Broadband and tunable PZT energy harvesting utilizing local nonlinearity and tip mass effects

International Journal of Engineering Science 118 (2017) 1–15

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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Broadband and tunable PZT energy harvesting utilizing local nonlinearity and tip mass effects Masoud Rezaei, Siamak E Khadem∗, Peyman Firoozy Department of Mechanical Engineering, Tarbiat Modares University, P.O. Box 14115-177, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 7 January 2017 Revised 17 March 2017 Accepted 27 April 2017

Keywords: Piezoelectric material Broadband energy harvesting Tip mass Nonlinear vibrations Local nonlinearity

a b s t r a c t In this paper, a broadband piezoelectric energy harvester with an applied restoring force is studied. The restoring force is modeled as a spring in the boundary conditions of the system. The system is composed of a clamped-free beam with a tip mass which is supported by a spring at the free end. A piezoelectric layer is bounded to the upper surface of the beam and the system is exposed to an external harmonic base excitation. A nonlinear distributed parameter model of the harvester is derived using the Hamilton’s principle utilizing the Euler–Bernoulli beam theory assumption. The nonlinear strain-displacement field is used to take the large transverse deflections effects into account. The reduced-order model is derived by implementing the Galerkin discretization method and the dimensionless form of the equations is used to analyze the system. The effect of the piezoelectric layer on the free vibrations of the structure is determined. Effects of tip mass and base acceleration on the frequency response of the system are investigated. The results show that, the tip mass can amplify the scavenged voltage and tune the resonance frequency. To study the influence of different types of the restoring force, linear, hardening, and pure nonlinear behavior are considered for the applied spring. The results demonstrate that by applying a pure nonlinear restoring force, the resonance bandwidth of the harvester increases which causes the harvester to generate energy in a larger frequency bandwidth and the output voltage increased remarkably in comparison to clamped-free energy harvester. This phenomenon enhances the harvester efficiency in the cases of the excitations with the time-varying frequency or the random excitations and the results in a broadband energy harvester. Finally, studying the frequency response curves for the different values of the spring location declares that, as the location of the attached spring changes, the nonlinear resonance frequency alters. The maximum increase in the bandwidth is accompanied by the case where spring is attached at the free end of the beam. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Harvesting wasted energy from the ambient vibrations of natural environments and converting it into useful energy, the so-called energy harvesting, has been of interest for the past decade. A variety of ambient sources such as solar, thermal, flow, acoustic energy, human motion, and mechanical vibrations have been studied as an additional energy supplier for the last decades (Penella & Gasulla, 2007) which their main goal is to feed the low power electronic devices specially in an

∗

Corresponding author. E-mail addresses: [email protected] (M. Rezaei), [email protected] (S.E. Khadem), p.fi[email protected] (P. Firoozy).

http://dx.doi.org/10.1016/j.ijengsci.2017.04.001 0020-7225/© 2017 Elsevier Ltd. All rights reserved.

