The effects of elastic edge constraints and fluid loading on the resonant response of thick...

The effects of elastic edge constraints and fluid loading on the resonant response of thick rectangular plates

Jamel Hammouda and Courtney B. Burroughs Engineering Science and Mechanics Department, The Pennsylvania State University, State College, Pennsylvania 16801

(Received 29 September 1992; revised 21 October 1993; accepted 14 December 1993)

An analytic model of the response of rectangular plates simply supported on two opposing edges and elastically supported on the other edges is derived. Transverse shear and rotary inertia in the plate and adjustable stiffness constants in the elastic supports are included in the model. The elastic stiffness constants are adjusted to simulate classical boundary conditions and results compared to published results. The variation of frequencies of resonance with the stiffness of the elastic supports is shown. The effect of fluid on the resonant responses of simply supported plates is presented.

PACS numbers: 43.40.Dx

INTRODUCTION

The vibration responses of finite plates to external excitation are often controlled by their resonant responses. The frequencies at which finite plates resonate and the am- plitudes of the resonant response of plates are affected by the conditions at the boundaries of the plates and by the presence of fluid loading. Many of the studies of the resonant response of plates have been limited to classical boundary conditions, i.e., simply supported, clamped, or free. Leissa • presents a comprehensive treatment of the resonant responses of rectangular plates with classical boundary conditions. Also, Leissa 2-7 references many pa- pers that treat plates with classical boundary conditions. Because plates are often attached to other structures, the boundary conditions often differ from ideal classical conditions. In addition, plates are sometimes subjected to heavy fluid loading, such as water loading in marine ves- sels. In this paper, the effects of both elastic boundary conditions and fluid loading on the resonant response of rectangular plates will be studied. Analytic models of the resonant vibrations of a moderately thick plate, simply supported on two opposing edges, and elastically restrained against translation and rotation along the other two edges, will be developed and solutions given. The effect of fluid loading on the response of the plate, simply supported on all edges, will also be modeled and solutions presented. The effect of the plate thickness, stiffness of the boundaries, and fluid loading on the resonant responses of the plate will be explored.

Among the first researchers to consider the effects of elastically restrained edges on the resonant response of plates were Joga Rao and Kantham, 8 who used beam functions as trial functions in the Rayleigh-Ritz method to obtain approximate solutions for the frequencies of resonance of rectangular plates with two opposing edges elastically restrained against rotation and rigidly supported against translational displacements. For a square plate, Joga Rao and Kantham showed the variation in the frequencies of resonance with the stiffness of the edge for the

first three modes. The frequencies approached constant values at low and high values of the edge stiffness. Carmichael 9 also used beam functions in the Rayleigh- Ritz method to obtain approximate solutions for the resonant responses of rectangular plates with the rotational motion at all four edges elastically restrained. Opposing edges had equal restraints.

In their research on resonant responses of plates, Laura and co-workers have considered the effects of elastic

restraints on the edges of rectangular plates on the frequencies of resonance. Laura and Romanelli 1ø used three-term polynomial expansions that satisfied the boundary conditions exactly in the Galerkin method to find approximate solutions for the frequencies of resonance for the first three even modes as a function of plate dimensions, static in- plane stress, and stiffness of the edge restraints. All four edges had symmetric rotational restraints with zero displacements. Nonsymmetric rotational elastic restraints on all four edges were considered by Laura, Luisoni, and Filipich 11 and Laura, Luisoni, and Ficcadenti. 12 The Galerkin method was employed with a five-term polynomial expansion. The frequencies of resonance for the fundamental mode 11 and the first three modes 12 were plotted as a function of the stiffness of one of the edge restraints with the other three edges free or clamped. An increase in the frequencies of resonance was shown with increasing rotational stiffness at the one edge. For small edge stiffness, the frequencies approached constant values corresponding to frequencies for simply supported edges. For large edge stiffness, the frequencies approached constant values corresponding to frequencies for clamped edges. The Galerkin method, with the approximate solution given by a product of a three-term polynomial and a cosine function, was employed by Filipich, Reyes, and Rossi 13 to develop approximate solutions for the frequencies of resonance for the fundamental mode of a plate with symmetric rotational and translational edge restraints. For rectangular plates with three edges with rotational elastic restraints and fixed displacements and the fourth edge free, Laura and Grossi•4 used four-term polynomial solutions in the Ritz method to

3350 J. Acoust. Soc. Am. 95 (6), June 1994 0001-4966/94/95(6)/3350/10/$6.00 ¸ 1994 Acoustical Society of America 3350

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obtain approximations for the frequencies of resonance for the first three modes. Plots of the frequencies of resonance as a function of the stiffness of the rotational restraint at

one edge were presented for different edge configurations. Again, increasing stiffness produced increases in the frequencies of resonance, with the frequencies approaching constant values at the low and high values of the stiffness. With rotational and translational elastic restraints on all

four edges, Laura and Grossi •5 applied the Rayleigh-Ritz method with five-term polynomials to obtain approximations to the frequencies of resonance for the fundamental modes. Comparisons were made to published results for 19 combinations of classical boundary conditions. Plots of the frequencies of resonance for one and two edges with translational restraints showed increases in the frequency with increases in the stiffness of the restraints, with the frequencies approaching constant values at low and high stiffnesses.

