A MULTIBODY DYNAMIC MODEL OF A CARDAN JOINT WITH EXPERIMENTAL VALIDATION

Transcription

1 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta MULIBODY DYNAMICS 2005, ECCOMAS hematc Conference J.M. Gocolea, J.Cuadrado, J.C.García Orden (eds.) Madrd, Span, June 2005 A MULIBODY DYNAMIC MODEL OF A CARDAN JOIN WIH EXPERIMENAL VALIDAION M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta Department of Mechancal Engneerng Unversty of Rome or Vergata Va del Poltecnco, Rome, Italy e-mal: vta@ng.unroma2.t Department of Structural Engneerng Unversty of Cassno Va D Baso, Cassno, Italy Keywords: multbody system dynamcs, mechancal effcency, dual algebra, clearance and dmensonal tolerances. Abstract. hs nvestgaton deals wth the mechancal effcency of Cardan jonts. he model ncludes also the effects due to manufacturng and mountng errors and the nfluence of rotaton speed and angular confguraton of the Cardan jont on the effcency. he jont has been modeled as an RCCC spatal lnkage and the full dynamc analyss has been performed. he equatons of moton have been deduced by means both of dual vectors algebra and of classc multbody approach. he results have been compared wth expermental ones. 1 INRODUCION Cardan jonts are common devces for transmttng the moton between msalgned ntersectng axes. A complete dynamc analyss has been presented n a seres of papers authored by F. Freudensten and hs coworkers [1,2,3]. However n the mentoned references, frcton s not ncluded. For what concerns the mechancal effcency analyss of Cardan jonts the frst contrbuton s due to Moreck [4]. hat model ncluded only the losses n the yoke bearngs. Consderng the wdespread ndustral applcatons of Cardan jonts, the desgn of these mechancal devces has to be conducted very carefully. For ths reason t s mportant to nvestgate the effects of manufacturng errors, msalgnments and clearances on the performances of the jonts. In ths nvestgaton the authors present a complete model of a Cardan jont takng nto account both the effects of frcton and the effects of mountng errors n all the knematc pars. In order to take nto account also the effects of msalgnment between the axes of two consecutve knematc pars, the Cardan jont has been modeled as an RCCC mechansm. A mechancal effcency analyss based on ths model has been developed. he obtaned results have been compared to those comng from the expermental test rg equpped at the laboratory of the Department of Mechancal Engneerng of Unversty of Rome or Vergata. he proposed methodology reveals as an useful tool for the desgner to predct the mechancal per-

2 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta formances of Cardan jonts n real workng condton. he nfluence of angular velocty and output shaft confguraton on mechancal effcency has been also nvestgated. 2 FRICION MODELLING In deal workng condton the resultant of the reacton forces at the knematc pars s null. hs s not necessarly true n presence of radal clearance or manufacturng tolerances. If the presence of frcton s consdered, frctonal forces (moments) arse along (about) the axs of the knematc par [5,6]. In order to completely model the effects of frcton t s necessary to defne the geometrc features of the journal bearng. Fgure 1: Modellng of frcton n revolute jont. 2.1 Revolute jont he frctonal forces are manly caused by the reacton forces ( Fx, Fy, F z) and the reacton moments ( M x, M y) at the knematc par. he effect of F z could be neglected consderng that there s no sldng velocty along the axs of the revolute jont. In partcular the effect of reacton moments could be taken nto account consderng the equvalent couple of forces (Fgure 1): M x F =, (1) L where F are orthogonal to the drecton of z axs. hus the frctonal torque due to the reacton moment M M x could be evaluated as follow x x τ f = f d, (2) L where f s the frcton coeffcent, d s the dameter of the journal bearng and L s the length of the bearng. In the same way the frctonal torque due to M y can be evaluated. herefore the whole frctonal torque about the axs of the journal bearng s () fd 2 2 M z = sgn( θ ) τ f + Fx + F y, (3) 2

