The document summarizes research on vibration energy harvesting under uncertainty. It discusses piezoelectric energy harvesting using cantilever beams and focuses on harvesting from single frequencies at resonance. It notes that most previous work assumes certain input but uncertainty exists in real environments. The document outlines sources of uncertainty in excitation and system parameters and presents linear and nonlinear models of piezoelectric energy harvesters. It transforms models into the frequency domain to analyze optimal harvester design under Gaussian excitation without an inductor.

1. Energy Harvesting Under Uncertainty
S Adhikari1
1 College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK
IIT Madras, India
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 1 / 43

2. Swansea University
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 2 / 43

3. Swansea University
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 3 / 43

4. Collaborations
Professor Mike Friswell and Dr Alexander Potrykus (Swansea
University, UK).
Professor Dan Inman (University of Michigan, USA)
Professor Grzegorz Litak (University of Lublin, Poland).
Professor Eric Jacquelin (University of Lyon, France)
Professor S Narayanan and Dr S F Ali (IIT Madras, India).
Funding: Royal Society International Joint Project - 2010/R2: Energy
Harvesting from Randomly Excited Nonlinear Oscillators (2 years from
June 2011) - Swansea & IIT Madras (Prof Narayanan).
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 4 / 43

5. Outline
1 Introduction
Piezoelectric vibration energy harvesting
The role of uncertainty
2 Single Degree of Freedom Electromechanical Models
Linear Systems
Nonlinear System
3 Optimal Energy Harvester Under Gaussian Excitation
Circuit without an inductor
4 Stochastic System Parameters
5 Equivalent Linearisation Approach
6 Conclusions
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 5 / 43

6. Introduction Piezoelectric vibration energy harvesting
Piezoelectric vibration energy harvesting
The harvesting of ambient vibration energy for use in powering
low energy electronic devices has formed the focus of much
recent research.
Of the published results that focus on the piezoelectric effect as
the transduction method, almost all have focused on harvesting
using cantilever beams and on single frequency ambient energy,
i.e., resonance based energy harvesting. Several authors have
proposed methods to optimize the parameters of the system to
maximize the harvested energy.
Some authors have considered energy harvesting under wide
band excitation.
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 6 / 43

7. Introduction The role of uncertainty
Why uncertainty is important for energy harvesting?
In the context of energy harvesting of ambient vibration, the input
excitation may not be always known exactly.
There may be uncertainties associated with the numerical values
considered for various parameters of the harvester. This might
arise, for example, due to the difference between the true values
and the assumed values.
If there are several nominally identical energy harvesters to be
manufactured, there may be genuine parametric variability within
the ensemble.
Any deviations from the assumed excitation may result an
optimally designed harvester to become sub-optimal.
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 7 / 43

8. Introduction The role of uncertainty
Types of uncertainty
Suppose the set of coupled equations for energy harvesting:
L{u(t)} = f(t) (1)
Uncertainty in the input excitations
For this case in general f(t) is a random function of time. Such
functions are called random processes.
f(t) can be stationary or non-stationary random processes
Uncertainty in the system
The operator L{•} is in general a function of parameters
θ1 , θ2 , · · · , θn ∈ R.
The uncertainty in the system can be characterised by the joint
probability density function pΘ1 ,Θ2 ,··· ,Θn (θ1 , θ2 , · · · , θn ).
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 8 / 43

9. Single Degree of Freedom Electromechanical Models Linear Systems
SDOF electromechanical models
Proof Mass
Proof Mass
- -
x x
Piezo- Piezo-
Rl v v
ceramic L Rl ceramic
+ +
Base Base
xb xb
Schematic diagrams of piezoelectric energy harvesters with two
different harvesting circuits. (a) Harvesting circuit without an inductor,
(b) Harvesting circuit with an inductor.
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10. Single Degree of Freedom Electromechanical Models Linear Systems
Governing equations
For the harvesting circuit without an inductor, the coupled
electromechanical behavior can be expressed by the linear ordinary
differential equations
¨ ˙
mx (t) + c x(t) + kx(t) − θv (t) = f (t) (2)
1
˙ ˙
θx(t) + Cp v (t) + v (t) = 0 (3)
Rl
For the harvesting circuit with an inductor, the electrical equation
becomes
1 1
¨ ¨ ˙
θx (t) + Cp v (t) + v (t) + v (t) = 0 (4)
Rl L
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 10 / 43

