Assessing different nonlinear analysis methods for free vibrations of initially stressed composite laminated plates

Transcription

1 ssessng dfferent nonlnear analyss methods for free vbratons of ntally stressed composte lamnated plates H. M. Panahha a and. Davar b,* a M.Sc. student, Malek shtar Unversty of echnology, Lavzan, ehran, , Iran b ssstant professor, Malek shtar Unversty of echnology, Lavzan, ehran, , Iran rtcle nfo: Receved: //5 ccepted: 3/8/5 Onlne: /9/5 Keywords: Nonlnear vbratons, Baxal ntal stress, Homotopy perturbaton method, Multple scales perturbaton method. bstract In ths paper, the nonlnear free vbratons of thn symmetrc and nonsymmetrc cross-ply composte plates subjected to baxal ntal stresses are nvestgated. Because of ther excellent propertes such as specfc strength and specfc stffness, composte plates have wde applcatons n aerospace and mechancal structures. Based on Von-Karman's stran-dsplacement relatons and usng Galerkn method, the nonlnear dfferental equaton of free vbratons of ntally stressed composte plate s obtaned. hs nonlnear equaton s solved usng two dfferent analytcal perturbaton methods, namely method of multple scales (MS and homotopy perturbaton method (HPM, to analyze the nonlnear vbratons of ntally stressed cross-ply composte plates. Effects of tensle and compressve baxal ntal stresses, ntal vbraton ampltude, thckness, and aspect ratos of the composte plates on the frequency behavor are nvestgated. he valy of the results s confrmed by makng a comparson wth those reported n the lterature. ccordng to the results, both analytcal solutons show ncreasng trends for natural frequency parameters by ncreasng normal ntal stresses. Regardless of the value of ntal baxal stresses, for both symmetrc and non-symmetrc plates, the results of MS and HPM are n close agreement for the smallest ntal ampltude. However, for compressve ntal stresses, by ncreasng ntal ampltude ratos, the dscrepances between the results of HPM and MS ncrease for symmetrc and non-symmetrc plates. lthough HPM ncludes less computatonal effort (smaller length of formulaton than MS, the lnear-to-nonlnear frequency ratos obtaned usng MS method become closer to those obtaned by HPM as ntal vbraton ampltude s decreased and ntal stress s ncreased.. Introducton Because of ther hgh strength and stffness to weght rato, composte lamnated plates are wdely used n many hgh technology ndustres * Correspondng author Emal address: a_davar@mut.ac.r such as aerospace, arcraft, and automoble structures. Dynamc behavors of these composte plates arealways very vtal for researchers n order to fnd an approprate materal wth the best response to the dynamc conons. Snce the accuracy of results n 5

2 JCRME H. M. Panahha, et al. Vol. 5, No., ut.wn. 5-6 dynamc analyss s so mportant, researchers use dfferent methods to fnd lnear and nonlnear results of the vbraton of composte plates. mong dfferent knds of composte plates, cross-ply composte plates have an mportant poston owng to ther specal strength and stffness propertes. Dfferent knds of materals are used to construct composte plates. he outstandng propertes of composte materals such as thermal nsulaton, wear resstance, and hgh strength and stffness to weght ratos have attracted consderable attenton n recent years. Sngh et al. [] used the method of drect numercal ntegraton of the frequency rato n the study of nonlnear free vbraton behavor of rectangular cross-ply lamnates. Radu [] dscussed Sngh's analyss and modfed ts formulaton accordng to the new equatons. Zhen et al. [3] used global-local hgher-order theory for the free vbraton of lamnated composte and sandwch