This paper considers the dynamics of an oscillatory dissipative system of two coupled pendulums in a magnetic field. The pendulums are coupled via an elastic element. The inertial components of the pendulums vary over a wide range, and in the analytical study the mass ratio is chosen as a small parameter. The magnetic forces are calculated using the Pade approximation, which best agrees with the experiment. This approximation describes the magnetic excitation to good accuracy. The presence of external inputs in the form of magnetic forces and various types of loads that exist in many engineering systems significantly complicates the mode shape analysis of nonlinear system. Nonlinear normal modes of this system are studied, one mode being coupled and the other being local. The modes are constructed by the multiple-scale method. Both regular and compound behavior is studied as a function of the system parameters: the pendulum mass ratio, the coupling coefficient, the magnetic intensity coefficient, and the distance between the axis of rotation and the center of gravity. The effect of these parameters is studied both at small and at sizeable initial pendulum inclination angles. The analytical solution is compared with the results of a numerical simulation based on the fourth-order Runge?Kutta method where the modes are calculated using the initial values of the variables found in the analytical solution. The numerical simulation, which includes the construction of phase diagrams and trajectories in the configuration space, allows one to assess the dynamics of the system, which may be both regular and compound. The stability of the coupled mode is studied using a numerical-analytical test, which is an implementation of the Lyapunov stability criterion. In doing so, the stability of a mode is determined by assessing the vertical off-trajectory deviation of the mode in the configuration space.
     

coupled pendulums, magnetic forces, nonlinear normal modes, multiple-scale method

1. Polczynski K. et al. Numerical and experimental study of dynamics of two pendulums under a magnetic field. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering. 2019. V. 233. No. 4. Pp. 441-453. https://doi.org/10.1177/0959651819828878

2. Wijata A., Polczynski K., Awrejcewicz J. Theoretical and numerical analysis of regular one-side oscillations in a single pendulum system driven by a magnetic field. Mechanical Systems and Signal Processing. 2021. V. 150. 107229. https://doi.org/10.1016/j.ymssp.2020.107229

3. Polczynski K. et al. Nonlinear oscillations of coupled pendulums subjected to an external magnetic stimulus. Mechanical Systems and Signal Processing. 2021. V. 154. 107560. https://doi.org/10.1016/j.ymssp.2020.107560

4. Surganova Y. E., Mikhlin Y. V. Localized and non-localized nonlinear normal modes in a system of two coupled pendulums under a magnetic field. International Journal of Non-Linear Mechanics. 2022. V. 147. P. 104182. https://doi.org/10.1016/j.ijnonlinmec.2022.104182

5. Mikhlin Y. V., Avramov K. V. Nonlinears Normal modes for vibrating mechanical systems. Review of theoretical developments. Applied Mechanics Reviews. 2010. V. 63. No. 6. 060802. https://doi.org/10.1115/1.4003825

6. Avramov K. V., Mikhlin Y. V. Review of applications of nonlinear normal modes for vibrating mechanical systems. Applied Mechanics Reviews. 2013. V. 65. No. 2. 020801. https://doi.org/10.1115/1.4023533

7. Modal Analysis of Nonlinear Mechanical Systems. G. Kersche (Ed.). Vienna: Springer, 2014. URL: https://link.springer.com/book/10.1007/978-3-7091-1791-0 (Last accessed on August 15, 2023).

8. Manevitch L. I., Smirnov V. V. Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains. Physical Review E. 2010. Vol. 82. No. 3. 036602. https://doi.org/10.1103/PhysRevE.82.036602

9. Vakakis A.F., Gendelman O.V., Bergman L., McFarland D.M., Kerschen G., Lee Y. S. Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems. Springer, 2008. 1033 pp.

10. Nayfeh A. H., Mook D. T. Nonlinear Oscillations. John Wiley & Sons, 1995. 720 pp. https://doi.org/10.1002/9783527617586

11. Mikhlin Y. V., Shmatko T. V., Manucharyan G. V. Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions. Computers & Structures. 2004. V. 82. No. 31-32. Pp. 2733-2742. https://doi.org/10.1016/j.compstruc.2004.03.082

Copyright (©) 2023 Surhanova Yu. E., Mikhlin Yu. V.

Copyright © 2014-2023 Technical mechanics