uncertainty in nist force measurements · uncertainty in nist force measurements ... expressing the...

1. Introduction

For more than 75 years NIST has maintained a forcelaboratory capable of disseminating force measurementstandards to government, industry, and academic facil-ities through the calibration of force transducers thatserve as transfer standards. The facilities available atNIST, the services provided, and the proceduresemployed have been described in previous publications[1-5]. The purpose of this paper is to develop an uncer-tainty estimate for NIST force measurements, based onan examination of the various uncertainty contributorsthat apply to the present primary force standard facili-ties.

The NIST primary force standards consist of sixmachines for applying discrete forces generated bystainless steel deadweights, spanning a range of 44 N to4.448 MN [1]. These machines were constructed aboutthe year 1965, becoming operational following thecompletion of the deadweight mass determinations in

1966. Automation of the weight-changing mechanismsof these machines was accomplished about 1989, alongwith the implementation of instruments for the preciseautomated measurement of the responses of straingauge load cells used for measuring force.

Section 2 of this paper presents the form of the trans-ducer calibration equation as a framework for proceed-ing with the examination of various force uncertaintycomponents. Following that are discussions of theuncertainties associated with the realization of force(Sec. 3), the measurement of transducer response (Sec.4), and the fit of the data to the calibration equation(Sec. 5). It is noted that the uncertainty components ofSec. 5, which are largely dependent upon the character-istics of the transducer being calibrated, are the domi-nant contributors of the overall measurement uncertain-ty.

Volume 110, Number 6, November-December 2005Journal of Research of the National Institute of Standards and Technology

589

[J. Res. Natl. Inst. Stand. Technol. 110, 589-603 (2005)]

Uncertainty in NIST Force Measurements

Volume 110 Number 6 November-December 2005

Tom Bartel

National Institute of Standardsand Technology,Gaithersburg, MD 20899-8222

[email protected]

This paper focuses upon the uncertainty offorce calibration measurements at theNational Institute of Standards andTechnology (NIST). The uncertainty of therealization of force for the national dead-weight force standards at NIST is dis-cussed, as well as the uncertainties associ-ated with NIST’s voltage-ratio measuringinstruments and with the characteristics oftransducers being calibrated. The com-bined uncertainty is related to the uncer-tainty of dissemination for force transferstandards sent to NIST for calibration.

Key words: calibration; deadweight forcestandard; expanded uncertainty; force;gravity; mass; transfer standard; metrolo-gy; uncertainty.

Accepted: November 3, 2005

Available online: http://www.nist.gov/jres

2. Expression of Uncertainty for theForce Calibration Equation

Current force transducer designs do not incorporatean absolute internal reference for the measure of force.Rather, a force transducer can achieve an accuracy of0.01 % or better only through calibration relative to aknown reference. To be fully useful, a transducer mustbe accompanied by its particular calibration equationrelating the transducer output response to the appliedforce.

A force transducer’s response is generally expressedin terms of the applied force by a polynomial equation:

(1)

where R is the transducer response, F is the appliedforce, and the Ai are coefficients characterizing thetransducer. In practice, the summation is usually carriedto an order of 2 or 3. The unit for R is appropriate forthe type of deflection-sensing system employed by thetransducer, which may be mechanical (in proving rings,for example), electronic (for strain gauge load cells), orhydraulic.

NIST provides a force calibration service wherebythe response Rj of a customer’s transducer is measuredfor each of several applied reference forces Fj, with theforces applied in a sequence in accordance with anappropriate test method such as ASTM E 74-04 [6].The coefficients Ai in Eq. (1) are then calculated from aleast-squares fit to the data set (Fj, Rj).

Thus the “disseminated result” of a force calibrationat NIST is the set of coefficients Ai for the particulartransducer being calibrated. The uncertainty in this dis-seminated result is attributable to the uncertainty in theapplied forces, the uncertainty in the calibration of theinstrumentation used to acquire the transducer respons-es, and the uncertainty of the fit of the measured data tothe equation chosen as a model, which can be attributedin part to certain characteristics of the transducer. Thesequantities are denoted as uf, uv, and ur in Eq. (2).

For each NIST force calibration report, this measure-ment uncertainty is given as the expanded uncertainty,U, which is calculated in accordance with NISTTechnical Note 1297, “Guidelines for Evaluating andExpressing the Uncertainty of NIST MeasurementResults” [7]. The NIST policy stated in this documentis based on an approach presented in detail by the ISOpublication, “Guide to the Expression of Uncertainty inMeasurement,” ISBN 92-67-10188-9 (1993) [8].

The expanded uncertainty U is reported in units ofthe transducer response, providing the uncertainty in

the response values calculated from the calibrationequation yielded by the NIST calibration measure-ments. Thus U defines an interval R ± U, within whichthe response of the transducer to a given applied forceis expected to lie, when R is calculated from the calibra-tion coefficients Ai according to Eq. (1).

The value of U is calculated by multiplying the com-bined standard uncertainty, uc, by a coverage factor, k,of 2. Thus the confidence level for the interval definedabove is about 95 %.

The combined standard uncertainty, uc, is determinedfrom

(2)

where:

uf is the standard uncertainty associated with theapplied force, due to uncertainties in the mass cali-bration and adjustment of the dead weights and touncertainties in the air density and the acceleration ofgravity. This component is explained in Sec. 3.

uv is the standard uncertainty in the calibration of thevoltage ratio measurement instrumentation used atNIST. This component does not apply if the forcetransducer being calibrated incorporates an indicat-ing instrument that is part of the calibrated device.This component is explained in Sec. 4.

ur is the standard deviation calculated according toASTM E 74-04 from the differences between theindividual measured responses and the correspon-ding responses computed from Eq. (1). An explana-tion of this calculation is given in Sec. 5.

