Design of Nonlinear Electromagnetic Energy Harvester Equipped with Mechanical Ampliﬁer and Spring Bumpers Are Often Non-Resonant

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 materials 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 33 times compared to a single mean generated power and the dimensions of the analysed xt(t) and x(t) are displacement of the tube and the relative dis- power in the case where enhancements are not implemented, mass system. Furthermore, the structure was working in a sig- EMEH device is also examined. For benchmark purposes di- placement of the magnet with reference to the electromagnetic and were listed in Table 1. nificantly wider frequency bandwidth. A system based on the mensions and properties of the coil and moving magnet adopted coil in the function of time t, respectively. The tube is equipped same principle, but different arrangement, is discussed in [14]. in this paper are similar to those of EMEH device shown in [21]. with two spring bumpers, which cause resilient collision. It is Table 1 Also, the structure with a stroke of a magnet with end wash- In this paper electromechanical coupling has been determined assumed that the length of these springs is very small and they Magnet properties ers or elastic rubber bumpers, increases the level of the gener- numerically using the Biot-Savart law for similarity between have high stiffness. Hence, when the magnet reaches extreme description symbol value ated power and significantly expands the range of operational the field of the magnet and the field induced by solenoid re- relative displacement with respect to the coil xmax, the influ- EMEH frequency [15, 16]. placement, as shown in [22]. The mathematical model of the ence of the spring bumpers on the moving magnet is modelled magnet material – NdFeB (N38) Another way to broaden the operational frequency bandwidth electrical circuit presented in [21] include only constant coil in- by the resilient collision described by the restitution coefficient magnet height hm [mm] 35 and increase harvester efficiency is a design of the so-called ductance, which does not depend on the position of the magnet c = 0.9. It follows that x [ x ,x ]. R max max d potential well, as shown in [17]. It has been observed that in (here as an iron core inside the coil). However, for large dis- ∈ − magnet diameter m [mm] 20 nonlinear systems greater amplitudes do not guarantee greater placements and velocities of the magnet, a mathematical model magnet mass m [g] 90 power generated by EMEH. Zhang et al. in [18] have shown of the electric circuit should be refined in order to describe its magnet direction of magnetisation – Axis (–) theoretically that a proper design of the potential well can guar- nonlinear dynamics. In this paper a mathematical model of the antee so-called global resonance, i.e. that work done by exci- EMEH device has been derived by using Hamilton’s princi- coil wire resistance Rc [Ω] 1 200 tation is always positive if some conditions related to the exci- ple of least action. Nonlinear behaviour is typical for the elec- coil inductance (without magnet inside) Lair [H] 1.463 tation are fulfilled. In engineering applications real time tuning tromagnetic devices and mathematically well-described [23]. number of coil turns N [–] 12 740 can be viable. However, for the EMEH devices the nonlinear terms depend Zhang et al. in [18] also indicate that the operational fre- also on the mechanical part of the system. In the present study quency bandwidth criterion can be misleading in considering it is assumed that, coil inductance depends on the magnet posi- 2.1. Dynamics of electromagnetic energy harvester. To the efficiency of energy harvesters, because in many cases tion, and force, derived from the reluctance of the magnetic cir- properly understand dynamic interaction between mechanical power generated at resonance decreases with expansion of the cuit, is also considered [24]. Based on the proposed mathemat- and electrical part of the proposed energy harvester equations of frequency bandwidth. Thus, many devices work more effi- ical model participation of the nonlinear terms in the dynamic motion of the overall system have to be assembled. The equa- ciently out of the resonance at the expense of available effi- response of EMEH has been studied. Additionally an exami- tions of motion were derived by employing Hamilton’s princi- ciency under resonance. It is true, however, that the cited pa- nation of the influence of the design parameters on the EMEH ple of the least action. The action I is expressed by the equa- per does not include considerations of the proportion between efficiency has been also performed. A quarter car model mov- tion below. generated power and amplitude of vibration, e.g. of a moving ing on randomly generated road profiles has been employed as Fig. 2. Scheme of EMEH device t2 magnet in the case of EMEH systems. Smaller efficiency at the source of vibration. The motivation is that vibration-based I = L dt, t1 < t2, (3) the resonance is the result of lower amplitudes, which deter- energy harvesters are often excited by random and nonperiodic Electrical circuit shown in Fig. 2 consists of: inductance of t1 mine smaller dimensions of the system. Considering this issue, forces. the coil depending on the relative displacement of the mag- one can show that expansion of the operational frequency band- Diagram presented in Fig. 1 shows all elements of the anal- where L is the Lagrangian: net L(x), resistance of the coil wire Rc, electromotive force width can still be profitable, which is one of the topics of the ysed dynamic system. The EMEH device and electrical load are εB(t) caused by the magnetic field of the moving magnet, self- present paper. described in detail in Section 2. The quarter car model moving L = T ∗ + M∗ V E (4) induced electromotive force εLx caused by change of the induc- − − The problem of extending the frequency bandwidth at the on randomly generated road profiles and the mechanical am- tance L(x) and voltage across the load uload(t). expense of the vibration at resonance can be avoided using self- plifier are discussed in Section 3. Section 4 contains results of written in displacement and electric charge formulation [25]. The electromotive force εB is generated in the coil turns by tunable energy harvesters [19]. However, the structure of such the parametric study and analysis of nonlinearities. The paper Here T ∗ is kinetic coenergy, M∗ is magnetic coenergy, V is me- change of the magnetic flux linkage Φ induced by the moving chanical potential energy and E is electrical potential energy. devices is usually complex, making the design procedure more is concluded with enumeration of the most important aspects of magnet (Faraday’s Law): difficult. The system parameters supplied also have the influ- EMEH in Section 5. These quantities are described by the following formulas:

d ∂Φ(x) 1 2 ε (t)= Φ(x(t)) = x˙(t), (1) T ∗ = m(x˙+ x˙t) , (5) input: B − dt − ∂x 2 output: 1 2 Gradient of the magnetic ﬂux linkage shown in the equa- M∗ = L(x)q˙ + Φ(x)q˙, (6) tion above is called electromechanical coupling. The mechani- 2 cal result of the electromechanical coupling is electrodynamic 1 (Lorenz) force: V = kx2, (7) 2 ∂Φ(x) F = q˙, (2) el ∂x E 0. (8) ≡ where q is electrical charge – its time-derivativeq ˙ is the elec- The second term (6) relates to the mechanical work of the force Fig. 1. The concept of vibration-based energy harvesting trical current in the electrical circuit. The reluctance force Frel Fel (see: Eq. (2)) done on electrical part of the system. It is also

