Energy harvesting from vibration of Timoshenko nanobeam under base excitation considering flexoelectric and elastic strain gradient effects

Journal of Sound and Vibration 421 (2018) 166e189

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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Energy harvesting from vibration of Timoshenko nanobeam under base excitation considering flexoelectric and elastic strain gradient effects S.A.M. Managheb, S. Ziaei-Rad, R. Tikani* Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 May 2017 Received in revised form 27 January 2018 Accepted 30 January 2018

The coupling between polarization and strain gradients is called flexoelectricity. This phenomenon exists in all dielectrics with any symmetry. In this paper, energy harvesting from a Timoshenko beam is studied by considering the flexoelectric and strain gradient effects. General governing equations and related boundary conditions are derived using Hamilton's principle. The flexoelectric effects are defined by gradients of normal and shear strains which lead to a more general model. The developed model also covers the classical Timoshenko beam theory by ignoring the flexoelectric effect. Based on the developed model, flexoelectricity effect on dielectric beams and energy harvesting from cantilever beam under harmonic base excitation is investigated. A parametric study was conducted to evaluate the effects of flexoelectric coefficients, strain gradient constants, base acceleration and the attaching tip mass on the energy harvested from a cantilever Timoshenko beam. Results show that the flexoelectricity has a significant effect on the energy harvester performance, especially in submicron and nano scales. In addition, this effect makes the beam to behave softer than before and also it changes the harvester first resonance frequency. The present study provides guidance for flexoelectric nano-beam analysis and a method to evaluate the performance of energy harvester in nano-dielectric devices. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Energy harvesting Flexoelectricity Electromechanical coupling Strain gradient Timoshenko beam theory Short circuit natural frequency Open circuit natural frequency

1. Introduction The flexoelectric effect was discovered in the mid-20th century. However, it was ignored by researchers for a long time because of its insignificant effect in macroscopic level. Piezoelectricity only exists in central asymmetric materials; however, strain gradients can locally break central symmetry and induce polarization even in centrosymmetric crystals. It is called flexoelectricity. It is noticeable that most of references and studies on the flexoelectricity have been concerned with its nature while the application of this effect, such as energy harvesting, crack detection in structures and etc. has been neglected to a large extent. The review part of this study was carried out in two parts. First, theoretical and experimental studies on flexoelectricity are mentioned. Second, a brief review of energy harvesting from piezoelectric materials is presented in terms of interest field of this study. In 1964, Kogan [1] introduced a phenomenological description for electric polarization due to strain gradient in solid crystals. Later, Indenbom proposed the flexoelectricity for liquid crystals [2]. In the 1980s, Tajantsev [3] conducted a broader

* Corresponding author. E-mail address: [email protected] (R. Tikani). https://doi.org/10.1016/j.jsv.2018.01.059 0022-460X/© 2018 Elsevier Ltd. All rights reserved.

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study on the flexoelectric effect based on lattice dynamics theory and he presented an explicit expression for the coefficients of flexoelectricity. Using lattice dynamics prediction and Tajantsev theory, numerous studies have been conducted concerning flexoelectricity. Marangati et al. [4] evaluated the effects of flexoelectricity and non-local size effects associated with nonuniform strain. Next, Majdoub [5] utilized molecular dynamics method to explain the flexoelectric effect, and he analyzed size-depended elastic and piezoelectric behavior which depend on the size. In contrast to previous developed theories on hard ceramic crystals, Deng et al. [6] extended the nonlinear theoretical framework of flexoelectricity due to soft materials and biological membranes. Experimental measurement of the flexoelectric coefficients of ferroelectric materials is another issue that can be referred to it [7]. The results suggested that perovskite ferroelectrics may exert stronger flexoelectric effect than lattice dynamics predictions. Theoretically and computationally, the variation principle is utilized for calculation of electromechanical problems for a long time. Hu and Shen [8] offered the variation principle basis of electric enthalpy for nano-sized dielectrics. With introducing flexoelectric effect, Shuling et al. [9] studied nano-scale dielectric with concurrent consideration of polarization gradient, strain gradient, and effects of electrostatic force. They obtained governing equations such as the electrostatic force using variation principle. Yan et al. [10] investigated the flexoelectric effect on the static bending and free vibration of a simply supported piezoelectric nanobeam. The process of obtaining energy from the environment and converting it into usable electrical energy is called energy harvesting. Among different mechanisms that convert mechanical energy (vibrations) into electrical energy, the piezoelectric mechanism has attracted attention due to ease of use. There are numerous literature reviews of this mechanism in terms of certain intrinsic properties of electromechanical coupling (e.g. high power density) [11,12]. Providing a discrete model consists of mass, spring and damper, Mikio et al. presented the first model for energy harvesting [13]. In next years, Roundy et al. improved the previous developed models [14]. Ajitsaria et al. investigated on a bimorph piezoelectric cantilever beam using assumptions of Euler-Bernoulli and Timoshenko beam and they presented an analytical solution for the harvesting problem [15]. According to studies of some researchers on the cantilever piezoelectric energy harvester, Erturk and Inman [16] presented an analytical solution for bimorph beams with series and parallel layers. The expression for closed form steady state was developed at arbitrary frequencies of harmonic excitation and electromechanical frequency response was defined based on from single- and multi-mode solutions. Junior et al. [17] presented an electromechanical finite element model for piezoelectric energy harvester plates. They expressed a finite element plate model to predict the output electric power of piezoelectric harvester plates based on Kirchhoff plate assumptions. Finally, they validated their finite element model by analytical solution. In piezoelectric energy harvesting topic, Cheng et al. [18] investigated the beam energy harvester made of an aluminum substrate surface bonded with piezoelectric patches and a stack actuator. They achieved an effective energy harvesting knowledge with a new frequency self-tuning method. The natural frequency of the harvester is tuned by the function of the stack actuator. They utilized an iterative numerical technique to solve the dynamic response and the produced electric charge in order to present the energy harvesting and the selftuning process. They studied the sizes of the piezoelectric patch and its effects on the energy harvester performance. Their results show that the self-tuning process is very important and significantly increases the power output by several times. Finally, the effectiveness of the proposed self-tuning method is validated with finite element method. De Paula et al. [19] illustrated influences of nonlinearities in the energy harvesting from a piezo-magneto-elastic structure subjected to random vibrations. In their research, numerical studies enable comparison of generated voltages by single-stable and bi-stable linear and nonlinear systems. Recently, Leng et al. [20] studied a tri-stable piezoelectric energy harvester with external magnets. They employed a method based on equivalent magnetizing current theory to evaluate the magnetic force and the potential function with triple wells. They explored that the method is appropriate for different magnet intervals. They applied Gaussian noise for excitation of the harvester (random excitation). Results show that the TPEH's frequency bandwidth is broader than BPEH and improves the output voltage compared to BPEH. On the other hand, development of technology in the field of electronic circuits enabled design and manufacturing of electronic devices with very low energy consumption and small dimensions. Therefore, the novel use of empowering micro and nano-systems without batteries is most favored. The use of vibrational energy to generate the power for low-power devices such as micro and nano-electromechanical systems has attracted a lot of attention [21,22]. Thus, many research activities have been conducted concerning piezoelectric energy harvesting. For example, Muralt and Xu studied piezoelectric energy harvesting of micro and nano-scales developing thin ferroelectric films and non-ferroelectric nanowires [23,24]. In another study, Deng et al. [25] addressed the Euler-Bernoulli beam energy harvesting based on a mathematical framework for flexoelectric coupling. In this study, linear constitutive equations are used to describe the elastic, dielectric and flexoelectric behavior of different materials. Wang et al. [26] developed an analytical model for unimorph Euler-Bernoulli piezoelectric energy harvester, which includes the flexoelectric effect. Their research was carried out on the nanoscale with the desired length and position of the piezoelectric layer. Approximate solution has been obtained to check the output voltage, power and load resistance optimum. The results showed that the maximum power output of the model with flexoelectric effect is several times more than the classic model that has only piezoelectric effect. Also, in their study, methods have been proposed to increase the efficiency such as appropriate length of the piezoelectric layer. Based on the review of previous studies, analysis of dynamic behavior of beams at nano-scale requires consideration of flexoelectricity effect and study of this effect in small scales is necessary due to larger strain gradient in such scales. Up to now, a few researches on flexoelectric effect and in particular its ability in energy harvesting has been carried out. Works are often concentrated on flexoelectric nature and conducted researches on its application such as energy harvesting have used simple

