Contents lists available at ScienceDirect

Journal of Sound and Vibration

Journal of Sound and Vibration 391 (2017) 35–49

http://d0022-46

n CorrE-m

journal homepage: www.elsevier.com/locate/jsvi

Integration of a nonlinear energy sink and a giantmagnetostrictive energy harvester

Zhi-Wei Fang a, Ye-Wei Zhang a,b,n, Xiang Li a, Hu Ding a,d, Li-Qun Chen a,c,d,n

a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, Chinab Faculty of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136, Chinac Department of Mechanics, Shanghai University, Shanghai 200444, Chinad Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China

a r t i c l e i n f o

Article history:Received 1 July 2016Received in revised form12 December 2016Accepted 12 December 2016Handling editor: L.G. ThamAvailable online 14 December 2016

Keywords:Nonlinear energy sinkGiant magnetostrictive materialVibration controlEnergy harvestingNonlinear pumping phenomenaTarget energy transfer

x.doi.org/10.1016/j.jsv.2016.12.0190X/& 2016 Elsevier Ltd. All rights reserved.

esponding authors at: Shanghai Institute ofail addresses: zhangyewei1218@126.com (Y.-

a b s t r a c t

This paper explores a promising novel approach by integrating nonlinear energy sink(NES) and giant magnetostrictive material (GMM) to realize vibration control and energyharvesting. The vibration-based apparatus consisting of a NES, a Terfenol-D rod, and alinear oscillator (the primary system) is proposed. The mathematical model of the pro-totype under displacement driven has been established and simulated by utilizing theRunge-Kutta algorithm. The exhibited responses and the obtained electric energy arecomputed. Furthermore, the Fast Fourier Transform (FFT) of the resonant responses isperformed. The distribution of the input energy is calculated to evaluate the designedstructure. The instantaneous transaction of the energy is then examined by consideringthe energy transaction measure (ETM). Lastly, a parametric study is conducted for furtheroptimization. The numerical simulations demonstrate that the nonlinear pumping phe-nomena occur, that is, the target energy transfer (TET) that leads to a very efficient vi-bration suppression. In addition, the results also illustrate that the localized vibrationenergy can be converted into magnetic field energy due to the Villari effect and thentransformed into electric energy.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Vibration extensively exists in vehicles, aircraft, aerospace, and other engineering conditions. Vibration can do damage tothe equipment and reduce the working life of the machine. However, vibration is also a renewable source which can be usedto supply wireless sensor nodes. In recent decades, there has been a surge of research in the area of vibration control andenergy harvesting.

A nonlinear energy sink (NES) is composed of a viscous damper, a small mass, and a spring with cubic stiffness non-linearity. The underlying capacity of a well-designed NES provides a powerful ability for vibration mitigation and extendsthe possibility for a wide range of applications of target energy transfer (TET) [1,2]. The essential nonlinearity in NES cancapture the external kinetic energy to itself where the pumped energy can be dissipated within the damper.

During the past years, NES has been widely studied and employed in practice. Gendelman et al. present the evidence ofenergy pumping in two and three-degrees-of-freedom coupled oscillators with essential nonlinearities [3]. Then, Vakakis

Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China.W. Zhang), lqchen@staff.shu.edu.cn (L.-Q. Chen).

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–4936

et al. focused on the two-degree-of-freedom damped system in an impulsively excited and investigated the phenomena bymeans of two analytical techniques [4]. The results illustrated tshat the energy-pumping phenomena are caused by tran-sient resonance captures on a 1:1 resonance manifold of the system. Georgiades et al. thoroughly examined the nonlinearresonant interactions between a linear elastic continuum and an ungrounded NES [5]. Their work numerically presented theintrinsic capacity of using NESs as passive broadband energy absorbers. Sapsis et al. dealt with a two-degree-of-freedomsystem of coupled oscillators of the Hamiltonian dynamics and performed direct analytical conduct to the governingnonlinear damped equations of motion [6,7]. The authors demonstrated that the structure of periodic and quasiperiodicorbits can strongly influence the damped transitions if the damping is small enough. Wierschem et al. experimentallyvalidated the broadband TET properties of a two-degree-of-freedom nonlinear energy absorber [8]. Luongo and Zulli ade-quately analyzed the dynamics and the aeroelastic instability of a multi-d.o.f. system with a NES excited by harmonicexternal force in 1:1 resonance via a mixed Multiple Scale/Harmonic Balance algorithm [9,10]. Apart from utilizing the NESto passively control the amplitude of vibrations of the main structure, Luongo and Zulli also extended the NES as a passivecontrol device to an internally resonant nonlinear elastic string [11]. Zhang et al. employed the parallel NES to suppressexcessive vibration of the beamwith varying axial speed [12]. Wang et al. utilized a NES to reduce the galloping vibrations ofan elastically-mounted square prism [13]. Nayfeh and co-workers did considerable amount of work in the efficiency of NES[45–47]. The authors assessed a NES on vortex-induced vibrations of a circular cylinder [46]. They also criticized the ef-fectiveness of NES in controlling the limit cycle oscillations of a nonlinear aeroelastic system [47].

In recent years, a multiplicity of vibration energy harvesting apparatus based on electromagnetic, electrostatic, andpiezoelectric are exploited to power standalone systems. Chen and Jiang designed an electromagnetic energy harvestingmechanism utilizing 2:1 internal resonance [14]. Blokhina et al. discussed the steady-state behavior of an electrostaticvibration energy harvester operating in a constant-charge mode [15]. Lefeuvre et al. introduced three new vibration-powered piezoelectric generators based on a particular processing of the voltage delivered by the piezoelectric element andcompared them to usual generators [16].

