© L.-Y. Dong et al., Published by EDP Sciences 2023

1 Introduction

In modern industry, centrifugal pumps play a vital role. In the past analysis of pumps or motors, a separate analysis was often performed for an individual device, making it difficult to consider the mutual coupling relationship between the devices.

The centrifugal pump transfers energy to the fluid through the impeller's rotation during operation, and the blades are subjected to the combined action of liquid pressure, gravity, and centrifugal force. The torque generated by the fluid pressure acting on the blades will produce an unsteady reverse torsional moment of the impeller, and the characteristics of this unsteady torsional moment will change with the operating conditions of the centrifugal pump and the changes in the internal flow state [1]. At present, many studies have carried out a lot of research based on the centrifugal pump speed is approximately constant; that is, the speed is regarded as a continuous value in the numerical simulation, and significant results have been achieved in the pressure pulsation, vibration, and noise in the centrifugal pump. Shojaeefard et al. [2] managed to reduce the loss of energy in the pump fluid flow by changing the pump outlet angle and channel width. It can be seen that the geometric design of centrifugal pumps can achieve the best results only when the pump conveys the right type of fluid. When the physical parameters of the transported fluid change greatly because of the type or temperature, the performance of the centrifugal pump will decline. The work done by the centrifugal pump is difficult to be converted into the energy of the fluid flow as much as possible, and the rest of the energy will be transferred to the structure in the form of pressure pulsation, causing vibration. Barrio et al. [3] performed a simulation calculation (URANS + k-ε model) on the unsteady flow behavior near the tongue-shaped domain of a single-suction volute centrifugal pump with a specific speed 0.47. The results show that the flow pulsation at medium and large flow rates is directly related to the flow passage in front of the blade at each reference position, and the main influencing factor of pressure pulsation in the pump is the interaction between the impeller and the pump shell.

The pressure pulsation of the centrifugal pump affects the operating conditions' stability, and the pressure pulsation always exists. Jiang et al. [4] studied the relative position's influence between the diffuser vane and the volute tongue (clocking effect) on the single-stage centrifugal pump. The results show that the clocking effect greatly influences the pressure fluctuation and the unsteady radial force imposed on the impeller. Zhang et al. [5] investigated the influence of rotational speed on pressure pulsation and found that pressure amplitude at the blade passing frequency and RMS value increase rapidly as rotational speed increases. Zuo et al. [6] observed in engineering practice that the pressure fluctuation in the vaneless space caused by the rotor-stator interaction affects the water pump's hydraulic stability. Zhang et al. [7] clarified the correlation between the flow structure and the pressure spectrum by analyzing the frequency spectrum of the pressure pulsation signal under different flow rates and the relative velocity distribution of the blade's internal passage. Gao et al. [8] analyzed the relationship between the profile of the trailing edge of the blade and the unsteady pressure pulsation. Through research, it is believed that the appropriately improved trailing edge of the blade weakens the interaction between the rotor and the stator, thereby reducing the pressure pulsation amplitude. Rotor-stator interference is the main influencing factor of pressure pulsation, and rotational speed fluctuation is closely related to rotor-stator interference. It can be predicted that when external factors cause the fluctuation of centrifugal pump impeller speed, it will cause an obvious coupling effect.

The centrifugal pump system is an inseparable whole, and the pump, shaft, and motor are coupled and interacted in pairs. The torque of an asynchronous motor is generated by the rotating magnetic field of the energized stator winding acting on the rotor. Due to the existence of stator teeth and rotor slots, the motor's torque fluctuation is inevitable. The change of the pump impeller's instantaneous speed will cause the fluctuation of the impeller torque, and the instability of the impeller torque will affect the motor through the shaft and finally make the speed of the impeller change. The asynchronous conversion of the motor and the pump's torque causes the instantaneous speed of the pump shaft to change all the time. The coupling simulation of the system composed of a pump, shaft and motor can better analyze the mutual influence relationship in the centrifugal pump system. Araste et al. [9] used a combination of numerical and analytical methods to study the electromechanical coupling of induction motor drives. The research results show that the pump's internal mechanical failure is related to the electrical signal of the motor current. Thorsen et al. [10] established a system model to predict the centrifugal pump system's electrical and mechanical conditions during the start-up phase and studied the influence of multiple factors such as design size and number of impellers on dynamic stress. Wu et al. [11] defined the impeller rotational speed and the inlet velocity as a piecewise function composed of three multi-polynomials by polynomial fitting the curve for the numerical simulation. Then, the centrifugal pump's transient characteristics during stopping periods are analyzed through experimental research and numerical research. Huang et al. [12] improved the transfer function method and built a nonlinear flexible model using Simulink for the simulation of hydraulic-mechanical-electrical coupling dynamics containing fault characteristics. Although Shi et al. [13] have not established the system coupling model, they have proposed that shafting is the core component of coupling problem in hydropower unit in their research on fault diagnosis. Yang et al. [14] studied the coupling mechanism in the hydraulic system of hydropower station, and the research results show that the coupling relationship between the pump and the motor in the system cannot be ignored. These studies provide valuable ideas for establishing the coupling simulation model in this paper. In this paper, the motor is connected with the centrifugal pump through the rotating shaft, and the instantaneous speed at different positions of the rotating shaft is calculated according to the torque balance relationship at both ends of the rotating shaft to obtain the coupling relationship between the motor, the shaft and the centrifugal pump.

