Identification of external harmonic force parameters

Abstract

The problem of determining the external force, which is given by the harmonic function of time and acts on a self-oscillating system of general type (Lienard oscillator) is considered. A general method of asymptotic estimation of oscillator velocity and force unknown parameters is proposed. Such problems of estimating the frequency, amplitude, and phase of an external force acting on a mechanical system are reflected in a sufficient number of publications both in past and present times. The reason for this interest lies in the use of appropriate techniques in various theoretical and engineering disciplines, for example, for mechanical systems for converting the kinetic energy of vibrations, in the problems of vibration isolation of periodic components of noise through rotating mechanisms, to compensate for harmonic disturbances in automatic control algorithms, in adaptive filtering during signal processing, and so on. In principle, the least squares method, Fourier analysis, and Laplace Transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. Despite the relative simplicity of the problem of determining the frequency, amplitude and phase of vibrations, approaches to solving them use a rather complex apparatus of modern methods of Applied Mathematics. The aim of this paper is to extend the method of synthesis of invariant relations to the problem of determining the parameters of external influence on a mechanical system. To obtain asymptotic estimates of the coefficients of external force, the method of invariant relations developed in analytical mechanics is used. Method was intended, in particular, to search for partial solutions (dependencies between variables) in problems of dynamics of rigid body with a fixed point. Modification of this method to the problems of observation theory made it possible to synthesize additional connections between known and unknown quantities of the original system that arise during the movement of its extended dynamic model. The asymptotic convergence of estimates of unknowns to their true value is proved. The results of numerical modeling of the asymptotic estimation process of oscillator velocity and external force parameters for the mathematical pendulum model are presented.