Numrical Analysis of Transint Rsponss for Elastic Structurs Connctd to a Viscolastic Shock Absorbr Using FEM with a Nonlinar Complx Spring Takao Yamaguchi, Yusaku Fujii, Toru Fukushima, Akihiro Takita, Suguru Ota, Norihisa Tomita and Tatsuya Noto Gunma Univrsity Tnjin-cho -5-, Kiryu, Gunma, Japan Graduat School of Gunma Univrsity Tnjin-cho -5-, Kiryu, Gunma, Japan yamagm@gunma-u.ac.jp Abstract This papr dscribs numrical simulation of impact rsponss for a viscolastic shock absorbr connctd with a S-shapd structur using a finit lmnt mthod. This S-shapd structur is a part of a forc transducr. And th shock absorbr is a spcimn to b masurd its impact rsponss whn th lvitatd is collidd with it. In this analysis, th viscolastic absorbr is modlld by using a nonlinar complx spring to dscrib its nonlinar hystrsis undr rlativly larg dformation. On nd of this nonlinar spring is connctd to th S-shapd structur. Th othr nd is connctd to th lvitatd block. Th S-shapd structur and th block ar modlld by thr-dimnsion finit lmnts. Th lvitatd block has initial vlocitis and collids with th shock absorbr. Acclration of th S-shapd lastic structur and th lvitatd block ar masurd using Lvitation Mass Mthod proposd by Fujii. Th calculatd acclrations from th proposd FEM, corrsponds to th xprimntal ons. Morovr, using this mthod, w also invstigat dynamic rrors of th S-shapd forc transducr du to lastic mods in th S-shapd structur. Introduction Various viscolastic shock absorbrs ar utilizd to diminish impacts from prcision instrumnts and so on. Undr rlativly larg dformation, th viscolastic absorbrs somtims hav nonlinarity btwn thir rstoring forcs and dformations [8-]. Morovr, thir rstoring forcs somtims hav nonlinarity in hystrsis. Thrfor, it is important to invstigat nonlinar dynamic charactristics of viscolastic absorbrs connctd to lastic structurs undr impact load. Using a fast finit lmnt mthod proposd by Yamaguchi [-], this papr dscribs numrical simulation of impact rsponss for a viscolastic shock absorbr connctd with an lastic structur. As an xampl of th lastic structur, w dal with an S-shapd structur, which is a part of a forc transducr as shown in Fig.. In this simulation, th viscolastic absorbr is modld by using a nonlinar complx spring. Th rstoring forc of th spring is xprssd as powr sris of its rlativ displacmnt btwn two nds. Th rstoring forc also involvs nonlinar hystrsis damping bcaus th hystrsis dpnds on its dformation. W introduc complx spring constants for not only th linar componnt but also nonlinar componnts of th rstoring forc. W xprss a finit lmnt of th nonlinar complx spring and th spring is connctd to an S-shapd lastic structur modld by linar finit lmnts. Th discrtizd quations of ths structurs ar transformd from physical coordinat to th nonlinar ordinary coupld quations using normal coordinat corrsponding to linar ignmods. Furthr, w intgrat th transformd quations numrically in drastically small dgr-of-frdom. In this papr, our proposd FEM is applid to clarify dynamic rrors in th S-shapd forc transducr whn w masur transint rsponss of th viscolastic shock absorbr. Th transint rsponss ar obtaind whn th lvitatd block is collidd with th absorbr. W chck th validity of th calculatd rsults in comparison with th xprimntal rsults.
