# Frequency Contour-Strip Method for Characterization of Damage in Structures under Noisy Conditions
## 4 Proposed Frequency Contour-Strip Method with Superiority in Characterizing Damage
### \(\varepsilon\)-Noise Cube  
The frequency contour-line method lacks a mechanism of reflecting the effect of random noise involved in measured eigen-frequencies on crack identification, giving rise to a specific estimate of possible crack scenarios. To investigate the effect of random noise on crack identification, a \(\varepsilon\)-noise cube is elaborated to depict the first three measured eigen-frequencies, shown as Figure 9. The \(\varepsilon\)-noise cube is defined by  
\[\omega_{1}^{*}=\omega_{1}(1+i\cdot\varepsilon),\ \omega_{2}^{*}=\omega_{2}(1+j \cdot\varepsilon),\ \omega_{3}^{*}=\omega_{3}(1+j\cdot\varepsilon) \tag{9}\]  
where \(\omega_{l}\), \(l\) = 1-3, are the exact first three eigen-frequencies, and \(\omega_{m}^{*}\), \(m\) = 1-3, are boundary values of measured eigen-frequencies under the random noise level \(\varepsilon\). According to the \(\varepsilon\)-noise cube, there could be eight directional deviations of \(\omega_{m}^{*}\) from \(\omega_{l}\), specified by the vertices of the cube. Consequently, eight groups of \(\omega_{m}^{*}\), \(m\) = 1-3, are required in light of the \(\varepsilon\)-noise cube to bound the possible variations of measured eigen-frequencies for each crack case. Without loss of generality, a \(\varepsilon\)-noise hypercube can be created when the first n measured eigen-frequencies are employed to identify a crack.  
\begin{table}
\begin{tabular}{c c l} \hline \hline
1 & \% & Generation of mesh \\
2 & & [xx, yy, \(\omega\omega\)] = _meshgrid_ (crl\_x, crd\_y, feq\_w); \\
3 & \% & Extract a series of values Val of the three-dimensional array Var into faces, verts and colors, where the rightmost xx represents the array size of colors \\
4 & & [faces, verts, colors] = _isosurface_(xx, yy, \(\omega\omega\), Var, Val, xx); \\
5 & \% & In the patch, the extracted faces, verts, colors are assigned to the three properties in the patch; \\
6 & & _patch_ (’Vertices’, verts, Faces’, faces, ’FaceVertexCData’, colors, ’FaceColor’, ’interp’, ’edgecolor’, ’interp’); \\
7 & \% & Finally, add the view, axis and colorbar (jet) to draw the three-dimensional map. \\
8 & & _view_ (30,15); \\
9 & & _axis vis3d_; \\
10 & & _colormap hsv_ \\ \hline \hline \end{tabular}
\end{table}
Table 1: Procedure of generating eigen-frequency panorama of a cracked beam.  
Considering the randomness of noise in measurements, the measured eigen-frequencies, i.e., \(\left(\omega_{1}^{*},\omega_{2}^{*},\omega_{3}^{*}\right)\), can be expressed in terms of the \(\varepsilon\)-noise cube as  
\[\begin{array}{l}\omega_{1}(1-\varepsilon_{1})\leq\omega_{1}^{*}\leq\omega_{1 }(1+\varepsilon_{1})\\ \omega_{2}(1-\varepsilon_{2})\leq\omega_{2}^{*}\leq\omega_{2}(1+\varepsilon_ {2})\\ \omega_{3}(1-\varepsilon_{3})\leq\omega_{3}^{*}\leq\omega_{3}(1+\varepsilon_ {3})\end{array} \tag{10}\]  
In practical crack identification, \(\omega_{1},\ \omega_{2},\ \omega_{3}\) and \(\varepsilon\) are determined by the mean and the standard variance of a large number of measured eigen-frequencies.  
Figure 7: Eigen-frequency panoramas generated using the function _isosurface._ (**a**) Cantilever beam; (**b**) simply-supported beam.  
Figure 8: Detailed illustration on eigen-frequency panorama generated by the function _isosurface_.