Design and optimization of lightweight bending strain energy harvesters using irradiation cross-linked polypropylene ferroelectret

In this work, a concept for bending strain energy harvesting with lightweight ferroelectrets subject to metallic host structures is proposed. The energy harvester (EH) works with the ferroelectret irradiation cross-linked polypropylene (IXPP) and uses the piezoelectric δ 33-mode by transferring in-plane bending strain energy of a host structure to a compression of an IXPP stack. This is achieved with a simple and lightweight transmission made of structural steel. In this way, a high ratio of generated power per used EH mass can be achieved, being the main design goal of this paper. To demonstrate this, miniature EHs as unit cells for larger EH systems are investigated. The strain magnitude of the excitation in an aircraft wing of 4.5⋅10−5mm−1 at 1.5 Hz from previous work is partly taken into account for calculations. In an analytical modeling approach, the energy conversion abilities of the presented concept is compared to concepts using lead zirconate titanate ceramics to stress the usefulness in cases where strain energy is the prominent source of vibration energy. Further, an optimization algorithm is presented for a static and a dynamic case without a host structure and for a dynamic case with an aluminum host structure. The optimized power output and power output per total EH mass for the three cases is calculated to 1.52μW and 271μWkg−1 , 0.7 mW and 80 mW kg−1 as well as 6.53μW (at 1 N excitation magnitude) and 10.8mWkg−1 (at 1 N excitation magnitude) respectively. Finally, experimental results for case three are presented to validate the model of the proposed and optimized EH topology up to 500 Hz. The results further show a good mechanical reproducibility of the measured transfer behavior and a fairly good reproducibility of the mechanical-electrical results due to deviations in material properties. A comparison of the model with the experimental results shows a good agreement. Therefore a linear ferroelectret model appears to be suitable to predict the system behavior in lower and higher frequency ranges as well as for high ferroelectret material strains. The optimized EH provides a comparably high ratio of power output per mass when added to a structure like an aircraft which is shown in a comparison to other research works. The performance in a real application can be further improved to 52 mW within a 1 m2 area using a clustering approach, discussed in the paper. Large deformations of lightweight structures like aircraft wings at low and high frequencies can thus be exploited to provide enough electrical power decentrally for low-power consumers.


Introduction
In the context of climate change, as a major challenge of modern society, flight traffic has to become more and more ecologically friendly.Based on calculations of Lee et al [1] it is causing 3.5% of the net anthropogenic effective radiative forcing.Therefore every aspect of an aircraft having an influence on the fuel demand has to be considered in designing modern aircrafts.Especially the weight of an aircraft has an influence on its fuel consumption.Therefore a central task is to reduce weight in every possible way.Energy harvesting (EH) describes the possibility to generate an electrical power decentrally by only using energy of the ambient environment.An advantage of this in the context of aircrafts is the replacement of cables with their associated weights or of batteries with their need of maintenance or recharging, while maintaining the ability to power devices like sensors [2].The wing is of special interest for energy harvesting, since a large part of it is fairly remote of the central energy supply and therefore a replacement of cables potentially saves a considerable amount of weight.Prior work [3,4] has shown that strain magnitudes, that could be used by the EH, are especially high in a low frequency range, apart from the eigenfrequency of the EH itself.Because of this, it can be useful to design an EH without an additional resonator mass for very low frequency applications to minimize its total mass.The focus of this work is on piezoelectric energy harvesting, that shows the advantage of a high power density of the used material [5].Piezoelectric energy harvesting is a research field covering a wide variety of use-cases and mechanisms with different piezoelectric materials.The most commonly used piezoelectric materials are piezoceramics, due to their high coupling coefficients and charge constants [5].Major drawbacks however are their brittleness, their high stiffness, that is not always desired and a relatively low internal resistance leading to nonnegligible parasitic current between the electrodes for lowfrequency applications.Further, many piezoceramics contain environmentally harmful substances like lead, which complicates recycling and disposal [6].Flexible piezoelectric materials are piezoelectric polymers, like polyvinylidene fluoride that however provides much lower charge coefficients than piezoceramics [7,8].Ferroelectrets, also called piezoelectrets, that are subject of this work were first developed at Tampere University of Technology in Finland [9].Already at that time the suitability for energy conversion was predicted.Since then, many different polymers were investigated as materials for ferroelectrets.Mostly polarized cellular polypropylene (PP) was used [10][11][12][13].To improve properties like thermal stability and linearity of charge coefficients, several works covered the usage and investigation of irradiation cross-linked PP (IXPP) [14][15][16].Other commonly used ferroelectret materials are based on fluorocarbon polymers [17][18][19][20][21].In terms of sensing and energy harvesting, especially fluoroethylenepropylene (FEP) ferroelectret films with cross-tunnel [22,23] and parallel-tunnel [8,[24][25][26] structure with comparably high δ 31 coefficient are promising.Ferroelectrets provide charge coefficients δ in the same order of magnitude like piezoceramics and furthermore provide much higher voltage coefficients g [27].This leads to a high figure of merit (FOM) in terms of the product of the two coefficients, which is advantageous in terms of energy harvesting [28,29].Mechanisms to increase the FOM using piezoceramics were discussed e.g. by Kim et al [28] using a structure for strain amplification and a more even distribution of the stresses.Other examples are mentioned in [30,31], where large strains of potential excitations are transformed to small strains with large forces, to fit the excitation to strain limitations of piezoceramics.A mechansim using more than one load direction of a piezoceramic material at the same time is mentioned in [32], exploiting the properties of an auxetic host structure.Since the used piezoceramics are very stiff, the mechanism has to become stiff as well in the mentioned works.This entails the need for a high material volume with its associated weight.
Several energy harvesting concepts using ferroelectret materials were discussed in the last years.Pondrom et al [33,34] and Zhang et al [35] proposed concepts of single and multilayer ferroelectret EHs with a seismic mass using the material IXPP, as it is used in this work.Anton et al [6] used PP ferroelectret samples by Emfit, Corp. subject to a strain excitation to provide several µW of continuous power.Further, Zhang et al [8] showed a ferroelectret nanogenerator using FEP ferroelectret with a tunnel structure.Ben Dali et al investigated the behavior of the latter material clamped mechanically in parallel to a cantilever beam with a tip mass, excited by a base acceleration [36].Prestressing effects were exploited in a subsequent work [37].
Currently no work from the literature is known, optimizing a ferroelectret EH for a bending strain excitation, exploiting its δ 33 -effect and calculating its power output subject to a realistic mechanical load.In this work, a mechanism is proposed, using an inverted effect of [30] or similar publications by transferring a low strain with a high stress to a high strain with a moderate stress, that fits the used ferroelectret material.Through the transmission, the high longitudinal δ 33 constants of the material can be exploited while multiplying the strain at the same time.A major advantage to prior research is created through the very light way this EH can be designed and hence its high power per mass that can be achieved even in a quasistatic use.It can further be shown that if space limitations are not an issue, the proposed concept leads to a higher power output per mass than piezoceramic patches.To investigate the concept, this work optimizes miniature EHs as unit cells for larger EH systems.Maximizing absolute power can result in high mass systems.Thus maximizing power per EH mass is the relevant goal for lightweight design.Fixed geometry parameters reduce the design parameters in this work and facilitate experimental validation and manual processing.To find a specific EH design, different geometric and material parameters of the system have to be considered.To save weight, the maximization of the power output per total mass is suggested in this work.Based on a parametric model, representing a permutation of the proposed EH design, an optimization process is conducted similar to [38].A comparison to other works is conducted.The resulting EH unit cells have a low absolute power due to their small size.This condition can finally be overcome by the design of EH arrays consisting of multiples of the mass-optimized unit cell EHs.In this way, the optimized power output per mass stays the same but the total power output is multiplied by the number of unit cells in the array.The novelty of the work compared to the state of the art can thus be summarized in the form of the following points • The investigation of the usage of ferroelectret EHs for a bending strain excitation similar to a piezoceramic patch has not been investigated yet.In this regard, no research has quantitatively shown the advantage of ferroelectret EHs to piezoceramics for a strain excitation.• The application of optimization algorithms to improve the ratio of power per mass of ferroelectret EHs is described for the first time in this work.• The topology and power output of an array design of the ferroelectret EH is firstly described as an extension to concepts of other works [25,39] by the authors.

