A Markovian Finishing Process

Addressed is the mechanism of finishing processes for a workpiece surface using hard abrasive tools such as grinding, abrasive paper, and filing. The mechanism is intended to monitor the gradual changes of the workpiece surface state roughness as the tool is applied for several strokes. Based on a number of common features, the present study simulates each rubbing stroke as a Markov process, and each set of several strokes as a Markov chain. In the simulating model, the discrete probabilistic properties of a specific tool abrasive surface can be expressed in terms of a corresponding Markov matrix operator. Thus, the tool action after one rubbing stroke is obtained via a matrix mapping from a given state roughness to a subsequent state roughness of the workpiece surface. Although the suggested model is capable to handle a comprehensive finishing mechanism, the study focuses on the simple case of zero feeding using a hard abrasive tool, in which the Markov matrix shrinks to a special triangular form. Main findings show that major aspects of the tool surface are transferred to the stepwise roughness state of the workpiece immediately after the first stroke. In addition, regardless of the initial roughness state of the workpiece surface, whether with flat or randomly distributed heights, the ultimate state roughness is unique and definitely features the theoretical case of a plain flat surface. However, this theoretical case is infeasible since it can only be reached after infinite number of strokes.