration of rectangular plates by the Ritz method", Jounal of Applied Mech. Trans. A.S.M.E., !950 Vol. 72, pp. 448-453. 7. S. Timoshenko and S. Woinowsky-Krieger, "Theorie des plaques et coques", Librairie Polytechnique B~ranger. Paris 1961. 8. G. B~zine "Etude des vibrations libres des plaques ~lastiques minces ~ventuellement multiplement connexes ou 'sandwich' ~ l'aide de m~thodes duales d'~l~ments finis", ThSse de Docteur-Ing~nieur. Universit~ de Poitiers, Mai !975. TABLE |. Clamped plate frequencies

Boundary Integral Equation MODES 4 x 4

Error ,

Ritz's

8 x 8

Error

Method (Young)

,

1

37.!57

3,2%

36.24]

0,7%

35.99

2

79.721

8,6%

74.768

1,8%

73.41

3

122.78

13,4%

!I!.33

2,8%

]08.3

4

164.07

25%

136.32

3,6%

]3].6

5

!69.05

28%

137.O2

3,6%

]32.3

6

20].03

22%

!72.7

4,6%

165.!

TABLE 2. Simply supported plate frequencies

Boundary Integral Equations MODES

Exact

4 x 4

Error

8 x 8

Error

solution TIMOSHENK~

I

20.252

2,6%

19.866

0,6%

19.74

2

52.491

6,4%

50.145

1,6%

49.34

3

86.565

9,6%

80.97!

2,5%

78.96

4

107.904

9,4%

10!.864

3,2%

98.69

5

!42.!5

!O.8%

]33.68!

4,2%

•

,

128.3

148

G. B E Z I N E

TABLE 3. Cantilever plate frequencies

Ritz's method (Young)

BOUNDARY INTEGRAL EQUATIONS _

TIME

Experimental results

FINITE ELEMENTS

,

50"

2' 30

9'

2' 50

40"

..

MODES

4 x 4

8 x 8

Error

5 x 5

Error

4 x 4

3 x 3

1

3.5;7

3.484

0,3%

3.494

3,43

3.470

0,7%

3.474

3.466

2

8.805

8.571

0,3%

8.547

7,9!

8.509

0,4%

8.523

8.529

3

24.488

22.525

0,5%

21.44

2|,19

2|.459

O,1%

.2~.538

2~o679

4

30.879

28.|04

2,4%

27.46

27,73

27.063

1,5%

26.994

26.852

5

33.537

3;.359

0,6%

31.|7

28,35

30.948

0,7%

30.915

30.802

.

Fig. 2.

First mode of vibration for a symmetric cantilever square plate.

.

PLATE VIBRATION PROBLEMS

Fig. 3. Second mode of vibration cantilever square plate.

149

for a symmetric

Z ~W ~

,

Fig. 4.

Third mode of vibration

,

for a symmetric

square plate with nodal curve represented

cantilever

as a broken line.

150

G. BEZINE

Fig. 5.

Fourth mode of vibration for a symmetric cantilever

square plate with nodal curve represented as a broken line.

Fig. 6.

Fifth mode of vibration for a symmetric cantilever

square plate with nodal curve represented as a broken line.