Analysis of discontinuous dynamical behaviors of a friction-induced oscillator with an elliptic control law.

*(English)*Zbl 1427.70037Summary: This paper develops the passability conditions of flow to the discontinuous boundary and the sticking or sliding and grazing conditions to the separation boundary in the discontinuous dynamical system of a friction-induced oscillator with an elliptic control law and the friction force acting on the mass \(M\) through the analysis of the corresponding vector fields and \(G\)-functions. The periodic motions of such a discontinuous system are predicted analytically through the mapping structure. Finally, the numerical simulations are given to illustrate the analytical results of motion for a better understanding of physics of motion in the mass-spring-damper oscillator.

##### MSC:

70F40 | Problems involving a system of particles with friction |

34A36 | Discontinuous ordinary differential equations |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

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\textit{J. Fan} et al., Math. Probl. Eng. 2018, Article ID 5301747, 33 p. (2018; Zbl 1427.70037)

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[1] | Luo, A. C.; Rapp, B. M., Flow switchability and periodic motions in a periodically forced, discontinuous dynamical system, Nonlinear Analysis: Real World Applications, 10, 5, 3028-3044, (2009) · Zbl 1168.70011 |

[2] | Luo, A. C.; Rapp, B. M., On motions and switchability in a periodically forced, discontinuous system with a parabolic boundary, Nonlinear Analysis: Real World Applications, 11, 4, 2624-2633, (2010) · Zbl 1198.37083 |

[3] | Hartog, J. P. D., Forced vibrations with coulomb and viscous damping, Transactions of the American Society of Mechanical Engineers, 53, 107-115, (1930) · JFM 56.1215.03 |

[4] | Levitan, E. S., Forced oscillation of a spring-mass system having combined Coulomb and viscous damping, The Journal of the Acoustical Society of America, 32, 1265-1269, (1960) |

[5] | Filippov, A. F., Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 42, 2, 199-231, (1964) · Zbl 0148.33002 |

[6] | Filippov, A. F., Differential Equations with Discontinuous Right-Hand Sides, (1988), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0664.34001 |

[7] | Popp, K.; Hinrichs, N.; Oestreich, M., Dynamical behaviour of a friction oscillator with simultaneous self and external excitation, Sadhana, 20, 2-4, 627-654, (1995) · Zbl 1048.70503 |

[8] | Oestreich, M.; Hinrichs, N.; Popp, K., Bifurcation and stability analysis for a non-smooth friction oscillator, Archive of Applied Mechanics, 66, 5, 301-314, (1996) · Zbl 0846.70016 |

[9] | Kunze, M.; Kupper, T.; You, J., On the application of KAM theory to discontinuous dynamical systems, Journal of Differential Equations, 139, 1, 1-21, (1997) · Zbl 0884.34049 |

[10] | Awrejcewicz, J.; Olejnik, P., Stick-slip dynamics of a two-degree-of-freedom system, International Journal of Bifurcation and Chaos, 13, 4, 843-861, (2003) · Zbl 1067.70021 |

[11] | Cid, J. A.; Sanchez, L., Periodic solutions for second order differential equations with discontinuous restoring forces, Journal of Mathematical Analysis and Applications, 288, 2, 349-364, (2003) · Zbl 1054.34012 |

[12] | Jacquemard, A.; Teixeira, M. A., Periodic solutions of a class of non-autonomous second order differential equations with discontinuous right-hand side, Physica D: Nonlinear Phenomena, 241, 22, 2003-2009, (2012) |

[13] | Kupper, T.; Moritz, S., Generalized Hopf bifurcation for non-smooth planar systems, Philosophical Transactions Mathematical Physical, 359, 1789, 2483-2496, (2001) · Zbl 1097.37502 |

[14] | Zou, Y.; Kupper, T.; Beyn, W. J., Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, Journal of Nonlinear Science, 16, 2, 159-177, (2006) · Zbl 1104.37031 |

[15] | Pascal, M., Sticking and nonsticking orbits for a two-degree-of-freedom oscillator excited by dry friction and harmonic loading, Nonlinear Dynamics, 77, 1-2, 267-276, (2014) · Zbl 1314.70028 |

[16] | Stefaïski, A.; Wojewoda, J.; Furmanik, K., Experimental and numerical analysis of self-excited friction oscillator, Chaos, Solitons and Fractals, 12, 9, 1691-1704, (2001) · Zbl 1048.70500 |

[17] | Andreaus, U.; Casini, P., Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle, International Journal of Non-Linear Mechanics, 37, 1, 117-133, (2002) · Zbl 1117.74303 |

[18] | Awrejcewicz, J.; Dzyubak, L., Conditions for chaos occurring in self-excited 2-DOF hysteretic system with friction, Proceedings of the Conference on Dynamical Systems Theory and Applications |

[19] | Csernak, G.; Stepan, G., On the periodic response of a harmonically excited dry friction oscillator, Journal of Sound Vibration, 295, 3-5, 649-658, (2006) · Zbl 1243.70022 |

