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[.]

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 AIP ADVANCES 6, 085310 (2016) Distributed parameter model and experimental validation of a compressive-mode energy harvester under harmonic excitations H.T Li,1,2 Z Yang,2 J Zu,2 and W Y Qin1,a Department of Engineering Mechanics, Northwestern Polytechnical University, Xian, China Department of Mechanical & Industrial Engineering, University of Toronto, Toronto, Ontario, Canada (Received 11 May 2016; accepted August 2016; published online 12 August 2016) This paper presents the modeling and parametric analysis of the recently proposed 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 Furthermore, 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 effect 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 otherwise 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 converting 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 frequencytuning scheme5 were proposed to expand the bandwidth Specially, many researchers have extensively 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 doubleclamped 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 2158-3226/2016/6(8)/085310/16 6, 085310-1 © Author(s) 2016 085310-2 Li et al AIP Advances 6, 085310 (2016) excitation Friswell et al.20 proposed 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, energy exchange or a coupling among the modes.22–24 Xiong et al.25 introduced 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 configurations hinder the potential for wide applications because piezoceramics have low fatigue strength in this mode The amount of energy generated by most bending-mode piezoelectric energy harvesters 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 amplitudes 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 generated 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 advantage 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.32 More 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 proposes 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 vibration 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 capabilities Additionally, a parametric study is carried out to obtain the harvester’s voltage response under 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 presents 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 longitudinal deformation are represented by w2(x,t) and u2(x,t) Furthermore, u1(L 1,t) and w1(L 1,t) are used to describe the vibration of the proof mass M in the x axis and y axis respectively 085310-3 Li et al AIP Advances 6, 085310 (2016) FIG Schematic of proposed energy harvester The Lagrangian of the system is expressed as L = T − U + We , where T is the kinetic energy, U is the potential energy and We is the electrical and electromechanical energy.33 T = T1 + T2 + T3 + TM 
L1 =2× m1 (w˙ + z˙)2 + u˙12 dx  L2
(u˙2 + z˙)2 + w˙ 22 d y + × m2 (1) + Mp (w˙ 1(L 1,t) + z˙)2
+ × M (w˙ 1(L 1,t) + z˙)2 + u˙1(L 1,t)2 where the overdot represents the derivative with respect to time, m1 and m2 are the mass per unit length of elastic beam and the bow-shaped beam, respectively m1 = ρ1 A1 [H(x) − H(x − (L − L b ))] + 2ρ1 A1 [H(x − (L − L b )) − H(x − L 1)] , m2 = ρ2 A2 (2) where H(x) is the Heaviside function to describe the varied cross section process L b are the length of the double cross section for connecting the proof mass ρi is mass density and Ai is cross section area; where ‘i’ denotes the member group of harvester Mp is the mass of the piezoelectric plate For a fixed-fixed slender beam as shown in Fig 2, when transversal deflection is large, stretching becomes important components As a result, the stretching component of fixed-fixed beam is FIG Double clamped beam with a point load at the center (a) The center deflection; (b) The expanded view of original neutral axis 085310-4 Li et al AIP Advances 6, 085310 (2016) expressed by ds =  [dx + u(x + dx) − u(x)]2 + [w(x + dx) − w(x)]2 (3) √ Employing the Taylor series ( + δ ≈ + δ2 ), we express the axial strain as εx = ( )2 