Theoretical modeling and nonlinear analysis of piezoelectric energy harvesters with different stoppers

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Theoretical modeling and nonlinear analysis of piezoelectric energy harvesters with different stoppers K. Zhou , H.L. Dai , A. Abdelkefi , Q. Ni PII: DOI: Reference:

S0020-7403(19)32160-5 https://doi.org/10.1016/j.ijmecsci.2019.105233 MS 105233

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

16 June 2019 10 October 2019 10 October 2019

Please cite this article as: K. Zhou , H.L. Dai , A. Abdelkefi , Q. Ni , Theoretical modeling and nonlinear analysis of piezoelectric energy harvesters with different stoppers, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105233

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Highlights 

Four stopper configurations for energy harvesting are proposed and theoretically modeled.



The derived exact mode shapes are superior in convergence compared to classical mode shapes.



The harvester’s behavior can change from softening to hardening due to effects of nonlinearities.



There are parameter regions where the bandwidth and generated average power are optimal.

1

Theoretical modeling and nonlinear analysis of piezoelectric energy harvesters with different stoppers K. Zhou1,2, H.L. Dai1,2, A. Abdelkefi3, Q. Ni1,2 1

Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China

2

Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Wuhan 430074, China

3

Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA

Abstract This study focuses on investigating the dynamic mechanism and nonlinear analysis of a piezoelectric energy harvester with different stoppers so as to determine optimal impact energy harvesting configurations. Based on the Hamilton’s principle, a set of coupled nonlinear governing equations for the energy harvesting system is established, which is then discretized by the Galerkin method. It is indicated that the energy harvester without stoppers displays a softening characteristic and this softening becomes more obvious with increasing the base acceleration. This is due to the considered geometric and inertia nonlinearities. By introducing stoppers to the harvester, however, the dynamic behavior changes from softening to hardening characteristic, owing to the induced impact force nonlinearity. Then four kinds of stoppers are compared and better stopper configurations for energy harvesting performance are picked out, following by the consideration of parametric analysis on stopper’s stiffness, placed position and spacing distance. It is noted that there exists optimal parameter regions where the energy harvester can strike a balance between bandwidth and generated average power depending on excitation frequencies in surroundings.

Keywords: Piezoelectric energy harvester; stopper; nonlinear dynamics; softening and hardening; resonance region

1. Introduction In the natural environment around us, there are many potential energy sources



Corresponding author. E-mail: [email protected] 2

that can be used, such as mechanical vibration energy, and flow energy [1-4]. Hence, investigate how to effectively convert these potential and wasted ambient energies into useful electrical power has received widespread concerns during the past decades [5-9]. Daqaq [7] proposed the concept of stiffness nonlinearities for broadband energy harvesting. That is because the designed energy harvesters can be possibly applied for realizing self-power operations of many electronic instruments, such as wireless sensors, data transmitters, medical implants, and health monitoring sensors due to their low-power consumptions [10-13]. According to the different transduction mechanisms, such as piezoelectric [14, 15], electromagnetic [16-18], magnetostrictive [19] and electrostatic [20], researchers proposed different types of energy harvesting systems to convert the ambient energy to electrical energy [21, 22]. In these different systems, the vibration-based energy harvesting system has attracted lots of interests due to the fact that mechanical vibration energy can be found everywhere. The mostly used vibration-based energy harvester consists of a piezoelectric cantilevered beam with a tip mass attached at free end. The main disadvantage of this harvesting system is that it can only obtain a considerable output power when the excitation frequency is near the natural frequency of the energy harvesting system. As long as the excitation frequency is slightly deviated, the output power will be dramatically reduced. As known that the excitation frequencies in practical engineering are random and time-varying, the proposed vibration-based energy harvesting system, therefore, needs to work under a wide operating frequency band. Consequently, many researches on proposing schemes to widen the band of energy harvesters for enhancing their output performance have been conducted in recent years. One of the simple methods is using the multimodal structures which have two or more resonant frequencies [23-24]. Erturk et al. [23] proposed an L-shaped beam-mass structure to produce a broader band energy harvesting system. The designed structure can be tuned to make the first two natural frequencies relatively close to each other. Afterwards, Wu and Yang [24] designed a 2-DOF vibration energy harvester composed of one main cantilevered beam and one secondary cantilevered 3

