Dynamics of a two-degree-of-freedom vibration system with bilateral rigid stops

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Journal of Vibration and Shock ›› 2019, Vol. 38 ›› Issue (14) : 37-47.

SHI Yuqing1,2,DU Sanshan2,YIN Fengwei1,L Xiaohong1,LUO Guanwei2

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Abstract

The dynamics of a two-degree-of-freedom vibration system with bilateral rigid stops was studied.The analytic experssion of the symmetrical n-1-1 motion and disturbed map of the Poincaré section at the fixed point of the periodically-forced system with symmetric rigid stops were derived.The Jacobi matrix of the disturbed map at the fixed point was determined.The instability and local bifurcation of the period n single-impact symmetrical motion were discussed.The distribution and formation of the fundamental period group of impact motions, the incomplete and complete chattering-impact motions of the vibration system in low frequency region were revealed by the multi-target and multi-parameter co-simulation analysis.The irreversible transition process between adjacent impact motions with fundamental period and the mechanism of hysteresis and tongue-shaped regions, as well as a series of singularities were discussed.Pattern types and characteristic laws of sub-harmonic impact motions were found in the tongue-shaped regions.Pattern types and existence regions of different periodic-impact motions of the vibration system with asymmetric bilateral stops under different clearance thresholds were analyzed.Pattern types, distribution and bifurcation characteristics of sub-harmonic impact motions in the tongue-shaped regions of the two kinds of periodically-forced vibration systems with symmetric and asymmetric stops were compared analytically.

Key words

clearance / stop / impact / low frequency vibration / bifurcation.

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SHI Yuqing1,2,DU Sanshan2,YIN Fengwei1,L Xiaohong1,LUO Guanwei2. Dynamics of a two-degree-of-freedom vibration system with bilateral rigid stops[J]. Journal of Vibration and Shock, 2019, 38(14): 37-47