Vibration-based energy harvesting can also be achieved in principle with the Villari effect of giant magnetostrictivematerial (GMM). Due to high energy density and long life cycles, a topic of scientific interest grows at a rapid pace recently.Lundgren et al. presented a prototype to convert vibrational energy to electric energy and performed numerical simulationand experimental verification [17]. Zhao and Lord investigated the GMM transducers under force driven and displacementdriven [18]. Then, the energy scavenged by Galfenol and Terfenol-D magnetostrictive rods are compared. Berbyuk et al.designed electric generators with Terfenol-D cores and used a linear magnetomechanical coupling model to describe thebehavior of GMM [19,20]. The simulation and experimental study showed that the maximum output power generated wasup to 242W. The authors also proposed a Galfenol-based transducer in which the device harvested about 0.45 W powerunder excitations with 50 Hz frequency [21]. Wang and Yuan set up a GMM-harvesting device with Metglas 2605SC andderived the model from the principle of Hamilton together with the Euler–Bernoulli beam theory [22]. In the experiments,the maximum electric power obtained can reach 200 mW at a low frequency and 576 mW at a high frequency. The quality ofmodeling can strongly influence the accuracy of the performance of the GMM-harvesting apparatus. To this end, Davinoet al. considered the effects of hysteresis, capacitive load, and vibration source [23,24]. Furthermore, the authors consideredthe eddy current induced by the Villari effect and dealt with the prestress and magnetic bias in the later works [25,26]. Adlyet al. experimentally tackled the analysis of the electrical power output which is strongly interrelated to the nonlinearity ofGMM [27]. Rezaeealam et al. looked into the anisotropy of Galfenol and employed the Armstrong model to characterize thebehavior of GMM [28]. Mohammadi and Esfandiari introduced a beam equipped with Metglas 2605SC in which the gen-erated power can reach 9.4 mW under excitation [29]. Viola et al. used the dynamic Preisach hysteresis model for GMM andexperimentally validated the capability of energy harvesting [30].

Utilizing passive TET to dissipate vibrational energy in the damper of NES and transform mechanical energy into electricenergy in harvester seemed to be very promising. In the past several years, there were a few papers combing NES andenergy harvesting device to mitigate vibration and scavenge energy. Kremer et al. integrated a NES with an electromagneticenergy harvester under transient responses [31]. Remick et al. numerically studied the single-impulsive dynamics of anenergy harvesting device composed of a NES and electromagnetic elements and experimentally validated the harvestingperformance [32]. Chtiba et al. collocated a simply supported beam with NES to absorb vibration and harvested energy byusing piezoelectric elements. [33]. Ahmadabadi et al. considered a NES and a piezoelectric-based harvester attached to afree–free beam and examined grounded and ungrounded configurations under shock excitation [34].

In this work, we present the first study of the integration of a nonlinear energy sink and a giant magnetostrictive energyharvester for vibration control and energy harvesting. The energy pumping phenomena can provide an effective approachfor vibration mitigation which is caused by 1:1 resonance captures of the dynamics. The intrinsic capacity of passive TET canlocalize external vibration energy from the linear structure to the nonlinear energy sink and dissipate that energy within thedamper. Meanwhile, the oscillation of the NES mass and the base will apply a mechanical stress to GMM. Then, the inducedmagnetic flux variation due to the Villari effect will be able to generate a current in the pick-up coils wounded around theGMM rod.

The rest of the paper is organized as follows. Section 2 introduces a vibration-based energy harvesting device consistingof a NES, a Terfenol-D rod, and a linear oscillator (the primary system) and formulates the mathematical model of theprototype. Section 3 numerically analyzes the exhibited responses and the obtained electric energy of the proposedstructure under displacement driven. Section 4 calculates the distribution of input energy in the system and investigates the

Fig. 1. Schematic diagrams of the prototype: (a) a linear oscillator with the NES-GMM system and (b) the cross section of a giant magnetostrictive energyharvester based on Terfenol-D rod.

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–49 37

instantaneous transaction of energy between the primary system and the NES-GMM system through energy transactionmeasure (ETM). Section 5 performs the parametric study for further optimization. Section 6 draws the conclusion of thework.

2. Formalizations

Fig. 1(a) depicts the schematic diagrams of the one-degree-of-freedom primary system with the NES-GMM system. TheNES-GMM system parallel with a spring with linear stiffness k1 and a viscous damper c1 is embedded between the rigidmass m1 and the base. The NES in the prototype is composed of a small mass m2, a moderate damper c2 and a spring withcubic stiffness k2. The GMM inset between the NES mass and the base is wounded by coils which make up a simple resistivecircuit together with the resistance R. The displacement x1 and x2 are used to describe the motion of masses m1 and m2

under external excitation u respectively. The proposed archetype is supposed to apply in the field of spacecraft. In con-sideration of the design of the GMM harvester, the structure may bring about more motions of the center of gravity whenthe GMM is placed in parallel to NES. A grounded GMM is more favorable to the attitude control of spacecraft as well as thereliability and the stability of the space vehicle. That is the reason why the priority is given to a grounded GMM instead ofplacing in parallel to NES. Anyway, if the parallel arrangement is found possible application in other engineering filed, it canbe similarly explored.

