Gram-Charlier Series for Enhanced Areal Roughness Assessment in End-Milled Low-Carbon Steel Surface Replication

Areal roughness significantly determines surface functionality. Traditional contact methods, though standard, are limiting surface evaluation to single profiles, rendering that evaluation inadequate for multi-tooth processes like milling. Non-contact techniques, however, rely on contact-based regression. This study proposes a framework that applies Gram-Charlier Probability Density Functions (PDFs) to enhance contact measurement accuracy for optimizing surface replication in machining operations. The approach models areal roughness using both Gaussian and Gram-Charlier PDFs. Analysis of 170 milled surfaces under varied machining conditions involves three-dimensional contact measurements that employ Hermite polynomials and cumulants for Gram-Charlier functions construction. The methodology determines roughness values through PDF peak analysis. The methodology determines roughness values through PDF peak analysis. Validation demonstrates that Gram-Charlier models achieve R² values exceeding 0.99 across all extraction densities (four to twenty-one linear extractions per surface), confirming superior performance. Response Surface Methodology links this statistical model to machining parameters, enabling replication similarity assessment. Application of Minkowski and Shannon's entropy family distance metrics identifies optimal combinations of spindle speed, depth of cut, and feed per tooth, revealing the depth of cut as the most sensitive parameter, and the carbon steel with 0.15% as the best material over the three selected for achieving precise surface replication control.