Nonlinear dynamic response of bimorph piezoelectric energy harvester with functionally graded porous core

Abstract

This study deals with piezoelectric energy harvesters carrying the magnet at the free end while a magnifier connected them to the base. The harvester consists of a cantilever beam with Functionally Graded (FG) porous foam core and two piezoelectric faces while taking into account the von-Karman strains and the magnetic interaction of two magnets. The governing equation has been developed using Hamilton’s principle and reduced-order via the Galerkin method. Frequency response of vibrations and voltage derived using harmonic balance method and the adjusted model has been confirmed by parametric studies of the lumped parameter model. A parametric study is also performed to expose the effects of porosity coefficient and pattern, magnifier ratios, and excitation level on the responses. Results show that by correct selection of the magnifier parameters, the proposed harvester can provide a higher output over a wider frequency band. Also, the core porosity increases the flexibility of the beam and raised the voltage by a factor of 1.8.

Keywords

piezoelectric energy harvester elastic magnifier FG porous core nonlinear vibrations time and frequency response

Introduction

During the last decade, with the progress of low-power electronic mechanisms and integrated energy harvesting systems like wireless sensors and medical implants,e.g. leadless peacemakers, numerous research studies have focused on piezoelectric energy harvester units, which converts mechanical energy into electricity using piezoelectric elements.¹ Between lots of energy sources, energy harvesting from mechanical vibration has developed wide-ranging because of availability,² besides harvesting from the ambient vibrations is much considered due to the fact that recent electrical devices require little power, energy.³

Most of the previous PEH is constructed based on resonance in linear systems, result in problems like high sensitivity to frequency band uncertainty. In linear systems, the device’s performance is limited to a narrowband frequency which was hard to achieve in practical operations. In order to broadband operating frequencies, scientists have developed several solutions including tuning method, multi-model and nonlinear method.^4–10 Between these approaches, the nonlinear method has mostly important effects on extending the operating bandwidth. This technique introduces nonlinear factors into a linear system, converts its motion equation of a single solution into a multi-solution nonlinear equation, subsequently moves the peaks, and significantly changes the bandwidth characteristic,^5,11 One of the advantages of a nonlinear system is the existence of multiple equilibrium points in them. Considering the nonlinearity in the system allows receiving a large amplitude response in a wider range of frequencies.

Newly scientists have used nonlinear methods to make energy harvesting available in a broadband frequency extensively.¹² Yang et al.¹³ analyzed PEH contain geometric nonlinear factors and proved that increasing the nonlinear coefficient rises harvested voltage. Jahani et al.¹⁴ considered PEH with nonlinear terms such as geometric nonlinearity and demonstrated the nonlinear factors lead to increasing the maximum harvested electrical power. Lin¹⁵ appealed that the harvesting power of a piezoelectric cantilever beam excited by a random source can be enhanced by repulsive magnetic force.

Between the nonlinear techniques, the BPEH can produce a large amplitude motion of chaotic motion at various excitation by means of broadening the bandwidth greatly.¹⁶ In general, beam-type bistable oscillators have a double-well restoring force potential that provides three separate inter-well dynamic regimes (e.g. low-energy vibration, chaotic and high-energy periodic vibration), depending on the input amplitude.¹⁷ One of the most familiar arrangements for BPEH is the magnetic repulsion harvester in which the nonlinearity governed by the repulsion force can greatly increase the harvesting power.¹⁸ Gammaitoni et al.¹⁹ presented bistable PEH configuration and calculated the effect of magnet spacing on the output voltage. Erturk et al.²⁰ showed that the bistable PEH excited by harmonic inputs can show large-amplitude periodic or chaotic motions. Bistable PEH improves eight times in harvested power compared with linear ones. Ferrari et al.²¹ compared the displacement response of a BPEH under band-limited excitation and confirmed the results by tests. Stanton et al.^22,23 considered the performance of a BPEH by numerical and experimental methods. They established a horizontally placed bistable model and studied the influence of different structural parameters on the system output, e.g., amplitude and frequency of the excitation. Karami and Inman²⁴ considered the perturbation method for BPEH in the initial resonance state. Erturk et al.²⁵ considered another BPEH structure with cantilever beams. The results illustrate that the output power is eight times the linear system.

