Metrology, Volume 6, Issue 2 (June 2026) – 11 articles

Train wheel wear is a critical factor affecting train operational safety, making the accurate and objective evaluation of wheel wear condition essential. However, current approaches are still constrained by inadequate measurement accuracy and incomplete evaluation methods. To address this issue, this study proposes an integrated method for the high-precision measurement and wear condition evaluation of train wheels. A multi-sensor data fusion-based measurement method is developed to synchronously acquire key wear-related parameters, including wheel diameter, flange height, and flange thickness. Based on the measured data, a matter-element model combined with game-theoretic weighting is established to evaluate wheel wear condition. Experimental results show that the proposed online measurement method for in-service wheels achieves standard deviations below 0.15 mm, and the measurement errors satisfy the requirements of Chinese railway industry standards. The evaluation results derived from the high-precision measurement data indicate that wheel wear condition gradually deteriorates with increasing service mileage, and that flange height wear is the dominant factor affecting the wear grade. These findings are consistent with actual operating conditions. The proposed method integrates high-precision multi-parameter measurements with wear condition evaluation, providing a reliable technical basis for wheel condition monitoring and predictive maintenance in rail transit. Full article

The main difficulty of pitch diameter calculation arises during the determination of the coordinates of the probing element and screw surface contact. This paper proposes a mathematical model for pitch diameter calculation of thread gauges using a two-ball stylus for internal thread calibration and three wires for external thread calibration. To describe the geometry of the thread and probing element, a non-linear equation system has been established and solved numerically. The solution of this system gives the actual contact points of the probing element with the thread profile. Pitch diameter is calculated directly without any further corrections. This mathematical model can be applied to parallel threads without any restrictions regarding lead and flank angles. Calculation of the rake correction is therefore avoided completely. The authors provide functional PHP/HTML code that can be easily integrated into any PHP-based website. Additionally, an open-access web tool has been developed that enables the direct calculation of thread pitch diameter from measured values, as well as the coordinates of the actual contact points between the thread profile and the measuring elements. Full article

A radial-polarization-based interferometric method is proposed for measuring object surface profiles. In the proposed approach, a radially polarized beam is generated by transmitting a linearly polarized beam through a zero-order vortex half-wave plate and is then introduced into a modified Twyman–Green interferometer, in which the test specimen is placed in one interferometric arm. By introducing a small variation in the wavelength illumination, two interferometric intensity patterns are recorded using a CMOS camera. The corresponding phase difference distribution is retrieved from the recorded intensities and subsequently used to reconstruct the surface profile of the specimen. The feasibility of the proposed method is experimentally validated by measuring a standard gauge block, and the results show good agreement with theoretical predictions. Owing to its simple optical configuration, ease of alignment, high measurement accuracy, and rapid measurement capability, the proposed method demonstrates strong potential for practical surface profile measurement applications. Full article

Acoustic emission (AE) monitoring in metal additive manufacturing (AM) requires compact sensors capable of high-frequency operation, broad bandwidth, and high sensitivity. However, increasing structural stiffness to achieve high resonance frequencies typically reduces electromechanical sensitivity. This work presents a finite element study of a coupled-resonator capacitive micromachined ultrasonic transducer (CMUT) designed to address this trade-off. The proposed architecture integrates three mechanically coupled silicon membranes with a stepped capacitive cavity that increases capacitance while preserving structural stiffness, enabling enhanced sensitivity without compromising high-frequency operation. COMSOL Multiphysics simulations were used to evaluate modal characteristics and frequency response under DC pre-stressed conditions. Modal coupling produced closely spaced resonances that broadened the effective bandwidth, while the stepped cavity significantly increased voltage output through improved electromechanical coupling. Compared to a single-resonator flat-cavity design, the coupled stepped-cavity configuration demonstrated nearly a threefold enhancement in output voltage while maintaining operation near 100 kHz. Additionally, adjusting the central resonator length enabled controlled frequency tuning for scalable array implementation. These results establish a proof of concept for a high-frequency, high-sensitivity micro-electro-mechanical systems (MEMS) CMUT architecture suitable for distributed AE monitoring in advanced manufacturing environments. Full article

In a perspective projection, a circular target appears as an ellipse for an oblique view. Herein, the ellipse center obtained from image coordinate measurement operators differs from the projection of the circle center. This discrepancy is called eccentricity and may lead to systematic errors. This article documents the significance of these discrepancies and discusses five different correction methods that can be applied in the image space or as a model adaptation. Two of the methods include the determination of the circle radius and thus also offer a possibility to define the scale. The eccentricity correction procedures are validated in a series of experiments, which proved that even extreme eccentricity effects can be fully compensated. In the experiment on the approaches including scale determination, the precision and accuracy of the scale definition is investigated, obtaining relative accuracies of 0.5–1%. Full article

For the calibration of surface plate, the classical Moody method is still commonly used. In this method the straightness of a number of lines over a surface plate in a union-jack configuration is measured and combined into a flatness measurement. The measurement of the two center lines is used to determine so-called closure errors. A shortcoming of this method is that it gives an ambiguous value for the central height and that the measurements of the central lines are not involved in the evaluation. This research shows how the lines can be incorporated in the measurement evaluation in a least-squares sense. This gives a measurement redundancy leading to an 18% reduction in the uncertainty. Also, it is shown that a further reduction in the uncertainty is possible when using the gravity vector as a common reference, as can be done when using electronic levels. A least-squares evaluation of measurements taken in this way gives an even further redundancy, leading to a reduction in the uncertainty of 29% relative to the traditional evaluation according to the Moody method. This is illustrated with an actual measurement example and additional Monte Carlo simulations. Full article