Piezoelectric energy harvesting
Piezoelectric energy harvesting describes the conversion of mechanical vibration energy into electrical energy using a piezoelectric material.The resulting electrical power can be conditioned and used in an electrical consumer device.For an alternating current as the output of a harmonically excited piezoelectric material apart from resonance with the frequency ω and with an optimal load resistance R opt , the rms power output can be calculated as showing a linear relation to frequency and capacitance but a quadratic relation to the electric voltage u el,R in the load resistance.R opt = 1/ωC p describes the optimal load resistance, which is typically used in the context of energy harvesting to compare EHs at different frequencies.The electric properties of piezoelectric materials are coupled with their mechanical properties via the piezoelectric constitutive equations and with compliances s ij , piezoelectric charge constants δ ij and permittivities ϵ ij .S is the strain tensor, T the stress tensor, E the electric field and D the electric flux density.For the use of the piezoelectric material in δ 33 -mode, equations ( 2) and (3) simplify to and Eliminating T 3 in equation ( 5) with equation ( 4) leads to the relation With ϵ T 33 − δ 2 33 s33 = ϵ S 33 (Index S for constant strain), equation ( 6) can be simplified to The relations D 3 = q el Ap and E 3 = u el tp are obtained using the surface area A p and the thickness t p of a piezoelectric patch.With the case of open electrodes (charge q el = 0), equation (7) becomes (8) with the Young's modulus of the piezo patch in thickness direction Y 3 .By means of a loss-free voltage divider, the voltage in an optimal load resistance calculates to With equations ( 1) and ( 9), the relation (10) for the power in an optimal load resistance results.As shown, the voltage is proportional and the power output is quadratically proportional to the local strain in the material, assuming all the other values to be constant.Because of this, an enhancement of strain in the material should be considered first when designing an EH for a specific usage scenario.Vibrating structures like aircrafts usually have the largest vibration strain Comparison of PZT and IXPP.Left: under same strain conditions (limit: maximum allowable strain of PZT); Right: same stress conditions (limit: maximum allowable stress of IXPP).W 1,i + W 2,i represents the total mechanical energy of a load cycle and W 1,i the electrically stored potential energy between the electrodes in the respective case.
amplitudes and energy at low frequencies (<100 Hz) [40].It therefore appears reasonable to use EHs in this low frequency range.In comparison to piezoceramics, the main differences of ferroelectret material properties are much higher compliances s 3 = 1/Y 3 and much lower permittivities ϵ 33 , while the charge constants δ 33 are in the same order of magnitude (cf table 1).Since the electromechanical coupling coefficient of a piezoelectric material in δ 33 -mode is defined as the general energy conversion efficiency in the ferroelectret material itself is lower than in piezoceramics.This is because they have a much lower elastic modulus (about five orders of magnitude lower), while the dielectric constant is only about three orders of magnitude lower.W 1,i + W 2,i represents the total mechanical energy of a load cycle and W 1,i the electrically stored potential energy between the electrodes, with and with equation ( 11), the relation follows. Figure 1 shows a comparison of lead zirconate titanate (PZT) and IXPP ferroelectrets for two scenarios.The left pictures show the behavior under same strain conditions where the maximum allowable strain of PZT is taken as a constraint.
In this case the PZT ceramic is performing better than the IXPP ferroelectret in terms of electrical energy conversion.In case two however (right pictures) at a constant stress where the maximum allowable stress of IXPP is taken as a constraint, the IXPP shows a much larger electrically converted energy than the PZT ceramic.This corresponds to the mentioned high FOM, since charge and voltage coefficients are defined with respect to the mechanical stress.Furthermore, a major advantage of ferroelectrets is their easy application and the resonant use in lower frequency ranges in connection with comparably small resonator masses.These can lead to increased material strains, that are beneficial to exploit the advantage of a high FOM.In use-cases where only a low strain excitation is present, a mechanical transmission can be used instead or additionally to make use of the advantages of ferroelectrets.