Nevertheless, despite the favorable profits of BPEHs, there is still a chance for improvement. One significant challenge is an operative technique to make the oscillations in a high-energy trajectory to increase the PEH performance. To progress the out power of PEH under low-level excitation, investigators have tried to make them vibrate with large-amplitude inter-well motion. Sebald et al.^26,27 initiate that an external intrusion can help BPEH return to a large-amplitude motion state, and rising the excitation level may help them to change weak low-amplitude orbits to a large-amplitude motion. Kim and Seok²⁸ planned multi-stable PEH, having thinner and broader potential wells compared with BPEH, thus allowing them to extract power in a broadband frequency, even at low excitation levels. Novel 2-DOF BPEH is presented by Zhao et al.²⁹ is composed of two cantilever beams and mass blocks, which could makes the oscillations supplementary stable.

There have been essential concerns paid to structures coupled with elastic substructures. The magnifier substructure is planned to supply an advance for the BPEH. Its tasks by leading high-energy motion and enlarged output performance. However, there are some arrangements of elastic magnifiers that are generally displayed as a linear mass-spring system positioned between the bimorph beam and the base. Vasic et al.³⁰ developed double beam harvesters with a magnifier to investigate the added harvesting power than a traditional harvester. Similarly, Wang et al.³¹ studied an effective PEH including an elastic substrate. Zhou et al.³² proposed an innovative PEH with a multi-mode magnifier, which is capable of significantly increasing the bandwidth and harvesting energy from ambient vibrations. Aldraihem et al.⁶ developed an energy harvester coupled with a mass-spring magnifier, their results are also consistent with increased power output and broader bandwidth process. In order to increase harvesting power, Aladwani et al.⁷ analyzed a PEH contain a simple spring magnifier by means of finite element theory, the suggested harvester enhanced power output and broad the bandwidth by about 21%. An improved prototype for cantilevered PEH with a tip mass offset and magnifier is reported by Tang et al.³³ They also investigated a series of MDOF harvesters that resemble an elastic magnifier attached with a bimorph beam.¹⁸ Wang et al.³⁴ investigated a BPEH with an elastic magnifier and derived the equations using a 2-DOF lumped parameter model.

In connection with vibration analysis of FG porous beam, Fazzolari³⁵ considered the vibration behavior of FG sandwich beams contained by two different types of porosity. Madenci et al.³⁶ examined the influence of porosity distributions on free vibration behavior of FG beams with different boundary conditions.

The effects of porosity coefficient and volume fraction index on the frequency responses of the FG piezoelectric bimorph cantilever beam are considered by AliAbbasi et al.⁸ for the case of near resonance frequencies. By means of porous piezoelectric thin films, a flexible dual-cantilever energy harvester is offered to harvest the mechanical energy from the motion of the pacemaker by Dong et al.³⁷ Shin et al.³² investigated porous sandwich structures based on ceramic piezoelectric materials for energy harvesting applications materials with high piezoelectric charge coefficient and low dielectric constant Fan³⁸ proposed a nano energy harvester with porous piezoelectric materials based on Biot’s porous elasticity and derived the analytical expressions for the resonant frequencies and the energy capturing ability of the porous energy harvester. A sandwich piezoelectric nano-energy harvester model under compressive axial loading with a core layer made-up FG porous material is presented by Zeng et al.³⁹ with the von Karman type assumption. The effects of various porosity forms, porosity coefficients, length scale, excitation frequencies and lumped on the natural frequency and voltage output of nanobeams were investigated. Moradi-Dastjerdi et al.⁴⁰ proposed a PEH made of an advanced porous nanocomposite substrate activated by two piezoceramic layers. The advanced passive layer is made of a lightweight polymeric foam which reinforced with carbon nanotubes. The results exposed that embedding pores leads to higher deflection/voltage peaks and higher vibration frequencies.

