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Piezoelectric vortex induced vibration energy harvesting in a randomflow fieldTo cite this article: Sondipon Adhikari et al 2020 Smart Mater. Struct. 29 035034

 

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Piezoelectric vortex induced vibrationenergy harvesting in a random flow field

Sondipon Adhikari1 , Akshat Rastogi2 and Bishakh Bhattacharya2

1College of Engineering, Swansea University, Bay Campus, Fabian Way, Crymlyn Burrows, Swansea,SA1 8EN, United Kingdom2Department of Mechanical Engineering, Indian Institute of Technology Kanpur, India

E-mail: S.Adhikari@swansea.ac.uk, akshat@iitk.ac.in and bishakh@iitk.ac.in

Received 29 June 2019, revised 12 September 2019Accepted for publication 28 October 2019Published 19 February 2020

AbstractVibration-based energy harvesters have significant potential for sustainable energy generationfrom ambience for micro-scale systems like wireless sensor networks and similar low powerelectronic devices. Vortex-induced vibration (VIV) is one of the richest sources for such powergeneration for devices installed within a fluid environment. However, uncertainty in the directionand magnitude of the free stream velocity could affect the performance of such systems. Wehave first developed the mathematical model of a piezoelectric cantilever beam with end massvibrating under the influence of VIV. The piezo patch is assumed to be in the unimorph andbimorph configurations. From the unimodal dynamic response of the system, an equivalentsingle degree of freedom mechanical model is developed. This is further integrated with theelectrical model of the piezoelectric system without and with an inductor. The energy harvestedfrom the deterministic harmonic excitation is estimated against the non-dimensional velocityparameter. A random process model is developed considering the excitation force due to vortexshedding to be a bounded, weakly stationary and narrowband random process. The powerspectral density of the random process is obtained using the Fourier transform of the auto-correlation function. The dynamic response of the energy harvester is obtained against suchrandom excitations. The expressions of the mean power are obtained in closed-formcorresponding to the cases without and with the inductor integrated to the electrical circuit. It isobserved that while for cases without the inductor, the average harvested power monotonicallydecreases with increase in damping ratio and decrease in the coupling factor; for models with theinductor, an optimal inductor constant exists corresponding to the maximum mean-powercondition. The extensive analytical modelling and initial representative results are expected topave the way for the practical design of VIV based piezoelectric energy harvesting systemsubjected to stochastic excitation.

Keywords: energy harvesting, piezoelectric, vortex induced vibration, stochastic, optimal design

(Some figures may appear in colour only in the online journal)

Nomenclature greek symbols

α dimensionless time constant a w= C Rn p l

a b,c c stiffness and mass proportional dampingfactors

β dimensionless inductor constant b w= L Cn i p2

DM ratio of the tip mass and the mass of the cantilever

γ uniform random variable between 0 to p2

gca constant for piezoelectric patch (bimorph orunimorph)

κ electromechanical coupling coefficient k =q kCp

2

lj natural frequencies of the beam

ν central vortex shedding frequency

Ω dimensionless frequency w wW = n

Smart Materials and Structures

Smart Mater. Struct. 29 (2020) 035034 (18pp) https://doi.org/10.1088/1361-665X/ab519f

0964-1726/20/035034+18$33.00 © 2020 IOP Publishing Ltd Printed in the UK1

ω frequency

wn natural frequency of the harvester

ws frequency of vortex shedding

fjmode shapes of the beam

ρ density of the fluid

rhdensity of the beam

i unit imaginary number = -i 1

σ strength of the random process

θ electromechanical coupling

c1 strain-rate-dependent viscous damping coefficient

c2 velocity-dependent viscous damping coefficient

ζ damping factor

A cross-section area of the beam

bc width of the piezoelectric patch

c damping of the harvester

CL lift coefficient

Cp capacitance of the piezoelectric layer

D diameter of the cylinder

d31 constant for piezoelectric material

F0 non-dimensional forcing amplitude =F0

u1

2C

M2 L

r

fL(t) vortex induced forcing to the harvester

h thickness of the beam

hc thickness of the piezoelectric patch

I second-moment of the cross-section ofthe beam

k equivalent stiffness of the harvester

L length of the beam

Lc length of the piezoelectric patch

Li inductance

m equivalent mass of the harvester

Mr dimensionless relative mass r=M m Dr3

Rl load resistance

S Strouhal number

t time

U free stream velocity

u non-dimensional free stream velocityw=u U D n

( )wV i Fourier transform of voltage v(t)

v(t) voltage

W(t) standard Wiener process

Wc work done by the piezoelectric patch

( )wY i Fourier transform of displacement y(t)

y(t) displacement of the mass

( )Z x t, transverse displacement of the beam

Other variables

( )• * complex conjugation

∣ ∣• absolute value

det • determinant of a matrix

[ ]E • expectation operator

( )wF• power spectral density

R• autocorrelation function

1. Introduction

Researchers over the years have been trying different ways toprepare for the energy-intensive lifestyle of mankind infuture. Significant research has been performed on harnessingenergy present in the environment such as kinetic energy ofthe wind, thermal energy which is abundant in tropicalcountries, high velocity of river water using which damsextract energy etc. The energy harvesting can be implementedon micro or macroscale depending on the application and thepower consumption. Of the micro-scale systems, the vibrationdependent energy harvesters, typically based on piezoelectricmaterials, are most common where the cause of vibration isthe differentiating factor. In this paper, we consider the energyharvesting from a cantilever beam with a piezo patch wherethe source of vibration is the vortex shedding in the wake of abluff body attached at the free-end of the cantilever beam(Park et al [1]).

Broadly, there are two approaches to model a vortexinduced vibration (VIV) problem. First is formulating it byusing the fluid mechanics intensive computational fluiddynamics (CFD) approach while the other is using the phe-nomenological models to simulate the behaviour of vortexshedding. In the CFD based approach, formulations likearbitrary Lagrangian–Eulerian (Hirt et al [2]), immersedboundary method (Wang [3]), accelerated reference framemethod (Mehmood et al [4]), moving immersed boundarymethod (Cai et al [5]) and spectral methods (Soti et al [6],Yieser and Townsend [7]) have been used. Due to theinherent complexity of the Navier–Stokes equations and highcomputational requirements, researchers have used phenom-enological models to simulate and study the inherent beha-viour of the vortex shedding phenomenon. Of thephenomenological models, there are two approaches forstudying VIV depending upon whether the feedback effect isincorporated or not. The simple linear harmonic model(Blevins [8]), also called the forced model by Païdoussis et al[9], does not incorporate the feedback effect while there arecoupled nonlinear models such as wake oscillator modelwhich do. The wake oscillator models predict the self-limitingand self-exciting characteristics (de Langre [10]) of theshedding phenomenon. Bishop and Hassan [11] first sug-gested the use of Van der Pol oscillator in modelling thebehaviour of the vortex shedding. Tamura and Matsui [12]proposed the Birkhoff model for modelling vortex shedding

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Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

phenomenon for circular cylinders. Amongst all the wakeoscillator models, the most commonly used is Van der Poloscillator which has been studied and refined over the years(see for example Skop and Balasubramaniam [13]) to matchclosely with the physical vortex shedding phenomenon.Facchinetti et al [14] analyzed coupled models (displacement,velocity, acceleration coupling) and demonstrated that theacceleration coupling closely simulates the VIV principle bymatching the behaviour near lock-in with the experimentalresults. Benaroya et al [15] provided a variational basedapproach to study the fluid-solid interaction problem. Far-shidianfar and Zanganeh [16] modelled the structural oscil-lator also as a Van der Pol oscillator reasoning that the modelthen accounts for the compliance with a wide range of massratios. Gabbai and Benaroya [17] in their review paper gavean overview of VIV for circular cylinders. Several authors(Pavlovskaia et al [18], Hoebeck and Inman [19], Bourguetet al [20], Antoine et al [21]) have extended the single degreeof freedom (SDOF) formulation of Van der Pol oscillator to acontinuous system such as a flow across a long cylinder orcables and Wu et al [22] reported a review paper on the same.