2

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15

inaccessible areas. Among them, vibration/kinetic driven power generator is more widely studied because of its ubiquity, high efficiency, potentials to miniaturization, and abundance (Belhaq and Hamdi, 2016; Harne, Sun, & Wang, 2016). The ambient and internal vibrations converts to electric power by using adequate transducers, such as electromagnetic (Penella & Gasulla, 2007), electrostatic (Dudka, Basset, Cottone, Blokhina, & Galayko, 2013; Perez et al., 2016), piezoelectric (Abdelkefi, Nayfeh, & Hajj, 2012, Abdelkefi, 2016; Aladwani, Aldraihem, & Baz, 2015; Leadenham & Erturk, 2015; Yang, Wu, & Soh, 2015), and magnetostrictive (Adly, Davino, Giustiniani, & Visone, 2010; Mohammadi & Esfandiari, 2015). Although the main focus is on macro scale energy harvesters, the micro and nano scale systems are of interest recently (Azizi, Ghodsi, Jafari, & Ghazavi, 2016; Dutoit, Wardle, & Kim, 2005; Mehrdad Pourkiaee, Khadem, & Shahgholi, 2015; Meitzler et al., 1988; Pourkiaee, Khadem, & Shahgholi, 2016). In view of copious advantages in piezoelectric transducers, many researchers have devoted their attention to piezoelectric energy harvesting from cantilever beams (Xie & Wang, 2015; Xie, Wang, & Wu, 2014; Xie, Wu, Yuen, & Wang, 2013). Sodano, Inman, and Park (2004) have reviewed some literature, up to 2004, in the field of vibration energy harvesting. The authors divided the researches of piezoelectric harvesters into three branches, first modeling/analysis (Bonello & Rafique, 2011; Dietl, Wickenheiser, & Garcia, 2010; Erturk & Inman, 2008a, Erturk & Inman, 2008b, Rafique & Bonello, 2010; Erturk, Tarazaga, Farmer, & Inman, 2009; Junior, Erturk, & Inman, 2009; Kim & Kim, 2011; Kim, Hoegen, Dugundji, & Wardle, 2010; Liao & Sodano, 2008; Mak, McWilliam, Popov, & Fox, 2011; Masana and Daqaq, 2011; Patel, McWilliam, & Popov, 2011), second energy harvesting efficiency (Anton, Erturk, & Inman, 2010; Challa, Prasad, & Fisher, 2009; Guan & Liao, 2007; Shen, Qiu, Ji, Zhu, & Balsi, 2010; Shu & Lien, 2006a, Shu & Lien, 2006b), and third broadband or tunable energy harvester (Abdelkefi, Nayfeh, Hajj, & Najar, 2012; Ayala-Garcia, Zhu, Tudor, & Beeby, 2010; Cammarano, Burrow, Barton, Carrella, & Clare, 2010; Challa, Prasad, Shi, & Fisher, 2008; Eichhorn, Tchagsim, Wilhelm, & Woias, 2011; Jung, Kim, Lee, & Seok, 2015; Liang, Xu, Ren, Luo, & Wang, 2014; Qi, Shuttleworth, Oyadiji, & Wright, 2010; Vijayan, Friswell, Khodaparast, & Adhikari, 2015; Youngsman, Luedeman, Morris, Anderson, & Bahr, 2010; Zhang & Cai, 2012). Among all of the models, single degree of freedom (SDOF) is the simplest one and conventional model (Erturk & Inman, 2008a). Erturk and Inman (2008b) developed a distributed-parameter model for a piezoelectric cantilevered beam and presented an exact analytical solutions of the harvester by considering harmonic and non-harmonic excitations. Liao and Sodano (2008) theoretically modeled a piezoelectric based energy harvesting system by considering a single mode of vibration and developed an expression for the optimal resistance and a parameter which describes the effective electromechanical coupling coefficient in order to enhance the power harvesting. They concluded that a lower permittivity and a higher electromechanical coupling lead to a more broadband power harvesting. Masana and Daqaq (2011) developed and experimentally validated an electromechanical nonlinear model of an axially loaded, tunable, and couple clamped end energy harvesting beam. They concluded that the axial static load can be used to tune the system over a wide range of frequency. Also, their results indicated that the axial load increases the steady-state response amplitude, output power and bandwidth of energy harvester by enhancing the electromechanical coupling and effective nonlinearity. Based on the literature there are many researches on adjusting the vibration characteristics of an energy harvester by tuning the excitation frequency. Vijayan et al. (2015) investigated a vibro-imapcting system in order to convert low frequency response to high frequencies. The authors concluded that using nonlinear system enhances the output power compared to linear one in similar conditions. Challa et al. (2008) designed and tested a tunable resonance frequency energy harvesting system using a magnetic force at beam’s tip. They observed that this enable resonance tuning to ± 20% of the untunable resonant frequency. Utilizing this technique, 26 Hz natural frequency is tuned over range of 22–23 Hz. Mann and Sims (2009) demonstrated that using magnetic restoring force can broaden the resonance frequency range of harvester and make it more efficient. Jung (2015) studied a bimorph piezoelectric cantilever beam with a tip magnet and two external rotatable magnets as an energy harvester. The authors concluded that the response amplitude could be enlarged and the frequency bandwidth be broadened by varying the magnets rotation angels. Later, to improve the power efficiency and to broad the bandwidth at low frequency range Liang et al. (2014) used double clamped-clamped and asymmetrically connected beams. Ayala-Garcia et al. (2010) developed and implemented a self-powering control system which autonomously adapts the resonance frequency of an electromagnetic vibration-based energy harvesting to ambient vibration frequency. They concluded that the system adjusts the resonant frequency of the harvester from 64 to 78 Hz. There were many attempts to make use of structural nonlinearities to broaden the bandwidth of energy harvester. Gammaitoni, Neri, and Vocca (2009) investigated a monostable/multistable nonlinear energy harvesters. They revealed that these systems as well as the bistable one can overcome the limitations of linear energy harvesters that are namely originate from tuning of the oscillators and wide frequency range of available vibrations. Firoozy, Khadem, and Pourkiaee (2017) found that considering geometric nonlinearity in the energy harvester, for both presence and absence of the magnets, and the gap distance between external magnets can affect and broadens the frequency range. Cottone, Vocca, and Gammaitoni (2009) maximized the harvested power of a cantilever piezoelectric beam using structural nonlinearities under random excitation. In order to broaden the resonance frequency, Barton, Burrow, & Clare, 2010) studied an electromagnetic energy harvesting device by considering a nonlinear cubic force in boundary conditions that are produced by using magnets. The authors concluded that their nonlinear system is able to overcome the issue of having a narrow resonant response of the linear systems. Friswell et al. (2012) both theoretically and experimentally investigated an inverted piezoelectric beam with a pointed tip mass, by considering nonlinearity in geometry, as an energy harvester. The results show that once the lower excitation frequencies are accounted, the maximum harvested power is significantly large. In addition, the excitation frequency can be tuned by the energy harvester frequency using a simple tip mass (Jiang & Hu, 2007; Jiang et al., 2005), mechanical preloads (Hu, Xue, & Hu, 2007), and varying the geometrical parameters of the structure (Hu, Cui, & Cao, 2007). Stanton, Erturk, Mann, Dowell, and Inman (2012) studied a cantilever piezoelectric beam by considering nonlinear elasticity

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15

3

-

R

Cp

V

DC

.

