Kaijun Yi

Mélodie Monteil

Manuel Collet
email: manuel.collet@ec-lyon.fr

Simon Chesné

S Chesne

Smart metacomposite-based systems for transient elastic wave energy harvesting

Keywords: piezoelectric material, negative capacitance, model reduction, wave focusing, energy harvesting, SSHI (synchronized switch harvesting on inductor) interface

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Introduction

Design more reliable, durable and comfortable products is a long-term objective in modern industries, such as aerospace, transport and civil engineering. Nowadays, with the rapid development in electronics, computer science and material science, smart structures are proposed in an attempt to achieve this goal [START_REF] Collet | Adaptive metacomposites for vibroacoustic control applications[END_REF]. Smart structures are hybrid systems composed of load-bearing materials and electronic units like but not limited to control and health monitoring units. An electronic unit typically contains sensors, actuators, computing devices and communication module, all these devices require power supply. Traditional batteries are not suitable choices due to their limited life spans, which will lead to maintenance problems. A more appropriate way is to integrate energy harvesting devices into the smart structures to realize self-powered systems.

Due to the ubiquitous presence of vibration in structures, extensive efforts have been made to harvest vibration energy [START_REF] Quyen | Review on energy harvesting for structural health monitoring in aeronautical applications[END_REF]. Among these studies, a lot of them were dedicated to obtain higher harvesting efficiency and/or to extend the operating frequency band of the harvesting system. Examples include (i) tuning the harvesting system through passive or active methods to match the operating frequency with the environment [START_REF] Zhu | Strategies for increasing the operating frequency range of vibration energy harvesters: a review[END_REF][START_REF] Jackson | Widening the bandwidth of vibration energy harvesters using a liquid-based non-uniform load distribution[END_REF]; (ii) exploiting nonlinear mechanical mechanisms to widen the operating frequency band [START_REF] Xiong | Internal resonance with commensurability induced by an auxiliary oscillator for broadband energy harvesting[END_REF]; (iii) using phononic crystals [START_REF] Chen | Broadband characteristics of vibration energy harvesting using one-dimensional phononic piezoelectric cantilever beams[END_REF], metamaterials [START_REF] Fan | A wave-based design of semi-active piezoelectric composites for broadband vibration control[END_REF] or acoustic black holes [START_REF] Zhao | Broadband energy harvesting using acoustic black hole structural tailoring[END_REF] to increase energy densities near the harvesters.

Apart from the mechanical approaches, nonlinear energy extraction circuits have been proposed to boost the harvested energy when piezoelectric transducers are used. The introduced nonlinear part is termed "synchronized switch harvesting on inductor" (SSHI) interface, which is composed of a switch and an inductor in series. The switch is turned off when the voltage of the piezoelectric transducer reaches a minimum or a maximum. Under this situation, the inductance and the capacitance constitute an oscillating circuit with a period much smaller than the mechanical one. After a very short time equals to a half of the period of the oscillating circuit, the switch is turned on again. Consequently, the voltage is inverted. The SSHI interface only requires a small amount of power to control the switch, it could be self-powered [START_REF] Guyomar | Recent progress in piezoelectric conversion and energy harvesting using nonlinear electronic interfaces and issues in small scale implementation[END_REF].

The SSHI interface is firstly proposed and studied by D. Guyomar et al. [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF], and has drawn considerable attention these years. It is connected with the piezoelectric element in parallel to form a parallel SSHI harvesting system [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF][START_REF] I C Lien | Revisit of series-SSHI with comparisons to other interfacing circuits in piezoelectric energy harvesting[END_REF][START_REF] Lefeuvre | A comparison between several vibration-powered piezoelectric generators for standalone systems[END_REF][START_REF] Shu | An improved analysis of the SSHI interface in piezoelectric energy harvesting[END_REF], or in series to obtain a series SSHI harvesting system [START_REF] I C Lien | Revisit of series-SSHI with comparisons to other interfacing circuits in piezoelectric energy harvesting[END_REF][START_REF] Lefeuvre | A comparison between several vibration-powered piezoelectric generators for standalone systems[END_REF]. Researches show that the SSHI interface can improve the converted energy by the piezoelectric transducer from mechanical to electrical form and reduce the backward conversion [START_REF] Liang | Energy flow in piezoelectric energy harvesting systems[END_REF]. In a weekly coupled electromechanical system, the harvested power is boosted by 4 to 9 times when compared to a standard DC technique proposed by G. Ottman et al [START_REF] Geffrey | Adaptive piezoelectric energy harvesting circuit for wireless remote power supply[END_REF]. In a strongly coupled case, these techniques will produce almost equivalent power but the required piezoelectric materials for the SSHI-based techniques are much less [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF]. Based on the SSHI interface, several improved techniques have been proposed, each of them addresses a particular concern. Such as the "double synchronized switch harvesting" (DSSH) [START_REF] Lallart | Double synchronized switch harvesting (DSSH): A new energy harvesting scheme for efficient energy extraction[END_REF] and the "enhanced synchronized switch harvesting" (ESSH) [START_REF] Shen | Enhanced synchronized switch harvesting: a new energy harvesting scheme for efficient energy extraction[END_REF] are proposed to obtain load independent techniques. The "adaptive synchronized switch harvesting" (ASSH) [START_REF] Shen | Adaptive synchronized switch harvesting: A new piezoelectric energy harvesting scheme for wideband vibrations[END_REF] technique is developed to harvest energy in multimodal vibration situations. The characteristics, advantages and drawbacks of these SSHI-based techniques are discussed in details by D. Guyomar et al. [START_REF] Guyomar | Recent progress in piezoelectric conversion and energy harvesting using nonlinear electronic interfaces and issues in small scale implementation[END_REF].

Harvesting vibration energy in structures is well studied, however limited effort has been devoted to harvest energy from traveling waves. Traveling waves are common in built-up structures, since the power in these structures transmits from one component to another especially at higher frequency bands [START_REF] Goyder | Vibrational power flow from machines into builtup structures, part I: Introduction and approximate analyses of beam and plate-like foundations[END_REF][START_REF] Fan | Energy flow prediction in built-up structures through a hybrid finite element/wave and finite element approach[END_REF]. In addition, waves will propagate away from the source when structures are under non stationary excitations, they are attenuated by damping and/or radiation thus are not reflected back to the source to form standing waves. It is important to develop harvesting systems to obtain energy from waves in those cases.

To harvest energy from traveling waves, one of the main challenges is that the amount of harvested energy could be very low. In recent years, to increase harvested energy from traveling waves, several innovative harvesting systems have been developed [START_REF] Carrara | Dramatic enhancement of structure-borne wave energy harvesting using an elliptical acoustic mirror[END_REF][START_REF] Carrara | Fourier transformbased design of a patterned piezoelectric energy harvester integrated with an elastoacoustic mirror[END_REF][START_REF] Carrara | Metamaterial-inspired structures and concepts for elastoacoustic wave energy harvesting[END_REF][START_REF] Tol | Gradient-index phononic crystal lens-based enhancement of elastic wave energy harvesting[END_REF]. The fundamental idea is to steer waves to increase the energy densities at particular positions and harvest there. For examples, an elliptical acoustic mirror or a parabolic acoustic mirror is used to focus waves in the systems proposed by [START_REF] Carrara | Dramatic enhancement of structure-borne wave energy harvesting using an elliptical acoustic mirror[END_REF][START_REF] Carrara | Fourier transformbased design of a patterned piezoelectric energy harvester integrated with an elastoacoustic mirror[END_REF][START_REF] Carrara | Metamaterial-inspired structures and concepts for elastoacoustic wave energy harvesting[END_REF]; in [START_REF] Carrara | Metamaterial-inspired structures and concepts for elastoacoustic wave energy harvesting[END_REF], an artificial periodic array with a defect is designed to localize energy at specific frequencies, an acoustic funnel formed by arrays of acoustic scatters is developed to guide waves into a narrow channel. However, the harvesting circuits in these studies are simply represented by resistive ones in order to focus analysis in the mechanical part of the system. This is acceptable from an academic point of view but not adequate in practice since most of the real-life low power electronic devices require DC, the actual harvesting circuit always contains a AC-to-DC converter and maybe some other nonlinear parts, which can't be simplified as a linear resistive load.

Different from the aforementioned methods to steer waves, recently a piezo-lens is proposed to focus waves [START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF]. The piezo-lens is composed of a host plate and several surfacebonded piezoelectric patches. These patches are shunted with negative capacitance (NC) circuits. The spatial variation of refractive index in the piezo-lens zone is designed to fulfill a hyperbolic secant profile. Results show that the piezo-lens can focus flexural waves near a designed point in a broad frequency band. Thus the piezo-lens has large potential to be exploited in developing advanced harvesting systems for waves.

In this paper, the piezo-lens is combined with SSHI-based harvesters to improve the harvested energy from transient waves. An analytical relationship which connects the effective refractive index of piezoelectric system to the shunting NC value is used to design the piezo-lens; corrected reduced models are developed to predict the transient responses of the piezoelectric systems. With these tools, the performances of the harvesting systems are studied and discussed.

Configuration of the harvesting system

The first part: a piezo-lens

The harvesting system proposed in this paper consists in the combination of a piezolens and a harvester. The concept and designing process of the piezo-lens are introduced in this section. The piezo-lens is obtained by periodically bonding piezoelectric patches on the surfaces of a host aluminum plate in a collocated fashion, as depicted in figure 1(a). The host plate is lying in the x -y plane and occupying the spatial region -h b /2 ≤ z ≤ h b /2. The piezo-lens zone could be divided into a 14-by-6 array of piezoelectric cell, the patches in each of these cells are shunted with a NC circuit and their bonding surfaces are grounded, as shown in figure 2.

