Virtual Work & d’Alembert’s Principle

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

Analytical Foundations for Modeling Interactions in Dynamic Systems Based on Impulsive Constraints

In this paper we outline the analytical foundations of an approach for modeling interactions in dynamic systems. The method is based on impulsive constraints which can be employed to represent time-varying interaction of dynamic subsystems, and the transition between different phases of motion. Besides impulsive constraints, the analysis is based on Jourdain’s principle, and a kinematic representation of constrained mechanical systems which is related to this principle. Both finite and impulsive constraints are considered in a general manner, assuming that those can be nonlinear in velocities. It will be shown that Jourdain’s principle can create a simple and physically clear basis for such constrained motion problems. A classification of motions constrained by finite or impulsive constraints is discussed. An impulse-momentum level form of Jourdain’s principle is presented to handle impulsive constraints. An example of two robotic arms in cooperation is employed to illustrate the material presented.