11. Single Degree of Freedom Electromechanical Models Nonlinear System
Simplified piezomagnetoelastic model
Schematic of the
piezomagnetoelastic device. The beam system is also referred to as
the ‘Moon Beam’.
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12. Single Degree of Freedom Electromechanical Models Nonlinear System
Governing equations
The nondimensional equations of motion for this system are
1
x + 2ζ x − x(1 − x 2 ) − χv = f (t),
¨ ˙ (5)
2
˙ ˙
v + λv + κx = 0, (6)
where x is the dimensionless transverse displacement of the beam tip,
v is the dimensionless voltage across the load resistor, χ is the
dimensionless piezoelectric coupling term in the mechanical equation,
κ is the dimensionless piezoelectric coupling term in the electrical
equation, λ ∝ 1/Rl Cp is the reciprocal of the dimensionless time
constant of the electrical circuit, Rl is the load resistance, and Cp is the
capacitance of the piezoelectric material. The force f (t) is proportional
to the base acceleration on the device. If we consider the inductor,
¨ ˙
then the second equation will be v + λv + βv + κx = 0. ¨
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 12 / 43

13. Single Degree of Freedom Electromechanical Models Nonlinear System
Possible physically realistic cases
Depending on the system and the excitation, several cases are
possible:
Linear system excited by harmonic excitation
Linear system excited by stochastic excitation
Linear stochastic system excited by harmonic/stochastic excitation
Nonlinear system excited by harmonic excitation
Nonlinear system excited by stochastic excitation
Nonlinear stochastic system excited by harmonic/stochastic
excitation
This talk is focused on application of random vibration theory to
various energy harvesting problems
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14. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit without an inductor
Our equations:
¨ ˙ ¨
mx (t) + c x(t) + kx(t) − θv (t) = −mxb (t) (7)
1
˙ ˙
θx(t) + Cp v (t) + v (t) = 0 (8)
Rl
Transforming both the equations into the frequency domain and
dividing the first equation by m and the second equation by Cp we
obtain
θ
−ω 2 + 2iωζωn + ωn X (ω) −
2
V (ω) = ω 2 Xb (ω) (9)
m
θ 1
iω X (ω) + iω + V (ω) = 0 (10)
Cp Cp Rl
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15. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit without an inductor
The natural frequency of the harvester, ωn , and the damping factor, ζ,
are defined as
k c
ωn = and ζ = . (11)
m 2mωn
Dividing the preceding equations by ωn and writing in matrix form one
has
θ
1 − Ω2 + 2iΩζ −k X Ω2 Xb
αθ = , (12)
iΩ Cp (iΩα + 1) V 0
where the dimensionless frequency and dimensionless time constant
are defined as
ω
Ω= and α = ωn Cp Rl . (13)
ωn
α is the time constant of the first order electrical system,
non-dimensionalized using the natural frequency of the mechanical
system.
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16. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit without an inductor
Inverting the coefficient matrix, the displacement and voltage in the
frequency domain can be obtained as
θ
X 1 (iΩα+1) k Ω2 Xb (iΩα+1)Ω2 Xb /∆1
V = −iΩ αθ (1−Ω2 )+2iΩζ 0
= −iΩ3 αθ Xb /∆1 , (14)
∆1 Cp C p
where the determinant of the coefficient matrix is
∆1 (iΩ) = (iΩ)3 α + (2 ζ α + 1) (iΩ)2 + α + κ2 α + 2 ζ (iΩ) + 1 (15)
and the non-dimensional electromechanical coupling coefficient is
θ2
κ2 = . (16)
kCp
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17. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit without an inductor
Mean power
The average harvested power due to the white-noise base
acceleration with a circuit without an inductor can be obtained as
|V |2
E P =E
(Rl ω 4 Φxb xb )
π m α κ2
= .
(2 ζ α2 + α) κ2 + 4 ζ 2 α + (2 α2 + 2) ζ
From Equation (14) we obtain the voltage in the frequency domain
as
αθ
−iΩ3 Cp
V = Xb . (17)
∆1 (iΩ)
We are interested in the mean of the normalized harvested power
when the base acceleration is Gaussian white noise, that is
|V |2 /(Rl ω 4 Φxb xb ).
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18. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit without an inductor
The spectral density of the acceleration ω 4 Φxb xb and is assumed to be
constant. After some algebra, from Equation (17), the normalized
power is
|V |2 kακ2 Ω2
P= = . (18)
(Rl ω 4 Φxb xb ) ωn ∆1 (iΩ)∆∗ (iΩ)
3
1
Using linear stationary random vibration theory, the average
normalized power can be obtained as
∞
|V |2 kακ2 Ω2
E P =E = dω (19)
(Rl ω 4 Φxb xb ) 3
ωn −∞ ∆1 (iΩ)∆∗ (iΩ)
1
From Equation (15) observe that ∆1 (iΩ) is a third order polynomial in
(iΩ). Noting that dω = ωn dΩ and from Equation (15), the average
harvested power can be obtained from Equation (19) as
|V |2
E P =E = mακ2 I (1) (20)
(Rl ω 4 Φxb xb )
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19. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit without an inductor
∞
Ω2
I (1) = dΩ. (21)
−∞ ∆1 (iΩ)∆∗ (iΩ)
1
After some algebra, this integral can be evaluated as
 
