Enhancing Energy Harvesting System Using ...

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Stanton et al. [7, 8] used recently the Melnikov theory to discuss the perfor- mance of a ..... [13] Samuel C. Stanton, Benjamin A.M. Owens, and Brian P. Mann.
Proceedings of the ASME 2013 Dynamic Systems and Control Conference DSCC2013 October 21-23, 2013, Palo Alto, California, USA

DSCC2013-4109

ENHANCING ENERGY HARVESTING SYSTEM USING MATERIALS WITH FRACTIONAL ORDER STIFFNESS

C.A. KITIO KWUIMY∗ Dept. of Mechanical Engineering Villanova University Villanova, PA19085, USA Email: [email protected]

G. LITAK Dept. of Applied Mechanics Technical University of Lublin Nadbystrzycka 36, Poland Email: [email protected]

the system should be considered during design task. Stanton et al. [7, 8] used recently the Melnikov theory to discuss the performance of a bistable harvester by analyzing the critical condition for homoclinic bifurcation and crossing-well dynamics as function of system parameters. Crossing-well dynamics is advantageous for harvester since the corresponding output voltage is higher. Stochastic resonance [9] has been demonstrated to lead to crossing well dynamics in bistable symmetric or asymmetric harvesters [2, 3]. Effects of linear and nonlinear transduction were discussed by Owens and Mann [10]. They demonstrated that with a proper design, nonlinear coupling is better than linear coupling, however nonlinear coupling can sometimes be detrimental for the system. Efforts have been devoted on developing composite materials with various physical properties to enhance the performance of the system. A direct approach is to design a material obeying a given physical law. These research contributions have demonstrated that nonlinear phenomena and design constraints highly influence the performance of the harvester. This paper considers the possibility of enhancing the output power by using materials which restoring force has a fractional order term.

ABSTRACT A bistable mechanical system having fractional order restoring force is considered for possible energy harvesting. The effects of the fractional order stiffness α on the crossing well dynamics (large amplitude motion) and the output electrical power are analyzed. The harvested electric power appears to be efficient for deterministic and random excitation, for small α (α ≈ 2). High level noise intensity was found to reduce the output power in the region of resonance and surprisingly increases the outup in other region of α. For larger enough amplitude of harmonic excitation this effect is realized in a stochastic resonance.

INTRODUCTION Energy harvesters are designed to transform available ambient energy into electrical energy through various transduction mechanisms such as piezoelectricity, electromagnetic induction, photoelectrochemical and biological processes. They are designed for small electronic devices, to recharge batteries and enable remote operation [1–3]. One of the ultimate goals of research in energy harvester is to use nonlinear phenomena, material science and system design to enhance the performance of the system, that is maximizing the output power harvests from environmental vibration of weak amplitude.

1 Mathematical model and characterization We discuss the efficiency of an energy harvester system with a mechanical part having a fractional order deflection. The generic physical set up is shown on Fig. 1 and the corresponding dimensionless mathematical model is given as [8, 11–13]

In achieving this goal, Shahruz [4], Ramlan et al. [5, 6] showed that a bistable configuration of the potential energy of ∗ Address

C. NATARAJ Mr. & Mrs. Robert F. Moritz Sr. Endowed Chair in Engineered Systems Dept. of Mech. Engineering Villanova University, Villanova, PA 19085 Email: [email protected]

x¨ + λx(1 ˙ + µx2) − x + x |x|α−1 − θv = f (t),

all correspondence to this author.

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(1)

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V(x)

0 −0.2 −0.4 −2

e d c b a −1

0 x

1

2

Figure 2. (color online) Potential energy for some value of the order deflection: α = 3 in green (c), α = 3/2 in blue (e), α = 2.4 in yellow (d) ,

α = 4 in red (b), α = 6 in black (a).

