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A magnetically levitated vibration energy harvester

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2013 Smart Mater. Struct. 22 055016

(http://iopscience.iop.org/0964-1726/22/5/055016)

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IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 22 (2013) 055016 (10pp) doi:10.1088/0964-1726/22/5/055016

A magnetically levitated vibration energyharvesterX Y Wang, S Palagummi, L Liu and F G Yuan

Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh,NC 27695, USA

Received 1 September 2012, in final form 23 February 2013Published 10 April 2013Online at stacks.iop.org/SMS/22/055016

AbstractIn this paper a novel electromagnetic vibration type energy harvester that uses a diamagneticlevitation system is conceptualized, designed, fabricated, and tested. The harvester uses twodiamagnetic plates made of pyrolytic graphite between which a cylindrical magnet levitatespassively. Two thick cylindrical coils, placed in grooves which are engraved in the pyrolyticgraphite plates, are used to convert the mechanical energy into electrical energy efficiently.The geometric configurations of the coils are selected based on the field distribution of themagnet to enhance the efficiency of the harvester. A thorough theoretical analysis is carriedout to compare with experimental results. At an input power of 103.45 µW and at a frequencyof 2.7 Hz, the harvester generated a power of 0.74 µW with a system efficiency of 0.72%.Both theoretical and experimental results show that this new energy harvesting system cancapture low frequency broadband spectra.

(Some figures may appear in colour only in the online journal)

1. Introduction

Wireless sensor networks are increasingly used in inaccessiblelocations or hostile environments to transmit data withoutthe cost and inconvenience of wiring. In most of theseapplications, a long lifetime network with a non-maintenancepower source is usually the best choice. However, traditionalsolutions such as conventional batteries, solar energy andwind energy cannot meet these needs because of their limitedlifetime or high maintenance cost.

Although advances in low power very large scaleintegration (VLSI) design, along with the low duty cyclesfor wireless sensors, have reduced power requirements byhundreds to thousands of µW (e.g., MICA) with projectionsinto tens of µW, such rapid improvements via Moore’slaw have not been matched by similar advances in batterytechnology. At the same time, owing to shrinking chip sizesand the ever-increasing integration of more functionalities,the expectations for powering wireless sensors continue tooutpace conventional battery energy storage capabilities. Sucha trend for the low power consumption of wireless sensorshas given rise to the development of energy harvesting(energy scavenging) techniques, which aim at powering thewireless sensors by scavenging ambient energy from the

environment, supplementing and even entirely replacing thebattery energy. Such self-powered wireless sensors could beemployed potentially in hostile or inaccessible areas with littleor no maintenance.

Typically, the energy harvester converts vibration energyinto useful electrical energy using electrostatic, electro-magnetic, magnetostrictive or piezoelectric transducers. Forexample, [1–3] discuss the designs based on electrostaticpowered harvesters, whereas [4–13] discuss designs basedon electromagnetic and piezoelectric based energy harvesters.The energy harvester designs mentioned above focus onenhancing the amplitude of the relative motion, increasingthe wideband response, and decreasing the size of the energyharvester.

Mann and Sims [5], however, explored an alternativeapproach using magnetic levitation for harvesting vibrationenergy. They introduced a novel energy harvester that usesmagnetic forces to levitate a central floating magnet, andemploys coils to generate energy. However, in this design, thecentral floating magnet has to be constrained by a tight-fit tubeas it would be unstable in the horizontal direction. Becauseof the physical contact between the central floating magnetand the tube, internal friction inevitably consumes usefulmechanical energy.

10964-1726/13/055016+10$33.00 c© 2013 IOP Publishing Ltd Printed in the UK & the USA

Smart Mater. Struct. 22 (2013) 055016 X Y Wang et al

Figure 1. Magnetic moment measurement device.

This is a consequence of Earnshaw’s theorem [14] whichstates that particles cannot have a stable equilibrium ifinteracted by forces which vary by the inverse square ofthe distance between them. However, Earnshaw’s theoremdoesn’t consider all kinds of magnetic materials. Unlike ferro-and para-magnetic materials which align with the externalmagnetic field, diamagnetic materials repel magnetic field andmove towards field minima, making levitation possible.