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theories. One study which is cited in the manuscript is conducted for energy harvesting of Euler-Bernoulli Beam. No work has been reported for energy harvesting of Timoshenko nanobeam considering flexoelectric effect. It is clear that the continuum models are approximate models for real nano-scale systems. The accurate model for such systems may obtain from Molecular dynamics (MD) or Ab initio models. Although the MD models consider the details of what happens in the reality in a large extent, they need large computation costs. In addition, at present, modeling a system containing wires, resistance and beam cannot be easily done by them. Therefore, nowadays nonlocal continuum model with considering size effect (such as strain gradient theory, coupled stress theory and so on) are widely used for modeling of systems in micro and nano-scale. The nonlocal continuum models are based on some parameters that should be measured in nano-scale or computed by MD or other similar methods. Using the measured data, we are sure that the nonlocal continuum model is able to predict the behavior of the energy harvester of current study in a good engineering degree of accuracy. In our work, in order to more precisely study the flexoelectric effect and the effective parameters in the energy harvesting, a model of Timoshenko beam has been developed with flexoelectric and strain gradient elasticity effects at both submicron and nanoscale. The model takes into account shear deformation, rotational bending effects and strain gradient elasticity, making it suitable for describing the behavior of short thick beams and beams subject to high-frequency excitation. All necessary coefficients of flexoelectricity and strain gradient elasticity which are ignored in other studies are considered here. It is noteworthy that in previous and similar works on energy harvesting, such coefficients were totally neglected. In other words, such comprehensive model for the beam, namely Timoshenko beam theory, is essential and more realistic in the study of flexoelectric energy harvester due to the strong effect of flexoelectricity at nanoscale and high frequency of vibration. The main objective of the current study is to develop a size-dependent flexoelectric nanobeam energy harvester model considering the elastic strain gradient effects based on the Timoshenko beam theory and Hamilton's principle, extending the work on the BernoullieEuler beam reported in Ref. [25]. In the current investigation, an appropriate flexoelectric energy harvester model was developed. The analytical model of the harvester was derived and then validated by some previous works. Since normal and shear strains are considered for the Timoshenko beam, the flexoelectric effect induced by the gradient of these strains leads to a more generalized model. Also, voltage and power density output of the harvester in response to the harmonic motion of the base was obtained analytically. 2. Derivation of the electromechanical model The proposed structure for flexoelectric energy harvesting is shown in Fig. 1. The relevant coordinate system is considered as x1 , x2 and x3 . The harvester system includes a cantilever beam under base excitation; the electrical circuit that consists of a load resistance and two thin conductive electrodes to collect the current. It is supposed that the electrodes cover the entire surfaces of the beam and they are connected to a resistive electrical load. The beam cross section is Ap with the width of b and height of h. The beam length is L and its material destiny is displayed by r. The beam is assumed to be thick and thus the Timoshenko beam theory is used for its modeling. The base motion includes both rotational and translational movement which is applied to the structure and it is expressed as wb ðx1 ; tÞ ¼ gðtÞ þ x1 hðtÞ where gðtÞ and hðtÞ are the transverse motion and the rotation of the beam cross section, respectively. The dynamic strain gradient under this excitation causes potential difference between electrodes that will be calculated in this study.

Fig. 1. Flexoelectric energy harvester of Timoshenko thick beam under base excitation. (a) 3D schematic; (b) 2D schematic.

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2.1. Extended Hamilton's principle considering flexoelectric electromechanical coupling In this section, the equations of motion will be extracted considering the effects of flexoelectricity. It should be noted that centrosymmetric harvester with a symmetric cross section could not harvest any power from the configuration shown in Fig. 1. This is due to the symmetric configuration of the harvester with a single layer. However, if one considers the flexoelectricity effect, the structure shown in Fig. 1 is capable of harvesting the energy and this is one of the advantages of flexoelectric energy harvesters. The extended Hamilton's principle is given by Ref. [27].

Zt2 ðdT  dH þ dWÞdt ¼ 0

(1)

t1

where T; H and W are kinematic energy, electric enthalpy density and the work done by external forces and fields respectively and can write:

Zt2

Z dt

t1

   Zt2 Z  Zt2 Z  1 1 a 2 2 0 0 L a F $du þ E $dP dV þ dt d rju_ j  W  ε0 jV∅j þ P$V∅ dV þ dt ðt$dua þ Dd∅ÞdA ¼ 0 2 2 t1

V

V

t1

vV

(2) where ua is the absolute displacement ðu ¼ fua1 ; ua2 ; ua3  wb ðtÞgT Þ, V∅ and P are the gradients of potential field and polarization density in the beam, t, E0 , and F0 are the external surface traction, electric field and body force, respectively. As it is mentioned before, H is the electric enthalpy density, and can be expressed as [28,29]:

 H¼

1 W L  ε0 jV∅j2 þ P$V∅ 2

 (3)

where the component of internal energy density, W L can be stated as [30]:

1 1 1 W L ¼ Cijkl Sij Skl þ dijk Pi Sjk þ fijkl Pi hjkl þ gijklmn hijk hlmn þ akl Pk Pl 2 2 2

(4)

where S and h are infinitesimal strain tensor and strain gradient tensor. The coefficients in Eq. (4) are defined as follows; in the first term (elastic potential term), C is the elastic modulus fourth-order tensor. The second term (the coupling between polarization and strain) contains a third-order piezoelectric tensor displayed with d. The flexoelectric coupling between the polarization and the strain gradient is through the third term with a fourth-order flexoelectric tensor  f. The fourth term in the above equation is the elastic potential of strain gradient with the sixth-order tensor g. Finally, the last term is associated with the electrostatic potential. 2.2. Flexoelectric model formulation for electromechanical Timoshenko beam Considering Timoshenko beam theory, the relative displacement field at any point x1 and time t can be expressed as [31]:

 u¼

 x3

T vjðx1 ; tÞ ; 0; wðx1 ; tÞ vx1

(5)

where jðx1 ; tÞ and wðx1 ; tÞ are the cross section rotation and transverse displacements of the neutral axis, respectively. According to this displacement field, the normal and shear strains can be obtained as follows and other strain components are zero.

