## 3 Generating Synthetic Bearing Fault Data via cycleGAN
### Simulating Bearing Vibration Signal  
The vibrational behavior of rolling bearings with localized damages was simulated using the analytical model originally proposed by [PERSON] and [PERSON] [74; 75; 76]. The model has gained significant recognition in scholarly works, despite being among the initial attempts to simulate the behavior of faulty bearings. According to recent research [77], incorporating more sophisticated modeling techniques such as three-dimensional finite element analysis and considering contact mechanics phenomena offers supplementary insights into the interactions among the constituents of the bearing system. Nevertheless, [PERSON] et al. [77] emphasize that that [PERSON] and [PERSON]'s model continues to effectively capture the overall behavior of the system.  
Also, the simple model of [PERSON] and [PERSON] was utilized in this study because it leaves out some details about the mechanical system being examined. For instance, the model does not consider the inertial interactions that occur between the rolling elements, lubricant, and races, nor does it account for the elasto-hydrodynamic behavior of the lubricant. In the context of this research, these limitations can be viewed as advantageous as they enable the generative AI model based on the cycleGAN architecture to assess its capacity to integrate all the absent information into the simulated model, particularly in the scenario where the complexity of the simulated model is significantly reduced. Consequently, if the generative model can generate synthetic data using minimal initial information, it is reasonable to assume that it can effectively evolve to more intricate simulation models with a higher level of detail.  
The model is shown in Equations (5)-(8), whereby the impulse response \(h(t)\) is derived from decay parameter \(\beta=500\) Hz and structural resonance \(f_{struct}=1700\) Hz. Function \(d(t)\) represents the Dirac comb function. Each pulse of the Dirac comb is separated by a distance equal to the reciprocal of defect characteristic frequency \(f_{defect}\). According to manufacturer SKF(r) the IR characteristic frequency related to the bearing under analysis is \(f_{defect}=10.824\cdot f_{r}\), where \(f_{r}\) represents the rotation frequency of the machine. The load operating on the rolling element at the angular coordinate \(\psi\), denoted as \(Q_{\psi}\), is determined using the well-known formulation by [PERSON] [82]. The load distribution factor is represented by \(\epsilon\). The convolution operator is denoted by symbol \(*\), and the characteristic frequency of the asynchronous motor, \(f_{motor}\), is defined as \(f_{motor}=6f_{stator}\), where \(f_{stator}\) represents the frequency of the stator power supply. Parameters \(\beta\), \(f_{struct}\), \(A_{1}\) and \(A_{2}\) were selected based on first attempt values. It was therefore intended for the generative algorithm to adjust the influence of these parameters by acquiring knowledge on how to modify the simulated data to align them with experimental observations.  
\[h(t)=e^{-\beta t}\sin(2\pi f_{struct}t), \tag{5}\]  
\[d(t)=\sum_{k}\delta(t-kT),\ \ T=1/f_{defect}, \tag{6}\]  
\begin{table}
\begin{tabular}{c c} \hline Radial load (kN) & 124.8 \\ Speed (rpm) & 607, 877, 997 \\ Duration (s) & 30 \\ Sampling frequency \(f_{s}\) (Hz) & 20,480 \\ Overlap & 0.7 \\ Chunk duration (s) & 0.8 \\ Chunks per signal & 250 \\ \hline \end{tabular}
\end{table}
Table 1: Test conditions and signal extraction for the bearing SKF 22240 CCK/W33.  
\[Q_{\psi}=Q_{max}\left[1-\frac{1-\cos(\psi_{i})}{2\epsilon}\right]^{\frac{10}{2}},- \psi_{m}<\psi<\psi_{m}, \tag{7}\]  
\[s(t)=\left[Q_{\psi}d(t)\right]*h(t)+A_{1}\text{sin}(2\pi f_{t}t)+A_{2}\text{sin }(2\pi f_{motor}t). \tag{8}\]