More importantly, in our previous experiments, we observed a unique phenomenon, that is, in the vibration signal of the pump casing, there is a noticeable peak at 120–125 Hz (see Fig. 1). The frequency corresponding to this peak is not a multiple of the rotation frequency or the passing frequency of the blades (the number of blades in the centrifugal pump is 6, the rotation frequency of the impeller is about 49 Hz, and the blade frequency is about 295 Hz). Because the asynchronous motor drives the pump, the blade speed gradually decreases as the flow increases, but the frequency corresponding to this peak shows an opposite trend. This unique phenomenon has not been effectively explained in the simulation of centrifugal pumps alone. We suspect that this phenomenon is caused by the coupling of the motor and rotating shaft connected to the centrifugal pump.

Therefore, the coupling simulation model of the centrifugal pump system is established in this paper. The centrifugal pump and the motor simulation calculation model are connected using a flexible shaft model in the coupled simulation calculation model. The shaft of the centrifugal pump system is connected with the pump and motor at the same time. The torque of the motor and pump causes the torsion deformation of the shaft. By calculating the dynamic response of the shaft (angle, speed, etc.), the pump and motor's instantaneous speed in the state of mutual influence can be obtained.

In the coupling simulation model established in this paper, the finite element method and finite volume method are used for the motor and centrifugal pump, respectively. The shaft's simulation calculation models can be divided into two categories: one is the continuous model with mass continuously distributed along the axis direction [15]; the other is the lumped parameter model, which only includes torsional stiffness and moment of inertia after simplification [16]. The lumped parameter model is the equivalent discretization of an actual structure according to certain simplification principles. It consists of disks with a centralized moment of inertia and flexible shafts without mass. Its advantages include clear concepts, convenient use, and fast calculation. In this paper, the lumped parameter model is used to simulate the centrifugal pump shaft. The centrifugal pump impeller torque and the motor output torque influence each other through the flexible shaft. The instantaneous speed fluctuation of the centrifugal pump impeller under the coupling effect can be solved. At the same time, the pressure pulsation of the centrifugal pump in the coupled state can also be calculated.

Fig. 1 Vibration response spectrum of pump casing at different flow rates. (a) 0–800 Hz; (b) 115–130 Hz; (c) 290–305 Hz.

2 Centrifugal pump system modelization

2.1 Simulation model of pump

The pump simulated in this paper is a six-blade centrifugal pump, and the main parameters are shown in Table 1. The inlet and outlet boundary conditions of the centrifugal pump during actual operation are unsteady and are jointly determined by the pump, piping system, and drive equipment. In conventional calculation and analysis, it is difficult to describe the unsteady boundary conditions of the pump accurately. In this paper, a semi-closed circulation pipeline system model consisting of pumps, pipelines, and reservoir is established, as shown in Figure 2.

In this simulation model of centrifugal pump piping system, a constant pressure boundary is set at the position corresponding to the free liquid surface of the stabilized reservoir, and a numerical self-coupling process solves boundary conditions such as flow and pressure at the inlet and outlet of the centrifugal pump.