ds − dx du dw = + dx dx dx (4) In this study, the longitudinal motion is trivial compared to the transversal one.34,35 Thus the strain in the longitudinal direction can be simplified as εx = L  L ( )2 dw dx dx (5) As shown in Fig 3, the longitude deformation along x axis is different from the clamped-guide case.36 The end of the elastic beam at point G still remains parallel to the x axis after the deformation Therefore, we express the elastic deformation of the elastic beams as ∆x Accordingly, ∆x is denoted as elastic deformation of the bow-shape beam in the flex-compressive center w2(L 2/2,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  L ′2 ∆x + ∆x = w1 dx, E1 A1 ∂ ((h0 + ∆x 2)s( y)) (∆x 1) = E2 I2 L1 ∂ y4 ( )2  E2 A2 ∂ ((h0 + ∆x 2)s( y)) L2 ∂ ((∆x 2)s( y)) − d y 2L ∂y ∂ y2 (6) FIG Geometric relationship of ∆x and ∆x (a) Flex-compressive center model, (b) Clamped guide model, (c) Magnification of ∆x and ∆x 2, (d) Thickness of beams at the joints and (e) Shape function of s(y) 085310-5 Li et al AIP Advances 6, 085310 (2016) The potential energy of the system is U = U1 + U2 + U3 + UM   L1 E1 A1 (∆x 1)2 E1 I1 (w ′′1) dx + =  2L    L1  L2  + m1gw1dx +  E2 I2 w2′′ d y  0  E2 A2 (2u2 (L 2/2,t))) + m2 L 2gw1(L 1,t) + 2L  * + MP gw1(L 1,t) + σ p y ε p y dv p y ,v p y  + + σ p x ε p x dv p x // + 2Mgw1(L 1,t) vp x - (7) where the prime indicates the differentiation with respect to the length coordinates E1 I1 and E2 I2 represent the flexural rigidity of the elastic beam and bow-shaped beam They can be expressed as E1b1 h13 [H(x) − H(x − (L − L b ))] 12
E1b1 h f (8) [H(x − (L − L b )) − H(x − L 1)] , + 12 E2b2 h23 E2 I2 = 12 where h f is the thickness of the fixing joints near to the proof mass σ p y , ε p y , σ p x and ε p x are the mechanical stress and strain of the piezoelectric plate, which comply with the linear constitutive relations33,36,38 E1 I1 = 1 σ p y − d 31 E x , εpx = σ p x − d 33 E x , E3 E3 = −d 31σ p y + e33 E x , D p x = −d 33σ p x + e33 E x εpy = Dp y (9) where E x and D p are the electrical field strength and electrical displacements, v p is the volume of the piezoelectric ceramics, the subscript x and y denote the direction along x and y axis respectively d 31 and d 33 are the piezoelectric constant, ε 33 is 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 (d 31), and the edges where are bonded to the bow-shaped beam experience compressive loads parallel to the poling direction (d 33)   We = (E x D p x )dv p x + (E x D p y )dv p y vp x  =2 vp y (−d 33σ x + ε 33 E x )E x dv p x v p x + (−d 31σ y + ε 33 E x )E x dv p y vp y V d 33 V ( )E1 A1∆x 1v p x +2ε 33( )2v p x A p x h3 h3  L2 d 31 V (∆x s( y))′ d y( )v p y − E2 A2 Ap y L2 h V + ε 33( ) v p y h3 = −2 (10) 085310-6 Li et al AIP Advances 6, 085310 (2016) where A p is 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,  L1  L2 1 w˙ 12dx + c2 w˙ 22d y + δQ (11) δW = c1 2 0 where c1 and c2 are the damping coefficients of elastic beam and bow-shaped beam, respectively; Q is 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) = ∞  qn(t)ψn(x),  − α1 x ′ u1(x,t) = w (x,t)dx, 0 L α1 w2( y,t) = s( y) w ′12dx, 2 y ′2 u2( y,t) = w2 ( y,t)d y n=1 (12) L where α1 = ∆x 2/ 12 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 (  )  πx ψ1 = − cos (13) 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 (  )  2π y s( y) = − cos (14) L2 By Lagrange’s equation, choosing q and V as the generalized coordinates, we can derive the dynamical equation by ( ) d ∂L ∂L δW − = , dt ( ∂ q˙ ) ∂q ∂q (15) d ∂L ∂L δW − = , dt ∂ V˙ ∂V ∂V Hence, Eq (15) gives the two-order nonlinear differential equations

