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Stability analysis for nonplanar free vibrations of a cantilever beam by using nonlinear normal modes

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Abstract

Stability analysis of nonplanar free vibrations of a cantilever beam is made by using the nonlinear normal mode concept. Assuming nonplanar motion of the beam, we introduce a nonlinear two-degree-of-freedom model by using Galerkin’s method based on the first mode in each direction. The system turns out to have two normal modes. Using Synge’s stability concept, we examine the stability of each mode. In order to check the validity of the stability criterion obtained analytically, we plot a Poincaré map of the motions neighboring on each mode obtained numerically. It is found that the maps agree with the stability criterion obtained analytically.

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References

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

    MATH  Google Scholar 

  2. Pak, C.H.: Nonlinear Normal Mode Dynamics. Inha University Press, Inha (1999)

    Google Scholar 

  3. Rosenberg, R.M., Atkinson, C.P.: On the natural modes and their stability in nonlinear two-degree of freedom systems. ASME J. Appl. Mech. 26, 377–385 (1959)

    MATH  MathSciNet  Google Scholar 

  4. Rosenberg, R.M., Kuo, J.K.: Nonsimilar normal mode vibrations of nonlinear systems having two degrees of freedom. ASME J. Appl. Mech. 31, 283–290 (1964)

    MathSciNet  Google Scholar 

  5. Rosenberg, R.M.: Steady-state forced vibrations. Int. J. Non-Linear Mech. 1, 95–108 (1966)

    Article  MATH  Google Scholar 

  6. Rosenberg, R.M.: On nonlinear vibration of systems with many degrees of freedom. Adv. Appl. Mech. 9, 155–242 (1966)

    Article  Google Scholar 

  7. Pak, C.H., Rosenberg, R.M.: On the existence of normal mode vibrations in nonlinear systems. Quart. Appl. Math. 24, 177–193 (1968)

    MathSciNet  Google Scholar 

  8. Pak, C.H.: On the stability behavior of bifurcated normal modes in coupled nonlinear systems. ASME J. Appl. Mech. 56, 155–161 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Pak, C.H., Rand, R.H., Moon, F.C.: Free vibrations of a thin elastica by normal modes. Nonlinear Dyn. 3, 347–364 (1992)

    Article  Google Scholar 

  10. Pak, C.H., Shin, H.J.: On bifurcation modes and forced responses in coupled nonlinear oscillators. J. Theor. Appl. Mech. 1, 29–67 (1995)

    Google Scholar 

  11. Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley, New York (1996)

    MATH  Google Scholar 

  12. Rand, R.H.: The geometrical stability of nonlinear normal modes in two degree of freedom systems. Int. J. Non-Linear Mech. 8, 161–168 (1973)

    Article  MATH  Google Scholar 

  13. Pak, C.H.: Synge’s concept of stability applied to nonlinear normal modes. Int. J. Nonlinear Mech. 41, 657–664 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Pak, C.H.: On the coupling of nonlinear normal modes. Int. J. Non-Linear Mech. 41, 716–725 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Synge, J.L.: On the geometry of dynamics. Philos. Trans. Roy. Soc. London Ser. A 226, 31–106 (1926)

    Google Scholar 

  16. Hamdan, M.N., Al-Qaisia, A.A., Al-Bedoor, B.O.: Comparison of analytical techniques for nonlinear vibrations of a parametrically excited cantilever. Int. J. Mech. Sci. 43, 1521–1542 (2001)

    Article  MATH  Google Scholar 

  17. 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, 585–596 (2003)

    Article  MATH  Google Scholar 

  18. Nayfeh, A.H., Pai, P.F.: Nonlinear nonplanar parametric responses of an inextensional beam. Int. J. Non-Linear Mech. 24, 139–158 (1989)

    Article  MATH  Google Scholar 

  19. Crespo da Silva, M.R.M.: Equations for nonlinear analysis of 3D motions of beams. Am. Soc. Mech. Eng. Appl. Mech. Rev. 44, s51–s59 (1991)

