{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,13]],"date-time":"2026-06-13T18:58:04Z","timestamp":1781377084293,"version":"3.54.1"},"reference-count":24,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,6,3]],"date-time":"2021-06-03T00:00:00Z","timestamp":1622678400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>With the rapid development of wearable electronics, novel power solutions are required to adapt to flexible surfaces for widespread applications, thus flexible energy harvesters have been extensively studied for their flexibility and stretchability. However, poor power output and insufficient sensitivity to environmental changes limit its widespread application in engineering practice. A doubly clamped flexible piezoelectric energy harvester (FPEH) with axial excitation is therefore proposed for higher power output in a low-frequency vibration environment. Combining the Euler\u2013Bernoulli beam theory and the D\u2019Alembert principle, the differential dynamic equation of the doubly clamped energy harvester is derived, in which the excitation mode of axial load with pre-deformation is considered. A numerical solution of voltage amplitude and average power is obtained using the Rayleigh\u2013Ritz method. Output power of 22.5 \u03bcW at 27.1 Hz, with the optimal load resistance being 1 M\u03a9, is determined by the frequency sweeping analysis. In order to power electronic devices, the converted alternating electric energy should be rectified into direct current energy. By connecting to the MDA2500 standard rectified electric bridge, a rectified DC output voltage across the 1 M\u03a9 load resistor is characterized to be 2.39 V. For further validation of the mechanical-electrical dynamical model of the doubly clamped flexible piezoelectric energy harvester, its output performances, including both its frequency response and resistance load matching performances, are experimentally characterized. From the experimental results, the maximum output power is 1.38 \u03bcW, with a load resistance of 5.7 M\u03a9 at 27 Hz, and the rectified DC output voltage reaches 1.84 V, which shows coincidence with simulation results and is proved to be sufficient for powering LED electronics.<\/jats:p>","DOI":"10.3390\/s21113861","type":"journal-article","created":{"date-parts":[[2021,6,3]],"date-time":"2021-06-03T21:03:32Z","timestamp":1622754212000},"page":"3861","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Analytical Modeling of a Doubly Clamped Flexible Piezoelectric Energy Harvester with Axial Excitation and Its Experimental Characterization"],"prefix":"10.3390","volume":"21","author":[{"given":"Jie","family":"Mei","sequence":"first","affiliation":[{"name":"Institute of Intelligent Manufacturing and Control, Wuhan University of Technology, Wuhan 430063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiong","family":"Fan","sequence":"additional","affiliation":[{"name":"Institute of Intelligent Manufacturing and Control, Wuhan University of Technology, Wuhan 430063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4630-7692","authenticated-orcid":false,"given":"Lijie","family":"Li","sequence":"additional","affiliation":[{"name":"College of Engineering, Swansea University, Swansea SA1 8EN, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dingfang","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Intelligent Manufacturing and Control, Wuhan University of Technology, Wuhan 430063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lin","family":"Xu","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, School of Materials Science and Engineering, Wuhan University of Technology, Wuhan 430070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qingyang","family":"Dai","sequence":"additional","affiliation":[{"name":"Institute of Intelligent Manufacturing and Control, Wuhan University of Technology, Wuhan 430063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qi","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Intelligent Manufacturing and Control, Wuhan University of Technology, Wuhan 430063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s10439-012-0623-3","article-title":"Energy harvesting from the beating heart by a mass imbalance oscillation generator","volume":"41","author":"Zurbuchen","year":"2013","journal-title":"Ann. Biomed. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1700341","DOI":"10.1002\/adfm.201700341","article-title":"In vivo self--powered wireless transmission using biocompatible flexible energy harvesters","volume":"27","author":"Kim","year":"2017","journal-title":"Adv. Funct. Mater."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1705195","DOI":"10.1002\/adma.201705195","article-title":"Shape memory polymers for body motion energy harvesting and self--powered mechanosensing","volume":"30","author":"Liu","year":"2018","journal-title":"Adv. Mater."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1016\/j.joule.2018.03.011","article-title":"High-performance piezoelectric energy harvesters and their applications","volume":"2","author":"Yang","year":"2018","journal-title":"Joule"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"041306","DOI":"10.1063\/1.5074184","article-title":"A comprehensive review on piezoelectric energy harvesting technology: Materials, mechanisms, and applications","volume":"5","author":"Liu","year":"2018","journal-title":"Appl. Phys. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7405","DOI":"10.1021\/nn502618f","article-title":"Harvesting broadband kinetic impact energy from mechanical triggering\/vibration and water waves","volume":"8","author":"Wen","year":"2014","journal-title":"ACS Nano"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1331","DOI":"10.1021\/nl104412b","article-title":"Enhanced piezoelectricity and stretchability in energy harvesting devices fabricated from buckled PZT ribbons","volume":"11","author":"Qi","year":"2011","journal-title":"Nano Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"10024","DOI":"10.1002\/adma.201603527","article-title":"Electric eel--skin--inspired mechanically durable and super--stretchable nanogenerator for deformable power source and fully autonomous conformable electronic--skin applications","volume":"28","author":"Lai","year":"2016","journal-title":"Adv. Mater."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1515\/ehs-2016-0028","article-title":"A review on piezoelectric energy harvesting: Materials, methods, and circuits","volume":"4","author":"Priya","year":"2019","journal-title":"Energy Harvest. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2020.3026462","article-title":"Power Conversion Efficiency of Cantilever-Type Vibration Energy Harvesters Based on Piezoceramic Films","volume":"70","author":"Quattrocchi","year":"2020","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"060003","DOI":"10.1063\/1.4952675","article-title":"Experimental characterization of cantilever-type piezoelectric generator operating at resonance for vibration energy harvesting","volume":"Volume 1740","author":"Montanini","year":"2016","journal-title":"AIP Conference Proceedings"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"035021","DOI":"10.1088\/0964-1726\/21\/3\/035021","article-title":"Piezoelectric buckled beams for random vibration energy harvesting","volume":"21","author":"Cottone","year":"2012","journal-title":"Smart Mater. Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"087101","DOI":"10.7567\/JJAP.53.087101","article-title":"A low frequency and broadband piezoelectric energy harvester using asymmetrically serials connected double clamped\u2013clamped beams","volume":"53","author":"Liang","year":"2014","journal-title":"Jpn. J. Appl. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1007\/s00542-014-2237-1","article-title":"Theoretical analysis of energy harvesting performance for clamped\u2013clamped piezoelectric beam","volume":"21","author":"Liang","year":"2015","journal-title":"Microsyst. Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.ymssp.2018.06.003","article-title":"Improving energy harvesting from random excitation by nonlinear flexible bi-stable energy harvester with a variable potential energy function","volume":"115","author":"Zhou","year":"2019","journal-title":"Mech. Syst. Signal Process"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Emad, A., Mahmoud, M.A., Ghoneima, M., and Dessouky, M. (2016, January 16\u201319). Modeling and analysis of stretching strain in clamped-clamped beams for energy harvesting. Proceedings of the 2016 IEEE 59th International Midwest Symposium on Circuits and Systems (MWSCAS), Abu Dhabi, United Arab Emirates.","DOI":"10.1109\/MWSCAS.2016.7870073"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kashyap, R., Lenka, T.R., and Baishya, S. (2016, January 17\u201319). Study of doubly clamped piezoelectric beam energy harvesters with non-traditional geometries. Proceedings of the 2016 7th India International Conference on Power Electronics (IICPE), Patiala, India.","DOI":"10.1109\/IICPE.2016.8079535"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"055028","DOI":"10.1063\/1.5023824","article-title":"Fabrication and characterization of a piezoelectric energy harvester with clamped-clamped beams","volume":"8","author":"Cui","year":"2018","journal-title":"AIP Adv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1087946","DOI":"10.34133\/2019\/1087946","article-title":"Buckled MEMS Beams for Energy Harvesting from Low Frequency Vibrations","volume":"2019","author":"Xu","year":"2019","journal-title":"Research"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Su, W.J., and Wang, Z.S. (2021). Development of a Non-Linear Bi-Directional Vortex-Induced Piezoelectric Energy Harvester with Magnetic Interaction. Sensors, 21.","DOI":"10.3390\/s21072299"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"045015","DOI":"10.1088\/1361-665X\/aaf5ae","article-title":"Simulation and experiment on bridge-shaped nonlinear piezoelectric vibration energy harvester","volume":"28","author":"Qin","year":"2019","journal-title":"Smart Mater. Struct."},{"key":"ref_22","first-page":"1","article-title":"Improved energy harvesting from a clamped\u2013clamped micro beam with cavity","volume":"4","author":"Yenuganti","year":"2020","journal-title":"Microsyst. Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"111807","DOI":"10.1016\/j.sna.2019.111807","article-title":"Analysis and test of a novel pre-compressed cruciform energy harvester","volume":"302","author":"Li","year":"2020","journal-title":"Sensors Actuators A Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"103639","DOI":"10.1016\/j.ijnonlinmec.2020.103639","article-title":"Dynamic analysis of straight stepped microbeams","volume":"128","author":"Alcheikh","year":"2020","journal-title":"Int. J. Non-Linear Mech."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/11\/3861\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:10:29Z","timestamp":1760163029000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/11\/3861"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,3]]},"references-count":24,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["s21113861"],"URL":"https:\/\/doi.org\/10.3390\/s21113861","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,3]]}}}