Egle •6 considered rectangular plates with three edges simply supported and the fourth edge elastically restrained against rotation and translational motion. Solutions to the point-driven plate were expanded in terms of functions for elastically supported beams. Wave numbers of resonance for the plate were presented as a function of the translation and rotational stiffnesses. Egle used the plate model to ex- plore the effects of edge restraints on the response of the plate to a point drive at resonance. Unequal rotational restraints on the edges of rectangular plates were treated by Nassar •7 by using beam functions as trial functions in the Rayleigh-Ritz method. Results for the fundamental mode were presented. Mukhopadhyay •8 used beam functions that satisfy boundary conditions in one direction to reduce the two-dimensional plate equation to an ordinary differ- ential equation, which was then solved with finite- difference methods in his treatment of rectangular plates with rotational edge restraints. Results were presented for one, two, and four elastically restrained edges with the other edges clamped or simply supported. Comparisons were made to results for classical boundary conditions for extremes of rotational stiffness values. Elishakoff and

Sternberg •9 employed models of beams to simulate elastic conditions at two opposing edges of rectangular plates with the other two opposing edges simply supported. Solutions for the frequencies of resonance for the fundamental mode were presented for different thicknesses of the two support- ing beams at the plate edges. Beam functions were used with the Rayleigh-Ritz method by Warburton and Edney 2ø on plates with four edges rotational restrained or with one edge with rotational and translational elastic restraints. For the lowest mode, frequencies of resonance were plotted versus the stiffness of the edge restraints, pro- ducing results similar to those reported by Laura and co-workers. •ø-•5 Bapat, Venkatramani, and Suryanarayan 2• considered rectangular plates with rotational and translational elastic restraints on all four edges to determine values of stiffness that effected classical clamped or simply supported boundary conditions. Approximate solutions were derived using beam functions in the Rayleigh-Ritz method. On plates with rotational and translational elastic

restraints on all four edges, Gorman 22 used the method of superposition, where solutions with a single edge elastically restrained were superimposed to produce solutions with all edges restrained. In each of the superimposed solutions, simple supported conditions were assumed on three edges with the fourth either rotationally or translationally restrained. Separation of variables was assumed with beam functions used with the elastically restrained edge. Solu- tions with rotational and translational elastic restraints

were superimposed to obtain final solutions. Results were presented for a square plate with different edge stiffnesses.

Because classical plate theory is easier to use than higher-order plate theory such as the Timoshenko-Mindlin plate theory 23 and the classical plate theory provides accurate results for the lowest-order modes of thin plates, classical theory was employed in all of the studies referenced above. However, for thick plates and higher-order modes, Timoshenko-Mindlin plate theory is required to obtain accurate results. The added complexity of using the Timoshenko-Mindlin plate theory on plates with elastic boundary conditions has limited its use. Magrab 24 applied the Galerkin method with sinusoidal trial functions that

satisfied the boundary conditions. All four edges rested on rigid supports with different rotational elastic restraints at each edge. Comparisons made to published results for classical plates with clamped boundaries showed that the Timoshenko-Mindlin theory produced the lower, and thereby the more accurate, estimates of the frequencies of resonance for the lowest-order mode. Results for the fun-

damental frequencies of resonance were presented for orthotropic plates with simply supported boundaries (zero elastic edge constant) and clamped boundaries (infinite elastic edge constant) as a function of plate thickness. Plots of the frequencies of resonance for the lowest-order mode as a function of the stiffness of the rotational edge restraints showed that for thicker plates higher spring constants are required to simulate clamped boundary conditions than those required for thinner plates.

The interaction of the plate with surrounding fluid has been ignored in all of the studies referenced above. For most plates vibrating in air, the effect of air on the vibration can be ignored. However, for plates vibrating in water, the interaction between the plate and water must be considered to obtain accurate estimates of resonant responses. Lindholm et al. 25 present measured frequencies of resonance for cantilevered plates in air and water. Davies 26 treated simply supported, fluid-loaded, baffled rectangular plates exposed to random excitation similar to that generated by turbulent boundary layer wall pressures. The intermodal coupling produced by the water at and of[' resonance was considered. When the plate damping was small, Davies showed that the water produced modal coupling. Pope and Leibowitz 27 developed expressions for modal coupling for fluid-loaded rectangular plates. By dividing the plate into preshaped segments, Nagaya and Takeuchi 28 developed solutions for an arbitrarily shaped plate, under tension, and supported from beneath by an elastic founda- tions, with the top surface in contact with a column of water in a tank. The effects of viscosity on the frequencies

3351 d. Acoust. Soc. Am., Vol. 95, No. 6, June 1994 d. Hammouda and C. B. Burroughs: Response of rectangular plates 3351


X x

FIG. 2. Plate with elastic edge restraints.

1,2 • • + O O FIG. 1. Geometry and coordinates for the plate.

(3)

of resonance were considered and found to be negligible. The translational displacements of the plate edges were fixed with elastic rotational restraints. Results were presented for an elliptic plate as a function of aspect ratio, in-plane tension, and elastic constant for the edge restraints, with and without the plate exposed to water in the tank. For rectangular plates with rounded corners, Nagaya and Takeuchi showed dramatic reductions in the frequencies of resonance of the plates when water loading was added.

