Modeling and power performance improvement of a piezoelectric energy harvester for low?frequency vibration environments

Dongxing Cao · Yanhui Gao · Wenhua Hu

Abstract A novel oscillator structure consisting of a bimorph piezoelectric cantilever beam with two steps of different thicknesses is proposed to improve the energy harvesting performance of a vibration energy harvester (VEH) for use in low-frequency vibration environments. Firstly, the piezoelectric cantilever is segmented to obtain the energy functions based on the Euler–Bernoulli beam assumptions, the n the Galerkin approach is utilized to discretize the energy functions. Applying boundary conditions and continuity conditions enforced at separation locations, the coupled electromechanical equations governing the piezoelectric energy harvester are introduced by means of the Lagrange equations. Furthermore, expressions for the steady-state response are obtained for harmonic base excitations at arbitrary frequencies. Numerical results are computed, and the effects of the ratio of lengths, ratio of thicknesses, end thickness, and load resistance on the output voltage, harvested power, and power density are discussed. Moreover, to verify the analytical results, finite element method simulations are also conducted to analyze the performance of the proposed VEH, showing good agreement. A ll the results show that the present oscillator structure is more efficient than the conventional, uniform beam structure, specifically for vibration energy harvesting in low-frequency environments.

Keywords Vibration energy harvesting · Piezoelectric cantilever beam · Stepped variable thicknesses · Finite element method simulation

1 Introduction

with the rapid development of integrated circuits, the size and power consumption of electronic devices have reduced dramatically [1, 2], making it possible to power devices using vibration energy harvesting techniques without an external power source. Over the last decade, energy harvesting from vibrating mechanical structures has been studied by several researchers [3–5]. Various transduction mechanisms have been reported for vibration energy harvesting, including electrostatic [6–8], electromagnetic [9–11], piezoelectric[4, 12], and magnetostrictive [13, 14] mechanisms, as well as the use of electronic and ionic electroactive polymers [15,16] or polymer electrets [17], and even flexoelectricity for energy harvesting [18, 19]. Among the basic transduction mechanisms that can be used for vibration-to-electricity conversion, piezoelectric transduction has received the most attention due to the high power density and ease of application of piezoelectric materials. Many researchers have focused their work on modeling and applications of piezoelectric energy harvesters in vibration environments.Various kinds of piezoelectric vibration energy harvester(VEH) were investigated by Erturk et al. [20–23] using analytical methods with experimental validation based on nonlinear dynamic theory. A systematic comparison between VEHs using Pb(M g1/3Nb2/3)O3-PbTiO3(PMN-PT) or Pb(Zn1/3Nb2/3)O3-PbTiO3(PZN-PT) single crystals and lead zirconate titanate (PZT) ceramics was presented by Yang and Zu [24]. A novel design for a rotational power scavenging system was presented by Febbo et al. [25] as an alternative to cantilever beams attached to a hub. Many studied on piezoelectric VEHs have been reported in Refs. [26–29].It is worth mentioning that Chen and Jiang [30, 31] also proposed use of the internal resonance of a nonlinear system to enhance VEH performance. The concept of nonlinear internal resonance was also introduced by Cao et al. [23]to study broadb and energy harvesting using an L-shaped beam–mass structure with quadratic nonlinearity. A large number of other studies have been devoted to improving the energy harvesting performance of piezoelectric VEHs, but we cannot list all the achievements here.

Note that most vibration oscillators in VEHs are currently uniform structures, for instance, uniform beam and plate structures. However, the re is no reason why the geometry should be lim ited to traditional, uniform configurations. In fact, use of nonuniform structures could increase the coupling performance, and energy harvesters with alternative geometries have been shown to be of interest. It was proposed by Baker and Roundy [32] that varying the width(trapezoidal shape) of a beam can increase the efficiency.Literature studies investigating the strain distribution in cantilever beams with various shapes, such as rectangular, triangular, and trapezoidal geometries, have revealed that use of a triangular cantilever beam can improve the strain distribution and generate more voltage compared with a rectangular beam under the same conditions [33–35]. An innovative design platform for a segment-type piezoelectric energy harvester was presented by Lee et al. [36]. A bimorph piezoelectric beam with periodically variable cross-sections was presented by Hajhosseini and Rafeeyan [37], offering three advantages over a uniform piezoelectric beam, i.e., greater voltage output over a wide frequency range, enhanced vibration absorption, and lower weight. A harvester based on a propped cantilever beam with variable overhangs having step sections was examined by Usharani et al. [38]. Flexible longitudinal zigzag energy harvesters were studied by Zhou et al. [39] with the aim of enhancing energy harvesting from low-frequency low-amplitude excitations.

