Structural Analysis and Optimization of Ultra-High-Speed Centrifugal Pump Rotor System Considering Fluid–Structure Interaction

Abstract

An ultra-high-speed centrifugal pump plays a crucial role as part of an aircraft engine’s fuel supply system. This paper focuses on the coupled vibration and optimization of a parallel double-stage ultra-high-speed centrifugal pump considering fluid–structure interaction (FSI). The accuracy of the numerical calculation is verified and compared with the experimental results. The steady and transient characteristics of the rotor system are analyzed to ensure the operational reliability of the rotor system. Moreover, an orthogonal test is conducted to explore the transient structural characteristics of the rotor system. The existing cross-support structure meets high-speed stability requirements and there is no resonance in the cantilevered rotor system. The maximum and minimum errors for the head of Pump 2 are 4% and 0.7%, respectively. The minimum values for maximum average deformation and maximum average stress are less than 0.31 mm and 245 MPa, respectively, at design conditions. The position of Bearing 1 near the multi-stage impeller has the greatest impact on the deformation and stress of the rotor system, and the deformation and stress increase as the distance increases. The results of this study can provide a valuable reference for the design of ultra-high-speed centrifugal pump rotor systems.

1. Introduction

An ultra-high-speed centrifugal pump is an important part of the oil supply system for an aero-engine, which provides high pressure to ensure optimal operating conditions [1,2]. However, higher speed places greater demands on the safety of the centrifugal pump system, and the complex transient flow and pressure pulsations at high-speed conditions increase system instability [3,4]. It is necessary to investigate the structure stability of the ultra-high-speed centrifugal pump to ensure the normal operation of the aero-engine.

The internal flow analysis of centrifugal pump is one of the main research fields because it can better present flow information and visualize the performance of centrifugal pump. Visser et al. [5] applied a laser Doppler velocimetry (LDV) measurement system to explore the multiple directional velocities of the relative flow. Their experimental results presented a good agreement with theoretical values for the core ow region. Jafarzadeh et al. [6] applied three known turbulence models to investigate the head and power coefficients. Jia et al. [7] researched the pressure distribution of the front casing, back casing, and volute casing according to the numerical calculation and pressure experiment at six different working conditions. The results indicated that the pressure fluctuation was affected by the tongue of the volute. Cui et al. [8] applied the sliding mesh to investigate the unsteady flow structures for a complex impeller. They found that the main reason for the unstable flow inside the pump was the interaction between the stationary volute and rotating impeller. Zhang et al. [9,10,11] carried out a series of studies and paid more attention to the unsteady flow characteristics. Wei et al. [12] studied the effect of the gap drainage structure on the steady and transient characteristics according to numerical calculation and experimental verification. They found that a small gap width was good for the pump performance.

Furthermore, with the advancement of numerical computations and experiments, the issue of pressure pulsation is receiving increasing attention. An increasing body of research suggests that elevated levels of pressure pulsation can lead to vibration and noise in centrifugal pumps, thereby exerting adverse effects on the secure and steady operation of these pumps [13,14,15,16]. Barrio et al. [17,18] investigated the relationship between impeller geometric parameters and pressure pulsation characteristics. They found that reducing the impeller–tongue gap led to a corresponding increase in pressure pulsation characteristics. González et al. [19] found that secondary flow played a main role in the blade-passing frequency. Chalghoum et al. [20] discovered that the relative position between the volute tongue and impeller resulted in periodic pressure fluctuations. Huang et al. [21] studied the characteristics of pressure fluctuations in pump mode. The results showed that there were differences in pressure fluctuations at different heights of the bladeless region, with the highest pressure fluctuations near the bottom ring.

