Modeling the Dynamics of a Gyroscopic Rigid Rotor with Linear and Nonlinear Damping and Nonlinear Stiffness of the Elastic Support

This study analytically and numerically modeled the dynamics of a gyroscopic rigid rotor with linear and nonlinear cubic damping and nonlinear cubic stiffness of an elastic support. It has been shown that (i) joint linear and nonlinear cubic damping significantly suppresses the vibration amplitude (including the maximum) in the resonant velocity region and beyond it and (ii) joint linear and nonlinear cubic damping more effectively affects the boundaries of the bistability region by its narrowing than linear damping. A methodology is proposed for determining and identification the coefficients of nonlinear stiffness, linear damping, and nonlinear cubic damping of the support material, at which jump-like effects are eliminated. Damping also affects the stability of motion; if linear damping shifts the left boundary of the instability region towards large amplitudes and speeds of rotation of the shaft, then nonlinear cubic damping can completely eliminate it. The varying amplitude (VAM) method used to determine the nature of the system response, supplemented with the concept of "slow" time, which allows us to investigate and analyze the effect of nonlinear cubic damping and nonlinear rigidity of cubic order on the frequency response at a nonstationary resonant transition.

本研究针对带有线性阻尼、非线性三次阻尼以及弹性支座非线性三次刚度的陀螺刚性转子的动力学特性，开展了解析与数值建模工作。研究表明：(1) 线性阻尼与非线性三次阻尼联合作用，可显著抑制共振转速区间及其以外范围内的振动幅值（含峰值幅值）；(2) 相较于单一线性阻尼，线性与非线性三次阻尼的联合作用可通过缩小双稳态区域的范围，更有效地调控该区域的边界。本研究提出了一种用于确定与辨识支座材料非线性刚度系数、线性阻尼系数以及非线性三次阻尼系数的方法，采用该方法可消除系统的跳跃现象。阻尼同样会对系统的运动稳定性产生影响：线性阻尼会将失稳区域的左边界向轴的更大幅值与更高转速方向偏移，而非线性三次阻尼则可完全消除该失稳区域。本研究采用变幅值法（Varying Amplitude Method, VAM）判定系统响应特性，并辅以"慢时间"概念，以此实现非平稳共振过渡工况下，非线性三次阻尼与三次非线性刚度对系统频率响应影响的研究与分析。