In the work presented herein, a bimorph piezoelectric cantilever beam with two steps with different thicknesses is proposed for high-output vibration energy harvesting.The remainder of this manuscript is organized as follows.Section 2 describes the structural model of the proposed VEH and the basic assumptions. Theoretical modeling of the VEH is the n established, and expressions for the steady-state response for harmonic base excitations at arbitrary frequencies are derived in Sect. 3. Numerical results based on the theoretical analysis are obtained and discussed in detail in Sect. 4. In Sect. 5, finite element method (FEM) simulations are conducted to validate the results of the theoretical analysis, where the effects of geometric parameters on the natural frequency, output voltage, harvested power, and power density of the harvester are analyzed and discussed. Finally, the main conclusions are drawn in Sect. 6.

Fig. 1 Structural model of piezoelectric energy harvester

2 Structural model and basic assumptions

A piezoelectric cantilever beam with two steps with different thicknesses is considered for vibration energy harvesting,being composed of two segments with different thicknesses and a tip segment, as shown in Fig. 1. A base excitationof the clamping mechanism is used to simulate environmental excitation.L is the total length of the beam,L1is the length of the first segment of the beam, and L2?L1is the length of the second segment of the beam.Hi(i=1,2,3)are the thicknesses of the first, second, and third segments of the substructure layer, respectively.Hpis the thickness of the piezoceramic layers, and the width of the whole structure is B. In Fig. 1, the x-, y- (perpendicular to the paper and pointing into it), and z-directions, respectively, are coincident with the 1-, 2-, and 3-directions of piezoelectricity, the former being preferred for mechanical derivations, whereas the latter are used in the piezoelectric constitutive relations.The piezoceramic layers of the bimorph are poled oppositely in the z-direction, so the configuration represents series connection of the piezoceramic layers. The output terminals of the electrodes of the first two segments of the piezoelectric beam are connected directly to load resistors RL1and RL2,respectively.

Before deriving the coupled electromechanical equation governing the VEH with two steps of different thicknesses,the following assumptions are introduced: (a) each segment of the piezoelectric cantilever beam is considered to be an Euler–Bernoulli beam; (b) the influence of the bonding layer is neglected, i.e., the piezoceramic layers and the substructure layer are ideally bonded, and the displacement and force on the bonding layer are continuous; (c) the electrode coated on the upper and lower surfaces of the piezoceramic layers is very thin compared with the total thickness of the harvester,so their contribution to the thickness dimension is negligible; (d) the piezoceramic layers produce an electric field perpendicular to the beam surface and distributed uniform ly along the thickness direction.

3 Theoretical modeling

3.1 Distributed?parameter electromechanical energy formulation

Based on assumption (a), the axial strain can be expressed as Eq. (1), where w(x,t)is the transverse displacement of the beam at point x and time t relative to the moving base

where the prime notation()is shorthand for α∕αx, and z is the position from the neutral axis of the piezoelectric cantilever beam.

The isotropic substructure layer obeys Hooke’s law

where Txx,Sxx, and Esare the axial stress, axial strain, and elastic modulus of the substructure layer, respectively.

Due to the transverse vibration of the piezoelectric cantilever beam system, the piezoelectric effect of the piezoelectric material is considered. The constitutive equations of the piezoceramic layers can be expressed as

where T1and S1are the axial stress and axial strain of the piezoceramic layers, respectively,is the elastic modulus under constant electric field,e31is the piezoelectric coupling coefficient, and E3and D3are the electric field strength and electric displacement in the z-direction, respectively.σSis

33 the permittivity under constant strain.

Based on assumption (d), the electric field distribution in the VEH with two steps of different thicknesses in series can be expressed as

where vR1(t) and vR2(t)are the voltages across the load resistances RL1and RL2, respectively.

The Lagrange equations are employed to establish the coupled electromechanical equations governing the piezoelectric cantilever beam with two steps of different thicknesses. The Lagrange function for the system can be expressed as

where T , U, and Weare the total kinetic energy, internal potential energy, and electrical energy of the system, respectively. The specific expressions for the various energies of the system are introduced as follows.

