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Modelling of hysteresis using Masing-Bouc-Wen’s framework and search of conditions for the chaotic responses. (English) Zbl 1221.70027
Summary: Hysteresis is simulated by means of internal variables. It is shown that Masing’s imitating mechanism of energy dissipation present in the differential equations of Bouc-Wen’s structure allows one to simulate hysteresis phenomena arising in very different fields. The constructed analytical models of different types of hysteresis loops are simple, allow one to reproduce major and minor loops and provide a high degree of agreement with experimental data. The models of such a structure are convenient for further investigation. Hysteretic systems under harmonic excitation described by models of such a structure may reveal chaotic behaviour. Using an effective algorithm based on the analysis of wandering trajectories, the evolution of regions of chaotic behaviour of oscillators with hysteresis is presented in various parametric planes. Substantial influence of a hysteretic dissipation value on the form and location of these regions, and also restraining and generating effects of the hysteretic dissipation on occurrence of chaos are ascertained. Conditions for pinched hysteresis are defined.
MSC:
70K05Phase plane analysis, limit cycles (general mechanics)
34C28Complex behavior, chaotic systems (ODE)
34C55Hysteresis
References:
[1]Awrejcewicz J, Mosdorf R. Numerical analysis of some problems of chaotic dynamics, WNT, Warsaw, 2003 [in Polish].
[2]Awrejcewicz, J.; Dzyubak, L.: Stick-slip chaotic oscillations in a quasi-autonomous mechanical system, Int J nonlinear sci numer simul 4, No. 2, 155-160 (2003)
[3]Awrejcewicz, J.; Dzyubak, L.: Regular and chaotic behavior exhibited by coupled oscillators with friction, facta universitatis, Ser mech autom control robot 3, No. 14, 921-930 (2003) · Zbl 1065.74555
[4]Awrejcewicz, J.; Dzyubak, L.; Grebogi, C.: A direct numerical method for quantifying regular and chaotic orbits, Chaos, solitons fract 19, 503-507 (2004) · Zbl 1086.37044 · doi:10.1016/S0960-0779(03)00062-6
[5]Mayergoyz, I. D.: Mathematical models of hysteresis, (1991) · Zbl 0723.73003
[6]Visintin, A.: Differential models of hysteresis, (1994)
[7]Vestroni, F.; Noori, M.: Hysteresis in mechanical systems – modelling and dynamic response, Int J non-linear mech 37, 1261-1459 (2002)
[8]Bernardini, D.; Rega, G.: Thermomechanical modelling, nonlinear dynamics and chaos in shape memory alloys oscillators, Math comput model dyn syst 11, No. 3, 291-314 (2005) · Zbl 1097.74528 · doi:10.1080/13873950500076404
[9]Kádár, G.; Szabó, G.: Hysteresis modelling, J magn magn mater 215 – 216, 592-596 (2000)
[10]Kolsch, H.; Ottl, D.: Simulation des mechanischen verhaltens von bauteilen mit statischer hysterese, Forsch ingenieurwes 4, 66-71 (1993)
[11]Koltermann, P. I.; Righi, L. A.: A modified jiles method for hysteresis computation including minor loops, Physica B 275, 233-237 (2000)
[12]Ktena, A.; Fotiadis, D. I.: A preisach model identification procedure and simulation of hysteresis in ferromagnets and shape-memory alloys, Physica B 306, 84-90 (2001)
[13]Makaveev, D.; Dupre, L.: Dynamic hysteresis modelling using feed-forward neural networks, J magn magn mater 254 – 255, 256-258 (2003)
[14]Ortin, J.; Delaey, L.: Hysteresis in shape-memory alloys, Int J non-linear mech 37, 1275-1281 (2002)
[15]Ossart, F.; Hubert, O.; Billardon, R.: A new internal variables scalar model respecting the wiping-out property, J magn magn mater 254 – 255, 170-172 (2003)
[16]Sapinski, B.; Filus, J.: Analysis of parametric models of MR linear damper, J theor appl mech 41, No. 2, 215-240 (2003)
[17]Masing, G.: Zur heynschen theorie der verfestigung der metalle durch verborgen elastische spannungen, Wiss veroffentl aus dem siemens-konzern 3, No. 1, 231-239 (1923)
[18]Masing G Eigenspannungen und Vertfestigung beim Messing. In: Proceedings of the second international congress of applied mechanics, Zurich, Switzerland. 1926. p. 332 – 5 [in German].
[19]Bouc R. Forced vibrations of mechanical systems with hysteresis. In Proceedings of the fourth international conference on nonlinear oscillations, Prague, Czechoslovakia, 1967.
[20]Wen, Y. K.: Method for random vibration of hysteretic systems, ASCE J eng mech 120, 2299-2325 (1976)
[21]Sapinski, B.: Dynamic characteristics of an experimental MR fluid, Eng trans 51, No. 4, 399-418 (2003)
[22]Awrejcewicz, J.; Dzyubak, L.: Influence of hysteretic dissipation on chaotic responses, J sound vibr 284, 513-519 (2005)
[23]Awrejcewicz, J.; Dzyubak, L.: Quantifying smooth and non-smooth regular and chaotic dynamics, Int J bifur chaos 15, No. 6, 2041-2055 (2005) · Zbl 1092.37529 · doi:10.1142/S0218127405013137
[24]Lacarbonara, W.; Vestroni, F.: Nonclassical responses of oscillators with hysteresis, Nonlinear dyn (2004)
[25]Li, H. G.; Zhang, J. W.; Wen, B. C.: Chaotic behaviors of a bilinear hysteretic oscillator, Mech res comm 29, 283-289 (2002) · Zbl 1024.70501 · doi:10.1016/S0093-6413(02)00266-5
[26]Capecchi, D.; Masiani, R.: Reduced phase space analysis for hysteretic oscillators of masing type, Chaos, soliton fract 7, 1583-1600 (1996) · Zbl 1080.70527 · doi:10.1016/S0960-0779(96)00062-8
[27]Jin, C.; Fan, L.; Qiu, Y.: The vibration control of a flexible linkage mechanism with impact, Comm nonlinear sci numer simul 9, 459-469 (2004) · Zbl 1040.70014 · doi:10.1016/S1007-5704(02)00134-X
[28]Luo, A.: A theory for non-smooth dynamic systems on the connectable domains, Comm nonlinear sci numer simul 10, 1-55 (2005) · Zbl 1065.34007 · doi:10.1016/j.cnsns.2004.04.004
[29]Levinson, N.: A second order differential equation with singular solutions, Ann math 50, 127-153 (1949) · Zbl 0045.36501 · doi:10.2307/1969357
[30]Coddington, E. A.; Levinson, N.: Theory of ordinary differential equations, (1955) · Zbl 0064.33002
[31]Leine RI. van de Vrande BL, vanCampen DH. Bifurcations in nonlinear discontinuous systems. Report WFW 99.010, Vakgroep Fundamentele Werktuigkunde, Eindhoven, 1999.
[32]Awrejcewicz, J.; Lamarque, C. -H.: Bifurcations and chaos in nonsmooth mechanical systems, (2003)