Qualitative Theory and Identification of Dynamic System with One Degree of Freedom

Abstract

Dynamic systems described by nonlinear differential equations of the second order are studied. It is assumed that certain preliminary information on the dissipative or elastic characteristics of systems is known. A new approach is demonstrated to obtaining full information on unknown or partially known characteristics of a system from measurements of not only displacements but also velocities and accelerations Keywords: dynamic system with one degree of freedom, nonlinear differential equations of the second order, identification, experimental observation, external excitation, graphical criteria of nonlinearity 1. Introduction. The identification of nonlinear dynamic systems is often more a subjective art than a direct application of some particular method in systems theory. A nonlinear problem is subjective because although there are many methods from which to choose, there is no general approach to detect and characterize the type of nonlinear systems.Identification problems differ in their purpose and initial data [6, 10, 12]. The following methods and approaches are applied to identify a mechanical system [1,8,11]: spectral analysis and reverse-path formulation, Volterra and Wiener series, nonlinear autoregressive moving average models, the restoring force method, the describing function methods, direct parameter estimation, Hilbert transforms and wavelet transforms, neural networks, and general nonlinear experimental testing techniques. Some of them apply only to particular types of systems [4].Despite the great number of investigations in this field of science, the development of new methods and approaches is currently central.Mathematical models of objects, constructed in one way or another, should meet two main requirements: they must be relatively simple and must make it possible to solve the corresponding problems.We will consider mechanical systems described by the scalar differential equation

Phase trajectories of non-linear oscillations of a tower structure with an attached damper in a uniform wind flow

“…As it is shown by the several authors (Volkova & Schneider, 2005;Volkova, 2013) the expansion of a phase space considering the phase planes   , y y  and   , y y   , substantially promotes the efficient analysis of a dynamic system behaviour.…”

Identification of several impulsive loads on a plate

“…Tikhonov's regularizing algorithm is used to solve the Volterra equations for the unknown loads Introduction. The need to ensure high efficiency of strength analysis of structural members contributes to the development of a branch of mechanics focused on solving direct and inverse problems to identify parameters of mechanical systems under static and dynamic loading [11][12][13][14][15][16][17].Of special interest are studies on ill-posed inverse problems. The possibility of solving such problems stems from the theory of ill-posed problems of mathematical physics, which is based on Tikhonov's fundamental works.…”