# Simulation-Based Model-Updating Method for Linear Dynamic Structural Systems
## 3 Methodology  
The inaccuracies or uncertainties within analytical models may or may not accurately represent the actual structural behavior. Given the reliability of experimental tests, it is advisable to update the analytical model of the building system by incorporating data derived from the measurement results. The modal updating methodology tries to correct differences between analytical models and experimental data by implementing numerical changes to the original analytical system matrices, such as mass, stiffness, and damping. The proposed model-updating technique employs simulated free vibration results to correct the mathematical model directly. To evaluate the efficacy of this approach, a simulation model depicting the free vibration behavior of the structural system has been developed within the MATLAB/Simulink environment. The structural representation of the building can be characterized as a five-degree-of-freedom dynamic system. The numerical example is based on correcting the damping and stiffness matrices inherent to the building model, which is achieved through the free vibration response. The dynamic parameters of the building model have been modified by employing a simulation-based model-updating approach to yield results approximating actual structural dynamics. The mechanical model representing the five-story shear frame in the simulation is shown in Figure 6.  
The equation of motion of the multi-degree-freedom system shown in Figure 6 can be derived from the free-body diagram of the masses and expressed in matrix form as  
\[M_{s}\ddot{x}(t)+C_{s}\dot{x}(t)+K_{s}x(t)=0 \tag{5}\]  
where \(M_{s},\ C_{s}\), and \(K_{s}\) denote mass, damping, and stiffness matrix, respectively. And also, \(\ddot{x},\  
displacements obtained from the experimental measurements are assigned to each story as the initial condition in the simulation. The system is operated under these initial conditions to give appropriate results with the experimental measurements. The proposed model-updating methodology is implemented utilizing the MATLAB/Simulink software to correct the dynamic parameters of the building model. The Simulink Design Optimization toolbar of the software is used for analyzing and tuning the simulation model parameters [21]. In this module, the effect of parameters on simulation model behavior can be evaluated by employing methodologies such as Monte Carlo simulation and Design of Experiments. The measured experimental data is integrated into the simulation model through the parameter estimation module, and selected model parameters are simultaneously estimated from multi-experiment data. The software formulates the parameter estimation as an optimization problem, and the solution of the problem is the estimated design variables. The elements of the optimization problem can be listed as design variables (\(x\)), objective function \((F(x))\), bounds (\(\underline{x}\leq x\leq x\)), and constraint function  
Figure 7: Block diagram representing the free vibration of the building model. \(C(x)\). This includes bounds to limit estimated parameter values and integrating a constraint function to specify limitations on design variables are both optional aspects. The software fine-tunes the model parameters to achieve a simulation response (_y_sim_) that closely follows the measured response (_y_ref_). The solver minimizes the estimation error or cost function to update these parameters, denoting the difference between simulated and measured responses. The formulation of the estimation error is presented in the equation below. The software also provides the user with various cost functions to handle the estimation error. \[e(t)=y_{ref}(t)-y_{sim}(t) \tag{7}\]  
In this study, the Parameter Estimation module in the MATLAB/Simulink software was employed in conjunction with the simulation-based model-updating approach, as illustrated in Figure 8. The design variables selected for the optimization problem include the stiffness and damping coefficients of the building model.