Prediction of lobe growth and decay in centreless grinding based on geometric considerations

The growth and decay of lobes during centreless grinding have been studied by previous researchers using physical tests, time-domain simulation and identification of the roots of the Laplace transform of the characteristic equation. In this paper, the authors have extended these latter two methods to generate complete stability diagrams encompassing the entire practical range of machine set-up angles. These diagrams indicate that by varying the set-up angles in a prescribed manner during grinding, rapid rounding of arbitrarily lobed components can be achieved. This is verified via time-domain simulation. Secondly, a novel and arguably more intuitive method of predicting the lobe growth and decay during centreless grinding is presented. The method considers the locations of the three points of contact between a lobed workpiece and the regulating wheel, the support plate and the grinding wheel. Axial symmetry is assumed. A unique circle can be drawn through these three points. The centre and radius of this circle vary continually as the workpiece rotates, in a manner dependent upon the workpiece's profile and the set-up angles. An above-average instantaneous radius leads, via machine stiffness, to a correspondingly larger grinding force and so to an increased instantaneous depth of cut. If this occurs when the trough of a lobe is being ground, the trough will become deeper and lobe growth will result. By contrast, if the instantaneous radius is below average when the trough is being ground, the lobe will decay. From this simple geometric consideration, the authors have calculated the rates of decay and growth of a range of numbers of lobes, across a wide range of set-up angles. The results are shown to agree well with those given using the previous methods.