Piezoelectric Energy Analysis on Diverse Buoy Coupling with Hydrodynamic Parameters

This paper mainly describes the influence factors of the captured energy power by huge wave energy harvesters, in which the vertical motion of buoy can transform ocean’s potential energy into piezoelectric energy power by undulating waves. Firstly, related environmental coefficients are analyzed by means of the incident wave theory. Besides, the geometric structural parameters are also analyzed and compared under optimal environmental coefficients with semi-analytical solutions. Thirdly, the numerical results also show the impact trend of hydrodynamic parameters and geometric volume on motion, voltage and power with qualitative agreement. The numerical simulation confirms that the improved structure parameters could markedly deliver sufficient power under the same conditions with long-time stability.


Introduction
As the global resource consumption per capita increases and ocean possesses the richest natural resources on earth, exploring ocean resources is becoming an inevitable topic and a promising research. At the same time, power supply devices for ocean exploration equipment become more and more complex. The pocket spontaneous electrical equipment cannot meet the requirements of high power due to its characteristics of lower power input, short endurance and disposable use. In order to deal with this current thorny problem, we proposed an large-sized piezoelectric cantilever beam that can convert a wide range of ocean energy into piezoelectric energy power and provide stable and sufficient power to detection equipment which does not have certain restrictions on enhanced energy power.
The pioneer work can be traced to Taylor et al. (2001) who designed an energy harvester called "eel" to capture energy from the vibration and brought up alternate shedding vortex by water currents at the beginning of 21st century. At the same time, Yao and Uchino (2001) adopted a lumpedmass modeling to analyze composite piezoelectric canti-lever and illustrated general trends associated with changing parameters. In addition to the improved algorithm, Anton and Sodano (2007) put forward a new piezoelectric material called PZT which further helped the development of completely self-powered devices. Besides, Erturk and Inman (2009) performed an experiment of basic harmonic excitation at an arbitrary excitation frequency for steady state voltage. Stanton et al. (2010), Dai et al. (2014), Viet et al. (2016) and Gu and Livermore (2011) also improved the capacity of Euler-Bernoulli piezoelectric cantilever beam from geometric construction and structure coefficient by experiments and simulations. Through a series of efforts the energy collection capability is obviously improved by increasing the structure size and optimizing boundary conditions, but more in-depth piezoelectric properties have never been analyzed. Abdelmoula and Abdelkefi (2017) gave guidelines for designing efficient galloping energy harvesters, and their energy harvester, with both mechanical and electrical characteristics, promoted the development of energy harvesters. In recent years, a large number of researchers have given feasible suggestions for the optimiza-tion of piezoelectric cantilever beam structure through simulation, experimental exploration and theoretical derivation, for example Xie et al. (2014), Mak et al. (2011) and Franco and Varoto (2017).
Owing to the large size, hydrodynamic calculation needs to be fully conducted for the harvester's stability and efficiency of generating capacity. The earliest Malenica et al. (1995) proposed a vertical cylinder and submerged it into regular waves in finite depth where calculation of mean drift forces, near-field, wave drift damping coefficients and farfield methods were analyzed. In the initial work, the researchers only analyzed a few important parameters of the simple floating body under limited conditions. Mavrakos (1985) presented linearized exciting wave forces and hydrodynamic parameters of vertical cylinders that were exposed to regular surface waves in finite depth water. Whereas what has been mentioned above only discusses the linear problem and infinite depth theoretically and numerically. But in later practice, Flocard and Finnigan (2009) presented from an experimental study the bottom-pivoted pitching point energy absorbers in intermediate water depth from regular and irregular waves; it showed that some parameters can be adjusted in response to captured power in various waves. The corresponding hydrodynamic problem analysis also becomes more and more complex from single motion to multidimensional, and the couplings of boundary conditions are extensive (Kara, 2010, Zhang et al., 2015and Caska and Finnigan, 2008. From the very beginning, the Frenchman Malenica proposed the wave energy conversion by a vertical circular cylinder in regular waves and finite depth. In addition, Greek, Australian, Japanese, British and Portuguese scholars provided a rich complement to wave energy conversion with theoretical and experimental studies respectively, and these countries have also placed experimental models in the sea. Wave energy pioneer countries such as Australia, Ireland and the United Kingdom have been carrying out demonstrative commercial construction on wave energy capturing devices. At present, Japan, Norway, India, China, the United Kingdom and Portugal have all done experiments on three common wave energy conversion devices in the sea, including oscillating water column, overtopping device and wave-activated bodies. Although China started to study wave energy relatively late, many researchers have made important theoretical contributions. For example, Zhang et al. (2016) adopted semi-analytical method of decomposing complex axisymmetric boundary into several ring-shaped and stepped surfaces based on the boundary discretization method.
From then on, many researchers have made much progress on harvesters, from the ideal structure to feasible piezoelectric cantilever beam. However, previous harvesters were usually pocket-sized models with lower output, short endurance, disposable use or achieved relatively higher energy power under very limited circumstances. On the other hand, there is not an acknowledged framework or a cost-effective model for its sustainable development. Be-sides, with boundary condition analysis, the environmental factors are not taken into account and analyzed. Furthermore, present structures of piezoelectric cantilever beam can hardly improve its power generation efficiency primarily because few people analyze the influence of the buoy's structure and lumped parameters from the piezoelectric energy curve.
In this paper, firstly we research the hydrodynamic parameters through the structural parameters in which the influence of ocean parameters are discussed then optimum values of environmental parameters are selected as reference. Then we set out to apply a spring-mass-damper system to simulate heaving motion of large-sized harvester, and the simplified model can ensure the accuracy and rationality based on control equation and semi-analytic method. Finally, numerical simulation results can also reveal the added mass, wave force and added damping in a complete logic algorithm chain. Besides, numerical graphs can comprehensively illustrate the influence of structural parameters and lumped coefficients on piezoelectric energy power. Furthermore, energy power capacity of piezoelectric cantilever beam has obtained dramatic improvement under optimized structural parameters which can transport higher power to exploration equipment with stable voltage output day and night.

