Spectral Finite Element Method for Free Vibration of Axially Moving Plates Based on First-Order Shear Deformation Theory

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1 Joural of Solid Mechaics Vol. 9, No. 3 (7) pp Spectral Fiite Elemet Method for Free Vibratio of Aiall Movig Plates Based o First-Order Shear Deformatio heor M.R. Bahrami, S. Hatami * Civil Egieerig Departmet, Yasouj Uiversit, Yasouj, Ira Received 3 Ma 7; accepted 6 Jul 7 ABSRAC I this paper, the free vibratio aalsis of moderatel thick rectagular plates aiall movig with costat velocit ad subjected to uiform i-plae loads is ivestigated b the spectral fiite elemet method. wo parallel edges of the plate are assumed to be simpl supported ad the remaiig edges have a arbitrar boudar coditios. Usig Hamilto s priciple, three equatios of motio for the plate are developed based o first-order shear deformatio theor. he equatios are trasformed from the time domai ito the frequec domai b assumig harmoic solutios. he, the frequec-depedet damic shape fuctios obtaied from the eact solutio of the goverig differetial equatios is used to develop the spectral stiffess matri. B solvig a o-stadard eigevalue problem, the atural frequecies ad the critical speeds of the movig plates are obtaied. he eactess ad validit of the results are verified b comparig them with the results i previous studies. B the developed method some eamples for vibratio of statioar ad movig moderatel thick plates with differet boudar coditios are preseted. he effects of some parameters such as the aiall speed of plate motio, the i-plae forces, aspect ratio ad legth to thickess ratio o the atural frequecies ad the critical speeds of the movig plate are ivestigated. hese results ca be used as a bechmark for comparig the accurac ad precisio of the other aaltical ad umerical methods.. All rights reserved. Kewords: First-order shear deformatio theor; Spectral fiite elemet method; rasverse vibratio; Aiall movig; Damic stiffess matri; Free vibratio. INRODUCION A XIALLY movig plates are widel used i idustr for variet applicatios. he badsaw blades, the steel strip i a thi steel sheet productio lie, the belt ad chai i power trasmissio, the high-speed magetic or paper tape, the movig wide bads ad belts, ad the coveor belts are some eamples of aiall movig plates. he aiall speed ma sigificatl affect the damic behavior of these structures. herefore, ivestigatig damic characteristics of movig plates is ecessar for the aalsis ad desig of these structures. * Correspodig author. el.: address: hatami@u.ac.ir (Sh.Hatami).. All rights reserved.

2 M.R. Bahrami ad S. Hatami 49 I geeral, methods for solvig damic problems of plates are divided ito two categories, amel approimate or umerical methods ad eact or aaltical methods. he fiite elemet method (FEM), fiite strip method (FSM) ad Raleigh-Ritz method are some of the approimate methods. Ulso ad Mote [] developed trasverse vibratio aalsis of wide badsaw blades b Raleigh-Ritz method. Legoc ad Mccallio [] studied cuttig coditios o the damic respose of wide badsaw blades usig of Galerki method. Based o Midli-Reisser plate theor, Wag [3] developed a FEM for a movig orthotropic thi plate. Luo ad Hutto [4] preseted formulatio of the movig triagular plate elemet. he free vibratio ad stabilit aalsis of aiall movig viscoelastic thi plate were studied usig of the differetial quadrature method b Zhou ad Wag [5]. ag ad Che [6] ivestigated oliear free trasverse vibratio of the aiall movig plates b multiple scales method. A ad Su [7] obtaied the damic respose of aiall movig orthotropic plates usig the geeralized itegral trasform techique. Eftekharia ad Jafari [8] proposed damic aalsis of the aiall movig orthotropic rectagular plates subjected to liearl varig i-plae stress b high order accurate mied fiite elemet-differetial quadrature method. O the other had, i the recet decade aaltical methods have bee used to etract the eact values for damic characteristics of the aiall movig plates. he closed-form solutio (CFS), eact fiite strip method (EFSM) ad damic stiffess method (DSM) are the eact aaltical methods that cosidered b ma researchers. Li [9] ivestigated the stabilit ad the vibratio of aiall movig plate subjected to uiform i-plae tesio i trasport directio usig of the closed form solutio. Equilibrium, membrae forces ad bucklig stabilit for thi plate aiall travelig with high-speed are studied b Luo ad Hamidzadeh []. he free vibratio of the aiall movig Lev-tpe viscoelastic thi plate was ivestigated b Marowski []. Hatami et al. developed a eact fiite strip method for aiall movig isotropic plates [], orthotropic plates o elastic foudatio [3], lamiated composite plates [4] ad viscoelastic plates [5] subjected to i-plae forces, based o classical plate theor (CP). B presetig a set of umerical eamples, the effect of speed of plate motio ad iteral supports o the free vibratio frequecies of the aiall movig plates were studied [-5]. he divergece velocities of aiall movig orthotropic thi plates are ivestigated aalticall via methods of comple aalsis b Saksa ad Jeroe [6]. Spectral Fiite Elemet Method (SFEM) is a combiatio of damic stiffess method, spectral aalsis method ad fiite elemet method. he SFEM provides damic resposes i frequec domai due to the use of damic shape fuctios. he first step i SFEM is trasformig the goverig differetial equatios from time domai ito frequec domai b assumig harmoic solutios. he, the frequec-depedet damic shape fuctios obtaied from the eact solutio of the goverig differetial equatios is used to develop the spectral stiffess matri. he SFEM was developed for oe-dimesioal structures icludig the aiall movig Berouli-Euler beam b Oh et al. [7], the aiall movig imosheko beam b Lee et al. [8] ad the aiall movig viscoelastic beam subjected to aial tesio b Lee ad Oh [9]. he SFEM also, was formulated for aiall movig thi plates with costat speed subjected to uiform i-plae aial tesio b Kim et al. [] ad aiall movig beam-plates subjected to sudde eteral thermal loads b Kwo ad Lee [] based o CP. Accordig to the kowledge of the authors, the damic resposes of moderatel thick rectagular plates with aiall movig speed usig eact aaltical methods have ot bee cosidered b a researcher, so far. I the preset paper, the free vibratio aalsis of moderatel thick rectagular plates aiall movig with costat velocit ad subjected to uiform i-plae loads is ivestigated b the spectral fiite elemet method. wo parallel edges of the plate are assumed to be simpl supported ad the remaiig edges have a arbitrar boudar coditios (simpl supported, clamped). he atural frequecies ad critical speeds of aiall movig plates are preseted for three combiatios of classical boudar coditios, amel CC, CS ad SS. he mode shapes are preseted for the trasverse displacemet of the plate. he effects of some parameters such as the i-plae forces, aspect ratio ad legth to thickess ratio o the atural frequecies ad the critical speeds of the movig plate are ivestigated. he obtaied damic resposes are compared with available aaltical or umerical solutios to cofirm the validit ad eactess of the results. GOVERNING EQUAIONS OF AXIALLY MOVING FSD PLAE Cosider a moderatel thick rectagular plate movig with costat velocit c i the -directio subjected to iplae forces. he plate has the legth L, width L ad uiform thickess h, as show i Fig.. he displacemet fields of plate i, ad z-directio are represeted b u(,,z,t), v(,,z,t) ad w(,,z,t), respectivel. Based o the first-order shear deformatio theor (FSD), the displacemet fields for moderatel thick rectagular plates, ma be epressed as:

3 49 Spectral Fiite Elemet Method for Free Vibratio of Aiall. u(,, z, t ) z (,, t ), v (,, z, t ) z (,, t ), w (,, z, t ) w (,, t ) () where ad are the rotatioal displacemets about the ad aes at the middle surface of the plate, respectivel, w is the trasverse displacemet ad t is the time variable. he strai-displacemet relatios for a moderatel thick rectagular plate based o Eqs. () are defied as follows []. w w z, z, zz, z ( ), z, z () Fig. Displacemet field, ad iteral forces ad momets for a aiall movig moderatel thick plate. he differetial equatios of motio ad the boudar coditios of the plate are derived b Hamilto s priciple, which is epressed i Eq. (3): t ( U V K ) dt (3) t w at t t, t where δu is the virtual strai eerg ad is defied as follows. h / U h / z z z z dz dd (4) Substitutig Eqs. () ito Eq. (4), the virtual strai eerg ca be epressed as follows. w w U M M M Q Q dd (5) where the momets ad trasverse forces resultig from iteral stress accordig to sig covetios show i Fig. are defied as follows. h/ h M M M z dz, Q Q k dz h/ h z z (6) where k is the shear correctio factor, which is obtaied b Reisser as 5/6 [3]. We itegrate b parts Eq. (5) to relieve the virtual geeralized displacemets ( w,, ) i Ω of a differetiatio. he the virtual strai eerg is give b:

4 M.R. Bahrami ad S. Hatami 493,,,,,, U M M Q M M Q w Q Q dd M M M M w Q Q ds (7) where the comma followed b or parameter deotes differetiatio with respect to the or, respectivel. Also ad are the compoets of the uit vector ormal to the correspodig edge i ad directios, respectivel. he virtual work doe b eteral forces δv is give as follows: V ˆ ˆ ˆ ˆ ˆ ˆ d Q w M M Q w M M d (8) where Qˆ, ˆ, ˆ, ˆ Q M M ad ˆ M are trasverse shear forces ad rotatioal momets per uit legth imposed o the boudaries =, L ad =, L. I the absece of N ad N, the work doe b the costat i-plae ormal force N (positive whe tesile) actig i the -directio is give as: w V N dd (9) where he virtual work due to Eq. (9) is rewritte as follows: V P w d N w dd () w w N N, P N () he kietic eerg of the aiall movig moderatel thick plate is epressed as: w w K I c c I c dd t t t () where I = ρh ad I = ρh 3 / ad ρ is the mass desit of plate material. he virtual kietic eerg due to Eq. () is obtaied as follows. w w K I w cw c I c c dd t t t (3) I cotiuatio, with respect to iitial coditios ad the itegratig b parts to relieve the virtual geeralized displacemets ( w,, ) i t of a differetiatio t t t t I w w cw c w I c c I dd K dt dt I w cw c w I c c ds (4)

