## 2 Dynamic Model of the System
### Analysis Modeling
As the impeller pulls up the fluid, continuous contact with the fluid occurs, and the force of the impeller pulling up the fluid increases with an increase in the density of the fluid. Therefore, the maximum reaction force of the pump was generated when the fluid is satiated, and the maximum excitation force of the pump, which can be confirmed through data, was 5083 N, as shown in the left graph of Figure 6.  
Based on the calculated reaction force of the pump, the maximum value of the force transmitted to the water tank is expressed as Equations (1)-(9). The pump mounted on  
Figure 5: Before and after pump operation. (right image, 4-s section). Figure 6: Reaction force data according to pump operation: (**a**) Pump reaction force in saturated state; (**b**) Water tank reaction force due to pump operation. the water tank generates a large vibration in the direction of gravity, and to meet the specifications of the scrubber, it is necessary to reduce the vibration of the water tank. Therefore, it is essential to verify the analysis data by comparing the maximum reaction force value of the water tank calculated for the direction of gravity with the theoretical value, such as the data on the right side of Figure 6. Equation (1) was used to calculate the kinetic equation with a 1 degree of freedom system to obtain the maximum force of the water tank transmitted in the direction of gravity. If the forced oscillation caused by the excitation of a one-degree-of-freedom system is expressed as a motion equation, it can be represented using Equation (1), and if the two sides are divided by m and the steady-state response is obtained, the amplitude can be calculated using Equation (2). In addition, as the pump was mounted on a water tank and a plate was placed between them, Equation (1) can be modified into Equation (3). Assuming that the transient response is 0, Equation (4) can be substituted into Equation (3) to obtain Equation (5), which can be summarized using sine synthesis to obtain Equation (6). Here, the maximum transfer force through which the reaction force of the pump is transmitted to the water tank can be expressed as Equation (7). By substituting Equation (2) into Equation (7) and applying and summarizing \(f_{0}=F_{0}/m\) Equation (8) can be obtained. Figure 3:0  
Figure 3:1  
Figure 3:2  
Figure 3:3  
Figure 3:4  
Figure 3:5  
Figure 3:6  
Figure 3:7  
where \(\left(F_{T}\right)_{max}/F_{0}\) is the ratio of the maximum value, \(\left(F_{T}\right)_{max}\) of the force transmitted to the floor to the maximum value, \(F_{0}\), of the reaction force \(f(t)\) applied to the water tank. To minimize the force transmitted by the reaction force of the pump to the water tank, the value of \(\left(F_{T}\right)_{max}/F_{0}\) should be minimized. The second term, \(mg/F_{0}\), on the right side of Equation (8) can be removed from the equation if the vibration in the horizontal direction of the pump is excluded. In addition, when the denominator and molecule are divided into \(\omega_{n}{}^{2}\), and making it dimensionless with the frequency ratio \(\gamma\), they can be summarized as expressed in Equation (9). The force transmission rate (\(TR\)) according to the reaction force of the pump in the direction of gravity can be expressed as Equation (9). Figure 3:8  
To calculate the maximum value of the force transmitted in the direction of gravity to the water tank, the coefficients of Equation (9) must be obtained. As the rotation speed of the pump is 1775 RPM, the exciting frequency, \(\omega\), is 185.8 rad/s. In addition, the natural frequency \(\omega_{n}\) is calculated as 49.9 rad/s when considering the stiffness value in Table 4 because the weight of the pump is 472 kg.