plates. nonlnear elastc foundaton was appled by Chen et al. [4] to analyze the nonlnear vbraton of lamnated plates. Matsunaga [5] nvestgated the free vbratons of cross-ply composte shells subjected to n-plane stresses usng hgher-order shear deformaton theory. Nonlnear free vbratons of lamnated composte plates on the elastc foundaton and random system propertes were nvestgated by Lal et al. [6]. closed form soluton was done for lnear and nonlnear free vbratons of composte and fber metal lamnated plates usng Galerkn and multple scales soluton methods by Shooshtar et al. [7]. he exstence of ntal stresses durng the applcaton of composte plates s very common n real applcatons, especally n aerospace structures. Consequently, researchers have a specal concern about ths mportant ssue and perform some analyses wth ntal stresses applyng to dfferent composte structures. Chen et al. [8] consdered an ntal stress n the analyss of nonlnear vbratons of hybrd composte plates. he dynamc response of ntally stressed functonally graded materal thn plate was nvestgated by Yang et al. [9]. Chen et al. [] conducted the nonlnear analyss of ntally stressed composte plates usng Runge-Kutta method to solve the man nonlnear equaton. Dfferent methods exst to fnd an approprate response of nonlnear dynamc equatons such as numercal method of Lndste-Poncare, averagng method, homotopy perturbaton method (HPM, and multple scales perturbaton technque (MS. Recently, HPM as an easy analytcal tool for solvng nonlnear problems has receved consderable attenton. new perturbaton technque coupled wth homotopy method to solve nonlnear equatons was presented by He []. n egenvalue problem of a lned duct usng homotopy method was studed by Sun []. Blendez [3] used He's HPM to show an approxmate soluton for conservatve nonlnear oscllatons. nonlnear vbraton analyss was nvestgated by Yazd [4] on the FGM plate usng homotopy perturbaton technque. Yongqang [5] analyzed nonlnear free vbratons of symmetrc rectangular honeycomb sandwch panels usng HPM. Nowadays, MS s a well-developed, wellaccepted, and very popular method to approxmate solutons of nonlnear dfferental equatons. hs method was developed from 935 to 96 by Krylov and Bogolubov, Kuzmak, Kevorkan, and Cole, Cochran, and Mahony. In early 97s, Nayfeh developed ths method by wrtng many papers and books on ths subject. Usefulness and mportance of the MS s related to the fact that the governng mechancs of deformng meda occur over the course of tme. lso, MS has good computatonal power to solve nonlnear dfferental equatons. Bhmaradd [6] used multple scales to solve the nonlnear flexural vbratons of rectangular plates subjected to nplane forces usng a new shear deformaton theory. Bucklng and post-bucklng of extensble rods were nvestgated by Mazzll [7] usng a multple scales soluton. In ths paper, the geometrcally nonlnear vbraton of thn composte cross-ply plates subjected to the baxal tensle and compressve ntal stresses s studed. HPM and MS perturbaton technques are used to compare two analytcal solutons n terms of accuracy and computatonal effcency. Baxal ntal 6

