Dynamic response of a finite length euler-bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force

Transcription

1 Journal of Mchanical Scinc and Tchnology 2 (1) (21) 1957~ DOI 1.17/s x Dynamic rspons of a finit lngth ulr-brnoulli bam on linar and nonlinar viscolastic foundations to a concntratd moving forc Alkim Dniz Snalp 1, Aytac Arikoglu 1, Ibrahim Ozkol 1,* and Vdat Ziya Dogan 1 1 Dpartmnt of Aronautical Enginring, Istanbul Tchnical Univrsity, Istanbul, 369,Turky (Manuscript Rcivd Fbruary 23, 29; Rvisd March 18, 21; Accptd Jun 28, 21) Abstract In this papr th dynamic rspons of a simply-supportd, finit lngth Eulr-Brnoulli bam with uniform cross-sction rsting on a linar and nonlinar viscolastic foundation actd upon by a moving concntratd forc is studid. Th Galrkin mthod is utilizd in ordr to solv th govrning quations of motion. Rsults ar compard with th finit lmnt solution for th linar foundation modl in ordr to validat th accuracy of th solution tchniqu. A good agrmnt btwn th two solution tchniqus is obsrvd. Th ffct of th nonlinarity of foundation stiffnss on bam displacmnt is analyzd for diffrnt damping ratios and diffrnt spds of th moving load. Th rsults for th tim rspons of th midpoint of th bam ar prsntd graphically. Kywords: Eulr-Brnoulli bam; FEM; Galrkin mthod; Moving forc; Vibration; Viscolastic foundation Introduction Rcntly, th invstigation of th dynamic rspons of bams on viscolastic foundations subjctd to moving loads has bn of grat significanc in railway nginring. Zhng t. al [1] gav a gnral solution dynamical problm of an infinit bam on an lastic foundation. [2] invstigatd th dynamic rspons of a Timoshnko bam on a Winklr foundation subjctd to a moving mass. Thambiratnam and Zhug [3] studid th dynamics of bams on an lastic foundation subjctd to moving loads by using th finit lmnt mthod. Thy invstigatd th ffct of th foundation stiffnss, travling spd and lngth of th bam on th dynamic magnification factor, which is dfind as th ratio of th maximum displacmnt in th tim history of th mid-point to th static mid-point displacmnt. Kim [] invstigatd th vibration and stability of an infinit Eulr-Brnoulli bam rsting on a Winklr foundation whn th systm is subjctd to a static axial forc and a moving load with ithr constant or harmonic amplitud variations. Th ffcts of load spd, load frquncy, damping on th dflctd shap, maximum displacmnt and critical valus of th vlocity, frquncy and axial forc ar also studid. Kargarnovin and Younsian [5] studid th rspons of a Timoshnko bam with uniform cross-sction and infinit lngth supportd by a gnralizd This papr was rcommndd for publication in rvisd form by Associat Editor Eung-Soo Shin * Corrsponding author. Tl.: , Fax: addrss: ozkol@itu.du.tr KSME & Springr 21 Pastrnak viscolastic foundation subjctd to an arbitrarily distributd harmonic moving load. Kargarnovin and Younsian [6] also analyzd th dynamic rspons of infinit Timoshnko and Eulr-Brnoulli bams on nonlinar viscolastic foundations to harmonic moving loads. In this study, th dynamic rspons of a simply-supportd, finit lngth, uniform cross-sction Eulr-Brnoulli bam rsting on a linar and nonlinar viscolastic foundation actd upon by a moving concntratd forc is studid. In xisting litratur, rsarch basd on th rspons of bams on foundations assums that th bam is infinit. In this study, an infinit track is rplacd by a finit track. Sinc th bam is placd ovr a vry stiff foundation, th moving load has a local ffct and it is sufficint to analyz a small portion of th bam. Th Galrkin mthod is usd to solv th initial boundary valu problm that govrns th transvrs vibration of th bam. Tim rspons historis of th bam ar graphically prsntd for various spds of forc. Th ffct of nonlinarity of th foundation stiffnss is also invstigatd. 