# Determining Optimal Parameters of Regular Microrelief Formed on the End Surfaces of Rotary Bodies
## 3 Results
\[\Delta_{y}=tg\Big{(}90-\frac{\alpha_{k1}}{2}\Big{)}\cdot\frac{\rho_{k}\cdot\left( \frac{1}{\sin\left(\frac{\alpha_{k2}}{2}\right)}+\frac{1}{\sin\left(\frac{\alpha _{k1}}{2}\right)}\right)-\left(S_{o}-A_{k1}-A_{k2}\right)}{tg\big{(}90-\frac{ \alpha_{k1}}{2}\big{)}+tg\big{(}90-\frac{\alpha_{k2}}{2}\big{)}}+\frac{S_{o}-A _{k1}-A_{k2}}{2}-\frac{\rho_{k}}{\sin\left(\frac{\alpha_{k1}}{2}\right)} \tag{19}\]  
The formula determines the area of the \(ACDB\) trapezoid:  
\[S_{t1}=\frac{a_{t1}+b_{t1}}{2}\cdot\Delta_{h1} \tag{20}\]  
where \(a_{t1}\) is the length of the lower base of trapezoid \(AB\) formed by the intersection of two grooves, mm, \(b_{t1}\) is the length of the lower base of trapezoid \(CD\) formed by the intersection of two grooves, mm; \(\Delta_{h1}\) is the height of the trapezoid formed by the intersection of two grooves, mm. \[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]0  
Formula (20) for determining the area of trapezoid \(ACDB\) is written as:  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]1  
Taking into account Formula (18), Formula (22) will be written as follows:  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]2  
Similarly, we find the area of trapezoid _ABLE_. \[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]3  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]4  
The formula found the area of the segment with the \(CD\) chord:  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]5  
The formula found the area of the segment with the \(EL\) chord:  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]6  
The area of existence for Formula (13) was determined by the limit value of the axial pitch of PRRM, which was determined by the dependence:  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]7  
Stage III. Consider the option when the intersecting grooves formed one sector on top of one of the grooves (Figure 5). The overlap area for this case was calculated by Formula (13), taking into account the absence of the circular sector area with radius \(\Delta_{t1}\), which was determined by the formula:  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]8  
where \(S_{s3}\) is the \(ABC\) sector area of the circle formed by the intersection of grooves 1 and 2, mm\({}^{2}\). The area of figures \(S_{t1}\) was determined by Formula (23), \(S_{t2}\)--by Formula (25), \(S_{s1}\)--by Formula (26), \(S_{s2}\)--by Formula (27)  
The formula determined the area of the \(ABC\) sector:  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2} \tag{13}\]9  
where \(\Delta_{t1}\) is the radius of sector \(ABC\), mm. The formula determined the sector radius:  
\[y_{i(a-a)}=tg\Big{(}90-\frac{\alpha_{k1}}{2}\Big{)}\cdot x_{i}+\frac{S_{o}-A_{ k1}-A_{k2}}{2} \tag{14}\]0  
Taking into account Formula (31), Formula (30) takes the following form:  
\[y_{i(a-a)}=tg\Big{(}90-\frac{\alpha_{k1}}{2}\Big{)}\cdot x_{i}+\frac{S_{o}-A_{ k1}-A_{k2}}{2} \tag{14}\]1  
The area of existence for this formula was determined by the limit value of the axial pitch of PRMR, which was determined from the dependence:  
\[\left\{ \begin{array}{cc}A_{k1}+A_{k2}+\rho_{k}\Big{(}1-\frac{1}{\sin(\alpha_ {k2}/2)}\Big{)}&>S_{o}>A_{k1}+A_{k2}+\rho_{k}\bigg{(}1-\frac{1}{\sin(\alpha_{k 1}/2)}\bigg{)}\\ \end{array} \right. \tag{33}\]  
Stage IV. Consider the option when the intersecting grooves form circular sectors at the top of each groove (Figure 6). Figure 5: The scheme of overlapping of V-shaped grooves of PRMR (stage III). The overlap area for this case was calculated by Formula (13), taking into account the absence of the circular sector area with radii \(\Delta_{t1}\) and \(\Delta_{t2}\), which was determined by the formula:  
\[S_{ov}=S_{s1}+S_{s2}+S_{t1}+S_{t2}-S_{s3}-S_{s4} \tag{34}\]  
where \(S_{s4}\) is the sector area of circle \(DEF\) formed by the intersection of grooves 1 and 2, mm\({}^{2}\). The area of figures \(S_{t1}\) is determined by Formula (23), \(S_{t2}\)--by Formula (25), \(S_{s1}\)--by Formula (26), \(S_{s2}\)--by Formula (27), \(S_{s3}\)--by Formula (32).