Ocean wave energy harvesting with a piezoelectric coupled buoy structure

Applied Ocean Research 50 (2015) 110–118

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Ocean wave energy harvesting with a piezoelectric coupled buoy structure Nan Wu b , Quan Wang a,b,∗ , XiangDong Xie b,c a b c

Khalifa University of Science, Technology & Research (KUSTAR), PO Box 127788, Abu Dhabi, UAE Department of Mechanical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6 School of Mechanical and Automotive Engineering, Hubei University of Arts and Science, Xiangyang, Hubei 441053, China

a r t i c l e

i n f o

Article history: Received 1 August 2014 Received in revised form 22 December 2014 Accepted 9 January 2015 Keywords: Energy harvesting Piezoelectricity Buoy Ocean energy Numerical model Optimal design

a b s t r a c t An expedient piezoelectric coupled buoy energy harvester from ocean waves is developed. The harvester is made of several piezoelectric coupled cantilevers attached to a floating buoy structure, which can be easily suspended in the intermediate and deep ocean for energy harvesting. In the buoy structure, a slender cylindrical floater is attached on a large sinker. The energy harvesting process is realized by converting the transverse ocean wave energy to the electrical energy via the piezoelectric patches mounted on the cantilevers fixed on the buoy. A smart design of the buoy structure is developed to increase the energy harvesting efficiency by investigation of the effects of the sizes of the floater and the sinker. A numerical model is presented to calculate the generated electric power from buoy energy harvester. The research findings show that up to 24 W electric power can be generated by the proposed expedient buoy harvester with the length of the piezoelectric cantilevers of 1 m and the length of the buoy of 20 m. The technique proposed in this research can provide an expedient, feasible and stable energy supply from the floating buoy structure. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction In the past decade, with the development of low powered devices and appliances, such as light-emitting diode (LED), low cost wireless sensors and wireless access points, the self-power technique and energy harvesting from ambient vibrations have attracted much attention [1–6]. Among the available mechanic-toelectric energy conversion mechanisms, such as electromagnetic, electrostatic and piezoelectric transductions, the energy density of the piezoelectric transduction is three times higher than electrostatic and electromagnetic transductions [7,8]. In addition, the structure of piezoelectric transducers is much simpler compared with the traditional generators. The piezoelectric materials can be easily attached to the electrical equipment and structures to provide the fast and convenient energy supply. Therefore, energy harvesting by piezoelectric materials has led to many different designs of piezoelectric energy harvesters [9–14]. To further improve the energy harvesting efficiency, different designs of electric circuits and structure optimisations were presented. By both numerical simulations and experimental studies, Liao and Sodano [15] studied a single mode energy harvester with different resistances to achieve a larger output electric power. Wang and Wang

∗ Corresponding author. Tel.: +1 204 474 6443. E-mail address: [email protected] (Q. Wang). 0141-1187/$ – see front matter © 2015 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apor.2015.01.004

[16] proposed an optimal design of a collocated pair of piezoelectric patch actuators surface bonded onto beams. Wang and Wu [17] developed an optimal design of a piezoelectric patch mounted on a beam structure to achieve a higher power-harvesting efficiency through both numerical simulations and experimental studies. Xie et al. [18] developed a design of a piezoelectric coupled cantilever structure attached by a mass subjected to base motion to achieve an effective energy harvesting. In addition, it is noted that there is a huge reservation of sustainable and clear energy on the earth, such as wind and ocean energy. The flowing power of winds is usually from a typical intensity of 0.1–0.3 kW/m2 to 0.5 kW/m2 on the earth surface along the wind direction, while the flowing power of ocean waves is round 2–3 kW/m2 under the ocean surface along the direction of the wave propagation [19]. Therefore, based on the direct piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force), harvesting of renewable nature energies by piezoelectric materials has been initiated recently to pursue a clean and expedient self-contained energy source. Some research works have been conducted on development of new energy conversion technologies using piezoelectric materials to absorb the wind and flowing water energies in ocean and rivers. Ovejas and Cuadras [20] developed a wind energy harvester with thin piezoelectric films in a laminar wind tunnel and studied the electric power generation by experiments. Li et al. [21] also proposed and tested a bioinspired piezo-leaf architecture converting wind energy

