## 3 Performance Analysis of Balance Actuator  
### Balancing Velocity  
For the liquid-transfer balance actuator, one connecting tube was used to connect two contra-positioned chambers, as shown in Figure 4. Here, we take the contra-positioned chambers A and C, as an example. Point 1 and Point 3 denote the two ends of the connecting tube. Between these two points, the Bernoulli equation is expressed by  
\[gz_{1}+\frac{p_{1}}{\rho}+\frac{{u_{1}}^{2}}{2}-\frac{\omega^{2}}{2}{r_{1}}^{2 }-k\frac{u^{2}}{2}=gz_{3}+\frac{p_{3}}{\rho}+\frac{{u_{3}}^{2}}{2}-\frac{ \omega^{2}}{2}{r_{3}}^{2} \tag{7}\]  
where \(u\) refers to the flow velocity of the balance liquid, and \(k\) stands for the drag coefficient along the transfer path. The radius \(r_{1}\) is identical to that of \(r_{3}\), and the velocity \(u_{1}\) starts from zero. Assuming gravity is neglected, Equation (7) is simplified to  
\[\frac{p_{1}-p_{3}}{\rho}=\frac{k+1}{2}u^{2} \tag{8}\]  
The gas pressure in Chamber A is zero, and pressure \(p_{3}\) is equal to the pressure formed by the balance liquid in Chamber A, that is:  
\[p_{3}=p_{0}+p_{V_{A}^{\prime}}=\frac{\rho\omega^{2}}{B\theta}V_{A}^{\prime} \tag{9}\]  
The gas pressure in Chamber C is identical to the pressure of the compressed air flowing into the chamber, and pressure \(p_{1}\) is expressed by  
\[p_{1}=p_{0}+p_{V_{C}^{\prime}}=p_{0}+\frac{\rho\omega^{2}}{B\theta}V_{C}^{\prime} \tag{10}\]  
During the liquid transmission procedure, the overall volume of the balance liquid \(V^{\prime}{}_{AC}\) in the two chambers is constant, so the relationship between them can be derived by  
\[V_{A}^{\prime}=V_{AC}^{\prime}-V_{C}^{\prime} \tag{11}\]  
Therefore, the flow velocity or balancing velocity in the connecting tube is obtained by  
Figure 4: Flow procedure of the balance liquid. \[u=\sqrt{\frac{2}{\rho(k+1)}[p_{0}+\frac{\rho\omega^{2}}{B\theta}(2V_{C}^{\prime}-V_{ AC}^{\prime})]} \tag{12}\]  
Assuming \(x_{c}\) is the variable expressing the relationship between the liquid volume \(V_{C}^{\prime}\) and the true volume of Chamber C, Equation (12) is deformed to  
\[u=\sqrt{\frac{2}{\rho(k+1)}[p_{0}+\frac{\rho\omega^{2}}{B\theta}V(2x_{c}-0.5)]} \tag{13}\]  
If the basic parameters of the liquid chamber and the gas pressure are known, the balancing velocity depends on the drag coefficient along the transfer path \(k\), which can be expressed by  
\[k=\lambda\frac{l}{d}+\zeta \tag{14}\]  
where \(\lambda\) is the friction coefficient, \(\zeta\) is the local drag coefficient, and \(d\) and \(l\) are the equivalent diameter and length of the connecting tube, respectively. From the experimental results, we can see that the flow condition in the connecting tube is laminar flow, so the friction coefficient can be expressed by \(\lambda=64/Re\), and the Reynolds number \(\mathrm{Re}\) can be obtained by  
\[gz_{1}+\frac{p_{1}}{\rho}+\frac{{u_{1}}^{2}}{2}-\frac{\omega^{2}}{2}{r_{1}}^{2 }-k\frac{u^{2}}{2}=gz_{3}+\frac{p_{3}}{\rho}+\frac{{u_{3}}^{2}}{2}-\frac{ \omega^{2}}{2}{r_{3}}^{2} \tag{7}\]0  
where \(\
u\) is the kinematic viscosity of the balance liquid. Finally, the flow velocity or balancing velocity in the connecting tube is presented by  
\[gz_{1}+\frac{p_{1}}{\rho}+\frac{{u_{1}}^{2}}{2}-\frac{\omega^{2}}{2}{r_{1}}^{2 }-k\frac{u^{2}}{2}=gz_{3}+\frac{p_{3}}{\rho}+\frac{{u_{3}}^{2}}{2}-\frac{ \omega^{2}}{2}{r_{3}}^{2} \tag{7}\]1  
For the balance actuator used in the following experiment, its basic parameters are given in \[gz_{1}+\frac{p_{1}}{\rho}+\frac{{u_{1}}^{2}}{2}-\frac{\omega^{2}}{2}{r_{1}}^{2 }-k\frac{u^{2}}{2}=gz_{3}+\frac{p_{3}}{\rho}+\frac{{u_{3}}^{2}}{2}-\frac{ \omega^{2}}{2}{r_{3}}^{2} \tag{7}\]2 Silicone oil was used as the balance liquid, and its kinematic viscosity \(\
u\) can be chosen from a wide range (10-100, 000 cSt). Figure 5 shows the relationship between the balancing velocity \(u\) and \(x_{c}\), with a gas pressure of 0.6 MPa and an operating speed of 15,000 rpm.