Asynchronous phase shifted electromagnetic energy harvester
Asynchronous phase shifted electromagnetic energy harvester Jinkyoo Parka , Soonduck Kwonb and Kincho H. Lawa a Stanford b Chonbuk
University, Stanford, USA National University, Chunju-si, South Korea ABSTRACT
A vibration-based energy harvester is built upon the idea of transforming mechanical vibration of an inertial frame into electrical power. When the excitation frequency matches the natural frequency of the harvester, the energy generated by mechanical vibration is maximized. However, the reliance on resonance inevitably poses a robustness issue, in that power production drops significantly when the excitation frequency is slightly off from the tuned natural frequency of the harvester. To reduce the sensitivity of power output on the resonance, this paper proposes a novel concept of vibration based energy harvester in which both the magnets and the coils are attached to the vibrating cantilevers whose natural frequencies are separated with an optimally chosen frequency band. Due to the relative motions between the coil and the magnet cantilevers, the proposed energy harvester generates higher power over a wider range of excitation frequency compared to a conventional inertial frame based energy harvester. The improvements in the power output and the robustness are validated by experiments in the laboratory and on a bridge. Keywords: Energy harvester, Electromagnetic transducer, Asynchronous vibration, Resonance
1. INTRODUCTION Energy harvesting systems refer to devices that capture and transform energy from the environment into electricity. Unlike conventional, large-scale renewable energy generating systems such as wind turbines, thermal generators and solar panels, energy harvesting devices are primarily targeted at powering small electronic devices. For example, many researchers are investigating how to supply power to wireless sensor modules using energy harvesters.1, 2 If such sensors can be operated solely on power generated from an energy harvester, the need for regularly changing batteries can be eliminated, which in turn can reduce maintenance costs of the sensor units. Vibration-based energy harvesters can be categorized into three main types, namely electromagnetic, piezoelectric, and electrostatic, depending on the medium of a transducer.3 Transducers require input vibration with a high frequency and, in addition, the excitation frequency needs to match the natural frequency of the vibrational component in the energy harvester in order to maximize the power output. The necessary conditions for input vibrations have restricted vibration-based energy harvesters from being applied to a real-world environment and, in particular, to civil structures that exhibit vibration with low frequency under 5 Hz and where its natural frequency varies depending on environmental and structural conditions. To overcome the low-frequency limitation, many researchers have studied how to convert low-frequency vibration in the environment (thus on the target inertial frame) into high-frequency vibration in a harvester. Mechanical devices such as gears and windmills have been proposed as secondary components to excite cantilevers vibrating with high frequencies.4, 5 To reduce the sensitivity of the power on the excitation frequency, researchers have proposed a number of frequency tuning methods. Frequency tuning is done by adapting the natural frequency of a harvester, for example, by modifying the structural or electrical properties of the harvester to the excitation frequency. Most of frequency tuning methods rely on the use of controllers, such as electromagnets and piezoelectric actuators, in modifying the characteristics of the harvester.6–8 Because active frequency tuning Further author information: Jinkyoo Park: E-mail: [email protected] Soonduck Kwon: E-mail: [email protected] Kincho H. Law: E-mail: [email protected]
methods require continuous power input, they are hard to implement in practice. In addition, multi-modal approaches that use a set of harvesters with different natural frequencies to capture energy from a broad-banded frequency input source have been proposed.9 However, the multi-modal approaches produce only a fraction of output energy since only a portion of vibrating components is reacted to excitations. This paper discusses a novel concept for a vibration-based electromagnetic energy harvester utilizing the relative motion between the magnets and the coils attached to separate cantilevers that are vibrating in an asynchronous manner. By optimally choosing the difference in natural frequencies between the coil and the magnet cantilevers, the electromagnetic energy harvester in designed to produce power output over a wide range of excitation frequencies. Numerical as well as experimental simulations with the prototypes are conducted to verify the concept. In addition, field test results are provided to investigate the feasibility of the asynchronous harvester in the real-world environment.
2. THEORETICAL BACKGROUND 2.1 Forced vibration Most energy harvesters with electromagnetic transducers use an inertial frame configuration, in which the relative movement between a magnet and a coil is induced by the vibration of the inertial frame. The vibration system can be described as a second-order differential equation represented as follows: m¨ z (t) + cT z(t) ˙ + kz(t) = −m¨ y(t)
(1)
where y(t) represents the inertial frame displacement, z(t) is the relative displacement between the magnet and the coil, and m, cT , and k are, respectively, the mass, the total damping coefficient including mechanical and electrical damping, and the stiffness of the vibration frame. Assuming that the the inertial frame vibrates with the single dominant frequency ω (i.e., y(t) = Y sin(ωt)), the relative displacement between the coil and the magnet can be determined as:10 Y z(t) = s
1−
w wn
w wn
2 2
2
sin(ωt − ϕ)
(2)
i2 h + 2ζT wwn
where ωn and ζT are, respectively, the natural frequency and the total damping ratio of the vibrating component in the energy harvester. Furthermore, ϕ is the phase difference between the excitation and the response of the vibrating system and is given as: ϕ = tan
−1
w wn
2ζT
1−
w wn
2
(3)
The phase shift angle ϕ is a function of the damping ratio ζT and the ratio between the excitation frequency ω and the natural frequency of the harvester ωn . The phase shift ϕ plays an important role in magnifying the power output and increasing the robustness of the harvester to be introduced in this paper.
2.2 Principle of electromagnetic energy harvesters An electromagnetic transducer converts the relative movement between a magnet and a coil into electrical energy. The voltage induced in the electromagnetic transducer can be expressed by Faraday’s law as; V = −N
dΦ dz dz dt
(4)
where N represents the number of coil turns, and Φ is the magnetic flux. That is, the induced voltage is proportional to the product of the magnetic flux gradient, which is determined by the geometries of the coil and the magnet as well as their relative configurations, and the relative velocity dz dt between the magnet and the coil.
11 The term −N dΦ dz in Eq. 4 can be replaced by an electrical damping coefficient cE approximated as:
cE =
(BN l)2 RL + Rc + jwLc
(5)
where B and N l are, respectively, the average flux density and the effective length of the coil. RL , Rc and Lc are, respectively, the load resistance, the coil resistance and the coil inductance.11 For a harmonic base excitation y(t) = Y sin(ωt), the average extracted power from the harvester is equivalent to the dissipated energy by the electrical damping and can be written as:12 3 w 2 w3 mζ Y E wn 1 Pavg (w) = cE |z| ˙2= 2 2 h i2 2 1 − wwn + 2ζT wwn
(6)
where the total damping ratio ζT (ζT = cT /2mwn ) includes both the structural (mechanical) and the electrical damping (ζT = ζS + ζE ). The structural damping ζS can usually be found by free vibration test, while the electrical damping coefficient ζE can be calculated as ζE = cE /2mwn . Note that the electrical damping coefficient term occurs in both the numerator and the denominator in Eq. 6, which means that the electrical damping contributes both in increasing the power through the inductive voltage and in decreasing the power by increasing the damping effect in the cantilever. Therefore, the power can be maximized by optimally choosing the load resistance RL . Furthermore, how close is the natural frequency ωn of the harvester to the base excitation frequency ω can significantly affect the power output; when ω ≈ ωn (resonance), the power is maximized.
3. METHODOLOGY Instead of fixing either a coil or a magnet to an inertial frame, as illustrated in Fig. 1, we attach the coils and the magnets to separate vibrating cantilevers with different natural frequencies. The relative motion between a coil and a magnet is exploited to generate power. If the natural frequencies of the coil and the magnet cantilevers are exactly same, no power will be produced since the two cantilevers will vibrate with the same phase angle. If the natural frequencies for the two cantilevers are intentionally separated, the two cantilevers vibrate with different amplitudes and phase angles, therefore inducing a relative motion (or speed) between the magnet and the coil. The energy harvester discussed in this paper utilizes this induced relative motion as a means to amplify the power as well as to decrease the sensitivity of the power output on the base excitation.
Figure 1: An electromagnetic energy harvester with multiple cantilevers, vibrating asynchronously
3.1 Design parameters For conventional vibration based energy harvesters, the frequency ratio ωωn , or equivalently ffn , between the base ω ) and the natural frequency fn (= ω2πn ) of a harvester, and the load resistance RL are excitation frequency f (= 2π the two of the most important design parameters. For the asynchronous harvester, the difference ∆f = fc − fm between the natural frequency fc of the coil cantilever and the natural frequency fm of the magnet cantilever is also an important design parameter affecting the level of power and the sensitivity of the power on the base excitation frequency. The following discussion examines the influences of the frequency difference ∆f and the load resistance RL on the power output characteristics.
