A Study on Nonlinear Forced Vibration of an Axial Moving Viscoelasticity Beam using the Multi-Scale Approach

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1 BAO-FU KOU t al: A STUDY ON NONLINEAR FORCED VIBRATION OF AN AXIAL MOVING A Study on Nonlnar Forcd Vbraton of an Axal Movng Vscolastcty Bam usng th Mult-Scal Approach Bao-Fu Kou *, Xao-L Hu Collg of Mchancal Engnrng, Tayuan unvrsty of scnc and tchnology, Tayuan, 3, Chna *Corrspondng author:koubaofu@63.com Abstract Th mult-scal mthod s usd to analys th stady-stat rspons ampltud of a nonlnar forcd vbraton of axal movng vscolastcty bam, and th Galrkn truncaton mthod s appld to solv ts numrcal dffrntal quaton by dscrtzaton. Th nfluncs on th stady-stat rspons ampltud du to varous factors such as: ) vard vscosty dampng coffcnt, ) xtrnal xctaton, ) nonlnar coffcnts and v) movng vlocty ar analysd. Th rsults agr wth thos n th ltratur,.. th xtrnal motvaton and vscous dampng coffcnt hav grat ffct on th dynamcs bhavour of th vbraton systm. Th rsults ar thn drvd for dffrnt ordr Galrkn truncatons and compard, and th nstablty boundary condtons ar dtrmnd. Th rsults show that th unstabl condton s manly nducd by th xtrnal xctaton and vscosty dampng, whch agrs wll wth th rsults n th ltratur. Our study provds rlabl thortcal bass for th opraton of flxbl dvcs wth scurty and stablty. Kywords: Axal movng bam, mult-scal mthod, Galrkn truncaton, stady-stat rspons. I. INTRODUCTION Axally movng vscolastc bam, as a smplfd physcal modl s wdly usd n a numbr of systms, such as host, powr transmsson blts, band saws, aral ropway ct.. Undr th acton of xtrnal xctatons, ths systms may produc unwantd vbratons and rsonancs, causng consdrabl damag n th normal opraton. Th vbraton s rlatd to th systm s flxbl structur, whch can b approxmatvly rgardd as th bam wth vscolastc proprts. Thrfor, studs on th nonlnar forcd vbraton of axal movng vscolastc bam has caught many scholars ys [-5]. Th govrnng quaton for th longtudnal vbratons of vscolastc movng body s a typcal contnuous systm, contanng mxd partal drvatv trms of both tm and spac [6]. Aftr th rsarch on transvrs vbraton of axally movng lastc bam by Pasn [7], a numbr of scholars dvotd thmslvs to focusng on vbraton problms. Among th studs rvwd, most vbraton problms ar smplfd basd on th Krchhoff thory. Oz [8] dscussd th plan vbraton of an lastomr n vscous mdum and obtand th rlatonshp btwn fr vbraton frquncy and dampng. Lu [9] stablshd th dynamc quaton of vbraton wth rod cntrln as any plan curv and dscussd th plan vbraton problms of lastc rod wth crcular cross scton n vscous mdum. Du to th lmtaton n statcs catgory, th Krchhoff thory s not suffcnt to calculat proprly whn solvng th contnuum movmnt problms. Thn, a srs of mthods such as th dffrntal quadratur, fnt dffrnc wr dvlopd and tstd, whch mak th nonlnar rsponss of axally movng bam undr paramtrc xctatons b acquantd by us. But thr s stll dsagrmnt btwn axal tnson of movng bam and spd pulsaton []. Galrkn truncaton mthod was rcntly usd to study th stablty and dynamcs of axally movng bam wth smpl support. Its basc prncpl s to ovrlap th fnt polynomal functon and gt a srs of algbrac quatons sutabl for crtan boundary condtons. Ghaysh [] dscussd th branch and chaotc charactrstcs of axal movng body undr th longtudnal forcd vbraton and plan nonlnar vbraton whch coupld th radal and transvrs drcton. Dng