There are several labs operating throughout the world that do not follow a designated guideline for calculating measurement uncertainty for force calibrations. Realizing the need for guidance, Morehouse decided to draft this document explaining how to calculate measurement uncertainty and how uncertainty propagation for force calibration systems works.

Figure 1. Measurement Uncertainty Pyramid

Calibration and utilization of measurement instruments will imply some level of uncertainty. As an instrument calibration is traced back to SI units, a higher number of intermediate calibration stages results in higher levels of measurement uncertainty (Figure 1). In other words, uncertainty of the unit under test is typically higher than the standard with which it was calibrated. It is not possible for the expanded measurement uncertainty of the unit being calibrated to be less than the machine or force measuring device that is used to calibrate the unit itself. This paper describes the propagation of uncertainties using Calibration and Measurement Capability (CMC) for force measurement instruments through the traceability chain to SI units.

Test Plan and Equipment

A 445 kN (100k lbf) Morehouse Ultra-Precision Load Cell was chosen for the testing plan. The calibration test setup is shown in Figure 2. The Morehouse load cell provides relatively high stability, resolution, and repeatability. Consequently, the testing plan represents an almost best-case scenario: the lowest level of Calibration and Measurement Capability (CMC) that a load cell user can achieve at each level of the traceability chain.

Figure 2. 445 kN (100k lbf) Load Cell in Deadweight Machine Being Calibrated

An 89 kN (20k lbf) test point was chosen for analysis based on historical data. This load point was chosen for studying the CMC propagation to follow the ILAC P14 requirements. Morehouse Ultra-Precision 445 kN (100k lbf) systems can often use this load cell in the Tier 2 group from 20% to 100% of capacity for force calibration purposes without switching standards. The reference standard of Tier 2 in this paper represents a load cell that is calibrated in accordance with ASTM E74 standard test method using other load cells with ASTM Class AA designation. Additionally, the 20% point represents a pivot point for achieving CMC of approximately 0.02% of applied force. At higher forces, the CMC is typically lower. However, at lower than 20% of capacity forces, CMC starts to increase; it continues to increase to the 10% and lower force points, where the CMC becomes higher than 0.05% of applied force. Therefore, it is often recommended that the end user in Tier 2 only uses the load cell from 20% through capacity in order to maintain CMC's better than 0.02% of applied force.

Tier 0: CMC for Primary Standards

Table 1. Uncertainty Propagation Analysis for Load Cell Calibrations

In this tier, CMC for Morehouse's deadweight calibration systems was determined. Table 1 contains the uncertainty contributors for this calculation, along with their appropriate divisors. Testing was conducted based on United States customary units, and then converted to SI units in Table 1 to make it more tangible for international users. Degrees of freedom and coverage factors were calculated separately using the Welch-Satterthwaite equation. In this tier, Morehouse had the reference deadweights calibrated directly by N.I.S.T. These weights, pictured in Figure 3, were adjusted for the local gravity, material density, and air buoyancy, and their traceability is derived from the international prototype kilogram (SI unit symbol kg).

When the calibration was performed in a Morehouse deadweight machine, CMC was calculated using these weights. A repeatability study was conducted with three Morehouse load cells (445 kN; 111 kN; and 44 kN capacities) throughout the entire range of the machine. Morehouse's CMC resolution for 89 kN (20k lbf) load was used for UUT resolution in Tier 0 only. This value was determined based on a 111 kN (25k lbf) load cell with 4 mV/V output at capacity and 0.00001 mV/V readability.

The environment was controlled by better than ±1.0 °C, while the stability of the weights was calculated using historical values for the material and years of wear history from our other deadweight machines. The resolution of the weights was zero since they are physical standards, and the resolution of a good measurement system was used as an uncertainty contributor for UUT resolution. Various technicians' tests were compared to determine the repeatability and reproducibility per point of the Morehouse deadweight calibration machine. All of these efforts, combined with continued process monitoring, yielded a CMC of better than 0.0016% of applied force.

Tier 1: Using Primary Standard Deadweights to Calibrate a Load Cell

Figure 3. View of Deadweight Machine

For Tier 1 calibration, the deadweight calibration machine was utilized to calibrate a load cell in accordance with the ASTM E74 standard. A Morehouse 445 kN (100k lbf) load cell was calibrated in this tier by deadweight primary standards known to have a CMC better than 0.016% of applied load.

To calculate the CMC of the calibration, a repeatability and reproducibility (R&R) study was done for Tier 1 using a 111 kN (25k lbf) Ultra-Precision load cell. Moreover, an environmental condition of ±1 degree Celsius, along with a stability value of 0.005% (50 parts per million), was used for calculating uncertainty values. The actual resolution of the UUT load cell 1.07 N (0.24 lbf) was employed for uncertainty calculations in Tier 1. It might be noteworthy to mention that the reference uncertainty used in Tier 1 already included the UUT resolution embedded in deadweight CMC calculations. Basically, UUT resolution is considered twice in the calculation of uncertainties for Tier 1–3. This method is on the conservative side of the uncertainty calculations, and there is ongoing debate about whether or not the resolution from CMC must be included in higher calibration tiers.

Load cell output stability is another of the uncertainty contributors when the cell is calibrated per ASTM E74. Stability is calculated by comparing the load cell output to the previous calibration data. Most Morehouse Ultra-Precision load cells provide a one year stability of around 0.005% through 0.01%. Typically, the actual numbers would be used for this evaluation; however, this test was controlled, and the experiment could not wait another year to obtain the actual UUT load cell's stability numbers.