A mechanical model of a balancing system is constructed and its stability analysis is presented. This model considers an interesting practical problem, the backlash. It appears in the system as a nonlinear spring characteristic with noncontinuous derivative. The upper equilibrium of the pendulum can be stabilized without backlash. Backlash causes oscillations around this equi- librium. Phase space diagrams are revealed based on simulations. Bifurca- tion analysis is carried out by the continuation method. The noncontinuous derivative of the spring characteristic causes problems during the calcula- tion, therefore different types of approximate characteristics are used. The conditions of the existence of stable stationary and periodic solutions are determined in case of the approximate systems and conclusions are obtained for the exact piecewise linear system

Hogan, S.J.

Kollar, László E.

Stepan, Gabor

English

University of Huddersfield Repository

University of Huddersfield RepositoryKollar, László E., Stepan, Gabor and Hogan, S.J.Backlash in balancing systems using approximate spring characteristicsOriginal CitationKollar, László E., Stepan, Gabor and Hogan, S.J. (2000) Backlash in balancing systems using approximate spring characteristics. In: 3rd European Nonlinear Oscillations Conference, 8-12 August 1999, Lyngby, Denmark.This version is available at http://eprints.hud.ac.uk/16654/The University Repository is a digital collection of the research output of theUniversity, available on Open Access. Copyright and Moral Rights for the itemson this site are retained by the individual author and/or other copyright owners.Users may access full items free of charge; copies of full text items generallycan be reproduced, displayed or performed and given to third parties in anyformat or medium for personal research or study, educational or not-for-profitpurposes without prior permission or charge, provided:• The authors, title and full bibliographic details is credited in any copy;• A hyperlink and/or URL is included for the original metadata page; and• The content is not changed in any way.For more information, including our policy and submission procedure, pleasecontact the Repository Team at: E.mailbox@hud.ac.uk.http://eprints.hud.ac.uk/Backlash in Balancing Systems UsingApproximate Spring CharacteristicsL. E. Kollar, G. Stepan, S. J. HoganAbstractA mechanical model of a balancing system is constructed and its stabilityanalysis is presented. This model considers an interesting practical problem,the backlash. It appears in the system as a nonlinear spring characteristicwith noncontinuous derivative. The upper equilibrium of the pendulum canbe stabilized without backlash. Backlash causes oscillations around this equi-librium. Phase space diagrams are revealed based on simulations. Bifurca-tion analysis is carried out by the continuation method. The noncontinuousderivative of the spring characteristic causes problems during the calcula-tion, therefore dierent types of approximate characteristics are used. Theconditions of the existence of stable stationary and periodic solutions aredetermined in case of the approximate systems and conclusions are obtainedfor the exact piecewise linear system.1 IntroductionUnstable equilibria of mechanical systems often have to be stabilized bycontrol force. A number of applications can be found in this eld, e.g. thebus running on icy road, the shimmying wheel or the balancing of standingand walking robots.A typical example of stabilization of unstable equilibria is the balancing.The simplest model of balancing is that of the inverted pendulum [1-6]. Theangle and the angular velocity of the pendulum are detected and a horizontalcontrol force at the lowest point of the pendulum is determined by themin a way that the stick should be balanced at its upper position. Controlparameters must be chosen from a bounded region for