The effect of a linear constraint on the small oscillations of a dynamical system with three degrees of freedom

Abstract

En
We consider the effect of a linear constraint on the small oscillations about an equilibrium position of a conservative dynamical system which has just three degrees of freedom. Explicit expressions are presented for the eigenfrequencies and eigenvectors of the constrained system in terms of the eigenfrequencies and eigenvectors of the unconstrained system. The unconstrained system is described in terms of two positive definite 3×3 matrices with which two concentric ellipsoids may be associated. A ‘plane’ corresponds to the linear constraint. It is seen that in general it is possible to choose two constraints such that the constrained motion has a double eigenfrequency,or equivalently, two central planes may be chosen which cut the two concentric ellipsoids in a pair of similar and similarly situated ellipses.