Exact Augmented Perpetual Manifolds Define Specifications for the Steady States of Similar Rigid Body Modes, a Corollary for Nonautonomous Systems

Abstract

Recently, the perpetual points in mathematics have been defined. In mechanical systems, the perpetual points for linear and some nonlinear systems are associated with rigid body motions and they form the perpetual manifolds. The mechanical systems that admit perpetual manifolds of rigid body motions are called perpetual mechanical systems, and these systems are not limited to linear (with a zero natural frequency) only but also nonlinear systems are considered. The concept of perpetual manifold is extended to the augmented perpetual manifold that is defined in externally forced mechanical systems when all the accelerations are equal but not necessarily zero. A proved theorem defines the external forcing conditions that a mechanical, and also a structural system can have solutions described by the exact augmented perpetual manifolds, which are the rigid body motions. Herein a corollary is proved defining the conditions that the steady states of a mechanical and also a structural system are associated with rigid body motions. The internal forces corresponding to the structural elements of the original system are separated to these forces that are associated with a perpetual mechanical (structural) system and the rest forces, which are associated with boundary structural elements, as external forces are considered. The developed theory is applied in a nonlinear perpetual structural system for the design of four types of boundaries, linear, nonlinear smooth, nonsmooth, and nonlinear generalized forces boundaries, and the numerical results verified the developed theory. This design can be considered as a passive control method for vibration mitigation of mechanical and structural systems, since in the exact augmented perpetual manifolds all the inertia elements move without oscillations and the sum of their internal forces is zero. This work is rather significant in engineering since it can be used for the structural design of systems.