含间隙振动系统低频周期冲击振动的模式类型及分岔特征
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摘要
研究了带有间隙-刚性约束(弹性约束)振动系统的低频振动特性,分析了系统低频范围内基本周期冲击振动和亚谐冲击振动的模式类型、多样性、参数平面内的分布规律及分岔边界特征。通过多目标、多参数协同仿真分析发现了相邻基本周期冲击振动相互转迁的不可逆性及其伴随的两类转迁区域-迟滞域和舌形域,研究了迟滞域和舌形域的形成和分布特征及舌形域内亚谐冲击运动的模式类型及规律特征,分析了基本周期冲击振动向非完整和完整颤冲击振动的转迁过程。
Two periodically-forced systems with single rigid stop and single elastic stop respectively were considered. Pattern types, diversity, regularity and bifurcation characteristics of the fundamental group of impact motions and subharmonic impact motions in the parameter plan, in low frequency range, were analyzed. The transition irreversibility of adjacent impact motions with fundamental period and two types of transition regions, narrow hysteresis and tongues-shaped regions, were found by using the multi-target and multi-parameter co-simulation analysis. The occurrence mechanism and distribution characteristics of the two types of transition regions(hysteresis and tongue-shaped regions), as well as the pattern types and regularity of subharmonic impact motions in the tongue-shaped regions were studied. The transition law from impact motions with fundamental period to incomplete and complete chattering-impact vibrations was studied.
Two periodically-forced systems with single rigid stop and single elastic stop respectively were considered. Pattern types, diversity, regularity and bifurcation characteristics of the fundamental group of impact motions and subharmonic impact motions in the parameter plan, in low frequency range, were analyzed. The transition irreversibility of adjacent impact motions with fundamental period and two types of transition regions, narrow hysteresis and tongues-shaped regions, were found by using the multi-target and multi-parameter co-simulation analysis. The occurrence mechanism and distribution characteristics of the two types of transition regions(hysteresis and tongue-shaped regions), as well as the pattern types and regularity of subharmonic impact motions in the tongue-shaped regions were studied. The transition law from impact motions with fundamental period to incomplete and complete chattering-impact vibrations was studied.
引文
[1]SHAW S W,HOLMES P J.A periodically forced piecewise linear oscillator[J].Journal of Sound and Vibration,1983,90(1):129-155.
[2]AIDANP■ J O,GUPTA B R.Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system[J].Journal of Sound and Vibration,1993,165(2):305-327.
[3]PETERKA F,VACIK J.Transition to chaotic motion in mechanical systems with impacts[J].Journal of Sound and Vibration,1992,154(1):95-115.
[4]HU Haiyan.Controlling chaos of a periodically forced nonsmooth mechanical system[J].Acta Mechanica Sinica,1995,11(3):251-258.
[5]JIN D P,HU H Y.Periodic vibro-impacts and their stability of a dual component system[J].Acta Mechanica Sinica,1997,13(4):366-376.
[6]DING W C,XIE J H,SUN Q G.Interaction of Hopf and period doubling bifurcations of a vibro-impact system[J].Journal of Sound and Vibration,2004,275(1/2):27-45.
[7]XIE Jianhua,DING Wangcai.Hopf-Hopf bifurcation and invariant torus T2of a vibro-impact system[J].International Journal of Non-Linear Mechanics,2005,40(4):531-543.
[8]WEN Guilin.Criterion to identify Hopf bifurcations in maps of arbitrary dimension[J].Physical Review E,2005,72:026201.
[9]LEINE R I.Non-smooth stability analysis of the parametrically excited impact oscillator[J].International Journal of Non-Linear Mechanics,2012,47(9):1020-1032.
[10]LIU Yang,PAVLOVSKAIA E,HENDRY D,et al.Vibroimpact responses of capsule system with various friction models[J].International Journal of Mechanical Sciences,2013,72:39-54.
[11]YUE Y,MIAO P,XIE J.Coexistence of strange nonchaotic attractors and a special mixed attractor caused by a new intermittency in a periodically driven vibro-impact system[J].Nonlinear Dynamics,2017,87(2):1187-1207.
[12]ZHANG Yongxiang,LUO Guanwei.Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibroimpact systems[J].International Journal of Non-Linear Mechanics,2017,96:12-21.
[13]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.
[14]WHISTON G S.Singularities in vibro-impact dynamics[J].Journal of Sound and Vibration,1992,152(3):427-460.
