Canonical Kane’s Equations of Motion for Discrete Dynamical Systems

Abstract

The generalized inversion-based canonical equations of motion (CEOMs) for ideally constrained discrete dynamical systems are introduced in the framework of Kane’s method. Upon selection of the inertia-scaled canonical generalized acceleration variables, the proposed formulation employs the acceleration form of constraints and the generalized inversion Greville formula for parameterized solutions of underdetermined algebraic equations together with the nonminimal constrained momentum balance equations. The resulting CEOMs are explicit and full order in the acceleration variables. Moreover, the geometry of constrained motion is revealed by the CEOMs intuitively by partitioning the canonical accelerations column matrix into two portions at every time instant: a portion that drives the dynamical system to abide by the constraints, and a portion that generates the momentum balance dynamics such that the system abides by the Newton–Euler laws of motion. Some insightful geometrical perspectives of the CEOMs are illustrated via vectorial visualizations, which lead to verifying Gauss’s principle of least constraints and its Udwadia–Kalaba interpretation. The procedure is illustrated by formulating a third-order CEOMs dynamic model of two particles experiencing a single degree of freedom, as well as a sixth-order CEOMs dynamic model of a three-degrees-of-freedom disk that is rolling without slipping on an inertial plane under the effect of friction.