Co-dimensional-two grazing bifurcation of two-degree-of-freedom vibro-impact systems

PDF(1175 KB)

Journal of Vibration and Shock ›› 2020, Vol. 39 ›› Issue (20) : 113-120.

WU Shuai，XU Jieqiong，WANG Zihan，YUAN Quan

Author information +

History +

Abstract

When grazing bifurcation and smooth bifurcation take place simultaneously, one kind of co-dimensional-two grazing bifurcation will occur. The co-dimensional-two bifurcation of a two-degree-of-freedom vibro-impact system and its dynamic behavior near the co-dimensional-two grazing bifurcation points were studied. Firstly, the existence conditions of grazing periodic motion were discussed. The global Poincaré mapping for the 1/n impact period motions was constructed by using the discontinuous mapping method, and the bifurcation conditions for the 1/n impact period motions were obtained. Then, an analytical expression satisfied for grazing bifurcation and smooth bifurcation was derived by combining the conditions of periodic grazing bifurcation and periodic impact bifurcation. Based on the expression, the distribution of co-dimensional-two bifurcation points of the system under different periods was analyzed by numerical simulation. Finally, the effectiveness of theoretical analysis was verified by comparing the bifurcation diagrams obtained by the global Poincaré map and the differential system.

Key words

vibro-impact system / discontinuous mapping method / co-dimensional-two grazing bifurcation

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

WU Shuai，XU Jieqiong，WANG Zihan，YUAN Quan. Co-dimensional-two grazing bifurcation of two-degree-of-freedom vibro-impact systems[J]. Journal of Vibration and Shock, 2020, 39(20): 113-120

References

[1] 陆启韶, 徐鉴, 徐伟. “非光滑系统动力学” 专题简介[J]. 力学学报, 2012, 45(1): 1-1.
Lu Qishao, Xu Jian, Xu Wei. Special introduction to "Dynamics of Non-smooth Systems" [J]. Journal of Mechanics, 2012, 45 (1): 1-1.
[2] Wen G , Chen S , Jin Q . A new criterion of period-doubling bifurcation in maps and its application to an inertial impact shaker[J]. Journal of Sound and Vibration, 2008, 311(1-2):212-223.
[3] Nordmark A B . Non-periodic motion caused by grazing incidence in an impact oscillator[J]. Journal of Sound and Vibration, 1991, 145(2):279-297.
[4] Nordmark A B . Universal limit mapping in grazing bifurcations[J]. Physical Review E Statistical Physics Plasmas Fluids & Related Interdisciplinary Topics, 1997, 55(55):266-270.
[5] Bernardo M , Budd C , Champneys A R . Piecewise-smooth Dynamical Systems[M]. Springer London, 2008.
[6] Di Bernardo M, Budd C J, Champneys A R. Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems[J]. Physica D: Nonlinear Phenomena, 2001, 160(3-4): 222-254.
[7] Kowalczyk P, di Bernardo M, Champneys A R, et al. Two-parameter discontinuity-induced bifurcations of limit cycles: Classification and open problems[J]. International Journal of bifurcation and chaos, 2006, 16(03): 601-629.
[8] Peterka F . Bifurcations and transition phenomena in an impact oscillator[J]. Chaos, Solitons and Fractals, 1996, 7(10):1635-1647.
[9] Zhao X , Dankowicz H . Unfolding degenerate grazing dynamics in impact actuators[J]. Nonlinearity, 2006, 19(2):399-418.
[10] Yin S, Wen G, Xu H, et al. Higher order zero time discontinuity mapping for analysis of degenerate grazing bifurcations of impacting oscillators[J]. Journal of Sound and Vibration, 2018, 437(2018): 209-222.
[11] Thota P, Zhao X, Dankowicz H. Co-dimension-two grazing bifurcations in single-degree-of-freedom impact oscillators[J]. Journal of Computational and Nonlinear Dynamics, 2006, 1(4): 328-335.
[12] Xu J , Chen P , Li Q . Theoretical analysis of co-dimension-two grazing bifurcations in n-degree-of-freedom impact oscillator with symmetrical constrains[J]. Nonlinear Dynamics, 2015, 82(4):1641-1657.
[13] Yin S, Shen Y, Wen G, et al. Analytical determination for degenerate grazing bifurcation points in the single-degree-of-freedom impact oscillator[J]. Nonlinear Dynamics, 2017, 90(1): 443-456.
[14] Xu H, Ji J. Analytical-numerical studies on the stability and bifurcations of periodic motion in the vibro-impact systems with clearances[J]. International Journal of Non-Linear Mechanics, 2019, 109(2019): 155-165.
[15] Dankowicz H , Zhao X . Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators[J]. Physica D, 2005, 202(3-4):238-257.
[16] Bernardo M, Budd C, Champneys A R, et al. Piecewise-smooth dynamical systems: theory and applications[M]. Springer Science & Business Media, 2008.
[17] Xu H , Yin S , Wen G , et al. Discrete-in-time feedback control of near-grazing dynamics in the two-degree-of-freedom vibro-impact system with a clearance[J]. Nonlinear Dynamics, 2017, 87(2):1127-1137.