BULLETIN OF THE POLISH ACADEMY OF SCIENCES MECHANICAL AND AERONAUTICAL ENGINEERING, TECHNICAL SCIENCES, Vol. 68, No. 6, 6, 2020 2020 THERMODYNAMICS DOI: 10.24425/bpasts.2020.135384 10.24425/bpasts.2020.XXXXXX DesignDesign of nonlinearnonlinear electromagnetic energy harvester equipped with mechanical amplifier amplifier and spring bumpers M. OSTROWSKI 1 , B. BŁACHOWSKI 1*, M. BOCHEŃSKI 2 , D. PIERNIKARSKI2, 1 1 2 2 M. OSTROWSKI , B. BŁACHOWSKI ∗, M. BOCHENSKI´ , D. PIERNIKARSKI , P. FILIPEK 3 , and W. JANICKI 4 P. FILIPEK3 and W. JANICKI4 1 Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Pawińskiego 5b, 02-106 Warsaw, Poland 1 Institute2 Faculty ofof Fundamental Mechanical Engineering, Technological Lublin Research, University Polish Academyof Technology, of Sciences, ul. Nadbystrzycka Pawinskiego´ 36, 5b, 20-618 02-106 Lublin, Warsaw, Poland Poland 2 Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland 3 Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, ul. Nadbystrzycka 38A, 20-618 Lublin, Poland 3 Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland 4 Faculty of Earth Sciences and Spatial Management, Maria Curie-Sklodowska University, Al. Kraśnicka 2d, 20-718 Lublin, Poland 4 Faculty of Earth Sciences and Spatial Management, Maria Curie-Sklodowska University, Al. Krasnicka´ 2d, 20-718 Lublin, Poland Abstract. TheThe main main drawback drawback of of vibration-based vibration-based energy energy harvesting harvesting is is its its poor poor efficiency efficiency due to small amplitudes of vibration and low sensi sensi-- tivity at frequencies far fromfrom resonantresonant frequency.frequency. The performance performance of electromagnetic energy harvester can be improved by using mechanical enhancements such as mechanical amplifiers amplifiers or spring bumpers. The The mechanical mechanical amplifiers amplifiers increase range range of of movement and velocity, improving also significantly significantly harvester efficiencyefficiency forfor thethe samesame levellevel ofof excitation.excitation. AsAs a result of this amplitude of motion is much larger comparing to the size of thethe electromagneticelectromagnetic coil.coil. ThisThis inin turnturn imposesimposes thethe needneed forfor modelling modelling of of electromagnetic electromagnetic circuit circuit parameters parameters as as the the function function of of the the moving mov- ingmagnet magnet displacement. displacement. Moreover, Moreover, high high velocities velocities achieved achieved by the by moving the moving magnet magnet reveal reveal nonlinear nonlinear dynamics dynamics in the in electromagnetic the electromagnetic circuit circuit of the ofenergy the energy harvester. harvester. Another Another source source of nonlinearity of nonlinearity is the collisionis the collision effect effect between between magnet magnet and spring and spring bumpers. bumpers. It has It been has shownbeen shown that this that effect this effectshould should be carefully be carefully considered considered during designduring processdesign ofprocess the energy of the harvesting energy harvesting device. device. The present paper investigates the influence of theThe above-mentioned present paper investigates nonlinearities the influence on power of level the above-mentioned generated by the nonlinearitiesenergy harvester. on power A rigorous level generatedmodel of the by theelectromagnetic energy harvester. circuit, A rigorousderived withmodel aid of of the the electromagnetic Hamilton’s principle circuit, derivedof the least with action, aid of thehas Hamilton’s been proposed. principle It includes of the least inductance action, hasof the been electromagnetic proposed. It includes coil as inductance the function of ofthe the electromagnetic moving magnet coil position. as the function Additionally, of the movingnonlinear magnet behaviour position. of the Additionally, overall electromagnetic nonlinear behaviour device ofhas the been overall tested electromagnetic numerically for device the casehas been of energy tested harvester numerically attached for the to case the ofquarter energy car harvester model moving attached on to random the quarter road carprofiles. model Such moving a source on random of excitation road profiles. provides Such wide a source band of excitation providesfrequencies, wide which band occur of excitation in variety frequencies, of real-life which applications. occur in variety of real-life applications. Key words: energy harvesting, velocityvelocity amplification,amplification, nonlinear nonlinear electromagnetic electromagnetic circuit, circuit, spring spring bumper, bumper, quarter quarter car car model. model. 