N. Wu et al. / Applied Ocean Research 50 (2015) 110–118

List of symbols Y0

length of the floater of the buoy under the average ocean wave level Y0 height of the buoy sinker fixed at the bottom of the floater length of the cantilever l h thickness of the cantilever distance from the cantilever to the ocean surface Y1 a amplitude of the ocean wave D0 cross section area of the floater D1 cross section area of the sinker b width of the cantilevers and the piezoelectric patches length of the piezoelectric patch a h1 thickness of the piezoelectric patch y vertical direction of the wave motion wo (y, t) water wave motion along the vertical direction. ωo angular frequency of the ocean wave ocean depth H k wave number  ocean wave length  density of the ocean water coefficients of the inertia forces of the added mass Cm when water particles pass through the buoy Y distance from the ocean surface to the bottom of the floater distance from the ocean surface to the bottom of the Y sinker CD coefficient of the drag force N number of the piezoelectric coupled cantilevers attached on the buoy harvester w(t) vibration displacement of the floating buoy wonb (t) relative vibration displacement between the ocean wave and the piezoelectric cantilevers attached on the floating buoy wb (x, t) deflection of the piezoelectric coupled cantilever subjected to the wave force P(t) water wave pressure applied on the cantilever 1 material density of the cantilever attached on the buoy flexural rigidity of the cantilever attached on the EI buoy Qgi charge on the surface of the ith piezoelectric patch attached on the cantilever voltage on the surface of the ith piezoelectric patch Vgi attached on the cantilever e31 piezoelectric constant CV electrical capacitance of the piezoelectric patches electrical capacitance per unit width of the piezoCV electric patches ωn natural angular frequency of the cantilever total electric power generated by all piezoelectric pe (t) patches on the cantilevers attached on the buoy

into electric energy by wind-induced fluttering motion. An electric power output of 0.61 mW was generated by the harvester with a size of 72 mm × 16 mm × 0.41 mm. Gao et al. [22] reported a flow energy harvester based on a piezoelectric cantilever (PEC) with a cylindrical extension. This device uses flow-induced vibration of the cylindrical extension to directly drive the PEC to vibrate for harvesting the energy from ambient flows such as wind or water stream. Wu et al. [23] developed an energy harvester made of a cantilever attached by piezoelectric patches and a proof mass

111

for wind energy harvesting from a cross wind-induced vibration of the cantilever. From the aforementioned research works, it is found that the harvested electric power from the wind energy is usually low due to the low energy density of the wind flows. In view of considerable large energy density from water flows and wave motions, for example ocean wave motions, which can easily exceed 50 kW per meter of wave front [24], harvesting energy from water flows and waves to electric energy by piezoelectric effects has been pursued as an alternative or self-contained power source. An energy harvester using a piezoelectric polymer ‘eel’ to convert the mechanical flow energy, available in oceans and rivers, to electric power was presented by Taylor et al. [25]. Zurkinden et al. [26] designed a piezoelectric polymer wave energy harvester from wave motions at a characteristic wave frequency and investigated the influences on generated energy by the free surface wave, the fluid-structure-interaction, the mechanical energy input to the piezoelectric material, and the electric power output using an equivalent open circuit model. Xie et al. [27,28] developed piezoelectric coupled plate structures, which are fixed on the sea bed and a base structure, to harvest the ocean wave energy. Burns [29] provided a piezoelectric device consisted of a buoy floating on the ocean surface, a few anchor chains fixed on the ocean-bed and an array of piezoelectric micro thin films between the buoy and chains, and showed that the device can generate electric power when the piezoelectric films bear tension and compression alternatively duo to the up and down motion of the buoy. Murray and Rastegar [24] presented a novel class of two-stage electric energy generators on buoyant structures. These generators use the interaction between the buoy and sea wave as a low-speed input to a primary system, which, in turn, to successively excite an array of vibratory elements (secondary system) into resonance. From the previous studies, it has proven that energy harvesting from ocean waves by piezoelectric materials is effective and is able to generate sufficient electric power for small electric appliances [27]. However, most of piezoelectric energy harvesting structures in current studies are designed to be fixed on the sea bed, and hence are costly and mostly applicable to shallow ocean. In addition, it is obvious that the amount of the ocean wave energy in the intermediate and deep ocean with larger wave heights is much larger than the one in the shallow-water. Therefore, an urgent request for a more efficient, convenient and economical energy harvesting by piezoelectric materials from intermediate and deep oceans call for challenging engineering designs. In this research, an expedient and economical floating buoyant energy harvester is developed for energy harvesting from the intermediate and deep ocean waves. Piezoelectric coupled cantilevers are attached to the buoy directly and located close to the ocean surface to absorb the transverse ocean wave energy. To achieve an efficient energy harvesting, the smart design of the buoy structure made of a slender cylindrical floater attached on a large sinker is proposed to reduce the vibration amplitude of the buoy for increasing the efficiency of the harvester. The floating buoy with the smart design will be more cost-effective and convenient than the traditional tension leg platform structures. In addition, to study the effects of different designs of the buoy structure on the energy harvesting efficiency, a numerical model is proposed to calculate the generated electric power.

