Dynamic Behavior of a Two-Degree-of-Freedom System With Electromagnetic Interaction via a Skew-Symmetric Matrix

Abstract

This paper analyzes the dynamic behavior of a two-degree-of-freedom system subjected to electromagnetic interaction modelled through a skew-symmetric coupling matrix. The system comprises two mechanically independent oscillators coupled by velocity-dependent electromagnetic forces. The equations of motion are formulated and analyzed in the modal domain, highlighting the effects of the antisymmetric interaction on natural frequencies and mode shapes. The classical orthogonality is broken, resulting in complex eigenvectors; nevertheless, the system remains conservative, as the interaction forces perform no work. The analysis is carried out using both configuration-space and state-space formulations, revealing modal frequency splitting and phase shifts induced by the skew-symmetric term. These modal features are further examined through time-domain simulations and frequency response functions. The main contribution of this study is the development and analysis of a deliberately simple yet general model that isolates the essential dynamic effects of skew-symmetric electromagnetic coupling. This minimal formulation, often hidden in more complex systems, reveals key phenomena such as modal frequency splitting, non-normal modes, and energy-conserving cross-effects. The model serves not only as a conceptual reference but also as a methodological framework applicable to a broad class of coupled electromechanical systems.

Keywords

frequency splitting in conservative systems / mechanical-electromagnetic mode coupling / non-orthogonal modes / skew-symmetric interaction matrix / time and frequency response

References

[1] M. Zhang, Modeling of Multiphysics Electromagnetic & Mechanical Coupling and Vibration Controls Applied to Switched Reluctance Machine (Université Paris Saclay, 2018), https://theses.hal.science/tel-02426016v2.

[2] X. Chen, H. Wei, T. Deng, Z. He, and S. Zhao, “Investigation of Electromechanical Coupling Torsional Vibration and Stability in a High-Speed Permanent Magnet Synchronous Motor Driven System,” Applied Mathematical Modelling64 (2018): 235-248, https://doi.org/10.1016/j.apm.2018.07.030.

[3] S. Wang, X. Zhao, Y. Xia, and J. Xiu, “Mechanical-Electromagnetic Coupling Elastic Vibration Instability of Symmetrical Three-Phase External Rotor Induction Motor,” Nonlinear Dynamics97, no. 1 (2019): 1-20, https://doi.org/10.1007/s11071-019-04901-1.

[4] J. Dou, Z. Li, H. Yao, M. Ding, and G. Wei, “Torsional Vibration Suppression and Electromechanical Coupling Characteristics of Electric Drive System Considering Misalignment,” Applied Mathematics and Mechanics45 (2024): 1987-2010, Published Online May, https://doi.org/10.1007/s10483-024-3179-6.

[5] A. Belahcen, R. Kouhia, and K. Fonteyn, “The Different Levels of Magneto-Mechanical Coupling in Energy Conversion Machines and Devices,” in IV International Conference on Computational Methods for Coupled Problems in Science and Engineering, ed. M. Papadrakakis, E. Oñate, and B. Schrefler (CIMNE, 2011), 472-483.

[6] Y. Chen, X. Xing, F. Sun, and J. Lin, “Electromagnetic-Mechanical Coupling Analysis and Optimization Method of Electromagnetic Vibroseis,” Measurement Science and Technology34, no. 9 (2023): 095901, https://doi.org/10.1088/1361-6501/accfb6.

[7] F. Cortés, O. Zarraga, I. Sarría, and M. J. Elejabarrieta, “Eddy Currents Damping Understood as Zener Viscoelastic Damping,” Journal of Sound and Vibration547 (2023): 117539, https://doi.org/10.1016/j.jsv.2022.117539.

[8] J. Li and G. Yang, “Equivalent Subdomain Method for Performance Prediction of Permanent Magnet Eddy Current Brakes,” IET Electric Power Applications15, no. 9 (2021): 1174-1186, https://doi.org/10.1049/elp2.12087.