2. Thory and formulation In Figs. 1 and 2, simply-supportd, homognous, isotropic and constant cross-sction bams of lngth ovr viscolastic foundations ar shown. inar and nonlinar foundation modls ar usd. Viscolastic foundations consist of dashpots and springs. In th litratur, th railway track is usually assumd to b linar in ordr to simplify th track modl, although th rail pad and ballast ar actually non-linar. Th bam is ini-

2 1958 A. D. Snalp t al. / Journal of Mchanical Scinc and Tchnology 2 (1) (21) 1957~1961 Fig. 1. Simply-supportd bam on linar viscolastic foundation. tially assumd to b at rst and undformd. Th forc f(x,t) is xprssd as follows: f ( xt, ) = δ ( x gt ( )) P (1) whr δ is th Dirac-Dlta function, P rfrs to th constant forc and g(t) rprsnts th kinmatics of th moving forc as follows: g() t = vt (2) whr v is th constant spd of th forc. Th Dirac-Dlta function, δ(x), is thought of as a unit concntratd forc acting at point x=. Th Dirac-Dlta function is dfind as: b δ ( x ξ ) f( x) dx = f( ξ ), for a< ξ < b. (3) a 2.1 inar foundation modl To compar th ffcts of th linarity and nonlinarity of th foundation, a linar foundation modl is considrd first. A linar viscolastic foundation modl is shown in Fig. 1. Th problm is govrnd by th following diffrntial quation: wxt ρ (, ) wxt (, ) EI + A + kw( x, t) wxt (, ) + c = δ ( x vt) P whr EI, ρ, A, c and w(x,t) ar th flxural rigidity, th dnsity, th cross-sctional ara, th damping cofficint and th transvrs dflction of th bam at point x and tim t, rspctivly. k is th linar foundation stiffnss and c is th viscous damping cofficint of th foundation. Th simplysupportd bam boundary and initial conditions ar: w(, t) = w(, t) = w(, t) w(, t) (5) = = wx (,) wx (,) = =. (6) () Fig. 2. Simply-supportd bam on a nonlinar viscolastic foundation. 2.2 onlinar foundation modl For this modl, th viscolastic foundation is modld by th combination of linar and cubic nonlinar springs, whr k is th nonlinar part of th foundation stiffnss and c is th damping cofficint. Th problm is govrnd by th following diffrntial quation: wxt ρ A (, ) wxt (, ) EI wxt (, ) kwxt (, ) + kw( xt, ) + c = δ ( x vtp ). (7) Th boundary and initial conditions ar th sam as givn in Eqs. 5 and 6. Eq. 7 rprsnts a nonlinar initial boundary valu problm. 3. Solution mthod 3.1 Galrkin mthod Th Galrkin mthod is applid to Eqs. and 7 to discrtiz th problm in a spatial coordinat and to obtain a systm of ordinary diffrntial quations in th tim domain. Th transvrs displacmnt is assumd in th following form: wxt (, ) = T( tsinn ) ( π x/ ). (8) n n= 1 Th basis functions ar slctd as Sin(nπx/) in ordr to satisfy th boundary conditions in Eq. 5. By using Eq. 8 and Eq., th Galrkin mthod can b applid for th linar foundation modl as follows: nπ EI k Tn() t ρ ATn''() t ctn'() t n= 1 nπx mπx mπx Sin Sin dx = P δ ( x vt) Sin dx which simplifis to th following form: mπ EI + k Tm() t + ρ ATm''() t + ctm'() t = 2P mπ vt Sin, for m = 1,2,3,..., (9) (1)