Due to the influence of the distributed mass of the cantilever structure on the exciting amplitude, the results of the lumped model are inaccurate. The present study attentions essentially on the development of a BPEH with a magnifier using distributed parameter model including von-Karman strains. The system is analytically solved to examine the harvested voltage and dynamics of the PEH consist of a bimorph beam with FG porosity.

Analytical modeling and equations

Harvester modeling

Figure 1 illustrates the configuration scheme of 2-DOF PEH involves a cantilever bimorph beam including an FG porous metallic core layer covered by two piezoelectric faces and resting on an elastic substructure, in which the porosities are distributed along with the core thickness. Additionally, $L, b$ , $h_{c}$ , and $h_{p}$ are beam length, width, core thickness, and piezoelectric layer height, respectively. Besides the magnifier substructure is modeled by the equivalent mass $M_{b}$ and equivalent stiffness $K_{b}$ . As shown in Figure 1(a) this bistable energy harvester comprises a couple of magnets of mass and inertia $M_{t}$ and $I_{t}$ , respectively. Also, $z_{b}$ and $z_{m}$ are the amplitudes of base excitation and magnifier motion respectively, also distributed transverse vibrations of the cantilever beam are $w (x, t)$ .

Figure 1..

(a) Configuration of bistable PEH with magnifier substructure, (b) Different pattern of porosity for metallic foam core.

Effective material properties

Figure 1(b) depicts the porosity distributions of the core layer along the thickness direction. The variation of Young’s modulus, shear modulus, and mass density through the thickness of metal foam core layer (-h_c/2 ≤ z ≤ h_c/2) corresponding to three different types of porosity distributions are explicitly formulated as⁴¹:

E_{c} (z) = E_{c} (1 - e_{0} λ (z))

G_{c} (z) = G_{c} (1 - e_{0} λ (z))

ρ_{c} (z) = ρ_{c} (1 - e_{1} λ (z))

(1)

In which $E_{c}$ and $ρ_{c}$ are the maximum values of Young’s modulus and the mass density of the metallic porous core, respectively. $λ (z)$ is formulated for each porosity pattern as follows⁴²:

λ (z) = {\begin{array}{c} λ_{0} F G - P (A) \\ \cos (π z / {2 h}_{c} + π / 4) F G - P (B) \\ 1 - \cos (π z / h_{c}) F G - P (C); (| z | \leq \frac{h_{c}}{2}) \\ \cos (π z / h_{c}) F G - P (D) \end{array}

(2)

Also, the porosity coefficient $e_{0} = 1 - \frac{E_{c}}{E_{c 0}}, (0 < e_{0} < 1)$ and $e_{1} = 1 - \frac{ρ_{c}}{ρ_{c 0}}$ is the coefficient mass density, where $E_{c 0}$ and $ρ_{c 0}$ are the minimum values of Young’s modulus and the mass density of the core, respectively. The relation between $e_{1}$ and $e_{0}$ can be expressed as $e_{1} = 1 - \sqrt{1 - e_{0}}$ . Furthermore, the term $λ_{0}$ which is implemented can be calculated by: $λ_{0} = \frac{1}{e_{0}} - \frac{1}{e_{0}} {[\frac{2}{π} \sqrt{1 - e_{0}} - \frac{2}{π} + 1]}^{2}$ . The variations of the Poisson’s ratio are small enough to be considered unimportant. So, the poison’s ratio is constant.