1. Introduction Despite relatively simple structure and easily accessible mate- rials mathematical modelling of the EMEHs is a quite com- Energy harvesting is a process of conversion of mechanical, plex task, because of inherent multi-physics associated with this thermal and other sources of energy into a usable form – usually problem [6]. Another advantage of EMEH devices is their high related to electrical energy [1]. Development of energy harvest- reliability. For that reason they can be successfully applied in ing is driven by modern energy-saving electronics and decreas- very demanding engineering tasks of such as structural health ing prices of modules used. Variety of DC/DC converters, e.g. monitoring, as it was shown in [7]. Another example is applica- highly efficient step-up or flyback converters allows us to easily tion of the energy harvester as self-powered magnetorheologi- adjust the voltage of the energy harvesting system to different cal dampers for vibration attenuation [8]. powered devices [2, 3]. Many solutions for vibration-based en- EMEHs exhibit relatively good performance, but the levels ergy harvesting devices are available. They can be classified of generated power are still not sufficient for many potential ap- based on their structure, topology or energy conversion mecha- plications where wireless power supply could be profitable. The nism utilising different type of motion and frequency range [4]. simple way to improve the efficiency of the energy harvesting In this study vibration-based electromagnetic energy harvesters process is based on amplification of the range of motion, e.g. (EMEHs) are taken into consideration. The main motivation for using pendulum-based mechanism [9]. It is also possible to use this choice is high potential of EMEH in many applications due simple mechanisms called mechanical amplifiers, which trans- to inexpensive and easily accessible materials used. Accord- fer the low amplitude motion between two bodies into the high ing to the Faraday’s law, only wire coil and a moving mag- amplitude one [10]. net are required to cause electromechanical coupling effect [5]. Another way to increase the level of generated power is known as frequency-up conversion. Frequency bandwidth of energy harvester can be increased by means of additional pre- ∗e-mail: [email protected] *e-mail: [email protected] liminary vibrating system, which absorbs low-frequency vibra- Manuscript submitted submitted 2020-07-01, 20XX-XX-XX, initially initially accepted accepted for publication for publication 2020-08-06, tion energy and transfers it into a high-frequency vibration of published20XX-XX-XX, in December published 2020 in ZZZZZZZZ 2020. secondary system. Energy harvesters based on this mechanism Bull. Pol.Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 2020 13731 M.M. Ostrowski, Ostrowski, B. B. Błachowski, Błachowski, M. Bocheński, Bochenski,´ D. Piernikarski, P.P. Filipek,Filipek, andand W.W. JanickiJanicki Design of nonlinear electromagnetic energy harvester equipped with mechanical amplifier and spring bumpers are often non-resonant. Several examples of conversion mecha- ence on the generated power level. Due to the modern trans- 2. Energy harvesting device results from the change of the reluctance with magnet displace- nisms are shown in paper [11]. Another example of this kind of ducers, the input resistance of the system can be widely mod- ment. The exact dependence of reluctance force Frel (mechan- mechanism is free mass moving between less-massive magnets ified, so the value of the load resistance can also be a design A schematic diagram of the studied model of EMEH is shown ical part of the system) and voltage εLx (electrical part) on the on springs. In this case collisions cause high-frequency vibra- parameter for the overall EMEH system [20]. in Fig. 2. This model represents the EMEH device consisting inductance in relation to x is derived in the next subsection. tions inside the coils [12]. High amplification of velocity can In the present paper the influence of the mechanical ampli- of a tube (casing) made of non-magnetic material, a spring with The magnet and coil properties used in the numerical study also be achieved by elastic collisions of the moving magnet fier and spring bumpers, imposing magnet stroke, on the EMEH stiffness k, a moving magnet with mass m, viscous damping in the paper are similar to the EMEH researched in [21]. These with other elements with larger masses [13]. In this way gen- work efficiency is discussed. Proportion between the level of with constant c and an electromagnetic coil attached to the tube. data are used in the present paper as a benchmark for generated erated power has been increased