3 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta where θ s the relatve angular speed of the journal w.r.t. the bearng and ( ) ( ) () x 2 y 2 f f f τ = τ + τ. (4) 2.2 Cylndrcal jont Followng the same methodology llustrated for the revolute jont, the frctonal moment can be evaluated. In ths case a frctonal force along the axs of the knematc par s also present. Its value s defned by the followng equaton 2 2 M 2 2 x + M y Fz = sgn( s ) f Fx + Fy + 2, (5) L where s s the sldng velocty along the z axs. 3 DYNAMIC ANALYSIS he Cardan jont has been modeled as an RCCC mechansm for two man reasons [1,2]. he frst one s to smplfy the equatons of moton avodng redundant constrants, the second reason s to model clearances and axes msalgnments n the knematc pars preservng the moblty of the whole mechansm. he dynamc analyss has been performed by means of two dfferent approaches. he frst s the classcal multbody approach, based on the systematc formulaton of the constrant equatons as proposed by E. J. Haug n hs textbook [7], n order to nvestgate the nfluence of angular velocty and output confguraton of the Cardan jont on ts mechancal effcency. he second s the dual algebra approach wth the Denavt Hartenberg method [8,9,10,11], n order to nvestgate the effects of dmensonal tolerances on the mechancal effcency. hs last model was also valdated by means of expermental tests. For both of the smulatons the nertal and geometrc features of the Cardan jont are the same of the one used n the laboratory of the Department of Mechancal Engneerng of Unversty of Rome or Vergata. In partcular n able 1 the tensor of nerta and statc moments, expressed n the local reference frame coordnates, of each element are summarzed. Input Shaft Spder Output Shaft ensor of Inerta [kg m 2 ] Statc Moments [kg m] Mass [kg] S = x S y = S = z S = x S y = S = 0 x z z S = able 1: Inertal features of the Cardan jont S y = S = 0 For the locaton of the local reference frame of each body see Fgure 2.

4 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta Fgure 2: Locaton of the local reference frames he journal bearngs between the nput (output) shaft and the frame have length L=0.01 m and dameter d=0.02 m; the journal bearngs between the nput (output) shaft and the spder have length L=0.006 m and dameter d=0.005 m. 3.1 Multbody model Accordng to the systematc deducton of constrant equatons proposed by E. J. Haug [7], the followng dfferental algebrac equatons system descrbes the dynamcs of the Cardan jont [ M ]{} q + Ψ q {} λ = { Fe} { Ψ ( qt, )} = 0 where [ M ] represents the global mass matrx, { }, (6) q the generalzed coordnates (seven for each body: three for the locaton of the centre of mass of the body and four Euler parameters for the atttude), Ψ q the Jacoban matrx of the constrant equatons { Ψ ( qt, ) }, { λ } the F the generalzed external forces. he system (6) s solved by Lagrange multplers and { } e means of the Radau5 code mplemented by E. Harer, G. Wanner [12]. For ths reason t s rearranged nto where { } [ K ]{ y} = φ ( y), (7) [ K] { y} φ ( y) In order to take nto account the effects of frcton t s necessary to evaluate the reacton forces and moments at the knematc pars. hus, solvng the equatons of moton for { λ }, t s possble to express the reacton forces and moments n the jont reference frame components as { F ( ) 0 k k } = C A Ψ r { λ } k { q } {} q {[ M]{} q q {} λ { Fe} } { } I q 0 I 0 0 q = = { } = q + Ψ λ Ψ (8)