Furthr, to valuat th dynamic rrors in th forc transducr, th rfrnc forc is also masurd using Lvitation Mass Mthod proposd by Fujii []. W chckd dynamic rrors btwn th rfrnc forc and th forc masurd by th transducr itslf in th prvious papr []. Th xprimntal dynamic rrors [] ar compard with th calculatd data from our proposd FEM. Morovr, w invstigat and find out th causs of th dynamic rrors as nonlinar oscillation phnomna. Exprimntal stup and rsults [] Figur shows a schmatic diagram of th xprimntal stup prformd by Fujii in th prvious papr []. An S-shapd forc transducr, shown in th photograph in Fig., is connctd with a viscolastic shock absorbr. To valuat impact rsponss of th viscolastic absorbr, a block lvitatd by linar pnumatic baring is collidd with th absorbr in z dirction. And thir transint rsponss ar masurd using th forc transducr. As shown in Fig., th S-shapd structur in th forc transducr contains two thin Figur : Outlin of xprimntal systm [] Figur : Photograph of th S-shapd forc transducr [] mmbrs in th vicinity of a cntral hol. Ths two mmbrs corrspond to a pair of paralll springs in z dirction. Strains ar masurd using strain gaugs attachd in th S-shapd structur whn xtrnal dynamic load is xrtd in z dirction. Using a bridg circuit and th masurd strains, forcs F of th impact rsponss ar trans masurd. To validat accuracy of th masurd forc F trans, Fujii additionally masurd th vlocity v of th lvitatd block using an intrfromtr as illustratd in Fig.. Firstly, acclration a of th lvitatd block can b idntifid by diffrntiating th masurd vlocity v with rspct to F M a is tim. Th rfrnc forc obtaind using th acclration a and M. 656kg of th lvitatd block if th block can b rgardd as a concntratd. In th prvious papr [], Fujii compard btwn th rfrnc Figur : Comparison btwn th masurd forc th S-shapd transducr and th rfrnc forc using Lvitation Mass Mthod [] forc F and th forc F masurd trans from th S-shapd transducr itslf as shown in Fig.. And h found out that small dynamic rrors ar hiddn in th masurd forc Ftrans Figur 4: Comparison btwn th masurd dynamic rror and th stimatd rror using acclration of th S-shapd structur [] of th S-shapd transducr. In Fig.4, a tim history of th diffrnc F F trans F btwn th masurd forc F by th transducr itslf and th trans rfrnc forc F. Fujii pointd out that th diffrnc F F trans F ar rlatd with M a. a is th acclration of th S-shapd structur and M is of a half portion of th S-shapd structur. Fujii proposd a corrction of th masurd forc F trans by th transducr itslf using an xprssion F M a. To stimat corrction forc F, Fujii fabricatd an acclromtr to th movabl half portion in th S-shapd
structur as illustratd in Fig.. Th masurd acclration a from th acclromtr is usd to corrct th masurd forc F in ral tim. In this papr, numrical analysis for this xprimntal systm ar carrid trans out to invstigat th corrction forc F which corrsponds to th diffrnc btwn F and trans F. Ths xprimntal data ar compard with our computd data in this papr to validat our proposd FEM with considration of nonlinar complx springs. To gt th rfrnc forc F, Lvitation Mass Mthod (i.. LMM) is utilizd. Th dtail stup and procdur of th LMM xprimntal systm ar notd as follows. As illustratd in Fig., th LMM xprimntal systm contains a lvitatd (i.. this corrsponds to a lvitatd rigid block), which is abl to mov along a guid way in z dirction du to a pnumatic linar baring. Th block is lvitatd du to air film at th intrfacs btwn th block and th guid. Thicknss of th air film is 8 μm. Prssur at th intrfacs is slf-controlld by orific ffct to kp th thicknss of th air film. Du to this systm, th block can travl in th z-dirction with xtrmly low friction. An xtnsion rod is fabricatd on th lvitatd. Initial vlocitis v ar givn with th lvitatd block manually. Th block is collidd with th viscolastic shock absorbr connctd with th S-shapd forc transducr. This sophisticatd xprimntal systm is namd as Lvitation Mass mthod by Fujii [4-6]. Nar th transducr, th acclromtr is fabricatd as w mntiond bfor. On nd of th S-shapd forc transducr is fixd on th rigid bas. Not that a gl is filld in th cntral hol of th S-shapd structur to incras damping as shown in Fig.. Numrical simulation As shown in Fig.5, w