t. al [] studd th chaotc dynamc bhavor of vscolastc bam wth paramtrc xctaton whch s causd by xtrnal harmonc xctaton and spd pulsaton, and obsrvd th doublng bfurcaton phnomnon of axally movng bam on dsturbanc vlocty ampltud by Poncar mappng. Th mult-scal approach (MSA) s appld to dtrmn th ampltud-frquncy charactrstcs and stady-stat rspons of nonlnar forcd vbraton whn approxmatly solvng th axally movng contnuum ssu n ordr to kp hgh accuracy and low computatonal cost. Th vbraton s a mult-scal coupld phnomnon and th man faturs of th objctv world complxty, namly, cohrnt structur and random fluctuatons appar smultanously or altrnatly and coupld wth ach othr, whch dtrmns that t cannot b xpland from a sngl pont of vw [3]. Currntly, many scholars nvstgatd th vbraton problms of movng body from th angl of mult-scal. Chn [] analyzd th stabl boundary of axal shft vscolastc bam undr clampd condtons through th avrag approach and got th stabl boundars of clampd bam by usng th mult-scal approach. Yao [5] studd th nonlnar dynamc bhavor of axal movng bam wth paramtrc xctaton undr mult-puls xctaton by combnng th multpl tm scals mthod and Galrkn truncaton mthod. Pakdmrl [6] appld th mult-scal approach to th partal dffrntal quatons and studd th stablty of acclratd bam whn th stffnss s small. DOI.53/IJSSST.a ISSN: 73-8x onln, prnt

2 BAO-FU KOU t al: A STUDY ON NONLINEAR FORCED VIBRATION OF AN AXIAL MOVING In ths papr, th mult-scal approach s appld to solv govrnng quatons and th dffrnt truncaton Galrkn mthod s usd to study th nonlnar forcd vbraton of axal movng vscolastc bam. Th nfluncs of vscous dampng, xtrnal ncntvs, nonlnar coffcnt and movng vlocty on stady-stat rspons ampltud ar studd and abundant ampltud rspons rsult ar obtand, manwhl, th boundary condtons whn nstablty s occurrd ar also analyzd. II. GOVERNING EQUATION Klvn modl has bn wdly usd n th axal movng bam. Aftr Mocknsturm t. al [7] dmd that th physcal tm should b takn nto consdraton to xprss th nrgy dsspaton durng stady movmnt, Chn t. al [8] appld th physcal tm drvatv n vscolastc proprts of axally movng bam. Thr was a smply clampd axally movng bam wth unform dnsty ρ and cross-sctonal ara A. And th lngth btwn supports L, ntal tnson P, axal vlocty v, xctaton ampltud F. Th modl was shown n Fg., n whch cross-sctonal ara momnt of nrtal I, lastc rato s E and vscosty coffcnt s η. Th control quaton can b show as Equaton () whn consdrng th latral vbraton n th plan only, whr th dsplacmnt X and forc F s th functon of axal coordnat X and tm T. 5 U U dvu U U U U ( v v ) P EI I F () T X T dtx X X X T X Boundary condtons of smply supportd bam at both nds: U (, T ), U ; X U ( L, T ), U () X X X L Brng n th dmnsonlss varabls and paramtrs: U u, L X x, L P t T, AL v A, P EI, P L I k, FL f, (3) 3 L AP P o whr s an small dmnsonlss paramtr. Th dmnsonlss form of Equaton () and () can b show as Equaton () and (5). 