3 JCRME ssessng dfferent... Vol. 5, No., ut.wn. 5-6 stresses, mum vbraton ampltude, thckness of the composte plates, and aspect rato of composte plates are consdered the parameters that affect the frequency rato. New results are presented as benchmark solutons n the state of the art.. Problem descrpton multlayered thn composte plate wth length a, wh b, and unform thckness h s consdered n Fg.. u, v, and w are the dsplacement components of the plate n x, y, and z drectons, respectvely. he orgn of the Cartesan coordnate system s placed on the md-surface of the plate. Fg.. Geometry of thn composte plate. ccordng to the assumptons of largeampltude deformaton of the plate n planestress state, the dynamc Von-Karman strandsplacement relatons are employed []: u w w xx ( z x x x v w w yy ( z y y y u v w w w xy ( z y x x y x y ( ( (3 xz yz zz (4 Based on the classcal lamnaton theory (CL and consderng plane-stress conons, the stress-stran relatons are defned as: xx Q Q xx yy Q Q yy Q xy 66 xy (5 where Qj are elements of reduced stffness matrx for each ply, defned n ppendx. he forces ( N x, N y, N xy, moment resultants ( M x, M y, M xy,pre-loads ncludng n-plane forces ( ( M x, M y, N x, N y, N xy, and out-of-plane moments M xy appled to the plate are gven by: N x N x xx xx h N y N y yy yy dz h N xy N xy xy xy M x M x xx xx h M y M y yy yy zdz h M xy M xy xy xy (6 (7 If the ntal stresses are consdered unform through the thckness, only n-plane ntal stresses (force pre-loads affect the nonlnear vbraton of composte plate and the moment pre-loads become zero. Hence, consderng nplane ntal stresses only, the governng equatons of the system are obtaned usng the Hamlton's prncple []: ( N N ( N xy Nxy x x x y ( N y Nx ( Nxy Nxy y x M M xy M x y w (( Nx Nx x xy y x x w w (( Nxy N xy (( N xy N xy x y y x w w (( N y N y I y y t (8 (9 ( where h I h ( z dz ( 7

4 JCRME H. M. Panahha, et al. Vol. 5, No., ut.wn. 5-6 he boundary conons are consdered smply supported defned as follows []: x, a, v w,n x=m x= ( y, b, u w,n y=m y= (3 Here, the set of admssble functons whch satsfy the smply supported boundary conons are consdered for the dsplacement components to solve the nonlnear free vbraton problem[]: m n u U ( t cos x sn y a b (4 m n v V ( t sn x cos y a b (5 m n w W ( t sn x sn y a b (6 U(t, V(t, and W(t are the dsplacement ampltudes n terms of generalzed coordnate, t. By substtutng the dsplacement components, Eqs. (4-(6, n the governng equatons (Eqs. (8-(, applyng Galerkn method, and smplfcaton, W(t s found n terms of U(t and V(t accordng to a sngle second-order ordnary dfferental nonlnear equaton as follows: d W ( t ( ( ( 3 W t W t W t (7 where, and are constant coeffcents dependng on the stffness and mass of the composte plate, mentoned n ppendx B. he ntal conons for the lateral deflecton of the plate are consdered as follows: W ( W, dw ( (8 wherew stands for the mum vbraton ampltude of the composte plate. 3. nalytcal solutons In order to solve Eq. (7 as a nonlnear dfferental equaton, many methods can be appled to get the approxmate analytcal result. In ths paper, HPM and MS perturbaton methods are consdered and dscussed by numercal results to fnd the effcency of each analytcal method. 3.. HPM soluton procedure HPM s used here to fnd the frequency results of the consdered cross-ply composte plate. w( r, p : [ :] defned as a homotopy small parameter whch satsfes the followng expresson s constructed []: H( w, p L( w L( u pl( u pn( w (9 Lnear and nonlnear parts of Eq. (7 are separated by L and N, respectvely, and u s the frst approxmaton whch satsfes the ntal conons []. Soluton of Eq. (7 s assumed as a power seres []:... w w pw p w ( Substtuton of Eq. ( n Eq. (9 yelds: L( w L( u ( 3 L( w L( u w w ( for whch the ntal conons are obtaned from Eq. (8: dw w W, (3 and, dw ( w ( (4 he frst approxmaton u s guessed as follows to satsfy the ntal conons[]: w ( t u( t W cos( t (5 So from Eq. ( we have: d w 3 w W ( W cos( t 4 3 W W cos( W t cos(3 t 4 (6 Usng varatonal teraton method [4], the soluton of Eq. (6 can be obtaned. Hence: w ( t t sn( t 3 [ W ( cos( 4 W t 3 W W W cos( t cos(3 t ] (7 4 8

