{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T04:47:09Z","timestamp":1777610829881,"version":"3.51.4"},"reference-count":49,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T00:00:00Z","timestamp":1684972800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62001210"],"award-info":[{"award-number":["62001210"]}]},{"name":"National Natural Science Foundation of China","award":["62201284"],"award-info":[{"award-number":["62201284"]}]},{"name":"National Natural Science Foundation of China","award":["BK20190789"],"award-info":[{"award-number":["BK20190789"]}]},{"name":"Natural Science Foundation of Jiangsu","award":["62001210"],"award-info":[{"award-number":["62001210"]}]},{"name":"Natural Science Foundation of Jiangsu","award":["62201284"],"award-info":[{"award-number":["62201284"]}]},{"name":"Natural Science Foundation of Jiangsu","award":["BK20190789"],"award-info":[{"award-number":["BK20190789"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Weak fault detection with stochastic resonance (SR) is distinct from conventional approaches in that it is a nonlinear optimal signal processing to transfer noise into the signal, resulting in a higher output SNR. Owing to this special characteristic of SR, this study develops a controlled symmetry with Woods-Saxon stochastic resonance (CSwWSSR) model based on the Woods-Saxon stochastic resonance (WSSR), where each parameter of the model may be modified to vary the potential structure. Then, the potential structure of the model is investigated in this paper, along with the mathematical analysis and experimental comparison to clarify the effect of each parameter on it. The CSwWSSR is a tri-stable stochastic resonance, but differs from others in that each of its three potential wells is controlled by different parameters. Moreover, the particle swarm optimization (PSO), which can quickly find the ideal parameter matching, is introduced to attain the optimal parameters of the CSwWSSR model. Fault diagnosis of simulation signals and bearings was carried out to confirm the viability of the proposed CSwWSSR model, and the results revealed that the CSwWSSR model is superior to its constituent models.<\/jats:p>","DOI":"10.3390\/s23115062","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T03:56:58Z","timestamp":1684987018000},"page":"5062","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Controlled Symmetry with Woods-Saxon Stochastic Resonance Enabled Weak Fault Detection"],"prefix":"10.3390","volume":"23","author":[{"given":"Jian","family":"Liu","sequence":"first","affiliation":[{"name":"College of Information Engineering, Nanjing University of Finance and Economics, Nanjing 210023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiaqi","family":"Guo","sequence":"additional","affiliation":[{"name":"College of Information Engineering, Nanjing University of Finance and Economics, Nanjing 210023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bing","family":"Hu","sequence":"additional","affiliation":[{"name":"College of Modern Posts, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiqing","family":"Zhai","sequence":"additional","affiliation":[{"name":"College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Can","family":"Tang","sequence":"additional","affiliation":[{"name":"College of Information Engineering, Nanjing University of Finance and Economics, Nanjing 210023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wanjia","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Information Engineering, Nanjing University of Finance and Economics, Nanjing 210023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2085","DOI":"10.1016\/j.matpr.2017.02.054","article-title":"Fault detection analysis in rolling element bearing: A review","volume":"4","author":"Gupta","year":"2017","journal-title":"Mater. Today Proc."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wei, Y., Li, Y., Xu, M., and Huang, W. (2019). A review of early fault diagnosis approaches and their applications in rotating machinery. Entropy, 21.","DOI":"10.3390\/e21040409"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4611","DOI":"10.1109\/TSP.2019.2931203","article-title":"Noise benefits in combined nonlinear bayesian estimators","volume":"67","author":"Duan","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"105245","DOI":"10.1016\/j.cnsns.2020.105245","article-title":"Stochastic Resonance in a delay-controlled dissipative bistable potential for weak signal enhancement","volume":"85","author":"Liu","year":"2020","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"085014","DOI":"10.1088\/0957-0233\/26\/8\/085014","article-title":"SVD principle analysis and fault diagnosis for bearings based on the correlation coefficient","volume":"26","author":"Qiao","year":"2015","journal-title":"Meas. Sci. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Xu, M., Zheng, C., Sun, K., Xu, L., Qiao, Z., and Lai, Z. (2023). Stochastic resonance with parameter estimation for enhancing unknown compound fault detection of bearings. Sensors, 23.","DOI":"10.3390\/s23083860"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"L453","DOI":"10.1088\/0305-4470\/14\/11\/006","article-title":"The mechanism of stochastic resonance","volume":"14","author":"Benzi","year":"1981","journal-title":"J. Phys. Math. Gen."