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Piezoelectric energy harvesting from hybrid vibrations
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2014 Smart Mater. Struct. 23 025026
(http://iopscience.iop.org/0964-1726/23/2/025026)
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Smart Materials and Structures
Smart Mater. Struct. 23 (2014) 025026 (14pp) doi:10.1088/0964-1726/23/2/025026
Piezoelectric energy harvesting from hybridvibrations
Zhimiao Yan, Abdessattar Abdelkefi and Muhammad R Hajj
Department of Engineering Science and Mechanics, MC 0219, Virginia Tech, Blacksburg, VA 24061,USA
E-mail: [email protected]
Received 28 August 2013, revised 2 December 2013Accepted for publication 9 December 2013Published 9 January 2014
AbstractThe concept of harvesting energy from ambient and galloping vibrations of a bluff body with atriangular cross-section geometry is investigated. A piezoelectric transducer is attached to thetransverse degree of freedom of the body in order to convert these vibrations to electricalenergy. A coupled nonlinear distributed-parameter model is developed that takes intoconsideration the galloping force and moment nonlinearities and the base excitation effects.The aerodynamic loads are modeled using the quasi-steady approximation. Linear analysis isperformed to determine the effects of the electrical load resistance and wind speed on theglobal damping and frequency of the harvester as well as on the onset of instability. Then,nonlinear analysis is performed to investigate the impact of the base acceleration, wind speed,and electrical load resistance on the performance of the harvester and the associated nonlinearphenomena that take place. The results show that, depending on the interaction between thebase and galloping excitations, and the considered values of the wind speed, base acceleration,and electrical load resistance, different nonlinear phenomena arise while others disappear.Short- and open-circuit configurations for different wind speeds and base accelerations areassessed. The results show that the maximum levels of harvested power are accompanied by aminimum transverse displacement when varying the electrical load resistance.
Keywords: energy harvesting, galloping oscillations, harmonic excitations, nonlineardynamics, piezoelectric material
(Some figures may appear in colour only in the online journal)
1. Introduction
Piezoelectric energy harvesting from mechanical vibrationshas received attention because of its ease of application andits suitability for designing MEMS devices [1] and wirelesssensors [2, 3]. Several studies have shown that harvestingenergy from ambient [3–7] and aeroelastic vibrations [8–17]can effectively operate some commercial sensors. The mostcommon configuration for piezoelectric energy harvestingfrom ambient vibrations has been either a unimorph or abimorph piezoelectric cantilever beam. In their review paper,Anton and Sodano [18] presented different strategies toenhance the level of the harvested power using more efficientpiezoelectric materials, distinct piezoelectric mode couplings,and thorough optimization of the power conditioning circuitry.
Other investigations have focused on the beam geometry inorder to maximize the level of the harvested power [19–24].Different studies [10, 12–18, 25] have focused on thegeneration of electrical energy from aeroelastic vibrationsof airfoil sections and vortex-induced vibrations of circularcylinders [26–28]. Another aeroelastic phenomenon thathas shown promise for harvesting energy is the gallopingof prismatic structures. Sirohi and Mahadik [15] proposedharvesting energy from transverse galloping of a structurethat has an equilateral triangle section and generates morethan 50 mW at a wind speed of 11.6 mph. Abdelkefiet al [29] developed a nonlinear distributed-parameter modelfor galloping-based piezoaeroelastic energy harvesters andvalidated their model with the experimental measurementsof Sirohi and Mahadik [15]. They reported that the
10964-1726/14/025026+14$33.00 c© 2014 IOP Publishing Ltd Printed in the UK
Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 1. A schematic of the piezoaeroelastic energy harvester.
maximum levels of harvested power are accompanied byminimum transverse displacement amplitudes for a band ofload resistances. The effects of the cross-section geometry,Reynolds number and ambient temperature on the onset speedof galloping and the performance of piezoaeroelastic energyharvesters were investigated by Abdelkefi et al [30–32].