5 u u d u u u u ( ) k f () t x t dt x x x x t u (, t), u,, u ( l, t) u x x x xl (5) Fg. Schmatc of axally movng bams. III. MULTI-SCALE APPROACH Th mult-scal approach can b usd n solvng th nonlnar partal dffrntal quaton problm approxmatly. Undr th smply supportd boundary condtons, th mult-scal approach s mployd to solv th partal dffrntal quatons (), wth th tm scal as T=εt. Th quaton can b wrttn by. u x, t; ) u ( x, T, T ) u ( x, T, ) (6) ( T whr u and u rprsnt th fast scal and slow scal dsplacmnt functon rspctvly. T =t and T =εt ar sparatly th fast tm scal and slow tm scal. Insrtng Equaton () nto Equaton (6) and sparatng th dffrnt ordrs, w gt, (7) t T T Whn Equaton (7) s substtutd n Equaton (), on obtans Equaton (8) and (9) by u and u. 5 u u d u u u u ( ) k f (8) T x T dt x x x x T u T u d u u u u u Th soluton of Equaton (8) s gvn by Wckrt and Mot [9] : u 5 ( ) k ( ) (9) x T dt x x x x T T T T T ( x) A ( T ) ( x) A ( T ) () whr ω and ϕ rprsnt th -th natural frquncy and th corrspondng complx modal functon of lnar homognous systm rspctvly. Th ϕ (x) that mt th condton of both-clampd bam can b xprssd as DOI.53/IJSSST.a ISSN: 73-8x onln, prnt

3 BAO-FU KOU t al: A STUDY ON NONLINEAR FORCED VIBRATION OF AN AXIAL MOVING ( x) ( ( r x ( r r3 r r )( r3 r r )( r3 r ( r r )( ) r x ( r r n )( r3 r r )( ) ( r r3 )( r r r ) ( r r )( ) r r r3 ) ( r r3 )( ) r r r x r ) r3 ) r3 x () whr r m (m =,, 3, ) s th soluton of th followng algbrac quaton: k r f m ( ) r r, () m Th -th vbraton modl s consstnt wth th nonlnar frquncy wthout consdrng th ntrnal rsonanc. Put Equaton () nto Equaton (9), w gt, u u u u da ( ) ( ) T T x x x dt 3 k A A ( ) T m cc NST T (3) whr cc rprsnts th complx conjugat of all trms abov and NST mans non-constant trm. Equaton (3) may hav th boundd soluton unlss t satsfy th followng condton. da 3 T T ( ) f k AA ( ) dt () () (5) Ak ( w ), whr th nnr product dfnton of complx functons g and h s ntroduc n th rang of [, ]. g, h ghdx, (5) Apply th nnr product dstrbuton rat to quaton () and th soluton of th quaton can b xprssd as, da k A A (6) dt whr k s dfnd by Equaton (7) and Equaton(8) as, 3 k ( A ' dx dx k dx dx) ' A s gvn by (7) (8) of th nonlnar fr vbraton sparatly. And th ral and magnary parts of A can b shown as: a f Im( )sn( ) -R( )cos( ) R( ) a ( an, n) 3 (9) f Im( )cos( ) R( )sn( a Im( k ) a ( an, n) n 3 a T whr σ s th harmonc paramtr. In ordr to analyz th ampltud rspons nar th rsonanc pont, th rlatonshp btwn dsturbanc frquncy ω and natural frquncy ω can b wrttn as Equaton (). And th rlatonshp btwn ampltud rspons and harmonc paramtr s obtand by lmnatng th phas, as shown Equaton (). / 3 ) () 3 3 R( ) a Im( k ) a a Im( k ) a f () In ordr to asly calculat, th Galrkn truncaton mthod s adoptd to convrt th dffrntal quaton of contnuum systm nto th ordnary dffrntal quaton wth fnt dmnson and th dynamc charactrstcs of axally acclratng vscolastc bam s analyzd numrcally, whr th dsplacmnt varabls can b rprsntd as, N u( x, t) q ( t) ( x) () N F( x) f ( x), (3) Th dscrtzaton quatons wll constran quckly wth th ncras of, whr q stands for th gnralzd dsplacmnt functon and f mans th -th forcd vbraton ampltud whch s a constant valu. Insrtng Equaton () and (3) nto Equaton (3), multply by th corrspondng charactrstc functon and convrt nto th dscrt functon [, ], whch can xprss as matrx: Iq Cq Kq Mq N () n whr a and rprsnt th ampltud and phas angl whr I C K M and N rprsnt th mass matrx th dampng matrx th stffnss matrx th xtrnal stffnss DOI.53/IJSSST.a ISSN: 73-8x onln, prnt

