## Initial model  
The damage distribution for Case 1 is reported in Figure 2 using Objective Function I. Furthermore, to check the effect of noise on frequencies and mode shapes, two noise cases were investigated, wherein noise was added both in frequency and mode shapes (Figure 2a) and noise in mode shapes only (Figure 2b). Likewise, Figures 3 and 4 represent the damage distribution for Case 1 using Objective Functions II and III. It has been noted that all the objective functions detected the damage correctly with 0% noise in the measurement. However, it can also be noticed that when the noise level increases, the damage detection capabilities of all the objective functions are degraded. Only Objective Function I performed better than the other two combinations. Wrong detection of damage in other elements is more pronounced in Objective Functions II and III at higher noise levels (above 1%).The results are nearly pointless in these cases, as the damages in other elements exceeded the original damage level in Element No. 6. Therefore, it can be concluded that Objective Function I (Figure 2) performed better than the other two (Figures 3 and 4). Figure 1: (**a**) Simulated healthy beam. (**b**) Damaged model (Case 1) withsingle damage scenario. (**c**) Damaged model (Case 2) with multiple damage scenario. MOI, moment of inertia. The results of Objective Function I are discussed further here (Figure 2). For different noise levels, the damage index is detected to be very near the actual damage index of 0.2 at 0%, 1% and 3% noise (Figure 2a,b). However, for noise at 5% in both frequencies and mode shapes (Figure 2a), the damage index was 0.24, which further degraded to 0.37 for 10% noise levels. Similarly, the damage index for noise in mode shapes only (Figure 2b) degraded to 0.22 and 0.17 as compared to the actual damage index of 0.2 in the case of 5% and 10% noise, respectively. It can be noticed that less degradation is seen in the mode shape noise than noise in both frequencies and mode shapes, which is rational. Likewise erroneous detection of damage in other elements especially close to damage elements is also more evident in noise in both the frequencies and mode shapes case than the noise in mode shapes only. Figure 2: Damage distribution for Case 1 using Objective Function I for (**a**) noise both in frequency and mode shapes and (**b**) noise in mode shapes only. It can be noted that all the objective functions have similar frequency-related residual included in their first part (see Equation (11)). Therefore, the differences in the results can be mainly attributed to the second part of the objective functions, and possible reasons related to the performance of each of the objective function are discussed now. Figure 3: Damage distribution for Case 1 using Objective Function II for (**a**) noise both in frequency and mode shapes and (**b**) noise in mode shapes only. although it is a good detector of the relative damage between the elements, it may also result in large numerical errors at elements where damage is not present. Lastly, Objective Function III related to strain energy also amplifies the errors as the expanded noisy mode shape gets multiplied with the stiffness matrix to calculate the strain energy. This discussion could be further associated with the objective function values obtained for each case. For example, the objective function values obtained for 10% noise in frequencies and mode shapes for all the objective functions are discussed here for comparison. The objective function values were 1 \(\times\) 10\({}^{-5}\), 1.03 and 0.004 for Objective Functions I, II and III, respectively, for 10% noise in frequencies and mode shapes. It can be seen that the minimum value has been obtained for Objective Function I and the maximum value for Objective Function II. This is also consistent with the damage indices obtained in Figures 2-4. As a result, the sparseness of the results with respect to different elements is more for Objective Functions II and III as compared to Objective Function I.