The galloping-based piezoaeroelastic system proposedby Abdelkefi et al [30–32] is composed of a beam–masssystem. As such, it is plausible to design a harvester that cangenerate energy from both ambient and galloping vibrations.Yet, the system’s response to both types of vibrations underdifferent conditions may not be a simple superposition of theindividual responses to both type of vibrations. In this work,we investigate the concept of harvesting energy from hybrid(ambient and galloping) vibrations and determine the effectsof the load resistance, wind speed, and base acceleration onthe performance of the harvester. To this end, a harvester thatconsists of a cantilever beam with a triangular cross-sectiongeometry tip mass attached to its end is considered. Details ofthe proposed harvester’s design and the governing equationsare presented in section 2. In section 3, a modal analysisis performed and a representative model for the hybridharvester is derived. Using the quasi-steady approximation,the galloping force and moment are presented in section 4.Linear and nonlinear analyses are performed in section 5 toinvestigate the effects of the load resistance, wind speed, andbase acceleration on the global frequency, electromechanicaldamping, and the performance of the harvester. Conclusionsare presented in section 6.
2. Representation of the energy harvester
The energy harvester under investigation consists of atriangular cross-section tip mass attached to a multilayeredcantilever beam. This energy harvester is subjected to twotypes of excitations, namely, base and galloping, as shown infigure 1. The cantilever beam is composed of one aluminumand two piezoelectric layers. The two piezoelectric sheets arebonded on both sides of the aluminum layer and connected inparallel with opposite polarity to an electrical load resistance.The geometric and material properties of the system arepresented in table 1.
The governing equation of motion of the relativetransverse displacement vrel = vrel(x, t) of a cantilever beamwhich is subjected to both base and galloping excitationsis obtained by using the Euler–Bernoulli beam assumptions,
Table 1. Physical and geometric properties of the cantilever beamand the tip body.
Es Aluminum Young’s modulus (GN m−2) 70Ep Piezoelectric material Young’s modulus
(GN m−2)62
ρs Aluminum density (kg m−3) 2700ρp Piezoelectric material density (kg m−3) 7800L Length of the beam (mm) 90b1 Width of the aluminum layer (mm) 38b2 Width of the piezoelectric layer (mm) 36.2hs Aluminum layer thickness (mm) 0.635hp Piezoelectric layer thickness (mm) 0.267mt Tip mass (g) 65Lstruc Length of the tip body (mm) 235bstruc Width of the tip body (mm) 30d31 Strain coefficient of piezoelectric layer
(pC N−1)−320
εs33 Permittivity component at constant strain
(nF m−1)27.3
which yields
∂2M(x, t)
∂x2 + ca∂vrel(x, t)
∂t+ m
∂2vrel(x, t)
∂t2= Ftipδ(x− L)
− Mtipdδ(x− L)
dx− [m+ mtδ(x− L)]
∂2vb(t)
∂t2
+ mtLc∂2vb(t)
∂t2dδ(x− L)
dx(1)
where δ(x) is the Dirac delta function, Ftip and Mtip are,respectively, the galloping aerodynamic force and moment atthe tip of the beam that are caused by the oscillation of thestructure and base excitation, L is the length of the beam, ca
is the viscous air damping coefficient, m is the mass of thebeam per unit length, vb is the base displacement, mt is the tipmass, Lc is the distance from the center of the tip mass to thetip of the cantilever beam and M(x, t) is the internal moment,which has three components. The first of these components
is the resistance to bending and is given by EI ∂2vrel(x,t)∂x2 . The
second component is due to the strain rate damping effect
and is represented by csI∂3vrel(x,t)∂x2∂t
. The third component is thecontribution of the piezoelectric sheets connected in parallel,which is represented by ϑp(H(x) − H(x − L))V(t), whereH(x) is the Heaviside step function, V(t) is the generatedvoltage and ϑp is the piezoelectric coupling term. The basedisplacement, vb, is written as vb = y0 cos(ωbt), where y0 isthe amplitude of the base displacement andωb is the excitationfrequency.
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
The piezoelectric coupling term, ϑp, is given by [4]
ϑp = −e31b2(hp + hs) (2)
where e31 = Epd31 is the piezoelectric stress coefficient, b2is the width of the piezoelectric layer, and hs and hp arethe thicknesses of the aluminum and piezoelectric layers,respectively.
Substituting the three components of the moment M(x, t)into (1), the equation of motion of the electromechanicalsystem is written as
EI∂4vrel(x, t)
∂x4 + csI∂5vrel(x, t)
∂x4∂t+ ca
∂vrel(x, t)
∂t
+ m∂2vrel(x, t)
∂t2+
(dδ(x)
dx−
dδ(x− L)
dx
)ϑpV(t)
= Ftipδ(x− L)−Mtipdδ(x− L)
dx
− [m+ mtδ(x− L)]∂2vb(t)
∂t2+ mtLc
∂2vb(t)
∂t2dδ(x− L)
dx(3)
where the stiffness EI and mass of the beam per unit length m
are given by: EI = 112 b1Eshs
3+
23 b2Ep[(hp +
hs2 )
3−
hs3
8 ] andm = b1ρshs+2b2ρphp, where Es and Ep are the Young’s mod-ulus of the aluminum and piezoelectric layers, respectively, ρsand ρp are the respective densities of these layers.
