The European Physical Journal Special Topics

, Volume 228, Issue 9, pp 1781–1792 | Cite as

Analytical solutions of periodic motions in a first-order quadratic nonlinear system

  • Guopeng Zhou
  • Peng JinEmail author
  • Fang Zhou
  • Xin Xia
Regular Article Topical issue
Part of the following topical collections:
  1. Periodic Motions and Chaos in Nonlinear Dynamical Systems


In this paper, analytical solutions of period-1 motion of a first-order quadratic nonlinear system with both parametric excitation and external excitation are investigated through the generalized harmonic balance method. The analytical solutions of the system are obtained via solving the coefficients of all harmonic terms at the equilibrium position. The precision of analytical solutions is guaranteed via convergence study of harmonic balance terms. Stability analysis is carried out via eigenvalue analysis. The analytical solutions are different from the perturbation analysis solutions. Moreover, the trajectories of periodic motions obtained from analytical solutions can better explain the dynamics of the system. To verify the accuracy of analytical solutions, numerical simulation are performed and the simulation results are compared with the analytical solutions. The harmonic spectra are also presented to illustrate of the contribution of each harmonic term on a periodic motion.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.L. Langrage, in in Mécanique analytique (Albert Blanchard, Paris, 1788), Vol. 2Google Scholar
  2. 2.
    H. Poincare, inMéthodes nouvelles de la mécanique céleste (Gauthier-Villars, Paris, 1899), Vol. 3Google Scholar
  3. 3.
    B. van der Pol, Radio Rev. 1, 701 (1920)Google Scholar
  4. 4.
    R.H. Rand, D. Armbruster, inPerturbation methods, bifurcation theory, and computer algebra (Springer-Verlag, New York, 1987), Vol. 65Google Scholar
  5. 5.
    S. Natsiavas, J. Sound Vib. 134, 315 (1989)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    V.T. Coppola, R.H. Rand, Acta Mech. 81, 125 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    K.L. Janicki, W. Szempliska-Stupnicka, J. Sound Vib. 180, 253 (1995)ADSCrossRefGoogle Scholar
  8. 8.
    A.C.J. Luo, R.P.S. Han, J. Franklin Inst. 334B, 447 (1997)CrossRefGoogle Scholar
  9. 9.
    Z.K. Peng, Z.Q. Lang, J. Sound Vib. 311, 56 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    A.C.J. Luo, Continuous dynamical systems (HEP-L/H Scientific, Beijing/Glen-Carbon, IL, 2012)Google Scholar
  11. 11.
    A.C.J. Luo, J.Z. Huang, J. Vib. Control 22, 1250093 (2012)Google Scholar
  12. 12.
    A.C.J. Luo, J.Z. Huang, Int. J. Bifurc. Chaos 23, 1350086 (2013)CrossRefGoogle Scholar
  13. 13.
    A.C.J. Luo, Int. J. Bifurc. Chaos 24, 1430013 (2014)CrossRefGoogle Scholar
  14. 14.
    A.C.J. Luo, B. Yu, Int. J. Bifurc. Chaos 25, 1550179 (2015)CrossRefGoogle Scholar
  15. 15.
    Y.Y. Xu, A.C.J. Luo, Z.B. Chen, Chaos Solitons Fractals 97, 1 (2017)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Hubei University of Science and TechnologyXianningP.R. China

Personalised recommendations