{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T03:26:10Z","timestamp":1771903570290,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T00:00:00Z","timestamp":1664323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004442","name":"National Science Centre, Poland","doi-asserted-by":"publisher","award":["2021\/40\/Q\/ST8\/00362"],"award-info":[{"award-number":["2021\/40\/Q\/ST8\/00362"]}],"id":[{"id":"10.13039\/501100004442","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004442","name":"National Science Centre, Poland","doi-asserted-by":"publisher","award":["52161135106"],"award-info":[{"award-number":["52161135106"]}],"id":[{"id":"10.13039\/501100004442","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021\/40\/Q\/ST8\/00362"],"award-info":[{"award-number":["2021\/40\/Q\/ST8\/00362"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52161135106"],"award-info":[{"award-number":["52161135106"]}],"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 subject of the research contained in this paper is a new design solution for an energy harvesting system resulting from the combination of a quasi-zero-stiffness energy harvester and a two-stage flexible cantilever beam. Numerical tests were divided into two main parts-analysis of the dynamics of the system due to periodic, quasiperiodic, and chaotic solutions and the efficiency of energy generation. The results of numerical simulations were limited to zero initial conditions as they are the natural position of the static equilibrium. The article compares the energy efficiency for the selected range of the dimensionless excitation frequency. For this purpose, three cases of piezoelectric mounting were analyzed-only on the first stage of the beam, on the second and both stages. The analysis has been carried out with the use of diagrams showing difference of the effective values of the voltage induced on the piezoelectric electrodes. The results indicate that for effective energy harvesting, it is advisable to attach piezoelectric energy transducers to each step of the beam despite possible asynchronous vibrations.<\/jats:p>","DOI":"10.3390\/s22197399","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T01:23:16Z","timestamp":1664414596000},"page":"7399","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Energy Harvesting in a System with a Two-Stage Flexible Cantilever Beam"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2249-4059","authenticated-orcid":false,"given":"Jerzy","family":"Margielewicz","sequence":"first","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\/0000-0002-9647-8345","authenticated-orcid":false,"given":"Grzegorz","family":"Litak","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, Lublin University of Technology, 20-618 Lublin, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8603-4647","authenticated-orcid":false,"given":"Piotr","family":"Wolszczak","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, Lublin University of Technology, 20-618 Lublin, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4034-193X","authenticated-orcid":false,"given":"Shengxi","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Aeronautics, Northwestern Polytechnical University, Xi\u2019an 710072, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1457","DOI":"10.1109\/JPROC.2008.927494","article-title":"Energy harvesting from human and machine motion for wireless electronic devices","volume":"96","author":"Mitcheson","year":"2008","journal-title":"Proc. IEEE"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"042901","DOI":"10.1063\/1.3679102","article-title":"Powering pacemakers from heartbeat vibrations using linear and nonlinear energy harvesters","volume":"100","author":"Karami","year":"2022","journal-title":"Appl. Phys. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"116886","DOI":"10.1016\/j.jsv.2022.116886","article-title":"Multistable vibration energy harvesters: Principle, progress, and perspectives","volume":"528","author":"Zhou","year":"2022","journal-title":"J. Sound Vib."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.ymssp.2016.07.047","article-title":"Enhancing ability of harvesting energy from random vibration by decreasing the potential barrier of bistable harvester","volume":"85","author":"Lan","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/j.sna.2017.08.039","article-title":"Improving efficiency of energy harvesting by a novel penta-stable configuration","volume":"265","author":"Zhou","year":"2017","journal-title":"Sens. Actuators A Phys."},{"key":"ref_7","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_8","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_9","doi-asserted-by":"crossref","first-page":"410003","DOI":"10.1063\/5.0081318","article-title":"Effect of stochastic excitation on sub-harmonic solutions in a bistable energy harvester","volume":"2425","author":"Cunha","year":"2022","journal-title":"AIP Conf. Proc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1109\/JETCAS.2014.2337172","article-title":"Challenges for Energy Harvesting Systems Under Intermittent Excitation","volume":"4","author":"Yang","year":"2014","journal-title":"IEEE J. Emerg. Sel. Top. Circuits Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"051019","DOI":"10.1115\/1.4036867","article-title":"Harvesting Energy from Time-Limited Harmonic Vibrations: Mechanical Considerations","volume":"139","author":"Brennan","year":"2017","journal-title":"J. Vib. Acoust."