Materials and methods
As shown in Fig. 1a, the outline of the cantilever beam is a rectangle with certain parameters. And PZT-5 is attached to the upper and lower surfaces of the beam for piezoelectric medium by equal density. Additionally, at the end of the cantilever beam is a conical buoy which can be arbitrarily shaped and vertically axisymmetric. The buoy can generate piezoelectric energy power excited by the heaving waves, with larger deformation, and it can generate greater energy power. In this paper the cantilever beam is assumed to have contact only with heaving buoy and no external factors are attached to the buoy. Accordingly, the motion of the harvester can be simplified as a lumped-parameter model with a linear spring-mass-damper system and a simple circuit under low frequencies as depicted in Fig. 1b. Furthermore, the water discharge of the buoy is equal to the concentrated mass and the buoy. Fig. 1c describes characteristics of motion of the buoy calculated to evaluate the velocity potentials, wave force and other parameters in the eigenfunction based on semi-analytical method. (r, θ, z) Meanwhile, the wetted surface of the buoy and the definition of fluid sub-domain are shown in Fig. 1. We define the cylindrical coordinate by its origin located at the center of the buoy and on the mean plane of the free surface. The wetted surface of the buoy is assumed to be a floating vertical axisymmetric curved surface. The outer radius and draught of the buoy are denoted as R and d, respectively. The structure parameters of the piezoelectric cantilever beam are shown in Table 1.

Boundary condition
According to the linear wave theory in frequency domain analysis, the spatial velocity potential can be decomposed as undisturbed incident wave velocity potential , scattered potential for fixed buoy and radiation potential induced by the buoy heave motion oscillation in otherwise calm water. Then we obtain: where, is the displacement amplitude in heave of buoy. In Eq. (1), , representing the velocity of undisturbed incident wave of amplitude and frequency , which propagates along the positivex-axis direction, can be expressed in cylindrical coordinates as: where, is the first kind Bessel function of order ; is the Neumann's symbol, meeting the condition of and for ; wave number comes from dispersion relation . According to the potential flow theory, the diffracted and radiated velocity potentials should satisfy, Hull boundary condition: on the wetted surface of buoy; Radiation condition: , ∂ n (·) where i is the imaginary unit and the symbol indicates the derivative in the normal vector pointing always outwards the wetted surface of the buoy. In the same way of incoming wave, diffraction velocity potential can be expressed as: The fluid flow caused by the body's force oscillation in θ π otherwise still water is symmetric about both =0°-and /2panels for heave. Thus the radiation velocity potential is: (4)