Concept of the EH
The ferroelectret material investigated in this work is IXPP.The material samples were provided by Tongji University in Shanghai, China.A sample is shown in figure 2. As described by Pondrom [34], it can be advantageous to use ferroelectrets in stacks to increase the equivalent compliance and therefore the effective piezoelectric coefficient.Since the ferroelectret sheets have a thickness of only around 180 µm this can be realized with very small amounts of space.For the use in aircrafts, the primary goal of the optimization of an EH is to maximize the power output per EH mass introduced into the aircraft As the EHs are comparably small, volume is not a major issue.
The strain energy of a bending mechanical structure is only available in the in-plane-direction.In a former work, the IXPP sheet was loaded in the δ 31 -mode as the easiest possible way of the patch to be attached to the structure [3].This is analogous to the use of a piezoceramic patch, but it is not the mode, that leverages the energy conversion properties of IXPP optimally.Furthermore, the strain is much lower than the strain the ferroelectret can withstand.Therefore, a mechanism is used in this work that converts the in-plane strain energy into a compression of a ferroelectret stack with a transmission rate i >> 1.
The basic principle is shown in figure 3.
The strain energy, represented by a force f x , is converted by a comparably stiff mechanism into a compression of the stack over a translatory bearing.Since a translatory bearing is relatively heavy and complicated to apply, a symmetric assembly with arms on both sides is suggested to compensate all resulting forces in x-direction and y-momentums.Further, the mechanism should be as simple as possible, i.e. made of as few parts as possible.The mechanism of the EH should consist of as few different materials as possible besides the ferroelectret to ensure a relatively easy recyclability and manufacturability.Since the EH is used in aircrafts, the goal is a maximization of equation ( 14).This work covers miniature size EHs as optimized unit cells of larger EH systems.The absolute power of an EH scales with the amount of energy converting material.The maximization of the absolute power could result in EH systems with high power output but unnecessary high mass.Therefore, the maximization of the power output per EH mass instead of absolute power is considered the relevant optimization goal for lightweight design.Since different EH sizes can have the same optimized ratio of power per mass, it is reasonable to set selected geometry parameters as constant in order  to reduce the optimization parameter space in its dimensions.As a size constraint in this work, the edge length of the ferroelectret stack is fixed to 1 cm.This is especially useful for the experimental validation of the proposed concept in order to realize stacks of many layers out of one ferroelectret sample.With this size, the samples can also still be processed by hand.Since the resulting EHs are small, the absolute power values are relatively low.The relevant metric P el,rel can however be optimized in the same way as with a larger EH.

Modeling approach
In a first step, an analytical model is described to show the influence of the main design parameters on the power output.Furthermore, a quantitative comparison to PZT is conducted to stress the usefulness and novelty of the presented concept.The second step is a parametric finite element model that additionally includes design parameters for the detailed geometry and therefore the quantitative influence on the total mass and the Young's modulus in 3-dir.0.7 MPa [35] dynamic behavior.For both approaches, small deflections and linear material properties are assumed similar to [3].For the IXPP ferroelectret, the properties shown in table 1 are assumed using a transversely isotropic material model.

