Nonlinear Hybrid Bistable Vibration-Energy-Harvester Modeling Considering Magnetostrictive and Piezoelectric Behaviors

Subject Areas : Computational Mechanics

Kamran Niazi ¹ , Mohammad-Javad Kazem Zadeh Parsi ² , Mehrdad Mohammadi ³

1 - Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran
2 - Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran
3 - Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran

Received: 2023-08-27 Accepted : 2023-09-18 Published : 2024-09-21

Keywords: Lumped parameter, Runge-Kutta method, Hybrid Energy harvesting, Harmonic balance method, Time and frequency responses,

Abstract :

The present study investigates a novel two degrees of freedom (2DOF) modeling of hybrid-bistable vibration energy harvester (VEH) considering nonlinear magnetic interaction and elastic magnifier to improve the efficiency and expand the action bandwidth. The main part of harvesting mechanism is a composite cantilever beam consists of three layers of magnetostrictive, piezoelectric and a metallic core with internal damping. Such a novel architecture generates more electrical power and operates at larger bandwidth than common piezoelectric or magnetostrictive energy harvesting systems. In the present work, a coupled 2DOF model is developed to investigate the vibration behavior and energy harvesting rate of the harvester. The harmonic balance method is used to obtain the frequency responses and then the Runge-Kutta method is utilized to calculate the dynamic responses. A parametric study is done to investigate the effects of the key features of the harvester such as magnets distances, base acceleration level and excitation frequency on the rate of electricity generation.

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Full-Text:



 

Journal of Solid Mechanics Vol. 15, No. 4 (2023) pp. 352-368 DOI: 10.60664/jsm.2024.3081788

Research Paper  Nonlinear Hybrid Bistable Vibration-Energy-Harvester Modeling Considering Magnetostrictive and Piezoelectric Behaviors

 

K. Niazi, M.J. Kazemzadeh-Parsi 1 , M. Mohammadi
Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran
Received 27 August 2023; accepted 20 September 2023
	ABSTRACT
	The present study investigates a novel two degrees of freedom (2DOF) modeling of hybrid-bistable vibration energy harvester (VEH) considering nonlinear magnetic interaction and elastic magnifier to improve the efficiency and expand the action bandwidth. The main part of harvesting mechanism is a composite cantilever beam consists of three layers of magnetostrictive, piezoelectric and a metallic core with internal damping. Such a novel architecture generates more electrical power and operates at larger bandwidth than common piezoelectric or magnetostrictive energy harvesting systems. In the present work, a coupled 2DOF model is developed to investigate the vibration behavior and energy harvesting rate of the harvester. The harmonic balance method is used to obtain the frequency responses and then the Runge-Kutta method is utilized to calculate the dynamic responses. A parametric study is done to investigate the effects of the key features of the harvester such as magnets distances, base acceleration level and excitation frequency on the rate of electricity generation.                                  © 2023 IAU, Arak Branch.All rights reserved.
	Keywords: Energy harvesting; Lumped parameter Time and frequency responses; Runge-Kutta method; Harmonic balance method.

1    INTRODUCTION

V

IBRATION Energy Harvester (VEH) systems are becoming popular due to need for low rated power sources for electronic devices and also accessibility of the vibrating sources [1]. With rapidly progress of VEHs, many researchers have been considered piezoelectric energy harvesters (PEHs), magnetostrictive energy harvesters (MEHs) [2], electromagnetic and electrostatic based VEH systems to be able to extract energy from broadband frequency extensively. Through the recent years, the nonlinear bistable energy harvesting (BEH) systems has received extensive attentions, since act over a further wide range of base excitation frequencies and can lead to extra output power [3]. The BEH can produce a large amplitude motion with high power by means of broadening the bandwidth greatly. Nonlinear magnetic repulsion load is one of the most conversant arrangements for BEH where can significantly growth the generated voltage and power [4].

