Dynamic Modelling of Hoisting Steel Wire Rope. Da-zhi CAO, Wen-zheng DU, Bao-zhu MA *

Transcription

1 17 nd Intrnational Confrnc on Mchanical Control and Automation (ICMCA 17) ISBN: Dynamic Modlling of Hoisting Stl Wir Rop Da-zhi CAO, Wn-zhng DU, Bao-zhu MA * and Su-bing LIU Xi an High chnology Rsarch Institut, Xi an, 715, China * Corrsponding author Kywords: Flxibl multibody dynamics, Stl wir rop, Absolut nodal coordinat formulation, Hrtz contact thory Abstract. Aiming at th stl wir rop and its conncting mchanism, th modling mthod and dynamic bhavior analysis was carrid out. h flxibl multibody dynamics modl of th rop was stablishd by using cabl lmnt basing on th absolut nodal coordinat formation, whil, othr parts in th conncting mchanism wr simplifid as rigid bodis. h contact impact forc btwn th flxibl rop and othr rigid bodis was incorporatd into th dynamic modl through th Hrtz contact thory. Finally th rigid-flxibl coupld dynamic modl which can dscrib th larg rotation and larg displacmnt of th whol stl rop systm was stablishd. h simulation of th stl rop s dynamic modl was prsntd. h rsults show that th dvlopd modl can b usd in th analysis of dynamic bhavior and th accurat control of th motion. h flxibl wir rop may undrgo larg rigid body displacmnt and flxibl dformation in th procss of hoisting th load. In addition, th wir rop is in contact and collision with th drum and th pully blocks in th cours of movmnt. hrfor, in th procss of stablishing th dynamic modl of th wir rop, it is ncssary to considr th xact rigid-flxibl coupld motion of th wir rop and th contact btwn th wir rop and othr parts in th conncting mchanism. At prsnt, th dynamic modl of flxibl multibody systm is usually stablishd by rlativ coordinat systm mthod and absolut coordinat systm mthod. Simo [1], Avllo [], Biaku [3] us rlativ local coordinat systm or lmnt coordinat systm to dscrib th larg displacmnt and lastic dformation of flxibl body. Shabana [4, 5] proposd th absolut nod coordinat formulation (ANCF). In th ANCF, coordinats that consist of displacmnt and gradint dgrs of frdom at th nodal points ar mployd. As th ANCF can accuratly dscrib th larg rotation, larg displacmnt and larg dformation of th flxibl body, th ANCF has bn widly applid to th dynamic analysis of flxibl multibody systm in diffrnt filds. In this papr, th flxibl wir rop is modld by th cabl lmnt basd on th ANCF. h contact btwn th wir rop and othr componnts of th systm is introducd into th dynamic modl through th Hrtz contact thory and th point-surfac dtction mthod. Finally, th rigid-flxibl coupld dynamic modl was stablishd. h rmaindr of th papr is organizd as follows. h ANCF modling mthod of th flxibl body and th contact dynamic modl was introducd in Sct.1. h multibody dynamics modl of th wir rop conncting mchanism is stablishd in Sct.. Modl s dynamic bhavior is analyzd basd on th slf-dvlopd solvr in Sct.3. Conclusions ar givn in Sct.4. 1