To focus flexural waves, the spatial variation of the refractive index of flexural wave inside the piezo-lens zone is designed to fulfill a hyperbolic secant function:

n(y) = n 0 • sech(α(y -β)) (1)
in which, n 0 represents the refractive index of the background plate, α is the gradient coefficient and β represents the y coordinate of the symmetry axis of the refractive index profile, as illustrated in figure 1(b). Due to this design, waves incident into the lens from the Ox direction will be focused at a focal point at the y = β line, with a focal length f = π/2α measuring from the left boundary of the lens [START_REF] Tsung | Focusing of the lowest antisymmetric Lamb wave in a gradient-index phononic crystal plate[END_REF]. By tuning the NC values, one can change the local dynamic properties where piezoelectric patches are bonded [START_REF] Fan | A wave-based design of semi-active piezoelectric composites for broadband vibration control[END_REF][START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF]. Thus, the variation of the refractive index inside the piezo-lens can be approximately realized in a piecewise form by designing the NC values of cells at different locations. According to equation (1), the refractive index only varies in the y direction. Thus, in a piezo-lens, the shunting NC values are equal in a same row (the x direction) but will differ in a same column (the y direction). To determine the required shunting NC value in each row, an analytical relationship between the effective refractive index of the cell and the NC value is developed.

The relationship is obtained in four steps. In the first step, the effective Young's modulus and effective Poisson's ratio of the piezoelectric patches shunted with NC are represented as functions of the NC value:

E p = E sh p C neg + C T p C neg + C T p (1 -k 2 31 ) µ p = µ sh p C neg + C T p (1 + k 2 31 /µ sh p ) C neg + C T p (1 -k 2 31 ) (2)
here, E sh p and µ sh p stand for the Young's modulus and the Poisson's ratio of the short-circuit piezoelectric material, respectively; k 31 is the coupling factor; C T p is the free intrinsic capacitance (namely the capacitance when the patch is under constant stress); C neg is the applied NC value.

In the second step, the effective parameters of the shunted piezoelectric sandwich structure highlighted by the dashed lines in figure 2 are determined according to the classical laminated plate theory [START_REF] Reddy | Mechanics of Laminated Composite Plates and Shells: Theory and Analysis[END_REF]. The effective area density and effective flexural rigidity of the piezoelectric sandwich structure are expressed as:

ρ A = ρ b h b + 2ρ p h p D A = D b + 2E p 3(1 -µ 2 p ) [( h b 2 + h p ) 3 -( h b 2 ) 3 ] (3)
here,

D b = E b h 3 b 12(1-µ 2 b )
is the flexural rigidity of the host plate and E b , µ b denote the Young's modulus and the Poisson's ratio of the host plate respectively; ρ b and ρ p are the densities of the host plate and the piezoelectric patches respectively.

In the third step, the effective area density and effective flexural rigidity of the entire shunted piezoelectric cell are derived, they are expressed as [START_REF] Zhang | Sound transmission loss of metamaterial thin plates with periodic subwavelength arrays of shunted piezoelectric patches[END_REF]:

ρ ef f = χρ A + (1 -χ)ρ b h b D ef f = D A D b χD A + (1 -χ)D b (4)
here, χ = (l p /l b ) 2 is the ratio of the surface covered by the piezoelectric patch to the surface of the unit cell.

In the last step, the effective refractive index of flexural wave incident from the background plate into the shunted piezoelectric cell is obtained:

n ef f = ( ρ ef f D b ρ b h b D ef f ) 1/4 (5)
With the relationship in equation ( 5), a piezo-lens is designed in three steps. Firstly, the parameters α and β in the refractive index function in equation ( 1) are chosen to design the location of the focal point. Then, the required refractive index for each row of piezoelectric cell in the lens zone is obtained by substituting the central y coordinate of each row into equation [START_REF] Collet | Adaptive metacomposites for vibroacoustic control applications[END_REF]. Lastly, the required refractive index for each row is fulfilled by choosing the NC value according to equation ( 5). The DC device involves a rectifier to convert the voltage of the transducer to a DC form, and a capacitance C s to store the harvested energy, as depicted in figure 3(a). In that figure, the piezoelectric patch is equivalently represented by a current source I eq (t) and a capacitance C h with the value equal to the blocked intrinsic capacitance (i.e. the capacitance when the patch is under constant strain condition) [START_REF] Liang | Energy flow in piezoelectric energy harvesting systems[END_REF]. The rectifier is assumed to be perfect. Thus, when the absolute value of V h (t) is larger than or equal to the output voltage V Cs (t), the rectifier conducts, the storage capacitance is charged. Under this circumstance, the harvesting circuit is governed by the equations below:

Vh (t) = - Qh (t) C s , V Cs (t) = |V h (t)| (6)
On the other hand, when the absolute value of V h (t) is smaller than the output voltage V Cs (t), the rectifier is blocked, no charge will flow to the storage capacitance, the piezoelectric element is under open-circuit condition:

Vh (t) = İeq (t) C h , VCs (t) = 0 (7)
in which, the exact expression of I eq is given in subsection 3.2.

The SSHI-based harvesting devices are obtained by integrating a SSHI interface into the DC one shown in figure 3(a). In the first SSHI-based device, the SSHI interface, consists of a switch S and an inductor L, is in parallel to the piezoelectric patch as illustrated in figure 3(b). It is called parallel SSHI-DC (P-SSHI-DC) device in this paper. The control law of the switch is similar to the one used in steady state cases. At most of the time, the switch is opened, the device works just like a standard DC one. The switch is triggered at the time t i when the current I eq is null (namely, the voltage of the piezoelectric patch is a local maximum or minimum). It is kept closed for a very short period of time, corresponding to a half of the period of the oscillating circuit: ∆t = π √ LC h . An inversion quality factor Q I [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF] is used to describe the energy loss mainly caused by the inductor in the SSHI interface (note that in some papers [START_REF] Shen | Enhanced synchronized switch harvesting: a new energy harvesting scheme for efficient energy extraction[END_REF][START_REF] Shen | Adaptive synchronized switch harvesting: A new piezoelectric energy harvesting scheme for wideband vibrations[END_REF], the voltage inversion coefficient γ = e

(-π 2Q I
) is used). In the simulations, this loss is taken into account by adding a resistive part R L = 1 Q I L C h with the inductor. Thus the governing equations for the circuit during the inversion process are:

L Qh (t) + R l Qh (t) + Q h (t) C h = 0 Vh (t) = Qh (t) C h , VCs (t) = 0 (8)
The second SSHI-based device contains a SSHI interface in series with the piezoelectric patch, as shown in figure 3(c). This device is called series SSHI-DC (S-SSHI-DC) device here. At most of the time, the piezoelectric element is under open-circuit condition. Different from the steady state cases [START_REF] I C Lien | Revisit of series-SSHI with comparisons to other interfacing circuits in piezoelectric energy harvesting[END_REF][START_REF] Lefeuvre | A comparison between several vibration-powered piezoelectric generators for standalone systems[END_REF], the switch is closed when I eq is null and

|V h (t i )| ≥ V Cs (t i ), the duration is ∆t = π L C h Cs C h +Cs .
An inversion quality factor Q I is also used to take into account the loss of the switch interface in this case, accordingly a resistive part R l = 1

Q I L C h +Cs
C h Cs is added to the inductor. The inversion process is governed by the equations below:

L Qh (t) + R l Qh (t) + Q h (t) C h C s /(C h + C s ) = 0 Vh (t) = Qh (t) C h , V l (t) = -L Qh (t) -R l Qh (t) V Cs (t) = |V h (t) -V l (t)| (9)
in which, V l (t) is the voltage difference between the two ends of the inductor.

Numerical models

Full finite element models

To study performances of the harvesting system, finite element (FE) models of piezoelectric systems are developed. In the FE models, the structures are discretized by 3D quadratic Lagrange elements. Each of the nodes corresponding to the piezoelectric patches has three mechanical degrees of freedom (DOFs) and a voltage DOF. The equilibrium equations for the discretized fully coupled piezoelectric system are:

M dd d + K dd d + K dV V = F -K T dV d + K V V V = Q (10)
here, d and V represent the structural and voltage DOFs, respectively; F and Q are the mechanical forces and charges respectively. The equations in [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF] are rewritten under following considerations: (i) the voltage DOFs in the piezoelectric patches can be partitioned into DOFs inside the patches, DOFs on the free electrodes of the patches and DOFs on the bonding surfaces; (ii) the voltage DOFs on the bonding surfaces are grounded, thus the corresponding equations and columns are directly removed; (iii) there is no charge source inside the piezoelectric patches, thus the internal voltage DOFs can be eliminated by exact static condensation; (iv) in the system, the piezoelectric patches are connected with different circuits -the patches constituting the piezo-lens are connected with NC circuits and the patch for harvesting is connected with one of the circuit illustrated in figure 3, therefore, the DOFs on the free electrodes are further separated into DOFs corresponding to the patches in the piezo-lens and DOFs of the patch for harvesting; (v) as the DOFs on one electrode have identical voltages, the voltage DOFs on the free electrode of each piezoelectric patch are reduced such that only one master voltage DOF remains on the free electrode per patch. Consequently, the governing equations can be rewritten as below:

  M dd 0 0 0 0 0 0 0 0      d VL Vh    +   H dd H dL H dh -H T dL C L 0 -H T dh 0 C h      d V L V h    =    F Q L Q h    (11)
in which, the matrices C L and C h are diagonal, each of their diagonal elements represents the blocked intrinsic capacitance of a piezoelectric patch; V L , V h are the master DOFs on the free electrodes of the patches in the lens and the patch for harvesting, respectively; Q L , Q h are the charges flowing to the patches in the lens and patch for harvesting respectively. More details about above process could be found in the Appendix or [START_REF] Becker | Finite element-based analysis of shunted piezoelectric structures for vibration damping[END_REF].

Model reduction

Instead of solving the full finite element model above, which has a large number of DOFs, a reduced model is used. For the sake of simplicity, let:

H dV = [H dL , H dh ], V = [V L , V h ] T Q = [Q L , Q h ] T , C Lh = C L 0 0 C h (12)
Using these notations and equations ( 11), the governing equations in time domain are written as:

M dd d + H dd d + H dV V = F (13a) -H T dV ḋ + C Lh V = Q (13b)
Equation (13b) and the following equations (17b), (20b) represent the Kirchhoff's current law which must be satisfied at the joints where circuits are connected with the piezoelectric patches.

