Homotopy Analysis Method for Large-Amplitude Free Vibrations of Strongly Nonlinear Generalized Duffing Oscillators

Affiliation(s)

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, China.

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, China.

ABSTRACT

In this study, the homotopy analysis method (HAM) is used to solve the generalized Duffing equation. Both the frequencies and periodic solutions of the nonlinear Duffing equation can be explicitly and analytically formulated. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HAM and numerical integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations which achieved between the results. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.

Cite this paper

Y. Qian, D. Ren, S. Chen and L. Ping, "Homotopy Analysis Method for Large-Amplitude Free Vibrations of Strongly Nonlinear Generalized Duffing Oscillators,"*Modern Mechanical Engineering*, Vol. 2 No. 4, 2012, pp. 167-175. doi: 10.4236/mme.2012.24022.

Y. Qian, D. Ren, S. Chen and L. Ping, "Homotopy Analysis Method for Large-Amplitude Free Vibrations of Strongly Nonlinear Generalized Duffing Oscillators,"

References

[1] I. Kovacic and M. J. Brennan, “The Duffing Equation: Nonlinear Oscillators and Their Behaviour,” John Wiley & Sons, Hoboken, 2011. doi:10.1002/9780470977859

[2] P. Amore and A. Aranda, “Improved Lindstedt-Poincaré Method for the Solution of Nonlinear Problems,” Journal of Sound and Vibration, Vol. 283, No. 3-5, 2005, pp. 1115-1136. doi:10.1016/j.jsv.2004.06.009

[3] R. R. Pu?enjak, “Extended Lindstedt-Poincare Method for Non-Stationary Resonances of Dynamical Systems with Cubic Nonlinearities,” Journal of Sound and Vibration, Vol. 314, No. 1-2, 2008, pp. 194-216. doi:10.1016/j.jsv.2008.01.002

[4] B. S. Wu and P. S. Li, “A method for Obtaining Approximate Analytic Periods for a Class of Nonlinear Oscillators,” Meccanica, Vol. 36, No. 2, 2001, pp. 167-176. doi:10.1023/A:1013067311749

[5] H. L. Zhang, “Periodic Solutions for Some Strongly Nonlinear Oscillations by He’s Energy Balance Method,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2480-2485. doi:10.1016/j.camwa.2009.03.068

[6] I. Mehdipour, D. D. Ganji and M. Mozaffari, “Application of the Energy Balance Method to Nonlinear Vibrating Equations,” Current Applied Physics, Vol. 10, No. 1, 2010, pp. 104-112. doi:10.1016/j.cap.2009.05.016

[7] L. Geng and X. C. Cai, “He’s Frequency Formulation for Nonlinear Oscillators,” European Journal of Physics, Vol. 28, 2007, pp. 923-931. doi:10.1088/0143-0807/28/5/016

[8] J. Fan, “He’s Frequency-Amplitude Formulation for the Duffing Harmonic Oscillator,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2473-2476. doi:10.1016/j.camwa.2009.03.049

[9] S. J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” Chapman & Hall, Boca Raton, 2003. doi:10.1201/9780203491164

[10] S. H. Hoseini, T. Pirbodaghi, M. T. Ahmadian and G. H. Farrahi, “On the Large Amplitude Free Vibrations of Tapered Beams: An Analytical Approach,” Mechanics Research Communications, Vol. 36, No. 8, 2009, pp. 892-897. doi:10.1016/j.mechrescom.2009.08.003

[11] Y. H. Qian, S. K. Lai, W. Zhang and Y. Xiang, “Study on Asymptotic Analytical Solutions Using HAM for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam with an Intermediate Lumped Mass,” Numerical Algorithms, Vol. 58, No. 3, 2011, pp. 293-314. doi:10.1007/s11075-011-9456-7

[12] Y. H. Qian, D. X. Ren, S. K. Lai and S. M. Chen, “Analytical Approximations to Nonlinear Vibration of an Electrostatically Actuated Microbeam,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 4, 2012, pp. 1947-1955. doi:10.1016/j.cnsns.2011.09.018

[13] R. A. Van Gorder and K. Vajravelu, “On the Selection of Auxiliary Functions, Operators, and Convergence Control Parameters in the Application of the Homotopy Analysis Method to Nonlinear Differential Equations: A General Approach,” Communication in Nonlinear Sciences and Numerical Simulation, Vol. 14, No. 12, 2009, pp. 4078-4089. doi:10.1016/j.cnsns.2009.03.008

