多体系统动力学指标-2超定运动方程HHT积分改进方法

Improved HHT integration method for index-2 over-determined equations of motion in multibody system dynamics

完整约束多体系统动力学运动方程是指标-3微分-代数方程(differential-algebraic equations,DAEs).速度约束方程通过对位置约束方程求导得到,将其并入指标-3运动方程,可得到指标-2超定DAEs形式的运动方程.对基于结构动力学中HHT(Hilber-Hughs-Taylor)方法的求解指标-2超定运动方程的HHT-SI2方法进行改进.首先,仅对HHT-SI2方法的校正项进行改进,仍引入两个未知数向量.改进后再求解离散得到的非线性方程组时,Jacobian矩阵的条件数减小,方法对步长的限制降低.其次,通过改变校正项的形式,仅引入一个未知数向量,使得非线性方程组的求解规模变小,方法的求解速度得到明显提高.随后,两种改进方法的有效性通过数值实验得到验证,并对改进方法与HHT-SI2方法从计算速度和步长限制角度进行了比较.数值实验还表明两种改进方法都具有二阶精度,数值阻尼是可以控制的.最后对两种改进方法与已有的类似方法进行了分析比较,指出了新方法的优势所在.

Equations of motion for multibody systems with holonomic constraints are index-3 differential-algebraic equations（DAEs）. Velocity constraint equations can be obtained by differentiating the position constraint equations with respect to time. When the velocity constraint equations are combined with index-3 equations of motion, the equations of motion in the form of index-2 over-determined DAEs can be obtained. HHT-SI2 method, which is extended from HHT（Hilbert-Hughs-Taylor） method in the community of structural dynamics and used for the numerical integration of equations of motion in the form of index-2 over-determined DAEs,is improved. Firstly, only the correction term of the HHT-SI2 method is improved, and two unknown vectors are introduced as before. The condition number of Jacobian matrix in Newton-Raphson method is decreased during the numerical solution of nonlinear equations from discretization, and then the limitation of step-size for the numerical integration method is reduced. Secondly, the form of the correction term in HHT-SI2 method is completely changed, and only one unknown vector is introduced. The quantity of equation in nonlinear equations form discretization is decreased, and the integration speed of the method is increased obviously.Then the validity of the two improved methods is verified by numerical experiments, and the improved methods are compared with HHT-SI2 method in the view of integration speed and step-size limitation. Numerical experiments also show that the two improved methods are second-order accuracy and numerical damping can be controlled. In the end, the two improved methods are also analyzed and compared with other existed similar methods in the conclusion section. The advantages of the new methods are pointed out.