Exact augmented perpetual manifolds as a tool for the determination of similar rigid body modes: Corollary for determining backbone curves of autonomous systems

Abstract

Perpetual points in mathematics have been defined recently, and their significance in describing the dynamics of systems is ongoing research. In mechanical engineering, the perpetual points of linear unforced mechanical systems are associated with rigid body motions, and they form the perpetual manifolds. The systems that admit perpetual manifolds of rigid body motions are called perpetual mechanical systems. The concept of perpetual manifolds has been extended to the exact augmented perpetual manifolds, whereas all the accelerations are equal but not necessarily zero. A theorem defines the conditions of external forces that lead to rigid body motion of externally forced mechanical systems. Herein, based on this theorem, a corollary is proven that, through a very simple framework, defines specifications to mechanical systems resulting in the existence of rigid body similar modes and can serve as a tool in the nonlinear normal modes (NNMs) theory for the determination of their backbone curves of dissipative mechanical systems. Further, on these, backbone curves can be used for passive vibration control design of N-degree-of-freedom (dof) mechanical systems. The verification of the theory has been done with the design of a 5-dof nonlinear perpetual mechanical system with three types of boundaries, linear, nonlinear smooth, and nonsmooth boundaries. The numerical simulations results are in excellent agreement with the theory. Although 5-dof mechanical systems have been examined, the corollary is valid for N-degree-of freedom systems. The corollary provides a simplified framework for the design of mechanical systems to admit rigid body modes for systems with a general configuration that is not obvious using traditional techniques. This work is rather significant in mechanical engineering because it provides a new tool firstly for determining similar NNMs of dissipative mechanical systems and secondly for the design of passive vibration control devices.