Skip to main content
SpringerLink
Log in

Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion

  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

A set of nonlinear differential equations is established by using Kane's method for the planar oscillation of flexible beams undergoing a large linear motion. In the case of a simply supported slender beam under certain average acceleration of base, the second natural frequency of the beam may approximate the tripled first one so that the condition of 3∶1 internal resonance of the beam holds true. The method of multiple scales is used to solve directly the nonlinear differential equations and to derive a set of nonlinear modulation equations forthe principal parametric resonance of the first mode combined with 3∶1 internal resonance between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability condition of the beam. The abundant nonlinear dynamic behaviors, such as various types of local bifurcations and chaos that do not appear for linear models, can be observed in the case studies. For a Hopf bifurcation, the 4-dimensional modulation equations are reduced onto the central manifold and the type of Hopf bifurcation is determined. As usual, a limit cycle may undergo a series of period-doubling bifurcations and become a chaotic oscillation at last.

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. Ho JYL. Direct path method for flexible multibody spacecraft dynamics.Journal of Spacecraft and Rockets, 1977, 14(1): 102–110

    Article  Google Scholar 

  2. Bodley CS, Devers AD, Park AC, et al. A digital computer program for the dynamic interaction simulation of controls and structure. NASA Technical Paper, 1978, 1219. 1–2

    Google Scholar 

  3. Simo JC, Quec VL. On the dynamics of flexible beams under large overall motions-the planer case: part I.ASME Journal of Applied Mechanics, 1986, 53(4): 849–854

    Article  MATH  Google Scholar 

  4. Simo JC, Quec VL. On the dynamics of flexible beams under large overall motions-the planer case: part II.ASME Journal of Applied Mechanics, 1986, 53(4): 855–863

    Article  MATH  Google Scholar 

  5. Boutaghou ZE, Erdman AG, Stolarski HK. Dynamics of flexible beams and plates in large overall motions.ASME Journal of Applied Mechanics, 1992, 59(4):991–999

    Article  MATH  Google Scholar 

  6. Ider SK Amirouche FML. Nonlinear modeling of flexible multibody systems dynamics subjected to variable constraints.ASME Journal of Applied Mechanics, 1989, 56(2): 444–450

    Article  MATH  Google Scholar 

  7. Wu SC, Haug EJ. Geometric non-linear substructuring for dynamics of flexible mechanical systems.International Journal for Numerical Methods in Engineering, 1988, 26(9): 2211–2226

    Article  MATH  Google Scholar 

  8. Kane TR, Ryan RR, Banerjee AK. Dynamics of a cantilever beam attached to a moving base.Journal of Guidance, Control, and Dynamics, 1987, 10(2): 139–151

    Google Scholar 

  9. Banerjee AK, Dickens JM. Dynamics of an arbitrary flexible body in large rotation and translation.Journal of Guidance, Control, and Dynamics, 1990, 13(2): 221–227

    Article  Google Scholar 

  10. Zhang DJ, Liu CQ, Huston RL. On the dynamic of an arbitrary flexible body with large overall motion: An integrated approach.Mechanics of Structures and Machines, 1995, 23(3): 417–435

    Google Scholar 

  11. Yoo HH, Ryan RR, Scott RA. Dynamics of flexible beams undergoing overall motions.Journal of Sound and Vibration, 1995, 181(2): 261–278

    Article  Google Scholar 

  12. Hyun SH, Yoo HH. Dynamic modeling and stability analysis of axially oscillating cantilever beams.Journal of Sound and Vibration, 1999, 228(3): 543–558

    Article  Google Scholar 

  13. Zavodney LD, Nayfeh AH. The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment.International Journal of Non-Linear Mechanics, 1989, 24(2): 105–125

    Article  MATH  Google Scholar 

  14. Nayfeh AH, Pai PF. Non-linear non-planar parametric responses of an inextensional beam.International Journal of Non-Linear Mechanics, 1989, 24(2): 139–158

    Article  MATH  Google Scholar 

  15. Esmailzadha E, Nakhaie-Jazer G. Periodic behaviors of a cantilever beam with end mass subjected to harmonic base excitation.International Journal of Non-Linear Mechanics, 1998, 33(4): 567–577

    Article  Google Scholar 

  16. Esmailzadha E, Jalili N. Parametric response of cantilever Timoshenko beams with tip mass under harmonic support motion.International Journal of Non-Linear Mechanics, 1998, 33(5): 765–781

    Article  MathSciNet  Google Scholar 

  17. Anderson TJ, Nayfeh AH, Balachandran. Experimental verfication of the importance of the nonlinear curvature in the response of a cantilever beam.ASME Journal of Vibration and Acoustics, 1996, 118(1): 21–27

    Google Scholar 

  18. Kar RC, Dwivedy SK. Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances.International Journal of Non-Linear Mechanics, 1999, 34(3): 515–529

    Article  MATH  Google Scholar 

  19. Dwivedy SK, Kar RC. Dynamics of a slender beam with an attached mass under combination parametric and internal resonances: Part I. Steady state response.Journal of Sound and Vibration, 1999, 221(5): 823–848

    Article  Google Scholar 

  20. Dwivedy SK, Kar RC. Dynamics of a slender beam with an attached mass under combination parametric and internal resonances: Part II. Periodic and chaotic responses.Journal of Sound and Vibration, 1999, 222(2):283–305

    Article  Google Scholar 

  21. Chin CM, Nayfeh AH. Three-to-one internal resonances in hinged-clamped beams.Nonlinear Dynamics, 1997, 12(2): 129–154

    Article  MATH  MathSciNet  Google Scholar 

  22. Chin CM, Nayfeh AH. Three-to-one internal resonances in parametrically excited hinged-clamped beams.Nonlinear Dynamics, 1999, 20(2): 131–158

    Article  MATH  MathSciNet  Google Scholar 

  23. Feng ZH, Hu HY. Dynamic stability of a slender beam with internal resonance under a large linear motion.Acta Mechanica Sinica, 2002, 34(3): 389–400 (in Chinese)

    Google Scholar 

  24. Feng ZH, Hu HY. Largest Lyapunov exponents and almost certain stability analysis of slender beams under a large linear motion of basement subject to narrow-band random parametric excitation.Journal of Sound and Vibration, 2002, 257(4): 733–752

    Article  Google Scholar 

  25. Feng ZH, Hu HY. Nonlinear dynamics modeling and periodic vibration of a cantilever beam subjected to axial movement of basement.Acta Mechanica Solida Sinica, 2002, 15(2): 133–139

    Article  Google Scholar 

  26. Hu HY. Applied Nonlinear Dynamics. Beijing: Aviation Industry Press, 2000

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016, Nanjing, China

    Feng Zhihua & Hu Haiyan

Authors
  1. Feng Zhihua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hu Haiyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The project supported by the Fund of Doctorial Program, Ministry of Education of China (20020287011)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhihua, F., Haiyan, H. Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion. Acta Mech Sinica 19, 355–364 (2003). https://doi.org/10.1007/BF02487813

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02487813

Key Words

Search

Navigation