Singularly perturbed formulation: Explicit modeling of multibody systems

Abstract

This paper details the development of a bond graph-based formalism for modeling multibody systems in a singularly perturbed formulation. As opposed to classical multibody modeling methods, the singularly perturbed formulation is explicit, which makes it suitable for parallel processing. Clearances of mechanical joints that couple rigid bodies are described by a set of differential equations with an order of magnitude smaller time scale than that of the system. The resulting increase in the number of integration steps is compensated for by eliminating the need to solve, at each integration step, an implicit system of equations for the accelerations. Moreover, singularly perturbed models of joints can be used to investigate nonlinear properties of joints, such as clearance and friction. The only restriction of this approach is that the simulation should be computed using 64 bit precision because of the two-time scale nature of the solution. The formalism detailed here is based on developing bond graph models of an elementary set of constraint functions. This set can be used to construct any type of mechanical joint. Complex models of multibody systems can now be built by graphically concatenating bond graphs of rigid bodies and bond graphs of joints. The dynamic equations of the resulting system are automatically generated in explicit form ready for numerical integration. Because bond graphs can be algorithmically translated into block diagrams, the formalism presented here can be directly stated in block diagram. We chose to present this work in bond graph notation because of its compactness and because of the insight it provides into the power structure of the system.