Analytical solution of piezoelectric energy harvester patch for various thin multilayer composite beams

Accepted Manuscript Analytical solution of piezoelectric energy harvester patch for various thin multilayer composite beams Ahmad Paknejad, Gholamhossein Rahimi, Amin Farrokhabadi, Mohammad Mahdi Khatibii PII: DOI: Reference:

S0263-8223(16)31072-8 http://dx.doi.org/10.1016/j.compstruct.2016.06.074 COST 7596

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

9 December 2015 28 June 2016 29 June 2016

Please cite this article as: Paknejad, A., Rahimi, G., Farrokhabadi, A., Khatibii, M.M., Analytical solution of piezoelectric energy harvester patch for various thin multilayer composite beams, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.06.074

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Analytical solution of piezoelectric energy harvester patch for various thin multilayer composite beams Ahmad Paknejad1, Gholamhossein Rahimi *1 , Amin Farrokhabadi1, Mohammad Mahdi Khatibii2 1

1 2

Department of Mechanical Engineering, Tarbiat Modares University, Tehran, 14115-111, Iran Department of Mechanical Engineering, Semnan University, Semnan, 14115-111, Iran

Abstract The study of piezoelectric cantilever beam as vibration based energy harvesting has been emerged over the past two decade. This paper presents distributed parameter electroelastic modeling of various multilayer composite beams with piezoelectric energy harvesters. This analytical model of beam with piezoceramic patch is developed base on classical laminate theory for the clamped-free boundary condition. For piezoceramic patch, two thin PZT-5A are symmetrically bonded on composite beams. The closed-form steady state response for coupled electrical output and structural vibration are obtained under harmonic time base excitation. The base motion vibration is applied in the form of translation in a transverse direction. The electrical output and vibration response are derived and generalized base on analytical electroelastic frequency response functions (FRFs) relating to base excitation for different layout composite laminate. Experiments are conducted to verify the first two natural frequencies for all specimens without active element layers. According to effect of damping coefficient on analytical electroelastic modeling, the parameter of damping ratio is given by experimental test. Finally, the effect of various composite laminated layout with piezoelectric patch harvester on power generated are discussed in details as well. Keywords: Smart composite beam, vibrational energy harvesting, piezoceramic patch-based harvesters, FRFs output analysis

*1Corresponding Author ([email protected])

1

1. Introduction Recently, there has been an explosion of research in the area of harvesting energy by using the direct piezoelectric effect from ambient vibrations [1], [2] and human activities [3] to generate self-power and wireless sensor monitoring systems [4]-[6]. The goal in this research field is to power small electronic components by using the vibrational energy available in their environment so that the demand for battery replacement and disposal can be minimized [3]. Basically rule some transformation mechanisms can be used for converting ambient vibrations into electricity such as electrostatic [7], [8], piezoelectric [9], [10], electromagnetic [11], [12] and magnetostrictive [13], [14]. The basic distinction of piezoelectric materials over the other transmit is their high power density, ease of application as well as relative ease of productivity at small scales. Therewith, no bias voltage input is needed (unlike the case of electrostatic transduction) and exploitable voltage levels can be acquired directly from the material itself without step-up conversion (unlike in electromagnetic induction) [9], [15]. On the other hand the application of composite structure shows that, theses structure are used in automotive, hydrospace, nuclear, defense and aerospace (such as aircraft wings, wind turbine blades and helicopter blades [16]) structural applications [17]. It should be emphases that, by embedded distributed piezoelectric sensors/actuators on advanced composites structure, the potential outputs with forming high-strength, high-stiffness and light-weight structures make them well suited for self-monitoring and self-controlling [18]. These advantages makes them to use in smart structure applications such as active vibration and buckling control, shape control, damage assessment, active noise control, structural health monitoring and energy harvesting. According to the research conducted, significant efforts in the field of smart composite structure have been focused in active vibration control and special type of energy harvesting. Due to the mentioned benefits and wide application of these type of material in various industries, active vibration control of composite structure are taken into account for the past recently decade. Using these structures under dynamic loads and control them is of paramount importance in safe and smooth functioning of the system [19]-[23]. These active layers can be embedded into or surface bonded with the structure. Many analytical models have been developed for active vibration control of composite beam [24]-[28]. Exact mathematical model for active damping [24] and an efficient new coupled one-dimensional model [25] were presented for Mindlin laminated beam theory [26], vibration shunt control [27] and using the frequency-domain spectral method [28] with finite element verification respectively. In addition to analytical models for vibration control of composite beam, the finite element method (FEM) [28], [29] consist of nonlinearities [30], [31], optimization [16] and active damping [32] has been extensively used, as reported in the literature. In what follows is a summary of selected studies focused on cantilever piezoelectric energy harvesting beam. In the most of conducted researches, the host structure with isotropic material properties were used [33]-[44]. Exact analytical solution and/or experimental response [33]-[37] for the coupled mechanical and the electrical outputs are then reduced for the particular case of harmonic transverse behavior in time [33] or in the radial direction [34] were presented for bimorph cantilever configurations [34] base on Timoshenko beam theory [36]. Furthermore the strain nodes of vibration modes presented with continuous electrodes [37]. It should be noted that approximate analytical formulation for cantilever piezoelectric 2