Analytical model
For an analytical model, the assumption is made, that the mechanical reaction of the EH on the host structure can be neglected because of the size and consequently the stiffness difference of a lightweight structure like an aircraft wing and the EH.Therefore, the assumed excitation of the EH is a strain with an amplitude Ŝx in x-direction, harmonically applied with a frequency ω.
A simplification of the ferroelectret material as well as the mechanism as spring elements (cf figure 3(c)) appears reasonable.The longer the mechanism, the lower its stiffness c mech becomes.The arrangement as a stack leads to an increase of the overall compliance of the used ferroelectret material.This results in a reduced force to reach a desired total stack compression.On the other hand, the ferroelectret strain is independent of the number of sheets n p in the stack when the force stays constant.Assuming the material properties of all the ferroelectret sheets to be equal, the equivalent stack stiffness can be expressed by Further, the assumption is made that the emerging strain magnitude in the host structure is constant over the application surface of the EH.A constant strain magnitude Ŝx in the structure leads to a deformation of the point A of the mechanism in xdirection of ∆x A = Ŝx l mech (17) where l mech is the projected length of the mechanism arm onto the x-axis.The x-displacement of point A leads to a zdisplacement of the point B being the contact point between press and ferroelectret stack.The distance of point B to the torsional spring is considered as negligibly small.By neglecting c ϕ , which is a good approximation for thin mechanism parts, the compression force of the piezoelectric stack can be derived from (18) with h p = n p t p .In equation ( 18), both forces f x,A and f z,B are unknown.The system of equations can be formulated using a unity displacement approach.With equations ( 18) and ( 19), the relation for ∆z ∆x A (20) can be derived.With ∆x A from equation ( 17), the strain in the piezoelectric stack can be calculated as The mechanical stress in the mechanism on the other hand can be derived using the displacements of points ∆x A and ∆z B as (22) Using equation ( 13), the electrically converted energy per load cycle calculates to The total mass of the EH can be calculated as With equations ( 23) and ( 24), the total electrical converted energy per used EH mass in a load cycle is obtained as As a measure to compare the concept to results of piezoceramics at the same strain, equations ( 23) and ( 25) can be used as a frequency independent formulation of equations ( 1) and ( 14).The materials in the mechanism have a yield strength that has to be considered in the design process.The maximum allowable strain of IXPP is about 5% to 10% [42].The latter value is used as a characteristic constraint in this work.A ferroelectret length of l p = 0.01 m and a mechanism cross section area of A mech = 2 • 10 −8 m 2 are chosen as A mech is later shown to become small.The yield strength of structural steel is shown as a horizontal line.
Results in terms of equations ( 23), ( 25), ( 22), ( 21) and ( 24) are shown in figures 4(a),(b),(d),(e),(f) respectively) for a mechanism area of A m = 1e − 8 (which is later shown to be a realistic order of magnitude), a number of piezoelectric sheets of n p = 20 and a strain excitation of Ŝx = 4.5 • 10 −5 m m −1 as presented in [3].The results are shown for the proposed topology using the introduced ferroelectret material.In comparison, results for the same topology using a PZT ceramic in δ 33 mode and a single PZT ceramic patch in δ 31 mode with same area are presented.The greyed out sections of the graphs represent the theoretical results that would be obtained if the ferroelectret strain limit was exceeded.The emerging strain in the piezoelectric material is therefore shown in the calculation shows, that the electrically converted energy and especially the energy per mass in a load cycle can become orders of magnitudes higher when using the proposed concept as when using a PZT ceramic, which is remarkable since PZT provides a much higher coupling coefficient than IXPP.Furthermore, it is shown that the stress limit of the mechanism material cannot be reached with the pure strain excitation used.Because of this fact and the low stiffness of the IXPP stack in the in-plane direction, the structural integrity of the metal structure is not significantly affected by the EH.By increasing the number of ferroelectret sheets n p , the electrically converted energy increases and by increasing the mechanism length l mech also the electrically converted energy per mass increases.Figure 4(f) also shows, that the ferroelectret concept is naturally much lighter than the same topology using PZT.The calculation does not consider low frequency electrical losses of piezoceramic patches or strain losses caused by a lower stiffness adhesive.Both loss types do not account for the presented ferroelectret concepts.The results can therefore be considered conservative.They motivate the use of the proposed EH concept instead of PZT-concepts.For a dynamic case, the strain amplification and therefore the power output of the EH strongly depends on the mass distribution.Hence, the detailed geometry has to be optimized.In this work this is not done analytically but in terms of an optimization algorithm using an FE model in the next part.

Parametric finite element model
As pointed out, the detailed design strongly depends on the excitation amplitude.The latter differs considerably depending on different load scenarios like smooth or rough cruise or under gust excitation [3,43,44] and further on the exact positioning of the EH.For this work, the excitation in [3] is  applied as it leads to a fairly conservative and therefore more reliable estimation of the power output.Under consideration of the yield strength, the analytical model suggests designs with a small angle, small piezo length, high mechanism length and a high number of sheets in the stack.For the design of the EH, conditions like strain losses and mass distributions, as well as maximum allowable material strains and stresses have to be considered.The detailed geometry especially has an influence on the dynamic behavior.Figure 5 shows the basic topology of the concept discussed in section 3.
It consists of two equal mechanism arms that are symmetrical to the y-z-plane to be able to work without a translatory bearing.The arms are subject to a strain excitation and transmit a z-displacement onto the ferroelectret stack beneath.The general case assumes, that the host structure observes no considerable change in deformation through the mechanism so the excitation can be directly modeled as a boundary condition.The ferroelectret stack is chosen to have a comparably low length to minimize the needed stress for compression.Between the mechanism arms and the stack, a steel plate is used to transmit the stress of the mechanism uniformly into the ferroelectret stack.Later results show that the latter one can be designed fairly thin for a quasistatic use.

