ANALYZING OF FREE TRANSVERSE VIBRATION OF AN ELASTICALLY CONNECTED RECTANGULAR PLATE- MEMBRANCE SYSTEM WITH A PASTERNAK LAYER IN-BETWEEN

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1 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co ANALYZIN OF FREE RANVERE VIBRAION OF AN ELAICALLY CONNECED RECANULAR PLAE- MEMBRANCE YEM IH A PAERNAK LAYER IN-BEEEN M. Nsrsho d N. Mohd Dpr of Mchcl Egrg Islc Azd Uvrs Prd Brch hr Ir E-Ml: hrddsrsho@gl.co ABRAC hs ppr horcll lzs fr rsvrs vros of lscll cocd rcgulr plr ss h Psr lr -. oluos of h prol r foruld usg h Nvr hod. Also url frqucs of h ss r drd. h ffc of Psr lr o h url frqucs of hs d ss s dscussd urcl pl. Icrsg shr foudo odulus of h Psr lr cuss crs h vlu of url frquc of h ss ω ; hovr hs fluc of h shr foudo odulus of h Psr lr s dffr so prculr frqucs. h s hs ffc s srogr ω frqucs rhr h h s s ω frqucs. h d ss hs rsg fur hch llos ch url frquc o chg s fuco of shr foudo odulus of h Psr lr hls ohr cosrucol d phscl prrs of h ss c r uchgd. h por rsul o hch hs ppr pus phsz s h h guds of h frqucs co lrgr h crsg shr foudo odulus of h Psr lr d h Psr lr c crs h gud of frqucs or h lr lsc lr c. hus h Psr lr c usd sd of lr lsc lr o or ffcvl supprss h cssv vros ssocd h plr sss. Nurcl rsuls of h prs hod r vrfd oc coprd h hos vll h lrur. Kords: fr rsvrs vro rcgulr pl-rc ss r so Psr lr Nvr hod.. INRODUCION rucurl rs such s pl-p p srucurs d of o prlll spl suppord pls s couousl jod lr lsc lr p lr r crsgl usd roucl chcl d cvl grg pplcos. As r of fc h phoo of rsvrs vro d dcl prols of such sss hs d pplco grg prcc. I clsscl vro pl hor o sc d ll o lcl hods r ppld for lzg fr vros of sgl rcgulr pl l h Lv d Nvr hods. uch srucurs hv svl covrd vsgors. Zhg l. 8 sudd vro d uclg of doul- ss udr coprssv l lodg. h vsgd h proprs of fr rsvrs vro d uclg of doul- ss udr coprssv l lodg o h ss of h Broull-Eulr hor. h ssud h h o s of h ss r spl suppord d couousl jod lr lsc lr. h drvd plc prssos for h url frqucs. h lso od h ssocd plud ros of h o s d h lcl soluo of h crcl uclg lod. h shod h h crcl uclg lod of h ss s rld o h l coprsso ro of h o s d h lr lsc lr d h proprs of fr rsvrs vro of h ss grl dpd o h l coprssos. h lso sudd ffc of coprssv l lod o forcd rsvrs vros of doul- ohr ppr Zhg l. 8. h ffcs of coprssv l lod o h forcd vros of h doul- ss r dscussd for o css of prculr co lodgs d h proprs of h forcd rsvrs vros of h ss r foud o sgfcl dpd o h coprssv l lod. ojovc d Kozc cosdrd forcd rsvrs vro d uclg of Rlgh d osho doul- ss couousl jod lr lsc lr udr coprssv l lodg. I hr ppr dflcos of h s sd o h osho hor r sho d grl soluos of forcd vros of s sujcd o rrrl dsru couous lods r foud. h lcl soluo of forcd vro h ssocd plud ros s drd. Also dc rsposs of h ss cusd rrrl dsrud couous lods r od. Vros cusd hroc cg forcs r dscussd d codos of rsoc d dc vro sorpo r foruld. Oszczu 3 horcll lzd udpd fr rsvrs vros of lscll cocd rcgulr pl-r ss h lr lsc lr -. oluos of h prol 585