Hamilton principle and equilibrium equation

To obtain equilibrium equations for the distributed parameter system using the Hamilton principle, one can write:

\int_{t_{1}}^{t_{2}} [δ (T - U) + δ W] d t = 0

(3)

$T, U$ , and $W$ are the total kinetic energy, total potential energy, and the work done by the external loads, respectively. The kinetic energy of the harvesting system can be written as $= T_{1} + T_{2} + T_{3}$ ; where $T_{1}$ , $T_{2}$ , and $T_{3}$ represent the kinetic energy of the beam, the elastic substructure, and the moving tip magnet respectively. These energy terms are as follows:

T_{1} = \frac{1}{2} \int_{V_{c}}^{0} ρ_{c} (z) {[\frac{\partial [z_{m} (t) + w (x, t)]}{\partial t}]}^{2} {d V}_{c} + \sum_{i = 1}^{2} \frac{1}{2} \int_{V_{p i}}^{0} ρ_{p i} {[\frac{\partial [z_{m} (t) + w (x, t)]}{\partial t}]}^{2} {d V}_{p i}

(4)

Where

ρ_{c} (z)

and

ρ_{p i}

are mass density of the core and piezoelectric faces, respectively and

i, (i = 1,2)

used for two faces layers of bimorph beam. Furthermore:

T_{2} = \frac{1}{2} M_{t} {[\frac{\partial [z_{m} (t) + w (L, t)]}{\partial t}]}^{2} + \frac{1}{2} I_{t} {[\frac{\partial w^{2} (L, t)}{\partial x \partial t}]}^{2}

(5)

T_{3} = \frac{1}{2} M_{b} \frac{\partial}{\partial t} {z_{m} (t)}^{2}

(6)

The total potential energies of the harvester are $U = U_{1} + U_{2} + U_{3} + U_{4}$ . In which $U_{1}$ and $U_{2}$ are the strain energies of the beam and elastic magnifier, respectively, besides $U_{3}$ and $U_{4}$ are stored energy of the electrical circuit and nonlinear potential generated by the repulsive magnetic forces, respectively. The strain energy of the beam and magnifier are:

U_{1} = \frac{1}{2} \int_{V_{c}}^{0} ε_{c} σ_{c} {(z) d V}_{c} + \frac{1}{2} \sum_{i = 1}^{2} \int_{V_{p i}}^{0} ε_{p i} σ_{p i} {d V}_{p i}

(7)

U_{2} = \frac{1}{2} K_{b} z_{m}^{2}

(8)

$σ_{c} (z)$ and $ε_{c}$ are stress and strain of the core, similarly, $σ_{p}$ and $ε_{p}$ are the stress and strain of both piezoelectric faces, respectively. Consuming Euler Bernoulli beam model and large-amplitude nonlinear von-Karman strains: $ε (x, t) = - z \frac{\partial^{2} w (x, t)}{\partial x^{2}} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}$ .

The main constructional relations of piezoelectric materials are as follows:

σ_{p} = C^{E} ε_{p i} - e_{31} E_{z}

D_{z} = e_{31} ε_{p i} + ε_{33}^{s} E_{z}

(9)

Where $E_{z}$ and $D_{z}$ characterize the electric field and displacement in the z-direction, correspondingly. Also, $e_{31}$ is the piezoelectric constant, $C^{P}$ is the elastic modulus of the piezo material and $ε_{33}^{s}$ is the piezoelectric permittivity. Substituting this relation in equation (4) the strain energy of piezoelectric layers is equals to:

\sum_{i = 1}^{2} \int_{V_{p i}}^{0} ε_{p i} σ_{p i} {d V}_{p i} = \sum_{i = 1}^{2} \int_{V_{p i}}^{0} ε_{p i} (C^{P} ε_{p i} - e_{31} E_{z}) {d V}_{p i}

(10)

The Electrical energy at both piezoelectric layers are:

U_{3} = \frac{1}{2} \sum_{i = 1}^{2} \int_{V_{p i}}^{0} E_{z} D_{z} d V_{p i} = \frac{1}{2} \sum_{i = 1}^{2} \int_{V_{p i}}^{0} (e_{31} {E_{z} ε}_{p i} + ε_{33}^{s} {E_{z}}^{2}) d V_{p i}

(11)

Besides the Magnetic potential created by the magnetic force are as follows, where $M_{A,}$ B, and $V_{A, B}$ are the magnetic moments and volumes of each magnet, respectively³:

U_{4} = \frac{μ_{0} M_{A} V_{A} M_{B} V_{B} [- w^{2} (x, t) + 2 d^{2} - 3 d w (x, t) + w^{'} (x, t)}{4 π \sqrt{w^{2} (x, t) + 1} [w^{2} (x, t) d^{2}]^{2.5}}

(12)

Reduced-order of equations

Dynamic response of the cantilever beam can be formulated using the Rayleigh-Ritz method as⁴³:

w (x, t) = \sum_{j = 1}^{N} ϕ_{j} (x) r_{j} (t) = {\vec{ϕ} (x)} {\vec{r} (t)}

(13)

In which $\vec{ϕ} (x)$ and $\vec{r} (t)$ are the mode shape function of the beam and generalized modal coordinates, respectively. Also, $N$ is the number of considered modes. Following calculation are considered the first mode vibrations (j = 1) and excitations are nearby the fundamental frequency, so $w (x, t) = ϕ (x) r (t)$ . A fundamental model for undamped vibration, for a cantilever beam with a tip mass, $ϕ (x)$ assumed to be:

ϕ (x) = [\cos \frac{λ}{L} x - \cos h \frac{λ}{L} x + σ (\sin \frac{λ}{L} x - \sin h \frac{λ}{L} x)]

σ = \frac{\sin λ - \sinh λ + λ (M_{t} / ρ A L) (\cos λ - \cosh λ)}{\cos λ - \cosh λ + λ (M_{t} / ρ A L) (\sin λ - \sinh λ)}

(14)

Where

ρ = \frac{\int ρ_{c} d z + 2 ρ_{p} h_{p}}{h_{c} + 2 h_{p}}

and

A = b (h_{s} + 2 h_{p})

λ

is the wavelength of the oscillations. After some mathematical operation the equations of the 2-DOF piezoelectric harvester obtained as:

M_{0} \ddot{r} + C_{0} \dot{r} + K_{0} r - α v - k_{1} r + k_{3} r^{3} + B_{2} \ddot{z_{m}} = - B_{2} \ddot{z_{b}}

B_{1} \ddot{z_{m}} + K_{b} z_{m} + B_{2} \ddot{r} = - B_{1} \ddot{z_{b}}

α \dot{r} + C_{p} \dot{v} + v / R = 0

(15)

The coefficients of above equations are:

M_{0} = \int_{V_{c}}^{0} [ρ_{c} (z) ϕ^{2} (x) {d V}_{c}] + \int_{V_{p i}}^{0} [ρ_{p i} ϕ^{2} (x) {d V}_{c}] + b [M_{t} ϕ^{2} (L) + I_{t} ϕ^{' 2} (L)]

K_{0} = \int_{V_{c}}^{0} C_{11} (z) [{- z ϕ}^{″} (x)]^{2} {d V}_{c} + \sum_{i = 1}^{2} \int_{V_{p i}}^{0} C_{11}^{P} {[{- z ϕ}^{″} (x)]}^{2} {d V}_{p i}

B_{2} = \int_{V_{c}}^{0} ρ_{c} (z) ϕ (x) d V_{c} + \sum_{i = 1}^{2} \int_{V_{p i}}^{0} ρ_{p i} ϕ (x) {d V}_{p i} + M_{t} ϕ (L)

α = \sum_{i = 1}^{2} \int_{V_{p i}}^{0} C_{11}^{P} {[z ϕ}^{″} (x) e_{31} [\nabla ψ (z)] {d V}_{p i}

C_{p} = \sum_{i = 1}^{2} \int_{V_{p i}}^{0} ε_{33}^{s} {[- \nabla ψ_{v} (z)]}^{2} {d V}_{p i}

k_{1} = \frac{μ_{0} M_{A} V_{A} M_{B} V_{B}}{2 π d^{5}} k [6 ϕ^{2} (L) + d^{2} ϕ^{' 2} (L) + 3 d ϕ (L) ϕ^{'} (L)]

k_{3} = \frac{{3 μ}_{0} M_{A} V_{A} M_{B} V_{B}}{8 π d^{7}} [30 ϕ^{4} (L) + 13 d^{2} ϕ^{2} (L) ϕ^{' 2} (L) + 20 d ϕ^{3} (L) ϕ^{'} (L)] + (\int_{0}^{L} φ φ^{″} d x) (\int_{0}^{L} {φ^{'}}^{2} d x)

