Energy harvesting from quasi-periodic vibrations

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DOI 10.1007/s11071-016-2668-6

O R I G I NA L PA P E R

Energy harvesting from quasi-periodic vibrations

Mohamed Belhaq · Mustapha Hamdi

Received: 27 September 2015 / Accepted: 2 February 2016

Abstract Quasi-periodic (QP) vibration-based energy harvesting is studied in this paper. The energy harvesting system consists in a delayed van der Pol oscillator with time-varying delay amplitude coupled to an electromagnetic energy harvesting device in which the vibration source is due to self-excitations. We consider the case of delay parametric resonance for which the frequency of the delay modulation is near twice the natural frequency of the oscillator. Application of the double-step perturbation method enables the approx- imation of the amplitude of the QP vibrations of the oscillator far from the resonance. This amplitude is used to extract the maximum and average QP powers from the harvester device. The influence of different system parameters on the performance of the QP vibration-based energy harvesting is reported and discussed. Results show that for appropriate values of parameters, QP vibrations can be more efficient for energy harvesting in pure self-excited systems not only in terms of power extraction, but also in terms of broad- ening the parameter range of energy extraction. Numer- ical simulation is systematically conducted to support the analytical predictions.

M. Belhaq (

B

⁾

Laboratory of Mechanics, University Hassan II of Casablanca, Casablanca, Morocco

e-mail: mbelhaq@yahoo.fr M. Hamdi

Faculty of Science and Technology-Al Hoceima, University Mohammed I, Oujda, Morocco

Keywords Quasi-periodic vibrations · Energy harvesting · Delayed van der Pol oscillator · Electro- mechanical coupling · Time-varying delay amplitude

1 Introduction

The design aspect and performance improvement of energy harvesting systems deeply depend on the optimal choice of mechanical and harvester system characteristics. To efficiently harvest energy from the ambient environment, the ultimate goal is to produce large- amplitude oscillations with broadband performance characteristics, keeping the inherent mechanical damping low [1].

When energy harvesting systems are subject to forced excitations, the resulting vibration responses usually have the disadvantage to strongly depend on the unknown frequency and amplitude of excitations.

In such circumstances, the efficiency of the energy harvesting operating in the linear resonance regime is achieved in a narrow region around the resonance peak limiting considerably its performance in the sense that the fundamental frequency of the harvester must always match the frequency of the forcing [2–5]. To overcome this limitation, nonlinearities were used to improve the energy harvesting capability by extend- ing the bandwidth of the harvester over broad range of excitation frequencies. In such, the harvesting system may be designed with hardening characteristic [6–10]

or with a bistable double well potential; see for instance

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[9,11–15] and references therein. However, bistability can reduce the system performance when it is attracted to the low-amplitude motions [16] or when it suffers jump phenomena. Likewise, in forced excitations the dependence of the response on the unknown excitation parameters may result in a complicated design realiza- tion of the excitation [17–19].

In the case where energy harvesting systems are self- excited, the dependence of the system response on the excitation parameters can be remedied. The induced limit-cycle (LC) oscillations produced by a destabilization of the system trivial solution through supercriti- cal Hopf bifurcation may indeed be exploited in harvesting power in the range of large-amplitude oscillations. Under certain circumstances (presence of forced excitations and bistability), the steady-state LC oscillations may lose stability via a secondary Hopf bifurcation producing QP vibrations and causing a substantial reduction in the harvested power away from the resonance [20–22]. To avoid this undesirable effect, focus was placed toward adjusting the parameter control to shift the secondary Hopf bifurcation to higher values of the control parameter [21]. As an example, in an energy harvester system under combined aerodynamic and base excitations it was observed that beyond the flutter speed, the response of the harvester is QP far from the resonance, leading also to a substantial drop of the output power [22].

In energy harvesting systems operating only in a pure self-induced oscillations regime (absence of forced excitations and bistability), LC oscillations born by destabilization of the system trivial solution can be exploited efficiently to enhance the power output using the largest vibrational amplitude. An open question is whether the QP vibrations born by destabilization of the LC may also be exploited to enhance the energy harvesting performance. A recent work reported on this issue in terms of vibrational amplitude by considering a delayed pure van der Pol oscillator with modulated delay amplitude [23]. More precisely, it was shown analytically that modulating the delay amplitude in the pure delayed van der Pol oscillator may produce large- amplitude QP vibrations (larger than the amplitude of the periodic response) performing in broader range of parameters.

Motivated by this finding, the present paper seeks to take advantage of such large-amplitude QP vibrations to enhance the power output in the QP vibration range

near and far from a delay resonance. The device under consideration consists in a delayed van der Pol equation with time-varying delay amplitude [23] coupled to an electromagnetic energy harvesting system. Atten- tion is paid to the effect of different system parameters on the performance of the QP vibration-based energy harvesting.