2. Design of the piezoelectric coupled buoy subjected to vertical wave motions A model is developed to examine the efficiency of the new developed expedient buoy attached by piezoelectric coupled cantilevers to absorb the energy from ocean waves. Fig. 1 illustrates schematically an energy harvesting buoy structure that is attached by piezoelectric coupled cantilevers floating

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Fig. 1. Energy harvesting buoy attached by piezoelectric coupled cantilevers.

on the ocean surface. The structure is very expedient and easy to be installed and suspended in the intermediate and deep ocean for energy harvesting owing to the buoy structure. Considering the water wave motion becomes weaker at the deeper location of the ocean, the buoy structure consists of a slender floater attached to a larger cylindrical sinker to reduce the vertical vibration from the ocean for increasing the efficiency of the harvester. Several piezoelectric coupled cantilevers are fixed on the slender floater close to the ocean surface. When a relative motion between the floating buoy and the transverse ocean wave is induced on the piezoelectric coupled cantilevers, the electric charge and voltage are generated on by piezoelectric effects. From the side view of the buoy energy harvester in Fig. 1, it is seen that the length of the floater of the buoy under the average ocean wave level is Y0 ; the height of the buoy sinker fixed at the bottom of the floater is Y0 ; and the cantilevers with an length of l and thickness of h are fixed on the floater of the buoy close to the ocean surface horizontally. The distance between the cantilevers and the ocean surface is Y1 .  a is the amplitude of the ocean wave. D0 and D1 are the cross section areas of the floater and the sinker of the buoy, respectively; b is the width of the cantilevers and the piezoelectric patches; a is the length of the piezoelectric patches that are attached/bonded on the cantilevers, and h1 is the thickness of the piezoelectric patches. The vibrations of the floating buoy and the piezoelectric coupled cantilevers subjected to transverse ocean waves are solved by a developed numerical model introduced below. The displacement function along the vertical direction of a transverse ocean wave at the depth of the ocean, y, is given by [30]: sinh(k(y + H)) , wo (y, t) = ςa cos(ωo t) sinh(k)

(1)

where ωo is the angular frequency of the ocean wave; H is the ocean depth, and k is the wave number. k is given as k = 2/, where  is the ocean wave length; and ωo can be calculated by ωo = 2/( 2/[g tanh(kH)]). To solve the vertical wave forces applied to the piezoelectric cantilevers, the relative motion between the ocean wave and the floating buoy should be obtained first by finding the exact

movement/vibration of the floating buoy on the vertical direction in the ocean. With the assumption that the displacement of the buoy structure is small and the ocean wave length is fairly large compared with the dimension of the buoy, the diffraction force could be neglected, and the hydrodynamic force is treated as a buoyancy force plus a force generated by the water particles passing a fixed body in the water, which include inertia and drag forces applied on the buoy and piezoelectric coupled cantilevers. Therefore, the total wave force applied to the buoy subjected to the vertical motion of the transverse ocean wave can be expressed as follows [30]: F(t) = Ffloat (t) + Fac1 (t) + Fdr1 (t) + Fac2 (t) + Fdr2 (t)

(2)

where Ffloat (t) is the buoyant force applied to the buoy due to the wave motion; Fac1 (t) and Fdr1 (t) are the inertia force and drag force caused by water particles passing the buoy, respectively; and Fac2 (t) and Fdr2 (t) are the inertia force and drag force caused by water particles passing the piezoelectric cantilevers. The calculation of the buoyant force and inertia forces applied to the buoy are listed as: Ffloat (t) = (ςa cos ωo t − w(t))D0 g,



0

Fac1 (t) =



−Y 0

−



−Y −Y

− −Y 

∂2 wo (y, t) ∂t 2

 dyD0 (1 + Cm )

d2 w(t) dyD0 Cm + dt 2



−Y

−Y 



∂2 wo (y, t) ∂t 2

 dyD1 (1 + Cm )

d2 w(t) dyD1 Cm , dt 2

 Fac2 (t) =



(3)

d2 wo (−YL , t) dt 2

 Lbh(1 + Cm ) −

(4)

d2 w(t) LbhCm , dt 2

(5)

where w(t) is the vibration function of the floating buoy in the vertical direction;  is the density of the ocean water and given as 1025 kg/m3 ; Cm is the coefficients of the inertia forces of the added mass when water particles pass through the buoy, Cm is set

N. Wu et al. / Applied Ocean Research 50 (2015) 110–118

as 0.6 for rectangular cross-section of the buoy shown in Fig. 1; Y and Y are the distances from the ocean surface to the bottom of the  floater and the sinker, respectively, which can be solved Y = (ςa cos ωo t − w(t)) + Y0 by . YL can be solved by, YL = Y  = (ςa cos ωo t − w(t)) + Y0 + Y0 (ςa cos ωo t − w(t)) + Y1 . The drag forces applied on the buoy and the cantilevers are given as:

113

is hence expressed as: EI

∂4 wb (x, t) ∂2 wb (x, t) + (C bh +  hb) m 1 ∂x4 ∂t 2 =





1 dwonb (t)  dwonb (t)  d2 wonb (t) + (1 + Cm )bh , CD b ·   2 dt dt dt 2

(11)

Fdr1 (t) =

CD D1 ((dwo (−Y  , t)/dt) − (dw(t)/dt))|(dwo (−Y  , t)/dt) − (dw(t)/dt)| , 2

(6)

Fdr2 (t) =

NCD Lb((dwo (−Y L , t)/dt) − (dw(t)/dt))|(dwo (−Y L , t)/dt) − (dw(t)/dt)| , 2

(7)

where CD is the coefficient of the drag force and is set as 2 for the given shape of the buoy in Fig. 1; N is the number of the piezoelectric coupled cantilevers attached on the buoy harvester. Based on the second Newton law, we have the relationship between the buoy motion and the total wave force applied to the buoy shown below: MB

d2 w(t) = F(t) dt 2

(8)

where MB = (D0 Y0 + D1 Y0 + Nbhl) is the total mass of the buoy. To obtain the vibration of the floating buoy, w(t), we need to solve the complex non-homogeneous differential equations, given in Eqs. (2)–(8). Since explicit solutions for the differential equations are hard to obtain, a numerical iterative algorithm is developed herein to find the accurate dynamic vibration displacement of the buoy under the ocean waves in the time period from 0 to T after the buoy is initially located in the ocean. First, a very short time step, t, is chosen to process the iterative algorithm. The initial condition is set to be (d2 w(t)/dt 2 ) = 0, (dw(t)/dt) = 0, w(t) = 0, w0 (0, t) = ςa cos ωt at t = 0 s. Based on the initial condition, the total wave force applied to the buoy at t = 0 s, F(t = 0), can be solved. The acceleration, velocity and displacement of the buoy under the force F(t = 0) at the first time step, (d2 w(t = 1 t)/dt 2 ), (dw(t = 1 t)/dt), w(t = 1 t), can be solved by Eq. (8). Accordingly, the acceleration, velocity and displacement of the buoy at the time of j t = T, (d2 w(t = j t = T )/dt 2 ), (dw(t = j t = T )/dt), w(t = j t = T ); j ≥ 2, can be obtained. Then, the relative vibration displacement between the ocean wave and the piezoelectric cantilevers attached on the buoy at any time between 0 to T can be solved below: wonb (i t) = ςa wo (−YL (i t), t) − w(i t),

(9)

where 0 ≤ i ≤ j. After the fitting of the discrete data curve of w(i t), the solution for the relative vibration displacement between the ocean wave and the piezoelectric cantilevers attached on the floating buoy, wonb (t), can be obtained. In the end, the vertical wave pressure applied to a single cantilever per unit length can be expressed as: P(t) =





voltage Vgi on the surface of the ith piezoelectric patch are hence obtained to be [31]: e31 b(h + h1 ) Qgi (t) = − 2

e31 (h + h1 ) Vgi (t) = − 2CV

where wb (x, t) is the vibration function of the piezoelectric coupled cantilever subjected to the wave force. It is assumed that the deflection and velocity of the beam vibration are small compared with the water wave motion. The effect of the piezoelectric patches on the dynamic response of the cantilevers is neglected in this research since the thickness of piezoelectric patches is very small compared to that of the host beam. According to the Euler–Bernoulli beam theory, the governing equation of the piezoelectric coupled cantilever



∂wb (x, t)   ∂x





x=ia

∂wb (x, t)  −  ∂x

, x=(i−1)a





∂wb (x, t)   ∂x





x=ia

∂wb (x, t)  −  ∂x

, (13) x=(i−1)a

where e31 is the piezoelectric constant; CV is the electrical capacitance of the piezoelectric patches; and CV is the electrical capacitance per unit width of the piezoelectric patches (CV = CV /b). From Eq. (13), it can be seen that the generated charge and voltage on the piezoelectric patches has a positive linear relationship with the thickness of the cantilever because the internal strain of the piezo-material depends on the internal shear stress field of the cantilever, which is increasing with the distance from the centerline. Therefore, thicker cantilever will lead to larger electric energy generation as long as same curvature integration or slope difference between the two ends of piezoelectric patch can be excited. To obtain the displacement field of the forced vibration of the piezoelectric harvester, wb (x, t), in Eq. (11), the free vibration of the harvester should be solved first to obtain its resonant frequencies and corresponding vibration modes. First, the vibration modes of the free vibration solution of the cantilever can be obtained: W (x) = C1 cosh sx + C2 sinh sx + C3 cos sx + C4 sin sx

