Journal of Sound and Vibration (1973) 28(1), 121-132

FORCED EXTENSIONAL VIBRATIONS OF PLATES

C. R. THOMAS

Benet Weapons Laboratory, Watervliet Arsenal, IVatervliet, New York 12189, U.S.A.

(Received 20 November 1972, and hz revised form 6 February 1973)

The forced extensional vibrations of plates is investigated within the framework of the Kane-Mindlin theory for the high frequency extensional vibrations of isotropic elastic plates. The solution to problems with time dependent surface and boundary conditions is determined according to Williams' method. The resulting equations are applied to the forced extensional vibrations of infinite plate strips with clamped edge conditions. Numerical results for a shear loading are presented.

I. INTRODUCTION

The free extensional vibrations of isotropic elastic plates have received extensive coverage in the literature in the last two decades. Plate theories resulting from the reduction of the three- dimensional theory of elasticity to a two-dimensional theory are of particular interest. Kane and Mindlin [1] effected a first-order power series approximation to derive a set ofdifferential equations and boundary conditions accounting for the coupling between extensional motion and the .first thickness mode of vibration and gave the solution for the axially symmetric vibrations of a circular disk. Mindlin [2] discussed the expansion of displacement in an infinite power series of plate thickness coordinate; Mindlin and Medick [3] derived a second- order approximation based on an expansion into a Legendre polynomial series of thickness coordinate. Anderson [4] applied the Kane-Mindlin equations for extensional motion to the infinite plate strip and compared the frequency spectra obtained with that of similar results derived from the theory of generalized plane stress.

The problem to be discussed in this paper is that of using the Kane-Mindlin [1] partial differential equations to obtain a solution for the forced extensional vibrations of isotropic elastic plates having time dependent body forces and surface tractions as well as time depend- ent, non-homogeneous boundary conditions. The solution of this problem will be found by the application of a superposition principle which resolves the original problem into a time dependent, non-homogeneous "static" problem and a homogeneous dynamic problem; the method outlined is commonly termed Williams' method [5]. The same principle was studied by Berry and Naghdi [6] who considered a transformation of variables to remove non- homogeneous boundary conditions, the final result being the solution of a time dependent, non-homogeneous "static" problem and a homogeneous dynamic problem. While the "static" problem is time dependent, the motivation for the designation "static" is based on the fact that in this segment of the solution significant parts of the inertia force are not present.

According to Leonard [7] two primary advantages of Williams' method are that for many loading conditions the modal solutions for the dynamic response converge rapidly and the discontinuities of the response function are isolated in the "static" portion of the response.

121

122 c.R. TttOMAS

Additional advantages of Williams' method discussed by Reismann [8, 9] are that it may be more expedient than solutions obtained by integral transforms and that solutions may be attainable in instances where transforms do not exist or transform inverses are difficult to obtain or are unknown.

Reismann [9] has applied Williams' method to the Mindlin equations for plate flexure to obtain the forced flexural motion of elastic plates under the action of time dependent trans- verse surface loads as well as time dependent boundary conditions; however, the effect of time dependent surface shear loads and body force were not included in the development.

In the present paper the author uses the Kane-Mindlin [1] equations for extensional vibrations of plates and develops a general representation for the solution of the forced extensional motion of plates via the Williams' method. The derived Williams' solution allows for the determination of the stresses and deformation in the plate when subjected to time dependent shear and transverse surface loads, time dependent body forces, and time dependent, non-homogeneous boundary conditions.

The solution to the homogeneous dynamic problem resulting from an application of Williams' method is sought in the form of modal expansion of the normal modes or "free vibrations" and a set of generalized coordinates; the basic advantage of the resulting modal expansion over the classical method of modal expansion is that the effect of the non-homo- geneous boundary conditions is transferred to the "static" problem. An orthogonality condition for the normal modes is derived from the equations of motion with the assumption of homogeneous surface and boundary conditions. The generalized coordinates are found by expanding the equations of motion in terms of the Williams' solution and utilizing the "static" solution, the normal modes, and the orthogonality condition to introduce appro- priate reductions.