Θy

+

(a)

(b)

Fig. 1. (a) Schematic of the energy harvester b Simplified electrical circuit.

in the piezoelectric materials and dissipation in the system both theoretically and the experimentally. They concluded that in the absence of proof mass, cubic nonlinearity can sufficiently model the nonlinear behavior of the system by a third order damping, and linear coupling. Even in the presence of proof mass, their model is sufficient for low base accelerations. However, if the excitation amplitudes are large, fifth-order nonlinearities in the modeling should be considered. Based on the literature, using nonlinear properties in an energy harvesting systems can remarkably broaden the frequency range of the system and affects the output power. In addition, it has been found that using a tip mass can tune the resonance frequency with the excitation frequency in low frequency ranges. In this work a comprehensive investigation on a piezoelectric beam by considering geometric nonlinearity and tip mass as a broadband energy harvester is presented. In addition, to overcome the issues of the non-stationary and random excitations and the more increase in the resonance bandwidth, a local nonlinearity which is imposed by a nonlinear spring located in somewhere along the length of cantilevered beam is utilized. Moreover, tunability of the resonance frequency is examined by attaching a tip mass. The rest of the paper is organized as follows: In Section 2, the dimensionless distributed-parameter electromechanical model is presented. In Section 3 the numerical analyses are performed to study the effects of the piezoelectric layer, tip mass, and base acceleration level on the frequency response of the system. Influences of different types of the restoring force on the response of the energy harvester and levels of harvested power are also investigated in this section. Finally the conclusions are given in Section 4. 2. Distributed parameter model The electromechanical system and simplified electrical circuit are shown in Fig. 1. The system is composed of a unimorph cantilever beam with a tip mass and an attached spring at x = lb where x and lb denote the distance along the neutral axis of the beam and the length of the beam, respectively. The Euler–Bernoulli beam theory is adopted in driving the equations of motion and the shear deformations are ignored due to the large length-tothickness ratio of the beam. The nonlinear strain-displacement field is utilized for considering the large deformation effects due to large amplitude vibrations. Based on Nayfeh and Pai (2008), the strain field is assumed to be:

  x = −z v − v u − v u − v v 2

(1)

where x is axial stress, u(x, t) and v(x, t) are the longitudinal and transverse displacements of a point of the beam at distance x from the clamped end,respectively, and primes denotes the partial derivatives with respect to x. Here, z is the vertical distance from the neutral axis of the beam. The inextensibility condition states that (Nayfeh & Pai, 2008):

u = −

1 2 v 2

(2)

by substituting Eq. (2) into Eq. (1), one may obtain the following equation as:

  1 x = −z v − v 2 v . 2

(3)

It is assumed that the whole system is subjected to a harmonic base excitation in the form of:

y = A.sin(ω.t ) where A and ω are excitation and frequency amplitudes, respectively.

(4)

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M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15

The extended principle of Hamilton states that (D’Souza & Garg, 1984):



t2

t1

(δ T − δU + δW )dt = 0

(5)

where t1 − t2 is an arbitrary time span, δ is the variation operator, T, U, and Wex are the total potential energy of the system, kinetic energy, and the work done by the non-conservative external forces, respectively. The total kinetic energy of the system (T) which is composed of the kinetic energy of the beam (Tbeam ) and tip mass (Ttip mass ), can be written as follows:

  2  ∂ lb 1  2 2 m (x ) − v dx + (v˙ + y˙ ) dx ∂t 0 2 0 

2   2 D   1 D 1 ∂ lb 1  2 2  + Mt − v dx − v lb v˙ lb i¯ + v˙ lb + v˙ lb + y˙ + j¯ + It vlb 2 ∂t 0 2 2 2 2

1 T = 2



l

(6)

where the first and second parts are the transverse kinetic energy of the beam and tip mass, respectively, and the third part shows the rotational kinetic energy of the tip mass. Here, m(x), Mt , and It are the mass per unit length of the unimorph, the mass of tip mass, and the mass moment of the inertia of the tip mass with respect to its center, respectively. m(x) is defined as:

m(x ) = ρbWbtb + ρ pWpt p [H (x ) − H (x − l p )]

(7)

here ρ , W, and t are density, width, and thickness of layers, respectively. Subscripts b and p are stand for substrate and piezoelectric layers, respectively. The total potential energy of the system (U) is the sum of the potential energy of the substrate layer (Ubeam ), piezoelectric layer (Upiezo ), electrical potential of piezoelectric layer (Uelectrical ), and potential energy of spring (U spring ), which can be written as follows:

U=

1 2



lb



EI (x )(v + v

0

2

   1 1 2 v ) + 2 ψ (x ) v + v 2 v RQ˙ R dx + Cp (RQ˙ R ) + U spring

2  2

2

2

(8)

where EI(x) is bending stiffness of the unimorph, ψ (x) is the electromechanical coupling, R is load resistance, QR is current, and Cp is the piezoelectric layer capacitance. EI(x), ψ (x), and Cp are defined as:

 3  3 1 1 3 3 Wb Eb z1 − z0 + Wp E p z2 − z1 [H (x ) − H (x − l p )] + W t 3 E [H (x ) − H (x − l p )] 3 12 b b b W E d  ψ (x ) = p p 31 z2 2 − z1 2 [H (x ) − H (x − l p )] 2t p EI (x ) =

Cp =

e33Wp L p tp

(9)

The spring force is assumed to be nonlinear and can be written as follows:

Fspring = K1 v + K3 v3

(10)

where K1 and K3 are the linear and nonlinear stiffness, respectively. Using the Dirac delta function, the potential energy of the spring due to the transverse displacement can be stated as:

Uspring = K1 v(lb )vlb − K3



lb 0

v(x )3 δ (x − lb )dx

(11)

The work done by the external forces (Wex ) can be written as (Bibo, Abdelkefi, & Daqaq, 2015):

