## 6 Conclusions  
This paper proposes a weak fault feature extraction method for rolling bearings under variable speed circumstances. In the proposed method, the measured vibration signal is transferred into the (cyclic) order frequency domain and visualized by OFSC. Then, The joint sparsity and low-rankness of the fault feature in the OFSC is then exposed, which is utilized to extract the fault feature. Moreover, an algorithm is developed based on the ADMM framework to optimize the sparsity and low-rankness jointly enforced fault feature in the noisy OFSC. Finally, using EEOS, the fault feature of the rolling bearing is improved and the fault type can be more easily justified. In the simulation study, the proposed method can extract the fault feature under the strong background noise with SNR = \(-\)16 dB. In addition, an experimental case is carried out to validate the practicality of the proposed method. It shows that the proposed method can effectively extract the rolling bearing fault feature under variable speed conditions in a complex gearing system. However, there are some shortages of the proposed model and the algorithm needs to be further improved. For example, the adaptive selection of parameters \(\lambda_{0}\) and \(\lambda_{1}\) is still worth studying. Figure 15: The extraction results of experimental data. The dash lines in red indicate the corresponding BPOO and its multipliers. (**a**) the noisy OFSC; (**b**) the noisy EEOS; (**c**) the denoised OFSC obtained by the proposed joint sparsity and low-rankness constraint; (**d**) the EEOS obtained by the proposed joint sparsity and low-rankness constraint; (**e**) the denoised OFSC obtained by single sparsity constraint; (**f**) the EEOS obtained by single sparsity constraint; (**g**) the denoised OFSC obtained by single low-rankness constraint; (**h**) the EEOS obtained by single low-rankness constraint. **Author Contributions:** Conceptualization, R.W., L.Y. and X.H. ; methodology, C.Z. ; software, C.Z., L.Y. and H.F.; validation, R.W. and C.Z. ; formal analysis, R.W., C.Z. and H.F.; resouces, L.Y. ; data curation, R.W. ; writing--original draft preparation, C.Z. ; writing--review and editing, R.W., L.Y. and H.F.; project administration, L.Y. and X.H. ; funding acquisition, R.W. and L.Y. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China grant numbers 51505277 and 12074254, National Key Research and Development Program of China grant number 2019 YFB2004600, Natural Science Foundation of Shanghai grant number 21 ZR1434100. This research was also sponsored by the Oceanic Interdisciplinary Program of Shanghai Jiao Tong University project number SL2021 MS009 and Open Fund of Laboratory of Vibration, Shock and Noise, Shanghai Jiao Tong University project number VSN202101. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Acknowledgments:** The authors would like to appreciate [PERSON] (Shanghai Jiao Tong University) for providing experimental data. At the same time, the author would like to thank [PERSON] and [PERSON] (Shanghai Maritime University) for their proof reading. **Conflicts of Interest:** The authors declare no conflict of interest. **Abbreviations**  
The following abbreviations are used in this manuscript: