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Two-span piezoelectric beam energy harvesting
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Two-span piezoelectric beam energy harvesting Ze-Qi Lu , Jie Chen , Hu Ding , Li-Qun Chen PII: DOI: Reference:
S0020-7403(19)34503-5 https://doi.org/10.1016/j.ijmecsci.2020.105532 MS 105532
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
27 November 2019 5 February 2020 11 February 2020
Please cite this article as: Ze-Qi Lu , Jie Chen , Hu Ding , Li-Qun Chen , Two-span piezoelectric beam energy harvesting, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105532
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Highlights
Nonlinear vibratory energy harvesting via a two-span piezobeam Benefits of nonlinear interaction between buckled beam spans Comparison of voltage output FRF between single-span, buckled, and two-span beam configurations A hybrid harvester of three structures is proposed to cover a demand of broadband voltages output
Two-span piezoelectric beam energy harvesting Ze-Qi Lu, Jie Chen, Hu Ding and Li-Qun Chen* Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200444, China
Abstract This paper reports an analytical study of nonlinear vibratory energy harvesting via a two-span piezoelectric beam. The distributed parameter electromechanical model is applied to explore the benefits of nonlinear interaction between buckled beam spans. The Galerkin truncation method and harmonic balance method are applied together to find forced responses of the energy harvesting system. The effects of axial pre-pressure on energy harvesting are also explored. The analytical results are supported by utilizing direct time integration of the equation of motion. The amplitude and voltage output frequency response functions (FRF) of single-span, buckled, and two-span piezoelectric beam configurations reveal some differences. For example, the voltage output FRF of the two-span buckled configuration is larger than that of the other two configurations at high frequencies, whereas at low frequencies, the voltage output FRF of two-span unbuckled configuration is larger than that of the other two configurations. Moreover, two-span buckled configuration has more natural frequencies lying at the concerned frequency range. Finally, a hybrid harvester is proposed, which combines harvested voltage from single-span buckled, two-span, and two-span buckled piezoelectric beam to cover a demand of broadband voltages output. Keywords: nonlinear vibration; energy harvesting; piezoelectric beam; two-span beam; Galerkin method
*
Corresponding author: School of Mechanics and Engineering Science, 99 Shangda Road, Shanghai 200444, China,
Tel.: +86 021 66132062; fax: +86 021 66134463. E-mail address: [email protected]
1. Introduction Energy scavenged from vibratory motion using piezoelectric materials is an important alternative to batteries or other sources of energy for powering electronic devices. In this article, we analyze, in particular, the scavenging of power from vibrations of a two-span piezoelectric beam. The transverse vibration of a two-span composite-beam presents a challenging problem [1], but there is a clear demand for utilizing scavenged energy from such sources [2-6]. Studies of transverse vibrations of two-span composite-beams have been conducted literally for centuries and remain an active area of research. Multi-span beams, our focus here, are widely used in many engineering applications, including aerospace, railway tracks, and bridges [2]. To analyze the transverse vibrations of multi-span structures, the transfer matrix method [7-10], the finite element method [11, 12], and the assumed modal method [13, 14] have been the most widely used. Li et al [7, 8] studied wave localization in disordered periodic multi-span beams and disordered periodic piezoelectric beams. Liu et al. [9] examined transient responses of a multi-span pipe conveying fluid. Wang et al. [11] studied vibrations of multi-span bridges under traffic loadings and vehicle dynamic interactions. Elishakoff and Santoro [12] studied random vibrations of a forced two-span beam with a linear elastic foundation. Recently, experiments have also been conducted to validate the theoretical results for vibration characteristics of multi-span structures [14, 15]. Although various methods have been applied to the study of vibrations of two-span beams, most studies, including those cited above, have been limited to the vibrations of single-material multi-span beams [16]. And even though the significant effect of piezoelectric laminations on the dynamics of two-span beams has been explored [17], little attention has been paid to energy harvesting via multi-span composite beams. The basic physical fact behind such energy harvesting is that multi-span beams with piezoelectric
film flexing can convert mechanical vibration into electrical energy. Among other applications, energy harvesting of waste vibrations to power wireless sensors in remote environments has been widely studied [18]. Nonlinear energy harvesting can effectively overcome the limitation of linear harvesting in a narrow bandwidth. Considerable progress has been achieved in the study of nonlinear energy harvesting in lumped-parameter systems [19-26]. Wiercigroch et al. [27] pioneered an investigation on vibratory energy harvesting via nonlinear pendulums system, and found the synchronized rotational motions being the desired response to harvest energy. Zhou et al. [28] broadened frequency responses of a nonlinear piezoelectric energy harvester by rotated magnets. The experiment confirms the broad low-frequency energy harvesting via rotatable external magnets. Internal resonance was proposed to enhance piezoelectric energy harvesting [29] and magnetoelectric energy harvesting [30], and analytical and numerical investigations reveal it as a promising approach to increase frequency responses. Xu et al. [31] experimentally examined 1:2 internal resonance in a two-direction piezoelectric energy harvester, and nonlinear multi-model harvesting can achieve two close resonant frequencies with significantly increasing power outputs [32, 33]. Xiong et al. [34], used an auxiliary oscillator to produce internal resonance in a piezoelectric energy harvester, and they observed rich nonlinear behaviors including multiple stable solutions, double jumping, and energy exchange between modes. Jiang et al. [35] designed an internal resonance energy harvesting system using a buckled beam and an auxiliary oscillator, and they found the improved harvesting performance. Chen et al. [36] implemented nonlinear modal-interaction in an L-shaped beam, piezoelectric energy harvester, and they demonstrated analytically, numerically, and experimentally that double-jumping extends bandwidth and enhances voltage output. These studies show that nonlinear energy harvesting systems have a broad-band capability for
energy transduction. However, in all the above-mentioned studies related to nonlinear energy harvesting, the distributed-parameter vibration of the harvesting structure has been neglected. Moreover, the ambient sources are not available at all the desired frequencies and each harvester has its own inherent limitation in power generation. Therefore, hybrid systems in multi-mode have become attractive [37]. In this paper, we propose using the voltage output frequency response function (FRF) of the nonlinear transverse vibrations of a two-span piezoelectric beam. Axial pre-pressure is applied to build a nonlinear energy harvesting system. The distributed-parameter electromechanical model is then utilized to analyze the amplitude and the voltage output of the FRF of the primary resonance. Finally, we present hybrid energy harvesting with simultaneous three-mode (single-span buckled, two-span, and two-span buckled piezoelectric beam configurations).
2. Formalization 2.1 Modeling of two-span buckled piezoelectric beam Fig.1 shows a two-span piezoelectric beam energy harvester. The harvester is connected to the electrical circuit through the electrodes, which bracket the piezoelectric layer. Axial pre-pressure is used to modify initial curvature of the two-span piezoelectric beam. The thicknesses of the piezoelectric layer and the substructure layer are hp and hs, respectively. The width and length of the respective layers are b and L .
Figure 1 Piezoelectric energy harvester of a two-span buckled beam. The two-span piezoelectric beam energy harvester is a uniform composite Euler-Bernoulli beam consisting of a PZT layer bonded to the substructure layer. The beam is connected to an electrical circuit through the electrodes, which are perfectly conductive and cover the entire surface of the PZT at the bottom and at the top. The simple electrical circuit consists of a resistive load only.
The governing equations for the transverse vibration of a two-span beam accounting for mid-plane stretching are given by EA L1 2 d ( x) d ( x L1 ) mw1 EIw1iv cw1 PL w1 w1 w1 dx 1 (t )[ ] F1 ( x, t ), 0 x L1 0 2 L1 dx dx mw2 EIw2iv cw2 PR w2
and, EI b[
(1a)
EA L2 2 d ( x) d ( x L2 ) w2 w2 dx 2 (t )[ ] F2 ( x, t ), 0 x L2 (1b) 0 2 L2 dx dx
Es (hb3 ha3 ) Ep (hc3 hb3 )) 3
] , EA Es ( ha hb )b Ep ( hc hp ) b ,
( x) is the Dirac delta function and it satisfies
Ep d31b 2hp
(hc2 hb2 ) ,
(n) d ( n ) ( x x0 ) ( x0 ) n df . f ( x ) dx ( 1) ( n) dx(n) dx
The overdot denotes differentiation with respect to t; the prime denotes differentiation with respect to x. Subscript 1 and 2 denote the left and right span. The boundary conditions and continuity conditions are given by
w1 (0) w1 ( L1 ) w2 (0) w2 (L2 ) 0 , w1 (0) w2 (0) 0 , w1 ( L1 ) w2 ( L2 ) , w1 ( L1 ) w2 ( L2 ) where the symbols have the following meanings: Table 1 Nomenclature
w( x, t )
(t )
the displacement of the piezoelectric energy harvester the generated voltage
t
the time
c
the viscous damping coefficient
m
the mass per unit length of the beam
EI
the bending stiffness
EA
the tension stiffness
Es
Young’s modulus of the substructure
(2)
Ep
Young’s modulus of the PZT
ha
the distance between the bottom of the structure layer and the neutral axis
hb
the distance between the bottom of the PZT layer and the neutral axis
hc
the distance between the top of the PZT layer and the neutral axis
the electromechanical coupling coefficient
d31
the piezoelectric constant
2.2 Buckled configurations The buckling configuration can be achieved from Eqs. (1) and (2) by neglecting time, damping, and excitation force terms. Denoting the buckled configuration by ψ(x) gives EI 1iv PL 1
EA L1 2 1 1 dx 0, 0 x L1 0 2 L1
(4a)
EA L2 2 EI 2iv PR 2 2 2 dx 0, 0 x L2 0 2 L2
(4b)
and
1 (0) 1 ( L1 ) 2 (0) 2 ( L2 ) 0 , 1 (0) 2 (0) 0 , 1 ( L1 ) 2 ( L2 ) , 1 ( L1 ) 2 ( L2 ) (5) Noting that the integrals in Eq. (4) are constant for a given ψ(x), we get
12
1 EA L1 2 ( PL 1 dx) , EI 2 L1 0
2 2
1 EA ( PR EI 2 L2
L2
0
22dx)
(6)
where constants λ12 and λ22 denote the critical buckled loads of the left and right span, respectively. Eq. (4) then reduce to
1iv 12 1 0 ,
2iv 2 2 2 0
(7)
The general solutions of Eq. (7) are assumed to be of the form
1 A1 B1x C1 cos 1x D1 sin 1x
(8a)
2 A2 B2 x C2 cos 2 x D2 sin 2 x
(8b)
Satisfying the boundary conditions by substituting Eq. (5) into Eq. (8) can give eigenvalues λ12 and λ22. For L1=L2=0.25m, it is achieved that λ1=λ2. Eigenvalues of the first three order buckled configurations are listed in Table 2.
