Vibration Analysis for Rectangular Plate Having a Circular Central Hole with Point Support by Rayleigh-Ritz Method

Transcription

1 Journl of Solid Mechnics Vol. 6, No. 1 (014) pp. 8-4 Vibrtion Anlysis for ectngulr Plte Hving Circulr Centrl Hole with Point Support by yleigh-itz Method K. Torbi *, A.. Azdi Deprtment of Mechnicl Engineering, University of Kshn, Kshn, Irn eceived September 013; ccepted 8 December 013 ABSTACT In this pper, the trnsverse vibrtions of rectngulr plte with circulr centrl hole hve been investigted nd the nturl frequencies of the mentioned plte with point supported by yleigh-itz Method hve been obtined. In this reserch, the effect of the hole is tken into ccount by subtrcting the energies of the hole domin from the totl energies of the whole plte. To determine the kinetic nd potentil energies of plte, dmissible functions for rectngulr plte re considered s bem functions nd it hs been tried tht the functions of the deflection of plte, in the form of polynomil functions proportionte with finite degrees, to be replced by Bessel function, which is used in the nlysis of the vibrtions of circulr plte. Considertion for vriety of edge conditions is given through combintion of simply supported, clmped nd free boundry conditions. In this study, the effects of incresing the dimeter of the hole nd the effects of number of point supported on the nturl frequencies were investigted nd the optimum rdius of the circulr hole for different boundry conditions re obtined. The method hs been verified with mny known solutions. Furthermore, the convergence is very fst with ny desirble ccurcy to exct known nturl frequencies. 014 IAU, Ark Brnch.All rights reserved. Keywords: ectngulr plte; Circulr plte; yleigh-itz method; Hole; Vibrtion; Point support 1 INTODUCTION ECTANGULA plte with rectngulr or circulr hole hs been widely used s substructure for ship, irplne, nd plnt. Uniform circulr, nnulr nd rectngulr pltes hve been lso widely used s structurl components for vrious industril pplictions nd their dynmic behviors cn be described by exct solutions. However, the vibrtion chrcteristics of rectngulr plte with n eccentric circulr hole cn not be nlyzed esily. Perforted pltes or pltes with cut-outs re commonly encountered in engineering prctice. Cut-outs re introduced to provide ccess, reduce weight, nd lter the dynmic response of structures. Furthermore, structureborne noise generted by mchinery such s the diesel engines, gerboxes, genertors, nd uxiliry mchinery re lso rdited by these plte structures nd should be suppressed in the vrious operting conditions. ectngulr pltes with point supports cn model severl structures of prcticl interest, such s slbs supported on columns, printed circuit bords or solr pnels supported t few points. The vibrtion chrcteristics of rectngulr plte with hole cn be solved by either the yleigh-itz method or the finite element method. The yleigh-itz method is n effective method when the rectngulr plte hs rectngulr hole. However, it cnnot be esily pplied to the cse of rectngulr plte with circulr hole since the * Corresponding uthor. Tel.: ; Fx: E-mil ddress: kvntrb@kshnu.c.ir (K. Torbi). 014 IAU, Ark Brnch. All rights reserved.