Stabilized magnetic levitation without external energyinputs at room temperature was discovered by Berry andGeim [15] utilizing diamagnetic materials. Diamagneticlevitation provides stability in all directions and virtuallyenables the possibility to design a friction-free mechanism.

This paper explores a novel magnetic levitation techniqueusing diamagnetic material [16]. Liu and Yuan [17] havediscussed the feasibility of using diamagnetic levitationfor energy harvesting. Here experimental and theoreticalinvestigations of their concept for energy harvesting areanalyzed.

This paper is organized as follows. Section 2 details thedesign of the energy harvester, its mathematical modelingand its optimization. In section 3, the frequency response ofthe harvester under harmonic base excitation is examined,and experimental results are compared with theoreticalpredictions. Section 4 gives the conclusion.

2. Modeling and theory

2.1. Measurement of the magnetic moment of a cylindricalmagnet

To accurately calculate the magnetic field intensity of thefloating magnet used in the energy harvester, its magneticmoment is needed. Hence, the magnetic moment is firstmeasured. In figure 1, the device consists of two identicalcylindrical magnets, a micrometer and a strain gage.

The cylindrical magnet used here is the floating magnetthat is used in the experiment. By placing two identically sized

Figure 2. Magnetic force F(x) versus distance x.

cylindrical magnets end to end, the force can be calculatedwith the equation given by [18]:

F(x) =πµ0

4M2a4

[1

x2 +1

(x+ 2h)2−

2

(x+ h)2

](1)

where the radius and height of the magnet are a and h,respectively; M is the magnetization of the magnets; F(x)is the force between the two magnets; µ0 = 4π × 10−7 isthe magnetic constant; and x is the distance between the twofaces of the magnets (see figure 1). The magnitude of themagnetic moment, p is given by:

p = MV (2)

where V = πa2h is the volume of the magnet.The force F(x) is calculated at different distances (at

different x) by adjusting the micrometer, and calculating viathe strain gage measurement. The magnetic moment can beobtained using equations (1) and (2):

p ≈

√4F(x)π

µ0hx

(1−

12

y+38

y2)

(3)

where y = x2

(x+2h)2−

2x2

(x+h)2.

The radius and the thickness of the magnet are 5.53 mmand 11.1 mm, respectively (K&J magnetics, Inc.). Figure 2shows the experimental results. Using the experimental resultsand equation (3), the magnetic moment is estimated as1.26 A m2 which is in good agreement with values from themanufacturer (K&J magnetics, Inc.).

2.2. Diamagnetic force generation

The magnetic behavior of materials may come about fromeither electron orbital motion or electron spin. Both theseorigins can be represented by atomic currents. The combinedeffect of all these atomic currents in a material may inducea magnetization in the material. For diamagnetic materials,such as water, protein, DNA, plastic, wood, copper and manyother common materials, a small magnetization is inducedthat opposes the applied external magnetic field. In idealconditions, pyrolytic graphite has the strongest diamagnetism,20 times stronger than water, and bismuth is the strongestdiamagnetic metal [16].

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Smart Mater. Struct. 22 (2013) 055016 X Y Wang et al

Figure 3. (a) 2D view of the proposed energy harvester based on magnetic levitation. (b) 3D view of the energy harvester.

2.3. Energy harvester model

The proposed mechanism is illustrated in figure 3. The deviceconsists of two N52 cylindrical (NdFeB) magnets, one as afloating magnet and the other as a lifting magnet of largersize. The floating magnet is positioned between two paralleldiamagnetic plates (pyrolytic graphite) used for providingvertical stability and a restoring force to the floating magnet.Two thick cylindrical coils made of copper are mounted ontocircular grooves which are engraved in the diamagnetic plates,to convert the mechanical energy of the floating magnet intoelectrical energy.