5 494 Spectral Fiite Elemet Method for Free Vibratio of Aiall. Fiall, substitutig the virtual strai eerg δu, virtual work δv ad virtual kietic eerg δk give b Eqs. (7), (8 ad ) ad (4), respectivel, ito the Hamilto s priciple i Eq. (3) ad collectig the coefficietsw, ad, the goverig differetial equatios of motio are obtaied as: w : Q Q N I ( w cw c w ),, : M, M, Q I ( c c ) : M M Q I,, (5) ad the atural boudar coditios o edges as: w : Q Q P I ( cw c w ) Qˆ Qˆ : M M I ( c c ) M M : M M Mˆ Mˆ ˆ ˆ (6) Based o Hook s law, the relatios betwee stress-strai are defied as follows []: E E,, G, z G z, z G z (7) where G=E/(+ν) is the shear modulus, E is the Youg s modulus, ad ν is the Poisso s ratio of the plate. Now, substitutig Eqs. () ad (7) ito Eqs. (6), the resultat trasverse shear forces Q, Q, bedig momets M, M, ad twistig momet M, ca be writte with respect to the trasverse displacemet w, ad rotatioal displacemets about the ad aes ad, as follows. w M Q M D, kgh Q w M (8) where D=Eh 3 /(-ν ) is the plate stiffess. Fiall, substitutig Eqs. (8) ito Eqs. (5), the goverig differetial equatios of motio i terms of displacemets w ad ca be epressed as:, w w w w w w kgh N h c c t t 3 w h D kgh c c t t 3 w h D kgh t (9) 3 SPECRAL FINIE ELEMEN MEHOD 3. Goverig equatios i the frequec domai Fig. shows a Lev-tpe plate i which two opposite edges (= ad L ) are simpl supported ad the remaiig

6 M.R. Bahrami ad S. Hatami 495 edges (= ad L ) of the plate ca have a arbitrar boudar coditios. he plate behavior i the -directio, accordig to eact aaltical solutio method, ca be represeted i the form of a Fourier series. It should be oted that these fuctios must satisf the boudar coditios alog two parallel edges at = ad L. From ow o, we ol eed to formulate a -D problem (alog ais) i SFEM. o obtai frequec-depedet damic shape fuctio, it is ecessar first to trasform goverig partial differetial equatios of motio from the time domai ito the frequec domai b discrete Fourier trasform (DF). he the spectral stiffess matri usig damic shape fuctio b the force-displacemet relatio method is achieved. Based o Lev-tpe solutio assumptio ad DF theor, displacemets w ad are represeted i spectral form as follows., N w (,, t ) W m (, k m, )si( k m ) e N m N (,, t ) (, k, )si( k ) e m m m N m N (,, t ) (, k, )cos( k ) e m m m N m i t i t i t () where k m =mπ/l ad m is the umber of half-wavelegths i directio ais correspodig to terms of Fourier series ad i is the imagiar uit, N umber of samples i the time domai ad ω =π/ is th discrete frequec, where is samplig time widow. W m, Ф m ad Ф m are the spectral-modal compoets related to the displacemets. B substitutig Eqs. () ito Eqs. (8), the resultat shear forces ad momets ca be rewritte i spectral forms as: N M (,, t ) M (, k, )si( k ) e m m m N m N M (,, t ) M (, k, )si( k ) e m m m N m N M (,, t ) M (, k, )cos( k ) e m m m N m i t i t i t N Q (,, t ) Q (, k, )si( k m m m N m m m m N m ) e N Q (,, t ) Q (, k, )cos( k ) e i t i t () where M m, M m, M m, Q m ad Q m are the spectral-modal compoets give b: dm M m (, k m, ) D k m m d dw m Q (, k, ) kgh d d m M m (, k m, ) D k m m d 3 Gh d Q (, k, ) kgh m k W M m (, k m, ) k m m d m m m m m m m m () Substitutig Eqs. () ito Eqs. (9) ields a set of coupled ordiar differetial equatios i the frequec domai as: d W m dw m dm m m m m kgh N hc c hi h kghk W kgh kghk d d d hc d d d m c h i m h m dw m D Dk m kgh m Dk m kgh d 6 d d d d D d 3 m h dm Dk m kgh m Dk m kgh k mw m d (3)

7 496 Spectral Fiite Elemet Method for Free Vibratio of Aiall. Sice = o edges = ad =L, b substitutig Eqs. () ad () ito Eqs. (6), the boudar values o these edges ca be rewritte i spectral forms as: N ˆ Q (,, t ) Qˆ (, k, )si( k ) e m m m N m N ˆ M (,, t ) Mˆ (, k, )si( k ) e m m m N m N ˆ M (,, t ) Mˆ (, k, )cos( k ) e m m m N m i t i t i t (4) where Q ˆ, M ˆ ad Mˆ m are the spectral compoets ad the force-displacemet relatioships i the frequec m m domai ca be epressed as follows. Q Q N dw d h ci W c M M h d ci c d Mˆ M ˆ m m m m ( m ) 3 ˆ m m m ( m ) m m dw d (5) 3. Spectral elemet equatio 3.. Geeral solutio of spectral-modal displacemets he damic (frequec-depedet) shape fuctios are obtaied from geeral solutio of the goverig differetial equatios. Assume the geeral solutios of Eqs. (3) is as follows. k m k m k m W b e, b e, b e (6) m m m m m m where k m are waveumbers i directio ais ad b m the costat coefficiet. Substitutig Eqs. (6) ito Eqs. (3) a eigevalue problem is obtaied. 3 b 3 m (7) where kgh N hc k c hi k h kghk, kghk, kghk m m m m 3 m hc c h i h kgh k D k k Dk kgh Dk k m, m m m, 3 m m 3 h 3 kgh k m, 3 Dk m k m, 33 D k m Dk m kgh (8) he algebraic Eq. (7) has a otrivial solutio, which is obtaied b settig determiat of the coefficiets matri as zero. Hece, after simplificatio, it ields a si order polomial i terms of waveumbers. m k m k m k m k m k m k m (9) m m 3 m 4 m 5 m 6 m 7

