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Hashemi, Gholamreza. (1402). A novel analytical approximation approach for strongly nonlinear oscillation systems based on the energy balance method and He's Frequency-Amplitude formulation. سامانه مدیریت نشریات علمی, 11(3), 464-477. doi: 10.22034/cmde.2022.52293.2189
Gholamreza Hashemi. "A novel analytical approximation approach for strongly nonlinear oscillation systems based on the energy balance method and He's Frequency-Amplitude formulation". سامانه مدیریت نشریات علمی, 11, 3, 1402, 464-477. doi: 10.22034/cmde.2022.52293.2189
Hashemi, Gholamreza. (1402). 'A novel analytical approximation approach for strongly nonlinear oscillation systems based on the energy balance method and He's Frequency-Amplitude formulation', سامانه مدیریت نشریات علمی, 11(3), pp. 464-477. doi: 10.22034/cmde.2022.52293.2189
Hashemi, Gholamreza. A novel analytical approximation approach for strongly nonlinear oscillation systems based on the energy balance method and He's Frequency-Amplitude formulation. سامانه مدیریت نشریات علمی, 1402; 11(3): 464-477. doi: 10.22034/cmde.2022.52293.2189

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
مقاله 3، دوره 11، شماره 3، شهریور 2023، صفحه 464-477    اصل مقاله (301.34 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2022.52293.2189
نویسنده
Gholamreza Hashemi*
Department of Civil and Environmental Engineering, Polytechnic University of Milan, Italy.
چکیده
Nonlinear oscillations are an essential fact in physical science, mechanical structures, and other engineering problems. Some of the popular analytical solutions to analyze nonlinear differential equations governing the behavior of strongly nonlinear oscillators are the Energy Balance Method (EBM), and He's Amplitude Frequency Formulation (HAFF). The lack of precision and accuracy despite needing several computational steps to resolve the system frequency is the main demerit of these methods. This research creates a novel analytical approximation approach with a very efficient algorithm that can perform the calculation procedure much easier and with much higher accuracy than classic EBM and HAFF. The presented method's steps rely on Hamiltonian relations described in EBM and the de ned relationship between frequency and amplitude in HAFF. This paper demonstrates the substantial precision of the presented method compared to common EBM and HAFF applied in different and well-known engineering phenomena. For instance, the approximate solutions of the equations govern some strongly nonlinear oscillators, including the two-massspring systems, buckling of a column, and duffing relativistic oscillators are presented here. Subsequently, their results are compared with the Runge-Kutta method and exact solutions obtained from the previous research. The proposed novel approach resultant error percentages show an excellent agreement with the numerical solutions and illustrate a very quickly convergent and more precise than mentioned typical methods.
کلیدواژه‌ها
Efficient algorithm؛ Nonlinear oscillators؛ EBM؛ HAFF؛ Analytical solution
مراجع
  • [1] M. Ahmadi, G. Hashemi, and A. Asghari, Application of IPM and HA for nonlinear vibration of Euler-Bernoulli beams, Latin American Journal of Solids and Structures, 11 (2014), 1049–1062.
  • [2] M. Ahmadi, G.  Hashemi,  and  D.  D.  Ganji,  A  Study  on  the  Motion  of  Mass  “m”  in  a  Model  for  Buckling  of a Column Using HAFF and EBM, International Journal on Numerical and Analytical Methods in Engineering (IRENA), 1(6) (2013), 273-280.
  • [3] M. Alam, M. Haque, and M. Hossian, A new analytical technique to find periodic solutions of nonlinear systems, International Journal of Non-Linear Mechanics, 42 (2007), 1035–45.
  • [4] M. Azimi and A. Azimi, Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator, Trends in Applied Sciences Research, 7 (2012), 514-522.
  • [5]A. Barari, A. Kimiaeifar, M. Nejad, M. Motevalli, and M. G. Sfahani, A closed form solution for nonlinear oscillators frequencies using amplitude-frequency formulation, Shock and Vibration, 19(6) (2012), 1415-1426.
  • [6] C. Bender, K. Milton, S. Pinsky, and L.Simmons, new perturbative approach to nonlinear problems, Journal of Mathematical Physics 30(7) (1989), 1447–1455.