energy harvesters was accounted for Euler–Bernoulli, Rayleigh, and Timoshenko models with axial deformations [38]. Moreover analytical and numerical formulations of piezoelectric energy harvesting were proposed from broadband random vibration [39]. On the other side, numerical research [40]-[44] for simple and complex structures include multiple modes of the system with the equivalent circuit model [40] as well as a linear system with a low natural frequency (non-buckled beam; and a buckled beam harvester) [41], [42] were prepared different aspects of cantilever beam harvesters with optimization for size and shape of piezoceramic layers [43]. Finally effect of enhancing the conversion abilities of piezoelectric materials derived based on initial energy injection [44]. The performed researches on vibration energy harvesting via composite structures has heavily focused on rectangular and nonrectangular bistable composite laminate [45], [46]. The bistable composite laminate due to asymmetric stacking sequences can exhibit large deformation and they have been identified as good candidates to integrate piezoelectrics for broadband energy harvesting [47], [48]. Significant power outputs over a wide bandwidth have been obtained using bistable composite laminates due to the mismatch in thermal properties and the temperature change experienced during production [49]. The study of bistable piezoelectric composite laminate for energy harvesting have been developed analytically, experimentally and optimization [50]-[54]. It is worth pointing out that one-dimensional analytical model of piezoelectric energy harvesters including three dimensional effects developed base on classical laminated beam theory (CLT). The equations of motion are calculated by the dissipative form of Euler– Lagrange equations in the field of Micro-Electro-Mechanical Systems (MEMS) [55]. In addition, an accurate parameterized analytical model was investigated a vibration-based active fiber composite (AFC) energy harvester material and interdigitated electrode. The analytical reduced-order model and study the dynamic response of the energy harvesting system are derived by the Galerkin procedure along with the Gauss’s law [56]. To the best knowledge of authors, the exact analytical solution of piezoelectric energy harvester layer for composite laminate host beam has not been reported in the literature yet. In present study, the electromechanical equations of motion (EOMs) will develop for composite laminate beam with bimorph piezoelectric layers. The EOMs would be derived base on Euler-Bernoulli cantilever beam theory and modal analysis procedure with small transverse direction at its base. Electrical and mechanical closed form steady state solution response will obtain by harmonic base excitation. The various composite laminate with four layers and different stacking sequence are considered in details. The electromechanical frequency response functions (FRFs) contain voltage, current, power and relative tip displacement will be presented for different composite structure and discussed in details as well. Furthermore, experimental modal vibration test is done to show the accuracy of analytical solution for all case studies.

3

2. Analytical Electromechanical Modeling 2.1.

Derivation of the Electromechanical Formulation

Fig. 1 represents a piezoelectric energy harvester consisting of a thin PZT layer which is structurally integrated to a laminated host beam. The harvester beam is assumed to be excited of its base by translation in the transverse direction. The beam’s thickness is assumed to be thin so that the displacement fields are developed based on the Euler–Bernoulli beam theory. The general governing equation of motion (EoM) of a beam with embedded piezoelectric layer can be written as [57]:


 , 

 , 
 , 
 ,  +     +  +  =
 

 


 

(1)

,  = −   !"! −  # !"!,

(2)

 % = !&'(( % )

(3)

− 

  ,   

.

where, ,  is the beam transverse deflection relative to its base,  ,  is the base excitation motion on the beam along z direction, ,  is the internal bending moment consist of substructure and piezoelectric layer effects,  is the equivalent damping term,  is the strain rate damping equivalent coefficient, I is the equivalent area moment of inertia,  is the coefficient of viscous air damping and  is mass per unit length. The total internal bending moment can be calculated by integrating the stress moment of host structure and piezoelectric layer individually through the thickness as



$

where  and # are the normal stresses of substructure and piezoelectric layer along x direction, respectively. The bending stress in the k-th layer of the composite laminate is defined by [58] where ) = −!

*  *

is the curvature of composite laminate. As a result, the normal

stress of host beam can be written as

 % = −&'(( % !

"  " 

(4)

4

Fig. 1. The schematic model of bimorph piezoelectric base on composite laminate beam a) 3D model b) Cross section c) 2D model

It is worth noting that the parameter &'(( is defined by [58]

&'(( = &(( cos . / + 2&( + 2&11  sin / cos  / + & sin. /

&(( =

4( 1 − 6(6(

& =

4 1 − 6( 6(

&( =

6(4 1 − 6(6(

(5)

&11 = 7(

where 4( and 4 are the Young’s moduli along the fiber direction and normal to the fiber direction, respectively, 7( is the shear modulus, 6( and 6( are the Poisson’s ratios and θ is the fiber orientation measured from the global x-axis. The other normal stress of Eq. (2) obtained by the constitutive equation of piezoelectric material [59] as

# = 4# 8( ,  − "( 4  #

(6) 5

Where, 4# is the modulus of elasticity of piezoelectric materials, "( is the piezoelectric strain constant and the electric field term of 4  is defined by 4  = −

9 :$

which is the division of the generated piezoelectric voltage ; by

piezoceramic patch thickness ℎ# . According to the Euler-Bernoulli beam assumption, the mechanical strain can be express as 8( ,  = −!

