Wet Modal Analyses of Various Length Coaxial Sump Pump Rotors with Acoustic-Solid Coupling

The dynamic characteristics of the rotor components were determined using a ﬁrst-order modal model of the rotor components for various sump pump shaft lengths for actual working environments. By employing ANSYS-Workbench software, this paper uses a ﬂuid-solid coupling analysis to calculate the reaction forces of the ﬂuid on the rotor with results, which is then used in dry and wet modal analyses of the rotor parts to calculate the vibration modal characteristics with and without prestresses. The diﬀerences between the wet and dry modal characteristics were compared and investigated by ANSYS. The results show that increasing the sump pump shaft length reduces the ﬁrst-order natural frequency of the prestressed rotor components. The structure also experiences stress stiﬀening, which is more obvious in the high-order modes. The natural frequency of the rotor in the wet mode is about 16% less than that in the dry mode for the various shaft lengths due to the added mass of the water on the surface which reduces the natural frequency. In the wet modal analysis, when the structure is in a diﬀerent ﬂuid medium, the inﬂuence of its modal distribution will also change, this is because the additional mass produced by the ﬂuid medium of diﬀerent density on the structure surface is diﬀerent. Thus, the wet modal analysis of the rotor is important for more accurate dynamic analyses.


Introduction
A sump pump is a vertical pump that all the flow components, such as the impeller and the pump body, are immersed in the fluid when connected to the motor [1]. e motor is fastened to the motor base and drives the rotor through a coupling [2].
ere are many kinds of sump pumps. Traditional sump pumps are used to transport corrosive materials such as strong acids, alkalis, salts [3], and strong oxidants [4] at various concentrations. New sump pumps are used to transport flammable and explosive materials. e liquid level in the pump can be changed by changing the shaft length for various applications. e working environment of sump pumps is complicated, and it is often affected by dynamic loads such as wave currents and shock vibrations [5]. e vibration of the structure not only affects the working stability but also affects the fatigue life [6]. erefore, understanding the natural frequency and critical speed of the structure can provide a certain basis for preventing the structure from resonance damage and vibration fatigue, evaluating the dynamic characteristics of the structure and optimizing the structure [7,8]. Rotating machinery often undergoes lateral bending vibration due to the eccentricity of the elastic rotor of the rotating shaft during operation. When the speed increases to a certain value, the amplitude will suddenly increase and the vibration will be extremely intense. When the speed exceeds this certain value, the amplitude will quickly decrease.
is specific value is called the critical speed of the rotor. Safe operation of the sump pump requires that the rotor speed avoids the critical speed, so the critical speed should not be within 20% of the operating speed [9]. Vibration modal analyses are used to analyze the complex dynamic characteristics. erefore, good dynamic characteristic analyses require accurate modal analyses for the actual operating conditions. e literature has numerous studies on the internal flow characteristics and structural optimization of sump pumps [10,11]. However, there are few modal analyses of sump pump rotor components. e sump pump rotor components are subjected to axial and radial forces during operation, including the fluid force, gravity, and centrifugal forces. e fluid force then leads to stress stiffening which changes the rotor modal distribution [12,13]. In addition, when the impeller rotates in the fluid, the additional mass produced by the fluid on the structure varies as the rotor rotates which also affects the modal characteristics [14][15][16]. erefore, sump pump designs must include modal analyses of the rotor components.
In this study, the reaction force of the fluid on the impeller of sump slurry pump was calculated using a coupled fluid-solid analysis [17,18]. e prestresses due to these forces were then used in the "dry" and "wet" modal analyses of the sump pump rotor [19] to determine the natural frequency and critical speed of the rotor. e natural frequency and critical speed of the sump pump rotor were then calculated for various shaft lengths. Comparisons with existing manufacturer data verified the accuracy of the simulation results. ese results provide a reference for analyses of the structural dynamic characteristics of sump pumps to improve the structural designs.

