# Bearing Severity Fault Evaluation Using Contour Maps--Case Study
## 2 Method
### Contour Maps Set-Up
#### 2.5.1 Lower Boundary  
[PERSON] [39] considered that the construction of a contour map should begin with the determination of the lowest isoline value, which is necessary to establish a lower bound (or threshold) that allows for the identification of representative amplitudes. As the main purpose of Envelope Analysis is to detect resonant zones modulated in amplitude by periodic impact forces, the representative information in the envelope spectrum will be observed at low frequencies. Figure 8 shows how the amplitudes of a section of the envelope spectrum have a significant amplitude (associated with a fault frequency and its harmonics); as the frequency increases, these amplitudes decrease.  
This makes it necessary to determine a section of frequencies to be analysed, whose amplitudes provide information of interest. In the estimation of the lower limit of the section (\(S_{low}\)), the frequency response of the piezo-ceramic accelerometer used in this study, whose linearity is above 0.4 Hz, was considered. On the other hand, the upper section limit (\(S_{up}\)) was determined by considering the maximum operating rotation frequency (\(F_{r}=8.33\) Hz) and the bearing characteristics, such as the number of rolling elements (\(N_{b}=15\)) and Ball Pass Frequency Inner (\(BPFI=8.8148\cdot F_{r}\)), given that it has the highest decay constant (Table 3):  
\[S_{up}=N_{b}\cdot BPFI. \tag{1}\]  
Thus, the estimated section considers all the possible fault frequencies of the bearing and its harmonics, within its operating range.  
This consideration allows the threshold to adapt to the change in spectrum amplitude, depending on the fault in the record to be evaluated. For this, a previous analysis was implemented, based on the distribution of amplitudes in the spectrum, using a frequency table. We make some definitions, as follows:  
* The sample (\(X\)) is a data set of length N, which is used in the analysis of the distribution of the amplitudes. These data correspond to the amplitudes of the frequencies from \(S_{low}\)-\(S_{up}\).
* The range (\(R_{a}\)) corresponds to the difference between the frequencies of maximum and minimum amplitude.
* The number of intervals (\(N_{int}\)) is defined by the range and the mean value of the amplitudes analysed: \[N_{int}=\frac{R_{a}}{\frac{\sum_{b=1}^{N}X_{b}}{\sum_{b=1}^{N}X_{b}}}.\] (2)  
Figure 8: Determination of the data sample used in the analysis of the distribution.  
* The width between intervals (\(I\)) is described by the division between the range and the number of intervals: \[I=\frac{R_{a}}{N_{int}}.\] (3)  
Once these criteria have been considered, the frequency distribution table is generated, from which the arithmetic mean (\(\overline{x}\)) of the amplitude distribution must be obtained, where \(x_{i}\) is the mean between the extremes of the interval, \(f_{i}\) the number of components that belong to each interval and \(N\) the number of elements in the sample:  
\[\overline{x}=\frac{\sum_{i=1}^{N}x_{i}\cdot f_{i}}{N}. \tag{4}\]  
Subsequently, in the final stage, the amplitudes that exceed the value of the arithmetic mean are identified; thus, establishing a new data set (\(X_{2}\)), whose mean will correspond to the threshold value:  
\[Threshold=\frac{\sum_{i=1}^{N_{2}}X_{2}}{N_{2}}. \tag{5}\]