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

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, then 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. All 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.

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 [3,4,5]. Various transduction mechanisms have been reported for vibration energy harvesting, including electrostatic [6,7,8], electromagnetic [9,10,11], 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. [20,21,22,23] using analytical methods with experimental validation based on nonlinear dynamic theory. A systematic comparison between VEHs using Pb(Mg_1/3Nb_2/3)O₃-PbTiO₃ (PMN-PT) or Pb(Zn_1/3Nb_2/3)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. [26,27,28,29]. 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 broadband 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, there is no reason why the geometry should be limited 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 [33,34,35]. 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 then 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.

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 excitation $\ddot{z}_{\text{b}} (t)$ of the clamping mechanism is used to simulate environmental excitation. $L$ is the total length of the beam, $L_{1}$ is the length of the first segment of the beam, and $L_{2} - L_{1}$ is the length of the second segment of the beam. $H_{i} {\kern 1pt} {\kern 1pt} (i = 1,{\kern 1pt} {\kern 1pt} 2,{\kern 1pt} {\kern 1pt} 3)$ are the thicknesses of the first, second, and third segments of the substructure layer, respectively. $H_{\text{p}}$ is 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 $R_{L1}$ and $R_{L2}$, respectively.

Fig. 1

Structural model of piezoelectric energy harvester

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 uniformly 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

$$ S_{xx} \left( {x,z,t} \right) = - zw^{\prime\prime}\left( {x,t} \right), $$

(1)

where the prime notation $(^{\prime } )$ is shorthand for ${\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-0pt} {\partial x}}$, and z is the position from the neutral axis of the piezoelectric cantilever beam.

The isotropic substructure layer obeys Hooke’s law

$$ T_{xx} = E_{\text{s}} S_{xx} , $$

(2)

where $T_{xx}$, $S_{xx}$, and $E_{\text{s}}$ are 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

$$ T_{1} = c_{11}^{\text{E}} S_{1} - e_{31} E_{3} , $$

(3)

$$ D_{3} = e_{31} S_{1} + \varepsilon_{33}^{\text{S}} E_{3} , $$

(4)

where $T_{1}$ and $S_{1}$ are the axial stress and axial strain of the piezoceramic layers, respectively, $c_{11}^{\text{E}}$ is the elastic modulus under constant electric field, $e_{31}$ is the piezoelectric coupling coefficient, and $E_{3}$ and $D_{3}$ are the electric field strength and electric displacement in the z-direction, respectively. $\varepsilon_{33}^{\text{S}}$ is 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

$$ E_{3} = \left\{ {\begin{array}{*{20}l} { - \frac{{v_{R1} \left( t \right)}}{{2H_{\text{p}} }},\quad 0 \leqslant x \leqslant L_{1} ,} \hfill \\ { - \frac{{v_{R2} \left( t \right)}}{{2H_{\text{p}} }},\quad L_{1} \leqslant x \leqslant L_{2} ,} \hfill \\ \end{array} } \right. $$

(5)

where $v_{R1} \left( t \right)$ and $v_{R2} \left( t \right)$ are the voltages across the load resistances $R_{L1}$ and $R_{L2}$, 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

$$ \ell \left( {x,t} \right) = T + W_{\text{e}} - U, $$

(6)

where $T$, U, and $W_{\text{e}}$ are 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 ($T_{\text{s}}$) and the piezoceramic layers ($T_{\text{p}}$) and can be written as

$$ T = \frac{1}{2}\sum\limits_{i = 1}^{3} {\int_{{V_{{{\text{s}}i}} }} {\rho_{\text{s}} \dot{w}^{2}_{0i} \left( {x,t} \right){\text{d}}V_{{{\text{s}}i}} } } + \frac{1}{2}\sum\limits_{j = 1}^{4} {\sum\limits_{i = 1}^{2} {\int_{{V_{{{\text{p}}j}} }} {\rho_{\text{p}} \dot{w}^{2}_{0i} \left( {x,t} \right){\text{d}}V_{{{\text{p}}j}} } } } = T_{\text{s}} + T_{\text{p}} , $$

(7)

where an overdot indicates a derivative with respect to time t. When $0 \leqslant x \leqslant L_{1}$, ${\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} j = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} 2$; when $L_{1} \leqslant x \leqslant L_{2}$, ${\kern 1pt} i = 2,{\kern 1pt} {\kern 1pt} j = 3,{\kern 1pt} {\kern 1pt} {\kern 1pt} 4$; when $L_{2} \leqslant x \leqslant L$, ${\kern 1pt} j = 3$. $V_{{{\text{s}}i}}$ is the volume of the i-th segment of the substructure layer, and $V_{{{\text{p}}j}}$ is the volume of the j-th piezoceramic layer. $\rho_{\text{s}}$ is the density of the substructure layer, and $\rho_{\text{p}}$ is the density of the piezoceramic layers. $w_{0i} \left( {x,t} \right)$ represents the absolute transverse displacement of the i-th segment at point x and time t.

The kinetic energy $T_{\text{s}}$ of the substructure layer can be expressed as

$$ T_{\text{s}} = \frac{1}{2}\rho_{\text{s}} A_{s1} \int_{0}^{{L_{1} }} {\left[ {\dot{w}_{1} \left( {x,t} \right) + \dot{z}_{\text{b}} \left( t \right)} \right]^{2} } {\text{d}}x + \frac{1}{2}\rho_{\text{s}} A_{{{\text{s}}2}} \int_{{L_{1} }}^{{L_{2} }} {\left[ {\dot{w}_{2} \left( {x,t} \right) + \dot{z}_{\text{b}} \left( t \right)} \right]^{2} } {\text{d}}x + \frac{1}{2}\rho_{\text{s}} A_{{{\text{s}}3}} \int_{{L_{2} }}^{L} {\left[ {\dot{w}_{3} \left( {x,t} \right) + \dot{z}_{\text{b}} \left( t \right)} \right]^{2} } {\text{d}}x, $$

(8)

where $A_{{{\text{s}}i}} = BH_{i}$ is the cross-sectional area of the substructure layer for the ith segment of the beam. $\dot{z}_{\text{b}} \left( t \right)$ is the vibration velocity of the base.

The kinetic energy $T_{\text{p}}$ of the piezoceramic layers can be expressed as

$$ T_{\text{p}} = \rho_{\text{p}} A_{{{\text{p}}1}} \int_{0}^{{L_{1} }} {\left[ {\dot{w}_{1} \left( {x,t} \right) + \dot{z}_{\text{b}} \left( t \right)} \right]^{2} } {\text{d}}x{ + }\rho_{\text{p}} A_{{{\text{p}}2}} \int_{{L_{1} }}^{{L_{2} }} {\left[ {\dot{w}_{2} \left( {x,t} \right) + \dot{z}_{\text{b}} \left( t \right)} \right]^{2} } {\text{d}}x, $$

(9)

where $A_{{{\text{p}}i}} = BH_{\text{p}}$ is 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

$$ U = \frac{1}{2}\sum\limits_{i = 1}^{3} {\int_{{V_{{{\text{s}}i}} }} {T_{xx} S_{xxi} } {\text{d}}V_{{{\text{s}}i}} } + \frac{1}{2}\sum\limits_{j = 1}^{4} {\int_{{V_{{{\text{p}}j}} }} {T_{1} S_{1} } {\text{d}}V_{{{\text{p}}j}} } . $$

(10)

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

$$ U = \frac{1}{2}\sum\limits_{i = 1}^{3} {\int_{{V_{{{\text{s}}i}} }} {E_{\text{s}} S_{xxi}^{2} } {\text{d}}V_{{{\text{s}}i}} } + \frac{1}{2}\sum\limits_{j = 1}^{4} {\sum\limits_{i = 1}^{2} {\int_{{V_{{{\text{p}}j}} }} {c_{11}^{\text{E}} S_{xxi}^{2} } {\text{d}}V_{{{\text{p}}j}} } } - \frac{1}{2}\sum\limits_{j = 1}^{4} {\sum\limits_{i = 1}^{2} {\int_{{V_{{{\text{p}}j}} }} {e_{31} S_{xxi} } E_{3j} {\text{d}}V_{{{\text{p}}j}} } } = U_{\text{s}} + U_{\text{ps}} - U_{\text{pe}} , $$

(11)

where $U_{\text{s}}$ and $U_{\text{ps}}$ only depend on the strain of the substructure layer and piezoceramic layers, respectively, while $U_{\text{pe}}$ depends on both the strain of the piezoceramic layers and the electric field.