Since the diamagnetic repelling forces are far too weakto support the weight of the floating magnet, of the orderof milli-Newtons, a lifting magnet is employed to balancethe weight. The lifting magnet is an axially magnetizeddisc-shaped permanent magnet which has a non-uniformmagnetic field in both the axial and the radial directions. Thesegradients can be used to stabilize the five degrees of freedom(except for the vertical direction which is unstable). Verticalstability is brought about by the pyrolytic graphite plates.In this energy harvester design, the equilibrium position ofthe floating magnet is located in the center between the twopyrolytic graphite plates. At this position the repulsive forcesfrom both the diamagnetic plates are equal and oppositein magnitude. Hence, the weight of the floating magnet isbalanced by the attractive force of the lifting magnet Flifting,

Flifting = mfloatingg (4)

where mfloating is the mass of the floating magnet, and g is theacceleration due to gravity. Ravaud et al [19] and Robertsonet al [20] presented analytical equations for calculating theforce between the two cylindrical magnets by modeling themas thin coils. For two cylindrical magnets, the force betweenthem is expressed as follows:

Fz =µ0k1k2

2Cd (5)

where k1 = I1/hmagnet1, k2 = I2/hmagnet2 are surface currentdensities for the thin coils. Cd is a scalar coefficient thatdepends on the geometry of the magnets and the spatialdistance between them [19, 20].

Figure 4. Magnetic force between the lifting magnet and thefloating magnet versus the distance between them.

Combining equations (4) and (5) yields

Flifting =µ0k1k2

2Cd = mfloatingg. (6)

In the harvester, the mass of the floating magnet is 8 g.The radius and thickness of the lifting magnet are 25.4 mmand 12.7 mm, respectively. Plotting the analytical solution forthe magnetic force given by equation (6) in figure 4, it canbe seen that if the distance between the lifting magnet andthe floating magnet is sufficiently large, the force betweenthe two magnets is almost constant for a minute change ofthe distance. Thus in the figure the attractive force from thelifting magnet is approximately equal to the weight of thefloating magnet near this equilibrium position. Hence, for theexperimental setup this approximate distance is used, alongwith the pyrolytic graphite plates, to smoothly levitate thefloating magnet around this equilibrium position.

Now we shall discuss the forces acting on the floatingmagnet in the dynamic state. To calculate the force betweenthe floating magnet and the diamagnetic plate we firstlycalculate accurately the magnetic field of the floating magnetat any arbitrary point in space. Derby and Olbert [21]introduced an analytical method to describe the magnetic fielddistribution of the thin coil, which can be approximated tothe magnetic field of a cylindrical magnet also. Hence, herea cylindrical magnet is modeled as a thin coil, with k asthe surface current density that equals NI/h, where N is thenumber of loops, I is the current in the coil and h is the

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Smart Mater. Struct. 22 (2013) 055016 X Y Wang et al

Figure 5. (a) Geometry for a thin cylindrical magnet in a cylindrical coordinate system. (b) A point (r, z) in the magnetic field of acylindrical magnet.

height of the cylindrical magnet. By using the Biot–SavartLaw, the magnetic field created by a thin coil is expressed ina cylindrical coordinate system as follows [19]:

B(˜r, ˜z) =µ0

4π

∫∫Sk ds×

{−∇G( ˜r, r)

}(7)

where G( ˜r, r) is the Green’s function, defined as follows (seefigure 5(a)):

G( ˜r, r) =1√

˜r2+ r2 − 2r ˜r cos(θ)+ (˜z− z)2

. (8)

With equation (7), the magnetic field strength in (˜r, ˜z)coordinates can be obtained [21]. After integrating withrespect to r and z through the entire magnet, the magneticinduction field components Br(r, z) and Bz(r, z) [21] (seefigure 5(b)) are given by:

Br(r, z)

= B0[α+C(k+, 1, 1,−1)− α−C(k−, 1, 1,−1)

](9)

Bz (r, z)

=B0a

a+ r[β+C(k+, γ

2, 1, γ )− β−C(k−, γ2, 1, γ )]

(10)

where B0 = (µ0/π) nI, (r, z) are the cylindrical coordinates,a is the radius of the magnet, and α±, β±, k±, C(k±, 1, 1,−1)and C(k±, γ 2, 1, γ ) are the scalar coefficients that depend onthe position of the point (r, z) with respect to the magnet [21].

