{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T20:14:06Z","timestamp":1787948046915,"version":"build-2784847793"},"reference-count":67,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T00:00:00Z","timestamp":1644278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100009558","name":"University Natural Science Research Project of Anhui Province","doi-asserted-by":"publisher","award":["KJ2020A0449 and KJ2020A0453"],"award-info":[{"award-number":["KJ2020A0449 and KJ2020A0453"]}],"id":[{"id":"10.13039\/501100009558","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12172001"],"award-info":[{"award-number":["12172001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"The University Synergy Innovation Program of Anhui Province","award":["GXXT-2021-019"],"award-info":[{"award-number":["GXXT-2021-019"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Micromachines"],"abstract":"<jats:p>For complex micro-active machines or micro-robotics, it is crucial to clarify the coupling and collective motion of their multiple self-oscillators. In this article, we construct two joint liquid crystal elastomer (LCE) spring oscillators connected by a spring and theoretically investigate their collective motion based on a well-established dynamic LCE model. The numerical calculations show that the coupled system has three steady synchronization modes: in-phase mode, anti-phase mode, and non-phase-locked mode, and the in-phase mode is more easily achieved than the anti-phase mode and the non-phase-locked mode. Meanwhile, the self-excited oscillation mechanism is elucidated by the competition between network that is achieved by the driving force and the damping dissipation. Furthermore, the phase diagram of three steady synchronization modes under different coupling stiffness and different initial states is given. The effects of several key physical quantities on the amplitude and frequency of the three synchronization modes are studied in detail, and the equivalent systems of in-phase mode and anti-phase mode are proposed. The study of the coupled LCE spring oscillators will deepen people\u2019s understanding of collective motion and has potential applications in the fields of micro-active machines and micro-robots with multiple coupled self-oscillators.<\/jats:p>","DOI":"10.3390\/mi13020271","type":"journal-article","created":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T23:37:48Z","timestamp":1644363468000},"page":"271","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Self-Sustained Collective Motion of Two Joint Liquid Crystal Elastomer Spring Oscillator Powered by Steady Illumination"],"prefix":"10.3390","volume":"13","author":[{"given":"Changshen","family":"Du","sequence":"first","affiliation":[{"name":"Department of Civil Engineering, Anhui Jianzhu University, Hefei 230601, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Quanbao","family":"Cheng","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Anhui Jianzhu University, Hefei 230601, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4898-2827","authenticated-orcid":false,"given":"Kai","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Anhui Jianzhu University, Hefei 230601, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1182-2248","authenticated-orcid":false,"given":"Yong","family":"Yu","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Anhui Jianzhu University, Hefei 230601, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ding, W.J. (2009). Self-Excited Vibration, Tsing-Hua University Press.","DOI":"10.1007\/978-3-540-69741-1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.physrep.2012.10.007","article-title":"Self-oscillation","volume":"525","author":"Jenkins","year":"2013","journal-title":"Phys. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1800207","DOI":"10.1002\/adom.201800207","article-title":"Optically driven soft micro robotics","volume":"6","author":"Nocentini","year":"2018","journal-title":"Adv. Opt. Mater."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"11758","DOI":"10.1002\/anie.201807495","article-title":"A multifunctional dyedoped liquid crystal polymer actuator: Light-guided transportation, turning in locomotion, and autonomous motion","volume":"57","author":"Ge","year":"2018","journal-title":"Angew. Chem. Int. Ed."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1603483","DOI":"10.1002\/adma.201603483","article-title":"Soft Actuators for Small-Scale Robotics","volume":"29","author":"Lindsey","year":"2017","journal-title":"Adv. Mater."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"19881","DOI":"10.1021\/jacs.1c09439","article-title":"Polyhedral Micromotors of Metal-Organic Frameworks: Symmetry Breaking and Propulsion","volume":"143","author":"Wang","year":"2021","journal-title":"J. Am. Chem. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"106894","DOI":"10.1016\/j.ymssp.2020.106894","article-title":"Self-excited vibrations due to viscoelastic interactions","volume":"144","author":"Papangelo","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1038\/s41565-019-0414-1","article-title":"Towards molecular motors in unison","volume":"14","author":"Huang","year":"2019","journal-title":"Nat. Nanotechnol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1860","DOI":"10.1016\/j.jsv.2010.11.005","article-title":"Self-excited oscillation under nonlinear feedback with time-delay","volume":"330","author":"Chatterjee","year":"2011","journal-title":"J. Sound Vib."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"eaar7986","DOI":"10.1126\/scirobotics.aar7986","article-title":"A soft bistable valve for autonomous control of soft actuators","volume":"3","author":"Rothemund","year":"2018","journal-title":"Sci. Robot."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1908842","DOI":"10.1002\/adfm.201908842","article-title":"An autonomous soft actuator with light-driven self-sustained wavelike oscillation for phototactic self-locomotion and power generation","volume":"30","author":"Yang","year":"2020","journal-title":"Adv. Funct. Mater."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"9020","DOI":"10.1126\/scirobotics.aas9020","article-title":"Electronic skins for soft, compact, reversible assembly of wirelessly activated fully soft robots","volume":"3","author":"Byun","year":"2018","journal-title":"Sci. Robot."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1904765","DOI":"10.1002\/adma.201904765","article-title":"Electronic skin: Recent progress and future prospects for skin-attachable devices for health monitoring, robotics, and prosthetics","volume":"31","author":"Yang","year":"2019","journal-title":"Adv. Mater."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Rihani, R.T., Kim, H., Black, B.J., Atmaramani, R., Saed, M.O., Pancrazio, J.J., and Ware, T.H. (2018). Liquid crystal elastomer-based microelectrode array for in vitro neuronal recordings. Micromachines, 9.","DOI":"10.3390\/mi9080416"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Lu, H., Zou, Z., Wu, X., Shi, C., Liu, Y., and Xiao, J. (2021). Biomimetic Prosthetic Hand Enabled by Liquid Crystal Elastomer Tendons. Micromachines, 12.","DOI":"10.3390\/mi12070736"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"25689","DOI":"10.1021\/acsami.8b07563","article-title":"Light or Thermally Powered Autonomous Rolling of an Elastomer Rod","volume":"10","author":"Ahn","year":"2018","journal-title":"ACS. Appl. Mater. Interfaces"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1039\/B916831A","article-title":"Liquid crystalline polymer cantilever oscillators fueled by light","volume":"6","author":"Serak","year":"2010","journal-title":"Soft Matter"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4986","DOI":"10.1002\/anie.200800760","article-title":"Photomobile Polymer Materials: Towards Light-Driven Plastic Motors","volume":"47","author":"Yamada","year":"2008","journal-title":"Angew. Chem. Int. Ed."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"eabi9704","DOI":"10.1126\/scirobotics.abi9704","article-title":"Electrospun liquid crystal elastomer microfiber actuator","volume":"6","author":"He","year":"2021","journal-title":"Sci. Robot."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"101116","DOI":"10.1016\/j.eml.2020.101116","article-title":"Self-sustained rolling of a thermally responsive rod on a hot surface","volume":"42","author":"Li","year":"2021","journal-title":"Extreme Mech. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2106175","DOI":"10.1002\/adma.202106175","article-title":"Liquid Crystal Elastomer Metamaterials with Giant Biaxial Thermal Shrinkage for Enhancing Skin Regeneration","volume":"33","author":"Wu","year":"2021","journal-title":"Adv. Mater."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2100139","DOI":"10.1002\/aisy.202100139","article-title":"Magnetic Arthropod Millirobots Fabricated by 3D-Printed Hydrogels","volume":"4","author":"Sun","year":"2021","journal-title":"Adv. Intell. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1038\/nature11223","article-title":"Synthetic homeostatic materials with chemo-mechano-chemical self-regulation","volume":"487","author":"He","year":"2012","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2629","DOI":"10.1126\/scirobotics.aar2629","article-title":"Hygrobot: A self-locomotive ratcheted actuator powered by environmental humidity","volume":"3","author":"Shin","year":"2018","journal-title":"Sci. Robot."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.apm.2021.08.013","article-title":"Deep neural networks for large deformation of photo-thermo-pH responsive cationic gels","volume":"100","author":"Su","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"043523","DOI":"10.1063\/1.4891520","article-title":"Chemomechanical oscillations in a responsive gel induced by an autocatalytic reaction","volume":"116","author":"Li","year":"2014","journal-title":"J. Appl. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1906233","DOI":"10.1002\/adma.201906233","article-title":"Kirigami-based light-induced shape-morphing and locomotion","volume":"32","author":"Cheng","year":"2019","journal-title":"Adv. Mater."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"258002","DOI":"10.1103\/PhysRevLett.124.258002","article-title":"Self-excited motions of volatile drops on swellable sheets","volume":"124","author":"Chakrabarti","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"111646","DOI":"10.1016\/j.chaos.2021.111646","article-title":"A light-fueled self-oscillating liquid crystal elastomer balloon with self-shading effect","volume":"155","author":"Cheng","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1702","DOI":"10.1038\/s41563-021-00931-6","article-title":"Coupled liquid crystalline oscillators in Huygens\u2019 synchrony","volume":"18","author":"Vantomme","year":"2021","journal-title":"Nat. Mater."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"108446","DOI":"10.1016\/j.ymssp.2021.108446","article-title":"Multistability and synchronization: The co-existence of synchronous patterns in coupled pendula","volume":"166","author":"Dudkowski","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1504","DOI":"10.1038\/s41467-017-01190-3","article-title":"Oscillators that sync and swarm","volume":"8","author":"Hong","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ymssp.2018.04.025","article-title":"Synchronization of two self-excited pendula: Influence of coupling structure\u2019s parameters","volume":"112","author":"Dudkowski","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1098\/rspa.2001.0888","article-title":"Huygens\u2019s clocks","volume":"458","author":"Bennett","year":"2019","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1119\/1.1475332","article-title":"Synchronization: A Universal Concept in Nonlinear Science","volume":"70","author":"Pikovsky","year":"2002","journal-title":"Am. J. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.physrep.2012.03.004","article-title":"Collective motion","volume":"517","author":"Vicsek","year":"2012","journal-title":"Physiol. Rep."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1594","DOI":"10.1038\/s41563-021-01159-0","article-title":"Synchronized dancing under light","volume":"20","author":"Yu","year":"2021","journal-title":"Nat. Mater."},{"key":"ref_38","unstructured":"Huygens, C. (1899). Oeuvres Compl\u00e8tes de Christiaan Huygens, Martinus Nijhoff."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"23580","DOI":"10.1038\/srep23580","article-title":"The sympathy of two pendulum clocks: Beyond Huygens\u2019 observations","volume":"6","author":"Ramirez","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"7248","DOI":"10.1016\/j.jsv.2014.08.030","article-title":"An improved model for the classical Huygens\u2019 experiment on synchronization of pendulum clocks","volume":"333","author":"Ramirez","year":"2014","journal-title":"J. Sound Vib."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"233906","DOI":"10.1103\/PhysRevLett.109.233906","article-title":"Synchronization of micromechanical oscillators using light","volume":"109","author":"Zhang","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"363","DOI":"10.3390\/cryst3020363","article-title":"Soft elasticity in main chain liquid crystal elastomers","volume":"3","author":"Dey","year":"2013","journal-title":"Crystals"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1906319","DOI":"10.1002\/adma.201906319","article-title":"Near-infrared photodriven self-sustained oscillation of liquid-crystalline network film with predesignated polydopamine coating","volume":"32","author":"Lan","year":"2020","journal-title":"Adv. Mater."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2000252","DOI":"10.1002\/adfm.202000252","article-title":"Light-Driven Liquid Crystalline Networks and Soft Actuators with Degree-of-Freedom-Controlled Molecular Motors","volume":"30","author":"Lan","year":"2020","journal-title":"Adv. Funct. Mater."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1606467","DOI":"10.1002\/adma.201606467","article-title":"Tunable photo controlled motions using stored strain energy in malleable azobenzene liquid crystalline polymer actuators","volume":"29","author":"Lu","year":"2017","journal-title":"Adv. Mater."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1701814","DOI":"10.1002\/adma.201701814","article-title":"Self-regulating iris based on light-actuated liquid crystal elastomer","volume":"29","author":"Zeng","year":"2017","journal-title":"Adv. Mater."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"11975","DOI":"10.1038\/ncomms11975","article-title":"Chaotic self-oscillating sunlight-driven polymer actuator","volume":"7","author":"Kumar","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2000609","DOI":"10.1002\/adma.202000609","article-title":"Blueprinting photothermal shape-morphing of liquid crystal elastomers","volume":"32","author":"Kuenstler","year":"2020","journal-title":"Adv. Mater."