## 3 Results and Discussion
### Mathematical Description of Pressure Self-Regulation Mechanism for Press Equipment  
Mathematical modeling of the pressure self-regulation mechanism in a screw press provides a tool for optimizing spring characteristics and operational parameters. By converting forces and deformation into equations, engineers can predict performance, determine the ideal spring stiffness, and minimize energy consumption. It also enables adjusting of the operating speed, press gap, and other variables to maintain optimal pressure for different materials. This systematic approach replaces trial-and-error with data-driven decisions, saving time, improving efficiency, and extending equipment life. Ultimately, it ensures a stable, energy-efficient operation while maintaining the desired product quality and throughput [33]. In the mechanism of pressure self-regulation, we use helical cylindrical compression springs of class III of the 2 nd grade made of steel of round cross-sections. The main parameters of the spring are accepted in accordance with the interstate standard GOST 13775-86 [34]. However, some parameters of pressure regulation require a coordinated calculation due to the specifics of the mechanism operation. The following forces are applied during pressing: _Fp_--pressure force, N; _Fs_--shear force, N; and _Fe_--elastic force, N (Figure 11). The pressure force, _Fp_, is the force generated by the pressure value during pressing. Consequently, its trajectory direction will be oriented in the direction of the product and if, for solids, we define it by a simple equation, as shown below:  
\[F_{\mathrm{P}}=P\cdot S,\mathrm{N} \tag{2}\]  
where  
_P_--pressing pressure, Pa;  
_S_--area affected by the force, m\({}^{3}\). Let us note that the area of the part of the body affected by the surface is equal to the area of the ring in our case:  
\[\Delta S=\pi\Big{(}\Delta R^{2}-\Delta r^{2}\Big{)},\mathrm{m}^{2} \tag{3}\]  
Taking into account the total pressure on the pressing surface of the screw coils, we determine the pressure force by the following Equation (4):  
\[F_{P}=\rho gQ\cdot\left[\frac{\hbar c}{D\omega^{\mathrm{C}}(1+\epsilon)}+\frac {\mu cL}{\delta c_{L}^{2}f}\Big{(}\frac{2\pm 1}{2n}\Big{)}\right]+\frac{g}{K^{ \prime}-\left(K^{\prime}-\frac{g}{\delta g}\right)\exp(\beta z)}\ \ \cdot\Delta S,\ \mathrm{N} \tag{4}\]The back-shear force, \(F_{S}\), is the back-shear force along the product screw channel caused by exceeding the pressing pressure during the pressing process. Hence, its trajectory direction is in the direction opposite to the pressure force. We determine the backward shear force based on [PERSON] force law:  
\[F_{S}=\rho_{\theta}\cdot g\cdot V_{\theta},\mathrm{N} \tag{5}\]  
where  
\(\rho_{\theta}\)--density of the sample, kg/m\({}^{3}\);  
\(V_{\theta}\)--volume of sample in the pressing channel, m\({}^{3}\). For the elastic force, \(F_{E}\), in our case we are referring to the spring back force due to the backward shear force. Consequently, its trajectory direction is directed by the pressure force opposite to the reverse shear force (Figure 12). We determine the elastic force based on [PERSON]'s law:  
\[F_{E}=k\cdot\Delta l,\mathrm{N} \tag{6}\]  
where  
\(k\)--spring stiffness, N/m;  
\(\Delta l\)--distance at absolute spring deformation, m.  
Figure 11: Scheme of acting forces in compression and extension of springs. \(l_{i}\)—length of spring coils along the longitudinal axis, m; \(l_{0}\)*—length of tensile spring without hooks in free state, mm; \(F_{i}\)—spring coil force along the longitudinal axis, N; D—spring outer diameter, m; \(D_{1}\)—spring center diameter, m; \(d\)—section diameter of steel wire, m; —spring rate, m.  
The absolute deformation of the spring can be characterized by both the length of the press screw channel along the longitudinal axis and the change in the gap between the gap nozzle and the screw coil.