2 Modling Mthod of Multibody Systm wo Nods ANCF Cabl Elmnt Grstmayr [6] has proposd th thr-dimnsional ANCF bam lmnt with nods, and 6 dgrs at ach nod, shown in Figur1. Figur 1. h ANCF-basd two nods cabl lmnt. In th ANCF, th global position vctor of an arbitrary point on th bam can b writtn as r [ r r r ] S (1) whr S is th global lmnt shap function, and is th vctor of nodal coordinats. h vctor is givn by [ ] () h vctor of nodal coordinat includs th global displacmnts r, r, r 1 x x 3 x r, r, r 7 x l 8 x l 9 x l and th global slop of th nods that ar dfind as (3) r r r 4, 5, 6 x x x x x x r r r 1, 11, 1 x x x x l x l x l h lmnt shap function S is dfind as (4) [ S S S S ] S I I I I (5) 4 whr I is a 3 3 idntity matrix and th shap function ar dfind as 3 3 S1 1 3ξ + ξ, S l( ξ ξ + ξ ) (6) 3 3 S3 3ξ ξ, S4 l( ξ + ξ ) and ξ x / l, l is th lmnt lngth. h global position vctor of an arbitrary point on th bam is giv by quation (1), by diffrntiating this quation with rspct to tim, th absolut vlocity vctor can b dfind as r S. his vctor can b usd to dfind th kintic nrgy of th lmnt as 1 1 ρr r dv M (7) V

3 Whr V is th volum, ρ is th mass dnsity of th bam matrial, and M ρs S dv is th mass matrix of th lmnt. h mass matrix is constant and V symmtric. h strain nrgy of th lmnt is dfind as 1 L 1 L U EAε dx + EJκ dx (8) Whr E is modul of lasticity, A is th bam cross ara, l is th lmnt lngth, J is th scond momnt of ara, ε is th longitudinal strain, and κ is th curvatur. Using th virtual work, th dynamic quations of th finit lmnt can b dfind as M + F Q (9) whr F U is th vctor of lastic forc, th dtail drivation can b found in th litratur [7]. Q is th vctor of th xtrnally applid forcs. Modling of Contact h contact btwn th stl wir rop and th rigid parts can b simplifid as th point-surfac contact whr th Gauss contact tst points ar distributd in th lmnt. h cabl- to-rigid contact is showd in Figur. Figur. h cabl-to-rigid contact. According th Hrtz contact thory [8], th contact forc at th contact points can b dfind as f f n + f t (1) n t whr f, n f τ ar rspct th normal contact forc and tangntial friction forc; n, t ar th contact normal and tangntial unit vctor. Using th principl of virtual work, th gnralizd nodal forcs contributd by th distributd contact forc can b drivd as Q S f (11) p p p whr S is th shap function of lmnt. h Motion Equations of Multibody Systm Using quation (7) and quation (8), th total kintic nrgy and strain nrgy of flxibl can b obtaind by summing thir lmnt countrparts, n i total total 1 n i 1 U U U (,, ) n i total total 1 n i 1 (,, ) (1) 3

4 whr n and n ar rspct th numbr of absolut nods and lmnts of flxibl body. Considring that th rigid body s motion quations can b dscribd by Eulr quations and by applying Lagrang s quation, dynamic quations of th total multibody systm can b xprssd as Mq + Cqλ + F Q( q, q, t) (13) C( q, t) whr M is th systm mass matrix, q is th vctor of systm gnralizd coordinats, λ is th vctor of Lagrang multiplirs, C(q, t) rprsnts th constraint conditions and C q is constraint Jacobian matrix, Q( q, q, t) is th gnralizd forc vctor including Coriolis forc, cntrifugal forc, xtrnal forcs, tc. h abov quations is th basis for th dynamic analysis of multibody systms, hr th implicit BDF [9] is adoptd to solv th motion quations. Multibody Dynamics Modl of th Stl Wir Rop Systm Dynamics modl of th Hoisting Systm h modl of th whol hoisting systm is composd of flxibl stl wir rop, hoisting drum, fixd pully and two movabl pullys, shown in Figur1. h stl wir rop is considrd as th flxibl body and th othrs ar considrd as rigid bodis. h main paramtrs of th modl ar listd in abl.1. Figur 3. Modling imag of hoisting systm. abl 1. Main paramtrs of modl. h abov mntiond nods cabl lmnt is usd to modl th stl wir rop and total 56 lmnts ar usd. For vry fiv gauss contact dtcting points ar arrangd in ach lmnt, and ach lmnt would b contactd with th othr four rigid parts, so total 56*5*411 point-surfac contact pairs ar includd. 4