3. Uncertainty in the Applied Force

The NIST deadweight force standards exert force bymeans of the earth’s gravitational attraction acting uponweights of calibrated mass. The downward force exert-ed on a static deadweight is given by

(3)

where F is the applied force in N, m is the mass of theweight in kg, g is the acceleration of gravity in m/s2, ρa

is the atmospheric density at the location of the weight,and ρw is the density of the weight in the same units asρa. The uncertainty in this force is dependent upon theuncertainties in the measured values of the mass, grav-


590

0 ,iiR A A F= + Σ

2 2 2 2c f v r ,u u u u= + +

a w[1 ( / )],F mg ρ ρ= −

itational acceleration, and the ratio of the air and weightdensities, which are discussed respectively in Secs. 3.1,3.2, and 3.3.

Uncertainties associated with transducer mounting inthe force machine, such as the placement of the point offorce application on the transducer or the alignment ofthe vertical gravity vector with the load cell axis, arediscussed in Sec. 5.

3.1 Uncertainty Associated with Mass

All of the weights for each of the six NIST dead-weight machines had their masses determined in 1965and 1966 by the mass laboratory at NIST, which wascalled the National Bureau of Standards prior to 1988.The organizational name for the mass laboratory at thetime was the Institute for Basic Standards, MetrologyDivision, Mass and Volume Section, with Paul E.Pontius serving as the section chief. Mass and forcemetrologies are currently organized at NIST within onegroup under the Manufacturing EngineeringLaboratory, Manufacturing Metrology Division, Massand Force Group [1].

The deadweight masses were determined by compar-isons with U.S. national mass standards, with the pro-cedure also incorporating adjustments of the weights toachieve the desired mass values. The reports of calibra-tion giving the results of the mass determinations per-formed in 1965 and 1966 provide the uncertainty foreach mass as a standard deviation representing “a limitto the effect of random errors of measurement plus sys-tematic error from known sources.” Those analysesthus incorporate all known Type A and Type B uncer-tainty components. The reported values yield standarduncertainties for the individual deadweight masses thatrange from 0.0001 % to 0.0003 % of the mass values.

Since the masses of the individual weights of eachmachine were determined similarly, the mass valuesmay be partially correlated; thus the combined massuncertainty of any combination of masses may moreappropriately be taken to be the sum of the individualuncertainties rather than the square root of their esti-mated variance. This combined uncertainty will then liein the range from 0.0001 % to 0.0003 % of the mass ofthe combination. Rather than compute separate com-bined uncertainties for different combinations ofweights, the upper end of the range, 0.0003 %, for therelative standard uncertainty of the individual masses isregarded to represent a reasonable value for the relativestandard uncertainty for any combination of masses.

Thus the standard uncertainty in the applied forcethat is associated with the uncertainty in the determina-

tion of the deadweight masses is no greater than0.0003 % of the applied force. The combined standarduncertainty given in Eq. (2) is expressed in transducerresponse units; thus the standard uncertainty in theapplied force must be transformed into equivalenttransducer response units. Since the determinedresponse R given in Eq. (1) is approximately a linearfunction of the applied force F, the standard uncertain-ty, ufa, in the response R that is associated with theuncertainty in the determination of the deadweightmasses is no greater than 0.0003 % of the transducerresponse R. Thus

(4)

This value represents an upper bound to the relativestandard uncertainty for any combination of weights.

The question of whether the deadweight masseschange with time must be addressed. Possible mecha-nisms for such mass change are the outgassing ofentrapped gases from the deadweight material, theoccurrence of oxidation or other chemical activity, orthe adsorption of contaminants. To minimize the possi-bility of such variation in the deadweight masses withtime, the weights were made of stainless steel. For the498 kN, 1.334 MN, and 4.448 MN machines, theAmerican Iron and Steel Institute (AISI) series 410alloy was chosen because of its superior strength andresistance to galling at the weight-bearing contact sur-faces. The design of the 2.2 kN, 27 kN, and 113 kNmachines, incorporating independent loading mecha-nisms for each weight, minimizes the possibility forgalling; thus the 300 series alloy was chosen for thesemachines.

The 498 kN machine was partially disassembled forservice in 1971 and again in 1989, providing opportu-nities for observing whether significant mass changesin its weights were taking place. New mass determina-tion measurements were conducted in each of thoseyears for the weights that were removed. These weightswere organized into two sets, with individual weightsof each set yielding forces of 4.448 kN and 44.48 kN,respectively. A comparison of the masses for theseweights for the 1965, 1971, and 1989 determinations isshown in Fig. 1. The points on this plot that are depict-ed with solid symbols represent the differencesbetween the 1971 and 1965 mass values, given in per-cent of each respective mass. Positive values representan apparent increase in mass since 1965. The corre-sponding uncertainty intervals represent the combinedstandard uncertainties for the 1971 and 1965 massdeterminations, given in percent of each respective


591

fa 0.000003 .u R≤

mass. The points on the plot that are depicted with opensymbols represent the relative mass differences, alongwith the combined standard uncertainties, for the 1989and 1965 measurements.

All but two of the points in Fig. 1 lie within±0.0003 %, which is the upper bound value for the rel-ative standard uncertainty in the determination of themass. The individual standard uncertainty intervalsdepicted by the error bars, having a confidence level ofapproximately 68 %, are seen to enclose the baselinefor fourteen of the twenty points. None of the devia-tions exceed their respective expanded uncertainties,for which the confidence level is approximately 95 %.The mean difference for the twenty points is–0.0001 %, which is not sufficient to establish a signif-icant systematic mass change phenomenon from theseobservations.

In order to more completely address the question ofstability of NIST’s deadweight masses, the 2.2 kNmachine was completely disassembled in 1996 and newmass determination measurements were performed forall of its weights. The 2.2 kN machine was selectedbecause it has the smallest weights, which provide thelargest ratio of surface area to mass. Under the assump-tion that any long term mass change involves a surfaceeffect, the relative change would be greater, and thusmore observable, for the smaller weights. In addition,

this effort enabled a check on the alloy used for thethree smaller machines.

A comparison of the 2.2 kN machine masses for the1965 and 1996 determinations is shown in Fig. 2. Thepoints represent the differences between the 1996 and1965 mass values, given in percent of the each respec-tive mass. As in Fig. 1, positive values represent anapparent increase in mass since 1965. The error barsrepresent the combined standard uncertainties for the1996 and 1965 mass determinations, given in percentof each respective mass. The uncertainty invervals dif-fer in length because the mass uncertainty for eachweight is calculated from the data for that weight.