In the flow field simulation, the flow field model can be divided into a closed system, semi-closed system, and non-closed system according to boundary conditions. For the flow field simulation of a closed system, in theory, the fluid only flows in the flow field simulation model, and there is no need to set the inlet or outlet of the fluid. The pressure and velocity at any position in the flow field are obtained through the iterative calculation of the simulation model. Theoretically, this simulation method is the closest to the actual situation. However, this requires not only a simulation model large enough to describe the entire system but also sufficient computational time to simulate the simulation model. A non-closed flow field simulation model requires specifying the inlet and outlet of the flow field model. Inlet and outlet boundary conditions need to be reasonably selected, and it is often difficult to set a reasonable and practical unsteady boundary. If steady boundary conditions are given inflow field simulation, the flow state of the flow field under ideal conditions is obtained, ignoring the influence of disturbance in the pipeline system. In the semi-closed model established in this paper, the inlet and outlet of the centrifugal pump are connected to the water tank. In the simulation model, the boundary condition of the upper surface of the water tank is the boundary condition of constant pressure, which simulates the boundary condition of the actual water tank at a certain height below the water's surface (as shown in Fig. 3). The simulation model shown in this paper simulates the unsteady boundary conditions at the inlet and outlet of the centrifugal pump to a certain extent and avoids the increased difficulty of calculation due to the two-phase flow field calculation method.

Besides, the flow rate is related to the size of the valve in the circulation piping system, so the flow condition of the centrifugal pump can be changed by changing the minimum diameter of the circulation piping system.

In this paper, the finite volume method is used for the simulation of the flow field. The sliding grid method is used for the impeller region to realize the division and connection of the dynamic and static domains, and the RNG k-ε model is used to close the Reynolds average equation [17]. The flow field simulation model is shown in Figure 4, and the grid type is the unstructured grid. The working fluid in this semi-closed model is liquid water. In the transient calculation process, the time step is 5 × 10⁻⁵ s, and within a time step, the centrifugal pump impeller rotates about 0.88°. The calculation time of the flow field in the initial state for preparation of the coupling calculation is 0.2 s, corresponding to the rotation of the centrifugal pump impeller is about 10 revolutions. The total calculation time of flow field calculation in the coupled calculation is 0.3 s, corresponding to the rotation of centrifugal pump impeller is about 15 revolutions.

The steady-state calculation was carried out by changing the number of grids, and total head was compared to judge whether the number of grids met the calculation requirements. The mesh grids independence analysis is shown in Figure 5. In the centrifugal pump simulation model, the number of grids except the reservior domain is about 4 million, composed of impeller domain, volute domain and circulation pipe domain.

Figure 6 shows the numerical steady Q-H curve for the circuit system model (centrifugal pump speed is about 2955 r/min, Q is equal to 110 m³/h). By changing the minimum pipe diameter in the circuit system, the flow rate of the circuit simulation model is adjusted. Similarly, by adjusting the valve opening in the actual centrifugal pump circuit, the centrifugal pump head corresponding to different flow rates was measured, and the calculation results of the flow field simulation model were verified.

It can be seen from Figure 6 that, with the increase of flow, the head of the centrifugal pump gradually decreases from 73.6 m to 62.5 m. The hydraulic efficiency of the centrifugal pump increases first with the increase of the flow rate. When the flow rate reaches the rated condition of 110 m³/h, the hydraulic efficiency reaches the maximum value of 78.2%. When the flow rate is greater than the rated flow rate, the hydraulic efficiency gradually decreases to 76%.By comparison, it is found that the test results are basically consistent with the simulation results, but slightly lower than the simulation results, mainly due to energy loss caused by fluid leakage and other reasons.

Fig. 2 Semi-closed simulation model of centrifugal pump piping system.

Fig. 3 The corresponding relationship between the simulation model and the actual structure of reservoir.

Fig. 4 Calculation grid of semi-enclosed centrifugal pump piping system.

Fig. 5 Mesh grids independence analysis.

Fig. 6 Calculation results of centrifugal pump external characteristics (centrifugal pump speed is about 2955 r/min, Q is equal to 110 m3/h).