valuat th FEM modl for th S-shapd structur in th forc transducr. Both th S-shapd structur and th lvitatd block ar modld as an lastic body using thr dimnsional finit lmnts. Th viscolastic shock absorbr is modlld by a nonlinar spring with nonlinar damping using nonlinar complx spring constants. W st th origin of this modl on th position whr th lvitatd block bgins to contact with th viscolastic shock absorbr. W giv an initial vlocity v = -8 (mm/sc) in z- dirction to th lvitatd block by a hand. Discrtizd quation for th viscolastic absorbr Using a concntratd nonlinar spring with nonlinar hystrsis, th viscolastic absorbr is xprssd. To xprss th nonlinar hystrsis, w propos and introduc a nonlinar complx spring. W st that th nonlinar complx spring with viscolasticity has principal lastic axis in z dirction as shown in Fig.5. W dnot th displacmnt in z dirction at th nodal point as U z whr th S-shapd forc transducr is attachd with on nd of th nonlinar complx spring (i.. th viscolastic absorbr). Uz is th displacmnt at th nodal point on anothr nd of th nonlinar complx spring. At this nodal point, th lvitatd block is connctd with th viscolastic absorbr. W us nonlinar function using powr sris for th nodal forc of th complx spring at th point. Thus, th rstoring forc of th complx spring is xprssd using th rlativ displacmnt Uz -U z btwn U z and U z as follows. R z Uz Uz) ( + ( Uz Uz) + ( Uz Uz) () Figur 5: Simulation modl
Firstly, w introduc convntional linar hystrsis damping as ( js ). j is imaginary unit. is th ral part of and is matrial loss factor of th concntratd spring. Furthr, w propos and s introduc nonlinar hystrsis damping as ( js ) and ( js) in th sam mannr. and ar th ral part of and,rspctivly. and s ar nonlinar componnts of matrial loss factor for s th concntratd spring, rspctivly. For th nonlinar complx spring, nonlinar spring constants hav complx quantity to rprsnt changs of hystrsis du to th dformation of th spring. Th rstoring forc of th nonlinar complx spring can b writtn in th matrix form as follows. U U Rz U { r}, { U s} U U Rz U x y z x y z, [ ] R z { r} [ ]{ Us} { d} (), ( U { d} ( U z z U U ) ( U z ) ( U z z z U U z z ) ) Whr { U s } is th nodal displacmnt vctor at th nods. {r} is th nodal forc vctor at th nods and. {d } is th vctor including nonlinar trms of th rstoring forc. [ ] is th complx stiffnss matrix containing only linar trm of th rstoring forc. Th following linar and nonlinar complx spring constants ar usd for th latr computations. =.7 4 (N/m), =. (N/m ), =.5 (N/m ), =., s =., s =.. s. Discrtizd quations of th S-shapd forc transducr and th lvitatd block For th lastic structur of th S-shapd forc transducr and th lvitatd block, w assumd that quations of motion ar xprssd undr infinitsimal dformation using convntional thr-dimnsional finit lmnts. Ths ar mad by aluminium. To add damping ffcts, a viscolastic gl is filld in th cntral hol of th forc transducr as shown in Fig.. Thus, w also crat th thr dimnsional finit lmnts for th gl as dpictd in Fig.5. Viscolasticity of th gl is takn into account using complx modulus of lasticity E ( g Eg jg ). Th ral part E of th g E stands for th storag modulus of lasticity g whil is th matrial loss factor of th gl. By suprposing all lmnts rlatd to th S-shapd transducr, g th lvitatd block and th gl, th following quations ar obtaind. M s ]{ u s} [ Ks ]{ us} { f } () M ]{ u } [ K ]{ u } { f } (4) [ s [ g g g g g ]{ u } [ K ]{ u } { f } (5) [ M L L L L L Whr, [ M s], [ K s ], { f s } and { u s } ar th matrix, complx stiffnss matrix, nodal forc vctor and displacmnt vctor for th lmnts of th S-shapd transducr dnoting by th subscripts s in Eq. (). Th subscripts g in Eq. (4) and subscripts L in Eq. (5) dnot th gl and th lvitatd block, rspctivly. For th thr dimnsional finit lmnts of ths solid structurs, isoparamtric hxahdral lmnts with non- conforming mods [-] ar mainly adoptd. Discrt quation for combind systm among th S-shapd forc transducr, th viscolastic absorbr and th lvitatd block In Eq. (), th rstoring forc } {r is addd to th nodal forc at th nds of th nonlinar complx spring on th nods and. On th nod, th nonlinar complx spring (i.. th viscolastic shock absorbr) is 4
