Augmented Perpetual Manifolds and Perpetual Mechanical Systems—Part I: Definitions, Theorem, and Corollary for Triggering Perpetual Manifolds, Application in Reduced-Order Modeling and Particle-Wave Motion of Flexible Mechanical Systems

Abstract

Perpetual points in mechanical systems defined recently. Herein are used to seek specific types of solutions of N-degrees of freedom systems, and their significance in mechanics is discussed. In discrete linear mechanical systems, is proven, that the perpetual points are forming the perpetual manifolds and they are associated with rigid body motions, and these systems are called perpetual. The definition of perpetual manifolds herein is extended to the augmented perpetual manifolds. A theorem, defining the conditions of the external forces applied in an N-degrees of freedom system lead to a solution in the exact augmented perpetual manifold of rigid body motions, is proven. In this case, the motion by only one differential equation is described, therefore forms reduced-order modelling of the original equations of motion. Further on, a corollary is proven, that in the augmented perpetual manifolds for external harmonic force the system moves in dual mode as wave-particle. The developed theory is certified in three examples and the analytical solutions are in excellent agreement with the numerical simulations. The outcome of this research is significant in several sciences, in mathematics, in physics and in mechanical engineering. In mathematics, this theory is significant for deriving particular solutions of nonlinear systems of differential equations. In physics/mechanics, the existence of wave-particle motion of flexible mechanical systems is of substantial value. Finally in mechanical engineering, the theory in all mechanical structures can be applied, e.g. cars, aeroplanes, spaceships, boats etc. targeting only the rigid body motions.

Augmented Perpetual Manifolds and Perpetual Mechanical Systems-Part II: Theorem and a Corollary for Dissipative Mechanical Systems Behaving as Perpetual Machines

“…In the second research direction, the perpetual points are used to advance nonlinear dynamics, such as locating hidden and chaotic attractors [5][6][7][8][9][10][11][12][13]. The third research direction is through perpetual points to identify dissipative systems [14][15][16][17][18] and the fourth one with their significance in mechanics [19][20][21][22]. This article is a continuation of the research relative to mechanics, that already, there are three theorems relevant to perpetual points, proved in [19][20][21].…”

“…The third research direction is through perpetual points to identify dissipative systems [14][15][16][17][18] and the fourth one with their significance in mechanics [19][20][21][22]. This article is a continuation of the research relative to mechanics, that already, there are three theorems relevant to perpetual points, proved in [19][20][21]. The first two correlate the perpetual points of linear mechanical systems with rigid body motions, in [19] for conservative mechanical systems and in [20] for dissipative systems.…”

“…Also, the perpetual points of mechanical systems that are forming perpetual manifolds in [20] is shown. In [21], based on the perpetual manifolds concept, the augmented perpetual manifolds are defined as those manifolds that arise in a multi-degrees of freedom flexible mechanical system that all the accelerations are equal but not necessarily zero. In the exact augmented perpetual manifolds, a multi-degree of freedom system is moving as a rigid body, since all inertia elements are having the same generalized coordinates.…”

“…In the exact augmented perpetual manifolds, a multi-degree of freedom system is moving as a rigid body, since all inertia elements are having the same generalized coordinates. A theorem is proved by indicating the conditions that a solution of a nonlinear non-autonomous mechanical system can be in the exact augmented perpetual manifolds [21]. More precisely, the forces in the equations of motion are separated as internal forces and external forces.…”

“…The standing and the longitudinal wave-particle motion of a multi-degrees of freedom mechanical system in [21] is shown. Another immediate outcome of the theorem in [21][22] is a corollary proved in [22] that in the exact augmented perpetual manifolds, the sum of the internal forces is zero.…”