The fault diagnosis of a planetary gearbox under the speed-varying condition without another speed information is always challenging, especially when the vibration signal has a low signal-to-noise ratio (SNR). To solve the problem, a novel angular-domain weighted allocation deconvolution (AWAD) method based on the encoder signal is proposed in this paper. First, the rotational speed information and fault-related information are simultaneously presented as different patterns of the single encoder signal, which provides the ideal tool for the non-stationary problem. Second, benefiting from the advantage of the encoder signal, the prominent filtering performance of deconvolution method (DM) is highlighted, and a new angular-domain DM is designed under the speed-varying condition. Meanwhile, an angular-domain weighted allocation strategy based on the second-order cyclostationarity indicator is introduced to further enhance the weak fault feature. Finally, the superiority of the proposed AWAD is demonstrated to adaptively and accurately extract the weak fault feature as well as be robust to the low SNR and speed-varying condition without any prior knowledge using simulated and experimental data collected from the encoder signal with a complex gearbox fault.
CabrelliCA. Minimum entropy deconvolution and simplicity: a noniterative algorithm. Geophysics1985; 50: 394–413. https://doi.org/10.1190/1.1441919
4.
McDonaldGLZhaoQZuoMJ. Maximum correlated Kurtosis deconvolution and application on gear tooth chip fault detection. Mech Syst Signal Process2012; 33: 237–255. https://doi.org/10.1016/j.ymssp.2012.06.010
5.
JiaXZhaoMDiY, et al. Investigation on the kurtosis filter and the derivation of convolutional sparse filter for impulsive signature enhancement. J Sound Vib2017; 386: 433–448. https://doi.org/10.1016/j.jsv.2016.10.005
6.
MiaoYLiCShiH, et al. Deep network-based maximum correlated kurtosis deconvolution: a novel deep deconvolution for bearing fault diagnosis. Mech Syst Signal Process2023; 189: 110110. https://doi.org/10.1016/j.ymssp.2023.110110
7.
MiaoYLiCZhangB, et al. Application of a coarse-to-fine minimum entropy deconvolution method for rotating machines fault detection. Mech Syst Signal Process2023; 198: 110431. https://doi.org/10.1016/j.ymssp.2023.110431
8.
MiaoYWangJZhangB, et al. Practical framework of Gini index in the application of machinery fault feature extraction. Mech Syst Signal Process2022; 165: 108333. https://doi.org/10.1016/j.ymssp.2021.108333
9.
DwyerR. Detection of non-Gaussian signals by frequency domain kurtosis estimation[C]//ICASSP’83. Proc IEEE Int Conf Acoust Speech Signal Process1983; 8: 607–610.
10.
AntoniJ. The spectral kurtosis: a useful tool for characterising non-stationary signals. Mech Syst Signal Process2006; 20: 282–307. https://doi.org/10.1016/j.ymssp.2004.09.001
AntoniJRandallRB. The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines. Mech Syst Signal Process2006; 20: 308–331. https://doi.org/10.1016/j.ymssp.2004.09.002
13.
MiaoYZhangBLiC, et al. Feature mode decomposition: new decomposition theory for rotating machinery fault diagnosis. IEEE Trans Ind Electron2023; 70: 1949–1960. https://doi.org/10.1109/TIE.2022.3156156
14.
HouBWangDXiaT, et al. Difference mode decomposition for adaptive signal decomposition. Mech Syst Signal Process2023; 191: 110203. https://doi.org/10.1016/j.ymssp.2023.110203
ChenYMaoZHouX, et al. Noise-robust adaptive feature mode decomposition method for accurate feature extraction in rotating machinery fault diagnosis. Mech Syst Signal Process2024; 211: 111213. https://doi.org/10.1016/j.ymssp.2024.111213
17.
ZhengXYangYHuN, et al. A novel empirical reconstruction Gauss decomposition method and its application in gear fault diagnosis. Mech Syst Signal Process2024; 210: 111174. https://doi.org/10.1016/j.ymssp.2024.111174
18.
WangZYuXGuoY, et al. Research on a novel weak fault detection method based on vibrational resonance in high-dimensional chaotic system, and variational mode decomposition. Chaos Solitons Fractals2026; 208: 118084. https://doi.org/10.1016/j.chaos.2026.118084.
19.