To relate the mechanical and electrical variables, theGauss law [33] is used as follows:
ddt
∫A
D · n dA =ddt
∫A
D2 dA =V
R(4)
where D is the electric displacement vector and n is the normalvector to the plane of the beam. The electric displacementcomponent D2 is given by the following relation [4]:
D2(x, t) = e31ε11(x, t)+ εs33E2 (5)
where ε11 is the axial strain component in the aluminum and
piezoelectric layers and is given by ε11(x, y, t) = −y ∂2vrel(x,t)∂x2
and εs33 is the permitting component at constant strain.
Substituting (5) into (4), we obtain the equationgoverning the strain–voltage relation:
− e31(hp + hs)b2
∫ L
0
∂3vrel(x, t)
∂t∂x2 dx−2εs
33b2L
hp
dV(t)
dt
=V(t)
R. (6)
3. Eigenvalue problem analysis
To characterize the nonlinear response of the harvesterand the effects of different parameters on its performance,we discretize the coupled governing equations by usingthe Galerkin procedure. This procedure requires the exactmode shapes of the mechanical system. These mode shapesare determined by dropping the damping, forcing, andpolarization from equation (3). Furthermore, the relative
transverse displacement, vrel(x, t), is expressed in thefollowing form:
vrel(x, t) =∞∑
i=1
φi(x)qi(t) (7)
where qi(t) are the modal coordinates and φi(x) are the modeshapes of a cantilever beam–mass system. The mode shapesare calculated as [29]:
φ(x) = A sinβx+ B cosβx+ C sinhβx+ D coshβx (8)
where the relation between β and ω is given by ω = β2√
EIm .
To obtain the relation between the different coefficientsin (8), we use the associated boundary conditions of thesystem and then normalize the eigenfunctions (orthogonalityconditions). These conditions are, respectively, given by:
φ(0) = 0, φ′(0) = 0 (9)
EIφ′′(L)− ω2MtLcφ(L)− ω2Itφ′(L) = 0 (10)
EIφ′′′(L)+ ω2MtLcφ′(L)+ ω2Mtφ(L) = 0 (11)
and∫ L
0φs(x)mφr(x) dx+ φs(L)Mtφr(L)+ φs
′(L)MtLcφr(L)
+ φs(L)MtLcφr′(L)+ φr
′(L)Itφs′(L) = δrs (12)∫ L
0φs′′(x)EIφr
′′(x)dx = δrsω2r (13)
where It is the rotary inertia of the tip mass mt and Lc is halfof the length of the tip mass. s and r are used to representthe vibration modes and δrs is the Kronecker delta, defined asunity when s is equal to r and zero otherwise.
Substituting equation (7) into equations (3) and (6) andconsidering the first mode, we obtain the following coupledequations of motion:
q(t)+ 2ξωq(t)+ ω2q(t)+ θpV(t) = f (t) (14)
V(t)
R+ CpV(t)− θpq(t) = 0 (15)
where ξ is the mechanical damping coefficient, f (t) isthe first mode of the external force due to galloping andbase excitations, which is expressed as: f (t) = φ(L)Ftip +
φ′(L)Mtip − [∫ L
0 mφ(x) dx + mtφ(L) + φ′(L)mtLc]∂2vb(t)∂t2
, ωis the fundamental natural frequency of the structure, and thecoefficients θp and Cp are the piezoelectric coupling term andthe capacitance of the harvester, which are given by θp =
φ′(L)ϑp and Cp =2εs
33b2Lhp
respectively.
4. Representation of the aerodynamic loads
The use of the quasi-steady hypothesis to evaluate theaerodynamic loads is justified by the fact that thecharacteristic time scale of the oscillations is much larger thanthe characteristic time scale of the flow [34, 35]. As such, thelift force FL and the drag force FD per unit length are writtenas
FL =12ρairU
2bstrucCL
FD =12ρairU
2bstrucCD(16)
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
where ρair is the density of air, U is the incoming wind speed,bstruc is the width of the bluff body at the tip, and CL and CD
are, respectively, the lift and drag coefficients. The tip forceand moment are determined by integrating the aerodynamicloads over the whole length of the galloping structure as
Ftip = −
∫ Lstruc
0(FL cosα + FD sinα) ds
Mtip = −
∫ Lstruc
0s(FL cosα + FD sinα) ds
(17)
where Lstruc is the length of the prismatic structure ands is the length coordinate along the tip body. Thesecoefficients depend on the angle of attack, α, as well as theReynolds number. The angle of attack is expressed as: α =
tan−1(vrel(L,t)+sv′rel(L,t)+vb(t)
U ).
