# Calibration of a 3-Component External Balance for MAVs Wind Tunnel Research
## 4 Balance Calibration
### Balance interactions  
The study of force interactions in balance measurements allows for an assessment of force decoupling achieved through the mechanical design of the balance. First-order interactions:  
The linear terms correspond to first-order interactions, which represent misalignments due to fabrication tolerances and the mechanical assembly of balance components. The terms \(A_{ij}\) with \(i=j\) indicate the balance's sensitivity to the three types of loads being measured. In contrast, elements \(A_{ij}\) with \(i\
eq\ j\) are the linear interaction coefficients that account for the influence of a load that does not produce an output signal. The linear part of the A matrix is as follows:  
\[\left[A\right]_{linear}= \begin{bmatrix}-5.536\cdot 10^{-4}&2.341\cdot 10^{-6}&-4.4 69\cdot 10^{-6}\\ 3.929\cdot 10^{-6}&-8.892\cdot 10^{-5}&-8.773\cdot 10^{-6}\\ -1.681\cdot 10^{-8}&-5.923\cdot 10^{-8}&5.733\cdot 10^{-6}\end{bmatrix}  \tag{16}\]  
The sensitivity \(S_{i}\) of each load cell, as installed in the balance, corresponds to the inverse of the diagonal elements of matrix \([A]\). It is defined as the variation in output signal produced by a variation in input load and is given as follows:  
\[S_{i}=\frac{\Delta U_{i}}{\Delta F_{i}}=\frac{1}{A_{ii}}(i=1,2,3) \tag{17}\]  
Table 4 shows the sensitivity of each load cell. The negative sign in the sensitivity indicates a negative slope of the sensitivity curve, as the load cells measure negative bending moments. The type of load cells used in this balance can measure both negative and positive bending moments. From the sensitivity values, it is observed that the sensitivity of load cell 2 is approximately 6 times that of load cell 1, while the sensitivity of load cell 3 is 97 times higher than that of load cell 1. To understand the contribution of each component to the overall load, the balance is treated as a \"black box\" sensor from a global perspective. In this approach, applied loads excite the balance, which then responds with output signals. The following analysis examines a representative case where three types of loads--lift, drag, and pitching moment--are applied to the balance. The values of these loads lie within the mid-range of the measurement spectrum, and the resulting data are analyzed to assess their impact. \[\left[F\right]_{real\ load}= \begin{bmatrix}L(\text{g})\\ D(\text{g})\\ M(\text{g}\cdot\text{m})\end{bmatrix} _{real\ load}= \begin{bmatrix}309.2\\ 141.1\\ 3.07\end{bmatrix}  \tag{18}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Load Cell i** & **Sensitivity** & **Units** & **Relative Sensitivity** \\ \hline
1 & \(-1.806\cdot 10^{3}\) & Bit/g & 1 \\ \hline
2 & \(-1.124\cdot 10^{4}\) & Bit/g & 6.226 \\ \hline
3 & \(1.744\cdot 10^{5}\) & Bit/g\(\cdot\)m & \(-9.657\cdot 10^{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Sensitivity of load cells. Estimated loads from the measured signals with linear term (first-order approximation) are as follows:  
\[\left[\hat{F}\right]=\left[A\right]\left[U\right]= \begin{bmatrix}-5.536\cdot 10^{-4}&2.341 \cdot 10^{-6}&-4.469\cdot 10^{-6}\\ 3.929\cdot 10^{-6}&-8.892\cdot 10^{-5}&-8.773\cdot 10^{-6}\\ -1.681\cdot 10^{-8}&-5.923\cdot 10^{-8}&5.733\cdot 10^{-6}\end{bmatrix} \begin{bmatrix}-5.656\cdot 10^{5}\\ -1.557\cdot 10^{6}\\ 5.505\cdot 10^{5}\end{bmatrix}  \tag{19}\]  
\[\left[\hat{F}\right]= \begin{bmatrix}L(\text{g})\\ D(\text{g})\\ M(\text{g}\cdot\text{m})\end{bmatrix} = \begin{bmatrix}304.0\\ 132.5\\ 3.2\end{bmatrix}  \tag{20}\]  
For instance, lift \(L\) is given by the following expression:  
\[L=A_{11}\cdot U_{1}+A_{12}\cdot U_{2}+A_{13}\cdot U_{3} \tag{21}\]  
and dividing the right part by the dominant term, result,  
\[\frac{L}{A_{11}\cdot U_{1}}=1+\frac{A_{12}\cdot U_{2}}{A_{11}\cdot U_{1}}+ \frac{A_{13}\cdot U_{3}}{A_{11}\cdot U_{1}} \tag{22}\]  
and in the same manner, for drag (\(D\)) and pitching moment (\(M_{y}\)), the contributions to the total load are obtained (Table 5).