# Bearing Fault Diagnostics Based on the Square of the Amplitude Gains Method
## 2 Condition Monitoring Based on the Squared Amplitude Gain of Signals
Based on the above condition, a parametric diagnostic model can be determined in the form of the quotient of the power density function (square of the amplitude gains), allowing to eliminate the environment:  
\[A_{T12T01}^{2}(\omega)=\frac{\frac{S_{yy}^{T12}}{S_{yy}^{T01}}}{\frac{S_{yy}^{ T01}}{S_{yy}^{T01}}}\frac{S_{yy}^{T12}}{S_{yy}^{T01}}\frac{S_{yy}^{T12}}{S_{yy}^{T01}} \frac{A_{0}^{*}+A_{1}^{*}s+A_{2}^{*}s^{2}+\ldots+A_{n}^{*}s^{n}}{B_{0}^{*}+B_{ 1}^{*}s+B_{2}^{*}s^{2}+\ldots+B_{n}^{*}s^{n}} \tag{4}\]  
where: \(s\)--complex variable, \(s=j\omega\); \(A_{i}\)*, \(i=0\), 1, \(\ldots\), \(n\)--numerator estimates parameters; \(B_{i}\)*, \(i=0\), 1, \(\ldots\), \(n\)--denominator estimates parameters; \(n\)--polynomial order. Despite the elimination of the environment, the parameters of the \(A^{2}{}_{T12T01}\) model are directly related to the change in the technical condition of the rotor system supported on bearings. Therefore, a characteristic feature of the \(A^{2}{}_{T12T01}\) model is that it does not require the measurement of environmental signals. However, it is indirectly considered by conducting diagnostic tests (two observation periods, determining the diagnostic model as a quotient of diagnostic models combining diagnostic signals and any environment with technical condition parameters). Parameters of the numerator (\(A_{i}\)) and denominator (\(B_{i}\)) of the model are determined for each cycle of the rotating shaft, thus creating a matrix of parameters, describing the technical condition of the bearings during the operation-a damage map. The mean value \(\mu\) and the standard deviation \(\sigma\) are determined for the corresponding model parameters from each cycle (e.g., \(A_{0}\) for each cycle--the first column in the matrix of parameters). On their basis, statistical diagnostic thresholds of the forms \(\mu\pm\sigma\), \(\mu\pm 2\sigma\), \(\mu\pm 3\sigma\) are determined. Then, the determined values of Ai and Bi parameters are compared to the determined diagnostic thresholds and changed to the appropriate color (Table 1), creating the so-called damage maps of the technical condition of the object:  
* _green_ color if the parameter value did not exceed \(\mu\pm\sigma\),
* _blue_ color if the parameter value exceeded \(\mu\pm\sigma\),
* _red_ color if the parameter value exceeded \(\mu\pm 2\sigma\),
* _black_ color if the parameter value exceeded \(\mu\pm 3\sigma\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**The Predominant Color in** & **Technical Condition** & **Stationary Condition** \\
**the Damage Map** & & \\ \hline _Black_ and _red_ & “Serious failure” & “Strong” changes of bearing \\ _Red_ & “Excessive wear” & “Weak” changes of bearing \\ _Green_ and _blue_ & “Slight or no wear” & “Slight” changes of bearing \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Legend for the damage maps. This way, a damage map of the bearing is obtained, showing its technical condition. For example, if there are many _black_ colors, the bearing is damaged, _red_ means increased bearing wear, and _green_ and _blue_ colors mean little or no wear. This approach shows an unambiguous picture of the bearing damage assessment.