Table 2 Eigenvalues of first three orders for the buckled configuration Configuration 1
Configuration 2
Configuration 3
12.576
17.974
25.133
1 =2
Then, the first critical loads of the span can be given by Pc1 =PLc1 =PRc1 1 EI
EA L1 2 1 dx 2 L1 0
(9)
Therefore, for given axial loads PL and PR, the corresponding buckled shape corresponding to any eigenvalues λ1 and λ2 can be determined.
2.3 Galerkin truncation and natural frequencies To investigate nonlinear dynamics of the two-span buckled beam, considering the vibrations around the stable buckled configuration shape, the solutions to Eq. (1) are assumed to be
w1 ( x, t ) 1 ( x) v1 ( x, t )
(10a)
w2 ( x, t ) 2 ( x) v2 ( x, t )
(10b)
where v(x,t) is the displacement around the buckled configuration ψ(x). Substituting Eq. (10) into Eq. (1), yields mv1 EIv1iv cv1 ( PL
L1 L1 EA L1 2 EA EA 1 dx )v1 1 v1 1dx v v1 1dx 0 0 2 L1 0 L1 L1
EA L1 2 EA L1 2 d ( x) d ( x L1 ) 1 v1 dx v1 v1 dx 1 (t )[ ] F1 ( x, t ) 0 0 2 L1 2 L1 dx dx
(11a)
EA 2 L2
mv2 EIv2iv cv2 ( PR
L2
0
22 dx)v2
L2 EA L2 EA 2 v2 2 dx v v2 2dx 0 0 L2 L2
EA L2 2 EA L2 2 d ( x) d ( x L2 ) 2 v2 dx v2 v2 dx 2 (t )[ ] F2 ( x, t ) 0 0 2 L2 2 L2 dx dx
(11b)
and v1 (0) v1 ( L1 )=v2 (0) v2 ( L2 ), v1 (0) v2 (0) 0, v1 ( L1 ) v2 ( L2 ), v1 ( L1 ) v2 ( L2 )
(12)
For the linearized free vibration, solutions can be found by using the following reduced equations: mv1 EIv1iv cv1 ( PL
EA L1 2 EA L1 1 dx)v1 1 v1 1 dx 0 2 L1 0 L1
(13a)
EA 2 L2
(13b)
mv2 EIv2iv cv2 ( PR
L2
0
22dx)v2
EA L2 2 v2 2 dx 0 L2
The solutions are assumed to be of the form v2 ( x2 , t ) ( x2 )eit
v1 ( x1 , t ) ( x1 )eit ,
(14)
where ω and φ(x) are, respectively, a natural frequency and a mode shape. Substituting Eq. (14) into Eq. (13) yields
1iv 21
m 21 EA L1 1 1 1 dx 0 EI L1 EI
(15a)
2iv 22
m 22 EA L2 2 2 2 dx 0 EI L2 EI
(15b)
and
1 (0) 1 ( L1 )=2 (0) 2 ( L2 ) , 1 (0) 2 (0) 0 , 1 ( L1 ) 2 ( L2 ) , 1 ( L1 ) 2 ( L2 )
(16)
Eq. (15) are nonhomogeneous fourth-order ordinary-differential equations with constant coefficients whose general solutions can be expressed as [38]
1 ( x1 ) h ( x1 )+ p ( x1 ) , 1
1
2 ( x2 ) h ( x2 )+ p ( x2 ) 2
2
(17)
where the homogeneous solutions φh1 and φh2 are governed by
1iv 21
m 21 0, EI
2iv 22
m 22 0 EI
(18)
And the nonhomogeneous solutions φp1 and φp2 are governed by
p
iv
p
iv
1
2
p1 2
m 2 p1
p2 2
EI m 2 p2 EI
EA L1 EA L1 1 h1 1 dx 1 p1 1 dx 0 0 L1 EI L1EI
(19a)
EA L2 EA L2 2 h2 2 dx 2 p2 2 dx 0 0 L2 EI L2 EI
(19b)
The general solution of Eq. (18) can be expressed as [38]
h x A1 sin s1 x B1 cos s1 x C1 sinh s2 x D1 cosh s2 x
(20a)
h x A2 sin s1 x B2 cos s1 x C2 sinh s2 x D2 cosh s2 x
(20b)
1
2
where A1, B1, C1, D1 and A2, B2, C2, D2 are constants and
1 1 4m 2 s1,2 2 4 2 2 EI
(21)
Because the second integral on the right-hand side of Eq. (19) are constants for given ψ1(x), φp1(x) and ψ2(x), φp1(x). The nonhomogeneous solutions of Eq. (19) has the form
p x E1 1 ,
p x E2 2
1
(22)
2
where E1 and E2 are coefficients. Substituting Eq. (22) into Eq. (19) respectively, yields E1 1vi 2 1iv E1
m 2 1 EA L1 11 E1 1 1 1 dx 0 EI L1 EI
(23a)
L2 m 2 2 EA 2 2 E2 2 2 2dx 0 EI L2 EI
E2 2vi 2 2iv E2
(23b)
where 1 =
EA L1 EI
L1
0
1h dx , 1
2 =
EA L2 EI
L2
0
2h dx
(24)
2
Substituting Eq. (7) into Eq. (23) yield 1 +E1 (
m 2 EA EI L1EI
L1
0
1 1dx) 0 ,
2 +E2 (
m 2 EA EI L2 EI
L2
0
2 2dx) 0
(25)
There are two possibilities: ⋀1,2=0 or ⋀1,2≠0 [38]. In the first case, the buckling mode is orthogonal to the vibration mode, so E1,2=0 or
2
EA L1 1 1 dx , mL1 0
2
EA mL2
L2
0
2 2dx
(26)
For E1=0 and E2=0, the vibration mode shapes are given by only the homogeneous solutions, these called the even mode. They are independent of the applied axial load. When ⋀1,2≠0, Eq. (25) provide with an extra equation for the constant E1,2. This possibility holds for all boundary conditions since it is derived from the equation of motion [38]. The general solution of Eq. (19) can be expressed as follows:
2,i x A2 sin s1 x B2 cos s1 x C2 sinh s2 x D2 cosh s2 x E2 2
(27a)
2,i ( x) A2 sin(s1 x) B2 cos(s1 x) C2 sinh(s2 x) D2 cosh(s2 x) E2 2
(27b)
Applying the boundary conditions given by Eq. (16), one finds the natural frequencies by balancing the coefficients.