successful balancing.The stability conditions have been calculated and the stability chart in theplain of the control parameters has been constructed in earlier works [9].A pendulum-cart system is considered in the subsequent chapters. Theinverted pendulum and the motor displaying the control force are placed on a1cart and the motor drives one of the wheels of the cart through a teeth-belt.Controlling is executed by a computer which is situated outside this cart.Considering the backlash at the driving-wheel of the motor, the pendulumwill swing with small amplitude around its equilibrium. The stability domainin the plain of the control parameters does not change, but it means thedomain where the upper position of the pendulum or the oscillation aroundit is stable.2 The pendulum-cart modelIn order to describing a digital balancing system, the inverted pendulum isplaced on a cart as it can be seen in Figure 1 [8,10]. The motor drives oneof the wheels of this cart through a teeth-belt with stiness s. The systemhas 3 degrees of freedom described by the general coordinates, x; ' and  .The angle ' of the pendulum and the displacement x of the cart are detectedtogether with their derivatives.Figure 1: The inverted pendulum on a cart and its stability mapThe control force is determined by the motor characteristic. The driving-torque is linearly proportional to the voltage Umof the motor and to theangular velocity_ :Mm= LUm K_ : (1)Considering PD controllers, we have:Um= P'+D _'+ Pxx +Dx_x : (2)2The system can be stabilized if the displacement of the cart is not detected(Px= 0) and the dierential gain Dxof the cart eliminates the damping Kof the motor. Then the control force has this simplied form:Q = L (P'+D _') : (3)The system is reduced to a system with 2 degrees of freedom if a newgeneral coordinate is introduced. This is , the alongation of the spring: = rm  rwRwx : (4)The nonlinear equations of motion assume the form:0@(m+M)mmrm2 mmmlrmrwcos'4Rw0ml23 m2l2cos2'4(m+M)1A '!+ (m+M)Krm00 0! __'!+0@mmmlrmrw4Rw_'2sin'm2l2cos'4(m+M)_'2sin' mgl2sin'1A+0@(m +M) rm+mmrmr2w2R2wmlrw2(m+M)Rw1ARs= (m+M)Q0!; (5)whereRs= srwRw0 srm0B@x' 1CA= s (6)is the force in the spring.The stability analysis is carried out by the Routh-Hurwitz criterion. Ifthe belt is ideally rigid, then  = 0, x determines  uniquely, so the systemhas 2 degrees of freedom, namely x and '. The '  0 trivial solution of thissystem is asymptotically stable if and only ifP > P0=1L" m+M +12mmr2wR2w!grmRwrw#and D > 0 : (7)If the belt is elastic, then the trivial solution of the linearized form of (5)is asymptotically stable if and only ifP > P0and H2> 0 : (8)3where H2is the maximum sized Hurwitz-determinant, not presented herealgebraically.The stability chart is constructed as it is shown in Figure 1. The stabilitydomain shrinks as the stiness of the driving-belt decreases and at a certaincritical value, it disappears. This critical value has this form:s > scrit=3 (m +M)mmgm+ 4M + 2mmr2wR2wl: (9)3 Numerical study of the phase-spaceBacklash appears in the system as a nonlinear spring characteristic. Theforce in the spring is the function of :Rs=8><>:s ( + r0)    r00 jj < r0s (  r0)   r0; (10)where r0is the value of backlash. This function is given in Figure 2(a).-10-50510-0.002 -0.001 0 0.001 0.002Rs[N ][m](a)-10-50510-0.002 -0.001 0 0.001 0.002Rs[N ][m](b)Figure 2: (a) The piecewise linear spring characteristic at s = 10[kNm] andr0= 1[mm], (b) The linear spring characteristic at s = 10[kNm]New constant expressions appear in the equations of motion, that meansshifting of the solutions. The stability domain does not change but it is validonly if jj > r0. Otherwise, the system is just in backlash, so it cannot bestabilized, because the control force is not displayed in this little domain.If the control parameters are chosen from the stability domain, then rootsof the characteristic equation are complex numbers with negative real parts.4Trajectories form stable focus around the'; _';;_= (0; 0;r0; 0) equi-libria.If the system is just in backlash, then the roots of the characteristicequation are positive and negative reals. Trajectories form saddle aroundthe (0; 0; 0; 0) equilibrium.