Ferrari et al. [5] obtained the displacement response of a BPEH (Bistable PEH) under band-limited excitation and confirmed the model with experimental results. Utilizing perturbation method, Karami and Inman [6] investigated the initial resonance response of BPEH. Kim and Seok [7] considered dynamic response of multi stable VEH to harvest voltage in a wideband frequencies, even at low excitation amplitudes. Jiang et al. [8] proposed a lumped parameter model with magnetic interaction for enlargement the range of resonance frequency of bistable VEH. Nguyen et al. [9] established the magnetic interaction for 2-DOF bistable VEH producing high-energy oscillation and improved the generating power. Also Wang et al. [10] used nonlinear magnetic force to creating the electro-magnetic BEH for improving the efficiency and broadening the harvester resonance bandwidth. Optimization of harvested power of both rectangular and trapezoidal bimorph piezoelectric cantilever beams with tip mass have been investigated by Kianpour and Jahani [11] analytically. Dynamic behavior of functionally graded carbon nanotube reinforced piezoelectric cantilever harvesters have been considered by Heshmati and Amini [12] by means of finite element method.

In addition to the bistable vibration energy harvesting systems, the tri-stable vibration energy harvesting systems has attracted the attention of researchers. The reason for using tri-stable oscillators compared to linear and bistable systams is that tri-stable oscillators perform better in terms of broadband resonance frequency and lower excitation threshold. Rezaei et al. used the tri-stable non-linear restoring force in order to increase the energy harvesting and efficiency of the piezoelectric harvester. [13, 14] They also investigated the simultaneous effects of two hard excitations for the piezoelectric energy harvester. The results showed that when superharmonic and combinations resonances exist simultaneously in the system response, the generated voltage and power increase. [15]

Elastic magnifiers are mostly modeled using a linear mass-spring system. Using finite element theory, Aladwani et al [16] analyzed a PEH with elastic magnifier. A double beam VEH with elastic magnifier reported by Vasic et al. [17] to obtain the harvesting power and vibrational behavior. Utilizing lumped parameter formulation, dynamic analysis of BPEH with Elastic Magnifier have been reported by Wang and Liao [18] with the aid of harmonic balance method. Large-amplitude oscillation of cantilever BPEH has been reported by Wang et al.[19] using an auxiliary mass-spring magnifier to overcome the potential wells. Bernard and Mann [20] coupled the structures of VEH and dynamic amplifier to increase the harvesting bandwidth and harvested voltage.

Because of high energy density, Vibration based Magnetostrictive energy harvester (MEH) have gradually developed in recent years. Nowadays the researchers has been attracted to MEH, since the MEH does not require high equivalent impedance and avoids the leakage problems and depolarization, in comparison with the PEH [21]. Ueno et al. [22] established a bimorph energy harvester based on two cantilever beams made of MsM. Kita et al. [23] analysed MEH made of Fe-Ga alloy (Galfenol) with 35% of conversion efficiency. Energy harvesting and Damping of the vibration based MEH has been analysed by Fang et al. [24]. Dynamic response of a magnetostrictive VEH has been investigated by Ahmed et al.[25], using the finite element model. Cao et al. [26] considered the nonlinear vibration analysis of MEH consist of a cantilever beam with elastic magnifier, analytically and examined the effects of magnifier on the harvested power. By means of harmonic balance method and COMSOL Multiphysics software, Zhang et al.[27] studied vibration control and energy harvesting of NES-magnetostrictive coupled model made by cantilever beam. Liu et al. [28] investigated the nonlinear vibration analysis of bistable vibration MEH with dynamic magnifier. In their study, the bistable motion was formed by nonlinear magnetic force between repulsive magnets. Goudarzi et al. [29] considered a hybrid piezoelectric-pyroelectric harvesting systems made by cantilever beam with PZT and lead magnesium niobate–lead titanate, were subjected to vibration and sinusoidal heat loads. A 2DOF hybrid piezoelectric-electromagnetic energy harvester has been considered by Wang et al.[30] to enhance the collected power from the electromechanic transducer. Sengha et al. [31] investigated stochastic dynamic analysis of the hybrid energy harvester with the nonlinear magnetic coupling. Jahanshahi et al. [32] studied dynamic behavior of hybrid piezo-magnetoelastic based harvester under low frequency excitations and multi-frequency excitations. Bistable rotational energy harvesting systems with hybrid piezoelectric - electromagnetic mechanisms have been reported by Fang et. al.[33] for growing output voltage at the low-frequency excitation. Lia et al. [34] considered mathematical Modeling of vortex shedding-induced vibration behavior of piezoelectric-electromagnetic hybrid energy harvesters. Magneto-rheological fluids are also another type of advanced materials which are considered in vibration analysis of smart materials and interested readers can refer to [35, 36].