The organization of the paper is as follows: In Sect. 2, the electromagnetic vibration-based harvesting system is presented. The steady-state periodic response and the harvested power near the delay resonance are derived using averaging method. The influence of different parameters of the harvesting system on the periodic response and the corresponding power output is examined. In Sect. 3, we approximate the QP response applying the second-step perturbation method. The harvested power extracted from the QP vibration is ana- lyzed and compared to the periodic-based power output. A summary of the results is provided in the con- cluding section.

2 Equation of motion and periodic energy harvesting

The energy harvester under consideration consists of a pure delayed van der Pol oscillator coupled to an electric circuit through an electromagnetic coupling, as shown in the schematic presented in Fig. 1. The quantities F

k

, F

d

and F

_vd P

are, respectively, the linear restoring force, the time delay in the position and the van der Pol damping component. The governing equations can be written as

Fig. 1 Schematic of the energy harvesting model

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m ¨ z − (¯ α − ¯ β z

²

) z ˙ + kz = λ

t¯

z ( t ¯ − τ) + ˆ i (1a) L di

d t ¯ + R

eq

i = − ˆ z ˙ (1b) where the overdot represents a derivative with respect to time t. The variable ¯ z represents the relative displace- ment of the mass m, α ¯ and β ¯ are damping coefficients, k is the linear stiffness of the restoring force, ˆ is the electromagnetic coupling coefficient, λ

t¯

is the delay amplitude assumed to be time-varying, τ is the time delay, while i and

^di_d¯_t

have been substituted for electri- cal charge coordinate q (i = ˙ q). The quantities R

eq

and L are the equivalent resistance load and the inductance of the harvesting coil, respectively.

To obtain a dimensionless form of Eq. (1), we introduce the following nondimensional parameters:

t = ω

n

t, ¯ z =

_z^z₀

, i =

_ˆ^L_z

0

i, γ

1

=

_{k L}^ˆ²

, γ

2

=

_L^R_ω^eq_n

, α =

_m^α^¯_ω_n

, β =

_m^β^¯^z_ω²⁰_n

and λ

t

=

_m^λ_ω^t^¯2

n

where z

0

is the initial elongation of the compressed spring charged by proof mass m and ω

n

= √

k / m is the natural frequency of the mechanical oscillator. Introducing a bookkeeping parameter ε and scalling the electromechanical coupling, γ

1

, the damping components, α, β and the delay amplitude λ

t

, Eqs. (1) can be rewritten in terms of the nondimensional parameters as

¨

z − (α − β z

²

)˙ z + z = λ

t

z ( t − τ) + γ

1

i (2a) di

dt + γ

2

i = −˙ z (2b)

where an overdot denotes now differentiation with respect to time t .

In a recent work [23], the mechanical oscillator part of the considered energy harvester system presented in Fig. 1 (i.e., the pure delayed van der Pol equation) has been studied in the case where the delay amplitude λ

t

is modulated harmonically around a mean value, such that

λ

t

= λ

1

+ λ

2

cos ωt (3)

where λ

1

is referred to as the unmodulated delay amplitude, while λ

2

and ω are the amplitude and frequency of the delay amplitude modulation. Specifically, it was shown [23] that modulating the delay amplitude in the delayed van der Pol oscillator induces larger QP vibration amplitude than the periodic one in broadband

of parameters. Here, we shall take advantage of such large-amplitude QP vibrations to improve energy harvesting performance in the QP vibration domain. In this study, one considers the case of primary delay parametric resonance for which the frequency of the delay modulation ω is near twice the natural frequency of the oscillator ( ω ≈ 2), such that the resonance condition reads 1 = (

^ω₂

)

²

+ εσ where σ is a detuning parameter.

Neglecting higher-order harmonics, the steady-state solution of Eqs. (2) can be sought in the form

z(t ) = R cos ω

2 t − θ

(4a) i (t ) = I cos

ω 2 t − ϕ

(4b) where R, I , θ and φ are the amplitudes and the phases of the responses.