beam within the main beam. This structure can be tuned to have two close resonant frequencies to broaden the resonant frequency band. Some other multimodal harvesters, such as multi-mass harvester [25, 26], 2-DOF energy harvesters [27, 28], and fractal-inspired energy harvester were also designed for broadening the harvester’s resonant region [29]. It is noted that another strategy to get a wider bandwidth is by introducing the nonlinearity to the energy harvesting systems. The most common concept is adding the magnets to make the energy harvesters become mono-stable [30, 31], bi-stable [32, 33], tri-stable [34], quad-stable [35], and buckled compressive-mode [36] harvesting system. Stanton et al. [30] experimentally validated a nonlinear energy harvester capable of bidirectional hysteresis. By tuning the magnetic interactions, both hardening and softening responses of the energy harvester can occur. However, they found only the mono-stable system has a good performance under certain conditions. Inspired by this, Tang et al. [22] designed a bi-stable energy harvester to improve the performance of the energy harvesting under various vibration situations. Recently, Li et al. [34] presented a tri-stable energy harvesting device, numerical simulations, and experimental validation show that the system can broaden the frequency bandwidth and achieve a high energy harvesting efficiency under a random excitation. In addition, a novel quad-stable energy harvester was developed by Zhou et al. [35] to improve the ability of energy harvesting under weak random excitations. In addition to the techniques mentioned above, employing the mechanical stoppers in the energy harvesting system to get a wider frequency bandwidth has also been extensively studied [37-43]. For example, the wideband vibration-based micro-power generators (MPGs) were reported by Soliman et al. [37] who added a mechanical stopper. Experimental results showed that the up-sweep bandwidth can be greatly widened compared to the traditional MPG. Furthermore, the wideband frequency responses of the piezoelectric energy harvester (PEH) system with one-side stopper and two-side stoppers were reported by Liu et al. [38]. More recently, a low frequency response of generator with a proper amplitude limiter was designed by Song et al. [39]. As a result, compared with the nano-generator without a stopper, the 4

frequency band can be significantly widened. In addition, Zhao et al. [40] proposed a broadband energy harvester subjected to base excitations and wind flows by utilizing a stopper. It is found that the mechanical stopper can convert the quasi-periodic oscillations into periodic oscillations. It is noted that the previous studies have proposed to use the mechanical stopper for broadband energy harvesting, but they did not consider properties of the stopper like stiffness, space distance and placed position which have critical effects on the output performance of energy harvester. With this in mind, four stopper types, namely, Type 1: one-side fixed stopper; Type 2: two-side fixed stopper; Type 3: one-side follow stopper; and Type 4: two-side follow stopper, are taken into account to investigate how to effectively use the stopper and which of the stopper types is the best option. In addition, it should be mentioned that Zhou et al. [41] experimentally investigated impacts of stopper type and material on broadband characteristics and power output of the piezoelectric energy harvester recently, but they did not determine the optimal parameters’ values of the stopper. This is because no accurate theoretical model was established and hence no parametric analysis was conducted. As a result, the piezoelectric energy harvester with four stopper configurations is theoretically modeled in the present study. Then, the optimal stopper configuration and the corresponding parameters are provided and discussed, including the physical mechanism and explanations. This is quite significant when choosing an optimal stopper for efficient energy harvesting. Furthermore, the accurate mode shapes for piezoelectric sheets partially covered on the cantilevered beam are derived and found to be superior in convergence compared to classical mode shapes. In this work, a set of coupled nonlinear governing equations of the energy harvesting system is first established, with consideration of four different stopper types by using cubic spring nonlinear terms to describe the produced impact force. This is elaborated in Section 2. In Section 3, a reduced-order model of the energy harvester is developed by virtue of the Galerkin discretization adopting the derived exact mode shapes. Linear analysis is performed in Section 4 to investigate the energy harvester’s coupled frequency and damping, and then nonlinear studies are carried out 5

in Section 5 to explore the output performance of the energy harvester with different stopper types. Finally, some important conclusions are presented in Section 6.

2. Theoretical modeling The considered energy harvesting system consists of a cantilevered beam with a bimorph partially covering piezoelectric sheets on it. A small tip mass mt is attached to the free end of beam, as shown in Fig. 1. Here, the inertia of tip mass is considered but its rotation is neglected. The energy harvester under investigation is subjected to harmonic base excitation, wb=Asinωt, where A represents the base amplitude and ω is the excitation frequency. As reported that when the excitation frequency is close to any-order mode frequency of the energy harvester, a steady-state oscillation occurs for the energy harvester, resulting in strains of the piezoelectric sheets and hence the alternate voltage. The piezoelectric sheets are connected in series to an electrical load resistance, R. In order to broaden the harvesting frequency bandwidth, four different types of stoppers are introduced, as shown in Figs. 1 (a)-(d). They are (a) Type 1: one-side fixed stopper; (b) Type 2: two-side fixed stopper; (c) Type 3: one-side follow

Base excitation

stopper; (d) Type 4: two-side follow stopper. (a)

d

Lf Stopper

Base excitation

(b)

Piezoelectric sheet Aluminum beam

Tip mass

(c)

(d)

Figure 1 Schematic of the piezoelectric energy harvester subjected to base excitations with four different stopper types. (a) Type 1: one-side fixed stopper; (b) Type 2: two-side fixed stopper; (c) Type 3: one-side follow stopper; and (d) Type 4: two-side follow stopper. The space distance 6

between stopper and piezoelectric beam is d; the placed position of stopper is at x=Lf along the beam length.

