Skip to main content
SpringerLink
Log in

The Extended Rayleigh–Ritz Method for Higher Order Approximate Solutions of Nonlinear Vibration Equations

  • Original article
  • Published:
Aerotecnica Missili & Spazio Aims and scope Submit manuscript

Abstract

An extension has been made with the popular Rayleigh–Ritz method by integrating the Lagrangian functional of a nonlinear vibration equation of motion over one period of vibrations to eliminate harmonics from the simplification. A set of successive nonlinear equations of coupled higher order amplitudes of deformation is obtained, and a nonlinear eigenvalue problem is presented for the frequency–amplitude dependence of nonlinear vibrations of successive displacements. The subsequent solutions of vibration frequencies and deformation are actually consistent with other successive approximate methods such as the harmonics balance method. This is an extension of the powerful Rayleigh–Ritz method which has broad applications for approximate solutions for vibration problems in solid mechanics. This extended Rayleigh–Ritz method can now be utilized for the analysis of free and forced nonlinear vibrations of structures as a new technique with significant advantages.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data availability

The data used and generated in this research will be publicly available from the publication’s website and author’s website.

References

  1. Oliveri, V., Milazzo, A.: A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels. Comput. Struct. 196, 263–276 (2018)

    Article  Google Scholar 

  2. Ghafari, E., Rezaeepazhand, J.: Two-dimensional cross-sectional analysis of composite beams using Rayleigh-Ritz-based dimensional reduction method. Compos. Struct. 184, 872–882 (2018)

    Article  Google Scholar 

  3. Ganesh, R., Ganguli, R.: Stiff string approximations in Rayleigh-Ritz method for rotating beams. Appl. Math. Comput. 219(17), 9282–9295 (2013)

    MATH  MathSciNet  Google Scholar 

  4. Chu, E.K.W., Fan, H.Y., Jia, Z.X., Li, T.X., Lin, W.W.: The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs. J. Comput. Appl. Math. 235(8), 2626–2639 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mokhtari, M., Permoon, M.R., Haddadpour, H.: Dynamic analysis of isotropic sandwich cylindrical shell with fractional viscoelastic core using Rayleigh-Ritz method. Compos. Struct. 186, 165–174 (2017)

    Article  Google Scholar 

  6. Hussien, H.S.: A spectral Rayleigh-Ritz scheme for nonlinear partial differential systems of first order. J. Egypt. Math. Soc. 24(3), 373–378 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  7. Monterrubio, L.E., Ilanko, S.: Proof of convergence for a set of admissible functions for the Rayleigh-Ritz analysis of beams and plates and shells of rectangular planform. Comput. Struct. 147, 236–243 (2015)

    Article  Google Scholar 

  8. Li, F.M., Kishimoto, K., Huang, W.H.: The calculations of natural frequencies and forced vibration responses of conical shell using the Rayleigh-Ritz method. Mech. Res. Commun. 36(5), 595–602 (2009)

    Article  MATH  Google Scholar 

  9. Lee, H.W., Kwak, M.K.: Free vibration analysis of a circular cylindrical shell using the Rayleigh-Ritz method and comparison of different shell theories. J. Sound Vib. 353, 344–377 (2015)

    Article  Google Scholar 

  10. Pradhan, K.K., Chakraverty, S.: Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method. Compos. Part B Eng. 51, 175–184 (2013)

    Article  Google Scholar 

  11. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (2011)

    MATH  Google Scholar 

  12. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  13. Hu, H.Y.: Applied Nonlinear Mechanics. Aviation Industry Press, Beijing (2000). (in Chinese)

    Google Scholar 

  14. Chen, S.H.: Quantitative Analytical Methods of Strong Nonlinear Vibrations. Science Press, Beijing (2009). (in Chinese)

    Google Scholar 

  15. Liu, Y.Z., Chen, L.Q.: Nonlinear Vibrations. Higher Education Press, Beijing (2001). (in Chinese)

    Google Scholar 

  16. Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC, Boca Raton (2003)

    Book  Google Scholar 

  17. Bao, S.Y., Cao, J.R., Wang, S.D.: Vibration analysis of nanorods by the Rayleigh-Ritz method and truncated Fourier series. Results Phys. 12, 327–334 (2018)

    Article  Google Scholar 

  18. Grabec, T., Sedlák, P., Seiner, H.: Application of the Ritz-Rayleigh method for Lamb waves in extremely anisotropic media. Wave Motion 96, 102567 (2020)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wang, J.: The extended Rayleigh-Ritz method for an analysis of nonlinear vibrations. Mech. Adv. Mater. Struct. 29(22), 3281–3284 (2021)