[15]IVANOV A P.Stabilization of an impact oscillator near grazing incidence owing to resonance[J].Journal of Sound and Vibration,1993,162(3):562-565.
[16]HU H Y.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.
[17]WEGER J G,MOLENAAR J.Grazing impact oscillations[J].Physics Review E,2000,62(2):2030-2041.
[18]DI B M,BUDD C J,CHAMPNEYS A R.Grazing and bordercollision in piecewise-smooth systems:a unified analytical framework[J].Physical Review Letters,2001,86(12):2553-2556.
[19]LONG X H,LIN G J,BALACHANDRAN B.Grazing bifurcations in an elastic structure excited by harmonic impactor motions[J].Physica D,2008,237(8):1129-1138.
[20]XU Jieqiong,LI Qunhong,WANG Nan.Existence and stability of the grazing periodic trajectory in a two-degree-offreedom vibro-impact system[J].Applied Mathematics and Computation,2011,217(12):5537-5546.
[21]CHILLINGWORTH D.Dynamics of an impact oscillator near a degenerate graze[J].Nonlinearity,2010,23(11):2723-2748.
[22]高尚晗,龙新华,孟光.主轴-滚动轴承系统三种分岔形式[J].振动与冲击,2009,28(4):59-65.GAO Shanghan,LONG Xinhua,MENG Guang.Three types of bifurcation in a spindle-ball bearing system[J].Journal of Vibration and Shock,2009,28(4):59-65.
[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]高尚晗,龙新华,孟光.主轴-滚动轴承系统三种分岔形式[J].振动与冲击,2009,28(4):59-65.GAO Shanghan,LONG Xinhua,MENG Guang.Three types of bifurcation in a spindle-ball bearing system[J].Journal of Vibration and Shock,2009,28(4):59-65.
[25]BUDD C J,DUX F.Chattering and related behaviour in impact oscillators[J].Philosophical Transactions Physical Sciences&Engineering,1994,347:365-389.
[26]TOULEMONDE C,GONTIER C.Sticking motions of impact oscillators[J].European Journal of Mechanics A/Solids,1998,17(2):339-366.
[27]WAGG D J.Multiple non-smooth events in multi-degree-offreedom vibro-impact systems[J].Nonlinear Dynamics,2006,43(1/2):137-148.
[28]NORDMARK A B,PIIROINEN P T.Simulation and stability analysis of impacting systems with complete chattering[J].Nonlinear Dynamics,2009,58(1/2):85-106.
[29]H■S C,CHAMPNEYS A R.Grazing bifurcations and chatter in a pressure relief valve model[J].Physica D Nonlinear Phenomena,2012,241(22):2068-2076.
[30]朱文骅,许跃敏,叶贵如.非线性边界约束下梁的弯曲振动[J].固体力学学报,1990,11(2):181-189.ZHU Wenhua,XU Yuemin,YE Guiru.Vibrations of beam under nonlinear boundary constrains[J].Act Mechanica Solida Sinica,1990,11(2):181-189.
[31]金栋平,胡海岩.两柔性梁碰撞振动类型的实验研究[J].实验力学,1999,12(2):129-135.JIN Dongping,HU Haiyan.An experimental study on possible types of vibro-impacts between two elastic beams[J].Journal of Experimental Mechanics,1999,14(2):129-135.
[32]国家自然科学基金委员会工程与材料科学部.机械工程学科发展战略报告(2011-2022)[M].北京:科学出版社,2011.
[2]AIDANP■ J O,GUPTA B R.Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system[J].Journal of Sound and Vibration,1993,165(2):305-327.
[3]PETERKA F,VACIK J.Transition to chaotic motion in mechanical systems with impacts[J].Journal of Sound and Vibration,1992,154(1):95-115.
[4]HU Haiyan.Controlling chaos of a periodically forced nonsmooth mechanical system[J].Acta Mechanica Sinica,1995,11(3):251-258.
[5]JIN D P,HU H Y.Periodic vibro-impacts and their stability of a dual component system[J].Acta Mechanica Sinica,1997,13(4):366-376.
[6]DING W C,XIE J H,SUN Q G.Interaction of Hopf and period doubling bifurcations of a vibro-impact system[J].Journal of Sound and Vibration,2004,275(1/2):27-45.
[7]XIE Jianhua,DING Wangcai.Hopf-Hopf bifurcation and invariant torus T2of a vibro-impact system[J].International Journal of Non-Linear Mechanics,2005,40(4):531-543.
