Experimental investigation of perpetual points in mechanical systems

Abstract

In dissipative dynamical systems, equilibrium (stationary) points have a dominant organizing effect on transient motion in phase space, especially in nonlinear systems. These time-independent solutions are readily defined in the context of ordinary differential equations, that is, they occur when all the time derivatives are simultaneously zero. However, there has been some recent interest in perpetual points: points at which the higher time derivatives are zero, but not necessarily the first. Previous work has focused on analytic work (including simulation) and some experimental studies of electric circuits. This paper focuses attention on the occurrence of these points in a simple mechanical system, including experimental verification. Thus, points of zero acceleration can be found in which the corresponding velocity is a maximum or minimum, but not zero. Specifically, the rigid-arm pendulum is used to generate data for which acceleration (and its derivative) can be evaluated. In this paper an experimental (mechanical) setup is described, specifically designed to investigate perpetual points, including a description of the data analysis approaches developed to identify their location.

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

“…defines the generalized coordinates− and their velocities−̇ in the exact augmented perpetual manifold as described by equation (2).…”

“…In the exact augmented perpetual manifolds, considering a solution in the form of equation (2), the kinetic energy is taking the form,…”

“…3) For ∈ (2,3], the external forcing is (2) , and based on analysis (II), the power is having alternated sign. As indicated in Table 3, the wave velocity is zero and therefore, the system returns after the excitation period to the original state.…”

“…The perpetual points have been defined recently by Prasad in [1] as the sets of points that arise when the accelerations and jerks, describing the motion in a mechanical system, for non-zero velocity vector, are equal to zero. In [2], the experimental investigation of perpetual points in mechanical systems in a tilted pendulum is reported. So far, relevant to perpetual points, there are four main research directions.…”

“…So far, relevant to perpetual points, there are four main research directions. The first one is directly related to the perpetual points with the strict mathematical formulation of them [1,[3][4], including the experimental research for identifying the perpetual points of mechanical systems [2]. 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].…”