13742 Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 Bull. Pol. Ac.: Tech. 68(6) 2020 3 DesignDesign of nonlinear of nonlinear electromagnetic electromagnetic energy energy harvester harvester equipped with with mechanical mechanical amplifier ampliﬁer and andspring spring bumpers bumpers

2. Energy harvesting device results from the change of the reluctance with magnet displacement. The exact dependence of reluctance force Frel (mechan- A schematic diagram of the studied model of EMEH is shown ical part of the system) and voltage εLx (electrical part) on the in Fig. 2. This model represents the EMEH device consisting inductance in relation to x is derived in the next subsection. of a tube (casing) made of non-magnetic material, a spring with The magnet and coil properties used in the numerical study stiffness k, a moving magnet with mass m, viscous damping in the paper are similar to the EMEH researched in [21]. These with constant c and an electromagnetic coil attached to the tube. data are used in the present paper as a benchmark for generated xt(t) and x(t) are displacement of the tube and the relative dis- power in the case where enhancements are not implemented, placement of the magnet with reference to the electromagnetic and were listed in Table 1. coil in the function of time t, respectively. The tube is equipped with two spring bumpers, which cause resilient collision. It is Table 1 assumed that the length of these springs is very small and they Magnet properties have high stiffness. Hence, when the magnet reaches extreme description symbol value relative displacement with respect to the coil xmax, the inﬂu- ence of the spring bumpers on the moving magnet is modelled magnet material – NdFeB (N38) by the resilient collision described by the restitution coefﬁcient magnet height hm [mm] 35 cR = 0.9. It follows that x [ xmax,xmax]. ∈ − magnet diameter dm [mm] 20 magnet mass m [g] 90 magnet direction of magnetisation – Axis (–)

coil wire resistance Rc [Ω] 1 200

coil inductance (without magnet inside) Lair [H] 1.463 number of coil turns N [–] 12 740

2.1. Dynamics of electromagnetic energy harvester. To properly understand dynamic interaction between mechanical and electrical part of the proposed energy harvester equations of motion of the overall system have to be assembled. The equations of motion were derived by employing Hamilton’s principle of the least action. The action I is expressed by the equation below. Fig. 2. Scheme of EMEH device t2

I = L dt, t1 < t2, (3) Electrical circuit shown in Fig. 2 consists of: inductance of t1 the coil depending on the relative displacement of the mag- where L is the Lagrangian: net L(x), resistance of the coil wire Rc, electromotive force εB(t) caused by the magnetic ﬁeld of the moving magnet, self- L = T ∗ + M∗ V E (4) induced electromotive force εLx caused by change of the induc- − − tance L(x) and voltage across the load uload(t). written in displacement and electric charge formulation [25]. The electromotive force εB is generated in the coil turns by Here T ∗ is kinetic coenergy, M∗ is magnetic coenergy, V is me- change of the magnetic ﬂux linkage Φ induced by the moving chanical potential energy and E is electrical potential energy. magnet (Faraday’s Law): These quantities are described by the following formulas:

d ∂Φ(x) 1 2 ε (t)= Φ(x(t)) = x˙(t), (1) T ∗ = m(x˙+ x˙t) , (5) B − dt − ∂x 2

1 2 Gradient of the magnetic ﬂux linkage shown in the equa- M∗ = L(x)q˙ + Φ(x)q˙, (6) tion above is called electromechanical coupling. The mechani- 2 cal result of the electromechanical coupling is electrodynamic 1 (Lorenz) force: V = kx2, (7) 2 ∂Φ(x) F = q˙, (2) el ∂x E 0. (8) ≡ where q is electrical charge – its time-derivativeq ˙ is the elec- The second term (6) relates to the mechanical work of the force trical current in the electrical circuit. The reluctance force Frel Fel (see: Eq. (2)) done on electrical part of the system. It is also

Bull. Pol.Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 2020 13753 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