The reduced model is obtained through a transformation between the displacement d and a set of modal coordinates η:

d = Φη ( 14
)
in which, Φ = [φ 1 , φ 2 , ..., φ m ]. φ i is the ith natural mode of the piezoelectric system under short-circuit condition with specific homogeneous Dirichlet boundaries, it is obtained by solving the following eigenvalue problem:

(-ω 2 i M dd + H dd )φ i = 0 (15)
here, ω i is the corresponding natural frequency. The modes are mass-normalized, resulting in:

Φ T M dd Φ = I, Φ T H dd Φ = Λ = diag(ω 2 i ) (16)
Using equations ( 14) and ( 16), the governing equations ( 13) are represented in modal coordinates as:

η + Λη + Φ T H dV V = Φ T F (17a) -H T dV Φ η + C Lh V = Q (17b)
Only the first m modes in modal matrix Φ will be retained, and the number is much smaller than that of the system's DOFs. Thus, the number of equations in ( 17) is largely reduced. However, the reduced model in equations ( 17) can't accurately describe the piezoelectric behaviors of the system [START_REF] Collet | Modal Synthesis and Dynamical Condensation Methods for Accurate Piezoelectric Systems Impedance Computation[END_REF] since the truncation of the higher order modes will lead to a static reduction error [START_REF] Becker | Finite element-based analysis of shunted piezoelectric structures for vibration damping[END_REF]. This static error will cause a non-negligible error of the electrostatic voltage relative to the full model's as indicated below:

T e err = T e f -T e r = H T dV (H -1 dd -ΦΛ -1 Φ T )H dV (18)
here, T e f and T e r are the electrostatic transfer matrices between V and Q of the full and reduced models, respectively.

To obtain more accurate voltage responses, the reduced model is corrected by modifying the blocked intrinsic capacitance matrix C Lh . The voltage responses are corrected by guaranteeing that the voltage outputs V (i) of the i th piezoelectric patch in the corrected reduced model and the full model are consistent when a same static electric input Q(i) is applied to this patch. According to equations ( 13) and ( 17), such requirement is fulfilled by modifying the capacitance matrix C * Lh in the corrected model as:

C * Lh = C * L 0 0 C * h = diag(H T dV (H -1 dd -ΦΛ -1 Φ T )H dV ) + C Lh (19)
C * Lh is still a diagonal matrix, the diagonal terms of C * L and C * h represent the modified blocked intrinsic capacitances of corresponding patches. The corrected reduced model is obtained by replacing the C Lh in equation (17b) with C * Lh :

η + Λη + Φ T H dV V = Φ T F (20a) -H T dV Φ η + C * Lh V = Q (20b)
The structural damping is taken into account by introducing a constant viscous damping coefficient 2ξω i into each remained mode. The NC circuits are connected to the piezoelectric patches through the following relation:

QL (t) = -C neg VL (t) (21)
The DC and SSHI-based circuits are implemented in the corrected reduced models through equations ( 6) to [START_REF] Guyomar | Recent progress in piezoelectric conversion and energy harvesting using nonlinear electronic interfaces and issues in small scale implementation[END_REF]. Note that, in the equivalent model of the piezoelectric patch for harvesting, the current source is I eq (t) = H T dV Φ η(t) and the blocked intrinsic capacitance is the modified one C * h .

Numerical results

In the simulations, the dimensions of the piezo-lens and host plate are illustrated in figure 1(a). The geometry parameters of the piezoelectric cells and piezoelectric patch for harvesting are given in table 1, meanings of the symbols used in table 1 are given in figure 2. The Young's modulus, Poisson's ratio and density of the aluminum plate are E b = 70 Gpa , µ b = 0.3 and ρ b = 2700 kg/m 3 , respectively. The material parameters for the PZTs are listed in table 2. The plate is clamped at the right end side and a 10-period Hanningwindowed tone burst excitation with central frequency 2000 Hz is applied at the left end side to generate transient waves. The excitation is a line transverse force, its waveform and spectrum are illustrated in figure 4. Modes that have natural frequencies smaller than five times of the maximum frequency of interest are remained in the reduced model; a small damping coefficient ξ = 0.001 is used to minimize the influence of the structural damping on The piezo-lens' parameters are chosen as α = π/0.6, β = 0, and the harvester is mounted at the corresponding focal point, which is located on the y = 0 line (namely the central axis) with a distance of 0.06 m from the right boundary of the lens. (Note that the focal lenght is 0.3 m counting from the left boundary.) These values are chosen because the energy will be more better concentrated if the focal point of the piezo-lens is designed on the central axis and close to the lens boundary [START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF].

Comparison between the standard DC and SSHI-based devices

The standard DC and SSHI-based harvesting devices have been well studied in steady state, however lack of knowledge of their applications in transient wave cases could be found. Thus first of all, the performances of the systems with these devices in transient wave energy harvesting are studied. The piezo-lens is applied in these studies. To assess the harvesting performances, two metrics are used. The first one is the harvested energy, which indicates how much energy can be yielded when a transient wave package passes through the harvester, it is the maximum energy that stored in the storage capacitance of the harvesting device:

E har = max( 1 2 C s V 2 Cs (t)) (22)
The second one is the harvesting efficiency, it is defined as [START_REF] Kim | Efficiency of piezoelectric mechanical vibration energy harvesting[END_REF]:

η E = E har E input (23)
in which, E input is the total input energy introduced by the excitation. The efficiency indicates how much a harvesting system can convert the input mechanical energy (E input ) into the output electric one (E har ). This metric is a useful criterion to compare the harvesting performances of systems with different configurations and/or under different operating conditions.

The performances of systems with different harvesting devices are compared in figure 5. In the simulations in figure 5 and those hereinafter, the inversion quality factor Q I of the SSHI-based device is equal to 3. Q I indicates the energy loss caused by the harvesting circuit, typically a larger value of it means less energy loss [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF]. Q I mainly depends on the involved piezoelectric material, the switch and the inversion inductance, its real value can only be obtained experimentally, herein the value of it is chosen according to the results in [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF]. From figure 5 it can be seen that the harvested energy and efficiency strongly depend on the storage capacitance, and the optimal capacitances corresponding to the maximum harvested energy and the maximum efficiency are nearly the same for each device. In addition, when the devices are all working at optimal conditions, the two SSHI-based devices have almost equal performances, and they both harvest 2.6 times more energy than the DC device and also have 2.6 times better efficiency.

To gain more insights into the harvesting performances of the systems with DC or SSHIbased devices, the typical waveforms and the converted power are compared in figure 6. The meanings of V h , I eq and V Cs are given in figure 3; the converted power is the product of V h and I eq , positive it represents the amount of power converted from mechanical to electrical, and negative it means the contrary. In the simulations in figure 6, the storage capacitances are all set as C s = 30C * h to guarantee an acceptable performance for all the devices according to the results in figure 5. To facilitate the comparison, the maximum V Cs in the DC case is normalized to unit, and all the other voltages are normalized according to this value; a similar normalization process is used to deal with the converted power.

When the transient wave package reaches the harvester, the charges begin to accumulate in the storage capacitance; the accumulation stops after the main part of the package passes through the harvester. The charges accumulated in the storage capacitance are converted from strain energy by the piezoelectric transducer, it can be seen that the SSHI-based devices can promote the conversion of power from mechanical to electrical part but suppress the contrary, thus they can harvest more energy than the DC device. It can also be observed that when the SSHI-based devices are used, the waveforms of I eq are distorted. I eq depends directly on the strains of the piezoelectric patches, which are caused by the waves in the media, the distortions indicate that the harvesting processes in those cases could have non-negligible influences on the wave propagation at the location where the harvester is mounted. Since the switch is trigged at the time when I eq = 0, the distortions result in multiple inversions, which cause the fluctuation of the harvesting performance as revealed in figure 5.

Effects of piezo-lens on transient waves and harvesting

It is demonstrated in [START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF] that the piezo-lens can focus harmonic waves near the designed focal point in a large frequency band, thus it is expected that the piezo-lens could also focus transient waves if the main frequency components of the waves are within the effective frequency band of the piezo-lens, and harvesting at the designed focal point can yield improved energy. This expectation is verified in this subsection.

According to the results in [START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF], the piezo-lens used in this paper is effective from about 100 Hz to 8000 Hz. Thus the major frequency components of the waves generated by the excitation in figure 4 are totally within the effective frequency band. Firstly to study the effect of the piezo-lens on transient waves, the harvester is removed from the system since its presence at the focal point will make the energy concentration effect less obvious. The instantaneous transverse displacement w f at the designed focal point and input power are shown in figure 7(a). The results corresponding to the case without lens are also given as references, the maximum amplitudes of the transverse displacement and input power in this case are both normalized to unity. The piezo-lens will reflects a part of the incident waves and these reflected waves will interact with the excitation forces, thus the input power is a little bit reduced when the piezo-lens is active. Even though with this reduction of input power, the maximum amplitude of w f is improved. This result indicates that the transient waves are focused by the lens. To further verify the focusing effect, the transverse displacement of the host plate at the time corresponding to the maximum amplitude of w f is depicted in figure 7(b) for the case without piezo-lens and in figure 7(c) for the case with piezo-lens. From the comparison, it can be clearly observed that the transient waves are focused near the designed focal point.

Figure 8 compares the harvesting performances between the cases with and without piezolens. When the piezo-lens is used and due to the focusing effect demonstrated above, the maximum harvested energy is enhanced about 2.5 times no matter which kind of harvester is used. The input power is reduced when the piezo-lens is applied, consequently the maximum harvesting efficiency is improved about 3 times for each device as illustrated in figure 8(b).

