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

Abstract

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.