Distributed parameter model and experimental validation of a compressive mode energy harvester under harmonic excitations

Distributed parameter model and experimental validation of a compressive mode energy harvester under harmonic excitations Distributed parameter model and experimental validation of a compressive mode[.]

energy harvester under harmonic excitations

H.T Li, Z Yang, J Zu, and W Y Qin,

Citation: AIP Advances 6, 085310 (2016); doi: 10.1063/1.4961232

View online: http://dx.doi.org/10.1063/1.4961232

View Table of Contents: http://aip.scitation.org/toc/adv/6/8

Published by the American Institute of Physics

Distributed parameter model and experimental validation

of a compressive-mode energy harvester

under harmonic excitations

H.T Li,1,2Z Yang,2J Zu,2and W Y Qin1, a

1Department of Engineering Mechanics, Northwestern Polytechnical University, Xian, China

2Department of Mechanical& Industrial Engineering, University of Toronto, Toronto,

Ontario, Canada

(Received 11 May 2016; accepted 5 August 2016; published online 12 August 2016)

This paper presents the modeling and parametric analysis of the recently pro-posed nonlinear compressive-mode energy harvester (HC-PEH) under harmonic excitation Both theoretical and experimental investigations are performed in this study over a range of excitation frequencies Specially, a distributed parameter electro-elastic model is analytically developed by means of the energy-based method and the extended Hamilton’s principle An analytical formulation of bending and stretching forces are derived to gain insight on the source of nonlinearity Fur-thermore, the analytical model is validated against with experimental data and a good agreement is achieved Both numerical simulations and experiment illustrate that the harvester exhibits a hardening nonlinearity and hence a broad frequency bandwidth, multiple coexisting solutions and a large-amplitude voltage response Using the derived model, a parametric study is carried out to examine the ef-fect of various parameters on the harvester voltage response It is also shown from parametric analysis that the harvester’s performance can be further improved

by selecting the proper length of elastic beams, proof mass and reducing the mechanical damping C 2016 Author(s) All article content, except where other-wise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).[http://dx.doi.org/10.1063/1.4961232]

I INTRODUCTION

With the recent advances in low-power wireless electronics, the research of energy harvesting

to provide power for these devices has arisen great interest because energy harvesters have long-life,

efficient and portable advantages The vibration energy harvester (VEH), which concerns convert-ing ambient vibrations into electrical energy, has received significant attention because ambient vibration is widely available in the real world

Among various vibration-to-electricity conversion mechanisms, piezoelectric transduction has been the most favored one due to its high efficiency and compact size Vibration energy harvesters generally utilize a cantilever beam and operate around the resonant frequency While such energy harvesters have a simple structure, they are incapable of working effectively under the ambient wide-continuous spectrum because of the narrow bandwidth.1 3

To overcome the narrow bandwidth problem, the multi-modal oscillators4 and the frequency-tuning scheme5were proposed to expand the bandwidth Specially, many researchers have exten-sively studied the performance improvement achieved by introducing nonlinear dynamic behaviors such as mono-stable,6 , 7 bi-stable,8 14 tri-stable15 , 16 and impact17 to vibration energy harvesters

Gafforelli et al.18 experimentally investigated the mono-stable Duffing oscillator with a double-clamped beam Masana et al.19 studied the primary resonance and super-harmonic resonance of

a buckled piezoelectric energy harvester to improve the harvesting efficiency under low-intensity

a Electronic mail: qinweiyang67@gmail.com

excitation Friswell et al.20proposed an inverted piezoelectric beam energy harvester and achieved frequency-tuning by a tip mass Zhu et al.21 installed a magnet at the end point of a piezoelectric buckled beam and harnessed the energy from the snap-through motion In multi-degree-of-freedom nonlinear vibration systems, the internal resonance may occur and lead to modal interactions, en-ergy exchange or a coupling among the modes.22–24Xiong et al.25introduced a auxiliary oscillator

to achieve the internal resonance, results showed that the working bandwidth increased by nearly

