The potential comb improves the efficiency of low-frequency energy harvesting

JOURNAL OF APPLIED PHYSICS 109, 114512 (2011)

The potential comb improves the efficiency of low-frequency energy harvesting B. Gendelman,1,a) O. Gendelman,2,b) R. Pogreb,1 and E. Bormashenko1,c) 1

Applied Physics Department, Ariel University Center of Samaria, P.O.B. 3, 40700, Ariel, Israel Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel

2

(Received 22 November 2010; accepted 14 April 2011; published online 15 June 2011) This paper introduces a nonlinear resonant magneto-elastic device with 2 degrees of freedom that can serve as a mechanical frequency multiplier effective for energy harvesting. The device allows energy harvesting at low and extra-low exciting frequencies. The concept of effective frequency multiplication is proposed. The frequency multiplication factor may vary in the range from 2 up to several dozens. The theoretical model, the results of the experiment, and a computer simulation are presented. With a frequency of 3.5 Hz and an amplitude of 12.5 mm of shaker oscillations, the experimental setup produces output oscillations that have the principal harmonic of the amplitude 3.55 mm at 14 Hz. The amplitude of the output acceleration is 27 m/s2, which is 4.5 times the initial acceleration provided by the shaker. The numerical simulation of the setup with different elastic coefficients yields a frequency multiplication factor of 14, and the amplitude of output acceleration is equal to 35 to 60 m/s2. The operating principle used in the device is supposed to be useful as a principle of inexpensively transforming the mechanical energy with C 2011 American Institute of Physics. [doi:10.1063/1.3592189] frequency-upconversion. V

I. INTRODUCTION

II. EXPERIMENTAL DEVICE

In the past decade, numerous attempts at working out effective energy harvesting devices were undertaken. Commonly, harvesters are devices that directly convert mechanical energy to electricity. For most compact harvesters, the shaker frequency is on the order of 10 to 102 Hz (see Refs. 1–8). In contrast to this, in nature and in technical applications one observes powerful signals at low frequencies such as 0.5 to 5 Hz. These slow ambient oscillations could not be effectively exploited for energy harvesting. The regeneration of the energy to be harvested at a higher frequency is a challenging task. The magnetic snapping and releasing of short cantilevers has been applied already for frequency-upconversion,9,10 increasing the harvesting efficiency.3,9,10 Several other methods, such as mechanical frequency rectification and magnetic coupling, were also reported.6,11,12 In this paper, we present a novel model of a mechanical device with 2 degrees of freedom. We also present experimental verification of energy regeneration under effective frequency multiplying. The model is based on a combshaped magneto-elastic potential and demonstrates a controllable output-input frequency ratio. The simulations show that the parameters can be easily fitted to provide the required output frequency for the given input one. The comb-shaped potential was employed in recent harvesters and sensors13–15 in order to improve the control of the resonant frequency,16 whereas in the present work the comb potential is exploited for effective frequency transformation.

The experimental setup was built on a horizontal aluminum plate driven by a low frequency shaker (see Fig. 1). The setup comprised a long light polymer rail with a compact sliding carriage with a ferrous mass of 0.08 kg attached. The total carriage mass m ¼ 0.25 kg. The vertical springs allowed y-displacements of both the rail and the carriage. Horizontal displacements of the rail with respect to the aluminum plate are prevented by the lateral holders. The position y ¼ 0 corresponds to equilibrium of the rail with the carriage disposed on the vertical springs. The x position of the sliding carriage is measured relative to the center of the vibrating plate. After the y ¼ 0 position has been established as mentioned above, two round permanent magnets with a diameter of 50 mm and a height of 12.5 mm were attached to the plate symmetrically relative to the equilibrium position of x. The identical vertically oriented springs were chosen to be strong enough to keep the vertical clearance h between the ferrous mass and the surface of the magnets in the range of 9-20 mm during vertical oscillations. The effective elastic coefficient of the pair of two vertical springs was ky ¼ 2949 6 392 N/m, and the horizontal spring keeping the carriage had an elastic coefficient kx ¼ 114 6 17 N/m.

