## 3 Designing a New Hydraulic Platform and Control Method  
### Designing a New Hydraulic Platform  
The newly designed hydraulic circuit diagram is shown in Figure 4. It includes control devices for constant back pressure and left and right tension. Considering the property of the rolling process where one side cylinder always extends while the other retracts, the piston sides of the two tension cylinders and the outlet of the constant back pressure control device are connected together. In this way, the working flow of the piston sides can complement each other. The constant back pressure device mainly consists of a check valve and two identical relief valves. The adjustment method and the working principle are: (1) The setting pressures of the two relief valves are manually adjusted to \(Py_{11}\) and \(Py_{12}\), and \(Py_{11}=Py_{12}\) (ignoring the pressure drop generated when the oil flows through the first check valve). Assuming the flows through the two relief valves are \(Qy_{11}\) and \(Qy_{12}\), then \(Qy_{11}=Qy_{12}\) because the two relief valves have the same specifications. (2) When tension cylinders work in position control mode, the mill is not in a rolling state, so the pressure fluctuations in the piston sides do not influence the tension control results. (3) In rolling state, both tension cylinders work in force control mode. When the rolling speed is zero, the output fluid of the fixed displacement pump will flow through two relief valves, and the pressure in the piston chambers of the tension cylinders will be \(Py_{11}\). When the rolling speed is not zero, taking the rolling direction from left to right as an example, the left tension cylinder is on the entry side and has speed \(v_{\rm H}=(1-S_{\rm H})v\). The right tension cylinder is on the exit side and has speed \(v_{\rm h}=(1+S_{\rm h})v\). Obviously, \(v_{\rm h}>v_{\rm H}\), and the speed difference is \((S_{\rm h}+S_{\rm H})v\). The speed difference causes excess fluid to flow back into the constant back pressure control device through K5, and this flow is \(Q_{\rm g}=(S_{\rm h}+S_{\rm H})vA_{\rm w}\), where \(A_{\rm w}\) is the effective area of the ring side. Assuming the flow through the first relief valve is \(Qy_{21}\), the inlet pressure of first relief valve is \(P_{21}\), the flow through the second relief valve is \(Qy_{22}\), the inlet pressure of the second relief valve is \(P_{22}\), and ignoring the internal and external leakage of the tension control cylinders, we obtain\[Q_{\rm P}=Qy_{11}+Qy_{12} \tag{10}\]  
where \(Q_{\rm P}\) is the output flow of the fixed displacement pump. Then,  
\[Qy_{21}+Qy_{22}=Q_{\rm P}+Q_{\rm g} \tag{11}\]  
With Equations (10) and (11), and \(Qy_{11}=Qy_{12}\), we obtain  
\[Qy_{22}-Qy_{12}=Q_{\rm g}+(Qy_{12}-Qy_{21}) \tag{12}\]  
Ignoring the pressure drop caused by the first check valve, we get \(P_{21}=P_{22}\) and  
\[Qy_{21}=Qy_{22} \tag{13}\]  
Figure 4: Hydraulic circuit diagram of tension control: (1) first pressure gauge, (2) first pressure hose, (3) second pressure gauge, (4) second pressure hose, (5) first pressure joint, (6) second pressure joint, (7) first check valve, (8) electric motor, (9) coupling, (10) pump, (11) tank, (12) first pressure relief valve, (13) second pressure relief valve, (14) first filter, (15) second check valve, (16) third check valve, (17) first accumulator, (18) first servovalve, (19) first pressure sensor, (20) first hydraulic hose, (21) first displacement sensor, (22) left cylinder, (23) second hydraulic hose, (24) first proportional pressure relief valve, (25) second pressure sensor, (26) third pressure sensor, (27) second accumulator, (28) second servovalve, (29) third hydraulic hose, (30) right cylinder, (31) second displacement sensor, (32) fourth hydraulic hose, (33) fourth pressure sensor, (34) second proportional pressure relief valve, (35) fourth check valve, (36) fifth check valve, (37) second filter, (I) constant back pressure control device, (II) left tension control device, and (III) right tension control device.