Substituting Eqs. (1) and (5) into Eq. (11), $U_{\text{s}}$ can be expressed as

$$ \begin{aligned} U_{\text{s}} & = \frac{1}{2}\int_{0}^{{L_{1} }} {\int_{{A_{{{\text{s}}1}} }} {E_{\text{s}} S_{xx1}^{2} {\text{d}}A_{{{\text{s}}1}} {\text{d}}x} } + \frac{1}{2}\int_{{L_{1} }}^{{L_{2} }} {\int_{{A_{{{\text{s}}2}} }} {E_{\text{s}} S_{xx2}^{2} {\text{d}}A_{{{\text{s}}2}} {\text{d}}x} } { + }\frac{1}{2}\int_{{L_{2} }}^{L} {\int_{{A_{{{\text{s}}3}} }} {E_{\text{s}} S_{xx3}^{2} {\text{d}}A_{{{\text{s}}3}} {\text{d}}x} } \\ & = \frac{1}{2}E_{\text{s}} I_{{{\text{s}}1}} \int_{0}^{{L_{1} }} {w_{1}^{\prime \prime 2} \left( {x,t} \right)} {\text{d}}x{ + }\frac{1}{2}E_{\text{s}} I_{{{\text{s}}2}} \int_{{L_{1} }}^{{L_{2} }} {w_{2}^{\prime \prime 2} \left( {x,t} \right)} {\text{d}}x{ + }\frac{1}{2}E_{\text{s}} I_{{{\text{s}}3}} \int_{{L_{2} }}^{L} {w_{3}^{\prime \prime 2} \left( {x,t} \right)} {\text{d}}x, \\ \end{aligned} $$

(12)

where $I_{{{\text{s}}i}} = {{BH_{i}^{3} } \mathord{\left/ {\vphantom {{BH_{i}^{3} } {12}}} \right. \kern-0pt} {12}}$ is the area moment of inertia of the substructure layer at segment i of the beam.

$U_{\text{ps}}$ is given by

$$ \begin{aligned} U_{\text{ps}} & = \frac{1}{2}\int_{0}^{{L_{1} }} {\int_{{H_{1} /2}}^{{H_{1} /2 + H_{p} }} {E_{\text{p}} z^{2} w_{1}^{\prime \prime 2} \left( {x,t} \right)B{\text{d}}z{\text{d}}x} } { + }\frac{1}{2}\int_{0}^{{L_{1} }} {\int_{{ - H_{1} /2 - H_{\text{p}} }}^{{ - H_{1} /2}} {E_{\text{p}} z^{2} w_{1}^{\prime \prime 2} \left( {x,t} \right)B{\text{d}}z{\text{d}}x} } \\ & \quad + \frac{1}{2}\int_{{L_{1} }}^{{L_{2} }} {\int_{{H_{2} /2}}^{{H_{2} /2 + H_{\text{p}} }} {E_{\text{p}} z^{2} w_{2}^{\prime \prime 2} \left( {x,t} \right)B{\text{d}}z{\text{d}}x} } { + }\frac{1}{2}\int_{{L_{1} }}^{{L_{2} }} {\int_{{ - H_{2} /2 - H_{\text{p}} }}^{{ - H_{2} /2}} {E_{\text{p}} z^{2} w_{2}^{\prime \prime 2} \left( {x,t} \right)B{\text{d}}z{\text{d}}x} } \\ & = E_{\text{p}} I_{{{\text{p}}1}} \int_{0}^{{L_{1} }} {w_{1}^{\prime \prime 2} } \left( {x,t} \right){\text{d}}x{ + }E_{\text{p}} I_{{{\text{p}}2}} \int_{{L_{1} }}^{{L_{2} }} {w_{2}^{\prime \prime 2} \left( {x,t} \right)} {\text{d}}x, \\ \end{aligned} $$

(13)

where $E_{\text{p}} = c_{11}^{\text{E}}$ is the elastic modulus of the piezoceramic layers, and $I_{{{\text{p}}i}} = {{BH_{\text{p}} \left( {4H_{\text{p}}^{2} { + }6H_{\text{p}} H_{i} + 3H_{i}^{2} } \right)} \mathord{\left/ {\vphantom {{BH_{\text{p}} \left( {4H_{\text{p}}^{2} { + }6H_{\text{p}} H_{i} + 3H_{i}^{2} } \right)} {12}}} \right. \kern-0pt} {12}}$ is the area moment of inertia of the piezoceramic layers for the i-th segment of the beam.

$U_{\text{pe}}$ is given by

$$ \begin{aligned} U_{\text{pe}} & = \frac{1}{2}\int_{0}^{{L_{1} }} {\int_{{H_{1} /2}}^{{H_{1} /2 + H_{\text{p}} }} {e_{31} zw^{\prime\prime}_{1} \left( {x,t} \right)\frac{{v_{R1} \left( t \right)}}{{2H_{\text{p}} }}B{\text{d}}z{\text{d}}x} } { + }\frac{1}{2}\int_{0}^{{L_{1} }} {\int_{{ - H_{1} /2 - H_{\text{p}} }}^{{ - H_{1} /2}} {e_{31} \left( { - z} \right)w^{\prime\prime}_{1} \left( {x,t} \right)\frac{{v_{R1} \left( t \right)}}{{2H_{\text{p}} }}B{\text{d}}z{\text{d}}x} } \\ & \quad + \frac{1}{2}\int_{{L_{1} }}^{{L_{2} }} {\int_{{H_{2} /2}}^{{H_{2} /2 + H_{\text{p}} }} {e_{31} zw^{\prime\prime}_{2} \left( {x,t} \right)\frac{{v_{R2} \left( t \right)}}{{2H_{\text{p}} }}B{\text{d}}z{\text{d}}x} } { + }\frac{1}{2}\int_{{L_{1} }}^{{L_{2} }} {\int_{{ - H_{2} /2 - H_{\text{p}} }}^{{ - H_{2} /2}} {e_{31} \left( { - z} \right)w^{\prime\prime}_{2} \left( {x,t} \right)\frac{{v_{R2} \left( t \right)}}{{2H_{\text{p}} }}B{\text{d}}z{\text{d}}x} } \\ & = \frac{{H_{1} + H_{\text{p}} }}{4}e_{31} v_{R1} \left( t \right)B\left. {w^{\prime}_{1} \left( {x,t} \right)} \right|_{0}^{{L_{1} }} + \frac{{H_{2} + H_{\text{p}} }}{4}e_{31} v_{R2} \left( t \right)B\left. {w^{\prime}_{2} \left( {x,t} \right)} \right|_{{L_{1} }}^{{L_{2} }} . \\ \end{aligned} $$

(14)

3.1.3 Electrical energy

The electrical energy of the harvester is defined as

$$ W_{\text{e}} = \frac{1}{2}\sum\limits_{j = 1}^{4} {\int_{{V_{{{\text{p}}j}} }} {D_{3} E_{3} } {\text{d}}V_{{{\text{p}}j}} } . $$

(15)

Substituting Eq. (4) into Eq. (15), $W_{\text{e}}$ can be written as

$$ W_{\text{e}} = \frac{1}{2}\sum\limits_{j = 1}^{4} {\sum\limits_{i = 1}^{2} {\int_{{V_{{{\text{p}}j}} }} {e_{31} S_{xxi} E_{3j} {\text{d}}V_{{{\text{p}}j}} } } } + \frac{1}{2}\sum\limits_{j = 1}^{4} {\int_{{V_{{{\text{p}}j}} }} {\varepsilon_{33}^{S} E_{3j}^{2} {\text{d}}V_{{{\text{p}}j}} } } = W_{{{\text{pe}}1}} + W_{{{\text{pe}}2}} , $$

(16)

where $W_{{{\text{pe}}1}}$ depends on both the strain and the electric field, while $W_{{{\text{pe}}2}}$ only depends on the electric field.

