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Voi.13(6), 349-357, 1986. Printed in the USA Copyright (c) 1987 Pergamon Journals Ltd.

FLEXURAL UIBRAT!ONS OF RECTANGULAR PLATES WITH FREE EOGES R~ O. Mind!in Grantlnam, New Hampshire 03?53,

U,S,A,

(Receiued 23 October i986; accepted for print 23 October i986)

i, Introduction. In t h i s p a p e r , e q u a t i o n s o f f l e x u r a ! vibrations of isotropic, elastic plates, with rotatory inertia and s h e a r d e f o r m a t i o n terms i n c l u d e d E l l , are s o l u e d f o r r e c t a n g u l a r p l a t e s w i t h a l l f o u r edges f r e e , The s o l u t i o n s a r e e x a c t and i n c l o s e d form, Two main r e s t r i c t i o n s aOplv~ ( 1 ) the ! e n g i t n . / ~ i d t h r a t i o must be a r a t i o o f i n t e g e r s ; ' ( 2 ) f o r each u a ! u e o f P o ! s s o n ' s r a t i o , a l l t h e a l l o w a b l e modes must haue the same f r e q u e n c y ° In each case, final frequenc 2 lies between about 1,9 and 2,7 times the f r e q u e n c y , say ~ o , o f the f u n d a m e n t a l t h i c k n e s s - s h e a r mode o f the infinite plate, Howeuer, Line equations, with t h e approp.r!ate shear-correction factor, a r e i n t e n d e d f o r use a t frequen-ties no h i g h e r t h a n a b o u t 1 . 2 ~ 0 . To improue the a c c u r a c 2 a t the higher frequencies, two additional correctlon factors are i n t r o d u c e d to make the e q u a t i o n s p r o d u c e the e x a c t u a l u e s o f f r e q u e n c y and w a u e - n u m b e r , o b t a i n e d from R a y l e i g h ' s [ 2 ] exact. dispersion reiation, at the frequency and for the Poisson's ratio requ!red bg t!ne solution for the rectangular plate. 2,

Equations

The notation ihe clreuiar

is that employed frequency,

in

[I]

except

for ~

whlch

is here

Strains;

where ~x and ~ a r e axes are

in the

the

the p l a n e o f

rectangular the p l a t e

thickness-curvatures,

components o f

and w i s ~is

3L:.9

the

rotation

the d e f l e c t i o n ,

thickness-twist

about l~

and

and I~zand

350

I,ig "

R.D.

are

thickness---shears

MINDLIN

The comma n o t a t i o n

for

deriL~atige

is

em-

',

p!oved.

Strain

in

and kinetic

which

/q i s

the

I)=d-~/6(/-p] i s

the plate

modulus,

thickness

the

//is

mass d e n s i t y are

three

hess

and

of the

correction

curuature

coffee'Lion

do n o t

appear

in

plate,

designates

factors;

I¢ f o r

I_'!]

and,

what

s!~ear m o d u l u s .

ratio,

time

inertia,

referred in

the

to

in

fo!lowe,

~) i s

derivatiue.

thickness-shear,

trangerse

factors

G is

Poisson's

ouer-dot

and [email protected] f o r

tional

Section

energ.v-densities;

!ntroduction~

are

taken

Tt~ere

~ for

i~ and I~ a r e the

the thick--

the

addithey

as u n i t y

in

4,

Stpesses:;

% ,

in

r, = -} I I-A o % , ,

R ?,

i.!)

which

Stress-equations

of

motlon~

I

i"l,~.~ r M%~ ~ Q ~

Substitution

of

(1)

in ( 3 )

(3)

*-~(~ ~°~ ~

produces

the

disp!acement-equations

VIBRATIONS

of

OF PLATES WITH FREE EDGES

351

mot. ion~ ,

I

2..3

c4)

where

~:~xr~.and

3, General the

stitute

(omitting

(4)

two-dimensional

Laplacian°

solution

To f i n d

in

V2 is t h e

flexure

and t h i c k n e s s - s h e a r a factore~rhere

to obtain

dispersion

and i n w h a t

relations,

sub

follows)

..

(~s

O,

- 6 =

,'~-t-RS~*)A

~'A-(~'-bsE)

(5)

~ = o,

where

R and 5

identify

transuerse the

time c o n t r i b u t i o n s

shear

deformation,

coefficients

quadratic

ofA

equation

of' t h e

rotatory

respectiuely,

and B i n

in & with

(5),

set

inertia

and

The d e t e r m i n a n t

equal

to zero,

is

of a bi-

roots,"

(6a) -In t e r m s of

the

of

the

ratio,

~O. , o f

thickness-shear

the

mode i n

frequency

an i n f i n i t e

.o. --

~

to

the

frequency ~ o

plate:

=

we have

-,.'W" These a r e (4)

of

the

flexure

the dispersion

"2..----~ branch

(~r)and

relation

cordin 9 to the t.wo-dimensiona! A/O;I-E,

4-

x

z--~e/ + -

:~

for

the

~

-J j"

thickness-shear

waues i n an i n f i n i t e

plate-equations,

where ~'=lCpb~5/6, Upon e m p l o y i n g

the

Also,

two u a l u e s

branch plate

ac-

from (5)., of5

from

(zt)

:(I)

puno; aJe sepow Jno~

lie

u~ ( 0 [ )

pue ( 8 )

Kq pe~nq~J~UOO s s e a ] s

gu~n~sqns

jo

4q

s~ueuodwoo e q l

• qoueJq :~s,~mq~-sseu~olq ] ~oe.~.~e ] o u op a>~ pue ~ ~eql.

