Nonlinear Dynamics

, Volume 49, Issue 1–2, pp 233–249 | Cite as

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

Original Article


An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.


Newton–harmonic balance (NHB) method Nonlinear stiffness Two-mass system Two-degree-of-freedom oscillation systems Duffing equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Huang, T.C.: Harmonic oscillations of nonlinear two-degree-of-freedom systems. J. Appl. Mech. 22, 107–110 (1955)MATHMathSciNetGoogle Scholar
  2. 2.
    Gilchrist, A.O.: The free oscillations of conservative quasilinear systems with two degrees of freedom. Int. J. Mech. Sci. 3, 286–311 (1961)CrossRefGoogle Scholar
  3. 3.
    Efstathiades, G.J.: Combination tones in single mode motion of a class of nonlinear systems with two degrees of freedom. J. Sound Vib. 34, 379–397 (1974)CrossRefMATHGoogle Scholar
  4. 4.
    Chen, G.: Applications of a generalized Galerkin's method to non-linear oscillations of two-degree-of-freedom systems. J. Sound Vib. 119, 225–242 (1987)CrossRefGoogle Scholar
  5. 5.
    Ladygina, Y.V., Manevich A.I.: Free oscillations of a non-linear cubic system with two degrees of freedom and close natural frequencies. J. Appl. Math. Mech. 57, 257–266 (1993)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cveticanin, L.: The motion of a two-mass system with non-linear connection. J. Sound Vib. 252, 361–369 (2002)CrossRefGoogle Scholar
  7. 7.
    Cveticanin, L.: Vibrations of a coupled two-degree-of-freedom system. J. Sound Vib. 247, 279–292 (2001)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Alexander, F.V., Richard, H.R.: Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities. Int. J. Non-Linear Mech. 39, 1079–1091 (2004)MATHCrossRefGoogle Scholar
  9. 9.
    Dimarogonas, A.D., Haddad, S.: Vibration for Engineers. Prentice-Hall, Englewood Cliffs, New Jersey (1992)MATHGoogle Scholar
  10. 10.
    Telli, S., Kopmaz, O.: Free vibrations of a mass grounded by linear and nonlinear springs in series. J. Sound Vib. 289, 689–710 (2006)CrossRefGoogle Scholar
  11. 11.
    Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)MATHGoogle Scholar
  12. 12.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)MATHGoogle Scholar
  13. 13.
    Mickens, R.E.: Oscillations in Planar Dynamic Systems. Word Scientific, Singapore (1996)MATHGoogle Scholar
  14. 14.
    Hagedorn, P.: Non-Linear Oscillations (Wolfram Stadler, trans.). Clarendon, Oxford (1988)Google Scholar
  15. 15.
    Bush, A.W.: Perturbation Methods for Engineers and Scientists. CRC Press, Boca Raton, Florida (1992)MATHGoogle Scholar
  16. 16.
    Agrwal, V.P., Denman, H.H.: Weighted linearization technique for period approximation in large amplitude nonlinear oscillations. J. Sound Vib. 99, 463–473 (1985)CrossRefGoogle Scholar
  17. 17.
    Cheung, Y.K., Chen, S.H., Lau, S.L.: A modified Lindstedt-Poincaré method for certain strongly non-linear oscillators. Int. J. Non-Linear Mech. 26, 367–378 (1991)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Qaisi, M.I.: A power series approach for the study of periodic motion. J. Sound Vib. 196, 401–406 (1996)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Liao, S.J., Chwang, A.T.: Application of homotopy analysis method in nonlinear oscillations. ASME J. Appl. Mech. 65, 914–922 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pathak, A., Mandal, S.: Classical and quantum oscillators of sextic and octic anharmonicities. Phys. Lett. A 298, 259–270 (2002)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lim, C.W., Wu, B.S., He, L.H.: A new approximate analytical approach for dispersion relation of the nonlinear Klein-Gordon equation. Chaos 11, 843–848 (2001)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lim, C.W., Wu, B.S.: A new analytical approach to the Duffing-harmonic oscillator. Phys. Lett. A 311, 365–373 (2003)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Wu, B.S., Lim, C.W.: Large amplitude non-linear oscillations of a general conservative system. Int. J. Non-Linear Mech. 39, 859–870 (2004)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lim, C.W., Lai, S.K., Wu, B.S.: Accurate higher-order analytical approximate solutions to large-amplitude oscillating systems with a general non-rational restoring force. Nonlinear Dyn. 42, 267–281 (2005)MATHCrossRefGoogle Scholar
  25. 25.
    Jackson, E.A.: Perspectives of Non-Linear Dynamics, Vol. 1. Cambridge Univeristy Press, Cambridge, UK (1989)Google Scholar
  26. 26.
    Lai, S.K., Lim, C.W.: Higher-order approximate solutions to a strongly nonlinear Duffing oscillator. Int. J. Comput. Methods Eng. Sci. Mech. 7, 201–208 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Building and ConstructionCity University of Hong KongKowloonP.R. China

Personalised recommendations