K. Artois (Head Instrumentation And Controls)I. Philipsen (Project Manager)

Uncertainty EstimatesApplication of an adapted definition for the description of the uncertaintyof a strain-gauge balance

Contents

Introduction Definition for Uncertainty Estimation Application Recipe Application on DNW balance stock Conclusions

Introduction

At the 6th International Symposium on Stain-Gauge Balances a definition was postulated to describe the uncertainty for a stain-gauge balance.

DNW, AEDC and NASA volunteered to evaluate the proposed uncertainty definition.

A paper representing the DNW contribution to the evaluation has been published. (AIAA-2010-4546)

The proposed method is now used for several DNW balances with the intention to use it for all DNW balances.

Definition for Uncertainty Estimation

Uncertainty of the calibration

If K is set to 1 UCAL will represent an estimation of the sigma (standard deviation)

If K is set to 3 UCAL should represent “3*sigma”. Assuming a Gaussian distribution of the errors this should be close to the maximum error

6

11000nj

j j

jnnmaxnn

maxF

FbaF

KUCAL

Application Recipe Confirm the residuals of each balance component are

Gaussian Verify maximum error is between 2 to 4 times the

standard deviation Take the absolute value of all residuals Take uncertainty coefficient an to be the maximum of

either (maximum deviation)/3 out of a single load component

calibration, or standard deviation of the of a single load component

calibration Use equation with the coefficient bn set to zero to

determine the initial uncertainty for each load components of the balance

Subtract 3 times the initial uncertainty from the residuals (per load component) and set values below zero to zero.

Divide the remainder (larger then zero) by for each load point individually

Take uncertainty coefficient bn to be the maximum value per component divided by 3

1000

max6

1 max

n

njj j

j F

F

F

Application on DNW balance stock

B668 Errors of back calculation (incl. check points)

 

T [˚] AF [kN] SF [kN] NF [kN] RM [kNm] PM [kNm] YM [kNm]

min max max max min max min max min max min max minmax

Full Range 5 45 -20 20 -12.5 12.5 -50 50 -9 9 -15 15 -9 9Cal. Range Ambient -20 20 -12.5 12.5 -50 50 -9 9 -15 15 -9 9

Application on DNW balance stock

Uncertainty coefficients and maximum errors for the balance load calibration

The uncertainty coefficients are rounded up to two digits. The formulation reasonably well predicts the maximum

error (K=3) over all loads. In the table the estimated maximum error is included. This is the value of UCAL when all forces and moments are simultaneously at their maximum (this is (ai+5bi) x 3/10). The maximum error during the calibration and the maximum error including the calibration check points is also included.

T [˚]F&M

componentRange

[N, Nm]an

[-]bn

[-]

Max. estimated

error [%FS]

Max. residual error of

calibration points[%FS]

Max. error of

all points (including check points)[%FS]

Ambient

AF 20000 0.26 0.00 0.08 0.06 0.13SF 12500 0.25 0.12 0.25 0.10 0.17NF 50000 0.31 0.00 0.09 0.07 0.07RM 9000 0.23 0.00 0.07 0.06 0.06PM 15000 0.18 0.00 0.05 0.04 0.08YM 9000 0.22 0.00 0.07 0.06 0.07

Application on DNW balance stock

Uncertainty as a function of the number of applied combined loads

Application on DNW balance stock

Better example

T [˚]F&M

component

Range[N, Nm]

an

[-]bn

[-]

Max. estimated

error [%FS]

Max. residual error of

calibration points[%FS]

Max. error of

all points (including check points)[%FS]

Ambient

AF 12500 1.06 0.00 0.32 0.25 0.25SF 12500 0.82 0.36 0.79 0.59 0.68NF 30000 0.72 0.00 0.22 0.15 0.20RM 9000 0.41 0.10 0.27 0.23 0.23PM 11500 0.76 0.06 0.32 0.30 0.30YM 9000 1.28 0.00 0.38 0.30 0.33

Application on DNW balance stock

Better example: Uncertainty as a function of the number of applied combined loads

Application on DNW balance stock

Uncertainty coefficients have been determined or all DNW-LLF balances and some DNW-LST and HST balances (in total 10 balances; 8 internal, 2 external so far) Automated & dead weight calibrations Moment & force balances OFAT & MDOE calibrations

For those balances which are not calibrated or validated for multiple load combinations the bn coefficients can only be determined for , therefore also the maximum error can only be estimated for single load components(this is (an+1bn) x 3/10).

16

1 max

nj

j j

j

F

F

Conclusions

The postulated uncertainty definition is not perfect but applicable and has the feature that it can capture the common notion that the uncertainty increases with the number of applied combined loads to a balance.

Check points are essential to validate estimates of the uncertainty of a balance.

Uncertainty as a function of the number of applied combined loads can only be estimated if combined loads are applied during calibration.

The used uncertainty definition can be used for error propagation purposes