8 M.R. Bahrami ad S. Hatami 497 where 3 m D ( ) Ghk N hc h c D 4 m hdci ( ) D h Ghk N hc 7 D ( ) 3Ghkk k N hc h h c Gk h Ghk N hc m3 G hk Gk N h c c h k h 44 6Dh m 4 m m 4 ( 3) m 4 h c hk m N ( 3) c N k m 4h 3h 3 44 D k m ( ) h Ghk N hc h Gk Dh Gk h k m h k m N hc h 4 hci 7 6 ( 3) ( ) 7D k m ( ) 3Ghkk m k m N hc h 4 4 h N 6 hc G k N h c h h Gk h 4N h 84c c h k m h m G hk k m h N k m ( 3) hc k m ( 3) h ( 3) 6Dh Gk h c k m 6 h ( 3) k m ( 3) 6N 6 hc h m m 6 7 m m G h k 4Dk h 3 Dk ( ) h hci 7 3 Ghk Dk m 8 h k m 6 h 36 h k m h Ghk 6 Dk ( ) h h Gk h k h D Gkk k m m m m (3) Eq. (9) is kow as the dispersio relatio or spectrum relatio. here is o geeral formula for computig the latet roots of a polomial of degree five or greater. hus, from a practical poit view, the approimate latet roots that are accurate eough for the desired precisio will be as useful as the eact oes [4]. he coefficiets α ad β correspodig to each waveumber obtaied from Eqs. (7) are as follows., (3) or Ghk h c k mi h ci k mi 6Dk mi 6Dk m Ghk h i D hc k mi D hci k mi DGhkk mi 6k mi G h k DGhkk m D h DN k mi G h k k mi k mi Ghk N hc hci k mi Ghkk m h k mi D h c h ci k mi Dk m Ghk h 6 i G h k k mk mi Dk mk mi k mi Ghk N hc hci k mi Ghkk m h (3)

9 498 Spectral Fiite Elemet Method for Free Vibratio of Aiall. Fiall, the geeral solutios of Eqs. (3) is preseted as follows. W E (, k, ) b, E (, k, ) b, E (, k, ) b (33) m m m p m m m m m m m m p m where E k e e e e e e k m k m k m 3 k m 4 k m 5 k m 6 m (, m, ) b b b b b b b m m m m 3 m 4 m 5 m 6 (34) ad diag[ ] diag, diag[ ] diag (35) p i p i Spectral-modal odal DOFs vector For a spectral fiite elemet show i Fig. (b), the geometric boudar coditios must be satisfied at the elemet odes ( = ad = ). he geometric boudar coditios iclude the trasverse displacemet ad rotatioal displacemets about the ad aes. he spectral-modal odal degrees of freedom (DOFs) for the aiall movig moderatel thick rectagular plate elemet are specified i Fig. (b). he spectral-modal odal DOFs vector d m ca be defied b applig the geometric boudar coditios, as follows. d W W m m m m m m m W ( ) ( ) ( ) W ( ) ( ) ( ) m m m m m m (36) where Substitutig Eqs. (33) ito Eq. (36) ields the relatioship as: d G ( k, ) b (37) m m m m G e e e e e e e e e e e e e e e k m k m km 6 6 k m k m km 6 k m k m km 6 6 m ( k m, ) k m k m km 6 e e 6e k m k m km 6 k m k m km (38) (a) dividig the plate to some elemets (b) spectral fiite elemet Fig. Sig covetios spectral-modal odal DOFs ad forces.

10 M.R. Bahrami ad S. Hatami Damic shape fuctios B elimiatig the costat vector b m from Eqs. (33) b usig Eq. (37), the geeral solutios of the goverig ordiar differetial equatios of motio i the frequec domai (relatios Eqs. (3)) i terms of the spectralmodal odal DOFs vector d m are epressed as follows. W N d, N d, N d (39) m Wm m m m m m m m where N, N ad Wm m N m are the damic (frequec-depedet) shape fuctios related to spectral-modal displacemets W m, Ф m ad Ф m, respectivel. he are obtaied from the eact solutio of the goverig differetial equatios ad are defied as follows. he damic shape fuctios are -b-6 matrices. N (, k, ) E G, N (, k, ) E G, N (, k, ) E G (4) Wm m m p m m m m m m m m p m 3..4 Spectral-modal odal forces vector Substitutig Eqs. (33) ito Eq s. () ad (5), the force-displacemet relatioships i the frequec domai ca be rewritte as follows. 6 Qˆ (, k, ) b kgh k N k h ci c k e m m mi i mi i mi i i mi i 6 3 ˆ h M m (, k m, ) bmi D k mi k mi ci c k mi e i 6 3 ˆ Gh M m (, k m, ) bmi k m i k mi e i kmi kmi kmi (4) For a spectral fiite elemet show i Fig. (b), the atural boudar coditios at the elemet odes ( = ad = ) must be satisfied. he atural boudar coditios i the frequec domai iclude the spectral-modal trasverse shear force Q ˆm, spectral-modal bedig momet M ˆ m ad spectral-modal twistig momet M ˆ m. he spectral-modal odal force ad momets for the moderatel thick rectagular plate elemet are show i Fig. (b). he spectral-modal odal forces vector f m ca be obtaied b applig the atural boudar coditios, ad give the sig covetios used i the theor of the plate (show i Fig. ), the are writte as follows. f Qˆ Mˆ Mˆ Qˆ Mˆ Mˆ m m m m m m m Qˆ ( ) Mˆ ( ) Mˆ ( ) Qˆ ( ) Mˆ ( ) Mˆ ( ) m m m m m m (4) Substitutig Eqs. (4) ito Eq. (4) ields the relatioship as: f R ( k, ) b (43) m m m m where R A e A e A e k m k m km 6 6 k m k m km 6 Be B e B 6e k m k m km 6 C e C e C 6e m ( k m, ) k m k m km 6 Ae Ae A6e k m k m km 6 Be B e B 6e k m k m km 6 Ce C e C 6e 66 (44)