5 Boundary Conditions h load conditions of th wir rop includ th gravity of th initial load and th angular vlocity applid to th drum. h angular vlocity curv is shown in Figur. 4 and th total simulation tim is 3s. Figur 4. h load boundary condition. Analysis of th Simulation Basd on th slf-dvlopd multibody dynamics solvr, th dynamic modl of th wir rop lifting mchanism is simulatd and th vlocity curv of th load is prsntd, as shown in Figur. 5. Figur 4. Vlocity curv of th load. h Calculatd tnsion forc of th stl rop is shown in Figur. 6. In s-1s, du to th application of th load, th axial tnsil vibration of th wir rop is obvious. h ratio of th dynamic load to th static load of th wir rop can rach about 1.8, and th strain of th wir rop can rach.4. Du to th damping of th wir rop and th numrical damping of th calculation procss, th wir rop s tnsion forc tnds to b smooth, and th amplitud of th axial vibration is gradually rducd. About 1s, th tnsion forc tnds to b stabl. During th acclratd raising tim (1s-s) and th stabl raising tim (s-3s), th chang of wir rop s tnsion forc is not much, and th dynamic load is about 1.3 tims of th static load. Figur 5. nsion forc curv of stl cabl. In th procss of hoisting th load, th wir rop contact with th drum and th pully blocks at tims. h associatd contact forc will chang th strss nvironmnt of th flxibl body, and finally it will affct th movmnt of flxibl body, th flxibl dformation and th strss distribution. h distribution of th contact impact forc btwn th wir rop and th pully is shown in Figur. 7. 5

6 Figur 6. h distribution of contact forc. Conclusions h flxibl stl wir rop was modld basing on ANCF cabl lmnt and its rigid-flxibl motion including big rotation, big displacmnt and flxibl dformation can b xactly dscribd. hrough th point-surfac contact tsting mthod and Hrtz contact thory, th contact-impact forc was incorporatd into th stl cabl dynamic modl, and th rigid-flxibl coupld dynamic modl including contact was stablishd. Basing on th stablishd modl, th dynamics bhavior of th stl rop was analyzd, and th dynamic rsponss such as th tnsional load, th contact forc distribution and th vibration frquncy wr calculatd. h prsntd rigid-flxibl coupld multibody dynamic modl can provid basis in th dynamic analysis of cabl structural systms and th accurat motion control. Rfrncs [1] Simo J C, Quoc V L. h rol of nonlinar thoris in transint dynamics analysis of flxibl structurs. Journal of Sound and Vibration, 1987, 119(3): [] Avllo A, Jalon J G D. Dynamic of flxibl multi-body systm using cartsian coordinats and larg displacmnt thory. Intrnational Journal of Numrical Mthods in Enginring, 1991, 3(6): [3] Biaku G, Jzqul L. Multibody formulation with larg bnding finit lmnts (LBFE). Nonlinar Dynamics, 6, 46(1-): [4] Shabana A A. Dfinition of th slops and th finit lmnt absolut nodal coordinat formulation. Multibody Systm Dynamics, 1997, 1(3): [5] Shabana A A, Yakoub R Y. hr dimnsional absolut nodal coordinat formulation for bam lmnts: hory. Journal of Mchanical Dsign, 1, 13(4): [6] Grstmayr J, Shabana A A. Analysis of thin bams and cabls using th absolut nodal co-ordinat formulation. Nonlinar Dynamics, 6, 45(1-): [7] Brzri M, Shabana A A. Dvlopmnt of simpl modls for th lastic forcs in th absolut nodal co-ordinat formulation. Journal of Sound and Vibration,, 35(4): [8] Johnson K L. Contact mchnics. Cambridg Nw York: Cambridg Univrsity Prss, [9] Hairr E, Wannr G. Solving ordinary diffrntial quations ii: Stiff and diffrntial-algbraic problms. nd d. Hidlbrg: Springr-Vrlag,