On a macro scale, it is the role of the designers andengineers to curb the harmful effect of the fluid-solidinteraction phenomenon as large amplitude motion of thestructure may cause serious damage. On micro-scale, thesame phenomenon can be advantageous in the sense that itcan be used to harvest the energy arising out of the vibrationof the structure. Dai et al [23] mounted a bluff body at theend of a piezoelectric beam and performed nonlinear ana-lysis to determine the number of modes needed to evaluatethe performance of harvester. Mehmood et al [4] discuss theenergy harvesting from VIV of a cylinder in the regime of96�Re�118 and also incorporates the effect of loadresistance. The results show that there is an optimumresistance for which the harvested energy is maximum. Sotiet al [6] considered the energy harvesting from the VIV of acylinder and used a conducting coil in a magnetic field toharness the energy from vibration. Hu et al [24] showed themeans to increase the unstable range of VIV by attachingtwo thin rods at circumferential location of q = 60 oneither side of the cylinder while Zhang et al ([25, 26]) usedan interference cylinder of various shapes [25] and nonlinearmagnetic forces [26] to enhance the harvested powerin VIV.

Extensive work has been done on studying othermechanisms of energy harvesting from flow-induced vibra-tions. Li et al [27], Hamlehdar et al [28], Daqaq et al [29],Zhao and Yang [30] and Abdelkefi [31] in a review paperdiscuss the existing research on harvesting energy from flow-induced vibration mechanisms such as VIV, turbulenceinduced vibration, galloping and wake galloping. Hoebeckand Inman [19] discuss the energy harvesting from a light-weight and highly robust piezoelectric grass subjected toturbulence induced vibration. Petrini and Gkoumas [32] dis-cuss the energy harvesting from the vibration arising due tothe simultaneous effect of vortex shedding and galloping.

Shan et al [33] have shown energy harvesting from micro-fibre piezoelectric energy harvester placed in the wake of avibrating cylinder in water flow.

Although extensive research has been carried out onharvesting power from VIV of a bluff body such as a cylinderin a steady and uniform flow field, not much focus has beengiven on the randomness which may occur in the flow field. Itis important in the context since although we can guarantee aconstant flow field in a laboratory setup, this cannot beensured in real-life situations. There will be inherent vari-abilities associated with naturally occurring fluid flow, whichin turn acts as the main excitation to the energy harvester.This paper develops new approaches for energy harvestingfrom VIV when the flow is random in nature. The dynamicsof cantilever energy harvesters with a bluff-body at the tip andreduced-order modelling approaches are discussed insection 2 (further details are in the appendix). Piezo patchwith unimorph and bimorph configurations have beenemployed. In section 3, the SDOF electromechanical model isbriefly reviewed, basic equations describing the system arederived and key non-dimensional parameters are introduced.The random process model describing the fluid flow isintroduced in section 4. General approaches to obtain themean of the harvested power due to random flow are outlinedin section 5. Analytical methods to quantify harvested powerare developed in section 6. A closed-form expression ofaverage harvested power for systems without an inductor isderived in section 6.1, while the same quantity for systemswith an inductor is derived in subsection 6.2. The analyticalexpressions derived in the paper are numerically applied toillustrative problems. Based on this study, the main conclu-sions are drawn in section 7.

2. Model-order reduction for vibration energyharvesters

2.1. The partial differential equation

Vibration energy harvesters are often implemented as a can-tilever beam with a piezoelectric patch and a mass attached atthe tip of the beam [34]. Due to the small thickness to lengthratio, Euler–Bernoulli beam theory is generally used to modelbending vibration of such cantilevers. The equation of motionof free-vibration of a damped cantilever modelled (see forexample [35]) using Euler–Bernoulli beam theory can beexpressed as

( ) ( ) ( )

( ) ( )

r¶

¶+

¶¶ ¶

+¶

¶

+¶

¶=

EIZ x t

xc

Z x t

x tA

Z x t

t

cZ x t

t

, , ,

,0. 1

h

4

4 1

5

4

2

2

2

In the above equation x is the coordinate along the length ofthe beam, t is the time, E is the Young’s modulus, I is thesecond-moment of the cross-section, A is the cross-sectionarea, ρh is the density of the material and Z(x, t) is the

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Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

transverse displacement. The length of the beam is assumedto be L. Additionally c1 is the strain-rate-dependent dampingcoefficient, c2 is the velocity-dependent viscous dampingcoefficient. The strain-rate-dependent damping can be used tomodel inherent damping property of the material of the can-tilever beam. The velocity-dependent viscous damping can beused to model damping due to external factors such as acantilever immersed in a fluidic environment. The presentmodel uses only an isotropic material property for the beam.Anisotropic/orthotropic materials, such as composite mate-rials, are increasingly being used along with piezoelectrictransducers for vibration energy harvesting. In order toincorporate such advanced materials, different equations willbe necessary. However, the reduced-order model in thosecases (anisotropic/orthotropic) would follow an approachsimilar to what is to be presented here. Schematic diagram ofcantilever vibration energy harvesters with unimorph andbimorph configurations are shown in figure 1. An equivalentSDOF model can be obtained from this equation. Thenecessary details are given in the appendix.

2.2. The electromechanical coupling

Equivalent mass, damping and stiffness considering the firstmode of vibration can be obtained following the approach inthe appendix. The analysis presented there does not incorporateany piezoelectric effect. Here we will take this into account.Suppose that piezoelectric layers added to a beam is either in aunimorph or a bimorph configuration as shown in figure 1.Then the moment about the beam neutral axis produced by avoltage V across the piezoelectric layers may be written as

( ) ( ) ( )g=M x t V t, . 2c

The constant γc depends on the geometry, configuration andpiezoelectric device and V(t) is the time-dependent voltage. Fora bimorph as in figure 1(b), with piezoelectric layers in the 31configuration, with thickness hc, width bc and connected inparallel

( ) ( )g = +Ed b h h . 3c c c31

Here h is the thickness of the beam and d31 is the piezoelectricconstant. For an unimorph as in figure 1(a), the constant is

⎜ ⎟⎛⎝

⎞⎠¯ ( )g = + -Ed b h

hz

2, 4c c

c31

where z is the effective neutral axis. These expressions assumea monolithic piezoceramic actuator perfectly bonded tothe beam.

