## Initial model
### Experimental Beam (Case 3)  
In actual structures, there are differences in the measured quantities as compared to their theoretical counterparts mainly due to differences in material and geometrical properties. Damage detection becomes more difficult in these problems, and it is imperative to check the effectiveness of the proposed approach on actual experimental test. Furthermore, by using this example, comparative analysis can be easily made with other studies. The experimental beam was taken from [PERSON] et al. [45]. The aluminum beam has a rectangular section of 0.05 m \(\times\) 0.006 m with two fixed ends. The length of the beam is 0.6 m. Other material  
Figure 10: Damage distribution for Case 2 using Objective Function II for (**a**) noise both in frequency and mode shapes and (**b**) noise in mode shapes only. specifications were the modulus of elasticity as 70 GPa, the Poisson's ratio as 0.3 and the density as 2700 kg/m\({}^{3}\). A saw cut damage was induced in the beam at the ninth element by cutting a quarter of the total thickness from the top and bottom surface of the intact specimen. The theoretical damage index for the cracked location was 0.875, i.e., an 87.5% reduction of MOI in y direction. The first three mode shapes were compared as these were measured experimentally. Figure 11: Damage distribution for Case 2 using Objective Function III for (**a**) noise both in frequency and mode shapes and (**b**) noise in mode shapes only. SAP 2000 was used to model and analyze the beam. For the experimental beam, the adopted mesh has been selected based on convergence analysis of the FE model. For this purpose, the beam was divided into 10, 20 and 40 nodes having a thickness of 0.5 cm, and the mesh convergence effect has been checked based on the first three natural frequencies. The % error between the 10- and 20-node beam was 0.0034, 0.0058 and 0.1641%, whereas the error between the 20- and 40-node beam was 0.0000, 0.0016 and 0.0070 for the first three frequencies respectively. Therefore, due to the maximum error of 0.1641% between 10 and 20 nodes and 0.0070% between 20 and 40 nodes, the beam with 20 nodes has been selected. The beam with 20 equal two-dimensional beam elements is also used to model the beam with 20 nodes. consistent with the measured DOF in the experiment. Fixed end supports were imposed on the corner elements of the beam. The mass and stiffness matrices were afterwards exported to MATLAB for optimization purposes. The model was updated considering the first three natural frequencies and mode shapes for damage detection. The three objective functions are applied to update the beam to check the suitability of the objective functions in the actual test. The damage indices identified by using Objective Functions I, II and II are shown in Figure 15a. It can be noticed that for this damage case, the damage  
Figure 13: Damage distribution for Case 2 using Objective Function II with regularization for (**a**) noise both in frequency and mode shapes and (**b**) noise in mode shapes only. index for Element No. 9 (actual damage index = 0.875) was found to be 0.55, 0.65 and 0.61 for Objective Functions I, II and III, respectively. However, as compared to Objective Function I, misdetection in other elements was of much greater magnitude in Objectives II and III. The results are also consistent with the earlier results of Case 1 and Case 2. Later on, model updating was again done with all three objective functions with regularization, and the results are given in Figure 15b. The damage index for Element No. 9 was found to be 0.61, 0.58 and 0.43 (actual damage index = 0.875) for regularized Objective Functions I, II and III, respectively. Likewise, wrong detections in other elements were less in the regularized Objective Function I than the  
Figure 14: Damage distribution for Case 2 using Objective Function III with regularization for (**a**) noise both in frequency and mode shapes and (**b**) noise in mode shapes only. other two.