Table 3 The natural frequencies of the first five modes with PL1=PR1=1.8Pc1
(rad/s)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
47.8
62.9
130.9
174.6
320.7
The electrical circuit equation can be written as
33s bL1 d (t ) (t ) hp
dt
Rl
33s bL2 d (t ) (t ) hp
dt
Rl
L1
0
L2
3v1 ( x, t ) d31Ep hpcb dx 0 x 2t
(28a)
3v2 ( x, t ) dx 0 x 2t
(28b)
d31Ep hpcb 0
where 33s is the permittivity constant at constant strain, hpc is the distance from the center to the PZT layer to the neutral axis, and Rl is the resistive load. The deflections of the two-span piezoelectric beam are assumed to be
N
i 1
i 1
N
i 1
i 1
v1 ( x, t ) 1,i ( x)q1,i (t ) 1,i ( x)q1,i (t ) , i 1, 2
(29a)
N
v2 ( x, t ) 2,i ( x)q2,i (t ) 2,i ( x)q2,i (t ) , i 1, 2
(29b)
N
where q1,i(t) and q2,i(t) are the ith modal amplitudes and φ1,i(t) and φ2,i(t) are the eigenfunctions of the vibration mode. Substituting Eqs. (29a) and (26a) into Eq. (11a), substituting Eqs. (29b) and (26b) into Eq. (11b), then multiplying the result with φ1,j(t) and φ2,j(t) and integrating from x=0 to x=L1 and x=0 to x=L2, respectively, we obtain the following nonlinear ordinary differential governing equations N
m i 1
q c1,ij q1,ij k1,ij q1,ij 1,ij q1,2ij 1,ij q1,3ij 1, j1 (t ) f1, j cos(t ), j 1, 2
1,ij 1ij
N
m i 1
q
2,ij 2,ij
N
c2,ij q2,ij k2,ij q2,ij 2,ij q2,2 ij 2,ij q2,3 ij 2, j2 (t ) f 2, j cos(t ), j 1, 2
(30a)
N
where L1
L2
L1
L2
0
0
0
0
m1,ij m 1,i1, j dx , m2,ij m 2,i 2, j dx , c1,ij =c 1,i1, j dx , c2,ij =c 2,i 2, j dx , L1
k1,ij =EI 1, j 0
L1 d4 EA L1 d d2 2 d x [ P ( ) dx ]( 1,i L 1 0 1, j dx 2 1,i dx) dx 4 2 L1 0 dx
L1 d EA L1 d2 d ( 1, j 2 1dx)( 1i 1dx) 0 0 L1 dx dx dx L2
k2,ij =EI 2, j 0
d4 EA 2,i dx [ PR 4 dx 2 L2
L2
0
(
L2 d d2 2 ) 2 dx]( 2, j 2 2,i dx) 0 dx dx
L2 d EA L2 d2 d ( 2, j 2 2 dx)( 2i 2 dx) 0 0 L2 dx dx dx
1,ij =
L1 d L1 d EA L1 d2 d EA L1 d2 ( 1, j 2 1,i dx)( 1,i 1dx) ( 1, j 2 1dx)( ( 1,i ) 2 dx) 0 dx 0 L1 0 dx dx 2 L1 0 dx dx
2,ij =
L2 d L2 d EA L2 d2 d EA L2 d2 ( 2, j 2 2,i dx)( 2,i 2 dx) ( 2, j 2 2 dx)( ( 2,i ) 2 dx) 0 0 0 0 L2 dx dx dx 2 L2 dx dx
L1 d L2 d EA L1 d2 EA L2 d2 2 1,ij = ( 1, j 2 1,i dx)( ( 1,i ) dx) , 2,ij = ( 2, j 2 2,i dx)( ( 2,i ) 2 dx) 0 0 2 L1 0 dx dx 2 L2 0 dx dx
(30b)
d1, j ( x)
1,i
dx
, 2,i
d2, j ( x)
x L1
dx
x L2
L1
L2
0
0
, f1,i mAa 1, j dx , f 2,i mAa 2, j dx
and Aa is the excitation acceleration of the two-span buckled beams. Substituting Eqs. (29a) and (29b) into Eqs. (28a) and (28b), respectively, gives N hp dq (t ) d1 (t ) ( t ) 1,i i 1 s dt Rl 33bL1 dt i 1
(31a)
N hp dq (t ) d2 (t ) ( t ) 2,i i 2 s dt Rl 33bL2 dt i 1
(31b)
where
1,i
d31Ep hpc hp
2,i
d31 Ep hpc hp
33s L1
L1
L2
0
33s L2
0
d E h h di ( x) d 2i ( x) dx 31 sp pc p 2 dx 33 L1 dx x L1
(32a)
d31Ep hpc hp di ( x) d 2i ( x) d x dx 2 33s L2 dx x L2
(32b)
3. Amplitude and voltage output frequency response function (FRF) 3.1 Harmonic balance method We apply the general harmonic balance method to analyze the amplitude and voltage output FRF. The displacement responses are denoted as q1i(t) and q2i(t), and voltage outputs are denoted as v1(t) and v2(t). As Fourier series, R
q1,i (t ) a1,0i (t ) [a1,in (t ) cos(nt ) b1,in (t ) sin(nt )]
(33a)
n 1 R
q2,i (t ) a2,0i (t ) [a2,in (t ) cos(nt ) b2,in (t ) sin(nt )]
(33b)
n 1
R
v1 (t ) (a1,n (t ) cos(nt ) b1, n (t ) sin(nt ))
(33c)
n 1 R
v2 (t ) (a2,n (t ) cos(nt ) b2, n (t ) sin( nt ))
(33d)
n 1
where i is the harmonic order, the quantities a1,0i, a1,in, b1,in and a2,0i, a2,in, b2,in are the coefficients to
be determined for the corresponding harmonic terms. From Eqs. (33a) and (33b), the velocity and acceleration responses are found to be R
q1,i (t ) a1,0i (t ) [(a1,in nb1,in ) cos(nwt ) (b1,in n a1,in ) sin(nt )]
(34a)
n 1 R
q1,i (t ) a1,0i (t ) [a1,in (n ) 2 a1,in 2nb1,in ]cos(nt ) [b1,in (n ) 2 2n a1,in ]sin(nt ) n 1
(34b)
R
q2,i (t ) a2,0i (t ) [(a2,in nb2,in ) cos(nwt ) (b2,in n a2,in ) sin(nt )]
(34c)
n 1 R
q2,i (t ) a2,0i (t ) [a2,in (n ) 2 a2,in 2nb2,in ]cos(nt ) [b2,in (n ) 2 2n a2,in ]sin(nt ) n 1
(34d)
From Eqs. (33c) and (33d), the time derivatives of voltage are R
v1 (t ) [(a1,n nb1,n ) cos(nt ) (b1,n n a1,n ) sin(nt )]
(35a)
n 1 R
v2 (t ) [(a2,n nb2,n ) cos(nt ) (b2,n n a2,n ) sin(nt )]
(35b)
n 1
Substituting Eqs. (33)-(35) into Eqs. (30)-(32) and balancing the coefficients for all harmonic terms of cos(nωt) and sin(nωt), we can get a series of algebraic equations for the coefficients of the harmonic terms. The amplitude and voltage output FRF can be achieved by the arc-length continuation method.
3.2 Results and discussion The geometrical, material, and electromechanical parameters of the harvester are listed in Table 4. The substructure layer is brass, with the elastic Young's modulus of 100 GPa and the density of 7165kg/m3. PZT layer is composed of PZT-5A piezoelectric elements, and its elastic Young's modulus is 66 GPa and its density is 7800 kg/m3. The PZT layer covers the entire length of the harvesting beam. The harvester is subject to the base motion Aacos(ωt). Let the frequency range of
interest be 0-250 rad/s. Then, for the parameters given in Table 4, the harvester has three natural frequencies located within this frequency range.
Table 4 Geometrical, material and electromechanical parameters of the energy harvester Item
Notation
Value
Length of the beam
L (mm)
500
Width of the beam
b (mm)
20
Thickness of the substructure
hs (mm)
0.5
Thickness of the PZT
hp (mm)
0.4
Young’s modulus of the substructure
Es (GPa)
100
Young’s modulus of the PZT
Ep (GPa)
66
Mass density of the substructure
s (kg/m3 )
7165
Mass density of the PZT
p (kg/m3 )
7800
Piezoelectric constant
d31 (pm/V)
-190
Permittivity
33s (nF/m)
15.93
(a)
(b)
(c)
(e)
(d)
(f)
Figure 2 Amplitude frequency response functions at mid-span point (a, c, e) and output-voltage frequency response functions (b, d, f). The parameters are Aa=0.0035m/s2, R=100Ω, and P1=1.8Pc1 (for (a) and (b)), 1.9Pc1 (for (c) and (d)), 2.0Pc1 (for (e) and (f)). Stable analytical solutions and unstable analytical solutions are respectively shown by the blue and the red dotted lines. black ‘o’ symbols denote the numerical results via the direct time integration of the equation of motion.
Fig. 2 shows the amplitude and voltage output FRF at different axial pre-pressures. The blue lines represent stable analytical approximate solutions of frequency response functions obtained by the method of harmonic balance. The red dotted lines represent unstable solutions. The black points represent numerical solutions calculated by using direct time integration of the equation of motion [39]. The numerical results agree well with the approximate analytical results of the harmonic
balance method. From Figs. 2 (a), (c), and (e), we find that the FRF amplitude shows nonlinearity at certain harmonic excitation. In particular, the magnitude of second peaks are very small and the third peak bends to left. Illustrations of resonance frequencies demonstrate that, as the axial pre-pressure increases, the resonance frequencies of the odd modes gradually increase and the resonance frequencies of the even modes remain almost unchanged. That is why the even-vibration modes are defined only by the homogeneous solution, as shown in Eqs. (20-27). The modes are independent of the axial pre-pressure. The voltage output FRF given by Eqs. (42) and (43) are shown in Figs. 2 (b), (d), and (f). The phenomenon of nonlinear bending occurs at the high frequencies of the voltage output FRF. Information other than the amplitude FRF is that the magnitude of the second peak is larger than that of the first peak.
3.3 Analysis of the different parameters Parameter studies on the FRF for the two-span piezoelectric beam energy harvester are illustrated in Figs. 3, 4, and 5. Fig. 3 shows effects on amplitude FRF at the mid-span point and the voltage output FRF when the axial pre-pressure increases. The vibration amplitude decreases at low frequencies and increases at high frequencies. For that reason, the static curvature of the beam can reduce the resonance response and shift the resonance peak to higher frequency. In addition, the voltage output decreases at low frequencies and increases at high frequencies. Fig. 4 shows the effects of the excitation amplitude. When the axial pre-pressure is fixed at P1=1.8Pc1 (the same static curvature), the amplitude FRF of two-span beams increases as the amplitude of excitation increases. However, the voltage output FRF remains almost unchanged except near the third resonance, where it shows nonlinearity. Fig. 5 shows effects of the load resistance. The values of load resistance Rl varied from 102 Ω to 104 Ω. As load resistance increases, the voltage output increases for the
frequency range of interest at every excitation frequency. Moreover, the current output FRF could readily be achieved by dividing the voltage output FRF by the load resistance. The magnitude of the current output FRF is shown in Fig. 6. Unlike the voltage output FRF shown in Fig. 5, the magnitude of the current decreases monotonically with increasing load resistance. Actually, the trend of the change of the current output is opposite to that of the voltage shown in Fig. 5. The power FRF output is easily the product of the voltage and the current output FRFs. Unlike the voltage and the current output FRFs, the power output FRF is defined as the power divided by the square of the base acceleration. The magnitude of the power output FRF is shown in Fig. 7. For the product of two FRFs that have the opposite trends with increasing load resistance, behavior of the power output FRF with load resistance is more interesting than the previous two electrical outputs. The power output FRF does not demonstrate a monotonic behavior with increasing (or decreasing) load resistance. And, switching between the curves of different load resistance causes overlapping between the FRFs around the resonance frequencies.