a metric of energy exchange between mechanical and electri- Here lc is the length of the curve along the wire turns of the coil like in the Fourier series approximation. It is a convenient ap- Here A is a component of the magnetic potential vector field cal part of the EMEH device through the magnetic field of the and Br is the radial component (normal to the coil wire) of the proach because of the orthogonality of the harmonic functions perpendicular to the longitudinal section, R denotes the radius moving magnet. magnetic field vector B. In further calculations it was assumed in period lx. Function α(x) is described by Eq. (17). from the axis of the coil, J is current density in the cross section The system has two degrees of freedom: relative displace- that the field of the magnetic induction B of the magnet can of the coil for unit current in the coil expressed by formula ment x and electric charge q. Conservative and dissipative be approximated by a field of a solenoid with the same dimen- Nh 2πn α(x)= An sin x . (17) forces should be taken into account. Hence, the Euler-Lagrange sions [22]. It is also accepted that the magnet is divided into Nm ∑ l N/((R3 R2)hc) if R2 R R3, n=1 x − ≤ ≤ equations have the following form: solenoid loops, each with diameter dm. The current in each such J(R,z)= z z z , (19)  1 ≤ ≤ 2 loop has a value calculated as follows: Here coefficients A minimise the least square error between  d ∂L ∂D ∂L n 0 if R, z elsewhere, + = F, the numerically calculated coupling coefficient (red dots) and dt ∂x˙ ∂x˙ − ∂x Brhm   (9) Im = , (15) the fitted curve. It has been calculated that the number of har- where N is number of the coil turns (see: Table 1). 0 and z5 µ0Nm  d ∂L ∂D ∂L monics Nh = 13 is sufficient to fit function α(x). The values of are limits of the whole domain, at which A = 0. It does not  + = u, dt ∂q˙ ∂q˙ − ∂q the coefficients An for n = 1,2...13 are listed in Table 3. determine dimensions of the EMEH device yet. Furthermore where Br is the residual magnetic field strength and µ0 is per-  0 z1 and z4 z5.  meability of the vacuum. Having Im magnetic induction B at the where: F is external force, u is external voltage and Table 3 Solution of Eq. (18) can be given as a series of the Bessel point on the curve along the coil wire has been calculated using Coefficients of Fourier series for electromechanical coupling functions and trigonometric functions as shown in detail in [24]. 1 2 1 2 the Biot-Savarat law: D = cx˙ + Rcq˙ (10) The results are shown in Fig. 5, where red dots denote a semi- 2 2 n A [Vm/s] n A [Vm/s] n A [Vm/s] µ0Im dl r n n n analytical solution for different values of x. Semi-analytical so- B = × , (16) is dissipation function expressed in terms of power of the en- r3 1 29.5578 6 0.5466 11 0.0453 lution has been obtained for 70 harmonics, z1 = 0.2475 m, z2 = 4π lm . = . ergy losses in the system. Only kinematic excitation acts on the 2 26.4298 7 0.4715 12 0.2148 0 3025 m, z5 0 550 m. z3 and z4 were varying according to magnet that is taken into account in equation (5), so F 0. For − the iron core (moving magnet) position. The blue line in Fig. 5 ≡ where dl is the vector differential length along curve lm of the 3 8.4275 8 1.3008 13 0.4254 the system shown in Fig. 2 u = uload. Substituting equations − − is the fitted curve described by Eq. (20), while Lair = 1.463 H (5)–(7) and (10) into (9) we receive equations of motion (11). solenoid, r is the vector from the point on the solenoid to the 4 2.5786 9 1.5718 – – is the inductance of the coil without core. point on the curve along the coil wire, r is the length of vector r. − − 5 2.1949 10 0.6452 – – 2 ∂Φ(x) − − x ∂Φ 1 ∂L σ c + q˙ Function has been calculated numerically for values L(x)=Lair + κLe− L . (20) m 0 x¨ − ∂x 2 ∂x x˙ ∂x +   Br = 1 T and Nm = 20 using MATLAB software. The assumed Function α(x) is used in equations of motion (11) during nu- 0 L(x)q¨ ∂L ∂Φ q˙ Parameters κ = 3.6877 H and σ = 0.02665 m have been de- q˙ + R dimensions of the coil are listed in Table 2. It is assumed that merical integration, where the α(x) values for the x outside the L L  c  termined by minimising the least square problem. Function in  ∂x ∂x  the turns of coils are evenly distributed in the coil’s cross sec- range shown in Fig. 3 are assumed equal to zero. Found α(x)   Eq. (20) is used in numerical integration of equations of mo- k 0 x mx¨ tion. The results are shown in Fig. 3, where the red dots denote function is in good agreement with the experimental results t L(x) + = − . (11) the numerical result and the blue line denotes the fitted curve shown in [21]. Electromechanical coupling is an odd function tion (11). The advantages of function are that it is easily 00q uload α(x). The fitting curve was assumed during these calculations of x, so oscillation of the magnet generates oscillations of the differentiable, decreases with displacement of the magnet and = achieves zero in infinity. The above equations contain nonlinearities associated with the that the range of the magnet motion is lx 200 mm (shown in current in the electrical circuit with doubled frequency. dependence of the inductance and electromechanical coupling Fig. 3). The function sought in interval lx can be assumed to on relative displacement of the moving magnet x. It is worth be one period with the length of lx of a periodic function. Then 2.3. Inductance as function of relative magnet displace- ( ) noticing that off-diagonal elements of the matrix next to first- curve α x can be expressed as a series of harmonic functions ment. To determine inductance L(x) a semi-analytical method order derivatives of the response vector are responsible for cou- shown in [24] was used. In this method calculation of the in- pling between mechanical and electrical part of the system. In Table 2 ductance is based on the Poisson partial differential Eq. (18) 1 ∂L ∂L Dimensions of the coil assumed in calculations describing magnetic vector potential on the longitudinal section Eq. (11) q˙2 = F (x,q˙) and x˙q˙ = ε (x,x˙,q˙). 2 ∂x rel ∂x Lx of the coil-iron core system shown in Fig. 4. Power P generated as a result of the load is expressed by the description symbol value 2 2 equation below. axial length of the coil hc [mm] 55 ∂ A 1 ∂A A ∂ A + + = µ0J(R,z). (18) P = uloadq˙. (12) 2 R R R2 2 − outer diameter Dc [mm] 28 ∂R ∂ − ∂z − Fig. 5. Coil inductance as a function of magnet position If the load is to be expressed as a resistance, the voltage across inner diameter dc [mm] 23 the load is expressed as follows: 3. Mechanical model of a vehicle moving uload = Rloadq˙. (13) − on uneven road ∂Φ(x) Functions L(x) and are calculated numerically in the ∂x This section presents the quarter car model and the method of next subsections. mounting a mechanical amplifier as well as the adopted method of road profile generation. These three parts constitute the envi- 2.2. Electromechanical coupling as the function of relative ronment of the EMEH device and allow for testing the device’s magnet displacement. The derivative of the magnetic flux efficiency. linkage over magnet displacement fulfils the equation below. 3.1. Quarter car model. In the present study the quarter car ∂Φ(x) = Br(l)dl (14) Fig. 3. Electromechanical coupling as a function of magnet displace- Fig. 4. Longitudinal section of the coil-core (magnet) considered in model is used, because it is one of the simplest models which ∂x ment x calculations of the inductance lc behaves similarly to the suspension of the whole vehicle. The