Due to the highly coupled nonlinear process of the interaction between fluids and liquids, it is necessary to use fluid–structure interaction methods to solve the problem [22]. Recently, the study of fluid–structure interaction (FSI) of centrifugal pumps has become a new hot topic in research. The positional variation of the impeller has been found to impact the internal flow field of the pump, subsequently altering the pressure pulsation and pump performance [23,24]. Baun et al. [25] investigated the impeller-to-volute position and tried to find the optimal position for different volutes. Cao et al. [26] conducted a study on centrifugal pumps under different eccentricity conditions; they observed a degradation in performance with increasing eccentricity. Tao et al. [27] examined pump performance and pressure pulsation under an eccentric condition. They discovered that considering eccentricity allowed for the capture of multiple frequency components, including the rotation frequency and its multiples. Zhou et al. [28,29] focused on the study of the fluid excitation characteristics in centrifugal pump impellers with compound motion. The calculated results were closer to the experimental results when the compound motion was in a certain range. Zhao et al. [30] determined that eccentricity could influence the head and radial force under cavitation conditions. Li et al. [31] explored the internal flow fields of mixed-flow pumps with eccentric impellers, identifying increased eccentricity as the primary factor for decreased pump efficiency. From the research in the aforementioned literature, it can be observed that the effect of FSI is crucial for centrifugal pump rotor systems and it should not be neglected. Yuan et al. [32] simulated impellers with different structures using a single fluid–structure coupling method and found that under the same flow rate, the maximum equivalent force value of the closed impeller was the highest, while the maximum equivalent stress value of the split impeller was the lowest. Shen et al. [33] considered the fluid–structure coupling effect and analyzed pump rotors with different blade installation angles. They found that under the same flow rate, the axial force increases with the increase in the blade installation angle. At the same flow rate, the maximum deformation equivalent stress increases with the increase in the blade installation angle.

For ultra-high-speed centrifugal pumps, the current research is very limited, and the impact of FSI on the structure characteristics of the rotor system is still an unknown field. Therefore, it is crucial to explore the structure characteristics of the pump rotor system considering FSI. In this paper, a parallel double-stage ultra-high-speed centrifugal pump rotor system model with a rotational speed of 28,000 rpm was regarded as the research object, and the structure characteristics considering the FSI effect were studied for different flow rates, bearing positions, and bearing stiffnesses. The simulation results show good agreement with the experimental results. In addition, the optimal design results were obtained through an orthogonal test and the transient characteristics were further analyzed.

2. Materials and Methods

2.1. Numerical Mode

The structure of the model pump is shown in Figure 1. The parallel double-stage ultra-high-speed centrifugal pump for this calculation mainly consists of two centrifugal pumps, transmission, and support components. The cylindrical roller bearings are installed between the two-stage supply pumps, and the right-end bearing of Pump 2 is a ball bearing. The design conditions for Pump 1 (left end) are rated speed n_d = 28,000 r/min and flow rate Q_d = 34.5 m³/h. The design conditions for Pump 2 (right end) are rated speed n_d = 28,000 r/min and flow rate Q_d = 15.9 m³/h. More details are listed in

Table 1. Parameters of pump model.

Parameters	Pump 1	Pump 2
Zr	16	16
Zr-long	8	8
Zr-short	8	8
Suction diameter D1	31.6 mm	20 mm
Outlet diameter D2	29 mm	36 mm
Balance hole number	8	8

.

It should be noted that the specific area of the balance hole is an important factor affecting the pump’s ability to resist cavitation, so the balance hole should be retained when processing the model [34]. The mesh of the overall fluid domain is shown in Figure 2. In order to reduce the iteration convergence time during numerical calculations, the local refinement of the mesh was performed at the volute tongue, impeller blade, front chamber, and rear chamber ring gaps during the non-structured mesh partitioning. After the mesh quality check, the average quality was 0.84, which meets the requirements for numerical calculations. The test results for grid independence are shown in

Table 2. Grid independence test.

Number of Grids (Million)	Head (m)
5	1155.39
6	1195.22
7	1235.06
8	1274.90
9	1288.18
10	1294.82

. A total of 6 grid schemes were designed for numerical calculation, and it was found that after 8 million grids, the head change was relatively small with the increase in the number of grids. Therefore, the 8 million grid scheme was ultimately chosen.

