## 3 Simulation Signal Analysis
### Multi-Fault Separation of Simulation Signal Analysis  
Rolling-bearings and gears are always the critical components for rotating machinery. If they are under running and heavy load for a long time, they are prone to fatigue damage. Commonly, the basic fault model of a gear is described as [43]  
\[s_{m}(t)=\sum_{k=1}^{K}A_{k}\cos(2\pi kf_{m}t+\phi_{k}) \tag{11}\]  
where \(f_{m}\) is gear-mesh frequency, \(k\) is the order of harmonic components, \(A_{k}\) and \(\phi_{k}\) correspond to the amplitude and phase of the \(k\)-th harmonic respectively. There are many types of the simulation signal models for bearing faults, with the most typical one having been proposed by [PERSON] [44]. Thus, the simplified model of the rolling-bearing inner race fault is expressed as follows:  
\[s_{b}(t)=\sum_{i=0}^{M}[A_{0}\cdot\cos(2\pi f_{r}t+\phi_{A})+C_{A}]\cdot e^{-B (t-iT-\tau_{i})}\cdot\cos[2\pi f_{n}(t-iT-\tau_{i})+\phi_{w}] \tag{12}\]  
where \(A_{0}\) is the amplitude of resonance; \(f_{r}\) is the rotational frequency; \(\phi_{A}\), \(\phi_{w}\), \(C_{A}\) are selected as arbitrary constants; \(B\) is the attenuation coefficient; \(T\) is defined as the average time between two impacts with \(T=1/f_{i}\); \(f_{i}\) is the inner race fault characteristic frequency; \(\tau_{i}\) is regarded as the time lag from its mean period due to the presence of slip and \(f_{n}\) is the resonance frequencies of the bearing system. In order to validate the proposed method without loss of generality, the following gear fault simulation signal based on Equation (11) is studied:  
\[s_{1}=\cos(2\pi f_{1}t+25)+0.55\cos(4\pi f_{1}t+60) \tag{13}\]  
where the mesh frequency is set as \(f_{1}=150\) Hz. Meanwhile, the inner race fault model \(s_{2}\) is examined according to Equation (12), with the parameter selection being listed in Table 2. Furthermore, the noise components \(s_{3}\) cannot be neglected and they can be defined as additive white Gaussian noise, whose variance is 0.5 and average is 0. In order to randomly mix the three simulated original signals, a random 3 \(\times\) 3 matrix (\(A\) is optionally employed by the computer as  
\[A=\left( \begin{array}{llll}-0.9371&0.3212&0.4948\\ 0.8352&0.0151&-1.2158\\ 1.0945&-1.1645&1.5908\end{array} \right) \tag{14}\]  
According to the instantaneous mixed signal model \(x(t)=As(t)\), where \(x=(x_{1},x_{2},x_{3})\) and \(s=(s_{1},s_{2},s_{3})\), three mixed signals can be obtained. We assumed that only a single-channel composite signal is measured, due to the limitation of monitoring conditions in the process of blind source separation of mechanical failure. Following this, the time-domain diagram of the original simulation signal \(s_{1}\), \(s_{2}\) and the analyzed single-channel composite signal \(x_{1}(t)\) are shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\mathbf{A_{0}}\) & \(\mathbf{f_{r}\) (Hz)} & \(\mathbf{f_{i}\) (Hz)} & \(\mathbf{\tau_{i}}\) & \(f_{n}\)(Hz) & \(C_{A}\) & \(\mathbf{\phi_{A}}\) & \(\mathbf{\phi_{w}}\) & \(B\) \\ \hline
0.003 & 29 & 156 & 0.01 & 2000 & 1 & 0 & 0 & 800 \\ \hline \end{tabular}
\end{table} 
Table 2: The parameter selection for inner race fault simulation signal \(s_{2}\). Figure 5 presents the IMF components obtained by RPSEMD. The IMFs provided by EEMD for the mixed fault simulation signal is plotted in Figure 6. When comparing Figures 5 and 6, the RPSEMD result has better correspondence with the ideal IMFs. For instance, the IMF2 has a greater similarity with the rolling-bearing fault signal. Moreover, IMF5 and IMF4 correspond closely to the gear fault signal. However, the IMFs provided by EEMD have less relevance to the original source signal. From the graph, it can be seen that the RPSEMD has obvious advantages in the signal decomposition for the simulation signal, which can be used to identify the source signal.