me qă + 2qq ă + 12q2q + 3qq z + 4q3 ză + 3(2qq ă + 4q2q) + ză + k 1q + k2q3 + k 3q7+ χ1q3V + χ2qV + λ + µ1q˙ + µ2q2q˙ = 0, χ1q3q˙ + χ2q q˙ + 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; k1, k2 and k2 are the linear and nonlinear stiffness, 085310-7 Li et al AIP Advances 6, 085310 (2016) TABLE I Model parameters used for numerical test Description Length of elastic beam Width of elastic beam Thickness of elastic beam Density of elastic beam Young’s modulus of elastic beam Young’s modulus of bow beam Length of bow beam Width of elastic beam Thickness of elastic beam Young’s modulus of piezoelectric Damping coefficient Damping coefficient Coupling coefficient Coupling coefficient Electrical permittivity The volume of PZT-5H Gravity acceration Symbol Value L1 b1 h1 ρ1 E1 E2 L2 b2 h2 E3 c1 c2 d 31 d 33 ϵ 33 vp g 0.05 m 0.0005 m 0.015 m 7800 kg/m3 69 Gpa 69 Gpa 0.04 m 0.0005 m 0.0005 m 63 Gpa 6.5 Nsm−1 90 Nsm−1 −285 × 10−12 CN−1 480 × 10−12 CN−1 × 108 FN−1 32 × 15 × 0.7mm3 9.81ms−2 respectively; χ1 and χ2 are 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 in Appendix Obviously, this distributed model is different to the lumped model in the existing works.29–31 The nonlinear restoring force can be described by a polynomial of degree 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 (a) Restoring force; (b) Potential energy 085310-8 Li et al AIP Advances 6, 085310 (2016) FIG 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 = 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 Matlab) for numerical simulation The base excitation is set as harmonic motion ză(t) = a cos(2 f t), where a is the amplitude and f is the excitation frequency in unit of Hz Fig shows 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 clearly 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 Basins 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 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 PL respectively 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 085310-9 Li et al AIP Advances 6, 085310 (2016) FIG 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 Experimental setup (a) signal generator and amplifier; (b)Shaker and Dropper micrometer; (c) HC-PEH; (d) Oscilloscope 085310-10 Li et al AIP Advances 6, 085310 (2016) using an oscilloscope (Tektronix 3014) and a laser Doppler micrometer(Polytec Inc OFV-534), respectively To demonstrate its improved functionalities, we study the relation between the load resistance ( ) and the generated power, in which Vpeak R are used for calculating the peak power for different resistances As shown in Fig 8(a), when the resistance equals 100 kΩ, the generated power reaches the peak This resistance is close to the resistance calculated by: R= Ce f (17) where Ce is the capacitance of the piezoelectric layer and f the first-order resonant frequency For our structure, the internal capacitance of the piezoelectric plate is 30 nF The validity of the theoretical model is further verified in Figs 8(b), 8(c), 8(d) where the experimental voltage-frequency matches the numerical voltage-frequency responses quite well for the resistance of 100 kΩ, MΩ and 10 MΩ Fig shows the comparisons between the experimental and numerical voltage-frequency response results under R=100 kΩ conditions In both forward and backward frequency sweeping experiments, hardening nonlinearity is observed with distinct jump phenomena and large amplitude in voltage response The results of experiment are in good agreement with the numerical simulation for different base excitation levels It should be noted that the theoretical model predicts the harvesters’ primary nonlinear behaviors quite well, in peak of voltage, jump frequency and bandwidth VI PARAMETRIC STUDY In this section, the parametric studies are carried out to investigate the electrical response of the harvester under base excitation More details of the effect of different parameters such as damping, mechanical coupling, proof mass and length of elastic beams on the harvester’s response will be discussed later FIG Theoretical and experimental voltage and power versus load resistance (a) Peak-peak voltage and power; ((b),(c),(d)) the comparison of theoretical and experimental voltage-frequency response for connecting a resistance of 100 KΩ , MΩ and 10 MΩ 085310-11 Li et al AIP Advances 6, 085310 (2016) FIG Comparisons between experimental and numerical voltage-frequency responses results under 100 KΩ conditions ((a),(b)) Forward and backward sweep simulation; ((c),(d)) Forward and backward