The work done by the piezoelectric patches in moving orextracting the electrical charge is

( ) ( ) ( )ò k=W M x t x x, d , 5c

L

c0

c

where Lc is the active length of the piezoelectric material, whichis assumed to be attached at the clamped end of the cantileverbeam. The quantity κc(x) is the curvature of the beam and this isapproximately expressed by the second-derivative of the dis-placement. Using the approximation for κc we have

( )q=W V , 6c

where the coupling coefficient

( ) ( )òq gf

=¶¶

x

xxd . 7c

L

0

2

2

c

Considering the change to the non-dimensional variable inequation (A.4) and noting that we have the second-order deri-vative within an integral, the above equation can be simplifiedas

( ) ( )qgf x= ¢

L, 8c

c

where x = L Lc c is the fraction of the length of the piezo patch.Using the first mode shape, as an example when the piezo patchcovers the full length of the beam, this can be evaluated as

( ) ( ( ) ( ) ( ) ( ))( ) ( )

( )fl l l l l

l l¢ =

++

12 cos sinh cosh sin

cosh cos. 9

Figure 1. Cantilever vibration energy harvesters with unimorph and bimorph configurations. The length of the cantilever is L and the mass ofthe bluff body attached at the tip is M.

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Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

3. The electromechanical model of the energyharvester

Various types of piezoelectric harvesting devices are available,integrating stack or patch transducers. Often these can berepresented mechanically as a SDOF system, particularly whenthe excitation is band-limited and the natural frequencies arewell separated. In this paper, we use a beam type harvester witha tip mass (in unimorph and bimorph configuration) as shownin figure 2. The tip mass is a cylinder that acts a bluff bodywhen placed in a running fluid to produce VIV in the system.The tip mass not only produces the effect of shedding ofvortices in its wake but also increases the strain in the piezo-electric material and the separation between the first and sec-ond natural frequencies. The two typical types of circuits usedin this paper, namely without and with an inductor, are shownin figures 2(a) and (b) respectively. A more detailed model ofthe cantilever beam harvester, along with correction factors fora SDOF model that accounts for distributed mass effects, wasgiven by Erturk and Inman [36–39]. This enables the analysisdescribed here to be used in a wide range of practical appli-cations, providing that the VIV does not excite higher vibrationmodes of the harvester. The SDOF model could be extended tomulti-degree of freedom mechanical systems by using a modaldecomposition of the response.

3.1. Circuit without an inductor

The coupled electromechanical behaviour of the energy har-vester [34] within the fluid flow can be expressed (see, forexample [40, 41]) by linear ordinary differential equations as

( ) ( ) ( ) ( ) ( ) ( ) q+ + - =my t cy t ky t v t f t 10L

( ) ( ) ( ) ( ) q+ + =C v tR

v t y t1

0. 11pl

The mechanical part of equation (10) is simply the SDOFmodel in (A.14) with ( ) ( )=y t z tj and

( ) ( )r qg

= + D = =m AL I M I kEI

LI

LI, and . 12h

c1 3 3 2 4

Here the integrals are given by

( ) ( ) ( )

( ) ( )ò òf x x f x x f

f x

= = =

= ¢

I I I

I

d , d , 1 and

, 13c

1 0

1 22 0

13

2

4

2

where f(ξ) is the assumed displacement shape expressed as afunction of the non-dimensional length ξ. In equation (10), fL(t)is the applied force on the SDOF harvester due to vortexshedding. This forcing will be considered random in this paper.The electrical load resistance is Rl and the mechanical force ismodelled as proportional to the voltage across the piezo-ceramic, v(t). Equation (11) is obtained from the electricalcircuit, where the voltage across the load resistance arises fromthe mechanical strain through the electromechanical coupling,θ, and the capacitance of the piezoceramic, Cp.

In equation (10), the lift force due to vortex shedding [8]is given by

( ) ( ) ( )r w=f t U DC t1

2cos , 14L L s

2

where ρ is the fluid density, U is the free stream velocity, D isthe diameter of the cylinder, CL is the non-dimensional liftcoefficient. Here ωs is circular frequency of vortex shedding,which can be expressed as

( )w p= =f fSU

D2 , where , 15s s s

where S is the Strouhal number. For a cylindrical shape, thisis normally between 0.2 and 0.3 [8].

For the initial discussion we consider deterministic har-monic excitation so that

( ) ( )r= wf t U DC e1

2. 16L L

t2 i

Figure 2. Schematic diagrams of vortex-induced piezoelectric vibration energy harvesters with two different harvesting circuits.

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Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

Here ω is the driving frequency which has the same mathe-matical form as given by (15). Transforming equations (10)and (11) into the frequency domain and dividing the firstequation by m and the second equation by Cp we obtain

( ) ( ) ( ) ( )w wzw w wq

w- + + - =Ym

VU C

D M2i i i

1

217n n

L

r

2 22

2

⎛⎝⎜

⎞⎠⎟( ) ( ) ( )w

qw w w+ + =

CY

C RVi i i

1i 0. 18

p p l

Here ( )wY i and ( )wV i are respectively the Fouriertransforms of y(t) and v(t). The natural frequency of theharvester (ωn), the damping factor (ζ) and the relative massare defined as

( )w zw r

= = =k

m

c

mM

m

D2, and . 19n

nr 3

Dividing the equation (17) by wn2 and equation (18) by ωn and

writing in a matrix form one has

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎧⎨⎩⎫⎬⎭

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪ { }

( )

( )( )( )

( )

z

a

- W + W -

W W +WW

= =

q

aqYV

u F

1 2i

i i 1

ii

1

20

0. 20

k

C

C

M

2

20

p

L

r

Here the non-dimensional forcing amplitude, the non-dimensional free stream velocity, the dimensionless fre-quency and the dimensionless time constant are defined as

( )w

ww

a w= = W = =F uC

Mu

U

DC R

1

2, , and .

21

L

r n nn p l0

2

α is the time constant of the first order electrical system, non-dimensionalized using the natural frequency of the mechan-ical system. Inverting the coefficient matrix, the displacementand voltage in the frequency domain can be obtained as

⎪⎪

⎪⎪

⎧⎨⎩⎫⎬⎭

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎧⎨⎩

⎫⎬⎭{ }

( )( ) ( )

( )

( )

( ) ( )( ) ( )

a

z

a

WW

=D W

W +

- W - W + W

=W + D W

- W D W

q

aq

aq

YV

F F

F

ii

1

i

i 1

i 1 2i

0

i 1 i

i i, 22

k

C

C

1

2

00 1

0 1

p

p

where the determinant of the coefficient matrix is

( ) ( ) ( )( )( )( ) ( )a z a

a k a zD W = W + + W

+ + + W +i i 2 1 i

2 i 1 231

3 2

2

and the non-dimensional electromechanical coupling coeffi-cient is

( )kq

=kC

. 24p

22

It is convenient to view the response quantities in a non-dimensional form. This can be achieved in various ways. Herewe propose to non-dimentionalise the dynamic response withthe response at zero frequency, that is, with the static force.This quantity can be obtained form equation (22) as

