## 4 Simulation Study
### Case 1: Outer Race Fault  
Two rotation speeds with linear function (\(40t+10\)) and quadratic function (\(40t^{2}+10\)) are used with the ball-pass-order on the outer race (BPOO) of fault feature cyclic orders \(\alpha_{\text{BPOO}}^{L}=2.5\) and \(\alpha_{\text{BPOO}}^{\mathcal{O}}=5\) respectively are studied in this case. Other parameters are arranged as follows: the sampling frequency \(f_{s}=51,200\) Hz, the sampling length \(N=51,200\) (a signal with one second), the resonance frequency \(f_{\text{resonance}}=4000\) Hz and coefficient of resonance damping attenuation \(\beta=2000\). The intervals between two transient impulses are decreasing with the increasing speed, which can be founded in Figure 3a,c. The Gaussian white noise (SNR = \(-16\) dB) is added in simulated signals, which can be seen that the fault transient impulses are covered by the strong background noise (seen in Figure 3b,d). Then, OFSC is computed to transfer the time-domain signal into the cyclic order-frequency domain. The time costs of the three methods are listed in Table 1, which can be seen that the computation burdens are similar. The OFSCs of the two simulated signals are shown in Figure 4. The fault cyclic orders (fault feature) can be found in the OFSCs of the pure transient impulses clearly. However, the strong background noise almost completely covers the fault feature as shown in Figure 4b,d. Then, the EEOSs of the two noisy OFSCs are tried to enhance the fault feature. Figure 5 shows that the fault cyclic orders are more pronounced compared with that in OFSCs, especially at the condition  
Figure 3: The simulated signals in Case 1. **(a)** the transient impulses and rotation speed with linear function (line in blue: the signal waveform adhering the left \(y\)-axis and line in black: the shaft rotating speed adhering the right \(y\)-axis); **(b)** the noisy signal with linear function; **(c)** the transient impulses and rotation speed with quadratic function (line in blue: the signal waveform adhering the left \(y\)-axis and line in black: the shaft rotating speed adhering the right \(y\)-axis); **(d)** the noisy signal with quadratic function. of \(\alpha_{\text{BPOO}}^{Q}=5\). The first order component at the condition of \(\alpha_{\text{BPOO}}^{L}=2.5\) is obvious. However, its multipliers are still covered by the strong background noise, which needs to be effectively extracted. Hereafter, the three methods are applied to the noisy OFSCs. The denoised results can be seen in Figures 6 and 7. The fault feature can be effectively extracted with the proposed method for its joint sparsity and low-rankness inducing capability, as shown in Figures 6a and 7a. The proposed method can effectively extract the outer race fault feature in OFSCs under variable speed conditions. As described in Section 3.1, the fault feature in OFSC is modeled as joint sparsity and low-rankness component. Thus, the components which mismatch the joint sparsity and low-rankness are unexpected (i.e., the strong back \begin{table}
\begin{tabular}{c c} \hline \hline
**Constraint** & **Time** \\ \hline Joint sparsity and low-rankness & 6.04 s \\ Single sparsity & 6.23 s \\ Single low-rankness & 6.11 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Time costs of the three methods. Figure 4: The OFSCs of simulated signals in Case 1. (**a**) the OFSC of pure signal with linearly changing rotation speed; (**b**) the OFSC of noisy signal with linearly changing rotation speed; (**c**) the OFSC of pure signal with quadratically changing rotation speed; (**d**) the OFSC of noisy signal with quadratically changing rotation speed. Figure 5: The EEOSs of noisy OFSCs in Case 1. The dash lines in red indicate the corresponding BPOO and its multipliers. (**a**) the EEOS of noisy OFSC with linearly changing rotation speed; (**b**) the EEOS with noisy OFSC with quadratically changing rotation speed. ground noise).