The effects of contact stiffness between the rotor and stator on the internal resonance of forward and backward whirls in the rotating frame have been investigated, that is the forward and backward modes are commensurate in the rotating frame. First, the equations for an overhung rotor system with two rotors with coupling stiffness are established by Lagrangian method. By the transformation of a general coordinate system to a rotational coordinate system, we obtained the Campbell diagram in these two coordinate systems. The underlying rotating speeds corresponding to the internal resonances are analyzed by this Campbell diagram in the rotating coordinate system. Accounting for the impact force due to the exceeding of the clearance between the rotor and stator, we applied an ordinary differential equation solver to acquire the solutions, in bifurcation diagrams, orbits, FFT, and spectral density plots. The difference of the current study is investigating the stiffness effects on the nonlinear behaviors. More rich nonlinear phenomena were observed, such as 7:1, 5:1, 4:1, or 3:1 internal resonance in the rotating frame, but quasiperiodic motions are observed in the stationary frame. These internal resonance induced by forward and backward modes in the rotating frame in the FFTs was discussed, which can provide novel explanations for the complicated internal resonance.

中文翻译：

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接触刚度对悬臂转子系统冲击引起的非线性运动的影响


研究了转子和定子之间的接触刚度对旋转框架中正向和反向涡动内部共振的影响，即旋转框架中正向和反向涡动是相称的。首先，采用拉格朗日法建立了具有耦合刚度的两个转子的悬臂转子系统方程。通过将一般坐标系变换到旋转坐标系，我们得到了这两个坐标系下的坎贝尔图。通过旋转坐标系中的坎贝尔图来分析与内部共振相对应的基本旋转速度。考虑到由于转子和定子之间的间隙超出而产生的冲击力，我们应用常微分方程求解器在分岔图、轨道、FFT 和谱密度图中获取解。当前研究的不同之处在于研究刚度对非线性行为的影响。在旋转坐标系中观察到了更丰富的非线性现象，如7:1、5:1、4:1或3:1内部共振，但在静止坐标系中观察到了准周期运动。对FFT中旋转系中的前向和后向模式引起的内部谐振进行了讨论，这可以为复杂的内部谐振提供新的解释。