## 2 Methodology
### Deterministic Gravity Response Analysis Based on ROM  
One of the drawbacks of 3D FEM is the enormous number of DOFs, which limit the application of the 3D FEM. The enormous number of DOFs will lead to large dimensional equations of motion that demand large computational efforts. Thus, to overcome the large computational efforts, the fixed-interface CMS [31] is employed in this paper to reduce the dimension of the equations of motion. During the reduction, the displacements of the generator rotor system are divided into interior displacements \(\mathbf{u}_{i}\) and boundary displacements \(\mathbf{u}_{\Gamma}\). Consequently, the equations of motion before the reduction can be expressed as:  
\[\left[ \begin{array}{c}\mathbf{M}_{ik}&\mathbf{M}_{i\Gamma}\\ \mathbf{M}_{\Gamma k}&\mathbf{M}_{\Gamma\Gamma}\end{array} \right]\left\{ \begin{array} []{c}\ddot{\mathbf{u}}_{i}\\ \ddot{\mathbf{u}}_{\Gamma}\end{array} \right\}+\left[ \begin{array}{cc} \mathbf{C}_{ik}&\mathbf{C}_{i\Gamma}\\ \mathbf{C}_{\Gamma k}&\mathbf{C}_{\Gamma\Gamma}\end{array} \right]\left\{ \begin{array} []{c}\dot{\mathbf{u}}_{i}\\ \dot{\mathbf{u}}_{\Gamma}\end{array} \right\}+\left[ \begin{array}{cc}\mathbf{ K}_{ik}&\mathbf{K}_{i\Gamma}\\ \mathbf{K}_{\Gamma k}&\mathbf{K}_{\Gamma\Gamma}\end{array} \right]\left\{ \begin{array} []{c}\mathbf{u}_{i}\\ \mathbf{u}_{\Gamma}\end{array} \right\}= \tag{6}\]  
where:  
\[\mathbf{M}=\mathbf{M}^{\prime},\] \[\mathbf{C}=\mathbf{C}_{b}+\Omega\mathbf{C}_{\alpha\alpha}^{\prime}, \tag{7}\] \[\mathbf{K}=\mathbf{K}_{b}+\mathbf{K}_{s}^{\prime}-\Omega^{2} \mathbf{K}_{d}^{\prime}\]  
The reduction by fixed-interface CMS is carried out by transforming the equations of motion in the physical space to the modal space. For the convenience of obtaining the gravity responses of the desired node, the physical coordinates of the desired node are treated as boundary DOFs that can be retained after the reduction. Thus, the physical coordinates of the generator rotor system by employing the fixed-CMS can be expressed as:  
\[\left\{ \begin{array}{c}\mathbf{u}_{i}\\ \mathbf{u}_{\Gamma}\end{array} \right\}=\left[ \begin{array}{cc}\mathbf{ \Phi}_{ik}&\mathbf{\Phi}_{i\Gamma}\\ \mathbf{\Phi}_{\Gamma k}&\mathbf{\Phi}_{\Gamma\Gamma}\end{array} \right]\left\{ \begin{array}{c}\mathbf{P}_{k}\\ \mathbf{u}_{\Gamma}\end{array} \right\}=\mathbf{\Phi}\mathbf{q} \tag{8}\]  
where \(\mathbf{\Phi}_{ik}\) is the retained normal modes matrix, which is composed of the first \(k\)th real eigenvectors when fixing the boundary DOFs and ignoring the damping matrix in Equation (6); and \(\mathbf{\Phi}_{i\Gamma}=-(\mathbf{K}_{ik})^{-1}\mathbf{K}_{i\Gamma}\) is the constrained modes matrix, which is composed of the displacement vectors by releasing the boundary DOFs one by one and fixing other boundary DOFs. By employing the transformation defined in Equation (8), the equations of motion of the ROM can be expressed as:  
\[\mathbf{\overline{M}}\mathbf{\dot{q}}+\mathbf{\overline{C}}\mathbf{\dot{q}}+ \mathbf{\overline{K}}\mathbf{q}=\mathbf{\overline{G}}_{c}\cos\Omega t+ \mathbf{\overline{G}}_{s}\sin\Omega t \tag{9}\]  
where:  
\[\mathbf{\overline{M}}=\mathbf{\Phi}^{\mathsf{T}}\mathbf{M}\mathbf{\Phi}, \mathbf{\overline{C}}=\mathbf{\Phi}^{\mathsf{T}}\mathbf{C}\mathbf{\Phi}, \mathbf{\overline{K}}=\mathbf{\Phi}^{\mathsf{T}}\mathbf{K}\mathbf{\Phi}, \tag{10}\]  
Since the gravity excitation is periodic, the gravity response of the generator rotor system is periodic as well. Therefore, the gravity response of the ROM can be assumed as:  
\[\mathbf{q}=\mathbf{Q}_{c}\cos\Omega t+\mathbf{Q}_{s}\sin\Omega t \tag{11}\]  
where \(\mathbf{Q}_{c}\) and \(\mathbf{Q}_{s}\) are the coefficient vectors of the cosine and sine functions of assumed gravity response, respectively. Substituting Equation (11) into Equation (9) and maintaining the balance between the cosine and sine function, the following equations can be expressed as:  
\[\left[