 S11 ¼

 vjðx1 ; tÞ vx1   vwðx1 ; tÞ ¼ 0:5  jðx1 ; tÞ vx1

 x3

S13 ¼ S31

Therefore, the non-zero strain gradient components can be written as:

(6)

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S11;1 ¼ S11;3

 x3

! v2 jðx1 ; tÞ

vx21   vjðx1 ; tÞ ¼  vx1

S13;1 ¼ S31;1 ¼ 0:5

(7)

v2 wðx1 ; tÞ vx21

! vjðx1 ; tÞ  vx1

Due to these strain gradients, the local centrosymmetry will be broken and as a result, the polarization will be induced at the beam. For a simple ionic lattice, a schematic representation that illustrates this phenomenon is shown in Fig. 2. The introduced strain gradients in Eq. (7) which break the inversion symmetry cause relative displacements of the centers of negative and positive charges (The blue and red particles in Fig. 2) and thus the polarization will be induced in the ionic lattice [32]. The polarization density field for the beam can be considered as:

Pðx1 ; x3 ; tÞ ¼ f0; 0; Pðx1 ; x3 ; tÞgT

(8)

Given the above assumptions, the internal energy density, W L , can be developed as:

2   !     14 vjðx1 ;tÞ 2 vjðx1 ;tÞ vwðx1 ;tÞ 2 a33 P þ C1111 x3 W ¼  jðx1 ;tÞ ðC1113 þC1131 þC1311 þC3111 Þ þ0:5k x3 2 vx1 vx1 vx1 (     2 vwðx1 ;tÞ vjðx1 ;tÞ vwðx1 ;tÞ þ0:5  jðx1 ;tÞ ðC1313 þC1331 þC3113 þC3131 Þþ 2P d311 x3 þ0:25k vx1 vx1 vx1 ! !!    v2 wðx1 ;tÞ vjðx1 ;tÞ v2 jðx1 ;tÞ  jðx1 ;tÞ ðd313 þd331 Þþ0:5  þf Þþ f ðx ðf þ f 3131 3311 3111 3 3113 vx1 vx21 vx21 ! ! )! vjðx1 ;tÞ v2 jðx1 ;tÞ v2 wðx1 ;tÞ vjðx1 ;tÞ þ 0:5ðx3  ðg111131 þg111311 þg131111 þg311111 ÞÞ  2 vx1 vx1 vx1 vx21 !2   v2 jðx1 ;tÞ v2 jðx1 ;tÞ vjðx1 ;tÞ vjðx1 ;tÞ 2  Þþð0:5ðx Þð Þðg þg ÞÞþðg Þ þðg111111 x3 3 111113 113111 113113 vx1 vx1 vx21 vx21 L

vjðx1 ;tÞ v2 wðx1 ;tÞ vjðx1 ;tÞ Þð  Þðg113131 þg113311 þg131113 þg311113 ÞÞ vx1 vx1 vx21 3 !2 v2 wðx1 ;tÞ vjðx1 ;tÞ  ðg131131 þg131311 þg311131 þg311311 ÞÞ5 þð0:25 vx1 vx21

þð0:5ð

(9) where k is the shear correction factor. To simplify the calculations, the following statements are defined:

Fig. 2. Schematic of induced polarization by the strain gradient theory.

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171

Κc1 ¼ ðC1113 þ C1131 þ C1311 þ C3111 Þ Κc2 ¼ ðC1313 þ C1331 þ C3113 þ C3131 Þ Κf ¼ ðf3131 þ f3311 Þ Κd ¼ ðd313 þ d331 Þ Κg1 ¼ ðg111131 þ g111311 þ g131111 þ g311111 Þ Κg2 ¼ ðg113131 þ g113311 þ g131113 þ g311113 Þ Κg3 ¼ ðg131131 þ g131311 þ g311131 þ g311311 Þ Κg ¼ ðg111113 þ g113111 Þ

The kinetic energy of the beam that includes rotary inertia effect can be written as T ¼

(10)

R V

1rju _ a j2 dV 2

and its variation is

expressed as follows:

Zt2

d

Z dt

t1

1 a2 rju_ j dV ¼  2

Zt2 t1

V

Z dt

ru€a dudV

(11)

V

By applying the variation of the internal energy density, and the kinetic energy, and considering the lack of external electric fields and body forces, Eq. (2) can be rewritten as:

Zt2

Z dt

t1

V



r 



        € ðx ; tÞ þ 0:5 1  x vjðx1 ; tÞ Κ c1 k þ 0:5 vwðx1 ; tÞ  jðx ; tÞ Κ c2 k þ PΚ d dðjðx ; tÞÞ  x23 j 1 3 1 1 2 vx1 vx1

8       Zt2 Z <" vjðx1 ; tÞ vwðx1 ; tÞ v2 wðx1 ; tÞ €þw € b dw dV þ dt þ 0:5  jðx1 ; tÞ Κ d þ 0:5 þ w a33 P þ d311  x3 : vx1 vx1 vx21 t1 V 8 ! ! # >      <  2 vjðx1 ; tÞ v j ðx ; tÞ v j ðx ; tÞ v∅ vjðx1 ; tÞ 1 vwðx1 ; tÞ 1 1  þ C  x þ  d P  x Κ f þ f3111  x3 þ f 3113 3 1111 3 > vx1 vx1 vx3 vx1 4 vx1 vx21 : ! ! #  "    1 v2 jðx1 ; tÞ vjðx1 ; tÞ 1 v2 wðx1 ; tÞ vjðx1 ; tÞ g g2 g jðx1 ; tÞ Κ c1 k þ Pd311 þ Pf3113 þ þ  þ  Κ Κ  x3 113113 2 vx1 4 vx1 vx21 vx21 9 ! ! # " >   = vjðx ; tÞ 1 v2 jðx1 ; tÞ 1 vjðx1 ; tÞ g2 v2 wðx1 ; tÞ vjðx1 ; tÞ 1 g1 g3  x3  Κ þ þ 0:5  d Κ Κ þ0:5 PΚ f þ > 2 2 vx1 vx1 vx1 vx21 vx21 ; "         1 vjðx1 ; tÞ c1 vwðx1 ; tÞ vwðx1 ; tÞ 1 v2 wðx1 ; tÞ  x3 Κ k þ 0:5  x3 Pf3111 þ  jðx1 ; tÞ Κ c2 k þ PΚ d d þ0:5 2 vx1 vx1 vx1 4 vx21 ! ! # " !   vjðx1 ; tÞ v2 jðx1 ; tÞ 1 vjðx1 ; tÞ g 1 v2 jðx1 ; tÞ   x3 Κ þ 0:5 PΚ f þ  þ Κ g1 þ  x3 g Κ g1 111111 2 vx1 2 vx 2 vx1 vx21 1 ! # !!       1 vjðx1 ; tÞ g2 v2 wðx1 ; tÞ vjðx1 ; tÞ v2 wðx1 ; tÞ v∅ v∅ g3  Κ þ 0:5 þ  d  P d Κ þ  ε 0 2 vx1 vx1 vx3 vx3 vx21 vx21 9   >  v∅ v∅ = d þ  ε0 dV (12) vx1 vx1 > ;

The contribution of the kinetic energy induced from the rotational inertia is taken into account in developing Eq. (10), but the proportional damping regarding mechanical dissipation mechanism will be considered later. Since P is an independent variable in Eq. (11), dP will be arbitrary; so its coefficient should be zero.