  • [7] N. Bildik and A. Konuralp, The use of variational iteration method, differential transform method and adomian decomposition method for solving  different  types  of  nonlinear  partial  differential  equations, International Journal of Nonlinear Sciences and Numerical Simulation 7(1) (2006), 65–70.
  • [8] Bota, Caruntu, and Babescu, Analytical approximate periodic  solutions based  on harmonic analysis, SYNASC’ 08.  In: 10th International symposium on symbolic and numeric algorithms for scientific computing, 2008, 177-182.
  • [9] S. Durmaz, S. Demirbag, and M. Kaya, Approximate solutions for nonlinear oscillation of a mass attached to a stretched elastic wire, Computers & Mathematics with Applications, 61 (2011), 578-585.
  • [10]S. Durmaz, and M. Kaya, High-Order Energy Balance Method to Nonlinear Oscillators, Journal of Applied Mathematics, 2012, Special Issue.
  • [11]A. M. El-Naggar and G. M. Ismail, Analytical solution of strongly nonlinear Duffing oscillators, Alexandria Eng. J., 55 (2016), 1581-1585.
  • [12]E. Esmailzadeh, D. Younesian, and H. Askari, Analytical Methods in Nonlinear Oscillations, Solid Mechanics and Its Applications, 252 (2019).
  • [13]S. S. Ganji, A. Barari, and D. D. Ganji, Approximate analysis of two-mass-spring  systems  and  buckling  of  a  column, Computers & Mathematics with Applications, 61 (2011) 1088-1095.
  • [14] S. S. Ganji, A. Barari, S. Karimpour, and D. D. Gamji, Motion of a rigid rod rocking back and forth and cubic- quintic Duffing oscillators, J Theor Appl Mech, 50 (2012), 215–29.
  • [15] D. D. Ganji and A. Sadighi, Application of He’s methods to nonlinear coupled systems of reaction-diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation, 7(4) (2006), 411–418.
  • [16] S. S. Ganji, G. M. Sfahani, S. Tonekaboni, A. Moosavi, and D. D. Ganji,  Higher-Order Solutions of Coupled  Systems Using the Parameter Expansion Method, Mathematical Problems in Engineering, 2009.
  • [17] A. Zolfagharian, and D. D. Ganji,  Study on motion of rigid rod  on a circular surface using MHPM, Propulsion      and Power Research, 3(3) (2014), 159–164.
  • [18] H. Tari and D. D. Ganji, and H. Babazadeh, The application of He’s variational iteration method to nonlinear equations arising in heat transfer, Physics Letters A, 363(3) (2007), 213–217.
  • [19] G. Hashemi and M. Ahmadi, On choice of initial guess in the variational iteration method and its applications to nonlinear oscillator, Proceedings of the Institution of Mechanical Engineers- Part E: Journal of Process Mechanical Engineering, first published on February 22, (2015), Publisher Sage.
  • [20] G. Hashemi, M. Ahmadi, and A. Sirous, Evaluation of the Radius of Curvature Beam Equation Using Variational Iteration Method, Asian Journal of Civil Engineering, Publisher: Building and Housing Research Center (BHRC), 16(3) (2015), 489-499.
  • [21] J. H. He, Hamiltonian approach to nonlinear oscillators, Physics Letters A, 374 (2010), 2312–2314.
  • [22] J. H. He, Preliminary report on the energy balance for nonlinear oscillations, Mechanics Research Communica-tions, 29(2–3) (2002) 107–111.
  • [23] J. H. He, Determination of Limit Cycles for Strongly Nonlinear  Oscillators.  Physical  Review  Letters,  90(17)  (2003).
  • [24] J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73–79.
  • [25] J. H. He, Addendum: New interpretation of homotopy perturbation method, International Journal  of  Modern Physics B, 20 (2012), 2561–2568.
  • [26] J. H. He, Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng., 178(3-4) (1999), 257–262.
  • [27] J. H. He, Variational iteration method: a kind of nonlinear analytical technique: some examples,  International Journal of Nonlinear Mechanics, 34(4) (1999), 699–708.
  • [28] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167(1–2) (1998), 57–68.
  • [29] J. H. He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20 (2006), 1141–1199.
  • [30] J. H. He, Comment on ‘He’s frequency formulation for nonlinear oscillators, European Journal of Physics, 29(4) (2008), L19–L22.