(a)
(b)
Figure 3 (a) Amplitude frequency response functions at mid-span point. (b) Voltage output frequency response functions for different axial pre-pressures. The parameters are the same as in Fig. 2. Black dashed line: P1=1.8Pc1; blue solid line: P2=1.9Pc1; red dashed line: P3=2.0Pc1.
(a) (b) Figure 4 (a) Amplitude frequency response functions at mid-span point. (b) Voltage output frequency response functions for different amplitudes of the excitation. The parameters are, P1=1.8Pc1, R=100Ω. Black dashed line: Aa=0.0015m/s2, blue solid line: Aa=0.0025m/s2; red dashed line: Aa=0.0035m/s2.
Figure 5 Voltage output frequency response functions for various values of the load resistance. The parameters are, P1=1.8Pc1, Aa=0.0035m/s2. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 6 Current output frequency response functions for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 7 Power output frequency response functions for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
3.4 Compared with other energy harvesting configurations This section illustrates the comparison of voltage output FRF between single-span buckled, two-span and two-span buckled piezoelectric beams via the distributed parameter electromechanical model. Let the frequency range of interest be 0-250 rad/s. Then, for the parameters given in Table 4, their performances in energy harvesting differ. Single-span buckled, two-span and two-span buckled configurations have their self-advantage and self-limitation in the whole frequency range. Fig. 8 (a)
shows the comparison between two-span and two-span buckled configurations. It is clear that, compared with the two-span configuration, two-span buckled configuration has better performance at high frequencies of 150-250 rad/s, worse performance at lower frequencies of 0-80 rad/s and more resonance peaks at the middle frequency range of 80-150 rad/s. Fig 8 (b) shows the comparison between single-span buckled and two-span buckled configurations, under the same pre-pressure and length of the beam. It is found that, compared with the single-span buckled configuration, two-span buckled configuration has much more resonance peaks at this frequency range. In conclusion, for lower frequencies, the two-span piezoelectric beam performs better; for high frequencies, the two-span buckled piezoelectric beam performs better. Moreover, the two-span buckled configuration has more natural frequencies lying at the middle frequency range of 80-150 rad/s.
(a) (b) Figure 8 (a) Comparison of voltage output frequency response functions between a two-span and a two-span buckled piezoelectric beam. Parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. (b) Comparison of voltage output frequency response functions between a single-span buckled and a two-span buckled piezoelectric beam. Parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω.
The distributions of natural frequencies for these three configurations are compared to explain the characteristics of the voltage output FRF. Under the parameters given in Table 4, we take the axial force to be P1=1.8Pc1. The first five natural frequencies are then shown in Table 5.
Table 5 The natural frequencies of the first five modes with PL1=PR1=1.8Pc1 configuration
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
single-span buckled
47.8 rad/s
130.9 rad/s
320.7 rad/s
427.6 rad/s
534.5 rad/s
two-span
37.8 rad/s
59.0 rad/s
151.2 rad/s
191.3 rad/s
340.1 rad/s
two-span buckled
47.8 rad/s
62.9 rad/s
130.9 rad/s
174.6 rad/s
320.7 rad/s
These results demonstrate that the natural frequencies of single-span buckled piezoelectric beams equal the odd-order natural frequencies of two-span buckled piezoelectric beams. More resonance peaks of two-span buckled piezoelectric beam are found in the relevant frequency range. The first three natural frequencies of the two-span piezoelectric beam are lower than those for the two-span buckled piezoelectric beam, and the latter natural frequencies are larger than those for the two-span buckled piezoelectric beam. 3.5 Hybrid energy harvesting A hybrid system composed of three structures is presented in Fig. 9. It combines scavenged power from single-span buckled, two-span, and two-span buckled piezoelectric beams to support higher loads at the output. The harvester outputs contribute in parallel to a power management circuit. Each harvester subsystem is terminated with an off-chip Schottky diode to avoid reverse current flow. The two separated piezoelectric beams are possibly out of phase at the early stage. Thus the overall electrical outputs from the two separated piezoelectric beam will decrease. However, all piezoelectric beam of the system can eventually be synchronized. The followed results concern the long-term steady solutions. Fig. 10 depicts the voltage output FRFs of a hybrid harvester and a harvester with single-span buckled, two-span, or two-span buckled piezoelectric beam. Using the hybrid system can achieve relatively stable high-voltage output of around 0.1 V per excitation acceleration. Fig. 11 shows
effects on the resultant voltage output of the hybrid energy harvester when the load resistance is changed. As load resistance increases, the voltage output increases for the all frequency range. Fig. 12 illustrates effects on the resultant current output of the hybrid energy harvester when the load resistance is changed. The change of current differs from that of voltage, the amplitude of the current output decreases with increasing load resistance for the all frequency range. However, as shown in Fig. 13, the power output of the hybrid energy harvester does not exhibit a monotonic behavior with increasing (or decreasing) load resistance. When the resistance increases from 10 2 Ω to 103 Ω, the power is increased. But the curves of power overlap in some frequency bands, when the resistance changes from 103 Ω to 104 Ω. Fig. 14 shows effects of the damping ratios. The damping has a significant effect on the voltage output around the resonance, and could reduce the peak value of voltage output.
Figure 9 A hybrid generator composed of three structures. The harvester outputs contribute in parallel to a power management circuit. Each harvester subsystem is terminated with an off-chip Schottky diode to avoid reverse current flow.
Figure 10 Voltage output frequency response functions for the hybrid harvester and the harvester with single-span buckled, two-span, or two-span buckled piezoelectric beam. Parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω.
Figure 11 Voltage output frequency response functions for the hybrid harvester for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 12 Current output frequency response functions for the hybrid harvester for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 13 Power output frequency response functions for the hybrid harvester for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 14 Voltage output frequency response functions for the hybrid harvester for various values of the damping ratios. The parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. Black dashed line: ζ=0.19, blue solid line: ζ=0.38; red dashed line: ζ=0.76.
4. Conclusions We have fully investigated nonlinear energy harvesting from the transverse vibrations of a two-span beam. A buckled two-span beam with piezoelectric film flexing is used to enhance the performance. Based on the distributed-parameter electromechanical model, the Galerkin truncation method and harmonic balance method are used together to solve forced responses of the energy harvesting system. In addition, the Galerkin truncation results are validated by using the finite difference method. We conclude that the results provided by the two methods are trustworthy. We also investigated effects of pre-pressure and two-span boundary conditions on energy harvesting. From this work, the following conclusions can be drawn: (1) Frequency-response-function analysis shows that axial pre-pressure can increase resonance frequencies of the odd modes, but cannot modify resonance frequencies of even modes. More resonance peaks of the two-span buckled piezoelectric beam are found in the relevant frequency range. Beyond the FRF information is the additional insight that the magnitude of the second peak is considerable. (2) This paper also demonstrates the general superiority of the two-span piezoelectric energy harvesting system. However, unlike the energy harvesting of discrete systems, at the higher order primary resonance of continuous systems, the two-span piezoelectric energy harvesting system may degrade harvesting at low frequencies. Interestingly, pre-pressure can be used to enhance energy harvesting from transverse vibration of continuous systems, because at low frequencies, the resonant peaks are crowded together. (3) Since single-span buckled, two-span, and two-span buckled piezoelectric beam configurations all have both advantages and disadvantages, we propose a hybrid energy harvesting system composed of all these configurations. This improves the efficiency of piezoelectric energy harvesting.
Acknowledgements：
This work was supported by the National Natural Science Foundation
of China (Nos. 11872037, 11872159 and 11572182) and the Innovation Program of Shanghai Municipal Education Commission (No. 2017-01-07-00-09-E00019)
Statement of author contribution Authors: Ze-Qi Lu , Jie Chen, Hu Ding, Li-Qun Chen Title: Two-span piezoelectric beam energy harvesting Statement: In this paper, we present the results of a detailed analytical study of nonlinear vibratory energy harvesting via a two-span piezoelectric beam. In particular, we study amplitude and voltage output frequency response functions (FRF) of single-span, buckled, and two-span beam configurations, showing how their properties differ. We then propose a triple-hybrid energy harvester that utilizes advantages of all three configurations. Prof. Li-Qun Chen modeled the nonlinear two-span piezoelectric beam energy harvester, which make us to do further study on the enhanced vibratory energy harvesting based on configuration-hybrid conveniently. Ze-Qi Lu deduced all the formulas, all authors have contributions on the completion of matlab program and article. We believe that the findings of this study are relevant to the scope of your journal and will be of interest to its readership. We have approved the manuscript and agree with submission to International Journal of Mechanical Sciences. There are no conflicts of interest to declare. This manuscript has not been published elsewhere and is not under consideration by another journal. The manuscript has been carefully reviewed by an experienced editor whose first language is English and who specializes in editing papers written by scientists whose native language is not English.