0≤x≤l (14)

where C1 , C2 , . . .,C4 are the corresponding unknown coefficients, and s is given as: s=

(10)



(12)



1 dwonb (t)  dwonb (t)  d2 wonb (t) CD b  dt  + (1 + Cm )bh 2 dt dt 2 ∂2 wb (x, t) − Cm bh , ∂t 2

where 1 and EI are the material density and flexural rigidity of the cantilever substrate, respectively. It is assumed that the piezoelectric patches are mounted tightly on the surface of the cantilever. Due to the bending motion of the cantilever, the two faces of the beam exhibit positive and negative axial strains alternatively. As a result, a generated charge Qgi and

4

(Cm bh + 1 hb)ω2 , EI

(15)

and ω is the angular frequency of the cantilever. Substituting Eq. (14) into boundary conditions of the cantilever leads to four linear equations, from which we can easily obtain the solution of the nth normal mode of the cantilever, Wn (x). The forced vibration of the piezoelectric coupled cantilever subjected to the vertical wave pressure can thus be provided as follows: wb (x, t) =

∞  n=1

l

Wn (x) ·

0

Wn (x)dx

bhωn Bn



t

P() sin ωn (t − )d, 0

(16)

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N. Wu et al. / Applied Ocean Research 50 (2015) 110–118

where ωn is the natural angular frequency of the cantilever, Bn is given as Bn =

l

Wn2 (x)dx. The integration in Eq. (16) can be 0

solved by a discrete numerical algorithm,

t/ 

t 0

P() sin ωn (t − )d =

P(i ∗ )sin ωn (t − i ) .  is a very short time step, and i=0 for each time step, the value of the term, (dwonb (t)/dt), can be determined to find the exact value of the wave pressure applied to the piezoelectric harvester, P(i ). Then, by substituting Eqs. (16) into Eqs. (12) and (13), the generated charge Qgi (t) and voltage Vgi (t) from the ith piezoelectric patch bonded on the piezoelectric energy harvester subjected to vertical wave pressure at time t can be solved: e31 b(h + h1 )  2 ∞

Qgi (t) = −

l 0

n=1

Wn (x)dx





n=1

Wn (x)dx

bhωn Bn



(17)

0 ∞

0



t

e31 (h + h1 )  2CV

l



dWn (x)  dWn (x)  −   dx dx x=ia x=(i−1)a

P() sin ωn (t − )d,

bhωn Bn

Vgi (t) = −









dWn (x)  dWn (x)  −   dx dx x=ia x=(i−1)a



t

P() sin ωn (t − )d,

(18)

0

where 1 ≤ i ≤ m, and m is the number of the piezoelectric patches mounted on one face of one of cantilever harvesters. From Fig. 1, it is seen discrete piezoelectric patches are distributed on the whole surface area of the cantilevers. Considering the whole cantilever will be deformed under the distributed forced due to the water wave motion, to obtain more electrical energy, we count the electrical energy generated from all of the piezoelectric patches, but it is noted the piezoelectric patches near the fixed end of the cantilevers will be subjected to larger strain and generate larger electric power with a relatively low frequency of the excitation compared with the natural frequencies of the cantilevers. Finally, the expression of the root mean square (RMS) of the total generated electric power can be obtained. When the ocean wave oscillates for a period of T, the RMS of the generated electric power from time 0 to T is given as:



prms = e

1 T



T

[pe (t)]2 dt,

(19)

0

where pe (t) is the total electric power generated by all piezoelectric patches on the cantilever harvesters, which are attached the buoy, at time t (0 < t < T) and it is given by pe (t) = on m 2N i=1 (dQgi (t)/dt)Vgi (t). 3. Simulations and discussions Based on the above developed numerical model, the generated charge and voltage from the piezoelectric patches as well as the RMS of the electric power generated by the buoy energy harvester can be obtained. To find an efficient design of the buoy structure, the effects of dimensions of the buoy floater and sinker on the RMS are investigated from numerical simulations. In our simulations studying the design of the buoy structure, the height and length of the ocean wave and the ocean depth are H = 3 m, Ol = 80 m and Od = 40 m for the waves in the intermediate and deep ocean. The frequency of the ocean wave can be solved by the description given after Eq. (1) as 0.16 Hz. There are four aluminum piezoelectric cantilevers (N = 4) attached on the buoy structure and fixed at 2 m (Y1 = 2 m) below the average ocean wave level. The material properties and the fixed

Table 1 Material properties of the piezoelectric coupled cantilever. Parameters

Host beam (aluminum)

Piezoelectric patches (PZT4)

Length, l/a (m) Width, b (m) Thickness, h/h1 (m) Young’s modulus (N/m2 ) Mass density (kg/m3 ) e31 (C/m2 ) Cv (nF)