11 5 Spectral Fiite Elemet Method for Free Vibratio of Aiall. ad A kgh k N k h ci c k i ( i mi ) i mi ( i i mi ) h B D k k ci c k i 3 Gh C i ( k m i k mi ) 3 i ( mi mi ) ( mi ),,,..., 6 (45) 3..5 Spectral stiffess matri Elimiatig the costat coefficiets vector b m from Eq. (43) usig Eq. (37) gives the relatioship betwee the spectral-modal odal forces vector f m ad the spectral-modal odal DOFs vector d m as follows. his relatioship is kow i the literature as spectral elemet equatio. where f m S md (46) m S ( k, ) R G (47) m m m m he matri S m is kow as eact damic (frequec-depedet) stiffess matri. I the literature it is ofte called spectral elemet matri or spectral stiffess matri. Also the matri S m is a si-b-si ad smmetric matri. he preset spectral stiffess matri is formulated for the aiall movig moderatel thick rectagular plate elemet based o FSD. 4 FREE VIBRAION ANALYSIS o aalze the vibratio of a plate b SFEM it is required to divide plate ito some elemets as show i Fig. (a). Followig that, the spectral-modal odal DOFs ad forces at each ode of the spectral fiite elemets could be defied. he spectral elemet equatio eplaied i Eq. (46) ca be assembled b usig a method similar to that used i the covetioal fiite elemet method. Fiall, global sstem equatio is give as follows. f S ( k, ) d (48) gm gm m gm where S gm is the global spectral stiffess matri, d gm is the global spectral-modal odal DOFs vector ad f gm is the global spectral-modal odal forces vector. he boudar coditios alog the edges parallel to the ais for simpl supported ad clamped cases ca be epressed b Eqs. (49) ad (5), respectivel. Simpl Supported (S) W m (, k m, ), M m (, k m, ), m (, k m, ) Q (, k, ), (, k, ), M (, k, ) (49) m m m m m m Clamped (C) W m (, k m, ), m (, k m, ), m (, k m, ) Q (, k, ), M (, k, ), M (, k, ) (5) m m m m m m Fig. 3 shows three differet combiatios of boudar coditios epressed i Eqs. (49) ad (5), amel CC, CS, ad SS. he boudar coditios of two parallel edges at = ad L that are simpl supported are ot writte for brevit.

12 M.R. Bahrami ad S. Hatami 5 Fig.3 hree combiatios of boudar coditios for aiall movig moderatel thick rectagular plate. After applig the associated boudar coditios o the assembled set of equatios, the fial form of the global sstem equatio is obtaied. f S ( k, ) d (5) gm gm m gm B settig f gm to zero, the eigevalue problem for free vibratio of the iteded plate ca be obtaied. S ( k, ) d (5) gm m gm B settig the determiat of S gm to zero, the atural frequecies, ω NA, are determied. det S ( k, ) (53) gm m Eq. (5) is a trascedetal eigevalue problem. herefore, the traditioal procedures for solvig the liear eigevalue problem are ot applicable. I earlier publicatios, various methods icludig Wittrick-Williams algorithm ad method of trial ad error to obtai the atural frequecies of Eq. (53) were suggested. I the preset paper a efficiet umerical algorithm usig drawig method described i Ref. [4] is used. I this procedure, the variatio of stiffess matri determiat det( S gm ) i logarithmic scale versus discrete frequecies ω is plotted for differet values of half-wavelegths, m. he poits o the horizotal ais i which the logarithmic fuctio teds to egative ifiit are atural frequecies. Substitutig these atural frequecies ito Eq. (5), correspodig mode shapes ca be computed. 5 NUMERICAL RESULS AND DISCUSSION For programig the formulatios of the preset SFEM ad etractig the free vibratio frequecies, critical speeds ad mode shapes mathematica software is used. Differet i-plae ormal forces, boudar coditios, aspect ratios, legth to thickess ratio of plate ad also differet aial speed are cosidered i the results preseted i this article. he dimesioless variables used i the results are itroduced as: L L N L h / D, c h / D, k D (54) where ad ψ are dimesioless atural frequec ad, travelig speed respectivel, ad k is i-plae load parameter alog -directio. he parameter m i the results implies the umber of half-wavelegths alog the - directio ad the parameter is the root umber i the characteristic fuctio of Eq. (53). It should be metioed that there is a ifiite umber of roots for this fuctio. 5. Validit he accurac ad ecellet performace of the SFEM agaist the results available i the literature are ivestigated. At first, eact frequecies of free trasverse vibratio of Lev-tpe rectagular plates without aiall speed (c=)