Next, the nonlinear governing equations for the energy harvester are derived using the extended Hamilton’s principle as follows:

  T  U dt    Wnc dt  0 t2

t2

t1

t1

(1)

where δ is the variational operator, T-U denotes the Lagrangian, and Wnc represents the work done by non-conservative forces. Before the derivation of the equations, it should be mentioned that in this study, the cantilever beam is assumed to be axially inextensible, and then the inextensible condition can be easily obtained as [39]:

 x   w      1  s   s  2

2

(2)

where x and w are the axial and the transverse coordinates, respectively, and s is the curvilinear coordinates along the length of beam. The inextensibility condition will be used to reduce the degree of freedom of the energy harvesting system. It also should be mentioned that the following property of integral is important and applied in deriving the governing equations [44]:

 g s L

0

s

0



f ( s) wds ds 

L

0

  g  s  ds  f (s) wds L

s

(3)

The kinetic energy of the energy harvester T involves the sum of the kinetic energies due to the beam and tip mass which can be expressed as: T

1 Lb 2  m  s  +mt (s  Lb )   x 2   w  wb   ds    0 2

(4)

where the over-dot denotes partial differentiation with respect to time t, and the mass per unit length of the harvester m(s) is given by:

m( s)  m1  H ( s)  H (s  Lp )  +m2  H (s-Lp )  H (s  Lb )  m1  bWbtb  2  pWpt p , m2  bWbtb

(5)

where ρb and ρp are the mass density of the beam and piezoelectric layers, respectively. Wb and Wp respectively denotes the width of the beam and piezoelectric layers; tb and tp, Lb and Lp are the thickness and length of the beam and piezoelectric layers; H(s) is the Heaviside function. 7

Using the variational operator δ, and then applying the inextensibility condition of Eq. (1) and property of integral Eq. (2), we can obtain the following kinetic energy variation: t2

t2

t1

t1

  Tdt    

t2

t1



Lb

0



Lb

0

s  m  s  +mt ( s  Lb )   w  wb  w  w2  ww  ds   wdsdt 0  

 w Lb  m  s  +m  ( s  L )  s  w2  ww  dsds   wdsdt t b  0  s  

(6)

As to the total potential energy U, it contains the strain energy of the piezoelectric beam and the electric potential stored in the capacitance. It can be expressed as: U

1 b b p p   V  x  x dVb  V  x  x dVp  V E3 D3dVp  b p p 2 

(8)

where Vb and Vp are volumes of the beam and piezoelectric sheets, respectively. ζx and εx are the stress and strain in the axial direction. With consideration of geometric nonlinearity, the strain εx in the axial direction can be written as [45]:

 xb   xp   x   z

w

  1    z  w 1  w2      2 1  w 2

(9)

Then, using the Hooke’s law and the linear constitutive relationships of piezoelectricity, the stresses for the beam and piezoelectric layers are, respectively, given by:

 x b =E b x  xp  E p x  e31E3  E p  x  d31E3 

(10)

where Eb and Ep denote the Young’s moduli of the beam and piezoelectric layers, respectively. d31 denotes the strain coefficient of the piezoelectric layers, e31=Epd31 is the piezoelectric stress coefficient, and E3 is the electric field produced in the piezoelectric layers. The electric field can be related to the voltage V(t) by E3=V(t)/2tp. In addition, D3 is the electric displacement given by the following piezoelectric constitutive relationship:

D3  d31E p x   33 E3 8

(11)

in which ε33 is the permittivity component at constant strain. By substituting Eqs. (9), (10), and (11) into Eq. (8), we can obtain the expression of the total potential energy of the energy harvester as:

U

Lb 1 Lb 1  1  EI ( s) w 1  w2  ds     s  RQw 1  w2  ds  C p RQ  0 2 0 2  2 

 

2

(12)

where EI ( s) ,   s  and C p are the bending stiffness, the electromechanical coupling, and the piezoelectric capacitance, respectively. They are given by: EI ( s )  EI1  H  s   H  s  L p    EI 2  H  s  L p   H  s  Lb  

p 3 2 2 E bWbtb3 E W p  4t p  3t p tb  6tbt p  E bWb tb3 EI1   , EI 2  12 6 12 p  tb  t p Wp d31E  H s  H s  L  , C  Wp Lp 33  s    p  p 2t    2 p

(13)

Then, applying the variational operator δ in Eq. (12), and using the property of integral Eq. (2), the variation of the potential energy can be expressed as: t2

t2

t1

t1

  Udt   

t2



t2

t1

t1



Lb



Lb

0



Lb

0

EI ( s )  w  w3  4www  w2 w   wdsdt

 ( s)RQww wdsdt  

t2

t1

0

 



Lb

0

 

1

 

 ( s )RQ 1  w2   wdsdt 2  

1 2

(14)

t2

 ( s) R  w  www  w2 w  ds Qdt  C p R 2  Q Qdt t1

The virtual work done by non-conservative forces consists of three parts, namely, the virtual work done by the produced impact force F, the mechanical viscous damping, c and the power delivered to an electrical load resistance R. Thus, the virtual work can be expressed as: t2

  Wnc dt   t1

t2

t1



Lb

0

F  s  L f  wdsdt  

t2

t1

 c  w  w  wdsdt   Lb

0

b

t2

t1

RQ Qdt (15)

where Lf means the placed position of the stopper. The parameters for physical dimensions and material properties of the energy harvester are given in Table 1. It should be stated that these parameter values are provided according to the experimental model in reference of Akaydin et al. [46].

