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Analysis and simulations of vibrations of a beam with a slider

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Abstract

A model for vibrations of a beam with a slider is derived, analysed and numerically simulated. It describes a viscoelastic beam that is clamped at one end to a vibrating device, while the other end moves between two stops attached to a slider. The contact is described by the normal compliance or by the Signorini conditions. The existence of weak solutions is established using the theory of set-valued pseudomonotone operators. The model is discretized using fourth-order spatial discretization, the solutions are numerically simulated and their results presented. The dynamics of the vibrations are depicted and so are the noise characteristics of the system.

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References

  1. K. L. Kuttler and M. Shillor, Vibrations of a beam between two stops. Dyn. Contin. Discrete Impuls. Sys. Ser B 8 (2001) 93–110.

    Google Scholar 

  2. Y. Dumont, Vibrations of a beam between stops: simulations and comparison of several numerical schemes. in Math. Comput. in Simulation, to appear.

  3. Y. Dumont and M. Shillor, Simulations of beam's vibrations between stops. (1999) Preprint.

  4. K. L. Kuttler, A. Park, M. Shillor and W. Zhang, Unilateral dynamic contact of two beams. Math. Comput. Modelling 34 (2001) 365–384.

    Google Scholar 

  5. A. Visintin, Differential Models of Hysteresis. Berlin: Springer-Verlag (1994) 407pp.

    Google Scholar 

  6. F. C. Moon and S. W. Shaw, Chaotic vibration of a beam with nonlinear boundary conditions. J. Nonlin. Mech. 18 (1983) 465–477.

    Google Scholar 

  7. F. C. Moon, Chaotic and Fractal Dynamics. New York: John Wiley & Sons (1992) 508pp.

    Google Scholar 

  8. M. Schatzman and M. Bercovier, Numerical approximation of a wave equation with unilateral constraints. Math. Comput. 53 (1989) 55–79.

    Google Scholar 

  9. L. Paoli and M. Schatzman, Resonance in impact problems. Math. Comput. Modelling 28 (1998) 385–406.

    Google Scholar 

  10. M. Schatzman, Uniqueness and continuous dependence on data for one-dimensional impact problems. Math. Comput. Modelling 28 (1998) 1–18.

    Google Scholar 

  11. G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics. New-York: Springer-Verlag (1976) 397pp.

    Google Scholar 

  12. N. Kikuchi and J. T. Oden, Contact Problems in Elasticity. Philadelphia: SIAM (1988) 495pp.

    Google Scholar 

  13. W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics 30, Rhode Island: Americal Mathematical Society and International Press (2002) 442pp.

    Google Scholar 

  14. K. T. Andrews, M. Shillor and S. Wright, On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle. J. Elasticity 42 (1996) 1–30.

    Google Scholar 

  15. A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance. Int. J. Engng. Sci. 26 (1988) 811–832.

    Google Scholar 

  16. J. A. C. Martins and J. T. Oden, A numerical analysis of a class of problems in elastodynamics with friction. Comput. Meth. Appl. Mech. Engnr. 40 (1983) 327–360.

    Google Scholar 

  17. K. T. Andrews, K. L. Kuttler and M. Shillor, On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle. Euro. J. Appl. Math. 8 (1997) 417–436.

    Google Scholar 

  18. K. L. Kuttler Time dependent implicit evolution equations. Nonlinear Anal. 10 (1986) 447–463.

    Google Scholar 

  19. K. L. Kuttler and M. Shillor Set-valued pseudomonotone maps and degenerate evolution inclusions. Commun. Contemp. Math. 1 (1999) 87–123.

    Google Scholar 

  20. M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction. J. Elasticity 51 (1998) 105–126.

    Google Scholar 

  21. R. Adams, Sobolev Spaces. New York: Academic Press (1975) 268pp.

    Google Scholar 

  22. J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications. Vol. 1, Berlin: Springer (1972) 357pp.

    Google Scholar 

  23. W. Han, K. L. Kuttler, M. Shillor and M. Sofonea, Elastic beam in adhesive contact. Int. J. Solids Structures 39 (2002) 1145–1164.

    Google Scholar 

  24. H. M. Hilber, T. J. R. Hughes and R. L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engng. Struct. Dyn. 5 (1977) 283–292.

    Google Scholar 

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Authors and Affiliations

  1. IREMIA- Université de La Réunion, Saint-Denis, Ile de La Réunion, France

    Y. Dumont

  2. Department of Mathematics, Brigham Young University, Provo, Utah, U.S.A

    K.L. Kuttler

  3. Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309, U.S.A.

    M. Shillor

Authors
  1. Y. Dumont
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  2. K.L. Kuttler
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  3. M. Shillor
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Dumont, Y., Kuttler, K. & Shillor, M. Analysis and simulations of vibrations of a beam with a slider. Journal of Engineering Mathematics 47, 61–82 (2003). https://doi.org/10.1023/A:1025599332143

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  • DOI: https://doi.org/10.1023/A:1025599332143

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