Utilizing Newton's second law, the governing equations can be developed as follows:

¨ + ( ̇ − ̇) + ( ̇ − ̇ ) + ( − ) + ( − ) =¨ + ( ̇ − ̇ ) + ( − ) − = ( )

m x c x u c x x k x u k x x

m x c x x k x x F

0

0 11 1 1 1 2 1 2 1 1 2 1 2

3

2 2 2 2 1 2 2 13

NES

where FNES is the force NES exerted on the energy harvester.The magnetic bias and the mechanical prestress must be considered in the practical application of GMM [17,20,24,26].

The magnetic bias is capable of magnetizing the giant magnetostrictive rod in its axial direction, and the mechanicalprestress is able to prevent tension stress which can damage GMM. As shown in Fig. 1(b), the Terfenol-D rod is compressedby a spring washer to create prestress and the permanent magnet is applied to generate magnetic bias. Deducing from thestate of stress can obtain the following equation:

σ σ π= −

= ( − ) ( )

F F F

d4 2

NES GMM 0

0GMM2

where FGMM is the force of GMM, F0 is the pre-applied force, s is the stress of GMM, s0 is the prestress, and dGMM is thediameter of the giant magnetostrictive rod.

The GMM element has ultra-high coupling coefficient and no depolarization problem when compared to the piezo-electric element. However, due to the nonlinear effect, it is difficult to integrate with micro-electromechanical systems [22].In the following, the quadratic moment- rotation model and the Jiles-Atherton model are combined to characterize themagnetostriction and the ferromagnetic hysteresis of GMM separately.

As detailed in [35,36], the relationship between the magnetostriction λ and the total magnetization M appears as:

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–4938

λ λ=( )M

M32 3

s

s2

2

where λs is the saturation magnetostriction and Ms is the saturation magnetization. Substituting Eq. (3) into the constitutivemodel

ε σ λ= + ( )E 4

yields

ε σ λ= +( )E M

M32 5

s

s2

2

where ε is the strain and E is the Young's modulus. Thus, another equation can be derived from the definition of strain andEq. (5),

σ λ Δ+ = − +( )E M

Mu x

l32 6

s

s2

2 2 0

GMM

where Δ0 represents the displacement caused by the prestress and lGMM is the length of GMM.The equations of the Jiles-Atherton model is given as follows [35–40]:

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

( )( )

α

δ α

= + ˜

= −

= −− ˜ −

= −= + ( )

H H M

M MHa

aH

dMdH

M Mk M M

M c M M

M M M

coth

7

e

an se

e

irr an irr

an irr

rev an irr

rev irr

where the parameter α̃ in the equations is defined as α α˜ ≡ + λ σ

μ M

9

2s 0

0 s2 and the directional parameter δ is determined as

( )δ = sign dHdt

. He is the effective magnetic field, H is the magnetic field intensity, α is the amount of domain interaction, μ0 is

the free space permeability, Man is the anhysteretic magnetization, a is the shape parameter, Mirr is the irreversible mag-netization, k is the average energy required to break pinning sites, t is the time, Mrev is the reversible magnetization and cdenotes the coefficient of proportionality. Eq. (7) can be manipulated and figured out to the following two equations [41,42]:

σσξ σ

δ α

= ( − ) ( − ) +

= ( − ) −− ˜( − )

+( )

dMd

cE

M M cdM

d

dMdH

cM M

k M Mc

dMdH

1

18

an irran

an irr

an irr

an

where ξ is the coefficient with dimensions of energy per unit volume. Taylor expansion is used to handle the secondequation in Eq. (7), and the higher order term is omitted as follows:

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

= −

= +

= ( )

M MHa

aH

MH

ao

Ha

M Ha

coth

3

3 9

an se

e

se e

3

3

s e

Substituting Eqs. (9) and (7) into Eq. (8) leads to

σα σξ α

σξ α

= ( ˜ − )( − ˜)

+( − ˜) ( )

dMd

M a ME a cM

M HE a cM

33 3 10s

s

s 0

s

where H0 represents the magnetic bias. Then, the relationship of M and s can be solved by the first order differentialequation with the initial condition (s¼0, M¼0) below:

α= −

− ˜ ( )

σ αξ α

( ˜ − )( − ˜)

MH M H M e

a M3 11

M aE a cM0 s 0 s

2 s 32 3 s

s

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–49 39

The magnetic flux density B is generally calculated as Eq. (12) in a giant magnetostrictive energy harvester [19,20]

σ μ= + ( )σB d H 1233

where d33 denotes the parameter of the magnetomechanical effect, μs is the permeability at constant stress. Based onFaraday's law of induction, the generated voltage U(t) and power P(t) can be derived below as

⎛⎝⎜

⎞⎠⎟

ϕ

σ μσ

σ

( ) =

=

= +

( ) = ( )( )

U t Nddt

N AdBdt

N A dddt

dMd

ddt

P tU t

R 13

coil

coil coil

coil coil 33 0

2

where Ncoil represents the coil turns number, ϕ denotes the magnetic flux, and Acoil is the cross-sectional area of the coil.Substituting Eq. (2) into (1) and rearranging Eqs. (6) and (11), the dynamical equations of the proposed device can be

finally established as

⎛⎝⎜⎜

⎞⎠⎟⎟( )

σ σ π

σ λα

Δ

¨ + ( ̇ − ̇) + ( ̇ − ̇ ) + ( − ) + ( − ) =

¨ + ( ̇ − ̇ ) + ( − ) − ( − ) =

+− ˜

− = − +

( )

σ αξ α

( ˜ − )( − ˜)

m x c x u c x x k x u k x x

m x c x x k x xd

EH

a Me

u xl

0

40

3

2 31

14

M aE a cM

1 1 1 1 2 1 2 1 1 2 1 23

2 2 2 2 1 2 2 13

0GMM2

s 02

s2

32 3

22 0

GMM

2 ss

which appeared to be differential algebraic equations (DAEs).