To ensure the one-to-one correspondence of the FSI interface, three-dimensional modeling should include both the fluid domain and the solid domain. The fluid region is suppressed when the solid domain grids are generated. Moreover, the grid division is performed separately for the impeller structure model and the shaft according to their different structural features. The impeller structure model uses the tetrahedral mesh as the main grid division method while the hybrid grid is applied for the shaft. The number of finite element mesh nodes and finite element mesh elements are 251,685 and 440,494, respectively. The calculation mesh is shown in Figure 3.

2.2. Calculation Method

The unsteady flow was simulated using the SST k-ω turbulence model. The expression for the shear stress transfer (SST) k-ω turbulence model for incompressible fluids can be represented by the following equations [35]:

$$\frac{\partial (\rho k)}{\partial t}+\frac{\partial (\rho k{u}_i)}{\partial x_i}=\frac{\partial }{\partial x_j}[(u+\frac{u_t}{{\sigma }_k})\frac{\partial k}{\partial x_j}]+G_k-\rho k\omega {\beta }^{*}$$

(1)

$$\frac{\partial \left(\rho \omega \right)}{\partial t}+\frac{\partial \left(\rho \omega {\overline{u}}_i\right)}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\omega }}\right)\frac{\partial \omega }{\partial x_j}\right]+\frac{\partial \omega }{k}{G}_k-\rho {\omega }^2\beta +\frac{2\left(1-F_1\right)\rho }{\omega {\sigma }_{\omega }}\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}$$

(2)

$${\mu }_t=\frac{\rho k}{\omega }$$

(3)

$$F_1=\tanh \left\{{\left\{\min \left[\max \left(\frac{\sqrt{k}}{{\beta }^{*}\omega y},\frac{500\nu }{y^2\omega }\right),\frac{4{\sigma }_{\omega }k}{C{D}_{k\omega }{y}^2}\right]\right\}}^4\right\}$$

(4)

$$C{D}_{k\omega }=\max \left(2\rho {\sigma }_{\omega }\frac{1}{\omega }\frac{\partial k}{\partial x_i}\frac{\partial \omega }{\partial x_i},{10}^{-10}\right)$$

(5)

The constant terms are σ_k = σ_ω = 2, α = 0.555, β = 0.075, and β* = 0.09.

The wall surfaces of the impeller fluid domain and the pump chamber wall surfaces in contact with the cover plates of the impeller are set as rotating walls. The steady-state calculation is performed with 3000 iterations. The results of the steady-state calculation are further applied to the unsteady calculation. The impeller is rotated by 5° as a time step, with a time step size of ∆t = 2.976 × 10⁻⁵ s, and the number of time steps is 360. In the numerical calculation, convergence is essential, as it represents whether the flow field tends to be stable, which is set to 1 × 10⁻⁶.

This study adopts one-way transient FSI for structural analysis of the rotor system. The coupled three-dimensional equations of the flow field and impeller structure can be obtained [36]:

$$\left[\begin{array}{cc}{M}_s& 0\\ M_{fs}& M_f\end{array}\right]\left[\begin{array}{c}\ddot{X}\\ \ddot{P}\end{array}\right]+\left[\begin{array}{cc}{C}_s& 0\\ 0& C_f\end{array}\right]\left[\begin{array}{c}\dot{X}\\ \dot{P}\end{array}\right]+\left[\begin{array}{cc}{K}_s& K_{fs}\\ 0& K_f\end{array}\right]\left[\begin{array}{c}X\\ P\end{array}\right]=\left[\begin{array}{c}{F}_0\\ 0\end{array}\right]$$

(6)

In order to obtain the structural vibration characteristics of the rotor system under transient excitation forces, the loads on the impeller mainly include inertia forces and surface forces, namely the centrifugal force load and gravity load generated by the mass of the impeller itself as well as the pressure load exerted by the fluid on the impeller. Due to the high speed and large inertia of the impeller, as well as the significant pressure on the blade surface, the weight of the impeller itself can be neglected compared to the aforementioned forces. The centrifugal force load is achieved by applying rotational velocity and density to the impeller. The rotational speed setting must be consistent with the rotational speed setting of the flow field, so the rotor system is set to rotate clockwise around the Z-axis at 28,000 r/min. The pressure load on the flow field is directly obtained from the results of the flow field analysis and is transmitted through the FSI interface, which is shown in Figure 4. The gyroscopic effect is considered for the motion of the rotating body.

