Florian Huet

Fabien Formosa
email: fabien.formosa@univ-smb.fr

Adrien Badel

Jean-Fabien Capsal

Mickaël Lallart

Vibration energy harvesting device using P(VDF-TrFE) hybrid fluid diaphragm

Keywords: Vibration energy harvesting, Piezoelectric polymeric membrane, Fluid-structure interaction

We designed and realized a novel vibration energy harvester based on a P(VDF-TrFE) membrane. The mechanical arrangement consists in an incompressible fluid confined between two thin P(VDF-TrFE) piezoelectric membranes. It is called piezoelectric hybrid fluid diaphragm (PHFD). Compared with conventional vibration harvester, this solution appears to be simple and suitable for miniaturization and integration. The fluid-structure interaction allows a drastic reduction of the resonant frequency of the membrane whose mechanical tension is used to generate electrical power. Consequently, the realization of compact generators for low frequencies excitation (typically under 100 Hz) using membranes are possible. Moreover, nonlinear hardening behavior offers wideband capability. A theoretical model is established and allows the performance of the generator to be estimated. A first prototype has been fabricated and tested. The influence of the electric load,the amplitude and the frequency of the excitation on the voltage and the generated power have been investigated. A maximum output power of 158.33 µW per cubic centimeter of active material was obtained at 119.56 Hz for 40 m/s 2 amplitude acceleration.

The potential implementation of these WSN in industrial, transportation or buildings applications could offer significant benefits: improved reliability integrating predictive maintenance, energy performance optimization using extensive collected information (temperature, humidity, electric consumption, ...). Because of the advancements of low-power WSN and MEMS technology, millimeter scale vibration harvesters (VEH) have attracted worldwide research interests [START_REF] Paradiso | Energy scavenging for mobile and wireless electronics[END_REF]. VEH have been proposed using three main transduction principles: piezoelectric, electromagnetic and electrostatic [START_REF] Beeby | Energy harvesting vibration sources for microsystems applications[END_REF][START_REF] Mitcheson | Human and Machine Motion for Wireless Electronic Devices[END_REF]. For the aforementioned applications, frequencies of typical ambient vibration sources are relatively low (typically less than 200 Hz) [START_REF] Roundy | A study of low level vibrations as a power source for wireless sensor nodes[END_REF], which hinders the development of miniature VEHs.

Diaphragms like geometry are attractive to aim at simple and robust VEH. Moreover, low cost fabrication process can be envisioned for their mass production. Some prototypes have been proposed in the literature for vibration energy harvesting [START_REF] Kuehne | A new approach for MEMS power generation based on a piezoelectric diaphragm[END_REF][START_REF] Minazara | Piezoelectric diaphragm for vibration energy harvesting[END_REF][START_REF] X.-R. Chen | Vibration energy harvesting with a clamped piezoelectric circular diaphragm[END_REF]. The 5 mm × 5 mm square membrane introduced in [START_REF] Kuehne | A new approach for MEMS power generation based on a piezoelectric diaphragm[END_REF] shows a 2.4 kHz resonant frequency, while the harvester from [START_REF] Minazara | Piezoelectric diaphragm for vibration energy harvesting[END_REF], based on a 20.5 mm radius PZT buzzer, operates at 1.7 kHz. The very high resonant frequency of these devices prevents their use in realistic applications. Thus, some attempts have been made to solve the frequency issue by the addition of a bulky mass to the membrane. Chen proposed a 30 mm radius PZT buzzer with a bonded inertial mass estimated at 100 g [START_REF] X.-R. Chen | Vibration energy harvesting with a clamped piezoelectric circular diaphragm[END_REF]. It reduced the resonant frequencies in the 113 Hz range, but the integration capability of this structure is then hindered.

The membrane structures are also especially suitable for pressure energy harvesting. In this case, the excitation is no longer associated to vibration of the supporting structure but by the pressure fluctuation of any fluid. The imposed pressure excitation could lead to enhanced energy generation using optimized electrode distribution. Kim et al. [START_REF] Kim | Piezoelectric Energy Harvesting with a Clamped Circular Plate: Experimental Study[END_REF][START_REF] Kim | Piezoelectric Energy Harvesting with a Clamped Circular Plate: Analysis[END_REF][START_REF] Mo | Experimental validation of energy harvesting performance for pressure-loaded piezoelectric circular diaphragms[END_REF][START_REF] Mo | Energy harvesting with piezoelectric circular membrane under pressure loading[END_REF] studied this route. They proposed analytical and experimental works to improve PZT piezoelectric membranes conversion when submitted to quasi-static pressure solicitations. The total stress due to bending and stretching is considered in the model to aim at a better anticipation of the generated power.

But only quasi-static excitation is dealt with here. PVDF (or similar materials P(VDF-TrFE), loaded PVDF, ...) is chosen as elec-tromechanical transducer [START_REF] Rammohan | Low frequency vibration energy harvesting using arrays of PVDF piezoelectric bimorphs[END_REF][START_REF] Kato | Processing and energy-harvesting ability of (Na, K) NbO3 particle-dispersed fi brous polyvinylidene fl uoride multilayer composite[END_REF][START_REF] Jiang | Low-frequency energy harvesting using a laminated PVDF cantilever with a magnetic mass[END_REF][START_REF] Lockhart | Flexible and Robust Multilayer Micro-Vibrational Harvesters for High Acceleration Environments[END_REF] in many VEH. The flexibility of this material is a major advantage for low frequency application, but its intrinsic performance is lower than PZT. The power density (considering the volume of piezoelectric material) from the studied literature [START_REF] Rammohan | Low frequency vibration energy harvesting using arrays of PVDF piezoelectric bimorphs[END_REF][START_REF] Kato | Processing and energy-harvesting ability of (Na, K) NbO3 particle-dispersed fi brous polyvinylidene fl uoride multilayer composite[END_REF][START_REF] Jiang | Low-frequency energy harvesting using a laminated PVDF cantilever with a magnetic mass[END_REF][START_REF] Lockhart | Flexible and Robust Multilayer Micro-Vibrational Harvesters for High Acceleration Environments[END_REF] presents a range from 0.15 µW/cm 3 [START_REF] Kato | Processing and energy-harvesting ability of (Na, K) NbO3 particle-dispersed fi brous polyvinylidene fl uoride multilayer composite[END_REF] to 195.12 µW/cm 3 for PVDF [START_REF] Jiang | Low-frequency energy harvesting using a laminated PVDF cantilever with a magnetic mass[END_REF]. Such spread of values come from the quality of the PVDF material but also on the way the mechanical stress is induced in the material by the external excitations. For PZT diaphragm structures, literature shows a power density ranging from 15.04 mW/cm 3 [START_REF] Minazara | Piezoelectric diaphragm for vibration energy harvesting[END_REF] to 122.44 mW/cm 3 [START_REF] X.-R. Chen | Vibration energy harvesting with a clamped piezoelectric circular diaphragm[END_REF]. As expected the power density is much larger due to better material performance, but low frequency operations imply the use of a very large proof mass (122 g in [START_REF] X.-R. Chen | Vibration energy harvesting with a clamped piezoelectric circular diaphragm[END_REF]).

This work aims at demonstrating the modeling, the fabrication and testing of a new piezoelectric hybrid fluid diaphragm structure VEH. A diaphragm architecture aiming at the drastic reduction of the resonant frequency was proposed in [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF]. It is based on an incompressible fluid encapsulated between two flexible membranes.

The fluid dynamic inertia, when submitted to external excitations, generates hydraulic pressure on the diaphragms, resulting in a global low natural frequency and allowing their optimal deformation. Compared with similar structures with central mass introducing a rigid zone [START_REF] X.-R. Chen | Vibration energy harvesting with a clamped piezoelectric circular diaphragm[END_REF], the mechanical deformation is maximized here as the whole membrane is deformed. The assembly process would be simplified by using fluid as inertial fluid. Indeed, membrane are usually fragile due to their small thickness which make any mass addition a critical step. In this paper, single layered P(VDFTrFE) membranes are used to allow integrated transduction. This straightforward approach is justified because mechanical tension effect can be easily obtained for polymer membranes. As a result, the piezoelectric effect can be used with low complexity with respect to realization and electrical connection. The first part of the paper describes the VEH architecture and presents a complete theoretical electromechanical model. Realistic applications call for an efficient evaluation of a VEH performance potential under constrained background. Analytical is preferred here to a FEM approach because of the complexity of the device. In the case of the Piezoelectric Hybrid Fluid generator, a strongly coupled multiphysics model is indeed required: nonlinear membrane dynamic, piezoelectric and fluid-structure interactions have to be dealt with. The analytical approach eases the optimization step: Geometrical characteristics are used to simplify the model, which eventually allow a low computation time evaluation of the generator performance. Complex effects, such as mechanical or fluid damping, geometrical flaws are not taken into account in the proposed analytical approach. These issues would however not be easily addressed by a FEM strategy, that would induce a larger simulation time. Simulation results for low and large excitation amplitudes are analyzed to underline the nonlinear behavior of the structure. In a second part, the realization and testing of a prototype are detailed. Finally, based on the comparison between the theoretical and the experimental results, an additional unexpected distortion effect is identified. The model is modified to allow future optimization. Further developments are finally suggested.