9

Table 1. Dimensions and material properties of the energy harvester. PZT elements

Physical properties

Beam

Tip mass

Length, 𝑳𝒑 , 𝑳𝒃 (mm)

31.8

267

-

Width, 𝑾𝒑 , 𝑾𝒃 (mm)

25.4

32.5

-

Thickness, 𝒕𝒑 , 𝒕𝒃 (mm)

0.267

0.635

-

Mass density, 𝝆𝒑 , 𝝆𝒃 (kg/𝐦𝟑 )

7800

2730

-

Mass, 𝒎𝒑 , 𝒎𝒃, 𝒎𝒕 (g)

1.68

15.05

10

Young’s modulus, 𝑬𝒑 , 𝑬𝒃 (Gpa)

66

70

-

Strain coefficient, 𝒅𝟑𝟏 (pm𝐕 −𝟏 )

-190

-

-

Permittivity at constant strain, 𝜺𝟑𝟑 (nF𝐦−𝟏 )

13.28

-

-

-

0.2

-

mechanical viscous damping, c (Ns/m)

Concerning the nonlinear impact force F produced by the stopper, Paidoussis et al. [47] suggested the so-called ‘smoothened’ trilinear spring model which is capable of adequately represents the impact force. Therefore, the ‘cubic’ trilinear model is employed in the present study. It should be noted that the stopper configuration of types 1 and 2 produces the impact force as follows: 1 F  k[ w  wb  ( w  wb  d  w  wb  d )]3 2

(16)

As to types 3 and 4, the stopper is moving with the energy harvester, so the impact force is expressed as: 1 F  k[ w  ( w  d  w  d )]3 2

(17)

where k represents the impact stiffness, d is the impact distance between the stopper and the energy harvester. Substituting Eqs. (7), (14), and (15) into Eq. (3), the coupled nonlinear governing equations of the energy harvester system are obtained as follows:

10

s  m  s  +mt ( s  Lb )   w  w  w2  ww  ds    0

 w  m  s  +mt ( s  Lb )    w2  ww  dsds  cw s 0 Lb

s

 EI  s   w  w2 w  4www  w3 

(18)

 1    s  V  t  ww    s  V  t  1  w2   2 

  F  s  L f   cwb   m  s  +mt ( s  Lb )  wb C pV  t  

Lb V t  1       s   w  www  w2 w ds 0 R 2  

(19)

3. Derivation of the exact mode shapes As known that the obtained nonlinear coupled governing equations (18-19) are partial differential equations, we first discretize those equations by using the Galerkin procedure. It is noted that most of previous studies used the classical mode shapes to describe the vibration mode of piezoelectric cantilevered beam. Considering the kinetic and potential energies of the piezoelectric beam, including the cantilevered beam and piezoelectric element, they directly substituted the classical mode shapes into expressions of kinetic and potential energies to integrate according to the length of beam and piezoelectric element. This is right for the beam fully covered by the piezoelectric element along the length, but it is not accurate for the beam partially covered by the piezoelectric element like in the present study. In this way, it is important to obtain exact mode shapes. Firstly, the transverse displacement of the beam w  s, t  is separated into spatial and time variables components as: n

w  s, t    i  s  qi  t 

(20)

i 1

where qi  t  are the generalized coordinates and i  s  are mode shapes of the piezoelectric cantilevered-beam. However, as depicted in Fig. 1, the piezoelectric layers are partially covered on the cantilevered beam. Therefore, the mode shape of the piezoelectric beam system should consist of two parts as follows [48]:

11

  s   1  s  0  s  Lp

(21)

  s   2  s  Lp  s  Lb Then, the mode shape functions for these two parts can be written as:

1  s   A1 sin 1 x  B1 cos 1 x  C1 sinh 1 x  D1 cosh 1 x

(22)

2  s   A2 sin  2 x  B2 cos  2 x  C2 sinh  2 x  D2 cosh  2 x where the eigenvalues 1 and  2 are related by 1 

4

 EI 2 m1 / EI1m2 2 .