2 K. Torbi nd A.. Azdi 9 dmissible functions for the rectngulr hole domin do not permit closed-form integrls. Mny studies hve been done on the subject, some of which re mentioned in this section. Monhn et l. [1] pplied the finite element method to clmped rectngulr plte with rectngulr hole nd verified the numericl results by experiments. Prmsivm [] used the finite difference method for simplysupported nd clmped rectngulr plte with rectngulr hole. There re mny reserch works concerning plte with single hole but few works on plte with multiple holes. Aksu nd Ali [3] lso used the finite difference method to nlyze rectngulr plte with more thn two holes. jmni nd Prbhkrn [4] ssumed tht the effect of hole is equivlent to n externlly pplied loding nd crried out numericl nlysis bsed on this ssumption for composite plte. jmni nd Prbhkrn [5] investigted the effect of hole on the nturl vibrtion chrcteristics of isotropic nd orthotropic pltes with simply-supported nd clmped boundry conditions. Ali nd Atwl [6] pplied the yleigh-itz method to simply-supported rectngulr plte with rectngulr hole, using the sttic deflection curves for uniform loding s dmissible functions. Lm et l. [7] divided the rectngulr plte with hole into severl sub res nd pplied the modified yleigh-itz method. Lm nd Hung [8] pplied the sme method to stiffened plte. Lur et l. [9] clculted the nturl vibrtion chrcteristics of simply-supported rectngulr plte with rectngulr hole by the clssicl yleigh-itz method. Skiym et l. [10] nlyzed the nturl vibrtion chrcteristics of n orthotropic plte with squre hole by mens of the Green function ssuming the hole s n extremely thin plte. The vibrtion nlysis of rectngulr plte with circulr hole does not lend n esy pproch since the geometry of the hole is not the sme s the geometry of the rectngulr Plte. Tkhshi (1958) used the clssicl yleigh-itz method fter deriving the totl energy by subtrcting the energy of the hole from the energy of the whole plte. He employed the eigenfunctions of uniform bem s dmissible functions. Jog-o nd Pickett [11] proposed the use of lgebric polynomil functions nd bihrmonic singulr functions. Kumi [1] Hegrty [13], Estep nd Hemmig [14], nd Ngy [15-16] used the point-mtching method for the nlysis of rectngulr plte with circulr hole. The point-mtching method employed the polr coordinte system bsed on the circulr hole nd the boundry conditions were stisfied long the points locted on the sides of the rectngulr plte. Lee nd Kim [17] crried out vibrtion experiments on the rectngulr pltes with hole in ir nd wter. Kim et l. [18] performed the theoreticl nlysis on stiffened rectngulr plte with hole. Avlos nd Lur [19] clculted the nturl frequency of simply-supported rectngulr plte with two rectngulr holes using the Clssicl yleigh-itz method. Lee et l. [0] nlyzed squre plte with two colliner circulr holes using the clssicl yleigh-itz method. A circulr plte with en eccentric circulr hole hs been treted by vrious methods. Khursi nd wtni [1] studied the effect of the eccentricity of the hole on the vibrtion chrcteristics of the circulr plte by using the tringulr finite element method. Lin [] used n nlyticl method bsed on the trnsformtion of Bessel Functions to clculte the free trnsverse vibrtions of uniform circulr pltes nd membrnes with eccentric holes. Lur et l. [3] pplied the yleigh-itz method to circulr pltes restrined ginst rottion with n eccentric circulr perfortion with free edge. Cheng et l. [4] used the finite element nlysis code, Nstrn, to nlyze the effects of the hole eccentricity, hole size nd boundry condition on the vibrtion modes of nnulr-like pltes. Lee et l. [5] used n indirect formultion in conjunction with degenerte kernels nd Fourier series to solve for the nturl frequencies nd modes of circulr pltes with multiple circulr holes nd verified the finite element solution by using ABAQUS. Zhong nd Yu. [6] Formulted wek-form qudrture element method to study the flexurl vibrtions of n eccentric nnulr Mindlin plte. Wng [7], the itz method is used to determine the minimum stiffness loction of the elstic point support for rising the fundmentl nturl frequency of rectngulr plte to the second frequency of the unsupported plte, which usully is the upper limit of the first frequency for single support. Joseph Wtkins et ll. [8] Studied the vibrtion of n elsticlly point supported rectngulr plte using eigensensitivity nlysis. Lorenzo [9] is employed the trigonometric functions s dmissible solutions in the itz method for generl vibrtion nlysis of rectngulr orthotropic Kirchhoff pltes. As it ws mentioned erlier, in most of the reserches done in this field, yleigh-itz method nd numericl methods hve been used nd with the help of reducing the hole energy compring to the energy of the whole rectngulr plte, the problem hs been nlyzed. Also for studying the issues considering the position conditions nd ngle, the quntity of so mny points in the edges of the rectngulr plte hve been used. In this study, the nlysis of trnsverse vibrtions of rectngulr plte with circulr centrl hole with different point support is studied nd the nturl frequencies nd nturl modes of rectngulr plte with circulr hole hve been obtined. In this method, simple polynomil functions, the desired frequency rnge, which cn replce the Bessel functions will be used, nd convergence the problem will be obtined esily. 014 IAU, Ark Brnch