13764 Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 Bull. Pol. Ac.: Tech. 68(6) 2020 5 DesignDesign of nonlinear of nonlinear electromagnetic electromagnetic energy energy harvester harvester equipped with with mechanical mechanical amplifier amplifier and andspring spring bumpers bumpers like in the Fourier series approximation. It is a convenient ap- Here A is a component of the magnetic potential vector field proach because of the orthogonality of the harmonic functions perpendicular to the longitudinal section, R denotes the radius in period lx. Function α(x) is described by Eq. (17). from the axis of the coil, J is current density in the cross section of the coil for unit current in the coil expressed by formula Nh 2πn α(x)= A sin x . (17) ∑ n N/((R3 R2)hc) if R2 R R3, n=1 lx − ≤ ≤ J(R,z)= z z z , (19)  1 ≤ ≤ 2 Here coefficients An minimise the least square error between 0 if R, z elsewhere, the numerically calculated coupling coefficient (red dots) and  the fitted curve. It has been calculated that the number of har- where N is number of the coil turns (see: Table 1). 0 and z5 monics Nh = 13 is sufficient to fit function α(x). The values of are limits of the whole domain, at which A = 0. It does not the coefficients An for n = 1,2...13 are listed in Table 3. determine dimensions of the EMEH device yet. Furthermore 0 z and z z . 1 4 5 Table 3 Solution of Eq. (18) can be given as a series of the Bessel Coefficients of Fourier series for electromechanical coupling functions and trigonometric functions as shown in detail in [24]. The results are shown in Fig. 5, where red dots denote a semi- n An [Vm/s] n An [Vm/s] n An [Vm/s] analytical solution for different values of x. Semi-analytical so- 1 29.5578 6 0.5466 11 0.0453 lution has been obtained for 70 harmonics, z1 = 0.2475 m, z2 = . = . 2 26.4298 7 0.4715 12 0.2148 0 3025 m, z5 0 550 m. z3 and z4 were varying according to − the iron core (moving magnet) position. The blue line in Fig. 5 3 8.4275 8 1.3008 13 0.4254 − − is the fitted curve described by Eq. (20), while Lair = 1.463 H 4 2.5786 9 1.5718 – – is the inductance of the coil without core. − − 5 2.1949 10 0.6452 – – 2 − − x σ L(x)=Lair + κLe− L . (20) Function α(x) is used in equations of motion (11) during nu- Parameters = 3.6877 H and = 0.02665 m have been de- merical integration, where the α(x) values for the x outside the κL σL termined by minimising the least square problem. Function in range shown in Fig. 3 are assumed equal to zero. Found α(x) function is in good agreement with the experimental results Eq. (20) is used in numerical integration of equations of mo- L(x) shown in [21]. Electromechanical coupling is an odd function tion (11). The advantages of function are that it is easily of x, so oscillation of the magnet generates oscillations of the differentiable, decreases with displacement of the magnet and current in the electrical circuit with doubled frequency. achieves zero in infinity.

2.3. Inductance as function of relative magnet displacement. To determine inductance L(x) a semi-analytical method shown in [24] was used. In this method calculation of the inductance is based on the Poisson partial differential Eq. (18) describing magnetic vector potential on the longitudinal section of the coil-iron core system shown in Fig. 4.

∂ 2A 1 ∂A A ∂ 2A + + = µ0J(R,z). (18) ∂R2 R ∂R − R2 ∂z2 − Fig. 5. Coil inductance as a function of magnet position

3. Mechanical model of a vehicle moving on uneven road

This section presents the quarter car model and the method of mounting a mechanical amplifier as well as the adopted method of road profile generation. These three parts constitute the envi- ronment of the EMEH device and allow for testing the device’s efficiency.

3.1. Quarter car model. In the present study the quarter car Fig. 4. Longitudinal section of the coil-core (magnet) considered in model is used, because it is one of the simplest models which calculations of the inductance behaves similarly to the suspension of the whole vehicle. The