 , 

. Hence, the internal

bending moment is obtained by substituting Eqs. (4) and (6) into Eq. (2), as


 ,  # ,  = −4 + =# ;
   ,  = −4

(7)


  ,  + = ;
 

(8)

where 4 is the flexural rigidity of the composite beam. The both typical piezoelectric energy harvesting configuration are investigated in the form of cantilevered beams with one or two piezoceramic layers (i.e., a unimorph or a bimorph). Note that its value depends on the types of electrical connection while subscripts “s” and “p” indicate the series as well as parallel connections, respectively. As a result, the flexural rigidity in the form of bimorph piezoelectric beam can be calculated, as  ℎ ℎ  4 = ?@&'((%  ! "! + 24# F + ℎ# G − I 3 2 8 BCDE A

BC

%



(9)

In addition, the electromechanical coupling term of the bending moment =  for bimorph piezoelectric beam in Eq.(7), is calculated by

4# "( ℎ  ℎ  =#   = JFℎ# + G − L ℎ# 2 4

(10)

=   = =#  /2

(11)

Furthermore, the mass per unit length of beam can be calculated as A

 = NO P = @NO % ℎ% − ℎ%Q(

(12)

%R(

Where, NO % is the mass density of the k-th layer of the composite. The governing partial differential equation in terms of  ,  and ;  is given by substituting Eq. (7) into Eq. (1). This EoM is derived for the composite host beam with piezoelectric coupling base on parallel connection:

S4 









 ,  

T+

S   





U ,  

T + 

S=# ;VWX  − WX − YZT = −

, 

  ,

6

 

+

 ,  

+

(13)

Similarity, for series connection using Eq. (7) it can be deduced 



 ,  

T+





S  

U ,  

T + 

[= ;VWX  − WX − YZ\ = − 





S4

, 

  ,

 

+

 ,  

+

(14)

Furthermore, according to the linear piezoelectric constitutive equations, another essential governing equation is necessary to determine the unknown variables of  ,  and ; as [59]: ^ ] ,  = "( # ,  + 8 4 

(15)

^ Where, the parameters ] and 8 are the electric displacement and the permittivity at constant stress respectively. Rewriting Eq. (15) in terms of ,  and ; gives

] ,  = −"(4# ℎ#_


,  ; − 8 
 ℎ#

(16)

Where, 8 is the permittivity at constant strain and ℎ#_ =

`: a:$ b 

is the distance from

the center of one of the piezoelectric patch layer to the mid-plane surface of composite beam. It should be noted that X̅( = "( 4# is the effective piezoelectric stress constant. The electric charge generated at piezoelectric layer is calculated by integrating the electric displacement over the electrode area as [59]:

d  =  ] . f"P

(17)

g

Where, f is the unit normal vector to the piezoelectric plane. Substituting Eq. (16) into Eq. (17) expresses the electric charge by integrating the electric displacement over the electrode length as

d  = − 

h

X̅( ℎ#_

i


 ,  ; + 8  " 
 ℎ#

(18)

Multiplying electric resistance j by electric current, which is the first derivative of electric charge with respect to time, jk = j

for the voltage of the piezoelectric layer as

; = −j 

h

i

*l *

yields the following expression

h
 ,  8 ";  X̅( ℎ#_ " + " 
 
 ℎ# i "

(19)

Eq. (19) represents the specific expression of the voltage for one layer piezoelectric energy harvester. Therefore, the general form of the governing equation related to ; = j

*l *

can be express as

7

h ; 8 Y ";
 ,  + = −  X̅(ℎ#_ " j ℎ# "
 
 i

(20)

Hence, Eqs. (13) and (20) express the distributed parameter electroelastic model of composite laminated beam with piezoelectric layer in physical coordinate, which can be solved by standard modal analysis method. 1.2.

General modal procedure solution

Using the modal analysis procedure [60], the vibration motion of beam can be expressed by implementing the eigen-function expansion procedure as o

 ,  = @ m n 

(21)

R(

Where, m  and n  are the mass normalized mode shapes and the modal coordinate (modal time response) of beam. To solve the governing equations, it is necessary to extract the normalized eigen-functions of the beam. These eigenfunctions are obtained by solving free vibration of the beam with clamped-free boundary conditions. The mass normalized mode shapes of the beam are calculated using the following formulations [60]:

mp   = / qcosh

where

p =

sp sp sp sp  − cos  − p Fsinh  − sin Gt Y Y Y Y

sinh sp − sin sp cosh sp + cos sp

(22)

(23)

These mass normalized mode shapes of eigen-functions satisfies the orthogonality condition (which can be used to change EoM in physical coordinate to modal space) as [61] h

 up  u  " = vp i



h

i

" up   "  u   4  = wp vp "  " 

(24)

Note that / and wp are the modal amplitude constant and undamped natural frequency for r-th vibration mode of the beam, which can be calculated by these orthogonality conditions respectively. As an alternative solution, / = 1y can be used in eigen√Y

function mass normalized. The first two natural frequencies from this analytical 8

solution are validated by experimental test for various composite beams without active element layers that is explained in section 2. 1.3. Harmonic base excitation solution Using the modal analysis procedure for beam structure, changing the governing electroelastic equations in physical coordinate into modal coordinates can be obtained by substituting Eq. (21) into coupled electromechanical Eq. (13) as

∑o R(

S4 

*

*

*  { 

 ∑o R( u  

*

T n  + ∑o R( |

*  ~ 

* 

+

*

* 

S 

*

*

* {  * 

T +  }

*~  *

+

[= VWX   − WX − YZ\; = −

  ,

 

(25)

.

Integrating Eq. (25) over the length of the beam after premultiplying it by mp   gives the EoM in modal space as follow

n p  + 2€p wp n p  + wp np  + ‚p ; = ƒp 

(26)

Here, €p is damping ratio of the r-th mode. Furthermore, the modal coupling and forcing terms of ‚p and ƒp  are defined by

"up   … " Rh

‚p = =# „ #

,

" † ƒp   = − J up  " L "  i h

"up   … " Rh

‚p = = „

(27)

(28)

It is necessary to calculate the unknown damping ratio parameter (€p ) from experimental test. Also to change the other coupled electromechanical equation into ordinary differential equation with respect to time, Eq. (21) is also substituted in Eq. (20) as follows

1 ˆl "; ; + ‡# = − @ ‰p n p  j " o

(29)

pR(

Where, ‡# and ‰p are the equivalent capacitance of bimorph piezoelectric and the modal electromechanical coupling term. Finally the second coupled electromechanical equation of electrical circuit can be written in modal spaces as ˆl

ˆl ‡# ; 

1 + ; = @ ‰p n p  j o

(30)

pR(

9

Table 1. The constant equivalent capacitance and modal electromechanical coupling term [59]

Parallel Connection

Ca eq p

s bL 2ε33 hp

2e31hpc b

γr

Series Connection

ε 33s bL 2h p

d φr ( x ) dx

e31hpc b x= L

d φr ( x ) dx

x= L

To find the harmonic solution, it is assumed that the base is moving translation harmonically in ! direction, which means that

† = Ši X ‹Œ

(31)

and

ƒp  = p X ‹Œ

(32)

Now, substituting Eq. (31) and Eq. (32) into Eq. (28) gives the parameter p as h

p = Šiw J up   " L 

(33)

i

It is worth noting that, the steady-state expression due to the harmonic excitation of the beam, the unknown terms of voltage ; and the modal response of np  are also expected to be harmonic as follows

; = Ži X ‹Œ

np  = Wp X ‹Œ

(34)

Where, the temporal term amplitude Wp and the voltage amplitude Ži are calculated by substituting Eqs. (33) and (34) into Eqs. (26) and (30) as

Wp =

p − ‚p Ži wp − w  + 2€p wp w

(35)

mp p wp − w + 2€p wp w Ži = mp ‚p 1 ˆl + w‡# + w ∑o pR( w  − w  + 2€ w w j p p p w ∑o pR(

10

(36)

electrical resistance ( = Ž yj ) as

Furthermore, the power output FRF is defined by the square of the voltage output per 

 mp p 1 wp − w + 2€p wp w i = ‘ ’ mp ‚p j 1 + w‡ˆl + w ∑o pR( w  − w  + 2€ w w # j p p p

w ∑o pR(

(37)

resistance ( = Žyj ) as

Finally, the current output FRF is defined by dividing the voltage output per electrical

mp p w ∑o pR( w  − w  + 2€ w w 1 p p p

i = ‘ ’ mp ‚p 1 ˆl j + w‡# + w ∑o pR( w  − w  + 2€ w w j p p p

(38)

2. Results and discussion Here the analytical formulation of composite laminate beam as a host structure with bimorph piezoelectrics discussed in this section as well. Regarding the lack of results on vibration composite laminate base piezoelectric harvester in the literature, to verify the analytical solution, a modal test is firstly conducted for all specimens and the results are compared with those from experimental. The modal test is done to obtain the first two natural frequency as well as damping ratio. The next stage of results is shown the effect of electromechanical bimorph piezoelectric consist of voltage, current, power output and tip displacement for various composite laminate beam. In the numerical examples, the glass/epoxy composite material with different stacking sequence and thickness is analyzed. The material properties of composite lamina are listed in Table 2. The properties of the lamina have already been done through characterization experimental tests [62]. The dimensions of the different composite laminate beam are shown in Table 3. The piezoelectric actuator is assumed to be PZT5A with the material properties in Table 4. Table 2. Material properties of lamina [62] Parameter

4((

4 7(

Unit

Value

7‡

26.5

7‡ 7‡

11

8.83 3.67

6( N

0.33

Ҡ/

1980

Table 3. Dimension of the beam Type

V60, −60Z − –1

Thickness (mm)

Length (mm)

Width (mm)

1.44

65

25

1.38

63

26

1.52

62

25

1.32

65

26.5

V60, −60Z − –2 V0, 90Z − –3 V0, 90Z − –4

Table 4. Material properties of piezoelectric Parameter

4 N ˜

"( 8

Unit

7‡

Value

Ҡ/

66

™/Ž

0.3

f/Ž

7800

-190 10.38

2.1. Experimental modal analysis

In present study, the natural frequencies as well as the damping coefficients of different composite laminates are obtained using ambient modal testing and Stochastic Subspace Identification method which is a well-known identification technique [63]. For this purpose, the boundary conditions of laminates are considered to be clampedfree and imposed to vibrate due to arbitrary excitation. Then the response amplitude is measured using a Laser Doppler Vibrometer type 8329 (Fig. 2). The obtained data are analyzed and sent to the computer using a B&K analyzer type 3560D. The SSI method of pulse software is used for modal parameter identification [64]. It is clear that the damping coefficient which depends on the geometry, material properties and boundary conditions, should be defined in different laminates with the same geometry and weight. Furthermore, it is necessary to clamp the laminates with an equal torque inside the fixture. By performing the modal testing on the composite specimens i.e. TC1, TC2, TC3 and TC4, and using the SSI method, the system matrix is evaluated and the system poles are defined from the eigenvalues 12

decomposition of the system matrix. The stability diagram of state space model is obtained by the performed test on the composite specimens (Fig. 3). The stable poles are shown by “+” notation which represent the natural frequencies and damping coefficients of the each laminate.

a)

b) Fig. 2. The schematic model of bimorph piezoelectric base on composite laminate beam a) 3D model b) Cross section c) 2D model

The obtained natural frequencies and damping coefficients of different composite specimens are depicted in Table 5. Furthermore the natural frequencies of two first modes of different specimens are compared with each other in Fig. 4. The obtained result reveal that, the V0/90Z –3 stacking sequence have higher natural frequency and lower damping ratio than the other laminates. However, the V60/−60Z –2 stacking sequence have lower natural frequency and higher damping ratio just in first mode frequency. Table 5. Results output of experimental test Frequency (Hz) Type