3. Simulations

The algorithm utilized to simulate the dynamical equations is the Runge-Kutta method. To handle the equations, thesecond-order DAEs are firstly reduced to the first-order DAEs. Then, the differential index of DAEs is checked and reduced,which is somewhat different from the process of the ordinary differential equations (ODEs). Finally, the equations of motioncan be worked out using MATLAB solver ode15s.

The vibration mode considered in the proposed structure is displacement driven

π= ( ) ( )u

A ftsin 2 15u

where f is the frequency and Au is the amplitude of the base excitation. Table 1 lists the parameters employed in thenumerical simulations.

As explained in the previous section, in contrast with the ultra-high compressive strength, the tensile strength of GMM isvery weak that mechanical prestress should be exerted to avoid damage to the structure. Fig. 2 plots the maximum andminimum stress of GMM in distinct frequencies to examine whether or not the GMM is out of action. With the increase offrequency, the curves change dramatically at 30 Hz, which appears to be the natural frequency of the primary system.

Table 1Parameters applied in simulations [18,19,34,35,42,43].

Parameters Values Parameters Values

m1 60 kg dGMM 0.0127 mm2 7 kg lGMM 0.115 mk1 2.1346�106 N/m s0 6.9�106 Pak2 500 N/m3 H0 1.592�104 A/mc1 10 N s/m Δ0 2.2352�10-5 mc2 1000 N s/m R 1 ΩAu 0.0002 m E 3�1010 N/m2

α̃ -0.01 d33 1.79�10-8 m/Aa 7012 A/m dcoil 0.0162 mξ 8�103 Pa Ncoil 600c 0.18 λs 1003�10-6

μ0 4π�10-7 N/ A2 Ms 7.65�105 A/m

Fig. 2. The maximum and minimum stress of GMM in distinct frequencies.

Fig. 3. The amplitude-frequency response curves: (a) with and without the NES-GMM system; (b) without the GMM and with the NES-GMM system.

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–4940

However, the pre-applied force still reaches its goal to prevent tensile stress.To validate the effectiveness of TET in suppressing vibration, the maximum amplitudes of the steady-state response

under various frequencies are computed. Fig. 3(a) compares the amplitude-frequency response curves with and without theNES-GMM system. The obtained results show that the designed apparatus can effectively reduce vibration, especially whenthe structure is vibrated at resonance. Fig. 3(b) depicts the amplitude-frequency response curves of the system with NES-GMM and the system with NES only. The amplitude of the system with NES-GMM at resonance is lower than that of thesystem with NES only. This graphic illustrates that the GMM is coupled with the two remaining components, which can beindicated from the governing equations.

Another concern in this work is the harvested electric energy of the proposed device under displacement driven. Fig. 4

Fig. 4. The maximum output (a) voltage and (b) power in different frequencies.

Fig. 5. The responses of a linear oscillator with the NES-GMM system at resonance.

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–49 41

(a) and (b) show the maximum output voltage and power in different frequencies separately. Both curves have a peak at thenatural frequency of 30 Hz. That is to say that more energy is generated when the proposed archetype is vibrated at re-sonance. The process of energy harvesting begins when the excitation starts. The maximum voltage induced is 1.79 V andthe maximum power output is 4.018 W.

Investigating the internal mechanism at the natural frequency of the primary system is more significant. Thus, thefrequency chosen for a later simulation is 30 Hz. Fig. 5 illustrates the responses of the linear oscillator with the NES-GMMsystem at resonance.

The Fast Fourier Transform (FFT) of the resonant responses is performed and revealed in Fig. 6(a) and (b) to explore thedominant frequency. Any peaks exhibited in the FFT diagram can indicate the frequency in which the system was excited[44]. The graphs of the frequency spectrum possess only one peak at the frequency of 30 Hz, indicating that the applicationof a NES does not increase the degree of freedom of the initial system when compared to the linear vibration absorbers.Emphasis should be given to the fact that NES does not entail the existence of a further frequency. The result of FFT iscorresponding to the former researchers’ work and can be explained by the perturbation method of Refs. [9–11], even if thepresent investigation does not perform the analysis.

4. Evaluations

To investigate the internal mechanism and evaluate the integration of a NES and a GMM harvester thoroughly, thedistribution of input energy in the system is primarily computed. The percentage of input energy absorbed in the primarysystem ηPS can be developed as

η ( ) = ( ) + ( ) + ( )( )

×( )

tT t V t W t

W t100%

16PSPS PS PS

in

where TPS and VPS denote the kinetic energy and the potential energy in the primary system respectively. WPS is the energydissipated within the damper of the linear oscillator, and Win is the input energy. The proportion of the input energyabsorbed in the NES-GMM system ηNES-GMM appears as follows:

Fig. 6. Fast Fourier Transforms (FFTs) of (a) x1(t) and (b) x2(t).

Fig. 7. Energy absorption of (a) the primary system and (b) the NES-GMM system.