[20] | Gegg, B. C.; Luo, A. C.; Suh, S. C., Grazing bifurcations of a harmonically excited oscillator moving on a time-varying translation belt, Nonlinear Analysis: Real World Applications, 9, 5, 2156-2174, (2008) · Zbl 1156.70312 |

[21] | Luo, A. C. J., Impact dynamics of a constrained mass-spring-damper system, International Mechanical Engineering Congress, 9, 1-8, (2012) |

[22] | Guo, Y.; Luo, A. C., Parametric analysis of bifurcation and chaos in a periodically driven horizontal impact pair, International Journal of Bifurcation and Chaos, 22, 11, 1-19, (2012) · Zbl 1258.34031 |

[23] | Luo, A. C.; Thapa, S., Periodic motions in a simplified brake system with a periodic excitation, Communications in Nonlinear Science and Numerical Simulation, 14, 5, 2389-2414, (2009) · Zbl 1221.70032 |

[24] | Luo, A. C. J.; Chen, L., Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos, Solitons & Fractals, 24, 2, 567-578, (2005) · Zbl 1135.70312 |

[25] | Fu, X.; Zhang, Y., Stick motions and grazing flows in an inclined impact oscillator, Chaos, Solitons & Fractals, 76, 218-230, (2015) · Zbl 1352.74134 |

[26] | Sun, J.; Xu, W.; Lin, Z., Research on the reliability of friction system under combined additive and multiplicative random excitations, Communications in Nonlinear Science and Numerical Simulation, 54, 1-12, (2018) |

[27] | Lu, Z. Q.; Li, J. M.; Ding, H.; Chen, L. Q., Analysis and suppression of a self-excitation vibration via internal stiffness and damping nonlinearity, Advances in Mechanical Engineering, 9, 12, (2017) |

[28] | Simpson, D. J., Grazing-sliding bifurcations creating infinitely many attractors, International Journal of Bifurcation and Chaos, 27, 12, (2017) · Zbl 1383.34061 |

[29] | da Silva, C. E.; da Silva, P. R.; Jacquemard, A., Sliding solutions of second-order differential equations with discontinuous right-hand side, Mathematical Methods in the Applied Sciences, 40, 14, 5295-5306, (2017) · Zbl 1385.34015 |

[30] | Makarenkov, O., A new test for stick-slip limit cycles in dry-friction oscillators with a small nonlinearity in the friction characteristic, Meccanica, 52, 11-12, 2631-2640, (2017) · Zbl 1380.34033 |

[31] | Ma, Y.; Yu, S.; Wang, D., Vibration analysis of an oscillator with non-smooth dry friction constraint, Journal of Vibration and Control, 23, 14, 2328-2344, (2017) · Zbl 1373.70013 |

[32] | Yu, S. D.; Wen, B. C., Vibration analysis of multiple degrees of freedom mechanical system with dry friction, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 227, 7, 1505-1514, (2013) |

[33] | Bernardo, M.; Kowalczyk, P.; Nordmark, A., Sliding bifurcations: a novel mechanism for the sudden onset of chaos in dry friction oscillators, International Journal of Bifurcation and Chaos, 13, 10, 2935-2948, (2003) · Zbl 1099.70504 |

[34] | Galvanetto, U., Sliding bifurcations in the dynamics of mechanical systems with dry friction - Remarks for engineers and applied scientists, Journal of Sound and Vibration, 276, 1-2, 121-139, (2004) |

[35] | Kowalczyk, P.; Bernardo, M., Two-parameter degenerate sliding bifurcations in Filippov systems, Physica D: Nonlinear Phenomena, 204, 3-4, 204-229, (2005) · Zbl 1087.34019 |

[36] | Kowalczyk, P.; Piiroinen, P. T., Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator, Physica D: Nonlinear Phenomena, 237, 8, 1053-1073, (2008) · Zbl 1151.34031 |

[37] | Guardia, M.; Hogan, S. J.; Seara, T. M., An analytical approach to codimension-2 sliding bifurcations in the dry-friction oscillator, SIAM Journal on Applied Dynamical Systems, 9, 3, 769-798, (2010) · Zbl 1201.37086 |

[38] | Luo, A. C., A theory for non-smooth dynamic systems on the connectable domains, Communications in Nonlinear Science and Numerical Simulation, 10, 1, 1-55, (2005) · Zbl 1065.34007 |

[39] | Luo, A. C., The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation, Journal of Sound and Vibration, 283, 3-5, 723-748, (2005) · Zbl 1237.70083 |

[40] | Luo, A. C., Imaginary, sink and source flows in the vicinity of the separatrix of non-smooth dynamic systems, Journal of Sound and Vibration, 285, 1-2, 443-456, (2005) · Zbl 1237.34017 |

[41] | Luo, A. C., Singularity and Dynamics on Discontinuous Vector Fields, (2006), Amsterdam, The Netherlands: Elsevier, Amsterdam, The Netherlands · Zbl 1182.37002 |

[42] | Luo, A. C., A theory for flow switchability in discontinuous dynamical systems, Nonlinear Analysis: Hybrid Systems, 2, 4, 1030-1061, (2008) · Zbl 1163.93016 |

