Quasiperiodic energy harvesting in a forced and delayed Duffing harvester device

Quasiperiodic energy harvesting in a forced and delayed Duffing harvester device

Zakaria Ghouli, Mustapha Hamdi

¹

, Faouzi Lakrad, Mohamed Belhaq

ⁿ

Faculty of Sciences Aïn Chock, University Hassan II-Casablanca, Morocco

a r t i c l e i n f o

Article history:

Received 25 February 2017 Received in revised form 2 June 2017

Accepted 4 July 2017 Handling Editor: Ivana Kovacic

Keywords:

Energy harvesting Quasiperiodic vibrations Delayed Duffing oscillator Piezoelectric coupling External excitation Perturbation method

a b s t r a c t

This paper studies quasiperiodic vibration-based energy harvesting in a forced nonlinear harvester device in which time delay is inherently present. The harvester consists of a delayed Duffing-type oscillator subject to a harmonic excitation and coupled to a piezo- electric circuit. We consider the case of a monostable system and we use perturbation techniques to approximate quasiperiodic responses and the corresponding averaged power amplitudes near the primary resonance. The influence of different system para- meters on the performance of the quasiperiodic vibration-based energy harvesting is examined and the optimal performance of the harvester device in term of time delay parameters is studied. It is shown that in the considered harvester system the induced large-amplitude quasiperiodic vibrations can be used to extract energy over broadband of excitation frequencies away from the resonance, thereby avoiding hysteresis and in- stability near the resonance.

&2017 Elsevier Ltd All rights reserved.

1. Introduction

In vibration-based energy harvesting (EH) systems subject to a harmonic excitation EH performance is considerably limited when the harvester device operates in a linear regime. This is because the natural frequency of the mechanical subsystem must always match the fundamental frequency of the excitation [1

–

3]. To overcome such a limitation nonlinear attachments is often used to substantially extend the bandwidth of the harvester over a broadband of excitation frequencies, either in the case of monostable harvester devices with hardening characteristic [4

–

7] or in the case of bistable ones [6,8

–

11]. However, exploiting nonlinear attachments in the harvester gives rise to hysteretic behavior in the frequency response near the resonance [12] and therefore the problem of instability of the response remains. To circumvent such instabilities near the nonlinear resonance, the idea of exploiting quasiperiodic (QP) vibration away from the resonance to extract energy in broadband of frequency has been proposed [13,14].

Yet, in certain harvester systems under aerodynamic and base excitations, it was shown that QP vibrations cause a substantial reduction in the harvested power [15,16] beyond the flutter speed and then extracting energy from such sys- tems, QP vibrations should be avoided. Nevertheless, it was demonstrated recently that in the presence of time delay QP vibrations can have a beneficial effect on the EH performance [14]. Indeed, in a delayed van der Pol-type harvester system under delay amplitude modulation, the induced large-amplitude QP vibrations occurring in broad range of parameters were

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

http://dx.doi.org/10.1016/j.jsv.2017.07.005 0022-460X/&2017 Elsevier Ltd All rights reserved.

nCorresponding author.

E-mail address:mbelhaq@yahoo.fr(M. Belhaq).

1FST-Al Hoceima, University Mohammed I Oujda, Al-Hoceima, Morocco

This paper will first present the harvester system in the next section. The periodic response and the output average power are then approximated using the multiple scales method. In Section 3, the QP response is obtained applying the second-step multiple scales method and the corresponding harvested power is examined. The influence of different system parameters of the harvester device on the EH performance is analyzed in Section 4 and a summary of the results is given in the concluding section.

2. Model description and periodic energy harvesting

Consider an energy harvester system consisting in an excited Duffing oscillator coupled to an electrical circuit through a piezoelectric device. We assume that the mechanical component of the harvester is under an inherent time delayed in the position such that the governing equation for the harvester can be written in the dimensionless form as

δ ω γ χ α τ λ

¨( ) + ̇( ) + ( ) + ( ) − ( ) = ( − ) + ( ) ( )

x t x t

02

x t x t

3

v t x t f cos t 1

β κ

̇( ) + ( ) + ̇( ) = ( )

v t v t x t 0 2

where

x

(

t

) is the relative displacement of the rigid mass

m

,

v

(

t

) is the voltage across the load resistance, δ is the mechanical damping ratio, γ is the stiffness parameter, χ is the piezoelectric coupling term in the mechanical attachment, κ ^{is the}

piezoelectric coupling term in the electrical circuit, β is the reciprocal of the time constant of the electrical circuit,

f

and λ

are, respectively, the amplitude and the frequency of the excitation, while α ^and τ are, respectively, the feedback gain and time delay. It is assumed that time delay is inherently present in the harvesting system, as in the milling and turning operations [18