The forced extensional vibrations of an infinite plate strip with clamped edge conditions and uniform applied shear load is considered as an example of the method. The natural frequencies and mode shapes are obtained as functions of the width-to-thickness ratio. The transcendental frequency equations corresponding to the boundary conditions are solved numerically for a given set of material and geometric constants and the orthogonality condition is used to obtain the corresponding eigenfunctions. The "static" solution is obtained by direct integration and the generalized coordinates are evaluated in terms of the specified time dependence. Numerical results are presented.

2. FORMULATION OF THE PROBLEM

For an isotropic plate of thickness, h, the Kane-Mindlin stress equations of motion [l] for linear elastic extensional vibrations are

OT~/Ox + OT=,./ay + F= + f= = ph 0 2 v~/Ot 2,

OT~,./Ox + OTy,./ay + Fy + fy = ph a 2 v,./Ot 2,

aT==/ax + OT,.=/Sy - T== + F= + f= = (pl f l /6)a 2 v j o t 2, (1)

and the corresponding stress-displacement relations are

T= = h[(;. + 2t~) avAOx + ). av,./Oy + (2/(2 ;./h) v=],

T , =/1[(;. + 210 av,./ay + ;. avJOx'+ (2K, ;./h) v~],

T= = h[;.K, avAOx + 2(;. + 2t0 K ~, v,/h + ;.Ks av,/Oy],

T~ , = hi,[av,.lax + avAay], 7"=, = (t, 5 p]6) av,/Ox, T,,= = (112 p/6) Ov=/ay, (2)

F O R C E D E X T E N S I O N A L V I B R A T I O N S OF P L A T E S 1 2 3

where ). and It are Lamd constants, K2 is the correction factor ~ /V '~ , and p is the density; the expressions

F~ = [,,"r l+h/2 F r = f ~ l+h/z F~ = 1"Trr l+h/2 (3) t~xzJ- -h /2 ~ t~Yz/--h/2 ' t--~,~ZI--h/2

are the ptate surface traction effects, where z is the plate thickness coordinate, and ats(x,y,z, t) is the elastic stress tensor; the expressions

+hi2 +hi2 +hi2

f ~ = f B~dx, s f Brdy , f~= f zB~dz (4) -hn -hi2 -h/Z

are the plate body force effects, where B~, B,., and B~ are the body forces. The boundary conditions associated with the dynamic problem are of the form

T=,=T=,, T:,B=7"~,e, T==T== (5) on boundary C, and

t'==~=, t'a=~B, v : = 0 . ( 6 )

on boundary C2, where the total boundary C is C = C, + C2, the coordinates ~ and fl are measured normal and tangential, respectively, to the boundary curve C, and the barred quantities are the actual values of stress or displacement specified on the boundaries. Addi- tionally, the initial conditions imposed on the problem are

v~(x,y, 0) = t,~ m, vr(x,y, O) = v~ m, v=(x,y, O) = vT ~, (7)

b:.(x.y.O) = b~ ~, b,(x.y.O) = b~ ~ b.(x.y.O) = b c~ (8)

3. ORTHOGONALITY CONDITION

In utilizing Williams' method to represent a solution to the forced motion problem, an eigenfunction expansion of the free vibrations will be made; in anticipation of this, it is now convenient to develop an orthogonality condition for the free vibration eigenfunctions. The standard orthogonality procedure involving the ith and j t h displacements and stress equations of motion (1) in the absence of body forces and surface tractions, after a term-wise integration by parts together with appropriate substitutions of the stress-displacement relations (2) and the assumption of a free vibration solution of the form

v ~ ~ = U ck~ sin 12~ t , t;~ ~ = V a~ sin f2k t, vr ~ = W m sin 12~ t , ( 9 )

yields the equation

(O]- f22) f f t u " ' u t i ' + V " ' V ~ 1 7 6 (10) xy

Clearly, if f2, # f2s, the orthogonality condition is

f [u m U~J~ + y m Vtj~ + IV,~ WO~/3]dxdy = 0 (11) X y

for i # j . Now, unique expressions for the eigenfunctions can be assured by the introduction of a mode normalization condition in the form

f [u m Urn+ V u~ V "~ + IV "~ Wm/3]dxdy = Nl, (12) x y

where Art is yet to be determined.