The

corresponding boundary conditions can be expressed as:

1  0   0, 1  0   0

1  Lp   2  Lp  , 1  Lp   2  Lp 

(23)

EI11 Lp   EI 22  L p  , EI11 L p   EI 22 L p  EI 22  Lb   0, EI 22 Lb    2 mt2  Lb   0

It is noted that the coefficients A1, B1, C1, D1, A2, B2, C2, and D2 can be determined by the boundary conditions of Eq. (23). Then the orthogonal mode shapes

  s  can be obtained after using the following orthogonality conditions: M ij  m1  1i1 j ds  m2  2i2 j ds  mt2i  Lb  2 j  Lb    ij Lp

Lb

0

Lp

Lp

Lb

0

Lp

Kij  EI1  1i1j ds  EI 2  2i2j ds  i 2 ij

(24) (25)

where Mij and Kij denote the linear part for the mass and stiffness matrices of the system, respectively.  ij is the Kronecker delta and equal to unity for i  j and equal to 0 for i  j . i denotes the undamped natural frequency for the ith mode. According to the geometric and material properties of the piezoelectric cantilever beam given in Tab. 1, after calculation by virtue of modified mode shapes, Tab. 2 offers the values of β1, β2, and ω for the first four mode shapes. Table 2. Values of β1, β2 and ω for the first four mode shapes. Mode number N

N=1

N=2

N=3

N=4

β1

4.797

14.560

25.350

38.850

β2

5.469

16.599

28.900

40.869

ω (rad/s)

27.766

255.760

755.267

1550.4

12

In the following, we plot in Fig. 2 the first four-order modified mode shapes for different values of length of the piezoelectric sheets (Lp). When Lp=0, in fact, the mode shape is the same as that of the classical mode. It is expected that the length has a big impact on the mode shape of the energy harvester, especially for the higher-order mode shapes. Clearly, it is indicated that the modified mode shapes depend on the covered length of piezoelectric sheets.

(a)

(b)

(d)

(c)

Figure 2 The first four-order modified mode shapes of the cantilevered piezoelectric beam with a tip mass at free end for different values of piezoelectric sheet length. (a) The first order mode shape; (b) The second order mode shape; (c) The third order mode shape; (d) The fourth order mode shape.

As a result, after determining the exact mode shape of the energy harvesting system, we first substitute Eq. (20) into Eq. (18) and multiply by i  s  in both sides with integrating from 0 to Lb , leading to the following matrix equation:

Mq  Cq  Kq  G1  q, q   G2  q, q   G3  q    E1  q   E2  q   V  t 

(26)

 f  q   B  wb , wb  where q   q1; q2 ;

qN  ,

q   q1; q2 ;

qN  13

and

q   q1; q2 ;

qN 

are

the

piezoelectric beam’s displacement, velocity, and acceleration vectors, respectively.

wb , wb , respectively, denotes the acceleration and velocity for the basic excitations. M , C and K denote the linear mass, linear damping, linear stiffness matrices of

the discrete system, respectively. G1 is the inertial nonlinear term; G2 and G3 indicate different the geometric nonlinear terms. E1 and E 2 are the linear and nonlinear electromechanical coupling terms, respectively;

f

is the nonlinear

impacting force; and B are the external base excitation coefficients. The detailed expressions of these matrices and vectors are given in Appendix. Next, by substituting Eq. (20) into Eq. (19), we can get the following equation governing the dynamics of the circuitry part:

C pV  t  

Lb Lb V t      s  1j dsq j     s  1j1k1l dsq j qk ql 0 0 R

1 Lb     s  1 j1k1l dsq j qk ql 2 0

(27)

Afterwards, the obtained dynamic equations of (26-27) are numerically solved using MATLAB (R2017a). It should be mentioned that the initial conditions for displacement and velocity are zero except for q1  0   0.0001 .

4. Convergence analysis for modified and classical mode shapes In general, the piezoelectric energy harvesting system is designed to oscillate at the first resonant frequency under base excitations. Using the classical mode shapes can gradually approach to the accurate model, but it requires at least six-mode number to arrive at convergence, as can be shown in Tab. 3. Most of previous studies did not consider the convergence. This states that it is not enough to use the first or first two-mode number for Galerkin discretization in the classical mode shapes, because it will first result in a big error in the detection of the first natural frequency. Indeed, this error is resulted from the piezoelectric sheets partially covered on the beam which has a great effect on the mode shape of the energy harvester system. Table 3. Values of the first natural frequency for different mode numbers (N) using the modified 14

and classical mode shapes. Mode number N

N=1

N=2

N=3

N=4

N=5

N=6

ω(rad/s) (modified)

27.766

27.766

27.766

27.766

27.766

27.766

ω(rad/s) (classical)

36.098

30.455

29.224

28.955

28.945

28.907

Error (%)

30

9.68

5.25

4.28

4.25

4.11

(b)

(a)

(c)

(d)

Figure 3 Convergence analysis and frequency response curves of the energy harvester in the case (a, b): without stopper and (c, d): with stopper configuration (Type 2) by using (a, c) modified and (b, d) classical mode shapes in the Galerkin discretization for different mode numbers.