2.2 Simulation model of other equipment in the centrifugal pump system

The normal operation of the centrifugal pump system is inseparable from the motor that produces torque and the shaft that drives the impeller to rotate. First, the centrifugal pump system starts. The current in the motor stator coil generates a rotating magnetic field. The motor rotor's induced current interacts with the stator rotating magnetic field to generate a driving torque. Then, the shaft drives the pump impeller to rotate together with the motor rotor. Due to the centrifugal force, the liquid is thrown from the impeller's center to the outer edge. The liquid's action produces a load moment on the pump impeller to hinder the impeller from rotating. Then, the load torque is transmitted to the motor rotor through the shaft so that the motor speed is reduced. Finally, the motor's drive torque in the centrifugal pump system and the load torque generated by the impeller reaches a balanced state. The motor's rotor and the impeller of the pump in the centrifugal pump system rotate steadily and continuously until the motor's power is cut off. To simulate the running state of the centrifugal pump system, the simulation models of the centrifugal pump, shaft, and motor were established respectively in this paper. Then the simulation model of pump and motor is connected according to the dynamic equation of shaft. Finally, the coupling simulation results of the centrifugal pump system are obtained.

The motor in the centrifugal pump system is an electromagnetic device that converts mechanical energy and electrical energy through a magnetic field. The energy stored in the stator and rotor inductance and the energy stored in the mutual inductance between the stator and rotor constitute the total magnetic field energy of the motor. This paper uses the finite element method to calculate the unsteady simulation calculation of the motor, which can calculate the total energy of the motor magnetic field corresponding to the different angular positions of the motor rotor. The calculation equation of the total energy of the magnetic field can be written as [18]:

(1)

where θ_me Is the electrical position angle of the rotor; θ_m is the spatial position angle of the rotor; L_ss, L_rr and L_sr are the stator self-inductance, rotor self-inductance, and stator-rotor mutual inductance of the motor respectively; i_s and i_r are the current in the stator winding and the current in the rotor guide bar respectively; p is the number of pole pairs of the motor.

The electromagnetic torque T_em is obtained by solving the derivative of the total magnetic field energy W_m to the space position angle θ_m.

(2)

The motor simulated in this paper is an asynchronous motor, and the main parameters are shown in Table 2. The two-dimensional cross-sectional schematic diagram of the motor in this paper is shown in Figure 7, and the finite element mesh is shown in Figure 8. A two-dimensional electromagnetic field model is selected to consider the accuracy of calculation and the consumption of computing resources. Because unit grid-scale to have some impact on the precision of electromagnetic field, the electromagnetic force is mainly based on the air gap flux density, so part of the air gap for grid meticulous division (maximum length is 2 mm) and for a yoke of iron core and other relative to the main parts adopt relatively coarse grid (maximum length of 12 mm).

The forced vibration equation of centrifugal pump shafting under excitation can be written as

(3)

[J] is the moment of inertia matrix, which can be expressed as

(4)

where J_i is the moment of inertia corresponding to the i th axis segment,n is the total number of axis segments.

The moment of inertia of an object about a rotating axis can be written as

(5)

where r is the distance between any tiny mass block on the object and the axis of rotation, and dm is any tiny mass block on the object.

When the centrifugal pump impeller rotates in the fluid, the fluid will produce a reaction force on the centrifugal pump impeller, which affects the dynamic characteristics of the centrifugal pump impeller. In other words, the fluid brings additional mass (moment of inertia) and additional damping to the impeller [19]. Additional water mass is assumed that the mass of water attached to the impeller is 0.4J_impeller to simplify the calculation.

[K] is the stiffness matrix, which can be expressed as

(6)

where k_i,j is the connection stiffness between the i-th inertia and the j th inertia, n is the total number of axis segments.

If torque T is applied at both ends of a shaft segment of L length, the torsional angular displacement of the shaft segment Δϕ is:

(7)

where G is the shear elastic modulus of the material (for the centrifugal pump shaft in this paper, G = 8.14933 ×10¹⁰N/m²), and J_P is the polar moment of inertia of the shaft segment

The stiffness k of the shaft segment can be written as

(8)

[C] is the damping matrix. There are many kinds of damping in the centrifugal pump system, such as fluid viscous damping, material damping, and electromagnetic damping. In this paper, more attention is paid to the influence of the coupling between multiple devices on the frequency components of signals (such as speed, pressure, vibration, etc.) in the centrifugal pump, so the damping matrix in the dynamic equation is ignored. (such as speed, pressure, vibration, etc.)

Combining the above-mentioned inertial parameter and stiffness parameter solution method, the lumped parameter model of the centrifugal pump shaft is obtained, as shown in Figure 9, and the calculated parameters are shown in Table 3.

{θ}, , are the rotation angle, rotation speed and angular acceleration of the shaft, which can be expressed as

(9)

(10)

(11)

where θ_i, and θ_i are respectively the rotation angle, speed and angular acceleration of the ith shaft segment.