'os[e

'e:~ON "[Z3 suo~'}enbe

eql.

[euo,~suetu,~p

eaJq:~ eq-~ Kq peJ~,nbeJ se 0=~3 ~e [ = ~ &ouenbeJj. ~ j o - l , no ].oeJJOO aq:~ seq ueq:~ ( 9 ) eql-

'os[d

uoT~e[eJ "(~£]

uo~sJads,~p eq~ j o "~2 qoueaq .~eeqs-sseu>lo~q-~

senem enoq >o eseo Ie,~oeds e)

-ueu~Tp-eaJq } e q l . . ~ o qoueaq }s~m].-sseu>toTq~ uo,t}e[ea

uo,~sJedst, p aq'} ~[:~oexe s t •

([[)

am ' ( # )

JapJo "~sJ~.~ eq~ JO.~

'~lZ.tl:24 j.,~ "~eq} Z

pu.~

suoTt~enbe [ e u o ~ s

u,~ ( O I )

}

~

e~ou efq

Z

Gu,~:}n:~,~Isqns

"0=.^¢ pue

:sepow :}s,~m1.-sseu>loTql. om-~ ppe am (8)

o±

(6) qo~qm u~

(8)

-ssau~oTq]

' u o ~ % n [ o s eq~ ~o sepow J e e q s ' e n e q am ' s % [ n s a o e s a q ] 6 u ~ [ q w e s s u

pue eanxeT9 eq~ ao9

6 u ~ p u o d s e J o o o eq~ o~ eonpeJ

(&)

" [ I ] uT ( 0 9 ) pue (6G) s e [ n ~ a o ~ pue ( 9 ) 'K~Tun aJe J>, pue %j ueqR

:~330 s e n I ~ n or.~ p u ~

NIqaNIN

"G'~

em

'(9)

Z~

VIBRATIONS OF PLATES WITH FREE EDGES

The free edge condltlons to be s a t i s f i e d ,

353

for a plate bounded by

X ' - ~ , ~(=Jb , are

The f i r s t

M~,= QxsM~=O or~ ~ : ± O~

(-13)

M~, : O~=M~ =0 o n H = ! b ,

(14)

two of (13) and (14) are s a t i ~ f l e d i f

i,e.

,f,~: ~Tr/s,

'~,b: n~'/z,

•")~.b ~. q ~17-,

The remaining condiLlons,

w,,% F', r~ ¢ v~n.

those ontlx and My, result

(15)

in

(is)

The determinant of lhe c o e f f i c i e n t s of the Bc in (16), set equal to zero,

reduces to

(I?)

•J~.--/I ")

.which, with

(ii),

requires

(18)

Hence, the l e n g t h / w l d t h r a t i o s of the plate and the mode shapes are restricted to

alb ~ ~/~ - rl~, If

sines and cosines of l l k e argument,

(19) In (8) and ( I 0 ) ,

are In-

terchanged, four solutions are obtained with m and n or p and q

354

R.D. MINDLIN

even o r any

odd,

Accordingly,

ratio of integers.

that

the length/width

It may also be seen,

fixed edge conditions,

or both x = ! o and

and d e f l e c t i o n , "simple

d=Ib.

from

a/b, can be (8),and

(10),

~=%=w=O could be satisfied on either

Of course,

mixed conditions

such as M~N~-w~O on x=~a

supports")

ratio,

can be satisfied

of moments

o r M = M ~ w ~ O on f f ~ b

without

restrictions

(i,e, like

(i?),

From ( 6 )

but,

and ( 9 ) ,

from

(11), 'Z-

Hence,

from

(20)

and ( 2 i )

/~={ mp:~m (18) and

(21),

?- ~-

1,

2.

"~

//

w-Lth ,~t/?(Z,

the

frequency

- [(I-/~V4-*bz2%/~ ] / ¢ J

the length/thickness

(21) ls gigen

,

ratio

b/ (22)

is

(23)

Finally, the Poisson found as follows.

ratio corresponding

First,

from

(S),

to a ratio m/p may be

(18) and

(15), (24)

Then,

from

(6) and

(24):

from which

(25) where

VIBRATIONS OF PLATES WITH FREE EDGES

355

4. Solutions without additional correction factors If,~=~tp=l,

(22) and (25)

n

reduce to

[27)

i

--

/ ~ = ( ~ - v ' - Z 4 R ) / ~ , T &-, -Z4),

(28)

and (23) remains unchanged. A set of compatible values of the parameters may be obtained as follows. Choose m and p first as they must be inteers.