4 BAO-FU KOU t al: A STUDY ON NONLINEAR FORCED VIBRATION OF AN AXIAL MOVING matrx and th xtrnal xctaton matrx sparatly. q q t) q ( t) q ( t), ( x) ( x) ( ) (5) ( x I T dx (6) T () T C dx k dx (7) T () T () K ( ) dx dx k dx (8) 3 T () T T M dx k d k dx (9) A. Modl Valdaton IV. RESULT ANALYSIS Fg. shows th chang of latral vbraton dsplacmnt n th cntr of axally movng vscolastc bam wth tm undr th systm paramtrs st as vscous dampng coffcnt k=.5, xtrnal xctaton f=.8, movng vlocty γ=.5, and nonlnar coffcnt v=. It s obsrvd that th vbraton dsplacmnt s stabl whn loadng prodcally. Howvr, th prodcal changng of dsplacmnt s not th ky to masur th vbraton charactrstcs. In ths papr, th nvstgaton focus s th varaton of ampltud, namly, th rlatv valu btwn th maxmum and mnmum dsplacmnt varaton. u(.5, t) t Fg. Dsplacmnt varaton of th vscolastcty bam cntr wth tm In ordr to vrfy th fasblty of th mult-scal modl, Fg.3 gvs th quanttatv comparsons among th mult-scal calculaton, dffrntal quadratur mthod (DQM) [] and fnt dffrnc mthod (FDM) [] wth paramtrs as k=., f=., γ=, and v=. It s clar that by th abov thr mthods, th curvs that rprsnt th chang of th stady-stat rspons ampltud wth harmonc paramtrs show grat agrmnt. Thus, t s fasbl to mploy th mult-scal mthod to smulat th vbraton of vscolastc bam. Undr th sam paramtrs, th valus of th stady stat rspons ampltud got by mult-scal mthod ar btwn that of DQM and FDM, whch rsults from th dffrnt dsposals at th nonlnar paramtr trms. And th mult-scal mthod can obtan rlatvly accurat rsult by contrast. a.3.. MSA FDM DQM Fg.3 Comparson of forcd vbraton btwn dffrnt numrcal mthods. B. Forcd Vbraton Analyss Th stady stat rspons of vbraton ampltud s affctd by macro prformanc, such as vscous dampng coffcnt k, xtrnal xctaton f, movng vlocty γ and nonlnar coffcnt v. Fg. (a) shows th rlatonshp btwn stady stat rspons ampltud and vscous dampng coffcnt wth f =., γ=, and v=. It can b sn that th stady-stat rspons ampltud dcrass wth th ncras of dampng coffcnt and ths varaton s obvous nar th natural frquncy ω (σ=). In othr words, th vscosty play a rol to absorb vbraton, and that s why flxbl structur s usd n many transportaton systms wth dynamc load. For xampl, th wr rop n hostng systm. Th nconspcuous rlatonshp btwn varaton of stady-stat rspons ampltud and nonlnar coffcnt s dpctd n Fg. (b) wth k=., f =., and γ=. Furthr, th law that th stady-stat rspons dcrass wth th ncras of movng vlocty s obtand n Fg. (c) wth k=., f=., and v=. It s found that hghr vlocty can mprov th dynamc balanc of systm and aggravat th bfurcaton phnomnon, whch agrs wll wth th rsults n ltratur [-]. Th rlatonshp btwn th stady-stat rspons ampltud and xtrnal xctaton s studd wth k=., γ=, and v=. As shown n Fg. (d), th stady stat rspons ampltud wll ncras whn th xtrnal DOI.53/IJSSST.a ISSN: 73-8x onln, prnt

5 BAO-FU KOU t al: A STUDY ON NONLINEAR FORCED VIBRATION OF AN AXIAL MOVING ncntv bcoms ntns, whch agrs wll wth th concluson n tst [3]..3 (a) ordr Galrkn truncaton k=.5.3. (d) f=.5 ordr Galrkn truncaton. f=. k=. a a.. f=.5 k=.3 a (a) Th vscous dampng coffcnt.3 (b) =5 = = ordr Galrkn truncaton (d) th xtrnal xctaton Fg. Influnc of dffrnt paramtrs on th stady-stat rspons ampltud. Th nflunc of xtrnal ncntvs on th stady-stat rspons ampltud s obvous whn th valus ar far away from th natural frquncy ω. Howvr, th xtrnal motvaton s nvtabl n th practcal ngnrng and how to rgulat th systm paramtrs bcoms a ky n systm dsgnng. To conclud that lowr xtrnal motvaton, hghr vscous dampng coffcnt and movng vlocty can ffctvly rstran vbraton, whch s usful for th movng of axal vscolastcty bam. a (c).. (b) Th nonlnar coffcnt =3 =5 ordr Galrkn truncaton = (c) Th opraton vlocty γ C. Vbraton Stablty Fg.5 (a) and (b) show th comparsons of th ral part of stady-stat rspons ampltud whn th ordr of Galrkn truncaton s,, undr th paramtrs as k=.5,f=.,γ=3,υ=5 and k=.3,f=.8, γ=,υ= rspctvly. It s obsrvd that th ordr Galrkn truncaton can produc rlabl prmary rsonanc, but and ordr Galrkn truncatons show clos rsults. That,s to say, th dffrnc among natural frquncs of axal movng bam can b obtand by Galrkn truncaton mthod, whch s rflctd by stady-stat rspons of dffrnt ampltuds n th crtcal rgon []. Thn, th nstablty boundary of man rsonanc s analyzd and Fg.6 (a) and (b) show th rsults of stady-stat rspons ampltud and th nstablty boundars undr th condton of k=.5,f=.5,γ=5,υ= and k=.3, f=.5,γ=5,υ= rspctvly, whr th sold ln rprsnts th stabl rgon, th dottd ln rprsnts th nstablty boundary, and th magnary ln rprsnts th nstablty rgon. By comparson, although th movng vlocty and nonlnar coffcnt s hghr, th rspons ampltud can always kp stabl f th vscous dampng coffcnt ncras and th xtrnal xctaton bcom DOI.53/IJSSST.a ISSN: 73-8x onln, prnt

6 BAO-FU KOU t al: A STUDY ON NONLINEAR FORCED VIBRATION OF AN AXIAL MOVING wak, whch s consstnt wth th concluson n rfrnc []. Th rsults ndcat that ovr small vscous dampng coffcnt and ovrlarg xtrnal xctaton wll caus nstablty..3 (a) stabl rgon nstablty boundary nstablty rgon.3 (a). ordr Galrkn truncaton ordr Galrkn truncaton ordr Galrkn truncaton a.. a (b) (b) ordr Galrkn truncaton ordr Galrkn truncaton ordr Galrkn truncaton a.. stabl rgon nstablty boundary a Fg.5 Comparson of dffrnt ordrs Galrkn truncaton Ths s bcaus th vscous has th powr to absorb vbraton to kp balanc, and th xtrnal ncntvs nhanc th occurrnc of chaos and bfurcaton [9]. It s known that nstablty wll nflunc th runnng stablty of th flxbl systms and th unwantd phnomna lk skddng may occur wth th ovrlarg movng dsplacmnt of nonlnar movng bam du to th aggravaton of nstablty. Thrfor, th choc of paramtrs n practcal applcatons should b rasonabl to avod th occurrnc of nstablty Fg.6 Stady-stat rspons and th nstablty boundars. V. CONCLUSIONS In ths papr, th nonlnar forcd vbraton of th axally movng vscolastc bams ar rsarchd by usng th mult-scal approach, togthr wth Galrkn truncaton mthod to numrcally solv ts dffrntal quaton, and obtan th rlabl conclusons. Frstly, th accuracy of th modl was vrfd through th quanttatv comparson wth th dffrntal quadratur mthod and fnt dffrnc mthod. Thn, th nfluncs of vscous dampng coffcnt, xtrnal motvaton, nonlnar coffcnt and movng vlocty on th stady-stat rspons ampltud wr analyzd. Th stady-stat rsponss ampltud s nvrsly proportonal to th valu of vscous dampng coffcnt and movng vlocty, and proportonal to xtrnal xctaton. All abov rsults ar consstnt wth th rfrncs. At last, by comparng th dffrnt ordr Galrkn truncaton, th nstablty boundary condtons ar analyzd, and found that th too small vscosty or too larg xtrnal motvatons caus nstablty. Th rsarch that mult-scal approach and Galrkn truncaton mthod ar appld to xplor th nonlnar forcd vbraton of th axally movng vscolastc bam s vry usful. Th DOI.53/IJSSST.a ISSN: 73-8x onln, prnt

