Abstract
In this paper, the flow switching theory of discontinuous dynamical systems is used as the main tool to study the dynamical behaviors of a two-degree-of-freedom friction oscillator with elastic impacts, where considering that the inequality of static and kinetic friction forces results in the existence of flow barriers at the relative velocity between the mass and conveyor belt tending to zero. The negative feedback is also taken into account, \(\mathrm {i.e.}\), when the direction of the relative velocity between the mass and conveyor belt changes, the constant force in the periodic excitation force will also change to adapt to the variation in speed. Due to the discontinuity resulted from the friction and non-smoothness resulted from impact, the motion of the two masses can be divided into the following kinds: non-sliding motion (or free motion), sliding motion, non-sliding–stick motion and sliding–stick motion, and the phase space in system is divided into different domains and boundaries, where the vector field is continuous in each region but discontinuous on the velocity boundary. The corresponding normal vectors and G-functions on the separation boundaries are introduced to acquire the switching conditions for all possible motions such as passable motion, sliding motion, grazing motion and stick motion. The mapping theory provides an effective method for the discussion of periodic motion in discontinuous dynamical systems. The four-dimensional switching sets are defined by the means of the form of direct product of two-dimensional switching sets, and the mapping structures are introduced based on the switching sets to describe periodic motions. Eventually, in order to explain more intuitively the above motions in such a friction–impact system, the velocity, displacement, G-functions time histories and the trajectories in phase plane for the masses are illustrated by simulation numerically.
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References
Levitan, E.: Forced oscillation of a spring-mass system having combined Coulomb and viscous damping. J. Acoust. Soc. Am. 32, 1265–1269 (1960)
Yeh, G.: Forced vibrations of a two-degree-of-freedom system with combined Coulomb and viscous damping. J. Acoust. Soc. Am. 39(1), 14–24 (1966)
Filippov, A.F.: Differential equations with discontinuous right-hand side. Am. Math. Soc. Transl. 2(42), 99–231 (1964)
Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, Dordrecht (1988)
Trinkle, J., Pang, J., Sudarsky, S., Lo, G.: On dynamic multi-rigid-body contact problems with Coulomb friction. Z. Angew. Math. Meth. 77(4), 267–279 (1997)
Natsiavas, S.: Stability of piecewise linear oscillators with viscous and dry friction damping. J. Sound Vib. 217(3), 507–522 (1998)
Natsiavas, S., Verros, G.: Dynamics of oscillators with strongly nonlinear asymmetric damping. J. Sound Vib. 20(3), 221–246 (1999)
Schlesinger, A.: Vibration isolation in the presence of Coulomb friction. J. Sound Vib. 63(2), 213–224 (1979)
Leine, R., Campen, D., Kraker, A., Steen, L.: Stick-slip vibrations induced by alternate friction models. Nonlinear Dyn. 16(1), 41–54 (1998)
Luo, A., Gegg, B.: Stick and non-stick periodic motions in periodically forced oscillators with dry friction. J. Sound Vib. 291(1–2), 132–168 (2006)
Gao, C.: Stick-slip motion in boundary lubrication. Tribol. Trans. 38(2), 473–477 (1995)
Galvanetto, U.: Bifurcations and chaos in a four-dimensional mechanical system with dry friction. J. Sound Vib. 204(4), 690–695 (1997)
Awrejcewicz, J., Olejnik, P.: Stick-slip dynamics of a two-degree-of-freedom system. Int. J. Bifurc. Chaos. 13(4), 843–861 (2003)
Pascal, M.: Dynamics of coupled oscillators excited by dry friction. ASME J. Comput. Nonlinear Dyn. 3(3), 20–26 (2008)
Pontes, B., Oliveira, V., Balthazar, J.: On stick-slip homoclinic chaos and bifurcation in a mechanical system with dry friction. Int. J. Bifurc. Chaos. 11(7), 2019–2029 (2001)