( )∣ ( )= W =W=Y Y Fi . 250 0 0

Therefore, the non-dimensional dynamic response is given by

( ) ( )( )

( ) aW =

W=

W +D W

YY

Yi

i i 1

i. 26

0 1

The harvested power is given by

( ) ( ) ( )W =W

PV

R. 27

l

2

As Rl is a constant, to obtain an expression of the power in anon-dimensional form, it is necessary to non-dimentionalisethe voltage. Again, several choices are possible. For analyticalconvenience, the voltage at Ω=1 when the damping is zerois considered to be used for the normalisation

( )∣ ( ){ }ak ak

= W =-

=-

z

aq aq

W= =V VF F

ii

i. 28

C C0 1, 0

0

2

0

2

p p

Using this, we obtain the non-dimensional voltage responseas

( ) ( )( )

( ) akW =W

=W

D WV

V

Vi

i i

i. 29

0

2

1

The non-dimensional power is therefore given by

( ) ( ) ∣ ( )∣ ( ) W =W

= WPP

PV i . 30

0

2

For the case of deterministic excitation, a key interest isthe variation of the dynamic response of the harvester andharvested power as function of the free stream velocity. Thenon-dimensional frequency parameter Ω can be related to thenon-dimensional velocity parameter u as

( )ww

pw

pW = = =SU

DSu2 2 . 31

n n

To gain physical insights, the results obtained so far is appliednumerically to an example problem. Table 1 gives the para-meters of the system for the simulations. In figure 3, the non-dimensional displacement amplitude of the harvester given byequation (26) is plotted as function of the non-dimensionalfree-stream velocity u.

The non-dimensional harvested power given byequation (30) is plotted as a function of the non-dimensional free-stream velocity u in figure 4. For both quantities, five values ofthe damping factors of energy harvesters have been considered.As expected, less damping in the harvester leads to more har-vested power from the VIV. For lightly damped systems, theharvested power peaks about Ω≈1. A mathematical optim-isation analysis shows that the optimal value of the non-dimen-sional frequency for the maximum power is given by

( )( )

( )k ak a

W »- -- -

1 1

1 2 11. 32max

22 2

2 2

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Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

Therefore, from equation (31), the non-dimensional velocity formaximum harvested power is obtained as

( )p

=W

uS2

. 33maxmax

The non-dimensional velocity for the maximum harvestedpower is therefore a function of the Strouhal number, dimen-sionless time constant and electromechanical coupling coeffi-cient only. Therefore, umax is effectively a fixed value for achosen circuit and the shape of the bluff body. One should adjustthe tip mass such thatWmax obtained from equation (32)matcheswith a given design flow velocity. The value of umax obtainedfrom equation (33) is shown in figure 4 by a ‘

*’. It can be

observed that it closely matches with the flow velocity for whichthe harvested power is maximum for all the five damping values.

3.2. Circuit with an inductor

Piezoelectric vibration energy harvester comprising a circuitwith an inductor is shown in figure 2(b). For this case theelectrical equation (see for example, [42]) becomes

( ) ( ) ( ) ( ) ( ) q+ + + =C v tR

v tL

v t y t1 1

0, 34pl i

where Li is the inductance of the circuit. The mechanicalequation is the same as given in equation (10).

Transforming equation (34) into the frequency domainand dividing by wCp n

2 one has

⎛⎝⎜

⎞⎠⎟( ) ( ) ( )q

wa b

w-W + -W + W + =C

Y Vi i1 1

i 0, 35p

2 2

where the second dimensionless constant is defined as

( )b w= L C 36n i p2

and is the ratio of the mechanical to electrical naturalfrequencies. Similar to equation (20), this equation can bewritten in matrix form with the equation of motion of themechanical system (17) as

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎧⎨⎩⎫⎬⎭ { }

( )

( )

( )( )( )

z

a b bww

- W + W -

-W - W + W=

q

abq

37

YV

F1 2i

1 i

ii 0

.k

C

2

2 20

p

Inverting the coefficient matrix, the displacement and voltagein the frequency domain can be obtained as

⎪⎪

⎪⎪

⎧⎨⎩⎫⎬⎭

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎧⎨⎩

⎫⎬⎭{ }

( )( )

( )

( )

( ( ) ) ( )( )

( )

ww

a b b

z

a b b w

w

=D

- W + W

W - W + W

=- W + W D

W D

q

abq

abq

YV

F F

F

ii

1 1 i

1 2i

0

1 i i

i,

38

k

C

C

2

2

2 2

02

0 2

20 2

p

p

Figure 3. The non-dimensional displacement amplitude of aharvester without an inductor as a function of the non-dimensionalfree stream velocity for five selected values of damping factors.

Figure 4. The non-dimensional power of a harvester without aninductor as a function of the non-dimensional free stream velocityfor five selected values of damping factors.

Table 1. Parameter values used in the simulation.

Parameter Value Unit

m 17.0×10−3 kgk 4.1×103 N m−1

c 0.218 Ns m−1

Rl 3×104 OhmCp 4.3×10−8 Fθ - ´ -4.57 10 3 N V−1

D 19.8×10−3 mρ 1×103 kg m−3

CL 1 —

Mr 2.19 —

S 0.2 —

α 0.8649 —

κ2 0.1185 —

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Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

where the determinant of the coefficient matrix is

( ) ( ) ( )( )( )( )( )( ) ( )

b a z b a bb a a z b k b ab z a a

D W = W + + W+ + + + W+ + W +

i i 2 i

2 i2 i . 39

24 3

2 2

We first consider the case of deterministic excitation. Likethe previous case, both the response quantities are expressed innon-dimensional forms. For the displacement response one has

( ) ( ) ( )( )

( ) a b bW =

W=

- W + WD W

YY

Yi

i 1 i

i. 40

0

2

2

As before, the voltage at Ω=1 when the damping is zero isconsidered to be used for the normalisation

( )∣ ( ){ }abk

= W =-

z

abq

W= =V VF

i . 41C

0 1, 0

0

2

p

Using this, we obtain the non-dimensional voltage response as

( ) ( ) ( )( )

( ) abkW =W

=W

D WV

V

Vi

i i

i. 42

0

22

2

The main interests are the variation of the dynamicresponse of the harvester and harvested power as a function ofthe free stream velocity. In figure 5, the non-dimensional dis-placement amplitude of the harvester given by equation (26) isplotted as function of the non-dimensional free-streamvelocity u. The non-dimensional harvested power given byequation (30) is plotted as function of the non-dimensionalfree-stream velocity u in figure 6. For both quantities, fivevalues of the damping factors of energy harvesters have beenconsidered. As expected, less damping in the harvester leads tomore harvested power from the VIV. For lightly dampedsystems, the harvested power peaks about Ω≈1. A detailedoptimisation analysis shows that the optimal value of the non-

dimensional frequency for the maximum power is given by

(( ) )(( ) )

( )

b b k b b a bb b k b b a b

W »- + + - + +- + + - + +

3 2 1

4 2 2 1.