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!    vjðx1 ; tÞ vwðx1 ; tÞ v2 wðx1 ; tÞ vjðx1 ; tÞ þ 0:5  jðx1 ; tÞ Κ d þ 0:5  Κf vx1 vx1 vx1 vx21 !   v2 jðx1 ; tÞ vjðx1 ; tÞ v∅  ¼0  x3 þ f3113  2 vx1 vx3 vx1

 a33 P þ d311 þ f3111

 x3

(13)

In the absence of free electrical charges, the Gauss's law can be considered as ε0 E3;3 þ P3;3 ¼ 0, where ε0 is the vacuum v∅ ¼ E , one can permittivity. After performing some mathematical operations, substituting and considering the relation vx 3 3

find the potential and the electric field as:

!! x h2 vjðx1 ; tÞ x h2 v2 jðx1 ; tÞ vðtÞ vðtÞ x þ þ d311 ð 3  Þ þ f3111 ð 3  Þð þ vx1 h 3 2 2 8 2 8 vx21 !! v∅ 1 vjðx1 ; tÞ v2 jðx1 ; tÞ vðtÞ E3 ¼  þ d311 ðx3 Þ ¼ þ f3111 ðx3 Þ  vx3 ða33 ε0 þ 1Þ vx1 h vx21 ∅¼

1 ða33 ε0 þ 1Þ

(14)

By substituting the polarization density P from Eq. (12) into Eq. (11), the electromechanical equation without this parameter can be obtained. Furthermore, the current, iðtÞ, is equal to the time rate of the change in the average of electric displacement 1 0 R ~ ¼ 1 D dVand common definition of electric displacement is D ¼ ε V∅ þ PA, and can be expressed as: @D 3 3 0 h V

iðtÞ ¼

vðtÞ 1 ¼ R h



"      ZL _ ðx ; tÞ _ 1 ; tÞ _ _ 1 1 vj vwðx vðtÞ 1  ε0 þ ðbLhÞ þ f þ 0:5Ap  jðx1 ; tÞ Κ d d311  Hp a33 a33 vx1 vx1 h 0

þ0:5Ap

! _ 1 ; tÞ vj_ ðx1 ; tÞ v wðx  Κ f þ f3111 vx1 vx21 2

! v2 j_ ðx1 ; tÞ  Hp þ f3113 vx21

1 !#) vj_ ðx1 ; tÞ  Ap dx1 A vx1

(15)

3. Assumed modes method Here, in this section, by applying the assumed modes method [33], the equations of motion will be achieved. In the first step, the distributed-parameter variables are written in the expanded form as:

wðx1 ; tÞ ¼

jðx1 ; tÞ ¼

N X k¼1 N X

ak ðtÞxk ðx1 Þ (16)

ck ðtÞbk ðx1 Þ

k¼1

where N is the number of modes in the series discretization, ak ðtÞ and ck ðtÞ are generalized variables and xk ðx1 Þ and bk ðx1 Þ are kinematically admissible trial functions that satisfy the geometric boundary conditions [34]. Since the problem has not an exact solution, its eigenfunctions cannot be simply determined. Therefore, the eigenfunctions of a Euler-Bernoulli beam is considered as admissible functions for the under study problem. The suitability and accuracy of results by selecting the EulerBernoulli beam eigenfunctions for a Timoshenko problem concerning the strain gradient elasticity theory or other nonlocal theories have been considered in different researches [35e37]. For ensuring the convergence of the solution, sufficient mode numbers of admissible functions should be used.





l l sinlk  sinhlk l l sin k x1  sinh k x1 xk ðx1 Þ ¼ cos k x1  cosh k x1 þ L L coslk þ coshlk L L   lk lk sinlk  sinhlk lk l cosh x1  cos k x1 bk ðx1 Þ ¼ sin x1  sinh x1 þ L L coslk þ coshlk L L

(17)

where lk is the kth root of the characteristic equation of a fixed-free Euler-Bernoulli beam. As a result, three coupled equations for three unknown generalized coordinates i.e. aðtÞ, cðtÞ and vðtÞ will be obtained. The first discrete equation of motion can be obtained in the form of Eq (17).

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€ þ Kaa aðtÞ þ Kac cðtÞ þ q1 vðtÞ þ q2 vðtÞ ¼ f maa aðtÞ

173

(18)

where maa and Kaa are the mass and the stiffness general matrices, respectively. Electromechanical coupling and force vectors are represented by q and f. The elements of these matrices are as:

ZL

maa kl ¼ rAp

ZL

ZL ZL d2 gðt d2 hðt € b ðtÞ rAp ðxl Þdx1 ¼  ðxk Þðxl Þdx1 fl ¼ w r A ð x Þdx  rAp x1 ðxl Þdx1 p l 1 dt 2 dt 2

0

0

0

0

o ZL 

o ZL  00 1 n 1 n 0 d f 0:5Ap Κ 0:5Ap Κ q1l ¼ xl dx1 q2l ¼ xl Þdx1 Kklaa a33 h a33 h 0

ZL  ¼



 0

þ 0:5Ap 0:5 xk Κ c2 k þ

0

0



  00  i   1 h  0 d 1 h  0 d 0:5 xk Κ þ 0:5 xk Κ f Κd 0:5 xk Κ   x0l þ þ 0:5Ap a33 a33

9  =  00  i  00  00 f f g3 Κ þ 0:5 xk Κ þ 0:5 xk Κ xl dx1 Kklac ;

ZL  ¼ 0

    0    0   0 1 1 h  Hp bk Κ c1 k  0:5Ap ðbk ÞΚ c2 k þ d311 Hp bk þ 0:5Ap ðbk ÞΚ d þ 0:5Ap bk Κ f þ 0:5 þ 2 a33

   00   0 i 

  0



1 h Κd d þ f3111 Hp bk þ Ap f3113 bk x0l þ ð0:5 þ Hp bk þ 0:5Ap ðbk ÞΚ d þ 0:5Ap b0k Κ f a33 311 9   =   00    00   0

i 



00 1 1 Κf  Hp bk Κ g1  Ap b0k Κ g2  0:5Ap b0k Κ g3 Þðxl Þ dx1 þ f3111 Hp bk þ f3113 Ap bk ; 2 2 (19)

The second discrete EulereLagrange equations for the undamped structural Timoshenko beam model can be calculated as:

mcc c€ðtÞ þ Kca aðtÞ þ Kcc cðtÞ þ q3 vðtÞ þ q4 vðtÞ þ q5 vðtÞ ¼ 0

(20)

where

mcc kl ¼ rIp

ZL

ðbk Þðbl Þdx1

(21)