[43] | Luo, A. C. J.; Gegg, B. C., Periodic motions in a periodically forced oscillator moving on an oscillating belt with dry friction, Journal of Computational and Nonlinear Dynamics, 1, 3, 132-168, (2006) · Zbl 1243.70025 |

[44] | Gegg, B. C.; Luo, A. C.; Suh, S. C., Sliding motions on a periodically time-varying boundary for a friction-induced oscillator, Journal of Vibration and Control, 15, 5, 671-703, (2009) · Zbl 1272.70104 |

[45] | Luo, A. C., On flow barriers and switchability in discontinuous dynamical systems, International Journal of Bifurcation and Chaos, 21, 1, 1-76, (2011) · Zbl 1208.34007 |

[46] | Luo, A. C.; Huang, J., Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator, Nonlinear Analysis: Real World Applications, 13, 1, 241-257, (2012) · Zbl 1238.70003 |

[47] | Zhang, Y.; Fu, X., On periodic motions of an inclined impact pair, Communications in Nonlinear Science and Numerical Simulation, 20, 3, 1033-1042, (2015) · Zbl 1328.70008 |

[48] | Zhang, Y.; Fu, X., Flow switchability of motions in a horizontal impact pair with dry friction, Communications in Nonlinear Science and Numerical Simulation, 44, 89-107, (2017) |

[49] | Chen, G.; Fan, J. J., Analysis of dynamical behaviors of a double belt friction-oscillator model, WSEAS Transactions on Mathematical, 15, 357-373, (2016) |

[50] | 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 |

[51] | Fan, J.; Li, S.; Chen, G., On dynamical behavior of a friction-induced oscillator with 2-DOF on a speed-varying traveling belt, Mathematical Problems in Engineering, 2017, (2017) |

[52] | Xue, S.; Fan, J., Discontinuous dynamical behaviors in a vibro-impact system with multiple constraints, International Journal of Non-Linear Mechanics, 98, 75-101, (2018) |

[53] | Zheng, S.; Fu, X., Periodic motion of the van der Pol equation with impulsive effect, International Journal of Bifurcation and Chaos, 25, 9, 1-11, (2015) · Zbl 1325.34019 |

[54] | 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 |

[55] | 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 and Learning Systems, 24, 6, 868-877, (2013) |

[56] | Zhang, X.; Lv, X.; Li, X., Sampled-data-based lag synchronization of chaotic delayed neural networks with impulsive control, Nonlinear Dynamics, 90, 3, 2199-2207, (2017) · Zbl 1380.34082 |

[57] | Lv, X.; Li, X., Finite time stability and controller design for nonlinear impulsive sampled-data systems with applications, ISA Transactions, 70, 30-36, (2017) |

[58] | Li, X.; Cao, J.; Perc, M., Switching laws design for stability of finite and infinite delayed switched systems with stable and unstable modes, IEEE Access, 6, 6677-6691, (2018) |

[59] | Li, X.; Wu, J., Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, Institute of Electrical and Electronics Engineers Transactions on Automatic Control, 63, 1, 306-311, (2018) · Zbl 1390.34177 |

[60] | Li, X.; Bohner, M.; Wang, C.-K., Impulsive differential equations: periodic solutions and applications, Automatica, 52, 173-178, (2015) · Zbl 1309.93074 |

[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 Journal on Control and Optimization, 55, 6, 3437-3457, (2017) · Zbl 1373.93249 |

[66] | Li, H.; Song, P.; Yang, Q., Pinning control design for robust output tracking of k-valued logical networks, Journal of The Franklin Institute, 354, 7, 3039-3053, (2017) · Zbl 1364.93190 |

[67] | Zheng, Y.; Li, H.; Ding, X.; Liu, Y., Stabilization and set stabilization of delayed Boolean control networks based on trajectory stabilization, Journal of The Franklin Institute, 354, 17, 7812-7827, (2017) · Zbl 1380.93205 |

[68] | Li, H.; Wang, Y., Robust stability and stabilisation of Boolean networks with disturbance inputs, International Journal of Systems Science, 48, 4, 750-756, (2017) · Zbl 1358.93136 |

[69] | Senqupta, P., Elliptic rendezvous in the chaser satellile frame, Journal of the Astronautical Science, 59, 216-236, (2012) |

[70] | Chang, D. E.; Chichka, D. F.; Marsden, J. E., Lyapunov-based transfer between elliptic Keplerian orbits, Discrete and Continuous Dynamical Systems - Series B, 2, 1, 57-67, (2012) · Zbl 1024.70013 |

[71] | Yan, H.; Ross, I. M.; Alfriend, K. T., Pseudospectral feedback control for three-axis magnetic attitude stabilization in elliptic orbits, Journal of Guidance, Control, and Dynamics, 30, 4, 1107-1115, (2013) |

[72] | Do, K. D., Formation control of multiple elliptic agents with limited sensing ranges, Asian Journal of Control, 14, 6, 1514-1526, (2012) · Zbl 1303.93011 |

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