124 C.R. THOMAS

4. THE WILLIAMS' SOLUTION

A solution to the forced motion problem will now be sought in terms of Williams' type modal solutions. The essence of Williams' method is to represent the total solution as a superposition of the time dependent "static" solution and of an eigenfunction expansion in terms of the free vibration eigenfunctions and a set of generalized coordinates. The Williams' solution for forced extensional motion is

vx(x,y, t) = u*(x,y, t) + ~ U~"~(x,y)q.(t), t l

vr(x,y, t) = vS(x,y, t) + ~. V~"~(x,y)q,(t), i i

vz(x,y, t) = ~ ( x , y , t) + ~ tVC")(x,y)q.(t), (13) i1

where u s, v ~, and w ~ are the "static" solutions, U ("), V ("), and IV (") are the free vibration eigenfunctions and q,(t) are the generalized coordinates. Since the Kane-Mindlin theory is a linear theory, the stresses may also be represented as a superposition of the "static" stresses and an eigenfunction expansion of the free vibration stresses and the generalized coordinates"

T~j(x,y, t) = T~(x ,y , t) + ~ T~ ' (x ,y )q , ( t ) , (14) i1

where the T]j(x,y, t) are the "static" stresses and the T~)(x ,y) are the free vibration stresses. The critical step in applying the Williams' solution to the boundary value problem (I) is a

proper choice of the "static" solution--the "static" solution must be formed to include the effects of body force, surface tractions, and non-homogeneous boundary conditions. Conse- quently, the "static" solution is picked to satisfy the "static" stress equations of motion,

Or~JOx + OT~,./Oy + F~ + f~ = O,

OT], /Ox + OT],/ay + Fy + fy = O,

OT~=/Ox + OT]./Oy - 7..5 + F, +f~ = 0, (15)

subject to the "static" boundary conditions,

TaS~ = T=,, T2e = T,a, T~S~ = T,~ (16)

on boundary Ca and vl = t3,, v~ = t5 a, t,~ = ~ (17)

on boundary C2. The stress equations of motion for free vibrations are

~"t"'Ov ph3 2v~,/0t2' OT~"]/Ox + o l ~ , ! ., = ~"~ = ph a z t,~"~/Ot z, (n)lOv OT~)/Ox + oT,:= t . , -- T ~ ) = ( hz p/6) 02 v?)/Ot 2, (! 8)

subject to homogeneous boundary conditions. Williams' solution (I 3) is substituted into the stress equations of motion (1) and then the

resulting relations are reduced by the "static" equations (15) and the free vibration equations, which are identically satisfied, to yield the expressions

~- [0"~ + 12Z.q.] U~"~ = -i~, n

E [~/. + g22.q.l V'11' = _/~s. I I

Y~ [0. + a~.q111 w , " , = -~(,,. n

(19)

F O R C E D E X T E N S I O N A L V I B R A T I O N S O F P L A T E S 125

The normal coordinates are found by, respectively, multiplying equations (19) by U ("), V ("), and Wt")]3, adding the resulting equations, integrating through the plate area, and applying the orthogonality conditions (11) and (12) to the result; the differential equation for the normal coordinates then becomes

where

iL + z P,(t), (20)