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]. b) Electronic mail: [email protected]. c) Electronic mail: [email protected]. 0021-8979/2011/109(11)/114512/6/$30.00

III. EXPERIMENTAL TESTS

In the experiment, we observed the 2D excitement of the magneto-elastic system exposed to 1D horizontal vibration. Under the horizontal oscillations, the carriage overrides the positions of both magnets. The vertical component of the magnetic forces leads to y-displacements in the system. The horizontal amplitude of the carriage with respect to the moving plate was 35 to 45 mm, whereas the shaker oscillation

109, 114512-1

C 2011 American Institute of Physics V

Downloaded 14 Jul 2011 to 132.68.17.202. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

114512-2

Gendelman et al.

FIG. 1. The experimental setup has 2 degrees of freedom and a 2-wells’ potential comb.

amplitude was only 12.5 mm. Two ultrasound displacement sensors (PASCO CI-6742) were attached to the stationary inertial base of the setup in order to measure both horizontal and vertical carriage displacements with an accuracy of 1 mm. Two flat, inflexible plates were fixed at the carriage and used as reflecting screens for ultrasound signals emitted by the sensors. We detected the resonant horizontal oscillations of the carriage at frequencies fh of 3.5 to 3.7 Hz. At the same time, we observed the excitation of the intensive vertical oscillations of the rail and carriage on vertical springs. The latter (vertical) oscillations will be proved to be the near-resonant ones. By this we have a system with both horizontal and vertical resonances. The frequency of the y oscillations was 4fh ¼ 14 Hz with light modulation at 2fh. The amplitude of the excited y oscillations was about 4 mm. The graph of the oscillations is shown in Fig. 2(a). IV. RESULTS AND DISCUSSION

Here a simplified mathematical model is presented in order to explain the observed phenomena. For the sake of simplicity, we consider an infinite number of equidistant permanent magnets. The discrepancy with the performed experiment is not significant as soon as the entire motion of the carriage occurs in the close vicinity of two “central” magnets. The empirical 2D magneto-elastic potential with a space-periodic term is thus defined as Uðx; y; dÞ ¼ kx

px x2 y2 d þ ky þ qy þ ða  byÞ cos ; (1) 2 2 p d

J. Appl. Phys. 109, 114512 (2011)

where a, b, and q are the positive constants, 2d is the space period of the potential comb, and kx and ky are the elastic coefficients related to carriage displacements in the x and y directions, respectively. The coefficient q takes account of both gravity and the constant component of magnetic forces acting on the carriage. The last term in Eq. (1) gives a primary qualitative approximation of the magnetic force acting on the ferrous mass attached to the carriage. The use of the cosine function is dictated by the further application of a larger number of magnets. The factors a and b describe the amplitude of the magnetic force of a single magnet on the ferrous mass and the spatial gradient of the amplitude with distance from the magnet, correspondingly. The coefficient b simultaneously characterizes the maximal magnetic force of a single magnet acting on the carriage when the carriage moves along the rails away from position of the magnet (see Eq. (2b)). The d/p multiplier was introduced for convenience in further calculations. The values of the parameters in “empiric” potential function (1) were fitted according to experimental measurements of the real magneto-elastic potential of the system. The parameters set in Eq. (1) are kx ¼ 114 N/m, ky ¼ 2949 N/m, d ¼ 0.028 m, a ¼ 1.68 N, b ¼ 534 N/m, and q ¼ 7.21 N. We needed to use an effective value of d ¼ 0.028 m for calculations due to our assumption that the magnetic force acting on the carriage has approximately the same value when the carriage moves over any of the magnets. We restrict the consideration in the last term of Eq. (1) by the linear approximation for the y dependence. The term describes the coupling of x and y variables. The Lagrange equations describing the system are px kx 1 x  ða  byÞ sin ¼ Ax2 cos xt; (2a) m m d px ky db q cos ¼ ; (2b) y€ þ 2fy y_ þ y  m pm d m

x€ þ 2fx x_ þ

with A ¼ 0.0125 m, m ¼ 0.25 kg, x ¼ 7p s1, fx ¼ 1.5 s1, and fy ¼ 4.4 s1. The acceleration term Ax2cos(xt) approximately describes the external excitement by the _ y_ shaker. At t ¼ 0, the values of x, y, and of the velocities x; are equal to zero. Introducing the new, dimensionless variables X, Y, s as X¼

px p ðy þ y 0 Þ ; Y¼ ; s ¼ xt; d d

(3)

where y0 ¼ q=ky , one obtains the following dimensionless dynamic equations: X€ þ cx X_ þ jx X  ða  bY Þ sin X ¼ v cos s;

(4a)

Y€ þ cy Y_ þ jy Y  b cos X ¼ 0;

(4b)

with the following set of parameters: a¼ FIG. 2. Portions of the curves x(t) and y(t) in the length of 1 s: (a) experimental, (b) model, numerical calculation.