WangZYuXGuoY, et al. Research on high-order Vese–Osher model based on partial differential equation and its application in planetary gearbox fault diagnosis. IEEE Trans Instrum Meas2026; 75: 1–9. https://doi.org/10.1109/TIM.2026.3676173
20.
ShiMDingCWangR, et al. Deep hypergraph autoencoder embedding: an efficient intelligent approach for rotating machinery fault diagnosis. Knowl Based Syst2023; 260: 110172. https://doi.org/10.1016/j.knosys.2022.110172
21.
ShiMDingCChangS, et al. Cross-domain class incremental broad network for continuous diagnosis of rotating machinery faults under variable operating conditions. IEEE Trans Ind Inform2024; 20: 6356–6368. https://doi.org/10.1109/TII.2023.3345449
22.
ZhangGWangYLiX, et al. Hypothetical order test enhanced fault diagnosis architecture for rotating machinery under variable speed conditions. Mech Syst Signal Process2025; 238: 113227. https://doi.org/10.1016/j.ymssp.2025.113227
WangKSHeynsPS. Application of computed order tracking, Vold–Kalman filtering and EMD in rotating machine vibration. Mech Syst Signal Process2011; 25: 416–430. https://doi.org/10.1016/j.ymssp.2010.09.003
25.
WangKSHeynsPS. The combined use of order tracking techniques for enhanced Fourier analysis of order components. Mech Syst Signal Process2011; 25: 803–811. https://doi.org/10.1016/j.ymssp.2010.10.005
26.
BorghesaniPPennacchiPChattertonS, et al. The velocity synchronous discrete Fourier transform for order tracking in the field of rotating machinery. Mech Syst Signal Process2014; 44: 118–133. https://doi.org/10.1016/j.ymssp.2013.03.026
27.
WangZYangJGuoY. Unknown fault feature extraction of rolling bearings under variable speed conditions based on statistical complexity measures. Mech Syst Signal Process2022; 172: 108964. https://doi.org/10.1016/j.ymssp.2022.108964
28.
LiangKZhaoMLinJ, et al. Maximum average kurtosis deconvolution and its application for the impulsive fault feature enhancement of rotating machinery. Mech Syst Signal Process2021; 149: 107323. https://doi.org/10.1016/j.ymssp.2020.107323
29.
MiaoYHuSLiuY, et al. Angle-domain feature mode decomposition for fault diagnosis under speed-varying condition. IEEE Trans Instrum Meas2025; 74: 1–9. https://doi.org/10.1109/TIM.2025.3529067
30.
LuSYanRLiuY, et al. Tacholess speed estimation in order tracking: a review with application to rotating machine fault diagnosis. IEEE Trans Instrum Meas2019; 68: 2315–2332. https://doi.org/10.1109/TIM.2019.2902806
31.
ZhaoMLinJWangX, et al. A tacho-less order tracking technique for large speed variations. Mech Syst Signal Process2013; 40: 76–90. https://doi.org/10.1016/j.ymssp.2013.03.024
32.
DuanRLiaoYYangL. Adaptive tacholess order tracking method based on generalized linear chirplet transform and its application for bearing fault diagnosis. ISA Trans2022; 127: 324–341. https://doi.org/10.1016/j.isatra.2021.08.039
33.
JiangZZhangKZhangX, et al. A tacholess order tracking method based on spectral amplitude modulation for variable speed bearing fault diagnosis. IEEE Trans Instrum Meas2023; 72: 1–8. https://doi.org/10.1109/TIM.2023.3280512
HuYTuXLiF, et al. An adaptive and tacholess order analysis method based on enhanced empirical wavelet transform for fault detection of bearings with varying speeds. J Sound Vib2017; 409: 241–255. https://doi.org/10.1016/j.jsv.2017.08.003
36.
LuSWangX. A new methodology to estimate the rotating phase of a BLDC motor with its application in variable-speed bearing fault diagnosis. IEEE Trans Power Electron2018; 33: 3399–3410. https://doi.org/10.1109/TPEL.2017.2703819
37.
PengDSmithWARandallRB, et al. Speed estimation in planetary gearboxes: a method for reducing impulsive noise. Mech Syst Signal Process2021; 159: 107786. https://doi.org/10.1016/j.ymssp.2021.107786
38.
PengDSmithWARandallRB, et al. Iterative improvement in tacholess speed estimation using instantaneous error estimation for machine condition monitoring in variable speed. Mech Syst Signal Process2024; 216: 111488. https://doi.org/10.1016/j.ymssp.2024.111488
39.