[8]WEN Guilin.Criterion to identify Hopf bifurcations in maps of arbitrary dimension[J].Physical Review E,2005,72:026201.
[9]LEINE R I.Non-smooth stability analysis of the parametrically excited impact oscillator[J].International Journal of Non-Linear Mechanics,2012,47(9):1020-1032.
[10]LIU Yang,PAVLOVSKAIA E,HENDRY D,et al.Vibroimpact responses of capsule system with various friction models[J].International Journal of Mechanical Sciences,2013,72:39-54.
[11]YUE Y,MIAO P,XIE J.Coexistence of strange nonchaotic attractors and a special mixed attractor caused by a new intermittency in a periodically driven vibro-impact system[J].Nonlinear Dynamics,2017,87(2):1187-1207.
[12]ZHANG Yongxiang,LUO Guanwei.Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibroimpact systems[J].International Journal of Non-Linear Mechanics,2017,96:12-21.
[13]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.
[14]WHISTON G S.Singularities in vibro-impact dynamics[J].Journal of Sound and Vibration,1992,152(3):427-460.
[15]IVANOV A P.Stabilization of an impact oscillator near grazing incidence owing to resonance[J].Journal of Sound and Vibration,1993,162(3):562-565.
[16]HU H Y.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.
[17]WEGER J G,MOLENAAR J.Grazing impact oscillations[J].Physics Review E,2000,62(2):2030-2041.
[18]DI B M,BUDD C J,CHAMPNEYS A R.Grazing and bordercollision in piecewise-smooth systems:a unified analytical framework[J].Physical Review Letters,2001,86(12):2553-2556.
[19]LONG X H,LIN G J,BALACHANDRAN B.Grazing bifurcations in an elastic structure excited by harmonic impactor motions[J].Physica D,2008,237(8):1129-1138.
[20]XU Jieqiong,LI Qunhong,WANG Nan.Existence and stability of the grazing periodic trajectory in a two-degree-offreedom vibro-impact system[J].Applied Mathematics and Computation,2011,217(12):5537-5546.
[21]CHILLINGWORTH D.Dynamics of an impact oscillator near a degenerate graze[J].Nonlinearity,2010,23(11):2723-2748.
[22]高尚晗,龙新华,孟光.主轴-滚动轴承系统三种分岔形式[J].振动与冲击,2009,28(4):59-65.GAO Shanghan,LONG Xinhua,MENG Guang.Three types of bifurcation in a spindle-ball bearing system[J].Journal of Vibration and Shock,2009,28(4):59-65.
[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]高尚晗,龙新华,孟光.主轴-滚动轴承系统三种分岔形式[J].振动与冲击,2009,28(4):59-65.GAO Shanghan,LONG Xinhua,MENG Guang.Three types of bifurcation in a spindle-ball bearing system[J].Journal of Vibration and Shock,2009,28(4):59-65.
[25]BUDD C J,DUX F.Chattering and related behaviour in impact oscillators[J].Philosophical Transactions Physical Sciences&Engineering,1994,347:365-389.
[26]TOULEMONDE C,GONTIER C.Sticking motions of impact oscillators[J].European Journal of Mechanics A/Solids,1998,17(2):339-366.
[27]WAGG D J.Multiple non-smooth events in multi-degree-offreedom vibro-impact systems[J].Nonlinear Dynamics,2006,43(1/2):137-148.
[28]NORDMARK A B,PIIROINEN P T.Simulation and stability analysis of impacting systems with complete chattering[J].Nonlinear Dynamics,2009,58(1/2):85-106.
[29]H■S C,CHAMPNEYS A R.Grazing bifurcations and chatter in a pressure relief valve model[J].Physica D Nonlinear Phenomena,2012,241(22):2068-2076.
[30]朱文骅,许跃敏,叶贵如.非线性边界约束下梁的弯曲振动[J].固体力学学报,1990,11(2):181-189.ZHU Wenhua,XU Yuemin,YE Guiru.Vibrations of beam under nonlinear boundary constrains[J].Act Mechanica Solida Sinica,1990,11(2):181-189.
[31]金栋平,胡海岩.两柔性梁碰撞振动类型的实验研究[J].实验力学,1999,12(2):129-135.JIN Dongping,HU Haiyan.An experimental study on possible types of vibro-impacts between two elastic beams[J].Journal of Experimental Mechanics,1999,14(2):129-135.
[32]国家自然科学基金委员会工程与材料科学部.机械工程学科发展战略报告(2011-2022)[M].北京:科学出版社,2011.