11TH

CONFERENCE

on

DYNAMICAL SYSTEMS

THEORY AND APPLICATIONS

December 5-8, 2011. Lodz, Poland

Energy harvesting by two magnetopiezoelastic oscillators

G. Litak, M.I. Friswell, C.A. Kitio Kwuimy, S. Adhikari, and M. Borowiec

Abstract: We examine an energy harvesting system of two magnetopiezoelasticoscillators coupled by electric circuit and driven by harmonic excitation. Wefocus on the effects of synchronization and escape from a single potential well.In the system with relative mistuning in the stiffness of the harvesting oscil-lators, we show the dependence of the voltage output for different excitationfrequencies.

1. Introduction

Ambient energy harvesting by autonomous electro–mechanical systems is an important

source of energy for small electronic devices and to recharge batteries or enable remote

operation [1,2]. Many of the proposed devices use the piezoelectric effect as the transduction

method [3-5]. These devices are usually implemented as patches on cantilever beams and

designed to operate at resonance conditions. The design of an energy harvesting device must

be tailored to the ambient energy available. For a single frequency excitation the resonant

harvesting device is optimum, provided it is tuned to the excitation frequency [6,7].

To optimise the harvesting system for harmonic excitation, the harvester is designed

with a natural frequency to match the excitation frequency [1,6]. For harmonic excitation

where the frequency varies, or for broadband excitation, the bandwidth of the device has to

be extended. Nana and Woafo [8] suggested the use of an array of two or more harvesters to

increase the power delivered into the load. Shahruz [9] analyzed a set of parallel single degree

of freedom harvesters tuned at slightly different resonant frequencies, whereas Erturk et al.

[10] considered a harvester as a serial set of two beams connected to each other to form an L-

shape. Ramlan et al. [11] considered a harvester made of two oblique springs and analyzed

the potential benefits of the hardening effects of the spring on the output energy. More

recently Kim et al. [12] introduced the idea of association of two piezoelectric harvesters

to produce more efficient electric power generation. Their model consisted of a proof mass,

two cantilever piezoelectric beams delivering the electric signal into an electrical load. They

showed through experimental analysis that a two degree of freedom energy harvester has

two peaks at different frequencies and also has a large frequency bandwidth in comparison

with the conventional single degree of freedom piezoelectric harvester. As suggested by Kim

et al. [12], connecting energy sources do not necessarily result in an increase in the power

generated. Therefore a rigorous mathematical analysis has to be performed to analyse the

synchronization condition of the harvesters. This is the main objective of this paper where

we consider a system of two magnetopiezoelastic harvesters delivering power in an electrical

circuit. Attention is focussed on the mistuning in the stiffness of the harvesting oscillators.

2. The Model and Simulation Results

A schematic picture of the parallel coupled harvesters is shown in Fig. 1. The mathematical

model may be written as the following dimensionless equations:

load resistor, R

N N

F t( )

Piezoceramicpatches

Magnets

x

v

N

Magnets

y

CP1 CP2

Figure 1. Schematic diagram of the harvester system.

x + 2ζx −1

2x(1 − x2) − χv = F (t), y + 2ζy −

1

2αy(1 − y2) − χv = F (t), (1)

and

v + λv + κx + κy = 0, (2)

where x and y are the dimensionless transverse displacements of the beam tips, 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/RCP is the reciprocal of the dimensionless time constant of the

electrical circuit, R is the load resistance, and CP = CP1 + CP2 is the capacitance of

the piezoelectric material. Finally, α is the stiffness mistuning parameter which should be

considered in any realistic system, and F (t) is the harmonic excitation of the following form

F (t) = F0 sin(ωt). (3)

Using the above equations (Eqs. (1-3)) we performed simulations of the dynamical sys-

tem. The system parameters used in the calculations were chosen to fit a realistic experiment:

χ = 0.05, κ = 0.5, λ = 0.01, F0 = 0.2, α = 1.1. (4)

(a)0.5 1 1.5 2

0

1

2

3

4

ω

<v2 >

(b)0.5 1 1.5 2

0

0.5

1

1.5

2

ω

σ (x

−y)

Figure 2. (a) Output power in terms of mean squared voltage < v2 > versus excitation

frequency ω; (b) relative difference in the oscillator displacements x − y in terms standard

deviation σ(x−y) versus excitation frequency ω. In the simulations the frequency was changed

quasi-statically (the system parameters are given in Eq. (4)).

