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Two-to-one internal resonances in a shallow curved beam resting on an elastic foundation

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Summary

Vibrations of shallow curved beams are investigated. The rise function of the beam is assumed to be small. Sinusoidal and parabolic curvature functions are examined. The immovable end conditions result in mid-plane stretching of the beam which leads to nonlinearities. The beam is resting on an elastic foundation. The method of multiple scales, a perturbation technique, is used in search of approximate solutions of the problem. Two-to-one internal resonances between any two modes of vibration are studied. Amplitude and phase modulation equations are obtained. Steady state solutions and stability are discussed, and a bifurcation analysis of the amplitude and phase modulation equations are given. Conditions for internal resonance to occur are discussed, and it is found that internal resonance is possible for the case of parabolic curvature but not for that of sinusoidal curvature.

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References

  • L. W. Rehfield (1974) ArticleTitleNonlinear flexural oscillations of shallow arches AIAA J. 12 91–93 Occurrence Handle0284.73060 Occurrence Handle10.2514/3.49158

    Article  MATH  Google Scholar 

  • P. N. Singh S. M. J. Ali (1975) ArticleTitleNonlinear vibration of a moderately thick shallow clamped arch J. Sound Vibr. 41 275–282

    Google Scholar 

  • Yamaki, N., Mori, A.: Nonlinear vibrations of a clamped beam with initial deflection and initial axial displacement, Part 1: Theory. J. Sound Vibr. 71, 333—346 (1980).

  • W.-M. Tien N. Sri Namachchivaya A. K. Bajaj (1994) ArticleTitleNon-linear dynamics of a shallow arch under periodic excitation-I. 1:2 internal resonance Int. J. Non-linear Mech. 29 349–366 Occurrence Handle0808.73043 Occurrence Handle10.1016/0020-7462(94)90007-8

    Article  MATH  Google Scholar 

  • W.-M. Tien N. Sri Namachchivaya A. K. Bajaj (1994) ArticleTitleNon-linear dynamics of a shallow arch under periodic excitation-II. 1:1 internal resonance Int. J. Non-linear Mech. 29 367–386 Occurrence Handle0808.73044 Occurrence Handle10.1016/0020-7462(94)90008-6

    Article  MATH  Google Scholar 

  • H. R. Öz M. Pakdemirli E. Özkaya M. Yılmaz (1998) ArticleTitleNon-linear vibrations of a slightly curved beam resting on a non-linear elastic foundation J. Sound Vibr. 212 IssueID2 295–309 Occurrence Handle10.1006/jsvi.1997.1428

    Article  Google Scholar 

  • Q. Bi H. H. Dai (2000) ArticleTitleAnalysis of non-linear dynamics and bifurcations of a shallow arch subjected to periodic excitation with internal resonance J. Sound Vibr. 233 IssueID4 557–571 Occurrence Handle10.1006/jsvi.1999.2813

    Article  Google Scholar 

  • M. Pakdemirli (1994) ArticleTitleA comparison of two perturbation methods for vibrations of systems with quadratic and cubic nonlinearities Mech. Res. Comm. 21 203–208 Occurrence Handle0811.73036 Occurrence Handle1262532 Occurrence Handle10.1016/0093-6413(94)90093-0

    Article  MATH  MathSciNet  Google Scholar 

  • M. Pakdemirli H. Boyacı (1995) ArticleTitleComparison of direct-perturbation methods with discretization-perturbation methods for nonlinear vibrations J. Sound Vibr. 186 837–845 Occurrence Handle1049.74609 Occurrence Handle10.1006/jsvi.1995.0491

    Article  MATH  Google Scholar 

  • M. Pakdemirli H. Boyacı (1996) ArticleTitleVibrations of continuous systems having arbitrary quadratic and cubic nonlinearities Appl. Mech. Engng. 1 445–463 Occurrence Handle0884.73032

    MATH  Google Scholar 

  • H. Boyacı M. Pakdemirli (1997) ArticleTitleA comparison of different versions of the method of multiple scales for partial differential equations J. Sound Vibr. 204 595–607 Occurrence Handle10.1006/jsvi.1997.0951

