Chaos in drive systems

Transcription

1 Applied nd Computtionl Mechnics 1 (2007) Chos in drive systems Ctird Krtochvíl, Mrtin Houfek, Josef Koláčný b, Romn Kříž b, Lubomír Houfek,*, Jiří Krejs Institute of Thermomechnics, brnch Brno, Czech Acdemy of Sciences, Technická 2896/2, Brno, Czech Republic b Institute of Power Electricl nd Electronic Engineering, Fculty of Electricl Engineering nd Communiction, Technická 8, Brno, Czech Republic Received 27 September 2007; received in revised form 4 October 2007 Abstrct The purpose of this rticle is to provide n elementry introduction to the subject of chos in the electromechnicl drive systems. In this rticle, we explore chotic solutions of mps nd continuous time systems. These solutions re lso bounded like equilibrium, periodic nd qusiperiodic solutions University of West Bohemi. All rights reserved. Keywords: chos, dynmicl system, drive systems 1. Introduction Dissiptive dynmic system cn be chrcterized s systems whose behviour with incresing time symptoticlly pproches stedy sttes if there is no energy dded from the outside. Such system description is in mny cses possible with reltively simple nonliner equtions of motion. For certin vlues of prmeters of those equtions the solution does not converge towrds expected vlues, but choticlly oscilltes. Strong dependency on smll chnges of initil conditions occurs s well. When nlyzing such phenomen its mthemticl essence cn be connected with existence of strnge ttrctor in phse plne. Possible cretion of chos cn be seen in repeted bifurction of solution, with so clled cumultion point behind which the strnge ttrctor is generted. Phse digrm of system solution then trnsfers from stble set of trjectories towrds new, unstble nd chotic set. Creting the globl trjectory digrms is of essentil importnce. When succesfull, the symptotic behvior of systems model is described [4], [1]. 2. Drive with DC motor with permnent mgnets Direct current drive with permnent mgnets is often used in mechtronic drive systems together with semiconductor trnsducers. Good properties of such drives re given by the fct tht vector of exciting mgnetic flux is perpendiculr to current direction in rmture circuit nd motor therefore lwys produces mximl torque. Mthemticl model of DC motor cn be expressed vi stte vribles of current I nd ngulr velocity ω by stte eqution: * Corresponding uthor. Tel.: , e-mil: houfek@fme.vutbr.cz 121

2 C. Krtochvíl et et. l. / / Applied nd Computtionl Mechnics X 1 (YYYY) (2007) XXX YYY di R = L d dt ω CΦ dt J C L 0 Φ 1 i + L ω 0 0 u 1 m J z. (1) Where prticulr symbols represent: Armture circuit current Motor ngulr velocity Motor feeding voltge Resistnce nd inductnce of rotor coil Mgnetic flux Rotor moment of inerti I ω u R, L Φ J used: To verify properties nd behvior of drive with DC motor in vrious stedy sttes we ) Theoreticl model of DC motor nd mthemticl model of chos module b) Rel systems of motor chos (for input signl genertion), mplifier nd smll DC motor MAXON with revolutions recording nd rmture current mesurement [2]. 3. Bifurction behvior of drive model [3] For evlution of DC drive behvior exhibiting possibilities of cretion of bifurction of stte vribles depending on smll chnge of control prmeter we hve used chos module which generted chotic input signl for DC motor module corresponding to MAXON Re 16. Block digrm of drive model mde in Mtlb / Simulink is shown on fig. 1. Fig. 1. Block digrm of drive model. Chos module is bsed on NJH 1101 chip, developed in Ymkow s Lb & FLSI for modeling nd nlysis of chotic sttes in discrete nonliner systems. Chotic signl 122

3 C. Krtochvíl et et. l. / / Applied nd Computtionl Mechnics X 1 (YYYY) (2007) XXX YYY generted by chos module in P Spice environment ws smpled, sved s vlue mtrix nd trnsferred to Mtlb / Simulink environment. We observed the courses of rmture voltge u (t), ngulr velocity ω(t) nd rmture current i(t), which corresponded to stedy sttes of the system for given vlues of control current vlue of extremely stbilized resistor R 1 in nonliner circuit, thus the chos module. Aprt from time dependencies of selected vribles we lso generted given trjectories (ttrctors) corresponding to bsic stble period, its double, qudruple nd following chotic stte when periodicity of the motion cn not be ensured. Fig. 2. Attrctors of model system. 4. Verifiction of rel system behviour Overll lyout of the system with MAXON Re-16 DC motor is shown on fig. 3. During the simultion experiments we used oscilloscope to record the courses of feeding voltge u (t), induced voltge of tchogenertor u i (t), corresponding ngulr velocities of motor ω(t) nd rmture current i(t) while chnging the control prmeter (resistor) R 1 of chos module. From given recordings on oscilloscope we cn see the existence of limit cycles with primitive period (fig. 4) nd when chnging the control prmeter with double period (fig. 4b), qudruple period (fig. 4c) nd subsequently with chotic ttrctor (fig. 4d). 123