One point in Fig. 2 lies outside ±0.0003 %, which isthe upper bound value for the standard uncertainty inthe determination of the mass. The individual standarduncertainty intervals are seen to lie outside of the base-line for four of the nine points. While two of the devia-tions exceed their respective expanded uncertainties fora coverage factor of two, the mean difference of+0.0001 % is not sufficient to establish a significantsystematic mass change phenomenon from these obser-vations. Since the larger NIST deadweight machineswould incur smaller relative mass changes than the 2.2kN machine, it is concluded that significant changes indeadweight mass are not evident in the NIST force lab-oratory facilities.


592

Fig. 1. Comparison of mass values determined in 1965, 1971, and 1989 for the NIST 498 kN deadweightmachine.

A diligent quality assurance program of inspections,maintenance, and security serves to provide confidencethat mass changes in the weights are not occurringthrough contamination, fluid leakage, extraneousobjects, or mechanical wear.

3.2 Uncertainty Associated with GravitationalAcceleration

The absolute value of the acceleration due to gravity,denoted as g in Eq. (3), was determined in 1965 at theNIST force laboratory in Gaithersburg, MD by meansof free-fall measurement apparatus constructed byDoug Tate [9]. The equipment consisted of a 1 m longfused silica tube that was allowed to fall freely within avacuum chamber; this vacuum chamber itself wasallowed to fall under the influence of gravity, restrainedonly by guide rods involving minimal friction. Theposition of the falling silica tube as a function of timewas determined by means of slits cut into the tube atcarefully measured positions, allowing light to passfrom an external light source horizontally through thetube. Transparent ports located in the falling vacuumchamber allowed detection by an external light sensorwhen any of the falling slits aligned momentarily witha stationary reference slit. The total height of the freefall was about 1.25 m.

D. R. Tate’s instrumentation enabled the gravitation-al acceleration to be established for a reference point

within the force laboratory, giving a value for g of9.801018 m/s2. This value is 0.0574 % less than thenominal sea level value for g of 9.806650 m/s2. Tatestated a measurement standard deviation of 0.000005m/s2, which is about 0.00005 % of the measurementresult. The measurement procedure also allowed adetermination of the gravity gradient, which was–0.000003 s–2. This measured value for the gravity gra-dient is about the same as that which can be calculatedfrom Newton’s gravitational equation if the earth wereassumed to be a sphere of radius Re, having a mass Me

of spherically symmetric distribution. At a distance rfrom the earth’s center, where r ≥ Re, the gravitationalacceleration would be

(5)

where G is the Newtonian gravitational constant. Thevalue for g at the earth’s surface is gs = GMe/Re

2. Over asmall differential ∆r in height at the earth’s surface, thegravity gradient can be computed from Eq. (5) as

(6)

Taking the earth radius as 6.379 × 106 m, Eq. (6) yields∆g/∆r ≈ –0.000003 s–2, about the same as observed byTate.

New determinations of the gravitational accelerationwere obtained from a gravity survey of the NIST force


593

Fig. 2. Comparison of mass values determined in 1965 and 1996 for the NIST 2.2 kN deadweight machine.

2e / ,g GM r=

s e/ 2 / .g r g R∆ ∆ ≈ −

laboratory in 1992, performed by the National Oceanicand Atmospheric Administration (NOAA), Office ofOcean and Earth Sciences, Gravity Section. This sur-vey was performed with portable equipment brought byNOAA personnel; the equipment employed an auto-mated short distance free-fall mechanism, sensed withlaser interferometry, that could be operated repeatedlyover a period of time. This survey yielded gravity val-ues at six locations throughout the laboratory. At thereference point characterized by Tate, the NOAA valuefor g was 9.80101353 m/s2 ± 0.00000008 m/s2; this issmaller than Tate’s value by about 0.000004 m/s2,which is about the same as the standard uncertainty inTate’s measurements. Thus the difference between theTate and NOAA measurements is within the expandeduncertainty interval.

The NIST deadweight masses were determined in1965 and 1966, following Tate’s gravity measurements.In preparation for this analysis, the value for g at themidpoint of each of the six NIST deadweight stackswas derived from the absolute gravitational accelera-tion at Tate’s reference location and the gravity gradi-ent. During the mass determination, each weight wasadjusted to exert its nominal force for the value of g atits stack’s midpoint. The only significant uncertaintyassociated with g in Eq. (3) is the variation of g withheight relative to the midpoint of each weight stack.The largest height variation relative to the stack mid-point in the NIST force laboratory is 5.5 m. Rather thanmake individual corrections for the location of eachweight, an associated uncertainty is estimated on thebasis of a rectangular probability distribution asdescribed in Sec. 4.6 of NIST Technical Note 1297,“Guidelines for Evaluating and Expressing theUncertainty of NIST Measurement Results” [7]. Thecorresponding relative standard uncertainty in g, andthus in F, is given by the (largest height variation) ×(relative gravity gradient) × 3–0.5, or 0.000001.Combining this uncertainty with the uncertainty inTate’s absolute gravity measurements yields a standarduncertainty in the applied force F, associated with theuncertainty in the gravitational acceleration, of about0.000001 F. The corresponding standard uncertainty,ufb, in the response R, resulting from the uncertainty inthe gravitational acceleration, is given by

(7)

This uncertainty component could be eliminatedthrough computation, by calculating g from the meas-ured height for each weight. Computation of the equiv-alent height of the weight frame of each machine would

require an integration over the distributed mass of theframe, which constitutes the first calibrated weight ofthe weight stack.

A discussion of the current state of the art in themeasurement of gravitational acceleration is given byJ. E. Faller [10], of the Quantum Physics Division ofthe NIST Physics Laboratory in Boulder, CO. Anonline model enabling the prediction of surface gravityfor any point within the continental United States canbe accessed from the tools section of the Internet website of the National Geodetic Survey (NGS), which hasthe address www.ngs.noaa.gov. The predictions are cal-culated by interpolation from observed gravity datacontained in the National Spatial Reference System ofthe NGS. The uncertainty in the interpolation, which isprovided for each location specified by the requester,has relative values that are typically about 0.000005.Application of this online model to the location of thereference point in the NIST force laboratory character-ized by Tate yields a value of 9.80102 m/s2 ± 0.00002m/s2, which is consistent with the fact that past gravi-metric measurements at NIST contribute to the NGSdatabase.