The total aerodynamic force per unit length, Fy, appliedto the prismatic structure in the direction normal to theincoming flow is directly related to the lift and drag forcesand is given by:
Fy =12ρairU
2bstrucCy
= −12ρairU
2bstruc[CL cos(α)+ CD sin(α)] (18)
where Cy is the total aerodynamic force coefficient in thedirection normal to the incoming flow. Barrero-Gil et al [35]showed that the total aerodynamic force coefficient can beexpressed by a polynomial function of tan(α) in the form
Cy = a1 tanα + a3(tanα)3 (19)
where a1 and a3 are empirical coefficients obtained bypolynomial fitting of Cy versus tan(α). A positive value fora1 indicates that the structure is susceptible to galloping [36].Turning to the nonlinear coefficient a3, it is always negativebecause Cy always has a maximum value, which decreases asa function of the angle of attack. Both the linear and nonlinearcoefficients depend on the geometry of the cross-section.Here, we consider isosceles triangles with δ = 30◦. Theempirical values of a1 and a3, as determined by Barrero-Gilet al [35], are 2.9 and -6.2, respectively. Using the aboveequations, the aerodynamic force and moment at the tip areexpressed as:
Ftip =12ρairU
2bstruc
×
∫ Lstruc
0a1
(vrel(L, t)+ sv′rel(L, t)+ vb(t)
U
)+ a3
(vrel(L, t)+ sv′rel(L, t)+ vb(t)
U
)3
ds
Mtip =12ρairU
2bstruc
×
∫ Lstruc
0s
(a1
(vrel(L, t)+ sv′rel(L, t)+ vb(t)
U
)+ a3
(vrel(L, t)+ sv′rel(L, t)+ vb(t)
U
)3)
ds.
(20)
5. Results and discussions
Substituting the discretized form into equation (18), theexternal force due to galloping and base excitations isrewritten as:
f (t) = φ(L)Ftip + φ′(L)Mtip −
∂2vb(t)
∂t2
×
(∫ L
0mφ(x) dx+ mtφ(L)+ mtLcφ
′(L)
)=
12ρairU
2bstruc
[k1
q
U+ k2
vb
U+ k3
(q
U
)3
+ k4vb
U
(q
U
)2
+ k5
(vb
U
)2 ( q
U
)+ k6
(vb
U
)3]
+ k7vb (21)
where k1, k2, k3, k4, k5, k6 and k7 are given by:
k1 = a1(φ2(L)Lstruc + φ(L)φ
′(L)L2struc +
13φ′2(L)L3
struc)
k2 = a1(φ(L)Lstruc +12φ′(L)L2
struc)
k3 = a3
(φ(L)
∫ Lstruc
0(φ(L)+ sφ′(L))3 ds
+ φ′(L)∫ Lstruc
0s(φ(L)+ sφ′(L))3 ds
)k4 = 3a3
(φ(L)
∫ Lstruc
0(φ(L)+ sφ′(L))2 ds
+ φ′(L)∫ Lstruc
0s(φ(L)+ sφ′(L))2 ds
)k5 = 3a3(φ
2(L)Lstruc + φ(L)φ′(L)L2
struc +13φ′2(L)L3
struc)
k6 = a3(φ(L)Lstruc +12φ′(L)L2
struc)
k7 = −
(∫ L
0φ(x)m dx+ mtφ(L)+ φ
′(L)mtLc
).