sweep experiment Fig 10 depicts the effect of mechanical damping on the response with 100 kΩ resistance-load A comparison between the figures reveals that the response is greatly influenced by c1 but slightly influenced by c2 Taking the forward frequency sweep for instance, one can observe from Fig 10 that the voltage response is largely influenced by c1, as can be seen from Figs 10(a) and 10(b) FIG 10 Numerical results of voltage response with 100 resistance-load condition in forward sweep for different damping coefficients: ((a),(b)) c ; ((c),(d)) c The excitation level is set as 0.2 g 085310-12 Li et al AIP Advances 6, 085310 (2016) where the increase of c1 (from to 3, and Nsm−1) in contrast with a decrease in the peak-peak voltage amplitude (from 222.98 to 142.84, 98, 74.5 V) and the half-power bandwidth (from 2.8875 to 2.13, 1.7088, 1.53 Hz) However, we can observe from Figs 10(c) and 10(d) that the great change of c2 from 1000 to 7000 Nsm−1 barely alters the peak-peak voltage (86.82 V) and the half-power bandwidth (1.67 Hz) It is noted that a large damping tending to alleviate the jumping phenomenon, reduces the amplitude of voltage response and decreases the frequency where the maximum voltage occurs This comparison also indicates that the energy loss of the system is main due to the viscous damping of elastic beams, rather than the viscous damping of bow-shape beam Fig 11 shows the effects of different coupling coefficients on the voltage response with R=100 KΩ load resistor From the results of forward sweeping excitation, we can see that the increases of d 31 and d 33 will decrease the critical frequency where the maximum voltage occurs, and the voltage response is considerably influenced by d 31 As shown in Fig 11(a), 11(b) as d 31 increase from −400 × 10−12CN−1 to −160 × 10−12CN−1, the peak-peak voltage increases from 116 V to 174 V However, we can observe from Figs 11(c), 11(d) that the change of d 33 (from 200 × 10−12CN−1 to 760 × 10−12CN−1) doesnt make a considerable change in peak-peak voltage These data in the figures reveal that the voltage generated from the piezoelectric material mainly comes from the d 31 mode rather than the d 33 mode The effect of length of the elastic beam on the output voltage with 100 KΩ resistance-load condition is illustrated in Fig 12 When other parameters remain constant, along with increases the length of elastic beams come an increase in the peak-peak voltage This phenomenon is believed to be caused by the bending and tensile effects When the length of elastic beams is fairly short, the bending effect becomes considerable As the length of elastic beams reaches a certain level, both bending and tensile result in the increase of stress in the piezoelectric plate Figs 12(c) and 12(d) show simulations of the frequency sweep results of the output voltage responses for L = 0.02 m and L = 0.06 m respectively It is shown that a longer elastic beam results in a lower resonant frequency, thus decreasing the frequency where the peak voltage occurs Fig 13 depicts the effect of proof mass M1 on the voltage response with 100 KΩ resistanceload condition It may be observed that the peak-peak voltage increases from 60 V to 135 V, and the half-power bandwidth extends from 2.5 Hz to 4.25 Hz, when the proof mass increases from FIG 11 The effect of the electromechanical coupling coefficients ((a),(b)) d 31; ((c),(d)) d 33 The excitation level is set as 0.5 g 085310-13 Li et al AIP Advances 6, 085310 (2016) FIG 12 The effect of length of elastic beams on the output voltage with 100 KΩ resistance-load condition (a) Peak- peak voltage; (b) Bandwidth; (c)L = 0.02 m; (d)L =0.06 m The excitation level is set as 0.3g 0.02 Kg to 0.09 Kg It is shown that a heavier proof mass results in a stronger nonlinearity and lower natural frequency, thus intensifying the jumping phenomenon and increasing the voltage amplitude at low-frequency excitation Unfortunately, the heavier proof mass will decrease the power density normalized to mass39 and increase the possibility of material fatigue As a result, the