3 A. D. Snalp t al. / Journal of Mchanical Scinc and Tchnology 2 (1) (21) 1957~ and th initial conditions in Eq. 6 bcom: T () = T '() = for m= 1,2,3,...,. (11) m m Th sam procdur for th nonlinar foundation modl is rpatd, and th govrning quations ar drivd. mπ EI + k Tm() t + ρ ATm''() t + ctm'( t) + k Ti() t Tj() t Tk() t [ 3 δ ( i+ j k m) i= 1 j= 1 k = 1 δ( i+ j+ k m) 3 δ( i+ j k + m) 2P mπ vt Sin for m = 1,2,3,..., = ] (12) Tabl 1. Proprtis of UIC6 rail. Young s modulus (/m 2 ) 21 x 1 1 Ara momnt of inrtia (m ) 3.55 x 1-5 Cross sctional ara (m 2 ) 7.69 x 1-3 Dnsity of th matrial (kg/m 3 ) 785 ngth of th bam (m) 5 inar spring stiffnss pr lngth (inar foundation) (/m 2 ) x 1 8 inar spring stiffnss pr lngth (onlinar foundation) (/m 2 ) 3.53 x 1 7 onlinar spring stiffnss pr lngth (onlinar foundation) (/m ).1 x 1 1 Moving load () 65 x 1 3 Th initial conditions in Eq. 11 apply to both linar and nonlinar foundation modls. 3.2 Finit lmnt mthod In ordr to validat th rsults obtaind with th Galrkin mthod, th displacmnt fild for th linar foundation modl is valuatd with FEM. For th Eulr-Brnoulli bam, th lmnt mass and stiffnss matrics ar givn as follows: M ρ A = (13) EI K = (1) Th lmnt forc vctor for a point load is givn by: tv / 3tv / tv + t v / 2t v / F = P t v / 2t v / tv / tv / (15) whr is th lngth of an lmnt. Th lmnt damping matrix and th lmnt stiffnss matrix for th foundation ar drivd from th mass matrix by rplacing th cofficint ρa with c and k, rspctivly. Th lmnt matrics ar joind to form th global matrics and thn th boundary conditions ar applid. Th lmnt forc vctor is takn, as in Eq. 15, if th load is on th lmnt, othrwis it is takn as zro. Fig. 3. Tim rspons of bam valuatd with Galrkin mthod and FEM (v=2 m/s, ξ=9.7).. Rsults and discussion A finit bam of 5 mtrs in lngth is considrd long nough to rplac an infinit bam. Th matrial proprtis for th rail, foundation and load ar prsntd in Tabl 1 [6]. For ach vlocity cas, dampd and ovr-dampd dynamic rsponss of linar and nonlinar viscolastic foundation modls ar invstigatd. For th linar foundation modl, damping ratios ar dtrmind by considring th critical damping cofficint (c cr ). Th damping ratio (ξ) is xprssd as follows: ξ = c/ ccr (16) c = 2 k ρ A (17) cr Fig. 3 shows th comparison btwn th rsults of th

4 196 A. D. Snalp t al. / Journal of Mchanical Scinc and Tchnology 2 (1) (21) 1957~1961 Fig.. Tim rspons diagrams of bams with two distinct foundation modls for th damping ratio ξ=.5. Fig. 5. Tim rspons diagrams of bams with two distinct foundation modls for th damping ratio ξ=5. Galrkin and finit lmnt mthods for a linar foundation. Sinc th bam modl is simply supportd at both nds, and bcaus of th symmtry, maximum dflctions will occur at th mid-span of th bam, /2. Thrfor, th rsults ar prsntd for this point. Th rsults show that a prfct agrmnt btwn ths two mthods is rachd as th numbr of lmnts considrd in th finit lmnt calculation incrass. Figs. -6 show a on scond tim portion of th mid-point vrtical dflctions with tim. Th vlocity is takn btwn 1 to 5 m/s. Fig. shows th ffct of moving load spd on th vrtical dflction for a rlativly small damping ratio. Th local ffct of th moving load is clarly sn from th figur, spcially at highr spds. Vrtical mid-point dflctions ar virtually ngligibl until th moving loads approach th midpoint. Anothr important point is that th transvrs vibration amplitud dcrass with incrasing spd. Th nonlinar and quivalnt linar modls ar in good agrmnt for small dflctions. Howvr, th ffct of nonlinarity starts to dominat for largr dflctions. It can also b sn from th figurs that for ξ=.5, th transvrs displacmnt of th bam is clos to bing symmtrical. Moving load spd and nonlinarity hav similar ffcts for highr damping ratios, as shown in Figs. 5 and 6. Howvr, as th damping ratio incrass, th symmtry of th displacmnt is distortd. Th magnitud of th vibration amplitud dcrass with incrasd damping du to th loss of kintic nrgy in th form of hat nrgy. Kargarnowin t al. [6] studid th ffct of two succssiv moving loads by using an FEM modl. Th rsults prsntd in Figs. -6 show vry good agrmnt with th rsults in [6] in trms of both th magnitud and th distribution of dflction. 