–

21] for which Eq. (1) with χ = 0 is commonly used to model such processes. The case where the time delay is included as a control within the electromagnetic coupling was considered recently in [30]. Note that the dynamics of the delayed Duffing oscillator in the absence of the piezoelectric coupling (Eq. (1) with χ = 0 ) has been examined in details [28,29], including bifurcation and stability of periodic and QP solutions Fig. 1.

To investigate the response of the harvester system (1), (2) near the primary resonance we suppose the resonance condition

λ=ω₀+σ

where

s

is a detuning parameter. The method of multiple scales [31] is implemented by introducing a bookkeeping parameter ϵ and scaling parameters as

δ= ϵ ˜δ γ, = ϵ ˜γ χ, = ϵ ˜χ α, = ϵ ˜α,f= ϵ ˜f

,

σ= ϵ ˜²σ

. Thus, Eqs. (1), (2) read

ω δ γ χ α τ λ

¨( ) + = ϵ( − ˜ ̇( ) − ˜ ( ) + ˜ ( ) + ˜ ( − ) + ˜ ( )) ( )

x t

02

x x t x t

3

v t x t f cos t 3

β κ

̇( ) + ( ) + ̇( ) = ( )

v t v t x t 0 4

Applying the multiple scales method [31] one obtains the modulation equations

Fig. 1. Schematic description of the EH system.

ω ω φ φ

ω ω ω φ

= + ( )

= − + ( )

( )

⎧ ⎨

⎩ da dt

C a f

a d dt

C a C

a f

2 2 sin

2 2 2 cos

5

1

0 0

2 0

3 0

3 0

where

= −δω −α (ω τ) − ^κβχω

β ω

( + )

C_{1 0} sin _{0 2} ⁰

02

,

= ω σ+α (ω τ) − ^κχω

β ω

( + )

C₂ 2 ₀ cos _{0 2} ⁰²

02

,

C₃= ³₄^γ

,

φ= ˜σT₂−θ

and

a

, θ are, respectively, the amplitude and the phase of the modulation. Detail of derivation of this slow flow is given in Appendix A.

The solution up to the first order can be written as

x T T T₀(₀, ₁, ₂) =acos(ω₀t+ )θ

and

v T T T₀(₀, ₁, ₂) =Vcos(ω₀t+θ+arctan_ω^β)

where the voltage amplitude

V

is expressed as

0

κω

β ω

=

+ ( )

V a

6

0 2

0 2

The steady-state response of system (5), corresponding to periodic oscillations of Eqs. (3), (4) are determined by setting

= ^φ =0

da dt

d

dt

. Eliminating the phase, we obtain the following algebraic equation in

a

− + ( + ) − = ( )

C a

32 6

2 C C a C C a f 0 7

2 3 4 12

22 2 2

An expression for the average power is obtained by integrating the dimensionless form of the instantaneous power

β

( ) = ( )

P t v t²

over the period of the excitation

T

. This is given by

∫ ^β

= ( )

P T 1 v dt

av

8

T

0 2

where

T=²_λ^π

. Then, the average power expressed by

P_av=^βV₂²

reads βκ ω

β ω

= (

+ )

( )

P 1 a

2 9

av

2 02 2

0 2

2

where the amplitude

a

is obtained from Eq. (7).

3. Quasiperiodic energy harvesting

To approximate the QP response it is necessary to apply the second-step perturbation method [32,33] on the slow flow system obtained up to second order. Hence, substituting the first and second order solutions (Eqs. (37), (38) and Eqs. (40), (41) into (35) (in Appendix A) and eliminating the secular terms yields

ω δ γ

ω

− − ( ) − ˜( ) − ˜ ¯

=

( )

D A i D A D A A A

2 3

8 0

1

10

2

0 2 1

2 3 2

02

Substituting the expression of A, introducing

φ= ˜σT₂−θ

and separating the real and imaginary parts, we obtain up to the second order the modulation equations

