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

OBRABOTKAMETALLOV Vol. 23 No. 2 2021 TECHNOLOGY where y L – is the length of the contact zone from the conditional outer surface of the tool to the main plane (Fig. 2), which can be calculated from the dependence:    ( ) y f e L t y D . (16) To calculate the probability of an event characterizing the chipping process of the workpiece material ck P during grinding, the following relationship was used [20]:            1 ( ) ck o f u P P t , (17) where 0 P – is the probabilistic characteristic of chipping of a brittle non-metallic material chipping;  – is the exponent in the probability equation. The indicated parameters of dependence (17) can be calculated by the method, proposed in [21]. When substituting the obtained expressions z b and ( ) f u into equation (15) and from equations (13) and (9), it takes the form:                      ( ) 2 1 1 0 ( ) ( , ) y m t z y z c b k u z f e L u u k C V V n z a y z t y u u dudz D V H                      ( ) 2 0 1 0 ( ) y m t z y z c b k u z f e L u u k C V V n P z t y u u dudz D V H                            ( ) 2 0 1 0 ( ) y m t z y z c b k u z f e f L u u k C V V n P z u t y u u dudz D t V H . (18) Results and discussion The previously adopted models of the grain tops and its depth distribution densities make it possible to proceed to the establishment of functional relationships between the probability of non-removal of the material and technological factors. After integrating the resulting equation over u we get:                         2 0 1 ( ) ( ) ( 1)(1 ) ( , ) ( 1) m z c b k u z f e L u u k C V V n m P z a y z t y dz D m V H à à à                           2 ( ) ( ) ( 1) ( 1) m z c b k u z f e L u u f k C V V n m z t y dz D m V H t à à à , (19) where Г (...) – are the corresponding gamma functions. Integration of equation (19) is possible only for particular values of the coef fi cients. For   1.5 ,  0.5 m ,   2 and    2 2 b z C we get:                              2 3 5 0 1 3 2 3 3 3 ( ) (1 )( ) 2 8 2 2 ( , ) 5 15 3 2 (3) c b k u z f y y y u u k C V V n P t y z z a y z z L L L V H à à à                               4 9 7 5 3 3 2 3 22 2 7 3 3 ( ) ( ) 4 6 4 8 2 2 5 20 3 9 2 (5) 7 c b k u z f y y y y u u f y k C V V n t y z z z z z L L L L V H t L à à à . (20)