Figure 2: Concept and design procedure for asynchronous energy harvester Considering a harmonic base excitation, we investigate first how the responses of the two vibrating cantilevers induce the relative motion between them. Fig. 2 shows the velocity response factor Rv (f ) and the phase angle ϕ(f ) curves, based on which the tip speeds of the coil cantilever z˙c (t) and the tip speed of the magnet cantilever z˙m (t) can be found. Based on the individually calculated responses (speed), the time series for the power P (t; f ) given the base harmonic excitation frequency f can be calculated as: P (t; f ) = cE (z˙m (t) − z˙c (t))2
(7)
where (z˙m (t) − z˙c (t)) represents the relative speed between the coil and the magnet. Fig. 2 shows two cantilevers having different natural frequencies as illustrated by the different peak locations in the velocity response curves as well as the different phase angle curves. Let’s denote ∆f = fc − fm as the difference in the frequencies between the coil and the magnet cantilevers. If the base excitation frequency f falls within the frequency band, the two cantilevers vibrate in an out of phase mode with magnified response amplitudes which can often lead to an increase in relative velocity between the coil and the magnet. When the excitation frequencies are slightly off from the frequency band, either the magnet or the coil cantilever vibrates with a magnified amplitude even though one of the the two cantilevers would act as an inertial frame and does not vibrate. Therefore, the power can be generated over an expanded range over the base excitation. By comparing the response curves for the two cantilevers, we can hypothesis how the frequency difference ∆f = fc − fm and the load resistance RL will affect the power output of the asynchronous harvester. 1. The frequency difference ∆f = fc −fm determines the width between the two peaks in the velocity response curves. If the difference ∆f is too large, the responses from the two cantilevers are isolated, making only
one of them to vibrate while the other remains stationary. On the other hand, if ∆f is small, the responses of the two cantilevers are synchronized, thus limiting the power production within a very narrow excitation frequency band. 2. The load resistance RL will change the shapes of Rv (f ) and ϕ(f ) curves by affecting the total damping (ζT = ζS + ζE = ζS + cE (RL )/2wn m) of the cantilevers. A lower value of RL will increase the total damping, thus making the Rv (f ) curve widen, which leads to an overlapping between the two Rv (f ) curves. A higher value of RL , on the contrary, will sharpen the Rv (f ) curves, making the curves narrower and separated with less overlapping of each other. The following section provides experimental results that demonstrate the effectiveness of the asynchronous electro magnetic energy harvester.
4. EXPERIMENTAL RESULTS 4.1 Experiment setting Fig. 3 shows a prototype that builds upon the concept of an asynchronous energy harvester. The prototype harvester has a total of 11 cantilevers: five cantilevers holding the coils and six cantilevers holding the magnets. The cantilevers have the same geometry and stiffness. Additional masses are included to the tips of the cantilevers holding the coils to alter their fundamental natural frequencies varying from 7 to 8.6 Hz. The six cantilevers holding the magnets are all connected through a magnet holder attached to the tip of cantilevers and their natural frequency is fixed at 6.8 Hz. The magnet holder is designed to (1) prevent the attractive force between the neighboring magnets from causing the lateral torsion in the magnet cantilevers, and (2) make the magnet cantilevers vibrate synchronously with the same phase so that the magnetic flux is concentrated in the inside of the coils instead of being radiated out in the air. The frequency band between the coil and the magnet cantilevers can easily be regulated by modifying the tip masses of the coil cantilevers. In this study, we investigate the power outputs from coil 1, coil 2 and coil 3 as shown in Fig. 3. The parameters for these coils are tabulated in Table 1.
Figure 3: Prototype for an asynchronous cantilever energy harvester As shown in Fig. 4, the experimental setting includes a modal shaker (K2007E01, a product from The Model Shop) to excite the energy harvester, a NI-USB 6008 DAQ driver to measure the voltage output time series from coils 1, 2 and 3, and load resistors RL ranging from 0.1 KΩ to 10.0 KΩ to vary the electrical damping coefficient cE . The influence of the different frequency bands can be investigated by comparing the power outputs from coils 1, 2 and 3. The influence of the electrical damping can be studied by comparing the power outputs from the harvester with the different resistances. A series of harmonic base excitations with frequencies f ranging
Figure 4: Shaker experiment setting Table 1: The structural parameters for the cantilevers holding the magnet, coil 1, coil 2, and coil 3, and the electrical parameters for the coils. Magnet
Coil 1
Coil 2
Coil 3
Cantilever width (m)
3.00 × 10−3
3.00 × 10−3
3.00 × 10−3
3.00 × 10−3
Cantilever length (m)
3.45 × 10−2
3.45 × 10−2
3.45 × 10−2
3.45 × 10−2
Cantilever thickness (m)
1.22 × 10−4
1.22 × 10−4
1.22 × 10−4
1.22 × 10−4
Tip mass (g)
3.30 × 10−3
3.20 × 10−3 1
3.00 × 10−3
2.65 × 10−3
Elastic Modulus E (N/m2 )
1.80 × 1011
1.80 × 1011
1.80 × 1011
1.80 × 1011
4.54 × 10−16
4.54 × 10−16
4.54 × 10−16
4.54 × 10−16
Stiffness K (N/m)
6.82
6.82
6.82
6.82
Natural frequency fn (Hz)
6.78
7.35
7.59
8.07
Coil resistance RC (Ω)
130
130
130
Coil inductance LC (mH)
18.3
18.3
18.3
Moment of inertia I (m4 )
from 5 to 10 Hz with an increment of 0.1 Hz are conducted for each combination of the frequency band ∆f and load resistance RL . Voltage output time series over a 30-sec duration are measured with a sampling rate of 3,000 Hz. For each excitation frequency f , the root mean square (RMS) power is calculated using the measured voltage time series. Figs. 5, 6, and 7 show the power response curves constructed using the RMS power values. Additionally, the figures also show the power outputs of the conventional approach where the magnet cantilevers are fixed with the inertia frame (attached to the shaker). There are a number of noteworthy observations from the experiments. For all frequency bands, ∆f = 0.5, 0.8, and 1.3 Hz, as the load resistance RL increases, the general RMS power level increases until reaching a certain load resistance. Comparing the conventional harvester and the asynchronous harvester, the shapes of the power response curves are different. For the conventional harvester with the magnets fixed to the inertial frame (shaker), the shape of the power response curve shows a sharper peak as the load resistance increases and the total damping (i.e., cT = cS + cE (RL )) decreases. For the asynchronous harvester, the shape of the power response curve changes from a single dominant peak to two dominant peaks as load resistance increases. The change in the shape of the power response curve is similar to the shapes of the velocity response factor curve Rv (f ) when considering both the magnet and the coil cantilevers. When the load resistance RL is low, the velocity response curves tend to be shallower in shape and overlap each other. When the load resistance RL is
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(d) RL =2.20KΩ
(e) RL =3.30KΩ
(f) RL =10.0KΩ
Figure 5: Power response curve with different RL over frequency band ∆f = 0.5Hz (Coil 1)
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(a) RL =0.10KΩ
(b) RL =0.33KΩ
(c) RL =1.00KΩ
(d) RL =2.20KΩ
(e) RL =3.30KΩ
(f) RL =10.0KΩ
Figure 6: Power response curve with different RL over frequency band ∆f = 0.8Hz (Coil 2) Magnet−fixed Asynchronous
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f(Hz)
f(Hz)
f(Hz)
f(Hz)
(a) RL =0.10KΩ
(b) RL =0.33KΩ
(c) RL =1.00KΩ
(d) RL =2.20KΩ
(e) RL =3.30KΩ
(f) RL =10.0KΩ
Figure 7: Power response curve with different RL over frequency band ∆f = 1.3Hz (Coil 3) high, each of the two velocity response curves becomes sharp with dominant and distinct peaks. Another interesting observation from the experimental results shown in Figs. 5 to 7 is that for low load resistance (i.e., high total damping) the maximum RMS power for the asynchronous harvester is larger than that of the conventional synchronous harvester. It can be seen from the results of the asynchronous harvester that the shape of the power response curve exhibits a single dominant peak as the load resistance is lower than 3.3 KΩ. This result is possibly caused by the different damping effects on the magnet and the coil cantilevers by the load resistance RL . The damping on the coil cantilever increases in a faster rate than that of the magnet cantilever. The result becomes more observable as the frequency band becomes larger.