Equations for the Timoshenko-Mindlin rectangular plate with elastic rotational and translational boundary conditions on opposing edges and simple supports on the other two edges are developed below in Sec. I. Also, in Sec. I, the equations for fluid loading of the simply supported Timoshenko-Mindlin plate are derived. Solutions to the equations derived in Sec. I are presented in Sec. II. In Sec. III, numerical results for in vacuo plates with elastic boundary conditions and simply supported plates with fluid loading are presented. Conclusions and a summary are given in Sec. IV.

I. EQUATIONS OF MOTION

Consider a rectangular plate of length 2a, width 2b, and thickness h with the middle surface of the plate in the x-y plane, as shown in Fig. 1. The equations of motion for the in oacuo Timoshenko-Mindlin plate theory are 29

(1)

where

W=Wl-}-W2,

•bx = --1 -•x-x+ •---1-•x-x-+-•y, (2)

( •22 )OWl (•;) Oto20H The transverse displacement is given by to and the rotational displacements from the normal to the middle surface are given by •Px and •py. In Eqs. (2),

and

fi= (Roo2--•)/D, a2=2fi/(1--v),

D=Eh3/12(1--v), Ro=ph3/12, (4)

•=k•2Eh/( l +v), al=ph,

for homogeneous, isotropic plates of constant thickness. In Eqs. (4), o• is the radian frequency, E is Young's modulus, p is the material density, v is Poisson's ratio, and • = •r2/12 is the shear correction factor.

In general, the boundary conditions along the edges x= +a can be written in the form 29

al•x=Mxx, a2•y=Mxy , a3w=Qx, (5)

where Mxx is the moment, Mxy is the twisting moment, Q• is the shear force per unit length, and a•, i= 1,2,3 are constants. For edges y= 4-b, x and y are interchanged in Eqs. (5). Like the classical plate theory, the Timoshenko- Mindlin plate theory will have an exact separable solution only if two opposite sides are simply supported. In addition to assuming that two opposing edges are simply supported, the other two opposing edges will be symmetric. This means that the constants in Eqs. (5) are identical at two opposing edges and that the characteristic equation will involve a 3 X 3 matrix instead of a 6 X 6 matrix. Therefore, the boundary conditions are written as follows:

w=-•x=•y=0, at x=+a, (6) and

__

f l to = q: O +'•y = Qy ,

K2•bx==v D 2 k Ox + =Mxy' at y= :• b,

(7)

(½ K31•y= • D + V & j =Myy, where K 1 is the linear spring constant, and K2 and K 3 are torsional spring constants for the elastic edge restraints illustrated in Fig. 2.



In the presence of fluid loading, the equation for the transverse displacement takes the form 3ø

DV 2 D(V2+812)(v2+822)w= 1--• Ro 02 ) 2 (8)

where p is the pressure of the fluid at the surface of the plate generated in response to the radial motion of the plate. For an inviscid acoustic fluid, the pressure satisfies the wave equation

=o, (9)

where k=co/c is the acoustic wave number. At the inter-

face between the plate and the fluid, the fluid pressure is related to the transverse motion of the plate and the rigid plate baffle by

poOj2W = • z=O

for Ixl <a, lYl <b,

w=0, for I xl>a, lyl>b, (lO)

where Po is the density of the fluid.

II. SOLUTIONS

A. Free vibration

With the plate simply supported at x= 4-a, the solutions to Eqs. (1) are separable and can be written as

Wl=y 1 (y) sin [ (rm•r/2a)x],

w2 --Y2 (Y) sin [ ( rm•r/2a ) x ], (11)

H= y3 (y ) cos [ ( rm•r/2a )x ] ,

where r m are even nonzero integers. Note that the term e køt has been dropped for convenience. Substituting Eqs. (11 ) into Eqs. ( 1 ) yields

y•' + [a 2-- (rm•r/2a)2]y3_--O,

Y'I' + ['5•--(rm•r/2a)2]yl =0, (12)

y•, = [ fi• _ ( rm•r/2 a) 2 ]Y2 -- 0,

where a prime denotes a derivative with respect to y. If we let

ct 2 _ ( rm•r/2 a ) 2= 02n,

812 ( rm•r/2 a ) 2 2 -- =/5., (13)

fi22-- ( rm•r/2a ) 2= y2n,

the solution to Eqs. (12) can be written, respectively, as

Y3 •

Y2 •

sin (0•) + C6, cos (0•), if a 2 > (rm•r/2a) 2, sinh (0•) + C6, cosh (0•), if a 2 < (rm•r/2a) 2;

Cln sin(/•v) +C2, cos(/•v), if fi12> (rm•r/2a) 2, Cln sinh (/3•v) + C2, cosh (/3•v), if • < (rm•r/2a) 2;

(14)

sin (y.y) + C4. cos (y.y), if 822 > (rm•r/2a) 2, sinh (y.y) + C4. cosh (y.y), if 8] < (rm•r/2a) 2.