LiangKZhaoMLinJ, et al. Toothwise health monitoring of planetary gearbox under time-varying speed condition based on rotating encoder signal. IEEE Trans Ind Electron2022; 69: 6267–6277. https://doi.org/10.1109/TIE.2021.3090713
40.
MiaoYZhaoMLiangK, et al. Application of an improved MCKDA for fault detection of wind turbine gear based on encoder signal. Renew Energy2020; 151: 192–203. https://doi.org/10.1016/j.renene.2019.11.012
41.
MiaoYZhaoMYiY, et al. Application of sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information. ISA Trans2020; 99: 496–504. https://doi.org/10.1016/j.isatra.2019.10.005
MiaoYZhangBLinJ, et al. A review on the application of blind deconvolution in machinery fault diagnosis. Mech Syst Signal Process2022; 163: 108202.
44.
GaoJWangYWangJ, et al. Blind deconvolution based on logarithmic cyclic content for early weak fault detection in gear degradation. Mech Syst Signal Process2025; 237: 113081. https://doi.org/10.1016/j.ymssp.2025.113081
45.
MiaoYLiCZhangB, et al. Application of a coarse-to-fine minimum entropy deconvolution method for rotating machines fault detection. Mech Syst Signal Process2023; 198: 110431. https://doi.org/10.1016/j.ymssp.2023.110431
46.
MiaoYZhaoMLinJ, et al. Application of an improved maximum correlated kurtosis deconvolution method for fault diagnosis of rolling element bearings. Mech Syst Signal Process2017; 92: 173–195. https://doi.org/10.1016/j.ymssp.2017.01.033
47.
MiaoYZhangBYiY, et al. Application of improved reweighted singular value decomposition for gearbox fault diagnosis based on built-in encoder information. Measurement2021; 168: 108295. https://doi.org/10.1016/j.measurement.2020.108295
48.
YangXGuoYWangH. Encoder signal analysis and its application in gear fault detection. IEEE Trans Instrum Meas2024; 73: 1–9. https://doi.org/10.1109/TIM.2023.3343798
49.
LiBZhangXWuJ. New procedure for gear fault detection and diagnosis using instantaneous angular speed. Mech Syst Signal Process2017; 85: 415–428. https://doi.org/10.1016/j.ymssp.2016.08.036
50.
ZhouYTaoTMeiX, et al. Feed-axis gearbox condition monitoring using built-in position sensors and EEMD method. Robot Comput Integr Manuf2011; 27: 785–793. https://doi.org/10.1016/j.rcim.2010.12.001
51.
HoffmanJDFrankelS. Numerical methods for engineers and scientists. CRC Press, 2018.
52.
RaadAAntoniJSidahmedM. Indicators of cyclostationarity: theory and application to gear fault monitoring. Mech Syst Signal Process2008; 22: 574–587. https://doi.org/10.1016/j.ymssp.2007.09.011
53.
AntoniJBonnardotFRaadA, et al. Cyclostationary modelling of rotating machine vibration signals. Mech Syst Signal Process2004; 18: 1285–1314. https://doi.org/10.1016/S0888-3270(03)00088-8
54.
MarsickAAndréHKhelfI, et al. Restoring cyclostationarity of rolling element bearing signals from the instantaneous phase of their envelope. Mech Syst Signal Process2023; 193: 110264. https://doi.org/10.1016/j.ymssp.2023.110264
55.
ZhangBMiaoYLinJ, et al. Adaptive maximum second-order cyclostationarity blind deconvolution and its application for locomotive bearing fault diagnosis. Mech Syst Signal Process2021; 158: 107736. https://doi.org/10.1016/j.ymssp.2021.107736
56.
BuzzoniMAntoniJD’EliaG. Blind deconvolution based on cyclostationarity maximization and its application to fault identification. J Sound Vib2018; 432: 569–601. https://doi.org/10.1016/j.jsv.2018.06.055
57.
DingCZhaoMLinJ, et al. Transient feature extraction of encoder signal for condition assessment of planetary gearboxes with variable rotational speed. Measurement2020; 151: 107206. https://doi.org/10.1016/j.measurement.2019.107206
58.
LiYFuHFengK, et al. Oscillatory time–frequency concentration for adaptive bearing fault diagnosis under nonstationary time-varying speed. Measurement2023; 218: 113177. https://doi.org/10.1016/j.measurement.2023.113177