Theoretical modelling

Architecture

The architecture is based on the work of Formosa et al. [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF] which demonstrates the possible use of fluid-membrane interaction to realize thick low resonant frequency membranes. It consists in an incompressible fluid confined between two thin membranes. The former is used as an inertial mass which allows a drastic reduction of resonant frequency for the global structure. The height of the device can be easily tuned to induce more or less inertia effect. The chosen fluid has to fulfill the trade-off between high density to enhance the fluid inertia effect and low viscosity to limit the potential mechanical dissipations. Here, the concept is developed to a further step by the integration of piezoelectric membranes for energy generation.

The hybrid fluid diaphragm (HFD) structure is defined in the Fig. 1, which describes the architecture and it geometrical parameters. The circular diaphragms are made from thin film P(VDF-TrFE). It is composed of a single piezoelectric material layer and top and bottom electrodes. The poling is or-thogonal to the film plane. Table 1 defines the geometrical parameters and material properties of the system.

Due to the film structure and the electrodes' locations, flexural deformations induce a same amount of compressed and stretched piezoelectric material across the thickness and over the radius so the generated voltage is theoretically zero. However, large deformation amplitude generates stretching through the thickness that can be the major mechanical stress. Then, it produce an electrical voltage. This nonlinear behavior is the underlying principle of the proposed VEH.

Analytical model

The model in [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF] was based on the Ritz method to obtain approximate solutions for both the fluid motion and the diaphragms' deformation. In the present work, this mechanical model is simplified and the piezoelectric membrane behavior is added.

Indeed, from the mechanical point of view, the final dynamic behavior (i.e the first resonant frequency) of the HFD is notably lower than the first natural frequency of a diaphragm alone. Therefore, a simple static behavior description of the membrane can be used for approximation of the mechanical solution instead of the first modal shapes as it was done in the previous model. Additionally, the frame displacement is introduced which is in line with the vibration energy harvesting application background of this work.

The modified modeling strategy is developed hereafter using the following assumptions:

• Clamped edge mechanical boundary condition.

• Same geometry and materials properties for each membranes.

• The initial prestress of the diaphragm is taken into account considering an initial uniform radial displacement u S0 applied on the membranes' rim.

• Large strain is considered.

• The shear stress is neglected.

• The fluid is incompressible and inviscid.

• The fluid motion is assumed to be irrotational [START_REF] Kim | Coupled vibration of partially fluid-filled cylindrical shells with ring stiffeners[END_REF].

• Axially symmetric motions.

• Diaphragms have the same displacement (in-phase motion).

• The polarization direction is along the normal of the diaphragms ( e z ).

Mechanical approximate static solution

The transverse and radial deformations of the diaphragm can be written as the quasi-static solution under uniform pressure [START_REF] Timoshenko | Theory of plates and shells[END_REF]:

w(t , r ) = x(t )w s (r ) = x(t ) 1 - r R 2 2 u(t , r ) = x(t ) 2 u s (r ) = x(t ) 2 126R 3 r (R -r ) (179 -89ν) + 3 R (-79 + 13ν) r (1)
In which w(t , r ) is the deflection of the diaphragms along the vertical direction ( e z ). u(t , r ) is the radial displacement along e r . x(t ) is the maximal relative with respect to its clamped rim vertical displacement at the center of one diaphragm.

Fluid equations and approximate solution

As the liquid motion is assumed to be irrotational, its velocity V f (t , r, z) can be associated to a velocity potential ψ(t , r, z) defined such as:

V f (t , r, z) = ∇ψ(t , r, z) (2)
In which ∇ is the gradient operator. In order to take into account the vertical velocity of the entire frame (V mec (t ) in Fig. 1) and in a case of small relative motion, a new velocity potential φ(t , r, z) is defined as:

V f (t , r, z) = V mec (t ) + ∇φ(t , r, z) = ∇ψ(t , r, z) = ∇(V mec (t )z + φ(t , r, z)) (3)
Finally, the equilibrium equation for the liquid results in:

∇ 2 φ(t , r, z) = 0, ∀ r ∈ [0, R] and ∀ z ∈ - H 2 , H 2 (4)
In which ∇ 2 is the Laplace operator. Based on the solution of the transcendental Eq. 4, the general form of the fluid velocity potential is:

φ(t , r, z) = N j =1 (C j (t ) sinh(k j z))J 0 (k j r ) (5)
In which N is the number of retained fluid modes J 0 (k j r ) is the Bessel first kind function and k j is the j t h root of Eq. 4. C j (t ) is a parameter which depends on the velocity of the diaphragm center ( ẋ).

The boundary conditions for the fluid can be expressed as:

V f . e r = 0, f or r = R and ∀ z ∈ - H 2 , H 2 V f . e z = d w d t + V mec , f or z = ± H 2 and ∀ r ∈ [0, R] (6)
Using Eq. 1 and Eq. 3, the boundary conditions from Eq. 6 are written again as:

∂φ ∂r = 0, f or r = R and ∀z ∈ - H 2 , H 2 ∂φ ∂z = ẋw s , f or z = ± H 2 and ∀r ∈ [0, R] (7)
For developing the weak form solutions of Eq. 7, we define the new function φ(t , r, z) = C 0 (t ) + φ(t , r, z) and integrate it over the whole fluid domain:

-

H 2 -H 2 R 0 ∂φ ∂r ∂ φ ∂r + ∂φ ∂z ∂ φ ∂z r d r d z + 2 R 0 ẋw s φ R, H 2 r d r = 0, ∀ C j , j = 0..N (8)
Solving Eq. 8, N is chosen such as the boundary condition at the diaphragm surface is satisfied with a sufficient accuracy. Using N = 3 actual deformed shape of the membrane is matched [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF].

The fluid pressure can be expressed using the Cauchy-Lagrange equation for unsteady flow:

ρ f ∂ψ ∂t + 1 2 ρ f V 2 f + p = const ant (9)
Where ρ f is the fluid density. Eq. 9 is developed using Eq. 3. For simplification purpose, the relative weights between the terms are assessed defining the dimensionless variables given in table 2.

Attribute Length Time Velocity potential Variable r z t φ Reference value R H Vmec h hV mec Dimensionless variable r * = r R z * = z H t * = ht Vmec φ * = φ hVmec

Table 2: Definition of the dimensionless variables

The dimensionless expression of Eq. 9 is then:

ρ f ∂φ * ∂t * + Vmec H z * h 2 + 1 2 ρ f V mec R ∂φ * ∂r * 2 + V mec H ∂φ * ∂z * 2 + V 2 mec h 2 + p h 2 = const ant (10)
Considering the size of the considered structure (Table 5) and Eq. 10, the first and second order terms ∂φ ∂r 2 , ∂φ ∂z 2 are neglected.

At the whole fluid domain frontier (r = R) the constant right hand term of Eq. 10 is null, so the pressure can be finally written as:

p = -ρ f ∂φ ∂t + Vmec z - 1 2 ρ f V mec 2 (11)
The joint effects of the global motion of the fluid (V mec ) and the local perturbation is clearly established by Eq. 11: the frame motion acts as a buoyancy force on the membranes.

Global dynamic equilibrium

The dynamic equilibrium equations are obtained using the Energy method applied to each of the piezoelectric diaphragms. The model is detailed for the bottom diaphragm. The procedure is same for the top one.

The piezoelectric constitutive relations are:

σ r = E 1 -ν 2 (ε r + νε θ -d 31 E z (1 + ν)) σ θ = E 1 -ν 2 (ε θ + νε r -d 31 E z (1 + ν)) D z = -d 31 E 1 -ν (ε r + ε θ ) + ε 33 ε 0 E z ( 12
)
The strain components can be expressed distinguishing the linear bending and the nonlinear stretching behaviors [START_REF] Haterbouch | Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates[END_REF] such as:

ε r = ε r L + ε r N L = -x(t )z d 2 w s (r ) d r 2 + x 2 (t ) d u s (r ) d r + 1 2 d w s (r ) d r 2 ε θ = ε θL + ε θN L = -x(t ) z r d w s (r ) d r + x 2 (t ) u s (r ) r ( 13
)
Using Eq. 12 and Eq. 13, the bending and stretching energies (E mecB and E mecS respectively) are derived. The linear and nonlinear stiffness coefficients k and b are finally defined as:

E mecB = 1 2 E 1 -ν 2 R 0 h 2 -h 2 2π 0 (ε 2 r L + 2νε r L ε θL + ε 2 θL )r d r d zd θ = 1 2 x 2 2πD B R 0 d 2 w s d r 2 2 + 1 r 2 d w s d r 2 + 2ν r d w s d r d 2 w s d r 2 r d r = 1 2 x 2 k (14) E mecS = 1 2 E 1 -ν 2 R 0 h 2 -h 2 2π 0 (ε 2 r N L + 2νε r N L ε θN L + ε 2 θN L )r d r d zd θ = 1 2 x 4 2πD S R 0 d u s d r + 1 2 d w s d r 2 2 + 2ν d u s d r + 1 2 d w s d r 2 u s r + u s r 2 r d r = 1 2 x 4 b (15)
In which,

