齿轮-轴承系统非线性混沌控制参数摄动与轨道偏差分析

针对齿轮-轴承系统混沌响应减振控制问题,建立了含多间隙的系统非线性振动模型,模型中考虑了齿侧间隙、轴承径向间隙等非线性激励因素。通过系统状态模型与变分转换求解了Jacobi矩阵与敏感度矢量,结合微分流形理论和OGY(Ott-Grebogi-Yorke)控制法对混沌吸引子高周期轨道控制不稳定维数变异情形改进控制条件;采用Newton-Raphson数值法搜寻到混沌吸引子内部镶嵌的P8和P10等不稳定周期轨道不动点,发现二者Jacobi矩阵特征值谱中均存在模为1的临界复共轭特征根,目标周期轨道表现非双曲性。以轴承预载荷为名义控制参数,对P1、P2、P4、P8和P10等周期的多阶段控制表明状态迁移点附近存在短暂混沌瞬态振荡,高周期轨道控制精度下降、轨道偏差增高,控制稳定后参数摄动按受控周期轨道状态规律演化。

Nonlinear chaos control parametric perturbation and orbital deviation of a gear-bearing system

In order to investigate the chaotic vibration control problems of gear-bearing system, nonlinear dynamic model with multiple clearances was performed, the nonlinear excitations, such as backlash, bearing radial clearances were contained. Jacobi matrix and sensitivity vectors were computed based on system state model and variational transformation, then associated with differential manifold theorem and OGY (Ott-Grebogi-Yorke) chaos control method, the dominating conditions for controlling chaotic attractors to higher periodic orbits when unstable dimension variability happens were improved. The P8 and P10 unstable periodic saddle points which embedded in the interior of chaotic attractors were calculated by means of Newton-Raphson numerical algorithm, and both of which were verified containing critical complex conjugate eigenvalues with modulus 1 inside Jacobi matrix eigenvalue spectra, consequently, the target periodic orbits of P8 and P10 were revealed to be non-hyperbolic. Taking bearing preload as nominal controlling excitation, chaotic transient oscillation takes place nearby the location of trajectory switching point according to the analysis of multi-stage controlling of P1, P2, P4, P8 and P10, for higher periodic orbits controlling, the accuracy decreases with high trajectory deviations, finally, the parameter excitation evolves and complies with the controlled periodic orbital state after stabilization.