Alternate Forms to Lagrange–D'Alembert Principle for Treatment of Rheonomic Constraints

Abstract

Lagrange–D'Alembert principle is based on the concept of the nonactual and nonmeasurable virtual displacement and the assumption that the system-constraint forces are workless. Because time is not considered in defining the virtual displacements, virtual changes in prescribed displacements that characterize rheonomic constraints, referred to as driving constraints, are zero. Consequently, Lagrange–D'Alembert principle does not account systematically, from the outset, for rheonomic constraints, which are not workless and have power associated with them. In multibody system implementations, rheonomic-constraint forces are considered as constraint forces and not as applied forces. Consequently, the statement of the virtual-work principle that virtual work of the system-inertia forces is equal to the virtual work of the system-applied forces because the virtual work of system-constraint forces is zero omits inclusion of rheonomic constraints forces. This paper discusses using alternate forms to Lagrange–D'Alembert principle to account for rheonomic constraints from the outset by using actual and measurable motion variables to replace the virtual displacements. The analysis presented in this paper, which is applicable to both holonomic and nonholonomic systems, shows that the power of the system-inertia forces is equal to the power of the system-applied forces plus the power of rheonomic-constraint forces. It is demonstrated that the existence of Lagrange multipliers is not rooted in the definition of the virtual displacement or virtual work, but in the independence of the constraint functions. It is shown that when redundant coordinates are used, the effect of the rheonomic constraints appears explicitly in the constraint equations while this effect appears as generalized inertia forces when using the independent coordinates.