Application of Wavelet Analysis for Crack Localization and Quantification in Beams Using Static Deflections

Wavelet and Lipschitz exponent based damage identification method for beams using mode shapes

“…The proposed damage identification method is developed based on the theory presented in References [1,21] for static deflections. In this section, the theory is briefly reviewed and extended to the application with mode shapes.…”

“…The stiffness change at x 0 reduces the bearing capacity of the cross-section, thereby modifying the moment distribution and displacement field along the beam. It can be demonstrated [22,23] that the damage induced variation of the displacement field, ∆u(x) (∆u(x) = u D (x) − u R (x)), is equivalent to the response of the damaged beam subjected to a pair of self-equilibrated bending moments that equals m R (x 0 ), applied at x 0 . This state of the beam is named the Incremental State (State I) and the corresponding response of the structure is denoted by u I (x), where u I (x) is equal to ∆u(x).…”

Wavelet analysis of static deflections for multiple damage identification in beams

“…The stiffness reduction at the damaged regions causes a moment redistribution along the beam. For notch type damage, assuming that there is no internal torsion created by the external loads, the deflection difference of the beam can be treated as the effect of applying a set of self-equivalent bending moments at the damage locations (Ma and Solís, 2018).…”

“…Wavelet analysis is a well-known mathematical tool for detecting singularities in a signal with the advantage of being able to identify the number of irregularities along with their locations simultaneously. Therefore, it has been used for detecting local damages in beams through the displacement data, which is treated as a space signal in the wavelet analysis (Chang and Chen, 2005;Zhu and Law, 2006;Jiang et al, 2012;Solís et al, 2013;Cao et al, 2016;Ma and Solís, 2018). Damages are located at the ridges of the wavelet coefficients in the spacescale domain.…”

“…First, the damage is located by using a damage locating index based on the number of local maximum values of the wavelet coefficient for all the considered scales (Zhu et al, 2019). Then, the damage severity is quantified by using a damage index based on the relationship between the wavelet coefficients at the damage positions and the scales (Ma and Solís, 2020). In contrast to the aforementioned approaches, this damage index is independent from the damage location, the load pattern and the boundary conditions.…”