0

ca Kkl ¼

      ZL    00  i 1 0 1 h  0 d 1 h  0 d 0:5 xk Κ þ 0:5 xk Κ f d311 þ Ap 0:5 xk Κ ð  Hp  xk Κ c1 k þ  4 a33 a33 0

9   =  00  i  00  i  00  0 1  00  g2 1 h  0 d f f f g3 f3113 þ 0:5 xk Κ þ 0:5 xk Κ Κ þ 0:5 xk Κ  Þðb Þ þ 0:5 xk Κ x Κ þ 0:5Ap ; l 4 k a33

    00  i   

1 h  0 d 1 h 0:5 xk Κ þ 0:5 xk Κ f Κd ðbl Þ þ ð  Hp    0:5Ap 0:5 x0k Κ c2 k þ a33 a33 9  =  00   00  i  0 d 00 1 f3111 þ þ 0:5 xk Κ þ 0:5 xk Κ f xk Κ g1 Þðbl Þ dx1 ; 4 

þ

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ZL  K cc kl ¼

 ð

0

   0  1   0  1 h d311  Ip bk þ 0:5Hp ððbk ÞÞΚ d C1111  Ip bk þ Hp ððbk ÞÞΚ c1 k þ  4 a33

3 2    0    00    0 i   0  1 h d311 5 þ 4 d311 Hp bk þ þ 0:5Hp  bk Κ f þ f3111  Ip bk þ f3113 Hp  bk a33

 0   00   0 i  00  1  Hp bk Κ g f3113 þ þ 0:5Ap ðbk ÞΚ d þ 0:5Ap bk Κ f þ f3111 Hp bk þ f3113 Ap bk 2 3 2   0    0   0 1  0 1 h d Hp bk þ 0:5Ap ðbk ÞΚ d þ 0:5Ap bk Κ f  Ap bk g113113  Ap bk Κ g2 5 þ 0:54 þ 4 a33 311 

 00 

þ f3111 Hp bk

39 =  0  0 i  0 1   00  g1 1  0  g2 f g3 5 Hp bk Κ  Ap bk Κ  0:5Ap bk Κ þ f3113 Ap bk Κ  Þ bl  ð ; 2 2

8 <

2   0    0  1 1 h  Hp bk Κ c1 k þ 0:5Ap ððbk ÞÞΚ c2 k þ d311  Hp bk þ 0:5Ap ððbk ÞÞΚ d   0:54 : 2 a33 39   =   0    00    0 i   0  1 h Κ d 5 Þðbl Þ þ  d311 Ip bk þ 0:5Ap  bk Κ f þ f3111  Hp bk þ f3113 Ap  bk þ ; a33  0    00   0 i 1  0 f3111  Hp bk Κ g1 bk Κ f þ f3111 Ip bk þ f3113 Hp bk 4 9   00 =   00  1  0  Ip bk g111111  Hp bk Κ g bl dx1 ; 2

þ 0:5Hp ðbk ÞΚ d þ 0:5Hp

o ZL  0  1 n f Hp d311 þ Ap f3113 þ 0:5Ap Κ q3l ¼  bl dx1 a33 h 0

q4l ¼ 

o ZL 1 n 0:5Ap Κ d ðbl Þdx1 a33 h 0

1 q5l ¼  Hp f3111 a33 h

ZL   00 bl dx1 0

The third Euler-Lagrange equation will be found as:

_ þ Cf vðtÞ

vðtÞ T_ T_  ðq1 þ q2 Þ aðtÞ  ðq3 þ q4 þ q5 Þ cðtÞ ¼0 R

(22)

  1 . where the parameter Cf is defined as Cf ¼ bL þ ε 0 a33 h Since most of the designed energy harvesting systems are working in the resonant condition to maximize the generated power, it is necessary to consider the dissipation in this system. In this paper, the Rayleigh damping matrix D is considered which is proportional to the mass and the stiffness matrices with m and g constant, respectively ðD ¼ mM þ gKÞ. By considering the proportional damping, the three coupled EulereLagrange equations for Timoshenko beam can be summarized as:

€ þ daa aðtÞ _ þ dac cðtÞ _ þ Kaa aðtÞ þ Kac cðtÞ þ ðq1 þ q2 ÞvðtÞ ¼ f maa aðtÞ _ þ dcc cðtÞ _ þ Kca aðtÞ þ Kcc cðtÞ þ ðq3 þ q4 þ q5 ÞvðtÞ ¼ 0 mcc c€ðtÞ þ dca aðtÞ

S.A.M. Managheb et al. / Journal of Sound and Vibration 421 (2018) 166e189

_ þ Cf vðtÞ

vðtÞ _  ðq3 þ q4 þ q5 ÞT cðtÞ _  ðq1 þ q2 ÞT aðtÞ ¼0 R

175

(23)

The electrical behavior induced from both piezoelectric and flexoelectric effects are coupled with the mechanical behavior of the cantilever beam through the parameter q in Eq. (22). In the following, since in such centrosymmetric beams with a symmetric cross section, the effect of piezoelectric vanishes, flexoelectric effect is purely studied. Assuming the harmonic base excitation, the force vector f will be obtained as:

f ¼ Fejut

(24)

where

Fk ¼ W0 u2

ZL 0

rAp xk dx1 þ V0 u2

ZL

rAp x1 ðxk Þdx1

(25)

0

Due to the linearity of the system, the steady-state response of harmonic base excitation is also assumed harmonic with the same frequency. Thus, the unknown generalized coordinates aðtÞ, cðtÞ and output voltage vðtÞ can be stated by the following harmonic forms

aðtÞ ¼ Aejut cðtÞ ¼ Cejut vðtÞ ¼ Vejut

(26)

Substituting Eq. (25) into Eq. (22), a set of algebraic equations can be obtained as follows:



  u2 maa þ judaa þ Kaa A þ ðjudac þ Kac ÞC þ ðQIÞV ¼ F   ðjudca þ Kca ÞA þ  u2 mcc þ judcc þ Kcc C þ ðQIIÞV ¼ 0   1 V  juðQIÞT A  juðQIIÞT C ¼ 0 juCf þ R

(27)

where QI ¼ q1 þ q2 and QII ¼ q3 þ q4 þ q5 are electromechanical coupling terms. The solution of algebraic equations in Eq. (26) can be calculated as:

1  A ¼ L aa  L ac L cc1 L ca F  1 F C ¼ L cc1 L ca L aa  L ac L cc1 L ca

(28)

   o 1 1 1 n ðQIÞT  ðQIIÞT L cc1 L ca L aa  L ac L cc1 L ca F V ¼ þju juCf þ R where

   1 1  u2 maa þ judaa þ Kaa þ ju juCf þ ðQIÞðQIÞT R   1 1 L ac ¼ ðjudac þ Kac Þ þ ju juCf þ ðQIÞðQIIÞT R   1 1 L ca ¼ ðjudca þ Kca Þ þ ju juCf þ ðQIIÞðQIÞT R     1 1 L cc ¼  u2 mcc þ judcc þ Kcc þ ju juCf þ ðQIIÞðQIIÞT R L aa ¼



(29)

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Fig. 3. Schematic of adding mass to the tip of flexoelectric energy harvester.