connctd with th S-shapd forc transducr in Eq.(). On th nod, th spring is also connctd with th lvitatd block in Eq.(5). Finally, th following xprssion in global systm can b obtaind using Eqs.()- (5). [ M]{ u } [ K]{ u} { dˆ} { f } (6) Whr, { f },[M ], [K] and {u} ar th displacmnt vctor, matrix, complx stiffnss matrix and xtrnal forc vctor in global systm, rspctivly. {d ˆ } is modifid from {d } to hav th idntical vctor siz to dgr-of-frdom of th global systm. Approximat calculation of modal damping Undr infinitsimal dformation, w us th following complx ignvalu problm of Eq. (6) by nglcting th trms of th nonlinar rstoring forc and th xtrnal forc to calculat modal loss factors as imaginary part of ignvalus. Ths paramtrs corrspond to linar modal damping. max ( ( ) ( j )[ M ] [ K ] ( j )){ } {} (7) tot (i) whr, ( ) is th ral part of complx ignvalu. { } is th complx ignvctor and is th modal loss factor. contains s and. Suprscript dnots th i-th ignmod. Nxt, th g following ar introducd using th maximum valu max among th lmnts' matrial loss factors, (,,,... ). max R / max, (8) Solutions of Eq.(7) ar xpandd using a small paramtr j on assumption of max max as follows [],[]. { } { } { } { },... (9) 4 ( ) ( ) ( ) ( ),... () 4 5 7 j,... () tot 5 7 Abov quations, w can obtain undr conditions of and max. Thrfor, also bcom small paramtr lik. In th quations,,,,... and,,,... hav ral quantitis. By Substitution of Eqs.(9)-() 5 into Eq. (7), w obtain approximat quations using and ordrs. As a rsult, th following quation can b drivd by arranging th approximat quations [],[]. max i ( i) T T S { } [ K ] { } / { } [ K ] { } () R From ths xprssions, w can calculat modal loss factor using matrial loss factors of ach (i) lmnt, shar S of strain nrgy of ach lmnt to total strain nrgy. Convrsion from th discrtizd quation in physical coordinat to th nonlinar quation in normal coordinat R 5
If w comput Eq.(6) dirctly in physical coordinat, it rquirs much computational tim bcaus of larg dgr-of-frdom. Thrfor, w introduc computational mthod to diminish th dgr-of-frdom for th discrtizd quations of motion Eq.(6). (i) ( ) It is assumd that w approximat linar natural mods { } of vibration to{ i }. Th nodal ( ) displacmnt vctor {u} can b xprssd using both { i } and b ~ by introducing normal i coordinats b ~ ( ) i corrsponding to th linar natural mods{ i } [4]. { u} ~ bi{ } () i By substitution of Eq.() into Eq.(6), w obtain th following nonlinar ordinary simultanous quations as to normal coordinats b ~. i ~ bi, tt tot i, t i ijk j k ijkl j k l i j k j k l ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ b ( ) b D b b E b b b P,( i, j, k, l,,,...) (4) ~ T T ~ Pi { } { f }, { } { ix, iy, iz, ix, iy, iz, ix,...}, Dijk ( iz iz)( jz jx)( kz kz ), ~ E ( )( )( )( ) ijkl iz iz jz jz kz kz lz lz (i) whr, is th i-th modal loss factor. rprsnts th i-th ignfrquncy. Subscript t following a tot comma dnots partial diffrntiation with rspct to tim t. is th z-componnt of th ignmod iz ( ) { i } at th connctd nod btwn th S-shapd forc transducr and th viscolastic shock absorbr. W can sav computational tim drastically bcaus Eq. (4) has much smallr dgr-of-frdom than that of Eq.(6). Nonlinar impact rspons An initial vlocity is givn to th lvitatd block attachd with th viscolastic shock absorbr. And nonlinar impact rsponss of th viscolastic absorbr with th S-shapd structur ar computd by applying Rung-Kutta-Gill mthod to Eq.(4). Numrical rsults and discussion Validity of th proposd FEM In this sction, w validat our proposd finit lmnt mthod in considration of nonlinar hystrsis Figur 6: Comparison of th vlocity btwn calculatd and xprimntal data [] Figur 7: Comparison of th acclration btwn calculatd and xprimntal data [] 6