(12)

where T_i is the torque acting on the i-th shaft segment. When i ≠ 6–8 or 28, T_i = 0; when i= 6–8, ; when i = 28, T_i = T_pump.

(13)

The solution of equation (13) can be written as θ_i = A_i cosω_it, where A_i is the ith main mode vector of the system, and ω_i is the natural frequency of the system. Let , equation (13) can be expressed as:

(14)

In equation (14), the inertia matrix [J] is symmetric positive definite, and the stiffness matrix [K] is symmetric positive definite or semi-positive definite matrix. Then equation (14) is a generalized eigenvalue problem, which is generally transformed into a standard eigenvalue problem when solving [20].

(15)

where [D] is the dynamic matrix, which can be expressed as [D] = [J]⁻¹ [K].

For the eigenvalue problem of high-order matrices, some scholars use numerical methods such as the Jacobi method, power method, and subspace iteration method to solve the problem. The solution process of these methods is generally more complicated, and the number of iterations is large. Therefore, this paper transforms the generalized eigenvalue problem of the centrifugal pump shafting vibration system with multiple degrees of freedom into a standard eigenvalue problem for solution. The first 3 mode of the centrifugal pump shaft system is shown in Figure 10.

As shown in Figure 10, the corresponding finite element calculation model is established by using the finite element method according to the geometric model (as shown in Fig. 11). Materials of various parts in the centrifugal pump system (as shown in Tab. 4), and the above modal calculation results are verified by the finite element method.

Comparing the modal calculation results of finite element calculation model and lumped parameter calculation model, the deviation of natural frequency calculation results is shown in Table 5. In this paper, the modal assurance criterion (MAC) is used to compare the quality of mode shapes or the similarity of two vectors in modal analysis. By comparing the modal calculation results of different calculation methods, it can be found that the calculation error of natural frequency is small, and the mode shapes are very similar. Therefore, the calculation results obtained by the lumped parameter model used in this paper are reliable.

The calculation of the shaft's dynamic response is the key to the simulation model in this paper. The Newmark-β method is a commonly used numerical calculation method in the field of structural dynamics. The Newmark-β method can calculate the dynamic response of the mechanical structure of a linear or nonlinear system under arbitrary excitation, and when γ ≥ 1/2, β ≥ γ/2, the algorithm is unconditionally stable [21]. If the dynamic response of the shaft at time t + Δt is required to be solved, the torque T_t+Δt corresponding to time t + Δt should be known in advance. However, the output torque of the motor and the torque of the pump impeller are determined by the instantaneous operation parameters of the centrifugal pump system at the present moment, so it is difficult to realize the coupling simulation calculation in the centrifugal pump system. Therefore, the Predictor-Corrector Newmark-β method (PC Newmark-β method) is used in this article to calculate the dynamic response of the centrifugal pump shaft.

The integral scheme of traditional Newmark-β method is

(16)

(17)

where Δt is the time step size defined by Δt = t_n+1 − t_n, and β and γ are the algorithmic parameters of the Newmark method. In this study, the time step size is selected as Δt = 5 × 10⁻⁵s. This study adopts the trapezoidal rule, which is the most widely used among the Newmark family algorithms. In the trapezoidal rule, the value of β and γ are 1/4 and 1/2, respectively, and the predictors of angle and speed, and are given as follows:

(18)

(19)

The correctors for angle and speed are given as follows:

(20)

(21)

Calculate T_motor and T_pump at time t + Δt according to and , and substitute the calculated torque T_i into equation (22) to obtain the correction value of the angular acceleration {...}_t+Δt at time t + Δt.

(22)

(23)

Flow chart of coupling calculation of concurrent pump system is shown in Figure 12.

Fig. 7 Schematic diagram of the two-dimensional cross-section of the motor.

Fig. 8 Two-dimensional finite element mesh of the motor.

Fig. 9 Lumped parameter model of the centrifugal pump shaft. (a) Geometric model; (b) lumped parameter model.

Fig. 10 The first 3 order mode of the centrifugal pump shaft system. (a) The first mode shape of the shaft (natural frequency: 140.66 Hz); (b) the second mode of the shaft (natural frequency: 419.31 Hz); (c) the third mode of the shaft (natural frequency: 857.20 Hz).