43

max2

2 2 2 2 2

2 2 2 2 2

The non-dimensional velocity for the maximum harvestedpower is, therefore, a function of the Strouhal number, thedimensionless time constant, electromechanical couplingcoefficient and the dimensionless inductor constant. The valueof umax is shown in figure 6 and we can observe that it matcheswith the flow velocity for which the harvested power is max-imum for all the damping values.

4. The random process model

We consider that the excitation force due to vortex sheddingfL(t) is a random process [43, 44] as this is time-varyingfunction. A wide range of random processes can be used tomodel uncertainties around the flow field which contributetowards external excitation to the piezoelectric vibrationenergy harvester. We consider a bounded, weakly stationary,narrowband random process η(t) such that the forcing func-tion is given by

( ) ( ) ( )r h=f t U DC t1

2, 44L L

2

where

( ) ( ( ) ) ( )h n s g= + +t t W tcos . 45

Here ν is the central vortex shedding frequency, W(t) isthe standard Wiener process [45] and σ is the strength of therandom process. The constant γ is a uniform random variablebetween 0 to 2π describing the phase of the vortex inducedexcitation. Zhu [46] used the function in (45) to model ran-domness in the flow velocity to investigate stability of flow-induced vibration. Cai and Wu [47] conducted a detailedstudy of a stochastic function similar to (45) in the context ofbounded stochastic processes. In the special case when both σ

and γ is zero, the excitation function in (44) becomesthe standard deterministic excitation as considered inequation (14). Therefore, comparing both the equations, theparameter ν in (45) would correspond to the frequency ωs inthe deterministic counterpart. The physical justificationbehind the choice of the function in (45) is that the randomflow is effectively generated by a random perturbation from amean flow given by (44).

A standard Wiener process W(t) (often called Brownianmotion) is a continuous-time stochastic process for t>=0with W(0)=0 and such that the increment ( ) ( )-W t W s isGaussian with mean 0 and variance t−s for any0<=s<t, and increments for nonoverlapping time inter-vals are independent. Therefore,

( ) ( )=W 0 0 46

( ) ( ) ( ) ( ) ( )- ~ - W t W s t s N s t0, 1 0 . 47

Figure 5. The non-dimensional displacement amplitude of aharvester with an inductor as a function of the non-dimensional freestream velocity for five selected values of damping factors.

8

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

The probability density function of γ is given by

( )p

=gp1

2. 48

In figure 7 five representative samples of the standardWiener process and forcing random process ( )h t have beenshown. The samples are generated using the Monte Carlosimulation approach (see for example [48]). Compared to thedeterministic case with perfect harmonic excitations con-sidered in the previous section, the forcing function η(t) israndomly perturbed from the baseline cosine function. For thenumerical results, 1001 points have been used in the MonteCarlo simulation with ν=1 and two different values of σ

have been used. Although the random function is boundedbetween ±1, significant variabilities can be observed at agiven point in time. The impact of this random perturbationon the energy generation will be quantified in the paper.

Since η(t) is a weakly stationary random process, itsautocorrelation function depends only on the difference in thetime instants

[ ( ) ( )] ( ) ( )h t h t t t= -hhE R . 491 2 1 2

After some algebra, the autocorrelation function of η(t)can be derived as

( ) ( ) ( )∣ ∣t nt=hhs t-R e

1

2cos , 50

12

2

where τ=τ1−τ2 denotes the difference between two timeinstants. The power spectral density of the random processcan be obtained using the Fourier transform of the auto-correlation function as

( ) ( ) ( )òw t tF =hh hhwt

-¥

¥-R e d . 51i

Substituting the autocorrelation function from (50) intothe above equation and after some algebraic simplification thepower spectral density [46, 47] is given by

( ) ( )[( ) ][( ) ]

( )ws w n s

w n s w n sF =

+ +- + + +

hh4

2 4 4. 52

2 2 2 4

2 4 2 4

It is convenient to express this in terms of the non-dimen-sional frequency Ω as

( ) ( )[( ) ][( ) ]

( )

s n s

w n s n sF W =

W + +W - + W + +

hh4

2 4 4,

53n

2 2 2 4

2 4 2 4

where the non-dimensional central frequency and the nor-malised standard deviation are given by

( ) nnw

ssw

= =and . 54n n

22

The power spectral density function of ( )F Whh inequation (53) is plotted in figure 8 for two representativevalues of the normalised standard deviation and four repre-sentative values of the non-dimensional central frequency. Asexpected, the power spectral density function is symmetricabout the y-axis. Some key observations from these plots are:

• The power spectral density function of the inputexcitation has a peak around the central vortex sheddingfrequency.

• The sharpness of the peak is governed by the standarddeviation of the standard Wiener process W(t). Thisquantifies the ‘amount’ of uncertainty in the excitationforce. The lower the standard deviation, the sharper thepeaks become around their respective central frequencies.In the limit when s 0, the power spectral densityfunction approaches a Dirac delta function and theexcitation becomes a deterministic harmonic excitationas discussed in the previous section. Conversely, when σ

becomes large, the spectral density function becomesflattered and in the limit, it approaches to a broad-bandexcitation (similar to white noise).

This random function physically models the uncertainty ofthe fluid flow exiting the harvester. The central vortex sheddingfrequency ν in equation (45) is considered as a free parameterfor the sake of generality. This should be as close to the naturalfrequency of the harvester as possible. Therefore, without anyloss of generality, we will consider n = 1 in our numericalcalculations. Previous works which considered random exci-tations only focused on broadband excitations [41, 49, 50].Next, we outline methods to derive expressions for dynamicresponse of the harvester subjected to random excitations withthe narrowband power spectral density as shown in figure 8.

5. Dynamic response of the energy harvester due torandom flow

Mechanical systems driven by random excitations have beendiscussed by Lin [45], Nigam [51], Bolotin [52], Roberts and

Figure 6. The non-dimensional power of a harvester with an inductoras a function of the non-dimensional free stream velocity for fiveselected values of damping factors.

9

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

Spanos [53] and Newland [54] within the scope of randomvibration theory. To obtain the samples of the randomresponse quantities such as the displacement of the mass y(t)and the voltage v(t), one needs to solve the coupled stochasticdifferential equations (10) and (11) or (10) and (34). How-ever, analytical results developed within the theory of randomvibration allows us to bypass numerical solutions because weare interested in the average values of the output randomprocesses. Here we extend the available results to the energyharvester subjected to VIV.

In this paper, we are interested in the average harvestedpower given by

⎡⎣⎢

⎤⎦⎥[ ( )] ( ) [ ( )] ( )= =E P t E

v t

R

E v t

R. 55

l l

2 2

For a damped linear system of the form ( ) ( ) ( )w w h w=V H , itcan be shown that [45, 51] the spectral density of V is relatedto the spectral density of η by

( ) ∣ ( )∣ ( ) ( )w w wF = FhhH . 56VV2

Here ( )h w is the Fourier transform of η(t) and Φηη(ω) is thepower spectral density of the random process η(t). For largevalues of t, taking the limit, we obtain

[ ( )] ( ) ∣ ( )∣ ( ) ( )ò w w w= = Fhh-¥

¥E v t R H0 d . 57vv

2 2

This expression will be used to obtain the average power forthe two cases considered.