},{"key":"ref_8","first-page":"230","article-title":"A review of stochastic resonance in rotating machine fault detection","volume":"116","author":"Lu","year":"2019","journal-title":"Sensors"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1006\/mssp.2002.1470","article-title":"The application of stochastic resonance theory for early detecting rub-impact fault of rotor system","volume":"17","author":"Hu","year":"2003","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjb\/e2009-00163-x","article-title":"Stochastic resonance: A remarkable idea that changed our perception of noise","volume":"69","author":"Gammaitoni","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Caccamo, M.T., and Magaz\u00f9, S. (2019). A physical\u2013mathematical approach to climate change effects through stochastic resonance. Climate, 7.","DOI":"10.3390\/cli7020021"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1929","DOI":"10.3390\/molecules17021929","article-title":"An adaptive single-well stochastic resonance algorithm applied to trace analysis of Clenbuterol in human urine","volume":"17","author":"Wang","year":"2012","journal-title":"Molecules"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Markina, A., Muratov, A., Petrovskyy, V., and Avetisov, V. (2020). Detection of single molecules using stochastic resonance of bistable oligomers. Nanomaterials, 10.","DOI":"10.3390\/nano10122519"},{"key":"ref_14","first-page":"1","article-title":"Enhancing active vibration control performances in a smart rotary sandwich thick nanostructure conveying viscous fluid flow by a PD controller","volume":"33","author":"Zhang","year":"2021","journal-title":"Waves Random Complex Media"},{"key":"ref_15","first-page":"469","article-title":"Influence of imperfection on the smart control frequency characteristics of a cylindrical sensor-actuator GPLRC cylindrical shell using a proportional-derivative smart controller","volume":"26","author":"Zare","year":"2020","journal-title":"Smart Struct. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ko, L.-W., Chikara, R.K., Chen, P.-Y., Jheng, Y.-C., Wang, C.-C., Yang, Y.-C., Li, L.P.-H., Liao, K.-K., Chou, L.-W., and Kao, C.-L. (2020). Noisy galvanic vestibular stimulation (stochastic resonance) changes electroencephalography activities and postural control in patients with bilateral vestibular hypofunction. Brain Sci., 10.","DOI":"10.3390\/brainsci10100740"},{"key":"ref_17","first-page":"1717","article-title":"A novel sequential spectrum sensing method in cognitive radio using suprathreshold stochastic resonance","volume":"63","author":"Li","year":"2013","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1016\/j.ymssp.2018.12.032","article-title":"Applications of stochastic resonance to machinery fault detection: A review and tutorial","volume":"122","author":"Qiao","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Dong, H., He, K., Shen, X., Ma, S., Wang, H., and Qiao, C. (2020). Adaptive intrawell matched stochastic resonance with a potential constraint aided line enhancer for passive sonars. Sensors, 20.","DOI":"10.3390\/s20113269"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1457","DOI":"10.1016\/j.physleta.2019.02.006","article-title":"Optimum adaptive array stochastic resonance in noisy grayscale image restoration","volume":"383","author":"Liu","year":"2019","journal-title":"Phys. Lett. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.neucom.2016.12.111","article-title":"Noise-modulated neural networks as an application of stochastic resonance","volume":"277","author":"Ikemoto","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"193108","DOI":"10.1063\/1.4766946","article-title":"Detecting signals buried in noise via nanowire transistors using stochastic resonance","volume":"101","author":"Nishiguchi","year":"2012","journal-title":"Appl. Phys. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.renene.2013.04.005","article-title":"A new noise-controlled second-order enhanced stochastic resonance method with its application in wind turbine drivetrain fault diagnosis","volume":"60","author":"Li","year":"2013","journal-title":"Renew. Energy"},{"key":"ref_24","first-page":"24","article-title":"Colored correlated noises induced regime shifts in a time-delayed lake eutrophication ecosystem","volume":"2","author":"Liu","year":"2022","journal-title":"Front. Sustain. Dev."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1109\/LCOMM.2014.011214.132465","article-title":"A novel parameter-tuned stochastic resonator for binary PAM signal processing at low SNR","volume":"18","author":"Liu","year":"2014","journal-title":"IEEE Commun. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"He, D., Chen, X., Pei, L., Jiang, L., and Yu, W. (2019). Improvement of noise uncertainty and signal-to-noise ratio wall in spectrum sensing based on optimal stochastic resonance. Sensors, 19.","DOI":"10.3390\/s19040841"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jsv.2018.03.007","article-title":"Stochastic resonance in a time-delayed feedback tristable system and its application in fault diagnosis","volume":"424","author":"Shi","year":"2018","journal-title":"J. Sound Vib."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"051008","DOI":"10.1115\/1.4030346","article-title":"Enhanced rotating machine fault diagnosis based on time-delayed feedback stochastic resonance","volume":"137","author":"Lu","year":"2015","journal-title":"J. Vib. Acoust."