φ β φ β φ

φ φ β φ β φ

= − ( ) + + ( ) + ( )

= − ( ) + + + ( ) − ( )

( )

⎧

⎨ ⎪⎪

⎩ ⎪

⎪ da

dt S a S a sin S a sin cos

a d

dt S a S a cos S a S a cos sin 3

11

1 3

2

4 3

1 2

2 3 2

5 3 6 5

1 2

where

S i_i( =1,…, 6)

, β

1

, and β

2

are given in Appendix B. Performing stability analysis of the steady-state solutions by calculating the eigenvalues of the Jacobian matrix of slow flow system (11) one obtains the stability chart delimiting the existence regions of stable and unstable periodic oscillations of the original system. Fig. 2 shows in the parameter plane

α τ

( , )

this classical chart where stable periodic (SP) and unstable periodic (UP) solutions exist. A similar stability chart was given in the parameter plane ( α ^{, Ω =}

²_τ^π

) for the uncoupled delayed Duffing oscillator (Eq. (1) with χ = 0 ) [28].

To approximate QP solutions of the slow flow it is convenient to transform the polar form (11) to the following Cartesian system using the variable change u = acos φ and

w= −asinφ

β η

β η

= + + { + + ( + )( + ) + ( + ) }

= − − + { + + ( + ) + ( − )( + ) − ( + ) }

( )

⎧

⎨ ⎪⎪

⎩ ⎪

⎪ du

dt S w S u S uw S u S w u w S u w w

dw

dt S u S w S w S u w S u S w u w S u w w

2

12

2 2 1 3 4 5

2 2

6

2 2 2

2 1 1 3 2

3 2 2

4 5 2 2

6 2 2 2

where η is a bookkeeping parameter introduced in damping and nonlinearity. A periodic solution of the slow flow (12) corresponding to QP solution of the original system (1), (2) can be expressed as

η η

( ) = ( ) + ( ) + ( ) ( )

u t u T T

0 1

,

2

u T T

1 1

,

2

O 13

2

η η

( ) = ( ) + ( ) + ( ) ( )

w t w T T

_{0 1}

,

₂

w T T

_{1 1}

,

₂

O

²

14

where

T₁=t

and T

₂

= η t . In terms of the variables

Ti

, the time derivatives become

^d =D +ηD +O( )η

dt 1 2 2

where

= ^∂

∂

D_i^j

T j ji

. Substituting (13) and (14) into (12), and equating coefficient of like powers of η , we obtain up to the second order

β

+ = − ( )

D u

1

S u S 15

2

0 2

2

0 1 2

β

= − ( )

S w

2 0

D u

1 0 2

16

+ = [ − + + + ( + ) + ( − )( + )−

( + ) ] − + + [ + ( + )( + ) + ( + ) ] ( )

D u S u S D w S w S w S u w S w S u u w

S u w u D D u S D u D 2 S u w S u S w u w S u w u 17

1 2

1 2

2

1 2 2 0 1 0 3 0

2 3 0

2 0 2

4 0 5 0 0

2 0 2

6 02 02 2

0 1 2 0 1 1 0 1 3 0 0 4 0 5 0 02

02 6 02

02 2 0

= + − [ + + ( + )( + ) + ( + ) ] ( )

S w

2 1

D u

1 1

D u

2 0

S u

1 0

2 S u w

3 0 0

S u

4 0

S w

5 0

u

02

w S u w w 18

02 6 02

02 2 0

The solution of the first-order can be written as

ψ α

( ) = ( ) ( + ( )) − ( )

u T T

_{0 1}

,

₂

R T cos S T

_{2 2 1}

T

_{2 1}

19

ψ α

( ) = − ( ) ( + ( )) − ( )

w T T

0 1

,

2

R T sin S T

2 2 1

T

2 2

20

where

α1= ^β_S¹

2

,

α2= ^β_S²

2

and

R

, ψ are, respectively, the amplitude and the phase of the slow flow limit cycle. Substituting (19) and (20) into (17) and removing secular terms gives the following autonomous slow-slow flow system on

R

and ψ

α α α

ψ α α α α α α α α

= + ( − + + )

= + ( + ( + ) ) + (( + ) + ( + + ) )

( )