130 % compared to the linear counterpart

Most vibration energy harvesters employ bending-mode configurations, but such configura-tions hinder the potential for wide applicaconfigura-tions because piezoceramics have low fatigue strength

in this mode The amount of energy generated by most bending-mode piezoelectric energy har-vesters is usually not sufficient to power electronic devices In order to improve the power output,

a flexible amplification mechanism such as “cymbal” transducers26 – 28 was designed to increase the electromechanical conversion rate and stress in piezoceramics The cymbal energy harvester has a high reliability in the tensile mode, which makes it suitable for using under high-frequency, large-amplitude excitations However, the ambient vibration usually associates with small ampli-tudes and low frequencies, so the cymbal structure cannot generate power efficiently due to its high fundamental frequency

To further improve the power output as well as the reliability, Yang et al.29 proposed a high-efficiency compressive-mode energy harvester (HC-PEH) This novel energy harvester was developed with a multi-stage amplification mechanism that effectively amplified the stress on a piezoelectric element Experimental results showed that a maximum power of 54.7 mW was gener-ated at 26 Hz under a harmonic excitation of 0.5 g, which was over one order of magnitude higher than other state-of-the-art systems Another advantage of HC-PEH is its small maximum amplitude compared with the bending-mode energy harvesters, so it satisfies the progressive miniaturization of electronic components and is suitable for space constrained applications It is challenging to model the HC-PEH due to the complicated mechanical conditions of the coupling system Yang30 , 31 has established a simplified lumped parameter model to conveniently describe the coupling between mechanical part of the harvester and a simple electrical harvesting circuit The most obvious advan-tage of the lumped parameter model is that it provides an initial insight into the problem by a simple close-form expression However, the model overlooks several important aspects of the physical meaning such as the strain distribution along the beam.32More over, the lumped parameter model can not use for parametric analysis, which restrict the optimization of energy harvesters

In order to address the lack of physical insight of the lumped parameter model, this paper pro-poses an accurately distributed model for the high compressive-mode energy harvester (HC-PEH) The Galerkin method is used to truncate the model to a single degree-of-freedom nonlinear vibra-tion system It is shown that the hardening nonlinearity results in a broadband frequency bandwidth and multiple coexisting solutions at an even small base excitation level The theoretical model is validated against experimental data to verify the harvester’s nonlinear response and enhanced capa-bilities Additionally, a parametric study is carried out to obtain the harvester’s voltage response un-der frequency sweeps excitations The effect of length of elastic beams, proof mass and mechanical damping, is examined on the harvester’s voltage response

II DISTRIBUTED MODEL

Fig.1presents the schematic diagram of HC-PEH The energy harvester is subjected to a base excitation z(t) In the modeling process, the whole system is divided into four parts: the elastic beams (subscript 1), the bow-shape beam (subscript 2), the piezoelectric plate (subscript 3) and the proof mass M We assume that all beams are slender and the deformations are small

The transverse deflection and the longitudinal deformation of the elastic beams are denoted by

w1(x,t) and u1(x,t) Correspondingly, the bow-shape beams’ transverse deflection and the longitu-dinal deformation are represented by w2(x,t) and u2(x,t) Furthermore, u1(L1,t) and w1(L1,t) are used to describe the vibration of the proof mass M in the x axis and y axis respectively

FIG 1 Schematic of proposed energy harvester.

