# Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions
## 3 Numerical Example
### Construction of Transform Matrices  
Since the GCS and ECS coincide, the rotation matrix \(R\) was the identity matrix. The measurement direction of the strain gauges was \(n_{S1}=\left[ \begin{array}{ccc}0.995&-0.100\end{array} \right]^{T}\), \(n_{S2}=\left[ \begin{array}{ccc}1.0&0.0\end{array} \right]^{T}\), \(n_{S1}=\left[ \begin{array}{ccc}0.995&0.100\end{array} \right]^{T}\) and the measurement direction (\(n^{\prime}=Rn\)) in the ECS became \(n^{\prime}_{S1}=n_{S1}\), \(n^{\prime}_{S2}=n_{S2}\), and \(n^{\prime}_{S3}=n_{S3}\). The measurement direction of the accelerometers was \(n_{S4}=n_{S5}=n_{S6}=\left[ \begin{array}{ccc}0.0&1.0\end{array} \right]^{T}\) and was \(n^{\prime}_{S1}=n^{\prime}_{S2}=n^{\prime}_{S3}=\left[ \begin{array}{ccc}0. 0&1.0\end{array} \right]^{T}\) in the ECS. The strain components of the two-dimensional Euler-Bernoulli beam element were the longitudinal strain component (\(e^{\prime}_{xx}\)) and the transverse strain component (\(e^{\prime}_{yy}\)) induced by the Poisson effect. Matrix \(B_{ei}\), which relates the strain (\(e^{\prime}=\left[ \begin{array}{ccc}e^{\prime}_{xx}&e^{\prime}_{yy}\end{array}  \right]^{T}\)) of the element and the displacement (\(\hat{d}^{\prime}_{e}=\left[ \begin{array}{ccc}\hat{u}^{\prime}_{1}&\hat{v}^ {\prime}_{1}&\hat{\theta}^{\prime}_{1}&\hat{u}^{\prime}_{2}&\hat{\theta}^{ \prime}_{2}&\hat{\theta}^{\prime}_{2}\end{array} \right]^{T}\)) of the node (subscripts 1 and 2 indicate, respectively, the first and second nodes of the element), can be obtained as follows using Equation (A3). \[B_{e1}=B_{e3}=\left[ \begin{array}{cccccc}0.2&0.0&-0.01&0.2&0.0&0.01\\ -0.04&0.0&0.002&-0.04&0.0&-0.002\end{array} \right] \tag{44}\]  
\[B_{e2}=\left[ \begin{array}{cccccc}0.2&0.0&-0.06&0.2&0.0&0.06\\ -0.04&0.0&0.012&-0.04&0.0&-0.012\end{array} \right] \tag{45}\]  
Transforming \(B_{ei}\) into \(\overline{B}_{ei}\), which relates the node and measurement directions in the GCS, gives  
\[\overline{B}_{e1}=\left[ \begin{array}{ccc}0.203&0.0&-0.01015&0.203&0.0&0.01 015\end{array} \right] \tag{46}\]  
Figure 3: Structure shape and sensor layout of numerical example (unit: m). \[\widetilde{B}_{e2}=\left[ \begin{array}{cccccc}0.2&0.0&-0.06&0.2&0.0&0.06\end{array} \right] \tag{47}\]  
\[\widetilde{B}_{e3}=\left[ \begin{array}{cccccc}0.195&0.0&-0.00975&0.195&0.0&0.0097 5\end{array} \right] \tag{48}\]  
Assembling the transform matrices obtained for the strain gauges with respect to the global DOFs provides the transform matrix \(\widetilde{B}\) for the whole set of strain gauges. \[\phi_{es}=\widetilde{B}\phi \tag{49}\]  
where  
\[\phi=\left[ \begin{array}{cccccc}\phi_{a_{1}}&\phi_{b_{1}}&\phi_{b_{1}}&\phi _{b_{2}}&\phi_{b_{2}}&\phi_{b_{3}}&\phi_{b_{3}}&\phi_{b_{3}}&\phi_{b_{4}}&\phi _{b_{4}}&\phi_{b_{4}}\end{array} \right]^{T} \tag{50}\]  
\[\widetilde{B}=\left[ \begin{array}{cccccc}0.203&0.0&-0.01015&0.203&0.0&0.0101 5&0.0&0.0&0.0&0.0&0.0\\ 0.0&0.0&0.0&0.2&0.0&-0.06&0.2&0.0&0.06&0.0&0.0&0.0\\ 0.0&0.0&0.0&0.0&0.0&0.0&0.195&0.0&-0.00975&0.195&0.0&0.00975\end{array} \right] \tag{51}\]  
In Equation (50), \(\phi_{a_{1}}\), \(\phi_{b_{1}}\), and \(\phi_{b_{4}}\), corresponding to the fixed DOFs, were 0 and could thus be discarded. The displacement components of the two-dimensional Euler-Bernoulli beam element were the longitudinal component (\(u^{\prime}\)) and the transverse component (\(v^{\prime}\)). Matrix \(N_{ei}\), which relates the element displacement (\(d^{\prime}=\left[ \begin{array}{cc}u^{\prime}&v^{\prime}\end{array} \right]^{T}\)) and the node displacement, was obtained as follows using Equation (A6). \[N_{e4}=N_{e5}=N_{e6}=\left[ \begin{array}{cccccc}0.5&0.15&0.125&0.5&-0.15&0.12 5\\ 0.0&0.5&0.625&0.0&0.5&-0.625\end{array} \right] \tag{52}\]  
Transforming \(N_{ei}\) into \(\widetilde{N}_{ei}\), which relates the DOFs of the GCS and the displacement in the measurement direction, gives:  
\[\widetilde{N}_{e4}=\widetilde{N}_{e5}=\widetilde{N}_{e6}=\left[ \begin{array}[ ]{cccccc}0.0&0.5&0.625&0.0&0.5&-0.625\end{array}