7 BAO-FU KOU t al: A STUDY ON NONLINEAR FORCED VIBRATION OF AN AXIAL MOVING conclusons hav grat valu n thory and also can gud us to control nstablty ffctvly through optmzng paramtrs. ACKNOWLEDGMENTS Th work dscrbd n ths papr was supportd by th Foundaton of Scntfc and Tchnologcal Innovaton Rsarch Projcts n Unvrsts of Shanx, Chna (Grant No. 665); and th Scntfc Rsarch Foundaton for th Doctors of Tayuan Unvrsty of Scnc and Tchnology, Chna (Grant No. 55). REFERENCES [] N.H. Zhang, L.Q. Chn, Nonlnar dynamcal analyss of axally movng vscolastc strngs, Chaos, Soltons & Fractals (5) [] J.R. Chang, W.J. Ln, C.J. Huang, S.T. Cho,Vbraton and stablty of an axally movng Raylgh bam, Appld Mathmatcal Modlng 3 () [3] J.A. Wckrt, C.D. MotJr., Currnt rsarch on th vbraton and stablty of axally-movng matrals, Shock and Vbraton Dgst (988) 3 3. [] F. Pllcano, F.Vstron, Complx dynamcs of hgh-spd axally movng systms, Journal of Sound and Vbraton 58 () 3. [5] S. Kazmrad, M.H. Ghaysh, M. Amabl, Thrmal ffcts on nonlnar vbratons of an axally movng bam wth an ntrmdat sprng-mass support, Shock and Vbraton (3) [6] Rdl C H, Tan C A. Coupld forcd rspons of an axally movng strp wth ntrnal rsonanc, Intrnatonal Journal of Non-lnar Mchancs, 37 () -6. [7] Pasn F, Ubr d stabltat dr bgschwngungn von n langsrchtung prodsch hn und hrbwgtn Stabn, 97, Ingnur-Archv, (97) [8] Oz H R. On th vbratons of an axally travlng bam on fxd supports wth varabl vlocty, Journal of Sound and Vbraton, 39 () [9] Y.Z. Lu, On dynamcs of lastc rod basd on xact cossrat modl, Chns Physcs B, 8 (9) 3 [] T.Z. Yang, B. Fang, Y. Chn, Y.X. Zhn, Approxmat solutons of axally movng vscolastc bams subjct to mult-frquncy xctatons, Intrnatonal Journal of Non-lnar Mchancs, (9) 3-38 [] M.H. Ghaysh, H.A. Kafabad, T. Rd, Sub-and supr-crtcal nonlnar dynamcs of a harmoncally xctd axally movng bam, Intrnatonal Journal of Solds and Structurs 9 () 7 3. [] H. Dng, L.Q. Chn, Nonlnar modls for transvrs forcd vbraton of axally movng vscolastc bams, Shock and Vbraton 8 () [3] B. Nayak, S.K.Dwvdy, K.S.R.K.Murthy, Mult-frquncy xctaton of magntorhologcal lastomr-basd sandwch bam wth conductv skns, Intrnatonal Journal of Solds and Structurs 7 () 8-6. [] L.Q. Chn, X.D. Yang, Stablty n paramtrc rsonancs of an axally movng vscolastc bam wth tm-dpndnt vlocty, Journal of Sound and Vbraton, 8(5) [5] M.H. Yao, W. Zhang, J.W. Zu, Mult-puls chaotc dynamcs n non-planar moton of paramtrcally xctd vscolastc movng blt, Journal of Sound and Vbraton. 33 () 6. [6] M. Pakdmrl, H.R. Oz, Infnt mod analyss and truncaton to rsonant mods of axally acclratd bam vbratons, Journal of Sound and Vbraton, 3(8) 5-7. [7] E.M. Mocknsturm, J. Guo, Nonlnar vbraton of paramtrcally xctd, vscolastc, axally movng Strngs, ASME J Appl Mch, 7(5) [8] L.Q. Chn, X.D. Yang, Bfurcaton and chaos of an axally acclratng vscolastc bam,chaos, Soltons and Fractals, 3(5) [9] J.A. Wckrt, C.D. Mot, Classcal vbraton analyss of axally movng contnua, ASME Journal of Appld Mchancs 57 (99) [] G.C. Zhang, H. Dng, L.Q. Chn, S.P. Yang, Galrkn mthod for stady-stat rspons of nonlnar forcd vbraton of axally movng bams at suprcrtcal spds, Journal of Sound and Vbraton 33 () [] M.H. Ghaysh, Nonlnar forcd dynamcs of an axally movng vscolastc bam wth an ntrnal rsonanc, Intrnatonal Journal of Mchancal Scncs 53 () 37. [] B. Wang, Asymptotc Analyss on Wakly Forcd Vbraton of an Axally Movng Vscolastc Bam Consttutd by Standard Lnar Sold Modl, Appld Mathmatcs and Mchancs,33() [3] Y.C. Duan, J.P. Wang, J.Q. Wang, Y.W. Lu, F. Shao, Thortcal and xprmntal study on th transvrs vbraton proprts of an axally movng nstd cantlvr bam, Journal of Sound and Vbraton. 333 () DOI.53/IJSSST.a ISSN: 73-8x onln, prnt