Pattern formation and linear stability analysis in centreless grinding

Abstract

Quite often a centreless ground surface has an undesired wavy shape instead of a circular shape owing to a geometric instability. In this paper the equations describing the kinematics of the grinding process obtained by Rowe et al.(1) are used to build a linearized model that describes the formation and evolution of the pattern on the manufactured surface. The profile of the workpiece is discretized and linearly interpolated. Stable or unstable patterns or eigenvectors are obtained for the configurations of the centreless grinding system. Circular profiles are shown to appear for configurations that have no unstable patterns while wavy profiles appear for configurations that have at least one unstable pattern. Some of the most important non-linear effects are considered as limits on the exponential growth of the unstable patterns predicted by the linear analysis. Specifically, the non-linear analysis is based on the limited allowed curvature of the profile of the workpiece. Although it is an approximation of the real phenomena, the non-linear analysis is able to predict patterns that appear on the ground surface that are very close to the experimental observations even if unstable patterns are present.