Variational Method for Percussion Motion of a Non-Ideal Constraint System

The purpose is to study the variational method for impact problem of a non-ideal constraint system. Compared to an ideal constraint system, a non-ideal constraint system has more practical engineering background, but it contains unknown non-ideal constraint forces, which usually can be written as the function of ideal constrained forces. The existing analytical method on ideal constraint systems does not contain constraint forces. The non-ideal constraint forces are considered for impact problems in the variational principle, and then by method of Lagrange multipliers, equations for impact motion of a holonomic system and an expression of the kinetic energy variation during the impact motion are obtained.

Constraint Reaction Forces and Lagrange Multipliers in Multi-Body Systems

The study performs the relation between the Lagrange multipliers and the constraint reaction forces in the multi-body systems. It helps to establish a simple and effective method to deal with the non-smooth factors including friction, contact and collision in multi-body systems. First, a general expression calculating the ideal constraint reaction forces is described by a form of complete Cartesian reference coordinate in a discrete dynamical system. Then, an explicit expression is given in the form of independent general generalized coordinates for a multi-body system. A condition under which the constraint reaction forces can be one-to-one corresponding to Lagrange multipliers is discussed. An example is used to demonstrate the process to realize the condition and the effectiveness of this method.

Dynamical Equations With Non-Ideal Constraints: New and Old Alternatives

This paper deals with the motion of mechanical systems with non-ideal constraints, defined as constraints where the forces associated with the constraint do work. The first objective of the paper is to show that two newly published formulations of equations of motion of systems with such non-ideal constraints are unnecessarily complex for situations where the non-ideal constraint force does not depend on the ideal constraint force, because they introduce and then eliminate these non-working constraint forces. We point out that a method already exists for nonideal constraints, namely, Kane’s equations, which are simpler because, among other things, they are based on automatic elimination of non-working constraints. The examples considered in these recent publications are worked out with Kane’s method to show the applicability and simplicity of Kane’s method for non-ideal constraints. A second objective of the paper is to present an alternative form of equations for systems where the non-ideal constraint force depends on the ideal constraint force, as in the case of Coulomb friction. The formulation is shown to lend itself naturally to also analyzing impact dynamics. The method is applied to the dynamics of a slug moving against friction on a moving ellipsoidal surface. Such a crude model may simulate, in essence, propellant motion in a tank in zero-g, or during docking of a spacecraft.