In the above studies, it is verified that the piezo-lens can enhance the harvested energy from transient waves when the harvester is located at the designed focal point. However in the studies in figure 7 the harvester is removed from the system, it is not clear that whether the placement of the harvester near the piezo-lens will has significant influence on the location where the waves are concentrated. This is important since if the influence is nonnegligible, the designed focal point may not be the optimal place for harvesting. The energy concentration location only depends on the incident waves when the parameters of the piezolens are specified [START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF], thus if the harvester has negligible effect on the excitation source, it will not affect the incident waves and the consequent energy concentration location. To study the influence of the harvester on the transient source, the instantaneous input power is used as a criterion. Figure 9 shows the input power induced by the transient excitation when different harvesters are used in the harvesting system. In these studies, the storage capacitance for each harvester is chosen to make the output energy maximum. The reference input power in the figure refers to the case without any harvesting device, and the maximum of it is normalized to unity. It can be observed that the harvester do not affect the input power, accordingly they won't influence the generated transient waves and the energy concentration location.

Energy balance

Since the NC circuits in the piezo-lens are active elements and they need to be powered, it is also important to consider the energy balance of the harvesting system. In the piezolens, the NC circuits need to stiffen the structure. To realize such effect, the circuits should be fully reactive [START_REF] Matten | Conception robuste d'actionneurs électromécaniques distribués pour le contrôle vibroacoustique de structures[END_REF], namely, they doesn't dissipate any energy. Thus, the active energy consumed by the NC circuit in a time interval from t 1 to t 2 is expressed as:

E Cneg = t 2 t 1 V neg (t)I neg (t) = 1 2 C neg (V 2 neg (t 2 ) -V 2 neg (t 1 )) (24)
For the transient wave cases here, the consumed energy is estimated in the interval between the time when the transient wave package first arrives at the piezo-lens and the time when the package leaves the lens. Figure 10 compares the consumed and harvested energy in the cases with different devices. The storage capacitance for each device is optimal (namely the output energy is maximum); the harvested energy of the system with piezo-lens and DC device is normalized to unity. When the piezo-lens is applied in harvesting systems, it is observed that the amount of consumed energy is really small compared with the harvested one, 11% with the DC device and less than 4% with the SSHI-based devices. Thus even though we take into account the energy consumed by the piezo-lens, the harvesting systems incorporating the piezo-lens still can yield considerably improved energy compared with the cases without piezo-lens.

Practical application considerations

In previous sections, performance of the harvesting systems are verified in ideal situations with very small structural damping (ξ = 0.001) and neglected forward voltages of diodes. In this section, influences of these factors on harvesting performance are considered, and applicability of the harvesting systems is discussed. In new examples, the structural damping is given as ξ = 0.05, the forward voltage of each diode is 0.6 V, the inversion quality factor Q I for SSHI interface is chosen as 5.6 according to the experimental results in [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF], and the storage capacitance C s for each harvester is chosen to make the output energy maximum, i.e., C s = 14C * h for the DC harvester, C s = 7.4C * h for the P-SSHI-DC harvester and C s = 30C * h for the S-SSHI-DC harvester (C * h = 22.0 nF ). Figure 11 shows the waveforms of displacements and voltages in harvesting systems incorporating a piezo-lens. The measure location of the displacement is the left-bottom corner on the upper surface of the piezoelectric patch for harvesting. It can be observed that even though the maximum amplitude of the mechanical response of the patch is only micrometerscale, the harvesting systems are adequate to yield energy from the transient waves. It is noted here that the fast switch actions illustrated in figure 11(b) and 11(c) perhaps are difficult to obtain in piezoelectric mechanical systems. However, the piezo-lens can work at lower frequencies as long as the wavelengths are smaller than the characteristic length of the lens [START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF]. In those cases, the switch actions could be more realistic.

The improvement of harvesting performance by using piezo-lens in these new cases are illustrated in figure 12, in which, the mean power P 1 for the case without lens and the mean net power P 2 for the case with piezo-lens are defined as:

P 1 = E har1 t har1 , P 2 = E har2 -E Cneg t har2 (25)
here, E har1 and E har2 are the harvested energy obtained by equation ( 22), E Cneg is the consumed energy by the piezo-lens evaluated by equation ( 24), t har1 and t har2 are the charging duration of the storage capacitances (see figure 11). From figure 12, it can be seen that the piezo-lens is also very effective to improve the harvesting performance in these more realistic cases. Indeed, to realize the harvesting systems, there are still challenges. The NC circuits required in the piezo-lens need to be totally reactive to stiffen the structure, but the existing circuits that could realize negative capacitance all contain resistive parts which more or less will dissipate some energy [START_REF] Benjamin S Beck | The power output and efficiency of a negative capacitance shunt for vibration control of a flexural system[END_REF][START_REF] Tateo | Design variables for optimizing adaptive metacomposite made of shunted piezoelectric patches distribution[END_REF], thus they are potentially not suitable to realize a piezolens. However, it is hopeful that approximate fully reactive NC circuits could be achieved to reach the required stiffening effect in the coming future by optimizing the existing circuits (see [START_REF] Benjamin S Beck | The power output and efficiency of a negative capacitance shunt for vibration control of a flexural system[END_REF]) or using synthetic circuits as proposed in [START_REF] Gaël Matten | Synthetic impedance for adaptive piezoelectric metacomposite[END_REF]. Since the NC circuits for the piezolens are not determined at present, giving a rigorous energy balance analysis which takes into account the dissipated power is impossible, but it is logical to predict that the dissipated power by the piezo-lens could be very low since the NC circuits need to be (approximately) fully reactive. Besides, in some applications, the energy balance is not a critical issue. For example, a potential application of the harvesting systems is in Structural Health Monitoring (SHM) in large-scale structures. Sensors are sometimes embedded in structures at different locations in those cases, the maintenance is very difficult or even impossible when traditional batteries are used to power the sensors. Thus, the harvesting systems can be used to power these sensors. In these applications, the piezo-lens can be placed on the surfaces of structures thus it can be powered by traditional batteries rather than harvesters.

Conclusions

The combination of a piezo-lens with a harvester to obtain enhanced energy from transient traveling waves is studied. The standard DC, parallel SSHI-based and series SSHI-based harvesters are used. The SSHI interface can promote the conversion of the power from mechanical to electric part but suppress the contrary, accordingly the SSHI-based harvesters are more efficient than the standard DC one for harvesting energy from transient waves. The piezo-lens can focus transient waves near a designed focal point, thus placing the harvesters at that point can enhance the harvested energy about 2.5 -3 times and improve the harvesting efficiency about 3 times as compared with the cases without lens. This improvement of harvesting performance is obtainable when the realistic damping effect of the host structure and the forward voltages of diodes are taken into account.

Acknowledgement

The authors acknowledge Carnot Institute Ingénierie@Lyon for granting this study. The first author also thanks a scholarship provided by the China Scholarship Council.

Appendix: Outlines of the derivation of the fully coupled finite element models for piezoelectric systems

The potential DOFs in the piezoelectric patches are partitioned into DOFs inside the patches V i , DOFs on the free electrodes of the patches V p and DOFs on the bonding surfaces V g . The potential DOFs on the bonding surfaces are grounded, thus the corresponding equations and columns are directly removed from the equations. Consequently, the governing equations in [START_REF] Guyomar | Toward energy harvesting using active materials and conversion improvement by nonlinear processing[END_REF]

K dd K di K dp -K T di K ii K ip -K T dp -K T ip K pp      d V i V p    =    F Q i Q p    ( 26
)
Since there is no charge source inside the piezoelectric patches (Q i = 0), the internal DOFs V i can be determined by exact static condensation from equation [START_REF] Tsung | Focusing of the lowest antisymmetric Lamb wave in a gradient-index phononic crystal plate[END_REF]:

V i = -K -1 ii (-K T di d + K ip V p ) (27)
After the condensation, the system equations become:

M dd 0 0 0 d Vp + G dd G dp -G T dp G pp d V p = F Q p ( 28
)
with

G dd = K dd + K di K -1 ii K T di G dp = K dp -K di K -1 ii K ip G pp = K pp + K T ip K -1 ii K ip (29)
The DOFs V p are further separated into DOFs corresponding to the patches in the piezolens V Lp and DOFs of the patches for harvesting V hp :

  M dd 0 0 0 0 0 0 0 0      d VLp Vhp    +   G dd G dL G dh -G T dL G LL 0 -G T dh 0 G hh      d V Lp V hp    =    F Q Lp Q hp    (30)
As the DOFs on one electrode have identical potentials, the potential DOFs on the free electrode of each piezoelectric patch are reduced such that only one master potential DOF remains per patch. The reduction is achieved by using an explicit transformation:

V Lp V hp = T L 0 0 T h V L V h (31)
here, T = T L 0 0 T h is the transformation matrix, it represents the Null space of the linear transformation above, the nonzero terms in it is either 1 or -1. Substituting equation [START_REF] Kim | Efficiency of piezoelectric mechanical vibration energy harvesting[END_REF] into equation [START_REF] Collet | Modal Synthesis and Dynamical Condensation Methods for Accurate Piezoelectric Systems Impedance Computation[END_REF] and then left multiplying the transposed transformation matrix yield the final fully coupled governing equations:

  M dd 0 0 0 0 0 0 0 0      d VL Vh    +   H dd H dL H dh -H T dL C L 0 -H T dh 0 C h      d V L V h    =    F Q L Q h    (32)
with

H dd = G dd , H dL = G dL T L , H dh = G dh T h C L = T T L G LL T L , C h = T T h G hh T h Q L = T T L Q Lp , Q h = T T h Q hp (33)
namely, equations (11) in section 3.1.

Figure 1 :

 1 Figure 1: (a) The harvesting system with piezo-lens and (b) the gradient variation profile of the refractive index n(y).

Figure 2 :

 2 Figure 2: Top and side view of one unit shunted piezoelectric cell

Figure 3 :

 3 Figure 3: Tested harvesting devices.

Figure 4 :

 4 Figure 4: The applied tone burst excitation.

  m 0.001 m the wave propagation. The classical Runge-Kutta method is used to solve the ODEs using a fixed time step equal to 1 × 10 -6 s.