 Wex = −

lb 0

ca v˙ dx − V.QR

(12)

where the first and second parts are the dissipative work done by the viscous air damping and load resistance, respectively (Bibo et al., 2015). Substituting Eqs. (6), (8), and (12) into the Hamilton’s principle which is given by Eq. (5) yields the following nonlinear electromechanical partial differential equations of motion:

  x    x  

   1 m (x ) v¨  v + v˙ 2 dx dx v EI (x )v v + θ (x ) 1 + v 2 V (t ) + v 2 lb 0   D = −Ca y˙ b − M (x )y¨ b − Mt δ (x − lb ) − δ  (x − lb ) − Fspring (δ (x − lb ) )

m(x )v¨ + Ca v˙ + EI (x )v



+

2

(13)

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15

C pV˙ (t ) +

1 V (t ) = R

∂ ∂t



5

 

 1 θ (x )v 1 + v 2 dx

lb

(14)

2

0

and the corresponding boundary conditions are:



v=0 v = 0  2 EIv = −Mt D2 v¨ − (It + Mt D4 )v¨  x = lb → EIv = Mt v¨ + Mt D2 v¨  + K1 v x=0→

(15)

The reduced-order model of PDEs are obtained by utilizing the Galerkin discretization. In this method, it is assumed that the transverse displacement can be stated as (Dai, Wang, & Wang, 2015; Farokhi, Ghayesh, & Hussain, 2016):

v(x, t ) =

∞ 

ϕi (x )qi (t )

(16)

i=1

ϕ i (x) are the mode shapes of a uniform clamped-free beam with tip mass, qi (t) are time-dependent coordinates, and i is mode number. According to Friswell et al. (2012) for a large tip mass, consideration of only the first mode is sufficient. The mode shapes of cantilever beam with tip mass are expressed as (Erturk & Inman, 2011):

   λ λ λ λ ϕi (x ) = C cos i x − cosh i x + τ sin i x − sinh i x lb

lb

lb

(17)

lb

where C is the normalized amplitude and can be derived using orthogonality expression as (Bibo et al., 2015):



lb 0



     D D2   ϕi (x )(mb + m p )ϕ j (x )dx + ϕi (x )Mt ϕ j (x ) + ϕi (x )Mt ϕ j (x ) + ϕi (x ) It + Mt ϕ j (x ) = δi j 2

4

(18)

lb

where mb and mp are the mass per unit length of substrate and piezoelectric layer, respectively and τ is defined as:

 (cosλi − coshλi ) − 2Dlb λi (sinλi + sinhλi )

 τ= (cosλi + coshλi ) − (mb +Mmt p )lb λi (sinλi − sinhλi ) + 2Dlb λi (cosλi − coshλi ) (sinλi − sinhλi ) +

(mb +m p )lb λi Mt

(19)

where λi are eigenvalues of the system, and can be obtained by solving the following frequency equation (Bibo et al., 2015):



Mt It

(mb + m p )2 lb 4 −2

(1 − cosλi coshλi )λi −

2

Mt D4

4

Mt D2

( mb + m p )lb

2

(sinλi sinhλi )λi 2 +

( mb + m p )lb 3 Mt

( mb + m p )lb

+

It

( mb + m p )lb 3



(sinλi coshλi + sinhλi cosλi )λi 3

(sinhλi cosλi − coshλi sinλi )λi + (1 + cosλi coshλi ) = 0

(20)

Using the above equations and considering only the first mode (i = 1 ) in the series, the following ordinary differential equations governing the system are obtained as:





q¨ + C1 q˙ + C2 q + C3V (t ) + C4 q2V (t ) + 2C5 q2 q¨ + qq˙ 2 + C6 q3 = C7 .A sin(2π .w.t ) + C11 q3

(21)

C pV˙ (t ) + C8V (t ) = C9 q˙ + 3C10 q2 q˙

(22)

where the coefficients are defined in Appendix. In order to obtain the dimensionless electromechanical equations of motion, the following dimensionless parameters are introduced (Bibo, Li, & Daqaq, 2011):

q¯ =

q t¯ = ωn t Lb

V¯ =

Cp

θ

V

yb =

yb

= Lb

ω ωn

(23)

where q¯ and V¯ are the dimensionless displacement and voltage, respectively, t¯ is the dimensionless time, and ωn is the fundamental frequency of the cantilever beam. Substituting dimensionless parameters introduced in Eq. (23) into Eqs. (21) and (22), the following dimensionless nonlinear ordinary differential equations of motion are obtained as:





   q¯ + N1 q¯  + κ1 q¯ + N3V¯ + N4 q¯ 2V¯ + 2N5 q¯ 2 q¯ + q¯ q¯ 2 + N6 q¯ 3 = N7 yb − κ3 q¯ 3

(24)

V¯  + N8V¯ = N9 q¯  + N10 q¯ 2 q¯ 

(25)

6

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15 Table 1. Properties of energy harvester (Bibo et al., 2015). Symbol

Value

Description

Lb Lp Wb Wp tb tp Eb Ep e33 d31 Mt D

14.5 cm 8.5 cm 1.4 cm 7 cm 0.51 mm 0.3 mm 190 Gpa 15.86 Gpa 19.36 nF/m −170 pm/V 102.3 gr 5 cm

