## 4 Simulation Study
### Case 2: Inner Race Fault  
Similar to Section 4.1, the inner race fault is simulated to further confirm the proposed method. The ball-pass-order on the inner race (BPOI) of fault feature cyclic orders are set to \(\alpha_{\text{BPOI}}^{L}=2.5\) and \(\alpha_{\text{BPOI}}^{O}=5\) respectively with the rotating shaft order 1. The rest of the parameter settings keep the same with Case 1. The simulated signals in Case 2 are displayed in Figure 8. One of the most obvious differences from Case 1 (Figure 8a,c) is that the transient impulses in Case 2 are modulated by the rotating frequency. The transient impulses are also covered by strong background noise from simulated signals. Then, OFSCs in Case 2 are calculated and shown in Figure 9. The cyclic order one appears on the OFSCs in Case 2 as shown in Figure 9a,c, which indicates the shaft rotating order. The side cyclic order with the width two (BPOI \(\pm\)1) is also appeared and can be viewed as a characteristic of inner race fault. The OFSCs of the noisy signals are shown in Figure 9b,d, where the fault features are almost completely covered as well. The EEOSs of the noisy OFSCs in Case 2 are shown in Figure 10. The cyclic order of the inner race fault is relatively obvious on the single and double BPOI. The amplitude on triple BPOI tends to be covered by the noise. Moreover, the side cyclic orders are covered by the strong background noise, which are hard to identify. Figure 7: The extraction results of Case 1 under the quadratically changing rotation speed condition. The dash lines in red indicate the corresponding BPOO and its multipliers. (**a**) the denoised OFSC obtained by the proposed joint sparsity and low-rankness constraint; (**b**) the EEOS obtained by the proposed joint sparsity and low-rankness constraint; (**c**) the denoised OFSC obtained by single sparsity constraint; (**d**) the EEOS obtained by single sparsity constraint; (**e**) the denoised OFSC obtained by single low-rankness constraint; (**f**) the EEOS obtained by single low-rankness constraint. The results after operating the three methods are shown in Figures 11 and 12. At the quadratically changing speed condition (\(\alpha_{\text{BPOI}}^{\text{Q}}=5\)), the proposed algorithm can effectively extract the fault feature (seen in Figure 12a, and the corresponding EEOS in Figure 12b shows it more clear. At the linearly changing speed condition (\(\alpha_{\text{BPOI}}^{\text{L}}=2.5\)), the fault cyclic orders and their multipliers are effectively extracted. In comparison, the side cyclic orders are not obvious in Figure 11a. This may be due to the relatively lower amplitude on the side cyclic orders. From the EEOS of the denoised OFSC with linearly changing speed (seen in Figure 11b), the amplitude on the BPOI and its sides are larger than the components of the original PEOS. Figure 8: The simulated signals in Case 2. (**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. Figure 9: The OFSCs of simulated signals in Case 2. (**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. around them, which is more obvious. Compared with the other two methods, the noise components in OFSCs are not greatly eliminated due to the inadequate constraints, which make the time-varying fault features not clear enough to identify.