基于G-P算法关联维数齿轮系统相空间吸引子数值特性
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摘要
为有效刻画齿轮系统相空间吸引子结构的数值特性,建立了直齿轮系统含间隙与综合传动误差的非线性动力学模型,基于G-P算法推导了等间隔的齿轮系统吸引子关联维数计算公式。对周期运动和混沌运动吸引子,采用Lyapunov指数与关联维数等手段定量表征其数值特性,利用Poincaré截面法定性分析了混沌吸引子的演化和迁移进程。通过关联维数对阻尼比和综合传动误差变化下的混沌吸引子演化行为进行了追踪刻画,结果表明,吸引子结构越复杂则关联维数越大,系统振动越敏感,混沌吸引子关联维数值介于整数1和2之间,具有分数维特征。
In order to depict the numerical characteristic of the attractor structure in phase space for gear system effectively, a nonlinear dynamical spur gear transmission model including backlash and general transmission error is established, the calculation formula of equispaced gear system correlation dimension is deduced based on G-P algorithm. The numerical characteristics are explored quantitatively for both of periodic attractor and chaotic attractor by employing largest Lyapunov exponent as well as correlation dimension, the evolution and migration process of chaotic attractors is analyzed by means of Poincaré section. The evolution of chaotic attractor under the excitation of damping ratio and general transmission error is described by correlation dimension. The results show that the more complex of attractor structure generates the larger correlation dimension values, and causing strong vibration to the system, correlation dimension of chaotic attractor is between integer 1 and 2 with fractal characteristic.
In order to depict the numerical characteristic of the attractor structure in phase space for gear system effectively, a nonlinear dynamical spur gear transmission model including backlash and general transmission error is established, the calculation formula of equispaced gear system correlation dimension is deduced based on G-P algorithm. The numerical characteristics are explored quantitatively for both of periodic attractor and chaotic attractor by employing largest Lyapunov exponent as well as correlation dimension, the evolution and migration process of chaotic attractors is analyzed by means of Poincaré section. The evolution of chaotic attractor under the excitation of damping ratio and general transmission error is described by correlation dimension. The results show that the more complex of attractor structure generates the larger correlation dimension values, and causing strong vibration to the system, correlation dimension of chaotic attractor is between integer 1 and 2 with fractal characteristic.
引文
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[11]党建武,黄建国.基于G.P算法的关联维计算中参数取值的研究[J].计算机应用研究,2004,21(1):48-51.
[12]张小明,刘建敏,乔新勇,等.时间序列关联维数快速算法及实现[J].装甲兵工程学院学报,2007,21(6):58-61.
[2]向玲,高雪媛,张力佳,等.支承阻尼对多自由度齿轮系统非线性动力学的影响[J].振动与冲击,2017,36(19):139-144.
[3]苟向锋,吕小红,陈代林.单级齿轮传动系统的Hopf分岔与混沌研究[J].中国机械工程,2014,25(5):679-684.
[4]KAHRAMAN A,SINGH R.Non-linear dynamics of a spur gear pair[J].Journal of Sound&Vibration,1990,142:49-75.
[5]林何,王三民,董金城.并车螺旋锥齿轮传动动力学参数二维域界结构分析[J].航空动力学报,2017,32(8):2017-2024.
[6]刘梦军,沈允文,董海军.单级齿轮非线性系统吸引子的数值特性研究[J].机械工程学报,2003,39(10):111-116.
[7]FARSHIDIANFAR A,SAGHAFI A.Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems[J].Nonlinear Dynamics,2014,75(4):783-806.
[8]WOLF A,SWIFT J B,SWINNEY H L,et al.Determining lyapunov exponents from a time series[J].Physica D Nonlinear Phenomena,1985,16(3):285-317.
[9]LIU X,HONG L,JIANG J,et al.Global dynamics of fractional-order systems with an extended generalized cell mapping method[J].Nonlinear Dynamics,2015,83(3):1419-1428.
[10]SAYAH M,BAPTISTA M S,ING J,et al.Attractor reconstruction of an impact oscillator for parameter identification[J].International Journal of Mechanical Sciences,2015,103(1):212-223.
[11]党建武,黄建国.基于G.P算法的关联维计算中参数取值的研究[J].计算机应用研究,2004,21(1):48-51.
[12]张小明,刘建敏,乔新勇,等.时间序列关联维数快速算法及实现[J].装甲兵工程学院学报,2007,21(6):58-61.