非线性传送带系统的Hopf分岔及极限环计算
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摘要
建立了一类单自由度非线性传送带系统的物理模型和动力学模型。通过理论推导得到传送带系统平衡点的存在条件,对系统在参数空间中的Hopf分岔存在条件进行判定,并利用增量谐波平衡法(IHB方法)计算得到了系统在纯滑动状态下的极限环及其近似解析表达式和振动频率,同时分析了系统的静平衡状态在参数空间中的稳定性。数值仿真结果表明,由IHB方法所计算出的系统极限环与四阶Runge-Kutta法计算结果十分吻合,IHB方法在求解此类非光滑系统时具有较高的可靠性和精度。
The physical model and mathematical model of a single-DOF nonlinear conveyor belt system are established in this paper.The existence conditions of equilibrium point of the belt system are derived by theoretical deduction.And the existence conditions of the hopf bifurcation of the system in parameter space is judged by numerical method.Then,the limit cycle and its approximate analytic expressions and vibration frequency of the system in pure slide condition can be calculated by applying the incremental harmonic balance method(IHB method),the stability of the static equilibrium state of the system in parameter space is analyzed simultaneously.The simulation results show that the limit cycle calculated by IHB method is very consistent with the four order Runge-Kutta method,the IHB method has high reliability and accuracy in solving such non-smooth systems.
The physical model and mathematical model of a single-DOF nonlinear conveyor belt system are established in this paper.The existence conditions of equilibrium point of the belt system are derived by theoretical deduction.And the existence conditions of the hopf bifurcation of the system in parameter space is judged by numerical method.Then,the limit cycle and its approximate analytic expressions and vibration frequency of the system in pure slide condition can be calculated by applying the incremental harmonic balance method(IHB method),the stability of the static equilibrium state of the system in parameter space is analyzed simultaneously.The simulation results show that the limit cycle calculated by IHB method is very consistent with the four order Runge-Kutta method,the IHB method has high reliability and accuracy in solving such non-smooth systems.
引文
[1]J.Awrejcewicz.Chaos in simple mechanical systems with friction[J].Journal of Sound and Vibration,1986,109(1):178-180.
[2]S.W.Shaw.On the dynamic response of a system with dryriction[J].Journal of Sound and Vibration,1986,108(2):305-325.
[3]B.L.van de Vrande,D.H.Van Campen,A.De Kraker.An approximate analysis of dry-frction-induced stick-slip vibrations by a smoothing procedure[J].NonlinearDynamics,1999,19(2):159-171.
[4]R.I.Leine,D.H.Van Campen,A.De Kraker.Stick-slip vibrations induced by alternate friction models[J].Nonlinear Dynamics,1998,16(1):41-54.
[5]A.C.Luo,J.Z.Huang.Discontinuous dynamics of a non-linear,self-excited,friction-induced,periodically forced oscillator[J].Nonlinear Analysis:Real World Applications,2012,13(1):241-257.
[6]李群宏,闫玉龙,韦丽梅.非线性传送带系统的复杂分岔[J].物理学报,2013,62(12):65-74.(Li Qun-hong,Yan Yu-long,Wei Li-mei.Complex bifurcations in a nonlinear system of moving belt[J].Acta Physica Sinica,2013,62(12):65-74.)
[7]丁旺才,张有强,张庆爽.含干摩擦振动系统的非线性动力学分析[J].工程力学,2008,25(10):212-217.(Ding Wang-cai,Zhang You-qiang,Zhang Qing-shuang.Nonlinear dynamic analysis of vibrate system with dry friction[J].Engineering Mechanics,2008,25(10):212-217.)
[8]丁旺才,张有强,谢建华.含对称间隙的摩擦振子非线性动力学分析[J].摩擦学学报,2008,28(2):155-159.(Ding Wang-cai,Zhang You-qiang,Xie Jian-hua.Analysisof nonlinear dynamicsof dry friction oscillatorswith symmetric clearance[J].Journal of Tribology,2008,28(2):155-159.)
