Аналітичне дослідження фрикційних автоколивань у системах з двома ступенями свободи

Abstract

Abstract Purpose of the study is to study the properties of frictional self-oscillations in systems with two degrees of freedom. As a research method, the asymptotic method of N.N. Bogolyubov and Y.A. Metropolitan. Research methods. The methodological basis of the work is the generalization and analysis of the known scientific results of the dynamics of systems in resonance modes and the use of a systematic approach. The analytical method and comparative analysis were used to form a scientific problem, goal and formulation of research objectives. When developing empirical models, the main provisions of the theory of stability of systems, methodology of system analysis and research of functions were used. The results of the study. A system with two degrees of freedom is considered, assuming that the friction function is approximated by a cubic polynomial in the sliding velocity, and friction is applied only to one of the masses. The exclusion of uniform rotation, corresponding to the third degree of freedom, leads to consideration not of the frictional moment, but the difference between the frictional moment and the moment of the moving forces. From the analysis of the results of the solutions of the equation, we can conclude that, with an accuracy up to the first approximation, inclusive, self-oscillations occur with constant frequencies equal to the natural frequencies of the system. This is consistent with the conclusions of other authors obtained using other methods. Stationary values of the amplitudes are found. The following four cases are possible: trivial solution corresponding to uniform rotation of the system without oscillations; single frequency oscillations with the first frequency; single frequency oscillations with a second frequency; two-frequency oscillatory mode. Conclusions. G. Boyadzhiev's method can be applied to study multi-mass self-oscillating systems and gives their general solution in the form of asymptotic expansions to any degree of accuracy. The obtained conditions for the stability of stationary regimes confirm the experimental results that in multi-mass systems, self-oscillations are possible only in the falling sections of the friction characteristics. The nature of the developing vibrations - their frequency and the ratio of the amplitudes of the constituent harmonics - is completely determined by the structure of the system, its elastic and inertial properties.