# Determining Optimal Parameters of Regular Microrelief Formed on the End Surfaces of Rotary Bodies
## 3 Results  
Consider the general case of the intersection of V-shaped grooves formed on rotary bodies' end surfaces (Figure 2). This will help us establish the dependent geometric parameters of the grooves and the relationship between them. The relationships between groove parameters were as follows:  
Figure 1: Partially regular microrelief formed on the disk: (**a**) Formation scheme with molding movements; (**b**) visual image; 1—deforming element; 2—end surface of the variator disk; \(D_{v}\)—rotational movement of the disk; \(D_{s}\)—tool feed movement; \(D_{i}\)—the oscillating reciprocating motion of the tool. The angular pitch of the groove is determined from the equation:  
\[\varphi_{k}=\frac{360^{\circ}}{N_{el}} \tag{1}\]  
where \(N_{el}\) is the number of PRMR elements located on the RBES within the range of \(360^{\circ}\). We write down the equation for determining the angle \(\alpha_{k1}\)  
\[\left\{ \begin{array}{l}AB=(R_{k1}-A_{k1})\cdot tg(\varphi_{k}/2)\\ AB=2\cdot A_{k1}\cdot tg(\alpha_{k1}/2)\end{array} \right. \tag{2}\]  
Having solved the system of equations to determine \(\alpha_{k1}\), we obtain:  
\[\alpha_{k1}=2\cdot arctg\bigg{[}\frac{(R_{k1}-A_{k1})\cdot tg(\varphi_{k}/2)} {2\cdot A_{k1}}\bigg{]} \tag{3}\]  
Similarly, we write down the equation to determine the angle \(\alpha_{k2}\):  
\[\left\{ \begin{array}{l}CD=(R_{k1}+S_{o}+A_{k2})\cdot tg(\varphi_{k}/2)\\ CD=2\cdot A_{k2}\cdot tg(\alpha_{k2}/2)\end{array} \right. \tag{4}\]  
Having solved the system of equations to determine \(\alpha_{k2}\), we obtain:  
\[\alpha_{k2}=2\cdot arctg\bigg{[}\frac{(R_{k1}+S_{o}+A_{k2})\cdot tg(\varphi_{ k}/2)}{2\cdot A_{k2}}\bigg{]} \tag{5}\]  
In addition, we can express angle \(\alpha_{k2}\) via angle \(\alpha_{k1}\). To do this, we make a system of equations from system of Equations (2) and (3). \[\left\{ \begin{array}{l}2\cdot A_{k1}\cdot tg(\alpha_{k1}/2)=(R_{k1}-A_{k1}) \cdot tg(\varphi_{k}/2)\\ 2\cdot A_{k2}\cdot tg(\alpha_{k2}/2)=(R_{k1}+S_{o}+A_{k2})\cdot tg(\varphi_{k} /2)\end{array} \right. \tag{6}\]  
Having solved this system of equations to determine \(\alpha_{k2}\), we obtain:  
\[\alpha_{k2}=2\cdot arctg\bigg{[}\frac{0.5\cdot(A_{k1}+A_{k2}+S_{o})\cdot tg( \varphi_{k}/2)+A_{k1}\cdot tg(\alpha_{k1}/2)}{A_{k2}}\bigg{]} \tag{7}\]  
Thus, the obtained dependencies allow us to express the required geometric parameters using known quantities. When analyzing the groove intersection process, seven stages of the intersection were found. In addition, the degree of overlap transforms from the intersection of the segmental parts of vertices to a complete intersection of rectilinear sections of the grooves. Analytical dependences to determine the overlap area will be different at each stage, thus we consider each case separately. It should also be noted that the overlap area of the grooves for cases I-IV was defined as the area of a complex figure formed by segments and trapezoids. The number of such areas will be equal to half the number of intersecting grooves. For cases V-VII, the overlap area was defined as the area of intersecting diamonds. The number of such areas will be equal to the number of intersecting grooves. Stage I. Consider the case when the groove vertices intersect within a circular segment with radius \(\rho_{k}\) and chord height \(\Delta_{\rho}\) (Figure 3). The overlap area of two grooves was defined as the sum of their segments' areas with the chord height \(\Delta_{\rho}\). \[S_{ov}=\rho_{k}^{2}\cdot\Big{[}\frac{\pi\cdot\alpha_{\rho}}{180^{\circ}}-\sin \alpha_{\rho}\Big{]} \tag{8}\]  
The formula determines the chord height:  
\[\Delta_{\rho}=\frac{2\cdot\rho_{k}-S_{o}+A_{k1}+A_{k2}}{2} \tag{9}\]  
where \(\rho_{k}\) is the segment radius, mm; \(S_{o}\) is the axial pitch of PRMR grooves, mm; \(A_{k}\) is the amplitude of PRMR grooves, mm.