Figure 6 shows that for the proposed electromagneticharvester the floating magnet is subjected to six externalforces in open-circuit, the four weak diamagnetic repulsiveforces Fupper graphite,Flower graphite from the pyrolytic graphiteplates and Fupper coil,Flower coil from the coils, Flifting fromthe lifting magnet and the weight of the floating magnet.Also, in this figure D is the distance between the twopyrolytic graphite plates, and z is the axial position ofthe floating magnet. In addition to this, two Lorentz

Figure 6. Schematic view of the floating magnet with the restoringforces.

forces Fupper coil(Lorentz),Flower coil(Lorentz) from the two series-wound coils act on the floating magnet in closed-circuit withload which will be discussed later in this section.

The force between the floating magnet and the pyrolyticgraphite plate can be found by taking the negative gradient ofthe magnetic potential energy. The potential energy for a pointdipole p in the presence of a uniform magnetic field B is givenby:

U = −p · B. (11)

In our case, we can assume that the complete pyrolyticgraphite plate is divided into small volume elements of sizedV each, with an induced dipole dp due to the magnetic fieldat that point. Also, similar to equation (2), the dipole momentcan be expressed by:

dp =M dV (12)

where M is the induced magnetization because of themagnetic field B for that volume element dV . As we havealready determined the magnetic field of a cylindrical magnetat any point in space by equation (7), we now write an

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Smart Mater. Struct. 22 (2013) 055016 X Y Wang et al

Figure 7. Restoring force versus displacement for the floatingmagnet.

expression for the force exerted on a volume element dV ofthe diamagnetic material in the z-direction:∫

dFz = ∇z

∫V(M · B) dV, M =

χ

µ0B (13)

where µ0 is the magnetic permeability of free air and χ is themagnetic susceptibility of the weak diamagnetic material [16].Here it should be noted that a part of the pyrolytic graphiteplate is replaced with copper coils, and for that region thecopper wire’s magnetic susceptibility has to be taken intoaccount. By calculating the force from the upper and the lowerpyrolytic graphite plate, and the upper and the lower coil onegets

Frestoring = Fupper graphite + Fupper coil − Flower graphite

− Flower coil + Flifting − mfloatingg (14)

where Frestoring is the total restoring force on the floatingmagnet (see figure 6).

Using the model given by equation (14), the restoringforce Frestoring versus the displacement of the floating magnetin between the two pyrolytic graphite plates is plotted infigure 7. The plot is unsymmetrical and nonlinear becauseof the nonlinear variation of the magnetic forces. Furtherdiscussion of a similar plot with such nonlinearity can befound in [22], and for this paper we deal with the restoringforce only in the linear region. The distance between the twopyrolytic graphite plates is D. The radius, r, and thickness, h,of the cylindrical floating magnet are 5.53 mm and 11.1 mm,respectively. The geometry of the coils, which is discussedin the next section, is such that they have an inner and outerdiameter of 7.15 mm and 12.9 mm respectively, and thethickness of the coils in the axial direction is 1 mm.

When there is relative motion between the magnet andthe pyrolytic graphite plate–coil assembly under ambientvibrations, eddy currents will be generated in the coil and thepyrolytic graphite plate due to the change in the magneticflux within them. These eddy currents produce a dampingforce on the floating magnet which is known as the Lorentzforce. Since the energy harvester is designed to work at lowfrequencies (less than 10 Hz), the eddy current in pyrolyticgraphite plates is very small, and therefore it can be ignored.

The eddy current in the coils, however, is harvested to powerthe external circuit.

In the proposed design, the two series-wound coilsare located over the upper face and the lower face of acylindrical magnet, respectively, thus the radial component ofthe magnetic field will play a key role in creating a voltagedifference. The total closed loop damping force based onequation (9) is given by:

Fdamping = Fupper coil(Lorentz) + Flower coil(Lorentz) (15)

Fupper coil(Lorentz)

= Icoil

n∑j=1

m∑i=1

Br(ri, zj + D/2− z)2πri (16)

Flower coil(Lorentz)

= Icoil

n∑j=1

m∑i=1

Br(ri, zj + D/2+ z)2πri (17)

Icoil =Vupper coil + Vlower coil

2Rcoil + Rload(18)

Vupper coil = z(t)n∑

j=1

m∑i=1

Br(ri, zj + D/2− z)2πri (19)

Vlower coil = z(t)n∑

j=1

m∑i=1

Br(ri, zj + D/2+ z)2πri (20)

where n is the number of coil layers in the axial direction, mis the number of turns in the radial direction, D is the distancebetween the two series-wound coils, ri is the radius of eachturn in the coil, and zj is the height of each layer of the coil.