(22)
Introducing the following state variables:
X =
X1
X2
X3
=q
q
V
, (23)
the equations of motion are rewritten as
X1 = X2 (24)
X2 = −
(2ξω −
ρairUbstruck1
2
)X2 − ω
2X1 − χX3
+ρairUbstruck2
2vb +
ρairbstruc
2U
× (k3X32 + k4vbX2
2 + k5v2bX2 + k6v3
b)+ k7vb (25)
X3 = −1
RCpX3 +
χ
CpX2. (26)
Clearly, these equations have the form
X = B(U)X+ Fb ++N(X,X,X)+G(U, vb)+ C(X, vb)
(27)
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 2. Variations of the (a) global frequency and (b) coupled damping as a function of the load resistance when U = 0 m s−1.
where
B =
0 1 0
−ω2n −
(2ξω −
ρairUbstruck1
2
)−χ
0χ
Cp−
1RCp
;
Fb =
0
k7vb
0
N(X,X,X) =
0
ρairbstruc
2Uk3X3
2
0
;
G(U, vb) =
0
12ρairUbstruck2vb
0
;
C(X, vb) =
0
ρairbstruc
2U(k4vbX2
2 + k5v2bX2 + k6v3
b)
0
.5.1. Linear analysis: effects of the load resistance and windspeed on the global frequency and electromechanicaldamping
The effects of the electrical load resistance on the naturalfrequency and damping of the hybrid harvester, andconsequently on the onset of galloping, are determined froma linear analysis of the coupled electromechanical problem.Inspecting the governing equations of motion (25), we notethat the matrix B gives a clear idea about the effects of thewind speed and load resistance on the overall damping andfrequency of the system.
The matrix B has a set of three eigenvalues λi, i =1, 2, 3. The first two eigenvalues are similar to those of a puregalloping problem in the absence of the piezoelectricity effect.These two eigenvalues are complex conjugates (λ2 = λ1). The
real part of these eigenvalues represents the electromechanicaldamping coefficient and the positive imaginary part representsthe global frequency of the coupled system. The thirdeigenvalue λ3 is a result of the electromechanical couplingand is always real and negative [6, 7]. It is noted that thestability of the trivial solution depends only on the real partof the first two eigenvalues, because λ3 is always real andnegative. The speed Ug for which the real part of λ1 iszero corresponds to the onset of instability or galloping andself-excited oscillations take place for higher values of windspeed.
Figure 2 shows the variation of the global frequency andthe coupled damping as a function of the load resistance whenthe wind speed is set equal to zero. It follows from figure 2(a)that when the load resistance is smaller than 104 � or largerthan 5 × 105 �, the global frequency is almost constant. Onthe other hand, the global frequency increases from 40.7 to44.2 rad s−1 as the load resistance is increased from 104 � to5×105 �. When R = 102 �, we refer to the global frequencyas the short global frequency. In contrast, when R = 108 �,we refer to the global frequency as the open global frequency.Turning to the coupled damping, figure 2(b) shows that thecoupled damping ratio of the system is smaller in the range ofload resistances between 102 and 103 � and between 106 and108 �. In the intermediate range of load resistances (104 � <
R < 105 �), maximum values of the coupled damping areobtained. The coupled damping ratio reaches its maximumvalue for a load resistance value of 3× 104 �.
To determine the effects of the load resistance on theonset of instability, we plot in figures 3(a) and (b) thevariations of the real part of the first two eigenvalues as afunction of the wind speed and the variations of the onsetspeed of galloping as a function of the load resistance,respectively. It follows from figure 3(a) that for a specificvalue of the wind speed, the real part of the first twoeigenvalues changes sign from negative to positive (onsetof instability). It is also noted that the load resistancesignificantly affects the onset of instability. This is clearer infigure 3(b), which shows that the onset speed of galloping
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 3. (a) Variations of the real part of the first two eigenvalues as a function of the wind speed for different values of the load resistanceand (b) variations of the onset speed of galloping as a function of the load resistance.
strongly depends on the value of the load resistance. In thelower and higher ranges of load resistance (102 � < R <
5×103 � and 5×105 � < R < 108 �), the effects of the loadresistance on the onset speed of galloping is negligible. On theother hand, in the intermediate range of load resistance, theonset speed of galloping changes significantly when varyingthe load resistance. This is expected, because the globaldamping of the system is higher in this intermediate range ofload resistance, as shown in figure 2(b).
The curves plotted in figure 4 show variations of theglobal frequencies as a function of the wind speed for differentload resistances. This analysis is helpful in determining theeffects of varying the wind speed on the global frequency ofthe harvester. It follows from this figure that, for different loadresistance values, the wind speed does not have much effecton the global frequency of the harvester. On the other hand,the variation of the global frequency as a function of the windspeed has a different tendency, depending on the associatedelectrical load resistance. In fact, when R = 103 and 106 �,the associated global frequencies decrease as the wind speedis increased. However, when R = 104 and 105 �, there is aspecific value of the wind speed at which the associated globalfrequency is maximum.