The calculation of the integral on the right-hand side ofequation (57) in general requires the calculation of integrals of

Figure 7. Samples of the standard Wiener process W(t) and the forcing random process η(t). For the numerical results 1001 points have beenused in the Monte Carlo simulation with ν=1 and two different values of σ. Five samples taken at random have been shown for t up to 2 s.

10

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

the following form

( )( ) ( )

( )òw ww w

=XL L-¥

¥I

d, 58n

n

n n*

where the polynomials have the form

( ) ( )w w wX = + + +--

--b b b 59n n

nn

n1

2 22

2 40

( ) ( ) ( ) ( )w w wL = + + +--a a ai i . 60n n

nn

n1

10

The evaluation of the integral in equation (58) requires con-tour integration techniques. Following Roberts and Spanos[53] this integral can be evaluated as

( )p=I

a

ND

det

det. 61n

n

n

n

Here the n×n matrices are defined as

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

( )

=

- -- -

-

-

- -

- - -

- - -

- -

b b ba a a a

a a aa a a

a a

N

00 00 00 00 0

62n

n n

n n n n

n n n

n n n

1 2 0

2 4 6

1 3 5

2 4

2 0

and

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

( )

=

- -- -

- --

-

- - - -

- - -

- - -

- -

a a a aa a a a

a a aa a a

a a

D

00 00 00 00 0

. 63n

n n n n

n n n n

n n n

n n n

1 3 5 7

2 4 6

1 3 5

2 4

2 0

These expressions will be used for the two cases considered.

6. The quantification of harvested power

6.1. Mean power for systems without an inductor

The main focus in this section is to obtain the mean of thenon-dimensional harvested power given by equation (30)when the flow is random in nature. From equation (29) andusing equation (45) we obtain the non-dimensional voltage inthe frequency domain as

( )( )

( ) ( )

( )

ak hW =W

D WW

W

V ii

i. 64

H

2

1

Here ( )h W is the Fourier transform of η(t) in equation (45)expressed in terms of the non-dimensional frequency para-meter Ω. Its power spectral density is given by equation (53).Setting the numerator to zero we note that there are four roots

( ) ( ) n sW = = j 1, ,4 i 2 65j2

( ) ( ) s nW = =jor i 1, ,4 2 i . 66j2

Therefore, the denominator can be factorised as

( )( )( )( ) ( )w a a a a+ W + W - W - W2 i i i i , 67n 1 2 1 2

where

( ) a s n a s n a= - = - =2 i and 2 i . 6812

22

1*

Using these, after some algebra, the power spectra density in(53) can be expressed as

( ) ( ){ ( ) ( )}

( ) sw

F W =W +W W

hhh h

c

F F2 i i, 69

n

2 2 2

*

where

( ) n s= +c 4 702 2 4

and

Figure 8. The power spectral density the forcing random process η(t) for four different values of non-dimensional central frequencyn .

11

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

( ) ( ) ( ) ( ) sW = + W + WhF ci i i . 712 2 2

The expression of the non-dimensional power is given byequation (30). We are interested in obtaining the mean of thisquantity

[ ] [∣ ( )∣ ] ( ) = WE P E V i . 722

Employing a change of variable ω=Ωωn and usingequation (72) we have

[ ] ∣ ( )∣ ( ) ( ) òw= W F W Whh-¥

¥E P H d . 73n

2

Using the expression of H(Ω) from equation (64) and Φηη(Ω)from equation (69), the non-dimensional mean power can beobtained by

[ ] ( )

( ) ( )( )

{ ( ) ( )}( )

òw ak

sw

=

WD W D W

W +W W

Wh h

-¥

¥E P

c

F Fi i 2 i id 74

n

n

2 2

2

1 1

2 2 2

* *

( ) ( )( ) ( )

( ) òak s=

W W +W W

W-¥

¥ c

G G

1

2 i id . 752 2

2 2 2

1 1*

Here the denominator function

( ) ( ) ( ) {( ) ( )( )( )( ) }

{ ( ) ( ) }( )

a z a

a k a z

s

W = D W W = W + + W

+ + + W +

+ W + W

hG F

c

i i i i 2 1 i

2 i 1

i i

76

1 13 2

2

2 2 2

is a fifth order polynomial in Wi . This can be expressed in theform of (60) with

( )

a k z a sz a a k s z s a sz a s a a k z a sz a a s

a

== + + += + + + + += + + + + += + +=

a c

a c c c

a c c

a caa

2

2 2 1

2 22 1

. 77

02

12 2 2 2

22 2 2

32 2

4

5

From equation (75), the average non-dimensional harvestedpower from VIV arising from randomly fluctuating flow canbe expressed as

[ ] ( ) ( )( ) ak s=E P I1

2, 782 2 1

where

( ) ( )( )( )

ò=W + W

W WW

-¥

¥I

c

G Gi id . 791

4 2 2

1 1*

Comparing I(1) with the general integral in equation (58) wehave

( )= = = = = =n b c b b b b5, , 1, 0. 8012

2 0 3 4

Now using equation (61), the integral can be evaluated as

Combining this with equation (78) we finally obtain theaverage harvested power as

where the constants aj, j=1, L, 4 are defined in (77). Themean of non-dimensional harvested power obtained formthis closed-form equation is plotted in figure 9 as functionsof α, ζ and κ. Since α and κ2 are positive, the averageharvested power is monotonically decreasing with dampingratio ζ. Thus the mechanical damping in the harvester shouldbe minimised. For fixed α and ζ the average harvested poweris monotonically increasing with the coupling coefficient κ2,and hence the electromechanical coupling should be as large

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

( )( )

( )( )

pa a

a a a= =

- + + ++ - + - - +

pa

a

a

a

a

- --

--

-- -

--

-

Ia a a c a a

c a a a a c a a a a a a a a. 81

ca a

a a ca a

a a c

a a ca a

a a ca a

a a c

1

det

0 0 1 00 0

0 00 0

0 0

det

0 00 0

0 00 0

0 0

2 3 42

1 42 4

2 3 32

42

1 22

12

42

1 2 3 4

2

3 1

4 22

3 1

4 22

4 22

3 1

4 22

3 1

4 22

[ ] ( ) ( )( )

( )

pak s

a aa a a

=- + + +

+ - + - - +E P

a a a c a a

c a a a a c a a a a a a a a2, 822 2 2 3 4

21 4

2 42 3 3

24

21 2

212

42

1 2 3 4

12

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

as possible. Additionally, we observe that for fixed κ and ζ

the average harvested power is monotonically increasingwith the coupling coefficient α.

