Equations of motion for constrained mechanical systems and the extended d’Alembert’s principle

Abstract

Abstract.Starting from the principle of virtual work, this paper states and establishes an extended version of D'Alembert's Principle. Using this extended principle and elementary linear algebra, it develops, from first principles, the explicit equation of motion for constrained mechanical systems. The results are compared with the authors' previous results. The approach points to new ways of extending these results.

A geometric approach to the extended D’Alembert principle of Udwadia-Kalaba-Hee-Chang

“…It is worth noting that, as expected, the virtual displacements as considered in [6] coincide with the classical ones if the constraints, whether holonomic or not, are linear in the velocity variables.…”

“…Throughout this section we recall the analysis of the concept of virtual displacements carried out in [6] and its corresponding extended version of the D'Alembert Principle.…”

“…It is shown in [6] that for any virtual displacement w(t + dt) there exists such a v(t) satisfying A(q(t),q(t), t)v(t) = 0 and vice-versa. Remark 1.…”

“…"The displacement is called virtual to distinguish it from an actual displacement of the system occurring in a time interval dt, during which the forces and constraints may be changing." [4] Henceforth, the virtual displacements as defined in [6] will be called dynamical virtual displacements.…”

“…The aim of this paper is to give a geometrical interpretation of such an approach and to show that the Extended D'Alembert Principle of [6] turns out to be equivalent to the version of the Principle of virtual work presented in [7].…”