[14] S. J. Liao, “An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations,” Communication in Nonlinear Sciences and Numerical Simulation, Vol. 15, No. 8, 2010, pp. 2003-2016. doi:10.1016/j.cnsns.2009.09.002

[15] Y. Davood, A. Hassan, S. Zia and K. Mohammad, “Frequency Analysis of Strongly Nonlinear Generalized Duffing Oscillators Using He’s Frequency-Amplitude Formulation and He's Energy Balance Method,” Computers and Mathematics with Applications, Vol. 59, No. 9, 2010, pp. 3222-3228. doi:10.1016/j.camwa.2010.03.013

[1] I. Kovacic and M. J. Brennan, “The Duffing Equation: Nonlinear Oscillators and Their Behaviour,” John Wiley & Sons, Hoboken, 2011. doi:10.1002/9780470977859

[2] P. Amore and A. Aranda, “Improved Lindstedt-Poincaré Method for the Solution of Nonlinear Problems,” Journal of Sound and Vibration, Vol. 283, No. 3-5, 2005, pp. 1115-1136. doi:10.1016/j.jsv.2004.06.009

[3] R. R. Pu?enjak, “Extended Lindstedt-Poincare Method for Non-Stationary Resonances of Dynamical Systems with Cubic Nonlinearities,” Journal of Sound and Vibration, Vol. 314, No. 1-2, 2008, pp. 194-216. doi:10.1016/j.jsv.2008.01.002

[4] B. S. Wu and P. S. Li, “A method for Obtaining Approximate Analytic Periods for a Class of Nonlinear Oscillators,” Meccanica, Vol. 36, No. 2, 2001, pp. 167-176. doi:10.1023/A:1013067311749

[5] H. L. Zhang, “Periodic Solutions for Some Strongly Nonlinear Oscillations by He’s Energy Balance Method,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2480-2485. doi:10.1016/j.camwa.2009.03.068

[6] I. Mehdipour, D. D. Ganji and M. Mozaffari, “Application of the Energy Balance Method to Nonlinear Vibrating Equations,” Current Applied Physics, Vol. 10, No. 1, 2010, pp. 104-112. doi:10.1016/j.cap.2009.05.016

[7] L. Geng and X. C. Cai, “He’s Frequency Formulation for Nonlinear Oscillators,” European Journal of Physics, Vol. 28, 2007, pp. 923-931. doi:10.1088/0143-0807/28/5/016

[8] J. Fan, “He’s Frequency-Amplitude Formulation for the Duffing Harmonic Oscillator,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2473-2476. doi:10.1016/j.camwa.2009.03.049

[9] S. J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” Chapman & Hall, Boca Raton, 2003. doi:10.1201/9780203491164

[10] S. H. Hoseini, T. Pirbodaghi, M. T. Ahmadian and G. H. Farrahi, “On the Large Amplitude Free Vibrations of Tapered Beams: An Analytical Approach,” Mechanics Research Communications, Vol. 36, No. 8, 2009, pp. 892-897. doi:10.1016/j.mechrescom.2009.08.003

[11] Y. H. Qian, S. K. Lai, W. Zhang and Y. Xiang, “Study on Asymptotic Analytical Solutions Using HAM for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam with an Intermediate Lumped Mass,” Numerical Algorithms, Vol. 58, No. 3, 2011, pp. 293-314. doi:10.1007/s11075-011-9456-7

[12] Y. H. Qian, D. X. Ren, S. K. Lai and S. M. Chen, “Analytical Approximations to Nonlinear Vibration of an Electrostatically Actuated Microbeam,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 4, 2012, pp. 1947-1955. doi:10.1016/j.cnsns.2011.09.018

[13] R. A. Van Gorder and K. Vajravelu, “On the Selection of Auxiliary Functions, Operators, and Convergence Control Parameters in the Application of the Homotopy Analysis Method to Nonlinear Differential Equations: A General Approach,” Communication in Nonlinear Sciences and Numerical Simulation, Vol. 14, No. 12, 2009, pp. 4078-4089. doi:10.1016/j.cnsns.2009.03.008

[14] S. J. Liao, “An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations,” Communication in Nonlinear Sciences and Numerical Simulation, Vol. 15, No. 8, 2010, pp. 2003-2016. doi:10.1016/j.cnsns.2009.09.002

[15] Y. Davood, A. Hassan, S. Zia and K. Mohammad, “Frequency Analysis of Strongly Nonlinear Generalized Duffing Oscillators Using He’s Frequency-Amplitude Formulation and He's Energy Balance Method,” Computers and Mathematics with Applications, Vol. 59, No. 9, 2010, pp. 3222-3228. doi:10.1016/j.camwa.2010.03.013