轴向变速运动弦线的非线性振动的稳态响应及其稳定性

Steady-State Responses and Their Stability of Nonlinear Vibration of an Axially Accelerating String

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性· 轴向运动速度为常平均速度与小简谐涨落的叠加· 应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程· 直接应用于多尺度方法求解该方程· 建立了避免出现长期项的可解性条件· 得到了近倍频共振时非平凡稳态响应及其存在条件· 给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响· 应用Liapunov线性化稳定性理论。

The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear partial-differential equation that governs the transverse vibration of the string was derived by use of the Hamilton principle. The method of multiple scales was applied directly to the equation. The solvability condition of eliminating the secular terms was established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one parametric resonance were obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation were presented. The Liapunov linearized stability theory was employed to derive the instability conditions of the trivial solution and the nontrivial solutions for the two-to-one parametric resonance. Some numerical examples highlighting influences of the related parameters on the instability conditions were presented.