Dynamics representation of mechanical systems for time-stepping problems

This paper introduces a novel two-stage time-stepping framework for mechanical systems. The method separates the update of configuration and dynamics: configuration is advanced using independent (joint) velocities, while the dynamic equations are formulated using a dependent set of velocities—specifically, the velocity of body-fixed points and the angular velocity of each rigid body. These dependent velocities are projected onto the joint space to ensure that kinematic constraints are satisfied at the position level, eliminating the need for constraint stabilization techniques. A key advantage of this decomposition is that it enables a compact and transparent expression of the nonlinear inertial terms, which can then be integrated either explicitly or implicitly within the same framework. This flexibility makes the approach particularly well suited for time-stepping schemes, including those used in challenging scenarios such as unilateral contact problems. Numerical results illustrate how different integration strategies and velocity representations affect the stability and accuracy of the solution, highlighting the robustness of the proposed method. • New formulation for time stepping. • Separation of dynamics (impulse–momentum), and kinematics (velocity–position) steps. • Constraints are satisfied without stabilization techniques. • Detailed analysis of nonlinear inertial terms in explicit and implicit time stepping. • Well suited for the development of stable time-stepping methods in systems.