## 2 Problem Statement and Active Damping Strategy Analysis  
For energy saving reasons, the pressure of the cylinder metering-out chamber was often designed at a lower level in the SMEHS. The system is easy to overshoot and oscillation when the load force drastically changes. The active damping strategy (ADS) is proposed in order to improve the system damping and suppress the oscillation in the SMEHS (Figure 1). The active damping strategy establishes the correlation between the pressure of the metering-out chamber and the acceleration of the load. In the following, the damping ratio of the SMEHS is mathematically derived to prove the effectiveness of ADS. Damping is the effect that tends to reduce the amplitude of vibrations [15], which means that the damping force always has the opposite direction with velocity. The pressure force at the metering-out chamber can be regarded as a damping force. As shown in Figure 2, when the rod is extending, the direction of the metering-out pressure force \(F_{p_{2}A_{2}}\) is opposite to the rod moving direction, and there is a tendency to hinder the movement of the piston, so that the metering-out pressure force can be considered as a kind of damping for the system. The viscous damping force \(F_{b\dot{x}}\) is also opposite to the speed direction and it is also the damping force of the system. Similarly, when the piston is retracted, as in Figure 3, the direction of the metering-out pressure force is opposite to the rod moving direction, so the metering-out pressure force \(F_{p_{1}A_{1}}\) is a damping force of the system. Theoretical analysis is carried out to illustrate the characteristics of the ADS. Several assumptions are made, as follows:  
1. The viscous damping coefficient and the leakages of the cylinder are ignored. 2. The friction force of piston in the cylinder is negligible. 3. The rod is extending. The fluid flow through the servo valve orifice is given by the following equations:  
\[q_{1}=K_{v1}x_{v1}\sqrt{\Delta P_{1}} \tag{1}\]  
\[q_{2}=K_{v2}x_{v2}\sqrt{\Delta P_{2}} \tag{2}\]  
where \(q_{1}\) and \(q_{2}\) are the flows through the valve, \(K_{v1}\) and \(K_{v2}\) are the valve coefficient of valve orifice, and \(\Delta P_{1}\) and \(\Delta P_{2}\) are pressure drops across the orifice, which are defined as:  
\[\Delta P_{i}=\left\{ \begin{array}{l}p_{s}-p_{i}\,\ x_{vi}\geq 0\\ p_{i}-p_{o}\,\ x_{vi}<0\end{array} \right.,i=1,2 \tag{3}\]  
where \(p_{s}\) is supply pressure and \(p_{o}\) is system tank pressure. Figure 1: Separating-metering system with active damping strategy (ADS). Figure 3: Rod is retracting. Figure 2: Rod is extending. Equations (1) and (2) are needed to be linearized for the mathematical deduction. The following equations can be given while using Taylor' series expansion. \[q_{1}=K_{v1}(K_{q1}x_{v1}-K_{c1}\Delta P_{1}) \tag{4}\]  
\[q_{2}=K_{v2}(K_{q2}x_{v2}-K_{c2}\Delta P_{2}) \tag{5}\]  
where \(K_{q1}\) and \(K_{q2}\) are the flow coefficient and \(K_{c1}\) and \(K_{c2}\) are the flow-pressure coefficient. The pressure of cylinder chambers can be obtained by the fluid flow balance equations:  
\[\dot{p}_{1}=(q_{1}-A_{1}\dot{x}_{L})\frac{\beta_{e}}{V_{L1}} \tag{6}\]  
where \(\beta_{e}\) is the effective bulk modulus of oil, \(p_{1}\) is the pressure in piston chamber, \(V_{L1}\) is the volume of cylinder chambers, \(A_{1}\) is the area of cylinder piston, and \(q_{1}\) is the flow rate of cylinder. The force balance equation following [PERSON]'s second law:  
\[\ddot{x}_{L}=\frac{1}{m}(p_{1}A_{1}-p_{2}A_{2}-b\dot{x}_{L}-f_{L}) \tag{7}\]  
where \(m\) is the total mass of the piston, \(f_{L}\) is the load force, and \(b\) is viscous damping coefficient.