Three cases for optimizing the parametric topology are investigated:
1. a quasistatic case without a host structure 2. a dynamic case without a host structure 3. a dynamic case with a bending aluminum plate as a host structure.Output power per mass P el /m EH For all cases, the parametric topology shown in figure 5 is used.For case 3 it is used together with a 0.14 • 0.11 • 0.001 m 3 aluminum plate in a four-point bending test condition.The parameters are chosen to be as few as possible and in a way that they can be changed independently from each other.The angle α is not used as an independent input parameter because it has a nonlinear influence on the mechanism length l mech when n p stays constant.Instead, l mech and n p together with the thickness of the pressure plate t press are used as input parameters.In table 2 all the independent parameters are shown.Further, dependent parameters and model outputs are shown in tables 3 and 4. In the latter one, especially the power output per mass P el,rel is of special interest as the main objective of optimization.
For the parametric finite element model, volume elements (solid 186 for metal parts and solid 226 for the ferroelectret stack) are used.An automatic mesh size always ensures 2 elements in thickness direction of the thin EH components.The ferroelectret stack is modeled by a pattern of patches.The pattern counter is set as a parameter which is used as the number of (additional) piezoelectric sheets.Figure 6 shows a meshed model of an example topology with aluminum plate (case 3).Using the geometry data of the setup, an FE model of the EH is set up, discretizing the aluminum plate and the IXPP ferroelectret by means of hexahedric elements with quadratic shape functions.A meshed geometry is shown in figure 6.
The equation of motion for the mechanical system can be expressed in its most general form as with the mass matrix M, the damping matrix B, the mechanical stiffness matrix K xx , the vector of deflections x(t) and the external force vector f(t).Expanding the system by additional degrees of freedom ϕ(t) for the electric potential on all nodes of the piezoelectric material, the system of equations results.K xϕ (δ) is the piezoelectric coupling matrix, K ϕϕ (ϵ) the electric stiffness matrix and q(t) the vector of external charges applied to the electrical degrees of freedom [45].The IXPP properties are defined by the data in table 1.The force excitation is induced with the help of an external point, coupling all degrees of freedom of the edges of excitation.The FE model regarding the generated voltage on the IXPP electrodes is shown in figure 6 when exposed to a static force.For the damping matrix B, the model of superposition of modal damping matrices is used as described in [46].For the simulation, the systems matrices are exported from the FE-Software to a numerical solver.In the solver, a model reduction of the piezomechanical structure and a conversion to a state-space representation is conducted as described in [45].For all simulations, the voltage of the stack in terms of equation ( 9) as a parallel circuit is used to calculate the maximum power output in an optimal load resistance.

Optimization of the EH
For the optimization, the different cases presented in section 4 are considered.The general workflow is shown in figure 7. Starting with a basic concept in the form of independent parameters and parameter relations, a parametric geometry is set up.The algorithm starts with a set of parameters defined by a sparse grid initialization algorithm to obtain a current design for a static or dynamic analysis.The result is used to initialize and update a response surface that can be seen as a surrogate model of the optimization problem.After an update, the algorithm checks, if a certain convergence criterion is achieved.If not, the algorithm updates the parametric geometry with another permutation of parameters.If convergence is reached, the response surface can be used for several kinds of optimization algorithms.In this work a multiobjective genetic algorithm and a mixed-integer sequential quadratic programming algorithm are used to find the best solution.The optimization also accounts for constraints like maximum strains or stresses of the materials used.In figure 8, a part of the response surface for case one is shown.The best result obtained shows a power output of 1.52 µW at a weight  271 µW kg −1 of 5.6 g.The parameters used as well as the resulting power output are shown in table 5.For the dynamic case two, the topology differs due to the strain amplification by a change of mass distribution.One part of the response surface is shown in figure 9.The best result obtained shows a power output of 0.7 mW at a weight of 8.2 g.The parameters used as well as the resulting power output are shown in table 6.This result however is just theoretically possible, since the excitation magnitude might not be as high as at 1.5 Hz in higher frequencies.Remarkable is that the resonator mass is automatically adjusted via t press by the algorithm.Results for case three with an aluminum plate are shown in figure 10 for all sheet numbers starting from 1 to 30.In this case the mechanism length of one arm is set to the maximum length, hence 0.045 m since it leads to an overall length of 0.1 m.The best result shows a power output of 2.4 µW at a total EH weight of 0.6 g.The corresponding parameter values are shown in table 7.

Experimental validation
The experimental investigation is used to show that the material model is valid in the frequency range of interest and subject to increased strains induced by the mechanism.Firstly the sample preparation and the experimental setup are explained, that are derived from the model shown in section 4.After that, the experimental results are shown with an updated model of the EH.