To approximate the steady-state solutions near the delay parametric resonance, we apply the averaging method [24]. To this end, we rewrite Eq. (2a) as

¨ z + ω

²

4 z = H (z, z, ˙ i, t) (5)

where

H ( z , ˙ z , i , t ) = −σ z + (α − β z

²

)˙ z

+(λ

1

+ λ

2

cos ω t ) z ( t − τ) + γ

1

i (6) The approximate solution of Eq. (5) is supposed to take from (4a) and (4b) where the amplitudes R and I and the phases θ and ϕ are governed by the equations R ˙ = −

2 π

⁴^π

ω

0

sin ω

2 t − θ

H(z, z, ˙ i, t )dt (7a)

R θ ˙ = 2π

⁴π ω

0

cos ω

2 t − θ

H (z, z, ˙ i, t)dt (7b) I ˙ + ˙ R cos(θ − ϕ) − R θ ˙ sin(θ − ϕ)

= −γ

2

I + ω

2 R sin(θ − ϕ) (7c)

I ϕ ˙ + ˙ R sin (θ − ϕ) + R θ ˙ cos (θ − ϕ)

= − ω 2 I − ω

2 R cos(θ − ϕ) (7d)

Integrating and truncating the Taylor series expansions

for those terms containing time delay [25] by assum-

ing the product τ is small, one obtains the following

modulation equations

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R ˙ = k

1

R + k

2

R

³

+ k

3

R cos 2 θ + k

4

R sin 2 θ + γ

1

ω I sin(θ − ϕ) (8a)

R θ ˙ = k

5

R + k

4

R cos 2θ − k

3

R sin 2θ + γ

1

ω I cos (θ − ϕ) (8b)

I ˙ + ˙ R cos(θ − ϕ) − R θ ˙ sin(θ − ϕ)

= −γ

2

I + ω

2 R sin(θ − ϕ) (8c)

I ϕ ˙ + ˙ R sin(θ − ϕ) + R θ ˙ cos(θ − ϕ)

= − ω 2 I − ω

2 R cos(θ − ϕ) (8d)

where k

1

=

^α₂

−

^λ_ω¹

sin(ωτ/2), k

2

= −

^β₈

, k

3

=

λ2

2ω

sin(ωτ/2), k

4

=

₂^λ_ω²

cos(ωτ/2) and k

5

= −

^σ_ω

+

λ1

ω

cos(ωτ/2). Equilibria of system (8), corresponding to periodic oscillations of Eq. (5), are determined by setting R ˙ = ˙ θ = ˙ I = ˙ φ = 0. Squaring, adding and eliminating θ and ϕ and define ρ = R

²

, we obtain the following system of equations

ρ( k

₁

+ k

2

ρ)

²

+ ρ( k

₅

+ k

6

ρ)

²

= ρ( k

₃²

+ k

₄²

) (9a) I = ω

4γ

₂²

+ ω

²

R (9b)

sin(θ − ϕ) = 2γ

2

4γ

₂²

+ ω

²

(9c) cos(θ − ϕ) = − ω

4γ

₂²

+ ω

²

(9d)

where k

₁

= k

1

+

₄²_γ^γ2¹^γ²

2+ω²

and k

₅

= k

5

−

₄_γ²^γ2¹^ω 2+ω²

. The first equation of this system, Eq. (9a), represents the amplitude–frequency response which has two solutions given by

R

1,2

=

B ± √

B

²

− AC

A (10)

where A = k

₂²

, B = −k

1

k

2

and C = k

²₁

+k

²5

−k

₃²

−k

₄²

. To extract the maximum and average output powers, we consider the nondimensional instantaneous power given by

P ( t ) = γ

2

i ( t )

²

(11)

It follows that the average power can be obtained by averaging over a single delay modulation period. This leads to

P

av

= ω 4π

⁴π ω

0

P ( t ) dt (12)

Using Eqs. (4b), (9b), (11) and the maximization procedure, one obtains the maximum power response as P

max

= γ

2

ω

²

4 γ

₂²

+ ω

²

R

²

(13)

and using Eqs. (4b), (11) and (12) the average power response reads

P

av

= γ

2

ω

²

2 ( 4 γ

₂²

+ ω

²

) R

²

(14)

Equations (9a), (13) and (14) are used to examine the influence of different system parameters on the steady- state response and on the maximum and average output powers of the harvester. In what follows, we fix the damping parameters α = 0 . 1, β = 0 . 7 and the electromechanical coupling coefficient γ

1

= 0.13.

The frequency–response curve, as given by Eq. (9a), is depicted in Fig. 2a near the delay parametric resonance for two different values of the time delay.

Figure 2b, c illustrates, respectively, the effect of the time delay on the maximum and average output powers, P

max

, P

av

, versus the frequency indicating that the periodic output powers increase by increasing τ and can be extracted only in a small range of the frequency ω near the resonance.

Figure 3a–c depicts, respectively, the variation of the amplitude of the steady-state response, the maximum power response P

max

and the average power response P

av

versus λ

1

. For validation, analytical predictions [solid (dot) lines for stable (unstable)] are compared to results obtained by numerical simulations (circles) using dde23 [26] algorithm. Inset in Fig. 3a, b are shown, respectively, time histories of the steady state and of the harvested power responses obtained by integrating numerically Eq. (2) and using Eq. (13).

