Analytical solutions to nonlinear conservative oscillator with fifth-order nonlinearity

Earthquake Engineering and Engineering Vibration

Volumn 9, Issue 3, 2010, Pages 367-374

  Sfahani, M G ^a   Ganji, S S ^b   Barari, Amin ^c   Mirgolbabaei, H ^d   Domairry, G ^a  

^a BABOL NOSHIRVANI UNIVERSITY OF TECHNOLOGY   (Iran)

^b ISLAMIC AZAD UNIVERSITY   (Iran)

^c AALBORG UNIVERSITY   (Denmark)

^d ISLAMIC AZAD UNIVERSITY   (Iran)

Author keywords

homotopy perturbation method (HPM); large amplitude free vibrations; max min approach (MMA); non linear oscillation; Rung Kutta method (R KM)

Indexed keywords

HOMOTOPY PERTURBATION METHOD; LARGE AMPLITUDE FREE VIBRATION; MAX-MIN; NONLINEAR OSCILLATION; RUNG-KUTTA METHOD (R-KM);

APPROXIMATION THEORY; NUMERICAL METHODS; OSCILLATORS (MECHANICAL);

PERTURBATION TECHNIQUES;

EID: 77957955495     PISSN: 16713664     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11803-010-0021-5     Document Type: Article

References (36)