Velocity potentials in each sub-domain
To develop velocity potentials of diffraction and radiation, due to floating vertical axisymmetric buoy the fluid domain is divided into sub-domains named E, as shown in Fig. 1c. In the sub-domain E, the velocity potentials of diffraction can be expressed in the form of Fourier series by where and are the first kind Hankel function and the modified second kind Bessel function of order , separately. Here and hereafter, a prime denotes taking the differentiation of a function with respect to its argument. The wave number comes from the dispersion relation . Similarly, the velocity potentials of radiation can be expressed as: . (6) In the sub-domain , the velocity potentials of diffraction can be expressed as: is the modified first kind Bessel function of order .
can be defined as . Similarly, the velocity potentials of radiation can be expressed as: In the sub-domain , the velocity potentials of diffraction can be expressed as: in which can be defined as . Similarly, the velocity potentials of radiation can be expressed as: In the eigenfunction expansions for velocity potentials , the sets of unknown Fourier coefficients are to be determined by taking advantage of orthogonality, in so-called Garrett's method, according to the matching of the potentials and its normal derivative on the juncture boundaries (matching surface) shared by subdomains.

Matching equations
On the matching surface between sub-domain and sub-domain in which ( ), the flowing series of functions are expressed as:

I P E
On the matching surface between sub-domain and sub-domain , the flowing series of functions are expressed as: By solving the above linear equations, the Fourier coefficients can be confirmed.

Excitation force and hydrodynamic coefficients
With the heave motion model applied in the wave energy conversion, the wave forcing and radiation damping coefficients in heave need to be obtained. Thus, by defining R 0 =0, the non-dimensional excitation force in heave can be calculated and defined by μ λ The hydrodynamic coefficients containing the added mass and the damping coefficients can be calculated and defined by

Governing equations
According to Newton's law, the dynamical equation of the heaving buoy in time domain can be written as: where m is the equivalent mass of the lumped-parameter model; , and are the vertical accelerated velocity, velocity and displacement of the buoy, respectively; , , and are excitation force, wave radiation force, restoring force, and force due to the oscillation of the linear springmass-damper system. The excitation force has been discussed in Section 2.4.
The wave radiation force can be written as: μ λ where is the added mass and represents the damping coefficients, which are functions of the potential flow frequency and will be determined by potential flow theory calculated in Section 3.
The restoring force can be written as: where is the restoring coefficients, ; is the density of the seawater; is the gravitational acceleration; and is the mean draught of the buoy.
The oscillation force of the lumped-parameter model is: c k η υ where , and are the equivalent damping coefficient, the equivalent spring coefficient, and the electromechanical coupling of the piezoelectric cantilever beam, respectively. is the voltage across the load resistance. Then, Eq. (16) can be obtained as: According to Kirchhoff's law, the electrical equation of the system in time domain can be written as: where is the resistance, and is the capacitance of piezoelectric material. Since the frequencies of wave are relatively low, and can be obtained from the fundamental vibration mode of the piezoelectric cantilever beam.
According to series piezoelectric layer, piezoelectric coupling of harvesting can be written as: where is the piezoelectric constant, is the distance between the center of the piezoelectric layer and the neutral axis.
According to the normalized quality eigenfunction of vibration cantilever beam, it can be written as: where , , , and are corresponding amplitude constants. is the dimensionless frequency of the model (eigenvalue) and . Electromechanical coupling of harvesting can be written as: where is the permittivity of constant force.

Average power
In the frequency domain analysis, in one period, the average power due to piezoelectric effect can be written as: with formula where is the symbol of a conjugate complex number, then we obtain 3 Numerical results and discussion In this section, related environmental parameters must be considered comprehensively because those are important boundary references in the calculation of piezoelectric energy power, which also means that the large-sized harvester could capture much more energy power under excellent conditions. First of all, the selected incident wave average frequency ranges from 4.0 rad/s to 6.0 rad/s with the water depth of 50 m in this paper. According to the linear wave diffraction theory can make sure the accuracy of the calculation of buoy-motion, where λ means the incident wave length and 2R represents the characteristic length. Secondly, in order to avoid the breaking of diffraction wave in processing of wave energy conversion, breaking wave theory can be expressed as , where A is the wave amplitude, and is the wave number. By introducing the dispersion equation can also simplify the related parameters. Besides, the viscous effect and potential flow effects may be important in determining wave-induced motions and loads on the harvester. In order to judge which one is more important, it is useful to refer to a simple picture like Fig. 2 (Faltinsen, 1993) in which the drawing is based on results for horizontal wave forces on a vertical circular cylinder standing on the sea floor and penetrating the free surface. Besides, based on Eq. (20) and boundary formula, we can also obtain variable coefficients including the geometrical volume , added mass , damping coefficient , wave diffraction force , spring stiffness and damping coefficient c to analyze the whole piezoelectric system. LIU Heng-xu et al. China Ocean Eng., 2019, Vol. 33, No. 3, P. 279-287 283 According to the analysis results shown in Table 2 based on the potential flow effect coupling with linear wave diffraction theory, the incident wave average frequency λ=4.0 rad/s satisfies the linear wave diffraction theory and also meets the requirements of the breaking wave theory under . Hence, the following will dissect the influence of structural parameters on energy power with certain environmental parameters. In analyzing simulation results, the optimal volume and structure coefficients among three basic models are firstly selected according to the semi-analytical method with the same hydrodynamic coefficient. Then the influence of the lumped parameters on the displacement, voltage and energy of the buoy is discussed when the cantilever beam is at the same wave energy condition. According to Eqs. (21) and (22) all details about the coefficients of harvester and their associated values are given in Table 3.

