Journal of Computational Applied Mechanics

Frequency–Amplitude Relationship in Damped and Forced Nonlinear Oscillators with Irrational Nonlinearities

Document Type : Research Paper

Authors

1 Yango University, No.99 Denglong Road, Fuzhou City, Fujian Province, China

2 School of Jia Yang; Zhejiang Shuren University, Hangzhou; Zhejiang, China

3 School of Mathematics and Big Data, Hohhot Minzu College, Hohhot, Inner Mongolia 010051, China

4 Department of Mathematics Abubakar Tafawa Balewa University Bauchi, Nigeria

Abstract

This paper undertakes an exhaustive investigation of nonlinear oscillators subject to damping and external excitation, with a particular emphasis on systems exhibiting irrational nonlinearities. Nonlinear oscillators play a foundational role in the modeling of a broad array of natural and engineering systems. These systems exhibit behaviors that are significantly different from linear systems. These behaviors include amplitude-dependent frequencies, subharmonic and superharmonic responses, bifurcations, and chaotic motions. The frequency-amplitude relationship, which is central to this research, is of significant importance in various fields, including vibration control, energy harvesting, and the study of biological rhythms. It is important to note that this relationship is subject to variation in accordance with amplitude changes. In this study, the frequency formulation is employed to meticulously analyze the response characteristics of damped and forced nonlinear oscillators. This analysis effectively validates the efficacy of frequency formulation in capturing the periodic behavior of these systems. The research findings not only validate the established results under optimal conditions but also extend the analytical scope to encompass the more intricate and nuanced dynamic phenomena encountered in real-world scenarios. The derivation of the frequency-amplitude relationship unveils the underlying mechanism through which damping and external forces influence the system's dynamic response, thereby facilitating a more profound comprehension of the behavior of nonlinear oscillation systems.