3.3 Uncertainty Associated with Density

The adjustment of the weights in 1965 and 1966,which incorporated the local value of g as describedabove, also incorporated the average local value for thebuoyancy factor [1 – (ρa/ρw)] that appears in Eq. (3).The value of ρa used in these adjustments was the year-round mean air density in Gaithersburg, MD, of 1.17kg/m3 as discussed below. The density ρw of the stain-less steel material of the weights was determined byassociates of Doug Tate at NIST, by determining themass for small cylindrical specimens of the material forwhich the volume was also determined from dimen-sional measurements. The values obtained were 7720kg/m3 for the AISI 410 alloy, used for the three largerdeadweight machines, and 7890 kg/m3 for the AISI 300alloy of the three smaller machines. The standarduncertainty in these measurements was less than 1 %.The application of the buoyancy correction involves arelative reduction, ρa/ρw, in the applied force of0.0152 % for AISI 410, and 0.0148 % for AISI 300.

Without corrections made to the applied force fordaily fluctuations in air density at NIST, the uncertain-ty associated with the use of the mean air density mustincorporate these normal weather related fluctuations.Paul Pontius [11] provides a compilation of the averageair densities, derived from Weather Bureau data, forselected cities throughout the continental United States.


594

fb 0.000001 .u R≈

The average air density for Washington, DC, correctedfor a constant temperature of 23 °C, is given as 1.185kg/m3 ± 0.04 kg/m3, where the limits define the rangeover which the actual air density may fluctuate throughthe year. The difference in elevation betweenGaithersburg, MD and Washington, DC is approxi-mately 120 m. According to documentation publishedjointly by the National Oceanic and AtmosphericAdministration, the National Aeronautics and SpaceAdministration, and the U.S. Air Force [12], this differ-ence in elevation reduces the air pressure, and thus theair density, by about 1.4 %. Employing this correctionyields the 1.17 kg/m3 mean air density used for theweight adjustments; the range of actual air density fluc-tuation remains as ± 0.04 kg/m3, giving an interval of1.13 kg/m3 to 1.21 kg/m3.

The variation of ± 0.04 kg/m3 in ρa corresponds to arelative change in the applied force of ± 0.0005 %,computed from Eq. (3) and using the density of eitheralloy for ρw. An associated uncertainty is estimated onthe basis of a rectangular probability distribution, giv-ing an estimated relative standard uncertainty of0.000005 × 3–0.5. Thus the standard uncertainty in theapplied force F, resulting from the variation of actualair density from the yearly mean air density, is about0.0003 F. The corresponding standard uncertainty, ufc,in the response R, resulting from the variation of actualair density from the yearly mean air density, is given by

(8)

This uncertainty component could also be eliminatedthrough computation, provided that the barometricpressure, humidity, and temperature are sampledthroughout each force calibration.

The NIST Mass and Force Group has some accumu-lated barometric pressure data that can corroborate theair density interval used in the above uncertainty calcu-lation. For the past eleven years NIST has been per-forming legal metrology load cell evaluations in accor-dance to specifications given by the OrganizationInternationale de Métrologie Légale (OIML) [13] andby the National Type Evaluation Program (NTEP) [14].Discussions of NIST’s conduct of these procedureshave been given previously [15]. During these meas-urements, the barometric pressure is recorded continu-ously, typically at 5 min intervals, for a period of two ormore days for each load cell evaluation.

The barometric pressure data from legal metrologyevaluations using the NIST 498 kN deadweightmachine have been extracted for a 5 year period begin-ning in 1998. These measurements involve evaluations

on forty load cells, spaced somewhat randomly over the5 year period, and incorporating a total accumulatedmeasurement time of 90 days. Of the 25 000 individualbarometric pressure samples taken in these measure-ments, the average, minimum, and maximum valuesare 100.13 kPa, 98.15 kPa, and 102.33 kPa, respective-ly.

The air density ρa can be calculated from the baro-metric pressure if other atmospheric parameters arealso known, using an internationally accepted equation[16] of the form

(9)

where p is the atmospheric pressure, T is the thermody-namic temperature, xv is the mole fraction of watervapor, Ma is the molar mass of dry air, Mv is the molarmass of water, Rg is the molar gas constant, and Z is thecompressibility factor. Necessary constants and supple-mentary relations are given in Ref. [16].

The temperature in the NIST force laboratory is reg-ulated to 23 °C ± 0.2 °C in the rooms where the loadcells are loaded, and to 23 °C ± 2 °C in the rooms hous-ing the deadweights. In addition, the relative humiditytypically ranges from 10 % to 60 %. Using Eq. (9) withthe extremes of the ranges for the barometric pressure,air temperature, and relative humidity as given above,the average, minimum, and maximum values for the airdensity in the vicinity of the NIST deadweights areobtained as 1.17 kg/m3, 1.14 kg/m3, and 1.21 kg/m3,respectively. These results are essentially identical tothe air density values derived from the Weather Bureaudata given in Ref. [11]. Thus Eq. (8) remains as an ade-quate estimator for the uncertainty in the force associ-ated with the air density.

The uncertainty in the applied force that is associat-ed with the material density of the weights is now to bediscussed. As indicated above, measurements at NISTof the density, ρw, of the stainless steel material of theweights was believed to be accurate to about one per-cent. The problem is to determine what error in F iscaused by an error in the value of ρw.