(21) WC"~13]dxdy. x y

The normal coordinates are given by the equation t

q. ( t )=q. (O)cos( f2 . t ) + [(l.(O)/f2.]sin(O.t)+ (l/f2.) f P . ( O s i n [ f 2 . ( t - O ] d ~ , (22) 0

which is the solution to differential equation (20) as found by the method of variation of parameters; q.(0) and r).(0) are yet to be determined with the aid of the initial conditions. The Williams' solution (13) is evaluated for the initial conditions on displacement (7) and velocity (8) at time zero. The initial conditions on displacement are multiplied, respectively, by U tin), V (m~, and IV (m), the equations are added and the resulting equation is integrated through the plate area, yielding the relationship for qn(O) :

q,(0) = (l/N,) f f [v~ ) U (") + v~ ~ V '") + v (~ W(")/3I dx dy + P,(0). (23) x y

In a similar manner, by beginning with the initial conditions on velocity, it can be shown that

d.(0) = (1/N.) f~ [,)~o, U'"' + e~o, V("' + t)~' IV'"'/31 dx dy + P,(0). (24) .x'y

With ,7,(0) and 0,(0) known, the formal solution for the normal coordinates represented by equation (22) is now finished. A complete solution to a specific problem requires the deter- mination of the "static" solution and the free vibration solution for that particular problem.

Some effort in the evaluation of equation (21) for P,( t ) can be saved by expressing it in a more convenient form. An appropriate utilization of the stress equations of motion for free vibrations (18) and the stress-displacement relations for the "static" and free vibration solutions, which have the same form as equations (2), along with several integrations by parts, yields

N, pld22pl(t) = f (0= T", + va T" ' 2~r,T~)/h)dl - - f (U~" T=. + U~ ~, T= B + 2 W u' T=z/h) dl C2 C 1

- I f I(r~ +Z) u, ' , + (r. +L) z,', + 2(F~ +L) w.,//,] dxdy, (25) x.V

which is far easier to evaluate for many applied problems than is equation (21).

5. FORCED VIBRATION OF AN INFINITE PLATE STRIP

The equations previously developed will now be applied to the forced extensional vibrations of an infinite plate strip with clamped edge conditions and a uniform applied shear load. The plate has width a and thickness h; the coordinate system is indicated in Figure 1. The displacements are chosen in the form [4]

vx = O, vy = Vy(y, t), t,~ = v~(y, t). (26)

126 c.R. THOMAS

A uniform, time dependent shear load of magnitude Qg(t) is applied to the upper face of the plate; the time function is chosen as triangular and is given by

l - - t ] T 0 < t < r ] , g(t) (27)

0 T < t ' j"

it might be noted in this case that the exact same problem results if the loading is equivalently considered as Qg(t)]2 on the upper face of the plate and -Qg( t ) ]2 on the lower face of the plate. The clamped edge conditions are chosen to be homogeneous and thus

vy(+a]2, t) = O, v,(• t) = 0 (28)

must be satisfied on the boundaries.

Z

Figure 1. Reference system and dimensions.

The "static" displacement equations of motion for the infinite plate strip with clamped edge conditions and a uniform applied shear load are

h(). + 210 02 vS/Oy 2 + 22K20~]Oy + Qg(t) = O,

(h 2/+/6) 02 wS/Oy 2 - 2(2 + 210 K~ w s - 2hK20vS/Oy = 0, (29)

where appropriate reductions have been made for the chosen displacements (26) and the shear load. The boundary conditions which must be satisfied are

vS(+a/2, t) = O, wS(• t) = 0. (30)

Taking one derivative of the second equation in (29), solving for OZv~]Oy 2, and substituting into the first equation in (29) gives the differential equation for w S as

0 3 WS/Oy 3 - - (A/h 2) Ow~/Oy + (B/h2)g(t) Q = 0, (31)

where

a = 48K~(2 + tOlO. + 2/~), B = 6).K2/[I,(2 + 21t)]. (32)

The differential equation for wL equation (31), is solved by the method of variation of parameters and the resulting solution is

w ~ =g( t ) QyB/A + dl + d2 cosh (V'-'Ay/h) + d3sinh(V'-'Ay/h), (33)

where d~, d2, and d3 are arbitrary constants. Substitution of wL equation (33), and its second derivative into the second equation of (29) together with one direct integration results in the solution

v ~ = -d2[2).K2/X/-A(). + 2/0] sinh (V '~y /h) - 2(2 + 210 K2 d~ y/).h + d4 -

- d3[2).K2/V'-'AO. + 21t)]cosh (V"-Ay/h) - (2 + 210g(t) Qy2/8tthO. + It), (34)

where d4 is an arbitrary constant.