harvesting efficiency and the longevity tend to decrease when the proof mass beyonds a certain level FIG 13 The effect of proof mass on the output voltage with 100 KΩ resistance-load condition (a) Peak-peak voltage; (b) Bandwidth; (c) sweeping results for M1 =0.04 Kg; (d) sweeping results for M1 =0.07 Kg The excitation level is set as 0.3g 085310-14 Li et al AIP Advances 6, 085310 (2016) FIG 14 The effects of materials of elastic beam on the output response with 100 KΩ resistance-load condition (a) Peak-peak voltage; (b) Bandwidth Fig 14 demonstrates the peak-peak voltage and half-power bandwidth for a range of acceleration and materials It is shown that as the base excitation increase from 0.2 g to 0.5 g, the peak-peak voltage of the Aluminum beam increase by 140V, which is higher than that of the copper beam and the steel beam Additionally, by comparing the bandwidth, one can find that a increase of base excitation extends the bandwidth of the aluminum beam from 1.5 Hz to 4.5 Hz, which outperforms other counterparts This phenomenon is considered to be caused by the varied material properties, such as Young’s modulus, density and material damping coefficient VII SUMMARY AND CONCLUSIONS This paper established a distributed parameter model for the compressive-mode energy harvester (HC-PEH) that synthesizes the structural nonlinearity and amplification effect The coupling equations are derived by the extended Hamiltion’s principle The governing equations are solved by numerical method for sweeping excitations Numerical simulations successfully predict the harvester’s nonlinear behaviors, including hardening type of nonlinearity, large-amplitude voltage output and broad frequency bandwidth From the basins of attraction, multiple coexisting solutions phenomenon is observed, and it demonstrates that the nonlinear response behavior of harvester is shown to be dependent on the initial condition The validation experiments were performed The experimental results are in good agreement with the simulation’s results in peak-peak voltage, jump frequency and bandwidth for various resistances The influences of parameters on the performance of the HC-PEH are studied The results reveal that the HC-PEH can attain the best performance by optimizing the mechanical and the electrical parameters ACKNOWLEDGMENT This work is financially supported by the Natural Science Foundation of China (Grant No 11172234), Natural Science and Engineering Research Council of Canada and the scholarship from China Scholarship Council (Grant No 201506290092) APPENDIX The coefficient of governing equations are given as following ) (  L1 m e = m1 ψ12dx + M1 + MP ,   y m2 L2 [(1 − α1) β1 = 0  ′2  L1 (ψ ′1(x))2dxs ( y)]′2 d y d y 085310-15 Li et al AIP Advances 6, 085310 (2016)  L2   y [(1 − α1) β2 = m2   L1 ′2 ′ (ψ 1(x)) dxs ( y)] d y d y, )2  L1 (  x ′ β3 = m1 (α1) (ψ 1(x)) dx dx 0 ( )2  L1 + M1 (α1) (ψ ′1(x))2dx )2  L2 (  L1 (ψ ′1(x))2dxs( y) d y, + m2 (1 − α1) ) (  L1 γ = m1 ψ1dx + M1 + MP , ) (  L1 (ψ ′′1(x))2dx , k = E1 I1 ( L ) ′′2  L2  (1 − α1) k =  E2 I2 (ψ ′1(x))2dxs( y) d y  0 ( )2  L1 α1 ′ + E1 A1 (ψ 1(x)) dx 2L ( )   L1 E1 A1 α1 ′ + (ψ 1(x)) dx v p x A p x 2L   L ( y ( E2 A2 (1 − α1) k3 = 2L 2 0 ) ′2  L1  (ψ ′1(x))2dxs( y) d y + d y     L ( -  (1 − α1) E2 A2 +2 A pz 2L 2 ) ′2     L1     (ψ ′1(x))2dxs( y) d y   d y       E2 A2 V χ1 = d 31 2L A pz h3 ) ′2   L1  L2 (  (1 − α1) ′ ψ 1(x) dxs( y) d y  v pz ,  0 ( )   L1 E1 A1 V χ2 = d 33 (α1) (ψ ′1(x))2dx v p x L1 Ap x h3 λ = (2M1 + MP ) g  L1 + 2m1g ψ1(x)dx + 2m2 L 2g, Energy Harvesting Technologies, edited by S Priya and D J Inman (Srpinger, New York, 2009) 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nonlinearity Furthermore, the analytical model is validated against with experimental data and a good agreement is achieved Both numerical simulations and experiment illustrate that the harvester. .. (a) Peak-peak voltage; (b) Bandwidth Fig 14 demonstrates the peak-peak voltage and half-power bandwidth for a range of acceleration and materials It is shown that as the base excitation increase