This problem can be addressed by noting that the1965 mass determinations were performed at NIST inair at ambient atmospheric pressure, with the tempera-ture and humidity controlled to the same values as stat-ed above. The mass determinations did not involve sep-arate density measurements; instead, they used as inputthe same values for ρw that were determined by associ-ates of Doug Tate as described above and used in Eq.(3) for all subsequent force measurements employing


595

fc 0.000003 .u R≈

a a g v v a( / )[1 (1 / )],pM ZR T x M Mρ = − −

these weights. Thus for any weight, the value of themass m of the weight is related to ρw by

(10)

where ms is the mass of the mass standard used to deter-mine m, and ρs is the density of this mass standard. Thisrelation assumes a simplified case of a single massstandard and a gas density equal to the mean air densi-ty at NIST. Thus

(11)

This mass value m is subsequently used in the forcelaboratory to determine the applied force, F, using Eq.(3). Since the uncertainty in F caused by a change in airdensity between the time of mass determination and thetime of force application has already been accountedfor, the same value for ρa may be used in both Eq. (3)and Eq. (11).

Suppose, however, that it is later discovered that thevalue ρw has significant error, and that the true value forthe density of the weight is ρw′.

The question is: what is the corresponding error inthe force; i.e., what is the true force F′ corresponding tothe true density ρw′? In order to answer that question,one must first ask: what is the true mass m′ based on themass determination performed earlier?

(12)

(13)

(14)

With the mass so corrected, the correct force may nowbe calculated as

(15)

(16)

(17)

(18)

Thus the answer is: there is no error in F caused by anerror in ρw.

3.4 Standard Uncertainty Associated with theApplied Force

The standard uncertainty, uf, in the transducerresponse, incorporating all significant uncertainty com-ponents in the applied force, may now be calculatedfrom

(19)

where ufa, ufb, and ufc are given by Eqs. (4), (7), and (8),respectively. For the forces applied by the NIST dead-weight machines, this calculation yields

(20)

4. Uncertainty in the Calibration ofNIST Voltage-Ratio Instrumentation

As discussed under Eq. (1), each force Fj applied toa transducer undergoing force calibration is paired witha response Rj of the transducer to that applied force.The uncertainty in acquiring each response datum Rj

results from the following two sources: (a) a “random”component related to the resolution of the transducerresponse indicating device and any variation in theresponses such as would be seen in successive readingsof the indicating device for a constant force input; and(b) a “systematic” component related to the calibrationof the instrumentation used to acquire the responses.

The uncertainties identified by item (a) contribute tothe deviations in the responses from the least-squaresfit to the data and are accounted for by the uncertaintyur discussed below in Sec. 5. The uncertainties of item(b) apply only if the responses are acquired by an indi-cating device that is not considered to be integratedwith the force transducer being calibrated.

Many transducers calibrated at the NIST force labo-ratory are combined with indicating systems that arenot separated from the transducers. Typical examplesare mechanical systems, such as the micrometer screwand precisely machined contact points that are integrat-ed within proving ring transducers, and electrical volt-age-ratio measuring instruments supplied by customersfor connection to strain gauge load cells. If an indicat-ing instrument accompanies a transducer and is used bythe customer in the same manner, without readjust-ment, as employed during calibration, then the indicat-ing instrument is considered to be part of the calibratedsystem. Any systematic characteristics of the instru-


596

a w s a s[1 ( / )] [1 ( / )],mg m gρ ρ ρ ρ− = −

s a s a w[1 ( / )] /[1 ( / )].m m ρ ρ ρ ρ= − −

s a s a w[1 ( / )] /[1 ( / )]m m ρ ρ ρ ρ′ ′= − −

s a s a w

a w a w

{[1 ( / )] /[1 ( / )]}

{[1 ( / )] /[1 ( / )]}

m m ρ ρ ρ ρ

ρ ρ ρ ρ

′ = − −

′⋅ − −

a w a w[1 ( / )] /[1 ( / )].m m ρ ρ ρ ρ′ ′= − −

a w[1 ( / )]F m g ρ ρ′ ′ ′= −

a w a w a w{[1 ( / )] /[1 ( / )]} [1 ( / )]F mg ρ ρ ρ ρ ρ ρ′ ′ ′= − − ⋅ −

a w[1 ( / )]F mg ρ ρ′ = −

.F F′ =

2 2 2 2f fa fb fc ,u u u u= + +

f 0.000005 .u R=

ment are then accounted for by the calibration relationreturned in the form of Eq. (1) by the procedure.

The NIST force laboratory maintains its own straingauge excitation and voltage-ratio measuring instru-ments for use in calibrating load cells that are notaccompanied by customer supplied indicating instru-ments. Because the calibration of NIST’s equipment isnot integrated with the transducer calibration, the NISTMass and Force Group must maintain a separate cali-bration of this instrumentation relative to national volt-age standards. The uncertainty uv of this electrical cali-bration must be incorporated into the combined stan-dard uncertainty uc of the force calibration procedure.

The NIST indicating system supplies direct currentexcitation to the load cell through the use of a DCpower supply, which applies voltages to the load cellexcitation input leads of ±5 V relative to the load cellground wire, thus giving 10 V between the leads. This10 V difference, serving as the excitation voltage, isstable to within ±5 µV over a time period of 15 s. Thispower supply was designed to internally switch thewires going to the ±5 V terminals by means of a com-puter command, thus reversing the polarity of the exci-tation signal to the load cell. This action makes it pos-sible to cancel out small thermal biases in the straingage bridge and connecting wires, as well as any zero-offsets in the rest of the indicating system. The switch-ing is not done if the load cell is not designed to accom-modate reversed polarity excitation.

The NIST indicating system simultaneously samplesthe excitation voltage and the load cell output voltagewith an 8½ digit computing multimeter operating involtage-ratio mode; the multimeter calculates the corre-sponding voltage ratio internally and returns that valuein digital form to the computer. The multimeter is readtwice, with the excitation voltage polarity reversedbetween readings; the final voltage ratio is taken as theaverage of the voltage ratios measured at each polarity.The meter sampling time at each polarity, and the delayafter switching polarity before resuming the sampling,are specified by the operator through the computer con-trol/acquisition program. A typical time for one com-plete voltage ratio reading is 10 s; this time can beshortened or lengthened as appropriate for the measure-ment being conducted.