Mode1

Mode2

13

Damping Ratio (%) Mode1

Mode2

V60/−60Z − –1 V60/−60Z − –2 V0/90Z − –3 V0/90Z − –4

133.43

821.07

0.978

1.716

121.67

771.33

1.168

1.464

222

1421

0.368

0.597

180.83

1128

0.514

1.752

a)

Fig. 3. Stabilization diagram of the specimens a) sample V0/90Z b) sample V60/−60Z

b)

14

Fig. 4. Comparison of natural frequencies

2.2. Electromechanical Result

In this section, the response of a cantilever bimorph energy harvester to transverse base excitation in different composite beams is analyzed. These electrodes are assumed to coat both the top and bottom of the piezoelectric layers evenly to assure a constantly distributed electric field within the dielectric. Each mode has a unique damping ratio (The damping ratios are calculated from the Eq. (39)). The beam is also considered to be excited harmonically. Frequencies of excitation range from below the first harmonic mode to above the second resonant. The results of first two natural frequency obtained by analytical and experimental are compared in Table 6. Table 6. Results output of analytical solution Frequency (Hz)

Error (%)

Type

V60, −60Z − –1

Mode1

Mode2

Mode1

Mode2

V60, −60Z − –2

128.3

804.3

2.3

4.1

130.9

820.5

6.9

5

V0, 90Z − –4

227.9

1428.3

2.6

0.5

180.1

1128.5

~0

~0

V0, 90Z − –3

15

The obtained results reveal that there are good agreements between the analytical and experimental results. Due to the impact of strain rate (š) as well as air viscosity damping (‡), the damping coefficient according to Eq. (26) can be defined as

€p =

  wp + 2Š 2 wp

(39)

Having the damping coefficient ζœ and natural frequency ωœ from the performed C I C experiment, the constant coefficients of Ÿ y2YI and ¢y2m will be obtained. By determining these factors, it is possible to define the damping coefficient for the other CI C natural frequencies. The evaluated constant coefficients of Ÿ y2YI and ¢y2m for each laminate specimen are mentioned in Table 7.

Table 7. Proportionality constants Type

V60, −60Z − –1 V60, −60Z − –2 V0, 90Z − –3 V0, 90Z − –4

¤¨ © ¦ª©

¤¥ ¦§

3.1 e-6

6.0203

2.708 e-6

7.3463

6.1938 e-7

3.928

2.4179 e-6

2.7187

Due to greater mass as well as better material properties of piezoelectric related to the composite laminates, the effects of piezoelectric material on the natural frequency and dynamics of structure should be considered. Furthermore, by changing the frequency, the damping coefficient is also altered. As a result, the modified natural frequencies and damping ratio of each composite laminates along with the attached piezoelectric layer is illustrated in Table 8. Table 8. Natural frequency and damping ratio coefficient of beam with active elements Frequency (Hz) / Damping Ratio Type

Mode1

Mode2

Mode3

Mode4

–1

226.8y 0.0086

1421.6y 0.0284

3980.6y 0.0778

7800.4y 0.1521

–3

289.9y 0.0033

1816.9y 0.0074

5087.5y 0.0199

9969.4y 0.0389

–2

234.1y 0.009

1466.9y 0.0258

16

4107.4y 0.0702

8048.8y 0.1371

–4

234.8y 0.0054

1471.3y 0.0226

4119.7y 0.0627

8073y 0.1227

In present study, the obtained results will be represented in three different categories. The results of first category consider the diagrams of voltage, current, power as well as the maximum deflection related to diverse applied frequencies for different mentioned composite beams. These results will be obtained for both of series and parallel connection separately. The second groups of diagrams involve the variation of voltage, current, power as well as maximum deflection related to a special frequency which is obtained for a special composite beam (TC1) with different resistances. Finally the third group of diagrams includes the variation of voltage, current and power in the terms of resistance in each composite beam for the series connection only. 2.2.1. The first group

The frequency response of different composite beam including the voltage, current, power as well as tip deflection are illustrated in Fig. 5 and 6 for series and parallel connection respectively. These frequency responses in Fig 5a are defined as the magnitude of the ratio of the voltage on the piezoelectric elements to the acceleration of the host structure. It is worth no note that there are three natural frequencies present in these frequency ranges, with the lowest frequency corresponding to the greatest magnitude and the highest corresponding to the lowest magnitude. Additionally, the apparent damping of the modes decreases as the frequency increases, (which was designed into the simulation).

(a)

17

(b)

(d)

(c)

Fig. 5. The different composite beam’s frequency responses for series connection: (a) voltage, (b) current, (c) power, (d) tip deflection

(a)

(b)

(c)

(d)

Fig. 6. The different composite beam’s frequency responses for parallel connection: (a) voltage, (b) current, (c) power, (d) tip deflection

2.2.2. The second group

For a broad range of load resistance including 10-100 kΩ, the frequency response of V60/−60Z composite beam (TC1) including the voltage, current, power as well as tip deflection are reported in Fig. 7. According to the obtained results in Fig 7a, as the load resistance is increased from short-circuit to open-circuit conditions, the voltage output at every frequency increases monotonically. Furthermore, with increasing the load resistance, the resonance frequency of each vibration mode moves from the short-circuit resonance frequency to the open-circuit resonance frequency. Contrary to the voltage output, Fig. 7b reveals that the amplitude of the current decreases with 18

increasing load resistance at every frequency. However, the behavior at every frequency is still monotonic. For every excitation frequency, the maximum value of the current is obtained when the system is close to short-circuit conditions. Referring the power output results, Fig. 7c shows that the power output does not necessarily exhibit monotonic behavior with increasing or decreasing the load resistance for a given frequency. Among the sample values of load resistance considered in present study, the maximum power output for the first vibration mode corresponds to the load of 100 kΩ at 226.8 Hz which is expectedly a frequency in between the fundamental short- and open-circuit resonance frequencies. For the second vibration mode, the maximum power output is obtained for 100 kΩ at 1421.6 Hz. It is worth noting that that the values of the load resistance used in this analysis are taken arbitrarily to observe the general trends. Therefore, the maximum power outputs obtained from each vibration mode are for these sample values and they are not necessarily for the maximum possible (or the optimized) power outputs.

