AIAA Recommended Practice 2-Component Balance Example
This Section will describe in detail the process used to determine the calibration matrix from a set of
calibration data for a 2-component balance example. While this example is illustrated with a 2component balance, this simplification involves no loss of generality, and the general approach
presented here is easily applicable to balances with more components. Further simplification was
made through the exclusion of both absolute value and third-order terms, thus only a quadratic
model is considered. It should be noted that a six-component example using all 96 mathematical
model terms discussed in Section 3.1.1 is available for downloading from the AIAA/GTTC web-site,
which can be reached from www.aiaa.org.
The calibration points for this example include two load components (A & B) and two outputs (r1
and r2). As described in the text and illustrated here, the matrix determination process consists of
four primary steps:
1.
2.
3.
4.
Determine the Zero-Load Output
Determine the Linear Matrix
Tare Load Iteration Process
Determine the Final Calibration Matrix
The process can begin once the calibration data and the zero-load output measurements have been
obtained. The data used for this example are shown in Tables 1 and 2 below. The data for the zeroload outputs were measured at roll angles indexed in 90 degree increments starting at 0 deg (i.e.,
0/90/180/270 deg). Several sets of zero-load output data were obtained, and the values shown in
Table 1 are averaged for each bridge output for all roll angles. The balance is considered to be level
in the pitch orientation during the acquisition of all zero-load outputs and calibration data.
Zero-Load Balance Measurements (ZLOS)
Table 1: ZLOS Measurements
Bridge Outputs
r1 (mV/V)
r2 (mV/V)
Roll Angle
0.1269
0.1235
0.1217
0.1247
0.1232
0.1257
0.1247
0.1283
0.1245
-0.7929
-0.7870
-0.7925
-0.7898
-0.7872
-0.7876
-0.7901
-0.7897
-0.7896
0
90
180
270
0
90
180
270
0
0.1248
-0.7896
= Average
0.0020
0.0021
= Std. Dev.
Balance Full-Scale Loads/Outputs
Table 2: Balance Full-Scale Outputs & Loads
Load Range
Component
Low
High
F.S. Output
A
-100
100
2.5
B
-500
500
3.5
Applied Calibration Loads & Raw Bridge Outputs
Table 3: Applied Calibration Loads (engineering units) and Raw Bridge Ouputs (mV/V)
Series
1
2
3
Point
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
Applied Loads (Eng. Units)
A
B
0.0
0.0
-50.0
0.0
-100.0
0.0
-50.0
0.0
0.0
0.0
50.0
0.0
100.0
0.0
50.0
0.0
0.0
0.0
0.0
0.0
0.0
-250.0
0.0
-500.0
0.0
-250.0
0.0
0.0
0.0
250.0
0.0
500.0
0.0
250.0
0.0
0.0
0.0
0.0
-100.0
-500.0
-100.0
500.0
0.0
0.0
100.0
-500.0
100.0
500.0
0.0
0.0
Raw Bridge Outputs
r1 (mV/V)
r2 (mV/V)
0.1219
-0.7906
-1.1304
-0.8112
-2.3834
-0.8356
-1.1274
-0.8148
0.1256
-0.7940
1.3718
-0.7633
2.6223
-0.7380
1.3672
-0.7565
0.1251
-0.7945
0.1190
-0.7956
0.1011
-2.5369
0.0671
-4.2848
0.0959
-2.5374
0.1223
-0.7856
0.1589
0.9625
0.1727
2.7074
0.1459
0.9622
0.1266
-0.8026
0.1253
-0.7800
-2.4293
-4.3354
-2.3248
2.6566
0.1319
-0.7903
2.5732
-4.2396
2.6727
2.7592
0.1269
-0.8025
Figure 1: Applied Loads vs. Point Number for each Load Component (A & B)
Step 1: Compute Delta Bridge Outputs (average of first/last zero load points in each series) & Linear
Matrix
The first step in the matrix determination process is to compute the initial linear matrix from the raw
calibration data. In order to do this, first the raw bridge outputs shown in Table 3 above must first be
modified in order to get them into a delta form. In order to get the data into this form, there are two
different ways to do this:
1.
2.
Take the first bridge output in each data series and subtract it from all other bridge outputs
within that series (for each bridge).
Take the average of the first/last bridge outputs in each data series and subtract this
average from all other bridge outputs within that series (for each bridge).