0 1 0
 
det  −α
 α + κ2 α + 2 ζ
0 

π 0 −2 ζ α − 1 1
I (1) =   (22)
α 2ζ α + 1 −1 0
 
det 
 −α α + κ2 α + 2 ζ 0 
0 −2 ζ α − 1 1
Combining this with Equation (20) we obtain the average harvested
power due to white-noise base acceleration.
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20. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Normalised mean power: numerical illustration
α2 1 + κ2 = 1 or in terms of physical quantities
Rl2 Cp kCp + θ2 = m.
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21. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit with an inductor
The electrical equation becomes
1 1
¨ ¨
θx (t) + Cp v (t) + ˙
v (t) + v (t) = 0 (23)
Rl L
where L is the inductance of the circuit. Transforming equation (23)
2
into the frequency domain and dividing by Cp ωn one has
θ 1 1
− Ω2 X + −Ω2 + iΩ + V =0 (24)
Cp α β
where the second dimensionless constant is defined as
2
β = ωn LCp , (25)
Two equations can be written in a matrix form as
(1−Ω2 )+2iΩζ −θ
k X = Ω2 Xb . (26)
−Ω2 αβθ
Cp
α(1−βΩ2 )+iΩβ V 0
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22. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit with an inductor
Inverting the coefficient matrix, the displacement and voltage in the
frequency domain can be obtained as
θ
1 α(1−βΩ2 )+iΩβ
X = k Ω2 Xb
V ∆2 Ω2 αβθ
Cp (1−Ω2 )+2iΩζ 0
(α(1−βΩ2 )+iΩβ )Ω2 Xb /∆2
= Ω4 αβθ Xb /∆2
(27)
C p
where the determinant of the coefficient matrix is
∆2 (iΩ) = (iΩ)4 β α + (2 ζ β α + β) (iΩ)3
+ β α + α + 2 ζ β + κ2 β α (iΩ)2 + (β + 2 ζ α) (iΩ) + α. (28)
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23. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Circuit with an inductor
Mean power
The average harvested power due to the white-noise base
acceleration with a circuit with an inductor can be obtained as
mαβκ2 π(β+2αζ)
E P = .
β(β+2αζ)(1+2αζ)(ακ2 +2ζ )+2α2 ζ(β−1)2
This can be obtained in a very similar to the previous case.
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24. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Normalised mean power: numerical illustration
Normalized mean power
1.5
1
0.5
0
4
3 4
2 3
2
α 1 1
0 0 β
The normalized mean power of a harvester with an inductor as a
function of α and β, with ζ = 0.1 and κ = 0.6.
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25. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Optimal parameter selection
1.5
Normalized mean power
1
0.5
0
0 1 2 3 4
β
The normalized mean power of a harvester with an inductor as a
function of β for α = 0.6, ζ = 0.1 and κ = 0.6. The * corresponds to
the optimal value of β(= 1) for the maximum mean harvested power.
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26. Optimal Energy Harvester Under Gaussian Excitation Circuit without an inductor
Optimal parameter selection
Normalized mean power
1.5
1
0.5
0
0 1 2 3 4
α
The normalized mean power of a harvester with an inductor as a
