## 2 Problem Description and Model Analysis
### Model Analysis  
#### 2.2.1 Model Establishment  
To quantitatively analyze the influence of key parameters on the unloader motion characteristics and the influence on the gas system caused by changes in the unloader and valve motion characteristics, we established a mathematical model of the flow control system, based on multi-system coupling, by considering the unloader dynamic model as the starting point. Furthermore, the coupling relationship between the mechanical, hydraulic, and compressor gas systems was analyzed. Figure 1: Stepless capacity control system for a reciprocating compressor. Figure 2: Coupling relationship between the suction valve and gas system. The cylinder gas pressure in the thermodynamic parameters of the reciprocating compressor is the power source for valve opening and closing, and its size affects the movement characteristics of the valve plate. Meanwhile, the effective flow area generated by the valve directly determines the flow rate of the gas, thus changing the compressor displacement and the thermodynamic characteristics of the gas in the cylinder. The unloader couples the hydraulic cylinder with the valve. The plunger of the hydraulic cylinder pushes the unloader up and down and maintains the ejection state. Consequently, the plunger of the hydraulic cylinder withdraws, and the unloader withdraws under the action of the reset spring. The mathematical model is built on the following assumptions:  
* The rebound during unloader impact is not considered. * Delays in oil supply or unloading due to the dynamic response of solenoid valves and the pulsation effect of hydraulic lines are not considered, i.e., the starting time of the unloader ejection or withdrawal is the same as that of the oil supply or pressure relief at the hydraulic oil station. According to the force analysis diagram of the mechanical actuator (Figure 3), during the ejection process, in the time range of \(t_{1}\leq t\leq t_{2}\) (Figure 2), the main power source is hydraulic pressure (\(F_{hy}\)), followed by gravity (\(G_{mu}\)); the main resistance source is reset spring force (\(F_{K}\)), followed by suction chamber gas force (\(F_{gs}\)), total friction force (\(f\)) and valve disc resultant force (\(F_{v}\)). In the withdrawal process, in the time range \(t_{3}\leq t\leq t_{4}\) (Figure 2), the main power source is the reset spring force (\(F_{K}\)), followed by suction chamber gas force (\(F_{gs}\)) and valve disc resultant force (\(F_{v}\)); the resistance source is hydraulic pressure (\(F_{hy}\)), followed by gravity (\(G_{mu}\)) and total friction force (\(f\)). The mechanical mechanism is stationary during maintenance, in the time range of \(t_{2}<t<t_{3}\). Therefore, the equation for the motion of the actuator is given by Equation (1):  
\[M_{mu}\frac{\mathrm{d}^{2}x_{mu}}{\mathrm{d}t^{2}}=\sum F \tag{1}\]  
where \(M_{mu}\) is the mass of the moving parts of the unloader; \(x_{mu}\) is the displacement of the unloader; and \(\sum F\) is the resultant force on the unloader. Figure 3: Diagram of force distribution related to the mechanical actuator. According to the time-division, we observe that the ejection and withdrawal differential equation of the unloader is given by Equation (2):  
\[M_{mu}\times\frac{\mathrm{d}^{2}x_{mu}}{\mathrm{d}t^{2}}= \begin{cases}F_{hy}-F_{ gs}+M_{mu}\times g\times\cos\alpha-f-F_{re}&t_{1}\leq t\leq t_{2}\\ 0&t_{2}<t<t_{3}\\ F_{hy}+M_{mu}\times g\times\cos\alpha-F_{re}+f-\varsigma F_{v}&t_{3}\leq t\leq t _{4}\end{cases}  \tag{2}\]  
where \(\varsigma\) is the force coefficient of interaction between the valve plate and unloader, and \(t_{1}\) is the starting point of the unloader ejection; \(t_{2}\) is the completion of unloader ejection; \(t_{3}\) is the beginning of unloader withdrawal; and \(t_{4}\) is the completion of unloader withdrawal. The forces in Equation (2) are provided by each subsystem, as follows:  
(i) Hydraulic pressure, supplied by the hydraulic system.