A momentum form of Kane’s equations for scleronomic systems

Abstract

Kane'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 presentfor scleronomic systems a momentum form of Kane's equations, fully compatible with portbased 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-N 3 to order-N. 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.

Corrigendum: a momentum form of Kane’s equations for scleronomic systems

“…The first worked example in [1] is an analysis of a spring pendulum, in which a solution for the pendulum tension is shown when the spring compliance is removed; this is said to demonstrate the solution process for nonworking constraint forces. The solution in the example is unaffected by the correction in Eq.…”

“…For a given problem, the generalized coordinate derivatives _ q r are obtained from the generalized speeds f i by means of the Jacobian matrix Q , as in Eqs. (1) and (2) of [1].…”

“…A bond graph interpretation of Eqs. (2) and (5) is shown in Figure 1; this is a corrected version of Figure 3 in [1]. The constraint is the source of zero flow on the left side of the figure, which is regarded as a passive component, since it cannot supply power to the system.…”

“…In a previously published paper [1], we develop a momentum form of Kane's equations [2] for scleronomic systems, and demonstrate its application, including (in Section 9) a procedure for calculating nonworking constraint forces. The procedure requires first, the definition of an additional generalized speed, f Sþ1 ð Þ , representing motion in the direction of the force of interest or rotation about the axis of the moment of interest.…”

Kane’s Equations for Nonholonomic Systems in Bond-Graph-Compatible Velocity and Momentum Forms

“…Bond graphs are a widely used graphical formalism for representing dynamic systems, which may encompass multiple energy domains, in a uniform fashion, using a small set of ideal elements [1,2]. Prior to the appearance of our 2018 paper [3], the most advanced methods for representing multibody systems in a concise bond-graph form were based on generalized momentum, using the IC-field bond-graph element, and these were limited to holonomic systems. In [3] we introduced a bond-graph-compatible momentum method for nonholonomic systems, based on Kane's equations [4,5,6], but it was limited to scleronomic systems.…”