3 30 Vibrtion Anlysis for ectngulr Plte Hving Circulr Centrl Hole with Point Support FOMULATION OF THE POBLEM.1 Applying the yleigh-itz pproch to rectngulr plte From the vibrtion theory of thick pltes, the strin energy U p nd kinetic energy T p of n elstic isotropic rectngulr plte in the crtesin coordinte cn be written s follows: 1 b W W W W W U (1 ) p D 0 0 dx dy x y x y xy (1) 1 b T p h W dx dy 0 0 (1b) where is the mss density of the mteril, nd D is the plte flexurl rigidity defined s: 3 Eh D 1(1 ) () Here, E is Young s modulus nd is Poisson s rtio. With side lengths in the X direction nd b in the Y, Tking the following non-dimensionl coordintes: x y,,. b b (3) The itz pproximtion is employed by ssuming the following solution: W(,, t) (, ) Q( t) (4) where (, ) 1... m is 1 m mtrix consisting of the dmissible functions nd Qt ( ) Q1 Q... Q m is m 1 vector consisting of generlized coordintes, in which m is the number of dmissible functions used for the pproximtion of the deflection. M N (, ) A ( ) ( ) (5) m1 n1 mn m n In which m ( ) nd n ( ) denote the ssumed dmissible functions in the x nd y with substituting Eq (4) into Eq (1) results in Eq (6). directions, respectively, 1 1 T T Tp Q M p Q, Up Q Kp Q (6) where M p hbm p, Db Kp K, 3 p (7) In which cse T M p d d (8) 014 IAU, Ark Brnch

4 K. Torbi nd A.. Azdi 31 K p 1 1 T T T T T (1 ) dd (8b) M nd K represent the non-dimensionlized mss nd stiffness mtrice. After substituting the plte displcement function in Eq (5) into the bove energy expressions, set of m nhomogeneous equtions of A mn is then formulted by differentiting the Lgrngin energy, defined byu T, with regrd to ech of the undetermined coefficient A derived s: mn. Afrer choosing set of pproprite dmissible function for nd, the eigenvlue eqution cn be Kp Mp A 0 (9) h where is the nturl frequency prmeter. Then, the non-dimensionlized mss nd stiffness mtrices D given by Eq (5) cn be expressed s [8-30]. 1 1 ( M ) d d, i, j 1,,... m p ij 0 i j 0 i j ( K p) ij i jd i jd i jd i jd i jd i jd i j i (1 ) j i j i j d d d d (10) (10b) Here, M nd K re digonl mtrices. In this section, by considering the following s dmissible function for the plte simply supported on ll side, the boundry mtrices will be pplied esily. nd i ( ) sin( i ), i 1,,... n for xdirection (11) j ( ) sin( j ), j 1,,... n for ydirection (11) In the cse of the clmped condition in the cn be used: x direction, the eigen function of clmped clmped uniform bem i( ) cosh( i ) cos( i ) i(sinh( i ) sin( i )), i 1,,... n (1) where i is obtined by solving the eqution of cosh( i) cos( i) 1 0 nd i 4.730, 7.853,... nd cosh( i) cos( i) i. sinh( i) sin( i) In similr mnner, expressions for plte with free edges in the x direction, the eigenfunction of free free uniform bem: 014 IAU, Ark Brnch

5 3 Vibrtion Anlysis for ectngulr Plte Hving Circulr Centrl Hole with Point Support i( ) cosh( i ) cos( i ) i(sinh( i ) sin( i )), i 1,,... n (13) where i nd i re the sme s the ones for the clmped clmped bem. For the dmissible functions in the y direction, the sme method cn be pplied. The frequency prmeter, is obtined by solving the generlized eigenvlue problem defined by Eq (9). det Kp Mp 0 (14). Applying the yleigh-itz pproch to circulr plte To obtin the nturl frequencies of rectngulr plte with circulr centrl hole, It is similr to the rectngulr plte, the mss nd stiffness mtrices re determined. From the vibrtion theory of circulr pltes, the strin energy U C nd kinetic energy T C of n isotropic uniform circulr plte with rdius nd thickness h cn be expressed s follows: 1 T c h 0 W rdr d 0 1 W 1 W 1 W W 1 W 1 W 1 W 1 W Uc D (1 ) rdr d 0 0 r r r r r r r r r r r (15) (15b) The itz pproximtion is employed by ssuming the following solution: W(, r,) t (, r ) Q () t (16) c where ( r, ) 1... m is 1 m mtrix consisting of the dmissible functions nd Q c ( t) Q1 c Q c... Q mc is m 1 vector consisting of generlized coordintes, in which m is the number of dmissible functions used for the pproximtion of the deflection [30]. With substituting Eq (16) into Eq (15) results in Eq (17). 1 1 T T T Q M Q, U Q K Q C c c c C c c c (17) where D M c h Mc, Kc K c (18) In which cse 0 c ij ij 0 ( M ) rdr d, (19) 014 IAU, Ark Brnch