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"105366","DOI":"10.1016\/j.ijmecsci.2019.105366","article-title":"Controllable vibration of liquid crystal elastomer beams under periodic illumination","volume":"170","author":"Zhao","year":"2020","journal-title":"Int. J. Mech. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"101547","DOI":"10.1016\/j.eml.2021.101547","article-title":"Thermally driven self-oscillation of an elastomer fiber with a hanging weight","volume":"50","author":"Li","year":"2021","journal-title":"Extreme Mech. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"061709","DOI":"10.1103\/PhysRevE.83.061709","article-title":"Measurement of electrically induced shear strain in a chiral smectic liquid-crystal elastomer","volume":"83","author":"Na","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1039\/b817340k","article-title":"Deformation and rotations of free nematic elastomers in response to electric fields","volume":"5","author":"Corbett","year":"2009","journal-title":"Soft Matter"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1002\/adma.201204406","article-title":"Liquid-crystalline elastomer-nanoparticle hybrids with reversible switch of magnetic memory","volume":"25","author":"Haber","year":"2013","journal-title":"Adv. Mater."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1857","DOI":"10.1021\/nl0514590","article-title":"Self-assembled polymer films for controlled agent-driven motion","volume":"5","author":"Harris","year":"2005","journal-title":"Nano Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"108140","DOI":"10.1016\/j.ymssp.2021.108140","article-title":"Light-powered self-excited oscillation of a liquid crystal elastomer pendulum","volume":"163","author":"Liang","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"5057","DOI":"10.1038\/s41467-019-13077-6","article-title":"Light-fuelled freestyle self-oscillators","volume":"10","author":"Zeng","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"2437","DOI":"10.1007\/s11071-021-06250-4","article-title":"Light-powered self-excited motion of a liquid crystal elastomer rotator","volume":"103","author":"Cheng","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"033004","DOI":"10.1103\/PhysRevE.103.033004","article-title":"Self-sustained eversion or inversion of a thermally responsive torus","volume":"103","author":"Li","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"3211","DOI":"10.1038\/s41467-021-23562-6","article-title":"Phototunable self-oscillating system driven by a self-winding fiber actuator","volume":"12","author":"Hu","year":"2021","journal-title":"Nat. Commun."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"106686","DOI":"10.1016\/j.ijmecsci.2021.106686","article-title":"Light-powered self-excited bouncing of a liquid crystal elastomer ball","volume":"208","author":"Xu","year":"2021","journal-title":"Int. J. Mech. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1038\/nature22987","article-title":"Making waves in a photoactive polymer film","volume":"546","author":"Gelebart","year":"2017","journal-title":"Nature"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1038\/425145a","article-title":"Photomechanics: Directed bending of a polymer film by light","volume":"425","author":"Yu","year":"2003","journal-title":"Nature"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"015501","DOI":"10.1103\/PhysRevLett.87.015501","article-title":"A new opto-mechanical effect in solids","volume":"87","author":"Finkelmann","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_64","unstructured":"Brannon, R. (2008). Kinematics: The Mathematics of Deformation, University of Utah. Course Notes, ME EN."},{"key":"ref_65","unstructured":"Warner, M., and Terentjev, E.M. (2007). Liquid Crystal Elastomers, Oxford University Press."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"041720","DOI":"10.1103\/PhysRevE.65.041720","article-title":"UV Manipulation of order and macroscopic shape in nematic elastomers","volume":"65","author":"Hogan","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"254102","DOI":"10.1063\/1.3670502","article-title":"Bending kinetics of a photo-actuating nematic elastomer cantilever","volume":"99","author":"Torras","year":"2011","journal-title":"Appl. Phys. Lett."}],"container-title":["Micromachines"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-666X\/13\/2\/271\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:16:07Z","timestamp":1760134567000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-666X\/13\/2\/271"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,8]]},"references-count":67,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["mi13020271"],"URL":"https:\/\/doi.org\/10.3390\/mi13020271","relation":{},"ISSN":["2072-666X"],"issn-type":[{"value":"2072-666X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,8]]}}}