Implementation of HHT algorithm for numerical integration of multibody dynamics with holonomic constraints

The Hilber–Hughes–Taylor (HHT) algorithm is implemented in this paper to solve the dynamic problem of flexible multibody system with holonomic constraints. The incremental formulas of governing equations of motion, a set of differential algebraic equations of index 3, are presented and integrated direct by current approach. The parameterization of rotation has been paid a special attention, which makes the formula of HHT algorithm much more complicated because of the nonlinearity of rotations. The associated nonlinear algebraic equations are solved with the application of simplified Newton’s iteration to simplify the linearization of finite rotation. Two options are available for solving the incremental linear equations, one is the direct method based on LU decomposition and the other GMRES-based matrix-free method. Based on the estimated truncation error, the time adaptive scheme is designed to enhance the integrating speed. Numerical simulations prove the modified HHT algorithm behavior as the second-order accuracy with numerical damping. The application of conformal rotation vector improves the robustness of the current approach which makes it possible to integral rotation of arbitrary magnitude.