Nonlinear Dynamics

, Volume 40, Issue 1, pp 61–102 | Cite as

Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging

Article

Abstract

I compare application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Three implementations of the method of multiple scales are considered, namely, application of the method to the system equations expressed as second-order equations, as first-order equations, and in complex-variable form. I show that all of these methods produce the same modulation equations.

I address the problem of determining higher-order approximate solutions of the Duffing equation in the case of primary resonance. I show that the conclusions of Rahman and Burton that the method of multiple scales, the generalized method of averaging, and Lie series and transforms might lead to incorrect results, in that spurious solutions occur and the obtained frequency–response curves bear little resemblance to the actual response, is the result of their using parameter values for which the neglected terms are the same order as the retained terms. I show also that spurious solutions cannot be avoided, in general, in any consistent expansion and their presence does not constitute a limitation of the methods. In particular, I show that, for the Duffing equation, the second-order frequency–response equation does not possess spurious solutions for the case of hardening nonlinearity, but possesses spurious solutions for the case of softening nonlinearity. For sufficiently small nonlinearity, the spurious solutions are far removed from the actual response. But as the strength of the nonlinearity increases, these solutions move closer to the backbone and eventually distort it. This is not a drawback of the perturbation methods but an indication of an application of the analysis for parameter values outside the range of validity of the expansion.

Also, I address the problem of obtaining non-Hamiltonian modulation equations in the application of the method of multiple scales to multi-degree-of-freedom Hamiltonian systems written as second-order equations in time and how this problem can be overcome by attacking the state-space form of the governing equations. Moreover, I show that application of a variation of the method of Rahman and Burton to multi-degree-of-freedom systems leads to results that do not agree with those obtained with the generalized method of averaging.

Keywords

higher-order approximations method of averaging method of multiple scales perturbation methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nayfeh, A. H., ‘The response of single-degree-of-freedom systems with quadratic and cubic nonlinearity to a subharmonic excitation’, Journal of Sound and Vibration 89, 1983, 457–470.Google Scholar
  2. 2.
    Nayfeh, A. H., ‘Combination resonances in the nonlinear response of bowed structures to a harmonic excitation’, Journal of Sound and Vibration 90, 1983, 457–470.Google Scholar
  3. 3.
    Nayfeh, A. H., ‘Combination tones in the response of single-degree-of-freedom systems with quadratic and cubic nonlinearities’, Journal of Sound and Vibration 92, 1984, 379–386.Google Scholar
  4. 4.
    Nayfeh, A. H., ‘Quenching of primary resonances by a superharmonic resonance’, Journal of Sound and Vibration 92, 1984, 363–377.Google Scholar
  5. 5.
    Nayfeh, A. H., ‘Quenching of a primary resonance by a combination resonance of the additive or difference type’, Journal of Sound and Vibration 97, 1984, 65–73.Google Scholar
  6. 6.
    Nayfeh, A. H., ‘Topical course on nonlinear dynamics’, in Perturbation Methods in Nonlinear Dynamics, Societa Italiana di Fisica, Santa Margherita di Pula, Sardinia, 1985.Google Scholar
  7. 7.
    Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973.Google Scholar
  8. 8.
    Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.Google Scholar
  9. 9.
    Nayfeh, A. H. and Khdeir, A. A., ‘Nonlinear rolling of ships in regular beam seas’, International Shipbuilding Progress 33(379), 1986, 40–49.Google Scholar
  10. 10.
    Nayfeh, A. H. and Khdeir, A. A., ‘Nonlinear rolling of biased ships in regular beam waves’, International Shipbuilding Progress 33(381), 1986, 84–93.Google Scholar
  11. 11.
    Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.Google Scholar
  12. 12.
    Nayfeh, A. H. and Chin, C.-M., Perturbation Methods with Mathematica, Dynamics Press, Virginia, 1999; http://www.esm.vt.edu/~anayfeh/.Google Scholar
  13. 13.
    Nayfeh, A. H. and Chin, C.-M., Perturbation Methods with Maple, Dynamics Press, Virginia, 1999; http://www.esm.vt.edu/~anayfeh/.Google Scholar
  14. 14.
    Rahman, Z. and Burton, T. D., ‘Large amplitude primary and superharmonic resonances in the Duffing oscillator’, Journal of Sound and Vibration 110, 1986, 363–380.Google Scholar
  15. 15.
    Rahman, Z. and Burton, T. D., ‘On higher order method of multiple scales in nonlinear oscillations-periodic steady state response’, Journal of Sound and Vibration 133, 1989, 369–379.Google Scholar
  16. 16.
    Luongo, A., Rega, G., and Vestroni, F., ‘On nonlinear dynamics of planar shear indeformable beams’, Journal of Applied Mechanics 53, 1986, 619.Google Scholar
  17. 17.
    Hassan, A., ‘Use of transformations with the higher order method of multiple scales to determine the steady state periodic response of harmonically excited nonlinear oscillations. Part I. Transformation of derivative’, Journal of Sound and Vibration 178, 1994, 21–40.Google Scholar
  18. 18.
    Hassan, A., ‘Use of transformations with the higher order method of multiple scales to determine the steady state periodic response of harmonically excited nonlinear oscillations. Part II. Transformation of detuning’, Journal of Sound and Vibration 178, 1994, 1–19.Google Scholar
  19. 19.
    Lee, C. L. and Lee, C. T., ‘A higher order method of multiple scales’, Journal of Sound and Vibration 202, 1997, 284–287.Google Scholar
  20. 20.
    Lee, C. L. and Perkins, N. C., ‘Nonlinear oscillations of suspended cables containing a two-to-one internal resonance’, Nonlinear Dynamics 3, 1992, 465–490.Google Scholar
  21. 21.
    Benedettini, F., Rega, G., and Alaggio, R., ‘Nonlinear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions’, Journal of Sound and Vibration 182, 1995, 775–798.Google Scholar
  22. 22.
    Pan, R. and Davies, H. G., ‘Responses of a nonlinearly coupled pitch-roll ship model under harmonic excitation’, Nonlinear Dynamics 9, 1996, 349–368.Google Scholar
  23. 23.
    Boyaci, H. and Pakdemirli, M., ‘A comparison of different versions of the method of multiple scales for partial differential equations’, Journal of Sound and Vibration 204, 1997, 595–607.Google Scholar
  24. 24.
    Luongo, A. and Paolone, A., ‘On the reconstitution problem in the multiple time-scale method’, Nonlinear Dynamics 19, 1999, 133–156.Google Scholar
  25. 25.
    Cartmell, M. P., Ziegler, S. W., Khanin, R., and Forehand, D. I. M., ‘Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems’, Applied Mechanics Reviews 56, 2003, 155–492.Google Scholar
  26. 26.
    Rega, G., Lacarbonara, W., Nayfeh, A. H., and Chin, C. M., ‘Multiple resonances in suspended cables: Direct versus reduced-order models’, International Journal of Non-Linear Mechanics 34, 1999, 901–924.Google Scholar
  27. 27.
    Nayfeh, A. H., Nonlinear Interactions, Wiley, New York, 2000.Google Scholar
  28. 28.
    Nayfeh, A. H., Arafat, H. N., Chin, C.-M., and Lacarbonara, W., ‘Multimode interactions in suspended cables’, Journal of Vibration and Control 8(3), 2002, 337–387.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Engineering Science and Mechanics, MC 0219Virginia Polytechnic Institute and State UniversityBlacksburgU.S.A.

Personalised recommendations