Modeling of the Process of Shaping Glass Beads According to the Method of Free Lapping

Mathematical modeling of  the processing of balls by the method of free lapping using a tool in the form of a hollow thin-walled cylinder (tool bushings) has been performed. An analytical expression has been obtained for calculating the cutting  

paths, which, according to the formula of F. Preston, are proportional to the amount of material removal from the workpiece being processed. The calculation of the cutting paths in the diametrical sections of the ball during its rotation by a total angle of 4000 radians was carried out. The calculation is performed for various values of such adjustment parameters of the base machine as the amplitude of the reciprocating rotational movement of the device with tool bushings fixed in it, the distance between the symmetry axes of the tool bushings and the axis of rotation of the device, the speed of rotation of the input link of the actuator mechanism of the base machine and a faceplate fixed on its spindle and serving to communicate relative rotation to the ball blanks. In this case, the section with the greatest divergence of cutting paths was considered and the relative value of these paths was determined, which is proportional to the accuracy of processing. Studies have been carried out that have made it possible to identify at least three locations of tool bushings, in which the minimum value of the relative cutting paths on the machined ball surface is achieved, i. e. maximum accuracy of processing, which makes it possible in practice to combine the operations of preliminary, main and final grinding of the parts under consideration. It is shown that the degree of correlation between experimental and theoretical results is at the level of 80–85 %, and the polishing performance according to the proposed scheme increases by about 30 % in comparison with the classical scheme for performing this operation.

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