5 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta for the reacton forces and ( ) 0 0 k k { } = C s p A Ψ r { λ} C [ G] Ψ p { λ} 2 (9) for the reacton moments. In the expressons (8) and (9) C represents the transform matrx from the local frame to k 0 the jont frame; A s the transform matrx from the nertal frame to the local one; Ψ r and Ψ p are the Jacoban matrx of the constrant equatons w.r.t. the Cartesan coordnates () and Euler parameters respectvely; s p k s the skew matrx of the locaton vector of the reacton force n local coordnates. Once known the reacton forces and moments, t s possble to evaluate the frctonal torque and force at each knematc par by means of equatons (3) and (5). he effects of frcton on the dynamcs of the system are evaluated by ntroducng the frctonal forces and torques as external forces nto (6). For a correct analyss at each tme step one has to evaluate the Lagrange multplers, then the reactons and the frctonal actons and ntroduce them nto the equatons of moton for evaluatng agan the Lagrange multplers and restart the procedure tll convergence. In order to smplfy the process, consderng that the () 1 k k k soluton of system (6) s not an easy task, choosng a lttle tme step t = s, the reactons evaluated at tme t have been used to compute the frctonal torques and forces at tme t+ t. hs approxmaton does not affect the correctness of the results n a senstve way but smplfy the evaluaton process avodng convergence problems. 3.2 Dual algebra approach hs second model of the Cardan jont has been mplemented by means of dual algebra. hs approach reveals sutable for the descrpton and the analyss of dmensonal tolerances and clearances usng a small set of equatons. hus the effects of mountng errors on the mechancal effcency of the Cardan jont has been nvestgated. Before deducng the equatons of moton n terms of dual algebra, some theoretcal recalls are heren presented. By ntroducng the dual vector as V = v + εw (10) where v s the geometrcal vector, ε s the dual operator ( 2 = 0) ε and w s the moment of vector v w.r.t a chosen pont, t s possble to defne the velocty screw and the wrench on a screw. he analytcal expresson of the velocty screw s V = ω+ εv u (11) ( ) where ωu s the angular velocty and vu represents the sldng velocty along the rotaton axs. hus the velocty screw descrbes the knematc features of a screw moton. he analytcal expresson of the wrench on a screw s F = F + εc (12) A A A

6 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta where F A s the resultant of external forces and C A s the resultant moment of the external forces w.r.t. a pont A belongng to the central axs 1. hus the wrench on a screw represents the resultant of the dynamc acton on a body. Another dual entty useful for ths nvestgaton s the dual angle between two axes n space θ = θ + εs (13) where θ s the geometrcal angle, ε s the dual operator and s s the mnmum dstance between the axes. he applcaton of trgonometrc functons to the dual angle n (13) leads to sn θ = snθ + εs cosθ cos θ = cosθ εs snθ tan θ = tanθ + εs 1+ tan θ 2 ( ) and all the propertes of the classcal algebra are stll vald. he dynamc equatons of the Cardan jont, modelled as an RCCC mechansm (see Fgure 3), are deduced by means of the dual momentum and the Denavt Hartenberg formulaton. he expresson of the dual momentum s { ()} ( ) () { } ( ) ( ) { ()} ( ) ( ) () k k { } ( ) k ( ) { } ( ) k k k k k H = m C v m C R ω + ε m R v + J ω C C (15) where the real part represents the momentum and the dual one the angular momentum w.r.t. a pont C(). By dfferentatng (15) w.r.t. tme one obtans the dual expresson of the equatons of moton d { ()} ( ) { ()} ( ) k k H = C F (16) C dt { } ( ) k where F C () s the wrench on a screw at each knematc par of each movng lnk. Consderng the equlbrum condton at the knematc pars, one obtans 1 A + where 2 (1) (2) (2) A { R } { } { C R } 1 C F 2 C2 1 A { R } { } { C R } 2 C F 3 C3 2 A { R } { } { C R } 3 C F 4 C4 = 3 (2) (3) (3) = 4 (3) (4) (4) = 3 (14), (17) s the dual transform matrx from the local reference system on lnk to the local one on lnk +1 and { } () R C are the dual reactons expressed n the reference frame of lnk ( () ) w.r.t. a pont C of the central axs of lnk. Substtutng (17) nto (16) one obtans the equatons of moton n terms of dual reacton forces. For more detals on how to evaluate the expresson llustrated see [10,11]. 1 he axs of the resultant of the external forces.

7 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta Fgure 3: Geometrc and knematc parameters of the RCCC model hus the dual reacton forces evaluated by means of (16) are used to compute the dual frctonal forces at the knematc pars usng (3) and (5). 4 MECHANICAL EFFICIENCY ANALYSIS he nstantaneous mechancal effcency of the RCCC mechansm for both of the approaches llustrated s deduced by the followng expresson Pout Pn Ploss Ploss η = = = 1 (18) P P P n n n where P loss s the power loss at the knematc pars and P n = n ω n s the nput power. he power loss could be easly evaluated multplyng the frctonal force by the sldng velocty along the axs of the cylndrcal jont and the frctonal moment by the angular velocty about the axs of the cylndrcal / revolute jont. 4.1 Multbody approach results he frst knd of analyss conducted refers to the nfluence of the nput / output angular confguraton on the mechancal effcency. Fgure 4: Mechancal effcency at 10 deg