[9] Z. Lu, B. Huang, Q. Zhang, and X. Lu, “Experimental and Analytical Study on Vibration Control Effects of Eddy-Current Tuned Mass Dampers Under Seismic Excitations,” Journal of Sound and Vibration421 (2018): 153-165, https://doi.org/10.1016/j.jsv.2017.10.035.

[10] B. Yan, Z. Wang, H. Ma, H. Bao, K. Wang, and C. Wu, “A Novel Lever-Type Vibration Isolator With Eddy Current Damping,” Journal of Sound and Vibration494 (2021): 115862, https://doi.org/10.1016/j.jsv.2020.115862.

[11] B. Yan, N. Yu, and C. Wu, “A State-of-the-Art Review on Low-Frequency Nonlinear Vibration Isolation With Electromagnetic Mechanisms,” Applied Mathematics and Mechanics43, no. 7 (2022): 1045-1062, https://doi.org/10.1007/s10483-022-2868-5.

[12] E. Diez-Jimenez, R. Rizzo, M. J. Gómez-García, and E. Corral-Abad, “Review of Passive Electromagnetic Devices for Vibration Damping and Isolation,” Shock and Vibration2019 (2019): 1250707, https://doi.org/10.1155/2019/1250707.

[13] N. Yu, X. Fei, C. Wu, and B. Yan, “Modeling and Analysis of Magnetic Spring Enhanced Lever-Type Electromagnetic Energy Harvesters,” Applied Mathematics and Mechanics43, no. 5 (2022): 743-760, https://doi.org/10.1007/s10483-022-2849-9.

[14] Z. Li, S. Zhou, and Z. Yang, “Recent Progress on Flutter-Based Wind Energy Harvesting,” International Journal of Mechanical System Dynamics2, no. 1 (2022): 82-98, https://doi.org/10.1002/msd2.12035.

[15] G. Xu, C. Li, C. Chen, J. Fu, T. Hou, and Y. Zi, “Dynamics of Triboelectric Nanogenerators: A Review,” International Journal of Mechanical System Dynamics2, no. 4 (2022): 311-324, https://doi.org/10.1002/msd2.12058.

[16] X. Guo, R. Avila, Y. Huang, and Z. Xie, “Flexible Electronics With Dynamic Interfaces for Biomedical Monitoring, Stimulation, and Characterization,” International Journal of Mechanical System Dynamics1, no. 1 (2021): 52-70, https://doi.org/10.1002/msd2.12017.

[17] A. Z. Hajjaj, N. Jaber, S. Ilyas, F. K. Alfosail, and M. I. Younis, “Linear and Nonlinear Dynamics of Micro and Nano-Resonators: Review of Recent Advances,” International Journal of Non-Linear Mechanics119 (2020): 103328, https://doi.org/10.1016/j.ijnonlinmec.2019.103328.

[18] L. A. Martinez, A. R. Castelli, W. Delmas, J. E. Sharping, and R. Chiao, “Electromagnetic Coupling to Centimeter-Scale Mechanical Membrane Resonators via RF Cylindrical Cavities,” New Journal of Physics18, no. 11 (2016): 113015, https://doi.org/10.1088/1367-2630/18/11/113015.

[19] M. Cui, H. Liu, H. Jiang, Y. Zheng, X. Wang, and W. Liu, “Active Vibration Optimal Control of Piezoelectric Cantilever Beam With Uncertainties,” Measurement and Control55 (2022): 359-369, Published online, https://doi.org/10.1177/00202940221091244.

[20] M. Kumar, B. Mukherjee, and S. Sen, “Analysis of Static Charge Induced Pull-In of an Electrostatic MEMS,” Communications in Nonlinear Science and Numerical Simulation96 (2021): 105690, https://doi.org/10.1016/j.cnsns.2021.105690.

[21] Y. Yang, N. Zhang, H. Liu, J. Ling, and Z. Tan, “Piezoelectric and Flexoelectric Effects of DNA Adsorbed Films on Microcantilevers,” Applied Mathematics and Mechanics44, no. 9 (2023): 1547-1562, https://doi.org/10.1007/s10483-023-3026-5.