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Two-span piezoelectric beam energy harvesting Ze-Qi Lu , Jie Chen , Hu Ding , Li-Qun Chen PII: DOI: Reference:
S0020-7403(19)34503-5 https://doi.org/10.1016/j.ijmecsci.2020.105532 MS 105532
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
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Please cite this article as: Ze-Qi Lu , Jie Chen , Hu Ding , Li-Qun Chen , Two-span piezoelectric beam energy harvesting, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105532
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Highlights
Nonlinear vibratory energy harvesting via a two-span piezobeam Benefits of nonlinear interaction between buckled beam spans Comparison of voltage output FRF between single-span, buckled, and two-span beam configurations A hybrid harvester of three structures is proposed to cover a demand of broadband voltages output
Two-span piezoelectric beam energy harvesting Ze-Qi Lu, Jie Chen, Hu Ding and Li-Qun Chen* Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200444, China
Abstract This paper reports an analytical study of nonlinear vibratory energy harvesting via a two-span piezoelectric beam. The distributed parameter electromechanical model is applied to explore the benefits of nonlinear interaction between buckled beam spans. The Galerkin truncation method and harmonic balance method are applied together to find forced responses of the energy harvesting system. The effects of axial pre-pressure on energy harvesting are also explored. The analytical results are supported by utilizing direct time integration of the equation of motion. The amplitude and voltage output frequency response functions (FRF) of single-span, buckled, and two-span piezoelectric beam configurations reveal some differences. For example, the voltage output FRF of the two-span buckled configuration is larger than that of the other two configurations at high frequencies, whereas at low frequencies, the voltage output FRF of two-span unbuckled configuration is larger than that of the other two configurations. Moreover, two-span buckled configuration has more natural frequencies lying at the concerned frequency range. Finally, a hybrid harvester is proposed, which combines harvested voltage from single-span buckled, two-span, and two-span buckled piezoelectric beam to cover a demand of broadband voltages output. Keywords: nonlinear vibration; energy harvesting; piezoelectric beam; two-span beam; Galerkin method
*
Corresponding author: School of Mechanics and Engineering Science, 99 Shangda Road, Shanghai 200444, China,
Tel.: +86 021 66132062; fax: +86 021 66134463. E-mail address: [email protected]
1. Introduction Energy scavenged from vibratory motion using piezoelectric materials is an important alternative to batteries or other sources of energy for powering electronic devices. In this article, we analyze, in particular, the scavenging of power from vibrations of a two-span piezoelectric beam. The transverse vibration of a two-span composite-beam presents a challenging problem [1], but there is a clear demand for utilizing scavenged energy from such sources [2-6]. Studies of transverse vibrations of two-span composite-beams have been conducted literally for centuries and remain an active area of research. Multi-span beams, our focus here, are widely used in many engineering applications, including aerospace, railway tracks, and bridges [2]. To analyze the transverse vibrations of multi-span structures, the transfer matrix method [7-10], the finite element method [11, 12], and the assumed modal method [13, 14] have been the most widely used. Li et al [7, 8] studied wave localization in disordered periodic multi-span beams and disordered periodic piezoelectric beams. Liu et al. [9] examined transient responses of a multi-span pipe conveying fluid. Wang et al. [11] studied vibrations of multi-span bridges under traffic loadings and vehicle dynamic interactions. Elishakoff and Santoro [12] studied random vibrations of a forced two-span beam with a linear elastic foundation. Recently, experiments have also been conducted to validate the theoretical results for vibration characteristics of multi-span structures [14, 15]. Although various methods have been applied to the study of vibrations of two-span beams, most studies, including those cited above, have been limited to the vibrations of single-material multi-span beams [16]. And even though the significant effect of piezoelectric laminations on the dynamics of two-span beams has been explored [17], little attention has been paid to energy harvesting via multi-span composite beams. The basic physical fact behind such energy harvesting is that multi-span beams with piezoelectric
film flexing can convert mechanical vibration into electrical energy. Among other applications, energy harvesting of waste vibrations to power wireless sensors in remote environments has been widely studied [18]. Nonlinear energy harvesting can effectively overcome the limitation of linear harvesting in a narrow bandwidth. Considerable progress has been achieved in the study of nonlinear energy harvesting in lumped-parameter systems [19-26]. Wiercigroch et al. [27] pioneered an investigation on vibratory energy harvesting via nonlinear pendulums system, and found the synchronized rotational motions being the desired response to harvest energy. Zhou et al. [28] broadened frequency responses of a nonlinear piezoelectric energy harvester by rotated magnets. The experiment confirms the broad low-frequency energy harvesting via rotatable external magnets. Internal resonance was proposed to enhance piezoelectric energy harvesting [29] and magnetoelectric energy harvesting [30], and analytical and numerical investigations reveal it as a promising approach to increase frequency responses. Xu et al. [31] experimentally examined 1:2 internal resonance in a two-direction piezoelectric energy harvester, and nonlinear multi-model harvesting can achieve two close resonant frequencies with significantly increasing power outputs [32, 33]. Xiong et al. [34], used an auxiliary oscillator to produce internal resonance in a piezoelectric energy harvester, and they observed rich nonlinear behaviors including multiple stable solutions, double jumping, and energy exchange between modes. Jiang et al. [35] designed an internal resonance energy harvesting system using a buckled beam and an auxiliary oscillator, and they found the improved harvesting performance. Chen et al. [36] implemented nonlinear modal-interaction in an L-shaped beam, piezoelectric energy harvester, and they demonstrated analytically, numerically, and experimentally that double-jumping extends bandwidth and enhances voltage output. These studies show that nonlinear energy harvesting systems have a broad-band capability for
energy transduction. However, in all the above-mentioned studies related to nonlinear energy harvesting, the distributed-parameter vibration of the harvesting structure has been neglected. Moreover, the ambient sources are not available at all the desired frequencies and each harvester has its own inherent limitation in power generation. Therefore, hybrid systems in multi-mode have become attractive [37]. In this paper, we propose using the voltage output frequency response function (FRF) of the nonlinear transverse vibrations of a two-span piezoelectric beam. Axial pre-pressure is applied to build a nonlinear energy harvesting system. The distributed-parameter electromechanical model is then utilized to analyze the amplitude and the voltage output of the FRF of the primary resonance. Finally, we present hybrid energy harvesting with simultaneous three-mode (single-span buckled, two-span, and two-span buckled piezoelectric beam configurations).
2. Formalization 2.1 Modeling of two-span buckled piezoelectric beam Fig.1 shows a two-span piezoelectric beam energy harvester. The harvester is connected to the electrical circuit through the electrodes, which bracket the piezoelectric layer. Axial pre-pressure is used to modify initial curvature of the two-span piezoelectric beam. The thicknesses of the piezoelectric layer and the substructure layer are hp and hs, respectively. The width and length of the respective layers are b and L .
Figure 1 Piezoelectric energy harvester of a two-span buckled beam. The two-span piezoelectric beam energy harvester is a uniform composite Euler-Bernoulli beam consisting of a PZT layer bonded to the substructure layer. The beam is connected to an electrical circuit through the electrodes, which are perfectly conductive and cover the entire surface of the PZT at the bottom and at the top. The simple electrical circuit consists of a resistive load only.
The governing equations for the transverse vibration of a two-span beam accounting for mid-plane stretching are given by EA L1 2 d ( x) d ( x L1 ) mw1 EIw1iv cw1 PL w1 w1 w1 dx 1 (t )[ ] F1 ( x, t ), 0 x L1 0 2 L1 dx dx mw2 EIw2iv cw2 PR w2
and, EI b[
(1a)
EA L2 2 d ( x) d ( x L2 ) w2 w2 dx 2 (t )[ ] F2 ( x, t ), 0 x L2 (1b) 0 2 L2 dx dx
Es (hb3 ha3 ) Ep (hc3 hb3 )) 3
] , EA Es ( ha hb )b Ep ( hc hp ) b ,
( x) is the Dirac delta function and it satisfies
Ep d31b 2hp
(hc2 hb2 ) ,
(n) d ( n ) ( x x0 ) ( x0 ) n df . f ( x ) dx ( 1) ( n) dx(n) dx
The overdot denotes differentiation with respect to t; the prime denotes differentiation with respect to x. Subscript 1 and 2 denote the left and right span. The boundary conditions and continuity conditions are given by
w1 (0) w1 ( L1 ) w2 (0) w2 (L2 ) 0 , w1 (0) w2 (0) 0 , w1 ( L1 ) w2 ( L2 ) , w1 ( L1 ) w2 ( L2 ) where the symbols have the following meanings: Table 1 Nomenclature
w( x, t )
(t )
the displacement of the piezoelectric energy harvester the generated voltage
t
the time
c
the viscous damping coefficient
m
the mass per unit length of the beam
EI
the bending stiffness
EA
the tension stiffness
Es
Young’s modulus of the substructure
(2)
Ep
Young’s modulus of the PZT
ha
the distance between the bottom of the structure layer and the neutral axis
hb
the distance between the bottom of the PZT layer and the neutral axis
hc
the distance between the top of the PZT layer and the neutral axis
the electromechanical coupling coefficient
d31
the piezoelectric constant
2.2 Buckled configurations The buckling configuration can be achieved from Eqs. (1) and (2) by neglecting time, damping, and excitation force terms. Denoting the buckled configuration by ψ(x) gives EI 1iv PL 1
EA L1 2 1 1 dx 0, 0 x L1 0 2 L1
(4a)
EA L2 2 EI 2iv PR 2 2 2 dx 0, 0 x L2 0 2 L2
(4b)
and
1 (0) 1 ( L1 ) 2 (0) 2 ( L2 ) 0 , 1 (0) 2 (0) 0 , 1 ( L1 ) 2 ( L2 ) , 1 ( L1 ) 2 ( L2 ) (5) Noting that the integrals in Eq. (4) are constant for a given ψ(x), we get
12
1 EA L1 2 ( PL 1 dx) , EI 2 L1 0
2 2
1 EA ( PR EI 2 L2
L2
0
22dx)
(6)
where constants λ12 and λ22 denote the critical buckled loads of the left and right span, respectively. Eq. (4) then reduce to
1iv 12 1 0 ,
2iv 2 2 2 0
(7)
The general solutions of Eq. (7) are assumed to be of the form
1 A1 B1x C1 cos 1x D1 sin 1x
(8a)
2 A2 B2 x C2 cos 2 x D2 sin 2 x
(8b)
Satisfying the boundary conditions by substituting Eq. (5) into Eq. (8) can give eigenvalues λ12 and λ22. For L1=L2=0.25m, it is achieved that λ1=λ2. Eigenvalues of the first three order buckled configurations are listed in Table 2.