Modal Analysis Theory
e general dynamics equation describing discrete vibrations with N degrees of freedom in the physical coordinate system [19] is where [M] is the mass matrix (symmetric and positive definite), [C] is the damping matrix, [K] is the stiffness matrix (symmetric and positive definite or semipositive definite), and F(t) { } is the N-dimensional excitation force vector. e excitation forces in this study include the fluid force, gravity, and centrifugal force.
Assuming that the initial system state is stationary, the Laplace transform of equation (1) is where [Z (s)] is which reflects the dynamic characteristics of the system and is called the system dynamics matrix or the generalized impedance matrix. In equation (3), s � jω and the impedance matrix is e frequency response function matrix in terms of the modal parameters is then e elements of [H(ω)] in row i and column j of the matrix are where ω 2 r � (k r /m r ) is the frequency of mode r, ξ r � (c r /2m r ω r ) is the damping ratio of mode r, and ϕ r is the mode shape of mode r. e frequency response of an N degree-of-freedom system is then equal to the linear combination of the frequency responses of N single-degree-of-freedom systems. All the modal parameters can then be determined from only one column or one row of the frequency response function matrix.

Modeling, Meshing, and Boundary Conditions
3.1. 3D Model and Mesh. Figure 1 shows the rotor components of the 100SP sump pump. e sump pump rotor consists of an impeller, a pump shaft, and two bearings. e impeller is an open impeller. Table 1 shows the main parameters of the sump pump. e pump assembly drawing is shown in Figure 2 where length A � D + 529, length B � D − 348, and D is the pump installation elevation which was varied to vary the pump shaft length (D � 1500 for the standard design pump). Table 2 lists the pump shaft lengths for various D. e sump pump flow region included the inlet extension, impeller, volute, and outlet extension. e flow region was modeled in a 3D model, as shown in Figure 3. e fluid domain model uses ANSYS-ICEM for unstructured meshing, and the mesh quality is above 0.3. e solid domain model is adaptive tetrahedral meshing, and the average mesh quality is above 0.7, as shown in Figure 4. In order to study whether the number of grids will affect the calculation results, it is also necessary to verify the grid independence. In this paper, 4 sets of different numbers of sump pump fluid domain model grids are selected, and a set of reasonable number of grids is selected through calculation and comparison of the head changes at Q � 216 m 3 /h. It can be seen from Table 3 that when the number of selected 4 sets of grids is very different, the fluctuation range of the calculated head is within 1.5%. Starting from the selected second set of grids, the calculated head value has become flat, so the second set of grids is considered appropriate. e fluid domain had 2,400,584 elements while the solid domain had 273,376 elements. e impeller and pump shaft material properties are listed in Table 4.

Boundary Conditions.
e pump components modal distribution was calculated in the presence of prestresses due to the fluid forces. is paper takes the clean water as the research medium, and the density and sound velocity properties of the required water will be defined when the acoustic-structure coupling method is used for the wet modal analysis later. e flow field in the submerged pump is unsteady, so the flow calculations should be unsteady. However, the natural frequencies for steady and unsteady flow conditions are known to be very similar [20], so a steady flow field was used here for the calculations. e flow calculations assumed that the fluid was a steady incompressible flow with the standard k − ε turbulence model [21]. e inlet was set as a total pressure inlet, the outlet was set as a mass flow outlet, and the walls were set as nonslip walls. In the solid domain, the submerged pump was assumed to have rigidly supported rolling bearings constrained by a cylindrical surface, so all the radial degrees of freedom were fixed. e shaft end and the coupling were connected with fixed restraints. e pump shaft and the impeller were connected by a key and a contact surface.

Experimental Validation.
Before the modal analysis, the full flow field in the sump pump was predicted for various working conditions to obtain the flow-head curve for the sump pump to compare with manufacturer data, as shown    Shock and Vibration 3 in Figure 5. e manufacturer data shown in Table 5 came from the characteristic curve released by Warman International [22]. e predicted heads agree well with the manufacturer data, as shown in Figure 5 with the same trends which shows that the steady flow field predictions are quite accurate. e mode analysis used the flow field at the rated flow of Q � 216 m 3 /h.

Dry Mode Analysis without the Prestresses.
e dry modal analysis without prestresses is a modal analysis in a vacuum without other factors to determine the natural frequency of the rotor system. e system constraints include the fixed constraint where the top of the rotor is connected to the coupling and the cylindrical constraint at the bearing, as shown in Figure 6 for a standard pump as an example, where A is the rolling bearing support, B is the cylindrical surface constraint, and C is the consolidation surface support. e dry modal analysis was applied to the standard pump rotor along with various other shaft lengths without the prestresses and with the same constraint conditions to calculate the first-order natural frequencies listed in Table 6.
Without the prestress, the first-order natural frequency of the sump pump rotor gradually decreases with increasing shaft length with the rate of change gradually decreasing. With the fixed support conditions, increasing the shaft length increases the shaft support span which reduces the strength and the natural frequency of the rotor.