 2fy p  b 2f a þ bq=ky ; b ¼ ; cx ¼ x ; cy ¼ 2 2 x x mx d mx ky kx p ; jy ¼ ; v ¼ A: jx ¼ mx2 mx2 d

(5a)

Downloaded 14 Jul 2011 to 132.68.17.202. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

114512-3

Gendelman et al.

J. Appl. Phys. 109, 114512 (2011)

The calculated values of the dimensionless parameters are a ¼ 2:77; b ¼ 4:42; cx ¼ 0:1364; cy ¼ 0:400; jx ¼ 0:943; jy ¼ 24:4; v ¼ 1:403:

(5b) Y0 ¼

The numerical simulation of x(t) and y(t) dynamics according to the dynamic equations (2a) and (2b) with the zero initial conditions yields quantitative agreement of the experimental results with the theoretical model. The experimental and simulated graphs of the dimension variables x(t) and y(t) shown in Fig. 2 correspond to x ¼ 7p (i.e., fh ¼ 3.5 Hz). Now let us describe the frequency multiplication analytically. The dimensionless dynamic equations (4a) and (4b) are coupled; to solve the system, we use the harmonic balance method. We suppose that the solution X(s) has the form XðsÞ ¼ XA cosðs þ UÞ:

Here the amplitudes Yn and phases un of the involved Y-harmonics are expressed as b J0 ðXA Þ; jy

2bJ4 ðXA Þ Y4 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðjy  16Þ2 þ 16c2y

YðsÞ ¼ Y0 þ Y2 cosð2ðs þ u2 ÞÞ þ Y4 cosð4ðs þ u4 ÞÞ (6b) þ Y6 cosð6ðs þ u6 ÞÞ þ   

u2 ¼ U 

¼ bðJ0 ðXA Þ  2J2 ðXA Þ cos 2ðs þ UÞ þ 2J4 ðXA Þ cos 4ðs þ UÞ  2J6 ðXA Þ cos 6ðs þ UÞ þ   Þ; where Jk(X) is the Bessel function of the first kind.On the basis of the analysis of the Bessel function values, our consideration is restricted by the 6th harmonics of the exciting frequency. Applying the harmonic balance method, we obtain the amplitude XA and phase U in the primary approximation as follows: 

v jx  1 a þ XA þ J1 ðXA Þ  2Y0  2 þ Y2 cosð2u2  UÞ b b b



þ J3 ðXA Þ  ðY2 cosð3U  2u2 Þ þ Y4 cosð4u4  3UÞÞ þ J5 ðXA Þ  ðY4 cosð5U  4u4 Þ þ Y6 cosð6u6  5UÞÞ ¼ 0;   jx  1 a XA sin U þ J1 ðXA Þ  sin U 2Y0  2 b b  þ Y2 sinð2u2  UÞ þ J3 ðXA Þ  ðY2 sinð3U  2u2 Þ þ Y4 sinð4u4  3UÞÞ þ J5 ðXA Þ  ðY4 sinð5U  4u4 Þ c (7a) þ Y6 sinð6u6  5UÞÞ þ x XA cos U ¼ 0: b

y

2cy cos d1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðjy  4Þ2 þ 4c2y u4 ¼ U 

jy  16 d2 p þ ; sin d2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 8 ðj  16Þ2 þ 16c2 y

y

(7c)

4cy cos d2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðjy  16Þ2 þ 16c2y u6 ¼ U 

jy  36 d3 p þ ; sin d3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 6 12 ðj  36Þ2 þ 36c2 y

y

6cy cos d3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðjy  36Þ2 þ 36c2y

k¼0

Y€ þ cy Y_ þ jy Y ¼ b cos XðsÞ

jy  4 d1 p þ ; sin d1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 4 ðj  4Þ2 þ 4c2 y

Using the Neumann series for the sine function in Eq. (4a), we obtain 1 X J2kþ1 ðXA Þ cosðð2k þ 1Þðs þ UÞÞ; sinðXA cosðs þ UÞÞ ¼ 2 and substituting Eq. (6a) in Eq. (4b), we get

2bJ6 ðXA Þ Y6 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðjy  36Þ2 þ 36c2y (7b)

(6a)

Also, we suppose that Y(s) is the sum of the even harmonics of the X(t) frequency because of the symmetry of the experimental system.