The effects of the frequency band on power output can be observed by comparing the results shown in Figs. 5 to 7. When the load resistance RL is high (the total damping is small), the two dominant peaks are further apart as the frequency band increases. When the load resistance is low (the total damping is large), the wider is the frequency band, the higher the maximum power level. Furthermore, as the frequency band become larger, the maximum power (i.e, the peak in the power response curve) occurs at a lower load resistance RL . The magnified power level and the widened frequency range for producing power are the benefits that are provided by an asynchronous harvester over the conventional inertial frame based vibrational energy harvester.
4.2 Comparison between analytical model and the experiment results
(a) ∆f = 0.5Hz
(b) ∆f = 0.8Hz
(c) ∆f = 1.3Hz
Figure 8: Comparison of power response curves obtained from the analytical model and from the experimental measurements for the low damping case (RL = 10.0KΩ). For lower damping case (RL = 10.0KΩ), Fig. 8 compares the power response curves obtained from the experiments and the numerical simulations. The simulated power response curves are calculated by using the dynamic response factor curve Rv (f ) and the phase curve ϕ(f ), which are shown in Fig. 8. In addition, the influence of the frequency band ∆f on the power response curves are studied. In general, the estimated power response curves from the analytical model (Eq. 7) accurately predicts the general trends in the power response curves measured from the experiments. For the conventional harvester case (magnet-fixed case), the shape of the power response curve resembles the velocity response curve Rv (f ) of the coil cantilever and the measured and simulated curves are in good agreement. For the asynchronous case, the power response curve has two peaks that resemble the union of the two dynamic response curves for the magnet and the coil cantilevers. The two power response curves from the analytical model (shaded area) and the experiment (line with square symbol) are also generally in good agreement. In addition, as the frequency band ∆f increases, the distance between the two peaks in a power response curve also increases (see Figs 8a, 8b and 8c). The result infers that, when the damping ratio is low, reducing the width of the frequency band between the magnet and the coil cantilevers seems to increase the robustness of the asynchronous harvester (Fig. 8a). When the electrical damping is high (i.e., the lower value of the RL is used), the analytical model fails to predict the power output trends observed during the laboratory experiments. In the analytical model, the total damping ratio for the magnet and the coil cantilevers are assumed to be affected by the load resistance in the same way (i.e., ζT = ζS + ζE (RL )). However, when the damping is high, this assumption does not explain the experimental trend. To account for the observations (large power amplification and asymmetric power curve in
(a) ∆f = 0.5Hz
(b) ∆f = 0.8Hz
(c) ∆f = 1.3Hz
Figure 9: Comparison of power response curves obtained from the analytical model and from the experimental measurements for the high damping case (RL = 1.0KΩ). the asynchronous harvester) found in the experiments, we modify the electrical damping ratio ζE of the magnet cantilever in the analytical model by reducing the electrical damping by a third, i.e., c′E (RL ) = 13 cE (RL ). Upon this modification, Fig. 9 compares the power response curves obtained from the experiments and the numerical simulations. The dynamic response curves for the magnet cantilever have sharper peak and the phase plots show a rapidly increasing curve due to the reduced total damping ratio. The modification results are in close agreement between the two power response curves obtained from the analytical model and the experiments. For the asynchronous case, the power response curves resemble the union of the two dynamic response curves for the magnet and the coil cantilevers. In addition, as the frequency band ∆f increases, the maximum power also increases (see Figs. 9a, 9b and 9c) possibly due to the large difference in the phase angles between the magnet and the coil cantilevers since a larger phase difference often introduces faster relative tip speed between the two cantilevers. For higher damping, increasing the frequency band is beneficial in magnifying the power output.
5. FIELD TEST To validate the feasibility of the concept of the asynchronous harvester, the prototype harvester has been tested on a bridge that exhibits vibration with low frequency.
5.1 Electro-mechanical coupling equation Unlike the laboratory shaker tests where pure sinusoidal input base vibrations are used to excite the prototype harvester, in a field test, the prototype harvester is excited by an arbitrary form of bridge vibration. Instead of using a forced vibration harmonic solution, therefore, a mechanical-electrical coupling state space equation is used to simulate the dynamic responses of cantilevers and the power outputs. In particular, we simulate the simultaneous vibrational and electrical responses of multiple cantilevers by accounting for their interactions. Accounting for the mechanical inertial force as well as the electro motive force, the motion of the jth coil cantilever can be described by the following second order differential equation:13, 14 ext em mc,j z¨c,j (t) + cc,j z˙c,j (t) + kc,j z(t) = Fc,j (t) + Fc,j (t),
j = 1, ..., n
(8)
where zc,j is the relative displacement of the jth coil cantilever with respect to the inertial frame, and mc,j , cc,j and kc,j are, respectively, the tip mass, the damping coefficient and the stiffness of the jth coil cantilever. In ext addition, Fc,j = −mc,j x ¨g (x¨g is the base excitation acceleration) is the inertial force exerted on the jth coil and em Fc,j is the electro-motive force caused by the interactions between the jth coil and the magnet and is expressed as em Fc,j = φIj (t) (9) where φ is the electro-mechanical coupling coefficient which is related to magnetic field. Since the magnetic field experienced by the coils depends on the relative location of the coil and the magnet, the electro-mechanical coupling coefficient is also a function of the relative location of the coil and the magnet. Expressing the electromechanical coupling coefficient φ as a function of the relative locations between the coils and the magnets requires complicated magnetic flux analysis. In this study, φ is simply assumed to be constant (i.e., φ = cE in Eq. 5) for a simplicity. Similar to the motion of the coil cantilever, the motion of the magnet cantilever is described by the following equation: em ext (t) + Σnj Fc,j (t) (10) mm z¨m (t) + cm z˙m (t) + km zm (t) = Fm where zm is the relative vertical displacement of the magnet cantilever, and mm , cm , and km are, respectively, the tip mass, the damping coefficient and the stiffness of the magnet cantilever. The external force exerted on ext ext the magnet Fm (t) can be expressed as Fm = −mm x¨g . In addition, the electro-motive force exerting on the n em magnet Σj Fc,j (t) is the sum of individual electro-motive force Fjem (t) because the magnet cantilevers, which are connected through the magnetic holder, interact with all the coils in the harvester. The electro-mechanical coupling coefficient φ relates the electrical energy and the mechanical energy. For the jth coil, the relationship between the rate of current I˙c,j (t) and the relative vertical speed between the jth coil and the magnet (z˙c,j − z˙m ) can be written as a first order differential equation in terms of electro-mechanical coupling coefficient φ, the resistance Rc,j including both the coil and load resistance, and the inductance Lc,j of the jth coil:14 Rc,j φ (zm ˙ (t) − zc,j ˙ (t)) − Ic,j (t) j = 1, ..., n (11) I˙c,j (t) = Lc,j Lc,j Expressing the states of the system as a vector z = [zm , z˙m , zc,1 , z˙c,1 , ..., zc,n , z˙c,n , Ic,1 , ..., Ic,n ]T , and combining Eqs. 8, 10, and 11 yield the mechanical and electrical coupling equation for the asynchronously vibrating multiple cantilevers as follows: z(t) ˙ = Az(t) + E x ¨g (t) (12) Suppose there are three coils in the example, the state dynamic matrix A can be expressed as 0
km − m m 0 0 0 A= 0 0 0 0 0
0
1
0 0 0
0 0 1
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0 0 0
0 0 0 0 0 0 1
0 0 0
0 0 0 0 0
cm −m m 0 0 0 0 0 0
− mc,1 c,1 0 0 0 0
− mc,1 c,1 0 0 0 0
− mc,2 c,2 0 0
− mc,2 c,2 0 0
− mc,3 c,3
− mc,3 c,3
− Lφc,1
0
φ Lc,1
0
0
0
0
c,1 − Lc,1
R
0
− Lφc,2
0
0
0
φ Lc,2
0
0
0
c,2 − Lc,2
− Lφc,3
0
0
0
0
0
φ Lc,3
0
0
k
c
k
c
k
c
− mφc,1 0 0 0 0
− mφc,2 0 0 R
0 0 0 0 0 0 0
− mφc,3 0 0
(13)
R
c,3 − Lc,3
with the state vector z = [zm , z˙m , zc,1 , z˙c,1 , zc,2 , z˙c,2 , zc,3 , z˙c,3 , Ic,1 , Ic,2 , Ic,3 ]T and E is an input excitation vector given as E = [0, −1, 0, −1, 0, −1, 0, −1, 0, 0, 0]T (14) Assuming that the electro-mechanical coefficient φ is a constant, Eq. 12 is easy to solve. The vibrational responses of all the cantilevers and the inductive currents in the coils can be extracted from the state trajectory z(t).
