Rotary Burnishing of Cylindrical Surfaces: Kinematic Layout, Relative Curvature Tensor, and Contact-Conformity Classification

Rotary and vibro-rotary burnishing create regular arrays of imprints whose geometric interaction depends on the pitch, indentation depth, and relative curvature of the tool–workpiece pair. This paper develops a unified kinematic–curvature model that maps machine settings to an imprint layout on the unwrapped cylindrical surface and constructs the relative curvature tensor. The tensor state and full normalized neighbor metric assign the contact state to one of four mutually exclusive categories: non-elliptic, cell-limited, near-conformal (warning), or isolated elliptic. The tensor formulation is invariant under rotation of the tangent basis and provides a common geometric mapping for external cylinders and internal tubes. The kinematic map and tensor classification are independent of the rigid-plastic mean-pressure approximation and are verified by the factor relation between the center-line slope and imprint orientation, tensor invariance, limiting cases, and a closed-form identity. The calibrated approximation is used solely to estimate the maximum-load penetration and an equivalent projected footprint. At a fixed normal force and effective hardness, the projected area is identical in all curvature cases, whereas the curvature changes the penetration and footprint aspect ratio. Neglecting the workpiece curvature underestimates the external cylinder indentation depth by about 5.8% relative to the full tensor calculation; this ratio is independent of the force and effective hardness within the approximation. For the stated internal tube row pitch and hardness, the axis-aligned cell-limited transition force is 5–8 N. Evaluation with the full metric shows that the consecutive-event vectors are separated well; the periodic-row neighbor nevertheless places the reference case in the cell-limited class. A closed-form expression for this transition force is derived. The model provides a transition criterion verified by analytical identity and consistency checks; residual geometry and post-threshold pressure redistribution require unloading calibration and a periodic unilateral contact formulation, respectively.