EH sample preparation
For the assembly of the EH system, the IXPP sheets with a 60 • 60 mm 2 edge length are divided into a 10 • 10 mm 2 -grid as shown in figure 11.The positive and negative electrode are marked to be able to identify them in the cut state.Afterwards, the sheet is cut along the grid to create a continuous strip of IXPP with a width of 10 mm.In this way positive and negative electrodes stay connected for the whole grid.In the next step the strip is folded like shown in figure 11 on the right to enable a stack of sheets connected in parallel.The electrical connection in the stack is verified for every stack in terms of the electrical resistance measurement between the lowest and the uppermost sheet for both the positive and negative electrode.Due to the elastic properties of IXPP ferroelectret, the material and electrodes stay intact after folding in most of the cases.Some samples had to be discarded however due to electrode damage.A connection cable for the positive electrode is attached in the lowest fold of the stack without a gluing in the fold itself to prevent an influence on the mechanical properties of the stack.The mechanism arms are made of structural steel with a width of 1 mm each.They are bent to meet the parameter values shown in table 7. To connect the arms to the plate, an epoxy glue is used, that is suitable for an aluminumsteel connection.The cable for the negative electrode cable (left in figure 12) is attached to the center of the aluminum plate after an insulation varnish is brought to the respective area to prevent a connection of the IXPP electrode to the aluminum plate.The insulation is also applied to both sides of the pressure plate.Finally the stack is clamped between mechanism arms and aluminum plate, as shown in the right picture in figure 12.For the experiments, four IXPP stacks (numbers 2, 3, 4 and 5) are combined with three mechanisms (1, 2 and 3), so a total amount of 12 permutations of EHs is investigated.

Experimental setup
The experimental setup used in this work is shown in figure 13.
The bearings and the contact edge of force excitation are made of aluminum rolls.The rolls for the excitation are mounted on a base plate with a mass of 0.93 kg.An impedance sensor underneath the base plate measures acceleration and force.Furthermore the electric voltage of the ferroelectret is measured.With an FFT-analyzer, complex transfer functions of the system are measured, more specifically the ones from force to acceleration and from force to voltage in open state and from force to charge in short circuit state are of interest.The measuring chain is shown in figure 14.
An electrodynamic shaker excites an aluminum plate with an applied EH with an acceleration of the base (cf figure 13).This causes a force f on the edges of excitation that bends the plate.The bending causes an electrical voltage u el and/or depending on the electrical network connected an electrical charge q el between the electrodes.The electrical voltage is measured with an electrometer (Keithley 6517B), the charge is measured with a charge amplifier (Kistler 5015A) and the frequency response functions are measured in an FFT analyzer (Ono Sokki DS2000).

Results and modelfit
The experimental results for equations ( 28)-( 30) are shown in figures 15(a)-(c) in the frequency range up to 500 Hz.Further for all measurements a matrix of the FRAC values is shown in the respective figure taking into account the complex transfer functions Z i in the way of For the mechanical transfer behavior in terms of equation ( 28), a very good linear dependency between all the measurements is reached, therefore the main deviations of the mechanicalelectrical transfer behaviors can be assigned to the different ferroelectret samples with varying coupling coefficient, as well as the mechanism parts and cable masses.For the model fitting process, small masses are introduced accounting for the glue, the insulation varnish and the cables.In the figures 15(a) and (b) the simulated transfer behaviors of the mechanical-mechanical (in terms of equation ( 28)) and mechanical-electrical (in terms of equation ( 29)) model are compared with the corresponding measured transfer behavior.
A good agreement for the mechanical case can be observed.
For the mechanical-electrical case a good agreement between model and the best performing samples can be seen.The experimental results slightly differ from the simulated ones.Additional resonance peaks observed between 200-250 Hz can be reasonably explained by minor asymmetry of the setup.The modal analysis reveals a tilting and a torsional mode in this range.These modes are not excited in the simulated harmonic response because of the perfectly parallel excitation and bearing edges in the model.The experimental setup however does not ensure perfectly parallel edges.Even very small angles between the edges leads to an excitation of these modes.This alone would not explain the voltage peaks.If the EH itself was perfectly symmetric w.r.t. the x-z-and y-z-planes, tilting and torsional modes would stress the stack symmetrically, leading to cancelation of charges.Therefore, it is assumed that also the EH itself is slightly unsymmetric due to one or more of the following reasons: • Slight differences in the manual preparation of the two steel mechanism parts.• Influence of cables, that are only on one side of the plate.This can lead to more stress in the front part of the stack than in the back part.• Difference in ferroelectret properties based on the yposition.This is rather unlikely, because the results of the different samples appear very similar in terms of magnitude.
The main difficulty in conducting the experiments is therefore to achieve a symmetrical test setup consisting of excitation and bearing edges as well as the EH itself.9.843 MΩ.Therefore a 10 MΩ resistor is chosen as a load resistance for the experiment.For 70 Hz, a mean power output of around 80 nW is observed at a shaker force magnitude of 0.25 N.With the linear relationship of the EH voltage with the excitation force shown in figure 16(b), much higher forces and therefore powers are possible.For 1 N excitation, the power output therefore calculates to 1.28 µW for this specific sample.Scaling this result by the used value of δ 33 = 400 pC N −1 in the model, a power output of 2.20 µW is obtained, being lower than in the optimized model, but in the same order of magnitude.The result is naturally sensitive to the respective ferroelectret sample coupling coefficient and imperfections in the manual sample and mechanism preparation as observable in the results of figures 15(b) and (c).The time data of voltage and (calculated) power at 1.5 Hz in figure 17(b) are also presented for different shaker voltage magnitudes.The theoretical load resistance calculates to 459.3 mΩ, that was not used explicitly in the experiment but can be used for calculations.The voltage is taken for the power calculation in an optimal load resistance.The shaker force excitation magnitude is 18.25 N which is much higher than before since the frequency is far from resonance.Figure 17(b) however shows a pronounced linearity, which substantiates the statement, that the shown time data in figure 16(a) are still far from the strain limit of the ferroelectret material.The maximum generated mean power for the low frequency excitation is 125 nW.A normalization with 1 N excitation magnitude is not conducted, since in the low frequency range high forces have to be applied to reach representative material strains of the wing excitation.This result is lower than in case one, since the mechanism length is much lower.The achieved voltage magnitudes are in the range of < 1 to 10 V and are therefore compatible with typical power management circuits.By the EH topology, this voltage can be adjusted if necessary.

