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1 Univesity of Huddesfield Repositoy Kolla, László E., Stepan, Gao and Hogan, S.J. Backlash in alancing systems using appoximate sping chaacteistics Oiginal Citation Kolla, László E., Stepan, Gao and Hogan, S.J. (2) Backlash in alancing systems using appoximate sping chaacteistics. In: 3d Euopean Nonlinea Oscillations Confeence, 8-2 August 999, Lyngy, Denmak. This vesion is availale at The Univesity Repositoy is a digital collection of the eseach output of the Univesity, availale on Open Access. Copyight and Moal Rights fo the items on this site ae etained y the individual autho and/o othe copyight ownes. Uses may access full items fee of chage; copies of full text items geneally can e epoduced, displayed o pefomed and given to thid paties in any fomat o medium fo pesonal eseach o study, educational o not-fo-pofit puposes without pio pemission o chage, povided: The authos, title and full iliogaphic details is cedited in any copy; A hypelink and/o URL is included fo the oiginal metadata page; and The content is not changed in any way. Fo moe infomation, including ou policy and sumission pocedue, please contact the Repositoy Team at: E.mailox@hud.ac.uk.

2 Backlash in Balancing Systems Using Appoximate Sping Chaacteistics L. E. Kolla, G. Stepan, S. J. Hogan Astact A mechanical model of a alancing system is constucted and its staility analysis is pesented. This model consides an inteesting pactical polem, the acklash. It appeas in the system as a nonlinea sping chaacteistic with noncontinuous deivative. The uppe equiliium of the pendulum can e stailized without acklash. Backlash causes oscillations aound this equiliium. Phase space diagams ae evealed ased on simulations. Bifucation analysis is caied out y the continuation method. The noncontinuous deivative of the sping chaacteistic causes polems duing the calculation, theefoe diffeent types of appoximate chaacteistics ae used. The conditions of the existence of stale stationay and peiodic solutions ae detemined in case of the appoximate systems and conclusions ae otained fo the exact piecewise linea system. Intoduction Unstale equiliia of mechanical systems often have to e stailized y contol foce. A nume of applications can e found in this field, e.g. the us unning on icy oad, the shimmying wheel o the alancing of standing and walking oots. A typical example of stailization of unstale equiliia is the alancing. The simplest model of alancing is that of the inveted pendulum [-6]. The angle and the angula velocity of the pendulum ae detected and a hoizontal contol foce at the lowest point of the pendulum is detemined y them in a way that the stick should e alanced at its uppe position. Contol paametes must e chosen fom a ounded egion fo successful alancing. The staility conditions have een calculated and the staility chat in the plain of the contol paametes has een constucted in ealie woks [9]. A pendulum-cat system is consideed in the susequent chaptes. The inveted pendulum and the moto displaying the contol foce ae placed on a

3 cat and the moto dives one of the wheels of the cat though a teeth-elt. Contolling is executed y a compute which is situated outside this cat. Consideing the acklash at the diving-wheel of the moto, the pendulum will swing with small amplitude aound its equiliium. The staility domain in the plain of the contol paametes does not change, ut it means the domain whee the uppe position of the pendulum o the oscillation aound it is stale. 2 The pendulum-cat model In ode to desciing a digital alancing system, the inveted pendulum is placed on a cat as it can e seen in Figue [8,]. The moto dives one of the wheels of this cat though a teeth-elt with stiffness s. The system has 3 degees of feedom descied y the geneal coodinates, x; ' and ψ. The angle ' of the pendulum and the displacement x of the cat ae detected togethe with thei deivatives. Figue : The inveted pendulum on a cat and its staility map The contol foce is detemined y the moto chaacteistic. The divingtoque is linealy popotional to the voltage U m of the moto and to the angula velocity _ ψ: Consideing PD contolles, we have: M m = LU m K _ ψ: () U m = P'+ D _' + P x x + D x _x: (2) 2