The variation of the amplitude of the steady-state

response, the P

max

and P

av

, versus the modulated delay

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0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.2 0.4 0.6 0.8 1 1.2

ω

R

(a)

τ=1.58

τ=π

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1 1.2

ω Pmax

(b)

τ=1.58

τ=π

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1 1.2

ω Pav

(c)

τ=1.58

τ=π

Fig. 2 Vibration and powers amplitude versusωforλ1 = −0.2,λ2 = 0.2,γ2 =1.33 and for different values ofτ;afrequency response,bmaximum power response,caverage power response.Solid linefor stable,dotted linefor unstable

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

λ₁ λ₁ λ₁

R ⁵⁰ ¹⁰⁰ ¹⁵⁰ ²⁰⁰

−1 0 1

Time, t

z(t)

(a)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Pmax

0 50 100 150 200

0 0.5 1

Time, t

P(t)

(b)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Pav

(c)

Fig. 3 Vibration and powers amplitude versusλ1forω=2.1,τ=1.58,λ2=0.2 andγ2=1.33;avibration amplitude,bmaximum power response,caverage power response.Solid linefor stable,dotted linefor unstable andcirclefor numerical solution

amplitude λ

2

is presented, respectively, in Fig. 4a–c for different values of the unmodulated delay amplitude λ

1

. The figures show clearly an increase of the output powers by increasing λ

2

and negative λ

1

in the range where the stable steady-state solution emerges.

Figure 5 depicts the variation of the amplitude of the periodic response and the output power amplitudes P

max

and P

av

versus the coupling term γ

2

for two values of λ

1

. It can be clearly seen that the amplitude response and the output powers increase substantially by increasing negative λ

1

and the powers can be extracted only in a certain range of the coupling coefficient γ

2

.

Similar plots are presented in Fig. 6 showing also an increase of the output powers by increasing negative λ

1

and indicating that the energy harvesting can be generated only in small alternate ranges of time delay.

3 Quasi-periodic energy harvesting

Next, we shall approximate the QP response, its modulation envelope as well as the QP vibration-based energy harvesting near and far from the delay parametric resonance using the second-step perturbation method [27–29]. Since the slow flow (8) is invariant under the transformation θ → −θ +

^π₂

, φ → −φ +

^π₂

, ω → −ω, k

3

→ −k

3

and k

5

→ −k

5

, system (8) can be rewritten in the form

R ˙ = k

1

R + k

2

R

³

+ sk

3

R cos 2θ + k

4

R sin 2θ + γ

1

sω I sin (θ − ϕ) (15a)

R θ ˙ = sk

5

R + sk

6

R

³

+ k

4

R cos 2θ − sk

3

R sin 2θ + γ

1

sω I cos(θ − ϕ) (15b)

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0 0.5 1 1.5 0

0.5 1 1.5 2 2.5 3

λ2 λ2 λ2

R

(a)

λ₁=−0.2 λ1=−0.9

0 0.5 1 1.5

0 0.5 1 1.5 2 2.5 3

Pmax

(b)

0 0.5 1 1.5

0 0.5 1 1.5 2 2.5 3

Pav

(c)

Fig. 4 Vibration and powers amplitude versusλ2forω=2.1,τ=1.58,γ2=1.33 and for different values ofλ1;avibration amplitude, bmaximum power response,caverage power response

0 0.5 1 1.5 2 2.5

γ2 γ

2 γ

2

R

λ1=−0.9

λ1=−0.2

(a)

0 0.5 1 1.5 2 2.5

Pmax

(b)

0 0.5 1 1.5 2 2.5

Pav

(c)

Fig. 5 Vibration and powers amplitude versusγ2forω =2.1,τ =1.58 andλ2=0.2;avibration amplitude,bmaximum power response,caverage power response

0 1 2 3 4 5 6 7 8 9

0 0.5 1 1.5 2 2.5 3

Time delay, τ

R

(a)

λ1=−0.2 λ₁=−0.9

0 1 2 3 4 5 6 7 8 9

0 0.5 1 1.5 2 2.5 3

Time delay, τ Pmax

(b)

0 1 2 3 4 5 6 7 8 9

0 0.5 1 1.5 2 2.5 3

Time delay, τ Pav

(c)

Fig. 6 Vibration and power amplitudes versusτforω =2.1,λ2=0.2 andγ2 =1.33;avibration amplitude,bmaximum power response,caverage power response

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I ˙ + ˙ R cos (θ − ϕ) − R θ ˙ sin (θ − ϕ)

= −γ

2

I + sω

2 R sin(θ − ϕ) (15c)

I ϕ ˙ + ˙ R sin(θ − ϕ) + R θ ˙ cos(θ − ϕ)