Comparison to the state of the art
A proper comparison to other energy harvesting concepts in the state of the art is not trivial, since this work focuses on a bending strain excitation which represents a novelty for ferroelectret energy harvesting.Most research work covering ferroelectret energy harvesting focuses on acceleration excitations without an excitation of a realistic application for reference.This, on the other hand, substantiates the novelty of the presented concept.Nevertheless a comparison of the here presented and also experimentally investigated principle is possible when neglecting the mentioned issues.Since this work focusses on the optimization of power per mass, a comparison to seismic mass harvesters is conducted by also deriving their total power per mass taking into account the seismic mass itself.The results for the comparison with different relevant research works are shown in figure 18.The comparison shows, that the power per mass of the here presented and investigated mechanism can be higher than the power per mass of works using the same or comparable ferroelectret materials.Other materials like parallel-tunnel FEP, as used in [8,25], are not taken into the comparison, since they provide a different principle and a much lower robustness.In case one, a similar order of magnitude for the power output per mass is achieved as for other concepts from the literature, even though it works at the very low frequency of 1.5 Hz.The other cases investigated in this work provide a much higher power output per mass than the research works from literature, while generally being used at a lower frequency.As stated, a higher structural energy can be expected at these lower frequencies, which adds up to the already depicted advantage of the presented EH concept.

Clustering
The condition of the low absolute power of a unit cell can be overcome by the design of EH systems consisting of multiples of the optimized miniature EHs.As stated, this work is about optimizing equation ( 14) and not the total power output of one EH, since its size can always be scaled for that purpose.A reasonable design goal for this size scaling can  [6,34,35,47,48] ( * strain is one order of magnitude higher than in the presented work).however be the optimization of the power output per area using the mass-optimized EH unit cells.This can be done using a clustering approach as shown in figure 19.In this way, the optimized power output per mass stays the same but the total power output is multiplied by the number of unit cells in the array.Since only one load direction is of interest when considering the use-case of low frequency deformations of an aircraft wing, the EHs only use mechanism arms in one space dimension.The arrangement of many EHs in rows using long stripes of ferroelectret stacks is therefore possible.By rotating the stripes to an angle, where the mechanism arms just do not touch each other, the EH number per area can be further increased.A maximum of 4000 unit cell EHs (of case 3 topology) can be applied in a 1 m 2 area with square shape.The total mass equals to 2.4 kg.With an excitation of case 3, a total power output of 52 mW can be obtained with the cluster.This order of magnitude is sufficient for the operation of different electrical consumers.For a single EH, the perpendicular load direction could in principle be exploited additionally to increase the power output.For use-cases like spherical shell structures and at higher eigenfrequencies with smaller vibration wavelengths, two load-directions could therefore be exploited.In a use-case like an aircraft wing at low eigenfrequencies, there is only one load direction expected to provide the major part of the mechanical strain energy.Another reason for not using two load directions is that by using steel, like in this work, the mechanism arms of the less loaded direction could potentially lead to an additional stiffness in thickness direction of the EH.This could potentially lower the overall efficiency of the concept.An alternative to this could be the use of steel wires or carbon fibre rovings.For the proposed cluster design, an additional direction of mechanism strings can lead to a loss of space for neighboring EHs.Hence, if there is only one major load direction of the host structure, the achievable power per area is potentially lowered in this way.The periodic application of the concept could be further used as a vibroacoustic metamaterial to optimize the power output in a certain frequency band while reducing vibration magnitudes simultaneously.This was in principle first discussed in [39] using a model based approach with EHs using resonator masses.

Conclusion
This work introduced a novel approach for the use of IXPP ferroelectret for energy harvesting in a bending strain condition in lightweight design.Based on miniature EHs, it was shown that the concept can be suitable for the use in lightweight design, since it provides a comparably high ratio of power output per mass when added to a structure like an aircraft.Especially for the wing, that is taken as an example in this work, the concept is very promising since the availability of space is not a major problem.Large deformations of the wing at low frequencies can be exploited to provide enough electrical power decentrally for low-power consumers.In an analytical modeling approach, the energy conversion abilities of the presented concept were quantitatively compared to concepts using PZT ceramics to stress the usefulness in cases where strain energy is the prominent source of vibration energy.This represents a novelty for the type of excitation.The application of optimization algorithms to improve the metric P el,rel for lightweight design is furthermore a novel approach in the field.Three cases for optimizing the concept were investigated.For a quasistatic case subject to a strain excitation without a defined host structure, a maximum power output of 1.52 µW and power output per mass of 271 µW kg −1 were achieved.For the corresponding dynamic case and for the dynamic case with a host structure, the power outputs 0.7 mW and 6.53 µW (at 1 N excitation magnitude) and the power outputs per mass of 80 mW kg −1 and 10.8 mW kg −1 (at 1 N excitation magnitude) were achieved respectively.Typical voltage magnitudes are in the range of 1 to 10 V and are therefore compatible with power management circuits.The experimental results show a good reproducibility in the mechanical transfer behavior up to 500 Hz, the electrical behavior still strongly depends on the material properties of the specific sample.The model fit shows, that the mechanical and electrical behavior of the material can be modeled with a linear approach.Additionally, the authors proposed a novel array design for the presented concept.The clustering study shows, how the EHs can be implemented for a real application.Achieving a power output of about 52 mW using a 1 m 2 area, several electrical consumers can be powered.Example consumers are sensors and signal processing units for structural health monitoring.In future works, other ferroelectret materials with even higher transducer coefficients are of interest for increasing the power output of every EH.A FoM definition for a more general comparison with other applications or EH topologies can also be considered, taking into account the excitation magnitude in terms of surface strain and bending radius.Investigating Clusters of the EHs experimentally with respect to the exploitation of the metamaterial effect is a possible next step in research.Multistep mechanisms are further of interest to increase the transmission ratio from strain excitations with high energy, yet low strains.