4 The system can e stailized if the displacement of the cat is not detected (P x =)and the diffeential gain D x of the cat eliminates the damping K of the moto. Then the contol foce has this simplified fom: Q = L (P'+ D _') : (3) The system is educed to a system with 2 degees of feedom if a new geneal coodinate is intoduced. This is, the alongation of the = m ψ w R w x: (4) The nonlinea equations of motion assume the fom: (m+m)m m m whee ml 2 3 m 2 l 2 cos ' 4(m+M) mm ml m w 4R w mmmlmw cos ' 4R w m2 l 2 cos 2 ' 4(m+M) _' 2 sin ' _' 2 sin ' mgl 2 sin ' A ψ ' A +! ψ (m + M) Q + ψ (m + M) K (m + M) m + mmm2 w 2R 2 w ml w! R s = s w B R w s 2(m+M)R w!ψ _ _'! A R s = + ; (5) x ' ψ C A = s (6) is the foce in the sping. The staility analysis is caied out y the Routh-Huwitz citeion. If the elt is ideally igid, then =, x detemines ψ uniquely, so the system has 2 degees of feedom, namely x and '. The ' tivial solution of this system is asymptotically stale if and only if P >P = L "ψ m + M + 2 m m 2 w R 2 w! g # mr w w and D> : (7) If the elt is elastic, then the tivial solution of the lineaized fom of (5) is asymptotically stale if and only if P >P and H 2 > : (8) 3

5 whee H 2 is the maximum sized Huwitz-deteminant, not pesented hee algeaically. The staility chat is constucted as it is shown in Figue. The staility domain shinks as the stiffness of the diving-elt deceases and at a cetain citical value, it disappeas. This citical value has this fom: s>s cit = 3(m + M) m m g w m +4M +2m 2 m R 2 l : (9) w 3 Numeical study of the phase-space Backlash appeas in the system as a nonlinea sping chaacteistic. foce in the sping is the function of : 8 >< R s = >: The s ( + )» j j < ; () s ( ) whee is the value of acklash. This function is given in Figue 2. () 5 5 R s [N ] R s [N ] Figue 2: The piecewise linea sping chaacteistic at s = [ kn m ] and = [mm], () The linea sping chaacteistic at s = [ kn m ] New constant expessions appea in the equations of motion, that means shifting of the solutions. The staility domain does not change ut it is valid only if j j >. Othewise, the system is just in acklash, so it cannot e stailized, ecause the contol foce is not displayed in this little domain. If the contol paametes ae chosen fom the staility domain, then oots of the chaacteistic equation ae complex numes with negative eal pats. 4

6 Tajectoies fom stale focus aound the '; _'; ; _ = (; ; ± ; ) equiliia. If the system is just in acklash, then the oots of the chaacteistic equation ae positive and negative eals. Tajectoies fom saddle aound the (; ; ; ) equiliium...5 ().5. _ [ m s ] _ [ m s ] (c) (d).6.4 _ [ m s ].2 _ [ m s ] Figue 3: Phase-diagams on _ plane, P =2[Nm];D =2[Nms]; () P =2[Nm];D =2[Nms]; (c) P = [Nm]; D =2[Nms]; (d) P = 2[Nm];D = 2[Nms] with changed initial conditions Simulations wee accomplished using Runge-Kutta method fo the study of the phase-space [7]. Results ae pesented in Figue 3 fo the given values of paametes: m = :69[kg];M = :36[kg];mm = :2[kg];g = 9:8[ m s 2 ];l = :5[m]; w = :2[m];R w = :3[m]; m = :[m];k = :[Nms];s = [ N m ]; =:[m]. Figue 3, 3() and 3(c) show the phase diagams on _ plane nea diffeent values of eithe of the contol paametes, P. The (; ; ± ; ) equiliia ae stale fo smalle values of P. A stale peiodic solution appeas fo geate values of P and its amplitude is lage and lage as P inceases. Its amplitude tends to infinity as P tends to the 5

7 ode of the staility domain. Now the staility domain means the domain whee stale stationay o peiodic solution can e found. The initial conditions ae the same in these figues. They ae changed in Figue 3(d) and P is the same as in Figue 3(). The (; ; ; ) equiliium is stale, ut if P is the same as in Figue 3(c), then it ecomes unstale. Fo cetain values of P all the (; ; ± ; ) equiliia and the limit cycle ae stale and the tajectoies spial to one of them depending on the initial conditions. Moe investigations ae needed fo the exact knowledge of the phase space. 4 Appoximate sping chaacteistics Fist, the linea chaacteistic is consideed, as if thee wee no acklash in the system. It is given in Figue 2(). The ifucation analysis is caied out y the continuation method using the (5) nonlinea equations of motion. The ifucation diagam fo D = 2[Nms] is sketched in Figue 4. A pitchfok ifucation is occued at P = :986[Nm] otained fom (7), whee the uppe equiliium ecomes stale. It maintains its staility till the supecitical Hopf-ifucation occued at P =39:7[Nm] otained fom the Huwitz-deteminant in (8). An unstale stationay solution appeas at the pitchfok ifucation and ' tends to ß as P inceases. A stale peiodic solution appeas at the supecitical Hopf-ifucation and its amplitude 2 tends to ß as P inceases. 2 2 () '[ad] '[ad] P [Nm] D[Nms] Figue 4: Bifucation diagam fo linea sping chaacteistic The ifucation paamete is P, D =2[Nms] () The ifucation paamete is D, P = 2[Nm] 6