(16)

The dimensionless form of equation (15) are as follows:

\ddot{X} (τ) + μ \dot{X} (τ) + (1 - r) X (τ) - X^{3} + {\ddot{Z}}_{m} - V (τ) = - {\ddot{Z}}_{b} (τ)

{\ddot{Z}}_{m} (τ) + a Z_{m} (τ) + β \ddot{X} (τ) = - {\ddot{Z}}_{b} (τ)

k^{2} \dot{X} (τ) + \dot{V} (τ) + θ V (τ) = 0

(17)

In which:

ω_{0} = \sqrt{K_{0} / M_{0}}; μ = \frac{C_{0}}{M_{0} ω_{0}}; τ = ω_{0} t; r = \frac{k_{1}}{K_{0}}; l = \sqrt{K_{0} / k_{3}}

r (t) = l X (τ); v (t) = \frac{K_{0} l V (τ)}{α}; {z_{m} (t), z_{b} (t)} = \frac{l M_{0}}{B_{2}} {Z_{m} (τ), Z_{b} (τ)};

a = \frac{r_{k}}{1 + r_{m}}; β = \frac{B_{2}^{2}}{B_{1} M_{0}}; θ = \frac{1}{C_{p} R_{L} ω_{0}}; κ^{2} = \frac{α^{2}}{C_{p} K_{0}}

(18)

Where

r_{m} = M_{b} / M_{e q}, r_{k} = K_{b} / K_{e q}

are the mass and stiffness ratios of 2-DOF harvester, Moreover

K_{e q} = 3 {E I}_{e q} / L^{3}

and

M_{e q} = \sum_{i = 1}^{2} \int_{V_{p i}}^{0} ρ_{p i} ϕ (x) {d V}_{p i} + \int_{V_{s}}^{0} ρ_{c} (z) ϕ (x) {d V}_{c}

, in which

{E I}_{e q} = \sum_{i = 1}^{2} \int_{V_{p i}}^{0} C_{11}^{P} z^{2} {d V}_{p i} + \int_{V_{c}}^{0} C_{11} (z) z^{2} {d V}_{c}

Steady-state solution

In this section, the frequency response of the nonlinear harvester was obtained using the harmonic balance method, analytically. Considering harmonic excitation of base as ${\ddot{Z}}_{b} = F C o s Ω τ$ , first-order periodic solution of the system could assume as follows:

X (τ) = a_{0} (τ) + a_{1} (τ) Cos Ω τ + a_{2} (τ) Sin Ω τ

V (τ) = b_{1} (τ) Cos Ω τ + b_{2} (τ) Sin Ω τ

(21)

Where

Ω

is the dimensionless excitation frequency and others are unknown coefficients. Rewriting

Z_{m} (τ)

in terms of

X (τ)

and

V (τ)

by means of equation (18), the fourth order differential equation of the harvester could simplify as:

\frac{1 - β}{a} X^{(4)} + \frac{μ}{a} X^{(3)} + (\frac{1 - r + a}{a}) \ddot{X} + μ \dot{X} + (1 - s) \dot{X} + X^{3} + \frac{1}{a} (3 \ddot{X} X^{2} + 6 X {\dot{X}}^{2}) - \frac{1}{a} \ddot{V} - V = - {\ddot{Z}}_{b}

(22)

Substituting the above relations in the main equation and performing some math operation, the Frequency response of the harvester are:

{(D_{1} + D_{3} \frac{κ^{2} Ω θ}{θ^{2} + Ω^{2}})}^{2} + {(D_{2} - D_{3} \frac{κ^{2} Ω^{2}}{θ^{2} + Ω^{2}})}^{2} = \frac{f^{2}}{A^{2}}

