A mechanical model of a digital balancing system is constructed and its stability analysis is presented. This model considers experimental problems like backlash and sampling delay. The conditions of existence of stable stationary and periodic solutions are determined for the case of the system without delay. Phase diagrams and bifurcation diagrams are revealed after simulations and bifurcation analysis. Adding sampling delay to the system, the stability conditions are changed and above a critical value of the delay, the balancing is impossible. The stability conditions and the stability chart are determined again and the critical sampling delay is calculated versus the parameters describing the system

Hogan, S.J.

Kollar, László E.

Stepan, Gabor

English

A mechanical model of a digital balancing system is constructed and its stability analysis is presented. This model considers experimental problems like backlash and sampling delay. The conditions of existence of stable stationary and periodic solutions are determined for the case of the system without delay. Phase diagrams and bifurcation diagrams are revealed after simulations and bifurcation analysis. Adding sampling delay to the system, the stability conditions are changed and above a critical value of the delay, the balancing is impossible. The stability conditions and the stability chart are determined again and the critical sampling delay is calculated versus the parameters describing the system.</p

Kollár, László E.

Stépán, Gábor

Hogan, S. John

Explore Bristol Research

Sampling delay and backlash in balancing systems

University of Huddersfield Repository

University of Huddersfield RepositoryKollar, László E., Stepan, Gabor and Hogan, S.J.Sampling Delay and Backlash in Balancing SystemsOriginal CitationKollar, László E., Stepan, Gabor and Hogan, S.J. (2000) Sampling Delay and Backlash in Balancing Systems. Periodica Polytechnica Mechanical Engineering, 44 (1). pp. 77-84. ISSN 0324-6051This version is available at http://eprints.hud.ac.uk/16842/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/PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 44, NO. 1, PP. 77–84 (2000)SAMPLING DELAY AND BACKLASHIN BALANCING SYSTEMSLászló E. KOLLÁR∗ , Gábor STÉPÁN∗ and S. John HOGAN∗∗∗ Department of Applied MechanicsTechnical University of BudapestH–1521 Budapest, Hungarye-mail: lakol@mm.bme.hu∗∗ Department of Engineering MathematicsUniversity of BristolBristol BS8 1TR, UKReceived: Sept. 5, 1999AbstractA mechanical model of a digital balancing system is constructed and its stability analysis is presented.This model considers experimental problems like backlash and sampling delay. The conditionsof existence of stable stationary and periodic solutions are determined for the case of the systemwithout delay. Phase diagrams and bifurcation diagrams are revealed after simulations and bifurcationanalysis. Adding sampling delay to the system, the stability conditions are changed and above a criticalvalue of the delay, the balancing is impossible. The stability conditions and the stability chart aredetermined again and the critical sampling delay is calculated versus the parameters describing thesystem.Keywords: sampling delay, backlash, bifurcation analysis.1. IntroductionUnstable equilibria of mechanical systems often have to be stabilized by controlforce. A number of applications can be found in this field, e.g. the bus running onicy road, the shimmying wheel or the balancing of standing and walking robots. Alot of problems occur during stabilization. Time delay, driving through elastic beltor backlash at the driving-wheel of the motor tend to destabilize dynamical systems.A typical example of stabilization of unstable equilibria is the balancing. Thesimplest model of balancing is that of the inverted pendulum [1,2,3,4,5]. The angleand the angular velocity of the pendulum are detected and a horizontal controlforce at the lowest point of the pendulum is determined by them in a way that thestick should be balanced at its upper position. Control parameters must be