Nonlinear free vibration of conical beams using He's frequency formula: Educational implications

Lingxing  Song; Xin  Wei; Cheligeer  Bai; Jiahui  Yu

doi:10.59400/sv3620

Nonlinear free vibration of conical beams using He's frequency formula: Educational implications

Lingxing Song
School of Mathematics and Big Data, Hohhot Minzu College, Hohhot 010051, China
Xin Wei
School of Mathematics and Big Data, Hohhot Minzu College, Hohhot 010051, China
Cheligeer Bai
School of Mathematics and Big Data, Hohhot Minzu College, Hohhot 010051, China
Jiahui Yu
School of Mathematics and Big Data, Hohhot Minzu College, Hohhot 010051, China

Article ID: 3620

Keywords: conical beams; nonlinear free vibration; He’s frequency formula; residual minimization; vibration analysis; educational implications; MEMS applications; resonance avoidance

Abstract

This study presents an analytical solution for the nonlinear free vibration of conical beams using He's frequency formula. This solution aims to efficiently address the challenges posed by their axially varying cross-sectional dimensions and associated nonlinear mechanical behaviors. Conical beams, with their customized stiffness and mass distribution, find wide application in aerospace, civil engineering, and micro-electromechanical systems (MEMS), where precise vibration analysis is imperative for ensuring structural stability and performance. The frequency formula, rooted in residual minimization, is employed to derive the frequency-amplitude relationship of conical beams. This method avoids complex iterative procedures and reduces computational complexity compared to traditional methods like Aboodh Transform-based variational iteration method (ATVIM) or homotopy perturbation. The validation of the method against ATVIM and numerical solutions (4th-order Runge–Kutta method) confirms its accuracy, with close agreement across moderate to large amplitudes and a frequency relative error of less than 5%. Beyond its practical utility in engineering design—enabling rapid parametric analysis for resonance avoidance—the study also highlights educational implications, as the conical beam case study bridges abstract nonlinear dynamics theory with real-world applications, aiding students in understanding frequency-amplitude coupling and method selection. This work demonstrates that He's frequency formula offers a robust, accessible framework for analyzing conical beam vibrations, linking theoretical nonlinear dynamics, engineering practice, and educational value.

Published

2025-11-27

How to Cite

Song, L., Wei, X., Bai, C., & Yu, J. (2025). Nonlinear free vibration of conical beams using He’s frequency formula: Educational implications. Sound & Vibration, 59(6). https://doi.org/10.59400/sv3620

Download Citation

Issue

Vol. 59 No. 6 (2025)

Section

Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Khudoynazarov K. Longitudinal-radial vibrations of a viscoelastic cylindrical three-layer structure. Facta Universitatis—Series: Mechanical Engineering. 2024; 22(3): 473–484.

[2]Anjum N, Rasheed A, He JH, et al. Free vibration of a tapered beam by the Aboodh transform-based variational iteration method. Journal of Computational and Applied Mechanics. 2024; 55(3): 440–450.

[3]Leissa A. Free vibration analysis of beams and shafts. Mechanism and Machine Theory. 1975; 10(5): 432–433. doi: 10.1016/0094-114X(75)90043-9

[4]Liu YP, He CH, Gepreel KA, et al. Clover-inspired fractal architectures: Innovations in flexible folding skins for sustainable buildings. Fractals. 2025; 33(5): 2550041. doi: 10.1142/S0218348X25500410

[5]Hendy MH, Ezzat MA, Al-Lobani EM, et al. A problem in fractional order thermo-viscoelasticity theory for a polymer micro-rod with and without energy dissipation. Advances in Differential Equations and Control Processes. 2024; 31(4): 583–607. doi: 10.17654/0974324324030

[6]Liu HW, Wang YF, Zhu C, et al. Design of 3D printed concrete masonry for wall structures: Mechanical behavior and strength calculation methods under various loads. Engineering Structures. 2025; 325: 119374. doi: 10.1016/j.engstruct.2024.119374

[7]Zhang LY, Gepreel KA, Yu J. He's frequency formulation for fractal-fractional nonlinear oscillators: A comprehensive analysis. Frontiers in Physics. 2025; 13: 1542758. doi: 10.3389/fphy.2025.1542758

[8]He JH. Periodic solution of a micro-electromechanical system. Facta Universitatis—Series: Mechanical Engineering. 2024; 22(2): 187–198. doi: 10.22190/FUME240603034H

[9]He JH, Bai Q, Luo YC, et al. Modeling and numerical analysis for an MEMS graphene resonator. Frontiers in Physics. 2025; 13: 1551969. doi: 10.3389/fphy.2025.1551969

[10]He JH, He CH, Qian MY, et al. Piezoelectric biosensor based on ultrasensitive MEMS system. Sensors and Actuators A: Physical. 2024; 376: 115664. doi: 10.1016/j.sna.2024.115664

[11]Zuo YT, Luo FF, Zeng SL. Gecko-inspired fractal buffer for passenger elevator. Facta Universitatis—Series: Mechanical Engineering. 2024; 22(4): 633–642.

[12]He CH, El-Dib YO. A heuristic review on the homotopy perturbation method for non-conservative oscillators. Journal of Low Frequency Noise, Vibration and Active Control. 2022; 41(2): 572–603. doi: 10.1177/14613484211059264

[13]Moussa B, Youssouf M, Abdoul Wassiha N, et al. Homotopy perturbation method to solve Duffing–Van der Pol equation. Advances in Differential Equations and Control Processes. 2024; 31(3): 299–315. doi: 10.17654/0974324324016

[14]Alshomrani NAM, Alharbi WG, Alanazi IMA, et al. Homotopy perturbation method for solving a nonlinear system for an epidemic. Advances in Differential Equations and Control Processes. 2024; 31(3): 347–355. doi: 10.17654/0974324324019

[15]He JH, He CH, Alsolami AA. A good initial guess for approximating nonlinear oscillators by the homotopy perturbation method. Facta Universitatis—Series: Mechanical Engineering. 2023; 21(1): 21–29.