Table 2 Eigenvalues of first three orders for the buckled configuration Configuration 1
Configuration 2
Configuration 3
12.576
17.974
25.133
1 =2
Then, the first critical loads of the span can be given by Pc1 =PLc1 =PRc1 1 EI
EA L1 2 1 dx 2 L1 0
(9)
Therefore, for given axial loads PL and PR, the corresponding buckled shape corresponding to any eigenvalues λ1 and λ2 can be determined.
2.3 Galerkin truncation and natural frequencies To investigate nonlinear dynamics of the two-span buckled beam, considering the vibrations around the stable buckled configuration shape, the solutions to Eq. (1) are assumed to be
w1 ( x, t ) 1 ( x) v1 ( x, t )
(10a)
w2 ( x, t ) 2 ( x) v2 ( x, t )
(10b)
where v(x,t) is the displacement around the buckled configuration ψ(x). Substituting Eq. (10) into Eq. (1), yields mv1 EIv1iv cv1 ( PL
L1 L1 EA L1 2 EA EA 1 dx )v1 1 v1 1dx v v1 1dx 0 0 2 L1 0 L1 L1
EA L1 2 EA L1 2 d ( x) d ( x L1 ) 1 v1 dx v1 v1 dx 1 (t )[ ] F1 ( x, t ) 0 0 2 L1 2 L1 dx dx
(11a)
EA 2 L2
mv2 EIv2iv cv2 ( PR
L2
0
22 dx)v2
L2 EA L2 EA 2 v2 2 dx v v2 2dx 0 0 L2 L2
EA L2 2 EA L2 2 d ( x) d ( x L2 ) 2 v2 dx v2 v2 dx 2 (t )[ ] F2 ( x, t ) 0 0 2 L2 2 L2 dx dx
(11b)
and v1 (0) v1 ( L1 )=v2 (0) v2 ( L2 ), v1 (0) v2 (0) 0, v1 ( L1 ) v2 ( L2 ), v1 ( L1 ) v2 ( L2 )
(12)
For the linearized free vibration, solutions can be found by using the following reduced equations: mv1 EIv1iv cv1 ( PL
EA L1 2 EA L1 1 dx)v1 1 v1 1 dx 0 2 L1 0 L1
(13a)
EA 2 L2
(13b)
mv2 EIv2iv cv2 ( PR
L2
0
22dx)v2
EA L2 2 v2 2 dx 0 L2
The solutions are assumed to be of the form v2 ( x2 , t ) ( x2 )eit
v1 ( x1 , t ) ( x1 )eit ,
(14)
where ω and φ(x) are, respectively, a natural frequency and a mode shape. Substituting Eq. (14) into Eq. (13) yields
1iv 21
m 21 EA L1 1 1 1 dx 0 EI L1 EI
(15a)
2iv 22
m 22 EA L2 2 2 2 dx 0 EI L2 EI
(15b)
and
1 (0) 1 ( L1 )=2 (0) 2 ( L2 ) , 1 (0) 2 (0) 0 , 1 ( L1 ) 2 ( L2 ) , 1 ( L1 ) 2 ( L2 )
(16)
Eq. (15) are nonhomogeneous fourth-order ordinary-differential equations with constant coefficients whose general solutions can be expressed as [38]
1 ( x1 ) h ( x1 )+ p ( x1 ) , 1
1
2 ( x2 ) h ( x2 )+ p ( x2 ) 2
2
(17)
where the homogeneous solutions φh1 and φh2 are governed by
1iv 21
m 21 0, EI
2iv 22
m 22 0 EI
(18)
And the nonhomogeneous solutions φp1 and φp2 are governed by
p
iv
p
iv
1
2
p1 2
m 2 p1
p2 2
EI m 2 p2 EI
EA L1 EA L1 1 h1 1 dx 1 p1 1 dx 0 0 L1 EI L1EI
(19a)
EA L2 EA L2 2 h2 2 dx 2 p2 2 dx 0 0 L2 EI L2 EI
(19b)
The general solution of Eq. (18) can be expressed as [38]
h x A1 sin s1 x B1 cos s1 x C1 sinh s2 x D1 cosh s2 x
(20a)
h x A2 sin s1 x B2 cos s1 x C2 sinh s2 x D2 cosh s2 x
(20b)
1
2
where A1, B1, C1, D1 and A2, B2, C2, D2 are constants and
1 1 4m 2 s1,2 2 4 2 2 EI
(21)
Because the second integral on the right-hand side of Eq. (19) are constants for given ψ1(x), φp1(x) and ψ2(x), φp1(x). The nonhomogeneous solutions of Eq. (19) has the form
p x E1 1 ,
p x E2 2
1
(22)
2
where E1 and E2 are coefficients. Substituting Eq. (22) into Eq. (19) respectively, yields E1 1vi 2 1iv E1
m 2 1 EA L1 11 E1 1 1 1 dx 0 EI L1 EI
(23a)
L2 m 2 2 EA 2 2 E2 2 2 2dx 0 EI L2 EI
E2 2vi 2 2iv E2
(23b)
where 1 =
EA L1 EI
L1
0
1h dx , 1
2 =
EA L2 EI
L2
0
2h dx
(24)
2
Substituting Eq. (7) into Eq. (23) yield 1 +E1 (
m 2 EA EI L1EI
L1
0
1 1dx) 0 ,
2 +E2 (
m 2 EA EI L2 EI
L2
0
2 2dx) 0
(25)
There are two possibilities: ⋀1,2=0 or ⋀1,2≠0 [38]. In the first case, the buckling mode is orthogonal to the vibration mode, so E1,2=0 or
2
EA L1 1 1 dx , mL1 0
2
EA mL2
L2
0
2 2dx
(26)
For E1=0 and E2=0, the vibration mode shapes are given by only the homogeneous solutions, these called the even mode. They are independent of the applied axial load. When ⋀1,2≠0, Eq. (25) provide with an extra equation for the constant E1,2. This possibility holds for all boundary conditions since it is derived from the equation of motion [38]. The general solution of Eq. (19) can be expressed as follows:
2,i x A2 sin s1 x B2 cos s1 x C2 sinh s2 x D2 cosh s2 x E2 2
(27a)
2,i ( x) A2 sin(s1 x) B2 cos(s1 x) C2 sinh(s2 x) D2 cosh(s2 x) E2 2
(27b)
Applying the boundary conditions given by Eq. (16), one finds the natural frequencies by balancing the coefficients.
