ODE vs DAE Formulations for Dynamic Simulation of a Four Bar Mechanism

Paper i proceeding, 2027

Reliable dynamic simulation of closed-chain mechanisms requires an appropriate treatment of loop closure constraints and time integration. This paper investigates how these choices influence the dynamic simulation exemplified by a classical four-bar mechanism. Three formulations are compared: (i) a Differential Algebraic Equation (DAE) model incorporating the implicit loop constraints via Lagrange multipliers, (ii) an explicitly-constrained Ordinary Differential Equation (ODE) model derived in minimal coordinates, and (iii) a penalty-based ODE model in which the coupler link is replaced by a highly stiff virtual spring. The ODE formulations are solved using Runge-Kutta method, while the DAE system is solved with the generalized-alpha method. Numerical experiments evaluate accuracy, constraint satisfaction, and computational cost. The explicit-constraint ODE approach achieves the highest accuracy and the lowest computational effort, owing to its minimal coordinate representation. The DAE formulation performs robustly, but is more expensive. The penalty-based ODE method exhibits the highest errors and computational costs due to the high numerical stiffness. This case study highlights the importance of constraint handling and time integration in achieving a stable and efficient simulation of closed-chain mechanisms.