For this example method 2 above was employed (average of first/last bridge outputs in each series
subtracted from all outputs in each respective series). Converting the bridge outputs into delta
outputs the tare loads cancel and thus have no influence on the linear matrix computation. The delta
bridge outputs are shown in Table 4 below, along with the applied calibration loads.
Table 4: Applied Calibration Loads vs. Delta Bridge Outputs
Series
1
2
Point
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
Applied Loads (Eng. Units)
A
B
0.0
-50.0
-100.0
-50.0
0.0
50.0
100.0
50.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-250.0
-500.0
-250.0
0.0
250.0
500.0
Bridge Outputs
r1 (mV/V)
r2 (mV/V)
-0.0016
-1.2539
-2.5069
-1.2509
0.0021
1.2483
2.4988
1.2437
0.0016
-0.0038
-0.0217
-0.0557
-0.0269
-0.0005
0.0361
0.0499
0.0019
-0.0186
-0.0430
-0.0222
-0.0014
0.0293
0.0546
0.0361
-0.0019
0.0035
-1.7378
-3.4857
-1.7383
0.0135
1.7616
3.5065
8
9
1
2
3
4
5
6
7
3
0.0
0.0
0.0
-100.0
-100.0
0.0
100.0
100.0
0.0
250.0
0.0
0.0
-500.0
500.0
0.0
-500.0
500.0
0.0
0.0231
0.0038
-0.0008
-2.5554
-2.4508
0.0058
2.4471
2.5466
0.0008
1.7613
-0.0035
0.0112
-3.5442
3.4479
0.0010
-3.4483
3.5505
-0.0112
Using the applied calibration loads and delta bridge outputs from Table 3, a simple linear regression
is performed in order to compute the linear matrix (mathematical model) for each bridge response
(r1 and r2). For this example, the resulting linear matrix includes and intercept and linear A & B
math terms, as shown in Table 5 below.
Compute Linear Matrix (determined from loads/outputs in Table X above)
Table 5: Linear Regression Coefficient Matrix (determined from data in Table 4)
r1 (mV/V)
r2 (mV/V)
Intercept
A
-0.000848
0.025005714
0.004912
0.000498857
B
0.000103886
0.006995029
Once the linear matrix is computed, the tare iteration process can begin.
Step 2: Average ZLOS measurements for each bridge component (r1 and r2)
The next step is to determine the zero-load outputs (ZLOS) for each bridge component. The zero-load
outputs are the averages of the bridge outputs contained within Table 1. The average bridge outputs for
each bridge are shown in Table 6 below.
Table 6: Average ZLOS for each bridge component
Bridge Outputs
r1 (mv/v)
r2 (mV/V)
0.1248
-0.7896
= Average
0.0020
0.0021
= Std. Dev.
Step 3: Tare Iteration Process (this step is iterated until tare loads converge on set tolerance value)
The next step is to begin the tare iteration process. In order to compute the tare loads, the first bridge
output within each load series shall have the average zero-load output (for that bridge) subtracted from it.
This should only be done for the initial point within each load series, not all bridge outputs. Table 7 below
contains the Modified Bridge Outputs for the first point in each load series.
Table 7: Modified Bride Outputs
Series
1
2
3
Modified Bridge
Outputs
r1 (mv/v)
r2 (mV/V)
-0.0029
-0.0010
-0.0058
-0.0060
0.0005
0.0096
<-- (Series 1/Point 1) - Average ZLOS
<-- (Series 2/Point 1) - Average ZLOS
<-- (Series 3/Point 1) - Average ZLOS
The iterations in the tare iteration process are shown in the following sections.
Tare Load Iteration Step 3.1
The tare loads are computed using the modified bridge outputs shown in Table 7 above, and the current
matrix – which for this step happens to be the initial linear matrix shown in Table 5 above. The computed
tare loads are shown below in Table 8, and the change in the tare loads are also computed and shown in
Table 8. Since this is the first tare load iteration step, the changes are all equal to the computed tare loads.
Looking at the changes in tare loads it is clear that deltas are all still greater than the established criteria of
0.002, therefore the tare iteration process will continue.
Table 8: Computed Tare Loads and Changes in Tare Loads
Tare Loads
Series
1
2
r1
-0.1154
-0.2276
r2
-0.1382
-0.8398
3
0.0131
1.3699
Series
Changes in Tare Loads
r1
r2
1
2
-0.1154
-0.2276
-0.1382
-0.8398
3
0.0131
1.3699
The next step in the tare iteration process is to adjust the applied calibration loads for the tare loads shown
in Table 8. These adjusted calibration loads are shown in Table 9, along with the XXXXXXXX.
