{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T06:29:01Z","timestamp":1769927341484,"version":"3.49.0"},"reference-count":34,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T00:00:00Z","timestamp":1681689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12072069"],"award-info":[{"award-number":["12072069"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The quantitative diagnosis of rolling bearings is essential to automating maintenance decisions. Over recent years, Lempel\u2013Ziv complexity (LZC) has been widely used for the quantitative assessment of mechanical failures as one of the most valuable indicators for detecting dynamic changes in nonlinear signals. However, LZC focuses on the binary conversion of 0\u20131 code, which can easily lose some effective information about the time series and cannot fully mine the fault characteristics. Additionally, the immunity of LZC to noise cannot be insured, and it is difficult to quantitatively characterize the fault signal under strong background noise. To overcome these limitations, a quantitative bearing fault diagnosis method based on the optimized Variational Modal Decomposition Lempel\u2013Ziv complexity (VMD-LZC) was developed to fully extract the vibration characteristics and to quantitatively characterize the bearing faults under variable operating conditions. First, to compensate for the deficiency that the main parameters of the variational modal decomposition (VMD) have to be selected by human experience, a genetic algorithm (GA) is used to optimize the parameters of the VMD and adaptively determine the optimal parameters [k, \u03b1] of the bearing fault signal. Furthermore, the IMF components that contain the maximum fault information are selected for signal reconstruction based on the Kurtosis theory. The Lempel\u2013Ziv index of the reconstructed signal is calculated and then weighted and summed to obtain the Lempel\u2013Ziv composite index. The experimental results show that the proposed method is of high application value for the quantitative assessment and classification of bearing faults in turbine rolling bearings under various operating conditions such as mild and severe crack faults and variable loads.<\/jats:p>","DOI":"10.3390\/s23084044","type":"journal-article","created":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T05:51:41Z","timestamp":1681710701000},"page":"4044","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Approach to the Quantitative Diagnosis of Rolling Bearings Based on Optimized VMD and Lempel\u2013Ziv Complexity under Varying Conditions"],"prefix":"10.3390","volume":"23","author":[{"given":"Haobo","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tongguang","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qingkai","family":"Han","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China"},{"name":"Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang 110819, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0701-6123","authenticated-orcid":false,"given":"Zhong","family":"Luo","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China"},{"name":"Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang 110819, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"8430","DOI":"10.1109\/TIE.2021.3108726","article-title":"Subdomain Adaptation Transfer Learning Network for Fault Diagnosis of Roller Bearings","volume":"69","author":"Wang","year":"2021","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"108018","DOI":"10.1016\/j.ymssp.2021.108018","article-title":"Bearing fault diagnosis method based on adaptive maximum cyclostationarity blind deconvolution","volume":"162","author":"Wang","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"109033","DOI":"10.1016\/j.ress.2022.109033","article-title":"Adaptive staged remaining useful life prediction method based on multi-sensor and multi-feature fusion","volume":"231","author":"Ta","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111393","DOI":"10.1016\/j.measurement.2022.111393","article-title":"Cyclic correlation density decomposition based on a sparse and low-rank model for weak fault feature extraction of rolling bearings","volume":"198","author":"Wang","year":"2022","journal-title":"Measurement"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/j.measurement.2017.12.012","article-title":"Hurst based vibro-acoustic feature extraction of bearing using EMD and VMD","volume":"117","author":"Mohanty","year":"2017","journal-title":"Measurement"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109852","DOI":"10.1016\/j.ymssp.2022.109852","article-title":"Stochastic resonance of a high-order-degradation bistable system and its application in fault diagnosis with variable speed condition","volume":"186","author":"Xu","year":"2023","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/j.apm.2022.08.023","article-title":"Stochastic resonance in a high-dimensional space coupled bistable system and its application","volume":"113","author":"Li","year":"2023","journal-title":"Appl. Math. Model."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"106549","DOI":"10.1016\/j.aap.2021.106549","article-title":"Fault diagnosis for train plug door using weighted fractional wavelet packet decomposition energy entropy","volume":"166","author":"Sun","year":"2021","journal-title":"Accid. Anal. Prev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5906","DOI":"10.1109\/TVT.2022.3158436","article-title":"Contactless Fault Diagnosis for Railway Point Machines Based on Multi-Scale Fractional Wavelet Packet Energy Entropy and Synchronous Optimization Strategy","volume":"71","author":"Sun","year":"2022","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"110417","DOI":"10.1016\/j.measurement.2021.110417","article-title":"Ensemble empirical mode decomposition energy moment entropy and enhanced long short-term memory for early fault prediction of bearing","volume":"188","author":"Gao","year":"2022","journal-title":"Measurement"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"108834","DOI":"10.1016\/j.ymssp.2022.108834","article-title":"Weak fault feature extraction of rolling bearings based on improved ensemble noise-reconstructed EMD and adaptive threshold denoising","volume":"171","author":"Yin","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"111060","DOI":"10.1016\/j.measurement.2022.111060","article-title":"Online chatter detection in milling process based on fast iterative VMD and energy ratio