Figure 1. Generic model of energy harvester. The magnetoeleastic beam is supposed to have fractional order restoring force [11].

v˙ + γv + θx˙ = 0,

Figure 2 shows the potential energy for some physical values of α. It appears that as α increases the deepness of the potential well increases. Thus an unperturbed system with a large value of α will hardly present complex phenomena such as snap through instability, since the corresponding threshold energy to cross the potential barrier is higher [8]. In other words, an energy harvester with α = 4 (yellow line) will easily present snap through instability than for example another one with α = 6 (black line). In this paper, the following values are used λ = 0.13, µ = 0.25, θ = 0.283, γ = 0.0078, and the initial conditions are x(t = 0) = −1, x(t ˙ = 0) = 0 and v(t = 0) = 0.

(2)

where x is a dimensionless defection of the magnetoelastic beam, v is a dimensionless voltage across the load resistor Rℓ , f (t) is the perturbation, λ and µ are linear and nonlinear damping coefficients, γ is the dimensionless electric impedance, α > 1 is the fractional order of deflection, whose value depends on the material properties. Equations (1)-(2) were considered by Stanton et al. [13] for α = 3. They used the method of harmonic balance for intra-well and inter-well oscillations. The analysis in this paper can be seen as a generalization of their contribution

2 Effects of fractional stiffness on the output power

Our motivation is based on two important points:

2.1

There are many materials whose elastic properties has α is larger than 1 and not an integer. Various physical values of α for some materials are given by Cvetivanin and Zukovic [11]. As example the soft non-linear model of beam dynamics is described by a second order differential equation with α = 3, the properties of suspension and tires in vehicles dynamics are done with α = 23 . Hammers in piano have α between 1.5 and 3.5. Geometrical nonlinearities leading to similar effects were discussed in Ref. [14]. In micro engineering, the restitution force for a wide class of micro systems is optimal for α in the interval 4 − 7. The experimental value of α for an open celled polyurethane foam automotive seat cushions is about 6. Advance in material science suggests to investigate the performance of the system for various values of α. Considering the “classic” harvester with Duffing potential, analysis of fractional order deflection gives and insight of the system response in case of alteration. It has been shown that, the fractional order deflection α strongly influence the criteria for crossing well dynamics [11, 12, 15] which leads to maximum output power in the electric load in the context of energy harvesting [8].

System under ambient harmonic excitation

Under harmonic ambient excitation, the maximum electric voltage is plotted in Fig. 3a as function of driven frequency Ω for different values of the fractional order α showing that, increasing the fractional order results in a decreasing of the output voltage and a forward shift of the resonant frequency. Similarly, Fig. 3b shows the output voltage as function of al pha. It is obtained a forward shift of the optimal value of the fractional order as the driven frequency increases. Optimal value of output voltage is obtained for small frequency. The bifurcation diagram from intra-well to inter-well dynamics is shown in Figure 4 for a deterministic perturbation f (t) = E0 cos(Ωt), E0 being the amplitude and Ω = 1 the frequency. This bifurcation leads to large amplitude dynamics which corresponds to higher output power [8]. From Figure 4a, it can be deduced that the lowest values of E0 for crossing well dynamics is obtained for α = αc ≈ 2.1 which is less than α = 3 generally used (Duffing potential) in Harvester model. The critical value of E0 increases as α increases from αc ; and decreases as α decreases toward αc . The bifurcation diagram of the maximum displacement is shown in Figure 4b. For E0 = 0.05 (intrawell dynamics, diagram in red), a pick of amplitude is obtained

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static equilibrium. For E0 = 0.4 (inter-well dynamics, diagram in blue), large amplitude chaotic motion is obtained (α < 2.18) followed by large amplitude periodic branch (2.18 < α < 3.65). An abrupt transition to small amplitude motion (intra-well motion) is obtained at α = 3.65.

0.4

v

max

a 0.2

b c d

0 0

vmax

0.2

1 a

e 2 Ω

3

To illustrate dynamics of phenomena happening in the system as α changes, the phase portrait are plotted (Fig. 5) for the values of Figure 3 and E0 = 0.4. Large amplitude chaotic motions are obtained for small values of α, this include chaotic behavior (Figure 4a) and periodic behavior (Figure 5b). Small amplitude motions are obtained for higher values of α. The system has periodic response (Figure 5c) and bi-periodic motion (Figure 5d). Thus for a fairly large value of alpha, the potential barrier is not overcame and the dynamics is confined within a single well.