(23)

Where

{A^{2} {= a}_{1}}^{2} + {a_{2}}^{2}

The unknown coefficients are as follows:

D_{1} = \frac{1 - β}{a} Ω^{4} - \frac{1 - r + a}{a} Ω^{2} + 1 - r + 3 {a_{0}}^{2} + \frac{3}{4} A^{2} - \frac{3}{a} Ω^{2} {a_{0}}^{2} - \frac{3}{4 a} Ω^{2} A^{2}

D_{2} = \frac{μ}{a} Ω^{3} - μ Ω; D_{3} = \frac{1}{a} Ω^{2} - 1

(24)

Also, the coefficients of voltage response are equals to:

b_{1} = - \frac{κ^{2} Ω}{θ^{2} + Ω^{2}} (Ω a_{1} + θ a_{2})

b_{2} = \frac{κ^{2} Ω}{θ^{2} + Ω^{2}} (θ a_{1} - a_{2} Ω)

(25)

Results and discussion

Validation

The physical properties of the harvester beam are: $L = 62 m m, b = 18 m m$ with $(h_{c}, h_{p i}) = (0.16,0.2) m m$ , $M_{t} = 5.5 g r$ and $d = 24 m m$ ; further $(E_{c}, E_{p i}) = (61,115) G P a$ , $ρ_{c} = ρ_{p i} = 7900$ , $ε_{33}^{s} = 1200$ and $e_{31} = 190$ .

The evaluation and justification of results based on Ref³³ are illustrated in Table 1. It can be concluded that considering the system with discrete parameters, the response value of the nonlinear response curve system is larger and results are more accurate.

Table 1.

Validation of frequency response curves with Ref.³³

Exc. Freq. ( $Ω$ )	Forward freq.sweep response		Backward freq. sweep response
Exc. Freq. ( $Ω$ )	Ref³³	Present	Ref³³	Present
0.2	0.56	0.58	0.24	0.59
0.3	0.65	0.68	0.16	0.24
0.4	0.71	0.74	0.14	0.20
0.5	0.82	0.86	0.12	0.18
0.6	0.93	0.98	0.11	0.17
0.7	1.09	1.14	0.12	0.16
0.8	1.26	1.32	0.12	0.18
0.9	1.98	2.14	0.17	0.28
1	0.55	0.64	0.55	0.64
1.1	0.94	1.04	0.20	0.19
1.2	0.07	1.16	0.07	0.08

Frequency response

In this subsection, the frequency-response curves and output voltage are plotted to investigate the effects of various parameters such as $r_{m}$ and $r_{k}$ on the harvester responses. Dimensionless constants are $μ = 0.04, κ^{2} = 0.0716, θ = 0.1349$ and $= 1.2158$ As shown in Figure 2, there are two resonance peaks in the frequency response of systems except in the case with no elastic substructure. Also, response curves bend to the right noticeably, because of the nonlinear magnetic force. Furthermore, by increasing $r_{m}$ and $r_{k}$ , the first peak moved to the right slowly and the amplitude of these two peaks amplified obviously. However, the lower frequency peak is shifted to the right more quickly, causing the frequency zone between the two peaks to have a lower frequency bandwidth. However, the lower frequency peak is shifted to the right more quickly, causing the frequency zone between the two peaks to have a lower frequency bandwidth.

Figure 2.

Frequency-response curves are plotted for vibration of tip mass and harvested voltage, at different values of mass and stiffness ratios.

In Figure 3, the frequency-response amplitudes are plotted for different values of porosity coefficient of metallic foam core when r_m = r_k = 10. According to Figure 4, by increasing the porosity of the beam core, the value of two frequency peaks is shifted to the left and occurs at lower frequency values. The two escape frequencies in the curves are also transmitted to the left. On the other hand, the rate of vibration response and surge voltage also increases with increasing porosity, which is due to reduced stiffness and increased flexibility. However, increased porosity will lead to increased stress and the possibility of beam failure for high values of vibration.

Figure 3.

