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Parametric and internal resonances of in-plane accelerating viscoelastic plates

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Abstract

This paper analytically and numerically investigates the nonlinear vibration in parametric and internal resonances of in-plane accelerating viscoelastic plates subjected to plane stresses. An approximate nonlinear plate theory was developed under the Kirchoff assumptions. The in-plane translating speed is characterized as a simple harmonic variation about the constant mean axial speed. The governing equation with the associated boundary conditions is derived from the generalized Hamilton principle and the Kelvin constitutive relation. The method of multiple scales is applied to establish the solvability conditions in principal parametric and internal resonances. The steady-state responses are predicted in three possible patterns: trivial, single-mode, and two-mode solutions. The stabilities of the steady-state responses are determined based on the Routh-Hurwitz criterion. The effects of the mean in-plane translating speed, the in-plane translating speed fluctuation amplitude, the viscosity coefficient, and the nonlinear coefficient on the steady-state responses are examined. The differential quadrature schemes are developed for the two-dimensional full plate model and the one-dimensional reduced plate model to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the trivial and single-mode solutions of the steady-state responses.

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References

  1. Chen L.Q.: Analysis and control of transverse vibrations of axially moving strings. Appl. Mech. Rev. 58, 91–116 (2005)

    Article  Google Scholar 

  2. Chen, L.Q.: Nonlinear vibrations of axially moving beams. Nonlinear Dyn. (ed. by Todd Evans, Intech) 145–172 (2010)

  3. Ulsoy A.G., Mote C.D. Jr: Vibration of wide band saw blades. J. Eng. Ind. Trans. ASME 104, 71–78 (1982)

    Article  Google Scholar 

  4. Lengoc L., McCallion H.: Wide band saw blade under cutting conditions, Part I: vibration of a plate moving in its plane while subjected to tangential edge loading. J. Sound Vib. 186, 125–142 (1995)

    Article  MATH  Google Scholar 

  5. Lengoc L., McCallion H.: Wide bandsaw blade under cutting conditions. II: stability of a plate moving in its plane while subjected to parametric excitation. J. Sound Vib. 186, 143–162 (1995)

    Article  MATH  Google Scholar 

  6. Lengoc L., McCallion H.: Wide bandsaw blade under cutting conditions: Part III: stability of a plate moving in its plane while subjected to non-conservative cutting forces. J. Sound Vib. 186, 163–179 (1995)

    Article  MATH  Google Scholar 

  7. Lee H.P., Ng T.Y.: Dynamic stability of a moving rectangular plate subject to in-plane acceleration and force perturbations. Appl. Acoust. 45, 47–59 (1995)

    Article  Google Scholar 

  8. Lin C.C.: Stability and vibration characteristics of axially moving plates. Int. J. Solids Struct. 34, 3179–3190 (1997)

    Article  MATH  Google Scholar 

  9. Lin C.C.: Finite width effects on the critical speed of axially moving materials. J. Vib. Acoust. 120, 633–634 (1998)

    Article  Google Scholar 

  10. Wang X.: Numerical analysis of moving orthotropic thin plates. Comput. Struct. 70, 467–486 (1999)

    Article  MATH  Google Scholar 

  11. Luo Z., Hutton S.G.: Formulation of a three-node traveling triangular plate element subjected to gyroscopic and in-plane forces. Comput. Struct. 80, 1935–1944 (2002)

    Article  Google Scholar 

  12. Kim J., Cho J., Lee U., Park S.: Modal spectral element formulation for axially moving plates subjected to in-plane axial tension. Comput. Struct. 81, 2011–2020 (2003)

    Article  Google Scholar 

  13. Hatami S., Azhari M., Saadatpour M.M.: Stability and vibration of elastically supported, axially moving orthotropic plates. Iran. J. Sci. Technol. B 30, 427–446 (2006)

    Google Scholar 

  14. Hatami S., Azhari M., Saadatpour M.M.: Exact and semi-analytical finite strip for vibration and dynamic stability of traveling plates with intermediate supports. Adv. Struct. Eng. 9, 639–651 (2006)

    Article  Google Scholar 

  15. Hatami S., Azhari M., Saadatpour M.M.: Free vibration of moving laminated composite plates. Compos. Struct. 80, 609–620 (2007)

