考虑弹簧阻尼作动器解析雅可比矩阵的多刚体动力学分析

弹簧-阻尼-作动器(spring-damper-actuator,SDA)是多体系统中常见的力元,在工程领域中有着广泛的应用.采用绝对坐标方法建立的多体系统动力学控制方程通常是复杂的非线性微分-代数方程组.为了保证数值解的精度和稳定性,通常需要采用隐式算法求解动力学方程,而雅可比矩阵的计算在隐式数值求解过程中至关重要.对于含有SDA的多体系统,SDA造成的附加雅可比矩阵是与广义坐标和广义速度相关的高度非线性函数.目前的很多研究工作专注于广义力向量的计算,然而对附加雅克比矩阵的计算则少有关注.针对含SDA的多刚体系统进行动力学分析,首先基于Newmark算法研究其在动力学方程求解中的雅可比矩阵的构成形式;然后推导SDA的广义力向量对应的附加雅可比矩阵,其中包括广义力向量对广义坐标和对广义速度的偏导数矩阵.最后通过两个数值算例研究附加雅可比矩阵对动力学分析收敛性的影响;数值分析表明:当SDA的刚度、阻尼和作动力数值较大时,SDA导致的附加雅可比矩阵对数值解的收敛性有重要影响;当考虑SDA对应的附加雅可比矩阵时,动力学分析可以以较少的迭代步实现收敛,从而减少分析时间.

RIGID BODY SYSTEM DYNAMIC WITH THE ACCURATE JACOBIAN MATRIX OF SPRING-DAMPER-ACTUATOR

The spring-damper-actuator (SDA) is a common force element in multibody system and widely used in the field of engineering. The governing equations of multibody dynamic system established by absolute coordinate methods are differential-algebraic equations which are usually nonlinear and complex. To ensure the stability and accuracy of the numerical solutions, the implicit algorithms are commonly used to solve the dynamic equations. While the calculations of Jacobian matrices are the crucial process in implicit algorithms. For a multibody system containing the SDA, the addi-tional Jacobian matrices induced by the SDA are highly nonlinear functions of the generalized coordinates and generalized velocities. A lot of current research works focus on the calculation of generalized force vector, however the calculations of additional Jacobian matrices are less concerned. This paper focuses on dynamic analysis of multi-rigid-body systems containing the SDA. Firstly, the construction of the accurate Jacobian matrices in solving the dynamic equations is investi-gated based on the Newmark algorithm. Then, the additional Jacobian matrices relating to the generalized force vector of the SDA are analytically derived. These matrices consist of the partial derivative of generalized force vector with respect to the generalized coordinates and the generalized velocities. Finally, the influence of additional Jacobian matrices on the convergence of dynamic analysis is investigated via two numerical examples. The numerical results indicate that when the values of stiffness, damping and active force are large, the additional Jacobian matrices induced by the SDA have a significant influence on the convergence of dynamic analysis. When the additional Jacobian matrices induced by the SDA are taken into account, the dynamic analysis can achieve convergence with less iteration steps and the computational time thus can be reduced.