= − sω 2 I − sω

2 R cos(θ − ϕ) (15d)

where s = ±1. This invariance property will be exploited later to determine the boundaries of the modulation envelope of the QP vibrations. To investigate the QP response, it is convenient to transform the polar form (15) to the following Cartesian system using the variable change u = R cos θ , v = −R sin θ, w = I cos ϕ, y = − I sin ϕ

du

dT

0

= (sk

5

− k

4

)v + η

(k

1

+ sk

3

)u +(k

2

u + sk

6

v)(u

²

+ v

²

) + γ

1

sω y

(16a) dv

dT

0

= −(sk

5

+ k

4

)u + η

(k

1

− sk

3

)v +(k

2

v − sk

6

u)(u

²

+ v

²

) − γ

1

sω w

(16b) dw

dT

0

= −γ

2

w − du

dT

0

− η sω 2 v + sω

2 y

(16c) d y

dT

0

= −γ

2

y − dv dT

0

+ η

− sω 2 u + sω

2 w

(16d)

where η is a new bookkeeping parameter introduced to implement a second perturbation analysis [27]. As usual, the parameter η is introduced in damping and nonlinearity so that the unperturbed system of Eq. (16) admits a basic solution.

The periodic solution of the slow flow (16), corresponding to the QP response of the original system (2), is obtained using the multiple scales method [24]. This can be done by expanding the solution as

u(T

0

, T

1

) = u

0

(T

0

, T

1

) + ηu

1

(T

0

, T

1

) + O(η

²

) v(T

0

, T

1

) = v

0

(T

0

, T

1

) + ηv

1

(T

0

, T

1

) + O(η

²

) (17) where T

0

= t and T

1

= ηt. Introducing D

i

=

_∂^∂_T

i

yields

_dt^d

= D

0

+ ηD

1

+ O(η

²

), substituting Eqs. (17) into Eqs. (16) and collecting terms, we get the following hierarchy of systems

D

₀²

u

0

+ ν

²

u

0

= 0 , (18) ( sk

5

− k

4

)v

0

= D

0

u

0

(19) D

0

w

0

+ γ

2

w

0

= −D

0

u

0

(20) D

0

y

0

+ γ

2

y

0

= −D

0

v

0

(21) D

₀²

u

1

+ ν

²

u

1

= (sk

5

− k

4

)

− D

1

v

0

+(k

1

− sk

3

)v

0

+(k

2

v

0

− sk

6

u

0

)(u

²0

+ v

0²

)

− γ

1

s ω w

0

− D

0

D

1

u

0

+ D

0

(k

1

+ sk

3

)u

0

+ (k

2

u

0

+ sk

6

v

0

) (u

²0

+ v

²0

) + γ

1

s ω y

0

(22) (sk

5

− k

4

)v

1

= D

0

u

1

+ D

1

u

0

− (k

1

+ sk

3

)u

0

−(k

2

u

0

+ sk

6

v

0

)(u

²₀

+ v

₀²

)

− γ

1

sω y

0

(23)

D

0

w

1

+ γ

2

w

1

= −D

1

w

0

− (D

0

u

1

+ D

1

u

0

)

− sω

2 v

0

− sω

2 y

0

(24)

D

0

y

1

+ γ

2

y

1

= − D

1

z

0

− ( D

0

v

1

+ D

1

v

0

)

− sω

2 u

0

+ sω

2 w

0

(25)

where ν =

k

₅²

− k

₄²

(26)

is the frequency of the periodic solution corresponding to the frequency of the QP modulation. Figure 7a, b shows the variation of this frequency modulation as function of τ and λ

1

, respectively. The plots indicate that the modulation occurs in certain alternate ranges of time delay, as depicted in Fig. 7a, while Fig. 7b determines the ranges of λ

1

where the modulation of the QP vibrations is more intense.

The solution to the first order is given by

u

0

(T

0

, T

1

) = r(T

1

) cos(νT

0

+ φ(T

1

)) (27a) v

0

( T

0

, T

1

) = − ν

(sk

5

− k

4

) r ( T

1

) sin (ν T

0

+ φ( T

1

)) (27b) w

0

(T

0

, T

1

) = S r (T

1

)

γ

2

sin(νT

0

+ φ(T

1

))

−ν cos(νT

0

+ φ(T

1

))

(27c) y

0

(T

0

, T

1

) = H r(T

1

)

γ

2

cos(νT

0

+ φ(T

1

)) +ν sin(νT

0

+ φ(T

1

))

(27d) where S =

_γ2^ν

2+ν²

, H =

_(γ2 ^ν²

2+ν²)(sk5−k4)