The rotor system structure of the model pump is shown in Figure 5. The rotor system structure mainly consists of two-stage impellers and a pump shaft. The impellers include front cover plates, rear cover plates, and blades. Additionally, the positions of the FSI interfaces are also identified in Figure 5. The red regions in the figure represent the FSI interfaces of the shroud, hub, and blades of the impeller. In the FSI simulation, these interfaces serve as the boundary surfaces for load transfer. During the calculation process, the mismatch between the fluid and structure grids at each fluid–structure interaction interface is less than 0.5%, indicating a high level of data transfer accuracy at the fluid–structure interfaces. The time step size for the impeller structure analysis is ∆t = 8.9286 × 10⁻⁵ s and the end time is t = 1.0714 × 10⁻² s, with a total of 120 steps.

3. Results

3.1. Critical Speed and Campbell Diagram

When the grid scheme is finished, the first and second critical speeds and mode shapes of the rotor system are calculated, respectively. The centrifugal pump rotor system uses rolling bearings, with the main stiffness set to 5.0 × 10⁷ N/m in the bearing constraint. The first 10 modes are set in the modal module, and both the damping vibration and the Coriolis effect are considered. The Campbell diagram for the centrifugal pump rotor system is plotted in Figure 6. When the disc rotates around the rotor’s central axis with an angular velocity of Ω₁, and at the same time the rotor rotates around the support central axis, it is called vorticity, with an angular velocity of Ω₂. If the rotation direction of the vortex angular velocity is the same as the rotation direction of the wheel disc, it is called forward whirl (FW), and vice versa, it is called backward whirl (BW).

As shown in Figure 6, the critical speeds of each modal of the rotor system refer to the values corresponding to the intersection points of the forward precession frequency curve of that modal with the auxiliary lines. Therefore, the first and second critical speeds of the rotor system can be obtained from the Campbell diagram, which is summarized in

Table 3. The first and second critical speed of the rotor system.

Modal Order	FW (rpm)	Frequency (Hz)
1	37,534	625
2	74,461	1241

. The first column represents the order of the critical speed, the second column represents the magnitude of the critical speed value, and the third column represents the frequency corresponding to the critical speed. The operating speed of this centrifugal pump is 28,000 rpm, which is lower than the first critical speed used for modal analysis. Therefore, it can be considered that this centrifugal pump can operate smoothly and safely under normal conditions without resonance. Additionally, the rotor system is rigid, which means that there is no phenomenon of intense vibration occurring at a specific moment during startup.

In Figure 7, the first two modal shapes are lateral vibrations, with the main deformation concentrated on the impeller. In the first modal shape, the maximum deformation occurs at the outlet of Impeller 1, with the front cover plate of Impeller 1 having greater overall deformation than the rear cover plate, and Impeller 2 having smaller deformation. The second modal shape is mainly characterized by the lateral oscillation of Impeller 2, with the maximum deformation occurring at Impeller 2 and the shaft segment in contact with Impeller 2, and slight deformation at the inlet of Impeller 1. The results are consistent with the structural characteristics of the multi-stage cantilever rotor system.

3.2. Stress and Deformation of the Rotor System

To demonstrate the accuracy of numerical calculations under different flow conditions, the relative errors between the experimental and simulated head of Pump 2 are compared in Figure 8. Q_d refers to the flow rate at which the centrifugal pump operates at its maximum efficiency, and departing from this point will result in a decrease in efficiency. It can be observed that the maximum and minimum errors are 4% and 0.7%, respectively. The maximum error is less than 5%, which indicates that the simulation calculation has good accuracy and can be used for further study.

In addition, a model with the positions of Bearing 1 at L₁ = 75 mm, Bearing 2 at L₂ = 210 mm, and stiffness K₁₁ = K₂₂ = 5.0 × 10⁷ N/m is selected for structure analysis under 3 different flow conditions. The total deformation contour and stress contour diagrams at four different times for different flow conditions are further obtained.

