Skip to main content
SpringerLink
Log in

Nonlinear dynamics of a three-beam structure with attached mass and three-mode interactions

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Nonlinear dynamics of elastic structures with two-mode interactions have been extensively studied in the literature. In this work, nonlinear forced response of elastic structures with essential inertial nonlinearities undergoing three-mode interactions is studied. More specifically, a three-beam structural system with attached mass is considered, and its multidegree-of-freedom discretized model for the structure undergoing planar motions is carefully studied. Linear modal characteristics of the structure with uniform beams depend on the length ratios of the three beams, the mass of the particle relative to that of the structure, and the location of the mass particle along the beams. The discretized model is studied for both external and parametric resonances for parameter combinations resulting in three-mode interactions. For the external excitation case, focus is on the system with 1:2:3 internal resonances with the external excitation frequency near the middle natural frequency. For the case of the structure with 1:2:5 internal resonances, the problem involving simultaneous principal parametric resonance of the middle mode and a combination resonance between the lowest and the highest modal frequencies is investigated. This case requires a higher-order approximation in the method of multiple time scales. For both cases, equilibrium and bifurcating solutions of the slow-flow equations are studied in detail. Many pitchfork, saddle-node, and Hopf bifurcations appear in the amplitude response of the three-beam structure, thus resulting in complex multimode responses in different parameter regions.

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

Similar content being viewed by others

References

  1. Apiwattanalunggarn, P., Shaw, S., Pierre, C.: Component mode synthesis using nonlinear normal modes. Nonlinear Dyn. 41, 17–46 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ashworth, R.P., Barr, A.D.S.: The resonances of structures with quadratic inertial nonlinearity under direct and parametric harmonic excitation. J. Sound Vib. 118, 47–68 (1987)

    Article  MathSciNet  Google Scholar 

  3. Bajaj, A.K., Chang, S.I., Johnson, J.M.: Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system. Nonlinear Dyn. 5(4), 433–457 (1994)

    Article  Google Scholar 

  4. Balachandran, B., Nayfeh, A.H.: Nonlinear motions of beam-mass structure. Nonlinear Dyn. 1, 39–61 (1990)

    Article  Google Scholar 

  5. Bux, S.L., Roberts, J.W.: Non-linear vibratory interactions in systems of coupled beams. J. Sound Vib. 104(3), 497–520 (1986)

    Article  Google Scholar 

  6. Cartmell, M.P., Roberts, J.W.: Simultaneous combination resonances in an autoparametrically resonant system. J. Sound Vib. 123(1), 81–101 (1988)

    Article  MathSciNet  Google Scholar 

  7. Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968)

    Article  MATH  Google Scholar 

  8. Crespo da Silva, M.R.M.: A reduced-order analytical model for the nonlinear dynamics of a class of flexible multi-beam structures. Int. J. Solids Struct. 35(25), 3299–3315 (1998)

    Article  MATH  Google Scholar 

  9. Doedel, E., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.: AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with Hom Cont) (1997). ftp://ftp.cs.concordia.ca/pub/doedel/auto/auto.ps.Z

  10. Dwivedy, S.K., Kar, R.C.: Nonlinear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions. Int. J. Non-Linear Mech. 36, 927–945 (2001)

    Article  MATH  Google Scholar 

  11. Dwivedy, S.K., Kar, R.C.: Simultaneous combination, principal parametric and internal resonances in a slender beam with a lumped mass: Three-mode interactions. J. Sound Vib. 242(1), 27–46 (2001)

    Article  Google Scholar 

  12. Dwivedy, S.K., Kar, R.C.: Simultaneous combination and 1:3:5 internal resonances in a parametrically excited beam-mass system. Int. J. Non-Linear Mech. 38(4), 585–596 (2003)

    Article  MATH  Google Scholar 

  13. Dwivedy, S.K., Kar, R.C.: Nonlinear dynamics of a cantilever beam carrying an attached mass with 1:3:9 internal resonances. Nonlinear Dyn. 31(1), 49–72 (2003)