Beam length Piezoelectric length Beam width Piezoelectric width Beam thickness Piezoelectric thickness Young’s modulus of beam Young’s modulus of piezoelectric Permittivity Piezoelectric constant Tip mass Tip mass dimensions

where the coefficients are defined as:

N1 = 2ζ , N6 =

κ1 =

C6 Lb 2

ωn

2

 C 2 2

ωn

, N7 =

, N3 =

θ C3 , N4 = Lb ωn 2 C p

θ C4 Lb , N5 = C5 Lb 2 , ωn 2 C p

A 1 L C9 L 3C10 = a0 , N8 = , N9 = b , N10 = b , Lb RC p ωn θ θ

κ3 =

Lb 3C11

(26)

ωn 2

3. Numerical results and discussions In this section, the numerical results are performed. The numerical simulations are presented for the case study of a harvester with MFC-M8514-P2 piezoelectric patch. Physical properties of the substrate and piezoelectric layer and tip mass are listed in Table 1. 3.1. Effects of the piezoelectric layer on the free vibration of the system To study the effects of the piezoelectric layer on the response of the system, RMS dimensionless displacement of system is plotted in Fig. 2. Fig. 2 depicts the influence of the presence and absence of the external damping on the tip displacement of the energy harvester and corresponding phase portraits, for the values of the Table 1, and under free vibration conditions. As it is expected and shown in Fig 2, in the presence and absence of external damping, the free vibration response of the system is oscillations around the equilibrium point (0, 0, and 0). It is also seen that when the air damping is zero (N1 = 0), the energy harvester annihilates free vibration response of the system, which means the piezoelectric layer acts as a damper. In addition, in the case of the zero air damping condition, the rate of decay is smaller than the case of presence of the air damping. These results are also shown in figure (c) and (d) where the phase portrait are plotted. It is seen that in the case of the non-zero air damping, the response reaches to the zero point very fast compared to the case of zero air damping condition. 3.2. Resonance tuning of the energy harvester using tip mass The frequency responses of the dimensionless tip displacement and voltage for different values of

Mt Mb

are shown in

Fig. 3, where Mb is the substrate layer mass. The value of external excitation is assumed to be equal to a0 = 0.02. Inspecting the Fig. 3, it is obvious that changing the value of the tip mass remarkably alters the resonance frequency of the energy harvester and tunes it with the ambient frequency. In other words, when the tip mass increases, the frequency at which the maximum displacement and voltage occur shifts to the lower frequencies. For example, the resonance frequency Mt of the system changes from 1.6 to 0.9 as the dimensionless mass M is increased from 5 to 15, respectively. It is also b

seen that, by increasing the value of tip mass, the response amplitude is increased from 0.3 to 0.8, and consequently the corresponding dimensionless output voltage is altered from 0.25 to 0.45. 3.3. Base acceleration effects on the system response To investigate the effect of the external excitation a0 on the frequency response of the system, the system response is plotted for different values of a0 in Fig. 4. As it is depicted in Fig. 4, when the excitation amplitude increases, the RMS of the tip displacement and the generated voltage are amplified. In other words, the pick of RMS displacement and voltage increases from 0.25 to 0.6 and 0.12 to 0.36, respectively. In addition, it is observed that, increasing a0 can cause the hardening effects to appear which widen the resonance bandwidth of energy harvester.

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15 -3

-3

x 10

2.5

2.5

N1>0

1.5 1 0.5 0 -0.5 -1 -1.5

1 0.5 0 -0.5 -1 -1.5

-2

-2 -2.5

500

1000

1500

2000

2500

3000

3500

4000

4500

N1=0

1.5

-2.5

0

x 10

2 Dimensionless Tip Displacement

Dimensionless Tip Displacement

2

5000

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

time (s)

time (s)

(a)

(b) -3

-3

1.5

7

x 10

1.5

x 10

N1=0

1

1

0.5

0.5 Velocity

Velocity

N1>0

0

-0.5

-0.5

-1

-1

-1.5 -1

0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Displacement

(c)

0.8

1 -3

-1.5 -1

-0.8

-0.6

-0.4

-0.2

0

Displacement

x 10

0.2

0.4

0.6

0.8

1 -3

x 10

(d)

Fig. 2. (a, b) Time history of the RMS dimensionless displacement for the case of non-zero and zero air damping conditions, respectively, and (c, d) phase portrait (x1-x2) for the case of non-zero and zero air damping, respectively.

3.4. Nonlinear dynamics of the energy harvester For a better comparison, numerical simulations have been carried out for the case of the clamped-free beam with an excitation amplitude of a0 = 0.02 and the frequency response is plotted in Fig. 5. Fig. 5 shows the stable solutions of Eqs. (24) and (25). It is seen that the frequency response curves show hardening behavior because of hardening stiffness term in the mentioned equations. Circle (o) represents forward path sweep and plus (+) shows the backward path sweep. Precisely, when the frequency increases from = 0.8, the RMS dimensionless displacement and voltage are increase up to point A which corresponds to the frequency of = 1.03, and in this point the response becomes unstable and it jumps to the lower amplitude branch (point B). An opposite trend is seen in the backward path where the frequency is decreased from = 1.4. In this path, decreasing frequency up to point C ( = 1.02), results in an increase in the RMS dimensionless displacement and voltage and at point C the response becomes unstable and jumps to the upper branch of stable solution (point D). On the upper stable branch, the more the frequency decrease the more the vibration amplitude and voltage are reduced. So, it can be said that at point A and C the bifurcation occurs which correspond to jump-up and jump-down, respectively. Initial conditions dictate being on which branch in the jump region. For the energy harvesting purposes, one must set the initial conditions such that the response take the higher value.

