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Applied Mathematics and Mechanics

, Volume 37, Issue 12, pp 1647–1668 | Cite as

Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions

  • Youqi TangEmail author
  • Dengbo Zhang
  • Mohan Rui
  • Xin Wang
  • Dicheng Zhu
Article

Abstract

The dynamic stability of axially accelerating plates is investigated. Longitudinally varying tensions due to the acceleration and nonhomogeneous boundary conditions are highlighted. A model of the plate combined with viscoelasticity is applied. In the viscoelastic constitutive relationship, the material derivative is used to take the place of the partial time derivative. Analytical and numerical methods are used to investigate summation and principal parametric resonances, respectively. By use of linear models for the transverse behavior in the small displacement regime, the plate is confined by a viscous damping force. The generalized Hamilton principle is used to derive the governing equations, the initial conditions, and the boundary conditions of the coupled planar vibration. The solvability conditions are established by directly using the method of multiple scales. The Routh-Hurwitz criterion is used to obtain the necessary and sufficient condition of the stability. Numerical examples are given to show the effects of related parameters on the stability boundaries. The validity of longitudinally varying tensions and nonhomogeneous boundary conditions is highlighted by comparing the results of the method of multiple scales with those of a differential quadrature scheme.

Key words

parametric resonance axially moving plate longitudinally varying tension nonhomogeneous boundary condition 

Chinese Library Classification

O323 

2010 Mathematics Subject Classification

70J40 

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Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Youqi Tang
    • 1
    Email author
  • Dengbo Zhang
    • 1
  • Mohan Rui
    • 2
  • Xin Wang
    • 1
  • Dicheng Zhu
    • 1
  1. 1.School of Mechanical EngineeringShanghai Institute of TechnologyShanghaiChina
  2. 2.Third Maintenance SquadronPeople’s Liberation Army 93256 TroopsShenyangChina

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