chosenfrom a bounded region for successful balancing. The stability conditions have beencalculated and the stability chart in the plane of the control parameters has beenconstructed in earlier works [6]. The stability domain shrinks as the time delayincreases and above a critical value of the delay, the successful balancing of theupper position of the pendulum is impossible.78 L. E. KOLLÁR et al.A digital balancing system is considered in the subsequent chapters. Theinverted pendulum and the motor displaying the control force are placed on a cartand the motor drives one of the wheels of the cart through a teeth-belt. Controllingis executed by a computer which is situated outside this cart. There are two generalfactors which influence the stability conditions: sampling delay and stiffness ofthe driving-belt. Increasing time delay or elasticity of the driving-belt tends todestabilize the examined system. Considering the backlash at the driving-wheel ofthe motor, the pendulum will swing with small amplitude around its equilibrium.The stability domain in the plane of the control parameters does not change, but anunstable zone appears in the phase diagram.2. The Mechanical Model and the Stability AnalysisIn order to describe a digital balancing system, the inverted pendulum is placedon a cart as it can be seen in Fig. 1 [7,8]. The motor drives one of the wheels ofthis cart through a teeth-belt with stiffness s. The system has 3 degrees of freedomdescribed by the general coordinates, x, ϕ and ψ . The angle ϕ of the pendulum andthe displacement x of the cart are detected together with their derivatives.Fig. 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 Um of the motor and to the angularvelocity ψ˙ :Mm = LUm − K ψ˙ . (1)Considering PD controllers, we have:Um = Pϕ + Dϕ˙ + Px x + Dx x˙ . (2)The system can be stabilized if the displacement of the cart is not detected(Px = 0)and the differential gain Dx of the cart eliminates the damping K of the motor.Then the control force has this simplified form:Q = L (Pϕ + Dϕ˙) . (3)SAMPLING DELAY 79The system is reduced to a system with 2 degrees of freedom if a new generalcoordinate is introduced. This is , the elongation of the spring: = rmψ − rwRw x . (4)The linearized equations of motion assume the form:((m+M)mmrm2 −mmmlrm rw4Rw0 ml23 − m2l24(m+M))(¨ϕ¨)+((m + M) Krm00 0)(˙ϕ˙)+(0 00 −mgl2)(ϕ)+((m + M) rm + mmrmr2w2R2wmlrw2(m+M)Rw)Rs =((m + M) Q0), (5)whereRs =( −s rwRw 0 srm )(xϕψ)= s (6)is the force in the spring.The stability analysis is carried out by the Routh–Hurwitz criterion. If thebelt is ideally rigid, then  = 0, x determines ψ uniquely, so the system has 2degrees of freedom, namely x and ϕ. The ϕ ≡ 0 trivial solution of this system isasymptotically stable if and only ifP > P0 = 1L[(m + M + 12mmr2wR2w)grm Rwrw]and D > 0 . (7)If the belt is elastic, then the trivial solution of (4) is asymptotically stable if andonly ifP > P0 and H2 > 0 , (8)where H2 is the maximum sized Hurwitz determinant, not presented here algeb-raically.The stability chart is constructed as it is shown in Fig. 1. The stability domainshrinks as the stiffness of the driving-belt decreases and at a certain critical value,it disappears. This critical value has this form:s > scrit = 3 (m + M)mmg(m + 4M + 2mm r2wR2w)l. (9)80 L. E. KOLLÁR et al.3. Numerical Study of the Phase-SpaceBacklash appears in the system as a nonlinear spring characteristic. The force inthe spring is the function of :Rs =s (+ r0)  ≤ −r00 || < r0s (− r0)  ≥ r0, (10)where r0 is the value of backlash. This function is given in Fig. 2.