Probabilistic model of surface layer removal when grinding brittle non-metallic materials

OBRABOTKAMETALLOV Vol. 23 No. 2 2021 TECHNOLOGY    1 ( ) h f u C u , (8) where h C – is the coef fi cient of proportionality of the distribution curve:    h u C H , where u H – is the thickness of the layer of the working surface of the tool in contact with the workpiece. Taking into account the above, dependence (8) can be represented as:      1 ( ) u f u u H , (9) where  – is the parameter of the distribution density function. Comparison of the values of the probability density of the distribution for different models (Fig. 3) indicates that the most signi fi cant difference from the dynamic distribution has a straight-line relationship. The best approximation is provided by the power-law dependence of the modi fi ed Г -distribution function. Fig. 3. Simulation the probability density of the distribution of the tops of grains when approximating their pro fi le: 1 – straight-line dependence; 2 – a parabola; 3 – modi fi ed function Г -distributions Power-law dependences are currently widely used not only for the mathematical description of the distribution of grain tops on the working surface of grinding tools. To characterize the shaping process, it is also of considerable interest to calculate the number of abrasive grains passing through an elementary surface area. The increment in the number of grains in general form is determined by equation (7), which, after substitution of values ( ) f u and transition from a discrete model to a continuous one, takes the form:                   2 0 ( ) ( ) t k u z f e u V V n z t t y d D H . (10)