2bJ2 ðXA Þ Y2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðjy  4Þ2 þ 4c2y

The equations in (7a) cannot be solved analytically; however, the numeric computation of the roots is straightforward. For the parameters fixed by Eq. (5b), we get the solution of dimensionless XA ¼ 4.26, U ¼ 0.32. This results in a dimensional amplitude of the x-oscillations of 0.038 m. The direct numeric solution of Eqs. (4a) and (4b) supplies the amplitude XA ¼ 4.9 (corresponding to 0.044 m). The experimentally observed amplitude of x(t) is 36 to 38 mm (see Fig. 2). The phase relations for the X and Y variables found in the numerical model show agreement with the experiment. Now the solution for Y is straightforward: YðsÞ ¼

bJ0 ðXA Þ 2bJ2 ðXA Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð2ðs þ u2 ÞÞ jy ðj  4Þ2 þ 4c2 y

y

2bJ4 ðXA Þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð4ðs þ u4 ÞÞ ðjy  16Þ2 þ 16c2y

(8)

2bJ6 ðXA Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð6ðs þ u6 ÞÞ þ    ðjy  36Þ2 þ 36c2y Thus, we explicitly obtain the frequency multiplication with even-numbered allowed values of a frequency multiplication factor. It is easy to see that the denominator in the amplitude expression for the component with a frequency of 4 is relatively small, and the component has maximal amplitude. Therefore, we reveal intense near-resonance oscillations under 4-fold frequency multiplication. With the parameters

Downloaded 14 Jul 2011 to 132.68.17.202. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

114512-4

Gendelman et al.

J. Appl. Phys. 109, 114512 (2011)

used in the experiment, the dimensional amplitudes of the y(t) harmonics predicted by Eq. (8) are y2 ¼ 1.1 mm at 7 Hz, y4 ¼ 2.9 mm at 14 Hz, and y6 ¼ 0.44 mm at 21 Hz. The experimental spectrum of the y(t) oscillations is shown in Fig. 3. The figure indicates quantitative agreement with the predicted spectrum. Here the experimental y2 ¼ 1.2 mm and y4 ¼ 3.55 mm at the predicted frequencies. The 6th harmonic (expected to be of an amplitude of about 0.4 mm at 21 Hz) may not be seen clearly while its amplitude is significantly less than the level of noises generated by the y sensor used. Now let us evaluate the efficiency of the mechanical energy regeneration at a higher frequency, values of the power during energy conversion under frequency multiplication, and the total efficiency of the frequency transformation. It is reasonable to supply the dimension value of the power. Let’s imagine an energy harvester that is fixed directly to the carriage and which vibrates with it. The harvester converts the energy of intensive rapid vertical vibrations to electricity. The energy absorption from the vertical oscillations may be described as some additional mechanical damping fyg applicable only to the y(t) vibrations. Along with the native “viscous” mechanical damping fyp in the setup, this will result in a total fy damping coefficient comprising the sum fy ¼ fyp þ fyg. So the “generative” damping power Pg associated with fyg equals the output mechanical power for the frequency convertor. The graphs in Figs. 2(a), 2(b), and 3 and Eq. (8) show that, provided with the actual parameters, the amplitudes of the second and sixths harmonics are much less than that for the fourth harmonic. So we may calculate the value of Pg as the approximate output power for clear harmonic oscillations at f ¼ 14 Hz with the same value of fy. Consider the harmonic oscillations hðtÞ ¼ Ah sinðxh tÞ þ C; where Ah is the amplitude and xh is the cyclic frequency. The power of the viscous mechanical damping is given by xh Pn ¼ 2p

