On the Moving Mass versus Moving Load Problem

Transcription

1 On he Movng Mass versus Movng Load Prolem Z. Dmrovová 1 1 Deparameno de Engenhara Cvl, Fauldade de Cênas e Tenologa, Unversdade Nova de Lsoa, Quna da Torre, Capara, and LAETA, IDMEC, Insuo Superor Téno, Unversdade de Lsoa, Lsoa, Porugal emal: zdm@f.unl.p ABSTRACT: In hs onruon, a new mehod for a defleon shape deermnaon of an nfne eam on a vso-elas foundaon raversed y unformly movng mass s presened. The mehod nvokes he dynam sffness marx onep and for he sake of smply he resuls are shown on Euler-Bernoull eams. The soluon s presened n he onex of a revew of some mehods for soluon of unformly movng mass and load prolems on fne and nfne eams. Advanages and dsadvanages of hese mehods are summarzed. KEY WORDS: Movng load; Movng mass; Egenvalue expanson; Dynam sffness marx. 1 INTRODUCTION Dynam analyses of eam sruures under movng loads have araed he engneerng and senf ommuny from he mddle of he 19h enury, when ralway onsruon egan. Inreasng demands on he ralway nework apay leads o a neessy of eer undersandng of dynam phenomena relaed o ran-rak-sol neraons and herefore quesons regardng he movng load and movng mass prolems are he sll mporan sues n nowadays nvesgaons. New modellng approahes, as well as her solvng mehods, are needed o perform smulaons ha ould refle mporan feaures of dynam sysems. In hs onex analyal and sem-analyal soluons have he undoued advanage of possly of dre sensly analyss on parameers nvolved n he prolem. Movng fore prolem s far smpler. I has a semanalyal or analyal soluon avalale for fne as well as nfne eams. Generalzaons affeng he eam heory and foundaon models, lke exenson from he Euler-Bernoull heory o he Tmoshenko-Raylegh heory, or generalzaons of Wnkler foundaon o Pasernak or oher foundaon models, nroduon of foundaons of fne deph or aleraons from vsous o hysere dampng models do no presen susanal dffuly [1], exep n ases when numeral soluon of omplex frequenes s neessary. In fne eams egenvalue expanson ehnques an e used and n nfne eams eher Fourer ransform or he onep of he dynam sffness marx an e exploed [, 3]. In he laer ase wo sem-nfne eams are onneed y he onnuy ondons a he load applaon pon. Suh a soluon an easly e exended o he movng fore wh harmon omponen [4] or non-unform foundaon [5, 6]. The neral effes of oh he eam and he movng vehle were suded as early as n 199 y Jeffo [7] y he mehod of suessve approxmaons. The movng mass prolem does no have fully analyal soluon. Analysng fne eams, s seen ha he governng equaons n modal spae reman oupled [3]. There s however a lassal work [8], whh s ofen aken as a enh-mark soluon, u hs soluon does no onsder all effes a he ona pon as already deped y ohers [9]. There are oher papers repeang he same error [1], some of hem orreed y Leers o he Edor [11]. If a seady-sae soluon exss for an nfne eam, hen exaly mahes he soluon for he movng fore and he mass has no onruon as ndaed n [, 1]. If he soluon s no seady, here s an osllaon around he seady-sae defleon and he amplude and frequeny of hs osllaon has o e deermned. In hs paper a new mehod for her deermnaon s presened. PROBLEM STATEMENT Le a unform moon of a onsan veral fore and a mass along a horzonal eam on a lnear vso-elas foundaon e assumed (Fgure 1). The foundaon s modelled as homogeneous dsrued sprng-and-dashpo ses. Smplfaons for he analyss of veral vraons are oulned as follows: () he eam oeys lnear elas Euler-Bernoull heory; () he eam dampng s proporonal o he veloy of vraon; () he eam and mass are n onnuous ona; (v) no oher loadng s added; (v) he veral dsplaemen s measured from he equlrum defleon poson aused y he eam mass; (v) he veloy s mananed onsan and no resron s mposed on s magnude. x M P w k, Fgure 1. Sruure under onsderaon. v EI, m