6 K. Torbi nd A.. Azdi 33 ( K ) c ij i i i 0 0 j j j r r r r r r r r 1 1 j i j i j i j i r r r r r r r r r i i j 1 j (1 ) r r r r r r rdr d. (19b) The origin of the polr coordinte system is t the center of the circulr plte. The boundry conditions possess symmetry with respect to the dimeter of the circulr plte. The deflection function in terms of Bessel functions nd trigonometric functions is written s [30, 31]: r r r r n n n n n n n n n n n n n n0 (, r ) A J ( ) B Y ( ) C I ( ) D K ( ) f () (0) where the coefficients An, Bn, Cnnd D n re determined from the boundry conditions nd Jn nd I n re the Bessel function nd the modified Bessel function of the first kind, Y n nd K n re Bessel function nd the modified Bessel function of the second kind of order n, respectively. Since the circulr hole is to be free of ll pplied stress, the boundry conditions to be stisfied long the edge of the hole t r re: 1 M r Mr 0, Qr 0. r (1) where M r is the bending moment norml to the hole, M r is the twisting moment in the sme plne, nd the Q r is the sher force cting t the edge of the hole. For instnce, if the boundry of the plte is considered to be clmped t the rdius of the plte then the boundry terms for solution (, r ) cn be written s: (, r ) (, r ) 0, 0. r (1b) Also, solution (, r ) must be finite t ll points within the plte. This mkes constnts B, Bessel functions of second kind Yn nd n Dnvnish since the K become infinite t r 0. As it hs been shown in Eq (0), deflection of n the intended plte, cn be expressed in terms of Bessel functions of the first kind. Due to the properties of the Bessel functions nd regrdless of terms with high degrees in Eq (0) nd lso obtined frequencies with the use of Finite Element method, in this section it hs been tried to cquire the nturl frequencies nd the mode shpes of the rectngulr plte with centrl hole, with the use of polynomil functions proportionte with finite degrees in the intended frequency limits insted of the mentioned Bessel functions. Bessel functions of the first kind, denoted s Jn ( ), re solutions of Bessel's differentil eqution tht re finite t the origin ( 0 ) for integer n, nd diverge s pproches zero for negtive non-integer n. The solution type (e.g., integer or non-integer) nd normliztion of Jn ( ) re defined by its properties below. It is possible to define the function by its Tylor series expnsion round 0. m ( 1) 1 mn n () m! ( m n 1) m 0 J ( ) ( ) P ( ) n 014 IAU, Ark Brnch

7 34 Vibrtion Anlysis for ectngulr Plte Hving Circulr Centrl Hole with Point Support where ( Z ) is the gmm function. The series expnsion for In ( ) is thus similr to tht for Jn ( ), but without the lternting ( 1) m fctor. 1 1 mn n (3) m! ( m n 1) m 0 I ( ) ( ) S ( ) n The series expnsion for Yn ( ) nd Kn ( ) using series expnsion of Bessel functions Jn ( ) nd In ( ) will be obtined esily. Here, Pn( ), Qn( ), Sn( ), Tn( ) re the polynomils with limited degree, will be sought in the form of series expnsions nd in the desired frequency rnge, will be replced by the Bessel functions Jn( ), Yn( ), In( ), Kn( ), respectively. Other reltionships, by substituting the polynomils functions nd simplify the equtions will be obtined. n n n n n n n n n n n n n (, r ) A P ( ) B Q ( ) C S ( ) D T ( ) f ( ) (4) n0 For exmple, if n 1, the following proposed polynomils, cn be replced by the Bessel functions J1 P1( ) , log( 1 /) log ( / ) Y1 Q1( ) 1 1, I1 S1( ) , log ( 1 / ) 3 K1 T1( ) log( 1 /) log ( / ) , (5) After choosing set of pproprite dmissible function for, the eigenvlue eqution for circulr plte cn be derived s: K c Mc A 0 (6) And the frequency prmeter for circulr plte, is obtined by solving the generlized eigenvlue problem defined by Eq (7). det K c Mc where h. D (7).3 Applying the yleigh-itz pproch for rectngulr plte with circulr hole For rectngulr plte with cutouts, the norml yleigh-itz method will require bem function tht is continuous over the plte domin while stisfying the inner nd externl boundry requirements. No such function hs been reported in the open literture, nd the nlysis of such problem using the yleigh-itz scheme will require some modifictions to the numericl procedures. To demonstrte the numericl procedures, rectngulr plte with centrlly locted circulr cutout is considered. The geometry nd dimensions of the plte re shown in Fig.1. In this cse, the totl kinetic nd potentil energies cn be obtined by subtrcting the energies to the hole from the totl energies for the rectngulr plte. 014 IAU, Ark Brnch