The results of the output power as well as the appearance of synchronization are illus-

trated in Fig. 2. As expected the resonance curve mirrors the mechanical hardening Duffing

type nonlinearity and the peak frequency is located at about ω ≈ 1.1 (Fig. 2a). Interestingly,

after passing through the maximum response the system switches from the resonant to the

non-resonant solution. By examining the standard deviation of oscillator’s relative displace-

ment σ(x− y) we observe that the mistuning parameter α = 1.1 breaks the synchronization

effect (Fig. 2b). Interestingly, synchronization (σ(x − y) ≈ 0) is fulfilled for ω ∈[0.60, 0.95]

and [1.55, 1.60], and the resulting power generated is low. However at frequency giving the

peak power σ(x − y) ≈ 1.6.

To investigate the solutions of Eqs. (1-3) further, Fig. 3 shows the simultaneously

estimated average values of < x > and < y >. By observing these parameters one can

distinguish the symmetric (usually double-well) and non-symmetric (usually single-well) so-

lutions. Apart from some synchronized motions where both averages (< x > and < y >)

have fairly close values, there are also regions with completely different averages. It is evident

that mistuning (see α in Eq. (1)) can lead to complicated mixed solutions where one of the

oscillators exhibits single well vibrations while the other exhibits double-well vibrations.

0.5 1 1.5 2−1

−0.5

0

0.5

1

ω

<x>

0.5 1 1.5 2−1

−0.5

0

0.5

1

ω<

y>

Figure 3. The average values of x and y displacements: < x > (a), < y > (b) versus

excitation frequency ω, obtained simultaneously with Fig. 2.

This effect of switching between different possible solutions, from single to double well

solutions and vice versa, can be also identified in Fig. 4, where we present the bifurcation

diagrams for the mistuned oscillators.

0.5 1 1.5 2

−2

−1

0

1

2

ω

x

0.5 1 1.5 2

−2

−1

0

1

2

ω

y

Figure 4. Simultaneously estimated bifurcation diagrams for x and y versus the excitation

frequency ω, which was changed quasi-statically.

The solution for ω = 1 is non-synchronized (see Fig. 2b). To explore this case more

carefully the corresponding phase portraits have been plotted with Poincare points projected

into different planes and corresponding time series. The initial conditions were chosen as

[x, x, y, y, v] = [0.01, 0, 0.01, 0, 0]). Figs. 5c-d clearly show that the discussed solution is

chaotic. Interestingly, the chaotic solution is induced by the second oscillator (with the

coordinate y) while the first oscillator (with the coordinate x) shows a more regular response

(Figs. 5ab). In the plane x–x, the attractor (Fig. 5a) resembles a smeared point of a regular

solution in the presence of noise-like disturbances. These disturbances are created by the

chaotically changing coordinate y coupled to the first oscillator through the linear electrical

circuit coupling (Eq. 2).

(a)−2 −1 0 1 2

−2

−1

0

1

2

x ve

loci

ty

x (b)4400 4420 4440 4460 4480 4500

−2

−1

0

1

2

t

x

(c)−2 −1 0 1 2

−2

−1

0

1

2

y ve

loci

ty

y (d)4400 4420 4440 4460 4480 4500

−2

−1

0

1

2

t

y

Figure 5. Phase portraits with Poincare points projected into planes x − x (a) and y − y

(c), and time series of x(t) (b) and y(t) (d) for ω = 1.

3. Summary and Conclusions

We have investigated the dynamical response of two magnetopiezoelastic harvesters with

mistuned stiffness connected in a parallel way via an electrical circuit. The total output

power versus the excitation frequency showed the typical resonance curve, however due to

mistuning the harvesters worked mostly in the unsynchronized regime. In the vicinity of the

resonance peak we found a chaotic solution which was driven by one of the oscillators.