5. Conclusions In this study, th dynamic rspons of a simply-supportd, finit lngth Eulr-Brnoulli bam with uniform cross-sction rsting on a linar and nonlinar viscolastic foundation actd upon by a moving concntratd forc is studid. An infinit track with nonlinar foundation is rplacd with a finit on. This boundary valu problm was solvd for linar and

5 A. D. Snalp t al. / Journal of Mchanical Scinc and Tchnology 2 (1) (21) 1957~ Winklr foundation subjctd to a moving mass, Applid Acoustics, 55 (3) (1998) [3] D. Thambiratnam and Y. Zhug, Dynamic analysis of bams on an lastic foundation subjctd to moving loads, Journal of Sound and Vibration, 198 (1996) [] Song-Min Kim, Vibration and Stability of axial loadd bams on lastic foundation undr moving harmonic loads, Enginring Structurs, 26 (2) [5] M. H. Kargarnovin and D. Younsian, Dynamics of Timoshnko bams on Pastrnak foundation undr moving load, Mchanics Rsarch Communications, 31 (2) [6] M. H. Kargarnovin and D. Younsian, Rspons of bams on nonlinar viscolastic foundations to harmonic moving loads, Computrs & Structurs, 83 (25) A. Dniz Snalp rcivd his B.S. dgr in Aronautical Enginring from Istanbul Tchnical Univrsity, Turky, in 25. H thn rcivd his M.S. dgr from th sam dpartmnt of ITU in 28. A. Dniz Snalp is currntly a Rsarch Assistant and PhD. Studnt in ITU Aronautics and Astronautics Dpartmnt. Fig. 6. Tim rspons diagrams of bams with two distinct foundation modls for th damping ratio ξ=1. nonlinar cass by applying th Galrkin mthod. Th tim rsponss of th bams with linar and nonlinar cass ar prsntd for various spds of moving forc. Th ffcts of nonlinarity in stiffnss can asily b obsrvd from th figurs. From Figs. -6, on can dduc th following rsults: As th forc spd incrass, th dynamic rspons of th bam dcrass for both linar foundation modls. As th damping ratio (ξ) incrass, th dynamic dflctions dcras for both linar and nonlinar cass. For th nonlinar foundation modl, th dynamic rspons of th bam is always gratr whn compard to th linar foundation modl. Th distribution of dflction is symmtrical for small valus of th damping ratio and this symmtry is distortd with incrasing damping. Rfrncs [1] D. Y. Zhng, Y. K. Chung, F. T. K. Au and Y. S. Chng, Vibration of multi-span non-uniform bams undr moving loads by using modifid bam vibration functions, Journal of Sound and Vibration, 212 (3) (1998) [2] H. P., Dynamic rspons of a Timoshnko bam on a Aytac Arikoglu rcivd his B.S and M.S. dgrs from Istanbul Tchnical Univrsity in 22 and 2 rspctivly. H is currntly a PhD candidat and a rsarch assistant in th Faculty of Aronautics and Astronautics in ITU. His rsarch aras ar applid mathmatics, thrmodynamics and vibration analysis of sandwich structurs. Ibrahim Ozkol is working in Istanbul Tchnical Univrsity at th faculty of aronautics and astronautics as a Profssor. His rsarch intrsts ar fluid mchanics, dynamics, kinmatics and control of paralll mchanisms. Vdat Z. Dogan rcivd th BS dgr from Istanbul Tchnical Univrsity, Turky; th M.S. and Ph.D. dgrs from Columbia Univrsity in w York, USA. H currntly works as an Associat Profssor in th Dpartmnt of Aronautical Enginring at Istanbul Tchnical Univrsity, Istanbul, Turky. His currnt rsarch intrsts includ structural dynamics, plats and shlls, and random vibrations.