Dry Modal Analysis with the Prestresses.
e dry modal analysis with the prestresses is a modal analysis with the rotor operating in a vacuum. e prestresses in the rotor components during operation include the gravitational force on the rotor itself, the centrifugal force generated by the     rotation, and the force of the water on the impeller. e system constraints are the same as without the prestresses. Since the sump pump is vertical, gravity acts directly along the pump axis. e centrifugal force is related to the rotational speed of the rotor and the fluid force is loaded on the impeller blades and hub. e specific results are shown in Figure 7 which shows that the maximum fluid force on the impeller appears at the blade outlet with the minimum fluid force at the blade root. In the figure, the imported pressure represents the fluid force loaded on the blade while imported pressure 2 and imported pressure 3 represent the fluid forces loaded on the impeller hub. e dry modal analysis of the pumps with various shaft lengths with the prestresses gave the first natural frequencies of the rotors listed in Table 7. Figure 8 shows that the rotor natural frequencies with and without the prestresses are basically the same. e first-order critical speed again decreases with increasing shaft length, while the first-order natural frequencies of the rotor components are slightly higher than without the prestresses, which shows that the prestresses make the pump rotor components and the structure more rigid. is stiffening is more obvious for the higher-order modal natural frequencies for D � 1500, as shown in Figure 9. e natural frequency of the third-order mode increases the most, reaching 16.82 Hz, while the natural frequencies of the other modes all increase more than the natural frequency of the first-order mode, indicating increased stress rigidity at higher-order modes. e higher-order natural frequencies for other D also showed this phenomenon.

Wet Modes of Rotors with Various Shaft Lengths with Prestresses.
e wet modal analysis with prestresses models the rotor running in a fluid. e modal analysis in the "wet" state is closest to the actual rotor operation. is not only considers the effect of the prestresses on the rotor but also considers the additional mass due to the fluid on the surfaces     when the rotor is running. In the dry modal analysis, the vacuum had negligible influence on the vibration modes and could be ignored. is study used an acoustic-solid coupling model to separately determine the flow in the impeller water region and the dynamic characteristics of the coupled fluidsolid model. e wet modal analysis of the various length rotors in the sump pump used water as the working medium with the results closer the results of the natural frequency and critical speed of the actual operation of the sump pump rotor [23,24] is shown in Table 8. e predicted and measured first-order critical speeds are compared, as shown in Figure 10 which shows little difference between the predictions and the manufacturer data with very similar trends. e differences are mainly due to the limitations of the flow field calculations, especially since the flow model cannot perfectly reproduce all the details of the actual geometry. e first-order critical speed of the rotor gradually decreases as the shaft length increases. e predicted natural frequencies for the wet mode are compared with those for the dry mode with the prestresses, as shown in Figure 11. e results show that the rotor natural frequencies are lower in the wet mode than in the dry mode for all the shaft lengths with an average decrease of about 16%. When the pump is submerged, the water adds additional weight on the surfaces which affects the dynamics and reduces the natural frequency. Additional calculations showed that changing the water to slurry reduced the natural frequency even more because the slurry is denser than water.
us, when the sump pump works in slurry, the additional    Shock and Vibration  Manufacturer data Simulated results Figure 10: Comparison of the predicted first critical speeds with measured data [22].  mass of the slurry on the rotor components further reduces the natural frequency. It can be inferred from this that when the structure works in different fluid media [25], due to the different density of the fluid media, the additional mass produced on the surface of the structure will also change, and the mode of the structure will be affected to different degrees.

Conclusions
e natural frequencies of various length shafts in a sump pump were analyzed with and without flow in the pump. e results show that the calculations showed that the first-order natural frequency and the critical speed of the sump pump rotor structure decreased with increasing pump shaft length.
is trend has been seen in previous manufacturer data. e dry mode analysis showed that application of the fluid forces and other prestresses to the sump pump rotor components resulted in stress stiffening of the components with higher natural frequencies of the rotor components for the various shaft lengths than with no prestresses. e stress stiffening was even more obvious in the higher-order modes. Acousticsolid coupling was used with a wet modal analysis of the sump pump rotor components with the prestresses. e results show that the natural frequency of the pump rotor components of the wet mode is about 16% less than that in the dry mode because the additional mass of the water on the surfaces reduces the natural frequency of the structure. us, changing the fluid medium in the pump, such as using slurries, will change the additional mass on the surfaces due to the different densities which will then affect the pump rotor modal distribution.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.