In our previous work, [37], we employed an analytical procedure to consider dynamic analysis of a novel 2DOF hybrid VEH based on both piezoelectric and magnetostrictive mechanisms considering a dynamic magnifier using lumped parameter model. The influence of magnifier parameters have been examined on the time and frequency response features, in detail. In the present study, a hybrid 2DOF VEH consists of a three-layered architecture of a cantilever beam with core and smart layers (piezoelectric and magnetostrictive) is considered while taking into account the nonlinear interaction of the magnets. Based on our best knowledge, the effect of metallic core damping and the distance between two magnets on the electromechanic behavior, have been considered here for the first time. A parametric study is done to provide a theoretical background to be used for practical design of hybrid MS-P VEH.

2 NOVEL HYBRID VIBRATION ENERGY HARVESTER  

A schematic representation of the main components of the Hybrid Magnetostrictive Piezoelectric (HMP) vibration harvesting system with elastic magnifier (EM) is shown in Fig.1. The hybrid power generation system consists of a composite cantilever beam made of three layers of base matalic core and magnetostrictive and piezoelectric layers at top and bottom of the core. The thickness of the piezoelectric, magnetostrictive and the metalic base are shon respectively by hp , hm and hb. The total efective legnth of the beam is l which consists of two parts,(as seen at Fig.1).

In the present study, the whole vibrating system is simplified to a 2-DOF lumped parameter vibration model, in which M0 and K0 denote the equivalent mass and stiffness of the elastic magnifier, respectively. The equivalent mass, stiffness and damping of the cantilever beam are Me , Ke, and Ce respectively. Where [38]:

 

(1)

 

Fig.1 Schematic structure of piezoelectric-magnetostrictive harvesting system with EM.

 

 

 

 

In which Mt is the tip mass and equals to the product of the density and the volume of the tip magnet. Also m, E and ζ are the composite beam’s mass, Young’s modulus, and damping ratio, respectively. Ib is the moment of inertia of the beam cross section about the neutral axis and EIb is the average stiffness of the beam. the mass ratio rm and stiffness ratio rk of the harvester are introduced as:

 

(2)	rm = M0 /Me ; rk = K0 /Ke

The surface of piezoelectric layer are completely enclosed with tinny electrode and an resistance RP is coupled to the piezoelectric energy harvester electrically. Also A pick-up coil , with resistance Rc , turns N, length lc ≈ lm, and cross-sectional area Sc ≈ gm hm, in which gm is beam width ,is bounded on the cantilever and linked in series with resistance RL the power collection part is simplified as the load impedance.

Two magnets A and C, provide a bias magnetic field Hb = 3580 A/m for the Ms layer. Magnets A, B, and C have the same volume , VA = VB = VC  , and their magnetization are MA, MC and MB. The beam’s longitudinal axis is x1, whereas the transverse axis is x3, so that the x1 - x3 plane is set on the neutral plane of the beam. The EM consisting of a mass, and a spring element and positioned between the BHEH and the base. An acceleration  is applied to the base  and the  displacement of the mass Me and M0 are written as xe(t) and y(t), respectively.

As shown in Fig.1 magnet A is attached to the beam free end; its magnetic field direction is reverse polarity to the field of another fixed magnet B. The two-part permanent magnets are mutually repellentand creates a bistable structure. The magnet interaction is the nonlinear vertical force FN , Based on the taylor series is [28]:

 

(3)

The distance between two magnets A, B are denoted by d ,measured acording to the undeformed shape of the beam. By adjusting the parameter d , the force between them is varied. When this distance is proper, the system is bistable. The system currently has two steady-state equilibria, and the harvester displays bistable features.

3    MATHEMATICAL MODELING OF HARVESTING SYSTEM  

3.1 Magnetostrictive Layer

When the harvester vibrates, because of the villari effect the magnetic induction Bz in Ms material (Galfenol) is varied, so the current i and magnetic field Hz in the coil are induced, In which:

(4)	Hc = Ni/lc

 In which N is the Number of coil turns. The constitutive equations for Ms layer are [36]:

(5)

Where  denotes the strain, Hz = Hb + Hc signifies the strength of magnetic field , is the stress applied to the Ms, Em is the Young’s modulus of the Ms; is piezomagnetic coefficient and μ is magnetic permeability. The average stress  associated with i is given as [39]:

(6)

where,  is the distance from the neutral axis of beam cross section to the center of Ms layer. x is the relative displacement of the beam tip and l is the beam effective length as mentiond before.