using nonlinar complx springs. First, w chck th calculatd rsults of th rfrnc forcs. Fig.6 shows tim historis of th masurd and calculatd vlocitis v at th cornr cub on th lvitatd block. In Fig.7, th xprimntal and calculatd acclrations a at th cornr cub ar compard. Furthr, th xprimntal and calculatd rfrnc forcs F using a of th lvitatd block ar shown in Fig.8. In ths graphs, w st th origin of tim whn th lvitatd block starts to contact with th viscolastic shock absorbr connctd with th S- Figur 8: Comparison of th rfrnc forc btwn calculatd and xprimntal data [] shapd structur. From ths graphs in Figs.6-8, th calculatd tim historis agr wll with th xprimntal ons for th vlocitis v, acclrations a and th rfrnc forc F. Thrfor, th validity of our proposd FEM is confirmd to comput th rfrnc forc in th impact rspons of th viscolastic shock absorbr. Furthr, w comput and invstigat dynamic rrors of th S-shapd forc transducr. Fujii point out in th prvious papr [] that thr xists a rlation of F M a for th dynamic rrors F in th masurd forc of th transducr. a is th Figur 9: Comparison of th acclrations at th S- shapd structur btwn calculatd and xprimntal data [] Figur : Elastic mod of th S-shapd structur in z-dirction Figur : Elastic mod of th nonlinar complx spring in z-dirction 7
signal of th acclration from th acclromtr fixd on th S-shapd structur, whil M is th half of th for th S-shapd structur in th transducr. From this viwpoint, w also comput th acclration a on th S-shapd structur in th transducr to clarify th dynamic rrors. As shown in Fig.9 (a), th xprimntal acclration a of th S-shapd structur in th transducr is much diffrnt qualitativly from th acclration a at th lvitatd block shown in th Fig.7 (a). Thr xist high frquncy componnts in th wavform of th acclration a in th transducr. Figur : Comparison of th vlocitis at th S- shapd structur btwn calculatd and xprimntal data [] Figur : Rsonant frquncis and Modal loss factors of lastic mods of th lvitatd block Thrfor, to clarify dynamic rrors of th forc transducr, w try to invstigat what happns in this systm. Th corrsponding calculatd tim history of th acclration a in th transducr is shown in Fig.9 (b). In comparison with th xprimntal data in Fig.9 (a), both tim historis agr wll. W can confirm that our proposd FEM in considration of nonlinar complx springs is valid. W can rcogniz that thr ar typical priods in th acclrations a at th S-shapd structur in th forc transducr. W can find a short priod.5(s) in th wavform. This corrsponds to th ign frquncy of th lastic mod for th S- shapd structur of th transducr in z-dirction. Th dformation in this ignmod is shown in Fig.. From this figur, th half portion of th S-shapd structur dforms. Anothr half portion nvr movs. This implis that th dynamic motion of th half portion for th S-shapd structur is dominant. W can rgard that th corrction forc F M a using th half M of th S-shapd structur is rasonabl. In Fig.9, a long priod. (s) is also found. W can find out an intrnal rsonanc du to th nonlinarity of th viscolastic shock absorbr in this priod. This long priod has a rlation with th suprharmonic componnt of ordr svn for th rigid translation of th lvitatd block in th z-dirction as shown in Fig.. Morovr, 8
this is rlatd with th subharmonic componnt of ordr /7 for th lastic mod of th S-shapd structur in Fig.. W also chck vlocitis at th S-shapd structur in Fig.. Th calculatd vlocity is consistnt with th xprimntal on. LMM Numrical analysis to confirm accuracy of th rfrnc forc using In this sction, by valuating masurmnt rrors du to undsirabl motions or dformation of th lvitatd block, w chck th accuracy [7] of th rfrnc forc using th Lvitation Mass Mthod. Elastic ignmods of th lvitatd block ar shown in Fig.. If ths mods ar gnratd whn th lvitatd block is collidd with th viscolastic shock absorbr with th S-shapd forc transducr, masurmnt rrors of th rfrnc forc incras. To chck ths rrors, w considr th consistncy btwn th displacmnt D at th cntr of gravity of th lvitatd block and th displacmnt D at th cornr cub. As a rsult, th g ratio D /.999747. is obtaind at t=.74 [sc]. Not that th displacmnts rach th local c D g maximal valu at t=.74 [sc]. From this ratio, th undsirabl dynamic motions and dformations of th lvitatd block can b rgardd as nough small. Thus, th rfrnc forc as shown in Fig.8 includs nough small masurmnt rrors du to th undsirabl bhaviours of th lvitatd block. Conclusion This papr dals with numrical analysis of impact rsponss for a viscolastic shock absorbr connctd with an lastic structur (an S-shapd structur) using a fast finit lmnt mthod proposd by Yamaguchi. In this analysis, th viscolastic absorbr is modlld by using a nonlinar complx spring. Th rstoring forc of th