Numerical simulation of nonholonomic rigid-body systems

Abstract

The paper proposes computer algebra system (CAS) algorithms for computer-assisted derivation of the equations of motion for systems of rigid bodies with holonomic and nonholonomic constraints that are linear with respect to the generalized velocities. The main advantages of using the D'Alembert-Lagrange principle for the CSA-based derivation of the equations of motion for nonholonomic systems of rigid bodies are demonstrated. Among them are universality, algorithmizability, computational efficiency, and simplicity of deriving equations for holonomic and nonholonomic systems in terms of generalized coordinates or pseudo-velocities Keywords: computer algebra system, equations of motion for systems of rigid bodies, holonomic and nonholonomic systems, D'Alembert-Lagrange principleIntroduction. Problems of nonholonomic mechanics are complicated regarding the formalization of description. They play a special role in studying mechanical systems and are important for engineering applications [2,[5][6][7][8][13][14][15]. There are different methods for describing, deriving, and integrating the equations of motion for nonholonomic systems [3,4,9,10]. These methods employ different representations of kinetic energy (or the Lagrangian function): as a function of generalized coordinates and velocities (equation with Lagrange multipliers); as a function of generalized coordinates and independent velocities (Chaplygin's and Voronets' equations); and as a function of generalized coordinates and pseudovelocities (Lagrange-Euler equations derived by Hamel). The acceleration energy function introduced by Appell [3] and the modified Hamiltonian function [10] are used with the same purpose. The listed methods are not universal-resulting expressions are rather awkward and do not guarantee correct results; and if the mechanical system changes its structure, then the equations should be derived anew. Therefore, the methods are of little use in the development of universal problem-oriented systems (POSs) for PCs with the purpose of research automation and solution of nonholonomic mechanics problems. The paper [12] proposed a theory and analytic algorithms for computer-algebra systems (CASs) used to develop a POS for PCs with the purpose of research automation and solution of basic problems in the mechanics of holonomic rigid-body systems with elastic and dissipative constraints based on the D'Alembert-Lagrange principle.In what follows, we will first calculate the number of operations to demonstrate the numerical efficiency of the equations of motion for holonomic systems and the fundamental possibility of deriving the equations of motion for nonholonomic systems using the method proposed in [12]. Then we will discuss algorithms (based on the D'Alembert-Lagrange principle) for computer-assisted derivation of the equations of motion for systems with nonholonomic constraints linear with respect to generalized velocities. We will also outline modifications of the algorithms from [12] that make it possible to use one CAS to find analytic and...