-0.2-0.15-0.1-0.0500.050.1-0.002 -0.001 0 0.001 0.002_[ms][m](a)-0.2-0.15-0.1-0.0500.050.10.15-0.002 -0.001 0 0.001 0.002_[ms][m](b)-0.8-0.6-0.4-0.200.20.40.60.8-0.005 -0.003 -0.001 0.001 0.003 0.005_[ms][m](c)-0.0004-0.000200.00020.00040.00060.000998 0.001 0.001002 0.001004_[ms][m](d)Figure 3: Phase-diagrams on_  plane, (a) P = 2[Nm]; D = 2[Nms];(b) P = 20[Nm]; D = 2[Nms]; (c) P = 100[Nm]; D = 2[Nms];(d) P = 20[Nm]; D = 2[Nms] with changed initial conditionsSimulations were accomplished using Runge-Kutta method for the studyof the phase-space [7]. Results are presented in Figure 3 for the given values ofparameters: m = 0:169[kg];M = 1:136[kg]; mm = 0:2[kg]; g = 9:81[ms2]; l =0:5[m]; rw= 0:02[m]; Rw= 0:03[m]; rm= 0:01[m]; K = 0:01[Nms]; s =10000[Nm]; r0= 0:001[m]. Figure 3(a), 3(b) and 3(c) show the phase diagramson_    plane near dierent values of either of the control parameters,P . The (0; 0;r0; 0) equilibria are stable for smaller values of P . A stableperiodic solution appears for greater values of P and its amplitude is largerand larger as P increases. Its amplitude tends to innity as P tends to the5border of the stability domain. Now the stability domain means the domainwhere stable stationary or periodic solution can be found.The initial conditions are the same in these gures. They are changed inFigure 3(d) and P is the same as in Figure 3(b). The (0; 0; r0; 0) equilibriumis stable, but if P is the same as in Figure 3(c), then it becomes unstable.For certain values of P all the (0; 0;r0; 0) equilibria and the limit cycle arestable and the trajectories spiral to one of them depending on the initialconditions. More investigations are needed for the exact knowledge of thephase space.4 Approximate spring characteristicsFirst, the linear characteristic is considered, as if there were no backlash inthe system. It is given in Figure 2(b). The bifurcation analysis is carriedout by the continuation method using the (5) nonlinear equations of mo-tion. The bifurcation diagram for D = 2[Nms] is sketched in Figure 4(a).A pitchfork bifurcation is occured at P0= 0:1986[Nm] obtained from (7),where the upper equilibrium becomes stable. It maintains its stability tillthe supercritical Hopf-bifurcation occured at P1= 139:7[Nm] obtained fromthe Hurwitz-determinant in (8). An unstable stationary solution appears atthe pitchfork bifurcation and ' tends to2as P increases. A stable peri-odic solution appears at the supercritical Hopf-bifurcation and its amplitudetends to2as P increases.-0.500.511.520 50 100 150 200'[rad]P [Nm](a)2 2rrrrrrrr-0.200.20.40.60.81-2 0 2 4 6 8'[rad]D[Nms](b)2 2rrrrrrrrrrrrrrrrrrrFigure 4: Bifurcation diagram for linear spring characteristic(a) The bifurcation parameter is P , D = 2[Nms](b) The bifurcation parameter is D, P = 20[Nm]6The bifurcation parameter is D and P = 20[Nm] in Figure 4(b). Theequilibrium is stable between the Hopf-bifurcation points. They occur atD =0:1991[Nms] and D = 5:916[Nms] obtained from the Hurwitz-determinantagain in (8). It seems that periodic solution exists only near the border ofthe stability domain at this value of P .