8

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15 0.8

0.45

Mt/Mb=15

Mt/Mb=15

Mt/Mb=10

0.4

0.7

Mt/Mb=10 0.35

RMS Dimensionless Voltage

RMS Dimensionless Tip Displacement

0.6

0.5

Mt/Mb=5

0.4

0.3

0.3

Mt/Mb=5 0.25

0.2

0.15

0.2 0.1

0.1

0.05

0 0.6

0.8

1

1.2

1.4

1.6

0 0.6

1.8

0.8

1

1.2

(a)

1.4

1.6

1.8

(b)

Fig. 3. (a) Frequency response of dimensionless tip displacement (b) frequency response of dimensionless voltage for different values of tip mass which are indicated on curves.

0.7

0.4

a0=0.002

a0=0.002

a0=0.006

0.6

a0=0.006

0.35

a0=0.02

a0=0.02 0.3

RMS Dimensionless Voltage

RMS Dimensionless Tip Displacement

0.5

0.4

0.3

0.25

0.2

0.15

0.2 0.1

0.1

0 0.8

0.05

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.8

0.85

(a)

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

(b)

Fig. 4. Effect of the excitation amplitude on the frequency response (a) RMS dimensionless tip displacement (b) RMS dimensionless voltage.

3.5. Effects of the linear restoring force on the response of the energy harvester To study the effects of the linear type restoring force on the harvester response, the frequency responses of the displacement and voltage are plotted in Fig. 6. It is assumed that κ1 > 1, κ3 = 0. Fig. 6 clearly shows that adding a linear restoring force shifts the resonance region to the higher frequencies. As it is seen in Fig. 6, the resonance frequency ( ) corresponding to the cases of κ1 = 1.5 and κ1 = 2, are about 1.2 and 1.4, respectively. Inspecting Fig. 6(a), it is obvious that, the vibration amplitude of the cases κ1 = 1.5 and κ1 = 2 are about 0.55 and 0.5, respectively. It demonstrates that, the amplitude of clamped-free beam which is about 0.6 is attenuated to the lower values by attaching the spring. It is worthy to mention that the voltage maybe decreased or increased at the resonance frequency of spring supported system. Inspecting Fig. 6(b), it is seen that the RMS output voltage of reference system is about 0.35 but it is 0.38 and 0.36 in case of the κ1 = 1.5 and κ1 = 2, respectively. This change in generated voltage is due to the fact that, voltage is also a function of frequency in addition to the vibration amplitude. But, this effect is not an accomplishment because of the fact that the generated voltage at the resonance frequency of the reference system ( = 1.1) is reduced. So, adding a linear restoring force cannot have a considerable effect on the output voltage.

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15 0.7

9

0.4

A A

0.6

0.35

D D 0.3

RMS Dimensionless Voltage

RMS Dimensionless Tip Displacement

0.5

0.4

0.3

0.25

0.2

0.15

0.2 0.1

B

C

B

0.1

C

0.05

0 0.8

0.9

1

1.1

1.2

1.3

0 0.8

1.4

0.9

1

(a)

1.1

1.2

1.3

1.4

(b)

Fig. 5. The frequency response curve of the clamped-free system; (a) RMS dimensionless tip displacement (b) RMS dimensionless voltage. 0.7

0.4

=1 1 0.6

=1.5

=1

=1.5 1

1

1

0.35

=2

1

=2 0.3

0.5

RMS Voltage

RMS Tip Displacement

1

0.4

0.3

0.25

0.2

0.15

0.2 0.1 0.1

0 0.7

0.05

0.8

0.9

1

1.1

1.2

(a)

1.3

1.4

1.5

1.6

1.7

0 0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

(b)

Fig. 6. (a) Frequency response of dimensionless tip displacement (b) frequency response of dimensionless voltage for different linear restoring force.

3.6. Effects of the nonlinear hardening type resorting force on the frequency response of energy harvester In this part, a nonlinear spring is considered to study its effect on the system behavior. In this conditions, the dimensionless frequency and stiffness are defined as:

κ1 > 1, κ3 > 0 Now the frequency response of the system for the case of hardening support is plotted in Fig. 7. In this case, it is assumed that κ1 = 1.5 and κ3 = 15. The dimensionless base acceleration is assumed to be a0 = 0.02. Frequency response of the system which depicts the stable branches of the solution is shown in Fig. 7. Comparing this figure with Fig. 5, it can be seen that, by applying a nonlinear hardening type restoring force, the resonance frequency shifts from resonance of free system ( = 1) to the higher values = 1.2, and simultaneously, the response curves bend to the right more and maximum amplitude occurs at = 1.3 (point A) which is larger than the corresponding frequency of the free system ( = 1.03). It also shows that the hardening behavior is intensified and the resonance bandwidth is increased. In the exact words, the linear and nonlinear parts of stiffness are responsible for the frequency shift hardening behavior. Although, the frequency shift is not favorable, the bending of curves are desirable because it increases the resonance bandwidth of the harvester which in turn increases the efficiency of the harvester when the excitation frequency is not constant or in case of random vibrations (Daqaq, 2012). It should be mentioned that, comparing Figs. 5 and 7, the maximum of RMS amplitude is attenuated from 0.6 to 0.3 and the voltage is increased from 0.35 to 0.17. Occurrence of a bifurcation at point

