多间隙耦合非线性动力系统的分叉与混沌

Bifurecation and Chaos in a Nonlinear Dynamic System with Multiple Clearances and Time-Varying Stiffness

在不考虑齿面摩擦的情况下 ,建立了齿轮 -转子 -轴承系统的 3自由度多间隙耦合的振动模型 ;利用数值方法 ,求得了系统的稳态响应 ,绘制了系统振动位移在不同支承条件下随激励频率的分叉图 ,借助响应的时间历程、相轨迹、Poincaré映射、Fourier谱及 Lyapunov指数等分析了响应的特性。由计算结果发现 :在支承刚度较大时 ,该 3自由度多间隙耦合系统经倍周期分叉进入混沌 ,而在支承刚度较小时 ,系统经拟周期分叉进入混沌。在增大支承间隙时 。

To our best knowledge, no paper in the open literature has taken multiple clearances and time varying stiffness into consideration as completely as our present paper in studying the bifurcation and chaos of the nonlinear frequency response of a geared rotor bearing system. We take into consideration time varying gear meshing stiffness and three different clearances: gear backlash and two bearing clearances. First, we develop a three degree of freedom nonlinear dynamic model for a geared rotor bearing system. Then we derive eqs.(1) through (6) in dimensionless form for calculating the steady state forced response to the internal sinusoidal excitation. After that, we analyze in section 2 the influence of the gear meshing stiffness, damper, and two bearing clearances on the chaos and bifurcation of this geared rotor bearing system by means of phase graphs, Poincaré maps, FFT spectra, and Lyapunov exponents (eqs. 8 and 9). Finally, on the basis of numerical results as presented in Figs. 2 through 7, we obtain the following preliminary conclusions: (1) Figs. 2, 3, and 4 show that the steady state solution of the system becomes chaotic through a sequence of period doubling bifurcation of the periodic response due to a change in excitation frequency at high gear meshing stiffness and low damping; (2) the quasi periodic route to chaos occurs at low gear meshing stiffness and high damping; (3) with increasing bearing clearances, the jump phenomena become more pronounced and the stability of frequency response decreases, so bearing clearances should be carefully controlled.