{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T18:11:52Z","timestamp":1777918312241,"version":"3.51.4"},"reference-count":29,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,5]],"date-time":"2022-01-05T00:00:00Z","timestamp":1641340800000},"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>The piezoelectric energy-harvesting system with double-well characteristics and hysteresis in the restoring force is studied. The proposed system consists of a bistable oscillator based on a cantilever beam structure. The elastic force potential is modified by magnets. The hysteresis is an additional effect of the composite beam considered in this system, and it effects the modal solution with specific mass distribution. Consequently, the modal response is a compromise between two overlapping, competing shapes. The simulation results show evolution in the single potential well solution, and bifurcations into double-well solutions with the hysteretic effect. The maximal Lyapunov exponent indicated the appearance of chaotic solutions. Inclusion of the shape branch overlap parameter reduces the distance between the external potential barriers and leads to a large-amplitude solution and simultaneously higher voltage output with smaller excitation force. The overlap parameter works in the other direction: the larger the overlap value, the smaller the voltage output. Presumably, the successful jump though the potential barrier is accompanied by an additional switch between the corresponding shapes.<\/jats:p>","DOI":"10.3390\/s22010381","type":"journal-article","created":{"date-parts":[[2022,1,9]],"date-time":"2022-01-09T23:08:26Z","timestamp":1641769706000},"page":"381","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9647-8345","authenticated-orcid":false,"given":"Grzegorz","family":"Litak","sequence":"first","affiliation":[{"name":"Faculty of Mechanical Engineering, Lublin University of Technology, 20-618 Lublin, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2249-4059","authenticated-orcid":false,"given":"Jerzy","family":"Margielewicz","sequence":"additional","affiliation":[{"name":"Faculty of Transport and Aviation Engineering, Silesian University of Technology, 40-019 Katowice, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2968-1626","authenticated-orcid":false,"given":"Damian","family":"G\u0105ska","sequence":"additional","affiliation":[{"name":"Faculty of Transport and Aviation Engineering, Silesian University of Technology, 40-019 Katowice, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9631-9785","authenticated-orcid":false,"given":"Andrzej","family":"Rysak","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, Lublin University of Technology, 20-618 Lublin, Poland"}]},{"given":"Carlo","family":"Trigona","sequence":"additional","affiliation":[{"name":"Dipartimento di Ingegneria Elettrica Elettronica e Informatica, University of Catania, 95125 Catania, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ka\u017amierski, T.J., and Beeby, S. (2011). Energy Harvesting Systems, Principles, Modelling and Applications, Springer.","DOI":"10.1007\/978-1-4419-7566-9"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Priya, S., and Inman, D.J. (2009). Energy Harvesting Technologies, Springer.","DOI":"10.1007\/978-0-387-76464-1"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Erturk, A., and Inman, D.J. (2011). Piezoelectric Energy Harvesting, John Wiley and Sons.","DOI":"10.1002\/9781119991151"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"254102","DOI":"10.1063\/1.3159815","article-title":"A piezomagnetoelastic structure for broadband vibration energy harvesting","volume":"94","author":"Erturk","year":"2009","journal-title":"Appl. Phys. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"080601","DOI":"10.1103\/PhysRevLett.102.080601","article-title":"Nonlinear energy harvesting","volume":"102","author":"Cottone","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/j.sna.2010.05.022","article-title":"Improved energy harvesting from wideband vibrations by nonlinear piezoelectric converters","volume":"162","author":"Ferrari","year":"2010","journal-title":"Sens. Actuators A Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1505","DOI":"10.1177\/1045389X12455722","article-title":"Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass","volume":"23","author":"Friswell","year":"2012","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.sna.2013.02.025","article-title":"Analysis of two dimensional, wide-band, bistable vibration energy harvester","volume":"202","author":"Baglio","year":"2013","journal-title":"Sens. Actuators A Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"023001","DOI":"10.1088\/0964-1726\/22\/2\/023001","article-title":"A review of the recent research on vibration energy harvesting via bistable systems","volume":"22","author":"Harne","year":"2013","journal-title":"Smart Mater. Struct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1177\/1045389X12444940","article-title":"Bistable vibration energy harvesters: A review","volume":"24","author":"Pellegrini","year":"2013","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.ymssp.2019.03.051","article-title":"Orbit jump in bistable energy harvesters through buckling level modification","volume":"128","author":"Huguet","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"012032","DOI":"10.1088\/1742-6596\/1736\/1\/012032","article-title":"Dynamics of a nonlinear energy harvester with subharmonic responses","volume":"1736","author":"Litak","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"40801","DOI":"10.1115\/1.4026278","article-title":"On the role of nonlinearities in vibratory energy harvesting: A critical review and discussion","volume":"66","author":"Daqaq","year":"2014","journal-title":"Appl. Mech. Rev."