Stability and dynamic behavior of in-plane transporting plates under internal resonance and two-frequency parametric excitation

The dynamic stability of in-plane transporting viscoelastic plates based on 1:3 internal resonance and two-frequency parametric excitation is investigated for the first time in this paper. Unnormal and peculiar phenomena of stability boundaries occur in 1:3 internal resonance and two-frequency parametric excitation. The governing equations and related inhomogeneous boundary conditions are obtained by the Newton's second law. The tension of axial variation is emphasized. The solvability conditions in principal parametric resonances are obtained by direct method of multiple scales. Based on the Routh-Hurwitz criterion, stability boundary conditions are derived. Numerical examples are presented to depict the influence of system parameters on the stability boundaries, such as, viscoelastic coefficients, viscous damping, and axial speed fluctuation amplitudes. In addition, the effects of internal resonance and inhomogeneous boundary conditions on the stability boundaries are compared. The approximation results show very interesting and exotic phenomena of unstable boundaries. Under 1:3 internal resonance and two-frequency parametric excitation, irregular instability boundary appears, and the instability boundary forms a fractured instability region, within which no stability region exists in the system. At the end of the paper, the accuracy of the numerical solution is verified by differential quadrature method.