Next, the convergence analysis of frequency responses for energy harvester in the case without and with stoppers is conducted by using the modified and classical modes shapes in the Galerkin discretization. The base acceleration is 0.1g, the load resistance is set equal to R=2  106  . The parametric values for the stopper (e.g. Type 2) are k=5e5N/m3, d=0.01m, and Lf=0.75Lb. Inspecting the plotted curves in Fig. 3(a, c) with the modified mode shapes, it is noted that the frequency response curves of the energy harvester’s tip vibration amplitude are almost the same when considering one-, two-, three- or four-mode shapes in the Galerkin discretization. When using the 15

classical mode shapes, as shown in Figs. 3(b, d), however, the tip amplitude of the energy harvester is changing with different mode numbers. Clearly, considering four modes in the Galerkin discretization is not sufficient to reach convergence. This is because the fundamental natural frequency of the energy harvester is varying when increasing the number of the modes in the Galerkin discretization, as shown in Tab. 3. The convergence analysis in this section has two purposes. One is to indicate that if using the classical mode shapes, it is not enough to just employ the first order mode, because it doesn’t arrive at convergence yet. And the choice of the number of modes in the Galerkin discretization strongly affects the resonance region, the amplitude, and strength of the nonlinear behavior. The other purpose is to verify that if utilizing the modified mode shapes, only the first order mode is enough to gain convergent and accurate results, which can greatly save the computation time in numerical calculation. Consequently, the following numerical calculations are based on one mode Galerkin discretization by virtue of the modified mode shapes.

5. Nonlinear analysis and output performance of the energy harvester Based on the derived fully nonlinear equations, the dynamic characteristics of the energy harvester are investigated in detail. Firstly, it is significant to determine the optimal load resistance where the output power is the highest. Fig. 4(a) shows variations of the harvested average power as a function of the load resistance for different values of the base acceleration. The excitation frequency is set equal to the first natural frequency of the energy harvester. The output average power is obtained by Pavg

2 Vrms  , where Vrms represents the root mean square value of the generated R

voltage. When the load resistance is around R=2  106  , the output average power reaches to the peak value for the considered base accelerations. It should be mentioned that this optimal load resistance is independent on the excitation frequency. Inspecting the plotted curves in Fig. 4(b), it is noted that when the base acceleration is small, i.e. a=0.05g, the tip vibration amplitude arrives at the maximum 16

value when the excitation frequency is equal to the first natural frequency of the energy harvester. However, as the base acceleration is increased to high values (i.e. a=0.15g), the range of resonant frequencies gradually shifts to left, showing a softening behavior. This is because the deflection of piezoelectric beam becomes large, the geometric nonlinearity produces the soften characteristic of the energy harvester, which is seldom reported before. (a)

(b)

Figure 4 Variations of (a) average power as a function of load resistance and (b) tip vibration amplitude with excitation frequency for different base accelerations in the case without stoppers.

According to the previous studies [33, 34], the introduced stopper can bring broadband energy harvesting effects for improving the output performance. However, most of them just considered one type of stopper configuration and did not conduct systematic analysis for the adopted stopper. For this reason, four stopper types shown in Fig. 1 are proposed and compared to find and understand which stopper configuration is better. With introducing the different stoppers, the tip vibration amplitude and harvested average power of energy harvester are obtained and compared in Fig. 5. The parameters are set as a=0.2g, k=5e5N/m3, d=0.02m, and Lf=0.75Lb. After inspecting Fig. 5(a-b), obviously, Type 1 and Type 2 not only cause much lower values of vibration amplitude but also brings much narrower resonant frequency region compared to those in the case of Type 3 and Type 4. This is because according to the impact force expressions in Eqs. (16-17), it is noted that the force is determined by the relative displacement between piezoelectric beam and stopper. As to the stopper configuration in types 1 and 2, the relative displacement is w+wb (w is the vibration displacement, wb is the base displacement), usually w and wb are in 17

different directions during oscillations. While for the stopper in types 3 and 4, the relative displacement is w. Therefore, types 3 and 4 will have a larger relative displacement and results in a stronger impact force. In this way, a larger strain occurs for the piezoelectric element and hence a higher output power is produced. In addition, a stronger impact force is followed by a clearer hardening characteristic of the piezoelectric beam, so it produces more broadband responses. As to Type 3, little change of the peak average power occurs but the bandwidth is much increased. It is noted from Type 4 that the bandwidth is much more increased compared to Type 3. However, the peak average power is decreased. Clearly, Type 1 and Type 2 are unadvisable. While both Type 3 and Type 4 promote the energy harvester to have a better output performance. Because Type 3 results in a higher peak average power while Type 4 produces a wider bandwidth, it is hard to say which is better. However, with regards to fabricate the real energy harvester with stoppers, the stopper’s mass and volume should be considered in evaluating the output power metrics of the energy harvester according to [49]. In the following, the theoretical outputs of energy harvester with different stoppers are qualitatively compared with those of experiments in [41]. The physical parameters like dimensions and permittivity of piezoelectric beam in the present theoretical study are totally different from those used in previous experimental research [41], including the type of piezoelectric layer (the present study is bimorph, the experiment is unimorph [41]). As a result, it is not possible to compare the results in quantitative. However, the considered four types of stopper configuration are the same for present theoretical study and previous experiments. So we can compare the results in qualitative, which are shown to be agreeable.

