×

Analysis of dynamical behaviors of a friction-induced oscillator with switching control law. (English) Zbl 1375.93064

Summary: In this paper, the dynamical behaviors of a friction-induced oscillator with switching control law are studied through the flow switching theory of discontinuous dynamical systems. The physical model consists of a mass on the conveyor belt and a spring-damping system with switching control law. Based on the switching control law and the friction between the oscillator and the conveyor belt, multiple domains and discontinuous boundaries are defined. The G-functions are introduced to illustrate the motion switching mechanism and the analytical conditions of the passable motion, stick motion, sliding motion and grazing motion are presented for motion switchability. The switching sets and mapping structures are adopted to describe the complex motions in this discontinuous system. The numerical simulations are also carried out from the analytical conditions and mapping structures in order to better understand the motion switching complexity of this oscillator.

MSC:

93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
37N35 Dynamical systems in control
70Q05 Control of mechanical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hartog, J., Forced vibrations with coulomb and viscous damping, Trans Am Soc MechEng, 53, 107-115 (1930) · JFM 56.1215.03
[2] Levitan, E., Forced oscillation of a spring-mass system having combined coulomb and viscous damping, J Acoust Soc Am, 32, 1265-1269 (1960)
[3] Hundal, M., Response of a base excited system with coulomb and viscous friction, J Sound Vib, 64, 3, 371-378 (1979) · Zbl 0398.73094
[4] Shaw, S., On the dynamical response of a system with dry-friction, J Sound Vib, 108, 2, 305-325 (1986) · Zbl 1235.70105
[5] Feeny, B.; Moon, F., Chaos in a forced dry-friction oscillator: experiments and numerical modeling, J Sound Vib, 170, 303-322 (1994) · Zbl 0925.70285
[6] Virgin, L.; Begley, C., Grazing bifurcations and basins of attraction in an impact-friction oscillator, Physica D, 130, 43-57 (1999) · Zbl 0964.70019
[7] Dankowicz, H.; Nordmark, A., On the origin and bifurcations of stick-slip oscillations, Physica D, 136, 280-302 (2000) · Zbl 0963.70016
[8] Leine, R.; Campen, D.; Vrande, B., Bifurcations in nonlinear discontinuous systems, Nonlinear Dyn, 23, 105-164 (2000) · Zbl 0980.70018
[9] Ko, P.; Taponat, M.; Pfaifer, R., Friction-induced vibration with and without external disturbance, Tribol Int, 34, 7-24 (2001)
[10] Kim, W.; Perkins, N., Harmonic balance/Galerkin method for non-smooth dynamical system, J Sound Vib, 261, 213-224 (2003) · Zbl 1237.70080
[11] Luo, A., Imaginary, sink and source flows in the vicinity of the separatrix of nonsmooth dynamic systems, J Sound Vib, 285, 443-456 (2005) · Zbl 1237.34017
[12] Luo, A., A theory for non-smooth dynamical systems on connectable domains, Commun Nonlinear Sci Numer Simul, 10, 1, 1-55 (2005) · Zbl 1065.34007
[13] Luo, A., The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation, J Sound Vib, 283, 723-748 (2005) · Zbl 1237.70083
[14] Luo, A.; Gegg, B., Stick and non-stick periodic motions in a periodically forced oscillator with dry-friction, J Sound Vib, 291, 132-168 (2006) · Zbl 1243.70025
[15] Luo, A.; Rapp, B., Switching dynamics of a periodically forced discontinuous system with an inclined boundary, ASME 2007 Int Des Eng Tech Conf Comput Inf Eng Conf, 5, 239-250 (2007), . Paper No. DETC2007-34863
[16] Luo, A., A theory for flow swtichability in discontinuous dynamical systems, Nonlinear Anal Hybrid Syst, 2, 4, 1030-1061 (2008) · Zbl 1163.93016
[17] Luo, A., Discontinuous Dynamical Systems on Time-varying Domains (2011), Higher Education Press: Higher Education Press Beijing China
[18] Luo, A., Discontinuous Dynamical Systems (2012), Higher Education Press: Higher Education Press Beijing, China · Zbl 1242.93001
[19] Chen, G.; Fan, J., Analysis of dynamical behaviors of a double belt friction-oscillator model, WSEAS Trans Math, 15, 357-373 (2016)
[20] Fan, J.; Li, S.; Chen, G., On dynamical behavior of a friction-induced oscillator with 2-DOF on a speed-varying traveling belt, Math Prob Eng, 2017 (2017) · Zbl 1426.70020
[21] Zhang, Y.; Fu, X., On periodic motions of an inclined impact pair, Commun Nonlinear Sci Numer Simul, 20, 3, 1033-1042 (2015) · Zbl 1328.70008
[22] Fu, X.; Zhang, Y., Stick motions and grazing flows in an inclined impact oscillator, Chaos, Solitons Fractals, 76, 218-230 (2015) · Zbl 1352.74134
[23] Zhang, Y.; Fu, X., Flow switchability of motions in a horizontal impact pair with dry friction, Commun Nonlinear Sci Numer Simul, 44, 3, 89-107 (2017) · Zbl 1465.70057
[24] Li, X.; Wu, J., Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64, 63-69 (2016) · Zbl 1329.93108
[25] Li, X.; Song, S., Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans Autom Control, 62, 1, 406-411 (2017) · Zbl 1359.34089
[26] Zhang, X.; Li, X., Input-to-state stability of non-linear systems with distributed-delayed impulses, IET Control Theory Appl, 11, 1, 81-89 (2017)
[27] Luo, A.; Gegg, B., On the mechanism of stick and non-stick, periodic motions in a periodically forced, ASME J Vib Acoust, 128, 97-105 (2005)
[28] Luo, A.; Rapp, B., Sliding and transversal motions on an inclined boundary in a periodically forced discontinuous system, Commun Nonlinear Sci Numer Simul, 15, 86-98 (2010) · Zbl 1221.70030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.