8 K. Torbi nd A.. Azdi T T T totl Q M pq Qc McQc 1 T 1 T Utotl Q KpQ Qc KcQc (8) Note tht the boundry condition round the circulr hole cn be stisfied exctly, while the boundry condition long the rectngulr outer edges of the plte must be hndled with some numericl procedure. By using the coordinte trnsformtion technique nd geometricl reltion between the Crtesin nd polr coordintes, the displcement mtching condition should be stisfied. Hence, the following condition should be stisfied inside the circulr hole domin [30]. mc m m W ( r, ) W(, ), ( r, ) Q ( t) ( r, ) Q ( t) ( ) ( ) Q ( t) (9) c cj cj l l l l l j1 l1 l1 In this section, with the use of wek solution nd lso with the use of orthogonlity properties of trigonometric functions ci (, r ) nd multiplying these functions in Eqs (9) nd integrtion of these equtions in the intervls of 0, the equtions will be obtined in the form of polynomil functions bsed one finite degrees of. mc m 0 ci cj cj 0 0 ci l l l 0 j1 l1 (, r ) (, r ) Q () t rdrd (, r ) ( ) ( ) Q () t rdrd (30) Using the orthogonl property of (, ), Eq (30) cn be rewritten s: ci r where m m ci 0 ci l l l c il l 0 l1 l1 Q () t (, r ) ( ) ( ) Q () t rdrd ( F ) Q () t (31) Eq (31) cn be expressed in mtrix form: Q F Q (3) c c F c is mc m trnsformtion mtrix [30]. By using the coordinte trnsformtion technique nd geometricl reltion between the Crtesin nd polr coordintes, the non-dimensionlized reltionship cn be written s: l1 rcos( ) l rsin( ),. b b (33) with substituting Eq (3) into Eq (9) results in Eq (34) T T T T T totl Q M pq Q Fc Mc Fc Q Q M pcq 1 T 1 T T 1 T Utotl Q KpQ Q Fc Kc Fc Q Q KpcQ (34) By using the Eqs (7) nd (18) nd simplifying the Eq (34) cn be written s follow: T M M F M F hb M (35) cp p c c c cp 014 IAU, Ark Brnch

9 36 Vibrtion Anlysis for ectngulr Plte Hving Circulr Centrl Hole with Point Support Db K K F K F K T cp p c c c 3 cp (35b) where T pc p ( ) c c c M M F M F (36) T Kpc Kp F, c Kc Fc (36b) which is the spect rtio given s /.The eigenvlue eqution for rectngulr plte with circulr hole cn be derived s: cp 0 (37) Kcp M A And the frequency prmeter for rectngulr plte with circulr hole, is obtined by solving the generlized eigenvlue problem defined by Eq (0) det Kcp 0 M cp (38) Fig. 1 ectngulr plte with circulr centrl hole..4 Applying the yleigh-itz pproch for rectngulr plte hving circulr hole with point support In this section, the trnsverse vibrtions of rectngulr plte with point supported hve been studied nd the nturl frequencies re obtined by the clssicl yleigh-itz method. Strin energy of the supporting springs given by 1 U k ( x x ) ( y y ) W ( x, y, t) (39) N ps A s s s s s s 1 where ks the stiffness of the sth is spring nd W( x, y) is the trnsverse displcement. The kinetic energy cn be expressed s: ps A s s s s s s 1 N T m ( x x ) ( y y ) W ( x, y, t) (40) By using the Eq (5) for the displcement of W( x, y ), the dynmic stiffness mtrix of the plte will be derived s [8, 9]: N (41) ( x ) ( x ) ( x ) ( x ) ijmn s1 i s j s m s n s 014 IAU, Ark Brnch

10 K. Torbi nd A.. Azdi 37 where ijmn is the product of the bsis functions nd is evluted where the springs nd msses re locted. In which cse the stiffness mtrix of the plte K T ps r r. Where r is the rnk of the support stiffness mtrix. In the cse of the plte with elstic point supports, r=1, nd ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 s1 1 s1 1 s1 s1 m s1 n1 s1 m s1 n s1 T (4) To determine the strin energy U PCS nd kinetic energy T PCS of rectngulr plte hving circulr centrl hole with n elstic point supports, using the Eqs (39), (40) nd Eq (34) results in Eq (43) T T T T T T PCS Q M pq Q MsQ Q Fc Mc Fc Q Q M pcsq, 1 T 1 T 1 T T 1 T UPCS Q KpQ Q KsQ Q Fc Kc Fc Q Q KpcsQ, (43) where N k s T pcs p c c c D s1 K ( K ) F K F, ms T M pcs ( M p ) ( ) Fc Mc Fc. hbm (44) (44b) Above m s is the sth discrete mss. The eigenvlue eqution for rectngulr plte hving circulr hole with severl point supports cn be derived s: Kpcs Mpcs A 0, (45) And the frequency prmeter for rectngulr plte with circulr hole, is obtined by solving the generlized eigenvlue problem defined by Eq (46) det Kpcs Mpcs 0. (46) 3 NUMEICAL ESULTS In this section, numericl results re presented for the derived pproximte closed-form results nd compred to results generted using the previous nd finite element method (FEM) for the elsticlly point supported plte. Tble 1 Fundmentl nturl frequency r 0 for simply supported squre plte with centrl hole Present ef[16] FEM Present ef[16] FEM Present ef[16] FEM IAU, Ark Brnch