Figure 9 shows the deformation contours of the rotor system for different times under a 0.8 Q_d flow condition. The maximum deformation of the rotor system is mainly concentrated at the outlet of Impeller 1. At the time of t = 1.0714 × 10⁻² s, the maximum deformation at Impeller 1 exhibits symmetric distribution. The maximum deformation is 0.5283 mm. Figure 10 shows the deformation contours of the rotor system at different times under a 1.0Q_d flow condition. The maximum deformation of the rotor system decreases with increasing time, and the deformation at t = 8.6602 × 10⁻³ s is the largest, mainly concentrated at the outlet of Impeller 1. The maximum deformation is 0.27734 mm. Circular-shaped minimal deformation regions appear at t = 9.6422 × 10⁻³ s and t = 1.0714 × 10⁻² s. The above results imply that the rotor system has smaller deformation for design conditions compared to low-flow conditions, i.e., the rotor system has better safety and stability under design conditions.

Figure 11 shows the deformation contours of the rotor system for different times under a 1.2Q_d flow condition. The maximum deformation of the rotor system is 0.47604 mm, which occurs at t = 9.6422 × 10⁻³ s. The maximum deformation region exhibits symmetric distribution. The max deformation at different moments shows a tendency to increase, then decrease, and then increase again, which is different from the results of the design condition and the low-flow condition. Moreover, the deformation for the high-flow condition is larger than that for the design condition.

It can be observed from Figure 9, Figure 10 and Figure 11 that the maximum deformation of the rotor system is mainly concentrated at Impeller 1. At most time points, the deformation gradient increases in the direction of the front and rear cover plates, appearing at the outlet of Impeller 1. At a few time points, the maximum deformation at Impeller 1 exhibits symmetric distribution. This is mainly because Impeller 1 is in a cantilever state and has less constraint from the bearings. The pressure fluctuation amplitude in the tongue area of the volute is larger, causing larger deformation near the tongue. The minimum deformation occurs at Impeller 2 and the right side of the Impeller 2 shaft, mainly because Impeller 2 is constrained by bearings at both ends, and it has a higher rigidity that is less prone to deformation.

Figure 12 shows the stress contours of the rotor system for different times under a 0.8Q_d flow condition. The maximum stress in the rotor system is mainly concentrated at the contact between Bearing 1 and the shaft and at the balance hole of the impeller. The maximum stress occurs at t = 1.0714 × 10⁻² s, reaching 395.65 MPa. Figure 13 shows the stress contours of the rotor system at different times under a 1.0Q_d flow condition. The maximum stress in the rotor system decreases as time increases, and the maximum stress occurs at t = 8.6602 × 10⁻³ s, reaching 211.92 MPa. As a whole, the maximum stress decreases and then increases at low-flow conditions while it monotonically decreases at design conditions.

Figure 14 shows the stress contours of the rotor system for different times under a 1.2Q_d flow condition. The maximum stress in the rotor system occurs at t = 1.0178 × 10⁻² s, with a maximum stress of 329.87 MPa. At t = 1.0714 × 10⁻² s, there is a difference in stress distribution on the front cover plate of Impeller 1 compared to the other 3 time points. The maximum stress for design conditions is smaller compared to the low-flow conditions and high-flow conditions. This result is similar to that of the maximum deformation.

It can be observed from Figure 12, Figure 13 and Figure 14 that the maximum stress in the rotor system is mainly concentrated at the contact between Bearing 1 and the shaft and at the balance hole of the impeller. The main reason for the phenomenon is that the balance hole has a smaller diameter, causing the impeller on both sides of Bearing 1 to oscillate and resulting in local high stress. The equivalent stress of the impeller exhibits a rotating cyclic distribution, with the number of cycles matching the number of blades. The equivalent stress at the connection between the cover plates and the blades is significantly higher than the equivalent stress of the front and rear cover plates at the flow channel. Besides, the equivalent stress of the front cover plate of the impeller is generally lower than that of the rear cover plate.

According to

Table 4. The effect of flow rate on the maximum average deformation and stress.

