Advertisement

Nonlinear Dynamics

, Volume 69, Issue 1–2, pp 159–172 | Cite as

Primary resonance in forced vibrations of in-plane translating viscoelastic plates with 3:1 internal resonance

  • You-Qi Tang
  • Li-Qun Chen
Original Paper

Abstract

Nonlinear forced vibrations of in-plane translating viscoelastic plates subjected to plane stresses are analytically and numerically investigated on the steady-state responses in external and internal resonances. A nonlinear partial-differential equation with the associated boundary conditions governing the transverse motion is derived from the generalized Hamilton principle and the Kelvin relation. The method of multiple scales is directly applied to establish the solvability conditions in the primary resonance and the 3:1 internal resonance. The steady-state responses are predicted in two patterns: single-mode and two-mode solutions. The Routh–Hurvitz criterion is used to determine the stabilities of the steady-state responses. The effects of the in-plane translating speed, the viscosity coefficient, and the excitation amplitude on the steady-state responses are examined. The differential quadrature scheme is developed to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the single-mode solutions of the steady-state responses.

Keywords

In-plane translating plates Nonlinearity Viscoelasticity Primary resonance Internal resonance Steady-state response Method of multiple scales Differential quadrature scheme 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chen, L.Q.: Analysis and control of transverse vibrations of axially moving strings. Appl. Mech. Rev. 58, 91–116 (2005) CrossRefGoogle Scholar
  2. 2.
    Chen, L.Q.: Nonlinear vibrations of axially moving beams. Nonlinear Dyn. 145–172 (2010) (ed. by Todd Evans, Intech) Google Scholar
  3. 3.
    Ulsoy, A.G., Mote, C.D. Jr.: Vibration of wide band saw blades. ASME J. Eng. Ind. Trans. 104, 71–78 (1982) CrossRefGoogle Scholar
  4. 4.
    Lengoc, L., Mccallion, H.: Wide bandsaw 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) zbMATHCrossRefGoogle Scholar
  5. 5.
    Lengoc, L., Mccallion, H.: Wide bandsaw blade under cutting conditions, Part II: Stability of a plate moving in its plane while subjected to parametric excitation. J. Sound Vib. 186, 143–162 (1995) zbMATHCrossRefGoogle Scholar
  6. 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) zbMATHCrossRefGoogle Scholar
  7. 7.
    Lin, C.C., Mote, C.D. Jr.: Equilibrium displacement and stress distribution in a two-dimensional, axially moving web under transverse loading. J. Appl. Mech. 62, 772–779 (1995) zbMATHCrossRefGoogle Scholar
  8. 8.
    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) CrossRefGoogle Scholar
  9. 9.
    Lin, C.C.: Stability and vibration characteristics of axially moving plates. Int. J. Solids Struct. 34, 3179–3190 (1997) zbMATHCrossRefGoogle Scholar
  10. 10.
    Lin, C.C.: Finite width effects on the critical speed of axially moving materials. J. Vib. Acoust. 120, 633–634 (1998) CrossRefGoogle Scholar
  11. 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) CrossRefGoogle Scholar
  12. 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) CrossRefGoogle Scholar
  13. 13.
    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) CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Hatami, S., Azhari, M., Saadatpour, M.M.: Nonlinear analysis of axially moving plates using FEM. Int. J. Struct. Stab. Dyn. 7, 589–607 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hatami, S., Azhari, M., Saadatpour, M.M.: Free vibration of moving laminated composite plates. Compos. Struct. 80, 609–620 (2007) CrossRefGoogle Scholar
  17. 17.
    Kartik, V., Wickert, J.A.: Parametric instability of a traveling plate partially supported by a laterally moving elastic foundation. J. Vib. Acoust. 130, 051006 (2008) CrossRefGoogle Scholar
  18. 18.
    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) CrossRefGoogle Scholar
  19. 19.
    Yang, X.D., Chen, L.Q., Zu, J.W.: Vibrations and stability of an axially moving rectangular composite plate. J. Appl. Mech. 78, 011018 (2011) CrossRefGoogle Scholar
  20. 20.
    Guo, X.X., Wang, Z.M., Wang, Y.: Dynamic stability of thermoelastic coupling moving plate subjected to follower force. Appl. Acoust. 72, 100–107 (2011) CrossRefGoogle Scholar
  21. 21.
    Hatami, S., Ronagh, H.R., Azhari, M.: Exact free vibration analysis of axially moving viscoelastic plates. Comput. Struct. 86, 1736–1746 (2008) CrossRefGoogle Scholar
  22. 22.
    Zhou, Y.F., Wang, Z.M.: Vibration of axially moving viscoelastic plate with parabolically varying thickness. J. Sound Vib. 316, 198–210 (2008) CrossRefGoogle Scholar
  23. 23.
    Marynowski, K.: Free vibration analysis of the axially moving Levy-type viscoelastic plate. Eur. J. Mech. A, Solids 29, 879–886 (2010) CrossRefGoogle Scholar
  24. 24.
    Luo, A.C.J., Hamidzadeh, H.R.: Equilibrium and buckling stability for axially traveling plates. Commun. Nonlinear Sci. Numer. Simul. 9, 343–360 (2004) zbMATHCrossRefGoogle Scholar
  25. 25.
    Luo, A.C.J.: Chaotic motions in resonant separatrix zones of periodically forced, axially travelling, thin plates. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 219, 237–247 (2005) Google Scholar
  26. 26.
    Nayfeh, A.H., Raouf, R.A.: Nonlinear forced response of infinitely long circular cylindrical shells. J. Appl. Mech. 54, 571 (1987) zbMATHCrossRefGoogle Scholar
  27. 27.
    Pai, P.F., Nayfeh, A.H.: Non-linear non-planar oscillations of a cantilever beam under lateral base excitations. Int. J. Non-Linear Mech. 25, 455–474 (1990) MathSciNetCrossRefGoogle Scholar
  28. 28.
    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) CrossRefGoogle Scholar
  29. 29.
    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) CrossRefGoogle Scholar
  30. 30.
    Pakdemirli, M., Özkaya, E.: Three-to-one internal resonances in a general cubic non-linear continous system. J. Sound Vib. 268, 543–553 (2003) CrossRefGoogle Scholar
  31. 31.
    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) CrossRefGoogle Scholar
  32. 32.
    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) CrossRefGoogle Scholar
  33. 33.
    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
  34. 34.
    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) CrossRefGoogle Scholar
  35. 35.
    Nayfeh, A.H., Mook, D.T., Sridhar, S.: Nonlinear analysis of the forced response of structural elements. J. Acoust. Soc. Am. 55, 281–291 (1973) CrossRefGoogle Scholar
  36. 36.
    Chen, J.C., Babcock, C.D.: Nonlinear vibration of cylindrical shells. AIAA J. 13, 868–876 (1975) zbMATHCrossRefGoogle Scholar
  37. 37.
    Chen, L.Q., Zu, J.W.: Solvability condition in multi-scale analysis of gyroscopic continua. J. Sound Vib. 309, 338–342 (2008) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghaiChina
  2. 2.School of Mechanical EngineeringShanghai Institute of TechnologyShanghaiChina
  3. 3.Department of MechanicsShanghai UniversityShanghaiChina
  4. 4.Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghaiChina
  5. 5.Modern Mechanics DivisionE-Institutes of Shanghai UniversitiesShanghaiChina

Personalised recommendations