基于随机摄动法的齿轮系统动态响应及灵敏度分析

考虑时变啮合刚度、齿轮综合误差及齿侧间隙等非线性影响因素，建立了多级齿轮传动系统的动力学模型.将齿轮系统中的相关物理参数、几何参数和载荷参数看做随机变量，得到随机振动模型.当随机参数的随机部分比确定部分小得多时，采用随机摄动理论，将振动微分方程在随机参数向量的均值处按Taylor级数展开定理展开至一阶项，并对整理后的方程组进行数值求解.求解过程中结合Kronecker积的代数和矩阵微分理论，得到系统响应的前四阶矩.为了衡量系统的平稳性，对齿对的动态传递误差的前两阶矩进行求解，并与MonteCarlo法进行比较，同时分析了动态传递误差对随机参数的无量纲均值灵敏度，分析结果为提高系统的平稳性提供了依据.

Analysis of Dynamic Response and Sensitivity of Gear Systems Based on Stochastic Perturbation Theory

A multi stage gear transmission dynamic model was developed， where the nonlinearity factors such as time varying stiffness， transmission error， and backlash were considered. This model took the relevant physical， geometrical and load parameters as random variables. When the random section of the variables was much smaller than the determining section， the kinetic equations were expanded to the first order at the mean value of the random parameter vector by Taylor series expansion theorem using the stochastic perturbation theory. Then the finishing equations were solved numerically， with the first fourth order moment of the system response obtained based on the Kronecker product algebra and matrix differential theory. To assess the stability of the system， the first second order moment of the dynamic transmission error of a gear pair was solved and compared with the Monte Carlo method. At the same time，the dimensionless mean sensitivity of dynamic transmission error to the random parameters was analyzed. The results provide a basis for improving the system stability.