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The added value of the present contribution is to demonstrate the benefits of the exploitation of advanced techniques, most often inherited from other fields of physics and engineering, to improve the performance of such systems. The review is focused on the modeling techniques that apply to the entire energy source\/mechanical oscillator\/transducer\/electrical load chain, describing mechanical\u2013electrical analogies to represent the collective behavior as the cascade of equivalent electrical two-ports, introducing matching networks enhancing the energy transfer to the load, and discussing the main numerical techniques in the frequency and time domains that can be used to analyze linear and nonlinear harvesters, both in the case of deterministic and stochastic excitations.<\/jats:p>","DOI":"10.3390\/computation11030045","type":"journal-article","created":{"date-parts":[[2023,2,27]],"date-time":"2023-02-27T03:23:36Z","timestamp":1677468216000},"page":"45","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Circuit Theory Perspective on the Modeling and Analysis of Vibration Energy Harvesting Systems: A Review"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9106-563X","authenticated-orcid":false,"given":"Michele","family":"Bonnin","sequence":"first","affiliation":[{"name":"Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, 10129 Torino, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0564-3905","authenticated-orcid":false,"given":"Kailing","family":"Song","sequence":"additional","affiliation":[{"name":"Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, 10129 Torino, Italy"},{"name":"IUSS, University School for Advanced Studies, 27100 Pavia, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5878-8115","authenticated-orcid":false,"given":"Fabio L.","family":"Traversa","sequence":"additional","affiliation":[{"name":"MemComputing Inc., San Diego, CA 92093-0319, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4964-0100","authenticated-orcid":false,"given":"Fabrizio","family":"Bonani","sequence":"additional","affiliation":[{"name":"Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, 10129 Torino, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Misra, S., Mukherjee, A., and Roy, A. (2021). Introduction to IoT, University of Cambridge ESOL Examinations.","DOI":"10.1017\/9781108913560"},{"key":"ref_2","unstructured":"(2023, February 22). IoT 2022: Connected Devices Growing 18% to 14.4 Billion Globally. Available online: https:\/\/www.iotforall.com\/state-of-iot-2022."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Munoz-Ausecha, C., Ruiz-Rosero, J., and Ramirez-Gonzalez, G. (2021). RFID Applications and Security Review. Computation, 9.","DOI":"10.3390\/computation9060069"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Penella-L\u00f3pez, M.T., and Gasulla-Forner, M. (2011). Powering Autonomous Sensors An Integral Approach with Focus on Solar and RF Energy Harvesting, Springer London, Limited.","DOI":"10.1007\/978-94-007-1573-8"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Roundy, S., Wright, P.K., and Rabaey, J.M. (2003). Energy Scavenging for Wireless Sensor Networks, Springer.","DOI":"10.1007\/978-1-4615-0485-6"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1109\/MPRV.2005.9","article-title":"Energy scavenging for mobile and wireless electronics","volume":"4","author":"Paradiso","year":"2005","journal-title":"IEEE Pervasive Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"R175","DOI":"10.1088\/0957-0233\/17\/12\/R01","article-title":"Energy harvesting vibration sources for microsystems applications","volume":"17","author":"Beeby","year":"2006","journal-title":"Meas. Sci. Technol."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1109\/COMST.2014.2368999","article-title":"Wireless Networks with RF Energy Harvesting: A Contemporary Survey","volume":"17","author":"Lu","year":"2015","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"110201","DOI":"10.35848\/1347-4065\/abbfa0","article-title":"Recent advances and future prospects in energy harvesting technologies","volume":"59","author":"Akinaga","year":"2020","journal-title":"Jpn. J. Appl. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"R1","DOI":"10.1088\/0964-1726\/16\/3\/R01","article-title":"A review of power harvesting using piezoelectric materials (2003\u20132006)","volume":"16","author":"Anton","year":"2007","journal-title":"Smart Mater. Struct."},{"key":"ref_12","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_13","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_14","doi-asserted-by":"crossref","unstructured":"Covaci, C., and Gontean, A. (2020). Piezoelectric energy harvesting solutions: A review. Sensors, 20.","DOI":"10.3390\/s20123512"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1867","DOI":"10.1177\/1045389X10390249","article-title":"Toward broadband vibration-based energy harvesting","volume":"21","author":"Tang","year":"2010","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1109\/TIE.2009.2024652","article-title":"Kinetic energy harvesting using piezoelectric and electromagnetic technologies\u2013state of the art","volume":"57","author":"Khaligh","year":"2009","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.rser.2017.01.073","article-title":"A comprehensive review on vibration energy harvesting: Modelling and realization","volume":"74","author":"Wei","year":"2017","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.chemphys.2010.08.012","article-title":"The benefits of noise and nonlinearity: Extracting energy from random vibrations","volume":"375","author":"Gammaitoni","year":"2010","journal-title":"Chem. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1080\/00107514.2011.647793","article-title":"There\u2019s plenty of energy at the bottom (micro and nano scale nonlinear noise harvesting)","volume":"53","author":"Gammaitoni","year":"2012","journal-title":"Contemp. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Daqaq, M.F., Masana, R., Erturk, A., and Dane Quinn, D. (2014). On the role of nonlinearities in vibratory energy harvesting: A critical review and discussion. Appl. Mech. Rev., 66.","DOI":"10.1115\/1.4026278"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1475-925X-13-79","article-title":"Energy harvesting for the implantable biomedical devices: Issues and challenges","volume":"13","author":"Hannan","year":"2014","journal-title":"Biomed. Eng. Online"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.rser.2016.12.073","article-title":"A review on performance enhancement techniques for ambient vibration energy harvesters","volume":"71","author":"Yildirim","year":"2017","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.ijengsci.2018.02.003","article-title":"Ambient vibration energy harvesters: A review on nonlinear techniques for performance enhancement","volume":"127","author":"Tran","year":"2018","journal-title":"Int. J. Eng. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Nguyen, V.T., Kumar, P., and Leong, J. (2018). Finite Element Modelling and Simulations of Piezoelectric Actuators Responses with Uncertainty Quantification. Computation, 6.","DOI":"10.3390\/computation6040060"},{"key":"ref_26","first-page":"102228","article-title":"Piezoelectric resonator design and analysis from stochastic car vibration using an experimentally validated finite element with viscous-structural damping model","volume":"52","author":"Khazaee","year":"2022","journal-title":"Sustain. Energy Technol. Assess."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"117492","DOI":"10.1016\/j.jsv.2022.117492","article-title":"Four-point bending piezoelectric energy harvester with uniform surface strain toward better energy conversion performance and material usage","volume":"548","author":"Khazaee","year":"2023","journal-title":"J. Sound Vib."},{"key":"ref_28","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_29","doi-asserted-by":"crossref","unstructured":"Erturk, A., and Inman, D.J. (2011). Piezoelectric Energy Harvesting, John Wiley & Sons.","DOI":"10.1002\/9781119991151"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"053903","DOI":"10.1063\/1.4790381","article-title":"Concise and high-fidelity predictive criteria for maximizing performance and robustness of bistable energy harvesters","volume":"102","author":"Harne","year":"2013","journal-title":"Appl. Phys. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s11071-021-06297-3","article-title":"Leveraging circuit theory and nonlinear dynamics for the efficiency improvement of energy harvesting","volume":"104","author":"Bonnin","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_32","unstructured":"Kuznetsov, Y.A., Kuznetsov, I.A., and Kuznetsov, Y. (1998). Elements of Applied Bifurcation Theory, Springer."},{"key":"ref_33","unstructured":"Guckenheimer, J., and Holmes, P. (2013). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1002\/cta.423","article-title":"Harmonic balance, Melnikov method and nonlinear oscillators under resonant perturbation","volume":"36","author":"Bonnin","year":"2008","journal-title":"Int. J. Circuit Theory Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/i2018-12076-0","article-title":"Efficient spectral domain technique for the frequency locking analysis of nonlinear oscillators","volume":"133","author":"Bonnin","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_36","unstructured":"Gardiner, C.W. (1985). Handbook of Stochastic Methods, Springer."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Ksendal, B. (2003). Stochastic Differential Equations, Springer. [6th ed.].","DOI":"10.1007\/978-3-642-14394-6"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1002\/j.1538-7305.1944.tb00874.x","article-title":"Mathematical Analysis of Random Noise","volume":"23","author":"Rice","year":"1944","journal-title":"Bell Syst. Tech. J."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1109\/25.492906","article-title":"A deterministic digital simulation model for Suzuki processes with application to a shadowed Rayleigh land mobile radio channel","volume":"45","author":"Patzold","year":"1996","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Le Ma\u00eetre, O., and Knio, O.M. (2010). Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, Springer.","DOI":"10.1007\/978-90-481-3520-2"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Xiu, D. (2010). Numerical Methods for Stochastic Computations, Princeton University Press.","DOI":"10.1515\/9781400835348"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Smith, R.C. (2013). Uncertainty Quantification: Theory, Implementation, and Applications, SIAM.","DOI":"10.1137\/1.9781611973228"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Kaintura, A., Dhaene, T., and Spina, D. (2018). Review of polynomial chaos-based methods for uncertainty quantification in modern integrated circuits. Electronics, 7.","DOI":"10.3390\/electronics7030030"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3917","DOI":"10.1109\/TCSI.2019.2914398","article-title":"Colored noise in oscillators. Phase-amplitude analysis and a method to avoid the Ito-Stratonovich dilemma","volume":"66","author":"Bonnin","year":"2019","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_45","first-page":"119","article-title":"Frequency domain analysis of a piezoelectric energy harvester with impedance matching network","volume":"100","author":"Bonnin","year":"2022","journal-title":"Energy Harvest. Syst."},{"key":"ref_46","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":"2017","journal-title":"Energy Harvest. Syst."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Costanzo, L., Lo Schiavo, A., Sarracino, A., and Vitelli, M. (2021). Stochastic thermodynamics of a piezoelectric energy harvester model. Entropy, 23.","DOI":"10.3390\/e23060677"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Costanzo, L., Lo Schiavo, A., Sarracino, A., and Vitelli, M. (2022). Stochastic Thermodynamics of an Electromagnetic Energy Harvester. Entropy, 24.","DOI":"10.3390\/e24091222"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Costanzo, L., Lo Schiavo, A., and Vitelli, M. (2022). Improving the Electromagnetic Vibration Energy Harvester Performance by Using a Double Coil Structure. Appl. Sci., 12.","DOI":"10.3390\/app12031166"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Pertin, O., Guha, K., and Jak\u0161i\u0107, O. (2021). Artificial Intelligence-Based Optimization of a Bimorph-Segmented Tapered Piezoelectric MEMS Energy Harvester for Multimode Operation. Computation, 9.","DOI":"10.3390\/computation9080084"},{"key":"ref_51","unstructured":"(2023, February 22). IEEE Standard on Piezoelectricity. Available online: https:\/\/ieeexplore.ieee.org\/document\/26560."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Jones, T.B., and Nenadic, N.G. (2013). Electromechanics and MEMS, Cambridge University Press.","DOI":"10.1017\/CBO9781139032605"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.jsv.2016.03.017","article-title":"Harmonic balance analysis of nonlinear tristable energy harvesters for performance enhancement","volume":"373","author":"Zhou","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.eml.2017.05.002","article-title":"On the efficiency of piezoelectric energy harvesters","volume":"15","author":"Yang","year":"2017","journal-title":"Extrem. Mech. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1007\/s11071-019-05005-6","article-title":"Analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit","volume":"97","author":"Huang","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s11071-020-06170-9","article-title":"Performance investigations of nonlinear piezoelectric energy harvesters with a resonant circuit under white Gaussian noises","volume":"103","author":"Yu","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.jsv.2008.06.011","article-title":"Energy harvesting from the nonlinear oscillations of magnetic levitation","volume":"319","author":"Mann","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"2314","DOI":"10.1016\/j.jsv.2010.11.024","article-title":"An experimentally validated electromagnetic energy harvester","volume":"330","author":"Elvin","year":"2011","journal-title":"J. Sound Vib."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"055007","DOI":"10.1088\/0964-1726\/22\/5\/055007","article-title":"Electromagnetic energy harvester with repulsively stacked multilayer magnets for low frequency vibrations","volume":"22","author":"Kwon","year":"2013","journal-title":"Smart Mater. Struct."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"2909","DOI":"10.1140\/epjst\/e2015-02597-1","article-title":"Energy harvesting in the nonlinear electromagnetic system","volume":"224","author":"Kucab","year":"2015","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1121\/1.1915605","article-title":"A new analogy between mechanical and electrical systems","volume":"4","author":"Firestone","year":"1933","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1088\/0960-1317\/6\/4\/006","article-title":"Modeling micropumps with electrical equivalent networks","volume":"6","author":"Bourouina","year":"1996","journal-title":"J. Micromech. Microeng."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Janschek, K. (2011). Mechatronic Systems Design: Methods, Models, Concepts, Springer Science & Business Media.","DOI":"10.1007\/978-3-642-17531-2"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1109\/JETCAS.2013.2265797","article-title":"A survey of fractional-order circuit models for biology and biomedicine","volume":"3","author":"Freeborn","year":"2013","journal-title":"IEEE J. Emerg. Sel. Top. Circuits Syst."