# Simulation-Based Model-Updating Method for Linear Dynamic Structural Systems
## 2 Modelling of the Structural System  
The methodology is demonstrated using a building model, as illustrated in Figure 1. The structural system has dimensions of 800 mm in the x direction and 600 mm in the y direction. The experimental specimen analyzed herein encompasses five stories with an inter-story height of 300 mm. For the material properties, the nominal yield stress of steel is selected as 235 MPa for the slabs and 750 MPa for the columns. The columns have a dimension of 6 \(\times\) 15 mm, and the floor slabs have a thickness of 15 mm.  
The parameters governing the structural system, encompassing mass and stiffness characteristics, are derived through an analytical methodology. The analytically estimated mass matrix that considers the weights of slabs, columns, and additional physical loads can be obtained as follows:  
\[[M_{\mathrm{s}}]=\begin{bmatrix}70.57&0&0&0&0\\ 0&70.57&0&0&0\\ 0&0&70.57&0&0\\ 0&0&0&70.57&0\\ 0&0&0&0&70.57\end{bmatrix}\mathrm{kg}\]  
In simple terms, the flexural stiffness of each column can be expressed by the following equation:  
\[k_{i}=\frac{12 EI}{L^{3}} \tag{1}\]  
In this equation, the modulus of elasticity (\(E=2\times 10^{11}\) N/m\({}^{2}\)), the section moment of inertia (\(I=2.7\times 10^{-10}\) m\({}^{4}\) ), and the length of the column (\(L=0.30\) m) are represented by the variables \(E\), \(I\), and \(L\), respectively. Based on this formulation, the stiffness matrix for the five-story building model is constructed by the following equation:  
\[[K_{\mathrm{s}}]=\begin{bmatrix}192000&-96000&0&0\\ -96000&192000&-96000&0&0\\ 0&-96000&192000&-96000&0\\ 0&0&-96000&192000&-96000\\ 0&0&0&-96000&96000\end{bmatrix}\mathrm{N/m}\]  
Figure 1: (**a**) Mathematical representation of the structure. (**b**) Test structure.  
Free vibration tests are widely used as an experimental method to determine the damping ratio of structural systems [19]. The logarithmic decrement method determines the damping ratio, utilizing data measured from free vibration test records. The damping ratio for lightly damped systems is mathematically expressed as follows:  
\[\xi=\frac{1}{2\pi\beta}ln\frac{u_{i}}{u_{i+j}} \tag{2}\]  
where, \(u_{i}\) and \(u_{i+j}\) are displacement amplitude values, respectively, with \(j\) indicating the number of successive pairs under consideration. This method is subsequently applied to the building model utilized within the experimental investigation. The building model is fixed to the shaking table and subjected to an initial displacement that keeps the structural response within the linear region. The free oscillation forms of the building model for each story are presented in Figure 2. The damping ratio of the structural system, \(\xi=0.0038\), is obtained from the free vibration measurement data using Equation (2).  
Using the Rayleigh damping model, the damping matrix of a structure is formulated as a linear combination of the mass and stiffness matrices, depicted by the following equations:  
\[C_{s}=\alpha M_{s}+\beta K_{s} \tag{3}\]  
\[\alpha=\xi\frac{\omega_{1}\omega_{2}}{\omega_{1}+\omega_{2}} \tag{4}\]  
\[\beta=\xi\frac{\omega_{1}+\omega_{2}}{\omega_{1}+\omega_{2}}\]  
Here, \(\xi\) represents the damping ratio of the structural system, and \(\alpha\) and \(\beta\) are real scalars. The values of alpha and beta, which are required to determine the damping matrix, have been calculated using Equation (4), as \(\alpha=0.3935\) and \(\beta=0.00057\). Additionally, \(\omega_{1}\) and \(\omega_{2}\) are circular frequencies. The damping matrix for the structural system is presented as follows:  
\[[C_{s}]=\left[\begin{array}{cccc}50.72&-23.59&0&0&0\\ -23.59&50.72&-23.59&0&0\\ 0&-23.59&50.72&-23.59&0\\ 0&0&-23.59&50.72&-23.59\\ 0&0&0&-23.59&27.108\end{array}\right]\text{Ns/m}\]