From Eqs. (5),(6) the magnetic field strengthcan be found as :

(7)

As mententiond before, Hb is the biased magnetic field providing for the magnetostrictive material. Considering the electrical part of the Ms layer, the induced current  in the pick-up coil with length lc using faraday law, is [40]:

(8)

Where RM =(RL+RC ), in which RC , RL  are the coil resistance and load resistance, respectively. Replacing Eq. (7) into Eq. (8) and participating the subsequent equation with respect to x1 yields the harvested voltage of the Ms harvester part as:

(9)

In which [39] :

(10)

So the electrical equation of the Ms is expressed as:

(11)

3.2 Piezoelectric layer

For the piezoelectric layer attached on the cantilever beam, the piezoelectric constitutive equation can be written as [38, 41, 42]:

(12)

Whereandcharacterize the electric field and displacement in the z-direction, consistently.is the strain in the x1-direction; s11  is the compliance coefficient under a constant electric field;is the stress in the x1-direction;is the piezoelectric coefficient; is the electric field strength in the x3-direction; Dz is the electric displacement in the x3-direction; εT33 is Dielectric coefficient under constant stress. The average stress  associated with v is given as follow [43]:

(13)

hb is the thickness of the cantilever beam, le is the length of the piezoelectric layer. Also is the piezoelectric constant. Consider the relationship between the displacement x and the generated, It can be obtained from Eq. (13):

(14)

where Ep is the elastic modulus of the piezoelectric layer, Then:

(15)

Considering the Piezoelectric part, the electrical equation based on the Kirchhoff's law is expressed as [36]:

(16)

Where v, Cp and Rp are the voltage, capacitance and resistance of the piezo part, respectively.

 

4    COUPLING MODEL AND EQUATION OF FORMATION 

The coupled nonlinear equations of the hybrid 2-DOF harvester may be found by combination of the ordinary differential equations of the Ms and piezoelectric parts as follows, Using relative motion x, .

(17a)
(17b)
(17c)
(17d)

This study uses Runge-Kutta method to solve coupled Eqs. (17a)-(17d) and investigate the temporal output behaviours of the harvester. 

 

5    FREQUENCY RESPONSE OF THE SYSTEM

Firstly, utilizing the four coupled equations, by omitting y(t), a 4th-order nonlinear differential equation is attained as:

 

(18)

In which:  

The harmonic base excitation is equals to , where ,are the excitation frequency and acceleration amplitude, respectively.

The frequency domain solution of the nonlinear hybrid harvester studied by HBM. The convergence of this method is conquered by the selected harmonics. Based on HBM, the periodic displacement, voltage and current response can be written in the form of a Fourier series expansion, as:

(19)

In which, n is the highest harmonic order and k is the modal order. Substitution of Eq. (19) into Eqs. (17a)- (17d) gives coupled functions of harmonic coefficients. The steady-state response of the harvester can be obtained after the harmonic coefficients are solved out.

Without losing the generality, only first harmonic has been considered for the following calculations since the nonlinearity in the coupled equations of system is weak and the lumped parametes with linear strains considered [44]. Based on HBM, first order periodic solution of the vibration, voltage and current are respectively defined by:

(20)

Substituting Eqs. (20) into Eqs. (17a)-(17d), equaling the coefficients of and, respectively, six equations around variable coefficients a1,1 , a2,1 , b1,1 , b2,1 , c1,1 , c2,1  can be obtained. After some mathematical operation this relations for frequency response are reported:

(21a)
(21b)

Where:

(22)

Also , represents the vibration amplitude of the beam tip. Utilizing Eqs. (21a) and (15b), frequency response relation can be derived as:

(23)
(24)

Obtaining  , ,the harvested power of each circuit can be obtained. Then the total output power is sum of them and equals to:

© 2023 IAU, Arak Branch. All rights reserved.

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[1] Corresponding author. Tel.: [phone]. 

E-mail address: [e-mail] (M.J. Kazemzadeh-parsi)