# Frequency Contour-Strip Method for Characterization of Damage in Structures under Noisy Conditions
###### Abstract
[34] presented a multiresolution finite element method capable of modal analysis for singularity problems and then plotted the first three calculated natural frequencies contour line to predict the damage, with the cracks quantitative identification method experimentally validated. [PERSON] et al. [35] conducted the performance assessment of the frequencies contour-line method in characterizing cracks in noisy conditions, a guideline for rational use of natural frequencies to identify cracks in actual beam-type structures provided. [PERSON] and [PERSON] [36] proposed a methodology to detect cracks in conjunction with improved cracked element for singularity problems. This methodology efficiently identified crack location and size based on the contour-line method. [PERSON] et al. [37] considered the vibration of cracked beams using both Timoshenko and Euler formulations. The frequency contour-line method identified the crack size and location properly, using the lower order frequencies. [PERSON] and [PERSON] [38] developed a p-version of finite element method to determine the location and size of an open edge crack in functionally graded material (FGM) beams and used the contour-line method to identify the crack parameters. The performance of the method was demonstrated efficiently by numerical experiments. [PERSON] et al. [39] proposed a contour-line method for damage identification from measured natural frequencies of a contiguously damaged reinforced concrete axial rod and beam, with the pros and cons of this method proved by experiments. [PERSON] et al. [40] applied the contour-line method in a static rotor with an open crack, based on the assumption that the cracked rotor was an Euler-Bernoulli beam with circular cross-section. Two experimental examples demonstrated the availability of the method. [PERSON] et al. [41] presented a contour-line method for identification in a breathing cracked beam, combined with the natural frequency equation and stiffness model. The feasibility of this method was verified. The above studies demonstrate that the frequency contour-line method functions as an active research branch in structural damage identification. Despite the prevailing of the frequency contour-line method, two remarkable deficiencies in this method can be observed in damage detection: (i) inefficiency in establishing the eigen-frequency panorama and (ii) incompetence to identify cracks in noisy conditions. These shortcomings fairly impair the efficiency, accuracy, and robustness of the method in identifying damage. To address these deficiencies, a more sophisticated method, termed frequency contour-strip method, is developed in this study. The frequency contour-stripmethod is developed based on promoting the contour-line method by extending the line to a strip, and the strip endows the new method with more efficiency, greater accuracy, and stronger robustness to noise. The proof of concept of the method is performed on an analytical model of a cracked Timoshenko beam and the method is experimentally validated on crack identification in a steel beam. The remainder of this paper is organized as follows. Section 2 presents the analytical model for transverse vibration of Timoshenko beam bearing a crack. Section 3 introduces the frequency contour-line method with its limitations in crack identification clarified. Section 4 develops the frequency contour-strip method with emphasis on its advantages over the frequency contour-line method. Section 5 validates the method via an experiment on detecting cracks in a beam. Conclusions are given in Section 6.