Spatial Static Gravity Balancing With Ideal Springs

This paper discusses mechanisms that allow for perfect static balancing of rotations about a fixed spherical joint by means of ideal springs. Using a potential energy consideration, balancing conditions of a spatial three-spring balancer will be derived. It will be shown that not satisfying these conditions causes non-constant terms in the potential energy expression of the spring-mechanism, which can be eliminated by coupling the spring-mechanism to an inverted pendulum.

On the Equivalence of the Galerkin and Rayleigh-Ritz Methods

The Galerkin method for the approximate solution of elasticity problems (see e.g. Ref. 1) is usually presented as an alternative to the Rayleigh-Ritz method. The main distinction between the two methods is stated to be that the former begins with an equation of equilibrium, whereas the latter begins with a total potential energy expression. The two methods are not always equivalent, and the conditions for equivalence are given in Duncan's lucid presentation of the method (Refs. 2 and 3, which unfortunately have been out of print for some time). These conditions, however, are generally omitted from discussions of the Galerkin method, and it is the purpose of this note to re-emphasise them in a slightly different form.