From Eqs. (3),

0•<822•<812 (15) and

R 1 (_D2/½< •j 12. ( 16 ) Also from Eqs. (3),

2 2 •Jl•J2 = (R lO)2/•)fl (17)

and

O• +O•=Rlo2/O+Roo2/D. (18) Therefore, if fi < 0, then

•> R•2/•+Ro•2/D (19) and fi• < 0. From Eq. (17),

and, from Eq. ( 1 6),

(R < 1, ( 21 ) so that

fl<fi• (22) and

• 2B/(l--v) =a 2. (23) Hence, for B• 0, we have

•2 • fi• • fi•. (24) Similarly for B• 0,

2 2 2 fi2• •fil' (25)

Four cases involving the solution given in Eqs. (11 )-(14) can be identified.

1. Case (1)

If

8•2 < (rm•r/2 a) 2, (26) then it follows that

822 < ( rm•r/2a ) 2, a 2 < ( rm•r/2a ) 2, (27) and the solutions given by Eqs. (14) are

Yl = C1 sinh (/5.y) + C2. cosh (/5.y),

Y2 ---- C3n sinh (y.y) + C4. cosh (y.y), (28)

Y3 • Csn sinh (0.y) + C6. cosh (0.y).



Using Eqs. (28) in Eqs. ( 11 ) yields

W = [ Cln sinh (tinY) + C2n cosh (tinY) + C3n sinh (ynY)

+C4n cosh(ynY)]sin[ (rmrC/2a)x].

Using Eqs. (28) in Eqs. (2) and ( 11 ) yields

•P•= [ 2a -- 1 [Cl• sinh (OnY) + C• cosh (OnY) ]

+(•--l)[C3nsinh(ynY)+C4ncosh(ynY)]]

(29)

q- O n [ C5n cosh (0nY) + C6n sinh (0nY) ] cos • x , 30)

•-- 1 [Jn[ Cln cosh(/JnY) +C2n sinh(/JnY) ]

+ ( ;---• - l ) yn [ C3n cosh ( ynY ) q- C4n sinh ( ynY ) ] r rn 71'

q- • [ C5n sinh (0nY) + C6n cosh (0nY) ] Substituting Eqs. (29) and (30) into the boundary conditions, given by Eqs. (7) for the elastic edges, gives the matrix equation

0

where

Cjn:O , for j=l,...,6, (31)

Equation (34) leads to

[ D1 sinh ([•n b) q- D 2 cosh ([•n b) ] sinh ( l'n b) sinh (Onb)

+ [ D3 sinh ([•n b ) q- D4 cosh ([•n b ) ]sinh ( l'n b)cOsh (Onb)

+ [ D5 sinh ([•n b) n t- D6 cosh ([•n b) ]sinh (l'n b)sinh (0nb)

+[D 7 sinh(/3nb)q- D8 cosh(/•nb) ]sinh(I'nb)cosh(Onb)

=0, (35)

for the odd modes in the y direction, and

[ D8 sinh (Bnb) + D7 cosh (Bnb) ] sinh (7nb) sinh (0nb)

q- [ D 6 sinh (/3nb)+ D5 cosh(/3b) ]sinh(I'nb)cosh(Onb)

q- [ D 4 sinh ([•n b) q- D3 cosh ([•n b) ] sinh (Ta b) sinh (0n b)

q- [ D 2 sinh ([•n b) n t- D1 cosh ([•n b) ]cosh (Ta b)cOsh (0nb)

=0, (36)

for the even modes in the y direction, and where a i, i= 1,...,8, are given in Ref. 31.

2. Case (2)

• > (rmrr/2 a) 2, (37) then the solution is the same as given in Eqs. (28)-(36) except that the hyperbolic trignometric functions are replaced with trignometric functions.

A1 sinh(/3nb)q-A2 cosh(/3nb) ] Sl,m: A 1 sinh (I'n b) q-A 3 cosh (I'nb) , A4 cosh (0nb)

A 5 sinh ( l'n b) n t- A 6 cosh ( l'n b) ] A7 sinh (l'n b) nt-A8 cosh (l'n b) , A 9 cosh (Onb) n t- A lO cosh (0nb)

(32a)

(32b)

3. Case (3)

•22 < (rmrr/2 a) 2 < a2, ( 38 ) then sinh(/•nY), cosh(/•,), sinh(0nY) are replaced with trigonometric functions in Eqs. (28)-(36).

[ A l• sinh ([•n b) q-.d 12 cosh ([•n b) ] S3,m=/-•13 sinh(7nb)+A•4 cosh(7nb) , L A15 cøsh(Onb) +A•6 sinh(Onb)

(32c)

and

A1 cosh(/•nb)q-A2 sinh(/•nb) ] Tl,m = A 1 cosh (l'n b) q-A 3 sinh (l'n b) , A4 sinh (Onb)

./15 cosh(/3nb) +./16 sinh(/3nb) ] T2,m = A7 cosh(I'n b) q-A8 sinh(I'n b) ,

./19 sinh ( Onb ) q-.d lO sinh ( Onb )

(33a)

(33b)

A 11 sinh ([•n b) n t- A 12 cosh ([•n b) ] T3,m= A13 sinh(ynb)+A14 cosh(ynb) , (33c) A•5 sinh(Onb) +A16 cosh(On b)

where -'•i, i= 1,...,16, are given in Ref. 31. Normal modes will be generated if

det(S)=0 or det(T)=0. (34)

4. Case (4)

at2 < ( rmrr/2 a) 2 < •, (39) then sinh (•nY) and cosh (•nY) are replaced by trigonometric functions in Eqs. (28)-(36). A complete listing of all of the solutions, along with the constants D i, are given in Ref. 31.