Verification of the solution method r/d
The bulk mass of theoretical model and confined sea conditions leads to the limited experiment. In order to verify the accuracy of the calculated model, as shown in Table 4, here list three groups of shaped buoys under the incident wave average frequency λ=4.0 rad/s that the radius to height ratio of the buoy ( ) is 1. Besides, the comparison between wave force and hydrodynamic coefficient is also made by the present semi-analytical method and Hstar. The numerical results show a good agreement which can be regarded as the validation of the present method.
3.2 Effect of geometrical volume V In theory, the bigger buoy can produce greater motion with the same wave force, which means that the piezoelectric potential difference between the upper and lower is large. We should pay attention to the deformation of piezoelectric cantilever beam while we are concerned about the large-volume float capturing more wave energy because the huge buoy may cause permanent failure of the cantilever beam. Besides, due to the effects of other factors, such as damping coefficient and the added mass, it is no longer linear. In this section we will discuss how the buoy's volume affects the energy power of the harvester. We build three different shape models to discuss the effect of buoy's volume ( ) on energy power with V ranging from 0.3 m 3 to 2.4 m 3 and other parameters being constants as listed in Table 3. The capacity of generating energy is based on the motion of buoy, captured voltage and converted power to ensure the accuracy of numerical simulation as shown in Fig. 3. First, we can figure out the motion of three buoys to present a convex parabolic curve with the increasing volume and bigger motion is generated with the increasing V ( ) in Fig. 3a. For example the motion of cone obtains maximum value 0.87 m, 14.1 W at . Cylinder and hemisphere generate 0.79 m and 0.55 m, 28.3 W and 11.3 W at , respectively. The main reason for this phenomenon is that the hydrostatic recovery stress of large V is bigger than that of the smaller ones.
On the other hand, due to the different shapes of buoy the diagram is divided into three sections with the increasing V ( ). The first part shows that the power of cone is larger than that of the hemisphere and the latter is larger than that of the cylinder between and . In the second part, the maximum of cylinder's power is more than others except when V is larger than , which is smaller than that of hemisphere because of the rapid decline and the increasing volume of the hemispherical has not yet reached resonance. In the final part, hemisphere power is obviously greater than others with the  Fig. 3c. The main reason for this is that various buoys resonate at different volumes. In summary, the optimal solution of piezoelectric energy power varies with the increasing V( ) under different geometric models and larger hemisphere produces more stable and efficient energy in this case. The selection of spring stiffness is related to the structure, installation, boundary conditions and other factors as shown in the simplified model diagram in Fig. 1b. In this section, the spring stiffness is used as sole variable to analyze its influence on piezoelectric energy power. By looking at these three graphs, we can find the trend of growing k. Here we can see clearly that the motion declines with increasing k when . For instance, the conical motion drops from 0.46 m to 0.11 m in a concave curve from to and that of the cylinder and hemisphere has similar trends under this condition in As mentioned before, the converted voltage and power highly depend on the motion of harvester, therefore, we can find that the voltage and power in Fig. 4 show a downward trend with an increasing k, and the produced voltage and power by conical ones decline from 3.1 kV to 0.8 kV, from 9.2 W to 0.9 W, respectively. The primary reason is that the stiffer spring stiffness is, the more deformation it will be, which leads to less motion of harvester by higher k under the same ω. Besides, the conical motion is larger than cylindrical motion and the cylindrical motion is larger than hemispherical motion in any spring stiffness. It is because the selected V of buoy belongs to the first part in Section 3.2 where all conical motion is larger than hemispherical motion, and the hemispherical motion is larger than cylindrical motion. In brief, small spring stiffness cannot guarantee the security of model connection when wave frequency is high. The larger spring stiffness can obtain the stability of the conversion but the piezoelectric power will obviously decrease. Hence, on the basis of the previous calculation, we can conclude that is the optimal value.

