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Analysis of dynamical behaviors of a 2-DOF vibro-impact system with dry friction. (English) Zbl 1442.74145
Summary: In this paper, the discontinuous dynamics in a 2-DOF (two-degree-of-freedom) vibro-impact system with dry friction is investigated by using the flow switchability theory for discontinuous dynamical systems. Multiple domains and discontinuous boundaries are defined according to friction and impact discontinuity. Based on above domains and boundaries, the onset and disappearance conditions of sliding-stick motion are developed and the analytical conditions of side-stick motion and grazing motion are obtained mathematically. The switching sets and mapping structures are adopted to describe the complex motions in such discontinuous system. The numerical simulations are also given to illustrate the analytical results of motion switching on different boundaries. This investigation can help us understand the motion switching mechanism in non-stick, sliding-stick or side-stick motions of the oscillator with friction and impact.
##### MSC:
 74M20 Impact in solid mechanics 74H45 Vibrations in dynamical problems in solid mechanics
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##### References:
 [1] Holmes, P. J., The dynamics of repeated impacts with a sinusoidally vibrating table, J Sound Vibr., 84, 2, 173-189 (1982) · Zbl 0518.73015 [2] Shaw, S. W.; Holmes, P. J., A periodically forced piecewise linear oscillator, J Sound Vibr, 90, 1, 129-155 (1983) · Zbl 0561.70022 [3] Heiman, M. S.; Sherman, P. J.; Bajaj, A. K., On the dynamics and stability of an inclined impact pair, J Sound Vibr, 114, 3, 535-547 (1987) · Zbl 1235.74256 [4] Heiman, M. S.; Bajaj, A. K.; Sherman, P. J., Periodic motions and bifurcations in dynamics of an inclined impact pair, J Sound Vibr, 124, 1, 55-78 (1988) · Zbl 1235.70059 [5] Han, R. P.; Luo, A. C.J; Deng, W., Chaotic motion of a horizontal impact pair, J Sound Vibr, 181, 2, 231-250 (1995) · Zbl 1237.70028 [6] Long, X.; Lin, G.; Balachandran, B., Grazing bifurcations in elastic structure excited by harmonic impactor motions, Physica D, 237, 1129-1138 (2008) · Zbl 1138.74031 [7] Chakraborty, I.; Balachandran, B., Near-grazing dynamics of base excited cantilevers with nonlinear tip interactions, Nonlinear Dyn, 70, 2, 1297-1310 (2012) [8] 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 [9] Zhang, Y.; Fu, X., Stick motions and grazing flows in an inclined impact oscillator, Chaos, Solitons Fractals, 76, 218-230 (2015) · Zbl 1352.74134 [10] Xue, S.; Fan, J., Discontinuous dynamical behaviors in a vibro-impact system with multiple constraints, Int J Non Linear Mech, 98, 75-101 (2018) [11] Filippov, A. F., Differential equations with discontinuous right-hand side, Am Math Soc Trans, 42, Series 2, 99-231 (1964) · Zbl 0148.33002 [12] Shaw, S. W., On the dynamic response of a system with dry-friction, J Sound Vibr, 108, 2, 305-325 (1986) · Zbl 1235.70105 [13] Awrejcewicz, J.; Delfs, J., Dynamics of a self-excited stick-slip oscillator with two degrees of freedom, Part I. Investigation of equilibrium, European J Mechanics-A/Solids, vol. 9, 269-282 (1990) · Zbl 0712.70043 [14] Feeny, B.; Moon, F. C., Chaos in a forced dry-friction oscillator: experiments and numerical modeling, J Sound Vibr, 170, 3, 303-323 (1994) · Zbl 0925.70285 [15] Oestreich, M.; Hinrichs, N.; Popp, K., Bifurcation and stability analysis for a non-smooth friction oscillator, Archive of Appl Mechanics, 66, 5, 301-314 (1996) · Zbl 0846.70016 [16] Natsiavas, S., Stability of piecewise linear oscillators with viscous and dry friction dampping, J Sound Vibr, 217, 3, 507-522 (1998) · Zbl 1236.70025 [17] Hong, H. K.; Liu, C. S., Non-sticking oscillation formulae for coulomb friction under harmonic loading, J Sound Vibr, 244, 5, 883-898 (2000) · Zbl 1237.70078 [18] Xia, F., Modeling of a two-dimensional coulomb friction oscillator, J Sound Vibr, 265, 5, 1063-1074 (2003) · Zbl 1236.74211 [19] Pilipchuk, V. N.; Tan, C. A., Creep-slip capture as a possible source of squeal during decelerating sliding, Nonlinear Dyn, 35, 3, 258-285 (2004) · Zbl 1071.70527 [20] Chin, W.; Ott, E.; Nusse, H. E.; Grebogi, C., Grazing bifurcations in impact oscillators, Physical Review E Statistical Physics Plasmas Fluids & Related Interdisciplinary Topics, 50, 6, 4427-4444 (1994) [21] Bapat, C. N., The general motion of an inclined impact damper with friction, J Sound Vibr, 184, 3, 417-427 (1995) · Zbl 0982.70521 [22] Cone, K. M.; Zadoks, R. I., A numerical study of an impact oscillator with the addition of dry friction, J Sound Vibr, 188, 5, 659-683 (1995) [23] Hinrichs, N.; Oestreich, M.; Popp, K., Dynamics of oscillators with impact and friction, Chaos, Solitons & Fractals, 8, 4, 535-558 (1997) · Zbl 0963.70552 [24] Virgin, L. N.; Begley, C. J., Grazing bifurcation and basins of attraction in an impact-friction oscillator, Physica, 130, 1-2, 43-57 (1999) · Zbl 0964.70019 [25] Blazejczyk-Okolewska, B., Study of the impact oscillator with elastic coupling of masses, Chaos, Solitons & Fractals, 11, 15, 2487-2492 (2000) · Zbl 0955.70506 [26] Andreaus, U.; Casini, P., Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle, Int J Non Linear Mech, 37, 1, 117-133 (2002) · Zbl 1117.74303 [27] Burns, S. J.; Piiroinen, P. T., The complexity of a basic impact mapping for rigid bodies with impacts and friction, Regular Chaotic Dynamics, 19, 1, 20-36 (2014) · Zbl 1353.70039 [28] Bazhenov, V. A.; Lizunov, P. P.; Pogorelova, O. S.; Postnikova, T. G.; Otrashevskaia, V. V., Stability and bifurcations analysis for 2-DOF vibroimpact system by parameter continuation method, Part I: Loading Curve J Appl. Nonlinear Dynamics, 4, 4, 357-370 (2015) [29] Bazhenov, V. A.; Lizunov, P. P.; Pogorelova, O. S.; Postnikova, T. G., Numerical bifurcation analysis of discontinuous 2-DOF vibroimpact system. part 2: frequency-amplitude response, J Appl. Nonlin Dyn, 5, 3, 269-281 (2016) [30] Luo, A. C.J, A theory for non-smooth dynamic systems on the connectable domains, Commun Nonlinear Sci Numer Simul, 10, 1, 1-55 (2005) · Zbl 1065.34007 [31] Luo, A. C.J, A theory for flow switchability in discontinuous dynamical systems, Nonlinear Analysis Hybrid Systems, 2, 4, 1030-1061 (2008) · Zbl 1163.93016 [32] Luo, A. C.J, Discontinuous dynamical systems on time-varying domains (2009), Higher Education Press: Higher Education Press Beijing · Zbl 1213.34001 [33] Luo, A. C.J; O’Connor, D., Mechanism of impacting chatter with stick in a gear transmission system, Int J Bifurcation and Chaos, 19, 1975-1994 (2009) [34] Luo, A. C.J; O’Connor, D., Periodic motions and chaos with impacting chatter with stick in a gear transmission system, Int J Bifurcation Chaos, 19, 2093-2105 (2009) [35] Luo, A. C.J; Thapa, S., Periodic motion in a simplified brake system with a periodic excitation, Commun Nonlinear Sci Numer Simul, 14, 2389-2414 (2009) · Zbl 1221.70032 [36] Luo, A. C.J; Mao, T. T., Analytical conditions for motion switchability in a 2-DOF friction-induced oscillator moving on two constant speed belts, Can Appl Math Q, 127, 1, 201-242 (2009) · Zbl 1380.70018 [37] Guo, Y.; Luo, A. C.J, Complex motions in horizontal impact pairs with a periodic excitation, ASME 2011 international design engineering technical conferences and computers & information in engineering conference, American Society of Mechanical Engineers, 1339-1350 (2011) [38] Luo, A. C.J, Discontinuous Dynamical Systems (2012), HEP/Springer: HEP/Springer Beijing/Heidelberg · Zbl 1242.93001 [39] Luo, A. C.J; Mosadman, F., Singularity, switchability and bifurcations in a 2-DOF periodically forced, frictional oscillator, Int J Bifurcation Chaos, 23, 1330009, 38 (2013) · Zbl 1270.34076 [40] O’Connor, D.; Luo, A. C.J, On discontinuous dynamics of a freight train suspension system, Int J Bifurcation Chaos, 24, 12, 44 (2014) · Zbl 1305.34077 [41] Chen, G.; Fan, J., Analysis of dynamical behaviors of a double belt friction-oscillator model, WSEAS Trans Math, 15, 357-373 (2016) [42] Fan, J.; Xue, S.; Chen, G., On discontinuous dynamics of a periodically forced double-belt friction oscillator, Chaos, Solitons Fractals, 109, 280-302 (2018) · Zbl 1390.74096 [43] Li, S.; Fan, J.; Chen, G., Passable motions and stick motions of friction-induced oscillator with 2-DOF on a