Flow Condition (Q/Qd)	Maximum Average Deformation (mm)	Maximum Average Stress (MPa)
0.8	0.3955	292.05
1.0	0.3047	243.94
1.2	0.3545	293.93

, the relationship between the maximum average deformation and stress under different flow conditions can be observed. It can be seen that the minimum values for the maximum average deformation and stress occur at a 1.0Q_d flow condition. In addition, both the maximum average deformation and stress in the rotor system under low-flow conditions are larger than those under design conditions. The main reason is that the higher pressure fluctuations can be induced for high-speed pumps under low-flow conditions, which can lead to larger deformation and equivalent stress in the rotor system. These results indicate that the flow conditions in high-speed pumps have a significant impact on the deformation and equivalent stress of the rotor system, which should be taken into consideration.

3.3. Orthogonal Test

An orthogonal test is a method to select a representative combination of factor levels from all levels of test factors for testing. The optimal combination of factor levels can be determined by reducing the number of experiments and analyzing the results according to the orthogonal test. Considering the factors affecting the stress and deformation of the multistage rotor system, the position of Bearing 1 (L₁), the position of Bearing 2 (L₂), the stiffness coefficient of bearing (K) and the flow rate of the centrifugal pump (Q) are selected as experimental factors. The different positions of Bearing 1 and Bearing 2 are shown in Figure 15. Three levels are set for each factor, which are listed in

Table 5. Orthogonal test factors and levels.

Levels	Factors
	A	B	C	D
	L1/mm	L2/mm	K/N·m−1	Q/Qd
1	50	160	5.0 × 107	0.8
2	75	180	7.5 × 107	1.0
3	100	210	1.0 × 108	1.2

.

For orthogonal design, range analysis [23] is adopted to seek the optimal parameter combination according to the results listed in Table 4.

$$k_i=\frac{1}{3}{K}_i=\frac{1}{3}{\displaystyle \sum _{j=1}^{N_i}{y}_{i,j}}$$

(7)

$$R=\max (k_1,k_2,\dots ,k_i)-\min (k_1,k_2,\dots ,k_i)$$

(8)

The maximum average deformation of each factor at different levels is listed in

Table 6. Orthogonal experimental results of maximum average deformation.

Number	A	B	C	D	Maximum Average Deformation/mm
	L1/mm	L2/mm	K/N·m−1	Q/Qd
1	50	160	5.0 × 107	0.8	0.1961
2	50	180	7.5 × 107	1.0	0.1249
3	50	210	1.0 × 108	1.2	0.1404
4	75	160	7.5 × 107	1.2	0.2494
5	75	180	1.0 × 108	0.8	0.3639
6	75	210	5.0 × 107	1.0	0.3051
7	100	160	1.0 × 108	1.0	0.5399
8	100	180	5.0 × 107	1.2	0.4929
9	100	210	7.5 × 107	0.8	0.5290

, and the analysis results are listed in

Table 7. The impact of each factor on the maximum average deformation.

	A	B	C	D
	L1/mm	L2/mm	K/N·m−1	Q/Qd
K1	0.4614	0.9853	0.9941	1.0889
K2	0.9184	0.9817	0.9032	0.9699
K3	1.5617	0.9745	1.0442	0.8827
k1	0.1538	0.3284	0.3314	0.3630
k2	0.3061	0.3272	0.3011	0.3233
k3	0.5206	0.3248	0.3481	0.2942
R	0.3668	0.0036	0.0470	0.0687
R Ranking	1	4	3	2

. The factors affecting the maximum average deformation, from the highest to lowest impact, are L₁, Q, K, and L₂. In order to observe the trend more intuitively, a trend diagram of the impact of each factor on the maximum average deformation is plotted, as shown in Figure 16. Based on the evaluation criteria, the optimal level combination is obtained as A1B3C2D3, which means that L₁ is 50 mm, L₂ is 210 mm, K is 7.5 × 10⁷ N/m, and the flow rate Q/Q_d =1.2. This indicates that the high-speed multistage rotor system under these parameters has the smallest numerical value for the maximum average deformation.

