РЕГУЛЯРНА ТА СКЛАДНА ПОВЕДІНКА МАЯТНИКОВОЇ СИСТЕМИ У МАГНІТНОМУ ПОЛІ

Abstract

The dynamics of an oscillatory dissipative system consisting of two connected pendulums in a magnetic field is considered. The connection of these pendulums is realized by some elastic element. The inertial components of pendulums vary widely, and the mass ratio is chosen in analytical investigation as a small parameter. For approximate calculations of magnetic forces, the Padé approximation which best satisfies the experimental data, is used. Such approximation permits to describe the magnetic excitation with good accuracy. The presence of external influences in the form of magnetic forces and various types of other loads that exist in many engineering systems leads to a significant complication in the analysis of vibration modes of nonlinear systems. Nonlinear normal modes (NNM) are analysed in the system where one mode is connected and the other is localized. These modes are constructed by the multiple scales method. It is studied as the regular, as well the complex behaviour when changing system parameters, including the pendulums mass ratio, the coupling coefficient, the magnetic impact intensity coefficient, and the distance between the axis of rotation and the center of gravity. The influence of these parameters is studied at both small and not small initial angles of the pendulums. The analytical solution is compared with results of numerical simulation which is based on the Runge–Kutta method of the fourth order, where initial values of variables defined in the analytical solution are used. Numerical simulation, which includes construction of phase diagrams and trajectories in the configuration space, permits to estimate the system dynamics which can be as regular, as well irregular one. The mode stability is studied by the numerical-analytical test which is a numerical realization of the Lyapunov stability criterion. Here the mode stability is determined by analysis of orthogonal deviations from the mode trajectory in the system configuration space. Keywords: connected pendulums, magnetic forces, nonlinear normal modes, multiple scales method.