⎧

⎨ ⎪⎪

⎩ ⎪

⎪ dR

dt S R S S S S R

R d

dt S R S S R S S R

2 2

6 2 2 3 3 6

21

4 3

1 2 3 12

4 22

4

6 5

5 1

2 2

2 6

3 1

2 2

2

5 1

4 2

4 1

2 2

2 6

Equilibria of this slow-slow flow determine the QP solutions of the original system (1), (2). The nontrivial equilibrium is obtained by setting

^dR_dt =0

and given by

α α α

= − + − −

( )

R S S S S

S

2 2

22

1 2 3 12

4 22

4 4

Consequently, the approximate periodic solution of the slow flow (12) is given by

φ α

( ) = ( ) − ( )

u t Rcos t

1

23

φ α

( ) = − ( ) − ( )

w t Rsin t

₂

24

The approximate amplitude

a

(

t

) of the QP response reads

Fig. 2.Stability chart of the periodic solution; SP: stable periodic, UP: unstable periodic solutions in the plane (α τ, ) for f=0.2,χ=0.05, ω₀=1,δ=0.1,γ=0.25 β=0.05,λ=0.7andκ=0.5.

α α α φ α φ

( ) = [ + + ] − [ ( ) − ( )] ( )

a t R

² 12

2 Rcos t 2 Rsin t 25

22

1 2

and the envelope of the QP modulation is delimited by

amin

and

amax

given by

{ ^{α α α α} }

= [ + + ] ± [ ± ] ( )

a

^min

min R 2 R 2 R 26

2 12

22

1 2

Fig. 3.Vibration (a) and power (b) amplitudes vsλforf=0.2,χ=0.05,κ=0.5,ω₀=1,δ=0.1,γ=0.25β=0.05,τ=4.8andα=0.2. Analytical prediction (solid lines for stable, dashed line for unstable), numerical simulation (circles).

Fig. 4. Time histories of the amplitudes for the parameter values ofFig. 3; (a)λ=0.7, (b)λ=1.25, (c)λ=1.7.

Fig. 5.Time histories of the average power responses for the parameter values ofFig. 3; (a)λ=0.7, (b)λ=1.25, (c)λ=1.7.

{ ^{α α α α} }

= [ + + ] ± [ ± ]

( )

a

^max

max R 2 R 2 R 27

2 1

2 2

2

1 2

The average QP power is obtained from βκ ν

β ν

= + ( )

⎛

⎝ ⎜ ⎞

⎠ ⎟

P 1 a

2 28

avQP

2 2

2 2

2

where ν = S

₂

is the frequency of the QP modulation and

a

is now derived from Eqs. (26), (27).

4. Results

Next, the influence of different parameters of the system on vibration and power amplitudes is examined. Figs. 3 and 6 show, respectively, the variation of periodic and QP amplitudes and the corresponding output average power amplitudes (P

_av,P_avQP

) versus the frequency of the external excitation λ for fixed values of parameters and for

α=0.2

(Fig. 3) and α = 0.4

Fig. 6.Vibration (a) and power (b) amplitudes vsλforf=0.2,χ=0.05,κ=0.5,ω₀=1,δ=0.1,γ=0.25β=0.05,τ=4.8andα=0.4. Analytical prediction (solid lines for stable, dashed line for unstable), numerical simulation (circles).

Fig. 7.Time histories of the amplitudes for the parameter values ofFig. 6; (a)λ=0.7, (b)λ=1.7.

(Fig. 6). The amplitude of the periodic response is given by (7) while the QP modulation envelope is obtained from Eqs. (26) and (27). Also, the average power amplitude for the periodic response is given by Eq. (9) and that corresponding to the QP response is deduced from (28). The analytical prediction of the QP modulation envelope (solid lines for stable and dashed line for unstable) are plotted along with numerical simulation (circles) showing a certain discrepancy in the QP envelope due to the accuracy of the approximate QP solution obtained here only up to the first order. The numerical results are obtained using "dde23" algorithm [34]. Figs. 4, 5 present, respectively, time histories of the amplitudes and the average power responses for different values of λ picked from Fig. 3

α=0.2. Similarly,

Figs. 7, 8 show, respectively, time histories of the amplitudes and the average power responses for different values of λ picked from Fig. 6 for α = 0.4 Fig. 7.

It can be observed that for

α=0.2, the periodic vibration-based EH can be extracted in a small range of the frequency

response near the resonance (solid line in the power response of Fig. 3b) with a good performance, as depicted in Fig. 5b.