2 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co r foruld usg h Nvr hod d url frqucs of h ss r drd h for of o f squcs. Also Norl od shps of vro prssg o ds of vro schroous d schroous r prsd. I urcl pl h ffc of r so o h url frqucs of hs d ss s dscussd. Forcd rsvrs vros of lscll cocd copl rcgulr spl suppord doulpl ss h lr lsc lr - s lso sudd Oszczu. Udpd oo of h ss cd rrrl dsrud couous lodgs ppld rsvrsl o oh pls s drvd sd o h Krchhoff-Lov pl hor. O h ss of grl soluos od hr prculr css of h co of cg sor hroc lods r cosdrd. h hvor of foudo rls grg prcc co rprsd foudo odl hch cosss of dpd lr lsc sprgs. I ordr o fd phscll clos d hcll spl foudo odl Psr proposd so-clld o-prr foudo odl h shr rcos. I p o fd phscll clos d hcll spl rprso of lsc foudo for hs rls Psr 95 proposd foudo odl cossg of lr foudo h shr rcos. hs ccoplshd cocg h ds of h vrcl sprgs o cossg of coprssl vrcl ls hch dfors ol rsvrs shr. I g l. 977 d D Ros 995 h url vro of osho o Psr-p foudo s sudd. Frquc quos r drvd for s h dffr d rsrs. A spcfc pl s gv o sho h ffcs of ror r shr dforo d foudo coss o h url frqucs of h. Fr rsvrs vros of lscll cocd rcgulr pl-r ss h Psr lr - r sudd h prs ppr. Fr rsvrs vros of hs d ss r sudd usg h Nvr hod.. FORMULAION OF HE PROBLEM Fgur- shos copl couous ss hch cosss of rcgulr hr-lrd srucur coposd of soropc pl d prlll r srchd uforl cos sos ppld h dgs d sprd h Psr lr. I s ssud h oh pls d r r h hoogous ufor d prfcl lsc. For h s of splc of vro lss s lso ssud h h pls s ll s h r r govrd spl suppord oudr codos. I h grl cs h ss s sujcd o rrrl dsrud rsvrs couous lods. Fgur-. h phscl odl of lscll cocd rcgulr pl-r copl ss h Psr lr -. Accordg o h Krchhoff-Lov pl hor rsvrs vros of lscll cocd rcgulr pl-r ss h Psr lr - r dscrd h follog dffrl quos: D N f f hr 3 D E h [ v ] h h suscrps d rfr o h pl d h r rspcvl. h oudr codos d l codos for h spl suppord pl d r r s follos: 586

3 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co d 3. OLUION OF HE FREE VIBRAION PROBLEM Fr vros of rcgulr pl-r ss s Fgur- r govrd h follog hoogous prl dffrl quos: N D 5 h ov quo ss h h oudr codos 3 c solvd h Nvr hod ssug soluos h follog for: s s s s 6 Also Y X Y X 3... s s s s 7 hr ssfs h corrspodg oudr codos 3 for h spl suppord pl d r s ll s h hoogous dffrl quos 5. Fgur-. h phscl odl of lscll cocd rcgulr pl-r ss h Psr lr - lzd for fr vros. usug soluos quos 6 o quos 5 gvs h follog prssos: ] [ ] [ N D 8 hr ] [. Fro hch s of ordr dffrl quos for h uo fucos s od 9 hr N D h soluos of quos 9 r s follos: D C Iroducg h o quos 9 rsuls ss of lgrc quos h uo coss of C d D : C D D C

4 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co For o-rvl soluos of h ov quos h dr of h ss coffc r s s qul o zro ldg h follog frquc quo: or [ D N ] { [ D N D N D N ] } 3 d hus h frquc quo hs o roos: { [ ] } h fucos c ssud o hv h follog fors C D C D C D 3 3 C D 5 or rgoorc for hr [ A s [ A s B B cos ] cos ] 6 D C D N A s B cos Fll h forcd vros of lscll cocd rcgulr pl r ss h Psr lr - c dscrd d s s [ A s B cos ] 8 Ad s s [ A s B cos ] 9 588

5 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co h uo coss A d B prssos 8 d 9 r clculd solvg h l-vlu prol. B ulplg h rlos 8 d 9 h g-fuco d usg orhogol codo l h grg h ldd s s d s s d c l l hr c cdd s d s d h susug soluos 8 d 9 o h l codos d h prforg h o usul rsforo procdur d pplg h orhogol codo h follog forul for vlug A d B r od: A B B A z z z z v v v v s s dd s s dd s s dd s s dd hr z z.5 c.5. NUMERICAL REUL AND DICUION I h follog h ffcs of shr foudo odulus of h Psr lr o h url frqucs of h ss r cosdrd. Vlus of h prrs chrcrzg proprs of h ss r sho l-. l-. Vlus of h prrs chrcrzg proprs of h ss. E h h 8 N 3 v N.3 5g g 5 3 g g 35 N N 3 Rsuls of h clculos for d r prsd ls- o 7 d Fgurs-3 o 8. I cs crsg N cuss crs. A vd fluc of shr foudo odulus of h Psr lr o h frqucs of h ss s osrvd d c s h h crsg h frqucs crs s ll. Hovr hs fluc of h r so d shr foudo odulus of h Psr lr s dffr o so prculr frqucs g srogr o frqucs hl r o h frqucs. If roduc h hol quos of hs ppr c o h vro quos of h pl-r ss h lr lsc lr - h rsuls of hch r vrfd coprg h hos vll Oszczu