Table 3 The natural frequencies of the first five modes with PL1=PR1=1.8Pc1
(rad/s)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
47.8
62.9
130.9
174.6
320.7
The electrical circuit equation can be written as
33s bL1 d (t ) (t ) hp
dt
Rl
33s bL2 d (t ) (t ) hp
dt
Rl
L1
0
L2
3v1 ( x, t ) d31Ep hpcb dx 0 x 2t
(28a)
3v2 ( x, t ) dx 0 x 2t
(28b)
d31Ep hpcb 0
where 33s is the permittivity constant at constant strain, hpc is the distance from the center to the PZT layer to the neutral axis, and Rl is the resistive load. The deflections of the two-span piezoelectric beam are assumed to be
N
i 1
i 1
N
i 1
i 1
v1 ( x, t ) 1,i ( x)q1,i (t ) 1,i ( x)q1,i (t ) , i 1, 2
(29a)
N
v2 ( x, t ) 2,i ( x)q2,i (t ) 2,i ( x)q2,i (t ) , i 1, 2
(29b)
N
where q1,i(t) and q2,i(t) are the ith modal amplitudes and φ1,i(t) and φ2,i(t) are the eigenfunctions of the vibration mode. Substituting Eqs. (29a) and (26a) into Eq. (11a), substituting Eqs. (29b) and (26b) into Eq. (11b), then multiplying the result with φ1,j(t) and φ2,j(t) and integrating from x=0 to x=L1 and x=0 to x=L2, respectively, we obtain the following nonlinear ordinary differential governing equations N
m i 1
q c1,ij q1,ij k1,ij q1,ij 1,ij q1,2ij 1,ij q1,3ij 1, j1 (t ) f1, j cos(t ), j 1, 2
1,ij 1ij
N
m i 1
q
2,ij 2,ij
N
c2,ij q2,ij k2,ij q2,ij 2,ij q2,2 ij 2,ij q2,3 ij 2, j2 (t ) f 2, j cos(t ), j 1, 2
(30a)
N
where L1
L2
L1
L2
0
0
0
0
m1,ij m 1,i1, j dx , m2,ij m 2,i 2, j dx , c1,ij =c 1,i1, j dx , c2,ij =c 2,i 2, j dx , L1
k1,ij =EI 1, j 0
L1 d4 EA L1 d d2 2 d x [ P ( ) dx ]( 1,i L 1 0 1, j dx 2 1,i dx) dx 4 2 L1 0 dx
L1 d EA L1 d2 d ( 1, j 2 1dx)( 1i 1dx) 0 0 L1 dx dx dx L2
k2,ij =EI 2, j 0
d4 EA 2,i dx [ PR 4 dx 2 L2
L2
0
(
L2 d d2 2 ) 2 dx]( 2, j 2 2,i dx) 0 dx dx
L2 d EA L2 d2 d ( 2, j 2 2 dx)( 2i 2 dx) 0 0 L2 dx dx dx
1,ij =
L1 d L1 d EA L1 d2 d EA L1 d2 ( 1, j 2 1,i dx)( 1,i 1dx) ( 1, j 2 1dx)( ( 1,i ) 2 dx) 0 dx 0 L1 0 dx dx 2 L1 0 dx dx
2,ij =
L2 d L2 d EA L2 d2 d EA L2 d2 ( 2, j 2 2,i dx)( 2,i 2 dx) ( 2, j 2 2 dx)( ( 2,i ) 2 dx) 0 0 0 0 L2 dx dx dx 2 L2 dx dx
L1 d L2 d EA L1 d2 EA L2 d2 2 1,ij = ( 1, j 2 1,i dx)( ( 1,i ) dx) , 2,ij = ( 2, j 2 2,i dx)( ( 2,i ) 2 dx) 0 0 2 L1 0 dx dx 2 L2 0 dx dx
(30b)
d1, j ( x)
1,i
dx
, 2,i
d2, j ( x)
x L1
dx
x L2
L1
L2
0
0
, f1,i mAa 1, j dx , f 2,i mAa 2, j dx
and Aa is the excitation acceleration of the two-span buckled beams. Substituting Eqs. (29a) and (29b) into Eqs. (28a) and (28b), respectively, gives N hp dq (t ) d1 (t ) ( t ) 1,i i 1 s dt Rl 33bL1 dt i 1
(31a)
N hp dq (t ) d2 (t ) ( t ) 2,i i 2 s dt Rl 33bL2 dt i 1
(31b)
where
1,i
d31Ep hpc hp
2,i
d31 Ep hpc hp
33s L1
L1
L2
0
33s L2
0
d E h h di ( x) d 2i ( x) dx 31 sp pc p 2 dx 33 L1 dx x L1
(32a)
d31Ep hpc hp di ( x) d 2i ( x) d x dx 2 33s L2 dx x L2
(32b)
3. Amplitude and voltage output frequency response function (FRF) 3.1 Harmonic balance method We apply the general harmonic balance method to analyze the amplitude and voltage output FRF. The displacement responses are denoted as q1i(t) and q2i(t), and voltage outputs are denoted as v1(t) and v2(t). As Fourier series, R
q1,i (t ) a1,0i (t ) [a1,in (t ) cos(nt ) b1,in (t ) sin(nt )]
(33a)
n 1 R
q2,i (t ) a2,0i (t ) [a2,in (t ) cos(nt ) b2,in (t ) sin(nt )]
(33b)
n 1
R
v1 (t ) (a1,n (t ) cos(nt ) b1, n (t ) sin(nt ))
(33c)
n 1 R
v2 (t ) (a2,n (t ) cos(nt ) b2, n (t ) sin( nt ))
(33d)
n 1
where i is the harmonic order, the quantities a1,0i, a1,in, b1,in and a2,0i, a2,in, b2,in are the coefficients to
be determined for the corresponding harmonic terms. From Eqs. (33a) and (33b), the velocity and acceleration responses are found to be R
q1,i (t ) a1,0i (t ) [(a1,in nb1,in ) cos(nwt ) (b1,in n a1,in ) sin(nt )]
(34a)
n 1 R
q1,i (t ) a1,0i (t ) [a1,in (n ) 2 a1,in 2nb1,in ]cos(nt ) [b1,in (n ) 2 2n a1,in ]sin(nt ) n 1
(34b)
R
q2,i (t ) a2,0i (t ) [(a2,in nb2,in ) cos(nwt ) (b2,in n a2,in ) sin(nt )]
(34c)
n 1 R
q2,i (t ) a2,0i (t ) [a2,in (n ) 2 a2,in 2nb2,in ]cos(nt ) [b2,in (n ) 2 2n a2,in ]sin(nt ) n 1
(34d)
From Eqs. (33c) and (33d), the time derivatives of voltage are R
v1 (t ) [(a1,n nb1,n ) cos(nt ) (b1,n n a1,n ) sin(nt )]
(35a)
n 1 R
v2 (t ) [(a2,n nb2,n ) cos(nt ) (b2,n n a2,n ) sin(nt )]
(35b)
n 1
Substituting Eqs. (33)-(35) into Eqs. (30)-(32) and balancing the coefficients for all harmonic terms of cos(nωt) and sin(nωt), we can get a series of algebraic equations for the coefficients of the harmonic terms. The amplitude and voltage output FRF can be achieved by the arc-length continuation method.
3.2 Results and discussion The geometrical, material, and electromechanical parameters of the harvester are listed in Table 4. The substructure layer is brass, with the elastic Young's modulus of 100 GPa and the density of 7165kg/m3. PZT layer is composed of PZT-5A piezoelectric elements, and its elastic Young's modulus is 66 GPa and its density is 7800 kg/m3. The PZT layer covers the entire length of the harvesting beam. The harvester is subject to the base motion Aacos(ωt). Let the frequency range of
interest be 0-250 rad/s. Then, for the parameters given in Table 4, the harvester has three natural frequencies located within this frequency range.
Table 4 Geometrical, material and electromechanical parameters of the energy harvester Item
Notation
Value
Length of the beam
L (mm)
500
Width of the beam
b (mm)
20
Thickness of the substructure
hs (mm)
0.5
Thickness of the PZT
hp (mm)
0.4
Young’s modulus of the substructure
Es (GPa)
100
Young’s modulus of the PZT
Ep (GPa)
66
Mass density of the substructure
s (kg/m3 )
7165
Mass density of the PZT
p (kg/m3 )
7800
Piezoelectric constant
d31 (pm/V)
-190
Permittivity
33s (nF/m)
15.93
(a)
(b)
(c)
(e)
(d)
(f)
Figure 2 Amplitude frequency response functions at mid-span point (a, c, e) and output-voltage frequency response functions (b, d, f). The parameters are Aa=0.0035m/s2, R=100Ω, and P1=1.8Pc1 (for (a) and (b)), 1.9Pc1 (for (c) and (d)), 2.0Pc1 (for (e) and (f)). Stable analytical solutions and unstable analytical solutions are respectively shown by the blue and the red dotted lines. black ‘o’ symbols denote the numerical results via the direct time integration of the equation of motion.
Fig. 2 shows the amplitude and voltage output FRF at different axial pre-pressures. The blue lines represent stable analytical approximate solutions of frequency response functions obtained by the method of harmonic balance. The red dotted lines represent unstable solutions. The black points represent numerical solutions calculated by using direct time integration of the equation of motion [39]. The numerical results agree well with the approximate analytical results of the harmonic
balance method. From Figs. 2 (a), (c), and (e), we find that the FRF amplitude shows nonlinearity at certain harmonic excitation. In particular, the magnitude of second peaks are very small and the third peak bends to left. Illustrations of resonance frequencies demonstrate that, as the axial pre-pressure increases, the resonance frequencies of the odd modes gradually increase and the resonance frequencies of the even modes remain almost unchanged. That is why the even-vibration modes are defined only by the homogeneous solution, as shown in Eqs. (20-27). The modes are independent of the axial pre-pressure. The voltage output FRF given by Eqs. (42) and (43) are shown in Figs. 2 (b), (d), and (f). The phenomenon of nonlinear bending occurs at the high frequencies of the voltage output FRF. Information other than the amplitude FRF is that the magnitude of the second peak is larger than that of the first peak.
3.3 Analysis of the different parameters Parameter studies on the FRF for the two-span piezoelectric beam energy harvester are illustrated in Figs. 3, 4, and 5. Fig. 3 shows effects on amplitude FRF at the mid-span point and the voltage output FRF when the axial pre-pressure increases. The vibration amplitude decreases at low frequencies and increases at high frequencies. For that reason, the static curvature of the beam can reduce the resonance response and shift the resonance peak to higher frequency. In addition, the voltage output decreases at low frequencies and increases at high frequencies. Fig. 4 shows the effects of the excitation amplitude. When the axial pre-pressure is fixed at P1=1.8Pc1 (the same static curvature), the amplitude FRF of two-span beams increases as the amplitude of excitation increases. However, the voltage output FRF remains almost unchanged except near the third resonance, where it shows nonlinearity. Fig. 5 shows effects of the load resistance. The values of load resistance Rl varied from 102 Ω to 104 Ω. As load resistance increases, the voltage output increases for the
frequency range of interest at every excitation frequency. Moreover, the current output FRF could readily be achieved by dividing the voltage output FRF by the load resistance. The magnitude of the current output FRF is shown in Fig. 6. Unlike the voltage output FRF shown in Fig. 5, the magnitude of the current decreases monotonically with increasing load resistance. Actually, the trend of the change of the current output is opposite to that of the voltage shown in Fig. 5. The power FRF output is easily the product of the voltage and the current output FRFs. Unlike the voltage and the current output FRFs, the power output FRF is defined as the power divided by the square of the base acceleration. The magnitude of the power output FRF is shown in Fig. 7. For the product of two FRFs that have the opposite trends with increasing load resistance, behavior of the power output FRF with load resistance is more interesting than the previous two electrical outputs. The power output FRF does not demonstrate a monotonic behavior with increasing (or decreasing) load resistance. And, switching between the curves of different load resistance causes overlapping between the FRFs around the resonance frequencies.