4.1 Calibration Relative to NIST VoltageStandards

Use of NIST instrumentation to obtain the load cellresponses during force calibrations mandates that thevoltage ratio measurements be traceable to U.S. nation-

al electrical standards. This is accomplished by period-ic “primary” calibration of the force laboratory’s com-puting multimeters by the Quantum ElectricalMetrology Division of the NIST Electronics andElectrical Engineering Laboratory. This procedure iscarried out in the multimeter’s voltage-ratio mode, byproviding direct current voltage signals simultaneouslyto both input channels, with the calibrated signalsderived from 1 V and 10 V Josephson-junction arrayvoltage standards (JVS) maintained by the QuantumElectrical Metrology Division [17,18]. Such a calibra-tion is performed by that division on at least one of theforce laboratory multimeters per year. Different multi-meters are selected for succeeding calibrations in orderto avoid bias that could be associated with the calibra-tion of the same meter repeatedly. The Mass and ForceGroup maintains calibration of all of its multimeters atleast quarterly by comparison with the multimetersmost recently calibrated by the Quantum ElectricalMetrology Division, as described in the next section.

During the multimeter voltage-ratio calibration theQuantum Electrical Metrology Division maintains a 10V signal from a solid-state dc voltage standard calibrat-ed against the 10 V JVS at the meter’s ratio referenceinput, while applying a sequence of reference signalsranging from 5 mV to 100 mV provided by the 1 V JVSto the meter’s primary input channel. The correspon-ding voltage-ratio range is from 0.5 mV/V to 10 mV/V.For most load cells calibrated at NIST, the output whenloaded to capacity is 2 mV/V to 4 mV/V.

The reference voltages derived from the Josephsonvoltage standard system are known with uncertaintiesof about 0.05 µV, provided that the sampling time ofthe multimeter is not greater than 10 s. This uncertain-ty corresponds to 0.00005 % of a 10 mV/V meter read-ing, or 0.0002 % of a multimeter reading of 2.5 mV/V.For a 10 s sampling time, the standard uncertainty inthe multimeter voltage-ratio readings is 0.00001 mV/V,corresponding to 0.0001 % at 10 mV/V, or 0.0004 % at2.5 mV/V.

From the Quantum Electrical Metrology Divisionmeasurements a meter calibration factor may be calcu-lated, taken as the quotient of the voltage-ratio indicat-ed by the multimeter and the ratio of the reference volt-ages applied to the meter inputs. A sufficient number ofrepetitions are conducted until the meter calibration canbe calculated with a standard uncertainty of about0.0003 %.

The measurements also establish the linearity of themultimeter, represented by the uniformity of the cali-bration factor over the range from 0.5 mV/V to 10mV/V. The multimeters used in the NIST force labora-


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tory demonstrate a linearity sufficient to enable a singlemeter calibration factor to be applied; the uncertaintyassociated with nonlinearity is about 0.0001 %.

The results of a typical calibration by the QuantumElectrical Metrology Division is shown in Fig. 3, plot-ted as the voltage-ratio indicated by the meter dividedby the ratio of the reference voltages.

The meter calibration factors for the eight computingmultimeters used by the force laboratory range fromabout 0.999985 to 1.000070. The standard uncertaintyin the load cell response R that is associated with theNIST Quantum Electrical Metrology Division determi-nation of these calibration factors is

(21)

The standard uncertainty associated with the multime-ter linearity is

(22)

4.2 Intercomparison of NIST Force LaboratoryInstruments

The NIST Mass and Force Group maintains eightidentical 8½ digit computing multimeters for voltage-ratio measurements at six deadweight machines, ensur-

ing that sufficient multimeters are available to accom-modate load cells with multiple strain gauge bridge net-works. While one of these multimeters is selected atleast yearly for a primary calibration by the NISTQuantum Electrical Metrology Division, a procedure isnecessary for frequent checks of the calibration of all ofthe multimeters. The method currently employed forthis purpose makes use of a precision load cell simula-tor to serve as a “voltage-ratio transfer standard”; thisdevice is used to transfer the primary calibration by theQuantum Electrical Metrology Division to the otherseven multimeters.

The load cell simulator is a passive electrical net-work with connections and impedances representativeof most load cells and an output providing a voltage-ratio that is selectable in steps from 0 mV/V to 10mV/V. It is stable within ±0.000005 mV/V over a 24 hperiod. Each multimeter is connected, one at a time, tothe load cell simulator output terminals and readingsare taken in voltage-ratio mode over a sampling timeinterval of 150 s. For this sampling time the standarddeviation of repeated measurements is ≤0.000003mV/V. Readings are taken for simulator output settingsof 10 mV/V, 2.5 mV/V, and 0 mV/V, and for each oftwo excitation conditions: +10 VDC and –10 VDC.These measurements are completed for all of the multi-meters within a half day’s time.


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Fig. 3. Plot of a typical multimeter calibration in voltage-ratio mode by the NIST Quantum Electrical MetrologyDivision.

va 0.000003 .u R=

vb 0.000001 .u R=

The multimeter which is most recently calibrated bythe Quantum Electrical Metrology Division is used todetermine the output of the simulator at the relevantvoltage-ratio settings. This simulator output is thenused to calculate a meter calibration factor for each ofthe other seven multimeters. This factor is calculatedseparately for each ratio setting from the +10 V and–10 V excitation values and again from the +10 V and0 V excitation values.

The results give a check on each meter’s linearity aswell as its proper functioning at both positive and neg-ative excitation voltage polarity. The calibration factorfor each multimeter is determined with a relative stan-dard uncertainty of 0.0003 % to 0.0004 %. Since thisfactor is a multiplier to all load cell response readingsR acquired for subsequent force calibration measure-ments, the standard uncertainty in R that is associatedwith the comparison calibrations of the multimeterswith the simulator is

(23)

Figure 4 shows plots of the repeatability of the metercalibration factors for six of the multimeters over an 8

year time period. The factors were determined from theprocedures described above. The multimeters are iden-tified on the plot by serial number. Six different meters,some repeatedly, were calibrated at intervals by theQuantum Electrical Metrology Division for use as ref-erences during this period.