    Article  MATH  Google Scholar 

  14. El-Bassiouny, A.F.: Three-mode interaction in harmonically excited system with cubic nonlinearities. Appl. Math. Comput. 139, 201–230 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hale, A.L., Meirovitch, L.: A general dynamics synthesis method for the dynamic simulation of complex structures. J. Sound Vib. 69, 309–326 (1980)

    Article  MATH  Google Scholar 

  16. Hurty, W.: Vibration of structural systems by component mode synthesis. Proc. Am. Soc. Civ. Eng. 85(4), 51–69 (1960)

    Google Scholar 

  17. Hurty, W.: Dynamic analysis of structural systems using component modes. AIAA J. 3(4), 678–685 (1965)

    Article  Google Scholar 

  18. Ibrahim, R.A.: Multiple internal resonance in a structure-liquid system. J. Eng. Ind. 98, 1092–1098 (1976)

    Google Scholar 

  19. Ibrahim, R.A., Barr, A.D.S.: Autoparametric resonance in a structure containing a liquid, part II: Three mode interaction. J. Sound Vib. 42(2), 181–200 (1975)

    Article  MATH  Google Scholar 

  20. Kar, R.C., Dwivedy, S.K.: Nonlinear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances. Int. J. Non-Linear Mech. 34, 515–529 (1999)

    Article  MATH  Google Scholar 

  21. Mahmoodi, S.N., Khadem, S.E., Rezaee, M.: Analysis of non-linear mode shapes and natural frequencies of continuous damped systems. J. Sound Vib. 275, 283–298 (2004)

    Article  Google Scholar 

  22. Meirovitch, L.: Principles and Techniques of Vibrations. McGraw Hill, New York (1997)

    Google Scholar 

  23. Meirovitch, L., Hale, A.L.: A general dynamics synthesis for structures with discrete substructures. J. Sound Vib. 85, 445–457 (1982)

    Article  MATH  Google Scholar 

  24. Nayfeh, A.H.: Nonlinear Interactions. Wiley Interscience, New York (2000)

    MATH  Google Scholar 

  25. Nayfeh, A.H.: Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn. 40, 61–102 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  27. Nayfeh, T.A., Asrar, W., Nayfeh, A.H.: Three-mode interaction in harmonically excited systems with quadratic nonlinearities. Nonlinear Dyn. 3, 385–410 (1992)

    Article  Google Scholar 

  28. Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164, 85–124 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Soares, M.E.S., Mazzilli, C.E.N.: Nonlinear normal modes of planar frames discretized by the finite element method. Comput. Struct. 77, 485–493 (2000)

    Article  Google Scholar 

  30. Wang, F.: Nonlinear normal modes and nonlinear model reduction for discrete and multi-component continuous systems. Doctoral Dissertation, Purdue University, West Lafayette, IN (2008)

  31. Wang, F., Bajaj, A.: Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure. Nonlinear Dyn. 47, 25–47 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Warminski, J., Cartmell, M.P., Bochenski, M., Ivanov, I.: Analytical and experimental investigations of an autoparametric beam structure. J. Sound Vib. 315, 486–508 (2008)

    Article  Google Scholar 

  33. Xie, W.C., Lee, H.P., Lim, S.P.: Nonlinear dynamic analysis of MEMS switches by nonlinear modal analysis. Nonlinear Dyn. 31, 243–256 (2003)

    Article  MATH  Google Scholar 

  34. Zavodney, L.D., Nayfeh, A.H.: The nonlinear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment. Int. J. Non-Linear Mech. 24, 105–125 (1989)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical & Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, 62026, USA

    Fengxia Wang

  2. School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN, 47907-2088, USA

    Anil K. Bajaj

Authors
  1. Fengxia Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anil K. Bajaj
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anil K. Bajaj.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, F., Bajaj, A.K. Nonlinear dynamics of a three-beam structure with attached mass and three-mode interactions. Nonlinear Dyn 62, 461–484 (2010). https://doi.org/10.1007/s11071-010-9734-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-010-9734-2

Keywords

Search

Navigation