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 2-component 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

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. Determine the Zero-Load Output
  2. Determine the Linear Matrix
  3. Tare Load Iteration Process
  4. 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 zero-load 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 / -0.7929 / 0
0.1235 / -0.7870 / 90
0.1217 / -0.7925 / 180
0.1247 / -0.7898 / 270
0.1232 / -0.7872 / 0
0.1257 / -0.7876 / 90
0.1247 / -0.7901 / 180
0.1283 / -0.7897 / 270
0.1245 / -0.7896 / 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)

Applied Loads (Eng. Units) / Raw Bridge Outputs
Series / Point / A / B / r1(mV/V) / r2(mV/V)
1 / 1 / 0.0 / 0.0 / 0.1219 / -0.7906
2 / -50.0 / 0.0 / -1.1304 / -0.8112
3 / -100.0 / 0.0 / -2.3834 / -0.8356
4 / -50.0 / 0.0 / -1.1274 / -0.8148
5 / 0.0 / 0.0 / 0.1256 / -0.7940
6 / 50.0 / 0.0 / 1.3718 / -0.7633
7 / 100.0 / 0.0 / 2.6223 / -0.7380
8 / 50.0 / 0.0 / 1.3672 / -0.7565
9 / 0.0 / 0.0 / 0.1251 / -0.7945
2 / 1 / 0.0 / 0.0 / 0.1190 / -0.7956
2 / 0.0 / -250.0 / 0.1011 / -2.5369
3 / 0.0 / -500.0 / 0.0671 / -4.2848
4 / 0.0 / -250.0 / 0.0959 / -2.5374
5 / 0.0 / 0.0 / 0.1223 / -0.7856
6 / 0.0 / 250.0 / 0.1589 / 0.9625
7 / 0.0 / 500.0 / 0.1727 / 2.7074
8 / 0.0 / 250.0 / 0.1459 / 0.9622
9 / 0.0 / 0.0 / 0.1266 / -0.8026
3 / 1 / 0.0 / 0.0 / 0.1253 / -0.7800
2 / -100.0 / -500.0 / -2.4293 / -4.3354
3 / -100.0 / 500.0 / -2.3248 / 2.6566
4 / 0.0 / 0.0 / 0.1319 / -0.7903
5 / 100.0 / -500.0 / 2.5732 / -4.2396
6 / 100.0 / 500.0 / 2.6727 / 2.7592
7 / 0.0 / 0.0 / 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. Take the first bridge output in each data series and subtract it from all other bridge outputs within that series (for each bridge).
  2. 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

Applied Loads (Eng. Units) / Bridge Outputs
Series / Point / A / B / r1(mV/V) / r2(mV/V)
1 / 1 / 0.0 / 0.0 / -0.0016 / 0.0019
2 / -50.0 / 0.0 / -1.2539 / -0.0186
3 / -100.0 / 0.0 / -2.5069 / -0.0430
4 / -50.0 / 0.0 / -1.2509 / -0.0222
5 / 0.0 / 0.0 / 0.0021 / -0.0014
6 / 50.0 / 0.0 / 1.2483 / 0.0293
7 / 100.0 / 0.0 / 2.4988 / 0.0546
8 / 50.0 / 0.0 / 1.2437 / 0.0361
9 / 0.0 / 0.0 / 0.0016 / -0.0019
2 / 1 / 0.0 / 0.0 / -0.0038 / 0.0035
2 / 0.0 / -250.0 / -0.0217 / -1.7378
3 / 0.0 / -500.0 / -0.0557 / -3.4857
4 / 0.0 / -250.0 / -0.0269 / -1.7383
5 / 0.0 / 0.0 / -0.0005 / 0.0135
6 / 0.0 / 250.0 / 0.0361 / 1.7616
7 / 0.0 / 500.0 / 0.0499 / 3.5065
8 / 0.0 / 250.0 / 0.0231 / 1.7613
9 / 0.0 / 0.0 / 0.0038 / -0.0035
3 / 1 / 0.0 / 0.0 / -0.0008 / 0.0112
2 / -100.0 / -500.0 / -2.5554 / -3.5442
3 / -100.0 / 500.0 / -2.4508 / 3.4479
4 / 0.0 / 0.0 / 0.0058 / 0.0010
5 / 100.0 / -500.0 / 2.4471 / -3.4483
6 / 100.0 / 500.0 / 2.5466 / 3.5505
7 / 0.0 / 0.0 / 0.0008 / -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 / -0.000848 / 0.004912
A / 0.025005714 / 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 / Modified Bridge Outputs
r1 (mv/v) / r2 (mV/V)
1 / -0.0029 / -0.0010 / <-- (Series 1/Point 1) - Average ZLOS
2 / -0.0058 / -0.0060 / <-- (Series 2/Point 1) - Average ZLOS
3 / 0.0005 / 0.0096 / <-- (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

Series / Tare Loads
r1 / r2
1 / -0.1154 / -0.1382
2 / -0.2276 / -0.8398
3 / 0.0131 / 1.3699
Series / Changes in Tare Loads
r1 / r2
1 / -0.1154 / -0.1382
2 / -0.2276 / -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) / Bridge Outputs
Series / Point / A / B / r1 (mv/v) / r2 (mV/V)
1 / 1 / -0.1154 / -0.1382 / -0.0016 / 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
2 / 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
3 / 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 / Tare Loads
r1 / r2
1 / -0.0652 / 0.2833
2 / -0.1534 / 0.5109
3 / -0.0383 / 1.6096
Series / Changes in Tare Loads
r1 / r2
1 / 5.02E-02 / 4.21E-01
2 / 7.43E-02 / 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) / Bridge Outputs
Series / Point / A / B / r1 (mv/v) / r2 (mV/V)
1 / 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
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
2 / 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
3 / 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 / 0.00287529 / 0.001638172
A / 0.025005667 / 0.000498698
B / 0.000103881 / 0.006995003
A*A / -6.05E-07 / -5.52E-07
A*B / -2.56E-08 / 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 / Tare Loads
r1 / r2
1 / -0.0652 / 0.2833
2 / -0.1534 / 0.5109
3 / -0.0383 / 1.6096
Series / Changes in Tare Loads
r1 / r2
1 / -3.66E-07 / -2.42E-06
2 / -1.26E-06 / -4.80E-06
3 / -1.02E-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) / Raw Bridge Outputs
Series / Point / A / B / r1 (mV/V) / r2 (mV/V)
1 / 1 / -0.0652 / 0.2833 / 0.1219 / -0.7906
2 / -50.0652 / 0.2833 / -1.1304 / -0.8112
3 / -100.0652 / 0.2833 / -2.3834 / -0.8356
4 / -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
2 / 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
3 / 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 / r2
Intercept / 0.126321298 / -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.