- 1 .Figure 5 :

 15 Figure 5: Performances of the harvesting system with piezo-lens when different devices are used, Q I = 3.

Figure 6 :

 6 Figure 6: Typical waveforms and converted energy when DC and SSHI-based devices are used to harvest energy from transient waves.

  (a) upper: normalized transverse displacement at the focal point; lower: normzlaized input powers (b) transvers dispalcement of the plate without lens at t = 3.14 ms. (c) transvers dispalcement of the plate with piezolens at t = 3.09 ms.

Figure 7 :

 7 Figure 7: Focusing transient waves.

Figure 8 :

 8 Figure 8: Comparison of the performances between the cases with and without piezo-lens. ---: with piezo-lens; --: without lens. Q I = 3.

Figure 9 :

 9 Figure9: Comparison of the input power when different devices are used to harvest the transient waves, the storage capacitance for each device is optimal, the reference input power refers to the case without any harvesting device.

Figure 10 :

 10 Figure 10: Comparison of the consumed and harvested energy in the harvesting system, the storage capacitance for each device is optimal, Q I = 3.

Figure 11 :

 11 Figure 11: Waveforms of transverse displacement and voltages when DC and SSHI-based devices are used to harvest energy from transient waves. The measure location of the displacement is the left-bottom corner on the upper surface of the piezoelectric patch for harvesting.

Figure 12 :

 12 Figure 12: Comparison of the mean harvested power between the cases without and with piezo-lens.

  are partitioned as below:

Table 1 :

 1 Geometry parameters

Table 2 :

 2 Material parameters of PZT26

	Symbol	Value	Property
	S E 11 = S E 22 , S E 33 S E 12 , S E 13 = S E 23 S E 44 = S E 55 , S E	1.30 × 10 -11 , 1.96 × 10 -11 (P a -1 ) -4.35 × 10 -12 , -7.05 × 10 -12 (P a -1 )	Compliance matrix under constant electric field

66

3.32 × 10 -11 , 3.47 × 10 -11 (P a -1 )

d 31 = d 32


Shaoyi Zhou

Claire Jean-Mistral

Simon Chesné
email: simon.chesne@insa-lyon.fr

Influence of internal electrical losses on optimization of electromagnetic energy harvesting

Keywords: vibration energy harvesting, electromagnetic device, shunt circuit optimization

A lot of effort has been made to optimize linear electromagnetic energy harvesters under harmonic or random excitation, connected to simple or more complicated electrical extraction circuits. Nevertheless, the internal electrical losses of electromagnetic coils are often neglected in these optimization analyses. To this end, the present paper investigates systematically and for the first time the influence of internal impedance of coils on energy harvesting performance under various types of ambient excitation sources, which are, respectively, external force acting directly on the seismic mass, base displacement-induced motion and disturbance generated by base acceleration. Our analysis highlights that under sinusoidal excitation, the resonant load outperforms its non-resonant counterpart in terms of energy harvesting performance when the internal resistance is very small, while its increase deteriorates significantly the broadband harvesting capability of resonant circuits. When subjected to random vibration, the resonant load presents no advantage compared to the latter one. The optimum design of non-resonant circuit is then carried out in each excitation scenario leading to well-known criteria and expanding to cases where no optimal conditions were defined or obtained. It is also reported that the neglect of internal losses underestimates the maximum available power.

Introduction

Energy harvesting from surrounding energy sources has attracted a lot of interest over the past few decades, and becomes a promising technique to supply low consumption of embedded electronic devices, such as actuators and wireless sensors [START_REF] Roundy | A piezoelectric vibration based generator for wireless electronics Smart[END_REF][START_REF] Anton | A review of power harvesting using piezoelectric materials (2003-2006[END_REF][START_REF] Harne | A review of the recent research on vibration energy harvesting via bistable systems Smart[END_REF][START_REF] Roundy | On the effectiveness of vibration-based energy harvesting[END_REF][START_REF] Lee | Robust segment-type energy harvester and its application to a wireless sensor Smart[END_REF]. Due to its high availability and ubiquity, mechanical vibration has received considerable attention and numerous transducers have been proposed and extensively studied in the literature, among which three most popular converting principles are: piezoelectric [START_REF] Erturk | Issues in mathematical modeling of piezoelectric energy harvesters Smart[END_REF], electromagnetic [START_REF] Wang | Vibration energy harvesting by magnetostrictive material Smart[END_REF] and electrostatic [START_REF] Boisseau | Optimization of an electret-based energy harvester Smart[END_REF].

As initially proposed by Williams and Yates [START_REF] Williams | Analysis of a micro-electric generator for microsystems[END_REF], a generic model of energy harvesters can be represented by a massspring-damper system of single degree of freedom (SDOF) coupled with an energy transducer. Tremendous research efforts have been devoted to energy harvesting optimization for this SDOF configuration under sinusoidal excitation [START_REF] Williams | Analysis of a micro-electric generator for microsystems[END_REF][START_REF] Roundy | A study of low level vibrations as a power source for wireless sensor nodes[END_REF][START_REF] Stephen | On energy harvesting from ambient vibration[END_REF][START_REF] Saha C R | Optimization of an electromagnetic energy harvesting device[END_REF][START_REF] Dutoit | Design considerations for mems-scale piezoelectric mechanical vibration energy harvesters[END_REF], leading to the well-known optimal condition which states that maximum power pumping is observed when the excitation frequency and electrical damping are matched to natural frequency and mechanical damping, respectively. However, as indicated by Tai and Zuo [START_REF] Tai | On optimization of energy harvesting from base-excited vibration[END_REF], a two-stage process has been adopted by a lot of authors to pursue the maximized energy transfer, in which the excitation and natural frequencies are matched prior to the matching between electrical and mechanical impedances. Nevertheless, in order to maximize the energy extraction from the mechanical to electrical domain, the electrical damping and excitation frequency should be simultaneously involved in the optimization analysis.

For that purpose, Tai and Zuo [START_REF] Tai | On optimization of energy harvesting from base-excited vibration[END_REF] conducted a twovariable optimization study for a SDOF energy harvester with either electromagnetic or piezoelectric transducer under baseexcited harmonic vibration, in which the coil inductance was omitted. The authors revealed that the long-believed optimal condition is correct only in the case of base excitation with constant acceleration amplitude, and is only an approximation for the constant displacement amplitude case under the condition of small mechanical damping (i.e. < 2%). They also underlined that in the latter excitation scenario, the optimal damping ratio and the excitation frequency are always greater than the mechanical damping and the natural frequency, respectively. Despite the novelty in their optimization process, the internal resistance of electromagnetic transducer was considered as a constant ratio of external resistive load instead of a fixed value, and the study was only conducted under specific harmonic cases. Besides, Tang and Zuo [START_REF] Tang | Vibration energy harvesting from random force and motion excitations Smart[END_REF] investigated the optimization of dual-mass and single-mass electromagnetic energy harvesters under random force and motion excitations with the coil inductance neglected. In [START_REF] Caruso | Broadband energy harvesting from vibrations using magnetic transduction Trans[END_REF], the electromagnetic energy harvester is connected with a resonant load and is subject to sinusoidal force excitation. With coil resistance being omitted, it was demonstrated that this specific energy converter exhibits the capability of harvesting maximum power over the whole bandwidth of frequencies. Mann and Sims [START_REF] Mann | On the performance and resonant frequency of electromagnetic induction energy harvesters[END_REF] studied a linear electromagnetic energy harvester connected with a simple resistive load with taking into consideration the total impedance of coil, and it was reported that the influence of coil inductance should not be neglected and its presence could alter the electrical resonance frequency. Wang et al [START_REF] Wang | Similarity and duality of electromagnetic and piezoelectric vibration energy harvesters[END_REF] investigated the similarity and duality between electromagnetic and piezoelectric energy harvesters by assuming that the internal resistance of coil is negligible. Tang et al [START_REF] Tang | Analytical solutions to h 2 and h ∞ optimizations of resonant shunted electromagnetic tuned mass damper and vibration energy harvester Trans[END_REF] derived the analytical formulae of electrical parameters for an energy harvester shunted with a resonant circuit under both harmonic and random force excitation, while the coil inductance was neglected. Zhang et al [START_REF] Zhang | Effects of electrical loads containing non-resistive components on electromagnetic vibration energy harvester performance[END_REF] investigated the effect of electrical loads containing non-resistive components (i.e. rectifiers and capacitors) on electromagnetic energy harvesting performance. In this work [START_REF] Zhang | Effects of electrical loads containing non-resistive components on electromagnetic vibration energy harvester performance[END_REF], the inductance of coil is again neglected and a conclusion has been drawn that optimum design for a purely resistive load can not be generalized to cases where non-resistive components are involved. However, ignoring a part of internal impedance could result in erroneous remarks for optimization analyses in certain excitation scenarios, which will be addressed in this work. Moreover, these aforementioned works only focused on energy harvesting performance under certain ambient vibration cases, and a complete and uniform optimization encompassing all excitation scenarios has not been yet proposed in literature.

Hence, the motivation and originality of our research is to propose a global optimization study for linear electromagnetic energy harvesting covering four ambient excitation cases and taking into account the internal impedance of electromagnetic transducer. The mechanical structure is shunted with three possible circuits: resistive (R), resistive-inductive (RL) and resistive-inductive-capacitive (RLC). And it undergoes two types of excitations: harmonic and random, which acts directly on the seismic mass (force type), or stems from the base movement (motion type).

The rest of this paper is organized as follows. Section 2 is dedicated to model the energy harvester associated with a generalized RLC circuit by taking into account the electrical losses of electromagnetic transducer. In this section, the Laplace transform method is adopted to yield the steady state frequency response. In section 3, the resonant circuit is considered and a three-variable optimization analysis is performed, which underlines the significant influence of electrical losses on its broadband energy harvesting performance. Section 4 considers the non-resonant circuit case, where some original results and optimal formulae of system parameters are presented. In this section, it is reported that the classic optimal condition is not always suitable for all excitation scenarios, and the exclusion of internal losses could result in a considerable underestimation of maximum attainable power, e.g. an underestimate of 10.4% is predicted for an energy harvester with mechanical damping ratio of 0.1 under force harmonic vibration.