1 0.2 0.006 78e9

0.05 0.2 0.0006 7.5e10

2800

7500 −2.8 0.375 for the piezoelectric patch with the geometry of 0.01 m, 0.01 m, 0.0001 m

dimensions of the cantilevers and the piezoelectric patches are given in Table 1. The buoy is designed as a hollow structure, whose mass and density can be changed by adjusting the quality of the filler, to keep the buoy floating at a certain position of equilibrium. Fig. 2(a) provides the vertical vibration displacements of the buoy and water particles at a position where the cantilevers are located. The dimensions of the buoy and the piezoelectric patches in the numerical simulation are set to be Y0 = 10 m, Y0 = 2 m, D0 = 0.04 m2 and D1 = 9 m2 . It is seen that the vertical displacement of ocean waves at the position, where cantilevers are located, changes from −1.4 m to 1.2 m with a height of 3 m given in Fig. 1. The vibration of the buoy is in its un-steady state with varying vibration amplitudes in the first 50 s after the buoy is initially located in the ocean. Then the vibration of the buoy becomes a steady one with amplitude of 0.32 m, and the frequency of the buoy being the one of the frequency of the ocean wave motion. In addition, it is noted that the amplitude of the steady state vibration of the buoy is much lower than the one of water particles and hence the large relative vertical vibration displacement between the ocean wave and the buoy can be obtained for an efficient energy harvesting. In simulations, only steady state vibration of the buoy is considered in calculating the RMS of the generated electric power. The results showing the first two vibration modes and the steady state deflection at the free end of the cantilevers subjected to the wave motion are given in Fig. 2(b) and (c). It is noted that the dynamic deflection of the cantilevers will be close to their first vibration mode shape due to the relatively low frequency of the excitation force induced by the water wave motion. Fig. 3 illustrates a variation of the steady state vibration amplitude of the buoy and the RMS versus the length of the floater of the buoy structure under the average ocean wave level. The dimensions of the buoy are set as Y0 = 2 m, D0 = 0.04 m2 and D1 = 9 m2 . In Fig. 3, it is found that the vibration amplitude is reduced significantly from 0.45 m to 0.16 m when the floater length changes from 5 m to 20 m. This phenomenon can be explained as follows. From Eq. (1), it is seen that the amplitude of vertical vibration of the ocean wave decreases with an increase in the water depth. Therefore, a longer floater of the buoy under the water leads to a deeper location of the sinker, where the ocean wave motion and drag force applied to the bottom of the buoy are smaller shown in Eq. (5). Correspondingly, the vibration amplitude of the buoy becomes smaller as well leading to a larger ocean wave force applied to the piezoelectric coupled cantilever and a corresponding higher generated electric power. It is seen from Fig. 3 that the RMS increases from 14.4 W to 24.3 W with a decrease in the vibration amplitude of the buoy from 0.45 m to 0.16 m when the length of the floater changes from 5 m to 20 m. Corresponding to the highest RMS of the generated electric power of 24.3 W, the largest vibration deflection of the piezoelectric coupled cantilever is 0.03 m, which is fairly small compared with length of the cantilever and will not lead to the materials failure.

N. Wu et al. / Applied Ocean Research 50 (2015) 110–118

115

Fig. 2. (a) The vibrations of the buoy (w(t)) and the water particles (wo (y,t)) at the ocean depth, where the cantilevers are located; (b) the first two vibration mode shapes of the cantilever; (c) deflection at the free end of the cantilevers.

It is hence concluded that a more effective energy harvesting from the ocean wave motions can be realized by the buoy energy harvester with a longer floater structure with an acceptable vibration deflection of the cantilevers. Fig. 4 provides a variation of the steady state vibration amplitude of the buoy and the RMS with respect to the cross section area of the floater of the buoy with the dimensions of the buoy and the piezoelectric patches being Y0 = 10 m, Y0 = 2 m and D1 = 9 m2 . It is found that the vibration amplitude of the buoy increase nonlinearly from 0.25 m to 0.4 m with an increase in the cross section area of the floater of the buoy from 0.01 m2 to 0.2 m2 . This phenomenon is mainly because of the increase in the buoyant force and the inertia forces applied to the buoy due to the increase in

the cross section area of the floater. As discussed before, with an increase in the vibration amplitude of the buoy, the amplitude of the relevant vibration displacement between the ocean wave and the piezoelectric coupled cantilever is reduced leading to a smaller ocean wave force on the cantilever and the lower electric energy generated by the piezoelectric patches. From Fig. 4, it is hence seen that the RMS decrease from 22.2 W to 16 W when the cross section area of the floater increases from 0.01 m2 to 0.2 m2 . Therefore, the buoy with a thinner floater structure can generate higher electric power. Fig. 5 illustrates the effect of the height of the sinker of the buoy on the amplitude of vibration of the buoy and the RMS with the dimension of the buoy and the piezoelectric patches being