The plots shown in Fig. 4 indicate how precisely thecalibrations of the multimeters can be maintained bythe established procedures. If linear least-squares com-putations are performed for the data for the multimetersshown here, the standard deviation of the individualdata points about the fitted line may be calculated foreach multimeter. These standard deviations range from0.000002 to 0.000004 for the multimeters shown.These results demonstrate that while the uncorrectedreadings from different multimeters may vary by0.0075 %, appropriate calibration procedures can main-tain agreement among all of these units to within0.0005 %

If the actual factor for the multimeter being used as areference for the other seven multimeters should beginto drift after its Quantum Electrical Metrology Divisioncalibration, the comparison calibrations would showthis drift as a simultaneous change in the factor for the


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vc 0.000004 .u R≈

Fig. 4. Plots of the repeatability of the calibration factors for six of the multimeters over an 8 year time period. The numbersshown are the instrument serial numbers.

other seven meters. Since an actual similar change inthe response of seven meters is statistically unlikely,the calibration procedures described above have someinherent safeguards against undetected data corruptionresulting from a single malfunctioning instrument.

4.3 Standard Uncertainty in Voltage-RatioInstrument Calibration

The standard uncertainty, uv, in the calibration of theNIST voltage-ratio instrumentation, incorporating allsignificant uncertainty components, may now be calcu-lated from

(24)

where uva, uvb, and uvc are given by Eqs. (21), (22), and(23), respectively. For the NIST instrumentation, thiscalculation yields

(25)

5. Deviations of Measurement Data fromthe Least-Squares Fit

A force calibration provides the transducer responseas a function of applied force in the form of Eq. (1) byderiving the coefficients Ai from a least-squares fit tothe calibration data. The uncertainty associated with thevariation in the measured data from the fitted curve isrepresented by the standard deviation ur in Eq. (2). Thisstandard deviation is calculated according to ASTM E74-04 from

(26)

where the dj are the differences between the measuredresponses, Rj, and the responses calculated from Eq.(1), n is the number of individual measurements in thecalibration data set, and m is the order of the polynomi-al plus one.

Many factors contribute to the standard deviation ur,including (a) random errors associated with the resolu-tion, instrument noise, and repeatability of the indica-tor; (b) variations caused by swinging of the weights;(c) deviations of the assumed transducer response mod-eled by Eq. (1) and the true transducer response; (d)irregularities due to the characteristics of the transduc-er being calibrated, such as creep, hysteresis, and sen-sitivity to placement in the force machine. Some ofthese factors can be minimized by procedural tech-

nique, such as choosing optimum indicator samplingparameters, achieving precise transducer alignment,maintaining of machine weight changing and motioninhibiting mechanisms, and properly selecting the orderof fit for the least-squares analysis.

The transducer related effects usually make up thelargest share of the deviations incorporated into ur andconstitute the dominant contributors of overall meas-urement uncertainty. Usual calibration practice enablesthese effects to be quantified by limiting the number offorces that are applied before returning to zero forceand by repeating the measurement sequence for sever-al reorientations of the transducer in the force machine.The dependence of the load cell response upon previ-ously applied forces and upon the degree of misalign-ment of the applied force relative to the load cell axisthen becomes evident, both in the quantity ur and in aplot of the deviations from the fitted curve. A detaileddiscussion of these uncertainty sources is given in C. P.Reeve [19].

An example of a load cell with a relatively high sen-sitivity to angular position with respect to the NIST4.448 MN deadweight machine loading platens isshown in Fig. 5. A measurement sequence consisting ofnine forces was conducted for each of six orientations.The ordinates represent the deviations in the data abouta least-squares fit in the form of a third-order polyno-mial. The fitted curve is represented on this plot as ahorizontal line of zero deviation. The standard devia-tion ur of these data about the fitted curve is 0.011 % ofthe load cell response at maximum force. The com-bined standard uncertainty uc, calculated from Eqs. (2),(20), and (25), is 0.011 % of the response at maximumforce; the expanded uncertainty, for a coverage factor,k, of 2, is thus 0.022 %.

An example of a load cell with a very low sensitivi-ty to orientation within the NIST 4.448 MN deadweightmachine is shown in Fig. 6. All measurement parame-ters were identical to the parameters used in the meas-urement depicted in Fig. 5. The scales of the axes arethe same in the two plots. The standard deviation ur ofthe data of Fig. 6 about the fitted curve is 0.0008 % ofthe load cell response at maximum force. The com-bined standard uncertainty uc, calculated from Eqs. (2),(20), and (25), is 0.0011 % of the response at maximumforce. The relative expanded uncertainty is thus0.0022 %.

For force calibrations that have been performed atNIST, the lower end of the uncertainty range associat-ed with load cell characteristics is represented by avalue of ur of 0.0003 % of the response at maximumforce, when ur is calculated from Eq. (26) using data


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v 0.000005 .u R=

( )2 2r /( ),ju d n m= ∑ −

2 2 2 2v va vb vc ,u u u u= + +


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Fig. 5. Deviations of individual data from a least-squares fit for a load cell with relatively high sensitivity to orientation withinthe NIST 4.448 MN deadweight machine.

Fig. 6. Deviations of individual data from a least-squares fit for a load cell with very low sensitivity to orientation within theNIST 4.448 MN deadweight machine.

from at least three orientations within the deadweightmachine. The corresponding expanded uncertainty is0.0015 % for calibrations performed with NIST’s volt-age-ratio instrumentation. If the load cell is paired witha dedicated indicator, thus eliminating the componentuv of Eq. (24), the relative expanded uncertainty is0.0012 %.

It is not practical to define the upper end of theuncertainty range, since this represents force transduc-ers of less precise design or with certain problems thatmay be correctable. Occasionally a transducer calibra-tion yields a value of ur in excess of 0.1 %. Dependingon the application, such a calibration may still be valu-able to the customer.

The plots in Figs. 5 and 6 demonstrate that thedependence of the measurement uncertainty upon thetransducer characteristics prevent predetermination ofthe final measurement uncertainty.