4 (a)

b c

0.1

d 0

Figure 3.

α 4

2

6

(b)

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(color online) Maximum output voltage as function of the sys-

tem parameters. (a): Effects of the fractional order: α = 3 in green (c), α = 3/2 in blue (e), α = 2.4 in yellow (d) , α = 4 in red (b), α = 6 in black (a). (b): Effects of the driven frequency: Ω = 1 in red (b), Ω = 3 − 4 in black (a), Ω = 1.6 in green (d) , Ω = 2.1 in yellow (b).

The previous paragraph gave useful information for the design of energy system. However, it many situations, the excitation is barely harmonic. Rather, it contains multi frequency, known as broadband random excitation. As shown in previous contributions [2, 3] under a well chosen amount of intensity a broadband random excitation can help enhance the performance of the system. In fact stochastic resonance yields to an overcome of the potential barrier which lead the system to large amplitude of motion and thus, higher output power. Such phenomena is considered in this paragraph by analysing the effects of the fractional order.

0.6 0.5 Inter−well behavior E0

0.4 0.3 Intra−well behavior

0.2 0.1 1

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α 4

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As illustration, we choose a random perturbation of the form f (t) = E0 cos(Ωt + Ψ(t)) where the random variable Ψ(t) is given by stationary Gaussian white noise with zero mean and standard deviation σΨ . Figure 5 is plotted for α = 4.5 (Values of Fig. 5d). In Fig. 5a, σ = 0.0, and the bi-periodic dynamics remain within a single well (Fig. 5d). For relatively large value of σ = 0.1, the system become more noisy, and for σ = 0.15 the potential well is overcame through the stochastic resonance.

(a)

E0=0.4

x

1 0 −1

E0=0.05

2

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α 4

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System under broadband random ambient excitation

Effects of the noise intensity on the average output power, which is proportional to < v2 >, are shown in Fig. 7. Note that increasing the noise level reduces the output power for α ≈ 2, which is already intra-well oscillation. However, for higher values of α, it works in the opposite direction. For given E0 , decreasing α causes the deterministic system to go from synchronized double well to single well oscillations (Fig. 4a). In the single well oscillations response, the noise is beneficial for the potential well and leads to the stochastic resonance. Furthermore, Fig. 7a-b show that, the highest amplitude is obtained for a system free of noise and for α < 3, this highest value of output power < v2 > is shifted forward as noise intensity increases (Fig 7b). As reported in Ref. [14] smaller α causes shallower

(b)

(color online) Single/double-well oscillations for λ = 0.13, µ = 0.25, θ = 0.283, γ = 0.0078.(a): Lower bound for inter-well dynamics in the plane. (α, E0 ). (b): Bifurcation diagram of the maximum amplitude as function of α. E0 = 0.4 in blue and E0 = 0.05 in red. The initial conditions used: (x(0), dx dt (0), v(0))=(-1,0,0). Figure 4.

at α = 2.01. Beyond this value of α the system tends to oscillate with amplitude which value progressively converges to the

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2

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dx/dt

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Illustration of stochastic resonance for α = 4.5. (a): σ = 0.1, σ = 0.15. The initial conditions: (x(0), dx dt (0), v(0))=(-1,0,0).

Figure 6.

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E0 = 0.4 and the value of Fig. 2.

(a):α

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´ Phase portraits with Poincarepoints for crossing well dynam-

ics as function of

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= 2.1, large

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amplitude chaotic motion. (b): α = 3.5, large amplitude periodic motion. (c): α = 3.65, small amplitude periodic motion. (d): α = 4.5,

1

small amplitude bi-periodic (single well) motion. The initial conditions:

Figure 7.

(x(0), dx dt (0), v(0))=(-1,0,0).

2

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α

4

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(color online) Average output power as function of α for various

noise intensity σ = 0 (1), 0.1 (2), 0.25 (3), 0.5 (4), 0.75 (5); and E0 = 0.05 (a), E = 0.4 (b). The initial conditions: (x(0), dx dt (0), v(0))=(1,0,0).

potential barrier (Fig. 2), and consequently easier to overcome. Simultaneously, such a shallow potential causes fairly small escape velocity and its range of variation. Finally, voltage fluctua-

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tion response (Eq. 2) is closely related with variation of velocity. Decreasing α resulting in competition between easier potential well escape, and smaller variation in velocity. These two effect on voltage response in the opposite way.

