# Research on Vibration Reduction in the Transmission System of Plastic Centrifugal Pump
## 3 Study of Vibration Reduction in the Pump Shaft  
### Forced Torsional Vibration Model of the Pump Shaft  
The shaft system of a single-stage centrifugal pump usually employs a single-span cantilever structure, as depicted in Figure 3. Due to the impeller's location at the extremity of the shaft system, its weight and load can induce bending deformations in the pump shaft. This deformation can lead to an increase in the reaction force on the bearings, potentially harming their normal functioning. During operation, the centrifugal pump's bearings are subjected to both radial and torsional vibrations caused by the system's inertia and elasticity. The phenomenon of resonance may arise when the vibration frequency of the pump shaft aligns with its internal vibration frequency. If the torsional stress exceeds the pump shaft's allowable stress, there is a risk of shaft fracture.  
Figure 1: Centrifugal pump model.  
Figure 2: The two-dimensional engineering drawing.  
The pump shaft of the plastic centrifugal pump incorporates an undamped single-mass system. Its structure can be referenced in Figure 4[20].  
By applying a continuous excitation torque, \(M_{1}\)\(sin\omega t\), the system transforms into a forced torsional vibration system. Disregarding the impact of damping, the motion differential equation of the system is expressed by Formula (11).  
\[\ddot{\varphi}+\omega^{2}\varphi=\frac{M_{1}sin\omega_{1}t}{I} \tag{11}\]  
where  
\(\omega_{1}\)--Excitation frequency, (\(Hz\));  
I--The moment of inertia of the disk, (kg\(\cdot\)m\({}^{2}\));  
\(\varphi\)--The angular displacement of the disk, (rad);  
\(M_{1}\)--The excitation torque, (N\(\cdot\)m).  
The comprehensive solution to this differential equation is  
\[\varphi=Asin(\omega t+\varphi_{0})+A_{1}sin(\omega_{1}t-\psi)\]  
where  
\(A\)--The maximum angular displacement of the disk during torsional vibration is the amplitude, (rad);  
\(\varphi_{0}\)--Initial phase, (rad).  
The influence of the excitation torque \(M_{1}\) on free vibration is akin to the impact of the initial excitation torque \(M_{0}\), typically represented by the static amplitude \(A_{0}\).  
\[A_{0}=\frac{M_{1}}{K} \tag{12}\]  
where  
\(K\)--The equivalent torsional stiffness of the system.  
The calculated result is  
\[A_{0}=5.41\times 10^{-4}\text{rad}\]  
Figure 4: Equivalent system diagram of the pump shaft.  
Figure 3: Structural diagram of the pump shaft system.  
The amplitude of forced vibration, \(A_{1}\), is computed using Formula (13), influenced by the excitation torque \(M_{1}\).  
\[A_{1}=\frac{M_{1}}{I\sqrt{\omega^{2}-\omega_{1}^{2}}}=\frac{M_{1}}{I\omega^{2} \sqrt{1-\left(\frac{\omega_{1}}{\omega}\right)^{2}}} \tag{13}\]  
where  
\[\frac{M_{1}}{I\omega^{2}}=\frac{M_{1}}{I}\cdot\frac{I}{K}=\frac{M_{1}}{K}=A_{0}\]  
Thus,  
\[\begin{array}{c}A_{1}=\frac{A_{0}}{\sqrt{1-\left(\frac{\omega_{1}}{\omega} \right)^{2}}}\\ \omega_{1}=\frac{2\pi n}{60}=151.84\text{ Hz}\\ A_{1}=1.40\times 10^{-3}\text{rad}\end{array}\]  
The phase angle \(\psi\) is a crucial parameter that illustrates the angle between the excitation torque \(M_{1}\) and the displacement \(\varphi\). In a non-damped forced vibration scenario, this angle \(\psi\) typically takes values of \(0\) or \(\pi\). Hence, the sign of the amplitude value can express it [21].