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1002\/cta.287","article-title":"Basic concepts of quantum systems versus classical networks","volume":"32","author":"Civalleri","year":"2004","journal-title":"Int. J. Circuit Theory Appl."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1002\/cta.2359","article-title":"Introduction to quantum electromagnetic circuits","volume":"45","author":"Vool","year":"2017","journal-title":"Int. J. Circuit Theory Appl."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Bonnin, M., Traversa, F.L., and Bonani, F. (2022). An Impedance Matching Solution to Increase the Harvested Power and Efficiency of Nonlinear Piezoelectric Energy Harvesters. Energies, 15.","DOI":"10.3390\/en15082764"},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Roundy, S., and Zhang, Y. (2005, January 28). Toward self-tuning adaptive vibration-based microgenerators. Proceedings of the Smart Structures, Devices, and Systems II. SPIE, Sydney, Australia.","DOI":"10.1117\/12.581887"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1016\/j.jsv.2005.08.018","article-title":"Design of mechanical band-pass filters for energy scavenging","volume":"292","author":"Shahruz","year":"2006","journal-title":"J. Sound Vib."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"015035","DOI":"10.1088\/0964-1726\/17\/01\/015035","article-title":"A vibration energy harvesting device with bidirectional resonance frequency tunability","volume":"17","author":"Challa","year":"2008","journal-title":"Smart Mater. Struct."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"104986","DOI":"10.1016\/j.nanoen.2020.104986","article-title":"Automatic resonance tuning mechanism for ultra-wide bandwidth mechanical energy harvesting","volume":"77","author":"Shin","year":"2020","journal-title":"Nano Energy"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"117577","DOI":"10.1016\/j.apenergy.2021.117577","article-title":"A flute-inspired broadband piezoelectric vibration energy harvesting device with mechanical intelligent design","volume":"303","author":"Wang","year":"2021","journal-title":"Appl. Energy"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1959","DOI":"10.1177\/1045389X09343218","article-title":"The effect of non-linear piezoelectric coupling on vibration-based energy harvesting","volume":"20","author":"Triplett","year":"2009","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"603","DOI":"10.3390\/vibration5030035","article-title":"On the Efficiency Enhancement of an Actively Tunable MEMS Energy Harvesting Device","volume":"5","author":"Aliasghary","year":"2022","journal-title":"Vibration"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"115332","DOI":"10.1016\/j.jsv.2020.115332","article-title":"Nonlinear multi-modal energy harvester and vibration absorber using magnetic softening spring","volume":"476","author":"Wang","year":"2020","journal-title":"J. Sound Vib."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1109\/JMEMS.2011.2170824","article-title":"Nonlinear springs for bandwidth-tolerant vibration energy harvesting","volume":"20","author":"Nguyen","year":"2011","journal-title":"J. Microelectromech. Syst."},{"key":"ref_77","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_78","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Duan, X., Shi, Y., and Yue, X. (2021). Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter. Appl. Sci., 11.","DOI":"10.3390\/app11219979"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"117384","DOI":"10.1016\/j.jsv.2022.117384","article-title":"Asymmetric tristable energy harvester with a compressible and rotatable magnet-spring oscillating system for energy harvesting enhancement","volume":"543","author":"Wang","year":"2023","journal-title":"J. Sound Vib."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Yan, Y., Han, J., Hao, S., and Wang, W. (2022). Dynamic Design of a Quad-Stable Piezoelectric Energy Harvester via Bifurcation Theory. Sensors, 22.","DOI":"10.3390\/s22218453"},{"key":"ref_81","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_82","doi-asserted-by":"crossref","first-page":"120047","DOI":"10.1016\/j.apenergy.2022.120047","article-title":"Bandwidth enhancement of a gimbaled-pendulum vibration energy harvester using spatial multi-stable mechanism","volume":"326","author":"Wang","year":"2022","journal-title":"Appl. Energy"},{"key":"ref_83","first-page":"1","article-title":"Analysis and Experiments of a Pendulum Vibration Energy Harvester with a Magnetic Multi-Stable Mechanism","volume":"58","author":"Wang","year":"2022","journal-title":"IEEE Trans. Magn."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"124028","DOI":"10.1016\/j.energy.2022.124028","article-title":"On a spring-assisted multi-stable hybrid-integrated vibration energy harvester for ultra-low-frequency excitations","volume":"252","author":"Yang","year":"2022","journal-title":"Energy"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"2223","DOI":"10.1140\/epjs\/s11734-021-00377-6","article-title":"Bursting analysis of multi-stable nonlinear mechanical oscillator and its application in energy harvesting","volume":"231","author":"Chen","year":"2022","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_86","unstructured":"Kundert, K.S., Sangiovanni-Vincentelli, A.L., and White, J.K. (2010). Steady-State Methods for Simulating Analog and Microwave Circuits, Springer."