Substituting Eqs. (1) and (5) into Eq. (16), $W_{{{\text{pe}}1}}$ can be expressed as

$$ W_{{{\text{pe}}1}} = \frac{{H_{1} + H_{\text{p}} }}{4}e_{31} v_{R1} \left( t \right)B\left. {w^{\prime}_{1} \left( {x,t} \right)} \right|_{0}^{{L_{1} }} + \frac{{H_{2} + H_{\text{p}} }}{4}e_{31} v_{R2} \left( t \right)B\left. {w^{\prime}_{ 2} \left( {x,t} \right)} \right|_{{L_{1} }}^{{L_{2} }} . $$

(17)

$W_{\text{pe2}}$ is given by

$$ \begin{aligned} W_{{{\text{pe}}2}} & = \frac{1}{2}\int_{0}^{{L_{1} }} {\int_{{H_{1} /2}}^{{H_{1} /2 + H_{\text{p}} }} {\varepsilon_{33}^{S} \left[ { - \frac{{v_{R1} \left( t \right)}}{{2H_{\text{p}} }}} \right]^{2} } } B{\text{d}}z{\text{d}}x + \frac{1}{2}\int_{0}^{{L_{1} }} {\int_{{ - H_{1} /2 - H_{\text{p}} }}^{{ - H_{1} /2}} {\varepsilon_{33}^{S} \left[ {\frac{{v_{R1} \left( t \right)}}{{2H_{\text{p}} }}} \right]^{2} } } B{\text{d}}z{\text{d}}x \\ & \quad + \frac{1}{2}\int_{{L_{1} }}^{{L_{2} }} {\int_{{H_{1} /2}}^{{H_{1} /2 + H_{\text{p}} }} {\varepsilon_{33}^{S} \left[ { - \frac{{v_{R2} \left( t \right)}}{{2H_{\text{p}} }}} \right]^{2} } } B{\text{d}}z{\text{d}}x + \frac{1}{2}\int_{{L_{1} }}^{{L_{2} }} {\int_{{ - H_{1} /2 - H_{\text{p}} }}^{{ - H_{1} /2}} {\varepsilon_{33}^{S} \left[ {\frac{{v_{R2} \left( t \right)}}{{2H_{\text{p}} }}} \right]^{2} } } B{\text{d}}z{\text{d}}x \\ & = \frac{{\varepsilon_{33}^{S} B}}{{4H_{\text{p}} }}\left[ {\int_{0}^{{L_{1} }} {v_{R1}^{2} \left( t \right)} {\text{d}}x + \int_{{L_{1} }}^{{L_{2} }} {v_{R2}^{2} \left( t \right)} {\text{d}}x} \right] = \frac{{\varepsilon_{33}^{S} B}}{{4H_{\text{p}} }}\left[ {v_{R1}^{2} \left( t \right)L_{1} + v_{R2}^{2} \left( t \right)\left( {L_{2} - L_{1} } \right)} \right]. \\ \end{aligned} $$

(18)

3.2 Spatial discretization of the energy equations

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

$$ w\left( {x,t} \right) = \sum\limits_{m = 1}^{N} {\phi_{m} \left( x \right)q_{m} \left( t \right)} , $$

(19)

where $\phi_{m} \left( x \right)$ and $q_{m} \left( t \right)$ are the unknown mode shape and generalized modal coordinate of the mth 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 mth mode can be written as

$$ \phi_{m} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\phi_{m1} \left( x \right),\quad 0 \leqslant x \leqslant L_{1} ,} \hfill \\ {\phi_{m2} \left( x \right),\quad L_{1} \leqslant x \leqslant L_{2} ,} \hfill \\ {\phi_{m3} \left( x \right),\quad L_{2} \leqslant x \leqslant L,} \hfill \\ \end{array} } \right. $$

(20)

where $\phi_{m1} \left( x \right)$, $\phi_{m2} \left( x \right)$, and $\phi_{m3} \left( x \right)$ can be expressed as

$$ \phi_{m1} \left( x \right) = C_{1} \sin \left( {k_{m1} x} \right) + C_{2} \cos \left( {k_{m1} x} \right) + C_{3} \sinh \left( {k_{m1} x} \right) + C_{4} \cosh \left( {k_{m1} x} \right), $$

(21)

$$ \phi_{m2} \left( x \right) = C_{5} \sin \left( {k_{m2} x} \right) + C_{6} \cos \left( {k_{m2} x} \right) + C_{7} \sinh \left( {k_{m2} x} \right) + C_{8} \cosh \left( {k_{m2} x} \right), $$

(22)

$$ \phi_{m3} \left( x \right) = C_{9} \sin \left( {k_{m3} x} \right) + C_{10} \cos \left( {k_{m3} x} \right) + C_{11} \sinh \left( {k_{m3} x} \right) + C_{12} \cosh \left( {k_{m3} x} \right). $$

(23)

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

$$ \begin{aligned} \phi_{m1} \left( 0 \right) & = 0,\quad \phi^{\prime}_{m1} \left( 0 \right) = 0, \\ EI_{3} \phi^{\prime\prime}_{m3} \left( L \right) & = 0,\quad EI_{3} \phi^{\prime\prime\prime}_{m3} \left( L \right) = 0. \\ \end{aligned} $$

(24)

The continuity conditions at $x = L_{1}$ can be written as

$$ \begin{aligned} \phi_{m1} \left( {L_{1} } \right) & = \phi_{m2} \left( {L_{1} } \right),\quad \phi^{\prime}_{m1} \left( {L_{1} } \right) = \phi^{\prime}_{m2} \left( {L_{1} } \right), \\ EI_{1} \phi^{\prime\prime}_{m1} \left( {L_{1} } \right) & = EI_{2} \phi^{\prime\prime}_{m2} \left( {L_{1} } \right),\quad EI_{1} \phi^{\prime\prime\prime}_{m1} \left( {L_{1} } \right) = EI_{2} \phi^{\prime\prime\prime}_{m2} \left( {L_{1} } \right). \\ \end{aligned} $$

(25)

The continuity conditions at $x = L_{2}$ can be written as

$$ \begin{aligned} \phi_{m2} \left( {L_{2} } \right) & = \phi_{m3} \left( {L_{2} } \right),\quad \phi^{\prime}_{m2} \left( {L_{2} } \right) = \phi^{\prime}_{m3} \left( {L_{2} } \right), \\ EI_{2} \phi^{\prime\prime}_{m2} \left( {L_{2} } \right) & = EI_{3} \phi^{\prime\prime}_{m3} \left( {L_{2} } \right),\quad EI_{2} \phi^{\prime\prime\prime}_{m2} \left( {L_{2} } \right) = EI_{ 3} \phi^{\prime\prime\prime}_{m3} \left( {L_{2} } \right), \\ \end{aligned} $$

(26)

where $EI_{i}$ is the flexural stiffness at segment i of the beam, $EI_{i} = E_{\text{s}} I_{{{\text{s}}i}} + 2E_{\text{p}} I_{{{\text{p}}i}} {\kern 1pt} {\kern 1pt} (i = 1,{\kern 1pt} {\kern 1pt} 2)$, and $EI_{3} = E_{s} I_{s3}$.

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

$$ k_{m1}^{2} \sqrt {\frac{{EI_{1} }}{{\rho A_{1} }}} = k_{m2}^{2} \sqrt {\frac{{EI_{2} }}{{\rho A_{2} }}} = k_{m3}^{2} \sqrt {\frac{{EI_{3} }}{{\rho A_{3} }}} = \omega_{m} , $$

(27)

where $\rho A_{i}$ is the mass of the i-th segment of the beam, $\rho A_{i} = \rho_{\text{s}} A_{{{\text{s}}i}} + 2\rho_{\text{p}} A_{{{\text{p}}i}} {\kern 1pt} {\kern 1pt} (i = 1,{\kern 1pt} {\kern 1pt} 2)$, and $\rho A_{3} = \rho_{\text{s}} A_{{{\text{s}}3}}$.

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

$$ {\mathbf{A}}\left( {k_{m1} ,k_{m2} ,k_{m3} } \right){\varvec{\eta}} = 0, $$

(28)

where ${\mathbf{A}}\left( {k_{m1} ,k_{m2} ,k_{m3} } \right)$ is a $12 \times 12$ coefficient matrix (see Appendix), and $k_{m1}$, $k_{m2}$, and $k_{m3}$ satisfy Eq. (27), ${\varvec{\eta}} = \left[ {C_{1} \;C_{2} \;C_{3} \;C_{4} \;C_{5} \;C_{6} \;C_{7} \;C_{8} \;C_{9} \;C_{10} \;C_{11} \;C_{12} } \right]^{\text{T}}$. Setting the determinant of ${\mathbf{A}}$ equal to zero, the natural frequencies of the system can be obtained. According to the natural frequency of each order, 12 unknown coefficients of the corresponding mode shape function can also be obtained.

