# Damage Diagnosis for Offshore Wind Turbine Foundations Based on the Fractal Dimension
## 3 Damage Diagnose Strategy
### Data Collection and Manipulation  
A total of 100 experimental tests have been conducted that include the four amplitudes that represent the different speed regions. More precisely:  
(i) 10 tests with the original healthy bar for each amplitude, i.e., 40 tests;  
(ii) 5 tests with the replica bar for each amplitude, i.e., 20 tests;  
(iii) 5 tests with the 5 mm crack bar for each amplitude, i.e., 20 tests; and  
(iv) 5 tests with the unlocked bolt for each amplitude, i.e., 20 tests.  
For each experimental test, the acceleration has been measured through 24 sensors during 59.51636719 s and with a sampling frequency of 275.28 Hz, which leads to 16,384 time instants and a time step of about \(\Delta=0.0036328125\) s.  
The raw data of the \(k\)-th experimental test, \(k=1,\ldots,100\), can be arranged as the matrix \(\mathbf{X}^{(k)}\) in Equation (1). Each of the 24 columns of matrix \(\mathbf{X}^{(k)}\) contain the 16,384 measures of each sensor:\[\mathbf{X}^{(k)}=\left(\begin{array}{cccccc}\text{sensor \#1}&\text{sensor \#2}&\text{sensor \#3}&\cdots&\text{sensor \#24}\\ x_{1,1}^{(k)}&x_{1,2}^{(k)}&x_{1,3}^{(k)}&\cdots&x_{1,24}^{(k)}\\ x_{257,1}^{(k)}&x_{258,1}^{(k)}&\cdots&x_{512,1}^{(k)}&\cdots&x_{257,24}^{(k)} &\cdots&x_{512,24}^{(k)}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ x_{16129,1}^{(k)}&x_{16130,1}^{(k)}&\cdots&x_{16384,1}^{(k)}&\cdots&x_{16130,24 }^{(k)}&\cdots&x_{16384,24}^{(k)}\end{array}\right)\in\mathcal{M}_{16384\times 2 4}(\mathbb{R}) \tag{1}\]  
Each column of the matrix \(\mathbf{X}^{(k)}\) in Equation (1) is reshaped into a 64-by-256 matrix to build a new matrix \(\mathbf{Y}^{(k)}\in\mathcal{M}_{64\times(256\cdot 24)}(\mathbb{R})\) in Equation (2):  
\[\mathbf{Y}^{(k)}=\left(\begin{array}{cccccc}\text{sensor \#1}&\cdots&\text{sensor \#24}\\ x_{1,1}^{(k)}&x_{2,1}^{(k)}&\cdots&x_{256,1}^{(k)}&\cdots&x_{1,24}^{(k)}&x_{2,24}^{(k)}&\cdots&x_{256,24}^{(k)}\\ x_{257,1}^{(k)}&x_{258,1}^{(k)}&\cdots&x_{512,1}^{(k)}&\cdots&x_{257,24}^{(k)}&x _{258,24}^{(k)}&\cdots&x_{512,24}^{(k)}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\ x_{16129,1}^{(k)}&x_{16130,1}^{(k)}&\cdots&x_{16384,1}^{(k)}&\cdots&x_{16129,2 4}^{(k)}&x_{16130,24}^{(k)}&\cdots&x_{16384,24}^{(k)}\end{array}\right) \tag{2}\]  
Two are the main reasons for reshaping the matrix \(\mathbf{Y}^{(k)}\) in Equation (2):  
(i) on the one hand, for a single experimental test, we create 64 rows. Each one of these rows is what we call a _sample_;  
(ii) on the other hand, each sample will contain time-history measures of the whole set of sensors.  
We will see in Section 4 that when we want to diagnose whether a wind turbine is healthy or not, we just need to measure these 24 sensors during 256 time instants, that is, during 256\(\Delta\approx 0.93\) s.  
To define the matrix that contains all the data, the matrices \(\mathbf{Y}^{(k)}\), \(k=1,\ldots,100\), from each experiment, are stacked to define  
\[\mathbf{Y}=\left(\begin{array}{c}\mathbf{Y}^{(1)}\\ \mathbf{Y}^{(2)}\\ \vdots\\ \mathbf{Y}^{(100)}\end{array}\right)\in\mathcal{M}_{(64\cdot 100)\times(256\cdot 2 4)}(\mathbb{R})=\mathcal{M}_{6400\times 6144}(\mathbb{R}). \tag{3}\]