8 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta he parameter chosen for ths nvestgaton are: Angular confguraton of the Cardan jont (whch s the angle between nput and output shaft): 10 deg. Input angular velocty (kept constant) 2000 r.p.m. Input torque 2 Nm. Coeffcent of frcton at the knematc pars of the spder f 1 =0.42; coeffcent of frcton at the knematc pars of the shafts f 2 =0.20. In Fgure 4 the plot of the nstantaneous mechancal effcency versus tme s reported. mean he average value s η = Fgure 5: Power loss at the knematc pars of the spder (left) and of the shafts (rght) In Fgure 5 the power loss at the knematc pars s shown. On the left there s the power loss at the knematc pars of the spder whch has the fluctuaton of the relatve angular veloctes between spder and shaft respectvely. On the rght hand sde of Fgure 5 the comparson between the power loss at the nput and the output shaft s reported. he hgher values at the output lnk are due to the angular confguraton of the Cardan jont whch affects the output angular velocty too. he fluctuaton at nput s caused by the eccentrcty of centre of mass of the shaft. Fgure 6: Comparson of mechancal effcency for dfferent confguratons

9 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta Followng the same crteron a comparson between the mechancal effcences at dfferent confguratons of the Cardan jont has been conducted. In Fgure 6 the plots concernng ths nvestgaton are reported. he ntersectons between plots of Fgure 6 could be explaned consderng that the plot at 35 deg s drastcally nfluenced by nertal effects at such confguraton and moreover that the plots represent the nstantaneous mechancal effcency of the jont. In fact the mean values of mechancal power loss ncrease wth the ncrement of angular confguraton (able 2). Angular Confguraton [deg] Mean Value able 2: Mechancal effcency mean values he nfluence of angular velocty on mechancal effcency s also nvestgated. he angular dsplacement for ths test case s 10 deg. Fgure 7 shows the decrease of effcency wth the ncrease of angular speed. Fgure 7: Comparson of mechancal effcency for dfferent angular speeds he angular veloctes consdered are 500 r.p.m., 1000 r.p.m. and 2000 r.p.m., the correspondng mean values of mechancal effcency are 0.991, and respectvely. 4.2 Dual algebra approach results he proposed approach has been valdated by means of expermental tests. At the Laboratory of the Department of Mechancal Engneerng of Unversty of Rome or Vergata a test rg has been bult up for ths purpose (Fgure 9). Actng on motor vrtual control panel and choosng the desred resstng torque exerted by the brake t s possble to smulate dfferent workng scenaros. Moreover, t s possble to smulate angular msalgnment of the nvestgated jont tltng one end of the test rg.

10 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta Fgure 9: Expermental test rg he test rg s equpped wth the followng nstruments: an adjustable steel table; two torque/speed transducers (model Magtrol MB 210 wth max torque: Nm; max speed: 4000 r.p.m.; torque senstvty 100mV/Nm; speed senstvty: 60 pulses per rev.); one brushless motor (two poles; peak torque: 110 Nm) wth a control panel and control software; one electromagnetc brake (model Merobel SA FRA 650; max torque 65 Nm; mn torque 0.63 Nm) wth a radal fan and a DG 200 MC dgtal controller; one personal computer wth an a/d converter and a Natonal Instrument multchannel acqurng system. For ths knd of analyss the mplemented code (see [10,11]) has been modfed n order to use the data acqured from the transducers. In partcular the nput torque and the nput angular poston, velocty and acceleraton have been used. he acqured data have been fltered n order to reduce nose. he average values of torque and angular speed on nput are 0.2 Nm and 6 rad/s respectvely. he angular confguraton of the Cardan jont s 15 and the dynamc frcton coeffcent s set equal to 0.42 for each knematc par. he manufacturng tolerances concernng the msalgnment of the axes of the knematc pars are set n a = 0.5 mm (=1,2,3) and the angular errors on α (=1,2,3) are equal to deg. In Fgure 10 the comparson between the expermental mechancal effcency and the computed one s reported. Even f two dfferent ways for evaluatng the nstantaneous mechancal effcency 2 have been necessarly used, the results ft qute well. Paper [11] reports n detals the nfluence of dmensonal tolerances and mountng errors on the mechancal effcency of a Cardan jont. 2 he nstantaneous mechancal effcency by means of the expermental data s obtaned evaluatng the nput and output power. Instead the code estmates the output power computng the power loss at the knematc pars.

