In this paper, linear vibrating systems, in which the inertia and stiffness
matrices are symmetric positive definite and the damping matrix is symmetric
positive semi-definite, are studied. Such a system may possess undamped
modes, in which case the system is said to have residual motion. Several
formulae for the number of independent undamped modes, associated with purely
imaginary eigenvalues of the system, are derived. The main results formulated
for symmetric systems are then generalized to asymmetric and symmetrizable
systems. Several examples are used to illustrate the validity and application
of the present results.