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"105890","DOI":"10.1016\/j.nanoen.2021.105890","article-title":"Integrated study of triboelectric nanogenerator for ocean wave energy harvesting: Performance assessment in realistic sea conditions","volume":"84","author":"Rodrigues","year":"2021","journal-title":"Nano Energy"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"012014","DOI":"10.1088\/1755-1315\/617\/1\/012014","article-title":"Research on Piezoelectric Energy harvesting from Multi-Direction Wind-Induced Vibrations","volume":"617","author":"Chen","year":"2020","journal-title":"IOP Conf. Ser. Earth Environ. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"012080","DOI":"10.1088\/1742-6596\/744\/1\/012080","article-title":"Energy harvesting from the vibrations of a passing train: Effect of speed variability","volume":"744","author":"Cleante","year":"2016","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"116140","DOI":"10.1016\/j.energy.2019.116140","article-title":"Piezoelectric energy harvesting from human walking using a two-stage amplification mechanism","volume":"189","author":"Qiana","year":"2019","journal-title":"Energy"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"114516","DOI":"10.1016\/j.apenergy.2020.114516","article-title":"Coupling mechanical and electrical nonlinearities: The effect of synchronized discharging on tristable energy harvesters","volume":"266","author":"Lallart","year":"2020","journal-title":"Appl. Energy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1002\/er.5643","article-title":"Vibration-based piezoelectric, electromagnetic, and hybrid energy harvesters for microsystems applications: A contributed review","volume":"45","author":"Iqbal","year":"2021","journal-title":"Int. J. Energy Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"103853","DOI":"10.1016\/j.ijnonlinmec.2021.103853","article-title":"Response of a stochastic multiple attractors wind-induced vibration energy harvesting system with impacts","volume":"138","author":"Zhang","year":"2021","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_19","first-page":"393","article-title":"Response analysis of the nonlinear vibration energy harvester with an uncertain parameter","volume":"234","author":"Huang","year":"2020","journal-title":"Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1950052","DOI":"10.1142\/S0219455419500524","article-title":"Uncertainty analysis of excitation conditions on performance of nonlinear mono-stable energy harvesters","volume":"19","author":"Li","year":"2019","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_21","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_22","doi-asserted-by":"crossref","first-page":"222","DOI":"10.5755\/j02.mech.31034","article-title":"Vibration energy harvester (veh) for a free body moving in space at vibro shock mode","volume":"28","author":"Gaidys","year":"2022","journal-title":"Mechanika"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"033405","DOI":"10.1088\/1742-5468\/ab0c15","article-title":"Novel multi-stable energy harvester by exploring the benefits of geometric nonlinearity","volume":"2019","author":"Yang","year":"2019","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"109291","DOI":"10.1016\/j.ymssp.2022.109291","article-title":"Promote performance of vibration energy harvesting by amplified inertial force and clamped piezoelectric beams","volume":"178","author":"Liu","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.ymssp.2018.01.024","article-title":"Dynamics identification of a piezoelectric vibrational energy harvester by image analysis with a high speed camera","volume":"107","author":"Wolszczak","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"040801","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_27","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/j.apenergy.2018.06.011","article-title":"Drastic bandwidth enhancement of bistable energy harvesters: Study of subharmonic behaviors and their stability robustness","volume":"226","author":"Huguet","year":"2018","journal-title":"Appl. Energy"},{"key":"ref_28","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_29","doi-asserted-by":"crossref","first-page":"122048","DOI":"10.1016\/j.energy.2021.122048","article-title":"A new hybrid piezo\/triboelectric SbSeI nanogenerator","volume":"238","author":"Mistewicz","year":"2022","journal-title":"Energy"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"113001","DOI":"10.1088\/1361-665X\/ab36e4","article-title":"A review of energy harvesting using piezoelectric materials: State-of-the-art a decade later (2008\u20132018)","volume":"28","author":"Safaei","year":"2019","journal-title":"Smart Mater. Struct."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.compositesb.2018.11.028","article-title":"New energy harvester with embedded piezoelectric stacks","volume":"163","author":"Keshmiri","year":"2019","journal-title":"Compos. Part B Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"112743","DOI":"10.1016\/j.sna.2021.112743","article-title":"A review on vibration-based piezoelectric energy harvesting from the aspect of compliant mechanisms","volume":"331","author":"Liang","year":"2021","journal-title":"Sens. Actuators A Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1016\/j.ymssp.2015.05.026","article-title":"Analysis and design of a nonlinear stiffness and damping system with a scissor-like structure","volume":"66-67","author":"Sun","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"106035","DOI":"10.1016\/j.cnsns.2021.106035","article-title":"Optimizing elastic potential energy via geometric nonlinear stiffness","volume":"103","author":"Gatti","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"020801","DOI":"10.1115\/1.4049946","article-title":"Bio-inspired vibration isolation: Methodology and design","volume":"73","author":"Yan","year":"2021","journal-title":"Appl. Mech. Rev."