# Rolling Bearing Incipient Fault Detection Based on a Multi-Resolution Singular Value Decomposition
## 4 Simulation Studies  
In order to verify the effectiveness of the MRSVD in the aspects of rolling bearing incipient fault detection, the mathematical model shown in Formula (8) was used to simulate the early fault vibration signal of a rolling bearing  
\[ \begin{array}{ll}x(t)&=s(t)+n(t)+h(t)\\ &=\sum\limits_{m=-M}^{M}A_{m}e^{-\beta(t-mT_{p}-\sum\limits_{i=-M}^{m}\tau_{i} )}\sin\left(\omega_{r}(t-mT_{p}-\sum\limits_{i=-M}^{M}\tau_{i})\right)u(t-mT_{p }-\sum\limits_{i=-M}^{m}\tau_{i})\\ &+n(t)+h(t)\end{array}  \tag{8}\]  
where \(n(t)\) is the noise signal added by the MATLAB function awgn.m,the signal-to-noise ratio (SNR) is set to \(-6\), and \(h(t)\) is the harmonic interference signal that contains \(N\) harmonic signals, which can be expressed as \(h(t)=\sum_{j=1}^{N}B_{j}\sin(2\pi f_{j}t)\), i.e., using the amplitude and the frequency of the \(j\)th harmonic signal, which are \(B_{j}\) and \(f_{j}\), respectively. The periodic shock signal caused by the fault contains 2M+1 impulses. In Formula (8), \(A_{m}\) is the amplitude of the \(m\)th shock, \(\beta\) is the decay coefficient caused by the structural damping, \(T_{p}\) is the time period corresponding to the fault characteristic frequency, i.e., the fault feature frequency, \(f_{C}=1/T_{p}\). \(u(t)\) is the unit impulse function, \(\omega_{r}\) is the excited frequency, and \(\tau_{i}\) is the random variable whose mean value is zero and whose variance is between (0.01\(T_{p}\), 0.02\(T_{p}\)), which is the effect of the random slippage of the rollers. The parameters of the simulated signal are displayed in Table 1. The noise was directly added to the simulated periodic impulse signal before the harmonic interference signals were added. Therefore, the signal-to-noise ratio (SNR) in Table 1 is relative to the simulated periodic signal. The simulation signal was sampled at 20,000 samples \(s^{-1}\), and the number of sampling points was set to 20,000. After sampling, the simulated signal, along with the noise, the harmonic interference signals, and their Hilbert demodulation spectrums, are plotted in Figure 3. Due to the influences of the noise and the harmonic interference, it is impossible to judge the fault of rolling bearing from Figure 3a. In Figure 3b, the frequency differences of the harmonic components contained in the harmonic signal are mistakenly identified as the demodulation frequencies. However, the actual fault characteristic frequencies are not highlighted in Figure 3b. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \(A_{m}\) & \(T_{P}\) & \(M\) & \(\beta\) & \(\omega_{r}\) & SNR & \(B_{1}\) & \(f_{1}\) & \(B_{2}\) & \(f_{2}\) & \(B_{3}\) & \(F_{3}\) \\ \hline \(1\) & 0.008s & 250 & \begin{tabular}{c} 1500 \\ Hz \\ \end{tabular} & \begin{tabular}{c} 2048 \\ Hz \\ \end{tabular} & \begin{tabular}{c} -6 \\ \end{tabular} & \begin{tabular}{c} 1.35 \\ \end{tabular} & \begin{tabular}{c} 60 Hz \\ \end{tabular} & \begin{tabular}{c} 1.4 \\ \end{tabular} & \begin{tabular}{c} 230 \\ Hz \\ \end{tabular} & \
\begin{tabular}{c} 1.5 \\ \end{tabular} &
\begin{tabular}{c} 800 Hz \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the simulated signal. Signal-to-noise ratio (SNR). Figure 3: Simulated signal (with noise and interference) and its Hilbert demodulation spectrum. (**a**) The waveform of the simulated signal in the time-domain. (**b**) The Hilbert demodulation spectrum of the simulated signal. The 2-row Hankel matrixes were constructed for the simulation signal and their periodic impact, noise, and resonance components are denoted as H, H\({}_{\text{s}}\), H\({}_{\text{n}}\), and H\({}_{\text{h}}\), respectively. The result singular values of the SVD of the matrix H, H\({}_{\text{s}}\), H\({}_{\text{n}}\), and H\({}_{\text{h}}\) are displayed in Table 2. The numerical results shown in Table 2 are consistent with the singular value analysis of the three kinds of signals (the periodic shock signal, the noise signal, and the harmonic signal) in Section 3.1 of this manuscript.