difference","volume":"194","author":"Zhang","year":"2022","journal-title":"Measurement"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1142\/S1793536910000422","article-title":"Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method","volume":"2","author":"Yeh","year":"2010","journal-title":"Adv. Adapt. Data Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1109\/TSP.2013.2288675","article-title":"Variational mode decomposition","volume":"62","author":"Dragomiretskiy","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2022.3216413","article-title":"Partial Transfer Learning of Multidiscriminator Deep Weighted Adversarial Network in Cross-Machine Fault Diagnosis","volume":"71","author":"Wang","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"108823","DOI":"10.1016\/j.measurement.2020.108823","article-title":"Intelligent fault diagnosis of diesel engine via adaptive VMD-Rihaczek distribution and graph regularized bi-directional NMF","volume":"172","author":"Wang","year":"2020","journal-title":"Measurement"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"111360","DOI":"10.1016\/j.measurement.2022.111360","article-title":"A VME method based on the convergent tendency of VMD and its application in multi-fault diagnosis of rolling bearings","volume":"198","author":"Li","year":"2022","journal-title":"Measurement"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"108185","DOI":"10.1016\/j.measurement.2020.108185","article-title":"An optimized VMD method and its applications in bearing fault diagnosis","volume":"166","author":"Li","year":"2020","journal-title":"Measurement"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"108216","DOI":"10.1016\/j.ymssp.2021.108216","article-title":"A fault information-guided variational mode decompose-tion (FIVMD) method for rolling element bearings diagnosis","volume":"164","author":"Ni","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.isatra.2019.04.031","article-title":"Impact fault detection of gearbox based on variational mode decomposition and coupled underdamped stochastic resonance","volume":"95","author":"Li","year":"2019","journal-title":"ISA Trans."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.jsv.2018.07.039","article-title":"Initial center frequency-guided VMD for fault diagnosis of rotating machines","volume":"435","author":"Jiang","year":"2018","journal-title":"J. Sound Vib."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"110798","DOI":"10.1016\/j.measurement.2022.110798","article-title":"Adaptive variational mode decomposition based on Archimedes optimization algorithm and its application to bearing fault diagnosis","volume":"191","author":"Wang","year":"2022","journal-title":"Measurement"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.isatra.2018.10.008","article-title":"Identification of mechanical compound-fault based on the improved parameter-adaptive variational mode decomposition","volume":"84","author":"Miao","year":"2018","journal-title":"ISA Trans."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1109\/TIT.1976.1055501","article-title":"On the complexity of finite sequences","volume":"22","author":"Lempel","year":"1976","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"108429","DOI":"10.1016\/j.ress.2022.108429","article-title":"Joint decision-making of parallel machine scheduling restricted in job-machine release time and preventive maintenance with remaining useful life constraints","volume":"222","author":"He","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2865","DOI":"10.1177\/01423312221088080","article-title":"Leakage detection method of natural gas pipeline combining improved variational mode decomposition and Lempel\u2013Ziv complexity analysis","volume":"44","author":"Zhu","year":"2022","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"109747","DOI":"10.1016\/j.ymssp.2022.109747","article-title":"Research on a remaining useful life prediction method for degradation angle identification two-stage degradation process","volume":"184","author":"Wang","year":"2023","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/j.measurement.2018.06.051","article-title":"Quantitative trend fault diagnosis of a rolling bearing based on Sparsogram and Lempel-Ziv","volume":"128","author":"Cui","year":"2018","journal-title":"Measurement"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.measurement.2019.02.011","article-title":"Fault diagnosis of bearing based on Symbolic Aggregate approXimation and Lempel-Ziv","volume":"138","author":"Yin","year":"2019","journal-title":"Measurement"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"055018","DOI":"10.1088\/1361-6501\/ac50e8","article-title":"Fault severity assessment for rotating machinery via improved Lempel\u2013Ziv complexity based on variable-step multiscale analysis and equiprobable space partitioning","volume":"33","author":"Zhou","year":"2022","journal-title":"Meas. Sci. Technol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.isatra.2021.01.042","article-title":"Fault severity assessment of rolling bearing based on optimized multi-dictionaries matching pursuit and Lempel\u2013Ziv complexity","volume":"116","author":"Dang","year":"2021","journal-title":"ISA Trans."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"82522","DOI":"10.1109\/ACCESS.2019.2923657","article-title":"Fault Feature Extraction of Reciprocating Compressor Based on AWD and LZC","volume":"7","author":"Tang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"108405","DOI":"10.1016\/j.ymssp.2021.108405","article-title":"An improved partial similitude method for dynamic characteristic of rotor systems based on Levenberg\u2013Marquardt method","volume":"165","author":"Li","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"104142","DOI":"10.1016\/j.mechmachtheory.2020.104142","article-title":"Partial similitude for dynamic characteristics of rotor systems considering gravitational acceleration","volume":"156","author":"Luo","year":"2021","journal-title":"Mech. Mach. Theory"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/8\/4044\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:17:17Z","timestamp":1760123837000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/8\/4044"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,17]]},"references-count":34,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["s23084044"],"URL":"https:\/\/doi.org\/10.3390\/s23084044","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,17]]}}}