(a)
(b)
Figure 3 (a) Amplitude frequency response functions at mid-span point. (b) Voltage output frequency response functions for different axial pre-pressures. The parameters are the same as in Fig. 2. Black dashed line: P1=1.8Pc1; blue solid line: P2=1.9Pc1; red dashed line: P3=2.0Pc1.
(a) (b) Figure 4 (a) Amplitude frequency response functions at mid-span point. (b) Voltage output frequency response functions for different amplitudes of the excitation. The parameters are, P1=1.8Pc1, R=100Ω. Black dashed line: Aa=0.0015m/s2, blue solid line: Aa=0.0025m/s2; red dashed line: Aa=0.0035m/s2.
Figure 5 Voltage output frequency response functions for various values of the load resistance. The parameters are, P1=1.8Pc1, Aa=0.0035m/s2. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 6 Current output frequency response functions for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 7 Power output frequency response functions for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
3.4 Compared with other energy harvesting configurations This section illustrates the comparison of voltage output FRF between single-span buckled, two-span and two-span buckled piezoelectric beams via the distributed parameter electromechanical model. Let the frequency range of interest be 0-250 rad/s. Then, for the parameters given in Table 4, their performances in energy harvesting differ. Single-span buckled, two-span and two-span buckled configurations have their self-advantage and self-limitation in the whole frequency range. Fig. 8 (a)
shows the comparison between two-span and two-span buckled configurations. It is clear that, compared with the two-span configuration, two-span buckled configuration has better performance at high frequencies of 150-250 rad/s, worse performance at lower frequencies of 0-80 rad/s and more resonance peaks at the middle frequency range of 80-150 rad/s. Fig 8 (b) shows the comparison between single-span buckled and two-span buckled configurations, under the same pre-pressure and length of the beam. It is found that, compared with the single-span buckled configuration, two-span buckled configuration has much more resonance peaks at this frequency range. In conclusion, for lower frequencies, the two-span piezoelectric beam performs better; for high frequencies, the two-span buckled piezoelectric beam performs better. Moreover, the two-span buckled configuration has more natural frequencies lying at the middle frequency range of 80-150 rad/s.
(a) (b) Figure 8 (a) Comparison of voltage output frequency response functions between a two-span and a two-span buckled piezoelectric beam. Parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. (b) Comparison of voltage output frequency response functions between a single-span buckled and a two-span buckled piezoelectric beam. Parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω.
The distributions of natural frequencies for these three configurations are compared to explain the characteristics of the voltage output FRF. Under the parameters given in Table 4, we take the axial force to be P1=1.8Pc1. The first five natural frequencies are then shown in Table 5.
Table 5 The natural frequencies of the first five modes with PL1=PR1=1.8Pc1 configuration
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
single-span buckled
47.8 rad/s
130.9 rad/s
320.7 rad/s
427.6 rad/s
534.5 rad/s
two-span
37.8 rad/s
59.0 rad/s
151.2 rad/s
191.3 rad/s
340.1 rad/s
two-span buckled
47.8 rad/s
62.9 rad/s
130.9 rad/s
174.6 rad/s
320.7 rad/s
These results demonstrate that the natural frequencies of single-span buckled piezoelectric beams equal the odd-order natural frequencies of two-span buckled piezoelectric beams. More resonance peaks of two-span buckled piezoelectric beam are found in the relevant frequency range. The first three natural frequencies of the two-span piezoelectric beam are lower than those for the two-span buckled piezoelectric beam, and the latter natural frequencies are larger than those for the two-span buckled piezoelectric beam. 3.5 Hybrid energy harvesting A hybrid system composed of three structures is presented in Fig. 9. It combines scavenged power from single-span buckled, two-span, and two-span buckled piezoelectric beams to support higher loads at the output. The harvester outputs contribute in parallel to a power management circuit. Each harvester subsystem is terminated with an off-chip Schottky diode to avoid reverse current flow. The two separated piezoelectric beams are possibly out of phase at the early stage. Thus the overall electrical outputs from the two separated piezoelectric beam will decrease. However, all piezoelectric beam of the system can eventually be synchronized. The followed results concern the long-term steady solutions. Fig. 10 depicts the voltage output FRFs of a hybrid harvester and a harvester with single-span buckled, two-span, or two-span buckled piezoelectric beam. Using the hybrid system can achieve relatively stable high-voltage output of around 0.1 V per excitation acceleration. Fig. 11 shows
effects on the resultant voltage output of the hybrid energy harvester when the load resistance is changed. As load resistance increases, the voltage output increases for the all frequency range. Fig. 12 illustrates effects on the resultant current output of the hybrid energy harvester when the load resistance is changed. The change of current differs from that of voltage, the amplitude of the current output decreases with increasing load resistance for the all frequency range. However, as shown in Fig. 13, the power output of the hybrid energy harvester does not exhibit a monotonic behavior with increasing (or decreasing) load resistance. When the resistance increases from 10 2 Ω to 103 Ω, the power is increased. But the curves of power overlap in some frequency bands, when the resistance changes from 103 Ω to 104 Ω. Fig. 14 shows effects of the damping ratios. The damping has a significant effect on the voltage output around the resonance, and could reduce the peak value of voltage output.
Figure 9 A hybrid generator composed of three structures. The harvester outputs contribute in parallel to a power management circuit. Each harvester subsystem is terminated with an off-chip Schottky diode to avoid reverse current flow.
Figure 10 Voltage output frequency response functions for the hybrid harvester and the harvester with single-span buckled, two-span, or two-span buckled piezoelectric beam. Parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω.
Figure 11 Voltage output frequency response functions for the hybrid harvester for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 12 Current output frequency response functions for the hybrid harvester for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 13 Power output frequency response functions for the hybrid harvester for various values of the load resistance. The parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. Black dashed line: R=102Ω, blue solid line: R=103Ω; red dashed line: R=104Ω.
Figure 14 Voltage output frequency response functions for the hybrid harvester for various values of the damping ratios. The parameters are P1=1.8Pc1, Aa=0.0035m/s2, R=100Ω. Black dashed line: ζ=0.19, blue solid line: ζ=0.38; red dashed line: ζ=0.76.
4. Conclusions We have fully investigated nonlinear energy harvesting from the transverse vibrations of a two-span beam. A buckled two-span beam with piezoelectric film flexing is used to enhance the performance. Based on the distributed-parameter electromechanical model, the Galerkin truncation method and harmonic balance method are used together to solve forced responses of the energy harvesting system. In addition, the Galerkin truncation results are validated by using the finite difference method. We conclude that the results provided by the two methods are trustworthy. We also investigated effects of pre-pressure and two-span boundary conditions on energy harvesting. From this work, the following conclusions can be drawn: (1) Frequency-response-function analysis shows that axial pre-pressure can increase resonance frequencies of the odd modes, but cannot modify resonance frequencies of even modes. More resonance peaks of the two-span buckled piezoelectric beam are found in the relevant frequency range. Beyond the FRF information is the additional insight that the magnitude of the second peak is considerable. (2) This paper also demonstrates the general superiority of the two-span piezoelectric energy harvesting system. However, unlike the energy harvesting of discrete systems, at the higher order primary resonance of continuous systems, the two-span piezoelectric energy harvesting system may degrade harvesting at low frequencies. Interestingly, pre-pressure can be used to enhance energy harvesting from transverse vibration of continuous systems, because at low frequencies, the resonant peaks are crowded together. (3) Since single-span buckled, two-span, and two-span buckled piezoelectric beam configurations all have both advantages and disadvantages, we propose a hybrid energy harvesting system composed of all these configurations. This improves the efficiency of piezoelectric energy harvesting.
Acknowledgements：
This work was supported by the National Natural Science Foundation
of China (Nos. 11872037, 11872159 and 11572182) and the Innovation Program of Shanghai Municipal Education Commission (No. 2017-01-07-00-09-E00019)
Statement of author contribution Authors: Ze-Qi Lu , Jie Chen, Hu Ding, Li-Qun Chen Title: Two-span piezoelectric beam energy harvesting Statement: In this paper, we present the results of a detailed analytical study of nonlinear vibratory energy harvesting via a two-span piezoelectric beam. In particular, we study amplitude and voltage output frequency response functions (FRF) of single-span, buckled, and two-span beam configurations, showing how their properties differ. We then propose a triple-hybrid energy harvester that utilizes advantages of all three configurations. Prof. Li-Qun Chen modeled the nonlinear two-span piezoelectric beam energy harvester, which make us to do further study on the enhanced vibratory energy harvesting based on configuration-hybrid conveniently. Ze-Qi Lu deduced all the formulas, all authors have contributions on the completion of matlab program and article. We believe that the findings of this study are relevant to the scope of your journal and will be of interest to its readership. We have approved the manuscript and agree with submission to International Journal of Mechanical Sciences. There are no conflicts of interest to declare. This manuscript has not been published elsewhere and is not under consideration by another journal. The manuscript has been carefully reviewed by an experienced editor whose first language is English and who specializes in editing papers written by scientists whose native language is not English.
Author Agreement
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We wish to confirm that there are no known conflicts of interest associated with this publication. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed.
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