4. Adding a mass to the tip of Timoshenko flexoelectric energy harvester In order to reduce the resonance frequency of the harvester, the system with an attached mass Mm at the free end is considered (Fig. 3). The rotary inertia of the mass is represented by Im. By considering the attached mass, the total kinetic energy and the effective force will be changed. So, mass matrices and effective force in the proposed model will be modified as follows:

maa kl ¼ rAp mcc kl

¼ rIp

ZL

0 ZL

ðxk Þðxl Þdx1 þ Mm xk ðLÞxl ðLÞ (30)

ðbk Þðbl Þdx1 þ Im bk ðLÞbl ðLÞ

0

€ b ðtÞ f l ¼ w

ZL

rAp ðxl Þdx1 ¼ 

0

0 L Z d2 gðtÞ @ dt2

0

1

1 0 L Z d2 hðtÞ @ A rAp ðxl Þdx1 þ Mm xl ðLÞ    rAp x1 ðxl Þdx1 þ Mm Lxl ðLÞA dt2

(31)

0

Another issue regarding to attach the proof mass is the necessity of considering new trial functions for more accuracy. Thus, the following trial functions are introduced:

Mm  ðcoslk  coshlk Þ  l l mL xk ðx1 Þ ¼ cos x1  cosh x1 þ sin k x1  sinh k x1 Mm L L L L ðsinlk  sinhlk Þ coslk þ coshlk  lk mL Mm ! ðcoslk  coshlk Þ  sinlk  sinhlk þ lk lk lk lk lk mL bk ðx1 Þ ¼  sin x1  sinh x1 þ cosh x1  cos x1 Mm L L L L ðsinlk  sinhlk Þ coslk þ coshlk  lk mL

lk

lk

sinlk  sinhlk þ lk

(32)

where lk is kth root of Eq. (32).

1 þ coslcoshl þ l

Mm l3 Im l4 Im Mm ðcoslsinhl  sinlcoshlÞ  ðcoshlsinl þ sinhlcoslÞ þ ð1  coslcoshlÞ ¼ 0 3 mL mL m2 L4

(33)

In Eq. (32), m is the mass per length of the beam. Using Eq. (31) as admissible trial functions in the presence of tip mass leads to a faster convergence in the discrete system (with less number of modes). As a result, the beam displacement wðx1 ; tÞ and the cross section rotation jðx1 ; tÞ, can be calculated as:

 1 wðx1 ; tÞ ¼ xT L aa  L ac L cc1 L ca Fejut   1  Fejut jðx1 ; tÞ ¼ bT L cc1 L ca  L aa  L ac L cc1 L ca

(34)

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177

Also, the voltage frequency response function can be written as (G is general form of gravitational acceleration):

n  o ju ðQIÞT  ðQIIÞT L cc1 L ca Bf V ¼  1 ¼ G 1 1  aa L  L ac L cc1 L ca juCf þ R   0 1  u2 mcc þ judcc þ Kcc þ 1 8 C 19 0B B C > > > > B  C >  > > @ A C> 1 B > > > > 1 T > > B C > j j u u C þ ð Q IIÞð Q IIÞ > > B C> f < R C= T TB B C ju ðQIÞ  ðQIIÞ B C Bf > > > B  1 C>  > > > B C> 1 > > ca ca > > @ A > > ðjud þ K Þ þ ju juCf þ > > > > R ; :  ðQIIÞðQIÞT    u2 maa þ judaa þ Kaa þ 

1 ju juCf þ 0 R B B B B B B 1 B  B 1 B juCf þ B R B B B B B B @

1

(35) ðQIÞðQIÞT Þ  ððjudac þ Kac Þþ

   1 1 ðQIÞðQIIÞT ju juCf þ R 0 B B B @



 u2 mcc þ judcc þ Kcc



11

C C C 1  A 1 T ðQIIÞðQIIÞ þju juCf þ R

11 C C C C C C C C C C C C C C C C A

ððjudca þ Kca Þþ    1 1 ðQIIÞðQIÞT ju juCf þ R   2 Using the above expression, the output power density can be calculated P ¼ VR and the optimal resistance for achieving maximum power density in the certain excitation frequency can be obtained via

vPout vR

¼ 0.

Table 1 Short circuit and open circuit resonance frequency values for flexoelectric beam. Resonance frequency (Hz)

Euler-Bernoulli theory [25] Timoshenko theory with L/h ¼ 100

3-mm thickness

0.3-mm thickness

Short circuit

Open circuit

Short circuit

Open circuit

7665 7661

7665 7663

74230 74117

75820 75802

Table 2 Maximum flexoelectric tip velocity for Timoshenko and Euler-Bernoulli theories (h ¼ 0.3 mm). Electrical resistance (MU)

Maximum of tip velocity of Euler-Bernoulli beam (micron s1 ) ]25[

Maximum of tip velocity of Timoshenko beam (micron s1 ) L/h ¼ 100

Short circuit 1 10 50 100 500 Open circuit

325 323 310 270 260 305 310

318.575 316.843 302.257 264.399 258.649 295.479 306.043

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value

Young modulus (GPa) Shear modulus (GPa) reciprocal dielectric susceptibility ð1010 N m2 C2 Þ Flexoelectric coefficient ð108 C m1 Þ Shear flexoelectric coefficient ð108 C m1 Þ Piezoelectric coefficient ð1012 m V1 Þ Density ð103 kg m3 Þ Strain gradient elastic constant ð107 NÞ Shear strain gradient elastic constant ð107 NÞ Damping ratio Relative permittivity Vacuum permittivity ð1012 C2  ðNm2 Þ1 Þ

3.7 1.4 1.38 1.3 1.04 20 1.78 5 1.9 0.05 9.2 8.854

5. Validation of the developed model After a discussion about the flexoelectric effect, energy harvesting from Timoshenko nanobeam and derivation of governing equations, the developed formulation was validated by comparing the results of special cases with those reported in the literature. For this purpose, the validation was conducted in two parts: at first, for short circuit and open circuit states, the resonance frequencies of the harvester for beams with thicknesses of 0.3 mm and 3 mm were calculated and compared with [25]. Second, the maximum tip velocity of the beam in various electrical resistances was calculated and compared with the reported values in literature. The good agreement between the results of the current model and those reported by other references shows the validity of the model. It is noteworthy that the results shown in Tables 1 and 2 are for a Timoshenko beam with length to thickness ratio 100, i.e. L/h ¼ 100. It means that for L/h ¼ 100, the Timoshenko beam is thin and thus it behaves like a Euler-Bernoulli beam.

6. Numerical results In the previous section, the equations of motion of the energy harvester were extracted. As a case study, polyvinylidene difluoride (PVDF) material is chosen. In general, solving equations are carried out for basic harmonic stimulation as it was mentioned before. The equations of motion are solved using a MATLAB program for different parameters values. First, the convergence of the results is studied by increasing in the number of modes. Then, the effects of various parameters such as electrical resistance, the effect of size, the influence of flexoelectric coefficients, elastic strain gradient, and adding proof mass are investigated. The aspect ratio of Timoshenko beam is selected as L/h ¼ 5 and b/h ¼ 2.5, where L, b, and h are the length, the width and the thickness of the beam, respectively. The properties of PVDF are presented in Table 3 [25]: In addition, due to the symmetry of the selected material in the strain tensor, the following relationship between the nonzero shear coefficients of flexoelectric and the elastic strain gradient constants can be considered.