6. Conclusions

The previous sections have explained the compo-nents of measurement uncertainty in NIST calibrationsof force transducers. The standard uncertainty in thetransducer response R due to the uncertainties in theforces applied by the NIST deadweight force standardsis given by Eq. (20) as uf = 0.000005 R. The standarduncertainty in R due to the uncertainty in the calibrationof the NIST voltage-ratio instrumentation is given byEq. (25) as uv = 0.000005 R. The standard deviation ur

of the measured transducer responses relative to the fit-ted curve derived from the calibration data is dependentupon the characteristics of the transducer being cali-brated; its value may range from 0.0003 % to more than0.1 % of the response at rated capacity. The finalexpanded uncertainty is U = 2uc, where the combinedstandard uncertainty is calculated according to Eq. (2)as uc = (uf

2 + uv2 + ur

2)1/2. The relative values for U mayrange from 0.0012 % for exceptionally precise trans-ducers to more than 0.2 %.

Certain aspects of the force calibration analysis andthe evaluation of uncertainty are under study for possi-ble refinement, such as may be appropriate as forcetransducer technology improves. The refinementsbeing considered at the force laboratory include (a)revisions to the least-squares algorithm employed forderiving the polynomial coefficients of Eq. (1) in orderfor the fitting computation to take adequate account ofthe uncertainty in applied force; (b) incorporation of anadditional error term, if necessary, when the order of fitrequested by the customer differs from the mathemati-

cally best possible order of fit; and (c) expression of theexpanded uncertainty in the transducer response as afunction of the response over the range of the transduc-er, rather than as a single number that is understood torepresent the uncertainty for points distributed random-ly throughout the range.

The values of uncertainty reported here are main-tained through a quality assurance program followed bythe NIST Mass and Force Group staff. This programincludes diligent mechanical inspection and mainte-nance of the deadweights and associated loading mech-anisms. The program also includes maintenance of thecalibration of the NIST voltage-ratio instrumentation,carried out through secondary calibration of the com-puting multimeters as described in Sec. 4.2 on a quar-terly basis, and primary calibration of a multimeter bythe NIST Quantum Electrical Metrology Division atleast yearly. In addition, intercomparisons of NIST’sdeadweight machines are conducted through the use ofa set of precision force transducers as transfer standardsamong the machines. While it is recognized that theuncertainty in the response of the force transducers isgreater than the uncertainty in the applied force, thisprocess is useful in maintaining assurance thatdetectable faults have not appeared in one or more seg-ments of the measurement system.

7. References

[1] Z. L. Jabbour and S. L.Yaniv, The Kilogram and Measurementsof Mass and Force, J. Res. Natl. Inst. Stds. Technol. 106 (1), 25-45 (2001).

[2] S. L.Yaniv and T. W. Bartel, Force Metrology at NIST,Proc.16th IMEKO World Congress, vol. 3, Vienna, Austria(2000) pp. 287-291.

[3] T. W. Bartel, S. L.Yaniv, and R. L.Seifarth, Force MeasurementServices at NIST: Equipment, Procedures, and Uncertainty,Proc.1997 Natl. Conf. of Stand. Lab. Workshop andSymposium, Washington, DC (1997) pp. 421-431.

[4] K. W. Yee, Automation of Strain Gage Load Cell ForceCalibrations, Proc. 1992 Natl. Conf. of Stand. Lab. Workshopand Symposium, Washington, DC (1992) pp. 387-391.

[5] R. A. Mitchell, Force Calibration at the National Bureau ofStandards, NBS Technical Note 1227 (1986).

[6] ASTM E 74-04, Standard Practice of Calibration of Force-Measuring Instruments for Verifying the Force Indication ofTesting Machines, available from the American Society forTesting and Materials, Philadelphia, PA (2002).

[7] B. N.Taylor and C. E. Kuyatt, Guidelines for Evaluating andExpressing the Uncertainty of NIST Measurement Results,NIST Technical Note 1297 (1994).

[8] ISBN 92-67-10188-9, Guide to the Expression of Uncertainty inMeasurement, International Organization for Standardization(ISO), Geneva, Switzerland (1993).

[9] D. R. Tate, Acceleration Due to Gravity at the National Bureauof Standards, NBS Monograph 107 (1967).


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[10] J. E. Faller, The current status of absolute gravimetry and someideas and suggestions for future improvements, Proc. Int.Symposium on Gravity, Geoid and Marine Geodesy, Tokyo,Japan (1996).

[11] P. E. Pontius, Mass and Mass Values, NBS Monograph 133(1974).

[12] U.S. Standard Atmosphere 1976, U.S. Government PrintingOffice, Washington, DC (1976).

[13] Organization Internationale de Métrologie Légale InternationalRecommendation R 60, Metrological Regulation for LoadCells, Edition 2000(E), Bureau International de MétrologieLégale, Paris, France (2000).

[14] Natl. Conf. Weights and Meas. Publication 14, WeighingDevices, Chap. 5, Checklist for Load Cells (2002).

[15] T. W. Bartel and S. L.Yaniv, Creep and Creep RecoveryResponse of Load Cells Tested According to U.S. andInternational Evaluation Procedures, J. Res. Natl. Inst. Stand.Technol. 102, 349-362 (1997).

[16] R. S. Davis, Equation for the Determination of the Density ofMoist Air (1981/91), Metrologia 29, 67-70 (1992).

[17] R. E. Elmquist, M. E. Cage, Y-h. Tang, A.-M. Jeffery, J. R.Kinard, Jr., R. F. Dziuba, N. M. Oldham, and E. R. Williams,The Ampere and Electrical Standards, J. Res. Natl. Inst. Stand.Technol. 106, 65-103 (2001).

[18] A. Hamilton and Y-h. Tang, Evaluating the Uncertainty ofJosephson Voltage Standards, Metrologia 36, 53-58 (1999).

[19] C. P. Reeve, A New Statistical Model for the Calibration ofForce Sensors, NBS Technical Note 1246 (1988).

About the author: Thomas W. Bartel is a physicist inthe Mass and Force Group of the ManufacturingMetrology Division in the NIST ManufacturingEngineering Laboratory. The National Institute ofStandards and Technology is an agency of theTechnology Administration, U.S. Department ofCommerce.


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