Application of higher order Hamiltonian approach to nonlinear vibrating systems

• Autorzy: Askari H. , Nia Z. S. , Yildirim A. , Yazdi M. K. , Khan Y.

Warianty tytułu

PL

Zastosowanie metody Hamiltona wyższego rzędu w zagadnieniu drgań układów nieliniowych

• Języki publikacji: EN

Abstrakty

EN

The higher order Hamiltonian approach is utilized to elicit approximate solutions for two nonlinear oscillation systems. Frequency-amplitude relationships and the model of buc kling of a column and mass-spring system are scrutinized in this paper. First, second and third approximate solutions of examples are achieved, and the frequency responses of the systems are verified by exact numerical solutions. According to the numerical results, we can conclude that the Hamiltonian approach is an applicable method for solving the nonlinear equations, and the accuracy of this method in the second and third approximates is very high and reliable. The achieved results of this paper demonstrate that this method is powerful and uncomplicated for solving of sophisticated nonlinear problems.

PL

W pracy przedstawiono zastosowanie metody Hamiltona wyższego rzędu do wyznaczania przybliżonych rozwiązań analitycznych dla dwóch nieliniowych układów drgających. Szczegółowej analizie poddano charakterystyki amplitudowo-częstościowe modelu ściskanej belki oraz dyskretnego układu sprężysto-inercyjnego. Otrzymano przybliżone rozwiązania pierwszego, drugiego i trzeciego rzędu, a odpowiedzi częstościowe układów porównano z dokładnymi rezultatami symulacji numerycznych. Na ich podstawie oceniono, że metoda Hamiltona jest stosowalna dla układów nieliniowych, a przybliżenia drugiego i trzeciego rzędu stanowią rozwiązania analityczne o wysokiej dokładności. Uzyskane w pracy wyniki przekonują, że zaproponowana metoda jest prostym i jednocześnie bardzo skutecznym narzędziem rozwiązywania nieliniowych problemów układów mechanicznych o dużym stopniu złożoności.

Słowa kluczowe

EN

higher order Hamiltonian approach; Duffing equation; analytical solutions

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2013
• Tom: Vol. 51 nr 2
• Strony: 287--296
• Opis fizyczny: Bibliogr. 36 poz., rys., tab.

Twórcy

autor

Askari H.

• Iran University of Science and Technology, Center of Excellence in Railway Transportation, School of Railway Engineering, Tehran, Iran

autor

Nia Z. S.

• Iran University of Science and Technology, Center of Excellence in Railway Transportation, School of Railway Engineering, Tehran, Iran

autor

Yildirim A.

• Ege University, Department of Mathematics, Bornova, Turkey

autor

Yazdi M. K.

• Iran University of Science and Technology, School of Mechanical Engineering, Tehran, Iran

autor

Khan Y.