-10-50510-0.002 -0.001 0 0.001 0.002Rs[N ][m]Fig. 2. The nonlinear spring characteristic at s = 10[ kNm] and r0 = 1[mm]New constant expressions appear in the equations of motion, that means shift-ing of the solutions. The stability domain does not change but it is valid only if|| > r0. Otherwise, the system is just in backlash, so it cannot be stabilized,because the control force is not displayed in this little domain.If the control parameters are chosen from the stability domain, then roots of thecharacteristic equation are complex numbers with negative real parts. Trajectoriesform stable focus around the(ϕ, ϕ˙,, ˙) = (0, 0,±r0, 0) equilibria.If the system is just in backlash, then the roots of the characteristic equationare positive and negative reals. Trajectories form saddle around the (0, 0, 0, 0)equilibrium.Simulations were accomplished for the study of the phase-space [9]. Resultsare presented in Fig. 3(a) and 3(b) near different values of either of the controlparameters, P . The (0, 0,±r0, 0) equilibria are stable for smaller values of P . Astable periodic solution appears for greater values of P and its amplitude is largerand larger as P increases. Its amplitude tends to infinity as P tends to the borderof the stability domain. Now the stability domain means the domain where stablestationary or periodic solution can be found. The physical meaning of the periodicsolution is the oscillation of the stick around its vertical equilibrium. The physicalmeaning of the stable fix points is that the control force does not push the stickfurther than the vertical line and it oscillates with less and less amplitude on eitherside of the vertical position. For certain values of P all the(0, 0,±r0, 0) equilibriaSAMPLING DELAY 81and the limit cycle are stable and trajectories spiral to one of them depending onthe initial conditions. More investigations are needed for the exact knowledge ofthe phase-space.-0.2-0.15-0.1-0.0500.050.10.15-0.002 -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)Fig. 3. Phase diagram on ˙− plane, (a) P = 2, D = 2, (b) P = 20, D = 24. The Bifurcation AnalysisThe bifurcation diagram was also examined numerically. The spring characteristichas a noncontinuous first derivative and this caused problems during determiningperiodic solutions. Therefore two approximate spring characteristics were appliedas they are shown in Fig. 4. Both of them include a parameter Ks , and approxima-tions are more and more accurate, as it tends to infinity.Rs1 = sKs ln1 + eKs(−r0)1 + e−Ks(+r0) ,Rs2 =s (+ r0)  ≤ −r0s (− r0) eKs(−r0) + s (+ r0) e−Ks (+r0) || < r0s (− r0)  ≥ r0. (11)A stable periodic solution can be obtained with both of the approximations.These solutions are closer and closer to each other as Ks increases. The limitbetween the two kinds of approximation is the solution with respect to the exactpiecewise linear system. The periodic solution appears at certain pairs of P, Dvalues. Approaching one of these pairs, the period of the solution increases as itcan be seen in Fig. 5(a), and a homoclinic orbit shows up for these pairs as it isshown in Fig. 5(b).The bifurcation diagrams are presented in Fig. 6(a) and 6(b), where the bifur-cation parameters are the control parameters, P and D. D = 2 in Fig. 6(a). Thereis a branch point at either border of the stability domain where stable fix pointsappear. A homoclinic orbit occurs at a certain value of P , so there is a homoclinic82 L. E. KOLLÁR et al.-10-50510-0.002 -0.001 0 0.001 0.002Rs1[N ][m](a)-10-50510-0.002 -0.001 0 0.001 0.002Rs2[N ][m](b)Fig. 4. Approximate spring characteristics at s = 10[ kNm ], r0 = 1[mm] and Ks = 104(a) Rs1, (b) Rs200.20.40.60.811.20 20 40 60 80 100 120T [s]P(a)-0.008-0.00400.0040.008-0.001 0 0.001_[ms][m](b)Fig. 5. (a) Period of the periodic solution at D = 2, (b) The homoclinic orbit at P = 3.46and D = 2bifurcation there, where a limit cycle appears. All the fix points and the limit cycleare stable in a domain for greater values of P . All of them have a domain of attrac-tion, so trajectories spiral to one of them depending on the initial conditions. Thereis another bifurcation point at another value of P , where the fix points become un-stable. The periodic solution retains its stability till the parameters reach the otherborder of the stability domain with increasing amplitude.P = 50 in Fig. 6(b). There is a stable limit cycle between the borders of thestability domain but the fix points are stable only along the continuous line betweenthe bifurcation points indicated with squares in the figure.After the bifurcation analysis the stability