D B = E h 3 12(1-ν 2 ) , D S = E h (1-ν 2 )
. The initial prestress of the diaphragm associated to the assembly procedure is taken into account. The ensuing in-plane radial and orthoradial stresses are:

σ r S0 = σ θS0 = E (1 -ν) u S0 R ( 16
)
From Eq. 16, the prestress mechanical energy is:

E mecS0 = 1 2 R 0 h 2 -h 2 2π 0 (σ r S0 ε r N L + σ θS0 ε θN L )r d r d zd θ = 1 2 x 2 2π E 1 -ν u S0 R R 0 d u s d r + 1 2 d w s d r 2 + u s r r d r = 1 2 x 2 n ( 17
)
The prestress linear stiffness n is deduced from Eq. 17. Finally, the mechanical work of the fluid dynamic pressure upon the diaphragm is:

W f = -x2πρ f R 0 w s ∂ φ ∂t - H 2 Vmec + 1 2 V mec 2 r d r = -x m m f ẍ + m t f H -H Vmec + 1 2 V mec 2 (18)
In which m m f is the fluid inertial mass and m t f is the fluid total (static) mass.

The electric field E z can be expressed as a function of the electric potential (V el ec ) such as: E z =

V el ec h . The stretching of the structure induces in-plane strain. The piezoelectric coupling energy E pi ezo is then expressed as:

E pi ezo = - 1 2 D s (1 + ν)d 31 V el ec h R 0 h 2 -h 2 2π 0 (ε r N L + ε θN L )r d r = - 1 2 2πD S (1 + ν) 31 V el ec h R 0 d u s d r + 1 2 d w s d r 2 + u s r r d r = 1 2 x 2 α V el ec h (19)
In which α is the piezoelectric force factor.

Finally, the dynamic equilibrium equation for the bottom diaphragm is given by:

∂ ∂x E mecB + E mecS + E pi ezo + E mecS0 + W f = 0 ( 20
)
Using Eq. (7-8), Eq. [START_REF] Mo | Energy harvesting with piezoelectric circular membrane under pressure loading[END_REF][START_REF] Rammohan | Low frequency vibration energy harvesting using arrays of PVDF piezoelectric bimorphs[END_REF][START_REF] Kato | Processing and energy-harvesting ability of (Na, K) NbO3 particle-dispersed fi brous polyvinylidene fl uoride multilayer composite[END_REF][START_REF] Jiang | Low-frequency energy harvesting using a laminated PVDF cantilever with a magnetic mass[END_REF][START_REF] Lockhart | Flexible and Robust Multilayer Micro-Vibrational Harvesters for High Acceleration Environments[END_REF] and Eq. [START_REF] Kim | Coupled vibration of partially fluid-filled cylindrical shells with ring stiffeners[END_REF][START_REF] Timoshenko | Theory of plates and shells[END_REF][START_REF] Haterbouch | Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates[END_REF] to write again Eq. 20, the dynamic equilibrium equation for the bottom diaphragm is:

m m f ẍ + d ẋ + (k + n)x + 2bx 3 + α x h V el ec = m t f Vmec - 1 2 m t f H V mec 2 (21)
In which the dissipation can be introduced as an equivalent viscous damping coefficient (d ) such as:

d = m m f Q ω 0 = m m f (k + n) Q ( 22
)
Where Q is the estimated mechanical quality factor of the structure in the case of small amplitude vibration. Applying the same procedure, the top diaphragm equilibrium equation is:

m m f ẍ + d ẋ + (k + n)x + 2bx 3 + α x h V el ec = m t f Vmec + 1 2 m t f H V mec 2 ( 23
)
From Eq. 21 the linear case (small amplitude and closed circuit condition) natural pulsation can be written as:

ω 0 = k + n m m f (24)
To support the quasi-static approximation, ω 0 is compared to the theoretical natural pulsation calculated from the previous model [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF] using the same geometry and material properties. The similarity of the results demonstrates that the simplification of previous model is relevant in the case of the proposed structure.

Electrical equation

The electric equilibrium equation can be derived from:

∂ ∂V el ec E el ec = -q (25)
Where q is the electrical charges so that q is the generated current (I ).

Using the piezoelectric constitutive relations Eq. 12, the electrical energy E el ec is written as:

E el ec = 1 2 2π R 0 D z E z r d r = 1 2 2π R 0 ε 33 ε 0 V el ec h -d 31 (σ r + σ θ ) V el ec h r d r = 1 2 CV 2 el ec -α E x 2 V el ec h (26)
E el ec is the addition of two terms: the stored energy in the capacitance (C 0 ) and the piezoelectric coupling related to the coupling factor (α E ). Using Eq. (12, 13) and

Eq. 25, one can demonstrate the equality α = α E . From Eq. 25 and Eq. 26 the electrical equilibrium equation is finally written as:

C 0 V el ec - 1 2 α x h x = -q (27)
Performing the time derivation of Eq. 27:

-C 0 Velec + α x h ẋ = I (28)
Eq. 28 underlines the voltage generation due to the large deformation of the membranes as small displacements (x) lead to second order values of the piezoelectric coupling term (second term of the left hand side of Eq. 28). It can also be deducted from Eq. 28 that in open circuit condition, the voltage is proportional to

x 2 .
The piezoelectric voltage frequency is then two times the excitation frequency, which is a another particular feature of the proposed concept.

Eq. 21 and Eq. 28 describe the electromechanical behavior of the PHFD. This

32πE h 3 9R 2 (1-ν 2 ) N/m Tension stiffness b πE h(7505+4250ν-2791ν 2 ) 39690R 2 (1-ν 2 ) N/m 3 Electromechanical coefficient α - 4πE d 31 3(1-ν) N/V Diaphragm capacitance C 0 ε 33 ε 0 πR 2 h F Prestress flexural stiffness n * 4u S0 πE h 3R(1-ν 2 ) N/m Damping coefficient d m m f (k+n * ) Q * N.s/m * Measured values

Acceleration

Fig. 3a shows the evolution of the LS and NLS forces according to the normalized displacement of the membrane center. The threshold from which the NLS effect is no more negligible compared to the LS one is estimated at 10 % of the LS force. It is reached when the displacement is about two times the thickness of the membrane (x ≈ 172.6 µm). Fig. 3b plots the displacement evolution with respect to the acceleration amplitude ( Vmec ). The nonlinear mechanical behavior appears at an acceleration of approximately 13 m/s 2 , which is the value required to reach the aforementioned threshold. The NLS effect becomes predominant at 34 m/s 2 (for 50 % of LS contribution). Between these two limits a transition area is identified.

Consequently, the acceleration value chosen for simulations is of 40 m/s 2 . It allows large strains to be reached and nonlinear effect to be clearly observed and compared to the theoretical model.

Displacement and power

The first results are based on the parameters defined in table 4 and table 5. In order to characterize the harvester performance, we carry out simulations using a slow frequency sweep ranging from 80 Hz to 140 Hz and a varying electrical load resistance r l (from 0.6 MΩ to 14 MΩ).

Fig. 4a shows the displacement of the membrane center (x) and the generated power with respect to the electrical load resistance and the excitation frequency. been performed yet as it is beyond the scope of the paper.

The next step is to put the two parts together when immersed into the chosen fluid to get a filled HFD without bubbles to guarantee incompressibility. The tightness is provided by an O-ring as can be seen in Fig. 5a. The internal fluid pressure, and consequently the flatness of the diaphragms can be tuned using a screw (Fig. 5b). Indeed, the slight squeeze of the O-ring associated to the incompressibility of the fluid turn any volume modification into static pressure variation.

Experimental setup

The piezoelectric PHFD was tested on an electrodynamic shaker driven by a slow varying increasing frequency signal similar to the used simulations excitation. Using an accelerometer and a dedicated control unit, a closed-loop control ensures constant acceleration amplitude of the shaker over the frequency range. The displacement and velocity of the top diaphragm center are measured using a differential laser vibrometer. The control unit also drives a set of electric load resistance (r l ) ranging from 0.6 MΩ to 14 MΩ. Different acceleration levels (from 30 m/s 2 to 60 m/s 2 ) allowing the nonlinear behavior to be triggered have been used.

Experimental results

Displacement and power

The experimental results are shown in Fig. 6 to be compared to Fig. 4 for an acceleration amplitude of 40 m/s 2 . The nonlinear effect associated to large displacements is clearly visible (Fig. 6a) while the electromechanical conversion has a negligible impact. The displacement amplitude at the resonant frequency is 376.5 µm, which is more than four times the thickness of the membrane. Fig. 6b plots the power response. Its maximum is about 4.18 µW at the resonant frequency and for a 2.88 MΩ load resistance. It is worthy of note that the mechanical hardening effect leads to a large bandwidth of about 40 Hz. The power density (calculated with respect to the volume of piezoelectric material) of the proposed generator is 158.33 µW/cm 3 , which is in the order of magnitude of P(VDF-TrFE) VEH performances reported in the literature.