Fig. 11 Finite element model of the shaft (Magenta: the rotor and shaft of the motor; purple: impeller and shaft of the pump; green: the rubber ring for a coupling; blue: the pin of the coupling).

Fig. 12 Flow chart of coupling calculation of centrifugal pump system.

3 Results and discussion

Figure 13 shows the torque curve of the centrifugal pump impeller when it rotates for one cycle. Due to rotation of the centrifugal pump impeller, the impeller blade repeatedly passes through the place near the volute tongue, and with the periodic change of the flow passage width, the boundary conditions at the outlet of the impeller change, resulting in the interaction between the impeller and the volute [22]. It can be seen that this interaction leads to the fluctuation of impeller torque with the change of impeller angle position.

Figure 14 is the calculation result of the shaft's transient speed on the pump and motor sides. Figure 15 is the calculation result of the outlet flow of the centrifugal pump. In this paper, the “motor-shaft-pump" coupling simulation model is used to calculate the operation state under the design load. Under this condition, the speed of the motor and the pump is about 2955 r/min, and the pump's flow rate is about 110 m³/h. To speed up the calculation, the motor and water pump are simulated, respectively. In the calculation process, the motor and pump speed is set at a fixed value of 2955 r/min, and the pump flow is set at a fixed value of 110 m³/h. The calculation time was 0.2 s, during which the motor and pump rotated for about 10 cycles. Then, the motor and water pump simulation results calculated separately are used as the initial state of the subsequent coupling calculation model.

There are apparent fluctuations in the calculation results of the speed and flow rate at the beginning of the calculation. The rate-torque relationship between the centrifugal pump and the motor is not perfectly matched at the beginning of the calculation. After about 0.35s, the calculated results tend to be stable, and the calculated results in the stationary stage are intercepted for subsequent analysis. In addition, it can also be found in Figure 14 that the rotational speed variation trend of the centrifugal pump impeller and the motor rotor is not consistent. According to the calculation results in Figure 10, it can be inferred that this phenomenon is due to the dynamic characteristics of the rotating shaft under the action of torque fluctuations at both ends of the rotating shaft.

It can be seen from Figures 14 and 15 that after the coupling calculation starts, the coupling simulation model of the centrifugal pump system needs a little time to transition to a stable state. The impeller's instantaneous speed and the centrifugal pump's flow rate are not changed to a specific constant value but fluctuate within a range. This phenomenon is consistent with the actual situation because the motor's torque fluctuation law is different from the pump's torque fluctuation law, which causes the rotational acceleration of the shaft to fluctuate up and down around a specific equilibrium position. Hence, the actual operating speed of the centrifugal pump is not a specific constant value.

Figure 16 is the frequency spectrum of the rotational speed at different positions of the shaft. It can be seen from Figure 16 that after the centrifugal pump is running stable, the main frequency component of the impeller speed fluctuation is about 124 Hz. Figure 17 shows the vector of amplitude and relative phase of the peak at 124 Hz. It is worth noting that the vector shown in Figure 17 is similar to the first-order mode shape calculated in the previous section. This calculation result confirms that the shaft's speed fluctuation calculated by the simulation model is related to the shaft's dynamic characteristics; that is, the torque excitation causes the occurrence of the speed fluctuation.

Figure 18 shows the calculation results of pressure pulsation at the outlet of the centrifugal pump. It can be seen that the pressure fluctuation of the centrifugal pump is gradually stable with the stability of rotating speed, and the calculation results of pressure fluctuation after stability are intercepted for subsequent analysis. By comparing the flow field calculation results obtained from the traditional calculation model (impeller speed is constant, impeller speed is 2955 r/min) and the coupled calculation model used in this paper (as shown in Fig. 19), it can be found that a new peak appears in the pressure pulsation at the outlet of the centrifugal pump. In the calculation results obtained from the traditional centrifugal pump calculation model, the peaks near 300 Hz, 600 Hz, and 900 Hz are the passing frequency of the blade and its multiplication frequency, respectively.

When the speed is constant, the frequency components of pressure pulsation include the passing frequency of the blade and its multiplication frequency. When the speed fluctuates, the time interval of impeller passing through the tongue is not equal, so the frequency component of pressure fluctuation has the peak value related to the speed fluctuation. The torque excitation of the motor and pump excites the natural frequency of the shaft near 124 Hz, making the rotating speed fluctuate when the shaft drives the centrifugal pump's impeller. The fluctuation of impeller instantaneous speed further affects the pressure pulsation at the pump outlet. Besides the passing frequency of the blade and its multiplication frequency, there are new peaks in the frequency components of pressure fluctuation, which are affected by the dynamic characteristics of the rotating shaft. This phenomenon confirms the coupling relationship of vibration between equipment.