5.2 Bridge test To investigate the feasibility of the concept, a field test has been conducted on a bridge (Nong-Ro Bridge), located in Kimhae-si, Korea. For the field test, only an approximate fundamental natural frequency of the bridge is known. Therefore, we adjust the natural frequencies of the coils and magnet cantilevers by adding additional masses to make the natural frequencies of the cantilevers close to the expected fundamental frequency of the bridge. Figs. 10a, 10b and 10c show the bridge, the installation of the harvester on the bridge with the data acquisition system, and a heavy vehicle passing through the bridge.
(a) Target Bridge
(b) Field test data accusation
(c) Heavy loaded vehicle
Figure 10: Field test setup (Nong-Ro Bridge, Kimhae-si, Korea) Fig. 11a shows the time series for the vertical acceleration of the bridge and Figs. 11b and 11c show the voltage time series measured form the harvester and the power spectrum density for the voltage time series, respectively. The measurement duration is 720 sec, during which two heavy trucks passed through the bridge. The instances of the passing vehicle are associated with the sharp peaks in the acceleration and voltage time series. Fig. 11c clearly shows that the asynchronous harvester produces the two peak in the power spectrum density curve, whose shape is similar to the power response curves from the laboratory test. This implies that the asynchronous harvester is excited over the broad frequency range in bridge vibration and produces the power.
0.8
1.5
0.018
15 ton dump truck
0.6
−0.2
PSD [V2/Hz]
Voltage (V)
Acceleration (m/sec2)
0
0.5
0
0.012 0.01
f
0.008
magnet
0.006
−0.4
0.004
−0.5 −0.6 −0.8 0
coil
0.014
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f
0.016
15 ton dump truck
1
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(a) Acceleration time series a(t)
−1 0
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(b) Voltage time series V (t)
0 2
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3 Frequency(Hz)
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(c) Power spectrum density for V (t)
Figure 11: Bridge acceleration measurement at the Nong-Ro Bridge (Kimhae-si, Korea) for the duration of 720 sec and voltage measurement (time series and power spectrum density) from the energy harvester coil 4 during the bridge field test (fc = 3.2Hz, fm = 3.4Hz) To further investigate how the dynamic responses of the bridge and the corresponding voltage output vary at the instance when a heavy vehicle passes, the portion of the acceleration and voltage measurements are studied in a narrow time window (10 ∼ 40 sec) as shown in Figs. 12 and 13. Figs. 12a and 13a show, respectively, the acceleration and the voltage time series for the 40-second duration when a 15-ton dump truck passes through the bridge. In addition, Figs. 12b and 13b show the wave form of the acceleration and the voltage output signal, respectively, between the 10 ∼ 15 sec period. Finally, Figs. 12c and 13c show, respectively, the power spectrum density functions for the acceleration and the voltage output over the 40 sec long period. The power spectrum density function, as shown in Fig. 12c, indicates that the bridge vibrates at a frequency around 3 Hz. As a result, the coil cantilever whose fundamental frequency is close to 3 Hz vibrates with a large
amplitude, which leads to a high level of voltage output. In this case, the voltage output signal resembles a pure periodic wave form with the frequency equal to the natural frequency of the coil cantilever (3.2 Hz). It should be noted that the voltage outputs were measured using a NI-6009 DAQ driver whose internal resistance is 144 KΩ, which is much higher than the optimum load resistance for the electromagnetic harvester. If the optimal load resistance is used, the power may increase considerably. −4
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(c) Power spectrum density for a(t)
Figure 12: Bridge acceleration measurement (when large vehicle passes) 0.6
1.5 Measured Simulated
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(c) Power spectrum density for V (t)
Figure 13: Voltage measurement from the energy harvester coil 4 during the bridge field test (fc = 3.2Hz, fm = 3.4Hz) (when large vehicle passes) We also compare the measured and simulated voltage outputs and the corresponding power spectra. The simulations are conducted based on Eq. 12 and the results are compared in Fig. 13. The general trend in the voltage output time series shows good agreement. However, the voltage amplitudes are slightly different over the 17 ∼ 20 sec period. This deviation is possibly due to the limitation of the prototype; when the cantilever reaches a displacement exceeding 1 cm, it collides against other cantilevers vibrating with different phases.
6. SUMMARY Conventional vibration-based electromagnetic energy harvesters rely heavily on resonance of vibration, which leads to a significant performance deterioration when the excitation frequency differ even slightly from the natural frequency of the vibrating system in the harvester. To overcome this robustness issue, we discuss a novel concept of a harvester utilizing asynchronously vibrating multiple cantilevers. In the asynchronous harvester, both a coil and a magnet are attached to the cantilevers that are vibrating separately with different natural frequencies and phase differences. The frequency difference between the natural frequencies of the coil and the magnet cantilevers, and the level of electrical damping affect the power response curve. The harvester is shown to have two dominant peaks in its power response curve. When the frequency difference between the two cantilevers is optimally chosen, the two power response curves overlap that leads to a widen frequency band in the power-response curve. The wider frequency band reduces
the sensitivity of the power production on the excitation frequency. Furthermore, asynchronously vibrating cantilevers produce more power than the conventional concept, especially when the electromagnetic damping coefficient is high. The increase in robustness as well as the increase in power producing capability are among the advantages that the asynchronous harvester provides. To validate the proposed concept, laboratory tests using modal shaker have been conducted, and the results show that the power as well as the robustness improve when comparing to the conventional type energy harvester. Finally, the prototype energy harvester is tested on a bridge that exhibits traffic induced vibration with a low frequency around 3 Hz. The field test reveals that the harvester vibrates and induces voltage when a heavy truck passes through the bridge during which the prototype produces a peak power of 30 µW . The amount of power can be significantly improved if the optimal load resistance is used and the frequency is tuned close to the structural fundamental frequency.
ACKNOWLEDGMENTS This research is partially supported by the US National Science Foundation under Grant No. CMMI-0824977. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.
REFERENCES [1] S. Roundy and P. Wright, “A Piezoelectric Vibration Based Generator for Wireless Electronics,” Smart Materials and Structures 13, pp. 1131–1142, 2004. [2] H. Jung and I. Kim, “An Energy Harvesting System Using the Wind-Induced Vibration of a Stay Cable for Powering a Wireless Sensor Node,” Smart Materials and Structures 20, 2011. doi:10.1088/09641726/20/7/075001. [3] S. Beeby, M. Tudor, and N. White, “Energy Harvesting Vibration Sources for Microsystems Application,” Measurement Science and Technology 17, pp. R175–R195, 2006. [4] S. Priya, “Modeling of Electric Energy Harvesting Using Piezoelectric Windmill,” Applied Physics Letter 87, 2005. http://dx.doi.org/10.1063/1.2119410 (accessed 15th Febuary 2013). [5] J. Rastegar and R. Murray, “Novel Two-Stage Piezoelectric Based Electrical Energy Generators for Low and Variable Speed Rotary Machinery,” Proc. SPIE 7643, 2009. doi:10.1117/12.847755. [6] S. Roundy and Y. Zhang, “Toward Self-Tuning Adaptive Vibration Based Micro-generators,” Proc. SPIE 5649, pp. 373–384, 2005. doi:10.1117/12.581887. [7] V. Challa, M. Prasad, and F. Fisher, “Towards an Autonomous Self-Tuning Vibration Energy Harvesting Device for Wireless Sensor Network Applications,” Smart Materials and Structures 20, 2011. doi:10.1088/09641726/20/2/025004. [8] M. Lallart, S. Anton, and D. Inman, “Frequency Self-Tuning Secheme for Broadband Vibration Energy Harvesting,” Jounal of Intelligent Material Systems and Structures 21, pp. 897–906, 2010. [9] S. Shahruz, “Design of Mechanical Band-Pass Filters for Energy Scavenging,” Journal of Sound and Vibration 292, pp. 987–998, 2006. [10] A. Chopra, Dynamics of Structures, Pearson Prentice Hall, 2007. [11] M. El-hami and P. Glynne-Jones, “Design and Fabrication of A New Vibration-based Electromechanical Power Generator,” Sensors and Actuators 92, pp. 335–342, 2001. [12] S. Beeby and R. Torsh, “A Mirco Electromagnetic Generator for Vibration Eergy Harvesting,” Journal of Micromechanics and Microengineering 17, pp. 1257–1265, 2007. [13] P. K. Sinha, Electromagnetic Suspension - Dynamics and Control, IEE, London, 1987. [14] J. S. Lee, S. D. Kwon, M. Y. Kim, and I. Yeo, “A Parametric Study on The Dynamics of Urban Transit Maglev Vehicle Running on Flexible GuidewaBridges,” J. of Sound and Vibration 328(3), pp. 301–317, 2009.