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N. Wu et al. / Applied Ocean Research 50 (2015) 110–118

0.5

24

RMS

0.45

buoy displacement amplitude

0.4

22 0.35 20

0.3

18

0.25 0.2

16

0.15 14 0.1 12

0.05

10

Vibraon amplitude of the buoy in stady state (m)

RMS of the generated electric power (W)

26

0 5

7

9

11

13

15

17

19

Length of the cylinder under the average ocean wave level, Y0 (m) Fig. 3. Variation of the steady state vibration amplitude of the buoy and the generated electric power versus different lengths of the floater of the buoy under the average ocean wave level (Y0 = 2 m, D0 = 0.04 m2 and D1 = 9 m2 ).

0.45 RMS

22

0.4

buoy displacement amplitude 21 20

0.35

19 0.3

18 17

0.25 16 15

Vibraon amplitude of the buoy in steady state (m)

RMS of the generated electric power (W)

23

0.2 0.01

0.03

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

Cross secon area of the cylinder, D0 (m^2) Fig. 4. Variation of the steady state vibration amplitude of the buoy and the generated electric power versus different cross section areas of the floater of the buoy (Y0 = 10 m, Y0 = 2 m and D1 = 9 m2 ).

Y0 = 10 m, D0 = 0.04 m2 and D1 = 9 m2 . It is seen clearly that the vibration amplitude of the buoy decreases nonlinearly from 0.54 m to 0.24 m with an increase in the height of the sinker of the buoy from 0.5 m to 10 m. Due to the increase in the height of the buoy sinker, the inertia force applied to the buoy is enlarged as shown in Eq. (4), but the increase is not obvious since the ocean wave motion at the buoy sinker in the deep water is weak with respect to the

0.6

RMS

22

buoy displacement amplitude

20

0.5

18

0.4

16 0.3

14 12

0.2

10 8

Vibraon amplitude of the buoy in stady state (m)

RMS of the generated electric power (W)

24

one at the ocean surface. On the other hand, the mass of the buoy is increased significantly with an increase in the height of the buoy sinker leading to a smaller acceleration of the buoy and reduction of the amplitude of the vibration of the buoy. Compared to the decrease of the vibration amplitude of the buoy motion, the RMS increases from 11.8 W to 22 W non-linearly, when the height of the sinker of the buoy changes from 0.5 m to 10 m. From Fig. 5, it is also

0.1 0.5

1.5

2.5

3.5

4.5 5.5 6.5 Height of the base, Y0' (m)

7.5

8.5

9.5

Fig. 5. Variation of the steady state vibration amplitude of the buoy and the generated electric power versus the height of the sinker of the buoy (Y0 = 10 m, D0 = 0.04 m2 and D1 = 9 m2 ).

N. Wu et al. / Applied Ocean Research 50 (2015) 110–118

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0.5

RMS of the generated electric power (W)

RMS 19

buoy displacement amplitude

18

0.45

0.4

17 16

0.35

15 0.3 14 13

Vibraon amplitude of the buoy in steady state (m)

20

0.25 2

4

6

8

10

12

14

16

18

20

Cross secon area of the base, D1 (m^2) Fig. 6. Variation of the steady state vibration amplitude of the buoy and the generated electric power versus different cross section areas of the sinker of the buoy (Y0 = 10 m, Y0 = 2 m and D0 = 0.04 m2 ).

RMS of the generated electric power (W)

30

25

20

15

10

5 60

70

80

90

100

110

120

130

140

150

160

Ratio of the ocean wave length to the length of piezoelectric couple cantilevers Fig. 7. Variation of the generated electric power versus different ratio of the ocean wave length to the length of the piezoelectric couple cantilevers (Y0 = 10 m, Y0 = 2 m, D0 = 0.04 m2 and D1 = 9 m2 ).

