Journal of Computational Applied Mechanics

Free Vibration of a Tapered Beam by the Aboodh Transform-based Variational Iteration Method

Document Type : Research Paper

Authors

1 Department of Mathematics, Government College University, Faisalabad, Pakistan

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

3 National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, Suzhou, China

4 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Abstract

Physical systems frequently exhibit nonlinear behavior that remains unresolved in the majority of cases. In this study, we employ the Aboodh transform-based variational iteration method (ATVIM) to resolve the nonlinear model of a tapered beam. In order to solve the governing equation, the periodic motion is sought, and the explicit relationship between frequency and amplitude is revealed. The outcomes of the ATVIM approach are compared with those of other prevalent techniques, and a satisfactory concordance is observed between them. This study also provides an analytical approximation of the tapered beam for a detailed understanding of the effects of factors on the nonlinear frequency, which can be beneficial to researchers and engineers working on the analysis and design of structural projects.

Keywords

Main Subjects

[1]          S. A. Faghidian, A. Tounsi, DYNAMIC CHARACTERISTICS OF MIXTURE UNIFIED GRADIENT ELASTIC NANOBEAMS, Facta Universitatis, Series: Mechanical Engineering, Vol. 20, pp. 539, 11/30, 2022.
[2]          Q. Ain, D. Tian, N. Anjum, Fractal N/MEMS: From pull-in instability to pull-in stability, Fractals, Vol. 29, 10/18, 2020.
[3]          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.
[4]          C. He, Q. Zhang, S. Tao, C. Zhao, C. Zhao, W. Su, Y. J. Dappe, R. J. Nichols, L. Yang, Carbon-contacted single molecule electrical junctions, Physical Chemistry Chemical Physics, Vol. 20, No. 38, pp. 24553-24560, 2018.
[5]          C.-H. He, T. S. Amer, D. Tian, A. F. Abolila, A. A. Galal, Controlling the kinematics of a spring-pendulum system using an energy harvesting device, Journal of Low Frequency Noise, Vibration and Active Control, Vol. 41, No. 3, pp. 1234-1257, 2022.
[6]          M. Bayat, I. Pakar, M. Bayat, Analytical study on the vibration frequencies of tapered beams, Latin American Journal of Solids and Structures, Vol. 8, pp. 149-162, 06/01, 2011.
[7]          J.-H. He, Variational approach for nonlinear oscillators, Chaos, Solitons & Fractals, Vol. 34, No. 5, pp. 1430-1439, 2007/12/01/, 2007.
[8]          Y.-T. Zuo, VARIATIONAL PRINCIPLE FOR A FRACTAL LUBRICATION PROBLEM, Fractals, 05/23, 2024.
[9]          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.
[10]        K.-L. WANG, C.-H. HE, A REMARK ON WANG’S FRACTAL VARIATIONAL PRINCIPLE, Fractals, Vol. 27, No. 08, pp. 1950134, 2019.
[11]        C.-H. He, C. Liu, Variational principle for singular waves, Chaos, Solitons & Fractals, Vol. 172, pp. 113566, 07/01, 2023.
[12]        J.-H. He, The simplest approach to nonlinear oscillators, Results in Physics, Vol. 15, pp. 102546, 08/01, 2019.
[13]        J.-H. He, The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, Journal of Low Frequency Noise, Vibration and Active Control, Vol. 38, No. 3-4, pp. 1252-1260, 2019.
[14]        H. Ma, SIMPLIFIED HAMILTONIAN-BASED FREQUENCY-AMPLITUDE FORMULATION FOR NONLINEAR VIBRATION SYSTEMS, Facta Universitatis, Series: Mechanical Engineering, Vol. 20, pp. 445, 07/28, 2022.
[15]        Y. Tian, Frequency formula for a class of fractal vibration system, Reports in Mechanical Engineering, Vol. 3, No. 1, pp. 55-61, 2022.
[16]        J.-Y. Niu, G.-Q. Feng, K. A. Gepreel, A simple frequency formulation for fractal–fractional non-linear oscillators: A promising tool and its future challenge, Frontiers in Physics, Vol. 11, 2023-March-16, 2023. English
[17]        G. Feng, A CIRCULAR SECTOR VIBRATION SYSTEM IN A POROUS MEDIUM, Facta Universitatis, Series: Mechanical Engineering, 2023.
[18]        Y. El-dib, A review of the frequency-amplitude formula for nonlinear oscillators and its advancements, Journal of Low Frequency Noise Vibration and Active Control, pp. 1-33, 04/05, 2024.
[19]        C.-H. HE, C. LIU, A MODIFIED FREQUENCY–AMPLITUDE FORMULATION FOR FRACTAL VIBRATION SYSTEMS, Fractals, Vol. 30, No. 03, pp. 2250046, 2022.
[20]        J.-H. He, Max-min approach to nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 9, No. 2, pp. 207-210, 2008.
[21]        J.-H. He, Variational iteration method–a kind of non-linear analytical technique: some examples, International journal of non-linear mechanics, Vol. 34, No. 4, pp. 699-708, 1999.
[22]        J.-H. He, Homotopy perturbation technique, Computer methods in applied mechanics and engineering, Vol. 178, No. 3-4, pp. 257-262, 1999.
[23]        J.-H. He, Y. O. El-Dib, The reducing rank method to solve third-order Duffing equation with the homotopy perturbation, Numerical Methods for Partial Differential Equations, Vol. 37, No. 2, pp. 1800-1808, 2021.
[24]        Y. El-dib, A heuristic review on the homotopy perturbation method for non-conservative oscillators, 05/07, 2022.
[25]        D. Goorman, Free vibrations of beams and shafts, Appl. Mech., ASME, Vol. 18, pp. 135-139, 1975.
[26]        N. Anjum, J.-H. He, Laplace transform: Making the variational iteration method easier, Applied Mathematics Letters, Vol. 92, pp. 134-138, 2019/06/01/, 2019.
[27]        K. Manimegalai, S. Zephania CF, P. Bera, P. Bera, S. Das, T. Sil, Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method, The European Physical Journal Plus, Vol. 134, pp. 1-10, 2019.
[28]        N. Al-Rashidi, Innovative approaches to fractional modeling: Aboodh transform for the Keller-Segel equation, AIMS Mathematics, Vol. 9, No. 6, pp. 14949-14981, 2024.
[29]        H. Ji-Huan, N. Anjum, H. Chun-Hui, A. A. Alsolami, BEYOND LAPLACE AND FOURIER TRANSFORMS Challenges and Future Prospects, Thermal Science, Vol. 27, 2023.
Journal of Computational Applied Mechanics
Volume 55, Issue 3
June 2024
Pages 440-450
  • Receive Date: 02 June 2024
  • Accept Date: 02 June 2024