5 JCRME ssessng dfferent... Vol. 5, No., ut.wn. 5-6 By ntegratng Eq. (7, the soluton s: w ( t W 3 ( ( cos 4 W ( t ( ( cos( t W (4 3 W W ( ( cos(3 t 4 (9 3 ( W [ 4 W W ( (4 3 W W ( ( ]cos( t (8 4 (9 In order to fnd, the coeffcent of the last term,.e. secular term, n Eq. (8 should be zero. herefore, a sxth-order ordnary dfferental equaton n terms of s obtaned whose coeffcents contan the geometrc and stffness parameters of composte cross-ply plate. Fnally, the frst-order approxmaton of Eq. (7 s obtaned as follows: W(t w ( t w ( t ( 3 W 4 W cos( t ( W ( ( cos( t (4 3 W W ( ( cos(3 t (9 4 (9 3.. MS soluton procedure Multple scales perturbaton method s used here to fnd the frequency results of the consdered composte plate. Soluton of Eq. (7 s wrtten as an asymptotc expanson and spatal scales are ntroduced nstead of tme scales: w( t w(,,,... w (,,,... 3 w (,,,... w3 (,,, (3 n n t, n,,,... (3 he functon wq wq (,,,..., q,,... s calculated when coeffcents of ncreasng orders of, extracted from Eq. (7, are solved and secular terms are elmnated. Dervatve operators are gven below: d D D D (3 d D D D ( D DD... (33 By substtutng Eqs. (3-(33 n Eq. (7 and retanng the terms of the coeffcents of the frst-order of, the followng equaton s obtaned: D w w (34 Soluton of Eq. (34 can be wrtten n a complex form as follows: w e +ComplexConjugate (35 where s a complex functon of,, 3. By retanng the terms of the coeffcents of the second-order of, the followng equaton s obtaned: D w w D D w w (36 For the convergence of expanson Eq. (3, the secular term on the rght-hand sde of Eq. (36 must be elmnated, whch leads to the so-called solublty conon equaton: D w (,,... (37 and w 3 e 3 L L +C.C. (38 Retanng the terms of the coeffcents of the thrd-order of, the followng equaton s obtaned : 3 3 D w w D D w D D w 3 D w w w w (39 In order to elmnate the secular term, the followng solublty conon equaton s obtaned: D 3 (4 3 9

6 JCRME H. M. Panahha, et al. Vol. 5, No., ut.wn. 5-6 where s the complex conjugate (C.C. of. Soluton of Eq. (4 s consdered n the followng form: B ae, a, B R (4 where a s a constant coeffcent and B s obtaned as follows: 9 B a 3 B 4 partcular soluton of Eq. (39 s: ( w3 ( e +C.C. (43 4 Hence, the natural frequency s obtaned as follows: 9 ( a (44 4 Fnally, from Eqs. (35, (38 and (43, the deflecton response up to terms of order 3 s obtaned as follows: w( t ( e e + ( 3 +C.C. +C.C ( ( e +C.C. (45 4 In the next secton, both HPM and MS are dscussed n detal by some numercal solutons for two multlayered cross-ply composte plates. 4. Numercal results he numercal results for cross-ply composte plates subjected to dfferent ntal stresses, ntal ampltude, aspect ratos of the plate, and length to thckness rato are provded. In order to valdate both of the proposed HPM and MS, the nonlnear vbraton of thn cross-ply lamnated plates wth dfferent ratos of ntal stresses s compared wth the one reported by Sngh [], Radu [], and Bhmaradd [6], as ndcated n ables and, respectvely. he values of ntal baxal stresses are consdered: N x = N y =N, N xy = (46 In the case of compressve ntal stresses, the crtcal value of N s obtaned and called, N, the crtcal bucklng stress. he nondmensonal rato of the baxal ntal stresses to the crtcal bucklng stress ( N s N consdered n ables to 4. he accuracy of HPM for nonlnear vbratons of cross-ply plates wthout ntal stresses s nvestgated n able. Results for HPM are compared wth those by both Sngh [] and Radu [] for two dfferent lay-ups of cross-ply composte plates, aspect ratos, and ntal ampltudes. Numercal ntegraton method s used to fnd the nonlnear vbratons of the consdered plates by Sngh []. Radu [] presented a dscusson on Sngh's results and modfed