10

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15 0.35

0.18

A 0.16

D

0.3

A 0.25

0.14

D RMS Dimensionless Voltage

RMS Dimensionless Tip Displacement

0.12

0.2

0.15

0.1

0.08

0.06 0.1 0.04

B B

0.05

C 0 0.9

1

1.1

1.2

C

0.02

1.3

1.4

0 0.9

1.5

1

1.1

1.2

(a)

1.3

1.4

1.5

(b)

Fig. 7. The frequency response curve of the harvester for the case of κ1 = 1.5 and κ3 = 15; (a) RMS dimensionless tip displacement (b) RMS dimensionless voltage. 0.35

0.18

=1.5

0.3

1

=1.5

1

=1.5

0.25

1

0.16

=15

=1.5

3

1

=20

3

=1.5

1

0.14

=25

=1.5

3

1

=15

3

=20

3

=25

3

RMS Dimensionless Voltage

RMS Dimensionless Tip Displacement

0.12

0.2

0.15

0.1

0.08

0.06 0.1 0.04 0.05 0.02

0 1.15

1.17

1.19

1.21

1.23

1.25

1.27

1.29

1.31

1.33

1.35

1.37

1.391.4

0 1.15

1.17

1.19

(a)

1.21

1.23

1.25

1.27

1.29

1.31

1.33

1.35

1.37

1.391.4

(b)

Fig. 8. The frequency response curves of the harvester for the case of κ1 = 1.5 and the various values of κ 3 (a) RMS dimensionless tip displacement (b) RMS dimensionless voltage.

A and C is also obvious in this case. To better consider the effects of the hardening restoring force, the frequency responses of RMS displacement and voltage are given in Fig. 8 for κ1 = 1.5 and different values of κ 3 . Inspecting Fig. 8, it is seen that, the nonlinear part of the stiffness (κ 3 ) can widen the frequency bandwidth and this is the predominant effect of this type of the stiffness. Precisely, in excitation frequency of = 1.33, the voltage response of free system is onits lower stable branch and the level of RMS of the harvested voltage is about 0.03 while in this frequency the response of the system with κ1 = 1.5 and κ3 = 25 is on its upper branch and the generated voltage is about 0.15. This means that the support spring makes the harvester efficient. 3.7. Effects of the hardening restoring force on the response of the energy harvester Based on the results of the linear and nonlinear hardening type stiffness, it can be deduced that, applying a pure nonlinear hardening type stiffness is the best choice because in this scenario the resonance frequency does not shift to the higher values, but, the resonance bandwidth increases and the harvester can generate voltage in larger frequency range. The frequency response curves of the RMS tip displacement and voltage corresponding to the case of pure nonlinear restoring force are shown in Fig. 9.

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15 0.35

0.16

=1

1 0.3

=1

0.25

=15

=1

3

=20

1

3

=1 1

=25 3

0.2

0.15

0.1

1

0.14

=1

1

=1

0.12

RMS Dimensionless Voltage

RMS Dimensionless Tip Displacement

11

1

=15

3

=20

3

=25

3

0.1

0.08

0.06

0.04 0.05

0.02

0 0.8

0.9

1

1.1

1.2

0 0.8

1.3

0.9

1

(a)

1.1

1.2

1.3

1.4

(b)

Fig. 9. The frequency response curves of the harvester for the cases of different pure hardening support (a) RMS dimensionless tip displacement (b) RMS dimensionless voltage. 0.5

0.35

Ls/Lb=0.25

0.45

Ls/Lb=0.5

0.4

Ls/Lb=0.75

0.35

Ls/Lb=1

0.3 0.25 0.2 0.15

RMS Dimensionless Voltage

RMS Di mensionl ess Tip Displacement

Ls/Lb=0.25 0.3

Ls/Lb=0.5 Ls/Lb=0.75

0.25

Ls/Lb=1 0.2

0.15

0.1

0.1 0.05 0.05 0 0.8

0.9

1

1.1

1.2

0 0.8

1.3

(a)

0.9

1

1.1

1.2

1.3

1.4

(b)

Fig. 10. Frequency response curves for different values of spring location and κ3 = 10 (a) dimensionless tip displacement (b) dimensionless voltage. 

As it is seen in Fig. 9, the pure nonlinear support bends the frequency response curve to the right which results in an increase in the bandwidth of the harvester. Of course, the amplitude of vibration attenuates and the generated voltage decreases. As the value of the κ 3 increases, the hardening behavior of the harvester intensifies and the curves bend to the right more. 3.8. Effects of the spring location on the harvester response Another case of study can be the effects of the spring location on the frequency response of the energy harvester. To study the effects of spring location, one must consider that

κ3 =

Lb 3C11

ωn 2

→ κ3 =

−Lb 3 Knonlinear ϕ1 (x )lb 4

ωn 2

(27)

or

κ3 (x ) = κ3 .ϕ1 (x )4

(28)

Fig. 10 indicates the frequency response curves of the dimensionless tip displacement and voltage for some special values of dimensionless location ( LLs ) values and κ3 = 10.Ls and Lb are the location of the spring from the fixed end and length of the beam, respectively.

b

12

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15 -12

-12

1.5

x 10

1.5

x 10

Clamped-Free System

Clamed Free System 1

=1.21 3=3

1

P*

P*

1

=5

3

=10

3

0.5

0

0 0

0.2

0.4

0.6

0.8

1

=1.21 3=5

1

0.5

1.2

1.4

1.6

1.8

R

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

R

11

x 10

*

(a)

1.6

1.8

2 11

x 10

*

(b) -12

1.5

x 10

Clamped-Free System

P*

1

=5

3

=10

3

0.5

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

R

*

1.8

2 11

x 10

(c) Fig. 11. Variation of the dimensionless power with the dimensionless load resistance for the different values of the dimensionless stiffness (a) linear restoring force (b) nonlinear hardening resorting force (c) pure nonlinear restoring force.