Table 9: Tare Adjusted Applied Loads (for Step 3.1 Tare Iteration)
Tare Adjusted Applied Loads (Eng.
Units)
Series
1
2
3
Bridge Outputs
Point
A
B
r1 (mv/v)
1
-0.1154
-0.1382
-0.0016
r2 (mV/V)
0.0019
2
-50.1154
-0.1382
-1.2539
-0.0186
3
-100.1154
-0.1382
-2.5069
-0.0430
4
-50.1154
-0.1382
-1.2509
-0.0222
5
-0.1154
-0.1382
0.0021
-0.0014
6
49.8846
-0.1382
1.2483
0.0293
7
99.8846
-0.1382
2.4988
0.0546
8
49.8846
-0.1382
1.2437
0.0361
9
-0.1154
-0.1382
0.0016
-0.0019
1
-0.2276
-0.8398
-0.0038
0.0035
2
-0.2276
-250.8398
-0.0217
-1.7378
3
-0.2276
-500.8398
-0.0557
-3.4857
4
-0.2276
-250.8398
-0.0269
-1.7383
5
-0.2276
-0.8398
-0.0005
0.0135
6
-0.2276
249.1602
0.0361
1.7616
7
-0.2276
499.1602
0.0499
3.5065
8
-0.2276
249.1602
0.0231
1.7613
9
-0.2276
-0.8398
0.0038
-0.0035
1
0.0131
1.3699
-0.0008
0.0112
2
-99.9869
-498.6301
-2.5554
-3.5442
3
-99.9869
501.3699
-2.4508
3.4479
4
0.0131
1.3699
0.0058
0.0010
5
100.0131
-498.6301
2.4471
-3.4483
6
100.0131
501.3699
2.5466
3.5505
7
0.0131
1.3699
0.0008
-0.0112
Using the tare adjusted applied calibration loads and the delta bridge outputs shown in Table 8 above, the
new interim regression coefficient matrix is computed, as shown in Table 10.
Table 10: Step 3.1 Iteration Interim Regression Coefficient Matrix
r1
r2
Intercept
0.004234222
0.007411703
A
0.025005623
0.000498501
B
0.000103868
0.006994936
A*A
-7.46E-07
-1.13E-06
A*B
-2.56E-08
3.35E-08
B*B
-2.79E-11
7.35E-09
Tare Load Iteration Step 3.2
The tare loads are computed from the modified bridge outputs (Table 7) and the interim matrix computed
during step 3.1 (shown in Table 10). The computed tare loads are shown below in Table 11, and the
change in the tare loads are also computed and shown in Table 11.
Table 11: Computed Tare Loads and Changes in Tare Loads
Series
1
2
3
Series
Tare Loads
r1
-0.0652
-0.1534
-0.0383
r2
0.2833
0.5109
1.6096
Changes in Tare
Loads
1
2
r1
5.02E-02
7.43E-02
r2
4.21E-01
1.35E+00
3
-5.14E-02
2.40E-01
Looking at the changes in tare loads in Table 11, it is clear that the changes in tare laods are all still greater
than the established criteria of 0.002, therefore the tare iteration process will continue. The next step in the
tare iteration process is to adjust the applied calibration loads for the tare loads shown in Table 11. These
adjusted calibration loads are shown in Table 12, along with the XXXXXXXX.
Table 12: Tare Adjusted Applied Loads (for Step 3.2 Tare Iteration)
Applied Loads (Coded)
Series
1
Point
Bridge Outputs
A
B
r1 (mv/v)
r2 (mV/V)
1
-0.0652
0.2833
-0.0016
0.0019
2
-50.0652
0.2833
-1.2539
-0.0186
3
-100.0652
0.2833
-2.5069
-0.0430
4
-50.0652
0.2833
-1.2509
-0.0222
2
3
5
-0.0652
0.2833
0.0021
-0.0014
6
49.9348
0.2833
1.2483
0.0293
7
99.9348
0.2833
2.4988
0.0546
8
49.9348
0.2833
1.2437
0.0361
9
-0.0652
0.2833
0.0016
-0.0019
1
-0.1534
0.5109
-0.0038
0.0035
2
-0.1534
-249.4891
-0.0217
-1.7378
3
-0.1534
-499.4891
-0.0557
-3.4857
4
-0.1534
-249.4891
-0.0269
-1.7383
5
-0.1534
0.5109
-0.0005
0.0135
6
-0.1534
250.5109
0.0361
1.7616
7
-0.1534
500.5109
0.0499
3.5065
8
-0.1534
250.5109
0.0231
1.7613
9
-0.1534
0.5109
0.0038
-0.0035
1
-0.0383
1.6096
-0.0008
0.0112
2
-100.0383
-498.3904
-2.5554
-3.5442
3
-100.0383
501.6096
-2.4508
3.4479
4
-0.0383
1.6096
0.0058
0.0010
5
99.9617
-498.3904
2.4471
-3.4483
6
99.9617
501.6096
2.5466
3.5505
7
-0.0383
1.6096
0.0008
-0.0112
Using the tare adjusted applied calibration loads and the delta bridge outputs shown in Table 12 above, the
new interim regression coefficient matrix is computed, as shown in Table 13.
