Free vibrations of a rectangular plate-cavity system

Apphed Acoustws 24 (1988) 49-61

Free Vibrations of a Rectangular Plate-Cavity System M a z e n I Qalsl Engineering Department, Mutah Umverslty, Karak, PO Box 7, Jordan (Recewed 16 July 1987, revised version received and accepted 7 October 1987)

ABSTRACT Thts paper presents a method for computmg the naturalfrequenctes and mode shapes of a rectangular plate backed by a closed rectangular cavtty The plate motton ts represented by a number of admtsstble functtons conststmg of tn v a c u o normal modes for stmply supported plates, and of beam[unctlons for fully clamped plates The acou~ttc motion ts descrlbed analyttcally m terms of ~uchfunctlons Thts allows the vtbratton of the enttre system to be adequately treated m terms of a small number of plate modes The free vtbratton solutton for the coupled system zs obtamed etther directly by solvmg the plate equatton of motion or by formulatmg the mass and sttffness matrices of each subsystem numertcally For practwal apphcatton, a stmphfied equation ts developed to calculate the fundamental natural frequency

NOTATION

E v

h P Po (~)

P Co

q~kt

plate transverse displacement modulus of elasticity Polsson ratio plate thickness sound pressure ambient static pressure acoustic displacement potential angular frequency air density speed of sound in air acoustic response functions 49

Apphed Acoustics 0003-682X/88/$03 50 © 1988 Elsevier Apphed Science Pubhshers Ltd, England Printed m Great Britain

50

lwl (I) n m

[c]

[My] C [M.] I/q]

t,1 I Q a t s t

plate modal vector containing ~kt constant coefficients of acoustic potential system coefficient matrix plate modal unknowns plate mass matrix plate stiffness matrix acoustic mass matrix acoustic stiffness matrix 1 INTRODUCTION

The dynamic interaction between a vibrating structure and the surrounding acoustic medium seldom lends itself to exact analysis and recourse must then be made to approximate methods Foxwell and Franklin t and later W a r b u r t o n 2 have used a normal mode technique to study the effects of an acoustic medium on the vibrations of a cyhndrlcal shell The shell motion was approximated by a series of normal modes, whereas the acoustics were described by an exact solution to the wave equation Pretlove s considered a one-dimensional model of a plate backed by a shallow cavity and showed that the cavity had a considerable effect on plate vibration However, the analysis 3 was restricted to the low-frequency regime where the acoustic wavelength is long compared with cavity dimensions and the pressure within the cavity is uniform Under these conditions, the cavity acts as an added stiffness to plate modes which are volume-displacing, the non-volumedisplacing modes being unaffected Similar results were obtained by Graggs, '~ who represented the plate by a number of finite elements and, using the acoustic volume-displacement theory, derived a cavity stiffness matrix which, when added to the plate stiffness matrix, coupled all the available degrees of freedom Although the volume-displacement theory gives adequate representation of the effects of the enclosure on the motion of structures at low frequencies, ~t does not provide a general representation of the acoustics A unified approach in which the acoustics and the structure are modelled by finite elements seems to afford greater flexlblhty and a more general means for treating coupled problems Graggs 5 used plate and acoustic finite elements to simulate the behavloul of a w i n d o w - r o o m system The plate was represented in terms of displacements and the acoustics in terms of pressures, but this mixed formulation gives unsymmetric system matrices, thus comphcatlng the elgenvalue problem This shortcoming was later overcome by Qalsl, 6 who presented a displacement formulation for acoustic finite elements by using a displacement potential to describe the acoustic motlon

Vtbrattons of a plate-camty system

51

However, for some special problems, it can be advantageous to combine analytical methods with numerical approaches rather than solving the whole problem numerically In this paper, the dynamic behavlour of a rectangular plate-cavity system is analysed in terms of a number of plate modes, and the acoustic effects are obtained analytically as cavity responses to such modes The free vibration problem of the coupled system is then formulated by considering the plate motion only An alternative variational formulation is also presented In which both the plate and acoustic motions are considered

2 E Q U A T I O N S OF M O T I O N The plate-cavity system considered is shown In Fig 1 A Cartesian coordinate system xs used with the origin at one corner The plate, covering one side of the cavity (x = a), is assumed to be of constant thickness h, and its material to be linearly elastic The remaining cavity walls are assumed acoustically hard, and the sound pressure acting on the plate from the exterior is ignored Under these assumptions, the plate motion is governed by the fourth-order partial differential equation