hs equatons by the same numercal ntegraton method. ccordng to able, for both symmetrc and non-symmetrc cross-ply lamnated plates, there s a good agreement between HPM results and those reported n the lterature for small ntal ampltudes. For nstance, the greatest percentage dscrepancy for W h =.5 s.8%. However, by ncreasng the ntal ampltude, the percentage dscrepances are ncreased and HPM shows less accuracy at hgh ntal ampltudes lke W h = by the percentage dscrepancy of.95% for the nonsymmetrc cross-ply plate. Lnear to nonlnear frequency ratos for twolayered cross-ply square plate that are calculated by both HPM and MS for dfferent ntal stresses and ntal ampltudes are compared wth those reported by Bhmaradd [6] n able. s can be seen n ths table, the mum percentage dscrepancy between the MS results and those of Bhmaradd [6] for ntal tensle stresses s 3.6%, whch occurs n the case of stress rato ( N equal to.5 and N ntal ampltude ( W h of. However, for compressve ntal stresses, MS shows less accuracy. he mum percentage dscrepancy s equal to -8.5%, whch occurs for stress rato ( N of -.5 and ntal N ampltude ( W h of. 3

7 JCRME ssessng dfferent... Vol. 5, No., ut.-wn. 5-6 ccordng to able, as compared to Bhmaradd's [6] results for both HPM and MS, the dscrepances are ncreased by ncreasng ( W h and decreasng ( N. N lso, the dscrepances are decreased by decreasng ( W h and ncreasng ( N. N Generally, the dscrepances correspondng to MS are less than those for HPM, as compared to Bhmaradd's [6] results. he results of frequency rato for both symmetrc and non-symmetrc cross-ply composte plates under dfferent ntal baxal stresses and ntal ampltudes are obtaned n able 3. ccordng to ths table, regardless of the value of ntal baxal stresses, for both symmetrc and non-symmetrc plates, the results of MS and HPM are n close agreement for the smallest ntal ampltude, W h =.5. However, for compressve ntal stresses, N N =.5, by ncreasng ntal ampltude rato from.5 to, the dscrepances between the results of HPM and MS ncrease from.67% to 33.74% for the symmetrc plate and.8% to 36.3% for nonthe symmetrc plate (a/h=75. Nevertheless, by alterng baxal ntal stresses from compressve to the tensle one, lower dscrepances are obtaned due to ncreasng ntal ampltudes. For N =, by ncreasng N ntal ampltudes from.5 to, the dscrepances between HPM and MS ncrease from.% to.98% for the symmetrc plates and.% to 3.47% for the non-symmetrc plates. ccordng to ths table, more accurate results are obtaned for both composte plates due to tensle ntal stresses. In contrast, for compressve ntal stresses, less accurate results are shown for HPM and more relable results from MS are provded as compared to Bhmaradd's [6] results. lso, t should be mentoned that symmetrc plates show less dscrepances between MS and HPM n comparson wth the non-symmetrc plates. able. Comparng nonlnear to lnear frequency rato ( for thn cross-ply lamnated plate, (/9 NL L and (/9 s, a/b=, no ntal stress ( N N =. E E =, G E =.5, =.5, a h = 75, n=m=. (/9 s (/9 W m a x h Sngh[] Radu[] HPM Sngh[] Radu[] HPM (.8* ( (.7 ( (.4 ( (.9 ( (5.6 ( (4.9 ( (7.7 ( (5.99 (-.95 n=, m=, ( NL L ( ( (-. ( (-.4 ( (-. (-. * : Percentage dscrepancy= (HPM-Ref []/Ref [] * + : Percentage dscrepancy= (HPM-Ref []/Ref []* (-. ( (-. ( (-. ( (-. (-.76 3