Then ~,

A and 2a/h are fixed from (28), (27) and (23). Finally, choose n a n d q so as to s a t i s f v p/q=m/n, a c c o r d i n 9 to (19% and this f i x e s the l e n g t h / w i d t h r a t i o , a / b , a l s o a c c o r d i n g to ( 1 8 ) , Typical

results over a range of Poisso~s ratios are listed in Table

I for m,n,o,Q all even. Table I 2a/h 32,18 0.1314 2.3910 16.4732

n, q ',aZb n,q ia/bil n/q a/b 8,4 4,2 I 8 ,,,~, 12,6 8/3 i)4,16 0.21451 2,2605 18.53501 34, 16 I S8~32 1.'2 !02_~__48 1/3 38,18 0.2878 2. iSi8 20.5550 18,8 2 36,16 i 54, 24! 2/3 38,16 0.3520 2,0848J 22,5391 38, 16 i 7S,32 I/2

114,481 1/3

40~16, 0.4087 2,0234124,4919 10,4

4

20,8

2 I 3o.1_2,_ ,31

The frequencies listed in Table I are all higher than the limit below which the dispersion relation (6), with~tb=l,

91yes close

approximations to the correspondln 9 branches of the Rayleigh dispersion relation [2], l,e, the first two branches of

(29) where

The~,

I=1,2 correspond to the6"~, i=1,2 of (6). An example with

356

R.D. MINDLIN

)~=0.3

is illustrated

in Fi9.~

Jn which

the dashed

[email protected] marked

,,/ / / , Y

,,

/

/"

/ //.,,

I ///-. I

3

FIS.

F'

and TS' are from ~j a n d S ,

marked F and TS [ f o r scissa of

respectiuely,

which ~ i s

the f i g u r e )

i

so t h a t

are from ( 2 9 ) ,

In p a r t i c u l a r ,

The errors

m,n,p,q.

As already

relation

(Ii),

dimensional

a t ~=2,

h~/~'---'i.04?iS,

h~,/fi = 2 . 3 0 2 1 5 ,

the a p p r o x i m a t e ~ , a n d Ci are

respectiuely,

lines

to be r e p l a c e d by ~ i n the ab-

Exact:: h~7;T =2,19288, Approx'

and the f u l l

increase

mentioned,

with i~=P~l~,

equations.

in e r r o r with

by about 5% and 6%,

increasingJO_and

the thickness-twist

is the same as that

Thai branch

is the curve

orders

dispersion

from

the three-

marked

TT in

F i g . 1. 5. Soluti'on with additional

correction

factors

To improve the a c c u r a c y o f the p r e c e d i n g s o l u t i o n a p p r o p r i a t e v a l u e s o f K andl~p proceed as f o l l o w s .

r'~ : h' ( (, ? ~,j,,,,, ''~ /,? ,

r,i~- h '~(,'~,/~"

by computing Set

(so)

VIBRATIONS OF PLATES WITH FREE EDGES

and f i n d

from (G),

357

with ~ and (~ r e p l a c i n g E, and ~z, (31)

f

A-i

J

_/

=-~---

Zi~'N

The c o m p u t a t i o n then proceeds: Step i , Choose m and p and f i n d ~ and ~ from (28) and (2?), Step 2. Find ~ and (z from (2~) and M and N from (30). Step 3, Findbp and ~ from (31). Step 4, Find n e w A ,

2a/h and/x from (22), (23) and (25),

Step 5, With the n e w A

and/x , return to Step 2 and repeat

the cycle as many times as are necessary to find

i~p,t~,El, 2a/h a n d s , Step 6. F i n a l l y , fixes

to the d e s i r e d accuracy.

the choice o f n and q ( s a t i s f y i n g

a/b according

n/q=m/p)

to ( 1 9 ) .

Table I I m, p

t~o

I

~P

/x

El

2a/h

32,16 0.5141 !0.8981 0.1691 2,0957 18.4258 34,16 0.5501 0.8779 0,2818 1.9585 22.3141 36,16 0.4122 0.8623 0.4506 1.8604 25.1121

Table I I contains the improved values,

to ~i0 "~, of the first three

rows of Table I, along with the correction factors ~ andre. T h e last two rows from Table I are omitted as the corrected values of the Polsson ratios approach unity. References I. R.O.Mlndlln, J. Appl. Mech, vol. 18, 23 (1851), 2. J.W.Strutt (Lord Rayleigh), Proc. Lond. Math. Soc, vol. 20, 225 (1888-1889). 3. A.E.H.Love, Some Problems of Geodynamics, Cambriage (1311).

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