From Table 7 and Figure 16, it can also be observed that the L₁ plays a vital effect on the maximum average deformation. The changing range of maximum average deformation for L₁ is obviously larger than that for the other parameters. Moreover, the L₁ and maximum average deformation present a positive relationship, which means that when Bearing 1 is close to Impeller 1, the rigidity of Impeller 1 can be significantly improved and the deformation of the rotor system decreases. The calculation results indicate that the distance between Bearing 1 and Impeller 1 can be minimized to improve the vibration characteristics of the cantilever rotor system.

The maximum average stress for each factor at each level is calculated and listed in

Table 8. Orthogonal test results of maximum average stress.

Number	A	B	C	D	Maximum Average Stress/MPa
	L1/mm	L2/mm	K/N·m−1	Q/Qd
1	50	160	5.0 × 107	0.8	192.7
2	50	180	7.5 × 107	1.0	223.6
3	50	210	1.0 × 108	1.2	215.6
4	75	160	7.5 × 107	1.2	224.9
5	75	180	1.0 × 108	0.8	304.5
6	75	210	5.0 × 107	1.0	244.1
7	100	160	1.0 × 108	1.0	327.8
8	100	180	5.0 × 107	1.2	291.3
9	100	210	7.5 × 107	0.8	323.7

, and the analysis results are listed in

Table 9. The impact of each factor on the maximum average stress.

	A	B	C	D
	L1/mm	L2/mm	K/N·m−1	Q/Qd
K1	631.9	745.4	728.1	821.0
K2	773.5	819.4	772.2	795.4
K3	942.8	783.4	848.0	731.8
k1	210.6	248.5	242.7	273.7
k2	257.8	273.1	257.4	265.1
k3	314.3	261.1	282.7	244.0
R	103.6	24.6	40.0	29.7
R Ranking	1	4	2	3

. It can be clearly seen that the factors affecting the maximum average stress, from the highest to lowest impact, are L₁, K, Q, and L₂. In order to observe the trend more intuitively, a trend diagram of the impact of each factor on the maximum average stress is plotted, as shown in Figure 17. Based on the evaluation criteria, the optimal level combination is obtained as A1B1C1D3, which means that L₁ is 50 mm, L₂ is 160 mm, K is 5 × 10⁷ N/m, and the flow rate Q/Q_d =1.2. This indicates that the multistage rotor system of the centrifugal pump under these parameters has the smallest numerical value for the maximum average stress.

From the data in Table 9 and Figure 17, it can also be observed that L₁ has a significant impact on the maximum average stress of the rotor system. As L₁ increases, the maximum average stress also increases, which is consistent with the previous calculation results of the maximum average deformation. As a whole, the position of Bearing 1 has a somewhat smaller influence on the maximum average stress compared to the influence on the maximum average deformation, but it is still the most important factor to be considered in the design of a multistage cantilever rotor system.

4. Conclusions

This paper investigates the structural characteristics of a parallel double-stage ultra-high-speed centrifugal pump rotor system considering the FSI. The steady characteristics of the coupled rotor system are analyzed using a Campbell diagram and modal shapes. The accuracy of numerical calculation is verified by the experimental data. The dynamic stress and deformation of the rotor system are obtained. Finally, the orthogonal test is conducted to explore the influence of flow rate, bearing position, and bearing stiffness on the transient structural characteristics of the system. The main conclusions are as follows:

1. The operating speed of the pump system is lower than the first critical speed. The existing cross-support structure meets high-speed stability requirements and there is no resonance in the cantilevered rotor system.

2. Compared with the numerical results and experimental data, the maximum and minimum errors for the head of Pump 2 are 4% and 0.7%, respectively. These small relative errors validate the accuracy of the proposed model.

3. The minimum values for maximum average deformation and maximum average stress are less than 0.31 mm and 245 MPa, respectively, under design conditions. Variable flow conditions can lead to an increase in the maximum average deformation and stress.

4. The influence of each factor on the maximum average deformation is in the order of L₁ > Q > K > L₂. The influence of each factor on the maximum average stress is in the order of L₁ > K > Q > L₂. L₁ has the greatest influence on the maximum average stress and stress of the rotor system. With the increase of L₁, the maximum average deformation and maximum average stress increase.