## 3 Mathematical Model
### Compressor  
In order to establish the mechanism's mathematical model of the compressor, some basic assumptions were made for the working process of the compressor:  
1. Ideal homogeneous medium is in the cylinder, that is, the pressure, temperature, and density are the same at any two points of the medium;
2. The flow process of gas in and out of the cylinder is adiabatic and stable;
3. Energy transfer in the working cylinder is uniform. The chamber of the compressor is taken as a control volume, and the inlet valve leakage is considered. It is modelled as an additional flow through an orifice in parallel to the normal valve flow [20]. The equation for the cylinder pressure \(P_{cy}\) can be derived from the continuity and the first law of thermodynamics as  
\[\frac{dp_{cy}}{d\theta}=\frac{1}{V_{cy}(\theta)}\Bigg{(} \begin{array}{c}c_{i }^{2}\frac{dm_{ci}}{d\theta}-c_{cy}^{2}\frac{dm_{ci}}{d\theta}-\gamma p_{cy} \frac{dV_{cy}}{d\theta}-\\ c_{i}^{2}\frac{dm_{ci}}{d\theta}-c_{cy}^{2}\frac{dm_{ci}}{d\theta}\end{array}  \Bigg{)}, \tag{1}\]  
where the instantaneous volume of the cylinder \(V_{cy}(\theta)\) and its change rate \(\frac{dV_{cy}}{d\theta}\) can be calculated by the clearance volume and piston motion, respectively. \[V_{cy}(\theta)=V_{c0}+S_{cy}x_{cy}(\theta), \tag{2}\]  
\[\frac{dV_{cy}(\theta)}{d\theta}=S_{cy}\frac{dx_{cy}(\theta)}{d\theta}, \tag{3}\]  
where \(\theta=\omega t\) represents the rotation angle of the crankshaft relative to the top dead center. \(\frac{dm_{ci}}{d\theta}\) and \(\frac{dm_{cd}}{d\theta}\) are the mass flow through the inlet and discharge valves, respectively. \(\frac{dm_{ci}}{d\theta}\) and \(\frac{dm_{cd}}{d\theta}\) are the mass flow rate of inlet and discharge valve leakage, respectively. \(\gamma=1.4\) is the ratio of specific heats. \(c_{i}^{2}=\gamma RT_{i}\) is the squared speed of sound in the inlet chamber. \(c_{cy}^{2}=\gamma RT_{cy}\) is the squared speed of sound in the cylinder. \(V_{c0}\) is the clearance volume. \(R\) is the gas constant. In full load condition, inlet and discharge valves of the reciprocating compressor are automatic valves. For the inlet valve, the driving force is the difference value between the suction pressure and the gas pressure in the cylinder. However, the additional force provided by the hydraulic system acts on the valve plate under the regulation condition. The movement of the suction valve plate can be described as:  
\[M_{se}\omega^{2}\frac{d^{2}x_{sp}}{d\theta^{2}}+c_{ss}\omega\frac{dx_{sp}}{d \theta}+k_{sp}x_{sp}-F_{lu}=\sum f_{sv}, \tag{4}\]  
where \(F_{lu}\) is the additional hydraulic force; \(M_{se}\) represents the total equivalent motion mass of suction valve and actuator; \(\frac{d^{2}x_{sp}}{d\theta^{2}}\), \(\frac{dx_{sp}}{d\theta}\), and \(x_{sp}\) are the valve plate acceleration, velocity, and displacement, respectively; \(c_{ss}\) is the damping coefficient; \(\sum f_{sv}\) represents the resultant force on the valve which is from three forces: gravity of valve, the preset force, and differential pressure force; and \(k_{sp}\) is equivalent spring stiffness. The mass flow rate through the inlet valve and discharge valve can be expressed as Equations (5) and (6), respectively. \[\frac{dm_{cin}}{d\theta}=C_{i}A_{fs}\frac{1}{\omega}\sqrt{2\rho_{cy}|p_{i}- p_{cy}|}, \tag{5}\]\[\frac{dm_{cd}}{d\theta}=C_{d}A_{fd}\frac{1}{\omega}\sqrt{2\rho_{cy}\big{|}p_{d}-p_{ cy}\big{|}}, \tag{6}\]  
where \(A_{fs}\) is the maximum flow area of all inlet valves; \(p_{i}\) and \(p_{d}\) represent the suction and exhaust pressures, respectively; \(A_{fd}\) is the maximum flow area of all discharge valves; \(\rho_{cy}\) is the density of the gas in the cylinder; \(C_{d}\) is a variable discharge coefficient which accounts for the reduced flow area resulting from the separated flows and changes with the valve lift; and \(C_{i}\) is a variable suction coefficient. The mass flow rate of inlet and discharge leakage can also be calculated according to Equations (5) and (6) by using the leakage flow area instead of the flow area of valve.