8 The ifucation paamete is D and P = 2[Nm] in Figue 4(). The equiliium is stale etween the Hopf-ifucation points. They occu at D = :99[Nms] and D = 5:96[Nms] otained fom the Huwitz-deteminant again in (8). It seems that peiodic solution exists only nea the ode of the staility domain at this value of P..4 () 5.2 R s [N ] R s [N ] Figue 5: Appoximate sping chaacteistics R s ;K s =25(thin line), K s = 5 (thick line) R s2 ;K s = 4 (thin line), K s = 5 (thick line) Backlash means nonlinea sping chaacteistic with noncontinuous deivative asitisshown in Figue 2. Two kinds of appoximation given in () ae applied. They have moe advantageous popeties fom view point of the calculations. The fist one, R s is diffeentiale any times, the second one, R s2 is diffeentiale once only, ut its fist deivative at =± is exactly the sping stiffness s. Both of these appoximations include a paamete K s, and appoximations ae moe and moe accuate as it tends to infinity. R s and R s2 in the vicinity of is depicted in Figue 5 and 5() fo diffeent values of K s. R s = s K s ln +eks( ) +e Ks( + ) ; 8 >< s ( + )» R s2 = s ( ) e Ks( ) + s ( + ) e Ks( + ) j j < : () >: s ( ) 7

9 5 The ifucation analysis () '[ad].4 '[ad] P [Nm] D[Nms] Figue 6: Bifucation diagams using R s, K s =25 The ifucation paamete is P, D =2[Nms] () The ifucation paamete is D, P = 2[Nm] () '[ad] '[ad] P [Nm] D[Nms] Figue 7: Bifucation diagams using R s, K s =5 The ifucation paamete is P, D =2[Nms] () The ifucation paamete is D, P = 2[Nm] The ifucation analysis in the vicinity of the uppe equiliium of the pendulum is implemented using the appoximate sping chaacteistics. The ifucation diagams in case of R s ae dawn in Figue 6, 6(), 7 and 7(). K s = 25 in Figue 6 and K s = 5 in Figue 7. The ifucation paamete is P in Figue 6 and 7 and D in Figue 6() and 7(). Changing P at a fixed value of D, the pitchfok ifucation is found at the same P whee in the linea case. Inceasing K s, the Hopf-ifucation point is close and close to P. A stale limit cycle appeas at this point 8

10 and its amplitude inceases as P appoaches P, the value whee the Hopfifucation occued in the linea case. Changing D at a fixed value of P, two Hopf-ifucation points ae indicated in Figue 6(). The equiliium is stale etween them, ut it cannot e stailized at the same value of P fo geate K s. Only the limit cycle is stale etween the odes of the staility domain calculated in the linea case. () kuk..8 kuk P [Nm] D[Nms] Figue 8: Bifucation diagams using R s2, K s = 5 The ifucation paamete is P, D =2[Nms] () The ifucation paamete is D, P = 75[Nm] () kuk.8 kuk P [Nm] D[Nms] Figue 9: Bifucation diagams using R s2, K s =3 6 The ifucation paamete is P, D =2[Nms] () The ifucation paamete is D, P = 75[Nm] The ifucation diagams in case of R s2 ae shown in Figue 8, 8(), 9 and 9(). K s = 5 in Figue 8 and K s = 3 6 in Figue 9. The ifucation paamete is P in Figue 8 and 9 and D in Figue 8() and 9

