Investigation of Stability of Nonlinear Normal Modes of Dissipative System Oscillations Under Influence of Magnetic Field

Abstract

In the paper the dynamics of the vibrating system consisting of two pendulums connected by elastic coupling and located in a magnetic field is studied. A case is considered when masses of the pendulums differ significantly. Under the influence of various external factors, such as magnetic forces and loads, which are present in engineering systems, the analysis of vibration regimes in non-linear systems becomes more complex. Here the nonlinear normal connected vibration mode in a system under consideration is analyzed. We have investigated the influence of changes in system parameters on the vibra- tion mode for both small and large initial angles of the pendulum deflection. Both the analytical method, namely, the method of many scales, and nu- merical simulations based on the fourth order Runge – Kutta method are used to analyze the vibration regimes. The initial conditions used to calculate this mode were determined analytically. The simulation includes construction of phase diagrams, trajectories in configuration space, and spectra, which allows us to evaluate the system dynamics, including both regular and complex regimes of vibrations. To study the stability of the vibration mode, a numerical-analytical method associated with the Lyapunov stability criterion is used. The stability of the mode is determined by assessing orthogonal deviations with respect to the corresponding modal trajectories in the configuration space. Regions of instability on planes and in the space of the system parameters are obtained.