chart in the plane of the controlparameters can be constructed as it can be seen in Fig. 7(a). It is bordered with thesame straight 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 the straightline. Stable limit cycle appears at the homoclinic bifurcation point indicated withthe dotted line. Fix points lose their stability at the other bifurcation point indicatedwith the smashed line, so all the fix points and the limit cycle are stable betweenSAMPLING DELAY 8300.20.40.60.810 20 40 60 80 100 120 140kukP(a)rrrrrrrrrrrrrrrrrrrrrrs s-0.500.511.522.5-1 0 1 2 3 4 5 6 7kukD(b)rrrrrrrrrrrrrrrrrrrrrrrrrrrs sFig. 6. Bifurcation diagrams, the bifurcation parameter is (a) P, (b) Dthe dotted and the smashed line, and only the limit cycle is stable in the remainingpart of the stability domain.-1012345670 20 40 60 80 100 120 140 160DP(a)bbbbbbbbb-1012345670 20 40 60 80 100 120 140 160DP(b)Fig. 7. (a) The stability chart with the bifurcation curves, (b) The stability charts withoutand with sampling delay5. Backlash and Sampling Delay TogetherRegarding the sampling delay τ the system becomes discrete. Simulations and in-vestigation of the phase-space are implemented again. The character of the stabilitydomain is the same as it was at the system without time delay, but it is smaller andsmaller as the time delay increases. The stability chart is given in Fig. 7(b) forτ = 0[s] and τ = 0.005[s].Balancing of the pendulum is impossible above a critical value of the samplingdelay as it was mentioned earlier. This critical value depends on the parametersdescribing the system. The connection between the critical sampling delay and thelength of the pendulum is shown in Fig. 8(a) and between the critical samplingdelay and the spring stiffness is shown in Fig. 8(b). An asymptote can be seen inthis figure, which is the same as the result of the examination of the system withrigid belt.84 L. E. KOLLÁR et al.00.050.10.150.20.250 0.2 0.4 0.6 0.8 1cr[s]l[m](a)00.050.10.150.20.250 2000 4000 6000 8000 10000cr[s]s[Nm](b)Fig. 8. (a) The critical sampling delay vs. the length of the pendulum, (b) The criticalsampling delay vs. the spring stiffness6. ConclusionsIncreasing time delay and decreasing spring stiffness of the driving belt tend todestabilize the digitally controlled dynamical systems. Backlash causes oscilla-tions of the pendulum around its upper equilibrium. It behaves as a spatial delay.The examined model is an example for the problems of stabilization of unstableequilibria of mechanical systems, but the main principles and methods are valid forstabilization of unstable equilibria of any other controlled mechanical systems.AcknowledgementThis research was supported by the Hungarian Scientific Research Foundation under grantno. OTKA T030762 and the Ministry of Culture and Education under grant no. FKFP 0380/97.References[1] HIGDON, D. T. – CANNON, R. H.: On the Control of Unstable Multiple-output MechanicalSystems, ASME Publications, Vol. 63-WA-48, (1963) pp. 1–12.[2] MORI, S. – NISHIHARA, H. – FURUTA, K.: Control of an Unstable Mechanical System, Int. J.Control, Vol. 23, (1976) pp. 673–692.[3] STÉPÁN, G.: A Model of Balancing, Periodica Polytechnica, Vol. 28, (1984) pp. 195–199.[4] HENDERS, M. G. – SONDACK, A. C.: ‘In-the-large’ Behaviour of an Inverted Pendulum withLinear Stabilization, Int. J. of Nonlinear Mechanics, Vol. 27, (1992) pp. 129–138.[5] KAWAZOE, Y.: Manual Control and Computer Control of an Inverted Pendulum on a Cart, Proc.1st Int. Conf. on Motion and Vibration Control, pp. 930–935, Yokohama, 1992.[6] STÉPÁN, G. – KOLLÁR, L. E.: Balancing with Reflex Delay, Mathematical and ComputerModelling, accepted in 1997.[7] ENIKOV, E. – STÉPÁN, G.: Micro-Chaotic Motion of Digitally Controlled Machines, J. ofVibration and Control, accepted in 1997.[8] KOLLÁR, L. E.: Backlash in Machines Stabilized by Control Force, Proc. of First Conferenceon Mechanical Engineering, pp. 147–151, Budapest, 1998.[9] LÓRÁNT, G. – STÉPÁN, G.: The Role of Non-Linearities in the Dynamics of a Single RailwayWheelset, Machine Vibration, Vol. 5, (1996) pp. 18–26.

