# Frequency Contour-Strip Method for Characterization of Damage in Structures under Noisy Conditions
## 4 Proposed Frequency Contour-Strip Method with Superiority in Characterizing Damage
### Instant Establishment of Eigen-Frequency Panorama  
In the frequency contour-line method, the establishment of eigen-frequency panorama needs repetitive manipulations of numerically solving Equation (8) to produce eigen-frequency values point by point, incurring huge cost and computation time. To overcome such a deficiency, the symbolic operation of solving implicit functions is studied to tackle Equation (8), since it is an implicit function of \(\omega\), \(x\), and \(y\). The symbolic operation is implemented in Matlab using the function _isosurface_ that has the advanced mechanism of evaluating and visualizing implicit functions in an instant mode. Herein, the function _isosurface_ is adopted to instantly solve Equation (8) by conquering the limitation of inefficiency to establish the eigen-frequency panorama in the frequency contour-line method. The procedure of using the function _isosurface_ to evaluate and visualize Equation (8) to produce the eigen-frequency panorama is tabulated in Table 1, where crl_x, crd_y, feq_w denote the variable domains of crack location, crack depth and eigen-frequency, respectively.  
Figure 6: Illustration on failure of the frequency contour-line method in identifying a crack under noise conditions.  
The function _isosurface_ is a highly integrated module featuring smooth symbol operation, totally different from the DDE method and other numerical methods used in the contour-line method. With this merit, the function _isosurface_ can produce an eigen-frequency panorama with a computational efficiency several orders of magnitude higher than general numerical methods in the frequency contour-line method.  
A steel cantilevered cracked beam with length 500 mm, thickness 19 mm, and width 12 mm is used to illustrate the use of the function _isosurface_ to produce the eigen-frequency panorama. For the beam, the elastic modulus, shear modulus, Poisson's ratio (\(\
u\)), shear coefficient, and density are 184 GPa, 86 GPa, 0.3, \(10(1+\
u)/(12+11\
u)\), and 7758 kg/m\({}^{3}\), respectively. With these parameters, Equation (8) provides a determined implicit function of \(\omega\), x, and y. The function _isosurface_ is used to solve this implicit function. The resulting eigen-frequency panorama involving the first 15 eigen-frequencies is shown in Figure 7a. To demonstrate the generality, the eigen-frequency panorama of a simply-supported beam with the same materials and the geometrical parameters is produced using the function _isosurface_, as shown in Figure 7b. The total computational duration using an in-home laptop is about 12 s for the above two cases, whereas the DDE method in the frequency contour-line method needs two and half hours to generate the eigen-frequency panoramas, with some numerical computation problems when the computation order of eigen-frequencies is higher than 12. To present the details of the eigen-frequency panorama, Figure 8 presents the first six order extracted eigen-frequency panoramas where the effect of a crack on each eigen-frequency is clear. As a result, the function _isosurface_ provides a means of instantly establishing the eigen-frequency panorama, overcoming the inefficiency in obtaining the eigen-frequency panorama through finite element modeling or experiments.