    Article  Google Scholar 

  20. Wang, J., Wu, R.X.: The extended Galerkin method for approximate solutions of nonlinear vibration equations. Appl. Sci. 12(6), 2979 (2022)

    Article  Google Scholar 

  21. Shi, B.Y., Yang, J., Wang, J.: Forced vibration analysis of multi-degree-of-freedom nonlinear systems with the extended Galerkin method. Mech. Adv. Mater. Struct. (2022). https://doi.org/10.1080/15376494.2021.2023922. (in press)

    Article  Google Scholar 

  22. Azzara, R., Carrera, E., Pagani, A.: Nonlinear and linearized vibration analysis of plates and shells subjected to compressive loading. Int. J. Nonlinear Mech. 141, 103936 (2022)

    Article  Google Scholar 

  23. Jing, H.M., Gong, X.L., Wang, J., Wu, R.X., Huang, B.: An analysis of nonlinear beam vibrations with the extended Rayleigh-Ritz method. J. Appl. Comput. Mech. 8(4), 1299–1306 (2022)

    Google Scholar 

  24. He, J.H.: Variational approach for nonlinear oscillators. Chaos Solitons Fract. 34(5), 1430–1439 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Chowdhury, M.S.H., Hosen, M.A., Ahmad, K., Ali, M.Y., Ismail, A.F.: High-order approximate solutions of strongly nonlinear cubic-quantic Duffing oscillator based on the harmonic balance method. Results Phys. 7, 3962–3967 (2017)

    Article  Google Scholar 

  26. Wang, L.J., Zhang, H.Z., Hu, H., Zhu, M.Q.: An improved KBM method for solving nonlinear vibration equations. J. Vib. Shock 37(3), 165–170 (2018). (in Chinese)

    Google Scholar 

  27. Amore, P., Aranda, A.: Improved Lindstedt-Poincaré method for the solution of nonlinear problem. J. Sound Vib. 283(3), 1115–1136 (2005)

    Article  MATH  Google Scholar 

  28. Pirbodaghi, T., Hoseini, S.H., Ahmadiana, M.T., Farrahi, G.H.: Duffing equations with cubic and quantic nonlinearities. Comput. Math. Appl. 57(3), 500–506 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yuan, P.X., Li, Y.Q.: Approximate solutions of primary resonance for forced Duffing equation by means of the homotopy analysis method. Chin. J. Mech. Eng. 24(3), 501–506 (2011)

    Article  Google Scholar 

  30. Zúñiga, A.E.: A general solution of the Duffing equation. Nonlinear Dynam. 45(3–4), 227–235 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sah, S.M., Fiedler, B., Shayak, B., Rand, R.H.: Unbounded sequences of stable limit cycles in the delayed Duffing equation: an exact analysis. Nonlinear Dyn. 103, 503–515 (2021)

    Article  Google Scholar 

  32. Mehri, B., Ghorashi, M.: Periodically forced Duffing’s equation. J. Sound Vib. 169(3), 289–295 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Bayat, M., Pakar, I., Domairry, G.: Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review. Lat. Am. J. Solids Struct. 9(2), 1–93 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

Not applicable.

Funding

This research is supported by the National Natural Science Foundation of China (Grant 11672142) with additional support through the Technology Innovation 2025 Program (Grant 2019B10122) of the Municipality of Ningbo and National Scientific Research Project Cultivation Project (NZ22GJ007) by Ningbo Polytechnic.

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, Ningbo Polytechnic, 1069 Xinda Road, Beilun District, Ningbo, 315800, Zhejiang, China

    Rongxing Wu

  2. Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, 818 Fenghua Road, Ningbo, 315211, Zhejiang, China

    Rongxing Wu & Ji Wang

Authors
  1. Rongxing Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

JW is responsible for the conception, revision, and final approval of the paper. The mathematical formulation, calculation, drafting, and validation were completed and checked by RW.

Corresponding author

Correspondence to Ji Wang.

Ethics declarations

Conflict of interest

The authors declared no potential conflicts of interest concerning the research, authorship, and publication of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, R., Wang, J. The Extended Rayleigh–Ritz Method for Higher Order Approximate Solutions of Nonlinear Vibration Equations. Aerotec. Missili Spaz. 102, 155–160 (2023). https://doi.org/10.1007/s42496-023-00153-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42496-023-00153-w

Keywords

Search

Navigation