},{"key":"ref_87","unstructured":"Webster, J. (2014). Wiley Encyclopedia of Electrical and Electronics Engineering, John Wiley & Sons, Inc."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1002\/cta.440","article-title":"A frequency-domain approach to the analysis of stability and bifurcations in nonlinear systems described by differential-algebraic equations","volume":"36","author":"Traversa","year":"2008","journal-title":"Int. J. Circuit Theory Appl."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.aeue.2011.09.002","article-title":"Improved harmonic balance implementation of Floquet analysis for nonlinear circuit simulation","volume":"66","author":"Traversa","year":"2012","journal-title":"AEU-Int. J. Electron. Commun."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1049\/iet-cds.2010.0138","article-title":"Frequency-domain evaluation of the adjoint Floquet eigenvectors for oscillator noise characterisation","volume":"5","author":"Traversa","year":"2011","journal-title":"IET Circuits Devices Syst."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"1451","DOI":"10.1007\/s11071-014-1527-6","article-title":"Stochastic averaging for nonlinear vibration energy harvesting system","volume":"78","author":"Xu","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s11071-020-05563-0","article-title":"Analysis of influence of nonlinearities and noise correlation time in a single-DOF energy-harvesting system via power balance description","volume":"100","author":"Bonnin","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_93","first-page":"260","article-title":"On the Principle of Averaging the Ito\u2019s Stochastic Differential Equations","volume":"4","author":"Khasminskii","year":"1968","journal-title":"Kibernetika"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1115\/1.3101980","article-title":"Recent developments and applications of the stochastic averaging method in random vibration","volume":"149","author":"Zhu","year":"1996","journal-title":"Appl. Mech. Rev."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1115\/1.2789009","article-title":"Stochastic averaging of quasi\u2013integrable Hamiltonian systems","volume":"64","author":"Zhu","year":"1997","journal-title":"J. Appl. Mech."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1115\/1.2787267","article-title":"Stochastic averaging of quasi-nonintegrable-Hamiltonian systems","volume":"64","author":"Zhu","year":"1997","journal-title":"J. Appl. Mech."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"R55","DOI":"10.1088\/0951-7715\/17\/6\/R01","article-title":"Extracting macroscopic dynamics: Model problems and algorithms","volume":"17","author":"Givon","year":"2004","journal-title":"Nonlinearity"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.physd.2005.11.010","article-title":"A comparative study of two stochastic mode reduction methods","volume":"213","author":"Stinis","year":"2006","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2140\/camcos.2006.1.1","article-title":"Problem reduction, renormalization, and memory","volume":"1","author":"Chorin","year":"2007","journal-title":"Commun. Appl. Math. Comput. Sci."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"2084","DOI":"10.1103\/PhysRevE.54.2084","article-title":"Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral","volume":"54","author":"Gillespie","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1137\/S0036144500378302","article-title":"An algorithmic introduction to numerical simulation of stochastic differential equations","volume":"43","author":"Higham","year":"2001","journal-title":"SIAM Rev."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1093\/imamat\/hxr016","article-title":"Stochastic ordinary differential equations in applied and computational mathematics","volume":"76","author":"Higham","year":"2011","journal-title":"IMA J. Appl. Math."},{"key":"ref_103","doi-asserted-by":"crossref","unstructured":"Kloeden, P.E., and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-3-662-12616-5"},{"key":"ref_104","doi-asserted-by":"crossref","unstructured":"Milstein, G.N. (1994). Numerical Integration of Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-94-015-8455-5"},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1137\/09076636X","article-title":"Runge\u2013Kutta methods for the strong approximation of solutions of stochastic differential equations","volume":"48","year":"2010","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_106","doi-asserted-by":"crossref","unstructured":"S\u00e4rkk\u00e4, S., and Solin, A. (2019). Applied Stochastic Differential Equations, Cambridge University Press.","DOI":"10.1017\/9781108186735"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"068701","DOI":"10.1103\/PhysRevE.74.068701","article-title":"Comment on \u201cNumerical methods for stochastic differential equations\u201d","volume":"74","author":"Burrage","year":"2006","journal-title":"Phys. Rev. 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