},{"key":"ref_36","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_37","doi-asserted-by":"crossref","unstructured":"Litak, G., Margielewicz, J., Gaska, D., Rysak, A., and Trigona, C. (2022). On Theoretical and numerical aspects of bifurcations and hysteresis effects in kinetic energy harvesters. Sensors, 22.","DOI":"10.3390\/s22010381"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2917","DOI":"10.1007\/s11071-022-07444-0","article-title":"Beneficial effects of antisymmetric nonlinear damping with application to energy harvesting and vibration isolation under general inputs","volume":"108","author":"Zhu","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"107890","DOI":"10.1016\/j.ymssp.2021.107890","article-title":"Development and experimental validation of self-powered wireless vibration sensor node using vibration energy harvester","volume":"160","author":"Rubes","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"121770","DOI":"10.1016\/j.energy.2021.121770","article-title":"Piezoelectric energy harvester for rolling bearings with capability of self-powered condition monitoring","volume":"238","author":"Zhang","year":"2022","journal-title":"Energy"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"116334","DOI":"10.1016\/j.jsv.2021.116334","article-title":"Dynamic responses of a two-degree-of-freedom bistable electromagnetic energy harvester under filtered band-limited stochastic excitation","volume":"511","author":"Li","year":"2021","journal-title":"J. Sound Vib."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/978-981-13-9463-8_3","article-title":"Nonlinear Characterization of a Bistable Energy Harvester Dynamical System","volume":"228","author":"Lopes","year":"2019","journal-title":"Springer Proc. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"119161","DOI":"10.1016\/j.apenergy.2022.119161","article-title":"A numerical-experimental dynamic analysis of high-efficiency and broadband bistable energy harvester with self-decreasing potential barrier effect","volume":"317","author":"Chen","year":"2022","journal-title":"Appl. Energy"},{"key":"ref_44","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_45","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.cnsns.2018.02.017","article-title":"Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting","volume":"61","author":"Zhou","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"110789","DOI":"10.1016\/j.engstruct.2020.110789","article-title":"A magnetic levitation-based tristable hybrid energy harvester for scavenging energy from low-frequency structural vibration","volume":"221","author":"Yang","year":"2020","journal-title":"Eng. Struct."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.ijmecsci.2019.03.034","article-title":"Dynamics and performance evaluation of a novel tristable hybrid energy harvester for ultra-low level vibration resources","volume":"156","author":"Yang","year":"2019","journal-title":"Int. J. Mech. Sci."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Zayed, A.A.A., Assal, S.F.M., Nakano, K., Kaizuka, T., and El-Bab, A.M.R.F. (2019). Design procedure and experimental verification of a broadband quad-stable 2-DOF vibration energy harvester. Sensors, 19.","DOI":"10.3390\/s19132893"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1140\/epjp\/s13360-022-02807-3","article-title":"Research on multi-valued response and bursting oscillation of series multi-stable piezoelectric energy harvester","volume":"137","author":"Qian","year":"2022","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.ymssp.2018.04.027","article-title":"A low-frequency, wideband quad-stable energy harvester using combined nonlinearity and frequency up-conversion by cantilever-surface contact","volume":"112","author":"Wang","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1665","DOI":"10.1140\/epjst\/e2013-01953-5","article-title":"Stationary response of nonlinear magneto-piezoelectric energy harvester systems under stochastic excitation","volume":"222","author":"Martens","year":"2013","journal-title":"Eur. Phys. J. Sp\u00e9c. Top."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Margielewicz, J., G\u0105ska, D., Litak, G., Wolszczak, P., and Trigona, C. (2022). Nonlinear dynamics of a star-shaped structure and variable configuration of elastic elements for energy harvesting applications. Sensors, 22.","DOI":"10.3390\/s22072518"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"118159","DOI":"10.1016\/j.apenergy.2021.118159","article-title":"Nonlinear dynamics of a new energy harvesting system with quasi-zero stiffness","volume":"307","author":"Margielewicz","year":"2022","journal-title":"Appl. Energy"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"114361","DOI":"10.1016\/j.enconman.2021.114361","article-title":"Design and characteristic analysis of magnetostrictive bistable vibration harvester with displacement amplification mechanism","volume":"243","author":"Liu","year":"2021","journal-title":"Energy Convers. Manag."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.egyr.2021.07.119","article-title":"Damping capacity of materials and its effect on the dynamic behavior of structures. Review","volume":"7","author":"Poddaeva","year":"2021","journal-title":"Energy Rep."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Litak, G., Margielewicz, J., G\u0105ska, D., Wolszczak, P., and Zhou, S. (2021). Multiple solutions of the tristable energy harvester. Energies, 14.","DOI":"10.3390\/en14051284"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/19\/7399\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:41:33Z","timestamp":1760143293000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/19\/7399"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,28]]},"references-count":56,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["s22197399"],"URL":"https:\/\/doi.org\/10.3390\/s22197399","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,28]]}}}