# Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions
## 2 Materials and Methods
### Measurement-Model Link
#### 2.1.2 The Case Where the Strain is Measured  
The method derived in Section 2.1.1 can be applied similarly for the strain. The strain \(\epsilon_{n}\) measured at the sensor location \(s\)(\(s_{x}\), \(s_{y}\), \(s_{z}\)) can be expressed in terms of the strain \(e^{\prime}=\left[\begin{array}{cccc}\epsilon_{xx}^{\prime}&\epsilon_{yy}^{ \prime}&\epsilon_{zz}^{\prime}&\gamma_{xy}^{\prime}&\gamma_{yz}^{\prime}& \gamma_{xz}^{\prime}\\ \end{array}\right]^{T}\) in the ECS at the sensor location and the direction vector \(n^{\prime}\) in the ECS.  
\[\epsilon_{n}=n_{x}^{\prime}n_{x}^{\prime}\epsilon_{xx}^{\prime}+n_{y}^{\prime }n_{y}^{\prime}\epsilon_{yy}^{\prime}+n_{z}^{\prime}n_{z}^{\prime}\epsilon_{ zz}^{\prime}+n_{x}^{\prime}n_{y}^{\prime}\gamma_{xy}^{\prime}+n_{y}^{ \prime}n_{z}^{\prime}\gamma_{yz}^{\prime}+n_{x}^{\prime}n_{z}^{\prime}\gamma_{ xz}^{\prime}=(\overrightarrow{n}^{\prime})^{T}\epsilon^{\prime} \tag{22}\]  
where  
\[\overrightarrow{n}^{\prime}=\left[\begin{array}{cccc}n_{x}^{\prime}n_{x}^{ \prime}&n_{y}^{\prime}n_{y}^{\prime}&n_{z}^{\prime}n_{z}^{\prime}&n_{x}^{ \prime}n_{y}^{\prime}&n_{y}^{\prime}n_{z}^{\prime}&n_{x}^{\prime}n_{z}^{ \prime}\end{array}\right]^{T} \tag{23}\]  
The strain \(\epsilon^{\prime}\) in the ECS can be rewritten in terms of the element node displacement \(\hat{d}_{e}^{\prime}\) using the displacement-strain transform matrix \(B_{e}\).  
\[e^{\prime}=B_{e}\hat{d}_{e}^{\prime} \tag{24}\]  
Matrix \(B_{e}\) also varies according to the element type. The displacement-strain transform matrix for the two-dimensional Euler-Bernoulli beam element is arranged in Appendix A. Following the process formulated in Equations (7)-(12), the strain \(\epsilon_{n}\) in the measurement direction and at the sensor location can be expressed in terms of the node displacement in the GCS.  
\[\epsilon_{n}=\overline{B}_{e}\hat{d}_{e} \tag{25}\]  
where  
\[\overline{B}_{e}=(\overrightarrow{n}^{\prime})^{T}B_{e}T \tag{26}\]  
The so-derived matrix \(\overline{B}_{e}\) relates to one strain gauge. Denoting \(\overline{B}_{e}\) for the \(i\)th sensor by \(\overline{B}_{ei}\), matrix \(\overline{B}\in\mathbb{R}^{n_{c}\times n_{d}}\) (\(m_{e}\) = the number of strain gauges) can be obtained for the whole set of sensors by assembling the \(\overline{B}_{ei}\) as follows.  
\[\overline{B}=\sum_{i}^{m_{e}}\overline{B}_{ei} \tag{27}\]  
The strain \(\epsilon_{s}\in\mathbb{R}^{m_{e}}\) in the measurement direction at the sensor location can be expressed by means of the node displacement \(\hat{d}\in\mathbb{R}^{n_{d}}\) in the GCS using this process. Moreover, the strain \(\epsilon_{s}\) can be expressed by means of the eigenvector \(\phi_{i}\) defined at the node DOF or the mode shape \(\phi_{esi}\in\mathbb{R}^{m_{e}}\) for the strain at the measurement location as shown in Equations (28) and (29).  
\[\epsilon_{s}=\overline{B}\tilde{d}=\overline{B}\sum_{i}^{n_{d}}\phi_{i}q_{di}= \sum_{i}^{n_{d}}\phi_{esi}q_{di} \tag{28}\]  
where  
\[\phi_{esi}=\overline{B}\phi_{i} \tag{29}\]  
The above process makes it possible to express the measured displacement, velocity, acceleration, and strain in terms of the node DOFs of the model. Moreover, the mode shapes \(\phi_{disi}\), \(\phi_{psi}\), \(\phi_{asi}\), and \(\phi_{esi}\) for the measured values corresponding to the eigenvector \(\phi_{i}\) of the node DOFs are also defined. The transform matrices \(\widetilde{N}\) and \(\overline{B}\) bridging the measurement and the analysis were derived solely using geometrical relations and are thus applicable, even in the case of change in the material's characteristics. For convenience in the following, \(\phi_{si}\) denotes the mode shape for the measured value and \(\widetilde{S}\) denotes the transform matrix to reduce Equations (19), (21), and (29) into the unique form expressed hereafter.  
\[\phi_{si}=\widetilde{S}\phi_{i} \tag{30}\]