Eh 3 /OCw ~'*w ~4w'~ ~2w 12(l_vZ)L~y4 + 2 Oy2c~z~+~z4)+ p~-~=p(a,y,z,t)

(1)

In which w IS the plate transverse displacement, E and Ps are the modulus of elasticity and the mass density of the plate material, respectively, v is the Polsson ratio, and p(a, y, z, t) IS the cavity sound pressure exerted on the plate at any time t a

.j

cavity plate

c t X

Fig. 1

Plate-cavity system and coordinate system

52

M I Qatar

In acoustics, it is n o r m a l practice to use sound pressure or v e l o o t y potential to represent the acoustm m o U o n Here it is m o r e convenient to use a displacement potential ~b(.x, v, z, t), the g r a & e n t vector o f whmh gives the acoustic d~splacement c o m p o n e n t s u, v, q m the v, v, z dlrectmns respectwely,

u - 8x

v = ~-

q = ?zz

(2)

The s o u n d pressure IS related to ~b simply by p = -pq~

(3)

where p d e n o t e s air density a n d ( ) m e a n s dlfferennatlon with respect to time t F o r free v l b r a n o n s at an a n g u l a r frequency co (rad/s), a rime dependence o f sm (cot) is assumed, a n d eqn (3) becomes p = pco2t/)

(4)

Thus, as far as acoustic m o d e s are concerned, the pressure a n d displacement potential m o d e s are identical, since one ~s related to the other by a c o n s t a n t m u l n p h e r The p o t e n u a l (h(x,y,z, t) m u s t satisfy the wave equauon ~aX~ + ~

d- ~pZ

C2 ?t 2

where c o is the speed o f sound In a d d m o n , q~ m u s t sansfy the a p p r o p r i a t e b o u n d a r y c o n d m o n s , n a m e l y (1)

[2)

The n o r m a l acoustic displacements on all hard walls vamsh, thus &b/Sx = 0

on t = 0

?(p/?)'= 0

on y = O, h

P4~/& = 0

on z = 0,

(6)

Air adjacent to the plate m u s t m o v e with a n o r m a l displacement identical to t h a t o f the plate ? v (a, -v, z, t) = wO', z, t)

(7)

A solution ~b(v,y, z, t) which satisfies the wave e q u a t i o n (5) and b o u n d a r y c o n d i t i o n s (6) can be expressed as

~(v,v,z,t)=sln(cot)EEOOn.,cosh(xgn,.x/a)cos(mD'/b)cos(m~z/c)( I

n=Om=O

f

Vibratton~ o f a p l a t e - c a v t t y s y s t e m

53

in which g.m = a x/(nrc/b)2 + (mTz/c)2 _ (O)/C0)2 and @.m are constant coefficients to be determined from the interface b o u n d a r y condition (7) The plate motion is approximated by a series solution of the type K

L

w(Y,Z,t)=sln(cot)ZZWklqk(Y)ql(2)

(9)

k=ll=l

In which W u are modal unknowns to be determined, and qr(Y) is a set of admissible functions satisfying the appropriate edge conditions For a simply supported plate, an obvious choice of solution IS the normal in vacuo modes, q r ( Y ) = sin(kr~y/b), qL(Z)----sin (Lrcz/c) For a plate with clamped edges, the beam function qk(Y) = (COSilk)' -- cosh flkY) -- CXk(Slnilk)' -- sinh flkY) may be used, where ~k = (COSflkb -- cosh flkb)/(Sln flk b -- sInh flkb), and fiR are the roots of the transcendental equation cos fib cosh fib = 1 q~(z) has similar expressions Substituting eqns (8) and (9) in (7) gives

¢.,.(~zg.,./a) slnh (ng.m) cos (nny/b) cos (mnz/c) = n=Om=O

Wuqk(y)qt(z) k =1/=1

(10) Now use is made of the orthogonal properties of trigonometric functions to express the constant unknowns • u in terms o f the modal parameters W u Multiplying eqn (10) throughout by cos (vzy/b) cos (lrtZ/C) and integrating over the plate area leads to K

a =

O,j

L

Z Z WktBk,Cu k=lt=l

(tog,j)slnh(rcg,j)b,cj

(11)

in which B k , = Sboqk(Y)COS(my/b)dy, and the constant b, = b for t = O, otherwise b, = b/2 C u and cj have similar expressions