Temporal results

The temporal results underline the frequency doubling voltage in accordance with Eq. 28. This is clearly visible comparing the blue line (displacement of the the Fourier transform of the voltage signal (red dashed line in Fig. 7b) reveals two dominant harmonics: the first is at the diaphragm displacement frequency (blue plain line in Fig. 7b) and can be related to a voltage produced by the bending motion whereas the second is at about twice this frequency (224.2 Hz) and is the expected nonlinear behavior. An ideal piezoelectric membrane does not present electrical generation associated to bending motion as stated by the theoretical model. Hence, the first harmonic is thought to be induced by an energy generation due to the membranes bending. This observation highlights what is thought to be the non-ideal flat geometry of the membranes, which was not taken into account in the model. Indeed, tiny residual wrinkles are likely to bring complex mechanical neutral plane shape which could induce piezoelectric voltage for flexural deformations. However, this effect is not detrimental as it tends to increase the generated voltage.

Model / experiment comparison and optimization

The aim of this section is to validate the theoretical model so it could be used as an optimization tool for the design of PHFD generators. Moreover, the identification procedure is mandatory for the dissipation of the prestress of the membrane.

Yet, the first prototype may present some specificity which are likely to generate discrepancies between the theoretical and experiment results.

Parameters identification

The mechanical behavior of the device is virtually not changed by the electric behavior considering the low coupling coefficient of the polymer piezoelectric film.

Therefore, the identification is performed in two steps. The first one is to identify the mechanical parameters m t f , m m f , b, n and Q. It is assumed that a reliable value of k is obtained from the theoretical approach. The prestress value n may be shifted with different tightenings of the membrane and has to be experimentally identified.

The identification procedure is to minimize the difference between the simulated and experimental displacement when varying the excitation frequency for a fixed given electrical load (r l = r peakpower ). The least square method is used.

The second step is to identify the electrical behavior by adjusting the two parameters α and C 0 . The chosen strategy is slightly different here as two sets of reference results are used: i) The generated power when varying frequencies for a given load resistance (r l = r peakpower ); ii) The power at a given frequency ( f = f peakpower ) when varying the electrical load resistance.

The choice of the power results provides a characteristic point (peak power) with more information than the voltage. Indeed, the voltage alone does not allow to accurately determine the α and C 0 parameters, because there may be several valid parameters' values sets for a given frequency. The sum of the normalized differences between experimental and simulated results defines the objective function to be minimized by the least square method.

This identification procedure has been firstly applied assuming that the "bending voltage" is negligible compared to the "stretch voltage". A rather good modeltesting correlation is obtained on both the mechanical (Fig. 8a) and electrical (Fig. 8b) responses. The power responses (Fig. 8b) present a larger divergence for values distant from the power peak. As mentioned before, the experimental measured signal is composed mainly of two harmonics (see Fig. 9a). In Fig. 9b, a Fourier transform allows these frequencies to be read as 119.56 Hz and 239.12 Hz. This voltage generation for bending motion is identified as the result of a slight initial bending of the membrane.

As it is considered here as a unexpected behavior, the model has not been deeply

modified. Yet, an additional linear piezoelectric electromechanical coefficient (β) is introduced and identified in the following to validate the interpretation. The modified model is defined by the set of coupled equations (Eq. 29).

m m f ẍ + d ẋ + (k + n)x + 2bx 3 + βV el ec + α x h V el ec = m t f Vmec -C 0 Velec + β ẋ + α x h ẋ = I (29)
A schematic representation of the model using for Eq. 29 is given in Fig. 10. Because of the negligible impact of the electrical behavior on the mechanical behavior, only the electrical parameters (α, C 0 and β) are identified again using the strategy described above. It can be seen from Fig. 11b that the overall difference between the simulated and experimental results has decreased. The bandwidth error decrease (table 7) is in accordance with this observation. Fig. 11a and11c show placement and voltages. The α and C 0 parameters are also closer to those determined by the theoretical model (Table 7).

Effect of the acceleration and identified model validation

For further investigations, we compared the simulated and experimental results

for two different accelerations (30 m/s 2 and 60 m/s 2 ) using the identified parameters. For the lowest acceleration case the correlation is slightly deteriorated, but the qualitative response is captured (Fig. 12b). The power is underestimated (Table 8).

For this level of acceleration, the generator response is close to a linear system behavior. The suspected "imperfections" of the membrane create important and com-390 plex effects associated with bending, for which coefficient β alone cannot represent. For the high acceleration case for which the nonlinear behavior dominates the response, the comparison is better. The performance criteria are validated (Ta-ble 9) with an error of 13.92 % for the power. The overall difference between is lower (Fig. 12c and Fig. 12d).

Conclusion

An original membrane generator matching low frequency vibration energy harvesting applications is proposed. A complete theoretical model has been developed.

A prototype was realized and is investigated aiming at the validation of the theoretical model. In this first prototype, single layered P(VDF-TrFE) film is used. The mechanical tension effect induced by large deformation has been obtained using 4 g excitation amplitude. The experimental results prove that this expected hardening effect and its ensuing electrical power generation is captured by the model.

The realized generator exhibits low resonant frequency (119.56 Hz), large bandwidth (40.28 Hz) and power density (158.33 µW/cm 3 ). However, unexpected additional voltage induced by the linear flexural mechanical response of the membrane is noted.

An identified linear piezoelectric electromechanical coefficient is then added to complete the model. In the targeted acceleration scope of the generator (4 g and more), the model gives rather good predictive results with an error lower than 15 % for the chosen comparison criteria (peak power amplitude, frequency and load at the optimal point). At lower acceleration or away from the resonance frequency, the deviations between analytical modeling and experimental results increase. In these cases, the influence of the linear electromechanical coefficient β, which is thought to be due to geometrical defects, on the membranes responses is indeed higher.

However, the origin of the linear coupling coefficient is not completely identified, but it explains the difference between the models and experiments at 3 g and 6 g acceleration levels. The electric generation for linear (low amplitude) responses calls for further investigations. The geometry of the membrane may present some flaws and has to be verified. The material constitutive relations were considered as linear.

Though, initial prestress associated to the assembly procedure and the large deformation dynamic solicitations could lead to additional material nonlinear effect.

Based on the obtained proof of concept, dedicated studies are envisioned. A better repeatability of the assembly will have to be reached. Plastic material was used for the frame realization to ease the prototyping. More rigid and nonconductive material may be used. It would also improve the prestress control of the membrane.

The damping will have to be lowered to increase the performance. To the authors' knowledge, most of the damping can be associated to the polymer membrane. However, different fluid will have to be tested to assess its contribution.

Furthermore, another architecture of electroactive membrane will be investigated to increase the power density. For instance, the diaphragms bending could be fully utilized using multi-layered piezoelectric film (bimorph structure) and two dissociated electrode rings on the external faces of membrane stack. By doing this, the bending deformation can be exploited and lower acceleration magnitude could be focused on. Though, the expected increased stiffness may lead to higher column of fluid to keep the low frequency response advantage of the proposed architecture.

A global optimization will then be necessary to aim at enhanced power density depending on the targeted application.

Nomenclatureb 2 z 3 σ

 23 Nonlinear stiffness, N/m 3 C 0 Capacitance, F C j Constant d Damping, N.s/m d 31 Piezoelectric coefficient, C/N D B Flexural stiffness, N.m D S Stretching stiffness, N/m D z Electric displacement, C/m 2 e i Coordinate system (i refer to subscript) E Diaphragm Young modulus, Pa E i Energy (i refer to subscript), J E z Electric field, V/m f Frequency, Hz h Diaphragm thickness, m H Fluid height, m I , q Current, A k Linear stiffness, N/m m mt Inertial mass, kg m t f Moving fluid mass, kg n Prestress stiffness, N/m p Pressure, Pa P Power, W q Electrical charge, C Q Quality factor r Radial position, m R Diaphragm radius, m r l Load resistance, Ω R opt i Optimal resistance, Ω t Time, s u, u s0 Radial displacement, m V Velocity, m/s V el ec Voltage, V Velec Voltage derivative, V/s V mec External velocity, m/s Vmec External acceleration, m/s 2 W f Fluid work, J w s , u s Unitary displacement w Vertical displacement, m x, X Membrane center displacement, m ẋ Membrane center velocity, m/s ẍ Membrane center acceleration, m/m Vertical position, m Greek Symbols α, α E , β Electromechanical coefficient, N/V ε 33 Relative permittivity ε Strain ε 0 Vacuum permittivity, F/m ν Diaphragm Poisson coefficient ρ f Fluid density, kg/m Stress, Pa ψ, φ, φ Velocity potential, m 2 /s ω Pulsation, rad/s Wireless sensors networks (WSN) developments have increased over recent years.