Electromechanical modeling

We consider an electromagnetic vibration energy harvester, as depicted in figure 1. As mentioned earlier, one can distinguish three possible configurations of harvesting circuitry in practice, which are: (i) R type; (ii) RL type; (iii) RLC type (see figure 2). It is noticeable that the two first circuits can be regarded as particular forms of RLC circuit by vanishing certain electrical components. Hence, the proposed generalized model is based on a SDOF energy converter shunted by a resonant RLC circuit.

Force excitation

Figure 1(a) shows a single-mass electromagnetic energy harvester, where an excitation of external forcing type is applied directly on the mass. The equations governing the electromechanical system can de described by

+++ =  () ( ) Mx Cx Kx F F t a ¨1 ssse m f ò ++ - =  () Li Ri C idt E b 1 01 emf
where M s is the seismic mass, C s denotes the parasitic damping, K s is the equivalent stiffness, x is the absolute displacement of the seismic mass, i stands for the current induced in the coil, C is the capacitance and a dot denotes differentiation with respect to time. Moreover, R=R i +R e and L=L i +L e stand for the total resistance and inductance in the circuit, in which the subscripts, i and e, refer to internal impedance of electromagnetic transducer and external electrical loads, respectively.

F emf =k f i represents the reaction force generated by the electromagnetic transducer and acts on the mechanical subsystem.

= Ė kx emf v indicates the voltage induced in the coil. k v and k f are voltage and force constant of electromagnetic transducer, respectively. The relationship between k v and k f can be simplified as k v =k f =k e .

Taking the Laplace transform of (1) yields to: Where ξ m is the mechanical damping ratio, f is the frequency tuning ratio, κ 2 is the stiffness ratio (the electromechanical coupling stiffness k L e 2 divided by the mechanical stiffness K s ), ξ e is the electrical damping introduced by resistive load R e , α represents the excitation frequency ω normalized by the natural frequency ω s , λ indicates the internal electrical loss of coil, w = KM ss s denotes the undamped natural frequency of mechanical structure and w = LC 1 e refers to the resonant frequency of electrical circuit.

= ++ + + + () ( )
(
== = == = () C M k KL R L MR k 2 ,, 2 ,, . 4 m

Motion excitation

The single-mass energy harvester is now subject to motion excitation from the base, as depicted in figure 1(b). And its dynamics in such a scenario can be described by the mathematical model • Acceleration profile:

+- +- + =  () () ( ) Mx C x x K x x F a ¨0 5
(a) R circuit ( ¹ R 0 i , L i =0, =¥ C ); (b) RL circuit ( ¹ R 0 i , ¹ L 0 i , =¥ C ); (c) RLC circuit ( ¹ R 0 i , ¹ L 0 i , ¹¥
== - ++ + + + =- () ( ¯) () ( ) ( ¯) 7
Gs P X Mk R s Ms Cs K L s R s k s MG s ¨.
x We firstly carry out the optimum design of a SDOF energy harvester which is shunted with a resonant circuit and is subjected to sinusoidal vibration. By squaring (3), ( 6) and (7), the normalized harvested power through the resistive load R e for three excitation scenarios can be expressed as

fx k ¶ ¶ = ¶ ¶ = ¶ ¶ = ¯¯¯( ) PPP 0, 0, 0. 12 e
It is obvious that the harvesting performance is the same under all three excitations and the only difference resides in the change rate of harvested power with respect to excitation frequency α. By solving the previous differential equations [START_REF] Saha C R | Optimization of an electromagnetic energy harvesting device[END_REF], two common optimal expressions are found:

fa k xa =- + () A A 1 4 ,1 3 opt m 2 2 2 2 x xka xa lk lx a k xa = + ++ + ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ () AA 4 4 4 .1 4 e opt m m m m , 2 22
2 2 2 2 24 24 2 2 2
It is worth noting that the optimal frequency ratio f opt is exactly the same as that in [START_REF] Caruso | Broadband energy harvesting from vibrations using magnetic transduction Trans[END_REF], while the optimal electrical damping ξ e,opt deviates from the one in [START_REF] Caruso | Broadband energy harvesting from vibrations using magnetic transduction Trans[END_REF], where the difference results from the internal resistive loss λ.

Regarding the stiffness ratio κ, no optimal value could be found. As reported in [START_REF] Caruso | Broadband energy harvesting from vibrations using magnetic transduction Trans[END_REF], there is no constraint on the choice of κ at an excitation frequency lower than its natural mechanical frequency (  a 1), while for α>1, the harvested power reaches its maximum if the stiffness remains lower than a maximal value:

 kk xa = + () A A 4 .1 5 max m 2 2 2
In this study, the stiffness tuning ratio is chosen as κ opt =1 for any  a 1; otherwise, κ opt =γκ max , where γ is a known parameter and is less than unity.

Figure 3 compares electromagnetic energy converters connected with R circuit

(i.e. ¹ R 0 i , L i =0, =¥ C ) and RLC circuit (i.e. ¹ R 0 i , ¹ L 0 i , ¹¥ C
), in terms of output power for various internal electrical losses. The metric of harvesting performance is chosen as the half-power bandwidth, which is defined as the frequency bandwidth between which the harvested power is superior to the half of maximum power, as illustrated in figure 3(a).W h e n λ=0.1, the half-power bandwidth of RLC circuit is 4.98 times as many as that of resistive load. If the internal loss increases to 1, the improvement of resonant load compared to its resistive counterpart is significantly reduced to 63% merely. As λ goes up to 10, the two curves of harvested power relevant to resonant and resistive loads are completely coincident with each other. Therefore, a conclusion can be drawn that compared to simple resistive load, the resonant circuit can harvest energy at higher magnitude over a larger bandwidth when the internal electrical loss λ is relatively small, while this outperformance disappears rapidly as λ increases. One can also remark that the maximum output power is the same for these two electrical extraction circuits (R and RLC).

Figure 4 summarizes energy harvesting performances of the resonant circuit under different values of internal losses λ. In the case of an idealized transducer (i.e. λ=0), the energy scavenger can harvest power at the same magnitude over the whole frequency range, which depends only on the mechanical damping ratio ξ m and is equal to 1/16ξ m complying with [START_REF] Caruso | Broadband energy harvesting from vibrations using magnetic transduction Trans[END_REF]. When a non-idealized transducer is considered, the half-power bandwidth of energy extraction narrows rapidly and the peak output power reduces considerably as the internal electrical loss λ increases. When λ=0.1, the dimensionless peak power Pfm a x , is 1.238 with its dimensionless half-power bandwidth being 1. As λ arrives at 1, the peak power and half-power bandwidth reduce by, respectively, 8.2% and 67% compared to the case of λ=0.1. If a considerable internal loss presents (i.e. λ=10), the decreases of peak power and half-power bandwidth become to, respectively, 49.5% and 85.9%.

Random excitation

3.2.1.

Force-induced vibration. The electromagnetic harvester is now subjected to a random force vibration which acts directly on the mass. In light of the stability, linearity and time invariance of this energy harvester, the power spectral density (PSD) of harvested power S p can be formulated as [START_REF] James | Theory of servomechanisms[END_REF]

= ( ¯)|( ¯)| (¯)( ) Ss Gs Ss 16 pf i 2
where ( ¯) S s i denotes the PSD of the input random force, considered as a stationary Gaussian process, which has a sufficiently large bandwidth and whose PSD is independent of frequency, i.e. = ( ¯) S sS if . The mean square value of output power under white noise force excitation of spectral density S f c a nb ef o r m u l a t e da s : with all coefficients given by xk The indefinite integral in (17) can be calculated by either using the residue theorem [START_REF] Bak | Applications of the Residue Theorem to the Evaluation of Integrals and Sums[END_REF] or applying directly the analytical formulae provided in [START_REF] James | Theory of servomechanisms[END_REF][START_REF] Jeffrey | 3-4-definite integrals of elementary functions Table of Integrals[END_REF]. For the sake of completeness, several formulae are annexed in the appendix A. Then, the performance index of harvested power with a resonant circuit is expressed as

òò p w p == - -¥ ¥ -¥ ¥ [] ( ¯) ( ¯) ( ¯)(¯) ¯() EP S
xx l k x x f lx k x lk x f f == = = ==+ + = + + ++ = + + = () () bb b b aa a aa 0, , 0, 0, 1, 2 2 , 1 4 12 , 2 2 ,
== -- [] () PI EP SM ab aa a a a a a 1 .2 0 f f fs 31 123 03 2 1 2 4
It is clearly seen from (20) that PI f is a function of electrical damping ξ e , frequency tuning ratio f and stiffness ratio κ,i . e . PI f =PI f (ξ e , f, κ). In order to optimize the energy harvesting performance, a similar approach as in the case of harmonic excitation is employed to locate the global maximum of PI f (ξ e , f, κ),n a m e l y :

xfk ¶ ¶ = ¶ ¶ = ¶ ¶ = () PI PI PI 0, 0, 0. 21 f e ff
Now special attention is paid to the optimal condition related to the frequency tuning ratio f, which leads to a polynomial function in f in the concise form of

ff x f x x l k -+ + + = () ( ) ( ) 1 2 0. 22 mm e 22 2
Given that ξ e , ξ m and λ are all positive, hence only two potential roots can be retained: f opt,1 =0a n df opt,2 =1:

(1) f opt,1 =0. The nullity of f opt,1 imposes that the electrical resonance does not exist, i.e. =¥ C . Therefore, the ultimate energy harvesting performance under random force vibration is attainable by connecting with a non-resonant circuit, which will be investigated in detail in section 4.