The Lagrangian of the system is expressed as L= T − U + We, where T is the kinetic energy,

Uis the potential energy and Weis the electrical and electromechanical energy.33

T = T1+ T2+ T3+ TM

= 2 ×1 2

 L1 0

m1 ( ˙w1+ ˙z)2+ ˙u2

1
dx + 2 ×1

2m2

 L2 0

( ˙u2+ ˙z)2+ ˙w2

2
dy

2Mp( ˙w1(L1,t) + ˙z)2

+ 2 ×1

2M ( ˙w1(L1,t) + ˙z)2+ ˙u1(L1,t)2

(1)

where the overdot represents the derivative with respect to time, m1 and m2are the mass per unit length of elastic beam and the bow-shaped beam, respectively

m1= ρ1A1[H(x) − H(x − (L1− Lb))]

+ 2ρ1A1[H(x − (L1− Lb)) − H(x − L1)] ,

m2= ρ2A2

(2)

where H(x) is the Heaviside function to describe the varied cross section process Lbare the length

of the double cross section for connecting the proof mass ρiis mass density and Aiis cross section area; where ‘i’ denotes the member group of harvester Mpis the mass of the piezoelectric plate For a fixed-fixed slender beam as shown in Fig.2, when transversal deflection is large, stretch-ing becomes important components As a result, the stretchstretch-ing component of fixed-fixed beam is

FIG 2 Double clamped beam with a point load at the center (a) The center deflection; (b) The expanded view of original neutral axis.

expressed by

Employing the Taylor series (√1+ δ ≈ 1 +δ

2), we express the axial strain as

εx=ds − dx

dx +1 2

( dw dx

)2

(4)

In this study, the longitudinal motion is trivial compared to the transversal one.34,35Thus the strain in the longitudinal direction can be simplified as

εx= 1 L



L

1 2

( dw dx

)2

As shown in Fig.3, the longitude deformation along x axis is different from the clamped-guide case.36The end of the elastic beam at point G still remains parallel to the x axis after the deforma-tion Therefore, we express the elastic deformation of the elastic beams as ∆x1 Accordingly, ∆x2

is denoted as elastic deformation of the bow-shape beam in the flex-compressive center w2(L2/2,t)

In this paper, we treat the bow-shaped beam as a shallow fixed-fixed arch In order to express the ratio of traversal displacement between the midpoint and other positions along y axis, we introduce

a function s(y) The stretch deformation of elastic beams and the bow-shaped beams follows the displacement and force compatibility equations.37

∆x1+ ∆x2= 1

2

 L1 0

w1′2dx,

E1A1

L1 (∆x1)= E2I2

∂4

((h0+ ∆x2)s(y))

∂ y4

−E2A2

2L2

∂2

((h0+ ∆x2)s(y))

∂ y2

 L2

0

(∂ ((∆x2)s(y))

∂ y

)2

d y

(6)

FIG 3 Geometric relationship of ∆x 1 and ∆x 2 (a) Flex-compressive center model, (b) Clamped guide model, (c) Magnifi-cation of ∆x and ∆x , (d) Thickness of beams at the joints and (e) Shape function of s (y).

The potential energy of the system is

U= U1+ U2+ U3+ UM

= 2



1 2

 L1 0

E1I1(w′′1)

2

dx+E1A1

2L1 (∆x1)2 +

 L1 0

m1gw1dx

 + 2



1

2E2I2

 L2 0

w2′′

2

d y + E2A2

2L2 (2u2(L2/2,t)))2+ m2L2gw1(L1,t)



+ MPgw1(L1,t) + *

,



v py

1

2σp yεp ydvp y

+



v p x

1

2σp xεp xdvp x

+ /

-+ 2Mgw1(L1,t)

(7)

where the prime indicates the differentiation with respect to the length coordinates E1I1and E2I2

represent the flexural rigidity of the elastic beam and bow-shaped beam They can be expressed as

E1I1= E1b1h1

12 [H(x) − H(x − (L1− Lb))]

+E1b1 hf

3

12 [H(x − (L1− Lb)) − H(x − L1)] ,

E2I2= E2b2h2

(8)

where hf is the thickness of the fixing joints near to the proof mass σp y, εp y, σp xand εp xare the mechanical stress and strain of the piezoelectric plate, which comply with the linear constitutive relations33 , 36 , 38