References

[1] S.W. Shaw, P.J. Holmes. A periodically forced piecewise linear oscillator [J]. Journal of Sound and Vibration, 1983, 90(1): 129–155.
[2] F. peterka, J. Vacik. Transition to chaotic motion in mechanical systems with impacts [J]. Journal of Sound and Vibration, 1992, 154(1): 95-115.
[3] J.O. Aidanpää, B.R. Gupta. Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system [J]. Journal of Sound and Vibration, 1993, 165(2): 305–327.
[4] Jin Dongping, Huhai Yan. Periodic vibro-impacts and their stability of a dual component system [J]. Acta Mechanica Sinica, 1997, 13(4): 366-397.
[5] J. Knudsen, A. R Massih. Dynamic stability of weakly damped oscillators with elastic impacts and wear [J]. Journal of Sound and Vibration, 2003, 263(1): 175–204.
[6] Li Qunhong, Lu Qishao. Coexisting periodic orbits in vibro-impacting dynamical systems [J]. Applied Mathematics and Mechanics, 2003, 24(3): 261-273.
[7] R.I. Leine. Non-smooth stability analysis of the parametrically excited impact oscillator [J]. International Journal of Non-Linear Mechanics, 2012, 47(9): 1020-1032.
[8] 李群宏，陆启韶. 一类双自由度碰振系统运动分析[J]. 力学学报, 2001, 33(6): 776-786.
LI Qunhong, LU Qishao. Analysis to motions of a two-degree-of-freedom vibro-impact system [J]. Acta Mechanica Sinica, 2001, 33(6): 776-786.
[9] G.L. Wen. Codimension-2 Hopf bifurcation of a two-degree -of-freedom vibro-impact system [J]. Journal of Sound and Vibration, 2001, 242(3): 475–485
[10] Ekaterina Pavlovskaia, Marian Wiercigroch, Celso Grebogi. Two -dimensional map for impact oscillator with drift [J]. Physical Review E, 2004, 70(3): 036201.
[11] Jianhua Xie, Wangcai Ding, E. H. Dowell, L. N. Virgin. Hopf–flip bifurcation of high dimensional maps and application to vibro-impact systems [J]. Acta Mechanica Sinica, 2005, 21(4): 402–410.
[12] 乐源. 一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学[J]. 力学学报, 2016, 48(1): 163-172.
YUE Yuan. Local dynamical behavior of two-parameter family near the neimark-sacker-pitchfork bifurcation point in a vibro-impact system [J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(1): 163-172.
[13] Krzysztof Czolczynski, Andrzej Okolewski, Barbara Blazejczyk -Okolewskaa. Lyapunov exponents in discrete modelling of a cantilever beam impacting on a moving base [J]. International Journal of Non–Linear Mechanics, 2017, 88: 74-84.
[14] Yongxiang Zhang, Guanwei Luo. Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems [J]. International Journal of Non-Linear Mechanics, 2017, 96: 12–21.
[15] A.B. Nordmark. Non-periodic motion caused by grazing incidence in an impact oscillator [J]. Journal of Sound and Vibration, 1991, 145(2): 279–297.
[16] G.S. Whiston. Singularities in vibro-impact dynamics [J]. Journal of Sound and Vibration, 1992, 152(3): 427–460.
[17] A.P. Ivanov. Stabilization of an impact oscillator near grazing incidence owing to resonance [J]. Journal of Sound and Vibration, 1993, 162(3): 562–565.
[18] H.Y. Hu. Detection of grazing orbits and incident bifurcations of a forced continuous, piecewise-linear oscillator [J]. Journal of Sound and Vibration, 1994, 187(3): 485–493.
[19] J. de Weger, W. van de Water, J. Molenaar. Grazing impact oscillations [J]. Physics Review E, 2000, 62 (2B): 2030–2041.
[20] M. di Bernardo, C.J. Budd, A.R. Champneys, Grazing and border-collision in piecewise-smooth systems: a unified analytical framework [J]. Physical Review Letters, 2001, 86(12): 2553–2556.
[21] Haibo Jiang, Antonio S.E. Chong, Yoshisuke Uedab, MarianWiercigroch. Grazing-induced bifurcations in impact oscillators with elastic and rigid constraints [J]. International Journal of Mechanical Sciences, 2017, 127: 204-214.
[22] Harry Dankowicz, Xiaopeng Zhao. Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators [J]. Physica D, 2005, 202(3–4): 238–257.
[23] 黄甫玉高，李群宏. 一类单侧碰撞悬臂振动系统的擦边分岔分析[J]. 力学学报, 2008, 40(6): 812-819 .
HUANGFU Yugao, LI Qunhong. Grazing bifurcation of a vibrating cantilever system with one-sided impact [J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(6): 812-819.
[24] N. Humphries, P.T. Piiroinen. A discontinuity-geometry view of the relationship between saddle–node and grazing bifurcations [J]. Physica D, 2012, 241(22): 1911–1918.
[25] Sergey Kryzhevich, Marian Wiercigroch. Topology of vibro-impact systems in the neighborhood of grazing [J]. Physica D, 2012, 241(22): 1919–1931.
[26] 冯进钤，徐伟. 碰撞振动系统中周期轨擦边诱导的混沌激变[J]. 力学学报, 2013, 45(1): 30-36.
FENG Jinqian, XU Wei. Grazing-induced chaostic crisis for periodic orbits in vibro-impact systems [J]. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(1): 30-36.
[27] C. Toulemonde, C. Gontier. Sticking motions of impact oscillators [J]. European Journal of Mechanics A/Solids, 1998, 17 (2): 339–366.
[28] Satoshi Ema, Etsuo Marui. Suppression of chatter vibration of boring tools using impact dampers [J]. International Journal of Machine Tools and Manufacture, 2000, 40(8): 1141–1156.
[29] D. J. Wagg. Multiple non-smooth events in multi-degree-of- freedom vibro-impact systems [J]. Nonlinear Dynamics, 2006, 43: 137–148.
[30] A.B. Nordmark, P.T. Piiroinen. Simulation and stability analysis of impacting systems with complete chattering [J]. Nonlinear Dynamics, 2009, 58: 85–106.
[31] D. Chillingworth, Dynamics of an impact oscillator near a degenerate graze [J]. Nonlinearity, 2010, 23(11): 2723–2748.
[32] Albert C.J. Luo, Dennis O'Connor. Mechanism of impacting chatter with stick in a gear transmission system [J]. International Journal of Bifurcation and Chaos, 2009, 19(6): 2093–2105.
[33] Csaba Hős, Alan R. Champneys, Grazing bifurcations and chatter in a pressure relief valve model [J]. Physica D, 2012, 241: 2068–2076.
[34] H.Y. Hu. Controlling chaos of a dynamical system with discontinuous vector field [J]. Physica D, 1997, 106(1): l-8.
[35] S.L.T. de Souza, I.L. Caldas. Controlling chaotic orbits in mechanical systems with impacts [J]. Chaos, Solitons and Fractals, 2004, 19: l7l-178.
[36] G. W. Luo, X. H. Lv. Controlling bifurcation and chaos of a plastic impact oscillator [J]. NonlinearAnalysis, Series B: Real World Applications, 2009, 10 (4): 2047-2061.