Table 13: Step 3.2 Iteration Interim Regression Coefficient Matrix
r1
r2
Intercept
A
B
A*A
A*B
0.00287529
0.025005667
0.000103881
-6.05E-07
-2.56E-08
0.001638172
0.000498698
0.006995003
-5.52E-07
3.35E-08
B*B
1.88E-09
-4.11E-09
Tare Load Iteration Step 3.3
The tare loads are computed from the modified bridge outputs (Table 7) and the interim matrix computed
during step 3.2 (shown in Table 13). The computed tare loads are shown below in Table 14, and the
change in the tare loads are also computed and shown in Table 14.
Table 14: Computed Tare Loads and Changes in Tare Loads
Series
1
2
3
Tare Loads
r1
-0.0652
-0.1534
-0.0383
r2
0.2833
0.5109
1.6096
Series
1
2
3
Changes in Tare Loads
r1
-3.66E-07
-1.26E-06
-1.02E-06
r2
-2.42E-06
-4.80E-06
-1.01E-05
Looking at the changes in tare loads in Table 14, it is clear that deltas are all less than the established
criteria of 0.002, therefore the tare iteration process is complete.
Step 4: Final Matrix Determination
The final step in the matrix determination process is to adjust the applied calibration loads for the final tare
loads shown in Table 14. The final adjusted calibration loads are shown in Table 15, along with the
XXXXXXXX.
Table 15: Final Tare Adjusted Applied Loads & Raw Bridge Outputs (for Step 3.3 Tare Iteration)
Applied Loads (Coded)
Series
1
2
3
Point
A
B
1
-0.0652
2
-50.0652
3
4
Raw Bridge Outputs
r1 (mV/V)
r2 (mV/V)
0.2833
0.1219
-0.7906
0.2833
-1.1304
-0.8112
-100.0652
0.2833
-2.3834
-0.8356
-50.0652
0.2833
-1.1274
-0.8148
5
-0.0652
0.2833
0.1256
-0.7940
6
49.9348
0.2833
1.3718
-0.7633
7
99.9348
0.2833
2.6223
-0.7380
8
49.9348
0.2833
1.3672
-0.7565
9
-0.0652
0.2833
0.1251
-0.7945
1
-0.1534
0.5109
0.1190
-0.7956
2
-0.1534
-249.4891
0.1011
-2.5369
3
-0.1534
-499.4891
0.0671
-4.2848
4
-0.1534
-249.4891
0.0959
-2.5374
5
-0.1534
0.5109
0.1223
-0.7856
6
-0.1534
250.5109
0.1589
0.9625
7
-0.1534
500.5109
0.1727
2.7074
8
-0.1534
250.5109
0.1459
0.9622
9
-0.1534
0.5109
0.1266
-0.8026
1
-0.0383
1.6096
0.1253
-0.7800
2
-100.0383
-498.3904
-2.4293
-4.3354
3
-100.0383
501.6096
-2.3248
2.6566
4
-0.0383
1.6096
0.1319
-0.7903
5
99.9617
-498.3904
2.5732
-4.2396
6
99.9617
501.6096
2.6727
2.7592
7
-0.0383
1.6096
0.1269
-0.8025
Table 16: Final Calibration Matrix
r1
Intercept
0.126321298
r2
0.793748404
A
0.025005837
0.00049865
B
0.000103856
0.00699501
A*A
-4.65E-07
2.80E-08
A*B
-2.51E-08
3.40E-08
B*B
3.77E-09
-1.56E-08
The Final Calibration Matrix is shown in Table 16 above. This completes the matrix determination
process.