# Research on the Multi-Angle Seismic Spectrum Response Law of Auxiliary Feedwater Electric Pumps
## 3 Seismic Analysis Methods for Auxiliary Feedwater Electric Pumps  
Seismic analysis methods for auxiliary feedwater electric pumps are traditionally classified into three main categories: the static analysis method, the response spectrum method, and the dynamic analysis method. In compliance with pertinent nuclear safety codes in China, seismic analysis calculations for nuclear power equipment may be performed utilizing the static analysis method, provided that the fundamental frequency of the nuclear power equipment surpasses 33 Hz. [27; 28; 29]. Despite its computational simplicity, this method tends to be overly conservative, leading to substantial margins and failing to accurately capture the structure's response characteristics. Conversely, the dynamic analysis method offers precision but demands extensive computations, resulting in complex data management challenges [30]. In contrast to these approaches, this study embraces  
Figure 3: Mesh independence verification. Figure 2: Finite element model of auxiliary feedwater electric pumps. (**a**) Rotor. (**b**) Stator. the response spectrum method to scrutinize the response characteristics of the auxiliary feedwater electric pump under seismic inputs from diverse angles. The core computational process is delineated as follows. The equations of motion for the auxiliary feedwater electric pump under seismic loading are expressed as follows:  
\[M\ddot{x}+C\dot{x}+Kx=M\ddot{x}_{g}(t) \tag{1}\]  
where \(M\) represents the mass matrix, \(C\) is the damping matrix, \(K\) stands for the stiffness matrix, \(x\) denotes the displacement of individual nodes, and \(x_{g}\) signifies the relative ground displacement. In examining the dynamic characteristics of the structure, the equation describing its undamped free vibration is formulated as follows:  
\[M\ddot{x}+Kx=0 \tag{2}\]  
Solutions for the generalized eigenvalue and eigenvector corresponding to Equation (2) can be obtained, revealing the intrinsic frequency and vibration pattern of the auxiliary feedwater electric pump for each order. The process of vibration mode decomposition entails simplifying the seismic response problem of a pump characterized by multiple degrees of freedom into n independent single-degree-of-freedom system seismic response problems. This simplification is achieved by expressing the dynamic response in modal coordinates, as illustrated in Equation (1). Through decoupling transformations and subsequent solving, the seismic excitation standard value for the pump's multi-degree-of-freedom system during an earthquake is derived. \[F_{ij}=a_{j}\gamma_{j}X_{ij}G_{i}\quad(i,j=1,2,\cdots,n) \tag{3}\]  
where \(F_{ij}\) represents the seismic excitation standard value corresponding to the \(j\)-th mode of the \(i\)-th mass point, \(a_{j}\) is the seismic influence coefficient of the \(j\)-th mode, \(\gamma_{j}\) is the participation coefficient of the \(j\)-th mode, \(X_{ij}\) is the displacement corresponding to the \(j\)-th mode of the \(i\)-th mass point, and \(G_{i}\) represents the representative value of the gravity load concentrated on the \(i\)-th mass point. The seismic response of the auxiliary feedwater electric pump involves intricate patterns where maximum values of vibration modes do not align simultaneously or in the same direction. Utilizing a simplistic summation of these maximum absolute values would result in an excessively conservative estimation of the total seismic effect, denoted as \(S\). To address this, our study adopts a sophisticated probabilistic analysis method, specifically the square root of the sum of the squares (SRSS). In this approach, we calculate the seismic response, denoted as \(S_{j}\), for each order of the \(i\)-th mass point of the \(j\)-th vibration mode in the multi-degree-of-freedom system based on the standard value of horizontal seismic excitation, \(F_{ij}\).