This paper considers the measurement of a quantity when a nominal value or previous estimate is available, which is the case with a quantity designed to be zero or which might be the case when a consensus value is to be calculated in a measurement comparison. If an upper bound can be placed on the magnitude of the difference between the nominal value and the true value, then the mean square error of the overall measurement procedure can be reduced by a statistical method known as shrinkage estimation. We describe the method for use in an individual measurement, but we give a deeper analysis assuming the context of a measurement comparison. Full article

The capacitance of the DC link is an important variable for the prediction of remaining useful life and failures in frequency inverters. The direct measurement of the DC-link capacitance in inverters operating under load is technically challenging and generally impractical. Recently, a great focus has been given to data-based soft sensors for estimating this variable. These methods, however, are evaluated based only on the estimate errors, and do not take into account the metrological aspects of these estimators. This paper proposes an uncertainty analysis method based on Monte Carlo simulations and bootstrapping that can be applied to all recently published methods for end-of-life (EOL) estimation based on data-driven regression and neural networks. A state-of-the-art model of EOL monitoring based on capacitance estimation was evaluated using the proposed framework, and an experimental study with a frequency converter drive for a brushless DC motor was performed, considering multiple output frequencies, loads and DC-link capacitance conditions. The output distributions are not symmetrical and show that the variable with the most significant impact in the propagated uncertainty is the DC link voltage. The results show confidence interval widths ranging from 12 μF to 61 μF, with wider confidence intervals obtained at higher power setpoints. Full article

This study presents a characterization method for High-Frequency Current Transformers (HFCTs) intended for partial discharge (PD) measurement in on-line acquisition systems designed for AI-based processing and clustering. The primary objective is to analyze how key design parameters, ferrite core material, and number of turns, influence HFCT frequency response, attenuation, and sensitivity, thereby providing a basis for optimized sensor design when data analysis is to be performed by means of AI-based algorithms. The investigation focuses on the influence of different ferrite core materials and varying secondary turn numbers on the frequency spectrum and the response to IEC 60270-compliant calibrator impulses Both concentrated and well-distributed HFCT secondary winding configurations are analyzed to evaluate their impact on signal behavior and sensitivity. The experimental results are compared with a simplified theoretical model to validate performance trends and identify key design factors. The HFCT response to IEC 60270-compliant calibrator impulses is examined to assess its suitability for PD measurement systems and monitoring. The results highlight the critical role of core selection and the number of turns in shaping HFCT bandwidth, attenuation, and impulse response, which are essential for accurate and reliable PD detection in continuous monitoring systems to perform the diagnostic of the electrical insulation condition. This diagnostic approach is based on the detection of partial discharge (PD) activity over time, with the objective of identifying evolving phenomena by monitoring the amplitude and characteristics of the signals associated with different defects. Therefore, accurate separation of signals originating from different defects and from noise is essential. These results provide a foundation for designing HFCT sensors suitable for integration into advanced diagnostic frameworks, AI-aided for Condition-Based Maintenance (CBM). Full article

A cost-effective and practical method for characterizing the dielectric properties of liquids at 1 MHz is presented in this article. A cylindrical parallel-plate capacitive sensor was developed, in which the circular end plates function as electrodes and the sidewall is formed by a thin polyvinyl chloride ring cut from a standard water pipe to enclose the liquid sample. Dielectric constant values of air, distilled water, ethanol, and methanol were determined through analytical calculations, electromagnetic simulations, and experimental measurements at 1 megahertz. Consistent results were obtained across all methods, and the extracted values were found to agree well with theoretical values, yielding extraction errors of 0.06% for methanol and 1.85% for ethanol with respect to theoretical values from the literature. A calibration technique was applied in which air and water were used as reference materials with known dielectric constants, effectively mitigating uncertainties associated with sensor geometry, spacer material, and fringing fields. Through this work, a practical and effective technique for dielectric characterization at low frequency has been demonstrated, with core validation of four reference materials (air, deionized water, ethanol, and methanol) at 1 MHz and an additional application example in which cow’s milk is characterized over 10–30 MHz. The 10–30 MHz measurement demonstrates the applicability of the proposed method in the low megahertz region, while the primary validation is conducted at 1 MHz. The technique is applicable to a wide range of applications in materials science, chemical, and biomedical engineering. Full article

Carbon capture, utilization, and storage (CCUS) plays an important role in meeting the European Union’s target to reduce greenhouse gas emissions by 55% by 2030 and become carbon neutral by 2050. Accurate flow metering is required throughout the carbon capture and storage (CCS) chain to meet fiscal and regulatory requirements. To establish accurate CO₂ flow metering, flow meters must be calibrated with traceability to international standards of measurement at relevant flow conditions. To ensure confidence, reliability, and comparability of calibration results, calibration facilities perform interlaboratory comparisons. However, there is currently a lack of CO₂ gas flow meter calibration facilities. The flow metering calibration facilities of VSL, NEL, INRIM, DNV, and FORCE participated in an interlaboratory comparison with CO₂ up to 400 m³/h and 31 bar(a) to compare the calibration results with several flow metering principles. At the intermediate-scale facilities of NEL, VSL, and INRIM, the difference in results between the VSL and INRIM facilities were within the facilities’ CMC values, while NEL’s facility showed a significant difference primarily due to vibrational relaxational effects of CO₂ with small critical flow Venturi nozzles. At the large-scale facilities of NEL, DNV, and FORCE, 91% of the test points passed the equivalency criteria in the range of 20 m³/h to 400 m³/h with a Coriolis meter, confirming traceability for carbon dioxide across the facilities. Overall, the interlaboratory comparison has made it possible for the CCUS industry to calibrate gaseous CO₂ flow meters with traceability to international standards. Full article