spring is xprssd as powr sris of its longation (.g. rlativ displacmnt btwn th nds of th complx nonlinar spring). Th rstoring forc also includs nonlinar hystrsis damping. Thrfor, complx spring constants ar introducd for not only th linar componnt but also nonlinar componnts of th rstoring forc. Finit lmnt for th nonlinar complx spring is xprssd and is connctd to an lastic structur modlld by linar solid finit lmnts. Th discrt quations of this systm in physical coordinat ar transformd into th nonlinar ordinary coupld quations using normal coordinat corrsponding to linar natural mods. Th transformd quations ar intgratd numrically in xtrmly small dgr-of-frdom. In this papr, w apply our proposd FEM to xamin dynamic rrors in an S- shapd forc transducr whn transint rsponss of a viscolastic shock absorbr ar masurd. Th transint rsponss ar obtaind whn a lvitatd block is collidd with th absorbr. Th lastic structurs in this study ar th lvitatd block and th S-shapd structur, which is a part of th forc transducr. Ths ar xprssd by using linar solid finit lmnts. Th viscolastic shock absorbr is modlld by using th complx nonlinar spring. Th nonlinar complx spring is attachd btwn th lvitatd block and th s- shapd structur. To chck th dynamic rrors in th transducr, th rfrnc forc is also masurd using Lvitation Mass Mthod proposd by Fujii. In this mthod, th block is lvitatd by a pnumatic baring. A cornr cub is fabricatd on th block to rciv a lasr bam from an intrfromtr. Th vlocity of th lvitatd block is masurd using th intrfromtr. Th xprimntal dynamic rrors ar wll simulatd with th calculatd ons from our proposd FEM. W find out that th dynamic rrors ar du to nonlinar dynamic rsponss of th ignmod of th S-shapd structur. Acknowldgmnts This work was supportd by JSPS KAKENHI Grant Numbr 4656. Rfrncs [] T. Yamaguchi, Y. Fujii, T. Fukushima, N. Tomita, A. Takita, K. Nagai and S. Maruyama, Damping rspons analysis for a structur connctd with a nonlinar complx spring and application for a fingr protctd by absorbrs undr impact forcs, Mchanical Systms and Signal Procssing, (in prss). c 9
[] T. Yamaguchi, Y. Fujii, K. Nagai and S. Maruyama, FEM for vibratd structurs with non-linar concntratd spring having hystrsis, Mchanical Systms and Signal Procssing,, Vol. (6), pp. 95-9. [] Y. Fujii, Masurmnt of th lctrical and mchanical rsponss of a forc transducr against impact forcs, Rviw of Scintific Instrumnts, Vol. 77, Aug. (6), pp. -5. [4] Y. Fujii and T. Yamaguchi, Impact rspons masurmnt of a human arm, Exprimntal Tchniqus,, May/Jun (6), pp. 64-68. [5] Y. Fujii and T. Yamaguchi, Mthod for valuating matrial viscolasticity, Rviw of Scintific Instrumnts, Vol. 75, No. (4), pp. 9-. [6] Y. Fujii and T. Yamaguchi, Proposal for matrial viscolasticity valuation mthod undr impact load, Journal of Matrial Scinc, Vol. 4, No.8 (5), pp. 4785-479. [7] T. Yamaguchi, Y. Fujii, K. Nagai and S. Maruyama, Dynamic analysis by FEM for a masurmnt systm to obsrv viscolasticity using lvitation mthod, Intrnational Journal of Advancd Manufacturing and Tchnology, Vol. 46, No.9- (), pp. 885-89. [8] B. F. Fny and R. Kappagantu, On th physical intrprtation of propr orthogonal mods in vibrations, Journal of Sound and Vibration, Vol., No.4 (998), pp. 67-66. [9] T. Kondo, T. Sasaki and T. Ayab, Forcd vibration analysis of a straight-lin bam structur with nonlinar support lmnts, Transactions of th Japan Socity of Mchanical Enginrs, Vol. 67, No.656C (), pp. 94-9. [] E. Pshck, N. Boivin, C. Pirr and S. W. Shaw, Non-linar modal analysis of structural systms using multi-mod invariant manifolds, Nonlinar Dynamics, Vol. 5 (), pp. 94-9. [] O. C. Zinkiwicz and Y. K. Chung, Th finit lmnt mthod in structural and continuum mchanics, MacGraw-Hill (967). [] E. L. Wilson, R. L. Taylor, W. P. Dohrty and J. Ghaboussi, Incompatibl displacmnt mthods, in numrical and computr mthods in structural mchanics, Acadmic Prss (97). [] B. A. Ma and J. F. H, A finit lmnt analysis of viscolastically dampd sandwich plat, Journal of Sound and Vibration, Vol. 5, No. (99), pp. 7-. [4] T. Yamaguchi and K. Nagai, Chaotic vibration of a cylindrical shll-panl with an in-plan lastic support at boundary, Nonlinar Dynamics, Vol. (997), pp. 59-77.