13 5 Spectral Fiite Elemet Method for Free Vibratio of Aiall. are calculated based o FSD with various boudar coditios. For this purpose, a isotropic square plate (L =L ) with legth to thickess ratio of (L /h=) is cosidered ad the shear correctio factor ad Poisso s ratio are assumed as 5/6 ad.3, respectivel. Also, plate ot subjected a i-plae forces (N =). Fig. 4 shows the variatio of Log (Abs (Det[ Sgm ( k m, ) ])) with respect to frequec parameter for differet values of halfwavelegths alog -directio, m. As metioed i Sec. 4, the poits o the horizotal ais i which the logarithmic fuctio teds to egative ifiit are atural frequecies. As show i the Fig. 4(a), for the case boudar coditio, CC of the first ad the third frequec parameter correspods to the first mode i directio ais (m=), the secod ad the fourth frequec parameter is associated with secod mode (m=) ad fifth frequec parameter of the third mode (m=3). (a) CC (b) SS Fig.4 Natural frequecies etractio of statioar Lev-tpe square plate for boudar coditios CC ad SS. I able. the first ie frequec parameters are compared with the results of eact closed form solutio ad damic stiffess method offered b Hosseii-Hashemi et al. [5], ad Boscolo ad Baerjee [6], respectivel, based o FSD for differet boudar coditios (CC, CS ad SS). I able frequec parameters are also compared with the results preseted b Liew et al. [7] that obtaied from a two dimesioal Raleigh-Ritz method. I this approach, the shear correctio factor is 5/6 for free vibratio aalsis of the moderatel thick plates. Maimum of two fiite elemets i -directio are used to obtai the results of spectral fiite elemet formulatio. As it could be observed, proposed SFEM with a miimum umber of fiite elemets produces eact results, compared with other eact aaltical methods available ad umerical methods. he dimesioless frequec parameter obtaied from the SFEM ad eact aaltical methods are i ecellet agreemet. herefore, this procedure could effectivel be used for the purpose of modelig plate structures with more tha oe fiite elemet (for eample, multi-spa plates, stepped thickess plates ad plates with iteral support). he damic stabilit of movig Lev-tpe rectagular plate subjected to uiform i-plae loads for differet boudar coditios (CC, CS ad SS) was studied. able. shows the dimesioless critical speed ψ cr of the movig Lev-tpe rectagular plate with differet aspect ratios L /L ad differet uiform i-plae load parameter k, obtaied b the spectral fiite elemet method based o FSD. For cofidece, these results are compared with the results reported i Ref. [] accordig to CP. Also, dimesioless fudametal frequecies of these plates for boudar coditios CC ad SS is preseted i able 3. he dimesioless fudametal frequecies of Levtpe rectagular plate subjected to uiform i-plae loads for two cases (statioar ad movig situatios) are ivestigated. he aiall speed of the movig plate is cosidered half of their critical speeds which are listed i able. he results are compared with those obtaied i Ref [] based o the classical plate theor.

14 M.R. Bahrami ad S. Hatami 53 able Compariso of the dimesioless atural frequecies a Lev-tpe square plate whe L /h=, ν=.3, k =5/6, N =, c=. B.C. Method Mode sequeces CC Preset SFEM (m,).353 (,).4889 (,).9996 (,) (,) (3,) (,3) (3,) (,3) (4,) CFS [5] DSM [6] Raleigh-Ritz Method [7] CS Preset SFEM (m,).34 (,).3863 (,).6493 (,) (,) (3,) 4.79 (,3) (3,) (,3) (4,) CFS [5] DSM [6] Raleigh-Ritz SS Method [7] Preset SFEM (m,) (,) (,) (,) (,) (,3) (3,) (,3) (3,) (,4) CFS [5] DSM [6] Raleigh-Ritz Method [7] able Compariso of the dimesioless critical speed ψ cr a travelig Lev-tpe rectagular plate subjected to uiform i-plae loads whe L /h=5, ν=.3, k =5/6. B.C. L /L k = k =4 Preset SFEM, FSD EFSM, CP [] Preset SFEM, FSD EFSM, CP [] CC / / CS / / SS / / able 3 Compariso of the dimesioless fudametal frequecies a statioar ad movig rectagular plate subjected to uiform iplae loads whe L /h=5, ν=.3, k =5/6. B.C. L /L ψ/ψ cr k = k =4 Preset SFEM, FSD EFSM, CP [] Preset SFEM, FSD EFSM, CP [] CC / / SS / /