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–4942

η ( ) = ( ) + ( ) + ( ) + ( ) + ( ) + ( )( )

×( )− t

T t V t W t W t W t W tW t

100%17NES GMM

NES NES NES h m s

in

where TNES and VNES represent the kinetic energy and the potential energy in the NES-GMM system separately. WNES is theenergy dissipated within the damper of NES, Wh is the energy harvested in GMM, Wm is the magnetic energy, and Ws isstrain energy of GMM. The input energy Win is calculated based on the conservation of energy

( ) ( ) ( ) ( ) ( )( ) = + ( ) + + + + + ( ) + ( ) + ( ) ( )W t T t V t W t T t V t W t W t W t W t 18in PS PS PS NES NES NES h m s

Substituting Eqs. (18) and (19) into Eqs. (16) and (17), the graphs of energy absorption can be obtained in Fig. 7(a) and (b).

⎡⎣ ⎤⎦

⎡⎣ ⎤⎦

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

( )

∫

∫

∫

( ) ( )

( )

( ) ( )

( ) ( ) ( ) ( )

τ τ τ

τ τ τ

τ

σ

( ) = ̇ ( )

( ) = ( ) − ( )

( ) = ̇ ( ) − ̇ ( )

( ) = ̇ ( )

( ) = −

( ) = ̇ ( ) − ̇ ( )

( ) =

( ) =

= − + Δ ( )

T t m x t

V t k x t u t

W t c x u d

T t m x t

V t k x t x t

W t c x x d

W tU t

Rd

W t B t H t A l

W t t A u t x t

1212

1214

1212 19

t

t

t

PS 1 12

PS 1 12

PS0

1 12

NES 2 22

NES 2 1 24

NES0

2 1 22

h0

2

m GMM GMM

s GMM 2 0

At the beginning of excitation, the percentage of energy absorbed in the primary system dramatically rises and thepercentage of energy absorbed in the NES-GMM system steeply declines, indicating that most of the input energy initiallylocalizes in the linear oscillator. Then, the energy absorption of the primary system decreases, whereas the energy ab-sorption of the NES-GMM system increases. Approximately 1.17% of input energy is distributed in the primary system and98.83% is stored in the NES-GMM system in 30 s. These demonstrate that the energy flowed from the linear oscillator to theNES-GMM system and the effective vibration suppression in Fig. 3 is caused by TET. The percentage of harvested energy inthe NES-GMM system is much lower than the proportion of damped energy in the NES-GMM system. However, the inducedvoltage and power (as show in Fig. 4) are sufficient for practical purposes, especially when compared to other vibration-based harvesters (piezoelectric, for example). In fact, the percentage of energy harvested in the NES-GMM system dependson the parameters chosen for simulation. That is the reason why a parametric study is performed for further optimization inSection 5.

Particularly, the percentage of the input energy damped ηNES,W and harvested ηGMM,h in the NES-GMM system is countedin Eq. (20) and plotted in Fig. 8(a) and (b), respectively. The distribution of the input that energy dissipated within the

Fig. 8. Energy distribution of input energy (a) damped and (b) harvested in the NES-GMM system.

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–49 43

damper of NES increases as time passed, reaching 97.45% in 30 s. The proportion of input energy harvested in the NES-GMMsystem steeply rises in a very short amount of time and then declines to 1.28% in 30 s. This may be due to the immediatemechanical response of GMM to excitation since it is embedded in the NES mass and the base.

( )

( )

η

η

= ( )( )

×

= ( )( )

×( )

tW tW t

tW tW t

100%

100%20

W

h

NES,NES

in

GMM,h

in

Apart from the distribution of input energy damped and harvested in the NES-GMM system, the percentage of the inputenergy damped in the primary system ηPS,W is calculated in Eq. (21) and depicted in Fig. 9(a). Here, the y-axis coordinaterange is from zero to 10%. The percentage of the input energy damped in the primary system is less than 1% in Fig. 9(a). Asshow in Fig. 8(a), the energy is mainly dissipated by the NES-GMM system in the steady regime. This outcome is in ac-cordance with the fact the added device strongly reduces the response.

( )η = ( )( )

×( )

tW tW t

100%21WPS,

PS

in

The distribution of the strain energy ηstrain, magnetic field energy ηmagnetic, kinetic energy ηNES,T and potential energyηNES,V is computed in Eq. (22) and is shown in Fig. 9(b).The obtained outcomes illustrate that their proportion is extremelysmall.

Fig. 9. The percentage of (a) damped energy in primary system and (b) strain energy, magnetic field energy, kinetic energy and potential energy in theNES-GMM system.

Fig. 10. (a) The energy transaction history and (b) the enlargement of the primary system, the NES system and the GMM system.

Fig. 11. (a) The energy transitions inside the NES system and (b) the enlargement and (c) the energy transitions inside the GMM system and (d) theenlargement.

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–4944

( )

( ) ( )( )

( )

( )

η

η

η

η

= ( )( )

×

= ×

= ( )( )

×

= ( )( )

×( )

tW tW t

tW t

W t

tT tW t

tV tW t

100%

100%

100%

100%22

T

V

strains

in

magneticm

in

NES,NES

in

NES,NES

in

Fig. 12. Color plots of the percentage of input energy (a) absorbed in the primary system (b) absorbed in the NES-GMM system (c) damped in NES and(d) harvested in GMM when the NES mass m2 varies. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–49 45

The energy transaction measure (ETM) can be used to deeply look into the energy exchange [5]. The positive value ofinstantaneous transaction of energy implies that the transient energy flowed into the system, and the negative value in-dicates that the transient energy is transferred out from the system. The transition of energy in the primary system Etrans,PS,the NES system Etrans,NES, and the GMM system Etrans,GMM is calculated in Eq. (23) and plotted in Fig. 10.

