# Frequency Contour-Strip Method for Characterization of Damage in Structures under Noisy Conditions
###### Abstract  
Eigen-frequency, compared with mode shape and damping, is a more practical and reliable dynamic feature to portray structural damage. The frequency contour-line method relying on this feature is a representative method to identify damage in beam-type structures. Although this method has been increasingly applied in the area of damage identification, it has two significant deficiencies: inefficiency in establishing the eigen-frequency panorama; and incompetence to identify cracks in noisy conditions, considerably impairing the effectiveness in identifying structural damage. To overcome these deficiencies, a novel method, termed the frequency contour-strip method, is developed for the first time. This method is derived by extending the frequency contour line of 1D to frequency contour strip of 2D. The advantages of the frequency contour-strip method are twofold: (i) it uses the isosurface function to instantly produce the eigen-frequency panorama with a computational efficiency several orders of magnitude higher than that of the frequency contour-line method; and (ii) it can accommodate the effect of random noise on damage identification, thereby thoroughly overcoming the deficiencies of the frequency contour-line method. With these merits, the frequency contour-strip method can characterize damage in beam-type structures with more efficiency, greater accuracy, and stronger robustness against noise. The proof of concept of the proposed method is performed on an analytical model of a Timoshenko beam bearing a crack and the effectiveness of the method is experimentally validated via crack identification in a steel beam. e-g:-frequency; crack detection; frequency contour-line method; frequency contour-strip method; noise robustness +
Footnote †: journal: Applied Sciences  
fundamental and reliable feature to portray damage from a practical perspective. Various eigen-frequency-based methods can be found in applications to damage identification in the literature [22; 23]. In particular, the frequency contour-line method has gained more attention in the field of structural damage identification in recent years [24; 25; 26; 27; 28]. Identifying damage based on the frequency contour-line method involves five components: eigen-frequency panorama, experimental eigen-frequencies, frequency contour lines, frequency projection lines, and intersection point indicating damage. The representative methods of using frequency contour-line method to detect the damage in a structure are briefed as follows. [PERSON] et al. [29] proposed a contour-line method on the identification of the damage location and depth in a beam, using only the value of the changes in the measured natural frequencies and the vectors of the curvature mode shapes of the intact structure. The effectiveness of the method was correctly verified in numerical and experimental cases. [PERSON] et al. [30] presented two techniques to detect damage in a beam with functionally graded materials, one using frequency contour-line method and the other using a genetic algorithm optimization technique. The crack detection techniques and the numerical analysis showed good agreement. [PERSON] et al. [31] introduced a contour-line method based on only post-damaged eigen-frequencies to estimate damage parameters. The capability of the method was demonstrated by the numerical and experimental results. [PERSON] et al. [32] utilized a wavelet finite element method to obtain the changes of natural frequency of a structure with multiple cracks and predicted the location and depth using the frequency contour-line method. The accuracy of the proposed method was fully verified. [PERSON] and [PERSON] [33] improved the contour-line method through an algorithm, making it capable to locate the suitable positions of two cracks. Satisfactory location of the cracks was achieved. [PERSON] et al.