University, Stanford, USA National University, Chunju-si, South Korea ABSTRACT
A vibration-based energy harvester is built upon the idea of transforming mechanical vibration of an inertial frame into electrical power. When the excitation frequency matches the natural frequency of the harvester, the energy generated by mechanical vibration is maximized. However, the reliance on resonance inevitably poses a robustness issue, in that power production drops significantly when the excitation frequency is slightly off from the tuned natural frequency of the harvester. To reduce the sensitivity of power output on the resonance, this paper proposes a novel concept of vibration based energy harvester in which both the magnets and the coils are attached to the vibrating cantilevers whose natural frequencies are separated with an optimally chosen frequency band. Due to the relative motions between the coil and the magnet cantilevers, the proposed energy harvester generates higher power over a wider range of excitation frequency compared to a conventional inertial frame based energy harvester. The improvements in the power output and the robustness are validated by experiments in the laboratory and on a bridge. Keywords: Energy harvester, Electromagnetic transducer, Asynchronous vibration, Resonance
1. INTRODUCTION Energy harvesting systems refer to devices that capture and transform energy from the environment into electricity. Unlike conventional, large-scale renewable energy generating systems such as wind turbines, thermal generators and solar panels, energy harvesting devices are primarily targeted at powering small electronic devices. For example, many researchers are investigating how to supply power to wireless sensor modules using energy harvesters.1, 2 If such sensors can be operated solely on power generated from an energy harvester, the need for regularly changing batteries can be eliminated, which in turn can reduce maintenance costs of the sensor units. Vibration-based energy harvesters can be categorized into three main types, namely electromagnetic, piezoelectric, and electrostatic, depending on the medium of a transducer.3 Transducers require input vibration with a high frequency and, in addition, the excitation frequency needs to match the natural frequency of the vibrational component in the energy harvester in order to maximize the power output. The necessary conditions for input vibrations have restricted vibration-based energy harvesters from being applied to a real-world environment and, in particular, to civil structures that exhibit vibration with low frequency under 5 Hz and where its natural frequency varies depending on environmental and structural conditions. To overcome the low-frequency limitation, many researchers have studied how to convert low-frequency vibration in the environment (thus on the target inertial frame) into high-frequency vibration in a harvester. Mechanical devices such as gears and windmills have been proposed as secondary components to excite cantilevers vibrating with high frequencies.4, 5 To reduce the sensitivity of the power on the excitation frequency, researchers have proposed a number of frequency tuning methods. Frequency tuning is done by adapting the natural frequency of a harvester, for example, by modifying the structural or electrical properties of the harvester to the excitation frequency. Most of frequency tuning methods rely on the use of controllers, such as electromagnets and piezoelectric actuators, in modifying the characteristics of the harvester.6–8 Because active frequency tuning Further author information: Jinkyoo Park: E-mail: [email protected] Soonduck Kwon: E-mail: [email protected] Kincho H. Law: E-mail: [email protected]
methods require continuous power input, they are hard to implement in practice. In addition, multi-modal approaches that use a set of harvesters with different natural frequencies to capture energy from a broad-banded frequency input source have been proposed.9 However, the multi-modal approaches produce only a fraction of output energy since only a portion of vibrating components is reacted to excitations. This paper discusses a novel concept for a vibration-based electromagnetic energy harvester utilizing the relative motion between the magnets and the coils attached to separate cantilevers that are vibrating in an asynchronous manner. By optimally choosing the difference in natural frequencies between the coil and the magnet cantilevers, the electromagnetic energy harvester in designed to produce power output over a wide range of excitation frequencies. Numerical as well as experimental simulations with the prototypes are conducted to verify the concept. In addition, field test results are provided to investigate the feasibility of the asynchronous harvester in the real-world environment.
2. THEORETICAL BACKGROUND 2.1 Forced vibration Most energy harvesters with electromagnetic transducers use an inertial frame configuration, in which the relative movement between a magnet and a coil is induced by the vibration of the inertial frame. The vibration system can be described as a second-order differential equation represented as follows: m¨ z (t) + cT z(t) ˙ + kz(t) = −m¨ y(t)
(1)
where y(t) represents the inertial frame displacement, z(t) is the relative displacement between the magnet and the coil, and m, cT , and k are, respectively, the mass, the total damping coefficient including mechanical and electrical damping, and the stiffness of the vibration frame. Assuming that the the inertial frame vibrates with the single dominant frequency ω (i.e., y(t) = Y sin(ωt)), the relative displacement between the coil and the magnet can be determined as:10 Y z(t) = s
1−
w wn
w wn
2 2
2
sin(ωt − ϕ)
(2)
i2 h + 2ζT wwn
where ωn and ζT are, respectively, the natural frequency and the total damping ratio of the vibrating component in the energy harvester. Furthermore, ϕ is the phase difference between the excitation and the response of the vibrating system and is given as: ϕ = tan
−1
w wn
2ζT
1−
w wn
2
(3)
The phase shift angle ϕ is a function of the damping ratio ζT and the ratio between the excitation frequency ω and the natural frequency of the harvester ωn . The phase shift ϕ plays an important role in magnifying the power output and increasing the robustness of the harvester to be introduced in this paper.
2.2 Principle of electromagnetic energy harvesters An electromagnetic transducer converts the relative movement between a magnet and a coil into electrical energy. The voltage induced in the electromagnetic transducer can be expressed by Faraday’s law as; V = −N
dΦ dz dz dt
(4)
where N represents the number of coil turns, and Φ is the magnetic flux. That is, the induced voltage is proportional to the product of the magnetic flux gradient, which is determined by the geometries of the coil and the magnet as well as their relative configurations, and the relative velocity dz dt between the magnet and the coil.
11 The term −N dΦ dz in Eq. 4 can be replaced by an electrical damping coefficient cE approximated as:
cE =
(BN l)2 RL + Rc + jwLc
(5)
where B and N l are, respectively, the average flux density and the effective length of the coil. RL , Rc and Lc are, respectively, the load resistance, the coil resistance and the coil inductance.11 For a harmonic base excitation y(t) = Y sin(ωt), the average extracted power from the harvester is equivalent to the dissipated energy by the electrical damping and can be written as:12 3 w 2 w3 mζ Y E wn 1 Pavg (w) = cE |z| ˙2= 2 2 h i2 2 1 − wwn + 2ζT wwn
(6)
where the total damping ratio ζT (ζT = cT /2mwn ) includes both the structural (mechanical) and the electrical damping (ζT = ζS + ζE ). The structural damping ζS can usually be found by free vibration test, while the electrical damping coefficient ζE can be calculated as ζE = cE /2mwn . Note that the electrical damping coefficient term occurs in both the numerator and the denominator in Eq. 6, which means that the electrical damping contributes both in increasing the power through the inductive voltage and in decreasing the power by increasing the damping effect in the cantilever. Therefore, the power can be maximized by optimally choosing the load resistance RL . Furthermore, how close is the natural frequency ωn of the harvester to the base excitation frequency ω can significantly affect the power output; when ω ≈ ωn (resonance), the power is maximized.
3. METHODOLOGY Instead of fixing either a coil or a magnet to an inertial frame, as illustrated in Fig. 1, we attach the coils and the magnets to separate vibrating cantilevers with different natural frequencies. The relative motion between a coil and a magnet is exploited to generate power. If the natural frequencies of the coil and the magnet cantilevers are exactly same, no power will be produced since the two cantilevers will vibrate with the same phase angle. If the natural frequencies for the two cantilevers are intentionally separated, the two cantilevers vibrate with different amplitudes and phase angles, therefore inducing a relative motion (or speed) between the magnet and the coil. The energy harvester discussed in this paper utilizes this induced relative motion as a means to amplify the power as well as to decrease the sensitivity of the power output on the base excitation.