Instead, QP vibration-based EH can be extracted over broad range of excitation frequencies away from the resonance but with lower performance, as shown in Fig. 5a,c. For increasing value of the feedback gain, α = 0.4 , the periodic vibrations turn into unstable, while the QP vibration-based EH can be extracted in broadband of frequencies away from the resonance with better performance, as shown by time histories of the average power responses in Fig. 8.

The vibration and average power amplitudes versus the feedback gain α are presented, respectively, in Figs. 9a,b for

λ=0.7, while

Fig. 10 shows time histories of amplitude and average power responses corresponding to a certain value of α

Fig. 9. Vibration (a) and power (b) amplitudes vsαforf=0.2,χ=0.05,κ=0.5,ω₀=1,δ=0.1,γ=0.25β=0.05,τ=4.8andλ=0.7. Numerical simulation is presented by circles.

Fig. 8.Time histories of the average power responses for the parameter values ofFig. 6; (a)λ=0.7, (b)λ=1.7.

chosen within the QP region. The plots show that QP vibration-based energy can be extracted beyond a certain threshold of delay amplitude α ^.

The influence of the damping δ on vibration and average power amplitudes is shown in Fig. 11 for

λ=0.7, while time

histories of amplitude and average power responses corresponding to

δ=0.1

chosen within the QP region are illustrated in Fig. 12. It can be observed from Fig. 11 that QP vibration-based energy can be extracted for small damping.

Figs. 13a,b show, respectively, the variation of the QP modulation envelope and the output average power versus the amplitude of the excitation

f

for

α=0.2. The corresponding time histories are presented in

Fig. 14 for

f¼

0.2. It can be clearly seen that the power can be extracted only from QP vibrations with substantial performance in the range of small values of

f

. Similarly, the variation of the QP modulation envelope and the average power versus χ is depicted in Fig. 15 showing that for values of λ taken relatively away from the resonance (here

λ=0.7), a good performance of QP vibration-based EH is

obtained for small values of piezoelectric coupling coefficient χ ^{Fig. 16.}

Fig. 10. Time histories and average power responses for the parameter values ofFig. 9and forα=0.14.

Fig. 11.Vibration (a) and power (b) amplitudes vsδforf=0.2,χ=0.05,κ=0.5,ω₀=1,γ=0.25β=0.05,τ=4.8,λ=0.7andα=0.4. Numerical simu- lation is presented by circles.

The evolution of the QP modulation envelope and the corresponding QP average power versus the coupling coefficient κ

is shown in Fig. 17 indicating the range of κ in which a substantial EH performance is achieved. The corresponding time histories are presented in Fig. 17 for

κ=1

Fig. 18.

In Figs. 19a,b is shown, respectively, the variation of the QP modulation envelope and the corresponding average power versus τ and the corresponding time histories are depicted in Figs. 20a,b for the value of

τ=6.32.

To guarantee the robustness of the QP solution during energy extraction operation it is important to perform its stability analysis by examining the nontrivial solution of the slow-slow flow (21) and determine the corresponding stability chart by calculating the eigenvalues of the Jacobian matrix of (21).

Fig. 21a shows this stability chart of the QP solution in the parameter plane (α τ

,

) for

λ=0.7

indicating the (grey) regions where stable QP (SQP) solutions take place and the (white) region corresponding to unstable QP (UQP) solutions. In the regions of SQP solution, UP solutions exists, while in that of UQP solution, SP solution is present which is coherent with the stability chart of the periodic solution given in Fig. 2. In Fig. 21b are shown time histories and the corresponding power output responses related to crosses labelled 1, 2, 3 in Fig. 21a. From cross 1 to cross 2 or 3 the system response bifurcates from SP to SQP oscillations via secondary Hopf bifurcation producing, as expected, a slight modulation of the amplitude response and a significant performance of the power output. At the bifurcation the SP response turns into unstable.

Fig. 22 depicts analytical results of the amplitude response and the average powers

Pav

,

PavQP

as function of λ for different values of τ ^and α picked from Fig. 21a. It can be clearly observed that for values of α ^and τ chosen inside the UQP region,

Fig. 12.Time histories and average power responses for the parameter values ofFig. 11and forδ=0.1.