  • [31] J. H. He, An improved amplitude-frequency formulation for nonlinear oscillators, International Journal of Non-  linear Sciences and Numerical Simulation, 9 (2008), 211–212.
  • [32] M. D. Hosen, Analytical approximate solutions of the duffing-relativistic oscillator, Third International Conference on Advances In Computing, Control And Networking - ACCN (2015), 71–74.
  • [33] A. Y. T. Leung, G. Zhongjin, and H. X. Yang, Residue harmonic balance analysis for the damped Duffing resonator driven by a van der Pol oscillator, International Journal of Mechanical Sciences, 63(1) (2012), 59–65.
  • [34] H. A. Navarro and L. Cveticanin, Amplitude–frequency relationship obtained using Hamiltonian approach for oscillators with sum of non-integer order nonlinearities, Applied Mathematics and Computation, 291 (2016), 162-171.
  • [35] A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981.
  • [36] Z. Odibat and S. Momani, Application of variational iteration method to nonlinear differential equations of  frac-  tional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7(1) (2006), 27–34.
  • [37] V. Prasolov and Y. Sovolyev, Elliptic Functions and Elliptic Integrals. Providence, (1997) RI: American Mathe- matical Society.
  • [38] M. Rafei, H. Daniali, and D. D. Ganji, Variational iteration method for solving the  epidemic  model and  the  prey   and predator problem, Applied Mathematics and Computation, 186(2) (2007), 1701–1709.
  • [39] A. Ranjbar, D. D. Ganji, M. Naghipour, and A. Barari, Solutions of nonlinear oscillator differential equations using the variational iteration method. Journal of Physics: Conference Series, 96 (2018), 012084.
  • [40] Z. Ren, A simplified He’s frequency–amplitude formulation for nonlinear oscillators. Journal of Low Frequency Noise, Vibration and Active Control, 2021.
  • [41] M. Sakar and O. Saldir, Improving Variational Iteration Method with Auxiliary Parameter for Nonlinear Time- Fractional Partial Differential Equations, Journal of Optimization Theory and Applications, (2017).
  • [42] W. P. Sun, B. S. Wu, and C. W. Lim, Approximate analytical solutions for oscillation of a mass attached to a stretched elastic wire, Journal of Sound and Vibration, 300 (2006), 1042-1047.
  • [43] M. Tatari and M. Dehghan, Improvement of He’s variational iteration method for solving systems of differential equations Citation data: Computers & Mathematics with Applications, 58 (2009), 2160-2166.
  • [44] M. Xiao, W. X. Zheng WX, and J. Cao, Approximate expressions of a fractional order van der Pol oscillator by the residue harmonic balance method, Math Comput Simulat 89 (2013), 1–12.
  • [45] L. Xu, Application of Hamiltonian approach to an oscillation of a mass attached to a stretched elastic wire, Mathematical and Computational Applications, 15 (2010), 901–906.
  • [46] L. Xu, Determination of limit cycle by He’s parameter-expanding method for strongly nonlinear oscillators, Journal of Sound and Vibration, 302 (2007), 178–184.
  • [47] L. Xu, Variational principles for coupled nonlinear Schr¨o dinger equations, Physics Letters A, 359 (2006), 627–629.
  • [48] D. Younesian, H. Askari, Z. Saadatnia, and M. K. Yazdi, Frequency analysis of strongly nonlinear generalized  Duffing oscillators using He’s frequency amplitude formulation and He’s energy balance method, Computers & Mathematics with Applications, 59 (2010), 3222–3228.
  • [49] D. Younesian, H. Askari, Z. Saadatnia, and M. K. Yazdi, Periodic solutions for nonlinear oscillation of a centrifu- gal governor system using the He’s frequency-amplitude formulation and He’s energy balance method, Nonlinear Science Letters A, 2 (2011), 143–148.
  • [50] D. Younesian, H. Askari, Z. Saadatnia, and M. K. Yazdi, Analytical approximate solutions for the generalized nonlinear oscillator, Applicable Analysis, 91(5) (2012), 965-977.
  • [51] H. L. Zhang, Application of He’s amplitude-frequency formulation to a nonlinear oscillator with discontinuity, Computers & Mathematics with Applications. 58 (2009), 2197–2198.
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