function of α for β = 1, ζ = 0.1 and κ = 0.6. The * corresponds to the
optimal value of α(= 1.667) for the maximum mean harvested power.
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27. Stochastic System Parameters
Stochastic system parameters
Energy harvesting devices are expected to be produced in bulk
quantities
It is expected to have some parametric variability across the
‘samples’
How can we take this into account and optimally design the
parameters?
The natural frequency of the harvester, ωn , and the damping factor, ζn ,
are assumed to be random in nature and are defined as
ωn = ωn Ψω
¯ (29)
¯
ζ = ζΨζ (30)
where Ψω and Ψζ are the random parts of the natural frequency and
¯
damping coefficient. ωn and ζ are the mean values of the natural
¯
frequency and damping coefficient.
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28. Stochastic System Parameters
Mean harvested power: Harmonic excitation
The average (mean) normalized power can be obtained as
|V |2
E [P] = E 2
(Rl ω 4 Xb )
¯
k ακ2 Ω2 ∞ ∞ fΨω (x1 )fΨζ (x2 )
= dx1 dx2 (31)
¯3
ωn −∞ −∞ ∆1 (iΩ, x1 , x2 )∆∗ (iΩ, x1 , x2 )
1
where
∆1 (iΩ, Ψω , Ψζ ) = (iΩ)3 α + 2ζαΨω Ψζ + 1 (iΩ)2 +
¯
ω
¯
αΨ2 + κ2 α + 2ζΨω Ψζ (iΩ) + Ψ2 (32)
ω
The probability density functions (pdf) of Ψω and Ψζ are denoted by
fΨω (x) and fΨζ (x) respectively.
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29. Stochastic System Parameters
The mean power
−4
10
−5
10
E[P] (Watt)
−6
10
σ=0.00
σ=0.05
σ=0.10
σ=0.15
σ=0.20
−7
10
0.2 0.4 0.6 0.8 1 1.2 1.4
Normalized Frequency (Ω)
The mean power for various values of standard deviation in natural
frequency with ωn = 670.5 rad/s, Ψζ = 1, α = 0.8649, κ2 = 0.1185.
¯
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30. Stochastic System Parameters
The mean power
100
90
80
Max( E[P]) / Max(Pdet) (%)
70
60
50
40
30
20
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Standard Deviation (σ)
The mean harvested power for various values of standard deviation of
the natural frequency, normalised by the deterministic power
(¯ n = 670.5 rad/s, Ψζ = 1, α = 0.8649, κ2 = 0.1185).
ω
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 30 / 43

31. Stochastic System Parameters
Optimal parameter selection
The optimal value of α:
2 (c1 + c2 σ 2 + 3c3 σ 4 )
αopt ≈ (33)
(c4 + c5 σ 2 + 3c6 σ 4 )
where
¯ ¯
c1 =1 + 4ζ 2 − 2 Ω2 + Ω4 , c2 = 6 + 4ζ 2 − 2 Ω2 , c3 = 1, (34)
¯
c4 = 1 + 2κ2 + κ4 Ω2 + 4ζ 2 − 2 − 2κ2 Ω4 + Ω6 , (35)
¯
c5 = 2κ2 + 6 Ω2 + 4ζ 2 − 2 Ω4 , c6 = Ω2 , (36)
and σ is the standard deviation in natural frequency.
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32. Stochastic System Parameters
Optimal parameter selection
The optimal value of κ:
1
κ2 ≈
opt (d1 + d2 σ 2 + d3 σ 4 ) (37)
(α Ω)
where
¯ ¯
d1 =1 + 4ζ 2 + α2 − 2 Ω2 + 4ζ 2 α2 − 2α2 + 1 Ω4 + α2 Ω6 (38)
¯ ¯
d2 =6 + 4ζ 2 + 6α2 − 2 Ω2 + 4ζ 2 α2 − 2α2 Ω4 (39)
d3 =3 + 3α2 Ω2 (40)
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 32 / 43

