Study of the Stability of the Mathematical Model of the Coupled Pendulums Motion

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher Anatolii Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 26, no. 4, 2023 (December)
Pages 50-58
Cited by J. of Mech. Eng., 2023, vol. 26, no. 4, pp. 50-58



Yuliia E. Surhanova, National Technical University “Kharkiv Polytechnic Institute” (2, Kyrpychova str., Kharkiv, 61002, Ukraine), e-mail:, ORCID: 0000-0002-6540-3025

Yurii V. Mikhlin, National Technical University “Kharkiv Polytechnic Institute” (2, Kyrpychova str., Kharkiv, 61002, Ukraine), e-mail:, ORCID: 0000-0002-1780-9346



The paper presents a study of the dynamics of the oscillatory dissipative system of two elastically connected pendulums in a magnetic field. Nonlinear normal vibration modes of the pendulum system are studied in the paper taking into account the resistance of the medium, and the damping moment created by the elastic element. A system with two degrees of freedom is considered. The masses of the pendulums in that system differ significantly, which leads to the possibility of localization of oscillations. In the following analysis, the mass ratio was chosen as a small parameter. For approximate calculations of magnetic forces, the Padé approximation, which satisfies the experimental data the most, is used. This approximation provides a very accurate description of magnetic excitation. The presence of external influences in the form of magnetic forces and various types of loads that exist in many engineering systems leads to a significant complication in the analysis of vibration modes of nonlinear systems. A study of nonlinear normal vibration modes in this system was carried out, one of the modes is a connected mode, and the second one is localized. Vibration modes are constructed by the multiples scales method. Both regular and complex behavior is studied when changing system parameters. The influence of these parameters is studied for small and significant initial angles of the pendulum inclination. An analytical solution, which is based on the fourth-order Runge-Kutta method, is compared to numerical simulation results. The initial conditions for calculating the vibration modes were determined by the analytical solution. Numerical simulation, which consists of constructing phase diagrams, trajectories in the configuration space and spectra, allows to estimate the dynamics of the system, which can be both regular and complex. The stability of vibration modes is studied using numerical analysis tests, which are an implementation of the Lyapunov stability criterion. The stability of vibration modes is determined by the estimation of orthogonal deviations of corresponding trajectories of vibration modes in configuration space.


Keywords: coupled pendulums, magnetic forces, nonlinear normal vibration modes, the multiples scales method, stability.


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Received 16 October 2023

Published 30 December 2023