Figure 1 .
Figure 1.Comparison of PZT and IXPP.Left: under same strain conditions (limit: maximum allowable strain of PZT); Right: same stress conditions (limit: maximum allowable stress of IXPP).W 1,i + W 2,i represents the total mechanical energy of a load cycle and W 1,i the electrically stored potential energy between the electrodes in the respective case.

Figure 3 .
Figure 3. Principle of the EH with transmission mechanism in undeformed (a) and deformed (b) state.Simplified spring model of the EH mechanism subject to a strain force without a host-structure (c).

Figure 4 .
Figure 4.Results for the analytical model for three types of EHs: proposed concept with IXPP ferroelectret and example topology and variable mechanism length (blue graphs); mechanism concept with same topology and PZT ceramic properties (red graphs); PZT ceramic patch of the same area with ideal strain excitation (purple graphs).The dash-dotted graphs in figure (c) show the relation of the result of equation (25) of the ferroelectret concept compared to the two PZT concepts.

Figure 5 .
Figure 5. Parametric geometry of the EH.

Figure 6 .
Figure 6.Mesh of the finite element model (shown for result of case 3).

Figure 7 .
Figure 7. Flow chart of the global design, modeling and optimization approach (description in the text).

Figure 9 .
Figure 9. Part of the response surface for case 2. The assumed number of IXPP sheets is 20 and tpress = 2 mm.

Figure 10 .
Figure 10.Part of the response surface for case 3 with an aluminum plate as host structure for l mech = 4.5 • 10 −2 m and tpress = 1 • 10 −4 m.

Figure 12 .
Figure 12.Mechanism preparation; left: mechanism glued onto plate with cable for negative electrode; right: EH with prestressed and electrically contacted ferroelectret stack.

Figure 13 .
Figure 13.Experimental setup; left: EH in four-point bending test setup before prestressing; right: EH in prestressed state for experimental test.

Figure 14 .
Figure 14.Measuring chain.ua and u f represent the signal voltages for acceleration and force.

Figure 15 .
Figure 15.Experimental results for the mechanical and mechanical-electrical transfer behavior corresponding to equations (28) (a), (29) (b) and (30) (c) for all investigated samples.The results show a good correlation of the respective transfer behavior with an average FRAC value of 0.93, 0.92 and 0.93 respectively (non diagonal elements).

Figure 16 .
Figure 16.Time data of voltage and power of the additionally tested sample at frequency of 70 Hz and load resistance of 10 MΩ for different shaker force magnitudes (a).Open circuit and 10 MΩ load voltage data magnitudes for different excitation magnitudes at 70 Hz (b).
Furthermore, time data of an additionally tested sample with δ 33 = 305 pC N −1 are shown in figures 16(a) and 17(a) for 70 Hz and 1.5 Hz excitation respectively for different shaker excitation voltages.The latter ones are proportional to the acceleration at a fixed frequency.The voltage magnitude is derived from the half of the mean peak to peak value of all maximum and minimum values.These magnitude values of the voltages are further shown in figures 16(b) and 17(b) for the two frequencies.The optimal load resistance for the excitation of 70 Hz with the EH capacitance of 231 pF results to

Figure 17 .
Figure 17.Time data of voltage and (calculated) power of the additionally tested sample at frequency of 1.5 Hz and theoretical load resistance of 459.3 MΩ for different shaker force magnitudes (a).Voltage data magnitudes for different excitation magnitudes at 1.5 Hz (open circuit) (b).

Figure 18 .
Figure 18.Comparison of generated power per mass with EHs of different research works[6,34,35,47,48] ( * strain is one order of magnitude higher than in the presented work).

Figure 19 .
Figure 19.Potential cluster arrangement of the EH topology to enhance the power output per area.

Table 1 .
Geometry and material properties of IXPP sheet for all simulations.

Table 3 .
Dependent parameters of the model.

Table 4 .
Outputs of the model.

Table 5 .
Parameter values and resulting mass, power output and power output per mass for optimized design in case 1.

Table 6 .
Parameter values and resulting mass, power output and power output per mass for optimized design in case 2.

Table 7 .
Parameter values and resulting mass, power output and power output per mass for optimized design in case 3.