PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 44, NO., PP. 77 84 () SAMPLING DELAY AND BACKLASH IN BALANCING SYSTEMS László E. KOLLÁR, Gáo STÉPÁN and S. John HOGAN Depatment of Applied Mechanics Technical Univesity of Budapest H Budapest, Hungay e-mail: lakol@mm.me.hu Depatment of Engineeing Mathematics Univesity of Bistol Bistol BS8 TR, UK Received: Sept., 999 Astact A mechanical model of a digital alancing system is constucted and its staility analysis is pesented. This model consides expeimental polems like acklash and sampling delay. The conditions of existence of stale stationay and peiodic solutions ae detemined fo the case of the system without delay. Phase diagams and ifucation diagams ae evealed afte simulations and ifucation analysis. Adding sampling delay to the system, the staility conditions ae changed and aove a citical value of the delay, the alancing is impossile. The staility conditions and the staility chat ae detemined again and the citical sampling delay is calculated vesus the paametes desciing the system. Keywods: sampling delay, acklash, ifucation analysis.. 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 lot of polems occu duing stailization. Time delay, diving though elastic elt o acklash at the diving-wheel of the moto tend to destailize dynamical systems. A typical example of stailization of unstale equiliia is the alancing. The simplest model of alancing is that of the inveted pendulum [,,3,4,]. 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 plane of the contol paametes has een constucted in ealie woks [6]. The staility domain shinks as the time delay inceases and aove a citical value of the delay, the successful alancing of the uppe position of the pendulum is impossile.
78 L. E. KOLLÁR et al. A digital alancing system is consideed in the susequent chaptes. The inveted pendulum and the moto displaying the contol foce ae placed on a 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. Thee ae two geneal factos which influence the staility conditions: sampling delay and stiffness of the diving-elt. Inceasing time delay o elasticity of the diving-elt tends to destailize the examined system. 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 plane of the contol paametes does not change, ut an unstale zone appeas in the phase diagam.. The Mechanical Model and the Staility Analysis In ode to descie a digital alancing system, the inveted pendulum is placed on a cat as it can e seen in Fig. [7,8]. 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,ϕ and ψ. The angle ϕ of the pendulum and the displacement x of the cat ae detected togethe with thei deivatives. Fig.. 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: M m = LU m K ψ. () Consideing PD contolles, we have: U m = Pϕ + D ϕ + P x x + D x ẋ. () 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 (Pϕ + D ϕ). (3)
SAMPLING DELAY 79 The system is educed to a system with degees of feedom if a new geneal coodinate is intoduced. This is, the elongation of the sping: The lineaized equations of motion assume the fom: = m ψ w R w x. (4) ( (m+m)mm m mm ml m w 4R w ml 3 m l 4(m+M) ) ( ϕ ) ( (m + M) K + m )( ϕ ) + ( mgl )( ϕ ) ( + (m + M) m + m m m w R w ml w (m+m)r w ) ( (m + M) Q R s = ), () whee R s = ( s ) ( ) x w Rw s m ϕ = s (6) ψ 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 degees of feedom, namely x and ϕ. The ϕ tivial solution of this system is asymptotically stale if and only if P > P = L [( m + M + m ) w m g ] m R w Rw w and D >. (7) If the elt is elastic, then the tivial solution of (4) is asymptotically stale if and only if P > P and H >, (8) whee H is the maximum sized Huwitz deteminant, not pesented hee algeaically. The staility chat is constucted as it is shown in Fig.. 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 ( ). (9) w m + 4M + m m l Rw
8 L. E. KOLLÁR et al. 3. Numeical Study of the Phase-Space Backlash appeas in the system as a nonlinea sping chaacteistic. The foce in the sping is the function of : s ( + ) R s = <, () s ( ) whee is the value of acklash. This function is given in Fig.. Rs[N ] - - -. -... Fig.. The nonlinea sping chaacteistic at s = [ kn m ] and = [mm] 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 >. 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. Tajectoies fom stale focus aound the ( ϕ, ϕ,, ) = (,, ±, ) 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. Simulations wee accomplished fo the study of the phase-space [9]. Results ae pesented in Fig. 3 and 3() 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 ode of the staility domain. Now the staility domain means the domain whee stale stationay o peiodic solution can e found. 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. Fo cetain values of P all the(,, ±, ) equiliia