2p=x ð h

_ Ff ðtÞhðtÞdt

0

where xh is the frequency of these oscillations, _ is the viscous dissipation force, and m is Ff ðtÞ ¼ 2mf  hðtÞ

the mass of the oscillating body. These assumptions result in the average mechanical power, given by Pf ¼ mfx2h A2h :

(9)

Assume that, as in the current setup, fy ¼ 4.4 s1 and fyp ¼ fyg ¼ fy/2. We mention here that the lower value of fyp may be practically achieved by improving the technical design of the available setup. The experimental value of y4 ¼ 3.55  103 m, so according to Eq. (9) we found the power associated with fyg for the fourth “harmonic” of fh present in y(t) oscillations: Pg ¼ 53.6 mW. The power consumed from the shaker is 262 mW at an x-oscillation amplitude of 0.038 m, fx ¼ 1.5 s1, and fh ¼ 3.5 Hz. Thus, the total efficiency of the mechanical energy conversion is 20.5%. Consider an electricity generator (harvester) attached directly to the moving carriage. The mass of the harvester thus becomes an integral part of the total carriage mass m. For example, in the aforementioned setup the total carriage mass had to be m ¼ 0.25 kg in order to provide the required inertness while the ferrous mass attached to the carriage was only 0.08 kg. The mass of the carrier itself was 0.02 kg, and the remaining 0.15 kg of the total carriage mass was the ballast. Replacing the ballast with a generating module (harvester) results in an allowed harvester mass of up to 0.15 kg. According to Ref. 4, 30% of the mechanical vibration power supplied from the environment may be converted to electricity by the resonant electromagnetic convertor mentioned in this work. The electrical power output of the harvester in this case equals 0.3Pg. Under these assumptions, the estimation for the resulting multiplier-harvester efficiency may be found as a product of both the above-mentioned mechanical conversion efficiency factor of 0.205 and the mechanical-to-electric conversion efficiency of 0.3 (proposed in Ref. 4). The overall converter-harvester efficiency will be about 6.15%. On the basis of the presented estimation method, we can reveal some possible ways to improve the efficiency by modifying the parameters of the setup. Generally, both a more efficient harvester and better mechanical transformation efficiency would result in better total efficiency of the harvesting. We would like to achieve better efficiency of the mechanical transformation under frequency-upconversion. The partial mechanical power of the harmonics of Y(s) with a frequency of 4 may be written from Eqs. (9) and (5a) as follows: P4g ¼ PD ðcy  cyp ÞY42 ;

(9a)

where PD ¼ mx34 d 2 =8p2 is the dimensional factor, x4 ¼ 4x; and cy  cyp ¼ cyg . The damping parameter cyp characterizes the setup itself. So we introduce the dimensionless generated power as p4g ¼

FIG. 3. The spectrum supplied by the portion of the experimental y(t) signal in the length of 4.096 s.

P4g ¼ ðcy  cyp ÞY42 : PD

(9b)

Here cyp ¼ 0.2, corresponding to the value of fyp ¼ 2.2 s1 assumed for the experimental setup. The numerical analysis of the mechanical conversion efficiency was performed in

Downloaded 14 Jul 2011 to 132.68.17.202. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

114512-5

Gendelman et al.

J. Appl. Phys. 109, 114512 (2011)

FIG. 4. The predicted dimensionless mechanic power p4g as a function of yelasticity and y-dissipation. More dissipation at elasticities of approximately 20 leads to the best efficiency of the energy transformation to a higher frequency (4x to excitation).