Bull. Pol.Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 2020 13775 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 quarter car model is shown in Fig. 6 and consists of the lower, 3.2. Mechanical amplifier. If the EMEH can be placed be- Table 5 EMEH device. In the following subsection an example of time so-called unsprung mass m1 simulating wheel and axle, the up- tween two relatively moving bodies, then a special kind of in- Degree of roughness for different road classes histories of the state of the EMEH device for randomly gener- per, so-called sprung mass m2 simulating quarter part of the car terface between the EMEH device and these moving bodies, ated road profile is shown. The third subsection addresses the ( ) 6 3 body, stiffness k1 representing tyre stiffness, suspension stiff- called mechanical amplifier, can be used. Due to the relative road Θ Ω0 , 10− m characteristics of mean generated power in relation to differ- class ness k2 and suspension viscous damping factor c2. y1(t) and motion between the bodies, the mechanical amplifier increases lower limit geometric mean upper limit ent system parameters. Finally, the influence of nonlinearities y (t) are displacements of the unsprung and sprung mass, re- displacements of the EMEH device per unit of time, thus also of the modelled system is discussed. 2 A – 1 2 spectively, w(t) is the height of the road profile at time instant increasing its velocities and accelerations. Examples of these B 2 4 8 t, V is the velocity of the vehicle. simple mechanisms used in energy harvesters are discussed 4.1. Power generated by EMEH device. In this subsection in [10]. Such a mechanism shown in Fig. 7 can be mounted C 8 16 32 behaviour of the EMEH equipped with the spring bumpers and between sprung and unsprung masses of the quarter car model D 32 64 128 the mechanical amplifier device is illustrated in exemplary time shown in Fig. 6. Taking into account the equation of motion histories. In this case the load resistance Rload = 2500 Ω, gain E 128 256 512 (21), the displacement of the energy harvester tube xt(t) from G = 20 and the stiffness k = 550 N/m. This value of stiffness Eq. (11) is as follows: provides similar undamped natural frequency of the EMEH de- In this approach the re-derived random series of the sinu- vice ω0 = k/m = 78.17 rad/s as the second natural frequency xt(t)=G[y2(t) y1(t)] + y1(t). (22) soidal functions has the form of Eq. (24). of the quarter car model ωu2. It allows the EMEH device to − get into resonance. In this subsection the road profile shown in In these example a gain of motion G can be defined as the NW Fig. 8(b) was used, whereas vehicle velocity was assumed to be output-input amplitude ratio. The input amplitude is a direct w(t)=∑Wi sin(ΩiVt+ φi), (24) equal V = 50 km/h to simulate behaviour of the EMEH while i=1 result of the relative motion of the bodies. driving the vehicle on urban roads. Figure 9 shows the displacement, corresponding accelera- ΩN Ω1 where Wi = Θ(Ωi) − , Ωi = iΩ1, ΩN = NW Ω1, φi tions of the EMEH tube (excitation), load voltage u , magnet π(N 1) ∈ load − displacement x and generated energy E for assumed maximum [0,2π) is a random variable with uniform probability distribu- displacement x = 5 cm and gain G = 20. Mean generated Fig. 6. Quarter car model max tion, NW is the number of harmonics. The values of parameters power has been calculated from the end value of the produced are as follows: Ω1 = 0.02π rad/m, ΩN = 6π rad/m and thus energy: The quarter car model is described by the linear equation of N = 300. These values cover the bandwidths suggested by W 1 motion (21), where tyre damping has been neglected because ISO 8608. The equation above is suitable only for V = const. P = E(t ), (25) mean t end of its small value relative to the damping of the suspension. Examples of generated random road profiles in relation to end the travelled distance s = Vt for the road classes A and C are ˙ m 0 y¨ c c y˙ shown in Fig. 8. It can be seen that for the lower road class the where: E(t) is equal to the generated power in Eq. (12). 1 1 + 2 − 2 1 roughness of the road is greater. The spatial harmonics content The frequency of the voltage is twice the vibration frequency 0 m2 y¨2 c2 c2 y˙2 − (21) is the same according to Eq. (24), where amplitudes Wi are not of the moving magnet, because the electromechanical coupling k + k k y k is an odd function of magnet relative displacement x. Voltage + 1 2 − 2 1 = 1 w. random variables. Due to this property of the generated profile occasionally achieves values greater than 200 V because of high k2 k2 y2 0 it is easy to determine the desired PSD value (see Eq. (23)). − number of coil turns. Mean power has a very rewarding value of Values of all coefficients from Eq. (21) are selected so as to Pmean = 1.85 W with relatively small dimensions of the EMEH simulate the lorry, as described in [26]. They are listed together device determined by xmax due to the use of the spring bumpers with modal parameters of the quarter car model in Table 4. Fig. 7. Mechanical amplifier concept and the motion range amplifier. It can be seen that in time intervals when collision of the magnet with the spring bumpers Table 4 does not occur, generated energy increments are significantly Properties and modal parameters of the quarter car model 3.3. Randomly generated road profiles. The International smaller. Also the level of generated voltage and electrical current are much smaller in these time intervals. description symbol value Organisation for Standardisation classified longitudinal road profiles according to their roughness (ISO 8608). Roughness unsprung mass m1 [kg] 500 is expressed by the power spectral density (PSD) values of the 4.2. Influence of selected parameters on EMEH perfor- sprung mass m2 [kg] 4500 profile. It allows us to generate road profiles by re-derivation of mance. This subsection discusses the characteristics describ- tire stiffness k [Nm/s] 2 106 ing dependences of the mean generated power and the values 1 · a random series of sinusoidal functions. This method has been 6 of the design parameters for selected road profiles and vehicle suspension stiffness k2 [Nm/s] 10 adopted from [27]. 4 Most road profiles roughness can be expressed by PSD in the velocities. shock absorber damping coeff. c2 [Ns/m] 2 10 · following way: The EMEH device is nonlinear and its characteristics have undamped natural ωu1 [rad/s] 12.10 Fig. 8. Examples of randomly generated road profiles: (a) for the road been determined by integration of equations of motion for each frequencies ω [rad/s] 77.95 2 class A, (b) for the road class C u2 Ω − value of the parameters which influence the generated power. ω [rad/s] 12.14 Θ(Ω)=Θ(Ω0) , (23) Furthermore, road profiles expressed by Eq. (24) are random d1 Ω0 damped natural frequencies and for each numerical test the time histories of the EMEH state ωd2 [rad/s] 74.33 (for example those shown in subsection 4.1) and the P val- ξ [%] 7.91 2π 4. Numerical study mean modal damping factors 1 where Ω = [rad/m] is angular spatial frequency, Ls is wave- ues will be different. The characteristics have been determined ∗ Ls ξ2 [%] 27.28 length [m], and Ω0 = 1 rad/m. The values of the PSD Θ(Ω0) This section presents a numerical study intended to calculate by calculating the average values of mean power from particular (n) ∗ % of critical damping for different road classes are listed in Table 5. the level of generated power and show the behaviour of the values of Pmean, each for n-th randomly generated road profile

13786 Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 Bull. Pol. Ac.: Tech. 68(6) 2020 7 DesignDesign of nonlinear of nonlinear electromagnetic electromagnetic energy energy harvester harvester equipped with with mechanical mechanical amplifier ampliﬁer and andspring spring bumpers bumpers

Table 5 EMEH device. In the following subsection an example of time Degree of roughness for different road classes histories of the state of the EMEH device for randomly generated road profile is shown. The third subsection addresses the 6 3 road Θ(Ω0), 10− m characteristics of mean generated power in relation to differ- class lower limit geometric mean upper limit ent system parameters. Finally, the influence of nonlinearities A – 1 2 of the modelled system is discussed. B 2 4 8 4.1. Power generated by EMEH device. In this subsection C 8 16 32 behaviour of the EMEH equipped with the spring bumpers and D 32 64 128 the mechanical amplifier device is illustrated in exemplary time histories. In this case the load resistance R = 2500 Ω, gain E 128 256 512 load G = 20 and the stiffness k = 550 N/m. This value of stiffness provides similar undamped natural frequency of the EMEH de- In this approach the re-derived random series of the sinu- vice ω0 = k/m = 78.17 rad/s as the second natural frequency soidal functions has the form of Eq. (24). of the quarter car model ωu2. It allows the EMEH device to get into resonance. In this subsection the road profile shown in NW Fig. 8(b) was used, whereas vehicle velocity was assumed to be w(t)=∑Wi sin(ΩiVt+ φi), (24) equal V = 50 km/h to simulate behaviour of the EMEH while i=1 driving the vehicle on urban roads. Figure 9 shows the displacement, corresponding accelera- ΩN Ω1 where Wi = Θ(Ωi) − , Ωi = iΩ1, ΩN = NW Ω1, φi tions of the EMEH tube (excitation), load voltage u , magnet π(N 1) ∈ load − displacement x and generated energy E for assumed maximum [0,2π) is a random variable with uniform probability distribu- displacement xmax = 5 cm and gain G = 20. Mean generated tion, NW is the number of harmonics. The values of parameters power has been calculated from the end value of the produced are as follows: Ω1 = 0.02π rad/m, ΩN = 6π rad/m and thus energy: N = 300. These values cover the bandwidths suggested by W 1 ISO 8608. The equation above is suitable only for V = const. Pmean = E(tend), (25) Examples of generated random road profiles in relation to tend the travelled distance s = Vt for the road classes A and C are ˙ shown in Fig. 8. It can be seen that for the lower road class the where: E(t) is equal to the generated power in Eq. (12). roughness of the road is greater. The spatial harmonics content The frequency of the voltage is twice the vibration frequency is the same according to Eq. (24), where amplitudes Wi are not of the moving magnet, because the electromechanical coupling random variables. Due to this property of the generated profile is an odd function of magnet relative displacement x. Voltage it is easy to determine the desired PSD value (see Eq. (23)). occasionally achieves values greater than 200 V because of high number of coil turns. Mean power has a very rewarding value of Pmean = 1.85 W with relatively small dimensions of the EMEH device determined by xmax due to the use of the spring bumpers and the motion range amplifier. It can be seen that in time intervals when collision of the magnet with the spring bumpers does not occur, generated energy increments are significantly smaller. Also the level of generated voltage and electrical current are much smaller in these time intervals.

