## 2 Operating Principle of the Active Balancing System
### Balancing Principle  
The oscillatory differential equation of a rotor system can be expressed by:  
\[m\ddot{x}+c\dot{x}+kx=F_{0}\cos(\omega t+\varphi) \tag{1}\]  
where the mass, the damping, the stiffness, and the rotating speed of the rotor are expressed by \(m,C,k\), and \(\omega\), respectively; and \(F_{0}\) is the unbalance force of the rotor system. By solving the differential Equation, the vibration amplitude of the rotor caused by imbalance is given by the following Equation:  
\[x=\frac{F_{0}}{\sqrt{\left(k-m\omega^{2}\right)^{2}+c^{2}\omega^{2}}} \tag{2}\]  
Before balancing the vibration of the rotor system with liquid actuator, a balancing disc rotating synchronously with the rotor should be installed on the rotor in advance. The balancing disc contains four liquid chambers to store the balance liquid. Through adjusting the liquid mass distribution in the balancing disc, a correction mass is formed, and then the active balancing system can on-line balance the rotor system. When the balancing disc rotates synchronously with the rotor, the balance liquid in the liquid chamber distributes annularly along the outer wall of the chamber under centrifugal force, as shown in Figure 3. Assuming \(V^{\prime}\) is the liquid volume in one chamber, through an integration calculation, the centrifugal force formed by the liquid in the chamber is derived as follows:  
\[F=\frac{2}{3}\rho B\omega^{2}\Bigg{[}R^{3}-\left(R^{2}-\frac{2 Vt}{B\theta} \right)^{\frac{3}{2}}\Bigg{]}\sin\frac{\theta}{2} \tag{3}\]  
Figure 2: Structure of the balance actuator. where \(\omega\) is the angular speed of the rotor, \(\rho\) is the density of balance liquid, \(R\) is the outer radius of the liquid chamber, \(B\) is the depth of the liquid chamber, and \(\theta\) is the central angle of the chamber. Therefore, the resultant force \(F_{x}\) in the \(x\)-axis direction can be derived by:  
\[ \begin{array}{ll}F_{x}&=F_{A}-F_{C}\\ &=\frac{2}{3}\rho B\omega^{2}\Bigg{[}(R^{2}-\frac{2V_{C}^{\prime}}{B\theta})^ {\frac{3}{2}}-\ (R^{2}-\frac{2V_{A}^{\prime}}{B\theta})^{\frac{3}{2}}\Bigg{]}\sin \frac{\theta}{2}\end{array}  \tag{4}\]  
For the liquid transfer type actuator, the same amount of balance liquid will be pre-filled into the four liquid chambers before balancing and the amount can by expressed by \(V^{\prime}\). In the working process, each pair of liquid chambers works at the same time. If the liquid in one chamber increases, then the liquid in the other chamber will inevitably decrease, thus the increase and decrease are equal. Assuming the volume of the transfer liquid is \(V_{x}\), Equation (4) can be rewritten to  
\[ \begin{array}{ll}F_{x}&=F_{A}-F_{C}\\ &=\frac{2}{3}\rho B\omega^{2}\Bigg{[}(R^{2}-\frac{2(V^{\prime}-V_{x})}{B\theta })^{\frac{3}{2}}-\ (R^{2}-\frac{2(V^{\prime}+V_{x})}{B\theta})^{\frac{3}{2}} \Bigg{]}\sin\frac{\theta}{2}\end{array}  \tag{5}\]  
The resultant force \(F_{y}\) in the vertical direction can be obtained in the same way, so the resultant force in the two directions can be orthogonally synthesized to obtain the composite force \(F_{1}\) formed by the actuator, that is \(F_{1}=\sqrt{F_{x}^{2}+F_{y}^{2}}\). The direction of the composite force is opposite to that of the unbalanced force. Therefore, the oscillatory differential equation of a rotor system can be rewritten as follows  
\[(m+m_{a})\ddot{x}+c\dot{x}+kx=(F_{0}-F_{1})\cos(\omega t+\varphi) \tag{6}\]  
where \(m_{a}\) is the mass of the balancing disc itself. In the design process, to minimize the effect of the additional mass on the dynamic characteristics of the rotor system, the mass of the balancing disc should be far less than the mass of the rotor. When the control unit of the balancing system drives the liquid transfer in the balancing disc, the compensation force \(F_{1}\) produced by the balancing disc will be approximately equal to the initial unbalance force \(F_{0}\), so the excitation force of the rotor system is minimized and the unbalance vibration amplitude of the system can be reduced to the set value.