6.2. Mean power for systems with an inductor

From equation (42) and using equation (45) we obtain thenon-dimensional voltage in the frequency domain as

( ) ( ) ( )( )

( ) ( )

( )

abk hW =W

=W

D WW

W

VV

Vi

i i

i. 83

H

0

22

2

Using the above H(Ω) and ( )F Whh from equation (69), thenon-dimensional mean power can be obtained by

( ) ( )( ) ( )

( ) òabk s=

W W +W W

W-¥

¥ c

G G

1

2 i id . 852 2

4 2 2

2 2*

Here the denominator function

( ) ( ) ( ) ( ) ( )( )

w wW =D W W = ++ + =

h --G F a a

a n

i i i i i

, and 6 86n

nn

n2 2 1

1

0

is a sixth order polynomial in Wi . The coefficients can be

obtained as

( )

aa z b a s

a b k a b b z aa s z b s a

a b z a b k s a b s bb s z a s a z b

a b a b k a b s z a bb s b z a

a b s a b z ba b

== + +

= + + ++ + +

= + + ++ + + +

= + + ++ + +

= + +=

a c

a c c

a c c c c

a c c

a c

a

a

,

2 ,

2

2 ,

2

2 2 ,

2

2 ,

2 ,

. 87

n

n

n

n

n n

n

n

n

02

12 2

22 2 2 2 2

32 2 2

42 2

5

6

From equation (85), the average non-dimensional harvestedpower from VIV arising from randomly fluctuating flow can

be expressed as

[ ] ( ) ( )( ) abk s=E P I1

2, 882 2 2

where

( ) ( )( )( )

ò=W + W

W WW

-¥

¥I

c

G Gi id . 892

6 2 4

2 2*

Figure 9. The mean of non-dimensional harvested power from the vortex induced vibration as functions of α, ζ and κ. The normalisedrandom field parameters are s = 1 andn = 1.

[ ] ( )( ) ( )

( ){ ( ) ( )}

( ) òw abk

sw

=W

D W D WW +W W

Wh h-¥

¥E P

c

F Fi i 2 i id 84n

n

2 24

2 2

2 2 2

* *

13

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

Comparing I(2) with the general integral in equation (58) we have

( )= = = = = = =n b c b b b b b6, , 1, 0. 9022

3 0 1 3 4

Now using equation (61), the integral can be evaluated as

Substituting this in equation (88), the average harvestedpower can be obtained using a closed-form expression. Theconstants aj, j=1, L, 5 are defined in equation (87). Themean of non-dimensional harvested power obtained form thisclosed-form equation is plotted in figure 10 as functions of α,ζ and κ. We observe that the average harvested power ismonotonically decreasing with damping ratio ζ and mono-tonically increasing with the time constant α. Thus themechanical damping in the harvester should be minimised.For a fixed α, β and ζ the average harvested power ismonotonically increasing with the coupling coefficient κ2,and hence the electromechanical coupling should be as largeas possible. The mean of non-dimensional harvested powerobtained form this closed-form equation is shown in figure 11

as functions of α, β and κ. A key difference is that thevariability in the mean power with respect to β is not amonotonically increasing function. There is a certain value forwhich the power reaches a peak and then it remains fairlyconstant. To investigate this further, the mean harvested

power as a function of β is shown in figure 12. This is plottedwith the other parameters fixed at ζ=0.01 and κ=0.3. Theoptimum value occurs at β=1, which is shown by the star infigure 12.

7. Conclusions

This paper developed the mathematical framework for theanalysis of VIV energy harvesters incorporating randomnessin the flow field. A cantilever beam with PZT layers and acylindrical bluff body at the tip facing the fluid flow is con-sidered. The excitation to the harvester arises due to vorticesoriginating from the bluff body. A reduced SDOF

Figure 10. The mean of non-dimensional harvested power from the vortex induced vibration as functions of α, ζ and κ. The value of theinductor constant is fixed at β=1.0. The normalised random field parameters are s = 1 andn = 1.

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

( )( )

( )

( )

pa a b a a b

a a b a b a a a a b

a b a b a b

=

=+ + - + -

+ - - - + +

- - + + + -

pab

a b a

a b a

a b a

a b a

a b a

a b a

- -- -

- --

- -

-- -

- -- -

-- -

I

c a a a a a a a a c a a a a

c a a a a a a a a a a a a a a c a

a a a a a a a a a a a a a a a a a a a a

3 2

2

. 91

ca a ca a a

a a ca a a

a a c

a a a

a a ca a a

a a ca a a

a a c

2

det

0 0 1 0 00 0

0 0 0

0 00 0 0

0 0

det

0 0 0

0 00 0 0

0 00 0 0

0 0

452

1 3 3 5 1 4 52

12

1 2 5

2 453 2

1 3 52

33

1 4 52

2 3 52

32

4 52 2 2

13

12

2 5 12

3 4 1 2 32

12

42

5 1 22

52

1 2 3 4 5

2

4 22

5 3 1

4 22

5 3 1

4 22

5 3 1

4 22

5 3 1

4 22

5 3 1

4 22

14

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

electromechanical model with fluid-structure interaction isemployed. The electromechanical coupling in the model isconsidered to be arising from unimorph or bimorph PZTconfigurations. Two cases of energy harvesting circuits,namely, one with an inductor and another without an induc-tor, have been used. For both the cases, the governingequations are in general represented by coupled second-orderordinary differential equations and they are expressed in termsof non-dimensional coefficients. Additionally, the displace-ment and voltage response is also transformed into non-dimensional forms for clarity and simplicity. The voltageresponse of the system has been studied under deterministicflow velocity and optimal velocity for maximum power

generation has been obtained analytically. The randomness inthe flow velocity is modelled using a narrowband Gaussianrandom process involving the Wiener process. The powerspectral density of the random process has a peak around acentral vortex shedding frequency representing the underlyingfluid flow with the mean velocity. Monte Carlo simulationresults of the samples of the flow velocity field have beenpresented. Analytical formulations exploiting the theory ofrandom vibration have been proposed to obtain the meanpower generated from the energy harvesters with the Wienerprocess flow velocity. Closed-form formulae for the averagenormalised harvested power for the energy harvesters withoutand with an inductor have been derived. Important aspects ofthis paper include:

• Simplified SDOF model. A reduced SDOF model withclosed-form expressions for mass, stiffness, damping andelectromechanical coupling is given in terms of non-dimensional integrals involving the assumed deformationshape of the cantilever harvester.

• Non-dimensionalisation of the voltage and power. A newsimple way of expressing the voltage of the system isproposed by normalising it with the maximum voltagearising from the underlying undamped system at theresonance frequency. This, in turn, simplified thestochastic analysis and made the results general in nature.

• Flow velocity for maximum power. For the case ofdeterministic flow, the non-dimensional velocity for max-imum harvested power is derived as p= Wu S2max max .Here Wmax is obtained using a mathematical optimisationprocess and is given by (32) or (43) depending on whetherthe harvesting circuit is without or with an inductor. Theconstant Wmax depends on the electrical parameters only.

• The random process model. The random process modeldescribing the flow field is used for the first time in the

Figure 12. The mean of non-dimensional harvested power for aharvester with an inductor as a function of β for two values of α withfixed values of ζ=0.01 and κ=0.3. The * corresponds to theoptimal value of β (=1) for the maximum mean harvested power.

Figure 11. The mean of non-dimensional harvested power from the vortex induced vibration as functions of α, ζ and κ. The value of thedamping factors is fixed at ζ=0.01. The normalised random field parameters are s = 1 andn = 1.