    Article  Google Scholar 

  16. Banichuk N., Jeronen J., Neittaanämki P., Tunvinen T.: On the instability of an axially moving elastic plate. Int. J. Solids Struct. 47, 91–99 (2009)

    Article  Google Scholar 

  17. Hatami S., Ronagh H.R., Azhari M.: Exact free vibration analysis of axially moving viscoelastic plates. Comput. Struct. 86, 1736–1746 (2008)

    Article  Google Scholar 

  18. Zhou Y.F., Wang Z.M.: Vibration of axially moving viscoelastic plate with parabolically varying thickness. J. Sound Vib. 28, 209–218 (2007)

    MATH  Google Scholar 

  19. Zhou Y.F., Wang Z.M.: Transverse vibration characteristics of axially moving viscoelastic plate. J. Sound Vib. 316, 198–210 (2008)

    Article  Google Scholar 

  20. Zhou Y.F., Wang Z.M.: Dynamic behaviors of axially moving viscoelastic plate with varying thickness. J. Sound Vib. 22, 276–281 (2009)

    Google Scholar 

  21. Luo A.C.J., Hamidzadeh H.R.: The nonlinear vibration and stability of axially traveling thin plates. Proc. ASME Des. Eng. Tech. Conf. 5, 1135–1143 (2003)

    Google Scholar 

  22. Hatami S., Azhari M., Saadatpour M.M.: Nonlinear analysis of axially moving plates using FEM. Int. J. Struct. Stab. Dyn. 7, 589–607 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Luo A.C.J.: Chaotic motions in resonant separatrix zones of periodically forced, axially travelling, thin plates. Proc. Inst. Mech. Eng. Part K J. Multi-body Dyn. 219, 237–247 (2005)

    Google Scholar 

  24. Luo A.C.J., Hamidzadeh H.R.: Equilibrium and buckling stability for axially traveling plates. Comm. Nonlinear Sci. Numer. Simul. 9, 343–360 (2004)

    Article  MATH  Google Scholar 

  25. Chen S.H., Huang J.L., Sze K.Y.: Multidimensional Lindstedt-Poincaré method for nonlinear vibration of axially moving beams. J. Sound Vib. 306, 1–11 (2007)

    Article  Google Scholar 

  26. Suweken G., Horssen W.T.: On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part II: the beam-like case. J. Sound Vib. 267, 1007–1027 (2003)

    Article  Google Scholar 

  27. Pakdemirli M., Özkaya E.: Three-to-one internal resonances in a general cubic non-linear continuous system. J. Sound Vib. 268, 543–553 (2003)

    Article  Google Scholar 

  28. Özhan B., Pakdemirli M.: A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: primary resonances case. J. Sound Vib. 325, 894–906 (2009)

    Article  Google Scholar 

  29. Sze K.Y., Chen S.H., Huang J.L.: The incremental harmonic balance method for nonlinear vibration of axially moving beams. J. Sound Vib. 281, 611–626 (2005)

    Article  Google Scholar 

  30. Huang J.L., Chen S.H.: Combination resonance of laterally nonlinear vibration of axially moving systems. J. Vib. Eng. 18, 19–23 (2005)

    Google Scholar 

  31. Tang Y.Q., Chen L.Q.: Nonlinear free transverse vibrations of in-plane moving plates: without and with internal resonances. J. Sound Vib. 330, 110–126 (2011)

    Article  Google Scholar 

  32. Chen L.Q., Zu J.W.: Solvability condition in multi-scale analysis of gyroscopic continua. J. Sound Vib. 309, 338–342 (2008)

    Article  Google Scholar 

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai, 200072, China

    You-Qi Tang & Li-Qun Chen

  2. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    Li-Qun Chen

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  1. You-Qi Tang
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  2. Li-Qun Chen
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Correspondence to Li-Qun Chen.

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Tang, YQ., Chen, LQ. Parametric and internal resonances of in-plane accelerating viscoelastic plates. Acta Mech 223, 415–431 (2012). https://doi.org/10.1007/s00707-011-0567-y

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  • DOI: https://doi.org/10.1007/s00707-011-0567-y

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