Figure 1 :

 1 Figure 1: Schematic of the proposed structure HFD

Figure 2 :

 2 Figure 2: Nonlinear inertial generator electromechanical modeling

Figure 3 :FrequencyFigure 4 : Theoretical simulation results 3 . Experimental approach 3 . 1 .

 34331 Figure 3: Linear / nonlinear limits

Figure 5 :

 5 Figure 5: HFD prototype

FrequencyFigure 6 :

 6 Figure 6: Experimental performances at 40 m/s 2

Figure 7 :

 7 Figure 7: Experimental displacement and voltages signals at 40 m/s 2

FrequencyFigure 8 :

 8 Figure 8: Optimized simulation (color scale) and experimental (red mesh) results comparison

Figure 9 :

 9 Figure 9: Optimized simulation and experimental signals comparison

Figure 10 :

 10 Figure 10: Nonlinear inertial generator modeling with flexural effect

Figure 11 :

 11 Figure 11: Optimized simulation with flexural effect and experimental results comparison

Frequency

Figure 12 :

 12 Figure 12: Comparison between experimental (red mesh) and optimized simulation (color scale) (a-b) at 30 m/s 2 and (c-d) at 60 m/s 2

Table 1 :

 1 Main parameters of the PHFD

	Diaphragm properties (P(VDF-TrFE))
	Radius (m)	R	Thickness (m)	h
	Young modulus (Pa)	E	Poisson coefficient	ν
	Piezoelectric coefficient (C/N)	d 31	Relative permittivity	ε 33
	Fluid Properties (Glycerin)
	Height (m)	H	Density (kg/m 3 )	ρ f

Table 3 :

 3 Table. 3 presents the frequencies and the discrepancies between the two models for various membrane thicknesses. Comparison of natural frequencies results from the two proposed models

	Attribute			Value
	Thickness h (µm)	20	25	40	50	110
	Dynamic model [16] f 0 (Hz)	5.486	7.664	15.494	21.637	70.298
	Static model f 0 (Hz)	5.479	7.658	15.498	21.659	70.677
	Difference (%)	-0.127	-0.079	0.029	0.103	0.539

Table 4

 4

	Parameter	Symbol	Relation	Unit
	Fluid inertial mass	m m f	See [16] f (ρ f , H , R)	kg
	Fluid excitation mass	m t f	H πρ f R 2 12	kg
	Flexural stiffness	k

details the whole model parameters, their units and relationship to the geometry and material properties. It is worthy of note that the stiffness property is independent to the mass property in the obtained structure, which is an advantage for optimization.

Table 4 :

 4 Model parameters relation2.2.5. Simulation resultsTable5gives the geometry and material properties values used in the simulations. They are those of the prototype presented in section 3.

	Diaphragm properties (P(VDF-TrFE))
	R	10 mm	h	84 µm
	E	1 GPa	ν	0.34
	d 31	-5 pC/N	ε 33	8.25
		Fluid Properties (Glycerin)
	H	20 mm	ρ f	1260 kg/m 3

Table 5 :

 5 Geometry and material properties of the PHFD

Table 6

 6 shows the principal features of the VEH with relative error between experimental and simulation results. The errors are acceptable (< 10 %), except for the bandwidth with a maximal discrepancy of 42.12 %.

		f 0	P max	X max	R opt i	B and wi d t h
	Exp	119.56 Hz	4.17 µW	376.5 µm	2.88 MΩ	40.28 Hz
	Simu	119.56 Hz	4.09 µW	373.6 µm	2.88 MΩ	23.31 Hz
	Error	0 %	-1.92 %	-0.77 %	0 %	-42.13 %

Table 6 :

 6 Comparison between experimental and simulation results with first identified parameter set

Table 7 :

 7 Comparison between experimental and second identified parameter set simulation results

		f 0	P max	X max	R opt i	B and wi d t h
	Exp	119.56 Hz	4.17 µW	376.5 µm	2.88 MΩ	40.28 Hz
	Simu	119.56 Hz	4.11 µW	373.6 µm	2.88 MΩ	28.39 Hz
	Error	0 %	-1.43 %	-0.77 %	0 %	-29.52 %

Table 8 :

 8 Comparison between experimental and optimized simulation results at 30 m/s 2 of acceleration

		f 0	P max	X max	R opt i	B and wi d t h
	Exp	112.91 Hz	3.01 µW	315.5 µm	2.88 MΩ	32.85 Hz
	Simu	111.29 Hz	2.13 µW	302.1 µm	3.58 MΩ	22.87 Hz
	Error	-1.43 %	-29.23 %	-4.25 %	+24.31 %	-30.38 %

Table 9 :

 9 Comparison between experimental and optimized simulation results at 60 m/s 2 of acceleration

		f 0	P max	X max	R opt i	B and wi d t h
	Exp	132.21 Hz	9.34 µW	475.8 µm	2.58 MΩ	38.28 Hz
	Simu	136.04 Hz	10.64 µW	487.5 µm	2.31 MΩ	28.12 Hz
	Error	+2.89 %	+13.92 %	+2.46 %	-10.45 %	-26.54 %

Acknowledgements

This work was supported by the Centre Technique des Industries Mécaniques (CETIM) and the Assemblée des Pays de Savoie (APS).


Florian Huet
email: florian.huet@univ-smb.fr

Fabien Formosa

Adrien Badel

Bimorph hybrid fluid diaphragm device for vibration energy harvesting

Keywords: Vibration energy harvesting, Piezoelectric membrane, Fluid-structure interaction, Linear generator

A low resonant frequency piezoelectric energy harvesting using a hybrid fluid diaphragm (HFD) is presented. This paper describes the design, fabrication and characterization of such a device for harvesting energy from vibrations. The PHFD consists in an incompressible fluid confined between two thin piezoelectric membranes. The output voltage and power of the P(VDF-TrFE) HFD are studied based on experimental and simulation results. This device resonates at 164 Hz for low acceleration (10 m/s²). The simulations showed the power production of about 6 µW with a power density of 78.98 µW/cm³ of electroactive material volume.

I. INTRODUCTION

An increasing interest in wireless sensors networks has been growing up over recent years. The implementation of communicating sensors networks in industrial application, transport or buildings offers possible improvements: productivity gains, reliability and energy performance are feasible by the exploitation of the extensive collected information (temperature, humidity, electric consumption...). With the rapid advancement of low-power wireless sensor nodes, energy harvesting has attracted worldwide research interests [START_REF] Paradiso | Energy scavenging for mobile and wireless electronics[END_REF].

Vibration energy harvesting (VEH) generators have been proposed using three main transduction principles: piezoelectric, electromagnetic and electrostatic [START_REF] Beeby | Energy harvesting vibration sources for microsystems applications[END_REF], [START_REF] Mitcheson | Human and Machine Motion for Wireless Electronic Devices[END_REF]. In order to extract the maximum power from the environment, the resonant frequency of a linear harvesting device has to match the dominant frequency of the excitation. Indeed frequencies of ambient available vibration sources are relatively low (typically less than 200 Hz) [START_REF] Roundy | A study of low level vibrations as a power source for wireless sensor nodes[END_REF]. As a result, many VEH piezoelectric energy harvesters of centimeter scale have been demonstrated to operate at low frequencies.

PVDF (or similar materials P(VDF-TrFE, loaded PVDF, ...) is chosen as piezoelectric transducer [START_REF] Rammohan | Low Frequency Vibration Energy Harvesting using Arrays of PVDF Piezoelectric Bimorphs[END_REF]- [START_REF] Lockhart | Flexible and Robust Multilayer Micro-Vibrational Harvesters for High Acceleration Environments[END_REF] in many VEH. The flexibility of this material is a major advantage for low frequency application, but its intrinsic performance is lower than PZT. The reached power density (considering the volume of piezoelectric material) from the studied literature shows a range from 0.15 µW/cm 3 [START_REF] Rammohan | Low Frequency Vibration Energy Harvesting using Arrays of PVDF Piezoelectric Bimorphs[END_REF] to 195.12 µW/cm 3 [START_REF] Jiang | Low-frequency energy harvesting using a laminated PVDF cantilever with a magnetic mass[END_REF].

This paper reports the modelling, fabrication and testing of a new piezoelectric fluid-membrane structure for vibration energy harvesting. Compared to classical cantilever architectures, this approach takes advantage of the mechanical strain of circular diaphragms. However, when miniaturized, they present high resonant frequency [START_REF] X.-R. Chen | Vibration energy harvesting with a clamped piezoelectric circular diaphragm[END_REF] and it hinders their direct use. The concept of piezoelectric hybrid fluid diaphragm (PHFD) consists in an incompressible fluid confined between two thin piezoelectric membranes. It allows the realization of low resonant frequency system as the fluid is used as an inerttial mass. Moreover, pressure fluctuation harvesting [START_REF] Mo | Experimental validation of energy harvesting performance for pressure-loaded piezoelectric circular diaphragms[END_REF] can be especially targeted.