54

'~1 I Qat~/

When eqn (11) is subsmuted in eqn (8). the acou,,tlc potential can be expressed as K

L

q~(x, v, z, t) = sm ((or)

~ x

Wkt k=l 1=1

ng.m smh (ng.,,,)b.~ ,. n=Om=O

nn)'

mrcz

x cosh rig.m-" cos h ~ cos - /~ = S l n (tO/) ~

L

(12)

~-~j WkI~)kI(X,y,Z)

k=l/=l

Clearly, ~bkLx,y,z ) represents the acoustic response to a normal plate mode Wkt of magmtude umty The total acoustic motion is thus made up from a comblnatmn of acoustic responses to individual plate modes Both 4~ and w are now written in terms of an arbitrary number of normal plate modes Furthermore, the potentml ~b, as given by eqn (12), satisfies the wave equation (5) and all the appropriate boundary c o n d m o n s (6) and (7)

3 FREE VIBRATIONS In the basic analysis, the acoustic motion has been shown to be dependent on plate vlbratlons Therefore, the free vibration problem of the coupled system can be formulated by considering the plate m o t m n only For h a r m o m c oscillations, the sound pressure is related to ~b by eqn (4) On using eqns (4) and (12), the equation of plate motion (1) becomes h

L

12(i -- vz) [ ? ) 4 + 2 ?),2~z~ + ~ . ~ j - psh(o211 = p(D 2

W k l ~ k l ( a , y , Z)

k=l/=l

(13) Subsmutlng eqn (9) m eqn (13), multiplying by qk(Ytq~(z) and Integrating over the plate area yields a system of hnear homogeneous algebraic equations which can be written in the matrix form

[ c ] { w } = I0}

(14)

m which [C] = [Cp] + [Ca] is a system m a m x conslsttng o f a plate m a m x [Cp] and a fully populated acoustic matrix [Ca] For a non-mvlal solution, the determinant of [ C ] is set equal to zero This condltlon provides the frequency equation which is solved numerically for the natural frequencies of the coupled system Having obtained the frequencies, eqn (14) is solved for

Vtbrattons of a plate-cavity system

55

the mode shapes of plate vibration { W} Thereafter, the corresponding sound pressure modes may be computed directly from eqn (12) 4 AN A L T E R N A T I V E F O R M U L A T I O N A more formal approach to free vibrations uses the assumed plate and acoustic motions to derive the mass and stiffness matrices for each subsystem Such property matrices are formulated directly from expressions of the potential and kinetic energies Again, since the motion of each subsystem is written in terms of plate modes, the free vibration problem will be defined within this framework The potential energy of the acoustic subsystem can be written as

U.(t)=½7Pojv\OX 2 +~y2 +~z2j dV=~yPo(og/Co)2

q~ZdV (15)

in whmh 7 IS the ratm of specific heats, and Po is the ambient static pressure With the aid ofeqn (12), eqn (15) may be transformed into the matrix form

U.(t) = ½{w}r[K.]{ W}

(16)

in which { W} is the modal unknown vector and [K.] is the acoustic stiffness matrix Similarly, the acoustic kinetic energy

Ta(t) ½P fV (0(~/t~f)2 "1-(0(])/0)') 2 Jr- (0~/02) 2 d V

(17)

can be reduced to the form

Ta(t)= ½{W} T[Ma]{ W}

(18)

In which { W} Is the time derivative of the modal vector and [M~] is the acoustic mass matrix Both [K.] and [ M j are fully populated matrices and depend on frequency of vibration Expressions for thmr elements are listed in Appen&x I In a similar manner, the plate mass and stiffness matrices [Mp] and [Kv] may be derived from the potential and kinetic energies of the plate, namely

Up(t)=212(1-v2) X

\0~-J

+\?~T~2j +2Vey2

/ 02u, \z 02W"~- 2 ( 1 - V ) [ - - ] dA 0z 2 \Oy&}

(19) 7.(,) =½p~h~ I~2dA =2(x-~l;V}r[Mp]( W} 3A

(20)

