# Research on Vibration Reduction in the Transmission System of Plastic Centrifugal Pump
## 4 Experimental Study on the Relationship between Pump Shaft Speed and Deflection  
This experiment aims to analyze the impact of rotational speed changes on the deflection of a centrifugal pump's shaft rotor system below its first critical speed. The test was conducted using a ZT-3-type rotor vibration test platform. The rotor at the cantilever end served as the experimental measurement point, with a span-to-cantilever ratio of 1:0.75. The test rig is depicted in Figure 14, and the actual setup is shown in Figure 15.  
The experimental platform utilizes the DH5910A data acquisition system, which offers flexibility in selecting the type of channels, supporting up to 128 measurement channels. The system operates in a frequency range from 10 Hz to 128 Hz, accommodating most types of measurement requirements. The eddy current sensor model used in the platform is HZ-891 YT08 HP-M10\(\times\)1-B-01-05-50. The testing software employed is DHDAS2013, a signal testing and analysis software featuring comprehensive analysis capabilities, including time-domain waveforms, bode plots, and shaft center trajectory representations. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & **Without Silicone Oil** & **With Silicone Oil** & \multirow{2}{*}{**Comparison**} \\  & **Damper** & **Damper** & \\ \hline Frequency (H\({}_{\text{z}}\)) & 151.84 & 135.53 & Decrease by 10.45\% \\ Amplitude (rad) & \(1.40\times 10^{-3}\) & \(0.71\times 10^{-3}\) & Decrease by 49.29\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of torsional vibration reduction effects of the pump shaft. Figure 12: Assembly diagram of silicone oil damper. Figure 13: Assembly diagram of silicone oil damper with coupling. The test rig is powered by a direct current (DC) motor, connected to the shaft via a rigid coupler, which drives the disc rotor to rotate. The rotational speed of the system is captured by a photoelectric speed sensor, while the vibrational displacement is measured by an eddy current sensor. The signals, after being filtered and shaped by the data collector, are transmitted to a monitoring computer through a 1394 interface card. The setup of the testing system is illustrated in Figure 16.  
Table 3 presents the experimental data correlating pump shaft rotational speed with deflection, as obtained from the test platform. The experimental data were fitted using MATLAB R2016a software, employing polynomial fitting methods ranging from the first to sixth order. The fitting results are shown in Figure 17.  
The mathematical expressions for the first- to sixth-order polynomials are shown in Table 4.  
Figure 16: Test platform measurement system setup. Figure 14: ZT-3 rotor simulation test platform. Figure 15: Actual ZT-3 rotor test rig. R-square is used to describe the degree of fit of a model to data, ranging in value from [0, 1]. Smaller values indicate poorer fit, and larger values indicate better fit, as shown in Equation (28). \[ \begin{array}{c}R-square=\frac{SSR}{SST}=\frac{SST-SSE}{SST}=1-\frac{SSE}{SST}\\ SSR=\sum\limits_{i=1}^{n}\left({{\hat{y}}_{i}}-{\overline{y}}\right)^{2}; \;SST=\sum\limits_{i=1}^{n}\left({{y}_{i}}-{\overline{y}}\right)^{2};\;SSE= \sum\limits_{i=1}^{n}\left({{y}_{i}}-{\hat{y}}\right)^{2}\end{array}  \tag{28}\]  
where  
\(\overline{y}\)--represents the mean value;  
\(y_{i}\)--The sample value;  
\(\hat{y}\)--The fitted value.