Perpetual points in natural mechanical systems with viscous damping: A theorem and a remark

Abstract

Perpetual points have been defined recently and their role in the dynamics of mechanical systems is ongoing research. In this article, the nature of perpetual points in natural dissipative mechanical systems with viscous damping, but excepting any externally applied load, is examined. In linear dissipative systems, a theorem and its inverse are proven stating that the perpetual points exist if the stiffness and damping matrices are positive semi-definite and they coincide with the rigid body motions. In nonlinear dissipative natural mechanical systems with viscous damping excepting any external load, the existence of perpetual points that are associated with rigid body motions is shown. Also, an additional type of perpetual points due to the added dissipation is shown that exists, and this type of perpetual points, at least in principle can be used for identification of dissipation in nonlinear mechanical systems. Further work is needed to understand the nature of this additional type of perpetual points. In all the examined examples the perpetual points when they exist, they are not just a few points, but they are forming manifolds in state space, the Perpetual Manifolds, and their geometric characteristics worth further investigation. The findings of this article are applied in all mechanical systems with no gyroscopic effects on their motion, e.g. cars, airplanes, trucks, rockets, robots, etc. and can be used as part of the elementary studies for basic design of all mechanical systems. This work paves the way for new design processes targeting stable rigid body motions eliminating any vibrations in mechanical systems.