EXACT AUGMENTED PERPETUAL MANIFOLDS: A COROLLARY FOR THEIR UNIQUENESS

Abstract

The perpetual points have been defined recently as characteristic points in a dynamical system. In many unexcited linear and nonlinear mechanical systems, the perpetual points are associated with rigid body motions and form the perpetual manifolds. The mechanical systems that admit rigid body motions as solutions are called perpetual. In the externally forced mechanical system, the definition of perpetual points to the exact augmented perpetual manifolds extended. The exact augmented perpetual manifolds are associated with the rigid body motion of mechanical systems but with externally excited. The definition of the exact augmented perpetual manifolds leads to a theorem that defines the conditions of an externally forced mechanical system to be moving as a rigid body. Therefore, it defines the conditions of excitation of only this specific type of similar modes, the rigid body modes. Herein, as a continuation of the theorem, a corollary is written and proved. It mainly states that the exact augmented perpetual manifolds for each mechanical system are not unique and are infinite. In an example of a mechanical system, the theory is applied by considering different excitation forces in two-time intervals. The numerical simulations with the analytical solutions are in excellent agreement, which is certifying the corollary. Further, due to the different solutions in the two-time intervals, there is a discontinuity in the vector field and the system's overall solution. Therefore, the state space formed by the exact augmented perpetual manifold is nonsmooth. This work is the first step in examining the exact augmented perpetual manifolds of mechanical systems. Further work is needed to understand them, which mathematical space they belong to, considering that nonsmooth functions might form them.