[2] [2] EI-AREM S.Nonlinear analysis,instability and routes to chaos of a cracked rotating shaft[J].Nonlinear Dynamics,2019,96(1):667-683.

[3] [3] KHORRAMI H,RAKHEJA S,SEDAGHATI R.Vibration behavior of a two-crack shaft in a rotor disc-bearing system[J].Mechanism and Machine Theory,2017,113:67-84.

[4] [4] OPPENHEIMER C H,LOPARO K A.Physically based diagnosis and prognosis of cracked rotor shafts［C］//Proceedings of Component and Systems Diagnostics,Prognostics,and Health Management,July 13-16,2002,Orlando,FL,United States.Washington:SPIE,2002,4733:122-132．

[5] [5] YU M, WANG D W, LUO M. Model-based prognosis for hybrid systems with mode-dependent degradation behaviors[J].IEEE Transactions on Industrial Electronics,2014,61(1):546-554.

[6] [6] LEAO L S,CAVALINI A A,MORAIS S,et al.Fault detection in rotating machinery by using the modal state observer approach[J].Journal of Sound and Vibration,2019,458:123-142.

[7] [7] ZHAO R,YAN R Q,CHEN Z H,et al.Deep learning and its applications to machine health monitoring[J].Mechanical Systems and Signal Processing,2019,115:213-237.

[8] [8] HOANG D T,KANG H J.A survey on deep learning based bearing fault diagnosis[J].Neurocomputing,2019,335:327-335.

[9] [9] SUN J D,YAN C H,WEN J T.Intelligent bearing fault diagnosis method combining compressed data acquisition and deep learning[J].IEEE Transactions on Instrumentation and Measurement,2018,67(1):185-195.

[10] [10] LIU D,XIAO Z H,HU X,et al.Feature extraction of rotor fault based on EEMD and curve code[J].Measurement,2019,135:712-724.

[11] [11] SAMPAIO D L,NICOLETTI R.Detection of cracks in shafts with the approximated entropy algorithm[J].Mechanical Systems and Signal Processing,2016,72-73:286-302.

[12] [12] GRADZKI R,KULESZA Z,BARTOSZEWICZ B.Method of shaft crack detection based on squared gain of vibration amplitude[J].Nonlinear Dynamics,2019,98(1):671-690.

[13] [13] LEI Y G,YANG B,JIANG X W,et al.Applications of machine learning to machine fault diagnosis:a review and roadmap[J].Mechanical Systems and Signal Processing,2020,138:106587.

[14] [14] WANG H Q,LI S,SONG L Y,et al.A novel convolutional neural network based fault recognition method via image fusion of multi-vibration-signals[J].Computers in Industry,2019,105:182-190.

[15] [15] YANG B,LEI Y G,JIA F,et al.An intelligent fault diagnosis approach based on transfer learning from laboratory bearings to locomotive bearings[J].Mechanical Systems and Signal Processing,2019,122:692-706.

[16] [16] SHAO H D,JIANG H K,LIN Y,et al.A novel method for intelligent fault diagnosis of rolling bearings using ensemble deep auto-encoders[J].Mechanical Systems and Signal Processing,2018,102:278-297.

[17] [17] OH H,JUNG J H,JEON B C,et al.Scalable and unsupervised feature engineering using vibration imaging and deep learning for rotor system diagnosis[J].IEEE Transactions on Industrial Electronics,2018,65(4):3539-3549.

[18] [18] GE W M,SUN M Y,WANG X F.An incremental two-dimensional principal component analysis for object recognition[J].Mathematical Problems in Engineering,2018,2018:3272809.

[19] [19] LIU J,WANG C,LUO Z W.High dimensional nonlinear spring characteristic modelling and vibration analyses of subharmonic resonance of a dual-rotor system based on energy tracks[J].Applied Mathematical Modelling,2021,91:390-411.