B. Fluid loading

For the simply supported plate,

W(X,y) = E E Wren sin(kmx)sin(knY), (40) m=l n=l

where

k m = m•r/2a, k n = mr/2a, (41 )

and rn and n are integers. By taking Fourier transforms of Eqs. (9) and (10), and using Eq. (41), the following ex

3354 J. Acoust. Soc. Am., Vol. 95, No. 6, June 1994 J. Hammouda and C. B. Burroughs: Response of rectangular plates 3354


p(x,y,O)-- (2r) 2 • • Wqr __ q=l 1 oo (k2•-ky2)( kr)(k 2 k•-k•) r= oo ky2_ 2 2 2 1/2 Using Eq. (40) in Eq. (8) gives

o• • ( D •2 Ro 02 • • • wm•D[g•-(k•+k2•)][g•-(k•+k2•)]sin(k•)sin(k•)=- 1-• m=l n=l

Operating on Eq. (43) with

f•• f•o sin(k•)sin(k•)dxdy yields

D[•--(k•+k2•)l[•--(kL+k2•)labw•= 1+• (km+kn)+• PO •2 Z Z Wqr q=l r=l

pression for the pressure on the surface on the plate can be obtained.

ipo 02 2kqkr[ (-- 1 )q½-2ik- xa-- 1 ][ (-- 1 )r½--2ik?__ 1 ] ei(k?+k? )

If we let

ikqkrkmkn 2

f, f, [ 1- ( - 1 )m COS 2kxa] [ 1 -- ( -- 1 )n COS 2kybldk,, dky X oo (k2x-k2m)(k2x-k2q)( k2n)(ky-kr)(k-kx-ky) oo ky2_ 2 2 2 2 2 1/2'

(42)

(43)

(44)

jrnnqr ikqkrkmkn f ø f o• [ l __ ( __ l )m cos 2kxa] [ l __ ( __ l )n cos 2kyb]dkx dky -- •T 2 ( 2 2 2 2 k 2 2 2 1/2' kx--km)( )(ky 2 kr2)( oo --oo kx-k2q)(ky 2 k n - - (45)

Eq. (44) can be written as

O[ i•-- ( k2m q- k2n) l [ $22-- ( k2m q- k2 n) ]abwmn /2002[ 1 q- (O/•)(k2m q- k2n) q- (Ro/•)w2l

= E Zjrnnqrwqr ß (46) q=l r=l

The terms jrnnqr are often referred to as the intermodal

coupling coefficients. 27 Equation (46) leads to four sets of infinite systems of equations, corresponding to the different possible combinations of even and odd mode numbers (m,q) and (n,r). In order to solve Eq. (46) for the resonance frequencies of the fluid-loaded simply supported plate, we need to evaluate the jrnnqr coefficients. At this point we shall assume that the fluid loading will affect mainly the modes of vibrations, which will be corner

modes, where km,kq>k and kn,kr>k. This assumption can be verified by using only the corner modes in evaluat- ing the resonance frequencies and then comparing the results to estimate modes by including higher order modes.

The evaluation of the jmnqr, which requires an integra- tion in the complex plane, has been approximated by Davies 26 as follows:

jrnnqr-J•Rnqr q- iC nqr, kn,kn,kq,kr> k, (47) where

--2k

J•Rnqr• 71.k•7qkr ( sin 2ka

1--(--1)m•--(--1) n 2ka

sin 2kb sin 2k(a 2 b 2 ) X 2kb --(--1)m+n q- ) • 2k(a2q-b2) (48)

and

ab 2a kink r

jrffnqr k2m q- k2n rr ( k2m q- k2n ) ( k2q q- kr2 ) •mq

2b kmk q ---- 2 k2r)•nr, 71' ( k2rn q- k2n ) ( k q q .

(49)

where t•mq and t•nr are Kronecker delta functions. The imaginary parts J•nxnqr of the coefficients are associated with the radiation damping of the plate. The real parts Jr•Rnqr of the coefficients can be associated with a virtual mass added

to the mass of the plate, which causes a decrease in the resonance frequencies.

Expanding Eq. (46) for rn, n, q, and r odd, we obtain

j1 jqrqr--Fq r •qr] 111_Fllj1113 ...... jllq }Vll ] j1313 F13 .... j13qr W13[ j3131 F31...j31qr W31 =0,

where

(5O)

2 2

O [ • • -- ( k 2q q- kr2 ) l [ • 22 -- ( k q q- kr ) l a b Fqr= . (51)

poW2 [ 1 q- (O/•)(k2qq-kr 2) q- (Ro/•)w2l Care must be exercised in assembling the above matrix. It should be assembled according to increasing modes, e.g., if the (3,1) mode is lower than the (1,3) mode, then the (3,1) coefficient should be first. The order of the modes

will depend on the geometry of the plate. The proper as- sembly of the matrix will guarantee that the proper modes are included in the series expansion, thereby reducing com-



TABLE I. Comparison of results from present model to published results for thick SSSS plate (h/a=0.1).

Mindfin plate Mode (m,n) theory 32 Present analysis

(1,1) 0.9300 0.9300 (1,2) 2.2176 2.2176 (2,2) 3.4018 3.4018 (1,3) 4.1440 4.1438 (2,3) 5.1974 5.1974 (3,3) 6.8208 6.8208

putations by increasing the rate of convergence for increasing number of modes.

Equation (50) will have a solution if the determinant of the matrix vanishes. The matrix in Eq. (50) must be truncated before computations are possible. We will start by truncating the matrix to one term and then increase the number of terms until convergence of the resonance frequencies is obtained. As will be seen in Sec. III, convergence is obtained using only the first mode. It will also be shown in Sec. III that the radiation terms have an insignificant effect on the resonance frequencies. Hence we can write down a simple equation that includes only the diag- onal term in Eq. (50) to solve for the resonance frequencies with good accuracy:

O[•i•- (k2m-f-•2n) ] [•- (•2m-f-k2n) ]aO

( ab 2(ak2m-f-bk2n) ) x : : : =0. •r( km -f- k n)

III. NUMERICAL RESULTS

In Sec. II, analytical solutions for both the free vibration and the fluid-loaded response of the plate are given. In this section, numerical results are presented. We shall start by presenting results for plates with classical boundary conditions, e.g., SSSS and SCSC, where S means simply supported and C means clamped. Comparison of these results to published results shall serve as validation of the current analysis. Then we shall present results featuring the effect of the different stiffness of the restraints on the

resonance frequencies of the plate. Finally, results on the convergence of the solutions with fluid loading will be presented and some of the features of the solutions will be

discussed.

A. Classical boundary conditions

The SSSS and SCSC boundary conditions are reached by, in Eqs. (7), letting K•-• •, K2-• •, and K 3 =0 for the SSSS, and K2-• •, K2-• •, and K3-• • for the SCSC boundary conditions. In Tables I and II the results for the frequencies of resonance for the first six flexural modes of square thick (h/a=O. 1) plates with two boundary conditions, SSSS and SCSC, are presented. For all of the results in Tables I and II, v=0.3 and k•2=•r2/12. For these plates,

TABLE II. Comparison of results from present model to published results for thick $CSC plate (h/a=0.1).

Mindlin plate Mindlin plate theory-- theorym

Rayleigh-Ritz finite strip Present Mode (rn,n) solution 33 solution 34 analysis

(1,1) 0.1411 0.1411 0.1400 (2,1) 0.2668 0.2668 0.2668 (1,2) 0.3377 0.3376 0.3376 (2,2) 0.4608 0.4604 0.4604 (3,1) 0.4979 0.4977 0.4977 (1,3) 0.6279 0.6279 0.6278

solutions based on Mindlin plate are available in Refs. 32- 34. As can be seen from the results in Tables I and II, there is a good agreement between results based on the present analysis and the results from Refs. 32-34 that were obtained by Rayleigh-Ritz and finite-strip methods.

B. Translational stiffness

Figures 3 and 4 feature the effect of the translation stiffness K• on the frequencies of resonance. The results presented in these figures are for a thick (h/a=O. 1) plate simply supported on two opposing edges, and restrained only against deflection on the other edges. The results in Figs. 3 and 4 are generated for different aspect ratios and modes of flexural vibration. The frequencies of resonance as a function of the translational spring constant K• are characterized by three regions. In the first region, for lower values of K•, the frequencies of resonance are constant; that is, up to a certain spring constant, the plate will respond as if the boundaries were free. The second region is characterized by an increase in the frequencies of resonance with increasing edge stiffness. Finally, the third region is a plateau, which means that there is a finite spring constant beyond which the plate will respond as if it were simply supported on the edge. Joga Rao and Kantham, 8 Laura and co-workers, •ø-•5 and Warburton and Edney 2ø observed similar variations in the frequencies of resonance with increasing stiffness of edge restraints with classical

0.9

10 '9 10 '6 10 '3 1 10 3 oo •

Free K 1/Ks 2 hG Simply Supported

a

a 2

b 3

FIG. 3. Frequencies of resonance for the ( 1,1 ) mode for the thick plate (h/a=O. 1) with translational edge restraints.



28

10-12

Free

a.•.. = 1 b

a 2

10 '9 10 '6 10 '3 1 103 oo •

K 1/Ks 2 hG Simply Supported

FIG. 4. Frequencies of resonance for the (10,10) mode for the thick plate with translational edge restraints.

38--

28

10-12

Simply Supported

LU

•' 34 -I-

a.•.= 1 b

a 2

10 '9 10 '6 10 '3 1 103 oo --•

K 1/Ks 2 hG Clamped

plate theory. By observing Figs. 3 and 4 it can be seen that the frequencies of resonance for the higher modes are less sensitive to the boundary conditions than those of the lower modes. For the higher modes, a larger number of waves exist across the plate, so that the boundary conditions affect a smaller percentage of the total number of waves in the plate at the higher modes than at the lower modes. Therefore, as the mode number increases, the effect of the boundary conditions on the frequencies of resonance decreases.

