## 2 Aircraft Wing Load Tests
### Solution of Coefficient Matrix
When regression analysis is applied, the load vector values are regression variables, while the strain vector values as the response values:  
\[\widetilde{\varepsilon}=\text{\boldmath{$K$}}_{\text{I}}\widetilde{\text{ \boldmath{$L$}}}. \tag{12}\]  
The least squares estimate from which the matrix \(\text{\boldmath{$K$}}_{\text{I}}\) can be obtained is as follows:  
\[\hat{\text{\boldmath{$K$}}}_{\text{I}}=\widetilde{\varepsilon}\widetilde{ \text{\boldmath{$L$}}}^{T}\bigg{(}\widetilde{\text{\boldmath{$L$}}}^{T} \bigg{)}^{-1}. \tag{13}\]  
To estimate the matrix \(\text{\boldmath{$K$}}_{\text{D}}\), in the most general case, when the number of strain signals exceeds the number of main load parameters, to obtain a relationship between the two, substituting Equations (4)-(6) yields the following:  
\[\widetilde{\text{\boldmath{$\varepsilon$}}}=\hat{\text{\boldmath{$K$}}}_{ \text{I}}\hat{\text{\boldmath{$K$}}}_{\text{D}}\widetilde{\text{\boldmath{$ \varepsilon$}}},\]  
\[\widetilde{\text{\boldmath{$\varepsilon$}}}\widetilde{\text{\boldmath{$ \varepsilon$}}}^{T}=\hat{\text{\boldmath{$K$}}}_{\text{I}}\hat{\text{\boldmath {$\Sigma$}}}\widetilde{\text{\boldmath{$\varepsilon$}}}^{\sim T},\]  
\[\hat{\text{\boldmath{$K$}}}_{\text{I}}\hat{\text{\boldmath{$K$}}}_{\text{D}}= \text{\boldmath{$I$}}.\]Thus,  
\[\left(\hat{\mathbf{K}}_{I}^{T}\hat{\mathbf{K}}_{I}\right)\hat{\mathbf{K}}_{D}=\hat{\mathbf{K}}_{I} ^{T}.\]  
Therefore,  
\[\hat{\mathbf{K}}_{D}=\left(\hat{\mathbf{K}}_{I}^{T}\hat{\mathbf{K}}_{I}\right)^{-1}\hat{\mathbf{K }}_{I}^{T}.\]  
I the matrix \(\mathbf{K}_{I}\) is a non-degenerate square, then  
\[\widetilde{\varepsilon}=\text{\boldmath{$K$}}_{\text{I}}\widetilde{\text{ \boldmath{$L$}}}. \tag{12}\]0  
The external loads sustained by the structure are generally the six spatial components. For an aircraft wing (including the tail wing), as shown in Figure 5, the effects of a \(y\)-directional load component of shear \(Q\), a load component around the \(x\)-axis of the bending moment \(M\), and a load component around the \(z\) axis of torque \(T\) are major, while the remaining three load components are relatively minor and can be neglected; that is, \(M\), \(Q\), and \(T\) are used as input variables, and the corresponding test strains \(\varepsilon_{M}\), \(\varepsilon_{Q}\), and \(\varepsilon_{T}\) serve as output variables. The tests demonstrate that the magnitude of the bridge output strain at each section is linearly related to the loads applied to that section, which can be expressed by the following equations:  
\[\left\{ \begin{array}{rcl}M&=&K_{\text{D11}}\varepsilon_{M}+K_{\text{D12}} \varepsilon_{Q}+K_{\text{D13}}\varepsilon_{T}\\ Q&=&K_{\text{D21}}\varepsilon_{M}+K_{\text{D22}}\varepsilon_{Q}+K_{\text{D23 }}\varepsilon_{T}\\ T&=&K_{\text{D31}}\varepsilon_{M}+K_{\text{D32}}\varepsilon_{Q}+K_{\text{D3 3}}\varepsilon_{T}\end{array} \right. \tag{14}\]  
\[\left\{ \begin{array}{rcl}\varepsilon_{\text{M}}&=&K_{\text{I11}}M+K_{\text{ I12}}Q+K_{\text{I13}}T\\ \varepsilon_{Q}&=&K_{\text{I21}}M+K_{\text{I22}}Q+K_{\text{I23}}T\\ \varepsilon_{T}&=&K_{\text{I31}}M+K_{\text{I32}}Q+K_{\text{I33}}T\end{array} \right. \tag{15}\]  
where \(K_{\text{D11}}\sim K_{\text{D33}}\) are the direct regression coefficients, which correspond to the applied load the bridge corresponds to when it outputs a unit strain; and \(K_{\text{I11}}\sim K_{\text{I33}}\) are an indirect regression coefficients, corresponding to the magnitude of the strain bridge output strain caused by a unit load. We define \(\mathbf{L}= \begin{bmatrix}M\\ Q\\ T\end{bmatrix} ,\varepsilon= \begin{bmatrix}\varepsilon_{M}\\ \varepsilon_{Q}\\ \varepsilon_{T}\end{bmatrix}