2 The oeve s o solve he me dependen defleon shape w(). 3 FINITE BEAMS Equaons If he prevous model has a fne lengh desgnaed as L, hen several oundary ondons an e onsdered. Here we wll show examples only for smply suppored eam and lef anlever. Thus: wl,, or x xl w,, x, xl w,, 3 x 3 x xl x x x,, respevely, and, for he sake of smply nal ondons are assumed as homogeneous w, x The equaon of moon for he unknown feld w() s wren as: 4 w EI m kw p 4 x For he onsan mass and load d w p P M x v d, w wv, (1) () (3) (4) (5) whh means ha he loadng erm an e wren n erms of he unknown feld w() as:,,, w x w x w x P M v v x v x x here EI and m represen he flexural rgdy and he mass per un lengh of he eam, and k are he dampng oeffen and Wnkler s onsan of he foundaon, P and M are he ravellng fore and mass. w() and w () sand for he veral dsplaemen of he eam and of he pon of load applaon (mass ona pon), v s he onsan veloy, x s he spaal oordnae, s he me and δ s he Dra funon. x has s orgn a he lef exremy of he sruure. Zero me orresponds o load poson a x=. In Equaon (6) he erms are as follows: he veral loadng fore, he mass neral fore ang along he dreon of defleon of he eam, he Corols fore relaed o he rae of nlnaon of he eam; and he enrfugal fore assoaed (6) wh he urvaure of he eam. The las wo erms are no used n [8, 1]. In some ases hey an e negleed, u generally no. Soluon an e oaned y mplemenng he Fourer mehod of varale separaon and assumng he exsene of free harmon vraons:, w x w x e, 1 (7) The frequeny ω of hese vraons s named as he naural frequeny and s deermned from he egenvalue prolem oaned from he homogeneous governng equaon. Then he ransen response n he me doman s expressed as nfne seres of hese modes, where eah vraon mode (funon of he spaal oordnae x) s mulpled y a generalzed dsplaemen (modal oordnae, amplude funon) ha s a funon of me. w x q w x, 1 (8) For he movng load prolem, he modal analyss s falaed y he unouplng n he modal spae. Neverheless, even n self-adon sysems, modal expanson s ommonly governed y undamped vraon modes, eause hs allows her deermnaon whn he real doman and ompleeness of he egenspae s guaraneed. In movng mass prolem he prevous saemens are no vald. Two mehods an e used: () usually he expanson s performed over eam modes alulaed whou he movng mass onruon; () he oher possly would e o nlude he movng mass n he eam mass. In he former ase modal equaons anno e unoupled; have he followng form: M q C q K q q (9) where mares M, C, K are defned y nroduon of vraon modes n her exa analyal form (whou any dsrezaon) as: M Mw v w v (1) C Mvw v w v (11) m K Mv w v w v (1) Here δ s he Kroneker dela and upper prmes sand for he dervaon wh respe o he spaal varale. Sandard ehnques [, 5, 6] an e used for wave numers λ /L deermnaon, s only realled ha 4 EI k L m m The modes n equaons aove are normalzed y L (13) N mw x dx (14)

3 The sysem anno e solved analyally, u numerally. Compuaonal me nreases exponenally wh he numer of modes nvolved. Preson of a soluon oaned for a eran numer of modes anno e smply nreased y nludng one more mode, u he whole sysem mus e ompleely realulaed. If here s no elas foundaon, usually low numer of modes s suffen (around 1). Wh foundaon nluded he numer of modes mus e muh hgher, dependng on several faors ranges around 1- or more [3]. Numeral soluon an e oaned n ommeral sofware, s onvenen o rewre he sysem as a se of frs order equaons and use Mala [1], u hen he numeral preson s ompromsed, [, 5, 6]. Doule preson may no e suffen for hgher numer of modes, espeally when hyperol funons are nvolved n mode shapes. As an example, soluon of he movng mass and s orrespondng wegh on a anlever s shown (Fgure ). Ths s he example presened n [8], [9]. I s seen ha n hs ase he effe of Corols and enrfugal fores s sgnfan. Numeral daa are aken from [9] as: L=7.6m, P=5.79kN, M=69kg, EI=948.6kNm, m=46kg/m, v=5.8m/s. Ths s however no a very good example, sne he defleon s que large and he valdy of he Euler-Bernoull eam heory s ompromsed. Anoher applaon s a smply suppored eam on an elas foundaon (Fgure 3). The npu daa are: L=1m, P=1kN, M=1on, EI=6.4MNm, m=6kg/m, k=4mn/m, v=1m/s. The eam and foundaon daa are relaed o ralway applaons. The eam sands for one sngle ral. In hs ase 15 modes were neessary for a good auray of he soluon, u for over 5 modes (even f n hs ase wh purely snusodal shape) aumulaed numeral errors aused unphysal exessve osllaons when he load approahed he rgh suppor. If he movng mass s added o he eam mass, hen s neessary o solve he vraon modes