noted that the RMS increases more significantly when the height of the buoy sinker changes from 0.5 m to 2.5 m. When the height of the buoy sinker is larger than 2.5 m, the increase in the generated electric power is not obvious. Thus, it can be concluded that an increase in the height of the sinker of the buoy harvester can lead to a more efficient energy harvesting, but further increasing of the height of the sinker structure above a certain value will not further enhance the energy harvesting effectiveness. The variation of the amplitude of the vibration of the buoy and the RMS versus different cross section areas of the buoy sinker is shown in Fig. 6. The dimensions of the buoy and piezoelectric patches are given as Y0 = 10 m, Y0 = 2 m and D0 = 0.04 m2 . Similar to the results showed in Fig. 5, the amplitude of the buoy vibration decreases nonlinearly from 0.45 m to 0.31 m with an increase in the cross section area of the buoy sinker from 2 m2 to 20 m2 . Although the drag force from the ocean wave motion applied to the buoy increases with the increase in the cross section area of the buoy sinker shown in Eq. (5), the increase in the mass of the buoy is more significant leading to a smaller acceleration and the amplitude of the vertical vibration of the buoy. Correspondingly, the RMS increases with the increase in the cross section of the buoy sinker. From Fig. 6, the RMS is found to increase significantly from 14.7 W to 17.6 W when the cross section area of the buoy sinker increases from 2 m2 to 7 m2 . However, the enhancement of the energy generation from the buoy harvester is not quite obvious when the cross section area of the buoy is greater than 7 m2 . In summary,

the increase in the cross section area of the sinker of the buoy harvester in a certain range will lead to a significant enhancement of the energy harvesting efficiency, but the effect is not very obvious when the cross section area of the sinker structure is beyond the range. It is noted that numerical simulations in this research just provide a guidance for the design of the proposed buoy harvester attached with piezoelectric coupled cantilevers. Higher electric energy can be generated by longer piezoelectric cantilevers subjected to stronger ocean waves. In addition, other than the design of the buoy harvester itself, the ocean wave condition also has significant effect on the energy generation. From Eqs. (1), (9), (10) and (16)–(18), it can be seen that with the increase of the ocean wave length, the frequency of the wave motion decrease nonlinearly leading to smaller current and electrical power generated on the piezoelectric coupled cantilevers. The result showing the relationship between the ratio of the wave length to the length of the cantilever and the generated electrical power is give in Fig. 7 (Y0 = 10 m, Y0 = 2 m, D0 = 0.04 m2 and D1 = 9 m2 ). 4. Conclusions In this research, an expedient and economic buoy energy harvester attached by horizontal piezoelectric coupled cantilevers is developed for an energy harvesting from wave motions in the intermediate and deep ocean. A smart design of the buoy structure,

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which is composed by a slender floater and a large sinker structure, is presented to reduce the vibration amplitude of the vertical motion of the floating buoy subjected to the ocean waves so that the larger wave force can be applied to the piezoelectric coupled cantilevers for a more efficient energy harvesting. A numerical model is developed to calculate the vibrations of the buoy and the piezoelectric coupled cantilevers subjected to ocean waves as well as the generated electric energy for the first time. From numerical simulations, it is concluded that higher electric power can be generated by the buoy harvester with a longer and thinner floater and a larger sinker. It is also found that the generated electrical power decreases non-linearly with the increase of the ratio of the wave length to the length of the cantilever. An electric power of 24 W can be generated by a single piezoelectric buoy harvester with the dimensions of Y0 = 20 m, Y0 = 2 m, D0 = 0.04 m2 , D1 = 9 m2 , Y1 = 2 m, l = 1 m, b = 0.2 m, h = 0.006 m, h1 = 0.0006 m, a = 0.05 m, N = 4 and m = l/a = 20 subjected to the ocean wave with the wave height, wave length and ocean depth of H = 3 m, Ol = 80 m and Od = 40 m, respectively. This electric power is sufficient to power small electrical appliances, such as high power LEDs and wireless signal access points. It is noted that the study is a preliminary work. Some further investigations, such as the three dimensional behavior of the proposed buoy structure and relevant experiment investigations, will be studied and conducted. In addition, the durability of the buoy design will be tested through experiments as well in the future research. For practical applications, the energy harvesting effectiveness and the generated electric power can be further improved by using larger buoys and longer cantilevers subjected to stronger ocean waves. The technique proposed in this research can provide an expedient, feasible and stable energy supply from the floating buoy. In the future works, the possible drifting motion of the buoy structure will be considered. For the practical application, some mooring devices such as an anchor with mooring spring will be applied to prevent the drifting of the buoy along the down-wave direction. References [1] Shu YC, Lien IC. Analysis of power output for piezoelectric energy harvesting systems. Smart Mater Struct 2006;15:1499–512. [2] Beeby SP, Tudor MJ, White NM. Energy harvesting vibration sources for microsystems applications. Meas Sci Technol 2006;17:175–95. [3] Erturk A, Inman DJ. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. J Vib Acoust 2008;130:041002. [4] Tanaka Y, Oko T, Mutsuda H, Popov AA, Patel R, Mcwilliam S. Forced vibration experiments on flexible piezoelectric devices operating in air and water environments. Int J Appl Electromagn Mech 2014;45:573–80. [5] Mutsuda H, Miyagi J, Doi Y, Yoshikazu T, Tanaka H, Sone Y. Flexible piezoelectric sheet for wind energy harvesting. Int J Energy Eng 2014;2:67–75.

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