Note that in this paper we used only one set of initial conditions (Fig. 5) and ω was

changed quasi-statically (to get the results in Figs. 2-4). However, to explain the problem

of multiple solutions in nonlinear systems (Eqs. (1-3)) and bifurcations one has to perform

more extended studies on initial conditions and has to estimate basins of attraction for the

given excitation frequency ω.

Acknowledgements The authors gratefully acknowledge the support of the Royal Society

through International Joint Project No. HP090343. C.A.K.K. was supported by the US of

Naval Reseach under the grant ONR N00014-08-1-0435. C.A.K.K. would like to thank C.

Nataray, A. Semamn, and S. Mastro for encouraging discussions.

References

1. Erturk A., Inman D.J.: Piezoelectric Energy Harvesting, (Wiley 2011).2. Anton S.R., Sodano H.A.: 2007, A review of power harvesting using piezo- electricmaterials (2003-2006), Smart Materials and Structures, 16, 2007, R1-R21.3. Arnold D.P.: Review of microscale magnetic power generation, IEEE Transactions onMagnetics, 43, 3940-3951, 2007.4. Beeby S.P., Torah R.N., Tudor M.J., Glynne-Jones P., O’Donnell T., Saha C.R., Roy S.: Amicro electromagnetic generator for vibration energy harvesting, Journal of Micromechanicsand Microengineering, 17, 1257- 1265, 2007.5. Litak G., Friswell M.I., Adhikari S.: Magnetopiezoelastic energy harvesting driven byrandom excitations, App. Phys. Lett., 96, 214103, 2010.6. Erturk A., Hoffmann J., Inman D.J., 2009,: A piezomagnetoelastic strucure forbroadband vibration energy harvesting, Appl. Phys. Lett., 94, 254102, 2009.7. Stanton S.C., McGehee C.C., Mann B.P.: 2010, Nonlinear dynamics for broadbandenergy harvesting: Investigation of a bistable piezoelectric inertial generator, Physica D,239, 640-653, 2010.8. Nana B., Woafo P.: Power delivered by an array of van der Pol oscillators coupled to aresonant cavity, Physica A, 387, 3305-3313, 2008.9. Shahruz S.M.: Design of mechanical band-pass filters for energy scavenging, Journal ofsound and Vibration, 292, 987-998, 2006.10. Erturk A., Renno J.M., Inman D.J.: Modeling of piezoelectric energy harvesting from anl-shaped beam-mass structure with an application to UAVs, J. Intell. Mater. Syst. Struct.,20, 529-544 2009.11. Ramlan R., Brennan M., Mace B., Kovacic I.: Potential benefits of a non-linear stiffnessin an energy harvesting device, Nonlinear Dynamics, 59, 545-558, 2010.12. Kim I., Jung H., Lee B.M., Jang S.: Broadband energy-harvesting using a two degree-of-freedom vibrating body, App. Phys. Lett., 98, 214102, 2011.

Grzegorz Litak, Professor: Lublin University of Technology, Department of Applied Mechan-ics, Nadbystrzycka 36, PL-20-618 Lublin, Poland (g.litak@pollub.pl).

Michael I. Friswell, Professor: Swansea University, College of Engineering, Singleton Park,Swansea SA2 8PP, United Kingdom (m.i.friswell@swansea.ac.uk).

Cedrick A. Kitio Kwuimy, Ph.D.: Villanova University, Department of Mechanical Engineer-ing, 800 Lancaster Avenue, Villanova, PA 19085, USA (kwuimy@yahoo.fr).

Sondipon Adhikari, Professor: Swansea University, College of Engineering, Singleton Park,Swansea SA2 8PP, United Kingdom (s.adhikari@swansea.ac.uk).

Marek Borowiec, Ph.D.: Lublin University of Technology, Department of Applied Mechanics,Nadbystrzycka 36, PL-20-618 Lublin, Poland (m.borowiec@pollub.pl).