# Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions
## 1 Introduction  
In general, the model updating for a structure proceeds by minimizing the difference in the responses of the structure and of the model. The response of the structure can be physical quantities measured by sensors or be eigenvalues and mode shapes obtained by modal identification. Except for the eigenvalues, the measured quantities and the mode shapes are values specific to the location of the sensors. Considering that the location of the sensors does not generally coincide with the mesh of the finite element model, it is necessary to perform a proper transform that makes the location of the sensors match with the finite element model. Figure 1 shows a case where the location of the acceleration and strain sensors does not coincide with the position of the nodes in the model. In addition, the type of the measured physical quantities as well as the measurement direction shall be considered.  
The modal identification extracting the dynamic characteristics of the structure from the measured data can be done in both the frequency domain and time domain. Frequency domain decomposition (FDD) [1] is a modal identification method in the frequency domain. The eigensystem realization algorithm (ERA) [2] and subspace identification method [3,4] are modal identification methods in the time domain. Such modal identification provides the natural frequencies of the structure at hand and the mode shapes for the measured physical quantities in the measurement direction at the location of the sensors.  
The parameters of the numerical model can be derived by model updating using the so-obtained eigenvalues and mode shapes. The model updating methods can be divided into those relying on a deterministic approach [5,6] and those relying on a statistical approach [7, 8, 9, 10]. The mode shapes obtained by eigen analysis in the model updating process are those related to the nodal degrees-of-freedom (DOFs) of the analytic model and differ from the measured mode shapes obtained by modal identification. Accordingly, it is necessary to provide a solution to remedy such a mismatch. [PERSON] and [PERSON] [11] introduced a transform matrix for the strain measurement to make the measured DOFs and model DOFs coincide and used the strain generated by static loading to perform model updating. [PERSON], et al. [12] defined a frequency response function for the strain to perform model updating. However, these studies are limited to the case where the measurement direction of the strain sensors coincides with the longitudinal direction of the member and cannot be applied for sensors installed in arbitrary directions.  
The survey shows that numerous studies were implemented on modal identification or model updating but stresses also the scarcity of studies on methods linking these two processes. Consequently, the present study intends to propose a systematic approach linking the modal identification results to model updating. Section 2 derives a method of achieving model updating using the data monitored by acceleration, velocity, displacement, and strain sensors installed in arbitrary locations and directions in the structure. Section 3 explains the proposed approach through a simple numerical example. In Section 4, the feasibility of the proposed approach is demonstrated through numerical examples for a bridge structure involving strain measurement with/without installation and measurement errors.