A revised treatment of the influence of the sample support on the measurement of line scales and the consequences for its use to disseminate the unit of length

Abstract

We revisited the treatment of the influence of the support conditions and the resulting bending of a scale on the positions of its graduation lines. In contrast to earlier publications, we did not calculate the position deviation with respect to the case without any bending. Due to the production processes used today, this is inappropriate and leads to an overestimation of the related uncertainty contribution. We also extended the treatment from two lines to all lines of the scale and included a finite starting point of the graduation. We verified our analytical model by means of FEM calculations. In addition, we showed that a first order Taylor expansion yields sufficiently accurate results for the position deviations and leads to simple equations for their size. Because line scale measurements are related to the zero line the constant term of the Taylor expansion cancels and the remaining coefficient is identical to the sensitivity coefficient required in the determination of the standard uncertainty contribution. If it is sufficient to suppress the position dependence and to consider only the case where the sample is supported at symmetric positions, then a new, simple equation is obtained for the resulting uncertainty contribution. Finally we showed that due to the position dependence of the deviations the scale coefficient of the scale, which is obtained by a linear fit to the deviations of the line positions of the scale from their nominal values, is also influenced, which has apparently not been noticed up to now. If the line standard used to disseminate the unit of length has not been designed carefully then the resulting change in the scale coefficient exposes a practical limit to the related achievable length-dependent uncertainty contribution.

Long-term dimensional stability and longitudinal uniformity of line scales made of glass ceramics

“…The measurement procedure was as follows. First, the scale to be measured was placed at the Bessel points [20] on the alignment stage of the calibration system with an orientation of 0 • and aligned with the measurement axis. Second, the scale was measured four times (two round trips of the moving microscope).…”

Development and Validation of a Calibration Gauge for Length Measurement Systems

“…A horizontal bar of great length requires two points of support in the direction of its length to be stable. The position of these standing legs determines the action of the gravity of this bar; depending on how it is supported, measurement errors can be caused [11]. Therefore, if the supports are positioned at the ends, it will warp in the center causing the ends to come closer and tilt upwards.…”

Measurement technologies for precision positioning

“…To obtain small uncertainties of only a few nm over measurement ranges of about 500 mm requires careful analysis of all relevant influences, including alignment errors, Abbe errors [116], dimensional sensitivity of substrates to variations of ambient pressure [204], ambient temperature [190] and mounting of the substrate [115].…”