= Δ + Δ + Δ= Δ + Δ + Δ

= Δ + Δ + Δ ( )

E T V W

E T V W

E W W W 23

trans,PS PS PS PS

trans,NES NES NES NES

trans,GMM h m s

The numerical results demonstrate that the energy in the primary system and the GMM system is flowing back and forth,highlighting that the instantaneous transaction of energy in the NES system is always positive (as shown in Fig. 10(a)) andclearly visible (as shown in Fig. 10(b)). Therefore, the nonlinear beating phenomena occur and the energy is passivelypumped from the linear oscillator to the NES-GMM system. This validates that the effective vibration mitigation in Fig. 3 iscaused by TET from another aspect. Furthermore, Fig. 11 depicts the energy transitions inside the NES and the GMM systemin detail separately. The instantaneous transaction of harvested energy is less than damped energy in the NES-GMM at theresonance which corresponds to the distribution of input energy in Fig. 8.

5. Parametric study

A parametric study to examine the performance of vibration control and the efficiency of energy harvesting is performedwhich further optimizes the proposed archetype. As mentioned before, a NES consists of a small mass, a spring with cubicstiffness nonlinearity, and a viscous damper. First, the NES mass of various values is considered in Fig. 12. Color plots of the

Fig. 13. Color plots of the percentage of input energy (a) absorbed in the primary system (b) absorbed in the NES-GMM system (c) damped in NES and(d) harvested in GMM when the NES stiffness k2 varies. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–4946

percentage of input energy absorbed in the primary system ηPS, absorbed in the NES-GMM system ηNES-GMM, damped in NESηNES,W and harvested in GMM ηGMM,h when the NES mass m2 varies. The results show that the mass of NES can dramaticallyinfluence the distribution of input energy at the beginning of excitation. As the mass of NES grows, the process of energypumped into the NES-GMM system is quicker and the percentage of the harvested energy increases, which is clearly visiblein Fig. 12(d). This indicates that more electric energy can be picked up in GMM with the increase of the NES mass.

Then, the NES spring of distinct cubic stiffness is investigated in Fig. 13. Color plots of the distribution of input energy ηPS,ηNES-GMM, ηNES,W and ηGMM,h when the stiffness of NES spring k2 differ. The changes are not obvious. This corresponds to theenergy transitions in Fig. 11, where the instantaneous transaction of potential energy in the NES system is rarely small.However, the cubic stiffness of NES does have a significant contribution to the system. As mentioned anteriorly, the non-linearity of NES spring did not increase the degree of freedom of the initial system, bringing the energy pumping phe-nomenon when compared to the linear vibration absorbers.

The NES damper of different values is studied in Fig. 14. Color plots of the distribution of input energy ηPS, ηNES-GMM,ηNES,W and ηGMM,h when the damper of NES c2 changes. The graphs illustrate that the values of NES damper can intenselyaffect the performance. The process of energy beating is faster and the energy harvested is higher with the growth of c2.

6. Conclusions

In this paper, a promising novel approach by integrating nonlinear energy sink (NES) and giant magnetostrictive material(GMM) is explored to achieve vibration suppression and energy scavenging. The design of the vibration-based energyharvesting device consisting of a NES, a Terfenol-D rod, and a linear oscillator is presented. The quadratic moment-rotationmodel and the Jiles-Atherton model are combined to characterize the magnetostriction and the ferromagnetic hysteresis ofGMM. The mechanical prestress and the magnetic bias are also considered to establish the mathematical model. The

Fig. 14. Color plots of the percentage of input energy (a) absorbed in the primary system (b) absorbed in the NES-GMM system (c) damped in NES and(d) harvested in GMM when the NES damper c2 varies. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–49 47

dynamical equations are simulated by utilizing the Runge-Kutta algorithm. The amplitudes of the steady-state response andthe output voltage and power under various frequencies are computed. The Fast Fourier Transform (FFT) of the resonantresponses is performed to explore the dominant frequency. To systematically evaluate the integration of a NES and a GMMharvester, the distribution of input energy in the system is primarily computed. Then, the energy transaction measure (ETM)is employed to examine the instantaneous transaction of energy. Finally, a parametric study is conducted for furtheroptimization.

The investigation demonstrates that the combination of NES and GMM is able to realize vibration control and energyharvesting. The prestress applied in the prototype has met the intended goal to prevent tensile stress under displacementdriven. The designed apparatus can effectively reduce vibration, especially when the structure is vibrated at resonance. Themaximum voltage induced is 1.79 V and the maximum power output is 4.018 W. Furthermore, the FFTs of the resonantresponses validate that the application of a NES does not increase the degree of freedom of the initial system when com-pared to the linear vibration absorbers.

The distribution of input energy illustrates that most of the input energy initially localizes in the linear oscillator. Then,the energy is flowed from the linear oscillator to the NES-GMM system due to TET which leads to effective vibrationsuppression. The instantaneous transaction of energy in the NES system calculated through ETM is always positive, whichimplies that the transient energy is flowed into the NES system all the time. This indicates that the nonlinear beatingphenomena caused by 1:1 resonance capturing of the dynamics occurs and the vibration energy is passively pumped fromthe linear oscillator to the NES-GMM system. These validate that the effective vibration mitigation is caused by TET fromanother aspect.