speed-varying belt, Int J Theor Appl. Mech, 1, 218-227 (2016) [44] Fan, J.; Li, S.; Chen, G., On dynamical behavior of a friction-induced oscillator with 2-DOF on a speed-varing traveling belt, Math Prob Eng 2017, 19 (2017) [45] Fan, J.; Xue, S.; Li, S., Analysis of dynamical behaviors of a friction-induced oscillator with switching control law, Chaos, Solitons Fractals, 103, 513-531 (2017) · Zbl 1375.93064 [46] Fan, J.; Liu, P.; Liu, T.; Xue, S.; Yang, Z., Analysis of discontinuous dynamical behaviors of a friction-induced oscillator with an elliptic control law, Math Prob Eng, 2018, 33 (2018) [47] Fan, J.; Liu, T.; Liu, P., Analysis of discontinuous dynamical behavior of a class offriction oscillators with impact, Int J Non Linear Mech (2018) [48] Zhang, Y.; Fu, X., Flow switchability of motions in a horizontal impact pair with dry friction, Commun Nonlinear Sci Numerical Simul, 44, 3, 89-107 (2017) [49] Li, X.; Wu, J., Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64, 63-69 (2016) · Zbl 1329.93108 [50] Li, X.; Song, S., Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans Automat Contr, 62, 1, 406-411 (2016) · Zbl 1359.34089 [51] Li, X.; Cao, J., An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Trans Automat Contr, 62, 7, 3618-3625 (2017) · Zbl 1370.34131 [52] Li, X.; Bohner, M.; Wang, C. K., Impulsive differential equations: periodic solutions and applications, Automatica, 52, C, 173-178 (2015) · Zbl 1309.93074 [53] Li, X.; Song, S., Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays, IEEE Transactions on Neural Networks & Learning Systems, 24, 6, 868-877 (2013) [54] Li, X.; Zhang, X.; Song, S., Effect of delayed impulses on input-to-state stability of nonlinear systems, Automatica, 76, 378-382 (2017) · Zbl 1352.93089 [55] Zhang, X.; Li, X., Input-to-state stability of non-linear systems with distributed-delayed impulses, Iet Control Theory & Applications, 11, 1, 81-89 (2017) [56] Li, X.; Fu, X., Effect of leakage time-varying delay on stability of nonlinear differential systems, J Franklin Inst, 350, 6, 1335-1344 (2013) · Zbl 1293.34065 [57] Stamova, I.; Stamov, T.; Li, X., Global exponential stability of a class of impulsive cellular neural networks with supremums, Int J Adaptive Control Signal Process, 28, 11, 1227-1239 (2014) · Zbl 1338.93316 [58] Li, X.; Liu, B.; Wu, J., Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans Automat Contr, 63, 1, 306-311 (2017) · Zbl 1390.34177 [59] Li, X.; Song, S.; Wu, J., Impulsive control of unstable neural networks with unbounded time-varying delays, Science China (Information Sciences), 61, 1, 12-23 (2018) [60] Sui, G.; Li, X.; Fan, J.; O’Regan, D., Sufficient conditions for pulse phenomena of nonlinear systems with state-dependent impulses, J Nonlinear Sci Appl, 9, 2649-2657 (2016) · Zbl 1348.49029 [61] Li, H.; Wang, Y., Further results on feedback stabilization control design of boolean control networks, Automatica, 83, 303-308 (2017) · Zbl 1373.93259 [62] Li, H.; Xie, L.; Wang, Y., Output regulation of boolean control networks, Institute of Electrical and Electronics Engineers Transactions on Automatic Control, 62, 6, 2993-2998 (2017) · Zbl 1369.93345 [63] Li, H.; Xie, L.; Wang, Y., On robust control invariance of boolean control networks, Automatica, 68, 392-396 (2016) · Zbl 1334.93057 [64] Li, H.; Wang, Y.; Xie, L., Output tracking control of boolean control networks via state feedback: constant reference signal case, Automatica, 59, 54-59 (2015) · Zbl 1326.93068 [65] Li, H.; Wang, Y., Lyapunov-based stability and construction of lyapunov functions for boolean networks, SIAM J Control Optim, 55, 6, 3437-3457 (2017) · Zbl 1373.93249 [66] Fu, X.; Zheng, S., New approach in dynamics of regenerative chatter research of turning, Commun Nonlinear Sci Numerical Simulation, 19, 2, 4013-4023 (2014) [67] Sun, X.; Fu, X., Synchronization of two different dynamical systems under sinusoidal constrain, J Appl Math, 9 (2014) [68] Liu, T.; Fan, J.; Xue, S., Synchronization of a duffing oscillator with a van der pol equation under sinusoidal constraints, Int J Math Comput, 29, 3, 1-25 (2018)
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