• Zheijiang University, Department of Mathematics, Hangzhou, China

Bibliografia

• 1. Askari H., Kalami Yazdi M., Saadatnia Z., 2010, Frequency analysis of nonlinear oscillators with rational restoring force via He’s energy balance method and He’s variational approach, Nonlinear Science Letters A, 1, 425-430
• 2. Bel´endez A., Arribas E., Franc´es J., Pascual I., 2011, Notes on application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms, Mathematical and Computer Modelling, http://dx.doi.org/10.1016/j.mcm.2011.06.024
• 3. Cai X.C., Wu W.Y., 2009, He’s frequency formulation for the relativistic harmonic oscillator, Computer and Mathematic with Application, 58, 2358-2359
• 4. Cveticanin L., Kalami Yazdi M., Saadatnia Z., Askari H., 2010, Application of Hamiltonian approach to the generalized nonlinear oscillator with fractional power, International Journal of Nonlinear Sciences and Numerical Simulation, 11, 997-1002
• 5. Durmaz S., Altay D., Kaya M.O., 2010, High order Hamiltonian approach to nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation, 11, 565-570
• 6. Ganji D.D., Esmaeilpour M., Soleimani S., 2010, Approximate solutions to Van der Pol damped nonlinear oscillators by means of He’s energy balance method, International Journal of Computer Mathematics, 87 2014- 2023
• 7. Ganji D.D., Ranjbar Malidarreh N., Akbarzade M., 2009, Comparison of energy balance period with exact period for arising nonlinear oscillator equations (He’s energy balance period for nonlinear oscillators with and without discontinuities), Acta Applicandae Mathematicae, 108, 353-362
• 8. Ganji S.S., Barari A., Ganji D.D., 2011, Approximate analysis of two mass-spring systems and buckling of a column, Computers and Mathematics with Applications, 61, 1088-1095
• 9. Ganji S.S., Ganji D.D., Davodi A.G., Karimpour S., 2010, Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max-min approach, Applied Mathematical Modelling, 34, 2676-2684
• 10. Ganji S.S., Ganji D.D., Ganji Z.Z., Karimpour S., 2009, Periodic solution for strongly nonlinear vibration systems by He’s energy balance method, Acta Applicandae Mathematicae, 106, 79-92
• 11. He J.H., 2002, Preliminary report on the energy balance for nonlinear oscillations, Mechanics Research Communications, 29, 107-111
• 12. He J.H., 2006, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20, 10, 1141-1199
• 13. He J.H., 2007, Variational approach for nonlinear oscillators, Chaos Solitons and Fractals, 34, 1430-1439
• 14. He J.H., 2008a, An improved amplitude-frequency formulation for nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation, 9, 2, 211-212
• 15. He J.H., 2008b, Max-min approach to nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation, 9, 2, 207-210
• 16. He J.H., 2010, Hamiltonian approach to nonlinear oscillators, Physics Letters A, 374, 2312- 2314
• 17. Khan Y., Wu Q., Askari H., Saadatnia Z., Kalami Yazdi M., 2010, Nonlinear vibration analysis of a rigid rod on a circular surface via Hamiltonian approach, Mathematical and Computational Applications, 15, 974-977
• 18. Mehdipour I., Ganji D.D., Mozaffari M., 2010, Application of the energy balance method to nonlinear vibrating equations, Current Applied Physics, 10, 104-112
• 19. Momeni M., Jamshidi N., Barari A., Ganji D.D., 2011, Application of He’s energy balance method to Duffing-harmonic oscillators, International Journal of Computer Mathematics, 88, 1, 135-144
• 20. Nayfeh A.H., Mook D.T., 1979, Nonlinear Oscillations, John Wiley & Sons, New York
• 21. Ozis T., Yildirim A., Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method, Computers and Mathematics with Applications, 54, 1184-1187
• 22. Ren Z.F., He J.H., 2009, A simple approach to nonlinear oscillators, Physics Letters A, 373, 3749-3752
• 23. Ren Z.-F., Liu G.-Q., Kang Y.-X., Fan H.-Y., Li H.-M., Ren X.-D., Gui W.-K., 2009, Application of He’s amplitude-frequency formulation to nonlinear oscillators with discontinuities, Physica Scripta, 80, 4, 045003
• 24. Rao S.S., 2006, Mechanical Vibrations Book, Fourth ed., ISBN: 978-964-9585-5-0
• 25. Shen Y.Y., Mo L.F., 2009, The max-min approach to a relativistic equation, Computers and Mathematics with Applications, 58, 2131-2133
• 26. Yazdi M.K., Khan Y., Madani M., Askari H., Saadatnia Z., Yildirim A., Analytical solutions for autonomous conservative nonlinear oscillator, International Journal of Nonlinear Sciences and Numerical Simulation, 11, 11, 979-984
• 27. Yildirim A., Saadatnia Z., Askari H., 2011a, Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms, Mathematical and Computer Modelling, 54, 697-703
• 28. Yildirim A., Askari H., Saadatnia Z., Kalami Yazdi M., Khan Y., 2011b, Analysis of nonlinear oscillations of a punctual charge in the electric field of a charged ring via a Hamiltonian approach and the energy balance method, Computers and Mathematics with Applications, 62, 486-490
• 29. Yildirim A., Saadatnia Z., Askari H., Khan Y., Kalami Yazdi M., 2011c, Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach, Applied Mathematics Letters, 24, 2042-2051
• 30. Yildirim A., Askari H., Kalami Yazdi M., Khan Y., 2012, A relationship between three analytical approaches to nonlinear problems, Applied Mathematics Letters, doi:10.1016/j.aml.2012.02.001
• 31. Younesian D., Askari H., Saadatnia Z., KalamiYazdi M., 2010a, Frequency analysis of strongly nonlinear generalized Duffing oscillators using He’s frequency-amplitude formulation and He’s energy balance method, Computer and Mathematic with Application, 59, 3222-3228
• 32. Younesian D., Askari H., Saadatnia Z., Kalami Yazdi M., 2011, Free vibration analysis of strongly nonlinear generalized duffing oscillators using He’s variational approach and homotopy perturbation method, Nonlinear Science Letters A, 2, 11-16
• 33. Younesian D., Askari H., Saadatnia Z., Yildirim A., 2010b, Periodic solutions for the generalized nonlinear oscillators containing fraction order elastic force, International Journal of Nonlinear Sciences and Numerical Simulation, 11, 1027-1032
• 34. Zeng D.Q., 2009, Nonlinear oscillator with discontinuity by the max-min approach, Chaos, Solitons and Fractals, 42, 2885-2889
• 35. Zhang Y.N., Xu F., Deng L.L., 2009, Exact solution for nonlinear Schr¨odinger equation by He’s frequency formulation, Computers and Mathematics with Applications, 58, 2449-2451
• 36. Zhao L., 2009, He’s frequency-amplitude formulation for nonlinear oscillators with an irrational force, Computers and Mathematics with Applications, 58, 2477-2479