15 54 Spectral Fiite Elemet Method for Free Vibratio of Aiall. 5. Eamples I Sec. 5., b the compariso of the atural frequecies of statioar plate obtaied usig the preset method ad the CFS, the accurac of the damic stiffess matri ca be esured. As well as, b the compariso of the atural frequecies ad critical speed of movig plate obtaied usig the preset method ad EFSM, we ca esure the accurac of the spectral fiite elemet formulatio to obtai the damic resposes. As a result, the preset method ca be used to obtai the damic resposes of the aiall movig moderatel thick rectagular plates. I this sectio several eamples are represeted to show the abilit of the proposed SFEM for solvig various problems of travelig FSD plates. Cosider a aiall movig moderatel thick square plate of Lev-tpe with boudar coditio CC. he geometric ad mechaical properties of the plate are L /h=5, ν=.3, k =5/6. I this eample, four values of,, 4 ad 6 for the o-dimesioal uiform i-plae tesio are selected. he o-dimesioal fudametal frequec is preset as a fuctio of the o-dimesioal aial speed ψ ad the uiform i-plae load parameter k i Fig. 5. he two spectral fiite elemets i -directio are used to obtai the eact results b the proposed spectral fiite elemet formulatio. As show i the Fig. 5, the atural frequec decreases with icreasig aial movig speed till the speed reach its critical value. he atural fudametal frequec i a costat speed, icreases with icreasig i-plae tesio. I additio, Fig. 5 also idicates that whe the uiform i-plae tesio icreases, the critical speed icreases as well. Fig.5 he effect of trasport speed o the fudametal frequec of a moderatel thick square plate subjected to uiform iplae load with boudar coditio CC (L /h=5, ν=.3, k =5/6). he first five dimesioless atural frequecies of the Lev-tpe square plate with boudar coditio SS for statioar ad movig situatios are show i able 4. he oe spectral fiite elemets i -directio are used to obtai the eact results. For assurace of the accurac of the results, the eact results for the statioar plate (c=) is compared with the results reported i Ref. [5]. As show i the able 4.,the atural frequec icreases whe iplae tesio icreases. I aother eample, a isotropic rectagular plate with legth to thickess ratio of 5 (L /h=5) is cosidered ad the shear correctio factor ad Poisso s ratio are assumed as 5/6 ad.3, respectivel. he results are etracted for three tpes of boudar coditios as CC, CS ad SS. he dimesioless aial speed ψ ad the dimesioless uiform i-plae tesio k are cosidered i the calculatio as costat values ad 5, respectivel. he variatio of the dimesioless fudametal frequecies agaist aspect ratio L /L for a aiall movig plate subjected uiform i-plae tesio is give i Fig. 6. he larger aspect ratio leads to the smaller the atural frequecies for a fied aial speed ad the uiform i-plae tesio. Fig.6 Variatio of the dimesioless fudametal frequec agaist the aspect ratio for a movig rectagular plate subjected to uiform i-plae load (L /h=5, ν=.3, k =5/6, k =5, ψ=).

16 M.R. Bahrami ad S. Hatami 55 able 4 he effects of legth to thickess ratio (L /h) o the dimesioless atural frequecies a Lev-tpe square plate with boudar coditio SS ad whe ν=.3, k =5/6. Dimesioless aial speed ψ Dimesioless aial tesio k L /h Method Dimesioless atural frequec Preset SFEM CFS [5] Preset SFEM CFS [5] Preset SFEM *.5 ψ cr 4 Preset SFEM L =5h: ψ cr =.84, L =h: ψ cr =.77, L =5h: ψ cr =.6368 o draw mode shapes of the moderatel thick plate, use is made of damic shape fuctio ( N N m ) correspodig to spectral-modal displacemets (W m, Ф m ad Ф m ), as preseted i Sec he mode shapes for the first mode (m=, =) ad the secod mode (m=, =) are plotted. I Fig. 7, mode shapes of the square plate for boudar coditio CC are show. Figs. 7(a)-7(c) show mode shapes for the trasverse displacemet of the statioar plate (k =, ψ=), the statioar plate subjected uiform i-plae loads (k =, ψ=) ad the aiall movig plate (k =, ψ=), respectivel. It should be oted that accordig to Eqs. (), variatio of W m ad Ф m alog the ais is i sie form ad variatio of Ф m is i cosie form. As show i Figs. 7(c) the mode shapes for the movig plate are the comple values. he Lev-tpe rectagular plate with width L ad thickess h equal to ad. m, respectivel, are cosidered ad the shear correctio factor ad Poisso s ratio are assumed as 5/6 ad.3, respectivel. he uiform i-plae tesio parameter k are cosidered i the calculatio as costat value 5. able 5. shows the dimesioless critical speed ψ cr of the aiall movig Lev-tpe rectagular plate subjected uiform i-plae tesio with differet aspect ratios L /L ad boudar coditios (CC, CS ad SS), obtaied b the spectral fiite elemet method based o FSD. he larger aspect ratio (L /L ) leads to the smaller the critical speed for a fied the uiform i-plae tesio. It ca also be see that the plates with boudar coditios CC ad SS have the highest ad lowest critical speed, respectivel. able 5 he first dimesioless critical speed ψ cr a travelig Lev-tpe rectagular plate subjected to uiform i-plae loads whe L = m, h=. m, ν=.3, k =5/6, k =5. B.C. Aspect ratio (L /L ) CC CS SS Wm, N m ad mode mode (a) statioar plate; k =, ψ=

17 56 Spectral Fiite Elemet Method for Free Vibratio of Aiall. mode mode (b) statioar plate subjected uiform i-plae loads; k =, ψ= mode mode Re [W m ] Im [W m ] mode mode Re [W m ] Im [W m ] (c) aiall movig plate; k =, ψ= Fig.7 First two mode shapes of the statioar ad movig Lev-tpe square plate with boudar coditios CC (L /h=, h=. m, ν=.3, k =5/6). 6 CONCLUSIONS he trasverse free vibratio of the Lev-tpe moderatel thick rectagular plates aiall movig with costat velocit ad subjected to uiform i-plae loads was studied b the spectral fiite elemet method based o firstorder shear deformatio theor. he spectral stiffess matri obtaied i the preset SFEM is a trascedetal fuctio of atural frequecies, aial speed, i-plae loads ad half-wavelegths. Sice the frequec-depedet damic shape fuctios i SFEM are obtaied from the eact solutio of the goverig differetial equatios of motio of the plate, icreasig the umber of spectral elemet to achieve the eact aswers is ot required. he results obtaied b the spectral fiite elemet formulatio eed maimum of two fiite elemets i -directio. As is oticeable, the proposed SFEM achieves eact results with miimal fiite elemets, compared with eact aaltical methods ad umerical methods. herefore, it is possible to effectivel use this procedure for the plate structures ad other plates i which more tha oe fiite elemet is eeded for modelig purposes.