4.2. Influence of selected parameters on EMEH performance. This subsection discusses the characteristics describing dependences of the mean generated power and the values of the design parameters for selected road profiles and vehicle velocities. The EMEH device is nonlinear and its characteristics have Fig. 8. Examples of randomly generated road profiles: (a) for the road been determined by integration of equations of motion for each class A, (b) for the road class C value of the parameters which influence the generated power. Furthermore, road profiles expressed by Eq. (24) are random and for each numerical test the time histories of the EMEH state 4. Numerical study (for example those shown in subsection 4.1) and the Pmean values will be different. The characteristics have been determined This section presents a numerical study intended to calculate by calculating the average values of mean power from particular (n) the level of generated power and show the behaviour of the values of Pmean, each for n-th randomly generated road profile

Bull. Pol.Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 2020 13797 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

Fig. 9. Time histories of: (a) displacement and accelerations of the EMEH tube (red dashed line), (b) voltage produced by the EMEH device, (c) relative displacement of the moving magnet and (d) generated energy

wn(t) according to the formula below. where the stroke value xmax is greater than amplitude of the Fig. 10. Average power in dependence on respective parameters; red line denotes the best Pmean/xmax ratio moving magnet motion. Thus no resilient collision with spring Nn 1 (n) bumpers appears. Harvested power does not strongly depend Pmean = Pmean . (26) ues of the Pmean/xmax ratio marked by the red line. Thus, the ∂Φ(x) N ∑ on randomly generated road profile when the magnet does not εB = x˙ α(x)x˙ – electromotive force caused by n n=1 characteristics falls very gently along the red line in Fig. 10(c) • − ∂x ≈− reach the bumpers, because the road profiles have the same har- the moving magnet [V]. For this research, the number of randomly generated road pro- monics content. The random variable in the randomly gener- and (d). Due to use of the spring bumpers the system does not To show the influence of changes of inductance L(x) caused by files N = 50. The results are shown in Fig. 10. For each iter- ated road profile is only a phase shift (see Eq. (24)). In the require the resonance effect for efficient operation. This obser- n the moving magnet, the values of ε are compared with self- ation t = 10 s. The numerical study shows that longer time cases when the magnet motion achieves sufficient amplitude to vation stays in good agreement with the research about non- L end induction of an air-core coil replacement ε . ε represents does not significantly affect the results. provide resilient collisions, the trajectory of the moving mag- resonant impact or collision-based energy harvesters discussed Lair Lair the self-induction which would appear for the actual state of the For comparison, the generated power characteristics have net changes significantly when a different phase shift of each in Section 1. EMEH device for the coil inductance with neglected influence been determined for two classes of the road: A and C and particular harmonic appears for different random road profiles. It can be shown that generated power is proportional to of the moving iron core (moving magnet). the corresponding values of vehicle velocity V: 110 km/h and This is the reason for the irregular shape of the plot of the char- square of amplitude of excitation for linear model of the energy harvester. Thus, similarly for x values greater than amplitude The influence of the nonlinear terms listed above was anal- 50 km/h, respectively, to simulate the system behaviour used on acteristic when the magnet reaches xmax position, even for cal- max of motion of the moving magnet in Fig. 10(e) the mean power ysed by the calculation of their participation in the time se- the urban roads as well as the motorways or expressways. For culation of the mean power Pmean for 50 randomly generated seems to be proportional to the square of the gain G. When ries describing EMEH behaviour. The equation of motion (11) each of these two cases three types of characteristics have been road profiles. The maximum of the Pmean is shifted into higher x is sufficiently small to provide collisions of the moving was used with harmonic excitation x (t)=A sinΩt (without values of the xmax parameter for worse road quality (outside the max t t determined: 2 x vs R : parameters x and R vary, others are con- plots). The reason is that for higher vibration levels it is more magnet with the spring bumpers, the square-like dependence employing the quarter car model), thusx ¨t(t)= AtΩ sinΩt. • max load max load − stant: k = 550 N/m, G = 20, expected that the moving magnet will have collision with the becomes disturbed by additional nonlinearity provided by these In this test the parameters have the following values: EMEH collisions. spring stiffness k = 550 N/m, load resistance Rload = 2500 Ω, k vs xmax: parameters k and xmax vary, others are constant: spring bumpers, even if they are set up with greater distance • Ω = k/m = 78.1736 rad/s, moving magnet is not limited by G = 20, and Rload = 2500 Ω, between them. 4.3. Nonlinearities present in the electromagnetic circuit. spring bumpers and parameter G is not used. Calculations for xmax vs G: parameters xmax and G vary, others are constant: The red line exposed on the characteristics shown in Fig. 10 • The question arises about the influence of the nonlinearities in amplitude At = 4 mm were performed. k = 550 N/m, Rload = 2500 Ω. corresponds to the optimal ratio Pmean/xmax. It is helpful in in- Generated power is greater for the case of smaller velocity dicating the optimal dimensions of the EMEH device provided Eq. (11) on the EMEH behaviour. The most important non- Results in the form of plots similar to the phase maps are linearity is collision of the moving magnet with the spring V and worse road quality. Higher vehicle velocity V does not by the parameter xmax. The characteristics show that use of the shown in Fig. 11. Values of nonlinear terms are related to points bumpers. Its influence was discussed in Subsections 4.1 and 4.2. ( , ) guarantee a significantly higher level of vibrations of the m1 bumpers enhances the EMEH device in the sense of the propor- x v determined by the steady-state vibration of the magnet, In this subsection the problem of nonlinearities resulting = and m2 masses of the model, but merely results in shifting the tion between Pmean and the size of the EMEH device. It can be where v x˙. from interaction of the moving magnet with the magnetic coil is excitation bandwidth provided by the random road profiles into seen that the optimal value of the xmax parameter is near to 5 Numerical results in Fig. 11(a) show that the difference be- considered. Below, the nonlinear terms from Eq. (11) are listed: the higher frequencies. The exact dependence of the generated cm for the considered EMEH device, for both vibration levels. tween εL and self-induction of an air-core coil replacement εLair ∂Φ(x) power Pmean on velocity V is difficult to analyse because the The xmax vs Rload characteristics shows that dependence on F = q˙ α(x)q˙ – electrodynamic force [N], is significant, compared to the load voltage uload of the harvester • el ∂x ≈ EMEH device is nonlinear system. Furthermore, in reality the the Rload parameter is similar to the characteristics of the so- in the Fig. 11(b). Trajectories of εLair and εL are strongly dis- 1 ∂L(x) 2 road profile is unknown. called matched condition in DC circuits. It is because of the Frel = q˙ – reluctance force [N], turbed because of the dynamics of electrical circuit. • 2 ∂x Areas where dependence of the Pmean value on the design pa- relatively steady RMS value of the εB for each given xmax value. ∂L(x) Figure 11(a) shows that total self-induced magnetic flux link- ε = L(x)q¨ + x˙q˙ – total derivative of magnetic flux rameters is more regular are visible in the characteristics shown The use of the spring bumpers significantly expands the oper- • L ∂x age change εL has significant values when the moving magnet in Fig. 10, cases (a), (c) and (e). These areas refer to situations ational frequency bandwidth – especially near the optimal val- linkage induced by the current in the coil over the time [V], is inside the coil as well as when it is near to the border of the