5.2. Nonlinear analysis
5.2.1. Analytical explanations and nonlinear phenomena.Before we perform nonlinear analysis of the governingequations of motion, we try to elaborate on a possiblenonlinear phenomenon that can arise when there are hybrid(base and galloping) excitations. To simplify the analysis,we assume that the external forcing terms (equation (19))that contain vb
U are relatively small compared to the puregalloping forcing term. This assumption allows us to neglectthe coupled terms in equation (19). Considering that the baseexcitation is a forced excitation and the galloping excitationis a self-excitation based on the above representation, thegoverning equations of motion of the harvester can besimplified and rewritten as the equation of motion of a
self-sustaining system with harmonic excitation [37]. Oneexample of such a system is the forced Rayleigh equation,which includes the self-excitation and the forced excitationand is written as
u+ ω20u = ε
(u− 1
3 u3)+ K cos�t (28)
where ω0 is the global frequency of the harvester dependingon the considered load resistance, � is the excitationfrequency, K is the forcing excitation and ε is a smallpositive parameter (for the galloping case). According toequation (23), it is noted that ε has the same sign asρairUbstruck1
2 − 2ξω, which is positive when the wind speed islarger than the onset speed of galloping.
Away from the possible resonances (primary, subhar-monic of order 1
3 , and superharmonic of order 3), the solutionof equation (26) is given by [37]:
u =
{4η
ω20 +
[(4η/a2
0)− ω20
]exp(−εηt)
}1/2
× cos(ω0t + β)+K
�2 − ω20
cos �t + O(ε) (29)
where a0 is the initial amplitude and η = 1 − 12 �
2K2(ω20 −
�2)−2.Inspecting equation (27), we note that the response of
the harvester is composed of a homogeneous solution, whichis the free-oscillation term due to galloping excitation, anda particular solution, which is a forced-oscillation term dueto harmonic direct excitation. Based on this analysis, weconclude that the harvester mainly oscillates due to thepresence of both excitations, with two harmonic frequencies,namely are ω0 and� (not at resonance). Thus, the response ofthe harvester is generally aperiodic, becoming periodic onlywhen the excitation frequency matches the global frequencyof the harvester.
Inspecting the free-oscillation term (homogeneous solu-tion) in equation (27), we note that its amplitude dependson the value of η = 1 − 1
2�2K2(ω2
0 − �2)−2, which
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 4. Variation of the global frequency as a function of the wind speed for different load resistances: (a) R = 103 �, (b) R = 104 �,(c) R = 105 �, (d) R = 106 �.
depends directly on the values of K and �. Therefore,the values of the forcing (acceleration) and frequencyexcitations significantly affect the value of η and hencethe oscillations due to galloping excitation. When η < 0(i.e. K >
√2�−1
∣∣ω20 − �2
∣∣), the free-oscillation term, whichis due to the galloping excitation, decays with time andonly the forced-oscillation takes place (periodic motion).The phenomenon associated with the increase of the forcingexcitation accompanied by a decay of the free-oscillations,which is due to galloping excitation, called quenching [37].For small values of the forcing excitation K, then η > 0and the steady-state response of the harvester contains boththe global frequency (ω0) and the excitation frequency (�).The free-oscillation term which is due to galloping excitationis expected to dominate when the excitation frequency isaway from the global frequency. This process of unlockingbetween both excitations is called pulling-out. On the otherhand, when the excitation frequency is almost equal tothe global frequency of the harvester, the response of theharvester changes significantly and the free-oscillation termwhich is due to galloping is entrained or locked onto theforced term which is due to direct excitation. Consequently,a synchronization of the response at the excitation frequency
takes place [37]. The frequency of external excitation whenK =√
2�−1∣∣ω2
0 −�2∣∣ is called the pull-out frequency.
