Abstract
Mechanisms are inherently constrained devices. Combining flexibility with mechanisms usually requires using Lagrange multipliers to handle the constraints. The added algebraic or numerical tedium, associated with the Lagrange multipliers, is well documented. Presented in this paper is a technique for obtaining the minimal set of hybrid parameter differential equations for a constrained device. That is, the set of equations that inherently incorporate the constraints.
The technique illustrated in this paper is a recently developed hybrid parameter multiple body (HPMB) system modeling methodology. The variational nature of the methodology allows rigorous equation formulation providing not only the complete nonlinear, hybrid differential equations, but also the boundary conditions. The methodology is formulated in the constraint-free subspace of the system’s configuration space, thus Lagrange multipliers are not needed for constrained systems, regardless of the constraint type (holonomic or nonholonomic).
To evince the utility of the method, a flexible four bar mechanism is modeled. Particularly, the inversion of the slider crank found in the quick return mechanism. A comparison of Hamilton’s principle and the described technique, as they are applied to the mechanism, is included. It is shown that the same equations result from either method, but the new technique is much more concise, more efficiently handles the constraints, and requires less algebraic tedium.
Theory of fractional hybrid differential equations