Galvanetto, U.: Some discontinuous bifurcations in a two-block stick-slip system. J. Sound Vib. 248(4), 653–669 (2001)
Capone, G., D’Agostino, V., Valle, S., Guida, D.: Influence of the variation between static and kinetic friction on stick-slip instability. Wear 161(1–2), 121–126 (1993)
Capone, G., D’Agostino, V., Valle, S., Guida, D.: Stick-slip instability analysis. Meccanica 27(2), 111–118 (1992)
Gaus, N., Proppe, C.: Bifurcation analysis of a stochastic non-smooth friction model. Proc. Appl. Math. Mech. 9(1), 281–282 (2009)
Awrejcewicz, J., Olejnik, P.: Analysis of dynamic systems with various friction laws. ASME Appl. Mech. Rev 58(6), 389–411 (2005)
Shaw, S., Holmes, P.: A periodically forced piecewise linear oscilator. J. Sound Vib. 90(1), 129–155 (1983)
Shaw, S., Holmes, P.: Periodically forced linear oscillator with impacts: chaos and long-period motions. Phys. Rev. Lett. 51(8), 623–626 (1983)
Lamba, H.: Chaotic, regular and unbounded behaviour in the elastic impact oscillator. Physical D 82, 117–135 (1995)
Balachandran, B., Zhao, M., Li, Y.: Dynamics of elastic sturcture subjected to impact excitations. In: Moon, F.C. (ed.) Appl. Nonlinear Chaotic Dyn. Mech. Kluwer, Dordrecht (1997)
Knudsen, J., Massih, A.: Dynamic stability of weakly damped oscillators with elastic impacts and wear. J. Sound Vib. 263, 175–204 (2003)
Andreaus, U., Placidi, L., Rega, G.: Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator. Commun. Nonlinear Sci. Numer. Simul. 15(9), 2603–2616 (2010)
Andreaus, U., Chiaia, B., Placidi, L.: Soft-impact dynamics of deformable bodies. Contin. Mech. Thermodyn. 25(2–4), 375–398 (2013)
Luo, G., Lv, X., Shi, Y.: Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance: diversity and parameter matching of periodic-impact motions. Int. J. Nonlin. Mech. 65, 173–195 (2014)
Senator, M.: Existence and stability of periodic motions of a harmonically forced impacting system. J. Acoust. Soc. Am. 47(5B), 1390–1397 (1970)
Peterka, F.: Bifurcations and transition phenomena in an impact oscillator. Chaos, Solitons Fractals 7(10), 1635–1647 (1996)
Wagg, D.: Multiple non-smooth events in multi-degree-of-freedom vibro-impact systems. Nonlinear Dyn. 43(1–2), 137–148 (2006)
Luo, G., Zhang, Y., Xie, J., Zhang, J.: Periodic-impact motions and bifurcations of vibro-impact systems near 1:4 strong resonance point. Commun. Nonlinear Sci. Numer. Simul. 13, 1002–1014 (2008)
Cao, J., Fan, J., Chen, S., Dou, C., Gao, M.: On discontinuous dynamics of a SDOF nonlinear friction impact oscillator. Int. J. Non-Linear Mech. 121, 103457 (2020)
Virgin, L., Begley, C.: Grazing bifurcations and basins of attraction in an impact-friction oscillator. Physica D 130(1–2), 43–57 (1999)
Cone, K., Zadoks, R.: A numerical study of an impact oscillator with the addition of dry friction. J. Sound Vib. 188(5), 659–683 (1995)
Hinrichs, N., Oestreich, M., Popp, K.: Dynamics of oscillators with impact and frictiont. Chaos, Solitons Fractals 8(4), 535–558 (1997)
Blazejczyk-Okolewska, B.: Study of the impact oscillator with elastic coupling of masses. Chaos, Solitons Fractals 11(15), 2487–2492 (2000)
Cheng, G., Zu, J.: Dynamics of a dry friction oscillator under two-frequency excitations. J. Sound Vib. 275, 591–603 (2004)
Zhu, S., Liu, Y., Lou, Y., Cao, D.: Stabilization of logical control networks: an event-triggered control approach. Sci. China. Inf. Sci. 63, 1–11 (2020)
Zhu, S., Lu, J., Liu, Y.: Asymptotic stability of probabilistic Boolean networks with state delays. IEEE T. Autom. Control 65(4), 1779–1784 (2020)
Yang, D., Li, X., Shen, J., Zhou, Z.: State-dependent switching control of delayed switched systems with stable and unstable modes. Math. Methods Appl. Sci. 41(16), 6968–6983 (2018)
Yang, D., Li, X., Qiu, J.: Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback. Nonlinear Anal-Hybri. 32, 294–305 (2019)