FORCED EXTENSIONAL VIBRATIONS OF PLATES 127

Now, for convenience, y will be made dimensionless with the transformation )7 --- 2 y / a and the dimensionless width-to-thickness ratio L = a/h will be introduced. The four arbitrary constants in equations (33) and (34) are evaluated with the aid of the "static" boundary conditions (30) and after simplification the "static" displacements are

v s = L a Q g ( t ) O. + 2it) (1 - .f,2)/[32p(). + p)] + a Q g ( t ) ).2[cosh (V 'ALf i[2) --

- cosh (V'-AL/2)]/[8X/-A sinh (VrAL/2) ItO. + It) O- + 2It)],

w s = [a Q g( t ) 2/16K2it() . +/t)][f i - sinh ( v ' ~ L p / Z ) / s i n h (V'~L/2)]. ('35)

The free vibration problem for the infinite plate strip with clamped edge conditions will now be considered. The eigenvectors and the natural frequencies will be determined. The free vibration displacement equations of motion for the infinite plate strip are

p 02 v~"~/Ot 2 = (2 + 2p) 02 v~"~/Oy 2 + (2;.K2[h) v~"',

p Oz v~.)/Ot 2 = It 0 z #~n) lOy2 - - (6/h) 2Kz Ov~"VOy - (I 2/I?) (2 + 2it) Kz"z .'z"~"~, (36)

and the corresponding boundary conditions are

v~"'(zka/2, t) = O, v~"~(• t) = 0, (37)

for clamped edges. A useful reduction can be accomplished by letting

v~"~(y, t ) = Odpt"'(y, t ) /ay; (38)

then, for free vibrations,

t,~ "~ = [O~"~(y)/Oy] sin f2. t, vt2 ~ = WC"~(y) sin I2. t. (39)

The wave velocities

C~ = (2 + 2It)lp, CZz = It/p, C~ = )./p = C 2, - 2C~ (40)

are introduced into the analysis and equations (39) are substituted into equations (36); some algebra together with suitable rearrangement of terms results in the differential equation

d 4 ~,t")/dy 4 + (Bt") /h2)d 2 IPtnVdy 2 -k- (CtnV]14) ~tn) = 0, (41)

where

B ' " ' = 12K.2[(1 + C~/C~)122.l~ 2 + 4(C22/C~ - 1)],

C ( . ) = 4 2 144K2(f2. C ~ / ~ z C 2) (t22./~ , - 1) . (42)

are dimensionless constants--appropriate use has been made ofthe Kane-Mindlin correction coefficient K 2 = rc2/12 and the fundamental thickness-stretch frequency ~ = =z C2[h 2, which has been used to form a dimensionless frequency ratio. Differential equation (41) has constant coefficients and may be written as

(D z + fi~""/h z) (D z + 6~"~'117)~,~") = 0, (43).

where the notation D = d /dy has been employed; the eigenvalues fie,) may be obtained by a superposition of the form

~t., = g,~., + ~.~, (44)

where 4] ") and ~t2") are solutions of the differential equations

(D 2 + a '?~ ' /h ' ) q,'?' = 0, ( D2 + a'~""//, ~) 4J?' = o, (45)

the coefficients being given by the equations

6].)' = (Bt.~/2) + (V'B c"~ _ 4Ct.)12),

6~ ")' = (B~")12) -- (a /B ~")~ - 4C~"~12). (46)