13808 Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 Bull. Pol. Ac.: Tech. 68(6) 2020 9 DesignDesign of nonlinear of nonlinear electromagnetic electromagnetic energy energy harvester harvester equipped with with mechanical mechanical amplifier ampliﬁer and andspring spring bumpers bumpers

Fig. 10. Average power in dependence on respective parameters; red line denotes the best Pmean/xmax ratio

ues of the Pmean/xmax ratio marked by the red line. Thus, the ∂Φ(x) εB = x˙ α(x)x˙ – electromotive force caused by characteristics falls very gently along the red line in Fig. 10(c) • − ∂x ≈− and (d). Due to use of the spring bumpers the system does not the moving magnet [V]. require the resonance effect for efficient operation. This obser- To show the influence of changes of inductance L(x) caused by vation stays in good agreement with the research about non- the moving magnet, the values of εL are compared with self- resonant impact or collision-based energy harvesters discussed induction of an air-core coil replacement εLair. εLair represents in Section 1. the self-induction which would appear for the actual state of the It can be shown that generated power is proportional to EMEH device for the coil inductance with neglected influence square of amplitude of excitation for linear model of the energy of the moving iron core (moving magnet). harvester. Thus, similarly for xmax values greater than amplitude The influence of the nonlinear terms listed above was anal- of motion of the moving magnet in Fig. 10(e) the mean power ysed by the calculation of their participation in the time se- seems to be proportional to the square of the gain G. When ries describing EMEH behaviour. The equation of motion (11) xmax is sufficiently small to provide collisions of the moving was used with harmonic excitation xt(t)=At sinΩt (without magnet with the spring bumpers, the square-like dependence employing the quarter car model), thusx ¨ (t)= A Ω 2 sinΩt. t − t becomes disturbed by additional nonlinearity provided by these In this test the parameters have the following values: EMEH collisions. spring stiffness k = 550 N/m, load resistance Rload = 2500 Ω, Ω = k/m = 78.1736 rad/s, moving magnet is not limited by 4.3. Nonlinearities present in the electromagnetic circuit. spring bumpers and parameter G is not used. Calculations for The question arises about the influence of the nonlinearities in amplitude At = 4 mm were performed. Eq. (11) on the EMEH behaviour. The most important non- Results in the form of plots similar to the phase maps are linearity is collision of the moving magnet with the spring shown in Fig. 11. Values of nonlinear terms are related to points bumpers. Its influence was discussed in Subsections 4.1 and 4.2. (x,v) determined by the steady-state vibration of the magnet, In this subsection the problem of nonlinearities resulting where v = x˙. from interaction of the moving magnet with the magnetic coil is Numerical results in Fig. 11(a) show that the difference be- considered. Below, the nonlinear terms from Eq. (11) are listed: tween εL and self-induction of an air-core coil replacement εL ∂Φ(x) air Fel = q˙ α(x)q˙ – electrodynamic force [N], is significant, compared to the load voltage uload of the harvester • ∂x ≈ in the Fig. 11(b). Trajectories of ε and ε are strongly dis- 1 ∂L(x) Lair L F = q˙2 – reluctance force [N], turbed because of the dynamics of electrical circuit. • rel 2 ∂x ∂L(x) Figure 11(a) shows that total self-induced magnetic flux link- ε = L(x)q¨ + x˙q˙ – total derivative of magnetic flux • L ∂x age change εL has significant values when the moving magnet linkage induced by the current in the coil over the time [V], is inside the coil as well as when it is near to the border of the

Bull. Pol.Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 2020 13819 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