Li, X., Yang, X., Huang, T.: Persistence of delayed cooperative models: impulsive control method. Appl. Math. Comput. 342, 130–146 (2019)
Yang, X., Li, X., Xi, Q., Duan, P.: Review of stability and stabilization for impulsive delayed systems. Math. Biosci. Eng. 15(16), 1495–1515 (2018)
Li, H., Ding, X.: A control Lyapunov function approach to feedback stabilization of logical control networks. SIAM. J. Control. Optim. 57(2), 810–831 (2019)
Li, H., Xu, X., Ding, X.: Finite-time stability analysis of stochastic switched Boolean networks with impulsive effect. Appl. Math. Comput. 347, 557–565 (2019)
Liu, Y., Zhang, Y., Li, H., Alsaadi, F., Ahmad, B.: Control design for output tracking of delayed Boolean control networks. J. Comput. Appl. Math. 327, 188–195 (2018)
Xue, S., Fan, J.: Discontinuous dynamical behaviors in a vibro-impact system with multiple constraints. Int. J. Non-Linear Mech. 98, 75–101 (2018)
Chen, S., Fan, J., Liu, T.: On discontinuous dynamics of a 2-DOF friction-influenced oscillator with multiple elastic constraints. Int. J. Non-Linear Mech. 110, 131–150 (2019)
Li, C., Fan, J., Yang, Z., Xue, S.: On discontinuous dynamical behaviors of a 2-DOF impact oscillator with friction and a periodically forced excitation. Mech. Mach. Theory 135, 81–108 (2019)
Luo, A.: A theory for non-smooth dynamic systems on the connectable domains. Commun. Nonlinear Sci. Numer. Simul. 10, 1–55 (2005)
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)
Luo, A., Gegg, B.: Grazing phenomena in a periodically forced, friction-induced, linear oscillator. Commun. Nonlinear Sci. Numer. Simul. 11, 777–802 (2006)
Luo, A., Gegg, B.: An analytical prediction of sliding motions along discontinuous boundary in non-smooth dynamical systems. Nonlinear Dyn. 49, 401–424 (2007)
Luo, A.: On flow switching bifurcations in discontinuous dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 12, 100–116 (2007)
Luo, A., Zwiegart, P.: Existence and analytical predictions of periodic motions in a periodically forced, nonlinear friction oscillator. J. Sound Vib. 309(1–2), 129–149 (2008)
Luo, A.: Flow switching bifurcations on the separation boundary in discontinuous dynamical systems with flow barriers. Proc. IMechE Part K: J Multi-Body Dyn. 221, 475–485 (2007)
Luo, A.: A theory for flow switchability in discontinuous dynamical systems. Nonlinear Anal. Hybrid Syst. 2, 1030–1061 (2008)
Luo, A.: Discontinuous Dynamical Systems. Higher Education Press, Beijing (2010)
Dou, C., Fan, J., Li, C., Cao, J., Gao, M.: On discontinuous dynamics of a class of friction-influenced oscillators with nonlinear damping under bilateral rigid constraints. Mech. Mach. Theory 147, 103750 (2020)
Fan, J., Liu, T., Chen, S.: Analysis of dynamical behaviors of a 2-DOF friction-induced oscillator with onesided impact on a conveyor belt. Nonlinear Dyn. 97(1), 797–830 (2019)
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)
Luo, A., Huang, J.: Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator. Int. J. Bifurc. Chaos. 13(1), 241–257 (2012)
Luo, A., Faraji Mosadman, M.: Singularity, switchability and bifurcations in a 2-DOF, periodically forced, frictional oscillator. Int. J. Bifurc. Chaos. 23(3), 30009 (2013)
Fan, J., Xue, S., Chen, G.: On discontinuous dynamics of a periodically forced double-belt friction oscillator. Chaos, Solitons Fractals 109, 280–302 (2018)
Acknowledgements
This research was supported by the National Natural Science Foundation of China (No. 11971275) and the Natural Science Foundation of Shandong Province, China (No. ZR2019MA048).
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Gao, M., Fan, J. Analysis of dynamical behaviors of a 2-DOF friction oscillator with elastic impacts and negative feedbacks. Nonlinear Dyn 102, 45–78 (2020). https://doi.org/10.1007/s11071-020-05904-z
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DOI: https://doi.org/10.1007/s11071-020-05904-z