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 were defined recently. Herein, they are used to seek specific solutions of N-degrees-of-freedom systems, and their significance in mechanics is discussed. In discrete linear mechanical systems, the perpetual points proved that they form the perpetual manifolds, they are associated with rigid body motions, and herein 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 led 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, it forms reduced-order modeling (ROM) of the original equations of motion. Further on, a corollary is proven that for harmonic motion in the augmented perpetual manifolds, 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. This research is significant in several sciences, mathematics, physics, and 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, airplanes, spaceships, and boats, targeting only the rigid body motions.