[22] A. S. Nemov, M. K. Matikainen, T. Wang, and A. Mikkola, “Analysis of Electromechanical Systems Based on the Absolute Nodal Coordinate Formulation,” Acta Mechanica233, no. 3 (2022): 1019-1030, https://doi.org/10.1007/s00707-022-03153-2.

[23] K. G. Aktaş and İ. Esen, “Dynamic Response Analysis and Active Vibration Control of the Smart Sandwich Composite Plate With FGM Core Layers and MIMO FGPM Actuators and Sensors,” International Journal of Mechanical System Dynamics5 (2025): 3-19, Published online March 1, https://doi.org/10.1002/msd2.70001.

[24] T. S. Amer, F. M. El-Sabaa, S. K. Zakria, and A. A. Galal, “The Stability of 3-DOF Triple-Rigid-Body Pendulum System Near Resonances,” Nonlinear Dynamics110, no. 2 (2022): 1339-1371, https://doi.org/10.1007/s11071-022-07722-x.

[25] T. S. Amer, G. M. Moatimid, S. K. Zakria, and A. A. Galal, “Vibrational and Stability Analysis of Planar Double Pendulum Dynamics Near Resonance,” Nonlinear Dynamics112, no. 24 (2024): 21667-21699, https://doi.org/10.1007/s11071-024-10169-x.

[26] T. S. Amer, S. A. Abdelhfeez, R. F. Elbaz, and M. K. Abohamer, “Investigation of the Dynamical Analysis, Stability, and Bifurcation for a Connected Damped Oscillator With a Piezoelectric Harvester,” Journal of Vibration Engineering & Technologies13, no. 2 (2025): 155, https://doi.org/10.1007/s42417-024-01641-4.

[27] T. S. Amer, E. H. Aly, and H. M. Gad, “Analysis and Stability Assessment of the Vibratory Motion of a Magnetic Mechanical System Near Resonance,” Journal of Vibration Engineering & Technologies13, no. 3 (2025): 201, https://doi.org/10.1007/s42417-024-01629-0.

[28] G. M. Moatimid, T. S. Amer, and M. H. Zekry, “Analytical and Numerical Study of a Vibrating Magnetic Inverted Pendulum,” Archive of Applied Mechanics93, no. 6 (2023): 2533-2547, https://doi.org/10.1007/s00419-023-02395-3.

[29] G. M. Moatimid and T. S. Amer, “Analytical Approximate Solutions of a Magnetic Spherical Pendulum: Stability Analysis,” Journal of Vibration Engineering & Technologies11, no. 5 (2023): 2155-2165, https://doi.org/10.1007/s42417-022-00693-8.

[30] R. R. Craig and A. J. Kurdila, “Vibration Properties of MDOF Systems: Modes, Frequencies, and Damping,” in Fundamentals of Structural Dynamics, 2nd ed. (John Wiley & Sons Inc., 2006), 281-324.

[31] W. Wang and J. Kirkhope, “New Eigensolutions and Modal Analysis for Gyroscopic/Rotor Systems, Part 1: Undamped Systems,” Journal of Sound and Vibration175, no. 2 (1994): 159-170, https://doi.org/10.1006/jsvi.1994.1320.

[32] W. Wang and J. Kirkhope, “New Eigensolutions and Modal Analysis for Gyroscopic/Rotor Systems, Part 2: Perturbation Analysis for Damped Systems,” Journal of Sound and Vibration175, no. 2 (1994): 171-183, https://doi.org/10.1006/jsvi.1994.1321.

[33] S. Adhikari, “Modal Analysis of Linear Asymmetric Nonconservative Systems,” Journal of Engineering Mechanics125, no. 12 (1999): 1372-1379, https://doi.org/10.1061/(ASCE)0733-9399(1999)125:12(1372).

RIGHTS & PERMISSIONS

2025 The Author(s). International Journal of Mechanical System Dynamics published by John Wiley & Sons Australia, Ltd on behalf of Nanjing University of Science and Technology.