11 38 Vibrtion Anlysis for ectngulr Plte Hving Circulr Centrl Hole with Point Support Fig. Modes of vibrtion for simply supported plte with circulr centrl hole. As it hs been presented in Tble 1, with incresing the rdius of the hole, the frequency vlues re first decresed nd then increses, this gined specil importnce in optimizing the hole rdius in nlysis of these types of problems. In ddition, the obtined results indicte tht with incresing the vlue of Poisson's rtio, the frequency vlues would decrese. In Fig., the first five modes of vibrtion for simply supported squre plte with centrl hole hve been shown. In Tble. lso the frequency prmeter of hs been shown bsed on the different rdius of the circulr hole nd different vlues of for clmped squre plte with centrl hole nd the results re very ner to the results of reference [16], which indicte the ccurcy of the suggested method. In this section lso with incresing the hole rdius, the vlues of frequencies re first decresed nd then incresed. Only with this difference tht when we re using clmped squre plte, the vlues of frequency prmeter shows bigger vlues compring to the cse of simply supported plte. Tble Fundmentl nturl frequency r 0 for clmped squre plte with centrl hole Present ef[16] FEM Present ef[16] FEM Present ef[16] FEM Tble 3 Fundmentl nturl frequency of simply supported squre plte with simply supported centrl hole. r Present ef[8] FEM Present ef[8] FEM Present ef[8] FEM In Fig.3, the first five modes of vibrtion for clmped squre plte with centrl hole hve been shown. In Tble 3. lso the frequency prmeter of hs been shown bse on the different rdius of the circulr hole nd different vlues of for simply supported squre plte with simply supported centrl hole. In this section lso with incresing the hole rdius, the vlues of frequencies re incresed. 014 IAU, Ark Brnch

12 K. Torbi nd A.. Azdi 39 Fig. 3 Modes of vibrtion for clmped squre plte with circulr centrl hole. In Tble 4. lso the frequency prmeter of hs been shown bsed on the different rdius of the circulr hole nd different vlues of for simply supported squre plte with clmped centrl hole. In this section lso with incresing the hole rdius, the vlues of frequencies re incresed. Tble 4 Fundmentl nturl frequency of simply supported squre plte with clmped centrl hole. r Present ef[16] FEM Present ef[16] FEM Present ef[16] FEM Tble 5 Fundmentl nturl frequency of clmped rectngulr plte with circulr centrl hole. b present ef[16] FEM present ef[16] FEM present ef[16] FEM r0 Tble 6 Fundmentl nturl frequency of simply supported rectngulr plte with circulr centrl hole. b r present ef[16] FEM present ef[16] FEM present ef[16] FEM In Tble 5. lso the frequency prmeter of hs been shown bsed on the different rdius of the circulr hole nd different vlues of b for simply supported rectngulr plte with circulr centrl hole. In this section lso with incresing the hole rdius nd length of plte, the vlues of frequencies re incresed. In Tble 6. lso the frequency prmeter of hs been shown bsed on the different rdius of the circulr hole nd different vlues of b for simply supported rectngulr plte with circulr centrl hole. In this section lso 014 IAU, Ark Brnch