A Model of Balancing,

Backlash in Machines Stabilized by Control Force,

Balancing with Reflex Delay, Mathematical and Computer Modelling, accepted in

Control of an Unstable Mechanical System,

In-the-large’ Behaviour of an Inverted Pendulum with Linear Stabilization,

Manual Control and Computer Control of an Inverted Pendulum on a Cart,

Micro-Chaotic Motion of Digitally Controlled Machines,

On the Control of Unstable Multiple-output Mechanical Systems,

Sampling Delay and Backlash in Balancing Systems Original Citation Kollar, László

The Role of Non-Linearities in the Dynamics of a Single Railway Wheelset,

Sampling Delay and Backlash in Balancing Systems

A mechanical model of a digital balancing system is
constructed and its stability analysis is presented. This model considers
experimental problems like backlash and sampling delay. The conditions
of existence of stable stationary and periodic solutions are determined for
the case of the system without delay. Phase diagrams and bifurcation
diagrams are revealed after simulations and bifurcation analysis. Adding
sampling delay to the system, the stability conditions are changed and
above a critical value of the delay, the balancing is impossible. The
stability conditions and the stability chart are determined again and
the critical sampling delay is calculated versus the parameters
describing the system

E. Kollár, László

John Hogan, S.

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 44, NO. 1, PP. 77–84 (2000)SAMPLING DELAY AND BACKLASHIN BALANCING SYSTEMSLászló E. KOLLÁR∗, Gábor STÉPÁN∗ and S. John HOGAN∗∗∗ Department of Applied MechanicsTechnical University of BudapestH–1521 Budapest, Hungarye-mail: lakol@mm.bme.hu∗∗ Department of Engineering MathematicsUniversity of BristolBristol BS8 1TR, UKReceived: Sept. 5, 1999AbstractA mechanical model of a digital balancing system is constructed and its stability analysis is presented.This model considers experimental problems like backlash and sampling delay. The conditionsof existence of stable stationary and periodic solutions are determined for the case of the systemwithout delay. Phase diagrams and bifurcation diagrams are revealed after simulations and bifurcationanalysis. Adding sampling delay to the system, the stability conditions are changed and above a criticalvalue of the delay, the balancing is impossible. The stability conditions and the stability chart aredetermined again and the critical sampling delay is calculated versus the parameters describing thesystem.Keywords: sampling delay, backlash, bifurcation analysis.1. IntroductionUnstable equilibria of mechanical systems often have to be stabilized by controlforce. A number of applications can be found in this field, e.g. the bus running onicy road, the shimmying wheel or the balancing of standing and walking robots. Alot of problems occur during stabilization. Time delay, driving through elastic beltor backlash at the driving-wheel of the motor tend to destabilize dynamical systems.A typical example of stabilization of unstable equilibria is the balancing. Thesimplest model of balancing is that of the inverted pendulum [1,2,3,4,5]. The angleand the angular velocity of the pendulum are detected and a horizontal controlforce at the lowest point of the pendulum is determined by them in a way that thestick should be balanced at its upper position. Control parameters must be chosenfrom a bounded region for successful balancing. The stability conditions have beencalculated and the stability chart in the plane of the control parameters has beenconstructed in earlier works [6]. The stability domain shrinks as the time delayincreases and above a critical value of the delay, the successful balancing of theupper position of the pendulum is impossible.78 L. E. KOLLÁR et al.A digital balancing system is considered in the subsequent chapters. Theinverted pendulum and the motor displaying the control force are placed on a