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 ChenEmail author
Original Paper


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.


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


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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

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