Implicit integration of the equations of multibody dynamics

Implicit integration approaches based on generalized coordinate-partitioning of the differential-algebraic equations of motion of multibody dynamics are presented. The methods developed are intended for simulation of stiff mechanical systems using the well-known Newmark integration method from structural dynamics and more recent implicit Runge-Kutta methods. Newmark integration formulas for second-order differential equations are first used to define independent generalized coordinates and their first time-derivative as functions of independent accelerations. The latter are determined as the solution of discretized equations obtained using the descriptor form of the equations of motion. Dependent variables in the formulation, including Lagrange multipliers, are determined to satisfy all the kinematic and kinetic equations of multibody dynamics. Next, the process is extended to implicit Runge-Kutta methods with variable step size and error control, The approach is illustrated by solving the constrained equations of motion for mechanical systems that exhibit stiff behavior. Results show that the approach is robust and has the capability to integrate differential-algebraic equations of motion for stiff multibody dynamic systems.