(a)

(b)

(c)

(d)

Fig. 7. Variation of the different parameter output with broad range of load resistance for [60/-60]s laminate (series connection): (a) voltage, (b) current, (c) Power, (d) tip displacement

Finally, the tip displacement response for the set of mentioned resistors and the frequency range can be observed in Fig. 8d. Obviously, it is not possible to distinguish between the curves of different load resistance on three vibration modes. 2.2.3. The third group

The behavior of different parameter output including that voltage, current as well as power with changing the load resistance for excitations at resonance frequencies of the first two vibration modes in V60/−60Z –1, –2 and V0/90Z –3, –4 composite 19

beams are given in Figs 8, 9, 10 and 11 respectively. As can be seen from 5a, while the peak voltage output for mode 1 is greater than mode 2, for values of load resistance less than 1000Ω, the voltage output increases gradually and reaches to a fixed amount in each vibration modes. The obtained results for current output in Fig. 8b reveals that the current output is very insensitive to the variations of the region of low load resistance. In this region, the slope is almost zero for R<1000Ω. Then, the current output starts decreasing with increasing load resistance. Finally Fig. 8c shows that the power output FRF does not exhibit a monotonic behavior with increasing or decreasing the load resistance. Among the sample values of the load resistance considered in present study, the value of maximum power output for the first and second vibration modes corresponds to R=1000Ω.

(a)

(b)

(c) Fig. 8. Variation of different parameter output with load resistance at resonance frequencies of the first two vibration modes in V60/−60Z g − –1 composite beams: (a) peak voltage; (b) peak current; (c) peak power, vs. load resistance.

(a) 20

(b)

(c) Fig. 9. Variation of different parameter output with load resistance at resonance frequencies of the first two vibration modes in V60/−60Z g − –2 composite beams: (a) peak voltage; (b) peak current; (c) peak power, vs. load resistance.

(a)

(b)

(c) Fig. 10. Variation of different parameter output with load resistance at resonance frequencies of the first two vibration modes in V0/90Zg − –3 composite beams: (a) peak voltage; (b) peak current; (c) peak power, vs. load resistance.

(a) 21

(b)

(c) Fig. 11. Variation of different parameter output with load resistance at resonance frequencies of the first two vibration modes in V0/90Zg − –4 composite beams: (a) peak voltage; (b) peak current; (c) peak power, vs. load resistance.

Furthermore, as it is expected, among the sample values of load resistance employed in the analysis, the maximum power output is assigned to the first vibration mode. The results of this research showed that the composite laminated beams with different stacking sequence have similar behavior by changing frequency and electrical resistance. Any contradictory behavior was not observed between V60/ −60Z and V0/90Z layerwise. In this case study piezoelectric energy harvesters, as observed in details, are poor current and power output generators [59]. For instance, in the series connection case, for unit base acceleration and first natural frequency (226.8 Hz and 1 kΩ) of composite beam TC1 the voltage output is around 2.2V while the current associated base on linear estimates with it is just 2.2mA. As a result, this voltage level is fairly good for charging a small device such as small battery, it is the current output that will make the duration of charging substantially long. The frequency response of isotropic aluminum beam including the voltage, current, power FRFs as well as tip deflection are presented in Fig. 12 for series connection with material properties of 4( = 70X9 7‡, =( = 0.3, N = 2700 “†/ and the same dimension by TC3 composite beam. For better comparison the FRFs results of TC3 composite beam are shown in Fig. 12.

(a)

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(b)

(c)

(d)

Fig. 12. . The isotropic and orthotropic beam’s frequency responses for series connection: (a) voltage, (b) current, (c) power, (d) tip deflection

According to Fig. 12, the natural frequencies of isotropic beam greater than composite for each mode but the peak voltage, peak current, peak power and tip deflection are almost equal. This feature shows that the electrical energy harvesting output in composite beam structure are obtained at lower frequencies.

3. Conclusion Composite beam structures with various boundary conditions are commonly used in industries application as well as discussed in the literature. However, the performed researches on energy harvesting due to vibration of composite structure, have been very limited. So the study of embedded piezoelectric energy harvester on composite laminate beam is very important and applicable. In present study, a distributed parameter electroelastic model was developed for piezoelectric energy harvester structurally integrated to cantilever composite laminate beam. Closed-form steady state solutions for the electrical output and structural response were derived for harmonic base excitation. The modal analysis as natural frequency and damping ratio were presented and experimentally verified for all case studies. The analytical FRFs response were derived and presented for voltage, current, power and tip displacement for bimorph piezoelectric energy harvester on composite laminate host beam. The electroelastic formulation were done herein can be utilized for every multilayer symmetric beam with bimorph piezoelectric energy harvester patch.