(2) f opt,2 =1. In order to facilitate the optimization analysis, f opt,2 is substituted in the expression of PI f . As a consequence, the other two optimal conditions in (21) yield two expressions briefly formulated as:

xx l k lx lk lx x k x x ++ ++ + - = () [( ) ( ) ]
() a It is noticeable that no rational solutions could be obtained for ξ e and κ. Therefore, one can reach a conclusion that under random force vibration, no optimal configuration of resonant circuit exists for electromagnetic energy harvesting. In other words, no additional benefits could be obtained by introducing electrical resonance into harvesting circuitry under random vibration. where S x 0 represents the spectrum of random base displacement, and the integral term can be regarded as the square value of H 2 norm of ( ¯) G s x 0 . The corresponding It is remarked that the output power is infinite, which is due to the fact that the rational function ( ¯) G s x 0 is not strictly proper, according to Plancherel and Parseval's theorem [START_REF] De Oliveira | Fundamentals of Linear Control: A Concise Approach[END_REF]. Therefore, it suggests that the output power is of ideal white noise type when subjected to random base vibration of displacement profile.

Base acceleration type. Equation [START_REF] Wang | Vibration energy harvesting by magnetostrictive material Smart[END_REF] implies that when shunted with RLC circuit, the optimization procedure of an harvester undergoing a random acceleration disturbance is exactly the same as in the case of random force vibration.

Non-resonant circuit

In this section, an optimization analysis of electromagnetic energy harvester connected to a non-resonant circuit ( =¥ C , L=L i , figure 2(b)) is performed in the scenarios of sinusoidal and random vibration, and some ready-to-use formulae of optimal resistive load are derived analytically.

Sinusoidal excitation

Given that the frequency tuning ratio f=0, the expressions of normalized output power under all three excitation scenarios (8)-( 10) can be further simplified into: with all aforementioned coefficients given by: The optimal expressions of normalized excitation frequency α under different excitation scenarios will be derived separately in the following context. In order to solve the simultaneous equations (29) and (31), two different approaches could be imagined. The first approach is described as follows: with (31) being regarded as a cubic equation in α 2 , the closed form of optimal α 2 can be then obtained analytically as a function of ξ e by using the cubic formula provided in [START_REF] Weisstein | [END_REF]; then substituting it into (29), yields an eventual function which depends only on the electrical damping ξ e , from which one can determine ξ e,opt and consequently α opt can be achieved by the back substitution. The second approach is exactly the reverse sequence of previous one. The authors remark that the latter strategy is more effective to yield optimal expressions in a concise way, which is then adopted in the following study.

xl k k l k kl x l k xl x l k x lk lk =+ -+ = = =+ + + - =+ - = - == = () (( ) ) () (
The optimal expression for α can be eventually given in the form of: One can observe that the polynomial expression (32) is of order 20 in α, the results demonstrated in the following work are then calculated numerically. Figure 5 depicts the optimization analysis of electromagnetic energy harvesting under harmonic force (or base acceleration) excitation. Two series of simulations have been performed for two extreme cases, namely no electrical losses (-: λ=0, κ=1 × 10 50 ) and considerable losses ( : λ=100, κ=1). It is noticeable that as the mechanical damping increases, the optimal α remains unchanged under these extreme circumstances, thus one can conclude that the optimal scenario takes place when an energy harvester is excited at its resonant frequency. Including internal electrical losses in the optimization process leads to the well-known results. Figure 5(b) shows the evolution of optimal normalized resistive load defined as xk = Re e 2 for these two extreme cases. One can notice that Re is dimensionless.

+- = () ( )
Finally, we conclude that the optimal Re is formulated as

x l =+ ¯() R 1 42 34 e opt m ,
which can be rewritten in a dimensional form as:

=+ () RR k C 35 e opt i e s , 2
which is consistent with the finding of Stephen [START_REF] Stephen | On energy harvesting from ambient vibration[END_REF].A s depicted in figures 6(a)-(c), the harvested power including internal losses (34) is always greater than the power harvested without including losses, namely with

x = R 14 e opt m ,
. Hence, one can remark that the maximum output power (marked by filled circles in figure 6) can be successfully predicted by using (34) and the exclusion of internal losses will reduce the maximum attainable power. The underestimation of maximum attainable power rises from 0.2% (ξ m =0.01) to 10.4% (ξ m =0.10) when internal losses are neglected. In other words, the internal electrical losses can be neglected for lightly damped system, while for moderately or highly damped structure, this simplification can lead to considerable underestimation. And the underestimation of maximum power rises as the mechanical damping ratio ξ m increases. One can also observe from figure 6 that the normalized power Pf and the optimal load R e, opt decreases dramatically as the mechanical damping ratio ξ m increases. Figure 6(d) demonstrates the frequency responses of harvested power in the optimal scenario for three different values of mechanical damping ratio ξ m . All peaks locate at 1, namely the forcing frequency is equal to the natural frequency and their magnitudes match with the ones found in previous contour plots. And a decrease of 85% is recorded for the maximum achievable power as the mechanical damping ratio ξ m increases from 0.01 to 0.05. Similar to the previous case, the optimal expression with respect to α can be formulated as: The aforementioned strategy is again adopted in this problem. Figure 7 depicts the contours of normalized output power with respect to normalized excitation frequency α and normalized resistive load Re for two cases, namely no electrical losses (λ=0, κ=1 × 10 50 ) andwithlosses(λ=5, κ=25).Itis clearly seen from figures 7(a) and (b) that unlike the case of force/base acceleration vibration, the optimal excitation frequency deviates from its resonant frequency and more the mechanical damping increases, more the deviation is important in both cases with and without electrical losses. And it is noted that without electrical losses, no local maximum can be found when the mechanical damping ratio exceeds a certain limit, as shown in figure 7(c). For the purpose of comparison, the maximum powers predicted by the classical formula (34) in the force excitation case and by the exact equation (37) are located for different mechanical damping ratios in figure 7. One can observe that there does exist an extremum of power in the rational range of mechanical damping ratio ξ m .I ti sa l s o noticeable that the maximum power predicted by (34) is close to the exact one when ξ m is small, while the prediction error becomes considerable as ξ m increases gradually (at ξ m =0.10, an underestimate of 7.2% for maximum power is observed and the optimal excitation frequency deviates by 4% from its natural frequency).

+- = () ( ) ff f f f f f 03 7
Without electrical losses, a critical damping ratio ξ cr has been underlined above which no local maximum can be found. Figure 8 plots the change of critical damping ratio ξ cr as a function of internal resistive loss λ, with excluding the influence of self-inductance of coil, i.e. k ¥(the derivation of critical damping is discussed in detail in appendix C).Wi th λ=0, the critical damping ratio ξ cr is equal to 0.0754, which agrees with the results in figures 7(a)-(c). Besides, our original optimization analysis highlights that when the resistive loss λ lies in the interval [1, 3.5], the critical damping ratio is always larger than 0.1, which means that there always exist an extremum for a typical mechanical system (i.e. ξ<10%).I f λ resides outside this range, the critical damping ratio ξ cr fluctuates significantly and no evident distribution rule could be distinguished. with a 0 , a 1 , a 2 and a 3 defined as previously. Thus, the squared transfer function from the force to the square root of power is simplified as:

It is noticed that a negative sign is present in its numerator, which is due to the fact that a =-s 22 . The mean square value of output power under white noise random force excitation of spectral density S f can be formulated as: The influence of self-inductance can not be excluded if the product ω s L i is not ignorable any more. When the selfinductance L i of coil is negligible, the optimal resistive load is then related to the internal resistance by the following expression Substituting the optimal electrical damping ratio (45) into (43), the maximum performance index reads If the self-inductance L i is sufficiently small, i.e. k ¥ ,t h e maximum performance index (48) could be further simplified as Remark 1. The previous expressions indicate that for an electromagnetic energy harvester under random force excitation, there does exist an optimal external electrical load (thus a maximum of output power can be observed), when one considers the presence of internal resistance and selfinductance of coil. This new result contrasts sharply with the conclusion drawn in [START_REF] Tang | Vibration energy harvesting from random force and motion excitations Smart[END_REF] that no optimal value can be found when internal losses are neglected. mechanical damping ratio ξ m is preferred for greater energy harvesting performance, as depicted in figure 9.

òò p w p == - -¥ ¥ -¥ ¥ () [] ( ¯) ( ¯) ( ¯)(¯) ¯41 EP S
=+ () RR k C R .
Remark 3. As indicated in (48) and (49), the maximum performance index is inversely proportional to the seismic mass M s . And it is clearly seen from the simplified expression (49) that if the product λξ m is constant, a lighter mass will then enhance the harvesting performance under random force excitation.

Remark 4. Considering an extreme case where the electrical losses and inherent mechanical damping are sufficiently small (i.e. l  0, x  0 m and k ¥ ), then the maximum performance index (48) can be written in an extremely concise form: PI f,max =1/2M s , which is in accordance with [START_REF] Tang | Vibration energy harvesting from random force and motion excitations Smart[END_REF]. In this idealized case, the maximum output is only dependent of the seismic mass. With this transfer function begin not strictly proper, we conclude that the mean square value of output power in this scenario is infinite, as remarked in section 3.2.2. Base acceleration type. When the base acceleration is chosen as excitation input, the output power of energy harvester can be related to that under force excitation by

= [] [ ] E PM E P x s f ¨2 0
, according to [START_REF] Wang | Vibration energy harvesting by magnetostrictive material Smart[END_REF]. Therefore, the optimization procedure and the optimal formulations are the same as in the circumstance of random force excitation. Besides, the previous remarks 1 and 2 will hold in this scenario, while the last two remarks related to seismic mass (remarks 3 and 4) have to be modified due to the presence of factor M s 2 . Thus the upper bound on the harvested power is reformulated as

= [] E PS M 2 xm a x x s ¨, 0 0
, which agrees with [START_REF] Halvorsen | Energy harvesters driven by broadband random vibrations[END_REF][START_REF] Langley | A general mass law for broadband energy harvesting[END_REF].

Conclusions

To the best knowledge of the authors, this paper investigated systematically for the first time the influence of internal electrical losses of electromagnetic transducer on the optimization of electromagnetic energy harvesting under various excitations.