SAMPLING DELAY 8 and the limit cycle ae stale and tajectoies spial to one of them depending on the initial conditions. Moe investigations ae needed fo the exact knowledge of the phase-space... ().... _ [ m s ] -. _ [ m s ] -. -. -. -. -. -. -. -... -. -. -... Fig. 3. Phase diagam on plane, P =, D =, () P =, D = 4. The Bifucation Analysis The ifucation diagam was also examined numeically. The sping chaacteistic has a noncontinuous fist deivative and this caused polems duing detemining peiodic solutions. Theefoe two appoximate sping chaacteistics wee applied as they ae shown in Fig. 4. Both of them include a paamete K s, and appoximations ae moe and moe accuate, as it tends to infinity. R s = s K s ln + ek s( ) + e K s( + ), s ( + ) R s = s ( ) e K s( ) + s ( + ) e K s( + ) <. () s ( ) A stale peiodic solution can e otained with oth of the appoximations. These solutions ae close and close to each othe as K s inceases. The limit etween the two kinds of appoximation is the solution with espect to the exact piecewise linea system. The peiodic solution appeas at cetain pais of P, D values. Appoaching one of these pais, the peiod of the solution inceases as it can e seen in Fig., and a homoclinic oit shows up fo these pais as it is shown in Fig. (). The ifucation diagams ae pesented in Fig. 6 and 6(), whee the ifucation paametes ae the contol paametes, P and D. D = infig. 6. Thee is a anch point at eithe ode of the staility domain whee stale fix points appea. A homoclinic oit occus at a cetain value of P, so thee is a homoclinic
8 L. E. KOLLÁR et al. () Rs[N ] Rs[N ] - - - -. -... - -. -... Fig. 4. Appoximate sping chaacteistics at s = [ kn m ], = [mm] and K s = 4 R s, () R s..8 ().4.8 T [s].6 _ [ m s ].4 -.4. 4 6 8 P -.8 -.. Fig.. Peiod of the peiodic solution at D =, () The homoclinic oit at P = 3.46 and D = ifucation thee, whee a limit cycle appeas. All the fix points and the limit cycle ae stale in a domain fo geate values of P. All of them have a domain of attaction, so tajectoies spial to one of them depending on the initial conditions. Thee is anothe ifucation point at anothe value of P, whee the fix points ecome unstale. The peiodic solution etains its staility till the paametes each the othe ode of the staility domain with inceasing amplitude. P = in Fig. 6(). Thee is a stale limit cycle etween the odes of the staility domain ut the fix points ae stale only along the continuous line etween the ifucation points indicated with squaes in the figue. Afte the ifucation analysis the staility chat in the plane of the contol paametes can e constucted as it can e seen in Fig. 7. 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
SAMPLING DELAY 83 () kuk.8.6.4. s s kuk... s s -. 4 6 8 4 P - 3 4 6 7 D Fig. 6. Bifucation diagams, the ifucation paamete is P, () D the dotted and the smashed line, and only the limit cycle is stale in the emaining pat of the staility domain. () 7 7 6 6 4 4 D 3 D 3 - - 4 6 8 4 6 P 4 6 8 4 6 P Fig. 7. The staility chat with the ifucation cuves, () The staility chats without and with sampling delay. Backlash and Sampling Delay Togethe Regading the sampling delay τ the system ecomes discete. Simulations and investigation of the phase-space ae implemented again. The chaacte of the staility domain is the same as it was at the system without time delay, ut it is smalle and smalle as the time delay inceases. The staility chat is given in Fig. 7() fo τ = [s] and τ =.[s]. Balancing of the pendulum is impossile aove a citical value of the sampling delay as it was mentioned ealie. This citical value depends on the paametes desciing the system. The connection etween the citical sampling delay and the length of the pendulum is shown in Fig. 8 and etween the citical sampling delay and the sping stiffness is shown in Fig. 8(). An asymptote can e seen in this figue, which is the same as the esult of the examination of the system with igid elt.
84 L. E. KOLLÁR et al. ()...... fic [s] fic [s]......4.6.8 l[m] 4 6 8 s[ N m ] Fig. 8. The citical sampling delay vs. the length of the pendulum, () The citical sampling delay vs. the sping stiffness 6. Conclusions Inceasing time delay and deceasing sping stiffness of the diving elt tend to destailize the digitally contolled dynamical systems. Backlash causes oscillations of the pendulum aound its uppe equiliium. It ehaves as a spatial delay. The examined model is an example fo the polems of stailization of unstale equiliia of mechanical systems, ut the main pinciples and methods ae valid fo stailization of unstale equiliia of any othe contolled mechanical systems. Acknowledgement This eseach was suppoted y the Hungaian Scientific Reseach Foundation unde gant no. OTKA T376 and the Ministy of Cultue and Education unde gant no. FKFP 38/97. Refeences [] HIGDON, D. T. CANNON, R. H.: On the Contol of Unstale Multiple-output Mechanical Systems, ASME Pulications, Vol. 63-WA-48, (963) pp.. [] MORI, S. NISHIHARA, H. FURUTA, K.: Contol of an Unstale Mechanical System, Int. J. Contol, Vol. 3, (976) pp. 673 69. [3] STÉPÁN, G.: A Model of Balancing, Peiodica Polytechnica, Vol. 8, (984) pp. 9 99. [4] HENDERS, M. G. SONDACK, A. C.: In-the-lage Behaviou of an Inveted Pendulum with Linea Stailization, Int. J. of Nonlinea Mechanics, Vol. 7, (99) pp. 9 38. [] KAWAZOE, Y.: Manual Contol and Compute Contol of an Inveted Pendulum on a Cat, Poc. st Int. Conf. on Motion and Viation Contol, pp. 93 93, Yokohama, 99. [6] STÉPÁN, G. KOLLÁR, L. E.: Balancing with Reflex Delay, Mathematical and Compute Modelling, accepted in 997. [7] ENIKOV, E. STÉPÁN, G.: Mico-Chaotic Motion of Digitally Contolled Machines, J. of Viation and Contol, accepted in 997. [8] KOLLÁR, L. E.: Backlash in Machines Stailized y Contol Foce, Poc. of Fist Confeence on Mechanical Engineeing, pp. 47, Budapest, 998. [9] LÓRÁNT, G. STÉPÁN, G.: The Role of Non-Lineaities in the Dynamics of a Single Railway Wheelset, Machine Viation, Vol., (996) pp. 8 6.