## 2 Mathematical Derivation of Applied Relationships  
The complexity of finishing the part \(T_{K}\) during the grinding process in one, two, three, or more operations can be calculated according to the generally accepted formula [14]:  
\[T_{K}=T_{M}+T_{B}+T_{eq}+T_{nn}+\frac{T_{nc}}{n_{pp}} \tag{1}\]  
where \(T_{M}\) is machine part processing time, min; \(T_{B}\) is auxiliary time, min; \(T_{eq}\) is time for organizational and technical maintenance of the workplace, min; \(T_{nn}\) is time for rest and natural needs, min; \(T_{nc}\) is preparatory and final time, min; \(n_{pp}\) is the number of parts in the package. Values \(T_{B}\), \(T_{eq}\), \(T_{nn}\), and \(T_{nc}\) can be found according to the normative data, and the machine grinding time of the part can be determined by the equation:  
\[T_{M}=f/Q_{\Omega} \tag{2}\]  
where \(f\) is the surface area to be processed, in mm\({}^{2}\); \(Q_{\Omega}\) is the grinding performance by processing area, mm\({}^{2}\cdot\)min\({}^{-1}\). Grinding performance with fine-grained wheels on the area processed per unit of time \(Q_{\Omega}\), can be expressed in terms of the grinding performance with regard to the volume of metal removed per unit of time, \(Q_{M}\), by the formula:  
\[Q_{\Omega}=Q_{M}/\Omega \tag{3}\]where \(\Omega\) is the radial allowance for grinding, in mm. During grinding, the metal is removed from the workpiece by single abrasive grains, so the grinding performance, \(Q_{M}\), depends on the number of slices and the volume of metal removed during a single slice, which are determined by the shape and size of the slice. The most typical cuts in finishing grinding processes are segmented cuts [14]. Figure 1 shows a diagram of a single mark applied to a polished surface, where \(S_{1}\) and \(S_{2}\) are areas of bulk formed on the surface of the material due to plastic deformation by its abrasive grain, in \(\upmu\)m\({}^{2}\). At the accepted coordinates, the elementary volume of one-quarter of the segmented mark is determined by the equation:  
\[dV_{p}^{\prime}=\phi(y,z)dydz \tag{4}\]  
where \(\phi(y,z)\) is the equation of the surface limiting the volume of the mark. Integrating Equation (4) and multiplying the result by four, we obtain the full volume of the segmented mark, \(dV_{p}\):  
\[dV_{p}=\int_{0}^{y}\int_{0}^{z}\phi(y,z)dydz \tag{5}\]  
The number of single slices per minute \(N\) can be determined by the equation:  
\[N=\Psi\cdot B\cdot V_{K}\cdot n_{g} \tag{6}\]  
where \(n_{g}\) is the number of grains located on a single site in 1 mm\({}^{2}\) of the wheel surface, \(\Psi\) is the scale factor (during processing with fine-grained wheels, \(\Psi=6\times 10^{4}\)), \(B\) is wheel height, and \(V_{K}\) is the circumferential speed of the wheel. It is shown in [15; 16; 17] that the abrasive grains on the surface of the wheel are located at different levels, and therefore are embedded in the processed metal at different depths. On a single site perpendicular to the radius of the wheel, the distribution of vertices can be expressed by the dependence:  
\[dn_{g}=f(y)dy \tag{7}\]  
where \(dn_{g}\) is the number of grains enclosed between the surface of the wheel in depth \(y\) and \(y+dy\) in the direction of the radius; and \(f(y)\) is the function of the distribution of the vertices of abrasive grains in depth. The volume of metal removed per minute by grains enclosed in a layer \(dy\), taking into account Equations (5)-(7), is determined by:  
\[dQ_{M}=2.4\cdot 10^{5}\cdot B\cdot V_{K}\bigg{[}f(y)\int_{0}^{y}\int_{0}^{z} \phi(y,z)dydz\bigg{]}dy \tag{8}\]  
Figure 1: Diagram of a single mark applied by a diamond cone to the polished surface of a steel sample. Integrating and substituting the value (8) into Equation (3) we obtain:  
\[Q_{\Omega}=\frac{2.4\cdot 10^{5}\cdot B\cdot V_{K}\left[f(y)\int_{0}^{y}\int_{0}^{z }\phi(y,z)dydz\right]dy}{\Omega} \tag{9}\]  
To use Equation (9), it is necessary to know the function of the distribution of abrasive grains in depth and the equation of the surface that limits the volume of the mark.