Fig. 13.Vibration (a) and power (b) amplitudes vsfforχ=0.05,κ=0.5,α=0.4,ω₀=1,γ=0.25β=0.05,τ=4.8,δ=0.1andλ=0.7. Analytical prediction (solid lines for stable, dashed line for unstable), numerical simulation (circles).

Fig. 14.Time histories of the amplitudes and the average power responses for the parameter values ofFig. 13and forf¼0.2.

Fig. 15.Vibration (a) and power (b) amplitudes vsχforf=0.2,κ=0.5,α=0.4,ω₀=1,δ=0.1,γ=0.25β=0.05,τ=4.8andλ=0.7. Numerical simulation is presented by circles.

Fig. 16. Time histories of the amplitudes and the average power responses for the parameter values ofFig. 15and forχ=0.5.

Fig. 17. Vibration (a) and power (b) amplitudes vsκforf=0.2,χ=0.05,α=0.4,ω₀=1,δ=0.1,γ=0.25β=0.05,τ=4.8andλ=0.7. Analytical prediction (solid lines for stable, dashed line for unstable), numerical simulation (circles).

Fig. 18.Time histories of the amplitudes and the average power responses for the parameter values ofFig. 17and forκ=1.

Fig. 19.Vibration (a) and power (b) amplitudes vsτforf=0.2,χ=0.05,κ=0.5,α=0.4,ω₀=1,δ=0.1,γ=0.25β=0.05andλ=0.7. Analytical prediction (solid lines for stable, dashed line for unstable), numerical simulation (circles).

Fig. 20.Time histories of the amplitudes and the average power responses for the parameter values ofFig. 19and forτ=6.32.

Fig. 21.(a) Bifurcation curves in the plane(α τ, ) forλ=0.7; time histories of the amplitudes and the average power responses corresponding to (b), (c) cross 1 (τ=1.5, α=0.08), (d), (e) cross 2 (τ=1.5, α= −0.2) and (f), (g) cross 3 (τ=4.8, α=0.6); UQP: unstable QP, SQP: stable QP;

χ ω δ γ

= = = = =

f 0.2, 0.05, ₀ 1, 0.1, 0.25β=0.05,λ=0.7andκ=0.5.

Fig. 22.Vibration and power amplitudes at (a), (b) cross 1, (c), (d) cross 2 and (e), (f) cross 3 vsλfor the parameter values ofFig. 21. (a), (b) at cross 1;

(c), (d) at cross 2; (e), (f) at cross 3. Analytical prediction (solid lines for stable, dashed line for unstable).

stable periodic response occurs (Fig. 22a,b at cross 1). For values of α ^and τ chosen inside the SQP region (Figs. 22c,d and Figs. 22e,f at crosses 2, 3, respectively) the QP modulation regime appears offering the possibility for harvesting energy from QP vibrations with enhanced performance.

5. Conclusion

In this study EH performance in a forced delayed Duffing oscillator coupled to a piezoelectric harvester is examined. It is assumed that time delay feedback in the position is inherently present in the mechanical component of the system and cannot be viewed as an additional input into the system. The analysis was carried out near the primary resonance and for a monostable oscillator. A two-step multiple scales method is performed to obtain the slow-slow flow of the system and approximations of QP solutions along with the corresponding output power are obtained. The effect of different system parameters on the EH performance is then explored. It was shown that the presence of inherent delay feedback in the harvester induces large-amplitude QP vibrations. For relatively large values of delay amplitude these QP vibrations can be used to extract energy over broadband of excitation frequencies away from the resonance thus circumventing hysteresis and instability near the resonance. On the other hand, for low values of delay amplitude periodic vibration-based EH can be extracted only in a narrow stable range of frequencies near the resonance. The results also provide optimal values of ex- ternal forcing, damping and mechanical/electrical coupling coefficients for which the output power obtained from QP vi- brations is maximum.

Acknowledgements

Part of the material of this paper was presented within the Third International Conference on Structural Nonlinear Dynamics and Diagnosis (CSNDD'2016), Marrakech, Morocco, May 23-25, 2016. The organizers are acknowledged for partial support.