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

$$ \begin{aligned} & \rho A_{1} \int_{0}^{{L_{1} }} {\phi_{m1} \left( x \right)} \phi_{n1} \left( x \right){\text{d}}x + \rho A_{2} \int_{{L_{1} }}^{{L_{2} }} {\phi_{m2} \left( x \right)\phi_{n2} \left( x \right)} {\text{d}}x + \rho A_{3} \int_{{L_{2} }}^{L} {\phi_{m3} \left( x \right)\phi_{n3} \left( x \right)} {\text{d}}x = \delta_{mn} , \\ & EI_{1} \int_{0}^{{L_{1} }} {\phi^{\prime\prime}_{m1} \left( x \right)\phi^{\prime\prime}_{n1} \left( x \right)} {\text{d}}x + EI_{2} \int_{{L_{1} }}^{{L_{2} }} {\phi^{\prime\prime}_{m2} \left( x \right)\phi^{\prime\prime}_{n2} \left( x \right)} {\text{d}}x + EI_{3} \int_{{L_{2} }}^{L} {\phi^{\prime\prime}_{m3} \left( x \right)\phi^{\prime\prime}_{n3} \left( x \right)} {\text{d}}x = \omega_{m}^{2} \delta_{mn} , \\ \end{aligned} $$

(29)

where $\delta_{mn}$ 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

$$ w\left( {x,t} \right) \approx \phi_{1} \left( x \right)q_{1} \left( t \right). $$

(30)

3.3 Coupled 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

$$ \begin{aligned} \ell & = \frac{1}{2}\dot{q}_{1}^{2} \left( t \right) - \frac{1}{2}\omega_{1}^{2} q_{1}^{2} \left( t \right) + \theta_{1} v_{R1} \left( t \right)q_{1} \left( t \right) + \theta_{2} v_{R2} \left( t \right)q_{1} \left( t \right) + \beta \dot{q}_{1} \left( t \right)\dot{z}_{\text{b}} \left( t \right) \\ & \quad + \frac{1}{2}C_{{{\text{p}}1}} v_{R1}^{2} \left( t \right) + \frac{1}{2}C_{{{\text{p}}2}} v_{R2}^{2} \left( t \right) + \frac{1}{2}M\dot{z}_{\text{b}}^{2} \left( t \right), \\ \end{aligned} $$

(31)

where $\omega_{1}$ is the first natural frequency of the piezoelectric cantilever beam, $\theta_{1}$ and $\theta_{2}$ are the model electromechanical coupling coefficients, $\beta$ is the model base excitation coefficient, $C_{{{\text{p}}1}}$ and $C_{{{\text{p}}2}}$ are equivalent capacitances, and $M$ is the total mass of the piezoelectric cantilever beam. These coefficients are given by

$$ \begin{aligned} \theta_{1} & = \frac{1}{2}\left( {H_{1} + H_{\text{p}} } \right)e_{31} B\phi^{\prime}_{11} \left( {L_{1} } \right),\quad \theta_{2} = \frac{1}{2}\left( {H_{2} + H_{\text{p}} } \right)e_{31} B\left[ {\phi^{\prime}_{12} \left( {L_{2} } \right) - \phi^{\prime}_{12} \left( {L_{1} } \right)} \right], \\ \beta & = \rho A_{1} \int_{0}^{{L_{1} }} {\phi_{11} } \left( x \right){\text{d}}x + \rho A_{2} \int_{{L_{1} }}^{{L_{2} }} {\phi_{12} } \left( x \right){\text{d}}x + \rho A_{3} \int_{{L_{2} }}^{L} {\phi_{13} } \left( x \right){\text{d}}x, \\ C_{{{\text{p}}1}} & = \frac{{\varepsilon_{33}^{S} BL_{1} }}{{2H_{\text{p}} }},\quad C_{{{\text{p}}2}} = \frac{{\varepsilon_{33}^{S} B\left( {L_{2} - L_{1} } \right)}}{{2H_{\text{p}} }}, \\ M & = \rho_{\text{s}} A_{{{\text{s}}1}} L_{1} + \rho_{\text{s}} A_{{{\text{s}}2}} \left( {L_{2} - L_{1} } \right) + \rho_{\text{s}} A_{{{\text{s}}3}} \left( {L - L_{2} } \right) + 2\rho_{\text{p}} A_{\text{p}} L_{2} . \\ \end{aligned} $$

(32)

Substituting Eq. (31) into the Lagrange equations yields

$$ \frac{\text{d}}{{{\text{d}}t}}\left( {\frac{\partial \ell }{{\partial \dot{q}_{1} }}} \right) - \frac{\partial \ell }{{\partial q_{1} }} = F_{1} \left( t \right), $$

(33)

$$ \frac{\text{d}}{{{\text{d}}t}}\left( {\frac{\partial \ell }{{\partial \dot{v}_{R1} }}} \right) - \frac{\partial \ell }{{\partial v_{R1} }} = Q_{R1} \left( t \right), $$

(34)

$$ \frac{\text{d}}{{{\text{d}}t}}\left( {\frac{\partial \ell }{{\partial \dot{v}_{R2} }}} \right) - \frac{\partial \ell }{{\partial v_{R2} }} = Q_{R2} \left( t \right), $$

(35)

where $F_{1} \left( t \right)$ is the generalized dissipative force, and $Q_{R1} \left( t \right)$ and $Q_{R2} \left( t \right)$ are the generalized output charges across the load resistances $R_{L1}$ and $R_{L2}$, respectively. Considering the energy function of the generalized dissipative force as the Rayleigh function yields $F_{1} \left( t \right) = - 2\xi_{1} \omega_{1} \dot{q}_{1} \left( t \right)$, where $\xi_{1}$ is the damping ratio of the first mode. Using Ohm’s law, the generalized current is $\dot{Q}_{R1} (t) = v_{R1} (t)/R_{L1}$ and $\dot{Q}_{R2} (t) = v_{R2} (t)/R_{L2}$ with purely resistive electrical loads $R_{L1}$ and $R_{L2}$, respectively. The coupled electromechanical governing equations of the piezoelectric cantilever beam with two steps of different thicknesses can then be obtained as

$$ \ddot{q}_{1} \left( t \right) + 2\xi_{1} \omega_{1} \dot{q}_{1} \left( t \right) + \omega_{1}^{2} q_{1} \left( t \right) - \theta_{1} v_{R1} \left( t \right) - \theta_{2} v_{R2} \left( t \right) = - \beta \ddot{z}_{\text{b}} (t), $$

(36)

$$ C_{{{\text{p}}1}} \dot{v}_{R1} \left( t \right) + \frac{{v_{R1} \left( t \right)}}{{R_{L1} }} + \theta_{1} \dot{q}_{1} \left( t \right) = 0, $$

(37)

$$ C_{{{\text{p}}2}} \dot{v}_{R2} \left( t \right) + \frac{{v_{R2} \left( t \right)}}{{R_{L2} }} + \theta_{2} \dot{q}_{1} \left( t \right) = 0. $$

(38)

3.4 Steady-state response

It is assumed that the base excitation is in the form of a harmonic acceleration $\ddot{z}_{\text{b}} (t) = Z{\text{e}}^{{{\text{j}} \omega t}}$, where $Z$ is the acceleration amplitude and $\omega$ is the excitation frequency. Based on the linear system assumption, the steady-state modal mechanical response of the piezoelectric cantilever beam and the steady-state voltage response across the load resistance are assumed to be harmonic with the same frequency as the excitation, viz. $q_{1} \left( t \right) = H_{1} {\text{e}}^{{{\rm j}\omega t}}$, $v_{R1} \left( t \right) = V_{R1} {\text{e}}^{{{{\text{j}}}\omega t}}$, and $v_{R2} \left( t \right) = V_{R2} {\text{e}}^{{{{\text{j}}}\omega t}}$, respectively, where the amplitudes $H_{1}$, $V_{R1}$, and $V_{R2}$ are complex valued, thus Eqs. (36)–(38) yield the following equations for $H_{1}$, $V_{R1}$, and $V_{R2}$:

$$ \left( {\omega_{1}^{2} - \omega^{2} + 2{{\text{j}}}\xi_{1} \omega_{1} \omega } \right)H_{1} - \theta_{1} V_{R1} - \theta_{2} V_{R2} = - \beta Z, $$