II. INERTIAL PIEZOELECTRIC HFD GENERATOR

A. Bimorph piezoelectric HFD concept

We propose here, an innovative architecture for vibration energy harvesters. Fig. 1 describes the geometry as well as the main geometrical parameters of the structure. The incompressible fluid is used as an inertial mass which allows a drastic reduction of the resonant frequency to effectively harvest ambient vibrations. The height of the structure can be easily tunned to provide more or less inertia effect. In a previously published work, the mechanical behavior of a HFD was presented [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF]. As an example of the frequency reduction, the use of glycerin allowed the resonant frequency of Kapton ® membranes (R = 10 mm, H = 20 mm and h = 100 µm) to decrease from 1200 Hz to 180 Hz. The circular diaphragms are made from thin P(VDF-TrFE) bimorph film. The film is composed of two piezoelectric material layer with structured electrodes. Table I defines the geometrical parameters and material properties of the system.

B. Theoretical model

A comprehensive theoretical model was proposed in [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF]. Based on the Ritz method, approximate solutions for the fluid and the mechanicals diaphragms behaviours are set. This prior model is modified and extended here. Because, the final investigated dynamic behavior of the HFD in the vicinity of its first resonant frequency is notably lower than the first As a result, a simple lumped model can be derived and presented by Eq.1-2 and the schematic representation in Fig. 2 and Fig. 3 .

m m f ẍ + d ẋ + (k + n)x + bx 3 + βV elec = m t f Vmec (1) β ẋ -C 0 Velec = I (2)
Table II details the lumped parameters, their units and relationship to the geometry and material properties.

m m f See [11] f (ρ f , H, R) kg Fluid excitation mass m t f Hπρ f R 2 12 kg Flexural stiffness k 32πEh 3 9R 2 (1-ν 2 ) N/m Tension stiffness b πEh(7505+4250ν-2791ν 2 ) 39690R 2 (1-ν 2 ) N/m 3 Electromechanical coefficient β - 64πEd 31 h 9(1-ν) N/V Diaphragm capacitance C 0 2ε 33 ε 0 πR 2 3h F Prestress flexural stiffness n * 4u S0 πEh 3R(1-ν 2 ) N/m Damping coefficient d √ m m f (k+n * ) Q * N.s/m * Measured values
The mechanical model (Fig. 2) consists in a moving fluid mass (m m f ) tied to a vertical spring (k + n) standing for the flexural diaphragm stiffness and two springs (b) standing for the stretching associated to large amplitude motions. The piezoelectric conversion (β ) is associated with the flexural stiffness. The diaphragm deformations induce a voltage between the diaphragm electrodes. The input force is m t f Vmec with Vmec the frame acceleration. The damping coefficient (d) embodies all the mechanical losses (the losses within the membrane material and the fluid dissipation).

Mass mm f V elec (t) The piezoelectric generator (Fig. 3) behaves as a voltage source whose value β ẋ is parallel to a capacitance (C 0 ) induced by the piezoelectric element. V elec is the electrical voltage across the load resistance. In the case of small displacement, the nonlinear behavior effects are minor and the nonlinear stretching (b) can be neglected. If a single layer (monomorph [START_REF] Li | Domain reorientation of piezoelectric bending actuators[END_REF]) of electroactive material is used, during a flexural motion, the same amount of film is compressed and stretched across the thickness so the global generated voltage is zero (Fig. 4).

I(t) x (

Electrode Traction Compression

Neutral surface

Figure 4: Flexural motion

To extract energy in linear behavior, a double layers (bimorph [START_REF] Li | Domain reorientation of piezoelectric bending actuators[END_REF]) of electroactive polymer is to be used. This bimorph structure use complex electrodes geometry (Fig. 5). So to extract energy, it is necessary to include an electrode on the neutral surface deformation splitting. Consequently, a membrane is composed of electro-active films with a common central electrode. The films are assembled with their polarity reversed. For differentiating the compression and the stretching on the radius, two areas have been delimitated, a disk and a ring. The model showed that the optimum radius of central electrode is r 1 = R 1 /3, and the inner radius of the external electrode is r 2 = R 2 /3. With these dimensions, the electrode surfaces are identical resulting in a same capacitance. A HFD prototype has been realized aiming at the validation of the proposed model (Fig. 6a). The geometrical and material properties of this prototype are presented in the table III. For this type of generator, the fluid used is glycerin and been used the membranes. This material is more flexible than a conventional PVDF and has similar piezoelectric properties. The Fig. 6b shows the structure of the membranes made. To connect the generator to an extraction circuit, the connection areas were performed. But the surface electrode surface is larger than the ideal surface which increases the the film capacitor. It is this actual capacitance which is used for the simulations. The first step for the assembly procedure is the clamping of the cylinders to obtain two equal halves of the final PHFD. This operation is realized without fluid. When tightening, the membranes are stretched over a low shoulder to ensure flat surfaces producing mechanical prestress (coefficient n in Eq.1). The next step is to put the two half-HFDs together immersed in a fluid to get a fluid-filled HFD without bubbles to ensure incompressibility. The tightness between the half-HFDs is assured by a o-ring. It is also possible to adjust the internal pressure, and consequently the flatness of the diaphragms, with a small tuning screw.

E. Experimental setup

The piezoelectric HFD was tested on an electro-dynamic shaker driven with a slow varying increasing frequency sweep from 100 Hz up to 250 Hz. Using an accelerometer and a dedicated control unit, the shaker is driven using a closed-loop control to ensure constant acceleration amplitude throughout the frequency range. The displacement and the velocity of the top diaphragm center are measured using a differential laser vibrometer. In parallel, the control unit drives a set of electrical load resistances (r l ). The generated power can be rigorously assessed varying the load between 0.1 MΩ and 12 MΩ.

III. THEORETICAL AND EXPERIMENTAL RESULTS

A. Model validation

Experiment using frequency sweep at 10 m/s 2 constant acceleration and various load resistances have been performed and the results (red grid in Fig. 7 and Fig. 8 ) are compared to simulation results (color scale). Fig. 7 shows the displacements with respect to the excitation frequencies and load resistances. The first resonant frequency of the system is 164.37 Hz. The displacement amplitude at resonant frequency is 71.45 µm.

The electric conversion has a negligible impact on the displacement, reflecting a weak electromechanical coupling. The simulations show a good correlation with the experimental results for the mechanical behavior.

B. Consideration of the parasitic resistance

Taking into account the parasitic resistance, the electrical behavior can be presented by Eq. 3 and the schematic representation in Fig. 9. The comparison between the experimental results and the simulation shows a good agreement. The resonant frequency and the optimal load resistance match perfectly. The simulated bandwidth is 29.31 Hz which is underestimated compared to the experimental bandwidth.

β ẋ -C 0 Velec - V elec r p = I (3)
I(t) V elec (t) r l C 0 Extraction circuit Piezoelectric transducer β ẋ(t) rp

IV. CONCLUSION

An original membrane generator matching low frequency vibration energy harvesting applications is proposed. A complete theoretical model has been developed. A prototype was realized and is thoroughly investigated aiming at the validation of the theoretical model. In this first prototype, custom P(VDF-TrFE) film is used. The experimental results show an unexpected parasitic resistance induced by the metallization protocol. This resistance is then added to complete the model. In th evicinity of the maximum power peak with respect to the excitation frequency and the electrical load resistance, the model gives good predictive results. The realized generator exhibits low resonant frequency of 164.37 Hz for low acceleration (10 m/s 2 ) and power density of 3.04 µW/cm 3 . Future approaches plan to improve the assembly procedure in order to increase the generated power and expect the theoretical power density of 78.98 µW/cm 3 .

Figure 1 :

 1 Figure 1: Architecture of the piezoelectric HFD

Figure 2 :

 2 Figure 2: Mechanical model of linear inertial generator

Figure 3 :

 3 Figure 3: Linear inertial generator equivalent circuit

Figure 5 :

 5 Figure 5: Bimorph diaphragm structure D. Prototype

Figure 6 :

 6 Figure 6: Bimorph PHFD prototype

6 Figure 7 :Figure 8 :

 678 Figure 7: Displacement comparison

Figure 9 :Figure 10 :

 910 Figure 9: Modified electrical model of linear inertial generatorThe piezoelectric transducer is represented by a voltage source in parallel of the capacitance (C 0 ) and the parasitic resistance (r p ). The extraction circuit is represented by a load resistance (r l ). The new modeling results incorporating r p effect are presented Fig.10.