Figure 1: An electromagnetic energy harvester with multiple cantilevers, vibrating asynchronously
3.1 Design parameters For conventional vibration based energy harvesters, the frequency ratio ωωn , or equivalently ffn , between the base ω ) and the natural frequency fn (= ω2πn ) of a harvester, and the load resistance RL are excitation frequency f (= 2π the two of the most important design parameters. For the asynchronous harvester, the difference ∆f = fc − fm between the natural frequency fc of the coil cantilever and the natural frequency fm of the magnet cantilever is also an important design parameter affecting the level of power and the sensitivity of the power on the base excitation frequency. The following discussion examines the influences of the frequency difference ∆f and the load resistance RL on the power output characteristics.
Figure 2: Concept and design procedure for asynchronous energy harvester Considering a harmonic base excitation, we investigate first how the responses of the two vibrating cantilevers induce the relative motion between them. Fig. 2 shows the velocity response factor Rv (f ) and the phase angle ϕ(f ) curves, based on which the tip speeds of the coil cantilever z˙c (t) and the tip speed of the magnet cantilever z˙m (t) can be found. Based on the individually calculated responses (speed), the time series for the power P (t; f ) given the base harmonic excitation frequency f can be calculated as: P (t; f ) = cE (z˙m (t) − z˙c (t))2
(7)
where (z˙m (t) − z˙c (t)) represents the relative speed between the coil and the magnet. Fig. 2 shows two cantilevers having different natural frequencies as illustrated by the different peak locations in the velocity response curves as well as the different phase angle curves. Let’s denote ∆f = fc − fm as the difference in the frequencies between the coil and the magnet cantilevers. If the base excitation frequency f falls within the frequency band, the two cantilevers vibrate in an out of phase mode with magnified response amplitudes which can often lead to an increase in relative velocity between the coil and the magnet. When the excitation frequencies are slightly off from the frequency band, either the magnet or the coil cantilever vibrates with a magnified amplitude even though one of the the two cantilevers would act as an inertial frame and does not vibrate. Therefore, the power can be generated over an expanded range over the base excitation. By comparing the response curves for the two cantilevers, we can hypothesis how the frequency difference ∆f = fc − fm and the load resistance RL will affect the power output of the asynchronous harvester. 1. The frequency difference ∆f = fc −fm determines the width between the two peaks in the velocity response curves. If the difference ∆f is too large, the responses from the two cantilevers are isolated, making only
one of them to vibrate while the other remains stationary. On the other hand, if ∆f is small, the responses of the two cantilevers are synchronized, thus limiting the power production within a very narrow excitation frequency band. 2. The load resistance RL will change the shapes of Rv (f ) and ϕ(f ) curves by affecting the total damping (ζT = ζS + ζE = ζS + cE (RL )/2wn m) of the cantilevers. A lower value of RL will increase the total damping, thus making the Rv (f ) curve widen, which leads to an overlapping between the two Rv (f ) curves. A higher value of RL , on the contrary, will sharpen the Rv (f ) curves, making the curves narrower and separated with less overlapping of each other. The following section provides experimental results that demonstrate the effectiveness of the asynchronous electro magnetic energy harvester.
4. EXPERIMENTAL RESULTS 4.1 Experiment setting Fig. 3 shows a prototype that builds upon the concept of an asynchronous energy harvester. The prototype harvester has a total of 11 cantilevers: five cantilevers holding the coils and six cantilevers holding the magnets. The cantilevers have the same geometry and stiffness. Additional masses are included to the tips of the cantilevers holding the coils to alter their fundamental natural frequencies varying from 7 to 8.6 Hz. The six cantilevers holding the magnets are all connected through a magnet holder attached to the tip of cantilevers and their natural frequency is fixed at 6.8 Hz. The magnet holder is designed to (1) prevent the attractive force between the neighboring magnets from causing the lateral torsion in the magnet cantilevers, and (2) make the magnet cantilevers vibrate synchronously with the same phase so that the magnetic flux is concentrated in the inside of the coils instead of being radiated out in the air. The frequency band between the coil and the magnet cantilevers can easily be regulated by modifying the tip masses of the coil cantilevers. In this study, we investigate the power outputs from coil 1, coil 2 and coil 3 as shown in Fig. 3. The parameters for these coils are tabulated in Table 1.
Figure 3: Prototype for an asynchronous cantilever energy harvester As shown in Fig. 4, the experimental setting includes a modal shaker (K2007E01, a product from The Model Shop) to excite the energy harvester, a NI-USB 6008 DAQ driver to measure the voltage output time series from coils 1, 2 and 3, and load resistors RL ranging from 0.1 KΩ to 10.0 KΩ to vary the electrical damping coefficient cE . The influence of the different frequency bands can be investigated by comparing the power outputs from coils 1, 2 and 3. The influence of the electrical damping can be studied by comparing the power outputs from the harvester with the different resistances. A series of harmonic base excitations with frequencies f ranging
Figure 4: Shaker experiment setting Table 1: The structural parameters for the cantilevers holding the magnet, coil 1, coil 2, and coil 3, and the electrical parameters for the coils. Magnet
Coil 1
Coil 2
Coil 3
Cantilever width (m)
3.00 × 10−3
3.00 × 10−3
3.00 × 10−3
3.00 × 10−3
Cantilever length (m)
3.45 × 10−2
3.45 × 10−2
3.45 × 10−2
3.45 × 10−2
Cantilever thickness (m)
1.22 × 10−4
1.22 × 10−4
1.22 × 10−4
1.22 × 10−4
Tip mass (g)
3.30 × 10−3
3.20 × 10−3 1
3.00 × 10−3
2.65 × 10−3
Elastic Modulus E (N/m2 )
1.80 × 1011
1.80 × 1011
1.80 × 1011
1.80 × 1011
4.54 × 10−16
4.54 × 10−16
4.54 × 10−16
4.54 × 10−16
Stiffness K (N/m)
6.82
6.82
6.82
6.82
Natural frequency fn (Hz)
6.78
7.35
7.59
8.07
Coil resistance RC (Ω)
130
130
130
Coil inductance LC (mH)
18.3
18.3
18.3
Moment of inertia I (m4 )
from 5 to 10 Hz with an increment of 0.1 Hz are conducted for each combination of the frequency band ∆f and load resistance RL . Voltage output time series over a 30-sec duration are measured with a sampling rate of 3,000 Hz. For each excitation frequency f , the root mean square (RMS) power is calculated using the measured voltage time series. Figs. 5, 6, and 7 show the power response curves constructed using the RMS power values. Additionally, the figures also show the power outputs of the conventional approach where the magnet cantilevers are fixed with the inertia frame (attached to the shaker). There are a number of noteworthy observations from the experiments. For all frequency bands, ∆f = 0.5, 0.8, and 1.3 Hz, as the load resistance RL increases, the general RMS power level increases until reaching a certain load resistance. Comparing the conventional harvester and the asynchronous harvester, the shapes of the power response curves are different. For the conventional harvester with the magnets fixed to the inertial frame (shaker), the shape of the power response curve shows a sharper peak as the load resistance increases and the total damping (i.e., cT = cS + cE (RL )) decreases. For the asynchronous harvester, the shape of the power response curve changes from a single dominant peak to two dominant peaks as load resistance increases. The change in the shape of the power response curve is similar to the shapes of the velocity response factor curve Rv (f ) when considering both the magnet and the coil cantilevers. When the load resistance RL is low, the velocity response curves tend to be shallower in shape and overlap each other. When the load resistance RL is
Magnet−fixed Asynchronous
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(a) RL =0.10KΩ
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(c) RL =1.00KΩ
(d) RL =2.20KΩ
(e) RL =3.30KΩ
(f) RL =10.0KΩ
Figure 5: Power response curve with different RL over frequency band ∆f = 0.5Hz (Coil 1)
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(a) RL =0.10KΩ
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(c) RL =1.00KΩ
(d) RL =2.20KΩ
(e) RL =3.30KΩ
(f) RL =10.0KΩ
Figure 6: Power response curve with different RL over frequency band ∆f = 0.8Hz (Coil 2) Magnet−fixed Asynchronous
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f(Hz)
f(Hz)
f(Hz)
f(Hz)
f(Hz)
(a) RL =0.10KΩ
(b) RL =0.33KΩ
(c) RL =1.00KΩ
(d) RL =2.20KΩ
(e) RL =3.30KΩ
(f) RL =10.0KΩ
Figure 7: Power response curve with different RL over frequency band ∆f = 1.3Hz (Coil 3) high, each of the two velocity response curves becomes sharp with dominant and distinct peaks. Another interesting observation from the experimental results shown in Figs. 5 to 7 is that for low load resistance (i.e., high total damping) the maximum RMS power for the asynchronous harvester is larger than that of the conventional synchronous harvester. It can be seen from the results of the asynchronous harvester that the shape of the power response curve exhibits a single dominant peak as the load resistance is lower than 3.3 KΩ. This result is possibly caused by the different damping effects on the magnet and the coil cantilevers by the load resistance RL . The damping on the coil cantilever increases in a faster rate than that of the magnet cantilever. The result becomes more observable as the frequency band becomes larger.