3.1.1 Kinetic energy

The kinetic energy T of the system is the sum of the kinetic energy of the substructure layer (Ts) and the piezoceramic layers (Tp) and can be written as

where an overdot indicates a derivative with respect to time t. When 0?x?L1,i=1,j=1,2; when L1?x?L2,i=2,j=3,4; when L2?x?L,j=3.Vsiis the volume of the i-th segment of the substructure layer, and Vpjis the volume of the j-th piezoceramic layer.psis the density of the substructure layer, and ppis the density of the piezoceramic layers.w0i(x,t)represents the absolute transverse displacement of the i-th segment at point x and time t.

The kinetic energy Tsof the substructure layer can be expressed as

where Asi=BHiis the cross-sectional area of the substructure layer for the i th segment of the beam.is the vibration velocity of the base.

The kinetic energy Tpof the piezoceramic layers can be expressed as

where Api=BHpis the cross-sectional area of the piezoceramic layers for the i-th segment of the beam.

3.1.2 Internal potential energy

The internal potential energy of the harvester can be defined as

Substituting Eqs. (2) and (3) into Eq. (10), the potential energy can be written as

where Usand Upsonly depend on the strain of the substructure layer and piezoceramic layers, respectively, while Upedepends on both the strain of the piezoceramic layers and the electric field.

Substituting Eqs. (1) and (5) into Eq. (11),Uscan be expressed as

Upsis given by

where Ep=is the ( elastic modulus of t)h/e piezoceramic layers, and Ipi=12 is the area moment of inertia of the piezoceramic layers for the i-th segment of the beam.

Upeis given by

3.1.3 Electrical energy

The electrical energy of the harvester is defined as

Substituting Eq. (4) into Eq. (15),Wecan be w ritten as

where Wpe1depends on both the strain and the electric field,while Wpe2only depends on the electric field.

Substituting Eqs. (1) and (5) into Eq. (16),Wpe1can be expressed as

Wpe2is given by

3.2 Spatial discretization of the energy equations

The Galerkin method is utilized to discretize the Lagrange function. The transverse displacement w(x,t) of the piezoelectric cantilever beam can be written as

where φm(x) and qm(t)are the unknown mode shape and generalized modal coordinate of the m th mode, respectively.

Due to the two steps with different thicknesses, the piezoelectric cantilever beam is a discontinuous laminated beam,with varying material and geometric characteristics. Therefore, the mode shape function of each segment is different, and piecewise calculation is required. The mode shape functions of the m th mode can be written as

where φm1(x),φm2(x), and φm3(x)can be expressed as

The coefficients in the mode shape function are determined by the boundary conditions and continuity conditions.The boundary conditions at x=0 and x=L can be written as

The continuity conditions at x=L1can be written as

The continuity conditions at x=L2can be written as

where EIiis the flexural stiffness at segment i of the beam,EIi=EsIsi+2EpIpi(i=1,2), and EI3=EsIs3.

The modal frequency of each segment of the beam is consistent, so the following equation applies:

where pAiis the mass of the i-th segment of the beam,pAi= psAsi+2ppApi(i=1,2), and pA3= psAs3.

Substituting Eq. (20) into Eqs. (24)–(26), homogeneous linear equations can be obtained as

To further standardize the mode shapes, the following orthogonality conditions are introduced:

where is the Kronecker delta.

When the piezoelectric cantilever beam with two steps of different thicknesses is operated in a low-frequency vibration environment, because of the sparsity of the structural modes,the first mode is often closer to the excitation frequency,playing the leading role in the displacement response of the structure, which can thus be simplified to

3.3 Coup led electromechanical equations for the VEH

Substituting Eq. (30) into Eq. (6) and using the orthogonality conditions in Eq. (29), the reduced Lagrange function can be obtained as

where w1is the first natural frequency of the piezoelectric cantilever beam,?1and ?2are the model electromechanical coupling coefficients,is the model base excitation coefficient,Cp1and Cp2are equivalent capacitances, and M is the total mass of the piezoelectric cantilever beam. The se coefficients are given by

Substituting Eq. (31) into the Lagrange equations yields

where F1(t)is the generalized dissipative force, and QR1(t) and QR2(t)are the generalized output charges across the load resistances RL1and RL2, respectively. Considering the energy function of the generalized dissipative force as the Rayleigh function yields F1(t)=?2