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References [1] N.S. Shenck, J.A. Paradiso, Energy scavenging with shoe-mounted piezoelectrics, IEEE Micro 21 (2001) 30–41. [2] S. Meninger, J.O. Mur-Miranda, R. Amirtharajah, A.P. Chandrakasan, J.H. Lang, Vibration-to-electric energy conversion, IEEE Transactions on the VLSI System 9 (2001) 64–76. [3] Paradiso J A and Starner T 2005 Energy scavenging for mobile and wireless electronics IEEE Pervas. Comput. 4 18–27 [4] Roundy S, Wright P K and Rabaey J 2003 A study of low level vibrations as a power source for wireless sensor nodes Comput. Commun. 26 1131–44 [5] Beeby S P, Tudor M J and White N M 2006 Energy harvesting vibration sources for microsystems applications Meas. Sci. Technol. 17 R175–95 [6] A. Alaimo A. Milazzo C. Orlando Numerical analysis of a piezoelectric structural health monitoring system for composite flange-skin delamination detection [7] Lee C et al 2009 Theoretical comparison of the energy harvesting capability among various electrostatic mechanisms from structure aspect Sensors Actuators A 156 208–16 [8] Chiu Y and Tseng V F G 2008 A capacitive vibration to electricity energy converter with integrated mechanical switches J. Micromech. Microeng. 18 104004 [9] Wang Z L and Song J 2006 Piezoelectric nanogenerators based on zinc oxide nanowire arrays Science 312 242–6 [10] Anton S R and Sodano H A 2007 A review of power harvesting using piezoelectric materials (2003–2006) Smart Mater. Struct. 16 R1–R21 [11] Beeby S P et al 2007 A micro electromagnetic generator for vibration energy harvesting J. Micromech. Microeng. 17 1257 [12] Zhu D et al 2012 Vibration energy harvesting using the Halbach array Smart Mater. Struct. 21 075020 [13] Wang L and Yuan F G 2008 Vibration energy harvesting by magnetostrictive material Smart Mater. Struct. 17 045009 [14] M.T. Piovan J.F. Olmedo R. Sampaio 2015 Dynamics of magneto electro elastic curved beams: Quantification of parametric uncertainties J. of Composite Structures 133 621–629 [15] Cook-Chennault K A, Thambi N and Sastry A M 2008 Powering MEMS portable devices—a review of nonregenerative and regenerative power supply systems with special emphasis on piezoelectric energy harvesting systems Smart Mater. Struct. 17 043001 [16] N. Zoric, A. Simonović, Z. Mitrović, S. Stupar 2013 Active Vibration Control of Smart Composite Beams Using PSOOptimized Self-Tuning Fuzzy Logic Controller J. of theorical and applied mechanics 51 2 275-281 [17] Robert M. Jones, 1999 Mechanics of composite materials Taylor & Frances Inc. [18] S. Narayanan, V. Balamurugan 2002 Finite element modelling of piezolaminated smart structures for active vibration control with distributed sensors and actuators J. of Sound and Vibration 262 529–562 [19] X. Q. Peng, K. Y. Lam and G. R. Liu 1998 Active vibration control of composite beams with piezoelectrics: a finite element model with third order theory J. of Sound and Vibration 209 4 635-650 [20] G. Song, P. Z. Qiao, W. K. Binienda and G. P. Zou 2002 Active Vibration Damping of Composite Beam using Smart Sensors and Actuators J. of Aerospace Engineering 15 3 [21] A. Mukherjee S.P. Joshi A. Ganguli 2002 Active vibration control of piezolaminated stiffened plates J. of Composite Structures 5 435–443 [22] Jin-Chein Lin M.H. Nien 2005 Adaptive control of a composite cantilever beam with piezoelectric damping-modal actuators/sensors J. of Composite Structures 7 170–176 [23] D. Panahandeh-Shahraki, H. R. Mirdamadi O. Vaseghi 2014 Fully coupled electromechanical buckling analysis of active laminated composite plates considering stored voltage in actuators J. of Composite Structures 118 94–105 [24] Lucy Edery-Azulay Haim Abramovich 2006 Active damping of piezo-composite beams J. of Composite Structures 74 458–466 [25] S. Kapuria, P.C. Dumir, A. Ahmed 2003 An efficient coupled layerwise theory for dynamic analysis of piezoelectric composite beams J. of Sound and Vibration 261 927-944 [26] D. Huang and B. Sun 2000 Approximate analytical solutions of smart composite MINDLIN beams J. of Sound and vibration 244 3 379-394 [27] M. Kerboua et al 2015 Vibration control beam using piezoelectric-based smart materials J. of Composite Structures 123 430–442 [28] Ilwook Park and Usik Lee 2012 Dynamic analysis of smart composite beams by using the frequency-domain spectral element method J. of Mechanical Science and Technology 26 8 2511-2521 [29] Levent Malgaca 2010 Integration of active vibration control methods with finite element models of smart laminated composite structures J. of Composite Structures 92 1651–1663 [30] Shun-Qi Zhang, Ya-Xi Li, Rudiger Schmidt 2014 Active shape and vibration control for piezoelectric bonded composite structures using various geometric nonlinearities J. Composite Structures 122 239–249 [31] Houssein Nasser et al 2012 Active vibration damping of composite structures using a nonlinear fuzzy controller J. of Composite Structures 94 1385–1390 [32] Seung-Chan Choi, Jae-Sang Park, Ji-Hwan Kim 2006 Active damping of rotating composite thin-walled beams using MFC actuators and PVDF sensors J. of Composite Structures 76 362–374 [33] Erturk A and Inman D J 2008 A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters J. Vib. Acoust. 130 041002