Fig. 4. Power density FRF for different number of assumed modes.

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Fig. 5. Convergence test for different number of modes. (a) Voltage FRF; (b) Tip velocity FRF.

m3131 ¼ m3311

g131131 ¼ g131311 ¼ g311131 ¼ g311311

(36)

The values of these coefficients are given in Table 3. Other coefficients for the selected material are equal to zero. Some coefficients of flexoelectricity and strain gradient elasticity for PVDF are equal to zero. 6.1. Convergence analysis Number of assumed modes can change the results, so, the convergence test is applied to the beam with a thickness of 0.3

mm and 10 MU resistance. Fig. 4 exhibits the power density frequency response function when using 1 to 4 assumed modes in

the model. It is clear that the maximum output power is related to the first natural frequency and the generated power decrease in the subsequent modes by increasing the excitation frequency. Hence, it is only sufficient to investigate the convergence for the voltage, the current, and the tip velocity responses around the first natural frequency. Output voltage and the tip velocity FRF graphs are illustrated by considering different numbers of assumed modes around the first natural frequency of the system in Fig. 5. It is obvious that the results show acceptable convergence for three modes

Fig. 6. Voltage FRF of flexoelectric Timoshenko cantilever beam with 3 mm thickness.

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Fig. 7. Voltage FRF of flexoelectric Timoshenko cantilever beam with 0.3-mm thickness.

Fig. 8. Resonance frequency versus load resistance for flexoelectric Timoshenko cantilever beam with 3 mm thickness.

Fig. 9. Resonance frequency versus load resistance for flexoelectric Timoshenko cantilever beam with 0.3 mm thickness.

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Table 4 Comparison of resonance frequency in micron and submicron thickness levels. Thickness (micron)

Open circuit resonance frequency ðkHzÞ

Short circuit resonance frequency ðkHzÞ

The difference between short circuit and open circuit ðkHzÞ

3 0.3

2993.9 29684

2993.2 29001

0.7 683

(N ¼ 3). Therefore, in the subsequent results and graphs, the numbers of modes are set to three unless otherwise is stated in the text and the results are reported around the first mode only. Since the flexoelectric effect is important at small scales generally, in these simulations, the size of the model is set to micron and smaller. The beam thickness in the convergence test simulations is three tenth of micron. The results are normalized by gravity (g).

6.2. Voltage FRFs The output voltage of FRFs for the beam with the thickness of 3 mm is displayed in Fig. 6 for different values of electrical resistance including conditions close to both open and short circuit conditions. For the thickness of 3 mm, the flexoelectric coefficients are small and a weak electromechanical coupling exists between the system governing equations. The peaks in

Fig. 10. Tip velocity FRF of flexoelectric Timoshenko cantilever beam with 3 mm thickness.

Fig. 11. Tip velocity FRF of flexoelectric Timoshenko cantilever beam with 0.3-mm thickness.

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this diagram represent the first resonance frequency of the system. In most energy harvester designs, the frequency range of the harvester coincides with this frequency. According to the results, it could be seen that the value of the voltage is directly related to that of the electrical resistance, i.e., the amount of the harvested voltage of the system increases with raising the electrical resistance and finally, it remains almost constant. To interpret this behavior, it can be said that the current in the low resistances (close to short circuit conditions) is almost constant and based on the relationship V ¼ RI, with enhancing the load resistance, the voltage also increases too. This issue can be better understood by considering the graph representing voltage in terms of different electrical resistance (Fig. 6). The lowest and highest curves corresponding to the open circuit and short circuit conditions of the system can be seen, respectively. It is also clear that the fundamental resonance frequency is almost constant and does not change for different values of the load resistance, thereby indicating that short and open circuit resonance frequency is almost the same which indicates a very low electromechanical coupling. Now, let us assume a value of 0.3 mm for the beam thickness with the same aspect ratio. In this case, it can be seen that there is an obvious difference between open and short circuit resonance frequency (as expected, with raising the electrical resistance, the resonance frequency is also increased (see Fig. 7)). This difference in the resonance frequency shows the improvement in the flexoelectric coupling with the size reduction. 6.3. Size effect To better compare the influence of the resonant frequency resulting from the electromechanical coupling, the resonant frequency graph in Figs. 8 and 9 has been plotted from short circuit to open circuit conditions versus electrical resistance, with the thickness of 3 mm and 0.3 mm, respectively. The results are summarized in Table 4.

Fig. 12. Power density FRF of flexoelectric Timoshenko cantilever beam with 3 mm thickness.

Fig. 13. Power density FRF of flexoelectric Timoshenko cantilever beam with 0.3-mm thickness.

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Fig. 14. Variation of maximum output power density versus thickness for flexoelectric Timoshenko cantilever beam.

Fig. 15. Variation of power density versus load resistance for flexoelectric Timoshenko cantilever beam (h ¼ 0.3 mm).

The difference between the open circuit and short circuit resonant frequency represents the effect of electromechanical coupling. Considering the flexoelectric energy harvester used in this study (see Fig. 1), the electromechanical coupling is only due to the flexoelectric effect. Hence, the difference between the open circuit and short circuit resonance frequencies represents the strength of the flexoelectric effect. Table 4 shows that for a beam with the thickness of 3 mm, the difference between the open and shot resonance frequencies is 0.7 kHz, while for a beam with 0.3 mm thickness is 683 kHz.

6.4. Tip velocity FRFs The size effect could be observed more clearly when the FRF of tip velocity is plotted in both micron and submicron thicknesses. According to Figs. 10 and 11, there was a visible difference between curves with various electrical resistances in the case of the 0.3-mm thickness. While there was almost identical curves category for the 3 mm thickness case, this could be due to the stronger flexoelectric coupling and the change of resonance frequency in the submicron and nanoscales. This effect cannot be seen in the piezoelectric materials. Furthermore, since the cantilever is centrosymmetric, the electromechanical coupling is due to flexoelectricity only, growing significantly with the reduction of device thickness. Note that as a non-centrosymmetric sample exhibits both piezoelectric and flexoelectric effects, the flexoelectric coupling can be comparable to the piezoelectric effects at much smaller thickness levels.

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Fig. 16. Variation of voltage and current versus load resistance for flexoelectric Timoshenko cantilever beam. (a) Current versus load resistance; (b) Voltage versus load resistance.

Fig. 17. Variation of power density FRF with increasing flexoelectricity coefficient for flexoelectric Timoshenko cantilever (h ¼ 3 mm).

Fig. 18. Variation of power density FRF with increasing flexoelectricity coefficient for flexoelectric Timoshenko cantilever (h ¼ 0.3 mm).

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Fig. 19. Variation of power density FRF with increasing strain gradient elastic constant for flexoelectric Timoshenko cantilever (h ¼ 3 mm).