Appendix A. Appendix

A solution to Eqs. (3), (4) can be sought in the forme

( ) = ( ) + ϵ ( ) + ϵ ( ) + (ϵ ) ( )

x t x T T T

0 0

,

1

,

2

x T T T

1 0

,

1

,

2 2

x T T T , , O 29

2 0 1 2 3

( ) = ( ) + ϵ ( ) + ϵ ( ) + (ϵ ) ( )

v t v T T T

0 0

,

1

,

2

v T T T

1 0

,

1

,

2 2

v T T T , , O 30

2 0 1 2 3

where

T₀=t

,

T₁= ϵt

and

T₂= ϵ²t

. In terms of the variables

Ti

, the time derivatives become

^d =D + ϵD + ϵD +O(ϵ )

dt 0 1 2

2 3

and

=D + ϵ (D +2D D) +2ϵD D +O(ϵ )

d

dt 0

2 2

1 2

0 2 0 1

2 3

2

where

= ^∂

D_i^j ∂ T j ji

. Substituting (29) and (30) into (3) and (4) and equating coeffi- cient of like powers of ϵ , we obtain up to the second order the following hierarchy of problems:

ω

+ = ( )

D x

02

x 0 31

0 02 0

β κ

+ + = ( )

D v

0 0

v

0

D x

0 0

0 32

ω δ γ χ α ω σ

+ = − − ˜ − ˜ + ˜ + ˜

_τ

+ ˜ ( + ˜ ) ( )

D x

02

x 2 D D x D x x v x f cos T T 33

1 02

1 0 1 0 0 0 03

0 0 0 0 2

β κ κ

+ = − − − ( )

D v

0 1

v

1

D v

1 0

D x

0 1

D x

1 0

34

ω δ δ χ γ α

+ = − − − − ˜ − ˜ + ˜ − ˜ + ˜

_τ

( )

D x

02

x 2 D D x D x 2 D D x D x D x v 3 x x x 35

2 02

2 0 1 1 12

0 0 2 0 0 1 1 0 1 1 02

1

β κ κ κ

+ = − − − − − ( )

D v

_{0 2}

v

₂

D v

_{1 1}

D v

_{2 0}

D x

_{0 2}

D x

_{1 1}

D x

_{2 0}

36

Up to the first order the solution is given by

( ) = ( )

^ω

+ ¯ ( )

^−ω

( )

x T T T

0 0

,

1

,

2

A T T e

1

,

2 ^{i 0 0T}

A T T e

1

,

2 ^{i 0 0T}

37

κ ω

β ω

κ ω

β ω

( ) = − ( )

+ + ¯ ( )

− ( )

ω −ω

v T T T i A T T

i e i A T T

i e

, , , ,

38

i T n i T

0 0 1 2 0 1 2

0

1 2 0

0 0 0 0

where A T T (

₁

,

₂

) and

A T T¯ (₁, ₂)

are unknown complex conjugate functions. Substituting Eqs. (37) and (38) into (33) and (34) and

eliminating the secular terms, one obtains:

Appendix B. Appendix

δ α τ α τ τ χκβ

β ω

χκα β τ ω τ

β ω

χ κ βω

β ω β ω

λ ω δ

α τ α τ α τ

χκω

β ω

χκα β τ ω τ

β ω

χ κ β ω

β ω β ω

γ

γδ αγ τ γκβ

β ω

γκβχ

β ω

γ αγ τ γκω

β ω

γκω

β ω

γ

β α τ χκω

β ω

β δ α τ χκβ

β ω

= − − ( ) − ( ) ( ) −

( + ) − ( ( ) − ( ))

( + ) +

[( − ) + ]

= − + + ( ) + ( ) − ( )

−

( + ) − ( ( ) + ( ))

( + ) − ( − )

[( − ) + ]

=

= + ( ) +

( + ) +

( + )

= − − ( ) +

( + ) −

( + )

=

= + ( ) −

( + )

= − ( ) −

( + )

S sin cos sin cos sin

S cos cos sin

sin cos

S f

S sin

S cos

S

f fcos f

f fsin f

2 1 2

1

4 2 8 4 4

8 1 2

1 8

1 8

2 8 4 4

3 8 3 16

3 8

3 8

3 16 3

8 3 16

3 8

3 16 15

256 1 2

1

8 8

1 8

1

8 8

1

2

2 02

0 2

02

2 2 0 2

02 2 2 02

2 0

2

2 2 2 2

0 2

02

0 2

02

2 2 2 0

2 2

02 2 2 02

3

4 2

02 2

02

5 0

2 0

2

0 2

0 2

6 2

1

0 2

0 2

2 2

02

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