cartand the motor drives one of the wheels of the cart through a teeth-belt. Controllingis executed by a computer which is situated outside this cart. There are two generalfactors which influence the stability conditions: sampling delay and stiffness ofthe driving-belt. Increasing time delay or elasticity of the driving-belt tends todestabilize the examined system. Considering the backlash at the driving-wheel ofthe motor, the pendulum will swing with small amplitude around its equilibrium.The stability domain in the plane of the control parameters does not change, but anunstable zone appears in the phase diagram.2. The Mechanical Model and the Stability AnalysisIn order to describe a digital balancing system, the inverted pendulum is placedon a cart as it can be seen inFig. 1 [7,8]. The motor drives one of the wheels ofthis cart through a teeth-belt with stiffnesss. The system has 3 degrees of freedomdescribed by the general coordinates,x, ϕ andψ . The angleϕ of the pendulum andthe displacementx of the cart are detected together with their derivatives.Fig. 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 voltageUm of the motor and to the angularvelocity ψ̇ :Mm = LUm − K ψ̇ . (1)Considering PD controllers, we have:Um = Pϕ + Dϕ̇ + Px x + Dx ẋ . (2)The system can be stabilized if the displacement of the cart is not detected(Px = 0)and the differential gainDx of the cart eliminates the dampingK of the motor.Then the control force has this simplified form:Q = L (Pϕ + Dϕ̇) . (3)SAMPLING DELAY 79The system is reduced to a system with 2 degrees of freedom if a new generalcoordinate is introduced. This is, the elongation of the spring: = rmψ − rwRwx . (4)The linearized equations of motion assume the form:((m+M)mmrm2 −mmmlrm rw4Rw0 ml23 − m2l24(m+M))(̈ϕ̈)+((m + M) Krm 00 0)(̇ϕ̇)+(0 00 −mgl2)(ϕ)+((m + M) rm + mmrmr2w2R2wmlrw2(m+M)Rw)Rs =((m + M) Q0), (5)whereRs =( −s rwRw 0 srm )(xϕψ)= s (6)is the force in the spring.The stability analysis is carried out by the Routh–Hurwitz criterion. If thebelt is ideally rigid, then = 0, x determinesψ uniquely, so the system has 2degrees of freedom, namelyx andϕ. Theϕ ≡ 0 trivial solution of this system isasymptotically stable if and only ifP > P0 = 1L[(m + M + 12mmr2wR2w)grm Rwrw]and D > 0 . (7)If the belt is elastic, then the trivial solution of (4) is asymptotically stable if andonly ifP > P0 and H2 > 0 , (8)whereH2 is the maximum sized Hurwitz determinant, not presented here algeb-raically.The stability chart is constructed as it is shown inF g. 1. The stability domainshrinks as the stiffness of the driving-belt decreases and at a certain critical value,it disappears. This critical value has this form:s > scrit = 3(m + M)mmg(m + 4M + 2mm r2wR2w)l. (9)80 L. E. KOLLÁR et al.3. Numerical Study of the Phase-SpaceBacklash appears in the system as a nonlinear spring characteristic. The force inthe spring is the function of:Rs =s (+ r0)  ≤ −r00 || < r0s (− r0)  ≥ r0, (10)wherer0 is the value of backlash. This function is given inFig. 2.-10-50510-0.002 -0.001 0 0.001 0.002Rs[N ][m]Fig. 2. The nonlinear spring characteristic ats = 10[ kNm ] andr0 = 1[mm]New constant expressions appear in the equations of motion, that means shift-ing of the solutions. The stability domain does not change but it is valid only if|| > r0. Otherwise, the system is just in backlash, so it cannot be stabilized,because the control force is not displayed in this little domain.If the control parameters are chosen from the stability domain, then roots of thecharacteristic equation are complex numbers with negative real parts. Trajectoriesform stable focus around the(ϕ, ϕ̇,, ̇) = (0,0,±r0,0) equilibria.If the system is just in backlash, then the roots of the characteristic equationare positive and negative reals. Trajectories form saddle around the(0,0,0,0)equilibrium.Simulations were accomplished for the study of the phase-space [9]. Resultsare presented inFig. 