[9]张瑞成,柳晓东,陈至坤.板带轧机AGC系统非线性参激振动机理研究[J].机械设计与制造,2010(1):85-87.(Zhang Rui-cheng,Liu Xiao-dong,Chen Zhi-kun.Study on nonlinear parametrically excited vibration in au tomatic gauge control system of the rolling mill[J].Machinery Design&Manufacture,2010,(1):85-87.)
[10]S.L.Lau,Y.K.Cheung.Amplitude incremental variational principle for nonlinear vibration of elastic system[J].Journal of Applied Mechanics,1981,48(4):959-964.
[11]陈树辉.强非线性振动系统的定量分析方法[M].北京:科学出版社,2007:171-178.(Chen Shu-hui.Quantitative Analysis Method for Strong Nonlinear Vibration System[M].Beijing:Science Press,2007:171-178.)
[12]王艾伦,龙清.IHB方法在含强非线性摩擦力失谐叶盘系统响应特性研究中的应用[J].机械强度,2012,34(2):159-164.(Wang Ai-lun,Long qing.Application of IHB method to study the response characteristics of mistuning bladed disks with strong non-linear friction[J].Journal of Mechanical Strength,2012,34(2):159-164.)
[2]S.W.Shaw.On the dynamic response of a system with dryriction[J].Journal of Sound and Vibration,1986,108(2):305-325.
[3]B.L.van de Vrande,D.H.Van Campen,A.De Kraker.An approximate analysis of dry-frction-induced stick-slip vibrations by a smoothing procedure[J].NonlinearDynamics,1999,19(2):159-171.
[4]R.I.Leine,D.H.Van Campen,A.De Kraker.Stick-slip vibrations induced by alternate friction models[J].Nonlinear Dynamics,1998,16(1):41-54.
[5]A.C.Luo,J.Z.Huang.Discontinuous dynamics of a non-linear,self-excited,friction-induced,periodically forced oscillator[J].Nonlinear Analysis:Real World Applications,2012,13(1):241-257.
[6]李群宏,闫玉龙,韦丽梅.非线性传送带系统的复杂分岔[J].物理学报,2013,62(12):65-74.(Li Qun-hong,Yan Yu-long,Wei Li-mei.Complex bifurcations in a nonlinear system of moving belt[J].Acta Physica Sinica,2013,62(12):65-74.)
[7]丁旺才,张有强,张庆爽.含干摩擦振动系统的非线性动力学分析[J].工程力学,2008,25(10):212-217.(Ding Wang-cai,Zhang You-qiang,Zhang Qing-shuang.Nonlinear dynamic analysis of vibrate system with dry friction[J].Engineering Mechanics,2008,25(10):212-217.)
[8]丁旺才,张有强,谢建华.含对称间隙的摩擦振子非线性动力学分析[J].摩擦学学报,2008,28(2):155-159.(Ding Wang-cai,Zhang You-qiang,Xie Jian-hua.Analysisof nonlinear dynamicsof dry friction oscillatorswith symmetric clearance[J].Journal of Tribology,2008,28(2):155-159.)
[9]张瑞成,柳晓东,陈至坤.板带轧机AGC系统非线性参激振动机理研究[J].机械设计与制造,2010(1):85-87.(Zhang Rui-cheng,Liu Xiao-dong,Chen Zhi-kun.Study on nonlinear parametrically excited vibration in au tomatic gauge control system of the rolling mill[J].Machinery Design&Manufacture,2010,(1):85-87.)
[10]S.L.Lau,Y.K.Cheung.Amplitude incremental variational principle for nonlinear vibration of elastic system[J].Journal of Applied Mechanics,1981,48(4):959-964.
[11]陈树辉.强非线性振动系统的定量分析方法[M].北京:科学出版社,2007:171-178.(Chen Shu-hui.Quantitative Analysis Method for Strong Nonlinear Vibration System[M].Beijing:Science Press,2007:171-178.)
[12]王艾伦,龙清.IHB方法在含强非线性摩擦力失谐叶盘系统响应特性研究中的应用[J].机械强度,2012,34(2):159-164.(Wang Ai-lun,Long qing.Application of IHB method to study the response characteristics of mistuning bladed disks with strong non-linear friction[J].Journal of Mechanical Strength,2012,34(2):159-164.)