In this study, the nonlinear vibration and stability of a simply supported axially moving Rayleigh viscoelastic beam equipped with an intermediate nonlinear support are investigated. The type of considered nonlinearity is geometric and is due to the axial stretching. The Kelvin–Voigt model is used to regard the beam internal damping. The Hamilton’s principle is employed to derive the governing equations and corresponding boundary conditions. The multiple scales method is applied to the dimensionless form of the governing equations and the nonlinear frequencies, time response of the system for two cases of the axial velocity fluctuation frequency are obtained. The stability of the system is investigated via solvability condition and Routh–Hurwitz criterion. Some case studies are accomplished to demonstrate the effect of rotary inertia, axial velocity and parameters of intermediate support on the system response, critical velocity and the system stability. Furthermore, the variation of the first two resonance frequencies with respect to mean axial velocities for different locations of the intermediate support are investigated. It is found that by moving the intermediate support from one end of the beam to its midpoint, the region in which the first mode undergoes static instability, shrinks. Moreover, although rotary inertia impressively decreases the natural frequencies, intermediate support has the dominant effect on increasing the natural frequencies.

中文翻译：

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柔性中间支座上轴向移动粘弹性瑞利梁的非线性振动与稳定性分析

在这项研究中，研究了配备中间非线性支撑的简支轴向移动瑞利粘弹性梁的非线性振动和稳定性。所考虑的非线性类型是几何的，是由轴向拉伸引起的。Kelvin-Voigt 模型用于考虑梁的内部阻尼。哈密​​顿原理用于推导控制方程和相应的边界条件。将多尺度方法应用于控制方程的无量纲形式，得到非线性频率，得到系统在轴向速度脉动频率两种情况下的时间响应。通过可解性条件和 Routh-Hurwitz 准则研究系统的稳定性。完成了一些案例研究来证明旋转惯性的影响，轴向速度和中间支撑参数对系统响应、临界速度和系统稳定性的影响。此外，研究了前两个共振频率相对于中间支撑不同位置的平均轴向速度的变化。发现通过将中间支撑从梁的一端移动到其中点，第一模态发生静态不稳定的区域会缩小。此外，尽管转动惯量显着降低了固有频率，但中间支撑对增加固有频率起主导作用。研究了前两个共振频率相对于中间支撑不同位置的平均轴向速度的变化。发现通过将中间支撑从梁的一端移动到其中点，第一模态发生静态不稳定的区域会缩小。此外，尽管转动惯量显着降低了固有频率，但中间支撑对增加固有频率起主导作用。研究了前两个共振频率相对于中间支撑不同位置的平均轴向速度的变化。发现通过将中间支撑从梁的一端移动到其中点，第一模态发生静态不稳定的区域会缩小。此外，尽管转动惯量显着降低了固有频率，但中间支撑对增加固有频率起主导作用。