εp y= 1

E3

σp y− d31Ex, εp x= 1

E3

σp x− d33Ex,

Dp y= −d31σp y+ e33Ex, Dp x= −d33σp x+ e33Ex

(9)

where Ex and Dp are the electrical field strength and electrical displacements, vp is the volume

of the piezoelectric ceramics, the subscript x and y denote the direction along x and y axis respectively d31and d33are the piezoelectric constant, ε33is the permittivity constant

The piezoelectric crystal can be divided into two parts, the center part of piezoelectric plate only experiences compressive load perpendicular to the poling direction (d31), and the edges where are bonded to the bow-shaped beam experience compressive loads parallel to the poling direction (d33)

We=



v p x

(ExDp x)dvp x+



v py

(ExDp y)dvp y

= 2



vp x

(−d33σx+ ε33Ex)Exdvp x +



v py

(−d31σy+ ε33Ex)Exdvp y

= −2d33

Ap x(V

h3)E1A1∆x1vp x+2ε33(V

h3)2vp x

− d31

Ap yL2

E2A2

 L2 0

(∆x2s(y))′2d y(V

h3)vp y

+ ε33(V

h )2vp y

(10)

where Apis the area subjected to stress, the subscript x and y denote the normal vector along x and

y axis respectively

A dissipation function δW is introduced to account for the effect of the primary mechanical and electrical damping phenomena,

2c1

 L1 0

˙

w2

1dx+1

2c2

 L2 0

˙

w2

where c1and c2are the damping coefficients of elastic beam and bow-shaped beam, respectively;

Qis the electric charge output of piezoelectric layer, and the time rate change of Q is the electric current passing through the resistive load, i.e ˙Q= V/R

III GALERKIN DISCRETIZATION

The transverse and longitudinal motions of sub-structures are expanded as a summation of trial function,

w1(x,t) =

∞



n =1

qn(t)ψn(x),

u1(x,t) = 1 − α1

2

 x 0

w′ 1 2

(x,t)dx,

w2(y,t) = α1

2 s(y)

 L1 0

w′

1dx,

u2(y,t) = 1

2

 y

0

w2′2(y,t)dy

(12)

where α1= ∆x2/1

2

L1

0 w1′2dx is defined as the axial stretch ratio

According to the frequencies of excitation considered here as well as the shape of the beam deflections induced in the experiments, it is safely assumed that the fundamental mode response contributes to the overall displacement dynamics in the elastic beam.20

ψ1=(1 − cos

πx

L1

 ) 

Set the relationship function of the mid-span and other positions on the bow-shaped beam as the fundamental mode of the fixed-fixed beam’s transverse vibration35

s(y) =

(

1 − cos 2π y

L2

 ) 

By Lagrange’s equation, choosing q and V as the generalized coordinates, we can derive the dynamical equation by

d dt

(∂L

∂ ˙q

)

∂q =

δW

∂q, d

dt

(∂L

∂ ˙V

)

∂V =

δW

∂V,

(15)

Hence, Eq (15) gives the two-order nonlinear differential equations

meq¨+ β1 2 ¨qq6+ 12 ˙q2q

+ β2 3 ˙qq2˙z+ 4q3¨z
+ β3(2 ¨qq2+ 4 ˙q2q) + γ ¨z + k1q+ k2q3+ k3q7+

χ1q3V+ χ2qV+ λ + µ1q˙+ µ2q2q˙= 0,

4 χ1q3q˙+ 2 χ2qq˙+ CeV˙ = V/R

(16)

where the me is the equivalent mass as considering the first vibration mode; β1, β2 and β3 are constants due to geometric nonlinearity of beams; k , k and k are the linear and nonlinear stiffness,

TABLE I Model parameters used for numerical test.