-10-50510-0.002 -0.001 0 0.001 0.002Rs[N ][m](a)-0.4-0.200.20.40 0.0005 0.001Rs[N ][m](b)Figure 5: Approximate spring characteristics(a) Rs1; Ks= 2500 (thin line), Ks= 5000 (thick line)(a) Rs2; Ks= 104(thin line), Ks= 105(thick line)Backlash means nonlinear spring characteristic with noncontinuous deriva-tive as it is shown in Figure 2(a). Two kinds of approximation given in (11)are applied. They have more advantageous properties from view point of thecalculations. The rst one, Rs1is dierentiable any times, the second one,Rs2is dierentiable once only, but its rst derivative at  = r0is exactlythe spring stiness s. Both of these approximations include a parameter Ks,and approximations are more and more accurate as it tends to innity. Rs1and Rs2in the vicinity of r0is depicted in Figure 5(a) and 5(b) for dierentvalues of Ks.Rs1=sKsln1 + eKs( r0)1 + e Ks(+r0);Rs2=8><>:s ( + r0)    r0s (  r0) eKs( r0)+ s ( + r0) e Ks(+r0)jj < r0s (  r0)   r0: (11)75 The bifurcation analysis00.0020.0040.0060.0080.01-20 0 20 40 60 80 100 120 140'[rad]P [Nm](a)2 2rrrrrrrrrrrrrrrrr00.0020.0040.0060.0080.01-1 0 1 2 3 4 5 6 7'[rad]D[Nms](b)2 2rrrrrrrrrrrrrrrrrrFigure 6: Bifurcation diagrams using Rs1, Ks= 2500(a) The bifurcation parameter is P , D = 2[Nms](b) The bifurcation parameter is D, P = 20[Nm]00.0020.0040.0060.0080.01-20 0 20 40 60 80 100 120 140'[rad]P [Nm](a)22rrrrrrrrrrrrrrrrrrrrrrrrr00.00020.00040.00060.00080.001-1 0 1 2 3 4 5 6 7'[rad]D[Nms](b)rrrrrrrrrrrrrrrrrrrrrrrrrrrrFigure 7: Bifurcation diagrams using Rs1, Ks= 5000(a) The bifurcation parameter is P , D = 2[Nms](b) The bifurcation parameter is D, P = 20[Nm]The bifurcation analysis in the vicinity of the upper equilibrium of thependulum is implemented using the approximate spring characteristics. Thebifurcation diagrams in case of Rs1are drawn in Figure 6(a), 6(b), 7(a) and7(b). Ks= 2500 in Figure 6 and Ks= 5000 in Figure 7. The bifurcationparameter is P in Figure 6(a) and 7(a) and D in Figure 6(b) and 7(b).Changing P at a xed value of D, the pitchfork bifurcation is found atthe same P0where in the linear case. Increasing Ks, the Hopf-bifurcationpoint is closer and closer to P0. A stable limit cycle appears at this point8and its amplitude increases as P approaches P1, the value where the Hopf-bifurcation occured in the linear case. Changing D at a xed value of P ,two Hopf-bifurcation points are indicated in Figure 6(b). The equilibrium isstable between them, but it cannot be stabilized at the same value of P forgreater Ks. Only the limit cycle is stable between the borders of the stabilitydomain calculated in the linear case.00.0020.0040.0060.0080.010.0120.014-20 0 20 40 60 80 100 120 140kukP [Nm](a)2 2bbbbbbbbbbbbbbbbbbbbbbbbbbbbbrrrrrrrr-0.100.10.20.30.40.5-1 0 1 2 3 4 5 6 7kukD[Nms](b)2 2b bb b b b b bb b b b b bb b brrrrrrrrrrrrrrFigure 8: Bifurcation diagrams using Rs2, Ks= 105(a) The bifurcation parameter is P , D = 2[Nms](b) The bifurcation parameter is D, P = 75[Nm]00.0020.0040.0060.0080.010.0120.014-20 0 20 40 60 80 100 120 140kukP [Nm](a)2 2bbbbbbbbbbbbbbbbbbbbbbbbrrrrrrrr-0.100.10.20.30.40.5-1 0 1 2 3 4 5 6 7kukD[Nms](b)2 2bb b bb bbbbbbbbbbbb b b b b b bbrrrrrrrrrrrrrrFigure 9: Bifurcation diagrams using Rs2, Ks= 3  106(a) The bifurcation parameter is P , D = 2[Nms](b) The bifurcation parameter is D, P = 75[Nm]The bifurcation diagrams in case of Rs2are shown in Figure 8(a), 8(b),9(a) and 9(b). Ks= 105in Figure 8 and Ks= 3  106in Figure 9. Thebifurcation parameter is P in Figure 8(a) and 9(a) and D in Figure 8(b) and99(b). Changing P at a xed value of D, the pitchfork bifurcation is foundat the same P0where in the linear case. Increasing Ks, the Hopf-bifurcationpoint moves towards less values of P , but it stops at P2= 69:89[Nm]. Anunstable limit cycle appears at this point and the numerical calculation isinterrupted at about P3= 46[Nm]. Its amplitude tends to 0 as Ksincreases.The L2-norm of the state variables can be seen in the gures, therefore thevalue where the stationary solution is indicated is r0. A bifurcation point issupposed at P3and the equilibrium is unstable for greater values of P . Astable limit cycle which was found with the other approximation is indicatedagain, but the calculation had stopped before the amplitude of the limitcycle would have decreased to 0. P is greater at the stopping point thanP0. A homoclinic orbit is showed up with simulations at this value, thus ahomoclinic bifurcation is assumed here. Changing D at a xed value of P ,two Hopf-bifurcation points are indicated. The equilibrium is stable betweenthem, but the unstable limit cycle is also found between them with decreasingamplitude as Ksincreases. The stable limit cycle is also indicated here.