11 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta Fgure 10: Comparson between the expermental and the computed mechancal effcency 5 CONCLUSIONS he authors presented a multbody dynamc model of a Cardan jont for the evaluaton of the nstantaneous mechancal effcency. wo dfferent approaches have been proposed. he frst, based on the classcal multbody methodology [7], has been used for the evaluaton of the nfluence of angular confguraton and angular speed on the mechancal effcency. he second, based on the dual algebra [8,9], allowed to model clearances n the knematc pars and consequently to evaluate ther nfluence on the mechancal effcency. For both of the methodologes heren llustrated the effects of frcton have been ncluded. he results reported show that an ncrease of angular velocty makes the mechancal effcency decrease of 1.4%; an ncrement of the angle between the nput and the output shafts of the Cardan jont causes a reducton of mechancal yeld of 8.6%. Both of the behavors could be explaned wth an ncrease of frcton effects at the knematc pars. hese are more evdent n concurrence wth a consderable nclnaton of the Cardan jont where the nertal effects are more relevant. he presence of small dmensonal errors does not affect so much the mechancal effcency (0.15%). hs s due to the not so relevant varaton of the equlbrum condton at the knematc pars caused by clearances [10,11]. Moreover an expermental test rg has been bult up and the results obtaned have been successfully compared wth the computed one. 6 ACKNOWLEDGEMENS he authors wsh to acknowledge the grant by Mnstry for Educaton, Unversty and Research (MIUR). REFERENCES [1] Fscher, I., Freudensten, F., Internal Force and Moment ransmsson n a Cardan Jont wth Manufacturng olerances, ASME Journal of Mechansms, ransmssons and Automaton n Desgn, 106, , December [2] Chen, C.K., Freudensten, F., Dynamc Analyss of a Unversal Jont wth Manufacturng olerances, ASME Journal of Mechansms, ransmssons and Automaton n Desgn, 108, , December 1986.

12 M. Cavacece, R. Stefanell, P.P. Valentn, L. Vta [3] Freudensten, F., Macey, J.P., he Inerta orques of the Hooke Jont, Proc. Of the 21st Bennal ASME Mechansms Conference, 24, , Chcago, [4] Dudtza, F., ransmssons par Cardan, Edtons Eyrolles, Pars, [5] Shh C.W., Shh M.Y., Haug E.J., Dynamcs of Mechancal Systems wth Coulomb Frcton, Stcton, Impact and Constrant Deleton - III, Mechansm and Machne heory, 21, pp , [6] Dhanaraj C., Sharan A.M., Effcent Modelng of Rgd Lnk Body Dynamc Problems wth Frcton, Mechansm and Machne heory, 30, pp , [7] E. J. Haug, Computer- Aded Knematcs and Dynamcs of Mechancal Systems, vol.i, Allyn and Bacon, [8] Fscher, I.S., Dual-Number Methods n Knematcs, Statcs and Dynamcs, CRC Press, Boca Raton, [9] Yang, A.., Applcaton of Quaternon Algebra and Dual Numbers to the Analyss of Spatal Mechansms, Doctoral Dssertaton, Columba Unversty, New York, [10] E. Pennestrì, L. Vta, Mechancal Effcency Analyss of a Cardan Jont wth Manufacturng olerances, Proceedngs RAAD th Internatonal Workshop on Robotcs, Cassno 7-10 maggo, 2003 [11] M. Cavacece, E. Pennestrì, P.P. Valentn, L. Vta, Mechancal Effcency Analyss of a Cardan Jont, Proceedngs of 2004 ASME Conference, Salt Lake Cty, Utah, USA, September 28-October 2, [12] E. Harer, G. Wanner, Solvng Ordnary Dfferental Equatons II: Stff and Dfferentalalgebrac Problems, Sprnger-Verlag, Berln Hedelberg New York, 1996.