6.5. Power density FRFs The power density FRF in 3 and 0.3-mm thickness cases has been shown in Figs. 12 and 13, respectively. As can be observed, in both cases, an optimal electrical resistance exists. Both cases showed that R ¼ 10 MU had the maximum output power. Again, by comparing Figs. 12 and 13, it could be found that the resonance frequency for the 3 mm thickness beam almost remained constant, while in the case of the 0.3-mm thickness beam; it was changed by changing the electrical resistance. The resonance frequency of the maximum power was lying between the short and the open-circuit conditions. Furthermore, regarding the effect of size, the performance of the flexoelectric harvester was evaluated by its power density. Fig. 14 shows the trends of power density output for the uniform flexoelectric energy harvester, with the continuous growth in the thickness of the beam; it was found that the influence of the flexoelectricity on the output power was reduced. In other words, there was some growth in the output power density by the reduction in the sample size; this was such that that by comparing Figs. 12e14, it could be observed that the maximum output power density of the beam with the thickness of 0.3 mm was about 7.5 and 15 times larger than that of the beam with the thickness of 3 and 6 mm, respectively. According to the results, a complete analysis on the effects of flexoelectricity and other effective parameters in energy harvesting are presented below. 6.6. Behavior analysis with load resistance changing In order to evaluate the effect of electrical resistance on the problems of energy harvesting and find an optimal electrical resistance, Figs. 15 and 16, show the variation of the harvester voltage, current and power density output versus the change in the electrical resistance. Fig. 15 shows that by increasing the electrical resistance, at first, the power density was increased and

Fig. 20. Variation of power density FRF with increasing strain gradient elastic constant for flexoelectric Timoshenko cantilever (h ¼ 0.3 mm).

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Fig. 21. Flexoelectric Timoshenko energy harvester with a tip mass.

Fig. 22. Power density FRF for 0.3-mm thickness flexoelectric Timoshenko cantilever beam with a tip mass.

then decreased after reaching its peak. In this graph, the power is dependent on the voltage and current; at first, due to the increase of the voltage and a constant current, the power output is ascending. After reaching a peak, there starts a descending trend; this is because of the constant voltage and the decreasing of the current (Fig. 16). The electrical resistance corresponding to the maximum power output is defined as optimal electrical resistance which is estimated to be about 17 MU for the current problem, thereby confirming the plotted power density FRFs curves in Fig. 13.

6.7. Analysis of flexoelectric and strain gradient elastic effects In order to provide more details regarding the flexoelectric effect and constant elastic strain gradient introduced in the internal energy density relation, the power density FRFs can be prepared as the performance of energy harvesting system. Simulation and geometry are chosen in a way not leading to large deformations in the beam and the use of linear theories would be permitted. Graphs in both micron and submicron thicknesses were drawn to show the importance of them and compare them different thicknesses. At first, the effect of flexoelectric coefficients is investigated. 6.7.1. Flexoelectric effect Figs. 17 and 18 show the power density FRFs for various flexoelectric coefficients in two thicknesses of 3 and 0.3 mm. The size effect and reduction in the hardness of the beam (decreasing the resonant frequency for a constant thickness) with increasing the flexoelectric coefficients are clear. Fig. 17 shows that by increasing the flexoelectric coefficient, energy harvesting was increased, mainly due to the increased in the coupling coefficients. The change in the resonance frequency for the beam with 3 mm thickness is very small and almost negligible. However, by reducing the size of the beam, the resonance frequency was shifted to the left by increasing the flexoelectric coefficient (see Fig. 18). This shows that the model with h¼0.3 mm has a softening behavior. In other words, the

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Fig. 23. Variation of power density FRF with increasing base acceleration for flexoelectric Timoshenko cantilever (h ¼ 3 mm).

Fig. 24. Variation of power density versus load resistance with increasing base acceleration for flexoelectric Timoshenko cantilever (h ¼ 0.3 mm).

bending stiffness of a classic Timoshenko beam is bigger than the bending stiffness of a model with considering the flexoelectricity. 6.7.2. Strain gradient elastic effect In this section, another parameter which is introduced in the internal energy density relation, namely, the strain gradient elastic constant (g), has been investigated. Figs. 19 and 20 show the corresponding graphs related to the analysis of this parameter. As expected, as this coefficient does not directly exist in the coupling vector for the case of h¼3 mm, increasing the strain gradient elastic constant does not affect the amount of the generated electric power (see Fig. 19). However, in the case of 0.3mm thickness, by increasing the strain gradient coefficient, the stiffness of the harvester is increased, leading to a resonance frequency growth (Fig. 20). Another issue is the increase of beam stiffness when strain gradient elastic constant is increased. Finally, two other factors including the adding of a lumped mass to the beam tip and the effect of base excitation change on the performance of the flexoelectric energy harvester system have been addressed briefly. 6.8. Tip mass effect on the performance of flexoelectric energy harvester In order to discuss the effect of adding mass and display the validity of the model in the presence of a tip mass, a rectangle cube mass was attached to the tip, as shown in Fig. 21. Due to the small dimensions of the beam, the information related to the lumped mass, with the density of about 1000 kg m3 , was calculated as Mm ¼ 1013 gr; Im ¼ 1:373  108 grðnmÞ2 . After utilizing mass information in the assumed modes

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model, a discussion for convergence was conducted, as mentioned earlier, and in this case, the number of modes was set to N ¼ 5 for more precise results. It is obvious from Fig. 22 that by adding the mass to the tip of the beam, the resonance frequency for open circuit was changed from 29 MHz (beam without tip mass) to 24 MHz, leading to a lower frequency for the maximum output power. 6.9. Base acceleration effect on the performance of the flexoelectric energy harvester Figs. 23 and 24 show the effect of base excitation on the electrical power output in terms of frequency and electrical resistance, respectively. It is obvious that by increasing the strength of the base excitation, the electrical power output was enhanced. Also, the frequency bandwidth of the energy absorbed was increased by the amplification of the base acceleration. 7. Conclusions In this study, a Timoshenko single-layer symmetrical beam was introduced as a candidate for flexoelectric energy harvester at submicron scales. The governing equations and the boundary conditions with flexoelectric and strain gradient elastic effect were obtained and then they validated by some simple previous works. Considering this comprehensive model of the beam is more realistic due to the taking account the normal and shear strains and all of the flexoelectric and strain gradient elasticity coefficients. Based on the obtained multi-degree freedom model, vibrations under harmonic base excitation and energy harvesting problem were investigated. The results, in addition to showing the effect of size, revealed that the influence of flexoelectricity, for small thickness and sub-micron flexoelectric layer, was high and important on the voltage, current and the output power. Also, the effect of the variation of flexoelectric and strain gradient elastic coefficients on the stiffness and resonance frequency of the beam was investigated. The results indicate that for the harvester with micron size by increasing the flexoelectricity coefficients, the power density increased and the resonance frequency remain almost constant. However, for the nano size harvester, by increasing the flexoelectricity coefficients, the power density increased and the resonance frequency decreased. The results show that by increasing the strain gradient elasticity constant, the power density of the micron size harvester remains almost unchanged. However, for the nano size harvester, by increasing the strain gradient elasticity constant, the maximum power density will decrease and the resonance frequency of the system increases. Aside from the effect of different coefficients, in order to achieve higher power output, using the mass connected to the end of the harvester or increasing the base excitation amplitude of cantilever beam can be useful ways.

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