M l Qatar

56

The potential and kinetic energies for each subsystem, eqns (16), (18), (19) and (20), may now be added up to give the potentml and kinetic energaes for the entire system, thus

u(t) = ½{

w} = ½{w} r[Ko +

w}

= k{ w}' [M]{ w} = ½{w}' [Mo + M.]{ w} Here, the free vlbratlon problem can be arranged m the form of eqn (14), where m th~s case [ C ] = [K. + Kp] - ooZ[M. + Mp] and the natural frequencies and m o d e shapes are obtained as described m the previous section

5 A NUMERICAL EXAMPLE A &gltal computer program has been written to compute the natural frequencies and mode shapes of vibration for the coupled plate-cavity system The program incorporates the basic and alternative formulations, as outlined m the preceding sections, as options It must be mentioned that the two formulations give ~dentlcal results, although the basic formulation reqmres less computing t~me In either case, a non-trivial solution ~s sought for eqn (14), for which the determinant of the matrix [ C ] must vamsh This provides a frequency equation, the roots of which represent the natural frequencies, which may be back-substituted m eqn (14) to yield the corresponding mode shapes of plate vibration { W} Then, the correspondmg sound pressure modes are computed from eqn (12) It must be noted that. m the evaluation of the determinant of [C], spurious roots may appear in ad&tlon to the actual roots These spurious roots correspond to the singular solution of the frequency equation For example, when the value of the denominator of eqn (11) is zero, the solution becomes singular However, away from the singular point, the solution is valid and provides the actual roots of the frequency equation Thus a change m the sign of the determinant over a small incremental interval does not imply that a root exists m such an interval, as shown m Fig 2 To demonstrate the apphcabfl~ty o f the analys~s and to illustrate the effect of cavity presence on plate v~bratlon, the formulation was apphed to a plate having the following &menslons b = 0 3 m, ~ = 0 4 m. h = 1 5 mm The plate materml was a l u m m m m having the following properties E = 7 0 G P a . v = 0 3. Ps = 2700kg/m~ Under simple support edge conditions, Table 1 gives the fundamental natural frequency for a cawty depth a = 1 m for various numbel s o f m va~ uo modes No slgmficant change ~s seen m the value

Vtbratlons o f a plate-cavtty system

57

i /-,0o°oo, _

actual

root

,,

z.,o

' ~

Frequency

:l Fig. 2.

Actual a n d spurious roots of frequency e q u a t i o n

of the fundamental frequency as the number of modes 1s increased This frequency, as shall be demonstrated, can be obtained accurately by using the first plate mode only (1 x 1) Table 2 shows the convergence of the third mode Both the pressure and plate modes are seen to converge to the correct solution to four decimal figures with a small number of modal unknowns Figure 3 shows, for different plate edge conditions, the effect of cavity depth on the fundamental frequency, which is seen to increase significantly with shallow cavities, the depths of which are smaller than typical plate dimensions This result is in agreement with the results obtained by Pretlove, 3 who considered a one-dimensional model of a plate backed by a shallow cavity, and with those of Graggs and Pretlove 7 For practical application, it is possible to obtain a simplified equation, the smallest root of which gives the fundamental natural frequency The most important acoustic effects on the fundamental frequency are due to the plane TABLE 1 F u n d a m e n t a l N a t u r a l Frequency, Cavity D e p t h a=10m In vacuo modes

K ×L 2 3 3 4 5

× × × × ×

2 2 3 4 5

Fundamental [requency (Hz) 65 080 65 088 65 088 65 092 65 089

58

M 1 Qatst

[ABLE 2 Convergence of the Third Mode (250 3 Hz), Cavity Depth a = 1 0 m In vacuo

P r e ~ u r e m o d e along ~ = O 1 5 m

:=0

2m

?#ioN(~

( K x L)

x =00

x = 0 25

(2 x 2) (3 x 3) (5 x 5)

1 0000 1 0000 1 0000

0 422 3 t) 422 3 0 422 3

~ =05

= 10 m

x = 0 75

- 0 643 2 - 0 643 2 0 643 2

- 0 9600 -0961 2 0961 2

0 2076 0 1240 0 1240

l = 0 2

I = O 25 m

0 865 6 0 987 6 0 987 6

0 499 1 0 709 7 0 709 7

Plate m o d e along : = 0 2 m

(2 x 2l (3 x 3) (5x5J

i =005

i =OlO

0 5002 0711 0 07110

08662 09878 09878

110

I

i =015

10000 1 0000 10000

I

[

1

I

lO0 N '-r

90 t,ea ,D 13-

80

L Ia_

-i -~ 7 0 F El "13 r-

,p 6o

50

I

04

I

08 Cavity

Fig. 3

I

I

I

12

16

2.0

Depth

(m)