(39)

$$ \left( {{j}\omega C_{{{\text{p}}1}} + \frac{1}{{R_{L1} }}} \right)V_{R1} + {{\text{j}}}\omega \theta_{1} H_{1} = 0, $$

(40)

$$ \left( {{j}\omega C_{{{\text{p}}2}} + \frac{1}{{R_{L2} }}} \right)V_{R2} + {{\text{j}}}\omega \theta_{2} H_{1} = 0. $$

(41)

Solving Eqs. (39)–(41), the magnitude of the steady-state vibration response and the voltage response expressions are as follows:

$$ H_{1} = \frac{{ - \beta Z\left( {{{\text{j}}}\omega C_{{{\text{p}}1}} R_{L1} + 1} \right)\left( {{{\text{j}}}\omega C_{{{\text{p}}2}} R_{L2} + 1} \right)}}{{\left( {\omega_{1}^{2} - \omega^{2} + 2{{\text{j}}}\xi_{1} \omega_{1} \omega } \right)\left( {{{\text{j}}}\omega C_{{{\text{p}}1}} R_{L1} + 1} \right)\left( {{{\text{j}}}\omega C_{{{\text{p}}2}} R_{L2} + 1} \right) + {{\text{j}}}\omega \theta_{1}^{2} \left( {{{\text{j}}}\omega C_{{{\text{p}}2}} R_{L2} + 1} \right)R_{L1} + {{\text{j}}}\omega \theta_{2}^{2} \left( {{{\text{j}}}\omega C_{{{\text{p}}1}} R_{L1} + 1} \right)R_{L2} }}, $$

(42)

$$ V_{R1} = \frac{{{{\text{j}}}\beta Z\omega \theta_{1} \left( {{{\text{j}}}\omega C_{{{\text{p}}2}} R_{L2} + 1} \right)R_{L1} }}{{\left( {\omega_{1}^{2} - \omega^{2} + 2{{\text{j}}}\xi_{1} \omega_{1} \omega } \right)\left( {{{\text{j}}}\omega C_{{{\text{p}}1}} R_{L1} + 1} \right)\left( {{{\text{j}}}\omega C_{{{\text{p}}2}} R_{L2} + 1} \right) + {{\text{j}}}\omega \theta_{1}^{2} \left( {{{\text{j}}}\omega C_{{{\text{p}}2}} R_{L2} + 1} \right)R_{L1} + {{\text{j}}}\omega \theta_{2}^{2} \left( {{{\text{j}}}\omega C_{{{\text{p}}1}} R_{L1} + 1} \right)R_{L2} }}, $$

(43)

$$ V_{R2} = \frac{{{\text{j}}\beta Z\omega \theta_{2} \left( {{{\text{j}}}\omega C_{{{\text{p}}1}} R_{L1} + 1} \right)R_{L2} }}{{\left( {\omega_{1}^{2} - \omega^{2} + 2{{\text{j}}}\xi_{1} \omega_{1} \omega } \right)\left( {{{\text{j}}}\omega C_{{{\text{p}}1}} R_{L1} + 1} \right)\left( {{{\text{j}}}\omega C_{{{\text{p}}2}} R_{L2} + 1} \right) + {{\text{j}}}\omega \theta_{1}^{2} \left( {{{\text{j}}}\omega C_{{{\text{p}}2}} R_{L2} + 1} \right)R_{L1} + {{\text{j}}}\omega \theta_{2}^{2} \left( {{{\text{j}}}\omega C_{{{\text{p}}1}} R_{L1} + 1} \right)R_{L2} }}. $$

(44)

For a harmonic signal, the respective average power dissipated by the resistors $R_{L1}$ and $R_{L2}$ can be expressed as

$$ P_{R1} = \frac{{V_{R1}^{2} }}{{2R_{L1} }}, $$

(45)

$$ P_{R2} = \frac{{V_{R2}^{2} }}{{2R_{L2} }}. $$

(46)

4 Discussion of the theoretical results

For the considered VEH (shown in Fig. 1), the geometric and material parameters [4] are listed in Table 1. Here, the substructure layer is taken to be beryllium bronze, which has a larger modulus of elasticity and can withstand greater deformation, while the piezoceramic layers are taken to be PZT-5H.

Table 1 Geometric and material parameters of VEH

In the following analysis, the acceleration amplitude $Z$ is chosen as 5 m/s², and the load resistances $R_{L1}$ and $R_{L2}$ are both chosen as 10 kΩ unless otherwise stated. When the ratio of the lengths of the first two segments $\left( {L_{2} - L_{1} } \right)/L_{1}$ is analyzed, the sum of the length of the first and second segments of the beam is 120 mm; when the ratio of the thicknesses of the first two substructure layers $H_{2} /H_{1}$ is analyzed, the thickness of the first segment of the substructure layer $H_{1}$ is 5 mm, while the thickness of the $H_{ 2}$ is reduced; i.e., only the case where the ratio of thicknesses is less than or equal to 1 is analyzed.

First, Figs. 2 and 3 display the voltage and power frequency response curves, respectively, for different ratios of lengths (0.5, 1.0, 1.5, 2.0, and 2.5). It can be seen that the natural frequency of the harvester decreases with increasing ratio of lengths. Meanwhile, increasing the ratio of lengths can increase the peak voltage and power across the load resistance $R_{L2}$, but reduce the peak voltage and power across the load resistance $R_{L1}$. The volume of the harvester is different for different ratios of lengths, thus the power density frequency response curves are plotted for different ratios of lengths in Fig. 4. An increase in the ratio of lengths causes an increase in the peak power density (PD₁) for the load resistance $R_{L1}$ but a decrease in the peak power density (PD₂) for the load resistance $R_{L2}$.

Fig. 2

Voltage frequency response curves across a$R_{L1}$ and b$R_{L2}$ for different ratios of lengths

Fig. 3

Power frequency response curves of a$R_{L1}$ and b$R_{L2}$ for different ratios of lengths

Fig. 4

Power density frequency response curves of the harvester for a$R_{L1}$ and b$R_{L2}$ for different ratios of lengths

The voltage, power, and power density frequency response curves are plotted for different ratios of thicknesses (0.2, 0.4, 0.6, 0.8, and 1.0) in Figs. 5, 6, and 7, respectively, where a ratio of thicknesses of 1 corresponds to a uniform beam. It can be observed that decreasing the ratio of thicknesses can reduce the natural frequency of the harvester. Decreasing the ratio of thicknesses can increase the peak voltage across the load resistance $R_{L2}$ but reduce the peak voltage across the load resistance $R_{L1}$, according to Fig. 5. Compared with a uniform beam, reducing the thickness $H_{2}$ can increase the peak voltage across the load resistance $R_{L2}$ but also reduce the peak voltage across the load resistance $R_{L 1}$. Also, it is clear from Figs. 6 and 7 that reducing the thickness $H_{2}$ can increase the peak power of the load resistance $R_{L2}$ and the peak power density of the corresponding harvester, but also reduce the peak power of the load resistance $R_{L1}$ and the peak power density of the corresponding harvester.

Fig. 5

Voltage frequency response curves across a$R_{L1}$ and b$R_{L2}$ for different ratios of thicknesses

Fig. 6

Power frequency response curves of a$R_{L1}$ and b$R_{L2}$ for different ratios of thicknesses

Fig. 7

Power density frequency response curves of the harvester for a$R_{L1}$ and b$R_{L2}$ for different ratios of thicknesses

For a total beam length L of 140 mm (i.e., a third segment with length of 20 mm), the voltage and power frequency response curves are plotted for different end thicknesses (H₃ = 5, 10, 15, 20, and 25 mm) in Figs. 8 and 9, respectively. It can be seen that, with increase of H₃, the natural frequency of the harvester gradually decreases, while the peak voltage and power gradually increase.