order to reveal values of cy and jy that would allow for improvement of the efficiency. These two parameters were chosen because they determine the dynamic response of the frequency-up converter, and therefore they strongly affect the harvester-to-converter mechanical compliance. We computed numerically the relative power equation (9b) in the range of the parameters cy e[0.2,1.0] and jy e[14,34] with a, b, cx, jx, and v fixed as in Eq. (5b). The cy,jy parameters of our experimental setup are covered by the analysis. The graph of p4g(cy,jy) is shown in Fig. 4. The power graph shows certain irregularities when the y-elasticity is lower than 19 and the dissipation is 0.5…1. The Y(s) realizations at these cy,jy values reveal the quasiperiodic oscillations, which in some small areas of the {cy,jy} plane include the harmonic with X ¼ 4. The situation corresponding to the parameters of the experimental setup is marked in Fig. 4 by a ring corresponding to the relative “generative” power p4g ¼ 0.034. Assuming that the model Eqs. (4a) and (4b) matches the reality for considered values of {cy,jy}, we can conclude that introducing larger “generative” y-dissipation via harvester and fitting the yelasticity of the frequency converter itself (to a value of jy ¼ 19…21.5) would result in the growth of the mechanical conversion efficiency and, according to the growth, the general harvesting efficiency. The calculations show a widening of the range of effective energy conversion at larger dissipation values. In the framework of the current research, these predictions were not verified; still, sufficient growth of the total harvesting efficiency over the abovementioned 6.15% may be expected for the aforementioned values of the system parameters. Now consider the details of the frequency multiplication. The simplified potential function for this consideration will be as shown follows: Uðx; y; dÞ ¼ kx

px x2 y2 d þ ky þ ða  byÞ cos ; 2 2 p d

FIG. 5. The 3D graph of the magneto-elastic potential, illustrating the nature of the frequency multiplication through the potential comb. The x and y ranges shown correspond to variations of the coordinates during vibrations. The parameters are m0 ¼ 0.15 kg, A0 ¼ 0.13 m, k0 x ¼ 4 N/m, k0 y ¼ 1170 N/m, d0 ¼ 0.035 m, a0 ¼ 3.5 N, b0 ¼ 60 N/m, x0 ¼ 2p s1, f0 x ¼ 1.4 s1, and f0 y ¼ 2 s1.

x€ þ 2fx x_ þ

px kx 1 x  ða  byÞ sin ¼ Ax2 cos xt; (11a) m m d px ky db y€ þ 2fy y_ þ y  cos ¼ 0: (11b) m pm d

According to Eq. (10), the magnetic component of the magneto-elastic potential is formed by multiple identical magnets separated by a distance 2d and situated along the OX axis. A new set of parameters will be used to illustrate the concept of effective frequency multiplication. The graph of the modified magneto-elastic potential is shown in Fig. 5. The amount of “potential wells” accessible to the carriage during one horizontal oscillation depends on the total mechanical energy of the carriage. The frequency of vertical exciting pulses coming from the magnets depends on the amount of potential wells crossed by the carriage. We carried out one more numerical experiment with a simplified model

(10)

and it does not include a “qy” term, which means the absence of gravity and constant magnetic fields. The second dynamic equations, then, will not include a  q/m term, as follows:

FIG. 6. Portions of the curves in the length of 1 s: (a) x(t), (b) y(t), (c) y¨(t). The graph of quick y oscillations (b) demonstrates visible beats that reveal regular bidirectional energy exchange between low-frequency and high-frequency oscillators. The beats of 1 Hz (which is exactly the excitation frequency) correspond to 2 peaks/s of the envelope curve. The amplitude of y(t) is 0.005 to 0.008 m, and the amplitude of x(t) is 0.20 m.

Downloaded 14 Jul 2011 to 132.68.17.202. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

114512-6

Gendelman et al.

[Eqs. (11a) and (11b)] in order to prove these dependences. With the parameters k0 x ¼ 4 N/m and x0 ¼ 2p s1 (i.e., with excitation at 1 Hz), we have found a frequency multiplication factor equal to 14. The x amplitude is 0.2 m. Thus, during horizontal oscillations with the new parameters, 6 potential wells are accessible to the carriage. The frequency of vertical pulses acting on the carriage then grows up to 12 pulses/s. One additional vertical oscillation occurs at each of 2 turning points of the carriage (see Fig. 6), when the carriage changes the direction of its horizontal movement. The resulting frequency of the vertical oscillations (14 Hz) lies close to the calculated value of the resonant frequency for yoscillations. The shaker amplitude in this simulation is A0 ¼ 0.13 m, providing the same value of excitation acceleration as we had in the experimental setup at A ¼ 0.0125 m. The x(t), y(t) and y¨(t) graphs are presented in Figs. 6(a), 6(b), and 6(c), respectively. The y-acceleration amplitude predicted by the model is 35 to 60 m/s2. V. CONCLUSION

