Augmented Perpetual Manifolds, a Corollary: Dynamics of Natural Mechanical Systems with Eliminated Internal Forces

Abstract

AbstractPerpetual points have been defined, in mathematics, recently, and their role in the dynamics of systems is on-going research. In unforced linear and some nonlinear mechanical dissipative systems, the perpetual points are associated with rigid body motions and they form the perpetual manifolds. Considering externally excited systems lead to the definition of exact augmented perpetual manifolds, that are associated with rigid body motions, whereas, all the generalized coordinates are coinciding and therefore, all their velocities too. In this article, a corollary is proven, that all the internal forces, depended by generalized coordinates and velocities, of natural mechanical systems that the dynamics described by exact augmented perpetual manifolds, are zero. The corollary in 2-dof nonlinear and the underlying linear systems has been applied, and its validity, analytically and with numerical simulations, is shown. Therefore, the dynamics of the mechanical systems in the exact augmented perpetual manifolds is rather important for mechanics since it is associated with eliminated internal forces that are the main reason for failures of mechanical systems.KeywordsAugmented perpetual manifoldsPerpetual mechanical systemsZero internal forcesPerpetual points