, and r and

φ are, respectively, the amplitude and the phase of

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0 1 2 3 4 5 6 7 8 0

0.05 0.1 0.15 0.2 0.25

τ

Frequency,ν

(a)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25

Frequency,ν

λ₁

(b)

Fig. 7 Variation of the QP frequency modulation versusaτwithλ1= −0.2 and versusbλ1withτ =1;ω =2.1,λ2 =0.2 and γ2=1.33

the slow flow limit cycle. Substituting (27) into (22)–

(25) and removing secular terms give the following autonomous slow-slow flow system on r and φ

dr

dT

1

= ( k

1

+ S

1

+ H

1

) r + sk

2

k

5

sk

5

− k

4

r

³

r dφ

dT

1

= − ν

γ

2

(S

1

+ H

1

)r (28)

where S

1

=

^γ¹^γ²_2s⁽^sk5_νω⁻^k4⁾

S and H

1

=

^γ_2s¹^γ_ω²

H . Equilibria in Eqs. (28) are obtained by setting d d

^T^r1

= 0 and given by

r = 0, r =

−(k

1

+ S

1

+ H

1

) sk

5

− k

4

sk

2

k

5

(29) I = ν

γ

₂²

+ ν

²

r (30)

with the condition λ

1

=

_cos^σ+χωτ

2

. Note that the expression for I in Eq. (30) is obtained using Eqs. (24), (27a) and maximization procedure. The nontrivial equilibrium in (29)–(30) corresponds to the amplitude of the slow flow limit cycle and consequently to the amplitude of the QP vibrations in the original Eq. (2).

On the other hand, the stability and bifurcation of the nontrivial QP solutions are obtained by calculat- ing the eigenvalues of the Jacobian matrix J of system (28) evaluated in the sth (s = 1, −1) branch (29).

The curves delimiting the regions of existence of the QP oscillations and the domains of their stability are given, respectively, by the conditions of saddle-node (Det(J) = 0 and T r (J) = −2k

₁

< 0) and Hopf bifur-

cations given in terms of λ

1

as λ

1

< ω

sin

^ωτ₂

α

2 + κ

(31a) λ

1±

= ( 4 − ω

²

) + 4 χ ± 4 ω k

4

4 cos

_ωτ

2

(31b)

where κ =

₄_γ²^γ2¹^γ²

2+ω²

and χ =

₄_γ^γ¹2^ω²

2+ω²

. The condition (31a) is written in terms of γ

2

as

γ

2

> −γ

1

+

γ

₁²

+ 4 ω

²

α

2 − λ

1

ω sin ωτ 2

(32)

Thus, the explicit expression of the QP response of the original Eq. (2) is written as

z(t ) = u (t ) cos ω

2 t

+ v(t ) sin ω

2 t

(33) and the QP solution of the current i (t), obtained by inserting the expression of z(t ) given by Eq. (33) into the second equation of system (2), can be extracted via a convolution integral with the boundary condition i(t) = i (0). This leads to

i(t) = −e

^−γ²^t

_t

0

˙

z(τ)e

^γ²^τ

dτ (34)

Consequently, the power, the average and the maximum

powers output in the QP modulation region are given,

respectively, by

(9)

P(t) = γ

2

e

^−γ²^t

t 0

˙

z(τ)e

^γ²^τ

dτ

2

(35) P

avQP

= γ

2

ν

²

2(γ

₂²

+ ν

²

) r

²

(36)

P

maxQP

= γ

2

ν

²

γ

₂²

+ ν

²

r

²

(37)

Figure 8a presents the bifurcation curves delimiting the regions of existence and stability of the QP vibrations in the parameter plane ( λ

1

, τ ), as given by Eqs. (31). The first condition, Eq. (31a), is presented by the dashed lines, while the second one, Eq. (31b), is presented by the solid lines. The regions in which the QP vibrations are stable are indicated by SQP (gray regions), while those where QP vibrations are unstable are designated by UQP (aqua regions). The limit-cycle oscillations exist in the domain indicated by LC (white regions).

Figure 8b shows time histories and the corresponding power output responses related to the different regions of Fig. 8a, and the transitions of solutions are shown by moving between the crosses 1, 2, 3 and 4 in Fig. 8a. Between cross 1 and cross 2 the response of the system undergoes a transition between SQP vibration and no oscillations which describes a bifurcation between a stable equilibrium point and the SQP solution. From cross 2 to 3 Hopf bifurcation occurs giving rise to LC oscillations. From cross 3 to 4 the system changes its behavior from LC to SQP oscillation hav- ing, as expected, a slight modulation just beyond the saddle-node bifurcation of periodic orbits.

