Implicit Numerical Integration for Design Sensitivity Analysis of Rigid Multibody Systems

Abstract

ABSTRACT An implicit Runge-Kutta numerical integration algorithm based on generalized coordinate partitioning is presented for solution of the differential-algebraic equations of motion and design sensitivity analysis for rigid multibody systems. The direct differentiation method for dynamic sensitivity analysis is employed to obtain differential-algebraic sensitivity equations. The implicit integration formula expresses design sensitivity of independent generalized coordinates and their first-time derivative as functions of sensitivities of independent accelerations at discrete integration times. The same integration Jacobian is used to determine generalized independent accelerations and their sensitivities, by iteratively solving the equations obtained from state-space, second-order ordinary differential equations in independent coordinates. Design sensitivities of dependent variables in the formulation, including design sensitivities of Lagrange multipliers, are recovered by satisfying the full design sensitivity system of kinematic and kinetic equations of motion. Design sensitivity analysis results are verified using finite differences, and a comparison with an explicit numerical integration algorithm is performed to validate the efficiency of the proposed algorithm for design sensitivity analysis of stiff vehicle models.