8 JCRME H. M. Panahha, et al. Vol. 5, No., ut.wn. 5-6 able. Comparng lnear to nonlnear frequency rato ( for thn two-layered cross-ply lamnated L NL plate, (/9, a/b=, E E = 5, G E =.5, =.5, a h =, n=m=. N Nc b HPM.764 (-7.3*.59 ( ( (-33. MS.97 ( ( ( (4.8 Bm[6] HPM.843( ( ( (-49. MS.957( (..75 ( (-8.5 Bm[6] HPM.8587 ( ( ( (-44.8 MS.963 ( (..739 ( (-.9 Bm[6] HPM.899 ( ( ( (-37.4 MS.9745 (..953 ( (-..75 (3.6 Bm[6] HPM.94 ( ( ( (-33.8 MS.988 (..973 (.4.85 (.3.76 (-.9 Bm[6] HPM.9455 ( ( ( (-9. MS.987 (..953 ( (.8.87 (. Bm[6] able 3. Comparng lnear to nonlnear frequency rato ( for thn four-layered cross-ply lamnated L NL plate,(/9 s and (/9 a/b=, E E = 5, G E =.5, =.5, a h = 75. (/9 s (/9 N N Methods MS HPM MS HPM MS HPM MS HPM MS HPM MS HPM

9 JCRME ssessng dfferent... Vol. 5, No., ut.wn. 5-6 able 4. Comparson of lnear to nonlnear frequency rato ( for thn four-layered cross ply lamnated L NL plate, (/9 s and (/9 a/b=, E E = 5, G E =.5, =.5, N N =, n=m=. (/9 s (/9 a h Methods MS HPM MS HPM MS HPM MS HPM Effects of dfferent ntal ampltudes and length to thckness ratos on the frequency ratos are nvestgated n able 4. ccordng to the results, the nfluence of thckness of the symmetrc and non-symmetrc cross-ply composte plates on the frequency ratos s neglgble. For hgh ntal ampltudes, by ncreasng the ntal ampltude, the dscrepances of HPM and MS are ncreased. For nstance, for a/h=, by ncreasng the ntal ampltudes from.5 to, the dscrepances between HPM and MS are.5% and 6.49%, respectvely, for the symmetrc plates and.5% and 6.45% for the non-symmetrc ones. In the case of four-layered plates, the effect of layup (symmetrc and nonsymmetrc on the frequency ratos s not sgnfcant for dfferent length to thckness ratos.some graphcal results are shown to nvestgate the effects of dfferent ntal stresses, ntal ampltudes, and aspect ratos on the frequency ratos. In Fg., the varaton of frequency rato versus ntal stress parameter ( N s shown for N MS and HPM frequency analyss methods. lso, the results are compared wth those of Bhmaradd [6]. Varaton of nonlnear to lnear frequency ratos versus dfferent ntal ampltudes for an sotropc plate wthout ntal stresses are nvestgated by both HPM and MS and valdated usng the results of Chen [] n Fg. 3. ccordng to Fgs. and 3, MS shows more accurate results, especally at hgh ntal ampltudes rather than HPM as compared to Chen's [] and Bmaradd's [6] results. ccordng to Fg. 3, both analytcal solutons show ncreasng trends by ncreasng normal ntal stresses, whch s analogous to the results by Chen [] that are obtaned usng Runge- Kutta soluton method. Varatons of lnear to nonlnear frequency ratos versus ntal ampltude for two specal compressve and tensle baxal ntal stresses are nvestgated n Fg. 4. In both cases, by ncreasng the ntal ampltudes, the dscrepances between two methods are ncreased. Frequency Rato ( L / NL Intal Stress Parameter (N/N c b HPM MS Bhmaradd[6] Fg.. Frequency rato versus ntal stress parameters are valdated wth the results of Bhmaradd[6] for two-layered cross-ply plate (,9, W/h=, a/b=, a/h=. 33

10 JCRME H. M. Panahha, et al. Vol. 5, No., ut.wn. 5-6 Frequency Rato ( NL / L Fg. 3. Frequency rato versus dfferent ntal ampltudes are valdated wth the results of Chen[] for an sotropc plate wth no ntal stresses, a/b=, a/h=, N N=. Frequency Rato ( L / NL Maxmum Vbraton mpltude Parameter (W /h Fg. 4. Frequency rato versus dfferent ntal ampltude for a four-layered symmetrc cross-ply plate (,9,9,, a/b=, N N=-.5, a/h= Conclusons HPM MS Chen[] HPM MS Maxmum Vbraton mpltude Parameter (W /h In ths work, the nonlnear vbratons of the symmetrc and non-symmetrc cross-ply composte plates ntally stressed by baxal stresses were nvestgated. Both homotopy perturbaton method and multple scales perturbaton technque as analytcal solutons were appled and the numercal results were compared to those avalable n the lterature. Investgatng the effects of dfferent ntal baxal stresses, ntal ampltudes, and plates' thckness rato on the frequency