[16]He JH. The simplest approach to nonlinear oscillators. Results in Physics. 2019; 15: 102546. doi: 10.1016/j.rinp.2019.102546

[17]He JH. The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators. Journal of Low Frequency Noise, Vibration and Active Control. 2019; 38(3–4): 1252–1260. doi: 10.1177/1461348419844145

[18]He JH, Adamu MY, Abdullahi I. Frequency-amplitude relationship in damped and forced nonlinear oscillators with irrational nonlinearities. Journal of Computational and Applied Mechanics. 2025; 56(2): 307–317.

[19]He JH. Frequency-amplitude relationship in nonlinear oscillators with irrational nonlinearities. Spectrum of Mechanical Engineering and Operational Research. 2025; 2(1): 121–129. doi: 10.31181/smeor21202535

[20]He JH. Frequency formulation for nonlinear oscillators (part 1). Sound and Vibration. 2025; 59(1): 1687. doi: 10.59400/sv1687

[21]He CH, Liu C. A modified frequency-amplitude formulation for fractal vibration systems. Fractals. 2022; 30(3): 2250046. doi: 10.1142/S0218348X22500463

[22]He CH, He JH, Ma J, et al. A modified frequency formulation for nonlinear mechanical vibrations. Facta Universitatis·Series: Mechanical Engineering. 2025. doi: 10.22190/FUME250526022H

[23]He JH, Iranian D. Point solution concept for analysis of reaction-diffusion in porous catalysts. Journal of Computational and Applied Mechanics. 2025; 56(4): 711–719. doi: 10.22059/jcamech.2025.398100.1535

[24]Liu YP, He JH. A fast and accurate estimation of amperometric current response in reaction kinetics. Journal of Electroanalytical Chemistry. 2025; 978: 118884.

[25]He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computation. 2004; 151(1): 287.

[26]He JH. Comparison of homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation. 2004; 156(2): 527.

[27]He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons and Fractals. 2005; 26(3): 695.

[28]He JH, Ma J, Alsolami AA, et al. Variational approach to micro-electro-mechanical systems. Facta Universitatis—Series: Mechanical Engineering. 2025; 23(4), 649. doi: 10.22190/fume250528023h

[29]Bakhtiari M, Lakis AA, Kerboua Y. Nonlinear vibration of truncated conical shells: Donnell, Sanders and Nemeth theories. International Journal of Nonlinear Sciences and Numerical Simulation. 2020; 21(1), 83–97. doi: 10.1515/ijnsns-2018-0377

This paper introduces a universal framework for understanding the vibration responses of systems subjected to harmonic excitation. By examining a simplified cylinder-spring-damper model, the study refurbishes traditional scaling methods for the excitation frequency ratio and critical damping ratio. The findings indicate that in damped systems, the maximum amplitude of vibration does not align with the natural frequency. This observation leads to the introduction of a new scaling method for reduced frequency. This new approach aligns resonance peaks at the new reduced velocity of 1.0 across different damping ratios, providing a consistent characterization of vibration behavior. A new critical damping ratio of 0.707 is identified for an excited system as opposed to the traditional damping ratio of 1.0 for an unexcited system. Key properties such as maximum amplitude, phase lag, bandwidth, and quality factor are analyzed, demonstrating that the proposed reduced frequency and critical damping ratio effectively capture the dynamics of both damped and undamped excited systems. The findings offer significant insights for practical applications in engineering and various scientific fields.

This study makes significant contributions to the field of ultrasonic testing (UT) by offering a novel approach to the identification of artificially introduced defects within Japanese cedar wood (Cryptomeria japonica). The findings are of particular relevance for the heritage conservation and construction sectors, where non-invasive defect detection is paramount. The study establishes a robust framework for assessing the structural integrity of timber by correlating ultrasonic wave velocity reductions with defect size and distribution. Big-sized defects led to more substantial decreases in wave velocity. The study establishes a robust framework for assessing the structural integrity of historical timber by correlating ultrasonic wave velocity reductions with defect size and distribution. This framework has the potential to be applicable to diverse wood species and defect types.

The control of vehicle interior noise has become a critical metric for assessing noise, vibration, and harshness (NVH) in vehicles. During the initial phases of vehicle development, accurately predicting the impact of road noise on interior noise is essential for reducing noise levels and expediting the product development cycle. In recent years, data-driven methods based on machine learning have gained significant attention due to their robust capability in navigating complex data mapping relationships. Notably, surrogate models have demonstrated exceptional performance in this domain. Numerous researchers have integrated diverse intelligent algorithms into the study of vehicle noise, leveraging advantages such as the elimination of precise modeling requirements, extensive solution space exploration, continuous learning from data, and robust algorithmic versatility. However, in NVH engineering applications, data-driven models face inherent limitations, particularly in interpretability and stability. To address these issues, this paper introduces an improved Long Short-Term Memory (LSTM) network that combines knowledge and data. Inspired by the physical information neural network concept, this approach incorporates values calculated through empirical formulas into the neural network as constraints. Comparative assessments with traditional LSTM networks highlight the advantages of this deep learning model. By integrating empirical formulas constraints, the model not only enhances interpretability but also achieves robust generalization with fewer data samples. The proposed method is validated on a specific vehicle model, showing significant improvements in prediction accuracy and efficiency.