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Advances in Mathematical Physics
Volume 2017 (2017), Article ID 5214616, 8 pages
https://doi.org/10.1155/2017/5214616
Research Article

The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

1College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
2College of Mathematics, Xiamen University of Technology, Xiamen 361024, China
3College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

Correspondence should be addressed to Y. H. Qian; nc.unjz@4002hyq

Received 2 October 2016; Revised 21 February 2017; Accepted 20 March 2017; Published 10 April 2017

Academic Editor: Zhi-Yuan Sun

Copyright © 2017 Y. H. Qian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. E. Mickens, Mathematical Methods for the Natural and Engineering Sciences, World Scientific, Singapore, 2004.
  2. O. Abdulaziz, N. F. M. Noor, and I. Hashim, “Homotopy analysis method for fully developed MHD micropolar fluid flow between vertical porous plates,” International Journal for Numerical Methods in Engineering, vol. 78, no. 7, pp. 817–827, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. Y. M. Chen and J. K. Liu, “Homotopy analysis method for limit cycle flutter of airfoils,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 854–863, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. R. R. Pušenjak, “Extended Lindstedt-Poincare method for non-stationary resonances of dynamical systems with cubic nonlinearities,” Journal of Sound & Vibration, vol. 314, no. 1-2, pp. 194–216, 2008. View at Google Scholar
  5. 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, pp. 1115–1136, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. M. Senator and C. N. Bapat, “A perturbation technique that works even when the nonlinearity is not small,” Journal of Sound and Vibration, vol. 164, no. 1, pp. 1–27, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. Y. K. Cheung, S. H. Chen, and S. L. Lau, “A modified Lindstedt-Poincar\'e method for certain strongly nonlinear oscillators,” International Journal of Non-Linear Mechanics, vol. 26, no. 3-4, pp. 367–378, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. Q.-Q. Hu, C. W. Lim, and L.-Q. Chen, “Nonlinear vibration of a cantilever with a Derjaguin-Müller-Toporov contact end,” International Journal of Structural Stability and Dynamics, vol. 8, no. 1, pp. 25–40, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. K. Huseyin and R. Lin, “An intrinsic multiple-scale harmonic balance method for nonlinear vibration and bifurcation problems,” International Journal of Non-Linear Mechanics, vol. 26, no. 5, pp. 727–740, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. L.-L. Ke, J. Yang, S. Kitipornchai, and Y. Xiang, “Flexural vibration and elastic buckling of a cracked timoshenko beam made of functionally graded materials,” Mechanics of Advanced Materials and Structures, vol. 16, no. 6, pp. 488–502, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. S. K. Lai, C. W. Lim, B. S. Wu, C. Wang, Q. C. Zeng, and X. F. He, “Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators,” Applied Mathematical Modelling, vol. 33, no. 2, pp. 852–866, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S. J. Liao, “An approximate solution technique not depending on small parameters: a special example,” International Journal of Non-Linear Mechanics, vol. 30, no. 3, pp. 371–380, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003. View at MathSciNet
  14. S. J. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1186–1194, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. S. J. Liao, “An optimal homotopy-analysis approach for strongly nonlinear differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 2003–2016, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. C. W. Lim, R. Xu, S. K. Lai, Y. M. Yu, and Q. Yang, “Nonlinear free vibration of an elastically-restrained beam with a point mass via the newton-harmonic balancing approach,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 5, pp. 661–674, 2009. View at Google Scholar · View at Scopus
  17. T. Pirbodaghi, S. H. Hoseini, M. T. Ahmadian, and G. H. Farrahi, “Duffing equations with cubic and quintic nonlinearities,” Computers and Mathematics with Applications, vol. 57, no. 3, pp. 500–506, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. F. F. Seelig, “Unrestricted harmonic balance II. Application to stiff ordinary differential equations in enzyme catalysis,” Journal of Mathematical Biology, vol. 12, no. 2, pp. 187–198, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. J. L. Summers and M. D. Savage, “Two timescale harmonic balance. I. Application to autonomous one-dimensional nonlinear oscillators,” Philosophical Transactions of the Royal Society B: Biological Sciences, vol. 340, no. 1659, pp. 473–501, 1992. View at Google Scholar