Fig. 8

Voltage frequency response curves across a$R_{L1}$ and b$R_{L2}$ for different end thicknesses

Fig. 9

Power frequency response curves of a$R_{L1}$ and b$R_{L2}$ for different end thicknesses

Assuming that the load resistances are equal ($R_{L1} = R_{L2}$), the voltage and power frequency response curves are plotted for different load resistors ($R_{L1} = R_{L2}$ = 10, 30, 70, 200, 400, and 800 kΩ) in Figs. 10 and 11, respectively. According to Fig. 10, it can be seen that, with an increase of the load resistance, the voltage output across the load resistances $R_{L1}$ and $R_{L2}$ increases monotonically at each frequency, respectively, and the resonance frequency moves up. From the results in Fig. 11, it can be observed that, with an increase of the load resistance, the peak power of $R_{L1}$ and $R_{L2}$ first increases then decreases, reaching a peak of 1.92 mW and 8.57 mW for $R_{L1}$ and $R_{L2}$, respectively, at 70 kΩ. The above-described phenomena are consistent with the trend of the distributed-parameter electromechanical model proposed by Erturk and Inman [40]. When the external load resistance is further enlarged, the output power decreases remarkably, as an increase of the load resistance will cause the current in the external circuit to decrease rapidly, and the influence of the current on the output power is greater than the voltage. The power output from a source is maximum when the external resistance is equal to the internal equivalent resistance, indicating that the internal equivalent resistance of a harvester of this size is close to 70 kΩ. Thus, to achieve the maximum energy output from the harvester, an external load resistor with the same value as the internal equivalent resistance is generally selected.

Fig. 10

Voltage frequency response curves across a$R_{L1}$ and b$R_{L2}$ for different load resistances

Fig. 11

Power frequency response curves of a$R_{L1}$ and b$R_{L2}$ for different load resistances

5 Verification by FEM simulation

To verify the theoretical model, FEM simulation is introduced to analyze the performance of the proposed VEH in this section. The anisotropic material parameters [4, 41] of the PZT-5H used in the simulation are as follows.

Relative permittivity matrix

$$ \varvec{\varepsilon}_{\text{r}} = \left[ {\begin{array}{*{20}c} {1700} & 0 & 0 \\ 0 & {1700} & 0 \\ 0 & 0 & {1470} \\ \end{array} } \right]. $$

Piezoelectric constant matrix (units: C/m²)

$$ \varvec{e} = \left[ {\begin{array}{*{20}c} 0 & 0 & { - 6.5} \\ 0 & 0 & { - 6.5} \\ 0 & 0 & {23.3} \\ 0 & 0 & 0 \\ 0 & {17} & 0 \\ {17} & 0 & 0 \\ \end{array} } \right]. $$

Piezoelectric elastic coefficient matrix (unit: N/m²)

$$ \varvec{C} = \left[ {\begin{array}{*{20}c} {12.6} & {7.95} & {8.41} & 0 & 0 & 0 \\ {7.95} & {12.6} & {8.41} & 0 & 0 & 0 \\ {8.41} & {8.41} & {11.7} & 0 & 0 & 0 \\ 0 & 0 & 0 & {2.3} & 0 & 0 \\ 0 & 0 & 0 & 0 & {2.3} & 0 \\ 0 & 0 & 0 & 0 & 0 & {2.3} \\ \end{array} } \right] \times 10^{10} . $$

The direct coupling method is applied for the coupled analysis of the piezoelectric cantilever beam. The three-dimensional (3D) eight-node hexahedral coupled-field element SOLID5 is used for the piezoceramic layer, the eight-node linear structural element SOLID45 is used for the substructure layer, and the piezoelectric circuit element CIRCU94 is used for the load resistor. The displacement degree of freedom is constrained to be zero at the end of the cantilevered beam. The polarization direction of the piezoelectric material is determined by the symbol of the piezoelectric constant. For this series connection of the piezoelectric bimorph, the piezoceramic layer on the upper and lower surface of the substructure layer should have the opposite sign of piezoelectric constant. The electrode connections are achieved by using the “couple” command. The voltage degree of freedom is coupled independently for each piezoceramic layer of the finite element model, so that each node has the same electric potential on each electrode surface. For the first two segments of the piezoelectric cantilever beam, the electrode surface between the substructure layer and the piezoceramic layer are coupled to make the potential equal; the load resistance is connected between the top surface of the upper piezoceramic layer and the bottom surface of the lower piezoceramic layer; the voltage at the bottom surface of the lower piezoceramic layer is constrained to be zero. The potential on the top surface of the upper piezoceramic layer in the first and second segments is the output voltage across the load resistance $R_{L1}$ and $R_{L2}$, respectively. A finite element model of the piezoelectric cantilever beam with two steps of different thicknesses is established according to the parameter values in Table 1, as shown in Fig. 12a. The modal deformation of the first mode of the piezoelectric cantilever beam can be obtained through modal analysis, as shown in Fig. 12b. The acceleration amplitude applied to the fixed end is 5 m/s², then the output voltage, power, and power density of the piezoelectric cantilever beam can be obtained at different excitation frequencies by harmonic response analysis.

Fig. 12

a Finite element model and b modal deformation of the piezoelectric cantilever beam

In this section, the geometric parameters are changed separately, and the variation trends of the natural frequency, output voltage, power, and power density of the piezoelectric cantilever beam studied. These results are significant to guide the optimization of the structure of the piezoelectric cantilever beam with two steps of different thicknesses, requiring in-depth study. The variation of the natural frequencies obtained from the FEM simulation and analytical model with the ratio of lengths, ratio of thicknesses, and end thickness are shown in Fig. 13a–c, respectively, revealing that the FEM simulation results are in close agreement with the results obtained from the analytical model. It can be seen from Fig. 13a that the natural frequency can be reduced by 49.4 Hz when the ratio of lengths is increased from 0.25 to 2.5, and increasing the ratio of lengths in the region below 1.25 can rapidly reduce the natural frequency. It can be observed from Fig. 13b that the natural frequency can be reduced by 84.8 Hz when the ratio of lengths is increased from 0.1 to 1.0, but increasing the ratio of thicknesses can rapidly reduce the natural frequency when the ratio of lengths is below 0.6. From these results, it can be concluded that decreasing the ratio of thicknesses can remarkably reduce the natural frequency compared with increasing the ratio of lengths. In other words, decreasing the thickness $H_{2}$ can remarkably reduce the natural frequency. When designing a piezoelectric cantilever beam with two steps of different thicknesses, one should thus prioritize decreasing the thickness $H_{2}$. As depicted in Fig. 13c, as the end thickness is increased, the natural frequency decreases and the rate of change of the natural frequency decreases, representing a widely used method to reduce the natural frequency of VEHs.

Fig. 13

Variation of natural frequency with a ratio of lengths, b ratio of thicknesses, and c end thickness

At the resonant frequency, the variations of the voltage across the load resistance, the power of the load resistance, and the power density of the harvester with the ratio of lengths are shown in Fig. 14a–c, respectively, comparing the results obtained from the FEM simulation and analytical model, with good agreement. As seen from Fig. 14a, when the ratio of lengths is increased from 0.25 to 2.5, the voltage $V_{R1}$ across the load resistance $R_{L1}$ is reduced by 8.4 V, and the voltage $V_{R2}$ across the load resistance $R_{L2}$ increases by 9.7 V, with the trend of decrease and increase being similar. Therefore, the effect of increasing the ratio of lengths on $V_{R2}$ is slightly greater than $V_{R1}$. In addition, it can be seen that the rate of decrease of $V_{R1}$ and increase of $V_{R2}$ is higher for ratios of lengths below 1.25, with $V_{R1}$ = $V_{R2}$ for a ratio of lengths of 0.5 and $V_{R1}$ < $V_{R2}$ for ratios of lengths greater than 0.5 (i.e., in most cases). From the results in Fig. 14b, when the ratio of lengths is increased from 0.25 to 2.5, the power $P_{R1}$ of the load resistance $R_{L 1}$ is reduced by 5.6 mW, while the power $P_{R2}$ of the load resistance $R_{L2}$ increases by 9.1 mW, and the rate of increase of $P_{R2}$ is obviously greater than the rate of decrease of $P_{R1}$. Thus, the effect of increasing the ratio of lengths on $P_{R2}$ is notably greater than $P_{R1}$. Moreover, it can be seen that the rate of decrease of $P_{R1}$ is higher for ratios of lengths below 0.75, that the rate of increase of $P_{R2}$ is higher for ratios of lengths below 1.25, and that $P_{R1}$ = $P_{R2}$ for a ratio of lengths of 0.5 but $P_{R1}$ < $P_{R2}$ for ratios of lengths above 0.5 (i.e., in most cases). According to Fig. 14c, when the ratio of lengths is increased from 0.25 to 2.5, the power density ${PD}_{1}$ of the harvester for the load resistance $R_{L 1}$ is reduced by 0.32 μW/mm³, while the power density ${PD}_{2}$ of the harvester for the load resistance $R_{L2}$ increases by 0.68 μW/mm³, with the rate of increase of ${PD}_{2}$ being obviously greater than the rate of decrease of ${{PD}_{1}}$. Thus, the effect of increasing the ratio of lengths on ${PD}_{2}$ is markedly greater than ${PD}_{1}$. Furthermore, it can be seen that the rate of decrease of ${PD}_{1}$ is higher for ratios of lengths below 1.0, that the rate of increase of ${PD}_{2}$ is higher for ratios of lengths below 1.25, and that ${PD}_{1}$ = ${PD}_{2}$ for a ratio of lengths of 0.5 but ${PD}_{1}$ < ${PD}_{2}$ for ratios of lengths above 0.5 (i.e., in most cases).