18 M.R. Bahrami ad S. Hatami 57 he eactess ad validit of the results is verified b comparig them with the results i the other studies. B the developed method some eamples for vibratio of statioar ad movig moderatel thick plates with differet boudar coditios are preseted. he effects of some parameters such as the aiall speed of plate motio, the iplae forces, aspect ratio ad legth to thickess ratio o the atural frequecies ad the critical speeds of the movig plate are ivestigated. REFERENCES [] Ulso A.G., Mote C.D.Jr., 98, Vibratio of wide bad saw blades, Joural of Egieerig for Idustr 4: [] Legoc L., McCallio H., 995, Wide badsaw blade uder cuttig coditios, part I: vibratio of a plate movig i its plae while subjected to tagetial edge loadig, Joural of Soud ad Vibratio 86(): 5-4. [3] Wag X., 999, Numerical aalsis of movig orthotropic thi plates, Computers ad Structures 7: [4] Luo Z., Hutto S.G.,, Formulatio of a three-ode travelig triagular plate elemet subjected to groscopic ad iplae forces, Computers ad Structures 8: [5] Zhou Y.F., Wag Z.M., 8, Vibratio of aiall movig viscoelastic plate with parabolicall varig thickess, Joural of Soud ad Vibratio 36: 98-. [6] ag Y.Q., Che L.Q.,, Noliear free trasverse vibratios of i-plae movig plates: without ad with iteral resoaces, Joural of Soud ad Vibratio 33: -6. [7] A C., Su J., 4, Damic aalsis of aiall movig orthotropic plates: Itegral trasform solutio, Applied Mathematics ad Computatio 8: [8] Eftekhari S.A., Jafari A.A., 4, High accurac mied fiite elemet-differetial quadrature method for free vibratio of aiall movig orthotropic plates loaded b liearl varig i-plae stresses, rasactios B: Mechaical Egieerig (6): [9] Li C.C., 997, Stabilit ad vibratio characteristics of aiall movig plates, Iteratioal Joural of Solids ad Structures 34(4): [] Luo A.C.J., Hamidzadeh H.R., 4, Equilibrium ad bucklig stabilit for aiall travelig plates, Commuicatios i Noliear Sciece ad Numerical Simulatio 9: [] Marowski K.,, Free vibratio aalsis of the aiall movig Lev-tpe viscoelastic plate, Europea Joural of Mechaics A/Solids 9: [] Hatami S., Azhari M., Saadatpour M.M., 6, Eact ad semi-eact fiite strip for vibratio ad damic stabilit of travelig plates with itermediate supports, Advaces i Structural Egieerig 9(5): [3] Hatami S., Azhari M., Saadatpour M.M., 6, Stabilit ad vibratio of elasticall supported, aiall movig orthotropic plates, rasactio B, Egieerig 3(B4): [4] Hatami S., Azhari M., Saadatpour M.M., 7, Free vibratio of movig lamiated composite plates, Composite Structures 8: [5] Hatami S., Roagh H.R., Azhari M., 8, Eact free vibratio aalsis of aiall movig viscoelastic plates, Computers ad Structures 86: [6] Saksa., Jeroe J., 5, Estimates for divergece velocities of aiall movig orthotropic thi plates, Mechaics Based Desig of Structures ad Machies: A Iteratioal Joural 43(3): [7] Oh H., Lee U., Park D.H., 4, Damic of a aiall movig Berouli-Euler beam: spectral elemet modelig ad aalsis, KSME Iteratioal Joural 8(3): [8] Lee U., Kim J., Oh H., 4, Spectral aalsis for the trasverse vibratio of a aiall movig imosheko beam, Joural of Soud ad Vibratio 7: [9] Lee U., Oh H., 5, Damics of a aiall movig viscoelastic beam subject to aial tesio, Iteratioal Joural of Solids ad Structures 4: [] Kim J., Cho J., Lee U., Park S., 3, Modal spectral elemet formulatio for aiall movig plates subjected to iplae aial tesio, Computers ad Structures 8: -. [] Kwo K., Lee U., 6, Spectral elemet modelig ad aalsis of a aiall movig thermoelastic beam-plate, Joural of Mechaics of Materials ad Structures (): [] Redd J.N., 7, heor ad Aalsis of Elastic Plates ad Shells, CRC Press, Boca Rato. [3] Reisser E., 994, O the theor of bedig of elastic plates, Joural Math Phsics 3(4): [4] Lee U., 9, Spectral Elemet Method i Structural Damics, Joh Wile & Sos, Ic. [5] Hosseii Hashemi Sh., Arsajai M., 5, Eact characteristic equatios for some of classical boudar coditios of vibratig moderatel thick rectagular plates, Iteratioal Joural of Solids ad Structures 4: [6] Boscolo M., Baerjee J.R.,, Damic stiffess elemets ad their applicatios for plates usig first order shear deformatio theor, Computers ad Structures 89: [7] Liew K.M., Xiag Y., Kitiporchai S., 993, rasverse vibratio of thick rectagular plates-i comprehesive sets of boudar coditios, Computers ad Structures 49(): -9.