Construction and analysis of nonlinear oscillation modes of a three-link pendulum by asymptotic methods

A. S. Smirnov ; S. A. Bulov ; E. A. Degilevich

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 24 (2024) no. 4, pp. 598-610

Résumé

This article is devoted to the study of nonlinear oscillations of one of the most common systems with three degrees of freedom – a three-link mathematical pendulum, the parameters of all weightless links and all end loads of which are assumed to be identical. The wide use of the three-link pendulum model in applied problems of robotics and biomechanics, as well as its important scientific significance in the problems of equilibrium stability, stabilization and motion control are discussed. The question of finding nonlinear oscillation modes of a three-link pendulum is considered, the knowledge of which makes it possible to implement single-frequency modes of its motion with sufficiently large deviations. For this purpose, asymptotic methods of nonlinear mechanics are used, which make it possible to determine the oscillation modes of the system in the first approximation within a weakly nonlinear model. The main features of the constructed nonlinear oscillation modes are discussed and their qualitative and quantitative differences from the traditional linear modes of small oscillations are revealed. In addition, it is noted that nonlinear oscillation modes can also be found on the basis of numerical simulation by accelerating the system under the action of collinear control from small deviations specified on a linear mode to finite amplitudes with access to single-frequency motion on a nonlinear mode. The obtained analytical expressions for the frequencies of nonlinear oscillations and the ratios of the oscillation amplitudes of the pendulum links for each of the nonlinear modes are compared with similar numerical dependencies by constructing graphic illustrations corresponding to them at the same level of total mechanical energy. It is established that the analytical and numerical results are in agreement with each other, which determines the value of the approximate solution constructed in the work. The formulas obtained and the conclusions drawn are of undoubted theoretical interest, and they may also be helpful for their use in specific practical purposes.