Fig. 14

Variation of a peak voltage, b peak power, and c peak power density of the harvester with the ratio of lengths

At resonant frequency, the variations of the voltage across the load resistance, the power of the load resistance, and the power density of the harvester with the ratio of thicknesses are shown in Fig. 15a–c, respectively, comparing the results obtained from the FEM simulation and analytical model, with good agreement. As seen from Fig. 15a, when the ratio of thicknesses is increased from 0.1 to 1, the voltage $V_{R1}$ across the load resistance $R_{L1}$ increases by 9.7 V, while the voltage $V_{R2}$ across the load resistance $R_{L2}$ is reduced by 11.6 V, with the trends of increase and decrease being similar and linear. Therefore, the effect of increasing the ratio of thicknesses on $V_{R2}$ is slightly greater than $V_{R1}$. In addition, it can be seen that $V_{R1}$ = $V_{R2}$ for a ratio of thicknesses of 0.62, while $V_{R1}$ < $V_{R2}$ for ratios of thicknesses above 0.62 (i.e., in most cases). From the results in Fig. 15b, when the ratio of thicknesses is increased from 0.1 to 1.0, the power $P_{R1}$ of the load resistance $R_{L1}$ increases by 6.3 mW, while the power $P_{R2}$ of the load resistance $R_{L2}$ is reduced by 12 mW, and the rate of decrease of $P_{R2}$ is obviously greater than the rate of increase of $P_{R1}$. Thus, the effect of increasing the ratio of thicknesses on $P_{R2}$ is notably greater than $P_{R1}$. Moreover, it can be seen that the rate of increase of $P_{R1}$ is always very small, while the rate of decrease of $P_{R2}$ is larger for ratios of thicknesses below 0.6, and $P_{R1}$ = $P_{R2}$ for a ratio of thicknesses of 0.62 but $P_{R1}$ < $P_{R2}$ for ratios of thicknesses above 0.62 (i.e., in most cases). According to Fig. 15c, when the ratio of thicknesses is increased from 0.1 to 1.0, the power density ${PD}_{1}$ of the harvester for the load resistance $R_{L 1}$ increases by 0.33 μW/mm³, while the power density ${PD}_{2}$ of the harvester for the load resistance $R_{L2}$ is reduced by 0.91 μW/mm³, and the rate of decrease of ${PD}_{2}$ is obviously higher than the rate of increase of ${PD}_{1}$. Thus, the effect of increasing the ratio of thicknesses on ${PD}_{2}$ is markedly greater than ${PD}_{1}$. Furthermore, it can be seen that the rate of increase of ${PD}_{1}$ is always very small, while the rate of decrease of ${PD}_{2}$ is higher for ratios of thicknesses below 0.6, and ${PD}_{1}$ = ${PD}_{2}$ for a ratio of thicknesses of 0.62 but ${PD}_{1}$ < ${PD}_{2}$ for ratios of thicknesses above 0.62 (i.e., in most cases).

Fig. 15

Variation of a peak voltage, b peak power, and c peak power density of the harvester with the ratio of thicknesses

According to the results described above, we finally select a ratio of lengths of 2.0, a ratio of thicknesses of 0.4, a total length L of 140 mm, H₃ of 20 mm, and a value for the external load resistance of 70 kΩ. All the piezoceramic layers are connected in series, and the voltage and power frequency response curves obtained from the analytical model and FEM simulation are shown in Fig. 16a and b, respectively. It is can be seen that the simulation results are in close agreement with the results obtained from the analytical model. When the external excitation frequency is 57 Hz, the voltage V_R across the load resistance and the output power P_R of the load resistance are the largest, reaching 67.2 V and 20.4 mW, respectively. Based on FEM simulations, the voltage and power frequency response curves for a uniform piezoelectric beam with the same size as the two-step piezoelectric beam (with a ratio of thicknesses is 1.0) are also plotted in Fig. 16a, b, respectively. When the external excitation frequency is 106 Hz, the voltage V_R across the load resistance and the output power P_R of the load resistance are the largest, reaching 39.8 V and 11.3 mW, respectively. Consequently, compared with the traditional uniform design, the resonant frequency of the two-step piezoelectric beam is decreased by 46.2%, while the peak voltage and power are 1.69 and 1.81 times higher, respectively. Therefore, the two-step piezoelectric beam can produce higher energy output in low-frequency environments.

Fig. 16

a Voltage and b power frequency response curves across the load resistance of the two-step piezoelectric beam and uniform piezoelectric beam

6 Conclusions

A novel structure consisting of a bimorph piezoelectric cantilever beam with two steps of different thicknesses is proposed for vibration energy harvesting. A coupled electromechanical model is derived via the Lagrange equations, and expressions for the steady-state response are obtained for harmonic base excitations at arbitrary frequencies. The frequency response curves of the output voltage, power, and power density for different values of the geometric parameters (ratio of lengths, ratio of thicknesses, and end thickness) and load resistances are plotted by using the proposed analytical model. It is shown that increasing the ratio of lengths or end thickness, or decreasing the ratio of thicknesses, can reduce the natural frequency of the harvester; increasing the ratio of lengths or decreasing the ratio of thicknesses can increase the performance of the harvester for the load resistance $R_{L2}$, but the opposite results are produced for $R_{L 1}$. Increasing the end thickness can increase the performance of the harvester. The internal equivalent resistance of the harvester is close to 70 kΩ.

Furthermore, the validity of the theoretical model is verified by FEM simulation. The effects of the geometric parameters on the natural frequency and the energy harvesting performance of the harvester are fully studied, revealing that decreasing the ratio of thicknesses can remarkably reduce the natural frequency compared with increasing the ratio of lengths. The effect of increasing the ratio of lengths or ratio of thicknesses on the energy harvesting performance of the harvester for the load resistance $R_{L2}$ outweigh the effect for $R_{L1}$. Eventually, compared with a uniform piezoelectric beam, the two-step piezoelectric beam can produce higher energy output in low-frequency environments.

