# Research on Vibration Reduction in the Transmission System of Plastic Centrifugal Pump
## 3 Study of Vibration Reduction in the Pump Shaft
### Vibration Reduction Method and Design  
The silicone oil damper mainly consists of four components: the end cover, shell, inertia body, and supporting bearing, illustrated in Figure 5. The end cover and shell form a sealed cavity, within which the inertia body is free to rotate. Uniform gaps between these elements are maintained by the supporting bearing, and these gaps are filled with high-viscosity silicone oil. Integrating the silicone oil damper, the pump shaft system model simplifies into a dual-mass equivalent structure, as shown in Figure 6. The system's movement equations are represented by Formula (14). \[ \begin{array}{c}I_{d}\tilde{\varphi}_{d}+C_{d}\big{(}\tilde{\varphi}_{d}- \tilde{\varphi}_{e}\big{)}=0\\ I_{e}\tilde{\varphi}_{e}+C_{d}\big{(}\tilde{\varphi}_{e}-\tilde{\varphi}_{d} \big{)}+K_{e}\varphi_{e}=M_{1}e^{i\omega_{1}t}\end{array}  \tag{14}\]  
where  
\(I_{e}\)--Equivalent moment of inertia of the pump shaft system (kg\(\cdot\)mm\({}^{2}\));  
\(K_{e}\)--Equivalent torsional stiffness of the pump shaft system;  
\(C_{d}\)--Damping coefficient of the silicone oil damper;  
\(\hat{\varphi}_{e}\)--Angular velocity of the equivalent concentrated mass (rad/s);  
\(\hat{\varphi}_{d}\)--Angular velocity of the damper's inertia block (rad/s). Figure 5: Silicone oil damper assembly diagram. The solution to Equation (14) is given by  
\[ \begin{array}{l}\varphi_{d}=\theta_{d}e^{i\omega_{1}t}\\ \varphi_{e}=\theta_{e}e^{i\omega_{1}t}\end{array}  \tag{15}\]  
Substituting the solution of Formula (15) into Equation (14) yields  
\[ \begin{array}{l}\big{(}-I_{d}\omega_{1}^{2}+i\omega_{1}C_{d}\big{)}\theta_{d} -i\omega_{1}C_{d}\theta_{e}=0\\ -i\omega_{1}C_{d}\theta_{d}+\big{(}K_{e}-I_{e}\omega_{1}^{2}+iC_{d}\omega_{1} \big{)}\theta_{e}=M_{1}\end{array}  \tag{16}\]  
According to the determinant method,  
\[ \begin{array}{l}\theta_{d}=\frac{D_{d}}{D}=A_{d}e^{i\omega_{1}}\\ \theta_{e}=\frac{D_{e}}{D}=A_{e}e^{i\omega_{1}}\end{array}  \tag{17}\]  
where  
\(A_{d}\)--Actual amplitude of the damper's inertia ring disc;  
\(A_{e}\)--Actual amplitude of the pump shaft's equivalent disc. \[\begin{array}{l}D\quad=\left| \begin{array}{ccc}-I_{d}\omega_{1}^{2}+i \omega_{1}C_{d}&-i\omega_{1}C_{d}\\ -i\omega_{1}C_{d}&K_{e}-I_{e}\omega_{1}^{2}+iC_{d}\omega_{1}\\ =\big{(}I_{e}I_{d}\omega_{1}^{4}-I_{d}K_{e}\omega_{1}^{2}\big{)}+i\omega_{1}C_{ d}\big{(}K_{e}-(I_{d}-I_{e})\omega_{1}^{2}\big{)}\\ D_{d}=\left|\begin{array}{ccc}0&-i\omega_{1}C_{d}\\ M_{1}&K_{e}-I_{e}\omega_{1}^{2}+iC_{d}\omega_{1}\end{array}\right|=i\omega_{1 }C_{d}M_{1}\\ D_{e}=\left|\begin{array}{ccc}-I_{d}\omega_{1}^{2}+i\omega_{1}C_{d}&0\\ -i\omega_{1}C_{d}&M_{1}\end{array}\right|=-I_{d}\omega_{1}^{2}M_{1}+i\omega_{1 }C_{d}M_{1}\end{array} \]  
where  
\[A_{d}=|\theta_{d}|=\frac{\omega_{1}C_{d}M_{1}}{\sqrt{\big{[}I_{d}I_{e}\omega_ {1}^{4}-I_{d}K_{e}\omega_{1}^{2}\big{]}^{2}+\omega_{1}^{2}C_{d}^{2}\big{[}K_{e }-(I_{d}+I_{e})\omega_{1}^{2}\big{]}^{2}}} \tag{18}\]  
\[A_{e}=|\theta_{e}|=\frac{\sqrt{\big{(}-I_{d}\omega_{1}^{2}M_{1}\big{)}^{2}+( \omega_{1}C_{d}M_{1})^{2}}}{\sqrt{\big{[}I_{d}I_{e}\omega_{1}^{4}-I_{d}K_{e} \omega_{1}^{2}\big{]}^{2}+\omega_{1}^{2}C_{d}^{2}\big{[}K_{e}-(I_{d}+I_{e}) \omega_{1}^{2}\big{]}^{2}}} \tag{19}\]  
Formula (19) shows that the vibration characteristics of the pump shaft change significantly with the use of the silicone oil damper. In this case, the damping coefficient \(C_{d}\) emerges as the critical factor in assessing the effectiveness of the damper. According to Formula (19), the amplitude curve of the silicone oil damper is plotted in Figure 7. The graph indicates that under extreme conditions of \(C_{d}=0\) and \(C_{d}=\infty\), the pump shaft's amplitude sharply increases, potentially leading to resonance. Thus, between \(C_{d}=0\) and \(C_{d}=\infty\), there must exist an optimal damping point \(P\), where energy dissipation is maximized and the pump shaft system experiences minimal vibration. Figure 6: Simplified model of the pump shaft system with a silicone oil damper.