Finally, the damping force on the floating magnet inopen-circuit is measured by calculating its ring-down aftergiving a mechanical impulse to the system. This dampingforce is due to the presence of eddy currents in the pyrolyticgraphite. The measured open-circuit Q factor was found tobe around 20, i.e. a damping ratio of 0.025. This factor ofequivalent mechanical damping has been neglected in thetheoretical model as it will be discussed later to be aroundtwo orders of magnitude lower when compared with dampingfor closed-circuit with load (Q factor closed-circuit withload = 0.42, i.e. damping ratio of 1.18). In future work, anattempt will be made to mitigate this damping force usinga laminated pyrolytic graphite plate, similar to the idea oflaminated cores in transformers.

2.4. Coil geometry

For a vibrational energy harvester, the electrical energy whichcan be extracted mainly originates from the mechanicalenergy of the relative motion in the harvester. The availablemechanical energy is determined by the mass and amplitudeof the moving object. The mechanical energy of themoving object can be transformed into electrical energydepending on the moving object working against the dampingforce. In particular, there are two kinds of damping inan electromagnetic vibrational energy harvester: electricaldamping and mechanical damping. The electromagnetic

5

Smart Mater. Struct. 22 (2013) 055016 X Y Wang et al

Figure 8. Radial magnetic flux density for a cylindrical magnet atdifferent distances above the flat surface of the cylindrical magnet.

vibrational harvester can transform mechanical energy intoelectrical energy, which is proportional to the electricaldamping. Electrical damping is in turn determined by themagnetic field gradient, the loops of coil, the impedance ofthe coil and the impedance of the load. Hence, the geometryof the coil and of the magnet play a key role in generatingmaximum power.

Using the accurate magnetic field model of the floatingmagnet developed earlier [21], a plot of the radial magneticfield distribution of a cylindrical magnet at various distancesabove the flat surface of the magnet is shown in figure 8. Itcan be seen that the maximum radial magnetic field intensityis at the radius of the magnet and decreases rapidly awayfrom the radius location. It can also be seen that the radialmagnetic field intensity decreases rapidly as the distance fromthe flat surface of the magnet increases. As the output voltageis proportional to the number of loops of the coil and themagnitude of the magnetic field intensity cutting throughit, the inner and outer diameters of the coil are selectedas 7.15 mm and 12.9 mm, respectively, and 1 mm as thethickness of the coils, which exceeds the maximum magneticfield intensity by around 60%. Two coils each containing 1300turns of AWG-48 enameled copper wire have a combinedresistance of 1870�. The AWG-48 enameled copper wire anda region exceeding 60% of the magnetic field intensity wasselected to accommodate a large number of turns of the coilin a reasonably high intensity region of the available magneticfield. However, this coil geometry led to excessive dampingwhich decreased the overall efficiency of the harvester,but it did restrict the floating magnet from impacting thepyrolytic graphite plates (not modeled theoretically), whichwas essential in providing a good proof of the concept for thetheoretical and experimental investigations carried out in thispaper.

2.5. Analytical model of the vibrational equation

The energy harvester can be modeled as a one degree offreedom (1-DOF) system with a spring, a mass and a damper,excited by a harmonic wave A sinωt as shown in figure 9. Itsequation of motion is governed by the following:

mz+ cz(t)+ kz(t) = mω2A sinωt (21)

Figure 9. A schematic diagram of the energy harvester.

where c is the damping constant, k is the stiffness, and z(t) isthe relative displacement between the floating magnet and thecoils. z(t) is the relative velocity between the floating magnetand the coil, m is the mass of the floating magnet, ω is theangular frequency of the harmonic excitation, and A is theamplitude of excitation.

In practice, the energy harvester in this paper is inherentlya nonlinear system since c and k vary with z(t) and z(t).Assuming the energy harvester operates in a vacuum, and thuswithout air friction, its differential equation of motion is givenby:

mz+ Fdamping(z(t), z(t))+ Frestoring(z(t))

= mω2A sinωt. (22)

where Fdamping is the electromagnetic damping force andFrestoring is the restoring force on the floating magnet, bothof which were discussed earlier.