C. Torsional stiffness

Figures 5 and 6 feature the effect of the torsional stiffness K 3 on the frequencies of resonance. The results presented here are for a thick plate simply supported on two opposing edges and fixed on the other two edges against deflection and restrained against rotation by a torsional spring constant K 3. As seen in these figures, all of the features discussed above for the translational boundary restraints are present here, except as K 3 is varied from zero to infinity the boundary conditions move from simply supported to clamped. We would like to emphasize the prac-

.2

.0

0.8--

0.6 I 10-10 10-8

Simply Supported

a__= 1 b

10 -6 10 -4 10 -2 1 102 104 oo •

K 1/Ks 2 hG Clamped

a 2

FIG. 5. Frequencies of resonance for the ( 1,1 ) mode for the thick plate with torsional edge restraints.

FIG. 6. Frequencies of resonance for the (10,10) mode for the thick plate with torsional edge restraints.

tical importance of the plateaus in Figs. 3-6. In general, it is very hard to simulate the simply supported classical boundary conditions, because there will always be some torsional stiffness present. However, this study shows that finite values of translational and torsional stiffnesses will

simulate simply supported and clamped boundary conditions. Also, one can determine the mode number or frequency above which any boundary condition can be used and still obtain fairly accurate solutions for the frequencies of resonance of plates.

D. Fluid loading

The frequencies of resonance for the first mode of water-loaded steel plates, having a/b= 3/4, and h/a=O.01 and 0.1, computed with and without the imaginary (radiation) terms of the determinant and with different num-

bers of terms in the determinant are presented in Table III. As seen in Table III, ignoring the radiation terms for the lower modes does not affect the accuracy of the solution. Also, it Can be noted that convergence of the solution to the third digit is attained by just taking a one-term expansion in the solution. Tables IV and V show the differences

between the frequencies of resonance for in vacuo and water-loaded thin and thick plates, respectively. Results are presented for the first eight modes for plates with an aspect ratio of a/b= 3/4. It can be seen that the water loading significantly lowered the lower frequencies of resonance of the thin plate (see Table IV). For both the thin

TABLE III. Solution for the frequencies of resonance for the ( 1,1 ) mode of the fluid-loaded plate.

Number of terms Thin plate with Thick plate with

and without radiation and without radiation

1 0.008 251 0.153 843 2 0.008 250 0.153 842 3 0.008 250 0.153 841



TABLE IV. Effect of fluid loading on the frequencies of resonance for the thin plate (h/a=O.01).

,

Mode (m,n) In vacuo Water loaded Percent difference

fected by the fluid loading than the lower modes. The effect of fluid loading on reducing frequencies of resonance is more pronounced for thinner plates.

( 1,1 ) 0.018 806 0.008 251 56.1 (1,2) 0.039 117 0.024 316 47.8 (2,1) 0.054 885 0.030 474 44.5 (1,3) 0.072 909 0.042 962 41.1 (2,2) 0.075 162 0.044 460 40.8 (2,3) 0.108 928 0.068 893 36.7 (3,1) 0.114 970 0.073015 36.5 (1,4) 0.120 176 0.077 465 35.5

and thick plates, the effect of the fluid loading becomes less pronounced at higher modes. Also, it can be seen that the thick plate is affected less by the fluid loading. This is because the fluid loading acts as an added mass that will lower the frequencies of resonance of the plate. Because the thin plate has less mass than the thick plate, the relative increase in the total effective mass for fluid loading is less for the thick plate than for the thin plate so that the decreases in the frequencies of resonance for the thin plate with fluid loading are greater than the decreases for the thick plate.

IV. CONCLUSIONS

Within the accuracy of the Mindlin plate theory, an exact solution is presented for the free vibration of a plate simply supported on two opposing edges and elastically supported on the other two edges. Increasing the stiffness of the restraints produced three regions for the frequencies of resonance. For lower values of the restraint stiffness, the frequencies of resonance were equal to the frequencies of resonance for free or simply supported plates, and were insensitive to the stiffness of the restraint. In the second

region, the frequencies of resonance increased with increasing stiffness of the edge restraints, reaching a plateau in the third region where the frequencies of resonance for classical simply supported or clamped boundary conditions were produced. Higher modes were found to be less affected by the kind of boundary conditions than lower modes. With fluid loading, it was shown that the radiation from the plate had an insignificant effect on the frequencies of resonance for the lower modes. The fluid loading lowered the frequencies of resonance. The higher modes are less af-

TABLE V. Effect of fluid loading on the frequencies of resonance for the thick plate (h/a-O. 1).

Mode (rn,n) In vacuo Water loaded Percent difference

(1,1) 0.182 982 0.153 843 15.9 (1,2) 0.370 169 0.329 702 10.9 (2,1) 0.509 215 0.461 398 9.4 (1,3) 0.661 919 0.609 519 7.9 (2,2) 0.680 576 0.626 739 7.9 (2,3) 0.949 961 0.888 594 6.5 (3,1) 0.996 138 0.932 612 6.4 (1,4) 1.035 996 0.972 885 6.1

1A. W. Leissa, "The free vibration of rectangular plates," J. Sound Vib. 31, 257-293 (1973).

2A. W. Leissa, "Recent research in plate vibrations: classical theory," Shock Vib. Dig. 9, 13-24 (1977).

3 A. W. Leissa, "Recent research in plate vibrations. 1973-1976: complicating effects," Shock Vib. Dig. 10, 21-35 (1978).

