Effect of edge thrust on the natural frequency of flexural vibration of rectangular plate

Reports of Research Institute f o r Appiied Mechanics Vol. X X , No. 66, 1973

NOTE

EFFECT OF EDGE THRUST ON THE NATURi^L

FREQUENCY OF FLEXURAL VIBRATION

OF RECTANGULAR PLATE

•

By Toyoji

A n i n v e s t i g a t i o n i n t o tlie effect o f edge cotnpres.sion or t e n t i o n upon tJie n a t u r a l frequencies o f f l e x u r a l v i b r a t i o n of tho rectangular plate is carried out as a f u n d a m e n t a l study o n the local v i b r a t i o n o f ship structure.

1. I n t r o d u c t i o n

A s one o f the v i b r a t i o n problems, i t should be noted t h a t the n a t u r a l f r e -quency o f the v i b r a t i o n o f a rectangular plate is d i f t e r e n t f r o m the f r e q u e n c y under the static action o f the edge compression or tension i n its neutral plane. I f the edge compression increases as h i g h as the c r i f i c a l load o f the rectangular plate, w h i c h is an extreme caso, there is no f u n d a m e n t a l natural frequency_of ihe v i b r a t i o n o f the plate. T h e i n v e s t i g a t i o n i n t o the extreme case o f t h e pro-b l e m m e n t i o n e d apro-bove v/as treated pro-b y Sezawa" f o r o pro-b t a i n i n g pro-b u c k l i n g load a n d n a t u r a l f r e q u e n c y o f the clamped rectangular plate, though the r e l a t i o t i between the n a t u r a l frequencies a n d edge compression is not considered i n his p a p e r .

-T h e present note shows the eft'ect o f edge compression or tension upon the n a t u r a l frequencies o f flexural v i b r a t i o n of the supported rectangular plate w i t h a n u m e r i c a l example. T h e re.sults thereby w i l l be applicable to the estima-t i o n o f estima-the n a estima-t u r a l frequencies of estima-the recestima-tangular panel o f estima-the w e b plaestima-te o f estima-t h e l o n g i t u d i n a l or transverse girder as a p r o b l e m o f the local v i b r a t i o n o f ship h u l l structure.

2. Basic Ekjuafioi? aitd Solutions

T h e basic equation o f the flexural v i b r a t i o n o f the rectangular plate under the a c t i o n o f edge compressions i n ,v a n d y direction and shear force, is w r i t t e n by

' ót'

'^'3/1

'

J-^^

* Professor, Rese;n-ch Institute for Applied Mechanics, Kyushu University, Fukuoka Japan.

74

where,

E, }' Young's modulus and Poisson's ratio respectively pit mass o f plate per unit area

H' deflection o f the plate

P,., Py edge conipresions p a r a l l e l to x and y axis respectively O shear f o r c e at f o u r edges

X , y cartesian co-ordinate

I n the above equation, i f there is no i n e r t i a f o r c e i n the above plate, the equation shov/s t h a t o f buclcling o f the plate, or i f there are no edge compres-sions a n d shear force, the equation shows t h a t o f flexural v i b r a t i o n o f a rectan-gular plate.

N o w w e put Q=0, a n d assume the deflection, t h a t

i r - s i n — ^ - ' y cosw/, /« = !, 2, 3 (2)

Substitute (2) i n t o ( 1 ) , and the equation w i l l become

r

V

fl' fl'

where, ^ = " 1 ? ^

T h e solutions o f the above equation are w r i t t e n by

Y=A cosqy + B sinqy + C Cospy + D Sin/;;' , (4)

„ l , , r e ,

J ] = ^ - r / ^

Q-j a l ' a' ^ D ^ 4 7 t ' D ' ~ ) a _

I f the t w o oposite edges y=0 and y=b are both assumed to be supported, the constants A, C, a n d D v a n i s h a n d the s o l u t i o n becomes

Y=^ B smtjy ,

where, ^ T ' " = 1. 2, 3 (6)

3. R e h i t i o n between Eigenvalue and Edge Contpressiou

F r o m the above tv/o equations (5) a n d ( 6 ) , the f o l l o w i n g equation is ob-tained,

™ . ^ . ^ ^ ^^^^^ ^^^^^^^^

ver, the b u c k l i n g load P or P i j T '''' P « ''=0 hov e fiiven. ' detenrund, p , - „ v i d e d t h a t the n n i o A / p ' -3

P u t t i n g = 0 i n equation (7) f o r the S . I - P •

°* .snnp,,c,ty, ,he eigen-vaiue w i l ] become,

^ P i > F (8) T h e bucicJitig load is obtait.ed, p r o v i d e d t h a t ;.=.o i n the al

O i n the above equation,

m b j (9)

^ : ™ r ; r ; r

-r»--the b u d d i n g load is w r i t t e n b y ^^^'"^ to a/b--. Hence

• b)

(10)

Since the edge compression w i l l eh ai^e f r n , .

sent p r o b l e m , the coefficent

c

..m^^^Z^U^

t^''^''

w h i c h . = P . y z , , , , , , ^ P i n equation (lo), i n by equation (8) becomes - ^'^^^ ^^^^ '^'gen-vaiue presented

the case o f the f u n d a m e n t a l mode, the eigenvalue is w r i t t e n by

When c becomes vpi-rs ; 1

I f V, e put P --^p or ('=-1 '

T, K U M A I r — 1

|/' 1

|/' 1

1.5 2.0

Tig. 1. Eigen-vnlue versus edge compression and tension by the ratio of buckling load of square plate supported f o u r edges.

4. Conclusions

T h e effect o f edge compression or tension u p o n the n a t u r a l frequences o f the rectangular plate supported fill edges was i n v e s t i g a t e d . I t is to be n o t e d t h a t the range o f v a r i a t i o n o f the eigen-value f o r t h a t o f the edge com-pression or tension is considerably w i d e . T h e resuhs w i l l be applied f o r estimat-i n g n a t o a l f r e q u e n c y o f w e b plate v estimat-i b r a t estimat-i o n estimat-i n a local structure o f shestimat-ip l estimat-i u l l . T h e r e is s t i l l m u c h to bo studied i n the present p r o b l e m f o r d i f f e r e n t types o f ex-ternal f o r c e , f o r instance, under the static action o f b e n d i n g m o m e n t a n d shear f o r c e i n its n e u t r a l plane.

References

1) Sezawa, K . , On the Lateral Vibration of a Rectangular Plate Clamped Four Edges and its Stability, Jour. Soc. N . A . Japan, V o l . 49, 1934, pp. 87-93.

2) Timosheuko, S., Theory of Elastic Stability, M c G r a w - H i l l , 1936, p. 330.