As it is depicted in Fig. 10, the location of the spring can alter the frequency response of the harvester. The highest vibration amplitude and voltage are present when the spring is placed near the clamped end ( LLs = 0.25). As the location b

of the spring get closer to the free end, the vibration is attenuated more and the generated voltage is get lower, but, the resonance bandwidth gets larger which and it makes the harvester more efficient. Inspecting Fig. 10(b), it is clear that, at frequency of = 1.23 the generated voltage of the system is about 0.06 and 0.03 for cases of the LLs = 0.25 and LLs = 1, b

b

respectively while the vibration amplitude is decreased remarkably. It demonstrates that, the system with a spring on its free end outperform the three other cases for energy harvesting purposes. 3.9. Simultaneous effects of the load resistance and stiffness on the generated power To study the effects of the load resistance on the generated power, the following dimensionless quantities are defined (De Marqui & Erturk, 2012):

R∗ =

RI∗ Lb mbWb 2 ωn 3

(29)

where R∗ and I∗ = 1 A are the dimensionless load resistance and reference current, respectively. Dimensionless power (P∗ ) is defined as:



P∗ =

RMS V¯ R∗

2

(30)

In Fig. 11, the graphs of the dimensionless power versus the dimensionless load resistance for the different cases of the restoring force are plotted at the frequency of = 1.

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15

13

As it is depicted in Fig. 11, regardless of type of the restoring force, the generated power drops as the spring is added but again it worth to mention that the previous results demonstrate that the applied restoring force can increase the frequency bandwidth.

4. Conclusions In this work, the concept of a broadband piezoelectric energy harvester under an applied restoring force is studied. The restoring force is modeled as a boundary condition spring. A nonlinear distributed parameter electromechanical model is derived utilizing the Hamilton’s principle and via the Galerkin method. To generalize the analyses, the dimensionless form of the governing equations is derived. To study the effects of the piezoelectric layer on the system response, the time history of the response is plotted for the cases of the zero and nonzero damping. Results show that, the piezoelectric layer can act as a damper. Effects of the tip mass on the frequency response curves are studied and it is demonstrated that, the attached tip mass can tune the resonance frequency of the system with the excitation frequency and enhance the vibrations amplitude which in turn increases the harnessed voltage. The frequency response curves for the different values of the base excitation amplitude indicate that, increasing the excitation amplitude can reveal the effects of nonlinear terms in the frequency response curves of the harvester. In the exact words, increasing the excitation amplitude, can cause the hardening effect to appear in the system response. Next, to investigate the influences of the applied resting force on the harvester, three scenarios of linear, hardening, and pure nonlinear cubic restoring force are considered. The results of the first scenario, i.e. linear, point out that, exploiting a linear restoring force shifts the resonance frequency to higher values and decreases the oscillations amplitude which results in voltage reduction, so, this scenario is not preferred. Next, the frequency response curves regarding the case of nonlinear hardening type restoring force depicted that although the linear part of the spring can have pointless effects, the nonlinear one can widen the resonance bandwidth of the harvester which is favorable in case of the energy harvesting purposes. As it is expected, the final case which is pure hardening restoring force does not suffer from the frequency shift issue and increases the resonance bandwidth which makes the harvester appropriate in case of excitation with time-varying frequency or random excitations. The effect of the spring location on the frequency response curve of the system demonstrates that, as the spring gets closer to the free end, the maximum tip displacement and the generated voltage decrease while the frequency bandwidth increases. In all of the considered cases, the voltage or power drop is seen because of the fact that the applied spring attenuates the vibrations amplitude.

Appendix

C1 = Ca  lb C2 = ϕ1 (x )[Y I (x )ϕ1 (x ) ] dx + Klinear ϕ1 (x )lb 2  C3 =

0

lb

ϕ1 (x )θ  (x )dx

0

 1 lb ϕ1 (x )[θ  (x )ϕ1 (x )ϕ1 (x )] dx 2 0    lb  s  s ϕ1 ( x )   C5 = ϕ1 (x ) (m(x ) ϕ1 (x )ϕ1 (x )dx dx] dx 2 0 lb 0  lb   C6 = ϕ1 (x )[ϕ1  (x ) EI (x )ϕ  1 (x )ϕ  1 (x ) ] dx C4 =

0

C7 = Mt β +

β=

 1 2



lb

ϕ1 (x )M (x )ds  D ϕ1 ( lb ) + ϕ1  ( lb ) 0

2 1 C8 = − R  lb C9 = θ (x )ϕ  (x )dx 0

C10

1 = 2

 0

lb



θ (x )ϕ1 (x )ϕ1 (x )ϕ1 (x )dx

C11 = −Knonlinear ϕ1 (x )lb 4

14

M. Rezaei et al. / International Journal of Engineering Science 118 (2017) 1–15

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