128 c.R. THOMAS

Care must be taken in considering equations (43) and (44); 6(1 ")" is always positive, but 3r ")2 may be positive, zero, or negative, depending on the value of (2,]~ in the form

/ +i} 3(2 "'~ = ~ . / ~ = , (47)

(> 0 >

and hence the form of the solution ~,~.") is frequency dependent. For the case where Y2 ")2 < 0, the substitution

6~ ")2 = - 6 7 ) ' (48)

is made with the result being 6] ")2 > 0 and the solutions to the differential equation

(D' - 5•)2/h 2) ~,c2") = 0 (49)

will be sought. While a third solution for the instant when g2,/s~ = 1 is possible, the con- tinuity of any desired data spectrums for the cases Q,/D --# 1 indicates that this third solution is not required for numerical results. The final solution, equation (44), will split into separate parts for symmetric and antisymmetric modes of vibration upon application of the boundary conditions (37) to evaluate the arbitrary constants; however, at this point $(") may be determined only to within an arbitrary multiplicative, frequency dependent constant.

Since the applied uniform, time dependent shear load Qg(t) is antisymmetric in space, only the antisymmetric part of the free vibration solution will be developed. An appropriate application of the boundary conditions (37) results in

b r [sin (5~")Y] h) _ 3r ") cos (6(;)L/2) s inh (6~")y/h!] 6(4 ") cosh (6(4 ") L/Z) ] (50)

I .

for ~. /D < 1 and

b c") [sin (6?)y/h) - 3]") cos (6c~') L/2) sin (6~2")y/h)] ~,(.) fit,.,) cos (6r ") LI2) J (51)

I.

for Q. /~ > I. The free vibration eigenfunctions are constructed from the equations

v( . ) = a r + aOg")lOy, W (") = (C~/2C 2 K 2 It) (6~ n)2 - - x 2 ~2/D2) ~ . ) + (C2/2C~ K2 h ) ~.u2tX(n)2 __ ~2,~ "~'nl "~'f'12/~221 'e'2dAn) (52)

and the results are gathered in the form

V <") = b ~") P(")lh, W c") = b r ff,'c")/h, (53)

where the b c") are constants yet to be determined. The orthogonality conditions (11) and (12) are reduced for the infinite plate strip; upon

using the dimensionless transformation k; = 2y]a, and further N. = a]2, the orthogonality condition becomes

+ t

f (V(")" + W(') ' ]3)dy-- 1. (54) -1

Now substituting equations (53) into equations (54) and solving for b c"~ gives these constants a s

,/J( b (") = (I 7("72 + ~Pt")2/3)dp; (55)

by carrying out the indicated operations these constants are easily determined.

FORCED EXTENSIONAL VIBRATIONS OF PLATES 129

The natural frequencies of free vibration are readily determined from the eigenfunctions (52) with the aid of the boundary conditions (37). The equations for determining the natural frequencies are

[6r n)2 -- rt2(f2n[~) z] 6~ n) - t anh (6c4 ") L]2) [6~ ")~ + ~z2(f2.l~) 216? ) = tan (3~ "~L/2) (56)

for -Q./-Q < 1 and [ 6 ~ ">2 - n2(O./~) 2] 6t2 ") +tan (6~ ">L/2)

= ( 5 7 ) [6t2 ")~ - r~2(f2./~) 2] 6tl "> tan (6~ "~ LI2)

for O . / ~ > 1, where the roots or natural frequencies of these equations can be found numerically.