rato ndcated that MS was more adequate than HPM n a wde range of ntal ampltude and ntal stresses. However, HPM had less computatonal formulaton than MS. Some new nterestng results were presented whch have not been prevously reported n the lterature. ppendx [6] Q = Q cos 4 θ + Q sn 4 θ + Q + Q sn θcos 66 θ ( Q = Q sn 4 θ + Q cos 4 θ + Q + Q sn θcos 66 θ (3 4 4 Q 66 = Q +Q - Q - Q 66 sn θcos θ +Q 66 sn θ +cos θ (4 ppendx B m n 8 Nx( Ny( 3 6 a b N h N h N (B3 h n 3 B (B4 b n m ( b a m n 66 (B5 (B 3 66 a b 3 4 n m n 4 ( 66 9mn b a b (B6 (B 3 (B7 m B (B8 a m n a b 3 4 n m n ( (B (B mn b a b 34

11 JCRME ssessng dfferent... Vol. 5, No., ut. Wn. 5-6 m m n n ( a a b b m n 3 4( ( ( ( m n 9mn a b m n 3 a b + m n ( 66 (B D D D66 D 9 B B 6 a b h j, Bj, Dj h Qj (, z, z dz, j,, 6 References (B (B (B4 [] G. Sngh, K. K. Raju and G.V. Rao, Nonlnear Vbraton of Smply Supported Rectangular Cross-Ply Plates, Sound. Vb., Vol. 3,No. 4, pp. 3-6, (99. [] G. R. dran, G. Sngh, K. K. Raju, G. V. Rao and N. G. R. Iyengar, Dscusson on: Non-lnear Vbraton of Smply Supported Rectangular Cross-Ply Plates, Sound. Vb., Vol. 33, No., pp , (. [3] W. Zhen and Wanj Chen, Free Vbraton of Lamnated Composte and Sandwch Plates Usng Global-Local Hgher-Order heory, Sound. Vb.,Vol.7, No. 49, pp , (6. [4] R. D. Chen, C. S. Chen and Rao Nonlnear Vbraton of Lamnated Plates on a Nonlnear Elastc Foundaton, Compos. Struct.,Vol. 8, No. 7, pp. 9-99, (5. [5] H. Matsunaga, Vbraton and Stablty of Cross-ply Lamnated Composte Shallow Shells Subjected to In-Plane Stresses, Compos. Struct.,Vol. 7, No. 78, pp , (7. [6]. Lal, B. N. Sngh and R. Kumar, Nonlnear Free Vbraton of Lamnated Composte Plates on Elastc Foundaton wth Random System Propertes, Int. J. Mech. Sc., Vol., No. 5, pp. 3-, (8. [7]. Shooshtar and S. Razav, Closed Form Soluton for Lnear and Nonlnear Free Vbratons of Composte and Fber Metal Lamnated Rectangular Plates, Compos. Struct.,Vol., No. 9, pp , (. [8] C. S. Chen and C. P. Fung, Nonlnear Vbraton of n Intally Stressed Hybrd Composte Plates, Sound. Vb.,Vol. 4, No. 74, pp. 3-9, (4. [9] J. Yang and H. S. Shen, Dynamc Response of Intally Stressed Functonally Graded Rectangular hn Plates, Compos. Struct.,Vol. 4, No. 54, pp , (. [] C. S. Chen,. J. Chen and R. D. Chen, Nonlnear Vbraton of Intally Stressed Functonally Graded Plates, hn-walled Struct., Vol. 3, No. 44, pp , (6. [] J. H. He and Raju, Homotopy Perturbaton echnque, Comput. Method. ppl. Mech., Vol., No. 78, pp. 57-6, (999. [] X. Sun, L. Du and V. Yang Homotopy Method for Determnng the Egenvalues of Locally or Non-Locally Reactng coustc Lners n Flow Ducts, Sound. Vb.,Vol. 3, No. 33, pp , (7. [3]. Blendez,. Blendez,. Marquez and. Nepp, pplcaton of He s Homotopy Perturbaton Method to Conservatve ruly Non-Lnear Oscllators, Chaos. Solton. Fract., Vol. 3, No. 37, pp , (8. [4].. Yazd, Homotopy Perturbaton Method for Nonlnear Vbraton nalyss of Functonally Graded Plate, Vb. coust.,vol., No. 46, pp. 7-, (3. [5] L. Yongqang, L. Feng and Z. Dawe, Geometrcally Non-Lnear Free Vbratons of the Symmetrc Rectangular Honeycomb Sandwch Panels Wth Smply Supported Boundares, Compos. Struct.,Vol. 5, No. 9, pp. -9, (. [6]. Bhmaradd, Nonlnear Flexural Vbratons of Rectangular Plates Subjected to In-Plane Forces Usng a New Shear Deformaton heory, hn-walled Struct., Vol. 8, No. 5, pp , (987. [7]. Bhmaradd, Nonlnear Flexural Vbratons of Rectangular Plates Subjected to In-Plane Forces Usng a New Shear Deformaton heory, hn-walled Struct., Vol. 7, No. 5, pp , (987. [8] E. N. C. Mazzll, Bucklng and Post- Bucklng of Extensble Rods Revsted: Multple Scales Soluton, Non-Ln. Mech., Vol., No. 44, pp. -8, (9. 35