[7] S. C. Stanton, Clark C. McGehee, and Brian P. Mann. Nonlinear dynamics for broadband energy harvesting: Investigation of a bistable piezoelectric inertial generator. Physica D: Nonlinear Phenomena, 239(10):640 – 653, 2010. [8] Samuel C. Stanton, Brian P. Mann, and Benjamin A.M. Owens. Melnikov theoretic methods for characterizing the dynamics of the bistable piezoelectric inertial generator in complex spectral environments. Physica D: Nonlinear Phenomena, 241(6):711 – 720, 2012. [9] L. Gammaitoni, P. Hnggi, P. Jung, and F. Marchesoni. Stochastic resonance. Review of Modern Physics, 70:223– 288, 1998. [10] Benjamin A.M. Owens and Brian P. Mann. Linear and nonlinear electromagnetic coupling models in vibrationbased energy harvesting. Journal of Sound and Vibration, 331(4):922 – 937, 2012. [11] L. Cveticanin and M. Zukovic. Melnikov’s criteria and chaos in systems with fractional order deflection. Journal of Sound and Vibration, 326(3-5):768–779, 2009. [12] C. A. Kitio Kwuimy and B. R. Nana Nbendjo. Active control of horseshoes chaos in a driven Rayleigh oscillator with fractional order deflection. Physics Letters A, 375(39):3442–3449, 2011. [13] Samuel C. Stanton, Benjamin A.M. Owens, and Brian P. Mann. Harmonic balance analysis of the bistable piezoelectric inertial generator. Journal of Sound and Vibration, 331(15):3617–3627, 2012. [14] G. Litak, M. Borowiec, and A. Syta. Vibration of generalized double well oscillators. Journal of Applied Mathematics and Mechanics, 87:590–602, 2007. [15] L. Cveticanin. Oscillator with fraction order restoring force. Journal of Sound and Vibration, 320(45):1064 – 1077, 2009.

3 Conclusion In summary, our analysis indicates that fractional order deflection of small order can be used to enhance energy harvester system. For these values, the depth of the potential energy of the mechanical part is lower which facilitate the large amplitude of vibration and correspond to optimal output power. Finally, for the given system parameters, we found the optimal value of the fractional order α = 2.1. Note that this results on fractional order deflection α should be reconsidered in a specific material design and for system parameters of energy harvester.

4 Acknowledgment This work has been funded by the US Office of Naval research (CAKK and CN) under the grant ONR N00014-08-10435 (Program manager: Mr. Anthony Seman III) and by the Polish National Science Center (GL) under grant agreement 2012/05/B/ST8/00080.

REFERENCES [1] A. Erturk. Piezoelectric energy harvesting for civil infrastructure system applications: Moving loads and surface strain fluctuations. Journal of Intelligent Material Systems and Structures, 22(17):1959–1973, 2011. [2] G. Litak, M. Borowiec, M. I. Friswell, and S. Adhikari. Energy harvesting in a magnetopiezoelastic system driven by random excitations with uniform and gaussian distributions. Journal of Theoretical and Applied Mechanices, 49:757, 2011. [3] C. A. Kitio Kwuimy, G. Litak, M. Borowiec, and C. Nataraj. Performance of a piezoelectric energy harvester driven by air fow. Applied Physics Letters, 100(2):024103– 3, 2012. [4] R. Ramlan, M.J. Brennan, B.R. Mace, and I. Kovacic. Potential benefits of a non-linear stiffness in an energy harvesting device. Nonlinear Dynamics, 59:545–558, 2010. [5] S. M. Shahruz. Design of mechanical band-pass filters for energy scavenging. Journal of Sound and Vibration, 292(35):987–998, 2006. [6] S. M. Shahruz. Increasing the efficiency of energy scavengers by magnets. Journal of Computational and Nonlinear Dynamics, 3(4):041001, 2008.

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