# Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions
## 2 Materials and Methods
### Measurement-Model Link
#### 2.1.1 The Case Where the Displacement, Velocity, and Acceleration are Measured  
The displacement \(d_{n}\) measured in the measurement direction \(n\) at the sensor location satisfies the following relationship with the displacement in the GCS at the sensor location. \[d_{n}=n^{T}d \tag{4}\]  
where \(u,\ v\), _and_\(w\) are the displacements in the directions \(x,\ y\), _and_\(z\) of the GCS, respectively. The displacement \(d\) in the GCS can be expressed in terms of the displacement in the ECS. \[d=R^{T}d^{\prime} \tag{5}\]  
where \(u^{\prime},\ v^{\prime},\ \textit{and}\ w^{\prime}\) are the displacements in the directions \(x^{\prime},\ y^{\prime},\textit{and}\ z^{\prime}\) of the ECS. The displacement \(d^{\prime}\) in the ECS can be expressed as the product of the nodal displacement \(\hat{d}^{\prime}_{e}\) of the element and the shape function \(N_{e}\). \[d^{\prime}=N_{e}\hat{d}^{\prime}_{e} \tag{6}\]  
The DOFs and shape functions of an element vary according to the type of element used in the analytic model. The shape function for the two-dimensional Euler-Bernoulli beam element is arranged in Appendix A.  
Figure 2: Relationship between the global coordinate system and element coordinate system. The displacement \(\hat{d}_{el}^{\prime}\) of the \(I\)th node of the element can be represented as follows by distinguishing the translational DOFs and the rotational DOFs. Some DOFs may vanish according to the element type. \[\hat{d}_{el}^{\prime}=\left[ \begin{array}{cccccc}\hat{u}_{I}^{\prime}&\hat{v}_ {I}^{\prime}&\hat{v}_{I}^{\prime}&\hat{\partial}_{xI}^{\prime}&\hat{\partial}_{ yI}^{\prime}&\hat{\partial}_{zI}^{\prime}\end{array} \right]^{T} \tag{7}\]  
\[\hat{d}_{e}^{\prime}=\left[ \begin{array}{cccccc}\cdots&\hat{d}_{el}^{\prime T }&\cdots\end{array} \right]^{T} \tag{8}\]  
where \(\hat{u}_{I}^{\prime}\), \(\hat{v}_{I}^{\prime}\), _and_\(\hat{w}_{I}^{\prime}\) are the translational DOFs in the \(x^{\prime},\ y^{\prime}\), _and_\(z^{\prime}\) directions of the \(I\)th node; and \(\hat{\partial}_{xI}^{\prime}\), \(\hat{\partial}_{yI}^{\prime}\), _and_\(\hat{\partial}_{zI}^{\prime}\) are the rotational DOFs with respect to axes \(x^{\prime},\ y^{\prime}\), _and_\(z^{\prime}\) of the \(I\)th node. The corresponding displacement \(\hat{d}_{el}\) of the \(I\)th node in the GCS can also be expressed similarly. \[\hat{d}_{el}=\left[ \begin{array}{cccccc}\hat{u}_{I}&\hat{v}_{I}&\hat{w}_{I}& \hat{\partial}_{xI}&\hat{\partial}_{yI}&\hat{\partial}_{zI}\end{array} \right]^ {T} \tag{9}\]  
\[\hat{d}_{e}=\left[ \begin{array}{cccccc}\cdots&\hat{d}_{el}^{T}&\cdots\end{array}  \right]^{T} \tag{10}\]  
where \(\hat{u}_{I}\), \(\hat{v}_{I}\), _and_\(\hat{w}_{I}\) are the translational DOFs in the \(x,\ y,\)_and_\(z\) directions of the \(I\)th node; and \(\hat{\partial}_{xI}\), \(\hat{\partial}_{yI}\), _and_\(\hat{\partial}_{zI}\) are the rotational DOFs with respect to axes \(x,\ y,\)_and_\(z\) of the \(I\)th node. The node displacement \(\hat{d}_{e}^{\prime}\) in the ECS satisfies the following relationship with the node displacement \(\hat{d}_{e}\) in the GCS. \[\hat{d}_{e}^{\prime}=T\hat{d}_{e} \tag{11}\]  
\[\hat{d}_{el}^{\prime}=T_{I}\hat{d}_{el}=\left[ \begin{array}{cccccc}R&0\\ 0&R\end{array} \right]\hat{d}_{el} \tag{12}\]  
The displacement \(d_{n}\) in the measurement direction at the sensor location can be expressed by the node displacement in the GCS using Equations (4)-(12). \[d_{n}=\widetilde{N}_{e}\hat{d}_{e} \tag{13}\]  
where  
\[\widetilde{N}_{e}=n^{T}R^{T}N_{e}T=n^{\prime T}N_{e}T \tag{14}\]  
The so-derived \(\widetilde{N}_{e}\) relates to one sensor. Denoting \(\widetilde{N}_{e}\) for the \(i\)th sensor by \(\widetilde{N}_{ei}\), matrix \(\widetilde{N}_{d}\in\mathbb{R}^{m_{d}\times n_{d}}\) (where \(m_{d}\) = the number of displacement sensors, \(n_{d}\) = the number of DOFs of the analytic model) can be obtained for the whole set of displacement sensors by assembling the \(\widetilde{N}_{ei}\) as follows.