The solution to the forced motion problem will be complete with the determination of the normal coordinates. For convenience in this example, the initial conditions are chosen to be homogeneous,

v~~ = Oty~ = v~)(y) = bc.~ = 0. + (58)

and hence equations (23) and (24) reduce to

q.(0) = e.(0), 0.(0) = P.(0). (59)

For the chosen problem with homogeneous boundary conditions and a shear load Qg(t), equation (25) reduces to

+1

P.(t) = --4Og(t) b ("> ~2/(phLZ n 2 C10z~) f (a P<")/2h) dp, (60) --1

which is easily integrated. Thus, by using equations (22), (59), and the integrated form of equation (60) along with the fact that P.(t) = 0 for a triangular time dependence,

q.(O) = P.(0) cos (0f2./~) + (~/I2.) P.(0) sin (0f2./~), (61)

where 0 is a dimensionless time defined as 0 = Ot. Upon considering the definition of the triangular time function, equation (27), and the equations for P.(t), equation (60), P.(0) and P.(0) are readily evaluated, and the problem is solved. All elements of the "static" solution, the free vibration problem, and the normal coordinates are known.

6. DISCUSSION AND CONCLUSIONS

The most meaningful results can be obtained by going to a dimensionless form of the displacement equations (16) and the stress equations (17), where the relationship

C21/C~ = (2 - 2v)/(l - 2v) (62)

is used to establish a dependence on Poisson's ratio. Thus, dimensionless stresses and dis- placements may be obtained by specifying Poisson's ratio v, width-to-thickness ratio L, width coordinate p, and dimensionless time 0. For a plate with pre-specified Poisson's ratio and width-to-thickness ratio, the values of dimensionless stresses and displacements may be determined at a fixed dimensionless width for variations of dimensionless time or at a fixed dimensionless time for variations of dimensionless width.

A number of graphs of dimensionless frequency versus width-to-thickness ratio, including those for clamped edges, have been discussed in detail by Anderson [4, 10] and hence such information for free vibrations will not be discussed here.

130 c . R . THOMAS

A width-to-thickness ratio of 1-25 and a Poisson's ratio of 0.3 are specified and typical results for the variations of stresses and displacements are presented. Since the plate is clamped, displacements are zero at the boundaries. An inspection of the respective series solutions for displacements and stresses shows that at the plate center, y = 0, and that the thickness displacement v, and the normal stresses T~x, T;r, and T= are all identically zero for this particular problem.

Eo

F i g u r e 2.

0"3 i I I I i

1"2

O I

0

- 0 ' 1

- 0 2

- 0 3 0 I ~ T ~ I 5 I0 15 20 25 30

D i m e n s i o n l e s s t i m e

T i m e v a r i a t i o n o f w i d t h d i s p l a c e m e n t . L = 1-25, v = 0-3, ~ = 0"5.

0"2

",, 0 - I

r (3 %

0 c

8 _o - O - I

i5

- 0 . 2

I I I I I

I I l I ] 0 5 I0 15 20 25 30

D i m e n s i o n I e s s t i m e 8

F i g u r e 3. T i m e v a r i a t i o n o f t h i c k n e s s d i s p l a c e m e n t . L = 1-25, v = 0"3, 2 = 0-5.

0"08 1 f

0 0 4

if) - 0 0 4

- 0 " 0 8 l . - - I J. 0 5 tO 15 20 25 30

D i m e n s i o n t e s s l i m e

Figure 4. Time variation of normal stress Tyy. L = 1 . 2 5 , v = 0 " 3 , .P = 0 - 5 .

o

F O R C E D E X T E N S I O N A L V I B R A T I O N S O F P L A T E S

0.6

04

02

0

-0-2

-04

-0 E

L

I 5 ~0

I i i

I ~ I 15 20 25 50

D i m e n s i o n l e s s t i m e 8

Figure 5. T i m e var ia t ion o f n o r m a l stress T==. L = 1 . 2 5 , v = 0 - 3 , )7 = 0 - 5 .

131

-x

o;

03

0"008 - - T I l r T - -

0004 O! ~

-0004

-0-008 ~ 1 1 5 tO 15 20 2,5 30 D i m e n s i o n l e s s t i m e 8

Figure 6. Time variation of shear stress Tr=. L = 1-25, v = 0"3, 37 = 0.5.