Figure 8c depicts analytical results of the amplitude response and the maximum powers P

maxP

, P

maxQP

as function of λ

1

for different values of τ (= 0.2, 1.58, 2.8) picked from Fig. 8a. The upper left and lower right boundaries of the QP modulation envelopes are given by Eq. (29) for s = −1, while the lower left and upper right ones are obtained for s = 1. Results obtained by numerical simulations are indicated by circles and correspond exactly to cross 1 (τ = 0.2, λ

1

= 0.1), cross 4 (τ = 1.58, λ

1

= −0.9) and cross 3 (τ = 2.8, λ

1

= 0 . 05) in Fig. 8a. The plots in Fig. 8c show clearly that for negative values of λ

1

, the maximum power output in the QP domain, P

maxQP

, is larger than that of the periodic response, P

maxP

. Instead, for positive λ

1

and values of τ taken inside the small gray area in Fig. 8a, P

maxQP

is lower than P

maxP

.

The variation of the stability domains of the QP solutions in the plane (λ

1

, τ) is presented in Fig. 9 (gray regions) for three different values of λ

2

. The domains

located in the positive range of λ

1

provide small- amplitude QP vibrations, while the large domains supply large-amplitude QP oscillations.

Figure 8c middle left is redrawn in Fig. 10a for clarity. The curve labeled M

⁺

corresponds to s = +1 in Eq.

(29) and that labeled M

⁻

corresponds to s = −1. The analytical predictions of the periodic response and the QP modulation envelope (solid lines) are compared to results obtained by numerical simulations (circles and time history inset in the figure) for validation. It can be clearly observed from Fig. 10a that the amplitude of the QP modulation reaches larger values comparing to the amplitude of the periodic response [23].

The variation of the maximum power output for periodic and QP responses versus λ

1

is illustrated in Fig. 10b. The corresponding power amplitudes are shown inset in the figure demonstrating clearly improvement of the energy harvesting performance by the QP vibrations.

Figure 11a shows the frequency response and the QP modulation envelope for fixed values of λ

1

, λ

2

, τ and for a critical value of the coupling coefficient γ

2cr

given by the condition γ

2

≥ γ

2cr

= ν

2 (ν +

| ν

²

− 4 |) (38) obtained from (37) by imposing the coefficient

γ2ν

γ2²+ν²

≥ 1 which is required to ensure optimal performance of the QP vibration-based energy harvesting. Under such a condition, it can be observed from Fig. 11a that the amplitude of the QP envelope is larger than that of the periodic response for a certain range of frequencies. In this range of frequencies, a time history corresponding to ω = 1 . 5 is presented inset in the figure. The connected circles in the figure reported for ω = 1.5 and for other values of ω are obtained by numerical simulation for validation. The area around the periodic frequency response is enlarged for clarity.

Figure 11b shows the maximum power response cor-

responding to periodic and QP vibrations. As expected,

the energy harvesting performance is achieved in

broadband of the QP response zone (aqua regions)

away from the resonance. Inset in the figure is shown

time history of the power response. To appreciate the

response in the area around the periodic output power,

an enlargement is shown inset in the figure. The average

power response is also reported in Fig. 11c illustrating

similar results.

(10)

τ

λ1

0 1 2 3 4 5 6 7 8

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

2 UQP

LC 1

3

4 SQP

(a)

100 150 200 250 300

−0.5 0 0.5

z(t)

100 150 200 250 300

0 0.2 0.4

P(t)

0 10 20 30

−0.4

−0.2 0 0.2

z(t)

0 10 20 30

0 0.2 0.4

P(t)

100 150 200 250 300

−1 0 1

z(t)

100 150 200 250 300

0 0.5 1

P(t)

100 150 200 250 300

−2 0 2

Time, t

z(t)

100 150 200 250 300

0 1 2 3

P(t)

λ₁=0.1 τ=0.2 at cross 1

λ₁=0.4 τ=1.58 at cross 2

λ1=0.05 τ=2.8 at cross 3

λ₁=−0.9 τ=1.58 at cross 4

(b)

Time, t

−0.50 0 0.5

0.5 1

R, r

conected points at cross 1

−0.50 0 0.5

0.5 1

PmaxP, PmaxQP

−1 −0.5 0 0.5 1

0 1 2 3

R, r

conected points at cross 4

−10 −0.5 0 0.5 1

1 2 3

PmaxP, PmaxQP

−0.50 0 0.5

0.5 1

R, r

circle point at cross 3

λ₁ ^−0.5 ⁰ ^0.5

0 0.5 1

PmaxP, PmaxQP

λ₁

τ=0.2

τ=1.58 τ=1.58

τ=2.8 τ=2.8

(c)

τ=0.2

Fig. 8 aBifurcation curves in the plane(λ1, τ),btime histories corresponding to different regions,cvibration amplitudes ver- susλ1;λ2 =0.2,ω=2.1 andγ2 =1.33.UQPunstable QP

solutions,SQPstable QP solutions,LClimit-cycle oscillations (color online)

