A momentum form of Kane’s equations for scleronomic systems

Abstract

ABSTRACTKane’s dynamical equations are an efficient and widely used method for deriving the equations of motion for multibody systems. Despite their popularity, no publication has appeared which adapts them for use with port-based modelling tools such as bond graphs, linear graphs or port-Hamiltonian theory. In this paper, we present – for scleronomic systems – a momentum form of Kane’s equations, fully compatible with port-based modelling methods. When applied to holonomic systems using coordinate derivatives, the momentum form of Kane’s equations is an efficient alternative to Lagrange’s equations, providing a momentum formulation without the need to assemble and differentiate the system kinetic co-energy function. When applied to holonomic or nonholonomic systems using generalized speeds, a rotational decomposition of the generalized forces leads to a convenient set of matrix equations of motion, for which a system-level multibond graph interpretation is given. Heuristics are provided for selection of generalized speeds which, for systems with open-chain kinematics, produce a block-diagonal mass matrix and reduce the complexity of the equations from order- to order-. For scleronomic systems, the momentum formulation retains all analysis capabilities offered by the original acceleration formulation. Two example problems are solved with the momentum formulation, including the nonholonomic rolling thin disk.