Table I :

 I Main parameters of the PHFD

	Diaphragm properties (P(VDF-TrFE))
	Radius (m)	R	Thickness (m)	h
	Young modulus (Pa)	E	Poisson coefficient	ν
	Piezoelectric coefficient (C/N)	d 31	Relative permittivity ε 33
	Fluid Properties (Glycerin)
	Height (m)	H	Density (kg/m 3 )	ρ f
	natural frequency of a single diaphragm, it is thought that a
	simple static behavior. This approximation allows to propose
	an approached solution simplifying the model. Moreover, the
	electromechanical nature of the piezoelectric membranes is
	added in the model through the usual linear piezoelectric
	constitutive relations. The modelling strategy is developed
	using the following assumptions:

• Ideal geometry and homogeneous material properties • Incompressible and inviscid fluid • Elastic cylindrical diaphragms are considered • The diaphragm prestress is taken into account • Axially symmetrical motions • In-phase motion of the top and bottom diaphragms

Table II :

 II Model parameters relation

	Parameter	Symbol	Relation	Unit
	Fluid inertial mass

Table III :

 III PHFD properties

	Geometrical properties	Material properties
	R	10 mm	E	1 GPa
	H	40 mm	ρ f	1260 kg/m 3
	h	125 µm	d 31	-5 pC/N

ACKNOWLEDGMENT

This work was supported by the CEntre Technique des Industries Mécaniques (CETIM) and the Assemblée des Pays de Savoie (APS). The authors gratefully acknowledge this support and the help of the LGEF laboratory of INSA de Lyon.


F Huet

F Formosa

A Badel

New nonlinear vibration energy harvesters based on PVDF hybrid uid diaphragm

A low resonance frequency piezoelectric energy harvesting using a hybrid uid diaphragm (HFD) is presented. This paper describes the design, fabrication and measurement of such device for harvesting energy from environmental vibrations. The HFD consists in an incompressible uid conned between two thin piezoelectric membranes. The output voltage and power of the PVDF HFD are studied based on experimental and simulation results. Compared with conventional vibration harvester, this proposed solution is very simple and suitable for miniaturization and integration. 1

Introduction

An increasing interest in wireless sensors networks has been growing up over recent years. The implementation of communicating sensors networks in industrial application, transport or buildings oers possible improvements : productivity gains, reliability and energy performance are feasible by the exploitation of the extensive collected information (temperature, humidity, electric consumption ...). With the rapid advancement of low-power wireless sensor nodes and MEMS technology, microscale energy harvesting has attracted worldwide research interests for powering [START_REF] Paradiso | Energy Scavenging for[END_REF]. Vibration-based energy harvesting has been exploited using transduction mechanisms including piezoelectric, electromagnetic and electrostatic [START_REF] Beeby | Energy harvesting vibration sources for microsystems applications[END_REF][START_REF] Mitcheson | Human and Machine Motion for Wireless Electronic Devices[END_REF]. In order to extract the maximum power from the environment, the resonant frequency of a linear harvesting device has to match the dominant frequency of the excitation. Indeed frequencies of ambient available vibration sources are relatively low (normally less than 200 Hz) [START_REF] Roundy | A study of low level vibrations as a power source for wireless sensor nodes[END_REF]. As a result, many vibrationbased piezoelectric energy harvesters of centimeter scale have been demonstrated to operate at low frequencies [START_REF] Liu | A MEMSbased piezoelectric power generator array for vibration energy harvesting[END_REF]. This paper reports the modelling, fabrication and testing of a new piezoelectric uid-membrane structure for vibration energy harvesting. Compared to classical cantilever architectures, this approach takes advantage of the mechanical stretch of circular diaphragms. However, when miniaturized, they present high resonance frequency [START_REF] X.-R. Chen | Vibration energy harvesting with a clamped piezoelectric circular diaphragm[END_REF] and it hinders their direct use. The concept of hybrid uid diaphragm (HFD) consists in an incompressible uid conned between two thin piezoelectric membranes, it allows the realization of low resonance frequency membranes. Moreover, Pressure uctuation harvesting [START_REF] Mo | Experimental validation of energy harvesting performance for pressure-loaded piezoelectric circular diaphragms[END_REF] can be especially targeted.

Piezoelectric HFD concept

We propose here, an innovative architecture for vibration energy harvesters. Fig. 1 describes the geometry as well as the main geometrical parameters of the structure. The incompressible uid is used as an inertial mass which allows a drastic reduction of the resonance frequency to eectively harvest ambient vibrations. The height of the structure can be easily tunned to provide more or less inertia eect. In a previously published work, the mechanical behavior of a HFD was presented [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF]. As an example of the frequency reduction, the use of glycerin as the inertial uid allowed the resonance frequency of Kapton R (R = 10 mm, H = 10 mm and h = 100 µm) membranes to decrease from 1200 to 180 Hz. A comprehensive theoretical model was proposed in [START_REF] Formosa | Development of low frequency, insulating thick diaphragms for power MEMS applications[END_REF]. Based on the Ritz method, approximate solutions for the uid and the mechanicals diaphragms behaviours are set. This prior model is modied and extended here. Because, the nal investigated dynamic behaviour of the HFD in the vicinity of its rst resonance frequency is notably lower than the rst natural frequency of a single diaphragm, it is thought that a simple static behaviour of the latter can be used for approximating the solution. By doing this, the model is simplied. Moreover, the electromechanical nature of the piezoelectric membranes is added in the model through the usual linear piezoelectric constitutive relations. The modelling strategy is developed using the following assumptions : Ideal geometry and homogeneous material properties Incompressible and inviscid uid Elastic cylindrical diaphragms are considered The prestress eect of the diaphragms is taken into account Large amplitude deformation The analysis is restricted to axially symmetrical motions The analysis is restricted to in-phase motion of the top and bottom diaphragms As a result, a simple lumped model can be derived presented by the schematic representation in Fig. 2 and Eq.1-2. This system is constituted by a moving uid mass (m f ) tied to a vertical spring (k + n) standing for the exural diaphragm stiness and two springs (b) standing for the non-linear stretching associated to large amplitude motions. The piezoelectric conversion (α) is associated with the latter (Fig. 2a). The diaphragm deformationsinduce a voltage between the diaphragm electrodes. The input force is m i γ with γ the frame acceleration. The damping coecient (c) embodies all the mechanical losses (the viscous losses within the membrane and the uid dissipation). The piezoelectric generator behaves as a voltage source whose value is proportional to the product of the diaphragm bending and velocity (Fig. 2b).

m f ẍ + c ẋ + (k + n)x + 2bx 3 + α x h V = m i γ (1)
α x h ẋ -C V = I (2)
If the displacement is small, the nonlinear behavior eects are minor. Because a single layer of PVDF is used, during a exural motion, the same amount of PVDF lm is compressed and A HFD prototype has been realized aiming at the validation of the proposed model. Fig. 3a introduces a section of the HFD design. Fig. 3b shows the prototype realized in PVC. The rst step for the assembly procedure is the clamping of the cylinders to obtain two equal halves of the nal HFD. This operation is realized without uid. When tightening, the membranes are stretched over a low shoulder to get a at surface producing mechanical prestress (coecient n in Eq.1). The next step is to put the two half-HFDs together plunged in a uid to get a uid-lled HFD without bubbles to ensure incompressibility of the cavity. The tightness between the half-HFDs is assured by a seal that can be seen in Fig. 3a. It is also possible to adjust the internal pressure, and consequently the atness of the diaphragms, with a small tuning screw on one half-body (Fig. 3b). The lm used to make the diaphragms was home-made from a solution of Poly(VinyliDeneFluoride-co-TriFluoroEthylen) also noted as P(VDF-TrFE). It is a copolymer with 70 % of VDF and 30 % of TrFE with piezoelectric property. The mechanical and piezoelectric properties of the fabricated lms are similar to industrial ones. The main advantage of this choice is the tailor-made structures. The thickness h of the lms is 80 µm with variations of some microns. For the uid, we used glycerin for its high density. The piezoelectric HFD was tested on an electro-dynamic shaker driven with a slow varying increasing frequency sweep from 80 Hz up to 140 Hz. The rst resonance frequency of the system is 119.6 Hz. Using an accelerometer and a dedicated control unit, the shaker is driven using a closed-loop control to ensure constant acceleration amplitude throughout the frequency range. The displacement and the velocity of the top diaphragm center are measured using a dierential laser vibrometer. In parallel, the control unit drives a set of electrical load resistances (r l ). The generated power can be rigorously assessed varying the load between 0.8 MΩ and 14 MΩ. Several tests were performed for dierent acceleration levels (around 40 m/s 2 ) allowing the nonlinear behavior to be triggered.

Temporal results

The model highlights a frequency doubling voltage, which is clearly visible comparing Fig. 4a (displacement of the diaphragm center at 119.6 Hz) and Fig. 4b (voltage). Two dominant harmonics are shown performing the Fourier transform of the voltage signal (Fig. 4c). The rst is at the excitation frequency and the second is about the double (224.2 Hz). This rst frequency shows the voltage produced by the membrane bending and the second from the nonlinear behavior. This observation highlight a non-homogeneity in the membranes thickness,which was not taken into account in the model. Experiments using frequency sweep and various load resistances have been performed and the results (red grid in Fig. 5) are compared to simulation results (color scale). Fig. 5a shows the displacements with respect to the excitation frequencies and load resistances. The nonlinear eects associated with large displacement are very marked. The electric conversion has a negligible impact on the displacement, reecting a weak electromechanical coupling.