6 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co l-. Nurl frqucs of h rcgulr pl-r ss s for. N l-3. Nurl frqucs of h rcgulr pl-r ss s for. N l-. Nurl frqucs of h rcgulr pl-r ss s for. N

7 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co l-5. Nurl frqucs of h rcgulr pl-r ss s for 3. 3 N l-6. Nurl frqucs of h rcgulr pl-r ss s for. N l-7. Nurl frqucs of h rcgulr pl-r ss s for 5. 5 N

8 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co Fgur-3. h url frqucs of h pl-r ss s fuco of r so N d. Fgur-5. h url frqucs of h pl-r ss s fuco of r so N d. Fgur-. h url frqucs of h pl-r ss s fuco of r so N d. Fgur-6. h url frqucs of h pl-r ss s fuco of r so N d

9 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co gr dc vro sorpo phoo. As s ll o vro sorpo c usd o supprss cssv forcd vro pluds. Fgur-7. h url frqucs of h pl-r ss s fuco of r so N d. 5. CONCLUDIN REMARK I hs sud fr rsvrs vros of lscll cocd rcgulr pl r ss h Psr lr - r horcll lzd. h vror ss odl cosdrd coprss hr-lrd srucur hch s coposd of h pl Psr lr - d prlll r srchd uforl sul cos sos ppld h dgs. h prol s solvd usg h Nvr hod. I should od h h url frqucs of h ss vrd chgg shr foudo odulus of h Psr lr hou h css o vr prrs chrcrzg phscl d gorcl proprs of h ss. hs possl s of gr prccl porc. Mguds of h frqucs co lrgr h h shr foudo odulus of h Psr crss d Psr lr c crs h guds of frqucs or h lr lsc lr c. hus ordr o supprss h cssv vros of corrspodg pl-r sss o c us Psr lr sd of lr lsc lr. h cosdrd ss hs rsg fur hch ls o chg ch url frquc h cr ld rvl s fuco of h r so d shr foudo odulus of h Psr lr ol. h propr corol of h r so d shr foudo odulus of h Psr lr s possl o vod rsoc phoo or o gr dc vro sorpo phoo for h ss sujcd o hroc lodgs. hs c hv sgfc pcs prccl pplcos. Noclur Fgur-8. h url frqucs of h pl-r ss s fuco of r so N d 5. I c s h h d ss dscussd hs rsg fur hch llos ch url frquc o chg s fuco of r so d shr foudo odulus of h Psr lr hls h ohr cosrucol d phscl prrs of h ss c r uchgd. lcg sul so of h r d shr foudo odulus of h Psr lr gvs dsrl vlus of h ss frqucs cr ld dos so h s possl for sc o vod rsoc pho or o f D E N rsvrs pl r dsplc Ecg dsrud lod pc co-ords Flurl rgd of h pl Youg s odulus of lsc of h pl Ufor cos so pr u lgh of h r lr foudo odulus hr foudo odulus 5853

10 VOL. NO. AUU 5 IN ARPN Jourl of Egrg d Appld ccs 6-5 As Rsrch Pulshg Nor ARPN. All rghs rsrvd..rpjourls.co v h l Pl r dsos Posso s ro Mss ds Uo fucos Ko od shp fucos Prl frquc of h ss Couplg frquc of h ss Nurl frquc of h ss Krocr dl fuco doul- ss. Irol Jourl of Mchcl ccs. 5: REFERENCE D Ros M.A Fr vros of osho s o o-prr lsc foudo Copurs rucurs. 57: Oszczu Z. 3. Fr rsvrs vros of lscll cocd rcgulr pl-r copl ss. Jourl of oud d Vro. 6: Oszczu Z.. Forcd rsvrs vros of lscll cocd copl rcgulr spl suppord doul-pl ss. Jourl of oud d Vro Psr P.L. 95. O hod of lss of lsc foudo s of o foudo coss. I Russ osudrsvo Izdlsvo Lrur Po rolsvu IArhur Mosco. ojovc V. Kozc P.. Forcd rsvrs vro of Rlgh d osho doul- ss h ffc of coprssv l lod. Irol Jourl of Mchcl ccs. 6: g.. phs J.E Nurl frqucs of osho s o Psr foudos. Jourl of oud d Vro. 5: Zhg Y.Q. Lu Y. g.l. Lu X. 8. Vro d uclg of doul- ss udr coprssv l lodg. Jourl of oud d Vro. 38: Zhg Y.Q. Lu Y. M.. 8. Effc of coprssv l lod o forcd rsvrs vros of 585