## 1 Introduction  
As an indispensable important energy conversion device, centrifugal pump is extensively applied in various agricultural and industrial fields. For the reduction of equipment investment and energy consumption, modern centrifugal pumps have developed towards the direction of higher flow capacity, rotation speed, head, and efficiency [1; 2; 3; 4; 5]. This means that the vibration problem of centrifugal pump tends to become more prominent due to the increased energy density, and it will inevitably bring about unfavorable effects on the operational safety, stability, and life. The hazard of the vibration problem is specially reflected in the rotor assembly of pump, and it may lead to the serious damage of high-speed running parts such as the shaft, impeller, and bearings [6]. There are mainly two kinds of factors influencing the vibration of centrifugal pump rotor assembly--the mechanical factors and the hydraulic factors. The mechanical factors generally relate to the unbalance mass of the rotating parts, misalignment of shaft system and radial rub between different parts, so those factors can be effectively weakened by improving the machining accuracy and assembly quality [7,8]. While for the hydraulic factors, they are predominated by the complicated fluid-structure interaction (FSI) due to inner wall pressure fluctuation caused by the unsteady flow in centrifugal pump. As a result, it is very necessary to obtain the induction mechanism of the rotor assembly vibration from the view of internal unsteady flow, so the vibration level can be restrained at the source via inner flow field optimization. The unsteady flow behaviors in centrifugal pump has been systematically investigated in the way of numerical calculation and experiment test. The intense rotor-stator interaction due to the relative movement between impeller and solute forms the primary character of internal unsteady flow, so the dominant frequencies of unsteady pressure fluctuation are usually the blade passing frequency (\(f_{\text{BPF}}\)) and multiple shaft frequency (\(f_{\text{n}}\)) [9]. Similar pulsation tendency is also fund in the hydraulic radial force on the impeller, because the hydraulic force mainly depends on the integration of pressure on the impeller surfaces. In addition, there are also other obvious frequency components rather than \(f_{\text{BPF}}\) or multiple \(f_{\text{n}}\), which are stimulated by a serious complex unsteady flow behaviors, such as the jet-wake pattern along the blade extension direction, the flow impact on the tongue, and the flow separation on the blade suction surface [10, 11, 12]. The above behaviors usually become more evident under off-design conditions and would take much greater risks for the operation of centrifugal pump [13,14]. Based on the research of internal unsteady flow in centrifugal pump, the structural dynamic response analysis has been introduced to perform the FSI study. Because of the complex structure and twisted shape of pump, currently the typical application for the FSI problem of pump is the loosely coupled (explicit) simulation between computational fluid dynamics (CFD) and computational structural dynamics (CSD) [15]. Considering the fluid-induced vibration of the pump brings very limited displacement of the fluid-solid interface, and their interaction acts as a sustained effect in the whole operation process, the FSI problem can be solved with the partitioned method, in which the equations governing the internal flow and structural deformation can be solved separately and expediently with independent field solvers. The principal advantage of the partitioned method is that the whole modeling and solving procedure for the FSI problem can be built on existing and mature CFD and CSD codes with the widely used arbitrary Eulerian-Lagrangian description for the materials, where the Eulerian (spatial) method is used to descript the fluid domain and the Largangian method for the solid component.