# Bearing Fault Diagnostics Based on the Square of the Amplitude Gains Method
###### Abstract  
The article presents an adaptation of a parametric diagnostic method based on the square of the amplitude gains model, which was tested in experimental studies on bearing damage detection (outer race, inner race, bearing balls damage). The described method is based on the shaft displacement signal analysis, which is affected by vibrations coming from the bearings. The diagnostic model's parameters are determined by processing the signal from the time domain to the frequency domain in a few steps. Firstly, the recorded signal is divided into two observation periods, next the analytical autocorrelation functions are determined and approximated by a polynomial. Then, the diagnostic thresholds are adopted, and the model parameters are converted into damage maps that are easy to interpret and assess the technical condition of the bearings. The presented method shows the technical condition of bearings in a qualitative way. Depending on the received color damage maps, it is possible to determine their level of wear. _Green_ and _blue_ indicate poor wear or no damage, _red_ indicates increased wear, and _black_ clearly indicates a damaged bearing. b 2023 13 2160 2023 13  
require techniques of advanced signal processing: fast Fourier transform (FFT) [40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50], cepstrum analysis (CA) [51; 52; 53], short time Fourier transform (STFT) [54; 55], Wigner-Ville Distribution (WVD) [54; 55; 56], the envelope analysis (EA) [57; 58], and wavelet transform (WT) [59; 60; 61; 62], and various advanced models based on artificial intelligence [63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80]. Vibration data analysis mainly includes the time domain, frequency domain, time-frequency domain, and other analysis methods, but the vast majority are based on time-frequency analysis. The authors also developed time-frequency methods. In their work, they proved the effectiveness of parametric methods: in the form of a square signal amplitude amplification and an original model in the form of a difference in phase shifts of signals for diagnosing shafts and compressors of aircraft engine turbines. A mathematical model of a square of amplitude gains for diagnostic purposes was first presented in 2004 [81]. [PERSON] and [PERSON] presented its application for detecting damage to an aircraft engine blade in stationary conditions (removal of the blade from the engine required). The vibration signal was generated by modal hammer impact. In 2007, [PERSON] and [PERSON] [82] presented a complete study with an analysis of the impact of this model's parameters on the damage type and location. The next stage of the method development took place at the Air Force Institute of Technology (ITWL) in Warsaw (Poland) in 2009 [83; 84]. The article presents the mathematical basis for adapting the method to diagnostics during a rotating machine operation. However, this approach has not been experimentally verified. In 2010, [PERSON] and [PERSON] [85] determined the parameters of the diagnostic model from the recorded signal from the inductive sensor. In 2012, [PERSON] [86] in his doctoral thesis, presented extensive research on the compressor blade of the S0-3 turbine engine during its operation. An analysis of the impact of changes in the environment represented by the rotational speed on the changes of model parameters was carried out, and color damage maps were used to analyze the technical condition. For verification, a new diagnostic model in the form of a difference in phase shifts of signals was also developed and tested. Endoscopic measurements additionally verified all results. In 2018, the authors [87], using a parametric diagnostic model of a difference in phase shifts of signals, presented an analysis of the technical condition of the entire rim of SO-3 compressor blades. In 2020 [88], they presented another model modification, which allowed both to examine the technical condition of the blade and its stationary condition during operation.