The displacement amplitude at resonance frequency is 376 µm. Fig. 5b shows the power, whose maximum is about 4.18 µW at the resonance frequency and for a load resistance of 2.88 MΩ. However the mechanical stiening eect allows to have a high bandwidth, about 41.58 Hz. The simulations show a good correlation with the experimental results near the peak of power, however for other results furthest from the peak, we nd the voltage error induced by the suspected membrane inhomogeneity. A complete model of a piezoelectric inertial generator has been developed and a prototype realized. Simulation results showed that the voltage from the exural mechanical eect is not negligible when the PVDF lm is not perfect, but in the immediate area of the maximum power, the model is rather predictive and it correlates well. The generator performances are low but it has a very wide bandwidth and the rst resonance frequency is about 100 Hz. Now the model can be used as a design tools to aim at a set requirements from a given applications. A future approach is to double the improved PVDF lms. By reversing the polarity, this will fully exploit the voltage from the diaphragms exion allowing performance gain.

Figure 1 .

 1 Figure 1. Architecture of the piezoelectric HFD

Figure 2 .

 2 Figure 2. Non-linear inertial generator a) mechanical model and b) electrical model

Figure 3 .

 3 Figure 3. HFD a) design and b) prototype

  p l a c e m e n t [m m ] E x p e r i m e n ta l d i s p la c e m e n ta g e [V ] E x p e r i m e n ta l v o lta g e

Figure 4 .

 4 Figure 4. Experimental results a) displacements, b) voltages during the time for 119 Hz at 40 m/s 2 and c) voltage Fourier transform of the voltage signal

5 F 5 FFigure 5 .

 555 Figure 5. Experimental and simulation results comparison, a) displacements and b) power

Table 1 .

 1 Table 1 details the lumped parameters, their units and relationship to the geometry and material properties. Model Parameters

		Parameter	Symbol	Value	Unit
		Fluid inertial mass	m f	See [8] f (ρ, H, R)	kg
		Fluid excitation mass	m i	1 3 HπρR 2	kg
		Flexural stiness	k	-16Eh 3 π 9R 2 (1-ν 2 )	N/m
		Large deformation stiening coecient	b	Ehπ(-7505-4250ν+2791ν 2 ) 19845R 2 (1-ν 2 )	N/m 3
		Electromechanical coupling coecient	α	2d 31 Ehπ 3(1-ν)	N/V
		Diaphragm capacitance	C	ε 33 ε 0	πR 2 h	F
		Prestress exural stiness	n	Measured	N/m
		Damping	c	Measured	N.m/s 2
	2.2	Prototype


F Huet

F Formosa

A Badel

Hybrid fluid diaphragm structures for vibration energy harvesting applications

Keywords: Vibration energy harvesting, membrane, fluidstructure I

This paper deals with the design of a compact, generic membrane structure suitable for vibration energy harvesting (VEH). A hybrid fluid diaphragm architecture is proposed. It allows the resonant frequency to match the classical low frequencies ambient excitations to be harvested. A semi analytical model has been developed and validated so the proposed structure can be easily tuned for a given application. The next step will consist in designing a piezoelectric structure which will be integrated to the HFD.

INTRODUCTION

Diaphragm geometries are interesting structures for VEH. Numerous sensors are based on diaphragms. Moreover, they are advantageous for in vivo energy harvesting applications [START_REF] Beeby | Energy harvesting vibration sources for microsystems applications[END_REF]. The VEH from air flow streams usually uses membrane structures [START_REF] Jing | Vibration characteristics of micromachined piezoelectric diaphragms with a standing beam subjected to airflow[END_REF].

However, most of the proposed membrane energy harvesters present high resonance frequencies which prevent these systems to be used in common applications [START_REF] Ericka | Energy harvesting from vibration using a piezoelectric membrane[END_REF]. Indeed, the later show frequencies lower than a few hundred hertz whereas the membrane geometry generally induces high resonance frequency. We propose here a new concept of a hybrid fluid diaphragm (HFD). It consists in an incompressible fluid confined between two thin membranes. This architecture aims to be easily manufactured with standard microelectronics methods. Figure 1 described its main features as well as its geometric parameters. The inertia effect of the fluid allows obtaining operation frequencies significantly lower than for a single membrane. Moreover, in addition to the membrane mechanical parameters (geometrical and material properties) the fluid characteristics (height of the cavity, density) can be used as design parameters for optimization.

Fluid structure interaction problems are extensively studied in engineering. The particular case of two circular plates vibrating in contact with fluid has recently been studied by Jeong [START_REF] Jeong | Free vibration of two identical circular plates coupled with bounded fluid[END_REF]. Weak solutions can be found for these problems [START_REF] Grandmont | Weak solutions for a fluid-elastic structure interaction model[END_REF]. The accuracy of the solution is related to the number of function used for approximation. As an example, the solution is sought using Bessel decomposition in [START_REF] Jeong | Free vibration of two identical circular plates coupled with bounded fluid[END_REF]. In the present application, only the first in phase mode (IPM) also called the piston mode is of interest and a reduced simple model can be deduced. In addition, the filling of the container can be performed at a given pressure which induces static deformations of the membranes. For thin membranes this prestress is associated to tension effect and raises the natural frequencies. This can be seen as an additional control parameter for frequency tuning.

In this paper, a simple fluid structure model is proposed. The model is compared to experimental results for which the static prestress effects due to the fluid filling process have been taken into account. Finally a piezoelectric structure to be added to the HFD is presented.

II. MODELLING

A. Fluid model

The modeling of the fluid structure problem is based on potential fluid flow theory. The filling fluid is assumed to be an incompressible and inviscid liquid. With the liquid motion assumed to be irrotational, the fluid velocity is defined as:

( , , ) U r z t     
in which (r, z, t) is the velocity potential. At the side wall, the velocity of the fluid is parallel to the wall and at the bottom and the top of the cavity, the fluid velocity matches the diaphragm's one.

B. Diaphragm model

Elastic cylindrical diaphragms are considered here and prestress from static deformations of the diaphragms are taken into account. Moreover, we consider large amplitude vibrations of the mechanical structures. The main assumptions used for the modeling of the membrane are the following:

-Membranes are submitted to mechanical solicitations such as their strains are axisymmetric only. -Plane stress is considered (shear stress is neglected).

-Rotational kinetic energy is neglected.

-The midplane displacements are in the z direction and denoted w b (r,t) for the bottom and w t (r,t) for the top diaphragms (see Fig. 1). -Top and bottom membranes are supposed to be identical.

-The diaphragm is clamped at the wall.

C. Fluid structure model

A Ritz method is used to find approximated solutions to the coupled fluid-structure problem. Provided that the spectral basis is limited to the first mode, this approach gives an analytical model of the nonlinear fluid coupled dynamical behavior. In the case of one single mode, and assuming small deflection, the first IPM frequency of the HFD was calculated as [START_REF] Beeby | Energy harvesting vibration sources for microsystems applications[END_REF], where :

1 1 1 1 ( )/( ) 2 fluid f k n m m p = + + (1)
In which k 1 and m 1 are the generalized mechanical stiffness and mass respectively. Eq. ( 1) underlines the inertia effect of the fluid m fluid is a function of its density, the height and diameter of the cavity. The prestress (n 1 ) increases the effective stiffness.

III. MODEL VALIDATION

Experimental validation was performed. Prototypes of HFD were made from a 25 µm thickness Kapton ® film glued on a rigid aluminum frame. The cavity was filled with water and sealed. The aluminum frame is 9.2 mm height. Two cavities diameters were tested (7 and 8 mm). The HFD can be clearly seen on the picture of the experimental mockup given in Fig. 2. The HFD is excited by an electrodynamical shaker and the displacement of the top membrane is measured using an embedded laser optical sensor. The experimental identified resonance frequencies values are reported in Table 1. The fluid inertia effect is very important and allows the natural frequencies to dramatically lessen by a factor of about 10. As expected, the frequencies decreases when the HFD diameters is larger.

Though high quality factor of the HFD was not sought, these tests exhibited pretty sharp resonance peaks, which is favorable for vibration energy harvesting purpose. With optimized assembling process, the quality factor is expected to increase.

IV. CONCLUSIONS AND PERSPECTIVES

A HFD concept suitable for micro fabrications techniques has been proposed. Based on fluid-structure interaction, a comprehensive model was proposed and validated. It is able to predict the important frequency characteristics for the first IPM relevant for VEH applications. Experimental prototypes have been obtained and tested. First results are in accordance with the model predictions. Based on this structure, electromechanical conversion will be integrated. Preliminary designs of spiral shaped piezoelectric material to be glued on the membranes are identified as potential favorable geometries. Figure 3 presents the concept of HFD generators clusters which would be obtained through batch-process fabrication.

Figure 1 :

 1 Figure 1: Hybrid fluid diaphragm schematic

Figure 2 :

 2 Figure 2: Experimental setup for model validation

Figure 3 :

 3 Figure 3: Array of HFD generators