We proposed and demonstrated an effective resonant frequency-multiplying device operating at a low input frequency and exploiting the comb-shaped potential. For the given input frequency by variation of the parameters, the response is formed within the required frequency range (about 10100 Hz). The device might be used for energy harvesting as an input frequency convertor supplying high acceleration to the attached harvester. The fundamental frequency of the attached harvester equals the principal output frequency of the converter. The harvester itself might be a carriage. The device could be applied to collect the sufficient total mechanical energy from relatively slow periodic motions ð1HzÞ. The size of the device is of the order of amplitude of the oscillations to be harvested. Appropriate oscillating bodies supplying large-amplitude, low-frequency fundamental oscillations are ships, skyscrapers, antenna masts, pendant bridges, etc. Also, if miniaturized, the device might be part of harvesting chips equipped with frequency converter.

J. Appl. Phys. 109, 114512 (2011)

ACKNOWLEDGMENTS

The authors are grateful to the Israel Ministry of Absorption. The authors are indebted to Mrs. Elena Bormashenko and Mrs. Anna Shapira for their irreplaceable and effective help during more than half a year of continuous experiments. The authors are thankful to Dr. Gene Whyman and Dr. Marcelo Schiffer for their fruitful discussions. 1

M. I. Friswell and S. Adhikari J. Appl. Phys. 108, 014901 (2010). N. S. Hudak and G. G. Amatucci, J. Appl. Phys. 103, 101301 (2008). 3 E. Romero, T. Galchev, E. Aktakka, N. Ghafouri, H. Kim, M. Neuman, K. Najafi, and R. Warrington, “Micro energy harvesters—An alternative source of renewable energy.” Presented at COMS 2008, Mexico. Available at http://www.azonano.com/details.asp?ArticleId¼2613. 4 S. P. Beeby, R. N. Torah, M. J. Tudor, P. Glynne-Jones, T. O’Donnell, C. R. Saha, and S. Roy, J. Micromech. Microeng. 17, 1257 (2007). 5 R. X. Gao and Y. Cui, Proc. SPIE 5765, 794 (2005). 6 M. Ferrari, V. Ferrari, M. Guizzetti, D. Marioli, and A. Taroni, Sens. Actuators, A 142, 329 (2008). 7 G. Despesse, T. Jager, J. Chaillout, J. Leger, and S. Basrour, in Conference on Ph.D. Research in Microelectronic and Electronics (PRIME) (IEEE, New York, 2005), Vol. 1, pp. 225–228. 8 G. Despesse, J. J. Chaillout, T. Jager, J. M. Le´ger, A. Vassilev, S. Basrour, and B. Charlot, in Proceedings of the Joint sOc-EUSAI Conference, Grenoble, France, October 2005; abstract available at http://www.soceusai2005.net/proceedings/node17.html#P2-1. 9 S.-M. Jung and K.-S. Yun, Appl. Phys. Lett. 96, 111906 (2010). 10 H. Kulah and K. Najafi, in Proceedings of the 17th IEEE International Conference on Micro Electro Mechanical Systems, (MEMS) (IEEE, New York, 2004), pp. 237–240. 11 H. Kulah and K. Najafi, IEEE Sens. J. 8, 261 (2008). 12 D. Lee, G. Carman, D. Murphy, and C. Schulenberg, in Proceedings of the 14th International Conference on Solid-State Sensors, Actuators and Microsystes (Transducers ‘07), Lyon, France, 10–14 June, 2007, p. 871. 13 S. G. Adams, F. M. Bertscht, K. A. Shawt, P. G. Hartwell, N. C. MacDonald, and F. C. Moon, in Proceedings of the 8th International Conference on Solid-State Sensors and Actuators, and Eurosensors IX, Stockholm, Sweden, 25–29 June, 1995, pp. 438–441. 14 D. Scheibner, J. Mehner, D. Reuter, U. Kotarsky, T. Gessner, and W. Do¨tzel, Sens. Actuators, A 111, 93 (2004). 15 D. Scheibner, J. Mehner, D. Reuter, T Gessner, and W Do¨tzel, Sens. Actuators, A 123–124, 63 (2005). 16 D. Zhu, M. J. Tudor, and S. Beeby, Meas. Sci. Technol. 21, 022001 (2010). 2

Downloaded 14 Jul 2011 to 132.68.17.202. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions