Unified framework for direct sensitivity analysis of smooth and non-smooth multibody systems

Sensitivity analysis of multibody systems is essential for quantifying the effect of design parameters on the dynamic output. The most straight-forward and often used approach to calculate dynamic sensitivities is through finite difference method. This procedure is computationally expensive when the number of parameters is large, and numerical errors can severely limit its accuracy. The current state-of-the-art has demonstrated efficient techniques for evaluating sensitivities of smooth multibody systems with ideal joints and friction. However, real-world applications have clearances within the joints where the dynamics is governed by intermittent formation of contacts. Though excluded in case of ideal joints, these clearances have substantial influence on the dynamic response of multibody systems. This article explores a general unified framework for evaluating direct sensitivities of ideal joints as well as joints with clearance using a non-smooth dynamics approach. The sensitivity equations are derived from the non-smooth dynamics formulation with half-implicit discretization scheme. The proposed methodology is demonstrated on a simple pendulum, and a control problem related to motorcycle’s gear-shifting mechanism. The results have been validated with central finite difference method (CFDM) for real valued perturbations. Limitation on CFDM’s accuracy, and noise content in the solution are emphasized, specially in case of systems with clearances. The sensitivities corresponding to a system with ideal joints were found to be independent on step-size as opposed to time-step varying sensitivities in system with joints with clearance. To the best of our knowledge, the results of such a study have not been reported in the literature.