respectively; χ1and χ2are the electromechanical coupling constant; λ is the constant to describe the gravity effect; µ1 and µ2 are the equivalent damping used to approximate the energy loss; Ce

is the capacitance of piezoelectric layer; γ is a constant representing the base excitation on the first mode The definition of these coefficients is available inAppendix Obviously, this distributed model is different to the lumped model in the existing works.29 – 31The nonlinear restoring force can

be described by a polynomial of degree 7 If the displacement is fairly small, the influence of high order terms can be neglected, then this model approximates to the lumped model in the existing studies

IV HARDENING NONLINEAR RESPONSE

The geometric, material and electromechanical parameters of harvester are given in Table I The variation of restoring force is shown in Fig 4(a)with the formula k3q7+k2q3+ k1q+ λ The linear stiffness resulting from the bending is the major part when the deflection is fairly small (1 mm) When the transversal deflection beyond this, the cubic and higher-order term nonlinearity caused by the stretching becomes significant and needs to consider in the process of computing the restoring force The corresponding elastic potential energy is depicted in Fig.4(b) Under the effect

FIG 4 (a) Restoring force; (b) Potential energy.

FIG 5 Numerical results of forward sweep frequency sweep, connected with an external resister of 100KΩ under excitations levels of 0.2 g, 0.3 g, 0.4 g and 0.5 g.

of gravitation, the minimum value at q= −0.4 mm, rather than q = 0 mm However, the system still can be classified to the mono-stable energy harvester, which implies the hardening effect due to the large deflections

The governing equation (16) is used in an ordinary differential equation solver (ode45 in Mat-lab) for numerical simulation The base excitation is set as harmonic motion ¨z(t) = a cos(2π f t), where a is the amplitude and f is the excitation frequency in unit of Hz Fig.5shows the linearly increasing frequency sweep excitation (forward sweep) simulation about the displacement response Acceleration values of 0.2 g, 0.3 g, 0.4 g and 0.5 g are selected as the excitation level and the sweep rate is set as 0.1 Hz/s in the simulation Nonlinear oscillation with a distinct jump is observed as the frequency of excitation is upward The jump-up frequency has a tendency to increase as the excitation level increases due to the hardening nonlinearity The nonlinear response in Fig.5clearly demonstrates that a hardening type of nonlinearity contributes to the large-amplitude response and the extended bandwidth

For a nonlinear vibration energy harvester, the resonant behavior of the system is greatly increased and covers a wide band of frequencies, resulting in a large hysteresis loop where two stable states coexist (high-energy branch and low-energy branch) Hence, it is of great significance

to find practicable strategies to ensure the constant manifestation of the high-energy attractor Ba-sins of attraction of the system are plotted for different frequencies of 19.8 Hz, 21.5 Hz,23.1 Hz and 23.7 Hz in Fig 6 The basins of attraction show that when beginning from a different set of initial conditions, the steady-state response tends different branches in which the high-energy part

is illustrated by purple color and the low-energy part is illustrated by green color The fixed points are named as F PH and F PLrespectively in the plot The size of basins of attraction associated with each solution is used for measuring the weighting factor of multiple coexisting responses With the increase of frequency, the occupation of high-energy branch initial condition is decreased as shown

in Figs.6(a)–6(d), which implies there is little probability for response staying in the high-energy branch

V EXPERIMENTAL VALIDATION

This section introduces a series of experiments conducted to confirm the nonlinear characters

of the HC-PEH by numerical simulation The experimental setup is shown in Fig 7, a vibration

FIG 6 Basin of attraction from the numerical model at a = 0.4 g (a)19.8 Hz; (b)21.5 Hz; (c)21.5 Hz; (d)24.7 Hz. shaker (Labworks ET-127) is used to supply mechanical vibration to the prototype and a power amplifier is used to drive the shaker and amplify the signal The effective volume of PZT-5H plate

is 32 × 15 × 0.7mm3 Other parameter values of geometric and material of the prototype are given

in Table.I The mechanical response of velocity and electrical response of voltage are monitored

FIG 7 Experimental setup (a) signal generator and amplifier; (b)Shaker and Dropper micrometer; (c) HC-PEH; (d) Oscilloscope.