3(a) and 3(b) near different values of either of the controlparameters,P. The(0,0,±r0,0) equilibria are stable for smaller values ofP. Astable periodic solution appears for greater values ofP and its amplitude is largerand larger asP increases. Its amplitude tends to infinity asP tends to the borderof the stability domain. Now the stability domain means the domain where stablestationary or periodic solution can be found. The physical meaning of the periodicsolution is the oscillation of the stick around its vertical equilibrium. The physicalmeaning of the stable fix points is that the control force does not push the stickfurther than the vertical line and it oscillates with less and less amplitude on eitherside of the vertical position. For certain values ofP all the(0,0,±r0,0) equilibriaSAMPLING DELAY 81and the limit cycle are stable and trajectories spiral to one of them depending onthe initial conditions. More investigations are needed for the exact knowledge ofthe phase-space.-0.2-0.15-0.1-0.0500.050.10.15-0.002 -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)Fig. 3. Phase diagram oṅ− plane, (a)P = 2, D = 2, (b) P = 20, D = 24. The Bifurcation AnalysisThe bifurcation diagram was also examined numerically. The spring characteristichas a noncontinuous first derivative and this caused problems during determiningperiodic solutions. Therefore two approximate spring characteristics were appliedas they are shown inFig. 4. Both of them include a parameterKs , and approxima-tions are more and more accurate, as it tends to infinity.Rs1 = sKsln1 + eKs(−r0)1 + e−Ks(+r0) ,Rs2 =s (+ r0)  ≤ −r0s (− r0) eKs(−r0) + s (+ r0) e−Ks (+r0) || < r0s (− r0)  ≥ r0. (11)A stable periodic solution can be obtained with both of the approximations.These solutions are closer and closer to each other asKs increases. The limitbetween the two kinds of approximation is the solution with respect to the exactpiecewise linear system. The periodic solution appears at certain pairs ofP, Dvalues. Approaching one of these pairs, the period of the solution increases as itcan be seen inFig. 5(a), and a homoclinic orbit shows up for these pairs as it isshown inFig. 5(b).The bifurcation diagrams are presented inFig. 6(a) and6(b), where the bifur-cation parameters are the control parameters,P andD. D = 2 in Fig. 6(a). Thereis a branch point at either border of the stability domain where stable fix pointsappear. A homoclinic orbit occurs at a certain value ofP, so there is a homoclinic82 L. E. KOLLÁR et al.-10-50510-0.002 -0.001 0 0.001 0.002Rs1[N ][m](a)-10-50510-0.002 -0.001 0 0.001 0.002Rs2[N ][m](b)Fig. 4. Approximate spring characteristics ats = 10[ kNm ], r0 = 1[mm] andKs = 104(a) Rs1, (b) Rs200.20.40.60.811.20 20 40 60 80 100 120T [s]P(a)-0.008-0.00400.0040.008-0.001 0 0.001_[ms][m](b)Fig. 5. (a) Period of the periodic solution atD = 2, (b) The homoclinic orbit atP = 3.46andD = 2bifurcation there, where a limit cycle appears. All the fix points and the limit cycleare stable in a domain for greater values ofP. All of them have a domain of attrac-tion, so trajectories spiral to one of them depending on the initial conditions. Thereis another bifurcation point at another value ofP, where the fix points become un-stable. The periodic solution retains its stability till the parameters reach the otherborder of the stability domain with increasing amplitude.P = 50 in Fig. 6(b). There is a stable limit cycle between the borders of thestability domain but the fix points are stable only along the continuous line betweenthe bifurcation points indicated with squares in the figure.After the bifurcation analysis the stability chart in the plane of the controlparameters can be constructed as it can be seen inFig. 