4A. W. Leissa, "Plate vibration research, 1976-1980: classical theory," Shock Vib. Dig. 13, 11-22 (1981).

5A. W. Leissa, "Plate vibration research, 1976-1980: complicating effects," Shock Vib. Dig. 13, 19-35 (1981).

6A. W. Leissa, "Recent studies in plate vibrations: 1981-1985 Part I. classical theory," Shock Vib. Dig. 19, 11-18 (1987).

?A. W. Leissa, "Recent studies in plate vibrations: 1981-1985 Part II. complicating effects," Shock Vib. Dig. 19, 10-24 (1987).

s C. V. Joga Rao and C. L. Kantham, "Natural frequencies of rectangular plates with edges elastically restrained against rotations," J. Aero- naut. Sci. 24, 855-856 (1957).

9T. E. Carmichael, "The vibration of a rectangular plate with edges elastically restrained against rotation," Q. J. Mech. Appl. Math. 12, 29-42 (1959).

10p. A. A. Laura and E. Romanelli, "Vibrations of rectangular plates elastically restrained against rotation along all edges and subjected to a bi-axial state of stress," J. Sound Vib. 37, 367-377 (1974).

l l p. A. A. Laura, L. E. Luisoni, and C. Filipich, "A note on the deter- mination of the fundamental frequency of vibration of thin, rectangular plates with edges possessing different rotational flexibility coefficients," J. Sound Vib. $$, 327-333 (1977).

•2p. A. A. Laura, L. E. Luisoni, and G. Ficcadenti, "On the effect of different edge flexibility coefficients on transverse vibrations of thin, rectangular plates," J. Sound Vib. $7, 333-340 (1978).

13C. P. Filipich, J. A. Reyes, and R. E. Rossi, "Free vibrations of rectangular plates elastically restrained against rotational and translation simultaneously at the four edges," J. Sound Vib. 56, 299-302 (1978).

•4p. A. A. Laura and R. Grossi, "Transverse vibration of a rectangular plate elastically restrained against rotation along three edges and free on the fourth edge," J. Sound Vib. $9, 355-368 (1978).

•Sp. A. A. Laura and R. O. Grossi, "Transverse vibrations of rectangular plates with edges elastically restrained against translation and rotation," J. Sound Vib. 75, 101-107 (1981).

16D. M. Egle, "The resonant response of a rectangular plate with an elastic edge restraint," J. Eng. Industry 94, 517-525 (1972).

l?E. M. Nassar, "Rapid calculation of the resonance frequency for rotationally restrained rectangular plates," AIAA J. 17, 6-11 (1979).

iSM. Mukhopadhyay, "A semi-analytic solution for free vibration of rectangular plates," J. Sound Vib. 60, 71-85 (1978).

•9I. Elishakoff and A. Steinberg, "Vibration of rectangular plates with edge-beams," Acta Mech. 36, 195-212 (1980).

2øG. B. Warburton and S. L. Edney, "Vibrations of rectangular plates with elastically restrained edges," J. Sound Vib. 95, 537-552 (1984).

2• A. V. Bapat, N. Venkatramani, and S. Suryanarayan, "Simulation of classical edge conditions by finite elastic restraints in the vibration analysis of plates," J. Sound Vib. 120, 127-140 (1988).

22D. J. Gorman, "A general solution for the free vibration of rectangular plates resting on uniform elastic edge supports," J. Sound Vib. 139, 325-335 (1990).

23R. D. Mindlin, "Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates," J. Appl. Mech. 18, 31-38 (1951).

24 E. B. Magrab, "Natural frequencies of elastically supported orthotropic rectangular plates," J. Acoust. Soc. Am. 61, 79-83 (1977).

25 U.S. Lindholm, D. D. Kama, W. Chu, and H. N. Abrambom, "Elastic vibration characteristics of cantilevered plates in water," J. Ship Res. 9, 11-22 (1965).

26 H. G. Davies, "Low frequency random excitation of water-loaded rectangular plates," J. Sound Vib. 15, 107-126 (1971).

27 L. D. Pope and R. C. Leibowitz, "Intermodal coupling coefficients for a fluid-loaded rectangular plate," J. Acoust. Soc. Am. $6, 408-415 (1974).

28 K. Nagaya and J. Takeuchi, "Vibrations of a plate with arbitrary shape in contact with a fluid," J. Acoust. Soc. Am. 75, 1511-1518 (1984).



29E. B. Magrab, Vibrations of Elastic Structural Members (Sijhoff & Noordhoff, Groningen, 1979).

3øM. C. Junger and D. Feit, Sound, Structures, and Their Interaction (MIT, Cambridge, MA, 1986), 2nd ed.

31j. Hammouda, "Effect of boundary conditions and fluid loading on the resonant response of thick plates," M.S. thesis, Pennsylvania State Uni- versity (August 1988).

32 R. D. Mindlin, A. Shacknow and H. Deresiewicz, "Flexural vibration of rectangular plates," J. Appl. Mech. 23, 430-436 (1956).

33 D. J. Dawe and O. L. Roufaeil, "Rayleigh-Ritz vibration analysis of Mindlin plates," J. Sound Vib. 69, 345-359 (1980).

34D. J. Dawe, "Finite strip models for vibration of Mindlin plates," J. Sound Vib. $9, 441-452 (1978).



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