Abstract: Aiming at the problem of solving DAEs'motion equation in multi-body system dynamics simulation, new methods were proposed based on Backward Differentiation Formulas (BDF). The existing algorithm of NStiff which solved index-3DAEs form motion equation was extended to the index-2DAEs form motion equation of non-holonomic mechanics system. A new algorithm of NStiff-v was proposed to solve index-3DAEs form motion equation of holonomic constraint multi-body system and index-2DAEs form motion equation of non-holonomic mechanical system. During the discretization of motion equations, the constraint equations were scaled to eliminate ill-conditioning of Jacobian matrix caused by small time steps on solving nonlinear equations with Newton-Raphson method. Through two examples of slider-crank mechanism and Snakeboard model, the effectiveness of proposed algorithm was validated by comparing NStiff with NStiff-v. In addition, experiments illustrated that the algorithm had second-order accuracy in solving DAEs.

Key words: multi-body system, constraint equation, differential-algebraic equations, backward differentiation formulas, algorithms

摘要： 针对多体系统动力学仿真中需要求解DAEs形式运动方程的问题，提出基于向后微分公式的DAEs算法。将已有的求解指标-3DAEs形式运动方程的NStiff算法推广到求解非完力学整系统指标-2DAEs形式的运动方程；提出了NStiff-v算法，分别求解完整约束多体系统指标-3DAEs和非完整力学系统指标-2DAEs形式的运动方程。离散过程中对约束方程进行缩放，消除了Newton-Raphson方法求解非线性方程组时由于步长较小而存在的Jacobian矩阵病态问题。通过曲柄滑块机构和Snakeboard模型两个实例，比较NStiff算法和NStiff-v算法的结果来验证算法的有效性，并比较了效率。算例说明算法求解DAEs时具有二阶精度。

关键词: 多体系统, 约束方程, 微分-代数方程, 向后微分公式, 算法