The effects of the frequency band on power output can be observed by comparing the results shown in Figs. 5 to 7. When the load resistance RL is high (the total damping is small), the two dominant peaks are further apart as the frequency band increases. When the load resistance is low (the total damping is large), the wider is the frequency band, the higher the maximum power level. Furthermore, as the frequency band become larger, the maximum power (i.e, the peak in the power response curve) occurs at a lower load resistance RL . The magnified power level and the widened frequency range for producing power are the benefits that are provided by an asynchronous harvester over the conventional inertial frame based vibrational energy harvester.
4.2 Comparison between analytical model and the experiment results
(a) ∆f = 0.5Hz
(b) ∆f = 0.8Hz
(c) ∆f = 1.3Hz
Figure 8: Comparison of power response curves obtained from the analytical model and from the experimental measurements for the low damping case (RL = 10.0KΩ). For lower damping case (RL = 10.0KΩ), Fig. 8 compares the power response curves obtained from the experiments and the numerical simulations. The simulated power response curves are calculated by using the dynamic response factor curve Rv (f ) and the phase curve ϕ(f ), which are shown in Fig. 8. In addition, the influence of the frequency band ∆f on the power response curves are studied. In general, the estimated power response curves from the analytical model (Eq. 7) accurately predicts the general trends in the power response curves measured from the experiments. For the conventional harvester case (magnet-fixed case), the shape of the power response curve resembles the velocity response curve Rv (f ) of the coil cantilever and the measured and simulated curves are in good agreement. For the asynchronous case, the power response curve has two peaks that resemble the union of the two dynamic response curves for the magnet and the coil cantilevers. The two power response curves from the analytical model (shaded area) and the experiment (line with square symbol) are also generally in good agreement. In addition, as the frequency band ∆f increases, the distance between the two peaks in a power response curve also increases (see Figs 8a, 8b and 8c). The result infers that, when the damping ratio is low, reducing the width of the frequency band between the magnet and the coil cantilevers seems to increase the robustness of the asynchronous harvester (Fig. 8a). When the electrical damping is high (i.e., the lower value of the RL is used), the analytical model fails to predict the power output trends observed during the laboratory experiments. In the analytical model, the total damping ratio for the magnet and the coil cantilevers are assumed to be affected by the load resistance in the same way (i.e., ζT = ζS + ζE (RL )). However, when the damping is high, this assumption does not explain the experimental trend. To account for the observations (large power amplification and asymmetric power curve in
(a) ∆f = 0.5Hz
(b) ∆f = 0.8Hz
(c) ∆f = 1.3Hz
Figure 9: Comparison of power response curves obtained from the analytical model and from the experimental measurements for the high damping case (RL = 1.0KΩ). the asynchronous harvester) found in the experiments, we modify the electrical damping ratio ζE of the magnet cantilever in the analytical model by reducing the electrical damping by a third, i.e., c′E (RL ) = 13 cE (RL ). Upon this modification, Fig. 9 compares the power response curves obtained from the experiments and the numerical simulations. The dynamic response curves for the magnet cantilever have sharper peak and the phase plots show a rapidly increasing curve due to the reduced total damping ratio. The modification results are in close agreement between the two power response curves obtained from the analytical model and the experiments. For the asynchronous case, the power response curves resemble the union of the two dynamic response curves for the magnet and the coil cantilevers. In addition, as the frequency band ∆f increases, the maximum power also increases (see Figs. 9a, 9b and 9c) possibly due to the large difference in the phase angles between the magnet and the coil cantilevers since a larger phase difference often introduces faster relative tip speed between the two cantilevers. For higher damping, increasing the frequency band is beneficial in magnifying the power output.
5. FIELD TEST To validate the feasibility of the concept of the asynchronous harvester, the prototype harvester has been tested on a bridge that exhibits vibration with low frequency.
5.1 Electro-mechanical coupling equation Unlike the laboratory shaker tests where pure sinusoidal input base vibrations are used to excite the prototype harvester, in a field test, the prototype harvester is excited by an arbitrary form of bridge vibration. Instead of using a forced vibration harmonic solution, therefore, a mechanical-electrical coupling state space equation is used to simulate the dynamic responses of cantilevers and the power outputs. In particular, we simulate the simultaneous vibrational and electrical responses of multiple cantilevers by accounting for their interactions. Accounting for the mechanical inertial force as well as the electro motive force, the motion of the jth coil cantilever can be described by the following second order differential equation:13, 14 ext em mc,j z¨c,j (t) + cc,j z˙c,j (t) + kc,j z(t) = Fc,j (t) + Fc,j (t),
j = 1, ..., n
(8)
where zc,j is the relative displacement of the jth coil cantilever with respect to the inertial frame, and mc,j , cc,j and kc,j are, respectively, the tip mass, the damping coefficient and the stiffness of the jth coil cantilever. In ext addition, Fc,j = −mc,j x ¨g (x¨g is the base excitation acceleration) is the inertial force exerted on the jth coil and em Fc,j is the electro-motive force caused by the interactions between the jth coil and the magnet and is expressed as em Fc,j = φIj (t) (9) where φ is the electro-mechanical coupling coefficient which is related to magnetic field. Since the magnetic field experienced by the coils depends on the relative location of the coil and the magnet, the electro-mechanical coupling coefficient is also a function of the relative location of the coil and the magnet. Expressing the electromechanical coupling coefficient φ as a function of the relative locations between the coils and the magnets requires complicated magnetic flux analysis. In this study, φ is simply assumed to be constant (i.e., φ = cE in Eq. 5) for a simplicity. Similar to the motion of the coil cantilever, the motion of the magnet cantilever is described by the following equation: em ext (t) + Σnj Fc,j (t) (10) mm z¨m (t) + cm z˙m (t) + km zm (t) = Fm where zm is the relative vertical displacement of the magnet cantilever, and mm , cm , and km are, respectively, the tip mass, the damping coefficient and the stiffness of the magnet cantilever. The external force exerted on ext ext the magnet Fm (t) can be expressed as Fm = −mm x¨g . In addition, the electro-motive force exerting on the n em magnet Σj Fc,j (t) is the sum of individual electro-motive force Fjem (t) because the magnet cantilevers, which are connected through the magnetic holder, interact with all the coils in the harvester. The electro-mechanical coupling coefficient φ relates the electrical energy and the mechanical energy. For the jth coil, the relationship between the rate of current I˙c,j (t) and the relative vertical speed between the jth coil and the magnet (z˙c,j − z˙m ) can be written as a first order differential equation in terms of electro-mechanical coupling coefficient φ, the resistance Rc,j including both the coil and load resistance, and the inductance Lc,j of the jth coil:14 Rc,j φ (zm ˙ (t) − zc,j ˙ (t)) − Ic,j (t) j = 1, ..., n (11) I˙c,j (t) = Lc,j Lc,j Expressing the states of the system as a vector z = [zm , z˙m , zc,1 , z˙c,1 , ..., zc,n , z˙c,n , Ic,1 , ..., Ic,n ]T , and combining Eqs. 8, 10, and 11 yield the mechanical and electrical coupling equation for the asynchronously vibrating multiple cantilevers as follows: z(t) ˙ = Az(t) + E x ¨g (t) (12) Suppose there are three coils in the example, the state dynamic matrix A can be expressed as 0
km − m m 0 0 0 A= 0 0 0 0 0
0
1
0 0 0
0 0 1
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0 0 0
0 0 0 0 0 0 1
0 0 0
0 0 0 0 0
cm −m m 0 0 0 0 0 0
− mc,1 c,1 0 0 0 0
− mc,1 c,1 0 0 0 0
− mc,2 c,2 0 0
− mc,2 c,2 0 0
− mc,3 c,3
− mc,3 c,3
− Lφc,1
0
φ Lc,1
0
0
0
0
c,1 − Lc,1
R
0
− Lφc,2
0
0
0
φ Lc,2
0
0
0
c,2 − Lc,2
− Lφc,3
0
0
0
0
0
φ Lc,3
0
0
k
c
k
c
k
c
− mφc,1 0 0 0 0
− mφc,2 0 0 R
0 0 0 0 0 0 0
− mφc,3 0 0
(13)
R
c,3 − Lc,3
with the state vector z = [zm , z˙m , zc,1 , z˙c,1 , zc,2 , z˙c,2 , zc,3 , z˙c,3 , Ic,1 , Ic,2 , Ic,3 ]T and E is an input excitation vector given as E = [0, −1, 0, −1, 0, −1, 0, −1, 0, 0, 0]T (14) Assuming that the electro-mechanical coefficient φ is a constant, Eq. 12 is easy to solve. The vibrational responses of all the cantilevers and the inductive currents in the coils can be extracted from the state trajectory z(t).