The parametric studies check how the distinct values of NES mass, stiffness, and damper influence the performance ofthe integration of a NES and a GMM harvester. The results show that the mass of NES can dramatically influence thedistribution of input energy at the beginning of excitation. As the mass of NES grows, the process of energy pumped into theNES-GMM system is quicker and the percentage of harvested energy increases. However, the impact of the cubic stiffness ofNES which brings about the energy pumping phenomenon is rarely small. In contrast, the damper of NES can intensely

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–4948

affect the performance. The process of energy beating is faster and the energy harvested is higher with the growth of thevalues of NES damper. The above-mentioned will be beneficial for further optimization of the proposed structure.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Project Nos. 11402151, 11572182 and11232009). The authors acknowledge the funding support Natural Science Foundation of Liaoning Province (Project No.2015020106).

References

[1] A.F. Vakakis, O.V. Gendelman, L.A. Bergman, D.M. McFarland, G. Kerschen, Y.S. Lee, Nonlinear Targeted Energy Transfer in Mechanical and StructuralSystems, Springer Science & Business Media, 2008, 161–229.

[2] O.V. Gendelman, Targeted energy transfer in systems with external and self-excitation, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 225 (2011)2007–2043.

[3] O. Gendelman, L.I. Manevitch, A.F. Vakakis, R. M’closkey, Energy pumping in nonlinear mechanical oscillators: part I—dynamics of the underlyingHamiltonian systems, J. Appl. Mech. 68 (2001) 34–41.

[4] A.F. Vakakis, O. Gendelman, Energy pumping in nonlinear mechanical oscillators: part II—resonance capture, J. Appl. Mech. 68 (2001) 42–48.[5] F. Georgiades, A.F. Vakakis, G. Kerschen, Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-

linear end attachment, Int. J. Non-Linear Mech. 42 (2007) 773–788.[6] D.D. Quinn, O. Gendelman, G. Kerschen, T.P. Sapsis, L.A. Bergman, A.F. Vakakis, Efficiency of targeted energy transfers in coupled nonlinear oscillators

associated with 1:1 resonance captures: part I, J. Sound Vib. 311 (2008) 1228–1248.[7] T.P. Sapsis, A.F. Vakakis, O.V. Gendelman, L.A. Bergman, G. Kerschen, D.D. Quinn, Efficiency of targeted energy transfers in coupled nonlinear oscillators

associated with 1:1 resonance captures: part II, analytical study, J. Sound Vib. 325 (2009) 297–320.[8] N.E. Wierschem, D.D. Quinn, S.A. Hubbard, M.A. Al-Shudeifat, D.M. McFarland, J. Luo, L.A. Fahnestock, B.F. Spencer, A.F. Vakakis, L.A. Bergman, Passive

damping enhancement of a two-degree-of-freedom system through a strongly nonlinear two-degree-of-freedom attachment, J. Sound Vib. 331 (2012)5393–5407.

[9] A. Luongo, D. Zulli, Dynamic analysis of externally excited NES-controlled systems via a mixed multiple scale/harmonic balance algorithm, NonlinearDyn. 70 (2012) 2049–2061.

[10] A. Luongo, D. Zulli, Aeroelastic instability analysis of NES-controlled systems via a mixed multiple scale/harmonic balance method, J. Sound Vib. 20(2014) 1985–1998.

[11] A. Luongo, D. Zulli, Nonlinear energy sink to control elastic strings: the internal resonance case, Nonlinear Dyn. 81 (2015) 425–435.[12] Y.-W. Zhang, Z. Zhang, L.-Q. Chen, T.-Z. Yang, B. Fang, J. Zang, Impulse-induced vibration suppression of an axially moving beamwith parallel nonlinear

energy sinks, Nonlinear Dyn. 82 (2015) 61–71.[13] H.L. Dai, A. Abdelkefi, L. Wang, Usefulness of passive non-linear energy sinks in controlling galloping vibrations, Int. J. Non-Linear Mech. 81 (2016)

83–94.[14] L.-Q. Chen, W.-A. Jiang, Internal resonance energy harvesting, J. Appl. Mech. 82 (2015) 31004.[15] E. Blokhina, D. Galayko, P. Basset, O. Feely, Steady-state oscillations in resonant electrostatic vibration energy harvesters, IEEE Trans. Circuits Syst. I:

Regul. Pap. 60 (2013) 875–884.[16] E. Lefeuvre, A. Badel, C. Richard, L. Petit, D. Guyomar, A comparison between several vibration-powered piezoelectric generators for standalone

systems, Sens. Actuators A: Phys. 126 (2006) 405–416.[17] A. Lundgren, H. Tiberg, L. Kvarnsjo, A. Bergqvist, G. Engdahl, A magnetostrictive electric generator, IEEE Trans. Magn. 29 (1993) 3150–3152.[18] X. Zhao, D.G. Lord, Application of the Villari effect to electric power harvesting, J. Appl. Phys. 99 (2006) 08M703.[19] V. Berbyuk, Towards dynamics of controlled multibody systems with magnetostrictive transducers, Multibody Syst. Dyn. 18 (2007) 203–216.[20] V. Berbyuk, J. Sodhani, Towards modelling and design of magnetostrictive electric generators, Comput. Struct. 86 (2008) 307–313.[21] V. Berbyuk, Vibration energy harvesting using Galfenol-based transducer, in: SPIE Smart Structures and MaterialsþNondestructive Evaluation and

Health Monitoring, International Society for Optics and Photonics, 2013, pp. 86881F–86881F.[22] L. Wang, F.G. Yuan, Vibration energy harvesting by magnetostrictive material, Smart Mater. Struct. 17 (2008) 45009.[23] D. Davino, A. Giustiniani, C. Visone, Analysis of a magnetostrictive power harvesting device with hysteretic characteristics, J. Appl. Phys. 105 (2009)

07A939.[24] D. Davino, A. Giustiniani, C. Visone, Capacitive load effects on a magnetostrictive fully coupled energy harvesting device, IEEE Trans. Magn. 45 (2009)

4108–4111.[25] D. Davino, A. Giustiniani, C. Visone, W. Zamboni, Eddy current induced by Villari-effect in magnetostrictive energy harvesting devices, in: Proceedings

of the Digests of the 2010 14th Biennial IEEE Conference on Electromagnetic Field Computation, 2010.[26] D. Davino, A. Giustiniani, C. Visone, A.A. Adly, Energy harvesting tests with Galfenol at variable magneto-mechanical conditions, IEEE Trans. Magn. 48

(2012) 3096–3099.[27] A. Adly, D. Davino, A. Giustiniani, C. Visone, Experimental tests of a magnetostrictive energy harvesting device toward its modeling, J. Appl. Phys. 107

(2010) 09A935.[28] B. Rezaeealam, T. Ueno, S. Yamada, Finite element analysis of galfenol unimorph vibration energy harvester, IEEE Trans. Magn. 48 (2012) 3977–3980.[29] S. Mohammadi, A. Esfandiari, Magnetostrictive vibration energy harvesting using strain energy method, Energy 81 (2015) 519–525.[30] A. Viola, V. Franzitta, G. Cipriani, V. Di Dio, F.M. Raimondi, M. Trapanese, A magnetostrictive electric power generator for energy harvesting from

traffic: design and experimental verification, IEEE Trans. Magn. 51 (2015) 1–4.[31] D. Kremer, K. Liu, A nonlinear energy sink with an energy harvester: transient responses, J. Sound Vib. 333 (2014) 4859–4880.[32] K. Remick, D.D. Quinn, D.M. McFarland, L. Bergman, A. Vakakis, High-frequency vibration energy harvesting from impulsive excitation utilizing in-

tentional dynamic instability caused by strong nonlinearity, J. Sound Vib. 370 (2016) 259–279.[33] M.O. Chtiba, S. Choura, A.H. Nayfeh, S. El-Borgi, Vibration confinement and energy harvesting in flexible structures using collocated absorbers and

piezoelectric devices, J. Sound Vib. 329 (2010) 261–276.[34] Z.N. Ahmadabadi, S.E. Khadem, Nonlinear vibration control and energy harvesting of a beam using a nonlinear energy sink and a piezoelectric device,

J. Sound Vib. 333 (2014) 4444–4457.[35] M.J. Dapino, R.C. Smith, A.B. Flatau, Structural magnetic strain model for magnetostrictive transducers, IEEE Trans. Magn. 36 (2000) 545–556.[36] F.T. Calkins, R.C. Smith, A.B. Flatau, Energy-based hysteresis model for magnetostrictive transducers, IEEE Trans. Magn. 36 (2000) 429–439.[37] D. Lederer, H. Igarashi, A. Kost, T. Honma, On the parameter identification and application of the Jiles-Atherton hysteresis model for numerical

modelling of measured characteristics, IEEE Trans. Magn. 35 (1999) 1211–1214.

Z.-W. Fang et al. / Journal of Sound and Vibration 391 (2017) 35–49 49

[38] F. Liorzou, B. Phelps, D.L. Atherton, Macroscopic models of magnetization, IEEE Trans. Magn. 36 (2000) 418–428.[39] A. Benabou, S. Clénet, F. Piriou, Comparison of Preisach and Jiles–Atherton models to take into account hysteresis phenomenon for finite element

analysis, J. Magn. Magn. Mater. 261 (2003) 139–160.[40] K. Chwastek, J. Szczygłowski, An alternative method to estimate the parameters of Jiles–Atherton model, J. Magn. Magn. Mater. 314 (2007) 47–51.[41] D.C. Jiles, J.B. Thoelke, M.K. Devine, Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of

ferromagnetic hysteresis, IEEE Trans. Magn. 28 (1992) 27–35.[42] D.C. Jiles, Theory of the magnetomechanical effect, J. Phys. D: Appl. Phys. 28 (1995) 1537.[43] K. Yang, Y.-W. Zhang, H. Ding, T.-Z. Yang, Y. Li, L.-Q. Chen, Nonlinear energy sink for whole-spacecraft vibration reduction, J. Vib. Acoust. http://dx.doi.

org/10.1115/1.4035377.[44] A. Wicks, Shock & vibration, Aircraft/aerospace, and energy harvesting, Volume 9, in: Proceedings of the 33rd IMAC, a Conference and Exposition on

Structural Dynamics, Springer, 2015.[45] P. Malatkar, A.H. Nayfeh, Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator, Nonlinear Dyn. 47 (2007)

167–179.[46] A. Mehmood, A.H. Nayfeh, M.R. Hajj, Effects of a non-linear energy sink (NES) on vortex-induced vibrations of a circular cylinder, Nonlinear Dyn. 77

(2014) 667–680.[47] Y. Bichiou, M.R. Hajj, A.H. Nayfeh, Effectiveness of a nonlinear energy sink in the control of an aeroelastic system, Nonlinear Dyn. (2016) 1–17.