13 40 Vibrtion Anlysis for ectngulr Plte Hving Circulr Centrl Hole with Point Support with incresing the hole rdius nd length of plte, the vlues of frequencies re incresed. In Tble 7. the three first fundmentl nturl frequencies for rectngulr plte with corner point support Fig. 4 hve been shown nd depict comprison of results between frequency coefficients vilble in reference [3, 33]. Tble 7 Fundmentl nturl frequency for SFSF nd CFCF rectngulr plte with corner point support. Boundry condition for rectngulr plte n SFSF CFCF present ef[3] ef[33] present ef[3] ef[33] Fig. 4 Plte considered in the present study. Tble 8 Frequency prmeters for fully free squre plte with four point supports on the digonls nd 0. n Plte without hole Plte with centrl hole present ef[7] FEM[34] present FEM The finl nlysis model is fully free squre plte without ny restrint on the boundry edges, s shown in Fig. 5 long with the new coordinte system. Four identicl elstic supports, locted symmetriclly long the plte digonls, re utilized to increse the fundmentl nturl frequency [7]. Tble 8. lists the three first nturl frequency prmeter for free squre plte with circulr centrl hole. The finl results of the optiml solutions of the supports re given in Tble 8 long with the result estimted by FEM [34,7]. With the respective optiml support solution, the fundmentl nturl frequency becomes doubly repeted frequency for the desired frequency prmeter , nd triply repeted frequency for the desired frequency prmeter In Fig. 6, some modes of vibrtion for A uniform squre plte of fully free edges is supported by four elstic point supports on the digonls (see Fig. 5) with centrl hole hve been shown. Fig. 5 A uniform squre plte of fully free edges is supported by four elstic point supports on the digonls (full points) or on the xes (hollow points). 014 IAU, Ark Brnch

14 K. Torbi nd A.. Azdi 41 Fig. 6 Modes of vibrtion for uniform squre plte of fully free edges is supported by four elstic point supports on the digonls with centrl hole. 4 CONCLUSIONS In this pper, the free vibrtion of rectngulr pltes with circulr centrl hole for vrious boundry conditions ws nlyzed nd nturl frequencies were derived nd compred with the reported results of other reserchers. To solve the problem, it is necessry both Crtesin nd polr coordinte system be used. For the vlidtion, using the finite element method nd modes of vibrtion for clmped nd simply supported squre plte with centrl hole hs been obtined. Comprison of the results obtined from the method used in this rticle, shows tht the results re sufficiently ccurte. Also to investigte the problem, long term nd complex reltionships, re not used nd the problem is simply desired convergence is reched. In this study, the effects of incresed the dimeter of the hole on the nturl frequencies were investigted nd the optimum rdius of the circulr hole for different boundry conditions re obtined. The optimum vlue of the rdius hole for simply supported squre plte t r 0 0.1nd in this cse will hve the lest frequency, lso the minimum vlue of the frequency for clmped squre plte t r On the other hnd, in this pper the free vibrtion of rectngulr plte with circulr centrl hole for point supported in different boundry condition ws nlyzed nd nturl frequencies were obtined nd compred with the reported result by finite element method. ACKNOWLEDGEMENTS I would like to express my very gret pprecition to Kshn University Fculty for his vluble nd constructive suggestions during the plnning nd development of this reserch work. His willingness to give his time so generously hs been very much pprecited. I would lso like to thnk the stff of the following orgniztions for enbling me to visit their offices to observe their dily opertions: SUNI Compny EFEENCES [1] Monhn L.J., Nemergut P.J., Mddux G.E., 1970, Nturl frequencies nd mode shpes of pltes with interior cutouts, The Shock nd Vibrtion Bulletin 41: [] Prmsivm P., 1973, Free vibrtion of squre pltes with squre opening, Journl of Sound nd Vibrtion 30: [3] Aksu G., Ali., 1976, Determintion of dynmic chrcteristics of rectngulr pltes with cut-outs using finite difference formultion, Journl of Sound nd Vibrtion 44: [4] jmni A., Prbhkrn., 1977, Dynmic response of composite pltes with cut-outs, Journl of Sound nd Vibrtion 54: [5] jmni A., Prbhkrn., 1977, Dynmic response of composite pltes with cut-outs, Journl of Sound nd Vibrtion 54: [6] Ali., Atwl S.J., 1980, Prediction of nturl frequencies of vibrtion of rectngulr pltes with rectngulr cutouts, Computers nd Structures 1(9): [7] Lm K.Y., Hung K.C, Chow S.T., 1989, Vibrtion nlysis of pltes with cut-outs by the modified ryleigh-ritz method, Applied Acoustics 8: IAU, Ark Brnch