15

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

context of VIV energy harvesting. The random process isa function of the Wiener process (Brownian motion) anda uniform distribution. This is a two-parameter narrow-band stationary Gaussian random process with oneparameter denoting a central vortex shedding frequencyand another parameter quantifying variabilities aroundthat frequency.

• New closed-form power expressions. Exact closed-formexpressions for average normalised harvested power forenergy harvesters without and with an inductor have beenderived using contour integration techniques. Theseformulae eliminate the need for expensive Monte Carlosimulation based direct computations and also providephysical insights.

Results obtained from the derived average power equationshave been illustrated numerically for selected parametervalues. There are several parameters in the problem and thechoice of a particular set qualitatively and quantitativelychanges the amount of harvested power. The closed-formexpressions derived in the paper provide an easier route forinvestigating this parametric landscape. The limited para-metric analysis is shown here highlights the fact that theaverage harvested power can be maximised for certainparameter values (β for example). Future work shouldconsider direct practical applications of the analyticalexpressions.

Experimental validations of the closed-form expressionswould also be of paramount importance. Potential benefitsarising from the design of VIV energy harvesters with randomflow velocity considerations should be explored further withsuitable case studies.

Acknowledgments

SA acknowledges the financial support from the GlobalChallenges Research Fund, grant number GCRF RIG1029-103. BB and AR acknowledge DST-IUSSTF and SPARC forpartial funding of the research through the project grants:IUSSTF/ME /2017400A and MHRD /ME /2018544respectively.

Appendix. Reduced-order model for Euler–Bernoullibeams

A.1. Free vibration analysis

The undamped natural frequencies (Hz) of the cantileverbeam in (1) can be expressed as

( )l

p r= =f

EI

ALj

2, 1, 2, 3, A.1j

j

h

2

4

where λj needs to be obtained by [55] solving the followingtranscendental equation

( )l l + =cos cosh 1 0. A.2

Solving this equation, the values of λj can be obtained as1.8751, 4.694 09, 7.8539 and 10.995 57. For larger values ofj, in general we have ( )l p= -j2 1 2j . The vibration modeshape corresponding to the jth natural frequency can beexpressed as

⎛⎝⎜

⎞⎠⎟( ) ( )

( )( )

f x l x l xl ll l

l x l x

= - --

+

-

cosh cossinh sin

cosh cos

sinh sin ,

A.3

j j jj j

j j

j j

where

( )x =x

LA.4

is the normalised coordinate along the length of the cantilever.For sensing applications we are primarily interested in the firstfew modes of vibration only.

Consider the attached bluff body of mass M at the end ofthe cantilevered resonator in figure 1. The boundary condi-tions with an additional mass of M at x=L can be expressedas

( ) ( ) ( )( ) ( )

( )

= ¢ = =

¢¢¢ - =

Z t Z t Z L t

EIZ L t MZ L t

0, 0, 0, 0, , 0, and

, , 0.

A.5

Here ( ) ¢• denotes derivative with respective to x and ( )•denotes derivative with respective to t. It can be shown that(see for example [56]) the resonant frequencies are stillobtained from equation (A.1) but λj should be obtained bysolving

( )( ) ( )l l l l l

l l- D

+ + =Mcos sinh sin cosh

cos cosh 1 0. A.6

Here

( )r

D =MM

ALA.7

h

is the ratio of the tip mass and the mass of the cantilever. Ifthe tip mass is zero, then one can see that equation (A.7)reduces to equation (A.2).

A.2. Equivalent SDOF model

The equation of motion of the beam in (1) is a partial diff-erential equation. This equation represents infinite number ofdegrees of freedom. The mathematical theory of linear partialdifferential equations is very well developed and the nature ofsolutions of the bending vibration is well understood. Con-sidering a steady-state harmonic motion with frequency ω wehave

( ) ( ) [ ] ( )w=Z x t z x t, exp i , A.8

where = -i 1 . Substituting this in the beam equation (1)

16

Smart Mater. Struct. 29 (2020) 035034 S Adhikari et al

we have

( ) ( ) ( ) ( )

( )

w r w w+ - + =EIz x

xc

z x

xA z x c z x

d

di

d

di 0.

A.9

h

4

4 1

4

42

2

Following the damping convention in dynamic analysis as in[57], we consider stiffness and mass proportional damping.Therefore, we express the damping constants as

( ) ( ) ( ) a b r= =c EI c Aand , A.10c c1 2

where αc and βc are stiffness and mass proportional dampingfactors. Substituting these, from equation (A.9) we have

⎛⎝⎜

⎞⎠⎟

( ) ( ) ( )

( ) ( )

w a b r

r w

+ +

- =

EIz x

xEI

z x

xAz x

A z x

d

di

d

d

0. A.11

c c h

h

4

4

4

4

2

The first part of the damping expression is proportional to thestiffness term while the second part of the damping expres-sion is proportional to the mass term. The general solution ofequation (A.11) can be expressed as a linear superposition ofall the vibration mode shapes (see for example [57]). Vibra-tion energy harvesters are often designed to operate within afrequency range which is close to first few natural frequenciesonly. Therefore, without any loss of accuracy, simplifiedlumped parameter models can be used to correspondingcorrect resonant behaviour. This can be achieved usingenergy methods or more generally using Galerkin approach.

Assuming a unimodal solution, the dynamic response ofthe beam can be expressed as

( ) ( ) ( ) ( )f= =Z x t z t x j, , 1, 2, 3, A.12j j

Substituting this assumed motion into the equation of motion(1), multiplying by fj(x) and integrating by parts over thelength one has

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

ò ò

ò ò

f a f

b r f r f

¢ + ¢

+ + =

¢ ¢EIz t x x EIz t x x

Az t x x Az t x x

d d

d d 0.

A.13

j

L

j c j

L

j

c h j

L

j h j

L

j

0

2

0

2

0

2

0

2

Using the equivalent mass, damping and stiffness, thisequation can be rewritten as

( ) ( ) ( ) ( )+ + =m z t c z t k z t 0, A.14eq j eq j eq jj j j

where the equivalent mass and stiffness terms are given by

( ) ( ) ( ) ò òr f r f x x= =m A x x ALd d A.15eq h

L

j h j

I

0

2

0

12

j

j1

( ) ( ) ( ) ò òf f x x= = k EI x x

EI

Ld d . A.16eq

L

j j

I

0 3 0

1

j

j

2 2

2

For the first mode of vibration ( j=1), substitutingλ1=1.8751, it can be shown that =I 111

and =I 12.362412.

If there is a point mass of M at the tip of the cantilever, thenthe effective mass becomes

( ) ( ) ( )r f r= + = + Dm ALI M AL I MI1 . A.17eq h j

I

h12

1 3j j

j

j j

3

For the first mode of vibration it can be shown that =I 431.

The equivalent SDOF given by equation (A.14) is used in therest of the paper. However, the expression derived here aregeneral and can be used if higher modes of vibration were tobe employed in energy harvesting.

ORCID iDs

Sondipon Adhikari https://orcid.org/0000-0003-4181-3457

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