Quantitative Analysis of Spherical Probe Radius Effects on Surface Roughness Measurements

Probe-surface convolution distorts the measurement results of tactile surface measuring instruments, complicating comparisons with results from other tactile or optical measurement systems. This paper presents a theoretical analysis of how the spherical probe radius affects the measured surface profile and investigates its impact on roughness measurements. The analysis reveals that the probe convolution not only introduces multiple harmonic components of the original frequency but also alters the amplitude of the original frequency itself. Among these effects, this amplitude reduction predominantly contributes to deviations in roughness measurements, resulting in a systematic underestimation and cannot be eliminated by conventional filtering. This study indicates that for filtered profiles, probe convolution causes a measurement error proportional to the square of the ratio of the probe radius to the maximum allowable radius rmax and can reduce the measured roughness R_q by up to 12.5%. To maintain a measurement error below 5%, the tip radius should be smaller than 60% of rmax. The analytical model is validated through both numerical simulations and experimental measurements.