-1012345670 20 40 60 80 100 120 140 160D[Nms]P [Nm]bbbbbbbbbFigure 10: The stability chart with the bifurcation curvesExaminations are accomplished for the systems with approximate springcharacteristics and conclusions can be obtained for the exact piecewise linearsystem, thus the stability domain in the plain of the control parameters canbe constructed as it is sketched in Figure 10. It is bordered with the samestraight line and parabola as it was bordered in case of the linear system (thesystem without backlash). Fix points are stable in a little domain near thestraight line. Stable limit cycle appears at the homoclinic bifurcation pointindicated with the dotted line. Fix points lose their stability at the otherbifurcation point indicated with the smashed line, so all the x points andthe limit cycle are stable between the dotted and the smashed line, and only10the limit cycle is stable in the remaining part of the stability domain.6 ConclusionsBacklash causes the decrease of stability domain of the equilibrium. The sizeof the stability domain found in case of the linear spring characteristic isthe same, but a stable periodic solution exists instead of a stable stationarysolution in the largest part of this domain. The physical meaning of theperiodic solution is the oscillation of the stick around its vertical equilibrium.The physical meaning of the stable x points is that the control force doesnot push the stick further than the vertical line and it oscillates with less andless amplitude on either side of the vertical position.Acknowledgements: This research was supported by the Hungarian Scien-tic Research Foundation under grant no. OTKA T030762 and the Ministryof Culture and Education under grant no. FKFP 0380/97.References[1] Mori, S., Nishihara, H., Furuta, K., Control of an unstable mechanicalsystem, Int. J. Control 23, (1976) 673-692.[2] Stepan, G., A model of balancing, Periodica Polytechnica 28, (1984)195-199.[3] Stepan, G., Retarded Dynamical Systems, Longman, Harlow, UK, 1989.[4] Henders, M. G., Sondack, A. C., 'In-the-large' behaviour of an invertedpendulum with linear stabilization, Int. J. of Nonlinear Mechanics 27,(1992) 129-138.[5] Kawazoe, Y., Manual control and computer control of an inverted pen-dulum on a cart, Proc. 1st Int. Conf. on Motion and Vibration Control,pp. 930-935, Yokohama, 1992.[6] Enikov, E., Stepan, G., Stabilizing an Inverted Pendulum - Alternativesand Limitations, Periodica Polytechnica, Vol. 38, pp. 19-26, 1994.11[7] Lorant, G., Stepan, G., The Role of Non-Linearities in the Dynamics ofa Single Railway Wheelset, Machine Vibration 5, (1996) 18-26.[8] Enikov, E., Stepan, G., Micro-Chaotic Motion of Digitally ControlledMachines, J. of Vibration and Control, accepted in 1997.[9] Stepan, G., Kollar, L. E., Balancing with Reex Delay, Mathematicaland Computer Modelling, accepted in 1997.[10] Kollar, L. E., Backlash in Machines Stabilized by Control Force, Proc.of First Conference on Mechanical Engineering pp. 147-151, Budapest,1998.AuthorsKollar, Laszlo E. Hogan, S. JohnDepartment of Applied Mechanics Dep. of Engineering MathematicsBudapest University of University of BristolTechnology and Economics Queen's Building, University WalkH-1521, Budapest Bristol BS8 1TRHungary UKkollar@galilei.mm.bme.hu S.J.Hogan@bristol.ac.ukhttp://www.mm.bme.hu http://www.fen.bris.ac.uk/engmathsStepan, GaborDepartment of Applied MechanicsBudapest University ofTechnology and EconomicsH-1521, BudapestHungarystepan@galilei.mm.bme.huhttp://www.mm.bme.hu12

Backlash in balancing systems using approximate spring characteristics

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