5.2.2. Effects of the load resistance, wind speed, andbase acceleration on the harvester’s performance. Weinvestigate first the effects of the base acceleration on thefrequency-response curves of the harvester for different windspeeds and when the electrical load resistance is set equal to103 � and 104 �, as shown in figures 5 and 6, respectively. Itfollows from figure 5 that, when R = 103 �, the gallopingexcitation takes place for the cases when U is larger than3 m s−1. This is expected, because the onset speed ofgalloping at this load resistance is smaller than 3 m s−1.Furthermore, we note that the galloping effect is moreimportant as the wind speed is increased. This is clearerin the off-synchronization regions. In the synchronization orresonance region, the system can harvest energy from bothfree- and forced-oscillation contributions. In addition, thequenching phenomenon is more pronounced when the forcingexcitation or base acceleration value is larger. For instance,when the base acceleration is 0.7g, the case of direct excitationwithout the galloping effect is better at resonance whencompared to the cases which include galloping. This result can
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 5. Frequency-response curves for the tip displacement ((a), (c), (e)) and harvested power ((b), (d), (f)) when the load resistance is setequal to 103 � and for different base excitations and wind speeds. (a), (b) a = 0.1g, (c), (d) a = 0.3g and (e), (f) a = 0.7g .
be explained by the nonlinear effects of the wind speed on theamplitude of the oscillations at resonance. For the case whenthe load resistance is set equal to 104 �, as shown in figure 6,the tendency of the frequency-response curves changes for thecases when the wind speed is larger than 11 m s−1. At smallerwind speeds, the tendency of the frequency-response curves
is similar to a base excitation case without galloping. Atwind speeds values larger than 11 m s−1, there is a gallopingoscillation contribution which is clearer in the off-resonanceregions. This result can be explained by the fact that theonset speed of galloping is around 10 m s−1 when the loadresistance is set equal to 104 �. This result is the same for
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 6. Frequency-response curves for the tip displacement ((a), (c), (e)) and harvested power ((b), (d), (f)) when the load resistance is setequal to 104 � and for different base excitations and wind speeds. (a), (b) a = 0.1g, (c), (d) a = 0.3g and (e), (f) a = 0.7g .
different values of the base acceleration. However, when thevalue of the base acceleration is increased, the range of theexcitation frequency at which the harvester is near resonanceincreases. Furthermore, at a = 0.7g, more power is generatedat U = 0 m s−1 than at U = 3 m s−1, and is closer to the powerlevels generated at U = 5 m s−1. Furthermore, an increase
in the value of the base acceleration results in significanteffects associated with the quenching phenomenon. Thisphenomenon takes place when the excitation frequency, �, isvery close to the global frequency of the harvester, ω0, making∣∣ω2
0 −�2∣∣ very small and hence η positive. Consequently, the
response is constituted of forced-oscillations.
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 7. Frequency-response curves for the tip displacement ((a), (c), (e)) and harvested power ((b), (d), (f)) when the base acceleration isset equal to 0.3g and for different load resistances and speeds. (a), (b) U = 0 m s−1, (c), (d) U = 5 m s−1 and (e), (f) U = 13 m s−1.
The curves plotted in figure 7 show the frequency-response curves for the tip displacement and harvested powerfor different values of the load resistance and three distinctvalues of the wind speed, when the base acceleration isset equal to 0.3g. The plots show that the range of higherharvested power depends on the considered value of the load
resistance. This is particularly clear when the wind speedis set equal to zero (without galloping). Furthermore, theplots show that minimum values of the tip displacement areobtained when the load resistance is set equal to 104 or105 �. This result is explained by the high values of theglobal damping in this range, as shown in figure 2(b). On
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 8. Frequency-response curves for the tip displacement ((a), (c), (e)) and harvested power ((b), (d), (f)) when the load resistance is setequal to 103 � and for different base excitations and wind speeds. (a), (b) U = 0 m s−1, (c), (d) U = 5 m s−1 and (e), (f) U = 13 m s−1.
the other hand, it is noted that these two values of the loadresistance give a wider range of excitation frequencies atwhich the system can harvest more energy. At U = 5 m s−1,the frequency-response curves when the load resistance isequal to 102, 103, 106, and 107 � are affected by thegalloping excitation. This is predicted because the associatedonset speed of galloping is smaller than 5 m s−1 for all
load resistances. Inspecting figures 7(b) and (d), it is notedthat the existence of the galloping effect (U = 5 m s−1)decreases the level of the harvested power in the resonanceregion when the load resistance is equal to 102, 103, 106, and107 �. The highest levels of harvested power at resonanceare observed for the base excitation cases (without galloping).The interesting result is that maximum levels of harvested
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 9. Frequency-response curves for the tip displacement ((a), (c), (e)) and harvested power ((b), (d), (f)) when the load resistance is setequal to 104 � and for different base excitations and wind speeds. (a), (b) U = 0 m s−1, (c), (d) U = 5 m s−1 and (e), (f) U = 13 m s−1.
power are accompanied by minimum displacement valueswhen the load resistance is set equal to 104 and 105 �. AtU = 13 m s−1, all frequency-response curves are affectedby the wind speed because the onset speed of galloping forall considered load resistances is smaller than 13 m s−1, asshown in figures 7(e) and (f). Furthermore, minimum valuesof the tip displacement are associated with maximum levelsof harvested power when R = 104 and 105 �. It is also noted
that the quenching phenomenon is more pronounced whenR = 104 and 105 �.