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1140\/epjb\/e2016-60699-0","article-title":"Multiple solutions and corresponding power output of a nonlinear bistable piezoelectric energy harvester","volume":"89","author":"Syta","year":"2016","journal-title":"Eur. Phys. J. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.cnsns.2018.09.025","article-title":"Theoretical analysis of multi-stable energy harvesters with high order stiffness terms","volume":"69","author":"Huang","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"107989","DOI":"10.1016\/j.ymssp.2021.107989","article-title":"Piezoelectric autoparametric vibration energy harvesting with chaos control feature","volume":"161","author":"Tan","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"113174","DOI":"10.1016\/j.enconman.2020.113174","article-title":"Bio-inspired bi-stable piezoelectric harvester for broadband vibration energy harvesting","volume":"222","author":"Qian","year":"2020","journal-title":"Energy Convers. Manag."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.1007\/s11071-020-05660-0","article-title":"Nonlinear multi-scale dynamics modeling of piezoceramic energy harvesters with ferroelectric and ferroelastic, hysteresis","volume":"100","author":"Montegiglio","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1140\/epjp\/i2016-16109-4","article-title":"Dynamics of a vibrational energy harvester with a bistable beam: Voltage response identification by multiscale entropy and \u201c0\u20131\u201d test","volume":"131","author":"Harris","year":"2016","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1140\/epjb\/e2016-70619-y","article-title":"Output response identification in a multistable system for piezoelectric energy harvesting","volume":"90","author":"Harris","year":"2017","journal-title":"Eur. Phys. J. B"},{"key":"ref_21","unstructured":"Strogatz, S.H. (2015). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Westview Press."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Margielewicz, J., G\u0105ska, D., Litak, G., Wolszczak, P., and Yurchenko, D. (2021). Nonlinear dynamics of a new energy harvesting system with quasi-zero stiffness. Appl. Energy, 118159.","DOI":"10.1016\/j.apenergy.2021.118159"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.apenergy.2014.07.077","article-title":"Broadband tristable energy harvester: Modeling and experiment verification","volume":"133","author":"Zhou","year":"2014","journal-title":"Appl. Energy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1007\/s40430-015-0453-y","article-title":"An overview of 0\u20131 test for chaos","volume":"38","author":"Bernardini","year":"2016","journal-title":"J. Braz. Soc. Mech. Sci. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/0375-9601(94)90991-1","article-title":"A robust method to estimate the maximal Lyapunov exponent of a time series","volume":"185","author":"Kantz","year":"1994","journal-title":"Phys. Lett. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov Exponens from a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Physica D"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Litak, G., Margielewicz, J., Gaska, D., Wolszczak, P., and Zhou, S. (2021). Multiple solutions of the tristable energy harvester. Energies, 14.","DOI":"10.3390\/en14051284"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1007\/s40430-021-03188-x","article-title":"Multiple solutions and transient chaos in a nonlinear flexible coupling model","volume":"43","author":"Margielewicz","year":"2021","journal-title":"J. Braz. Soc. Mech. Sci. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1961","DOI":"10.1007\/s11012-015-0140-1","article-title":"Experimental analysis of the dynamical response of energy harvesting devices based on bistable laminated plates","volume":"50","author":"Syta","year":"2015","journal-title":"Meccanica"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/1\/381\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:00:23Z","timestamp":1760364023000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/1\/381"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,5]]},"references-count":29,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["s22010381"],"URL":"https:\/\/doi.org\/10.3390\/s22010381","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,5]]}}}