PSI - Issue 78

Antonio Sánchez López-Cuervo et al. / Procedia Structural Integrity 78 (2026) 1791–1798

1794

(a)

(c)

(d)

(e)

4

1

4

1

4

1

3

3

3

6

6

2

2

2

5

5

7

7

(b)

10

8

9

11

Nb. Damage scenario Damaged joint

X

Fig. 2: (a) Undamaged joint; (b) damaged joint; (c) damaged scenarios D.S.1 - D.S.4; (d) damaged scenarios D.S.5 - D.S.7; (e) damaged scenarios D.S.8 - D.S.11.

aiming to simultaneously minimize the error between the model parameters and the modal data obtained from both the accelerometers and the SGs. The cost function to be minimised was defined as:

    f i − f i    

    f i − f i     f i

α a aver + a max    a aver

i   

f i      

m  i = 1

CF =

+ a max max

(1)

1 − MAC  ϕ i ,ϕ i  m

β b aver + b max    b aver

m  i = 1

i 

1 − MAC  ϕ i ,ϕ i   

+ b max max

+

where f i and f i are the experimental and numerical frequencies of the i -th mode, respectively, and ϕ i and ϕ i are the corresponding experimental and numerical mode shapes. The weights were set to α = 2 . 5, β = 1 . 0, a aver = 1 . 25, a max = 1 . 0, b aver = 1 . 0, b max = 1 . 0, andMAC  ϕ i ,ϕ i  is the Modal Assurance Criterion, defined as: MAC( ϕ i ,ϕ i ) =    ϕ H i ϕ i    2  ϕ H i ϕ i   ϕ H i ϕ i  (2) where ϕ H i denotes the Hermitian transpose of the mode shape ϕ i . The results of the calibration process are presented in Section 3.2.

3. Results and discussion

3.1. E ff ect of damage on modal properties

This section presents the results regarding the influence of structural damage on the modal parameters of the frame. The initially undamaged condition (D.S.0) and a total of 11 progressive damage scenarios (D.S.1 to D.S.11) are considered, as detailed in Section 2. For each scenario, the first five flexural modes along the structure weak axis (Fig. 3) and their associated natural frequencies were identified. The e ff ect of damage on natural frequencies is shown in Fig. 4. As observed, the loss of sti ff ness at the joints generally results in a reduction of the modal frequencies. This trend is particularly noticeable between D.S.4 and