24

V a n a h o n of fundamental lrequency with cavity depth -- , slmphfied equauon, © O, plate simply supported, A A , plate fully clamped

Vtbratlons of a plate-cavtty system

59

sound wave component (n = 0, m = 0). Thus, by neglecting other wave components, the first equation corresponding to the fundamental frequency in the system of equations (13) can be slmphfied as (-02 1

_ ~ p c cot (oga/Co)" ~ _ ~o21 Psh co ] 1

0

(21)

in which 0911 IS the plate m v a c u o fundamental frequency, and ~ is a plate support constant having the following values = 64/rc4 = 0 42

for simple supports for clamped supports

A comparison between the frequencies calculated using the numerical solution and those calculated from eqn (21) is presented m Fig 3 Excellent agreement ~s shown between the results obtained from the simphfied equatxon and those from the computer program, over a w~de range of cavity depths The formulation was also checked by obtaining the pressure modes for shallow and deep cavmes Figure 4 shows the pressure variation along the hne of symmetry (y = b/2, z = c/2) for two cases, the first case related to the fundamental mode (98 3Hz) of a shallow cavity (a = 0 1 m), where the pressure was, as expected, essentially uniform throughout the cavity since the sound wavelength was long (2 = 3 5 m) compared with cavity I

I

I

I

1 o 6

i

I

Io

o5

-o 5 0

-Io

I 02

04

I 08

I 1o

(x/a) Fig.

4

S o u n d p r e s s u r e m o d e a l o n g the line o f s y m m e t r y ()' = b/2, z = ~/2), / ~ - - A , m o d e (98 3 Hz), a = 0 1 m, ( 3 - - ( 3 , third m o d e (250 3 Hz), a = 1 0 m

first

60

M 1 Qat~l

&mensmns The second case related to the third mode (250 3 Hz) of a deep cavity (a = 1 0 m), in which the pressure mode consisted mainly of a plane sound wave, m the &rectlon of cavity depth, except m the vlcm~ty of the plate where the sound field became three-dlmensmnal This near-plane wave had a wavelength of approximately 0 69 m, which agrees favourably w a h the exact value for the plane wave of 0 693 m ()/2 = Co/2/'1

6 CONCLUSION A simple method xs presented for analysing the free vibrations of a rectangular plate backed by a closed cavity under &fferent plate edge conditions The acoustic and plate motions are described m terms of a small number of admissible plate modes, thus reducing the computatmnal effort s~gnlficantly Such simple representation is also adequate for the first few modes which are of great practical interest F o r practical apphcatton, a simplified equation to calculate the fundamental natural frequency is presented

REFERENCES 1 J Foxwell and R E Franklin, The vibrations of a thin walled stiffened cylinder In an acoustic field, Aeronaut Q, l0 (1959), pp 47-64 2 G B Warburton, Vibration of a cylindrical shell in an acoustic medium, J Mesh Engng S¢t, 3 (1961), p 69 3 A J Pretlove, Acousto-elastlc effects m the response of large windows to sonic bangs, J Sound Vlbr, 9 (1969), pp 487-500 4 A Graggs, Computation of the response of coupled plate-acoustic systems using plate finite elements and acoustic volume-displacement theory, J Sound Vthr, 18 (1971Lpp 235-45 5 A Graggs, The transient response of a coupled plate-acoustic system using plate and acoustic finite elements, J Sound Vtbr, 15 (1971), pp 509-28 6 M I Qalsl, Finite element analysis of acoustic and acousto-structural systems, PhD thesis, Imperial College, University of London, 1976 7 A J Pretlove and A Graggs, A simple approach to coupled panel-cavity vibrations, J Sound Vtht, !1 (1970), pp 207-15

Vtbrattonsof a plate-cavttysystem

61

APPENDIX I The elements o f the acoustic matrices are as follows

n=Om=O

K.(kl, k'l') = 7Po ~o

2~Z2gnm2slnh 20Zgnm)bncm 1 4 n=Om=O

~

~

M.(kl, kT) = n=Om=O

paaBknBknC~mCrm 27z2g2msmh2 (rtgnm)b.cm

27zgn,~ J