Finally, the variation of periodic and QP responses as well as the corresponding maximum powers amplitude P

max

, P

maxQP

versus γ

2

is depicted, respectively,

in Fig. 12a, b for fixed values of ω and delay para-

meters. Again, the analytical prediction (solid lines) is

compared to numerical simulations (circles) for valida-

(11)

τ

λ1 λ1 λ1

0 1 2 3 4 5

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(a)

0 1 2 τ3 4 5

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(b)

0 1 2 τ3 4 5

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(c)

Fig. 9 Domains of stability (gray regions) of QP vibrations in the plane(λ1, τ)forω=2.1,γ2=1.33 andaλ2=0.4,bλ2=0.2,c λ2=0.02

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0 0.5 1 1.5 2 2.5 3

λ₁

R, r

50 100 150 200

−2

−1 0 1 2

50 100 150 200

−2

−1 0 1 2

Time, t

z(t)

(a)

M+

M−

at cross 4

z(t)

Time, t

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0 0.5 1 1.5 2 2.5 3

λ₁ P maxP, P maxQP

0 50 100 150 200

0 1 2 3

Time, t

P(t)

0 50 100 150 200

0 1 2 3

(b)

at cross 4

P(t)

Time, t

Fig. 10 Vibration (a) and power (b) amplitudes versusλ1forτ=1.58,ω=2.1 andγ2=1.33; analytical (solid lines) and numerical (circles) approximations

tion. The boxes inset in the figures show time histories for the QP response (Fig. 12a) and for the variation of the maximum QP power output (Fig. 12b). It can be observed from Fig. 12a that for the chosen set of parameters, the periodic and the QP vibrations have com- parable amplitudes, while Fig. 12b shows that there exists optimal values of the parameters for which the QP response may supply a broadband energy harvesting with good energy harvesting performance.

4 Conclusions

We have studied the QP vibration-based energy harvesting in a delayed van der Pol oscillator with time-

periodic delay amplitude coupled to an electromagnetic

energy harvesting device. The only vibration source

arises in such a system can be produced by self-induced

vibrations. By modulating the delay amplitude around

a nominal value near a delay parametric resonance, a

QP behavior may occur. Using the second-step pertur-

bation method, analytical expression of the QP vibra-

tions amplitude was obtained and exploited to obtain

the corresponding maximum and average output pow-

ers. The influence of the system parameters on the

performance of the QP vibration-based energy har-

vesting was reported and discussed. In particular, it

was shown that the modulation of the delay amplitude

near the delay parametric resonance gives rise to large-

amplitude QP vibrations in broadband of parameters

(12)

0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.5 1 1.5 2 2.5 3

ω

R, r

1.9 1.95 2 2.05 2.1

1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1

50 100 150 200

−2

−1 0 1 2

Time, t

z(t)

zoom

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.5 1 1.5 2 2.5 3

ω

P maxP, P maxQP ^1.9 ^1.95 ² ^2.05 ^2.1

0 0.2 0.4 0.6 0.8 1

50 100 150 200

−1 0 1

Time, t

P(t)

zoom

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.5 1 1.5 2 2.5 3

ω P avP, P avQP

1.9 1.95 2 2.05 2.1

0 0.1 0.2 0.3 0.4 0.5

(c)

zoom

Fig. 11 Frequency responses (a) and powers output (b) andcforτ=1.58,λ1= −0.53,λ2=0.2 andγ2=γ2cr(color online)

0 0.5 1 1.5 2 2.5 3

R, r

γ₂

50 100 150 200

−2

−1 0 1 2

Time, t

z(t)

(a) (b)

0 0.5 1 1.5 2 2.5 3

γ2

P maxP, P maxQP

50 100 150 200

0 1 2 3

Time, t

P(t)

Fig. 12 Vibration (a) and power (b) amplitudes versusγ2forω=2.1,τ=1.58,λ1= −0.6 andλ2=0.2. harvesting

(13)

near and far from the resonance. For negative values of the unmodulated delay amplitude, such stable large- amplitude QP vibrations are found to be more efficient to improve energy harvesting performance than the periodic vibration. More importantly, it was reported that in the presence of linear stiffness only, the QP vibration-based energy harvesting may be extracted in a broad range of system parameters providing a reli- able alternative to extend the bandwidth of the harvester.

It is clear from this study that in the absence of forced excitations and bistable double well potential, QP vibrations induced by delay amplitude modulation in pure delayed self-excited systems can improve the energy harvesting performance in broadband of parameters. This result is not limited only to van der Pol- type equation, but can also be obtained in general for other pure self-excited systems, as Froude pendulum or others.

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