## 2 Mathematical Modeling
### Mathematical Modeling of Valves  
The role and location of the suction and discharge valves can be seen in Figure 2, where the pump is shown in isometry and in section. The first phase of the mathematical modeling of the valve refers to the determination of the sizes of the flow sections of the valve. For the valve separation of fluid in radial-piston pumps, the size of the geometric flow section depends on the stroke of the valve parameters and then the differences are highlighted in terms of their functions.  
The size of the geometrical flow pressure of the valve is determined by the expression:  
\[A_{v}=\pi d_{v}x_{v} \tag{1}\]  
With the connection \(d\varphi=\omega dt\), the differential equation of movement of the moving mass of the valve is arrived at, to which \(1/3\) of the mass of the valve spring is added, in the form:  
\[\frac{dv_{v}}{d\varphi}=\frac{F_{v}}{\omega m_{v}} \tag{2}\]  
where \(F_{v}\) is the resultant of all forces acting on the moving mass of the valve \(m_{v}\), which depend on the position of the moving mass. All elements are shown in Figure 3.  
Figure 2: Radial piston pump. 1. pressure valve, 2. piston, 3. suction valve, 4. the pump shaft, 5. intake space of the pump, 6. cylinder block, 7. discharge space of the pump.  
During valve stroke \(0<x_{v}<h_{v}\), the resultant of all forces acting on the valve consists of fluid pressure force, spring force, friction force proportional to the first degree of speed and constant friction force:  
\[F_{v}=A_{v}(p_{c}-p_{v})-c_{v}(x_{v}+x_{0})-k_{v}v_{v}-F_{tr}\sigma \tag{3}\]  
where:  
\(\sigma=1\) for \(v_{v}>0\)  
\(\sigma=-1\) for \(v_{v}<0\)  
\(h_{v}\)--maximum valve stroke.  
When the stroke of the valve is \(x_{v}>h_{v}\), the valve is additionally acted upon by a force originating from the deflection of the valve stop and the damping force that additionally occurs during the deformation of the stop (it is assumed that this force is proportional to the first degree of the deformation rate):  
\[F_{v}=A_{v}(p_{c}-p_{v})-c_{v}(x_{v}+x_{0})-k_{v}v_{v}-F_{tr}\sigma-\sigma_{gv }\big{[}c_{gv}(x_{v}-h_{v})+k_{gv}v_{v}+F_{trgv}\sigma\big{]} \tag{4}\]  
where:  
\(\sigma_{gv}=0\) for \(x_{v}\leq h_{v}\)  
\(\sigma_{gv}=1\) for \(x_{v}>h_{v}\).  
In the case when \(x_{v}<0\), the valve is additionally acted upon by a force originating from the deformation of the valve seat and the damping force that additionally occurs during the deformation of the valve seat (proportional to the first degree of deformation rate):  
\[F_{v}=A_{v}(p_{c}-p_{v})-c_{v}(x_{v}+x_{0})-k_{v}v_{v}-F_{tr}\sigma-\sigma_{gv }[c_{gv}x_{v}+k_{sv}v_{v}+F_{trsv}\sigma] \tag{5}\]  
where:  
\(\sigma_{gv}=0\) for \(x_{v}\leq 0\)  
\(\sigma_{gv}=1\) for \(x_{v}>0\).  
The differential equation of the valve path is:  
\[\frac{dx_{v}}{d\varphi}=\frac{v_{v}}{\omega} \tag{6}\]  
Figure 3: Dynamic model of the valve of the radial-piston pump.  
The effective flow section of the valve through which the fluid flows in or out is obtained as a product of the geometric flow section and the corresponding flow coefficient, i.e.,  
\[A_{\mu v}=A_{v}\mu_{u(i)} \tag{7}\]  
When it is a suction or discharge valve, all the expressions (1) to (7) given above receive the corresponding indices (\(u\)--when it is a suction, that is, when it is a discharge valve).  
This is the first phase of modeling and represents the basis for all other phases, both modeling and design of the valves of the suction and discharge lines of the radial-piston pump.