Corollary for the exact augmented perpetual manifolds of linear and nonlinear mechanical systems

Abstract

Perpetual points in mathematics defined recently, their significance, in systems dynamics is ongoing research. In linear unforced mechanical systems, the perpetual points are associated with rigid body motion, and they are not just a few points, but they form the perpetual manifolds. The mechanical systems with perpetual manifolds the rigid body motion are called perpetual mechanical systems. The concept of perpetual manifolds has been extended to the augmented perpetual manifolds of multidegree of freedom system, that the accelerations are equal but not necessarily zero. This is the case of externally forced perpetual mechanical systems that are moving in rigid body motion. A theorem defines the conditions that an externally forced mechanical system moves as a rigid body with state–space the exact augmented perpetual manifolds. Herein following this theorem, a corollary is written and proved. The corollary is about the perpetual mechanical system with nonlinear forces, and the perpetual one with linear internal forces. More precisely for the same time and state-dependent inertia matrix, and the same external forcing they have the same solution or otherwise stated their exact augmented perpetual manifolds that define their state space are the same. Therefore for the same initial conditions, they have the same motion. The theory, analytically and numerically, with two examples, is verified with excellent agreement. The significance of this work is that there is no need for complicated modeling and model update of nonlinear internal forces of mechanical systems when rigid body motions are the target.