In this paper, the dynamic stability of axially transporting viscoelastic beams with two-frequency parametric excitation and 1:3 internal resonance is investigated for the first time. The governing equation and corresponding inhomogeneous boundary conditions are developed by the Newton’s second law. The viscoelastic characteristic of the transporting Euler-Bernoulli beam obeys the Kelvin-Voigt model. The axial tension is considered to vary longitudinally. Direct method of multiple scales is employed to obtain the solvability conditions in principal parametric resonances. The stability boundary conditions are obtained by the Routh-Hurwitz criterion. Numerical examples are shown to illustrate the effects of relevant parameters on the stability boundaries. Unusual and interesting phenomena of stability boundaries occur in two-frequency parametric excitation and 1:3 internal resonance. The accuracies of the approximate analytical results are verified by comparing with the numerical results, which obtained by a differential quadrature method.

本文首次研究了具有两频参量激励和1：3内部共振的轴向传输粘弹性梁的动力稳定性。牛顿第二定律提出了控制方程和相应的非均匀边界条件。Euler-Bernoulli传输梁的粘弹性特性服从Kelvin-Voigt模型。认为轴向张力是纵向变化的。采用多尺度的直接方法来获得主要参数共振中的可溶性条件。稳定性边界条件通过Routh-Hurwitz准则获得。数值例子表明了相关参数对稳定性边界的影响。稳定边界的异常现象和有趣现象都出现在两频参量激励和1：3内部共振中。通过与微分求积法获得的数值结果进行比较，验证了近似分析结果的准确性。