Direct Sensitivity Analysis of Spatial Multibody Systems with Joint Friction

Sensitivity analysis is one of the most prominent gradient based optimization techniques for mechanical systems. Model sensitivities are the derivatives of the generalized coordinates defining the motion of the system in time with respect to the system design parameters. These sensitivities can be calculated using finite differences, but the accuracy and computational inefficiency of this method limits its use. Hence, the methodologies of direct and adjoint sensitivity analysis have gained prominence. Recent research has presented computationally efficient methodologies for both direct and adjoint sensitivity analysis of complex multibody dynamic systems. Multibody formulations with joint friction were developed in the recent years and these systems have to be modeled by highly non-linear differential algebraic equations (DAEs) that are difficult to solve using numerical methods. The sensitivity analysis of such systems and the subsequent design optimization is a novel area of research that has been explored in this work. The contribution of this work is in the development of the analytical methods for computation of sensitivities for the most commonly used multibody formulations incorporated with joint friction. Two different friction models have been studied, capable of emulating behaviors of stiction (static friction), sliding friction and viscous drag. A case study has been conducted on a spatial slider-crank mechanism to illustrate the application of this methodology to real-world systems. The Brown and McPhee friction model has been implemented using an index-1 formulation for computation of the dynamics and sensitivities in this case study. The effect of friction on the dynamics and model sensitivities has been analyzed by comparing the sensitivities of slider velocity with respect to the design parameters of crank length, rod length, and the parameters defining the friction model. Due to the highly non-linear nature of friction, it can be concluded that the model dynamics are more sensitive during the transition phases, where the friction coefficient changes from static to dynamic and vice versa.