    Article  Google Scholar 

  • M. Pakdemirli H. Boyacı M. Yılmaz (1997) ArticleTitleContinuous systems with odd nonlinearities: a general solution procedure Math. Comput. Appl. 2 85–90 Occurrence Handle0913.73043

    MATH  Google Scholar 

  • M. Pakdemirli H. Boyacı (1999) ArticleTitleA comparison of different versions of the method of multiple scales for an arbitrary model of odd nonlinearities Math. Comput. Appl. 4 273–282 Occurrence Handle1071.74716

    MATH  Google Scholar 

  • M. Pakdemirli H. Boyacı (1997) ArticleTitleA generalized approach to coupled nonlinear vibrations of continuous systems Math. Comput. Appl. 2 141–154 Occurrence Handle0920.73216

    MATH  Google Scholar 

  • M. Pakdemirli (2001) ArticleTitleVibrations of continuous systems with a general operator notation suitable for perturbative calculations J. Sound Vibr. 246 841–851 Occurrence Handle10.1006/jsvi.2001.3691

    Article  Google Scholar 

  • M. Pakdemirli (2001) ArticleTitleA general solution procedure for coupled systems with arbitrary internal resonances Mech. Res. Comm. 28 617–622 Occurrence Handle1016.70015 Occurrence Handle10.1016/S0093-6413(02)00213-6

    Article  MATH  Google Scholar 

  • W. Lacarbonara (1999) ArticleTitleDirect treatment and discretizations of non-linear spatially continuous systems J. Sound Vibr. 221 849–866 Occurrence Handle1677937 Occurrence Handle10.1006/jsvi.1998.2049

    Article  MathSciNet  Google Scholar 

  • A. H. Nayfeh (1998) ArticleTitleReduced order models of weakly nonlinear spatially continuous systems Nonlinear Dyn. 16 105–125 Occurrence Handle0904.73044 Occurrence Handle1630145 Occurrence Handle10.1023/A:1008281121523

    Article  MATH  MathSciNet  Google Scholar 

  • A. H. Nayfeh (2000) Nonlinear interactions Wiley-Interscience New York Occurrence Handle0995.70001

    MATH  Google Scholar 

  • M. Pakdemirli E. Özkaya (2004) ArticleTitleTwo-to-one internal resonances in continuous systems with arbitrary quadratic nonlinearities Ara. J. Sci. Engng. 29 99–112

    Google Scholar 

  • W. Lacarbonara H. N. Arafat A. H. Nayfeh (2005) ArticleTitleNonlinear interactions in imperfect beams at veering Int. J. Non-Linear Mech. 40 987–1003 Occurrence Handle10.1016/j.ijnonlinmec.2004.10.006 Occurrence Handle05138642

    Article  MATH  Google Scholar 

  • A. H. Nayfeh (1981) Introduction to perturbation techniques Wiley-Interscience New York Occurrence Handle0449.34001

    MATH  Google Scholar 

  • A. H. Nayfeh D. T. Mook (1979) Nonlinear oscillations John Wiley New York Occurrence Handle0418.70001

    MATH  Google Scholar 

  • A. H. Nayfeh (1989) Application of the method of multiple scales to nonlinearly coupled oscillators J. O. Hirschfelder R. E. Wyatt (Eds) Lasers, molecules and methods John Wiley and Sons New York

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Gaziantep, 27310, Turkey

    H. R. Öz

  2. Department of Mechanical Engineering, Celal Bayar University, Muradiye Manisa, 45140, Turkey

    M. Pakdemirli

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  1. H. R. Öz
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  2. M. Pakdemirli
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Correspondence to M. Pakdemirli.

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Öz, H.R., Pakdemirli, M. Two-to-one internal resonances in a shallow curved beam resting on an elastic foundation. Acta Mechanica 185, 245–260 (2006). https://doi.org/10.1007/s00707-006-0352-5

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  • DOI: https://doi.org/10.1007/s00707-006-0352-5

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