# Frequency Contour-Strip Method for Characterization of Damage in Structures under Noisy Conditions
## 3 Existing Frequency Contour-Line Method with Limitations in Portraying Damage
### Frequency Contour-Line Method for Crack Identification  
From the perspective of forward problem, an eigen-frequency panorama comprehensively depicts the effect of all possible crack scenarios on the dynamic property of a beam. For an arbitrary crack at the point (x, y), the associated eigen-frequencies can be specified from the eigen-frequency panorama arising from Equation (8).  
From the perspective of an inverse problem, an eigen-frequency panorama along with experimentally obtained eigen-frequencies provides a strategy of identifying a crack in a beam. Consider a beam of the prior knowledge: bearing an unknown crack and experimentally acquired eigen-frequencies, \(\alpha_{i},\ i=\)1,2, \(\cdots,n\), with the question to identify the crack. One can obtain the eigen-frequency panorama of the beam in terms of Equation (8a), and \(\alpha_{i},\ i=\)1,2, \(\cdots,n\) are discrete points of the eigen-frequency panorama in the \(\omega\) axis. Thus, \(\alpha_{i}\) can provoke a horizontal plane cutting through the \(\omega_{i}\)-labeled eigen-frequency surface of the eigen-frequency panorama, leading to a frequency contour line. All the points of this frequency contour line designate all possible crack scenarios specified by \(\alpha_{i}\). The frequency contour line is further projected onto the \(x\)-\(y\) plane, giving the frequency projection curve indicating the possible locations and depths of the crack. In general, for \(\alpha_{i},\ i=\)1,2, \(\cdots,n\), all the frequency projection curves can be produced from frequency contour lines. The common intersection points of all the frequency projection curves designate the location and depth of the crack under inspection. Such a crack identification method is termed frequency contour-line method that uses measured eigen-frequencies with eigen-frequency panorama to identify a crack in a beam.  
The procedure of identifying a crack using the frequency contour-line method is illustrated on a beam with a cantilever crack. (i) Eigen-frequency panorama. Figure 3 presents the first three eigen-frequencies-led eigen-frequency panorama, generated by Equation (8). (ii) Experimental eigen-frequencies. Three eigen-frequencies, \(\alpha_{1},\alpha_{2},\alpha_{3}\), are acquired through experimental measurements. (iii) Frequency contour lines. \(\alpha_{1},\alpha_{2},\alpha_{3}\) provoke three frequency contour lines laying on the eigen-frequency panorama, as shown in Figure 3. The extracted frequency contour lines are also presented in Figure 4. (iv) Frequency projection lines. Frequency contour lines in Figure 4 are projected on the \(x\)-\(y\) plane, leading to the frequency projection lines as shown in Figure 5a, in which a common intersection point O can be observed. (iv) Crack parameter. The point O, as zoomed-in in Figure 5b, designates the dimensionless crack location \(x\) and the dimensionless depth \(y\) of the crack under investigation.  
Figure 2: The eigen-frequency panorama comprising the first three eigen-frequencies for a certain cantilever beam bearing a crack.  
Figure 4: Extracted frequency contour lines.  
Figure 3: Eigen-frequency panorama with frequency contour lines. \(\omega_{1},\omega_{2},\omega_{3}\).