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[34] Park J C, Khym S and Park J Y 2013 Micro-fabricated lead zirconate titanate bent cantilever energy harvester with multidimensional operation Appl. Phys. Lett. 102 043901 [35] Erturk A and Inman D J 2009 An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations Smart Mater. Struct. 18 025009 [36] J M Dietl, A M Wickenheiser and E Garcia 2010 A Timoshenko beam model for cantilevered piezoelectric energy harvesters Smart Mater. Struct. 19 055018 [37] Erturk A et al 2009 Effect of strain nodes and electrode configuration on piezoelectric energy harvesting from cantilevered beams J. Vib. Acoust. 131 011010 [38] Erturk A 2012 Assumed modes modeling of piezoelectric energy harvesters: Euler–Bernoulli, Rayleigh, and Timoshenko models with axial deformations Comput. Struct. 106/107 214–27 [39] Zhao S and Erturk A 2013 Electroelastic modeling and experimental validations of piezoelectric energy harvesting from broadband random vibrations of cantilevered bimorphs Smart Mater. Struct. 22 015002 [40] Yang Y and Tang L 2009 Equivalent circuit modeling of piezoelectric energy harvesters J. Intell. Mater. Syst. Struct. 20 2223–35 [41] Cottone F et al 2012 Piezoelectric buckled beams for random vibration energy harvesting Smart Mater. Struct. 21 035021 [42] Friswell M I et al 2012 Nonlinear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass J. Intell. Mater. Syst. Struct. 23 1505–21 [43] Friswell M I and Adhikari S 2010 Sensor shape design for piezoelectric cantilever beams to harvest vibration energy J. Appl. Phys. 108 014901 [44] Lallart M and Guyomar D 2010 Piezoelectric conversion and energy harvesting enhancement by initial energy injection Appl. Phys. Lett. 97 014104 [45] Sergio P Pellegrini et al 2012 Bistable vibration energy harvesters: A review J. of Intelligent Material Systems and Structures 24(11) 1303–1312 [46] R L Harne and KW Wang 2013 A review of the recent research on vibration energy harvesting via bistable systems J. of Smart Mater. Struct. 22 (2013) 023001 [47] Christopher J. Brampton et al 2013 Sensitivity of bistable laminates to uncertainties in material properties, geometry and environmental conditions J. of Composite Structures 102 276–286 [48] Jong-Gu Lee et al 2015 Effect of initial tool-plate curvature on snap-through load of unsymmetric laminated cross-ply bistable composites J. of Composite Structures 122 82–91 [49] S. Mehdi Tavakkoli et al 2015 An analytical study on piezoelectric-bistable laminates with arbitrary shapes for energy harvesting 7th ECCOMAS Thematic Conference on Smart Structures and Materials [50] David N. Betts et al Modelling the Dynamic Response of Bistable Composite Plates for Piezoelectric Energy Harvesting American Institute of Aeronautics and Astronautics [51] A. F. Arrieta et al 2010 A piezoelectric bistable plate for nonlinear broadband energy harvesting J. of APPLIED PHYSICS LETTERS 97, 104102 [52] David N. Betts et al 2012 Static and Dynamic Analysis of Bistable Piezoelectric Composite Plates for Energy Harvesting 20th ASC Structures, Structural Dynamics and Materials Conference AIAA 2012-1492 [53] Betts, D. N. et al 2012 Optimal configurations of bistable piezo-composites for energy harvesting J. of Applied Physics Letters, 100 (11). [54] David N. Betts et al 2012 Preliminary Study of OptimumPiezoelectric Cross-Ply Composites for Energy Harvesting J. of Smart Materials Research Volume 2012, Article ID 621364 [55] G. Gafforelli R. Ardito A. Corigliano 2015 Improved one-dimensional model of piezoelectric laminates for energy harvesters including three dimensional effects J. of Composite Structures 12 369–381 [56] Ahmed Jemai et al 2016 Modeling and parametric analysis of a unimorph piezocomposite energy harvester with interdigitated electrodes J. of Composite Structures 135 176–190 [57] Erturk, A., and Inman, D. J., 2008 On Mechanical Modeling of Cantilevered Piezoelectric Vibration Energy Harvesters, J. Intell. Mater. Syst. Struct. 19 11 1311-1325 [58] Jack R. Vinson, Robert L. Sierakowski 2008 The Behavior of Structures Composed of Composite Materials (Netherlands, Springer) [59] Erturk A and Inman D J 2011 Piezoelectric Energy Harvesting (Chichester: Wiley) [60] Caughey, T. K., and O’Kelly, M. E. J., 1965 Classical Normal Modes in Damped Linear Dynamic Systems, ASME J. Appl. Mech., 32, pp. 583–588. [61] Erturk, A., and Inman, D. J., 2007, “Mechanical Considerations for Modeling of Vibration-Based Energy Harvesters,” Proceedings of the ASME IDETC 21st Biennial Conference on Mechanical Vibration and Noise, Las Vegas, NV, Sep. 4–7. [62] H. Hosseini-Toudeshky, A. Farrokhabadi, B. Mohammadi 2012 Consideration of concurrent transverse cracking and induced delamination propagation using a generalized micro-meso approach and experimental validation J. of Fatigue Fract Engng Mater Struct 35, 885–901 [63] B. Peeters, 2000 System identification and damage detection in civil engineering, PhD thesis, Department of Civil Engineering, K.U.Leuven. [64] PULSE, Version8.0, 1996-2003 Brüel & Kjær, Sound &Vibration Measurement A/S.

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