The optimization analysis related to energy harvesting coupled with a RLC circuit indicates that, under sinusoidal excitation, the energy converter can exhibit ultra-wide bandwidth harvesting performance in the absence of internal electrical losses, while this capability is considerably influenced by the presence of internal losses. The performance of energy harvester connected with resonant circuits ceases to outperform its counterpart coupled to a simple resistive circuit when the internal loss exceeds a certain threshold. When subjected to random excitation, the authors demonstrate that no additional gain can be attained by introducing electrical resonance in the circuit, i.e. the optimum design under random vibration is to connect electromagnetic transducers with non-resonant circuits.

When shunted by a non-resonant circuit, it is shown that the long-believed optimization condition always holds regardless of magnitude of internal losses under the circumstance of force or base acceleration vibration, while the optimal resistive load should be equal to the sum of internal resistance of coil and the electrical analog of the mechanical damping. In the case of base displacement vibration, the optimal excitation frequency is always greater than the natural frequency, and the degree of deviation depends on the mechanical damping. Besides, there exists a critical damping ratio ξ cr , beyond which no optimum can be achieved. And this threshold is 7.54% in the absence of electrical losses. An evolution of ξ cr with respect to the resistive loss λ is depicted, which suggests that there always exist a local maximum for a typical mechanical system (i.e. ξ<10%) if λ satisfies l Î [] 1, 3.5 . When subjected to random excitation, an optimal expression of resistive load is derived analytically in the scenario of force or base acceleration vibration, which contrasts sharply with the existing literature in which the influence of internal losses is excluded and a conclusion has been drawn that no optimum could be obtained. It is also noted that the upper bound of attainable power is only a function of the seismic mass and the PSD of excitation sources.

It should be emphasized that the current work focuses on linear electromagnetic energy harvesting so that the conclusions made and optimal parameters derived can not be extended to cases where nonlinearity presents in the electromechanical system.

Finally, the optimization analyses performed in this paper is based on maximizing the power harvested by the resistive load. Meanwhile, optimum design of harvesters can be also carried out according to other metrics, such as energy harvesting efficiency, which is defined as the ratio between the power harvested by the electrical load and the total input energy. The proposed optimization framework can be easily adapted to analyses according to optimizing the energy harvesting efficiency. It is observable that dP I f /d ξ m <0 always holds for any positive value of ξ m .

Appendix C. Derivation of critical damping ratio ξ cr

By omitting the influence of self-inductance of coil, the eventual optimal expression (37 As mentioned in [START_REF] Tai | On optimization of energy harvesting from base-excited vibration[END_REF], the necessary and sufficient condition of absence of local maximum for a polynomial equation is that its corresponding discriminate is equal to zero. Given that (C.1) is of order 5 in α 2 , the critical damping ξ cr at each value of λ is estimated numerically by using Maple. For λ=0, the previous quintic equation (C.1) is reduced to cubic one, and its discriminate can be analytically expressed as  which yields the value at ξ cr,λ=0 =0.0754, validating the finding in [START_REF] Tai | On optimization of energy harvesting from base-excited vibration[END_REF].

)

ORCID iDs

Shaoyi Zhou https:/ /orcid.org/0000-0002-1739-5729

Figure 1 .

 1 Figure 1. Electromagnetic energy harvester: (a) external force excitation; (b) base excitation.

Figure 2 .

 2 Figure 2. Circuit configurations corresponding to three possible approximations: (a) R circuit ( ¹ R 0

Figure 3 .

 3 Figure 3. Normalized power Pf versus normalized frequency α for ξ m =5% and γ=0.5 under harmonic force excitation: (a)-(c) harvesting performance of a SDOF energy scavenger shunted with R and RLC circuits for different internal electrical losses λ=0.1, 1 and 10.

3. 2 . 2 .

 22 Base motion vibration. Base displacement type. When the energy harvester undergoes a random motion excitation of displacement type, the mean square value of output power is expressed as:

Figure 4 .

 4 Figure 4. Harvesting performance of a SDOF energy harvester shunted with RLC circuit for ξ m =5% and γ=0.5 under harmonic force excitation. (-: λ=0, LLL: λ=0.1, ---: λ=1, -•-•-: λ=10).

where f 1

 1 =e 1 and f 2 =e 2 , and other coefficients are expressed as:

Figure 5 .

 5 Figure 5. Optimization analysis harmonic force or base acceleration-induced vibration for two extreme cases (-: λ=0, κ=1 × 10 50 ; : λ=100, κ=1): (a) optimal normalized excitation frequency α opt against mechanical damping ξ m ; (b) optimal normalized resistive load Re opt , versus mechanical damping ξ m .

4. 2 .

 2 Random excitation 4.2.1. Force-induced vibration. The SDOF electromagnetic energy harvester attached with RL circuit, as shown in figure 2(b), is now subject to random force excitation. The frequency tuning ratio f is equal to zero and the generalized

Figure 6 .

 6 Figure 6. Harvesting performance under harmonic force or base acceleration-induced vibration for different mechanical damping ratios with λ=5 and κ=25: (a)-(c) contour of normalized power Pf (or Px ¨0) as a function of α and Re (filled blue square markers: maximum power without electrical losses, filled red circle markers: maximum power predicted by (34) with consideration of internal losses); (d) frequency responses of harvested power with optimal parameter (34)( black line: ξ m =0.01, red line: ξ m =0.05, cyan line: ξ m =0.10).

Figure 7 .

 7 Figure 7. Contours of normalized output power Px0 as a function of α and Re under harmonic base excitation of displacement profile for two series of parameters: (a)-(c) no electrical losses (λ=0, κ=1 × 10 50 ); (d)-(f) with electrical losses (λ=5, κ=25). Filled blue square marker: maximum power predicted by (34) with consideration of internal losses; Filled red circle marker: maximum power obtained by (37).

transfer function ( 3 )

 3 can be further reduced to:

4. 2 . 2 .

 22 Motion-induced vibration. Base displacement type. When the base displacement is again characterized by a stationary random process, we focus on the corresponding transfer function which leads to:

Figure 8 .

 8 Figure 8. Critical damping ratio ξ cr versus internal resistive loss λ with k ¥under harmonic base excitation of displacement profile.

Figure 9 .

 9 Figure 9. Maximum performance index PI f,max predicted by (48) as a function of mechanical damping ratio ξ m and internal electrical loss λ under random force excitation with M s =1 and κ=25.

For4

  Appendix A. Formulae of indefinite integralsSeveral formulae are provided here for indefinite integrals described by the highest power of h n (x), with the two functions being in the general form =++ n=3 and 4, the integrals are analytically formulated by Derivative of PI f with respect to ξ mThe denominator function of optimum performance index (Therefore, the derivative of optimum performance index PI f with respect to mechanical damping ratio ξ m is formulated as

  It should be mentioned that the normalized power Pf , Px 0 and Px ¨0 are all dimensionless. And their common denominator D A=α 2 -1 and B=f 2 -α 2 . It is noted from (8) to (10) that the output power is now controlled by electrical damping ratio ξ e , frequency tuning ratio f and stiffness tuning ratio κ. Mathematically speaking, the global maximum of

	is given by
	=+ + [( DA B 12	l x ka ) mm e x xka + ] 8 22 2 24
	lx x k a ++ 16 m e 2 24	16	x x a me 224
	xl k a ++ () 24 + AB x a e m 22 2 2 2 22	-	4	l x k a m 22	A B
						() 11
	with xf k ¯() P ,,
	• Force excitation:
	w == ¯() xka P FM D 2 .8 ss 2 f e P 24
	• Displacement excitation:
	w == ¯() xka P 28 P KX D ss 0 2 .9 e x 2 0
	• Acceleration excitation:
	w == ¯() xka P 24 P MX D 2 ss ¨.1 0 x e ¨0 2 0

e is attained at points satisfying the following conditions

  .

	01	e	2	23
	01		me	2	2	m e	2
			me 2 3		m 22	4	2
							19

  =0, b 1 =-ξ e κ 2 and b 2 =0. The performance index of energy harvester is then expressed as:

							One can also transform (45) into a dimensional form of
							w =++ RL k M L 22 ++ C M LR k s s s C RR .4 6 () e opt s i e i s ii i , i e 22 2 2
	1 fp 2	sd	S jM f s		gs hsh 3 33	s	ds
	where E[•] denotes the mean square value, and these two
	functions are defined as, respectively:
	( ¯) gs 3	=++ ¯( bs b s 0 4 1 2	b	2
	=++ ¯¯¯() + as as as a 42 30 ¯) h s 3 1 2 23
	with b 0 From (43), the mean square value of output power E[P f ] is
	now controlled by the electrical damping ratio ξ e . The global
	maximum of PI(ξ e ) is located at points satisfying the
	following condition:
			x PI ¶ f ¶ e	=	() 04 4
	which yields the following optimal expression
	xl x k =+ + ⎡ ⎣ ⎢ () 1 4 11 2 e opt mm l x + + () 1 2 , 2 2	lk x m 2 4	⎤ ⎦ ⎥	( ) .4 5

  Remark 2. Given that the internal electrical losses R i and L i are constant, the optimum energy performance index (48) is a monotonically decreasing function of ξ m (a detailed derivation is provided in the appendix B). It suggests that a smaller

	|( ¯)| Gs 2 f	w =´-+++ ¯¯)[ ( ¯)( ¯)( ¯)] xk -+ -+ -+ ( M s as as as a a s a s a s a 22 2 .4 0 () e ss 0 3 1 2 23 0 3 1 2 23
		PI fm a x ,	=	M 1 s	kl x kx ++ 28 88 + m m 22 2	x	k 1 mm l x k + + () 2 1 2	l k 22 4 +	+	lk x 2 m 4	() .4 8
	PI f	== [] EP SM 1 f fs	xk + + ( 12 2 e m e x x 22 lx k + () [ x x lk ++ ) ] ( -2 2 14 me m	2	x	e	+	lk	2	)	() .4 3


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