# Damage Identification of Stay Cables Based on a Small Amount of Deflection Monitoring Data
## 2 Theory Related to the Reduction in Measurement Points
### Location Identification  
Using the singularity detection feature of the wavelet transform, the first derivative of the girder deflection difference before and after cable damage is transformed by one-dimensional continuous wavelet transform [21].  
\[W_{f}(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}\Delta w^{\prime}{}_{ij} (x_{k})\overline{\psi}\bigg{(}\frac{x_{k}-b}{a}\bigg{)}dx \tag{10}\]  
where \(\Delta w^{\prime}{}_{ij}(x)\) is the first derivative of deflection difference of the girder before and after the cable \(i\) and \(j\) damage (it can be double-cable damage or single-cable damage).  
The Mexican hat Mexh wavelet formed by the second derivative of the Gauss function is chosen as follows:  
\[\psi(t)=\frac{-1}{\sqrt{2\pi}}\Big{(}1-t^{2}\Big{)}e^{\frac{-t^{2}}{2}} \tag{11}\]  
The function satisfies the requirements of wavelet permittivity and compact support, and its vanishing moment is 2.  
Taking working condition 4 listed in Table 1 as an example, the damage cable location identification method in the case of double-cable damage is studied. The cubic spline interpolation function (spline) is used to properly encrypt the original data of girder deflection difference before and after the cable M16 damage of 15% and M7 damage of 40% (working condition 4); then, the first derivative of the encrypted girder deflection difference data is obtained according to Equation (9). The scale a = 11 is selected, and the first derivative of deflection difference is transformed by one-dimensional continuous  
Figure 9: Comparison of beam deflection difference distribution curves before and after the reduction in the measurement points in the case of double-cable damage (different damage conditions).  
wavelet according to Equation (10). Figure 9 shows the distribution curves of the first derivative of the deflection difference of the beam before and after the double-cable damage (M16 damage 15%, M7 damage 40%).  
It can be seen from Figure 10 that the distribution curve of the wavelet coefficient after wavelet transformation changes monotonically from negative to positive twice through the 0 point. The two 0 points are located at the anchorage points of cable M16 and M7, which are the same as the preset position of the damage cable.  
According to the above method, the damage cable position is identified for other preset conditions of single-cable damage and double-cable damage (results are shown in Figures 11 and 12).  
As can be seen from Figures 11 and 12, the distribution curve of the wavelet coefficients changes from negative to positive across the X axis, and the location of the 0 point is the location of the damaged cable; in addition, the identification results are exactly the same as for the preset damaged cable location.  
Figure 11: Wavelet coefficients distribution curves of the first derivative of beam deflection difference before and after cable M16 damage.  
Figure 10: Wavelet coefficient distribution curves of the first derivative of beam deflection difference data encrypted before and after double-cable damage (M16 damage 15%, M7 damage 40%).