7(a). It is bordered with thesame straight 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 the straightline. Stable limit cycle appears at the homoclinic bifurcation point indicated withthe dotted line. Fix points lose their stability at the other bifurcation point indicatedwith the smashed line, so all the fix points and the limit cycle are stable betweenSAMPLING DELAY 8300.20.40.60.810 20 40 60 80 100 120 140kukP(a)rrrrrrrrrrrrrrrrrrrrrrs s-0.500.511.522.5-1 0 1 2 3 4 5 6 7kukD(b)rrrrrrrrrr r r r r r rrrrrrrrrrrrs sFig. 6. Bifurcation diagrams, the bifurcation parameter is (a)P, (b) Dthe dotted and the smashed line, and only the limit cycle is stable in the remainingpart of the stability domain.-1012345670 20 40 60 80 100 120 140 160DP(a)bbbbbbbbb-1012345670 20 40 60 80 100 120 140 160DP(b)Fig. 7. (a) The stability chart with the bifurcation curves, (b) The stability charts withoutand with sampling delay5. Backlash and Sampling Delay TogetherRegarding the sampling delayτ the system becomes discrete. Simulations and in-vestigation of the phase-space are implemented again. The character of the stabilitydomain is the same as it was at the system without time delay, but it is smaller andsmaller as the time delay increases. The stability chart is given inFig. 7(b) forτ = 0[s] andτ = 0.005[s].Balancing of the pendulum is impossible above a critical value of the samplingdelay as it was mentioned earlier. This critical value depends on the parametersdescribing the system. The connection between the critical sampling delay and thelength of the pendulum is shown inFig. 8(a) and between the critical samplingdelay and the spring stiffness is shown inFig. 8(b). An asymptote can be seen inthis figure, which is the same as the result of the examination of the system withrigid belt.84 L. E. KOLLÁR et al.00.050.10.150.20.250 0.2 0.4 0.6 0.8 1cr[s]l[m](a)00.050.10.150.20.250 2000 4000 6000 8000 10000cr[s]s[Nm](b)Fig. 8. (a) The critical sampling delay vs. the length of the pendulum, (b) The criticalsampling delay vs. the spring stiffness6. ConclusionsIncreasing time delay and decreasing spring stiffness of the driving belt tend todestabilize the digitally controlled dynamical systems. Backlash causes oscilla-tions of the pendulum around its upper equilibrium. It behaves as a spatial delay.The examined model is an example for the problems of stabilization of unstableequilibria of mechanical systems, but the main principles and methods are valid forstabilization of unstable equilibria of any other controlled mechanical systems.AcknowledgementThis research was supported by the Hungarian Scientific Research Foundation under grantno. OTKA T030762 and the MinistryofCulture andEducation undergrantno. FKFP 0380/97.References[1] HIGDON, D. T. – CANNON, R. H.: On the Control of Unstable Multiple-output MechanicalSystems,ASME Publications, Vol. 63-WA-48, (1963) pp. 1–12.[2] M ORI, S. – NISHIHARA, H. – FURUTA, K.: Control of an Unstable Mechanical System,Int. J.Control, Vol. 23, (1976) pp. 673–692.[3] STÉPÁN, G.: A Model of Balancing,Periodica Polytechnica, Vol. 28, (1984) pp. 195–199.[4] HENDERS, M. G. – SONDACK, A. C.: ‘In-the-large’ Behaviour of an Inverted Pendulum withLinear Stabilization,Int. J. of Nonlinear Mechanics, Vol. 27, (1992) pp. 129–138.[5] K AWAZOE, Y.: Manual Control and Computer Control of an Inverted Pendulum on a Cart,Proc.1st Int. Conf. on Motion and Vibration Control, pp. 930–935, Yokohama, 1992.[6] STÉPÁN, G. – KOLLÁR, L. E.: Balancing with Reflex Delay,Mathematical and ComputerModelling, accepted in 1997.[7] ENIKOV, E. – STÉPÁN, G.: Micro-Chaotic Motion of Digitally Controlled Machines,J. ofVibration and Control, accepted in 1997.[8] K OLLÁR, L. E.: Backlash in Machines Stabilized by Control Force,Proc. of First Conferenceon Mechanical Engineering, pp. 147–151, Budapest, 1998.[9] L ÓRÁNT, G. – STÉPÁN, G.: The Role of Non-Linearities in the Dynamics of a Single RailwayWheelset,Machine Vibration, Vol. 5, (1996) pp. 18–26.

SAMPLING DELAY AND BACKLASH IN BALANCING SYSTEMS

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Sampling Delay and Backlash in Balancing Systems

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