a eah mass poson. The modes are orhogonal a eah suh a poson and an e deermned followng [3, 5, 6]. Due o adapale numeral preson, Maple [13] s one of he mos adequae sofware. Then he modes hange her shape and frequeny wh he movng mass poson and n fa only movng fore prolem should e solved [14]. I s neessary o nrodue suffenly small me dsrezaon and s possle o assume lnear modes varaon eween dsree fore posons. For unform dsrezaon Δ= - -1 s onsan. An nermedae value reads as: w v w v 1 w v w v 1 (15) where τ s a loal me sarng a -1 and w (v ) desgnaes -h mode deermned for mass poson a x =v. I holds: wh 1 q Pw v e d m sn m 1 os, 1 d q q e m d q e m 1 1 sn, (16) Fgure. Defleon of he anlever free end, paral means ha some erms were omed as n [8], full means ha all erms are nluded. Afer some manpulaons: m os, sn, q e D E d e e, (17) (18) Fgure 3. Defleon of he smply suppored eam on an elas foundaon, nal 4m of he full lengh, defeleons relaed o mass poson a eah m. d q e D os, E sn, d e sn os e D E where (19)

4 d Pw v 1, e d D q w v w v P 1 1 e 1 d E q 1 q 1 d 1 3 d e d e () (1) () Expressons aove are only approxmaons o he analyal soluon, u her advanage s ha hey are very quk o evaluae. The oher advanage s ha he preson an smply e mproved y addng more modes. There are some aleraons ha mus e nrodued. One of hem s swhng he modes n he way ha he same order moves smoohly s shape and does no ump o he oher sde of he sruure. Some resuls are shown n Fgure 4, numeral daa are relaed o he prevously nrodued anlever. where w s he unknown defleon feld and he rgh hand sde smplfes o: w P M x (4) If M=, hen he prolem an e solved followng [] or oher known leraure. Oher possly s o use he dynam sffness marx. The defnon s usually nrodued n sruures separaed no n-sudomans. The loal dynam sffness marx of he n-h su-doman an e alulaed n he followng way. The degrees of freedom are represened n Fgure 5a). Exaon wh un amplude and gven rular frequeny s assumed n he dreon of one of he degrees of freedom, whle he oher degrees of freedom are kep fxed. Fgure 5) exemplfes mplemenaon of he frs degree of freedom and orenaon of he orrespondng erms of he sffness marx. n n w n K n,1 n h su doman a) n1 w n1 K n,41 n 1 n K n,11 n h su doman ) w n 1 n 1 K n,31 Fgure 4. Defleon under he movng mass. I s seen, however, ha even for fne dsrezaon, here s an error n he defleon under he load, whh does no ge eer wh fner dsrezaon. These values are governed y he frs vraon modes a eah separae poson. 4 INFINITE BEAMS If an nfne eam s under onsderaon, a movng oordnae sysem an e nrodued y x x v, =. Then he lef hand sde of Equaon (4) an e wren as: Fgure 5. a) Degrees of freedom, ) onsruon of he loal dynam sffness marx of he n-h su-doman. For suh an exaon, memer-end generalzed harmon fores n he seady-sae regme an e alulaed. The proedure s repeaed for he oher degrees of freedom. More deals an e found n [5]. If sem-nfne su-domans are onsdered, hen only wo degrees of freedom have o e onsdered, as shown n Fgure 6. a) w 4 w w w EI m v v 4 x x x v kw x (3) w ) Fgure 6. Degrees of freedom of sem-nfne su-domans: a) negave, ) posve.

5 Wave equaon s deermned ased on Equaon (3). Four roos are separaed aordng o he sgn of he real par. Only he real negave-valued ones are used n he posve semnfne su-domans o ensure vanshng of he dsplaemens and roaons for x endng o posve nfny, and ve versa. General form of all nvolved erms for movng harmon f fore Pe where f s he exaon frequeny s presened n [4]. Four onsans defnng defleon shapes of he wo sem-nfne su-domans are hen deermned from he four onnuy equaons, whh are he onnuy of dsplaemens, roaons and endng momens and, n addon, nernal shear fore mus e n equlrum wh he exernally appled fore. In he ase of he harmon fore, he onsans are me-dependen. For exaon dreed y sne, magnary par of he soluon orresponds o he eam defleon. The ase wh a movng mass an e solved n a smlar way. Aordng o [15], f he fnal soluon s seady, hen he mass effe s anelled and he soluon has he same form as f only movng fore was nrodued. When he soluon s no seady, u sale, hen perod osllaon our n he defleon shape around he seady sae lne. The frequeny and