18

(a)

No stopper

(c)

3 1 2

(b)

(d)

4

Figure 5 Variations of (a) tip vibration amplitude and (b) average power with the excitation frequency for different types of stopper configuration by virtue of the present theoretical analysis; (c) real-time measured output voltage and (d) the envelope curves in the experiments [41] for different stopper configurations which are the same as those in (a-b).

5.1 Parametric analysis of the stopper configuration In this subsection, parametric analysis of the stopper in the cases of Type 3 and Type 4 is performed to investigate the effects of stiffness k, spacing distance d, and placed position Lf on the bandwidth and peak average power of energy harvesting system.

19

(a)

(b)

Type 3

Type 4

Type 4

Type 3 II

III

(c)

(d) I

Figure 6 (a, b): Average output power of the energy harvester varying with the excitation frequency for different values of stopper stiffness and (c, d): contour plots for average output power varying with stiffness and excitation frequency in the case of stopper configuration in (a, c): Type 3 and (b, d): Type 4.

Firstly, the effect of stopper stiffness (k) on output average power of the energy harvester is explored with consideration of Type 3 (Fig. 6(a, c)) and Type 4 (Fig. 6(b, d)). Different values of stiffness, namely, k=5e4N/m3, 5e5N/m3, 5e6N/m3 are considered. Other parameters are kept constant as a=0.2g, d=0.02m, Lf=0.75Lb. It follows from Fig. 6 (a, b) that increasing stiffness results in a broadening of energy harvesting bandwidth but a decrease in the peak average power for both of Type 3 and Type 4. Indeed, the bandwidth is increased not so much but the peak average power is reduced a little in the case of Type 3. While using Type 4, it produces a much wider bandwidth but the peak average power is reduced a lot. For example, when the output average power is beyond 1mW, the resonance frequency region for the energy harvester without stopper is between 26.4Hz and 28.5Hz and the peak average power is 4.95mW. With the stopper of Type 3, the resonance region is increased to be between 26.4Hz and 31.6Hz and the peak power is reduced to 4.8mW when the 20

stiffness is 5e5. In the case of Type 4, the resonance region is between 26.4Hz and 37.4Hz and the peak power is 4.55mW. In order to further understand the variation trend of average power with the stiffness and excitation frequency, the contour plots are depicted in Fig. 6(c, d). In this case, after inspecting Fig. 6(c, d), it is noted that if the external excitation frequency changes from 26Hz to 32Hz, using the stopper of Type 4 is better for energy harvesting and the value of stiffness can be at region I in Fig. 6(d). When the excitation frequency is between 32Hz and 34Hz, Type 3 is better and the stiffness can be region II in Fig. 6(c). As the excitation frequency is beyond 34Hz, it needs the stopper of Type 4 to enhance the output performance. The stiffness values are at region III depending on the excitation frequency, as shown in Fig. 6(d). (a)

Type 3

Type 4

(b)

Type 3 I

Type 4 (c)

(d) II

III

Figure 7 (a, b): Average output power of the energy harvester varying with the excitation frequency for different values of spacing distance and (c, d): contour plots for average output power varying with spacing distance and excitation frequency in the case of stopper configuration in (a, c): Type 3 and (b, d): Type 4.

In the following, the impact of spacing distance (d) on output average power of the energy harvester with stopper in the cases of Type 3 (Fig. 7(a, c)) and Type 4 (Fig. 21

7(b, d)) is investigated. The other parameters are a=0.2g, k=5e5N/m3, Lf=0.75Lb. It follows from Fig. 7(a, b) that when the spacing distance is increased, the bandwidth of energy harvesting is narrowed but the peak average power is increased for both types. Under the same spacing distance, the reduction in the peak average power in the case of Type 3 is much less than that of Type 4. While the increment of bandwidth for Type 4 is much more than that of Type 3. For instance, when d=0.01m, the resonance region is between 26.4Hz and 28.6Hz and the peak power is 4.85mW in the case of Type 3. As for Type 4, the resonance region is between 26.4Hz and 30Hz but the peak power is 4.26mW. In this way, the contour plots are shown in Fig. 7(c, d) to further pick out which type is better according to external excitation frequencies. It is found that when the excitation frequency is between 26Hz and 29Hz, Type 3 is better used for energy harvesting and the values of spacing distance can be at region I shown in Fig. 7(c). When the excitation frequency is between 29Hz and 35Hz, using Type 4 can arrive at a better effect of energy harvesting, the spacing distance can be at region II in Fig. 7(d). As the excitation frequency is beyond 35Hz, it is required to use Type 4 to get better energy harvesting, as shown in region III of Fig. 7(d). Type 3

(a)

(c)

(b)

Type 4

(d)

Type 3

Type 4

III

I

II

(c)

22

(d)

Figure 8 (a, b): Average output power of the energy harvester varying with the excitation frequency for different values of placed position and (c, d): contour plots for average power varying with placed position and excitation frequency in the case of stopper configuration in (a, c): Type 3 and (b, d): Type 4.

Finally, the placed position of stopper configuration is also considered to affect the output power of the energy harvester system, as shown in Fig. 8.The other parameters are set as a=0.2g, d=0.02m, k=5e5N/m3. Inspecting Fig. 8(a, b), it is noted that when the placed position is closer to the free end, a wider bandwidth is obtained but a lower peak average power is produced for both types. It remains that Type 3 renders a higher peak average power but a narrower bandwidth compared to Type 4. In fact, when the stopper of Type 3 is placed at 0.8Lb, the resonance region is between 26.4Hz and 33Hz and the peak power is 4.4mW. While for Type 4, the resonance region is greatly increased to be between 26.4Hz and 43.7Hz but the peak power is decreased to 3.4mW. in this way, after analyzing the contour plots in Fig. 8(c, d), it is found that when the excitation frequency is between 26Hz and 30Hz, it’s better to choose Type 3 to enhance the output level of energy harvester. The placed position can be at region I in Fig. 8(c). As the excitation frequency is between 30Hz and 35Hz, the harvester with stopper of Type 4 produces better output performance, as shown in region II in Fig. 8(d). When the excitation is beyond 35Hz, in the same way, only Type 4 renders the harvester to have a wide bandwidth of energy harvesting. So the placed position can be at region III depending on excitation frequency, as shown in Fig. 8(d).

6. Conclusions In this study, a piezoelectric energy harvester with four different types of stopper configuration has been proposed, modeled, and compared for improving its harvesting performance. The coupled nonlinear governing equations of the energy harvesting system are derived based on the extended Hamilton’s principle, and the exact mode shapes are derived and employed to discretize the equations in the Galerkin process. After linear and nonlinear analyses by virtue of numerical calculations, some 23

important conclusions are given below: 1.

The convergence analysis for present energy harvesting system indicates that just one mode number is enough to reach convergence of the results when using exact mode shapes, but it requires at least four mode numbers to get convergence using classical mode shapes.

2.

With consideration of geometric and inertia nonlinearities, results show that the energy harvester without stoppers exhibits a softening behavior which becomes more apparent with an increase in the base acceleration. With introducing the stopper configuration, however, the softening characteristic is changed to hardening, which clearly enhances the energy harvesting performance.

3.

Comparison results show that Type 3 and Type 4 of stopper configuration give a better effect on output performance of energy harvester, displaying that using Type 3 can produce a higher peak average harvested power while using Type 4 offers a wider harvesting bandwidth.

4.

Parametric analysis demonstrates that increasing the stopper’s stiffness, decreasing the spacing distance and getting closer to the free end are followed by an increase in bandwidth but a reduction in harvested peak average power.

24

Declaration of competing interests The authors declare that they have no conflict of interest. All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. This article does not contain any studies with animals performed by any of the authors. Informed consent was obtained from all individual participants included in the study.

Acknowledgments The authors acknowledge the support provided by Fundamental Research Funds for the Central Universities, HUST (2017KFYXJJ135), Natural Science Foundation of Hubei Province (2017CFB429), and National Natural Science Foundation of China (No. 11672115 and No. 11602090).

25

Appendix The detailed expressions for corresponding matrices are given as follows:

M ij    m  s  +mt ( s  Lb )  i j ds   ij 0 Lb

K ij   EI  s  i j ds  i2 ij

Lb

Lb

Cij  c  i j ds 0

0

  m  s  +m  ( s  L )     s   dsds  t b  i j 0 k l  0   G1i   L  q j qk ql Lb s b   i j   m  s  +mt ( s  Lb )    kldsdsds  s 0  0   Lb  m  s  +m  ( s  L )     s   dsds  t b  i j 0 k l  0   G2i   L  q j qk ql Lb s b   i j   m  s  +mt ( s  Lb )    kldsdsds  s 0  0  Lb Lb  EI  s       ds  EI  s      ds  i j k l i j k 0  0  G3i    q j qk ql Lb 4  EI  s  i j klds   0  Lb E    s   ds Lb

1i



i

0

Lb  1  E2i     s  i j k    s  i j k  dsq j qk 0 2   Lb  N  f i    i F    j q j    s  L f  ds 0  j 1 

Bi  c  i dswb    m  s  +mt ( s  Lb )  i dswb 0 0 Lb

Lb

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