The pressure signal and the vibration signal of the centrifugal pump are collected to verify the calculation results of the simulation model. The experimental condition is the design condition of the centrifugal pump (flow rate is 110 m³/h). Two fast-response piezoresistive pressure sensors (CYG-401, SQ) were installed at the inlet and outlet of the pump to record the static pressure signal and then calculate the pump head. The maximum sampling frequency of the sensor was 20 kHz with an uncertainty of 0.2%. Flow rate is measured by a turbine flowmeter (LDG-32S, XE) mounted on the outlet pipe with a maximum sensor deviation of 1.5%. The pressure fluctuation signal is collected at the outlet pipe of the centrifugal pump, and the vibration signal is collected near the installation position of the pump frame (directly above the vibration isolator). The pressure sensor installed in the outlet pipe is a piezoelectric pressure sensor (106B51-PCB) with a resolution of 0.1% FS and an uncertainty of 0.1%, which is used to collect pressure pulsation signals. The piezoelectric acceleration sensor (ENDEVCO 2258a-100) was used to measure the acceleration signal of the structure. When the frequency of the signal is greater than 20 Hz and less than or equal to 100 Hz, the uncertainty of the sensor is 1.5%. When the frequency of the signal is greater than 100 Hz and less than or equal to 2500 Hz, the uncertainty of the sensor is 1.2%. All signals are collected and stored by the acquisition instrument (3560C-B&K).

The spectrum results of pressure fluctuation are shown in Figure 20. It can be seen that there is a noticeable peak at 124 Hz, which is consistent with the simulation results. Because the pressure pulsation in the centrifugal pump directly acts on the volute, the vibration of the centrifugal pump casing is transmitted to the pump bracket installation position through the mechanical structure. Finally, the peak can be found on the spectrum of the vibration signal, as shown in Figure 21.

Comparing the result of the experiment and the simulation results, the shaft frequency peak of measured pressure and vibration signal is obvious. This phenomenon is due to the shaft frequency peak value associated with the processing and installation precision of the centrifugal pump. In the simulation process, the installation accuracy of centrifugal pump is ideal, so the experimental results are different from the simulation results. In addition, the damping effect in the pumps and the motors is very complex. Still, this paper focuses more on the influence of the coupling effect of the centrifugal pump system on the frequency characteristics, so the influence of the damping on the centrifugal pump system is ignored in the calculation process, which also leads to the difference in the amplitude of the peak between the simulation results and the test results.

Combined with the simulation model results in this paper, the phenomenon in Figure 1 is further analyzed and discussed. The reason for the different changes is that the average speed of the centrifugal pump impeller increases, resulting in the decrease of the attached water mass from the centrifugal pump impeller, which further affects the dynamic characteristics of the shaft. The change of the equivalent mass of the centrifugal pump impeller causes the natural frequency of the shaft to change, resulting in a change in the frequency position of the peak near 124 Hz. This phenomenon indirectly explains the cause of this peak and verifies the rationality of the results obtained by the calculation model in this paper.

However, there are still some aspects of this paper that can be further improved. In the experiment, only the pressure signal at the pump outlet and the vibration signal at the pump bracket installation position is detected. If other types of signals (shaft speed, torque and motor power) in the centrifugal pump system can be detected and compared with the calculation results, it will be more convincing. Moreover, if the vibration simulation calculation of the centrifugal pump system can be further carried out, it will be more practical. This stage of the coupling simulation model is still in the very initial stage if somebody can simulate the running state of other conditions, and even to the “motor-shaft-pump” system from the start to the running stability of the complete process simulation in detail, believe that this model can get more meaningful analysis. In short, more abundant experimental data and more comprehensive simulation calculation will help to obtain more analysis and conclusions.

Fig. 13 Curve of impeller torque changing with rotation angle (centrifugal pump speed is 2955 r/min, flow rate is 110 m3/h).

Fig. 14 Calculation result of impeller's transient speed.

Fig. 15 Calculation result of pump outlet flow.

Fig. 16 Rotational speed fluctuation spectrum at different positions of the shaft.