3. Experimental investigation

This section primarily describes the experimental tests, alongwith a comparison of the results with theoretical predictions.The experimental tests were performed at different excitationfrequencies while keeping the given input power of the energyharvester and other experimental variables constant.

3.1. Experimental apparatus and instrumentation

The entire layout of the experimental equipment, the electricaland the mechanical components of the system are shown infigure 10(a). To avoid magnetic field interference from theshaker, a plastic cylindrical extension was used to keep theenergy harvester far enough away from the electromagneticshaker (VTS100-8, Vibration Test Systems, Inc.). The lowerfrequency excitations were performed on a servohydraulicload frame (INSTRON 8501), because of the frequencylimitation of the shaker. For frequencies at which the shakerwas used, a digital signal generator (DPO 2024, Tektronix)and an amplifier (7602 model, Krohn-hite) were used tocontrol the frequency and the amplitude of the shaker,respectively. A laser displacement sensor (OptoNCDT 1607,Micro-Epsilon) and a high speed camera (MotionScope PCI8000S, Redlake Imaging) were mounted on a base to obtainthe displacement of the floating magnet. Two tiny beams of

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Smart Mater. Struct. 22 (2013) 055016 X Y Wang et al

Figure 10. (a) A schematic of the experimental setup. (b) The experimental system for the diamagnetic levitation energy harvester.(c) A schematic diagram of the pick-up coil used in the experiment.

balsa wood were glued on the floating magnet to measurethe absolute displacement of the floating magnet via the laserdisplacement sensor.

For a system as shown in figure 10(b), if the input poweris held constant, it can absorb the maximum energy that willbe generated around its resonance frequency. For a harmonicbase excitation A sinωt, if the mass of the vibrator is muchless than the mass of the energy harvester system, the inputpower of the energy harvester is approximated by:

Pinput ≈ω

π

∫ π2ω

0mharvesterω

3A2 sin 2ωt dt

=1π

mharvesterω3A2 (23)

where mharvester is the mass of the energy harvester system,ω is the angular frequency of the base excitation, A is theamplitude of the base excitation, and Pinput is the averageinput power of the energy harvester system. As shown inequation (23), a constant input power to the energy harvesteris maintained by keeping ω3A2 constant.

The model parameters for the experimental system aregiven in table 1. m is the mass of the cylindrical floatingmagnet, mharvester is the total mass of the energy harvester,a is the radius of the floating magnet, h is the height of thefloating magnet, rw is the radius of the wire used for makingthe coils, hc is the thickness of the coil, ri is the inner radius ofthe ring-shaped coil, ro is the outer radius of the ring-shaped

7

Smart Mater. Struct. 22 (2013) 055016 X Y Wang et al

Figure 11. (a) Experimental results and theoretical predictions for the output voltage (RMS) (b) Comparison of experimental response withtheoretical predictions for the relative displacement of the floating magnet. (c) Comparison of experimental average output power withtheoretical predictions. (d) Comparison of the efficiency of the energy harvester with theoretical predictions.

Table 1. The model parameters for the experimental system.

Parameter Value Units

m 8 gmharvester 650 ga 5.53 mmh 11.1 mmrw 0.0158 mmhc 1 mmri 3.575 mmro 6.45 mmRcoil 935 �Rload 1870 �D 13 mmN 1300 Turnsω3A2 0.0005 m2 s−3

coil (see figure 10(c)), Rcoil is the resistance of each coil, Rloadis the resistance of the load, D is the perpendicular distancebetween the two pyrolytic graphite plates, N is the numberof turns in each coil and ω3A2, as previously discussed, isthe factor proportional to the average power input that ismaintained as a constant. With these model parameters, testsare performed at different excitation frequencies by keepingthe input power of the energy harvester fixed at around103.45 µW.

3.2. Comparisons between analytical prediction andexperimental results

The experimental tests were performed for a frequency sweepof 58 points to show the response behavior of the energy

harvester. The excitation frequency was varied from 0.2 Hzto 7 Hz with increments varying from 0.1 Hz to 0.4 Hz.Hence, the base accelerations were varied from 0.025 m s−2 to0.148 m s−2 and the input amplitudes from 15 mm to 76 µmas the frequency is varied from 0.2 to 7 Hz to keep the inputpower to a constant.