33. Equivalent Linearisation Approach
Nonlinear coupled equations
¨ ˙
x + 2ζ x + g(x) − χv = f (t) (41)
˙ ˙
v + λv + κx = 0, (42)
The nonlinear stiffness is represented as g(x) = − 1 (x − x 3 ).
2
Assuming a non-zero mean random excitation (i.e., f (t) = f0 (t) + mf )
and a non-zero mean system response (i.e., x(t) = x0 (t) + mx ), the
following equivalent linear system is considered,
x0 + 2ζ x˙0 + a0 x0 + b0 − χv = f0 (t) + mf
¨ (43)
where f0 (t) and x0 (t) are zero mean random processes. mf and mx are
the mean of the original processes f (t) and x(t) respectively. a0 and
b0 are the constants to be determined with b0 = mf and a0 represents
2
the square of the natural frequency of the linearized system ωeq .
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 33 / 43

34. Equivalent Linearisation Approach
Linearised equations
We minimise the expectation of the error norm i.e.,
(E 2 , with = g(x) − a0 x0 − b0 ). To determine the constants a0 and
b0 in terms of the statistics of the response x, we take partial
derivatives of the error norm w.r.t. a0 and b0 and equate them to zero
individually.
∂ 2 2
E =E [g(x)x0 ] − a0 E x0 − b0 E [x0 ] (44)
∂a0
∂ 2
E =E [g(x)] − a0 E [x0 ] − b0 (45)
∂b0
Equating (44) and (45) to zero, we get,
E [g(x)x0 ] E [g(x)x0 ]
a0 = 2
= 2
(46)
E x0 σx
b0 = E [g(x)] = mf (47)
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 34 / 43

35. Equivalent Linearisation Approach
Responses of the piezomagnetoelastic oscillator
30 (a) λ = 0.05 6 (b) λ = 0.05
λ = 0.01 λ = 0.01
σx /σf
f
20
σ /σ
4
v
10
0 2
0 0.05 0.1 0 0.05 0.1
σ σ
f f
(c)
0.2
λ = 0.05
2
σv
0.1 λ = 0.01
0
0 0.05 0.1
σf
Simulated responses of the piezomagnetoelastic oscillator in terms of the
standard deviations of displacement and voltage (σx and σv ) as the standard deviation of the random excitation σf varies. (a)
gives the ratio of the displacement and excitation; (b) gives the ratio of the voltage and excitation; and (c) shows the variance of
the voltage, which is proportional to the mean power.
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 35 / 43

36. Equivalent Linearisation Approach
Phase portraits
1 (a) 1 (b)
dx/dt
dx/dt
0 0
−1 −1
−2 −1 0 x 1 2 −2 −1 0 x 1 2
1 (c)
dx/dt
0
−1
−2 −1 0 x 1 2
Phase portraits for λ = 0.05, and the stochastic force for (a) σf = 0.025,
(b) σf = 0.045, (c) σf = 0.065. Note that the increasing noise level overcomes the potential barrier resulting in a significant
increase in the displacement x.
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 36 / 43

37. Equivalent Linearisation Approach
Voltage output
2.5
2 1
1.5
1 0.5
0.5
Voltage
dxdt
0 0
−0.5
−1 −0.5
−1.5
−2 −1
−2.5
0 1000 2000 3000 4000 5000 −2 −1 0 1 2
Time x
Voltage output due to Gaussian white noise (ζ = 0.01, χ = 0.05, and
κ = 0.5 and λ = 0.01.
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 37 / 43

38. Equivalent Linearisation Approach
Voltage output
2.5
2 1
1.5
1 0.5
0.5
Voltage
dxdt
0 0
−0.5
−1 −0.5
−1.5
−2 −1
−2.5
0 1000 2000 3000 4000 5000 −2 −1 0 1 2
Time x
´
Voltage output due to Levy noise (ζ = 0.01, χ = 0.05, and κ = 0.5 and
λ = 0.01.
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 38 / 43

39. Equivalent Linearisation Approach
Inverted beam harvester
150
L 100
Top velocity (mm/s)
50
ρA 0
y −50
−100
x
−150
−80 −60 −40 −20 0 20 40 60 80
Top displacement (mm)
(a) Schematic diagram of inverted beam harvester, (b) a typical phase
portrait of the tip mass.
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 39 / 43