Influence of core porosity on: (a) A-Ω curves, (b) V-Ω curves of harvester.

Figure 4.

Response versus excitation level for different porosity pattern of core.

Frequency response curves for different forms of porosity distribution along the core thickness are presented in Figure 4. The porosity types A and C had the highest and lowest amplitude of vibration response and the harvested voltage, respectively. The first peak of all curves has nearly the same states, but for the second peak of the response, the foam core with type C has the lowest frequency and the highest curvature due to the nonlinear behavior.

Transient response

In this section as shown in Figure 5, the voltage-time response of 2-DOF harvester and phase plane diagram of bimorph beam has been plotted when r_m = r_k = 10. Figure 5 shows the graphs for various excitation levels. As demonstrated in Figure 5(a), if the excitation level is very small, the system oscillates about an equilibrium point with very small tip velocity and vibrations which, cause the inter-well motion and low harvested voltage. Because of the weak excitation level, in this case, the performance of the 2-DOF harvester is not enough to overcome the potential barrier, so the harvester shows bistable motion. As shown in Figure 5(b), when the value of excitation is raised, the harvested voltage has high-level non-periodic motion since the tip mass oscillates between potential barriers with chaotic motion. Considering Figure 5(c), by rising the excitation level more, the beam tip shows a high-energy inter-well motion, considered by a periodic oscillation with high amplitude, because the substantial increase of the vibration amplitudes and the harvested power.

Figure 5.

Time response of harvested voltage and phase portrait diagram of tip mass when $Ω = 0.8$ for: (a) f = 0.025, (b) f = 0.1, (c) f = 0.2.

Figure 6 presents the phase portrait diagrams of the harvester tip mass and the response of harvesting voltage for three different values of the porosity coefficient of foam core. Accordingly, with increasing porosity in the beam core, the vibration amplitude of the tip increases, and consequently the harvesting voltage rises. In addition, the energy of bimorph for the oscillation between the two potential wells has increased.

Figure 6.

Harvesting voltage and oscillation phase plane of when $f = 0.05, r_{m, k} = 10, Ω = 0.8$ for various porosity: $(a) e = 0, (b) e = 0.3, (c) e = 0.6$

At the end of this section, the dynamic phase diagram of the harvester with metal core and the temporal changes of the harvested voltage with Ω = 0.8, Ω = 1 are presented in Figure 7, with different porosity porosities.

Figure 7.

Voltage-time and deflection-velocity curve of the tip mass oscillation when $f = 0.1; r_{m, k} = 10,$ for various porosity parameter and $Ω$ (a) e = 0.1, $Ω = 0.8$ (b) e = 0.3, $Ω = 0.8$ ; (c) e = 0.1, $Ω = 1$ ; (d) e = 0.3, $Ω = 1$

Conclusions

Based on distributed parameter model, the present study analyzes 2-DOF bistable PEH with an elastic substructure in order to amplify the harvesting voltage and broadband the frequency range, considering both nonlinear strains and porosity of metallic foam core.

Nonlinear dynamic analysis of the bistable piezoelectric harvester with high energy inter-well motion is the main purpose of this research.

After validating the numerical results, results of frequency and time response analysis in terms of various system parameters presents, the most important of which are summarized below:

• By tuning the system parameters, such as the mass and stiffness ratios of substructure to the harvester, the magnified excitation level should be expressively reinforced to supply the harvester oscillation and cause both large-amplitude motion and harvested power over broadband frequencies. Also, by rising the elastic substructure stiffness ratio, the harvester oscillations may transform from a low-energy and tight-fitted motion into a bistable and high-energy one that increases the harvesting power. It can be concluded that for BPEH, inter-well motion with a high energy level is crucial.

• Increasing the porosity of the beam core leads to a decrease in frequency and an increase in flexibility and increases the amplitude of the oscillation, and the peak frequency of the response curve is shifted to the left and lower frequencies are transferred. On the other hand, increasing porosity leads to a decrease in beam strength and increases beam stress in wide-range oscillations.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Narges Akbar

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