The voltage on the resistance load connected to theenergy harvester was measured at each frequency. Aprediction of the output voltage for the energy harvester wasalso made using the theoretical model via the experimentallyidentified parameters given in table 1. The theoreticalpredictions are compared with the experimental results infigure 11. The floating magnet can only vibrate in a smallrange because of the limitations of the design of thisdiamagnetic levitation energy harvester, with the maximumgap spacing between the floating magnet and the pyrolyticgraphite plate being around 0.95 mm in the experiment [16].Thus the output power of the energy harvester is relativelylow. Meanwhile, at relatively low excitation levels, thefrequency response behavior of the system behaves likea linear system. There are no jump response phenomenanear the primary resonance frequency. For frequencies below0.8 Hz, there is contact between the magnet and thediamagnetic plate and hence there is a moderate deviationbetween the analytical predictions and experimental resultsfor these frequencies. This is because the input amplitude atthese frequencies is much greater than the distance betweenthe floating magnet and the pyrolytic graphite plate. Alsowe can see an increase in the effect of eddy currents onthe pyrolytic graphite plates with frequencies close to 7 Hz,as the estimated theoretical value is slightly greater than

8

Smart Mater. Struct. 22 (2013) 055016 X Y Wang et al

the experimental value. However, there is a good matchfor the remaining frequencies which validates the analyticalprediction.

From figure 11(c), the quality factor for the harvester canbe calculated by measuring the ratio of the natural frequencyand the half-power bandwidth. The quality factor was found tobe around 0.42. This low quality factor is due to the selectionof the coil geometry, which was not optimum, and led tothe high magnitude of electrical damping in the coils. Theexcessive damping is possibly due to some of the turns inthe coil which are contributing more to the damping than tothe flux linkage. This led to a two orders of magnitude higherelectrical damping in closed-circuit with load when comparedwith open-circuit (eddy currents in pyrolytic graphite plate).An improvement to the coil geometry via an optimizationstudy will be pursued in future work to decrease the electricaldamping in the harvester.

4. Conclusions

This paper theoretically and experimentally investigated anovel magnetically levitating energy harvester. The energyharvester uses diamagnetic levitation theory to passivelylevitate a magnet between two diamagnetic plates, along witha lifting magnet. The diamagnetic plates also provide therestoring force to the floating magnet. For efficient energyharvesting, coils needed to be placed in a location where theycan capture the maximum magnetic flux from the floatingmagnet. Hence for this, circular grooves were engraved in thediamagnetic plates in which the coils were incorporated. Atheoretical model was made of this setup and its results werecompared with experimental results.

It was seen that the magnetic field intensity of the magnetchanges rapidly in both the radial and the axial directions,hence optimizing the coil geometry was necessary to makean effective energy harvester. Also, with this accurate modelof the magnet, the restoring forces and the damping forcesthat were used in the theoretical model were calculated. It wasseen here that the forces were weakly nonlinear for these lowlevel excitations and would become significantly nonlinear forhigher level excitations.

Unlike energy harvesting devices that operate at a highfrequency range, the energy harvester proposed in this paperprimarily works at a low frequency range (0.1–10 Hz). Bychanging the dimensions of the floating magnet along withthe geometry of the coil, the resonance frequency can betuned to a certain frequency for specific applications. Thislow working frequency range is much more suited to civilstructures whose natural frequency is very low. However, thework here doesn’t focus on maximizing the output power orthe efficiency, which would require a change in the design,such that the motion of the floating magnet is not constrained,and also a decrease in the overall mass of the harvester. Alsoan optimization study can be made of the coil geometry,which is the key in capturing maximum output power. Thefocus of this paper was to introduce an energy harvesterwhich can harvest energy from low frequency vibrations,

avoid mechanical frictions, and has the capability of beingnonlinear.

Further studies are being conducted to reduce the sizeof the harvester, reduce the eddy currents, increase thebandwidth of the system and give a better coil geometryfor a more effective energy harvester, while maintaining thealready achieved goals of low resonance frequency and zeromechanical friction.

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