# Bearing Fault Diagnostics Based on the Square of the Amplitude Gains Method
## 2 Condition Monitoring Based on the Squared Amplitude Gain of Signals  
The mathematical foundations of the method are described in detail in [89]. To use the proposed diagnostic model to monitor the condition of the bearings, signals of rotor displacements near the supporting bearings during machine operation are measured and sampled. It is assumed that the rotor is undamaged, its state remains unchanged, and various bearing damage variants are introduced simultaneously. In this way, researchers determine the condition of the bearings, using the method described in the article based on the rotor displacement signals. Finally, as a result of using the method, damage maps characterizing the technical condition of the bearings are obtained. In the first step of analyzing the signals obtained from the experimental studies, the time interval \(T_{02}\), hereinafter referred to as a cycle, is determined for every single revolution. This interval corresponds to a full revolution of the shaft reduced on the outer parts of the intervals by several samples so that successive cycles do not overlap. Then the interval \(T_{02}\) is divided into two separate time intervals \(T_{01}\) and \(T_{12}\) (sample sets). Point \(T_{1}\) is common for both ranges and marks the moment when the shaft surface is farthest from the sensor face (Figure 1). Each cycle's number of signal samples must be identical and selected to ensure the proper statistical evaluation of the measured operational signals. For the tested rotational speed, the intervals \(T_{01}\) and \(T_{12}\) contain the same number of signal samples, and these intervals do not overlap. Frequency leakage was reduced by scaling each interval by a Hanning window. For the displacement signal \(y(t)\) in the assumed time intervals \(T_{01}\) and \(T_{12}\), estimates of the autocorrelation functions \(R_{yy}\)[701] and \(R_{yy}\)[712], are determined, which are then approximated by analytical expressions (polynomials), ensuring a fit above 0.997. The correlation function takes the form:  
\[R_{yy}(\tau)=a_{n}\tau^{n}+\ldots+a_{4}\tau^{4}+a_{3}\tau^{3}+a_{2}\tau^{2}+a _{1}\tau+a_{0} \tag{1}\]  
where: \(a_{0}\), \(a_{1}\),... \(a_{n}\)--coefficients of the polynomials, \(n=0\), \(1\), \(2\), \(3\),..., \(r\). The order \(n\) of the polynomials should be selected carefully--a too low order will result in inaccurate approximations, a too high order will result in an excessive number of polynomial coefficients and longer calculation times. The described parametric diagnostic model is based on the functions of the spectral density of the power of the registered rotor vibration signal \(y(t)\) in two time intervals (\(T_{01}\), \(T_{12}\)) and the environment \(x(t)\). It is assumed that the rotor work signals \(y(t)\) and the envi  
Figure 1: Dividing the signal into two sub-periods \(T_{01}\) and \(T_{12}\). ronment \(x(t)\) are stochastic and the disturbed time courses are expressed by autocorrelation functions \(R_{xx}(\tau)\) and \(R_{yy}(\tau)\). Based on the analytical forms of the auto-correlation function, the corresponding power auto-spectral density functions \(S_{yy}{}^{T01}(j\omega)\) and \(S_{yy}{}^{T12}(j\omega)\) are determined using the Fourier transform:  
\[S_{yy}^{T01}(j\omega)=F(R_{yy}^{T01}(\tau)) \tag{2}\]  
\[S_{yy}^{T12}(j\omega)=F(R_{yy}^{T12}(\tau)) \tag{3}\]  
In the next step, it is assumed that the power densities of the environment signal \(x(t)\) in time \(T_{01}\), \(T_{12}\)-\(S_{xx}{}^{T01}(j\omega)\), and \(S_{xx}{}^{T12}(j\omega)\) have been determined (in the same way as \(S_{yy}\)). Since the observation times are very close to each other, it can be assumed that the environment has not changed at that time; therefore \(S_{xx}{}^{T01}(j\omega)\cong S_{xx}{}^{T12}(j\omega)\).