0 . 2

o o

0 -0 -2

-0-4

I I "1

I I I -0 '5 0 0"5

D i m e n s i o n l e s s w ~ c l t h Y

Figure 7. Variation of displacement versus width. , pC~vr/Qa; - - - , p C ~ v J Q a .

The time variations o f the dimensionless plate displacements at the point 9 = 0.5 are indicated in Figures 2 and 3. The displacement in the z direction, v., is the displacement at the surface z = h]2 . The displacements used in the Kane-Mindlin equations are actually the result o f the assumption that the plate displacements are reasonably approximated as

tt~ = v.,, uy = vy, u= = ( 2 z / h ) t,:, (63)

where the actual displacements u,, r = 1, 2, 3, have been approximated as truncated infinite series o f powers o f the thickness coordinate z and the result o f this truncation for extensional

132 c. R. THOMAS

vibrations [I] is equation (63). The time variations of the dimensionless plate stresses at the point p = 0.5 are indicated in Figures 4, 5, and 6. As for the displacements, the stresses involving the z-coordinate direction are the stresses at the outer extremes of thickness or the maximum stresses. The dimensionless plate displacement in the y-direction (see Figure 2) reaches its first maximum value at dimensionless time 0 = 3 and the displacement in the z-direction becomes maximum slightly later. The displacement across the width at dimension- less time 0 = 3 is shown in Figure 7 and, as one would expect for the clamped plate, the displacement along the width is a maximum at the plate center. In a similar manner, the dimensionless times of the first peak in the time variation of stresses as determined from Figures 4, 5, and 6 were used to respectively calculate the variation of shear and normal stress along plate width; the absolute values of the shear and normal stresses are maximum at the plate boundaries.

A check on the accuracy of the numerical results was carried out in terms of the homo- geneous initial conditions by observing to how many significant figures they were satisfied. For the displacement in the y-direction, the first term in the series expansion for displacement satisfies the initial conditions for two significant digit accuracy and the first ten terms in the series expansion satisfy the initial conditions for three significant digit accuracy. The series for the displacement in the z-direction was slower to converge; fifteen terms resulted in satisfaction of the initial conditions for two significant digit accuracy with only several additional terms being necessary to give three significant digit accuracy. All numerical results presented are based on thirty term series expansions for the variables involved.

ACKNOWLEDGMENT

The author would like to thank his colleague, Dr Gary L. Anderson, for his advice and several helpful suggestions given during the course of this investigation.

REFERENCES

I. T. R. KANE and R. D. MINDLIN 1956 Journal of Applied Mechanics 23, 277-283. High frequency extensional vibrations of plates.

2. R. D. MINDLIN 1955 Signal Corps Engineering Laboratories, Fort Monmouth, New Jersey (AD-88471). An introduction to the mathematical theory of vibrations of elastic plates.

3. R.D. MXNDUN and M. A. MEDXCK 1959 Journal of Applied Mechanics 26, 561-569. Extensional vibrations of elastic plates.

4. G. L. ANDERSON 1970 Journal of Sound and Vibration 11, 309-323. Free extensional vibrations in plate strips.

5. D. WILLIAMS 1946 British Royal Aircraft Establishment Report S. M. E. C[7219/DW/19. Dis- placements of a linear elastic system under a given load.

6. J. G. BERRY and P. M. NAGHDI 1956 Quarterly of Applied Mathematics 14, 43-50. On the vibra- tion of elastic bodies having time-dependent boundary conditions.

7. R. W. LEONARD 1959 NASA TR R-21. On solutions for the transient response of beams. 8. H. REmMANN 1967 Applied Science Research 18, 156-165. On the forced motion of elastic solids. 9. H. RElSMANN 1968 Journal of Applied Mechanics 35, 510-515. Forced motion of elastic plates.

10. G.L. ANDERSON 1971 Journal of Sound and Vibration 15, 545-556. Regular and singular perturba- tion expansions in the analysis of extensional vibrations of plates.