4 C.Krtochvíl Krtochvíletet.l.l/ Applied / Appliednd ndcomputtionl ComputtionlMechnics MechnicsX1(YYYY) (2007) XXX C. YYY Fig. 3. Lyout of DC motor rel system. Fig. 4. Attrctors of rel system: Primitive period (), double period (b), qudruple period (c) nd chotic ttrctor (d). 124

5 C. Krtochvíl et et. l. / / Applied nd Computtionl Mechnics X 1 (YYYY) (2007) XXX YYY Fig. 5. Time courses of voltge, ngulr velocity nd current chos. Time courses of feeding voltge u (t), ngulr velocity ω(t) nd rmture current in chotic stte re recorded on fig. 5. For completeness' ske let us show the clssicl bifurction digrm s ws shown on oscilloscope screen [3]. fig. 6. Fig. 6. Bifurction digrm. 5. Vlution of current results nd possibilities of its utiliztion If we compre results obtined on model system chos module + DC motor model nd rel system chos module + mplifier + DC motor, we cn stte tht results re very similr. 125

6 C. Krtochvíl et et. l. / / Applied nd Computtionl Mechnics X 1 (YYYY) (2007) XXX YYY From performed simultions nlysis it is cler tht in both cses the existence of stedy limit cycles precede the occurrence of chos. Those cycles cn be uniquely determined in phse spce using typicl trctors (cycles with primitive period, double period nd qudruple period). Chotic behvior in both cses is uniquely described by open stte trjectories, which cn not be predicted precisely. Those trjectories re situted in certin limiting res. This hs prticulr technicl product: Both ngulr velocity of the motor ω nd rmture current i will not exceed certin limits nd observed DC motor is not in direct dnger by exceeding the through current even in chotic stte. Bifurction nlysis cn be used s men for identifiction of limit vlues of prmeters, or s men for drive dignostic. Bifurction nlysis could extend lgorithms used of dptive controllers design. We cn not exclude the use of chotic signls s control signls for specil industril mechnisms requiring irregulr mechnicl motion depending on control prmeter. 6. Conclusion Chos becme phenomenon in vriety of engineering problems in lst yers. Therefore we focused on it lso in nlysis of drive systems. Bsed on performed nlysis we cn stte following recommendtions: when evluting the properties nd behvior of dynmic system it is useful to define such prmeters of models, which cn influence the occurrence of prsitic motion including chotic one (fluctution of initil conditions, links gps, control prmeters), to observe the evolution of responses in phse plnes bsed on chnges of selected prmeters nd to identify typicl chos effects, if such effect occur then evlute Fourier spectrum of responses. Chotic motion corresponds to brodbnd spectr, even when exciting spectr is nrrowbnd. With respecting given recommendtions it is not difficult to identify the res of possible occurrence of chos in technicl systems using mthemticl modelling. However, we do not wnt to disvlue the nlyticl pproches with bove described lterntive pproch. Acknowledgement Published results were cquired with the support of the Institute of Thermomechnics ASCR reserch pln AVO Z nd Grd Agency CR in the frme of grnd project number 101/06/0063. References [1] C. Krtochvíl, L. Houfek, M. Houfek, J. Krejs, Chos in Drive Systems, Wisl, [2] C. Krtochvíl, F. Procházk, J. Pulkrábek, Drive systems in mechtronics subjects, Proceedings of Computionl Mechnics 2005, Nečtiny Cstle, 2005, pp [3] R. Kříž, Electric Drive Exmple of Complex Dynmic System, M.Sc. Thesis, FEEC BUT Brno, [4] F. Procházk, C. Krtochvíl, Introduction to Mthemticl Modelling of Drive Systems, CERN s.r.o., Brno,