11 9(). Changing P at a fixed value of D, the pitchfok ifucation is found at the same P whee in the linea case. Inceasing K s, the Hopf-ifucation point moves towads less values of P, ut it stops at P 2 = 69:89[Nm]. An unstale limit cycle appeas at this point and the numeical calculation is inteupted at aout P 3 =46[Nm]. Its amplitude tends to as K s inceases. The L 2 -nom of the state vaiales can e seen in the figues, theefoe the value whee the stationay solution is indicated is. A ifucation point is supposed at P 3 and the equiliium is unstale fo geate values of P. A stale limit cycle which was found with the othe appoximation is indicated again, ut the calculation had stopped efoe the amplitude of the limit cycle would have deceased to. P is geate at the stopping point than P. A homoclinic oit is showed up with simulations at this value, thus a homoclinic ifucation is assumed hee. Changing D at a fixed value of P, two Hopf-ifucation points ae indicated. The equiliium is stale etween them, ut the unstale limit cycle is also found etween them with deceasing amplitude as K s inceases. The stale limit cycle is also indicated hee D[Nms] P [Nm] Figue : The staility chat with the ifucation cuves Examinations ae accomplished fo the systems with appoximate sping chaacteistics and conclusions can e otained fo the exact piecewise linea system, thus the staility domain in the plain of the contol paametes can e constucted as it is sketched in Figue. It is odeed with the same staight line and paaola as it was odeed in case of the linea system (the system without acklash). Fix points ae stale in a little domain nea the staight line. Stale limit cycle appeas at the homoclinic ifucation point indicated with the dotted line. Fix points lose thei staility at the othe ifucation point indicated with the smashed line, so all the fix points and the limit cycle ae stale etween the dotted and the smashed line, and only

12 the limit cycle is stale in the emaining pat of the staility domain. 6 Conclusions Backlash causes the decease of staility domain of the equiliium. The size of the staility domain found in case of the linea sping chaacteistic is the same, ut a stale peiodic solution exists instead of a stale stationay solution in the lagest pat of this domain. The physical meaning of the peiodic solution is the oscillation of the stick aound its vetical equiliium. The physical meaning of the stale fix points is that the contol foce does not push the stick futhe than the vetical line and it oscillates with less and less amplitude on eithe side of the vetical position. Acknowledgements: This eseach was suppoted y the Hungaian Scientific Reseach Foundation unde gant no. OTKA T3762 and the Ministy of Cultue and Education unde gant no.fkfp 38/97. Refeences [] Moi, S., Nishihaa, H., Fuuta, K., Contol of an unstale mechanical system, Int. J. Contol 23, (976) [2] Stépán, G., A model of alancing, Peiodica Polytechnica 28, (984) [3] Stépán, G., Retaded Dynamical Systems, Longman, Halow, UK, 989. [4] Hendes, M. G., Sondack, A. C., 'In-the-lage' ehaviouofaninveted pendulum with linea stailization, Int. J. of Nonlinea Mechanics 27, (992) [5] Kawazoe, Y., Manual contol and compute contol of an inveted pendulum on a cat, Poc. st Int. Conf. on Motion and Viation Contol, pp , Yokohama, 992. [6] Enikov, E., Stépán, G., Stailizing an Inveted Pendulum - Altenatives and Limitations, Peiodica Polytechnica, Vol. 38, pp. 9-26, 994.

13 [7] Lóánt, G., Stépán, G., The Role of Non-Lineaities in the Dynamics of a Single Railway Wheelset, Machine Viation 5, (996) [8] Enikov, E., Stépán, G., Mico-Chaotic Motion of Digitally Contolled Machines, J. of Viation and Contol, accepted in 997. [9] Stépán, G., Kollá, L. E., Balancing with Reflex Delay, Mathematical and Compute Modelling, accepted in 997. [] Kollá, L. E., Backlash in Machines Stailized y Contol Foce, Poc. of Fist Confeence on Mechanical Engineeing pp. 47-5, Budapest, 998. Authos Kolla, Laszlo E. Depatment of Applied Mechanics Budapest Univesity of Technology and Economics H-52, Budapest Hungay kolla@galilei.mm.me.hu Hogan, S. John Dep. of Engineeing Mathematics Univesity of Bistol Queen's Building, Univesity Walk Bistol BS8 TR UK S.J.Hogan@istol.ac.uk Stepan, Gao Depatment of Applied Mechanics Budapest Univesity of Technology and Economics H-52, Budapest Hungay stepan@galilei.mm.me.hu 2