5.2 Bridge test To investigate the feasibility of the concept, a field test has been conducted on a bridge (Nong-Ro Bridge), located in Kimhae-si, Korea. For the field test, only an approximate fundamental natural frequency of the bridge is known. Therefore, we adjust the natural frequencies of the coils and magnet cantilevers by adding additional masses to make the natural frequencies of the cantilevers close to the expected fundamental frequency of the bridge. Figs. 10a, 10b and 10c show the bridge, the installation of the harvester on the bridge with the data acquisition system, and a heavy vehicle passing through the bridge.
(a) Target Bridge
(b) Field test data accusation
(c) Heavy loaded vehicle
Figure 10: Field test setup (Nong-Ro Bridge, Kimhae-si, Korea) Fig. 11a shows the time series for the vertical acceleration of the bridge and Figs. 11b and 11c show the voltage time series measured form the harvester and the power spectrum density for the voltage time series, respectively. The measurement duration is 720 sec, during which two heavy trucks passed through the bridge. The instances of the passing vehicle are associated with the sharp peaks in the acceleration and voltage time series. Fig. 11c clearly shows that the asynchronous harvester produces the two peak in the power spectrum density curve, whose shape is similar to the power response curves from the laboratory test. This implies that the asynchronous harvester is excited over the broad frequency range in bridge vibration and produces the power.
0.8
1.5
0.018
15 ton dump truck
0.6
−0.2
PSD [V2/Hz]
Voltage (V)
Acceleration (m/sec2)
0
0.5
0
0.012 0.01
f
0.008
magnet
0.006
−0.4
0.004
−0.5 −0.6 −0.8 0
coil
0.014
0.4 0.2
f
0.016
15 ton dump truck
1
0.002 100
200
300 400 Time (sec)
500
600
700
(a) Acceleration time series a(t)
−1 0
100
200
300 400 Time (sec)
500
600
700
(b) Voltage time series V (t)
0 2
2.5
3 Frequency(Hz)
3.5
4
(c) Power spectrum density for V (t)
Figure 11: Bridge acceleration measurement at the Nong-Ro Bridge (Kimhae-si, Korea) for the duration of 720 sec and voltage measurement (time series and power spectrum density) from the energy harvester coil 4 during the bridge field test (fc = 3.2Hz, fm = 3.4Hz) To further investigate how the dynamic responses of the bridge and the corresponding voltage output vary at the instance when a heavy vehicle passes, the portion of the acceleration and voltage measurements are studied in a narrow time window (10 ∼ 40 sec) as shown in Figs. 12 and 13. Figs. 12a and 13a show, respectively, the acceleration and the voltage time series for the 40-second duration when a 15-ton dump truck passes through the bridge. In addition, Figs. 12b and 13b show the wave form of the acceleration and the voltage output signal, respectively, between the 10 ∼ 15 sec period. Finally, Figs. 12c and 13c show, respectively, the power spectrum density functions for the acceleration and the voltage output over the 40 sec long period. The power spectrum density function, as shown in Fig. 12c, indicates that the bridge vibrates at a frequency around 3 Hz. As a result, the coil cantilever whose fundamental frequency is close to 3 Hz vibrates with a large
amplitude, which leads to a high level of voltage output. In this case, the voltage output signal resembles a pure periodic wave form with the frequency equal to the natural frequency of the coil cantilever (3.2 Hz). It should be noted that the voltage outputs were measured using a NI-6009 DAQ driver whose internal resistance is 144 KΩ, which is much higher than the optimum load resistance for the electromagnetic harvester. If the optimal load resistance is used, the power may increase considerably. −4
0.8
x 10
0.5
0.2 0 −0.2 −0.4
PSD [(m/s)2/Hz]
0.4
Acceleration (m/sec2)
Acceleration (m/sec2)
0.6
0
2
1
−0.6 −0.8 0
10
20 Time (sec)
30
−0.5 10
40
(a) Acceleration time series a(t)
11
12 13 Time (sec)
14
0 2
15
(b) Wave form of a(t)
2.5
3
3.5 4 Frequency(Hz)
4.5
5
(c) Power spectrum density for a(t)
Figure 12: Bridge acceleration measurement (when large vehicle passes) 0.6
1.5 Measured Simulated
0.4
0.35
Measured Simulated
Measured Simulated
0.3
1
0
0.25 PSD [V2/Hz]
Voltage (V)
Voltage (V)
0.2 0.5
0 −0.2
0.2 0.15
−0.4
0.1
−0.6
0.05
−0.5
−1 0
5
10
15 20 25 Time (sec)
30
35
(a) Voltage time series V (t)
−0.8 10
11
12 13 Time (sec)
14
(b) Wave form of V (t)
15
0 2
2.5
3
3.5 4 Frequency(Hz)
4.5
5
(c) Power spectrum density for V (t)
Figure 13: Voltage measurement from the energy harvester coil 4 during the bridge field test (fc = 3.2Hz, fm = 3.4Hz) (when large vehicle passes) We also compare the measured and simulated voltage outputs and the corresponding power spectra. The simulations are conducted based on Eq. 12 and the results are compared in Fig. 13. The general trend in the voltage output time series shows good agreement. However, the voltage amplitudes are slightly different over the 17 ∼ 20 sec period. This deviation is possibly due to the limitation of the prototype; when the cantilever reaches a displacement exceeding 1 cm, it collides against other cantilevers vibrating with different phases.
6. SUMMARY Conventional vibration-based electromagnetic energy harvesters rely heavily on resonance of vibration, which leads to a significant performance deterioration when the excitation frequency differ even slightly from the natural frequency of the vibrating system in the harvester. To overcome this robustness issue, we discuss a novel concept of a harvester utilizing asynchronously vibrating multiple cantilevers. In the asynchronous harvester, both a coil and a magnet are attached to the cantilevers that are vibrating separately with different natural frequencies and phase differences. The frequency difference between the natural frequencies of the coil and the magnet cantilevers, and the level of electrical damping affect the power response curve. The harvester is shown to have two dominant peaks in its power response curve. When the frequency difference between the two cantilevers is optimally chosen, the two power response curves overlap that leads to a widen frequency band in the power-response curve. The wider frequency band reduces
the sensitivity of the power production on the excitation frequency. Furthermore, asynchronously vibrating cantilevers produce more power than the conventional concept, especially when the electromagnetic damping coefficient is high. The increase in robustness as well as the increase in power producing capability are among the advantages that the asynchronous harvester provides. To validate the proposed concept, laboratory tests using modal shaker have been conducted, and the results show that the power as well as the robustness improve when comparing to the conventional type energy harvester. Finally, the prototype energy harvester is tested on a bridge that exhibits traffic induced vibration with a low frequency around 3 Hz. The field test reveals that the harvester vibrates and induces voltage when a heavy truck passes through the bridge during which the prototype produces a peak power of 30 µW . The amount of power can be significantly improved if the optimal load resistance is used and the frequency is tuned close to the structural fundamental frequency.
ACKNOWLEDGMENTS This research is partially supported by the US National Science Foundation under Grant No. CMMI-0824977. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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