The curves plotted in figures 8 and 9 show thefrequency-response curves for the tip displacement andharvested power for different wind speeds and base excitationsfor the load resistance values 103 � and 104 �, respectively.Inspecting these curves, we note that the tip displacementand harvested power increase as the acceleration of the base
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Smart Mater. Struct. 23 (2014) 025026 Z Yan et al
Figure 10. Short- and open-circuit configurations for different wind speeds when a = 0.3g: (a) displacement and (b) harvested power.
excitation is increased. It follows from figure 8 that theharvester is affected by the galloping excitation for differentforcing excitations when U = 5 and 13 m s−1. In addition, atthese speeds, the existence of the galloping effect decreasesthe level of the harvested power and displacement at theresonance region. On the other hand, it increases the level ofthe harvested power and tip displacement in the off-resonanceregion. It is also noted that the difference between thepull-out frequency and global frequency increases as theforcing excitation is increased. Inspecting the plotted curvesin figure 9, we note that the galloping effect takes place onlyat U = 13 m s−1 when the load resistance is set equal to104 �. The quenching phenomenon is clearer when increasingthe value of the forcing excitation when both the free- andforced-oscillations are present.
The short- and open-circuit configurations of the tipdisplacement and harvested power for different wind speedvalues, namely, U = 0, 5 and 13 m s−1 have differentresponses. These configurations are defined by setting theexcitation frequency equal to the short- and open-circuitglobal frequencies, which are 40.7 rad s−1 and 44.2 rad s−1,respectively. For the short-circuit configuration, maximumvalues of the tip displacement are obtained in the low rangeof load resistances. At higher values of the load resistance,a significant decrease in the tip displacement values isobserved. This is due to the fact that, when increasing theload resistance, the global frequency increases, as shownin figure 2(a), and then the harvester is not at resonancefor these resistances. For the open-circuit configuration, aninverse tendency is obtained. Furthermore, there is a range ofload resistances when minimum values of the tip displacementare accompanied by maximum levels of the harvested powerfor both the short- and open-circuit configurations. At U = 5and 13 m s−1, the tendencies of the short- and open-circuitconfigurations change significantly. We note that high valuesof the tip displacement are obtained in the low and high rangeof load resistances. The appearance of new branches in theshort- and open-circuit configurations for the tip displacementare due to the presence of the galloping oscillations. The
associated range of load resistances of these new branches issmaller when U = 5 m s−1 than that when U = 13 m s−1.Furthermore, in the intermediate range of load resistances,minimum values of the tip displacement are always obtained.This is due to the associated maximum global damping valuesin this range; hence, no galloping effect takes place. For theharvested power, it follows from figure 10(b) that new peaksof maximum levels of harvested power take place for bothcircuit configurations. The range of load resistances whenthe harvested power is maximum for both the short- andopen-circuit configurations is totally different from the rangeof load resistances when the tip displacement is maximized.
6. Conclusions
We have investigated the concept of harvesting energyfrom hybrid vibrations, namely, base and galloping of abluff body with a triangular cross-section. In order toconvert the associated oscillations to usable electrical power,a piezoelectric transducer is attached to the transversedegree of freedom of the prismatic mass. A nonlineardistributed-parameter model that takes into consideration thegalloping force and moment nonlinearities and the baseexcitation effect is derived. The effects of the load resistanceand wind speed on the overall damping, global frequency,and onset of instability were investigated through a linearanalysis of the coupled equations of motion. Then, a nonlinearanalysis was performed to investigate the effects of thebase acceleration, wind speed, and electrical load resistanceon the performance of the harvester and the associatednonlinear phenomena that take place. The results show that,depending on the interaction between the base and gallopingexcitations and the considered values of the wind speed, baseacceleration, and load resistance, new nonlinear phenomenatake place while others disappear. Furthermore, the existenceof both types of excitations leads to the presence of new peaksin the maximum levels of harvested power for both short- andopen-circuit configurations. The range of load resistances overwhich the harvested power is maximum for both the short-
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and open-circuit configurations is totally different from therange of load resistances over which the tip displacement ismaximized.
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