amplude of hs addonal movemen has o e deermned. The frequeny an e solved y explong he fa ha n he pon of mass ona he veral fore exered on he eam s equal o w M M w M x x (5) where M s he orrespondng frequeny of hese osllaons. Therefore, he nfne eam an e solved for an exernally appled harmon fore and esed ha he fore value s orre. One an us sar wh some frequeny esmae, solve he wave numers, esalsh he four onnuy equaons and ompare he fore wh he expeed resul. Sne he prolem s lnear, he aual esmaed fore value s no mpor, eause for he same frequeny he rao of he nrodued and oaned value wll e he same. Ths allows defnng a smple eraon proedure, where afer he frs eraon he forng frequeny s realulaed, and so on. Ths proedure has very quk onvergene rao and s also very sale. Frequenes do no have o e searhed n a omplex doman as suggesed n [15]. Fgure 7. Defleon of he smply suppored eam on an elas foundaon, nal 4m of he full lengh, defleons relaed o mass poson a 18m and 36m. Naurally, a good nal esmae wll always help. In suh a ase one an use resuls on fne eams. In he ase shown n hs paper (Fgure 3) s seen ha he lowes defleon shapes happen when he mass s a posons equal o 18m, 36m, e. In hs parular ase s seen ha he defleon praally vanshes (Fgure 7).Ths predon ndaes ha he frequeny should e around 34.91rad/s. In wo seps hs value an e orreed o rad/s, wh dfferene eween he wo approxmaons less han.3%. Ths resul also fs well he relaon esalshed [15]. Oher ases were esed and same onlusons were aken. Then he soluon proedure s fnshed followng [15]. 5 CONCLUSION In hs onruon several aspes relaed o he dynam analyss of eam sruures under movng loads were summarzed. Dfferenes n soluon ehnques and resuls were gven for movng fore and movng mass prolems, as well as for fne and nfne eams. The onep of he dynam sffness marx was posed as a general prnple for fne, sem-nfne and nfne eams. Ths forms he ase for he new soluon ehnque for movng mass prolem on nfne eams. REFERENCES [1] Z. Dmrovová, Enhaned Formula for a Cral Veloy of a Unformly Movng Load, European Congress on Compuaonal Mehods n Appled Senes and Engneerng (ECCOMAS 1), Sepemer 1, Venna, Ausra. [] L. Frýa, Vraon of solds and sruures under movng loads. 3rd edon, Thomas Telford, London, [3] Z. Dmrovová, Dynam Analyss of Beam Sruures under Movng Loads: A Revew of he Modal Expanson Mehod, B.H.V. Toppng and Y. Tsompanaks, (eds.), Cvl and Sruural Engneerng Compuaonal Tehnology, Saxe-Courg Pulaons, Srlngshre, UK, Chaper 4 (11) [4] Y-H. Chen, Y-H. Huang, C.-T. Shh, Response of an Infne Tmoshenko Beam on a Vsoelas Foundaon o a Harmon Movng Load, Journal of Sound and Vraon, 41(5), 89-84, 1. [5] Z. Dmrovová, A general proedure for he dynam analyss of fne and nfne eams on pee-wse homogeneous foundaon under movng loads, Journal of Sound and Vraon, 39, , 1. [6] Z. Dmrovová, A.S.F. Rodrgues, Cral Veloy of a Unformly Movng Load, Advanes n Engneerng Sofware, 5 (Augus), 44 56, 1. [7] H.H. Jeffo, On he vraons of eams under he aon of movng loads, Phl. Magazne, Ser. 7, 8(48), 66 97, 199. [8] J.E. Akn, M. Mofd, Numeral soluon for response of eams wh movng mass, ASCE Journal of Sruural Engneerng, 115, 1 31, [9] C.J. Bowe, T.P. Mullarkey, Unsprung wheel-eam neraons usng modal and fne elemen models, Advanes n Engneerng Sofware 39, 911 9, 8. [1] E. Esmalzadeh, M. Ghorash Vraon Analyss of Beams Traversed y Unform Parally Dsrued Movng Masses, Journal of Sound and Vraon, 184(1), 9-17, [11] Y-H. Ln, LETTERS TO THE EDITOR, Commens on Vraon Analyss of Beams Traversed y Unform Parally Dsrued Movng Masses, Journal of Sound and Vraon, 199(4), 697-7, [1] Release R1a Doumenaon for MATLAB, The MahWorks, In., 1. [13] Release 11 Doumenaon for MAPLE, Maplesof a dvson of Waerloo Maple, In., 7. [14] Z. Dmrovová, Transversal Vraons n Infne Beams Suppored y Pee-Wse Homogeneous Vso-Elas Foundaon, 7h EUROMECH Sold Mehans Conferene (ESMC9), pp , 7-11 Sepemer, 9, Lson, Porugal, J.A.C. Amróso, M.P.T. Slva (edors), APMTAC, Porugal, ISBN [15] A.V. Merkne, H.A. Deerman, Insaly of Vraons of a Mass Movng Unformly along an Axally Compressed Beam on a Vso- Elas Foundaon, Journal of Sound and Vraons, 1, , 1997.