*2.2. Balance Calibration*

In order to provide quality measurements, the aerodynamic balance was calibrated before each experiment. The calibration routine employed for this work assumed a linear response of the load cells, so a direct (exact solution) method was chosen to determine the correlation coefficients. The calibration, which implied a two-stage procedure to determine the calibration matrix, was completed under "no wind" conditions. In the first stage, the prototype was just mounted in the balance and a measurement was performed at free load, defining the zero-loading state. In the second stage, several measurements were performed with the balance loaded with known weights. Specifically, 3 load cases (LC) were carried out:


From each load case, three outputs (one for each load cell) were obtained providing a 9 equation and 9 unknowns system to represent the direct correlation between loads and measured components. Matrix algebra can be applied to streamline this process in the following way.

First, a force matrix *FLC* is defined with the three load cases, with one column for each component and one row for each load case.

$$F\_{LC} = \begin{pmatrix} F\_{x\_1} & 0 & 0 \\ 0 & F\_{y\_2} & 0 \\ 0 & F\_{y\_3} & F\_{y\_3}b \end{pmatrix} \tag{1}$$

where *F* is the applied load, *x* and *y* are the horizontal and vertical directions, respectively, in the balance coordinate system, and *b* is the horizontal distance to the axis in load case 3.

With the output of the single load cell being associated with the measurements in the horizontal direction in these experiments, referred to as "signal 1" (*s*1), and the other two, associated with the measurements in the vertical direction, referred to as "signal 2" (*s*2) and "signal 3" (*s*3), the output of the balance in the zero-loading state can be posed as a vector *SLC*<sup>0</sup> containing the values recorded in each load cell.

$$S\_{LC\_0} = \begin{pmatrix} S\_{1\_0} & S\_{2\_0} & S\_{3\_0} \end{pmatrix} \tag{2}$$

Following this, the matrix *SLC* is defined with the load cells' output for the three load cases, yielding:

$$S\_{LC} = \begin{pmatrix} s\_{1,1} & s\_{1,2} & s\_{1,3} \\ s\_{2,1} & s\_{2,2} & s\_{2,3} \\ s\_{3,1} & s\_{3,2} & s\_{3,3} \end{pmatrix} \tag{3}$$

Then, the calibration matrix *K* with the coefficients that relate the output of the three load cells with the forces and moment is:

$$K = \begin{pmatrix} k\_{1,1} & k\_{1,2} & k\_{1,3} \\ k\_{2,1} & k\_{2,2} & k\_{2,3} \\ k\_{3,1} & k\_{3,2} & k\_{3,3} \end{pmatrix} \tag{4}$$

Finally, applying the linear response assumption, the equation system is thus stated as:

$$F\_{LC} = \left[S\_{LC} - S\_{LC\_0}\right] \cdot K \tag{5}$$

where the no-load signals are discounted as the system offset. From this matrix system, matrix *K* can be directly deduced as *K* = [*S* − *S*0] −1 ·*Fxyz*, thus obtaining the direct relation between load cell outputs and measured forces.

Once the calibration matrix is determined, it can be employed to obtain the forces acting on the models from the signals measured during the operation of the wind tunnel using a generalization of Equation (5) for a single-point measurement:

$$F\_{xyz} = \left[\mathcal{S} - \mathcal{S}\_0\right] \cdot \mathcal{K} \tag{6}$$

where *Fxyz* and *S* are now row vectors with three columns.

As the calibration used is a linear, two-point method, the balance was additionally tested before the aerodynamic measurements' campaign to evaluate its accuracy. Figure 2 shows the response of the calibrated balance (y-axis) to 5 different known weights in the

lower part of the balance range (x-axis), where the linearity of these types of sensors is mostly critical.

**Figure 2.** Balance response to a range of known weights using a two-point, linear calibration method.

The figure shows that the calibration method used provides a sufficiently accurate linear response, even for the lower part of the balance range. The response in this range is better in the horizontal direction because its range is half than the others, so it is better prepared to measure small forces. This is especially interesting to the case of airfoil testing, as drag forces are much lower than lift ones.

During the aerodynamic measurements campaign, the balance was calibrated before each experiment and tested after with known weights to validate the balance calibration. This reduces the influence of random errors produced by differences in the testing environment temperature, differences in the set-up assembly, etc. Up to 13 calibrations were performed during the campaign, providing useful statistical data of the balance performance. Table 2 shows the mean, standard deviation, maximum and median value of the relative errors between the known weights and the measured weights those 13 calibrations.


**Table 2.** Relative error statistics of 13 calibrations performed during the measurements campaign.

The mean relative error obtained was around 1%, slightly higher in the horizontal direction and slightly lower in the vertical direction. However, as the standard deviation reveals, there was some variability in the quality of the calibrations; hence, the mean is not very representative of the real performance of the balance. Note that although all 13 calibrations have been included in this analysis, a quality requirement was established in 1%. Thus, calibrations with errors above this, such as the one that achieved the maximum error shown in Table 2, were discarded and repeated. Nevertheless, as the median indicates, these discarded cases were not common and the typical balance calibration errors were about 0.8% for the horizontal direction, 0.4% for the vertical direction, and 0.3% for the moment. In these calibrations, the loads were adjusted to the expected measured forces; hence, in contrast with the previous figure, the vertical and moment errors are lower. The higher error in the horizontal force is probably due to the higher difficulty to produce a pure horizontal load, as opposed to the simplicity of vertical loading.