15 4 Vibrtion Anlysis for ectngulr Plte Hving Circulr Centrl Hole with Point Support [8] Lm K.Y., Hung K.C., 1990, Vibrtion study on pltes with stiffened openings using orthogonl polynomils nd prtitioning method, Computers nd Structures 37: [9] Lur P.A., omnelli E., ossi.e., 1997, Trnsverse vibrtions of simply-supported rectngulr pltes with rectngulr cutouts, Journl of Sound nd Vibrtion 0(): [10] Skiym T., Hung M., Mtsud H., Morit C., 003, Free vibrtion of orthotropic squre pltes with squre hole, Journl of Sound nd Vibrtion 59(1): [11] Jog-o C.V., Pickett G., 1961, Vibrtions of pltes of irregulr shpes nd pltes with holes, Journl of the Aeronuticl Society of Indi 13(3): [1] Kumi T., 195, The flexurl vibrtions of squre plte with centrl circulr hole, Proceedings of nd Jpn Ntionl Congress on Applied Mechnics [13] Hegrty.F., Arimn T., 1975, Elsto-dynmic nlysis of rectngulr pltes with circulr holes, Interntionl Journl of Solids nd Structures 11: [14] Estep F.E., Hemmig F.G., 1978, Estimtion of fundmentl frequency of non-circulr pltes with free,circulr cutouts, Journl of Sound nd Vibrtion 56(): [15] Ngy K., 195, Trnsverse vibrtion of plte hving n eccentric inner boundry, Journl of Applied Mechnics 18 (3): [16] Ngy K., 1980, Trnsverse vibrtion of rectngulr plte with n eccentric circulr inner boundry, Interntionl Journl of Solids nd Structures 16: [17] Lee H.S., Kim K.C., 1984, Trnsverse vibrtion of rectngulr pltes hving n inner cutout in wter, Journl of the Society of Nvl Architects of Kore 1(1):1-34. [18] Kim K.C., Hn S.Y., Jung J.H., 1987, Trnsverse vibrtion of stiffened rectngulr pltes hving n inner cutout. Journl of the Society of Nvl Architects of Kore 4(3):35-4. [19] Avlos D.., Lur P.A.,, Trnsverse vibrtions of simply supported rectngulr pltes with two rectngulr cutouts, Journl of Sound nd Vibrtion 67: [0] Lee H.S., Kim K.C., 1984, Trnsverse vibrtion of rectngulr pltes hving n inner cutout in wter, Journl of the Society of Nvl Architects of Kore 1(1):1-34. [1] Khursi H.B., wtni S., 1978, Vibrtion nlysis of circulr pltes with eccentric hole, Journl of Applied Mechnics 45(1): [] Lin W.H., 198, Free trnsverse vibrtions of uniform circulr pltes nd membrnes with eccentric holes, Journl of Sound nd Vibrtion 81(3): [3] Lur P.A., Msi U., Avlos D..,006, Smll mplitude, trnsverse vibrtions of circulr pltes elsticlly restrined ginst rottion with n eccentric circulr perfortion with free edge, Journl of Sound nd Vibrtion 9: [4] Cheng L., Li Y.Y., Ym L.H., 003, Vibrtion nlysis of nnulr-like pltes, Journl of Sound nd Vibrtion 6: [5] Lee W.M., Chen J.T, Lee Y.T.,007, Free vibrtion nlysis of circulr pltes with multiple circulr holes using indirect BIEMs, Journl of Sound nd Vibrtion 304: [6] Zhong H., Yu T., 007, Flexurl vibrtion nlysis of n eccentric nnulr mindlin plte, Archive of Applied Mechnics 77: [7] Wng D., Yng Z.C., Yu Z.G.,010, Minimum stiffness loction of point support for control of fundmentl nturl frequency of rectngulr plte by yleigh itz method, Journl of Sound nd Vibrtion 39: [8] Joseph Wtkins., Brton Jr O., 010, Chrcterizing the vibrtion of n elsticlly point supported rectngulr plte using eigensensitivity nlysis, Thin-Wlled Structures 48: [9] Dozio L., 011, On the use of the trigonometric ritz method for generl vibrtion nlysis of rectngulr kirchhoff pltes, Thin-Wlled Structures 49: [30] Kwk M.K., Hn S.,007, Free vibrtion nlysis of rectngulr plte with hole by mens of independent coordinte coupling method, Journl of Sound nd Vibrtion 306:1-30. [31] Seedi K., Leo A., 01, Vibrtion of circulr plte with multiple eccentric circulr perfortions by the yleigh-itz method, Journl of Mechnicl Science nd Technology 6 (5): [3] Fn S.C., Cheung Y.K., 1984, Flexurl free vibrtions of rectngulr pltes with complex support conditions, Journl of Sound nd Vibrtion 93: [33] Utjes J.C., Lur P.A., 1984, Vibrtions of thin elstic pltes with point supports: comprtive study, Second Ntionl Meeting of Users of the Method of Finite Elements. [34] Wng D., Jing J.S., Zhng W.H., 004, Optimiztion of support positions to mximize the fundmentl frequency of structures, Interntionl Journl for Numericl Methods in Engineering 61: IAU, Ark Brnch