1. Anderson DA and Tannehill JC (1984), "Computational Fluid Mechanics and Heat Transfer," Hemisphere Publishing Corp.
2. Barari A, Ghotbi Abdoul R, Farrokhzad F and Ganji DD (2008b), "Variational Iteration Method and Homotopy-Perturbation Method for Solving Different Types of Wave Equations," Journal of Applied Sciences, 8: 120-126.
3. Barari A, Omidvar M, Ghotbi Abdoul R and Ganji DD (2008a), "Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations," Acta Applicanda Mathematicae, 104: 161-171.
4. 1/3 Force Nonlinear Oscillator," Physics Letters A, 371: 421.
5. Catal S (2008), "Solution of Free Vibration Equations of Beam on Elastic Soil by Using Differential Transform Method," Applied Mathematical Modeling, 32(9): 1744.
6. Dehghan M (2004), "The Use of Adomian Decomposition Method for Solving the One-Dimensional Parabolic Equation with Non-local Boundary Specifications," International Journal of Computer Mathematics, 81(1): 25.
7. Dehghan M. and Shakeri F (2008), "The Use of the Decomposition Procedure of Adomian for Solving a Delay Differential Equation Arising in Electrodynamics," Physica Scripta, 78: 1.
8. Ganji SS, Ganji DD, Babazadeh H and Sadoughi N (2009a), "Application of Amplitude-frequency Formulation to Nonlinear Oscillation System of the Motion of a Rigid Rod Rocking Back," Mathematical Methods in the Applied Sciences, 33(2): 157-166.
9. Ganji SS, Ganji DD and Karimpour S (2009b), "He's Energy Balance and He's Variational Methods for Nonlinear Oscillations in Engineering," International Journal of Modern Physics B, 23(3): 461.
10. Ganji SS, Ganji DD, Karimpour S and Babazadeh H (2009c), "Applications of He's Homotopy Perturbation Method to Obtain Second-order Approximations of the Coupled Two-degree-of-freedom Systems," International Journal of Non-Linear Sciences and Numerical Simulation, 10(3): 303.
11. Ganji DD, Mirmohammadsadeghi SE and Safari M (2009d), "Application of He's Variational Iteration Method and Adomian's Decomposition Method to Prochhammer-Chree Equation," International Journal of Modern Physics B, 23(3): 435.
12. Ghotbi Abdoul R, Barari A and Ganji DD (2008), "Solving Ratio-dependent Predator-prey System with Constant Effort Harvesting Using Homotopy Perturbation Method," Journal of Mathematical Problems in Engineering, Article ID 945420, 1-8.
13. Hamdan MN and Shabaneh NH (1997), "On the Large Amplitude Free Vibrations of a Restrained Uniform Beam Carrying an Intermediate Lumped Mass," J. Sound Vib, 199: 711.
14. He JH (1999a), "Some New Approaches to Duffing Equation with Strongly and High Order Nonlinearity (II) Parameterized Perturbation Technique," Communications in Nonlinear Science and Numerical Simulation, 4: 81.
15. He JH (1999b), "Variational Iteration Method - a Kind of Nonlinear Analytical Technique: Some Examples," International Journal of Non-linear Mechanics, 34: 699.
16. He JH (2001), "Modified Lindstedt-Poincare Methods for Some Strongly Nonlinear Oscillations. Part III: Double Series Expansion," International Journal Non-linear Sciences and Numerical Simulation, 2: 317.
17. He JH (2002), "Modified Lindstedt-Poincare Methods for Some Strongly Nonlinear Oscillations, Part I: Expansion of a Constant," International Journal Non-linear Mechanic, 37: 309.
18. He JH (2005), "Homotopy Perturbation Method for Bifurcation on Nonlinear Problems," International Journal of Nonlinear Sciences and Numerical Simulation, 6: 207.
19. He JH (2006a), "Non-Perturbative Methods for Strongly Nonlinear Problems," Dissertation. de-Verlagim Internet GmbH.
20. He JH (2006b), "New Interpretation of Homotopy Perturbation Method," International Journal of Modern Physics B, 20(18): 2561.
21. He JH (2006c), "Determination of Limit Cycles for Strongly Nonlinear Oscillators," Physics Review Letter, 90: 174.
22. He JH (2007), "Variational Approach for Nonlinear Oscillators," Chaos, Solitons and Fractals, 34: 1430.
23. He JH (2008), "Max-Min Approach to Nonlinear Oscillators," International Journal of Nonlinear Sciences and Numerical Simulation, 9(2): 207.
24. He JH and Wu XH (2006), "Construction of Solitary Solution and Compaction-like Solution by Variational Iteration Method," Chaos, Solitons & Fractals, 29: 108.
25. Konuralp A (2009), "The Steady Temperature Distributions with Different Types of Nonlinearities," Computers & Mathematics with Applications, 58(11-12): 2152-2159.
26. Lai SK, Lim CW, Wu BS, Wang C, Zeng QC and He XF (2009), "Newton-harmonic Balancing Approach for Accurate Solutions to Nonlinear Cubic-quintic Duffing Oscillators," Applied Mathematics Modelling, 33(2): 852.
27. Lim CW and Wu BS (2003), "A New Analytical Approach to the Duffing-harmonic Oscillator," Phys. Lett. A, 311: 365.
28. Mirmoradi H, Mazaheripour H, Ghanbarpour S and Barari A (2009), "Homotopy Perturbation Method for Solving Twelfth Order Boundary Value Problems," International Journal of Research and Reviews in Applied Sciences, 1(2): 163-173.
29. Nayfeh AH (1973), "Perturbation Methods," New York: Wiley-Interscience.
30. Nayfeh AH and Mook DT (1979), Nonlinear Oscillations, New York: Wiley.
31. Okuizumi N and Kimura K (2004), "Multiple Time Scale Analysis of Hysteretic Systems Subjected to Harmonic Excitation," Journal of Sound and Vibration, 272: 675.
32. Omidvar M, Barari A, Momeni M and Ganji DD (2010), "New Class of Solutions for Water Infiltration Problems in Unsaturated Soils," Geomechanics and Geoengineering: An International Journal, 5(2): 127-135.
33. 1/3 Force by He's Modified Lindstedt-Poincare' Method," Journal of Sound and Vibration, 301: 415.
34. Ramos JI (2009), "An Artificial Parameter-Linstedt-Poincaré Method for Oscillators with Smooth Odd Nonlinearities," Chaos, Solitons & Fractals, 41(1): 380-393.
35. Wang SQ and He JH (2008), "Nonlinear Oscillator with Discontinuity by Parameter-expanding Method," Chaos, Solitons & Fractals, 35: 688.
36. Xu L (2008), "Variational Approach to Solitons of Nonlinear Dispersive K(m, n) Equations," Chaos, Solitons & Fractals, 37: 137.