The proposed EMEH with its enhancements offers additional [12] M.A. Halim, H. Cho, and J.Y. Park, “Design and experiment of independent power supply in a vehicle and can be critical for re- a human-limb driven, frequency up-converted electromagnetic liable operation of its safety and navigation systems. It could be energy harvester”, Energy Conv. Manag. 106, 393–404 (2015), mounted on the vehicle without significant interference to struc- doi: 10.1016/j.enconman.2015.09.065. ture of the lorry. The above aspects make that the proposed en- [13] F. Cottone, R. Frizzell, S. Goyal, G. Kelly, and J. Punch, “En- ergy scavenging system can find a broad range of applications hanced vibrational energy harvester based on velocity amplifica- in automotive industry. tion”, J. Intell. Mater. Syst. Struct. 25(4), 443–451 (2014), doi: 10.1177/1045389X13498316. [14] R. Frizzell, G. Kelly, F. Cottone, E. Boco, V. Nico, D. O’Dono- Fig. 11. Values of nonlinear terms of equation of motion for each state of the magnet in steady-state vibration for At = 4 mm and Rload = 2500: Acknowledgements. The research has been financed with the ghue, and J. Punch, “Experimental characterisation of dual-mass (a) self-induction of the coil εL compared with self-induction εLair of air-core coil replacement, (b) electromotive force εB compared with voltage participation of the Smart Growth Operational Programme as a vibration energy harvesters employing velocity amplification”, across the load uload, (c) electrodynamic force Fel compared with reluctance force Frel part of the project “Innovative security and logistics manage- J. Intell. Mater. Syst. Struct. 27(20), 2810–2826 (2016), doi: ment system in transportation of goods using geoinforma- 10.1177/1045389X16642030. tion technologies” [contract no. POIR.01.01.01-00-0371/17- [15] A. Haroun, I. Yamada, and S. Warisawa, “Micro electromagnetic electromagnetic coil. It follows that both greater values of the this part of motion like near-constant velocity. In this situation 00], financed by the European Regional Development Fund, vibration energy harvester based on free/impact motion for low coil inductance with when the magnet is inside as well as its amplitude of motion of the moving magnet is not important – managed by the National Centre for Research and Development frequency – large amplitude operation”, Sens. Actuator A-Phys. gradient significantly affect the EMEH device behaviour. rather the speed of the magnet crossing the coil plays impor- in Warsaw, Poland. 224, 87–98 (2015), doi: 10.1016/j.sna.2015.01.025. In this case voltage uload takes smaller values than εB be- tant role. U-shaped curve relates to the near-optimal velocity [16] K. Pancharoen, D. Zhu, and S.P. Beeby, “Design optimization of cause of the losses in the coil resistance Rc and high participa- of the magnet during crossing the coil. 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Warminski, “Energy equipped with the mechanical amplifier and spring bumpers on cantly with the coil in relatively short interval we can consider (2017), doi: 10.1016/j.enconman.2016.11.026. harvesting from a magnetic levitation system”, Int. J. Non- the energy harvesting process has been analysed. Numerical in- [6] Spreemann D, Manoli Y. Electromagnetic Vibration Energy Har- Linear Mech. 94, 200–206 (2017), doi: 10.1016/j.ijnonlinmec. vestigation has been performed using the mathematical model vesting Devices. Dordrecht: Springer; 2012, doi: 10.1007/978- 2017.03.021. of EMEH derived with aid of the Hamilton’s principle of the 94-007-2944-5. [22] N.G. Elvin and A.A. Elvin, “An experimentally validated elec- least action. This model takes into account the dependence of [7] S. Yang, S.-Y. Jung, K. Kim, P. Liu, S. Lee, J. Kim, and H. Sohn, tromagnetic energy harvester”, J. Sound Vibr. 330(10), 2314– the electromagnetic coil inductance on the moving magnet po- “Development of a tunable low-frequency vibration energy har- 2324 (2011), doi: 10.1016/j.jsv.2010.11.024. sition. Additionally, parametric study of the influence of the dif- vester and its application to a self-contained wireless fatigue [23] T. Sobczyk and M. Radzik, “Improved algorithm for peri- ferent design parameters of the EMEH device on the recovered crack detection sensor”, Struct. Health Monit. 18(3), 920–933 odic steady-state analysis in nonlinear electromagnetic devices”, (2019), doi: 10.1177/1475921718786886. energy has been performed. Quarter car model moving on ran- Bull. Pol.Ac.: Tech. 67(5), 863–869 (2019), doi: 10.24425/ [8] J. Snamina and B. Sapinski, “Energy balance in self-powered mr domly generated road profiles has been employed to simulate bpasts.2019.130878. damper-based vibration reduction system”, Bull. Pol. Ac.: Tech. [24] T. Lubin, K. Berger, and A. Rezzoug, “Inductance and force cal- common work conditions for the energy scavenging device. 59(1), 75–80 (2011), doi: 10.2478/v10175-011-0011-4. culation for axisymmetric coil systems including an iron core of Numerical analysis has shown that use of the mechanical am- [9] S. Lenci, “On the production of energy from sea waves by a ro- finite length”, Prog. Electromagn. Res. B 41, 377–396 (2012), plifier together with the spring bumpers increase level of the tating pendulum: A preliminary experimental study”, J. Appl. doi: 0.2528/PIERB12051105. produced energy significantly. Additionally, the spring bumpers Nonlinear Dyn. 3(2), 173–186 (2014), doi: 10.5890/JAND. limit the stroke of the moving magnet resulting in relatively 2014.06.008. [25] A. Preumont, Mechatronics – Dynamics of Electromechanical [10] I. Shahosseini and K. Najafi, “Mechanical amplifier for transla- and Piezoelectric Systems, Netherlands: Springer, 2006, doi: small dimensions of the EMEH device. Moreover, it has been 10.1007/1-4020-4696-0. shown that use of spring bumpers expands operating frequency tional kinetic energy harvesters”, J. Phys. Conf. Ser. 557, 012135 (2014), doi: 10.1088/1742-6596/557/1/012135. [26] I. Zaabar and K. Chatti, “Identification of localized roughness bandwidth of the EMEH device. It is valuable property for features and their impact on vehicle durability”, in 11th Interna- many engineering applications, in which excitation parameters [11] B. Maamer, A. Boughamoura, A.M. Fath El-Bab, L.A. Fran- cis, and F. Tounsi, “A review on design improvement sand tional Symposium on Heavy Vehicle Transportation Technology, are varying in time. techniques for mechanical energy harvesting using piezoelec- 2010. Fig. 12. (a) amplitude of the relative magnet displacement Ax and It has been shown that the nonlinearities included in the pro- tric and electromagnetic schemes”, Energy Conv. Manag. 199, [27] T.F. Tyan, Y.-F. Hong, R. Shun, H. Tu, and W. 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138210 Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. 68(6) 68(6) 2020 Bull. Pol. Ac.: Tech. 68(6) 2020 11 DesignDesign of nonlinear of nonlinear electromagnetic electromagnetic energy energy harvester harvester equipped with with mechanical mechanical amplifier ampliﬁer and andspring spring bumpers bumpers

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