Appendix

$$ \varvec{A}\left( {k_{m1} ,k_{m2} ,k_{m3} } \right) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {k_{11} } & 0 & {k_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {a_{39} } & {a_{310} } & {a_{311} } & {a_{312} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {a_{49} } & {a_{410} } & {a_{411} } & {a_{412} } \\ {a_{51} } & {a_{52} } & {a_{53} } & {a_{54} } & {a_{55} } & {a_{56} } & {a_{57} } & {a_{58} } & 0 & 0 & 0 & 0 \\ {a_{61} } & {a_{62} } & {a_{63} } & {a_{64} } & {a_{65} } & {a_{66} } & {a_{67} } & {a_{68} } & 0 & 0 & 0 & 0 \\ {a_{71} } & {a_{72} } & {a_{73} } & {a_{74} } & {a_{75} } & {a_{76} } & {a_{77} } & {a_{78} } & 0 & 0 & 0 & 0 \\ {a_{81} } & {a_{82} } & {a_{83} } & {a_{84} } & {a_{85} } & {a_{86} } & {a_{87} } & {a_{88} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {a_{95} } & {a_{96} } & {a_{97} } & {a_{98} } & {a_{99} } & {a_{910} } & {a_{911} } & {a_{912} } \\ 0 & 0 & 0 & 0 & {a_{105} } & {a_{106} } & {a_{107} } & {a_{108} } & {a_{109} } & {a_{1010} } & {a_{1011} } & {a_{1012} } \\ 0 & 0 & 0 & 0 & {a_{115} } & {a_{116} } & {a_{117} } & {a_{118} } & {a_{119} } & {a_{1110} } & {a_{1111} } & {a_{1112} } \\ 0 & 0 & 0 & 0 & {a_{125} } & {a_{126} } & {a_{127} } & {a_{128} } & {a_{129} } & {a_{1210} } & {a_{1211} } & {a_{1212} } \\ \end{array} } \right], $$

where $a_{39} = - EI_{3} k_{m3}^{2} \sin \left( {k_{m3} L} \right)$, $a_{310} = - EI_{3} k_{m3}^{2} \cos \left( {k_{m3} L} \right)$, $a_{311} = EI_{3} k_{m3}^{2} \sinh \left( {k_{m3} L} \right)$, $a_{312} = EI_{3} k_{m3}^{2} \cosh \left( {k_{m3} L} \right)$, $a_{49} = - EI_{3} k_{m3}^{3} \cos \left( {k_{m3} L} \right)$, $a_{410} = EI_{3} k_{m3}^{3} \sin \left( {k_{m3} L} \right)$, $a_{411} = EI_{3} k_{m3}^{3} \cosh \left( {k_{m3} L} \right)$, $a_{412} = EI_{3} k_{m3}^{3} \sinh \left( {k_{m3} L} \right)$, $a_{51} = \sin \left( {k_{m1} L_{1} } \right)$, $a_{52} = \cos \left( {k_{m1} L_{1} } \right)$, $a_{53} = \sinh \left( {k_{m1} L_{1} } \right)$, $a_{54} = \cosh \left( {k_{m1} L_{1} } \right)$, $a_{55} = - \sin \left( {k_{m2} L_{1} } \right)$, $a_{56} = - \cos \left( {k_{m2} L_{1} } \right)$, $a_{57} = - \sinh \left( {k_{m2} L_{1} } \right)$, $a_{58} = - \cosh \left( {k_{m2} L_{1} } \right)$, $a_{61} = k_{m1} \cos \left( {k_{m1} L_{1} } \right)$, $a_{62} = - k_{m1} \sin \left( {k_{m1} L_{1} } \right)$, $a_{63} = k_{m1} \cosh \left( {k_{m1} L_{1} } \right)$, $a_{64} = k_{m1} \sinh \left( {k_{m1} L_{1} } \right)$, $a_{65} = - k_{m2} \cos \left( {k_{m2} L_{1} } \right)$, $a_{66} = k_{m2} \sin \left( {k_{m2} L_{1} } \right)$, $a_{67} = - k_{m2} \cosh \left( {k_{m2} L_{1} } \right)$, $a_{68} = - k_{m2} \sinh \left( {k_{m2} L_{1} } \right)$, $a_{71} = - EI_{1} k_{m1}^{2} \sin \left( {k_{m1} L_{1} } \right)$, $a_{72} = - EI_{1} k_{m1}^{2} \cos \left( {k_{m1} L_{1} } \right)$, $a_{73} = EI_{1} k_{m1}^{2} \sinh \left( {k_{m1} L_{1} } \right)$, $a_{74} = EI_{1} k_{m1}^{2} \cosh \left( {k_{m1} L_{1} } \right)$, $a_{75} = EI_{2} k_{m2}^{2} \sin \left( {k_{m2} L_{1} } \right)$, $a_{7 6} = EI_{2} k_{m2}^{2} \cos \left( {k_{m2} L_{1} } \right)$, $a_{77} = - EI_{2} k_{m2}^{2} \sinh \left( {k_{m2} L_{1} } \right)$, $a_{78} = - EI_{2} k_{m2}^{2} \cosh \left( {k_{m2} L_{1} } \right)$, $a_{81} = - EI_{1} k_{m1}^{3} \cos \left( {k_{m1} L_{1} } \right)$, $a_{82} = EI_{1} k_{m1}^{3} \sin \left( {k_{m1} L_{1} } \right)$, $a_{83} = EI_{1} k_{m1}^{3} \cosh \left( {k_{m1} L_{1} } \right)$, $a_{84} = EI_{1} k_{m1}^{3} \sinh \left( {k_{m1} L_{1} } \right)$, $a_{85} = EI_{2} k_{m2}^{3} \cos \left( {k_{m2} L_{1} } \right)$, $a_{86} = - EI_{2} k_{m2}^{3} \sin \left( {k_{m2} L_{1} } \right)$, $a_{87} = - EI_{2} k_{m2}^{3} \cosh \left( {k_{m2} L_{1} } \right)$, $a_{88} = - EI_{2} k_{m2}^{3} \sinh \left( {k_{m2} L_{1} } \right)$, $a_{95} = \sin \left( {k_{m2} L_{2} } \right)$, $a_{96} = \cos \left( {k_{m2} L_{2} } \right)$, $a_{97} = \sinh \left( {k_{m2} L_{2} } \right)$, $a_{98} = \cosh \left( {k_{m2} L_{2} } \right)$, $a_{99} = - \sin \left( {k_{m3} L_{2} } \right)$, $a_{910} = - \cos \left( {k_{m3} L_{2} } \right)$, $a_{911} = - \sinh \left( {k_{m3} L_{2} } \right)$, $a_{912} = - \cosh \left( {k_{m3} L_{2} } \right)$, $a_{105} = k_{m2} \cos \left( {k_{m2} L_{2} } \right)$, $a_{106} = - k_{m2} \sin \left( {k_{m2} L_{2} } \right)$, $a_{107} = k_{m2} \cosh \left( {k_{m2} L_{2} } \right)$, $a_{108} = k_{m2} \sinh \left( {k_{m2} L_{2} } \right)$, $a_{109} = - k_{m3} \cos \left( {k_{m3} L_{2} } \right)$, $a_{1010} = k_{m3} \sin \left( {k_{m3} L_{2} } \right)$, $a_{1011} = - k_{m3} \cosh \left( {k_{m3} L_{2} } \right)$, $a_{1012} = - k_{m3} \sinh \left( {k_{m3} L_{2} } \right)$, $a_{115} = - EI_{2} k_{m2}^{2} \sin \left( {k_{m2} L_{2} } \right)$, $a_{116} = - EI_{2} k_{m2}^{2} \cos \left( {k_{m2} L_{2} } \right)$, $a_{117} = EI_{2} k_{m2}^{2} \sinh \left( {k_{m2} L_{2} } \right)$, $a_{118} = EI_{2} k_{m2}^{2} \cosh \left( {k_{m2} L_{2} } \right)$, $a_{119} = EI_{3} k_{m3}^{2} \sin \left( {k_{m3} L_{2} } \right)$, $a_{1110} = EI_{3} k_{m3}^{2} \cos \left( {k_{m3} L_{2} } \right)$, $a_{1111} = - EI_{3} k_{m3}^{2} \sinh \left( {k_{m3} L_{2} } \right)$, $a_{1112} = - EI_{3} k_{m3}^{2} \cosh \left( {k_{m3} L_{2} } \right)$, $a_{125} = - EI_{2} k_{m2}^{3} \cos \left( {k_{m2} L_{2} } \right)$, $a_{126} = EI_{2} k_{m2}^{3} \sin \left( {k_{m2} L_{2} } \right)$, $a_{127} = EI_{2} k_{m2}^{3} \cosh \left( {k_{m2} L_{2} } \right)$, $a_{128} = EI_{2} k_{m2}^{3} \sinh \left( {k_{m2} L_{2} } \right)$, $a_{129} = EI_{3} k_{m3}^{3} \cos \left( {k_{m3} L_{2} } \right)$, $a_{1210} = - EI_{3} k_{m3}^{3} \sin \left( {k_{m3} L_{2} } \right)$, $a_{1211} = - EI_{3} k_{m3}^{3} \cosh \left( {k_{m3} L_{2} } \right)$, $a_{1212} = - EI_{3} k_{m3}^{3} \sinh \left( {k_{m3} L_{2} } \right)$.