## 2 Problem Description and Model Analysis
### Model Analysis
#### 2.2.1 Model Establishment
 \tag{5}\]  
where \(\frac{\mathrm{d}V_{pump}}{\mathrm{d}t}\) is the flow of the hydraulic pump; \(\frac{\mathrm{d}V_{rr,in}}{\mathrm{d}t}(p_{in})\) is the flow rate of oil inlet relief valve at inlet pressure setting of \(p_{in}\); \(\frac{\mathrm{d}V_{rr,in}}{\mathrm{d}t}(p_{bf})\) is the flow of the oil return relief valve at a return pressure setting of \(p_{bf}\); \(p_{in}\) is the set pressure of the oil inlet overflow valve; and \(p_{bf}\) is the set pressure of the oil return overflow valve. (ii) Return spring force, provided by the mechanical system. The equation for calculating the return spring force is as follows:  
\[F_{re}=K_{re}\times(x_{mu,0}+x_{mu}) \tag{6}\]  
where \(K_{re}\) is the spring stiffness of the unloader; \(x_{mu,0}\) is the spring pre-compression; and \(x_{mu}\) is the displacement of the unloader. (iii) Gas force, supplied by the gas system in the suction chamber of the compressor. Gas force can be calculated using the following equation:  
\[F_{gs}=p_{s}\times A_{hp} \tag{7}\]  
where \(p_{s}\) is the oil inlet pressure of the suction chamber. (iv) Valve force, indirectly supplied by the gas system in the compressor cylinder. As shown in Figure 2, the movement state of the valve plate affects the flow characteristics of the gas, which enters the cylinder from the suction chamber in the suction stage and returns to the suction chamber from the cylinder in the reflux stage. Therefore, the mass flow rate of the gas through the suction valve at different times can be calculated using the following equation [11]:  
\[\frac{\mathrm{d}m_{s}}{\mathrm{d}t}= \begin{cases}\frac{\alpha_{s}\times A_{s} \times p_{s}}{R\times T_{s}}\times\left(\frac{p_{\alpha}}{p_{s}}\right)^{\frac {1}{k}}\times\sqrt{\frac{2\times k}{k-1}\times R\times T_{s}\left[1-\left( \frac{p_{xy}}{p_{s}}\right)^{\frac{k-1}{k}}\right]}&t_{a}\leq t\leq Tcy/2\\ \\ \frac{\alpha_{s}\times A_{s}\times p_{\alpha}}{R\times T_{s}}\left(\frac{p_{s} }{p_{\alpha}}\right)\times\sqrt{\frac{2\times k}{k-1}R\times T_{s}\left[\left( \frac{p_{xy}}{p_{s}}\right)^{\frac{k-1}{k}}-1\right]}&Tcy/2<t\leq t_{4}\end{cases}  \tag{8}\]  
A reciprocating compressor is a thermodynamic circulating system. According to the first law of thermodynamics, the energy conservation equation of gas in cylinder can be obtained. In order to simplify the calculation, the kinetic energy and potential energy change of gas are ignored. The equation is as follows [41]:  
\[\frac{\mathrm{d}Q_{xy}}{\mathrm{d}t}+\sum\frac{\mathrm{d}m_{s}}{\mathrm{d}t} \times h_{s}=\sum\frac{\mathrm{d}m_{d}}{\mathrm{d}t}\times h_{d}+\frac{\mathrm{ d}}{\mathrm{d}t}(m\times u)_{cy}+p_{cy}\times\frac{\mathrm{d}V_{cy}}{\mathrm{d}t} \tag{9}\]  
where \(\frac{\mathrm{d}Q_{xy}}{\mathrm{d}t},\sum\frac{\mathrm{d}m_{s}}{\mathrm{d}t} \times h_{s},\sum\frac{\mathrm{d}m_{d}}{\mathrm{d}t}\times h_{d},\frac{\mathrm{ d}}{\mathrm{d}t}(m\times u)_{cy}\) and \(p_{cy}\times\frac{\mathrm{d}V_{cy}}{\mathrm{d}t}\) represent the heat exchanged between the compressor and the outside world, the enthalpy of the gas sucked in, the enthalpy of the exhaust gas, the energy change of the gas in the compressor cylinder and the output power, respectively. The gas used in this model is normal-temperature and low-pressure air, and the properties of the gas are close to those of an ideal gas [42]; thus, the gas model is that of an ideal gas [43]:  
\[u=c_{p}\times T,\;h=c_{p}\times T,\;R=c_{p}-c_{p},\;P\times V=n\times R\times T \tag{10}\]  
The value of the interaction force coefficient (\(\zeta\)) in the mathematical model is determined by the coupling and the mutual interaction between the motion of the unloader and valve plate. This study further established that this is determined by the stress state and the relative position of the unloader and valve plate. In this paper, the relative position and state coefficients are defined; the final interaction coefficient is related to the values of the two coefficients.