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1007\/s10955-009-9864-4","article-title":"The effects of time delay on stochastic resonance in a bistable system with correlated noises","volume":"137","author":"Mei","year":"2009","journal-title":"J. Stat. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1007\/s11071-014-1774-6","article-title":"Dynamical complexity and stochastic resonance in a bistable system with time delay","volume":"79","author":"He","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"111503","DOI":"10.1016\/j.chaos.2021.111503","article-title":"Echo state network activation function based on bistable stochastic resonance","volume":"153","author":"Liao","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_32","first-page":"1","article-title":"Rolling bearing fault diagnosis based on adaptive multiparameter-adjusting bistable stochastic resonance","volume":"2020","author":"Lai","year":"2020","journal-title":"Shock Vib."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.neucom.2021.09.074","article-title":"Low-power-consumption physical reservoir computing model based on overdamped bistable stochastic resonance system","volume":"468","author":"Liao","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.isatra.2018.09.022","article-title":"Weak feature enhancement in machinery fault diagnosis using empirical wavelet transform and an improved adaptive bistable stochastic resonance","volume":"84","author":"Zhang","year":"2019","journal-title":"ISA Trans."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1016\/j.ymssp.2016.08.030","article-title":"An adaptive unsaturated bistable stochastic resonance method and its application in mechanical fault diagnosis","volume":"84","author":"Qiao","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2635","DOI":"10.1007\/s11277-017-3945-5","article-title":"A novel spectrum sensing method based on tri-stable stochastic resonance and quantum particle swarm optimization","volume":"95","author":"Lu","year":"2017","journal-title":"Wirel. Pers. Commun."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1016\/j.ymssp.2013.12.004","article-title":"Stochastic resonance with Woods-Saxon potential for rolling element bearing fault diagnosis","volume":"45","author":"Lu","year":"2014","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_38","first-page":"9","article-title":"Hybrid tri-stable stochastic resonance system used for fault signal detection","volume":"38","author":"Zhang","year":"2019","journal-title":"J. Vib. Shock"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"112692","DOI":"10.1016\/j.chaos.2022.112692","article-title":"Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities","volume":"164","author":"Khan","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Santos-Garc\u00eda, G., Noor, M.A., and Soliman, M.S. (2022). New Hermite\u2013Hadamard inequalities for convex fuzzy-number-valued mappings via fuzzy riemann integrals. Mathematics, 10.","DOI":"10.3390\/math10183251"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"113274","DOI":"10.1016\/j.chaos.2023.113274","article-title":"On fuzzy fractional integral operators having exponential kernels and related certain inequalities for exponential trigonometric convex fuzzy-number valued mappings","volume":"169","author":"Khan","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4266","DOI":"10.3934\/math.2022236","article-title":"Hermite-Hadamard inequalities for generalized convex functions in interval-valued calculus","volume":"7","author":"Khan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Treanta, S., and Soliman, M.S. (2022). Generalized preinvex interval-valued functions and related Hermite\u2013Hadamard type inequalities. Symmetry, 14.","DOI":"10.3390\/sym14091901"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1856","DOI":"10.2991\/ijcis.d.210616.001","article-title":"Higher-order strongly preinvex fuzzy mappings and fuzzy mixed variational-like inequalities","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_45","first-page":"1","article-title":"New Hermite\u2013Hadamard-type inequalities for -convex fuzzy-interval-valued functions","volume":"149","author":"Khan","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s00500-016-2474-6","article-title":"Particle swarm optimization algorithm: An overview","volume":"22","author":"Wang","year":"2018","journal-title":"Soft Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4862","DOI":"10.1109\/TCYB.2019.2943928","article-title":"Triple archives particle swarm optimization","volume":"50","author":"Xia","year":"2019","journal-title":"IEEE Trans. Cybern."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"10031","DOI":"10.1109\/ACCESS.2022.3142859","article-title":"Particle swarm optimization: A comprehensive survey","volume":"10","author":"Shami","year":"2022","journal-title":"IEEE Access."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.swevo.2018.01.011","article-title":"MPSO: Modified particle swarm optimization and its applications","volume":"41","author":"Tian","year":"2018","journal-title":"Swarm Evol. Comput."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/11\/5062\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:41:46Z","timestamp":1760125306000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/11\/5062"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,25]]},"references-count":49,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["s23115062"],"URL":"https:\/\/doi.org\/10.3390\/s23115062","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,25]]}}}