40. Equivalent Linearisation Approach
Energy harvesting from bridge vibration
15
10
Energy (µJ)
u = 10 m/s
u = 15 m/s
u = 20 m/s
5 u = 25 m/s
P u
y(x,t)
0
0 0.5 1 1.5 2
x
L α
(a) Schematic diagram of a beam with a moving point load, (b) The
variation in the energy generated by the harvester located at L/3 with
α for a single vehicle traveling at different speeds, u
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 40 / 43

41. Equivalent Linearisation Approach
Energy harvesting DVA
0
0
.0
F0 ei"t
8
0
6
0,s
/X2
4
max
h !
P
h 2
p
.h l
0
2
h
1.5
1
!"#$%#&#'()"'*#&#+#,(- 1
α Ω
0 0.5
/,#)01*23)4#-(",0*51,3+"'*4"6)3("%,*36-%)6#)
(a) Schematic diagram of energy harvesting dynamic vibration
absorber attached to a single degree of freedom vibrating system,
(b)Harvested power in mW /m2 for nondimensional coupling coefficient
κ2 = 0.3
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 41 / 43

42. Conclusions
Summary of the results
Vibration energy based piezoelectric and magnetopiezoelectric
energy harvesters are expected to operate under a wide range of
ambient environments. This talk considers energy harvesting of
such systems under harmonic and random excitations.
Optimal design parameters were obtained using the theory of
linear random vibration
Nonlinearity of the system can be exploited to scavenge more
energy over wider operating conditions
Uncertainty in the system parameters can have dramatic affect on
energy harvesting. This should be taken into account for optimal
design
Stochastic jump process models can be used for the calculation of
harvested power
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 42 / 43

43. Conclusions
Further details
1 Ali, S. F., Friswell, M. I. and Adhikari, S., ”Analysis of energy harvesters for highway bridges”, Journal of Intelligent
Material Systems and Structures, 22[16] (2011), pp. 1929-1938.
2 Jacquelin, E., Adhikari, S. and Friswell, M. I., ”Piezoelectric device for impact energy harvesting”, Smart Materials and
Structures, 20[10] (2011), pp. 105008:1-12.
3 Litak, G., Borowiec, B., Friswell, M. I. and Adhikari, S., ”Energy harvesting in a magnetopiezoelastic system driven by
random excitations with uniform and Gaussian distributions”, Journal of Theoretical and Applied Mechanics, 49[3] (2011),
pp. 757-764..
4 Ali, S. F., Adhikari, S., Friswell, M. I. and Narayanan, S., ”The analysis of piezomagnetoelastic energy harvesters under
broadband random excitations”, Journal of Applied Physics, 109[7] (2011), pp. 074904:1-8
5 Ali, S. F., Friswell, M. I. and Adhikari, S., ”Piezoelectric energy harvesting with parametric uncertainty”, Smart Materials
and Structures, 19[10] (2010), pp. 105010:1-9.
6 Friswell, M. I. and Adhikari, S., ”Sensor shape design for piezoelectric cantilever beams to harvest vibration energy”,
Journal of Applied Physics, 108[1] (2010), pp. 014901:1-6.
7 Litak, G., Friswell, M. I. and Adhikari, S., ”Magnetopiezoelastic energy harvesting driven by random excitations”, Applied
Physics Letters, 96[5] (2010), pp. 214103:1-3.
8 Adhikari, S., Friswell, M. I. and Inman, D. J., ”Piezoelectric energy harvesting from broadband random vibrations”, Smart
Materials and Structures, 18[11] (2009), pp. 115005:1-7.
Under Review
9 Ali, S. F. and Adhikari, S., ”Energy harvesting dynamic vibration absorbers”.
10 Friswell, M. I., Ali, S. F., Adhikari, S., Lees, A.W. , Bilgen, O. and Litak, G., ”Nonlinear piezoelectric vibration energy
harvesting from an inverted cantilever beam with tip mass”.
Adhikari (Swansea) Vibration Energy Harvesting Under Uncertainty January 2012 43 / 43