Fig. 17 Relative speed at different positions of the shaft at 124 Hz.

Fig. 18 Calculation results of outlet pressure pulsation of centrifugal pump.

Fig. 19 Calculation results of pressure fluctuation in water pump outlet flow field.

Fig. 20 Spectrogram of pressure pulsation (centrifugal pump speed is about 2955 r/min, flow rate is about 110 m3/h).

Fig. 21 Spectrogram of vibration response (centrifugal pump speed is about 2955 r/min, flow rate is about 110 m3/h).

4 Conclusions

In this paper, the multi-field coupling simulation model of the motor-shaft-pump system is proposed. The conclusions of this study can be summarized as follows:

• Comparing the calculation results and the experimental results, it can be found that the coupled simulation model proposed in this paper can reflect the relationship between the vibration of the motor, the water pump, and the flexible shaft. In the calculation process, the fluctuation of the motor's torque and the pump impeller makes the rotation speed of the driveshaft not constant.

• It can be seen from the calculation results that not all the frequency components of the pressure pulsation of the water pump come from the changes in the internal flow field of the centrifugal pump, and the dynamic characteristics of the drive shaft will also affect the pressure pulsation in the centrifugal pump. Therefore, it is necessary to conduct coupling calculations and analysis on the motor-shaft-pump system.

• Other equipment in the centrifugal pump system, such as the rotating shaft and the drive motor, can affect the running state of the centrifugal pump system. The pressure pulsation in the centrifugal pump directly acts on the centrifugal pump casing to cause the vibration of the pump casing, so the analysis of the vibration signal of the centrifugal pump system should not be limited to a single device of the centrifugal pump.

Conflict of interest

The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Financial support

This work was supported by the National Natural Science Foundation of China (Grant number 51805106), the authors would like to sincerely express their appreciation.

References

All Tables

All Figures

	Fig. 1 Vibration response spectrum of pump casing at different flow rates. (a) 0–800 Hz; (b) 115–130 Hz; (c) 290–305 Hz.
In the text

	Fig. 2 Semi-closed simulation model of centrifugal pump piping system.
In the text

	Fig. 3 The corresponding relationship between the simulation model and the actual structure of reservoir.
In the text

	Fig. 4 Calculation grid of semi-enclosed centrifugal pump piping system.
In the text

	Fig. 5 Mesh grids independence analysis.
In the text

	Fig. 6 Calculation results of centrifugal pump external characteristics (centrifugal pump speed is about 2955 r/min, Q is equal to 110 m3/h).
In the text

	Fig. 7 Schematic diagram of the two-dimensional cross-section of the motor.
In the text

	Fig. 8 Two-dimensional finite element mesh of the motor.
In the text

	Fig. 9 Lumped parameter model of the centrifugal pump shaft. (a) Geometric model; (b) lumped parameter model.
In the text

	Fig. 10 The first 3 order mode of the centrifugal pump shaft system. (a) The first mode shape of the shaft (natural frequency: 140.66 Hz); (b) the second mode of the shaft (natural frequency: 419.31 Hz); (c) the third mode of the shaft (natural frequency: 857.20 Hz).
In the text

	Fig. 11 Finite element model of the shaft (Magenta: the rotor and shaft of the motor; purple: impeller and shaft of the pump; green: the rubber ring for a coupling; blue: the pin of the coupling).
In the text

	Fig. 12 Flow chart of coupling calculation of centrifugal pump system.
In the text

	Fig. 13 Curve of impeller torque changing with rotation angle (centrifugal pump speed is 2955 r/min, flow rate is 110 m3/h).
In the text

	Fig. 14 Calculation result of impeller's transient speed.
In the text

	Fig. 15 Calculation result of pump outlet flow.
In the text

	Fig. 16 Rotational speed fluctuation spectrum at different positions of the shaft.
In the text

	Fig. 17 Relative speed at different positions of the shaft at 124 Hz.
In the text

	Fig. 18 Calculation results of outlet pressure pulsation of centrifugal pump.
In the text

	Fig. 19 Calculation results of pressure fluctuation in water pump outlet flow field.
In the text

	Fig. 20 Spectrogram of pressure pulsation (centrifugal pump speed is about 2955 r/min, flow rate is about 110 m3/h).
In the text

	Fig. 21 Spectrogram of vibration response (centrifugal pump speed is about 2955 r/min, flow rate is about 110 m3/h).
In the text