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Main Subjects

[1]          L. Yang, J. Zhang, J. Xia, S. Zhang, Y. Yang, Sound Transmission Loss of Helmholtz Resonators with Elastic Bottom Plate, Sound & Vibration, Vol. 58, No. 1, pp. 056968, 10/21, 2024.
[2]          Z. Zhong, Y. Li, Y. Zhao, P. Ju, A Method of Evaluating the Effectiveness of a Hydraulic Oscillator in Horizontal Wells, Sound \& Vibration, Vol. 57, No. 1, pp. 15--27, 2023.
[3]          J.-H. He, Frequency-Amplitude Relationship in Nonlinear Oscillators with Irrational Nonlinearities, Spectrum of Mechanical Engineering and Operational Research, Vol. 2, pp. 121-129, 03/30, 2025.
[4]          J.-H. He, Frequency formulation for nonlinear oscillators (part 1), Sound & Vibration, Vol. 59, No. 1, pp. 1687, 11/15, 2024.
[5]          L.-H. Zhang, C.-F. Wei, A powerful analytical method to some non-linear wave equations, Thermal Science, Vol. 28, pp. 3553-3557, 01/01, 2024.
[6]          M. Khater, S. Alfalqi, Analytical solutions of the Caudrey–Dodd–Gibbon equation using Khater II and variational iteration methods, Scientific Reports, Vol. 14, 11/14, 2024.
[7]          Z.-J. Liu, M. Adamu Yunbunga, E. Suleiman, J.-H. He, Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations, Thermal Science, Vol. 21, pp. 78-78, 01/01, 2017.
[8]          B. Moussa, M. Youssouf, N. A. Wassiha, P. Youssouf, HOMOTOPY PERTURBATION METHOD TO SOLVE DUFFING-VAN DER POL EQUATION, Advances in Differential Equations and Control Processes, Vol. 31, No. 3, pp. 299-315, 05/15, 2024.
[9]          Y. El-dib, A heuristic review on the homotopy perturbation method for non-conservative oscillators, 05/07, 2022.
[10]        C.-H. He, D. Tian, G. M. Moatimid, H. F. Salman, M. H. Zekry, Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller, Journal of Low Frequency Noise, Vibration and Active Control, Vol. 41, No. 1, pp. 244-268, 2022.
[11]        B. X. Zhang, J. L. Huang, W. D. Zhu, System response tracking based on the Runge–Kutta method and the incremental harmonic balance method, Nonlinear Dynamics, 2025/01/20, 2025.
[12]        A. A. Rossikhin, V. I. Mileshin, Application of the Harmonic Balance Method to Calculate the First Booster Stage Tonal Noise, Mathematical Models and Computer Simulations, Vol. 16, No. 1, pp. 63-75, 2024/02/01, 2024.
[13]        C.-H. He, A variational principle for a fractal nano/microelectromechanical (N/MEMS) system, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 33, No. 1, pp. 351-359, 2022.
[14]        Q. Ain, D. Tian, N. Anjum, Fractal N/MEMS: From pull-in instability to pull-in stability, Fractals, Vol. 29, 10/18, 2020.
[15]        D. Tian, C.-H. He, A fractal micro-electromechanical system and its pull-in stability, Journal of Low Frequency Noise, Vibration and Active Control, Vol. 40, No. 3, pp. 1380-1386, 2021.
[16]        A. H. Nayfeh, 1981, Introduction to Perturbation Techniques, Wiley,
[17]        C.-H. HE, C. LIU, A MODIFIED FREQUENCY–AMPLITUDE FORMULATION FOR FRACTAL VIBRATION SYSTEMS, Fractals, Vol. 30, No. 03, pp. 2250046, 2022.
[18]        G.-Q. Feng, He’s frequency formula to fractal undamped Duffing equation, Journal of Low Frequency Noise, Vibration and Active Control, Vol. 40, No. 4, pp. 1671-1676, 2021.
[19]        K. Tsaltas, An improved one-step amplitude–frequency relation for nonlinear oscillators, Results in Physics, Vol. 54, pp. 107090, 2023/11/01/, 2023.
[20]        Periodic Solutions of Strongly Nonlinear Oscillators Using He’s Frequency Formulation, European Journal of Pure and Applied Mathematics, Vol. 17, No. 3, pp. 2155-2172, 07/31, 2024.
[21]        Y. O. El-Dib, The frequency estimation for non-conservative nonlinear oscillation, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 101, No. 12, pp. e202100187, 2021.
[22]        Y. O. El-Dib, N. S. Elgazery, N. S. Gad, A novel technique to obtain a time-delayed vibration control analytical solution with simulation of He’s formula, Journal of Low Frequency Noise, Vibration and Active Control, Vol. 42, No. 3, pp. 1379-1389, 2023.
[23]        M. A. Kawser, M. A. Alim, N. Sharif, Analyzing nonlinear oscillations with He's frequency-amplitude method and numerical comparison in jet engine vibration system, Heliyon, Vol. 10, No. 2, 2024.
[24]        G. Hashemi, A novel analytical approximation approach for strongly nonlinear oscillation systems based on the energy balance method and He's Frequency-Amplitude formulation, Computational Methods for Differential Equations, Vol. 11, No. 3, pp. 464-477, 2023.
[25]        M. Mohammadian, Application of He's new frequency-amplitude formulation for the nonlinear oscillators by introducing a new trend for determining the location points, Chinese Journal of Physics, Vol. 89, pp. 1024-1040, 2024/06/01/, 2024.
[26]        J.-G. Zhang, Q.-R. Song, J.-Q. Zhang, F. Wang, APPLICATION OF HE’S FREQUENCY FORMULA TO NONLINEAR OSCILLATORS WITH GENERALIZED INITIAL CONDITIONS, 2023, pp. 12, 2023-12-16, 2023.
[27]        A. Elías-Zúñiga, Exact solution of the cubic-quintic Duffing oscillator, Applied Mathematical Modelling, Vol. 37, No. 4, pp. 2574-2579, 2013/02/15/, 2013.
[28]        B. I. Lev, V. B. Tymchyshyn, A. G. Zagorodny, On certain properties of nonlinear oscillator with coordinate-dependent mass, Physics Letters A, Vol. 381, No. 39, pp. 3417-3423, 2017/10/17/, 2017.
Journal of Computational Applied Mechanics
Volume 56, Issue 2
April 2025
Pages 307-317
  • Receive Date: 05 March 2025
  • Revise Date: 24 March 2025
  • Accept Date: 30 March 2025