    MathSciNet  Google Scholar 

  20. Crespo da Silva, M.R.M., Zaretzky, C.L.: Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics, I: primary resonance. Nonlinear Dyn. 5, 3–23 (1994)

    Google Scholar 

  21. Zaretzky, C.L., Crespo da Silva, M.R.M.: Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics, II: combination resonance. Nonlinear Dyn. 5, 161–180 (1994)

    Google Scholar 

  22. Crespo da Silva, M.R.M., Glynn, C.C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams, I: equation of motion. J. Struct. Mech. 6, 437–448 (1978)

    Google Scholar 

  23. Crespo da Silva, M.R.M., Glynn, C.C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams, II: forced motions. J. Struct. Mech. 6, 449–461 (1978)

    Google Scholar 

  24. Lee, W.K., Kim, C.H.: Combination resonances of a circular plate with three-mode interaction. ASME J. Appl. Mech. 62, 1015–1022 (1995)

    Article  MATH  Google Scholar 

  25. Yeo, M.H., Lee, W.K.: Corrected solvability conditions for nonlinear asymmetric vibrations of a circular plate. J. Sound Vib. 257, 653–665 (2002)

    Article  Google Scholar 

  26. Lee, W.K., Yeo, M.H.: Nonlinear interactions in asymmetric vibrations of a circular plate. J. Sound Vib. 263, 1017–1030 (2003)

    Article  Google Scholar 

  27. Lee, W.K., Yeo, M.H., Samoilenko, S.B.: The effect of the number of the nodal diameters on nonlinear interactions in two asymmetric vibration modes of a circular plate. J. Sound Vib. 268, 1013–1023 (2003)

    Article  Google Scholar 

  28. Lee, W.K.: Modal interactions in asymmetric vibrations of circular plates. In: Sun, J.-Q., Luo, A.C.J. (eds.) Advances in Nonlinear Science and Complexity, vol. 1: Bifurcation and Chaos in Complex Systems, Elsevier, Amsterdam (2006)

    Google Scholar 

  29. Yeo, M.H., Lee, W.K.: Evidences of global bifurcations of an imperfect circular plate. J. Sound Vib. 293, 138–155 (2006)

    Article  Google Scholar 

  30. Samoylenko, S.B., Lee, W.K.: Global bifurcations and chaos in a harmonically excited and undamped circular plate. Nonlinear Dyn. 47, 405–419 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  32. Nayfeh, A.H., Lacarbonara, W.: On the discretization of spatially continuous systems with quadratic and cubic nonlinearities. JSME Int. J. Ser. C 41, 510–531 (1998)

    Google Scholar 

  33. Nayfeh, A.H., Lacarbonara, W.: On the distributed-parameter systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 13, 203–220 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  34. Nayfeh, A.H., Nayfeh, J.F., Mook, D.T.: On methods for continuous systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 3, 145–162 (1992)

    Article  Google Scholar 

  35. Nayfeh, A.H.: Reduced-order models of weakly nonlinear spatially continuous systems. Nonlinear Dyn. 16, 105–125 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  36. Abe, A.: On nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. Int. J. Non-Linear Mech. 41, 873–879 (2006)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mechanical Engineering, Yeungnam University, Gyongsan, 712-749, Republic of Korea

    W. K. Lee

  2. Mobile Communication Division, Telecommunication Network Business, Samsung Electronics Co. Ltd., Kumi, 730-350, Republic of Korea

    K. S. Lee

  3. Department of Mechanical Engineering, Inha University, Inchon, 402-751, Republic of Korea

    C. H. Pak

Authors
  1. W. K. Lee
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  2. K. S. Lee
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  3. C. H. Pak
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Correspondence to W. K. Lee.

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Lee, W.K., Lee, K.S. & Pak, C.H. Stability analysis for nonplanar free vibrations of a cantilever beam by using nonlinear normal modes. Nonlinear Dyn 52, 217–225 (2008). https://doi.org/10.1007/s11071-007-9273-7

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  • DOI: https://doi.org/10.1007/s11071-007-9273-7

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