  20. 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,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 12, pp. 4078–4089, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. H. Wagner, “Large-amplitude free vibrations of a beam,” Journal of Applied Mechanics, vol. 32, no. 4, pp. 887–892, 1965. View at Google Scholar · View at MathSciNet
  22. C. Wang and I. Pop, “Analysis of the flow of a power-law fluid film on an unsteady stretching surface by means of homotopy analysis method,” Journal of Non-Newtonian Fluid Mechanics, vol. 138, no. 2-3, pp. 161–172, 2006. View at Publisher · View at Google Scholar · View at Scopus
  23. B. Wu and P. Li, “A method for obtaining approximate analytic periods for a class of nonlinear oscillators,” Meccanica, vol. 36, no. 2, pp. 167–176, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. K. Yabushita, M. Yamashita, and K. Tsuboi, “An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method,” Journal of Physics. A. Mathematical and Theoretical, vol. 40, no. 29, pp. 8403–8416, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. S. S. Ganji, D. D. Ganji, H. Babazadeh, and N. Sadoughi, “Application of amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back,” Mathematical Methods in the Applied Sciences, vol. 33, no. 2, pp. 157–166, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. D. D. Ganji, H. B. Rokni, M. G. Sfahani, and S. S. Ganji, “Approximate traveling wave solutions for coupled Whitham-Broer-Kaup shallow water,” Advances in Engineering Software, vol. 41, no. 7-8, pp. 956–961, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. S. S. Ganji, A. Barari, and D. D. Ganji, “Approximate analysis of two-massspring systems and buckling of a column,” Computers and Mathematics with Applications, vol. 61, no. 4, pp. 1088–1095, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. S. S. Ganji, A. Barari, L. B. Ibsen, and G. Domairry, “Differential transform method for mathematical modeling of jamming transition problem in traffic congestion flow,” Central European Journal of Operations Research, vol. 20, no. 1, pp. 87–100, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. M. N. Hamdan and N. H. Shabaneh, “On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass,” Journal of Sound and Vibration, vol. 199, no. 5, pp. 711–736, 1997. View at Publisher · View at Google Scholar · View at Scopus
  30. S. Telli and O. Kopmaz, “Free vibrations of a mass grounded by linear and nonlinear springs in series,” Journal of Sound and Vibration, vol. 289, no. 4-5, pp. 689–710, 2006. View at Publisher · View at Google Scholar · View at Scopus
  31. S. K. Lai and C. W. Lim, “Accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness,” Journal of Sound and Vibration, vol. 307, no. 3, pp. 720–736, 2007. View at Publisher · View at Google Scholar · View at Scopus
  32. S.-S. Chen and C.-K. Chen, “Application of the differential transformation method to the free vibrations of strongly non-linear oscillators,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 881–888, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. S. S. Ganji, D. D. Ganji, M. G. Sfahani, and S. Karimpour, “Application of AFF and HPM to the systems of strongly nonlinear oscillation,” Current Applied Physics, vol. 10, no. 5, pp. 1317–1325, 2010. View at Publisher · View at Google Scholar · View at Scopus
  34. 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, pp. 104–112, 2010. View at Publisher · View at Google Scholar · View at Scopus
  35. 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, pp. 293–314, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. Z. Guo and W. Zhang, “The spreading residue harmonic balance study on the vibration frequencies of tapered beams,” Applied Mathematical Modelling, vol. 40, no. 15, pp. 7195–7203, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. Z. Guo and X. Ma, “Residue harmonic balance solution procedure to nonlinear delay differential systems,” Applied Mathematics and Computation, vol. 237, pp. 20–30, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  38. P. Ju and X. Xue, “Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators,” Applied Mathematical Modelling, vol. 38, no. 24, pp. 6144–6152, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. P. Ju and X. Xue, “Global residue harmonic balance method for large-amplitude oscillations of a nonlinear system,” Applied Mathematical Modelling, vol. 39, no. 2, pp. 449–454, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. Y. Y. Lee, “Free vibration analysis of a nonlinear panel coupled with extended cavity using the multi-level residue harmonic balance method,” Thin-Walled Structures, vol. 98, pp. 332–336, 2016. View at Publisher · View at Google Scholar · View at Scopus