## 2 Methodology  
### Physical Model and Equations of Motion  
The schematic diagram of a generator rotor system is shown in Figure 1. To meet the demand of electricity generation, the generator rotor is slotted in the circumferential direction. As a result, the generator rotor loses its axial symmetry and is called a two-pole generator rotor. The two-pole generator rotor has two unequal principal second moments of area. The vertical stiffness of the rotor changes twice in every rotational period, which makes the gravity response change twice and leads to double frequency vibration, i.e., a gravity response. The generator rotor system is composed of a rotary part and a stationary part, i.e., a rotor and bearings. If the equations of motion of the generator rotor system are established in the fixed frame, the time-varying coefficients will occur in the equations of motion, which makes the equations difficult to solve. Considering that the time-varying coefficients are caused by the asymmetry of the generator rotor and the bearings in this paper are isotropic, the rotating frame is selected where the time-varying coefficients disappear. The aim of this paper is to study the gravity response of the generator rotor system with interval uncertainties. Thus, the only external excitation is gravity, and other external excitations such as unbalance forces are ignored. Figure 1: The schematic diagram of a generator rotor system. In the rotating frame, the equations of motion of a 3D FEM of a generator rotor system excited by gravity can be expressed as:  
\[\mathbf{M}^{\prime}\ddot{\mathbf{u}}+(\mathbf{C}_{b}+\Omega\mathbf{C}_{cor}^{ \prime})\dot{\mathbf{u}}+\Big{(}\mathbf{K}_{b}+\mathbf{K}_{s}^{\prime}-\Omega^{ 2}\mathbf{K}_{d}^{\prime}\Big{)}\mathbf{u}=\mathbf{G}_{c}\cos\Omega t+\mathbf{G} _{s}\sin\Omega t \tag{1}\]  
where \(\Omega\) is the rotational speed; \(\mathbf{M}^{\prime}\) and \(\mathbf{K}_{s}^{\prime}\) are the mass and stiffness matrices of the generator rotor, respectively; \(\Omega\mathbf{C}_{cor}^{\prime}\) is the Coriolis matrix; \(\mathbf{K}_{s}^{\prime}\) is the spin-softening matrix; \(\mathbf{C}_{b}\) and \(\mathbf{K}_{b}\) are the damping and stiffness matrices of bearings, respectively; \(\mathbf{G}_{c}\) and \(\mathbf{G}_{s}\) are the coefficient vectors of the cosine and sine functions of gravity vector, respectively; and \(\mathbf{u}\) is the displacement vector of the generator rotor system in the rotating frame. The stiffness and damping matrices of bearing in the rotating frame are different from those in the fixed frame. If the bearing is anisotropic, the time-varying coefficients are inevitable in the rotating frame. However, the anisotropic bearing is out of the scope of this study, and the bearing discussed in this paper is isotropic. The cross-coefficients of isotropic bearing are equal to zero. The direct coefficients are identical. Therefore, in the rotating frame, the coefficients of the equations of motion of the generator rotor supported by isotropic bearing are time-independent. The relationship between the physical variables in the rotating frame and in the fixed frame can be expressed as:  
\[\{\cdot\}^{rot}=\mathbf{T}\{\cdot\}^{fix}=\left[ \begin{array}{cc}\cos\Omega t &\sin\Omega t\\ -\sin\Omega t&\cos\Omega t\end{array} \right]\left[\cdot\right]^{fix} \tag{2}\]  
where \(\{\cdot\}^{rot}\) and \(\{\cdot\}^{fix}\) are the vectors of physical variables in the rotating frame and in the fixed frame, respectively, and \(\mathbf{T}\) is the transformation matrix. The stiffness and damping matrices of bearing in the rotating frame can be transformed from those in the fixed frame. The supporting forces of bearing in the fixed frame can be expressed as:  
\[\left\{ \begin{array}{c}f_{x}\\ f_{y}\end{array} \right\}=\left[ \begin{array}{cc}k&0\\ 0&k\end{array} \right]\left\{ \begin{array}{c}x\\ y\end{array} \right\}+\left[ \begin{array}{cc}c&0\\ 0&c\end{array} \right]\left\{ \begin{array}{c}\dot{x}\\ \dot{y}\end{array}