Influence of damping coefficients
Damping coefficient is another significant parameter of harvester based on the diagram in Fig. 1b. The wave in contact with harvester generates wave damping on the surface of buoy which mainly affects the selection of piezoelectric cantilever material and structure. The following will discuss the influence of damping coefficient on large-sized modified piezoelectric cantilever beam under while other related variables remain unchanged as listed in Table 3.
Various buoys perform differently under the linear wave action, and damping coefficient varies with the changing incident wave frequency, so in this section we mainly analyze the influence of damping coefficient on energy power with the incident wave frequency . It can be seen that the captured motion, generated voltage and power obviously decline with the increasing for cone, hemisphere and cylinder when in Fig. 5. For example, the value of conical motion goes straight down from 0.41 m to 0.07 m as evenly increases from 0.0 kN·m/s to 30.0 kN·m/s and it drops slowly after 20.0 kN·m/s. Meanwhile, other basic models have similar trends that less wave energy is captured with the increasing c, and other buoys capture less energy under the same c and volume than conical, for example the captured energy of hemispherical and cylindrical floats are 4.5 W to 0.6 W, 2.7 W to 0.3 W, respectively, when c changes from 0 to 30 kN·m/s. In such case, the influence of increasing damping can be seen from Fig. 5. The displacement, voltage and piezoelectric energy of the buoy are obviously decreased with the increasing structural damping coefficient c under the same wave conditions. Thus too much c does not cushion the motion of buoy, which leads to less heaving motion just like two connected objects seen as rigid connection. Furthermore, harvester converts wave energy through the motion of buoy and the less motion results in lower efficiency of piezoelectric energy conversion. Besides, the cone produces more energy in each of the same damping coefficients than cylinder and hemisphere do, which has been explained above. As a result, small damping coefficient makes the piezoelectric beam at high energy power while excessive damping coefficient obviously attenuates the efficiency of harvester, for example the over-damping only generates less than 0.1 W after =20.0 kN·m/s, hence, the reasonable small needs to be calculated accurately in accordance with geometric model.

Conclusions and expectation
In this paper, we mainly provide a high-efficient energy conversion device for deep-sea detection equipment with a whole framework, and we investigates the influence of structural parameters based on the hydrodynamic characteristics with semi-analytical solution and discuss the effects of structural parameter on large-sized piezoelectric cantilever beam. In theoretical analysis, a linear model is used to compute numerical results about motion and output power of harvester. Potential flow theory shows that the selected incident wave frequency can not only guarantee the diffraction of waves but also avoid the breaking of waves at λ=4.0 rad/s. V = 1.8 m 3 In numerical results, it turns out that the enlarged volume does enhance piezoelectric energy power in such case and the hemispherical buoy can produce the maximum power 33.8 W at and superior to others, meanwhile, the volume of buoy is obviously different among three different basic models when those produce the maximum energy power due to the resonance. Besides, the lumped parameters play an important role in conversion efficiency of harvester, the capacity index of power decreases sharply with the increasing spring stiffness and damping coefficient when the incident wave frequency and other parameters are constant. Through the coupling with the modified buoy's structure and parameter under optimized hydrodynamic coefficient, it obviously shows the enhanced energy power by the large-sized piezoelectric cantilever beam. These findings provide valuable advice for the piezoelectric cantilever beam for establishment, installation, processing and material selection of cantilever beam model.

Author Contributions
All the authors have worked hard to discuss the harvester, optimize the calculations and analyze the numerical results. LIU Heng-xu proposed the ideal model and de-

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LIU Heng-xu et al. China Ocean Eng., 2019, Vol. 33, No. 3, P. 279-287 duced the formula. LIU Ming mainly analyzed the environmental effects and put forward reasonable parameters. CHAI Yuan-chao and SHU Guo-yang performed numerical simulation preparation and processed numerical results. JING Feng-mei and WANG Li-quan provided advices for this paper. We wrote the manuscript together.