Computation of Direct Sensitivities of Spatial Multibody Systems with Joint Friction

Friction exists in most mechanical systems and may have a major influence on the dynamic performance of the system. The incorporation of friction in dynamic systems has been a subject of active research for several years owing to its high nonlinearity and its dependence on several parameters. Consequently, optimization of dynamic systems with friction becomes a challenging task. Gradient-based optimization of dynamical systems is a prominent technique for optimal design and requires the computation of model sensitivities with respect to the design parameters. The novel contribution of this paper is the derivation of the analytical methodology for the computation of direct sensitivities for smooth multibody systems with joint friction using the Lagrangian index-1 formulation. System dynamics have been computed using two different friction models; the Brown and McPhee, and the Gonthier et al. model. The methodology proposed to obtain model sensitivities has also been validated using the complex finite difference method. A case study has been conducted on a spatial multibody system to observe the effect of friction on the dynamics and model sensitivities, compare sensitivities with respect to different parameters and demonstrate the numerical and validation aspects. Since design parameters can have very different magnitudes and units, the sensitivities have been scaled with the parameters for comparison. Finally, a discussion has been presented on the interpretation of the case study results. Due to the incorporation of joint friction, ‘jumps’ or discontinuities are observed in the model sensitivities akin to those observed for hybrid dynamical systems.