Hysteresis suppression for primary and subharmonic 3:1 resonances using fast excitation

(1)

DOI 10.1007/s11071-008-9438-z

O R I G I N A L PA P E R

Hysteresis suppression for primary and subharmonic 3:1 resonances using fast excitation

Mohamed Belhaq·Abdelhak Fahsi

Received: 9 June 2008 / Accepted: 17 September 2008 / Published online: 9 October 2008

Abstract We analyze the effect of a fast harmonic

excitation on hysteresis and on entrainment area in a forced van der Pol–Duffing oscillator near the primary and the 3:1 subharmonic resonances. Analytical treatment based on perturbation techniques is performed to capture the entrainment zone, the quasiperiodic modulation domain and the hysteresis area in the vicinity of the two resonances. Specifically, it is shown that a fast harmonic excitation can suppress hysteresis for a certain range of the fast excitation leading to a smooth transition between the quasiperiodic and the frequency-locked responses near these resonances.

Furthermore, the influence of different system parameters on the hysteresis area has been investigated. In particular, the results reveal that the amplitude of the fast excitation and the nonlinear damping significantly affect the domain of hysteresis suppression near the primary and the 3:1 subharmonic resonances.

Keywords Fast harmonic excitation

· Frequency locking · Hysteresis suppression · Subharmonic resonance · Quasiperiodic modulation · Perturbation analysis

M. Belhaq (

⁾

University Hassan II-Aïn Chock, Casablanca, Morocco e-mail:mbelhaq@hotmail.com

A. Fahsi

FSTM, University Hassan II-Mohammadia, Mohammadia, Morocco

1 Introduction

Frequency locking behavior can occur in various me- chanical systems. In a such phenomenon, self-excited oscillations synchronize by periodic forcing of the system leading to frequency-locked oscillations for which the response follows the forcing frequency. Out of the resonance but nearby, the response is quasiperiodic.

In nonlinear systems, the transition between quasiperiodic and entrained motions can take place at two different specific frequencies when the forcing frequency is swept backward and forward leading to hysteresis effect. This effect, producing jumps in the system response is considered as one of the serious problems in the design of devices like resonant microsensors [1,

2].

Hysteresis occurring inside entrainment area, referred to as entrained hysteresis, has been reported and an- alyzed in optically driven MEMS resonators; see [3]

and references therein. The presence of such a hystere-

sis near a resonance is associated with the coexistence

of two distinct stable states (periodic and quasiperi-

odic) and, hence, a jump between these states may oc-

cur. Therefore, the control of hysteresis is an important

issue to realize a high functionality of systems and to

improve their specific performance. In [4], the control

of hysteresis in a directly modulated semiconductor

laser using delayed optoelectronic feedback was pro-

posed. In [2], the attenuation of hysteresis in MEMS

resonators was performed by acting on the quality fac-

tor which is related to the damping.

(2)

In a recent work [5], the suppression of hysteresis in a forced van der Pol–Duffing oscillator was studied near the fundamental resonance 1:1. It was shown that adding a fast harmonic excitation (FHE) can suppress hysteresis for a certain range of the fast frequency.

The purpose of the present paper is twofold. First, further investigation will be carried out for the forced van der Pol–Duffing oscillator near the 1:1 resonance.

In particular, the influence of different system parameters on the suppression of hysteresis will be examined.

Second, we perform a careful analysis of the effect of an FHE on frequency locking and on hysteresis suppression near the subharmonic resonance 3:1.

The idea of using FHE to study the stabilization of an inverted pendulum on a vibrating support has been reported in [6–8]. Other effects of FHE on mechani- cal systems have been examined intensively in the recent years. These include equilibrium stability [9], linear stiffness [10], natural frequencies [11], resonance behavior [12], symmetry breaking [13], and limit cycle [14–16].

Consider the following forced van der Pol–Duffing oscillator subjected to an FHE in the dimensionless form:

¨ x + x −

α − βx

²

˙ x − γ x

³

= h cos ωt + aΩ

²

cos x cos Ωt (1) where damping α, β , nonlinearity γ and excitation amplitudes h and a are small. Dots denote differentiation with respect to time t . We assume that the FHE frequency Ω is high compared to the frequency of the external forcing ω such that the resonance between the two frequencies cannot occur. In a previous work [17], a van der Pol–Mathieu–Duffing equation was investigated near the 2:1 and the 1:1 resonances. It was shown that varying the FHE shifts the backbone curve and changes the nonlinear characteristic behavior of the system near these resonances from softening to hardening or vice versa.

In the present work, we focus our attention on the effect of an FHE on the frequency locking area and on the suppression of the entrained hysteresis in (1) near the fundamental 1:1 and the subharmonic 3:1 resonances.

The rest of the paper is organized as follows. In Sect.

2

we average the oscillator (1) over the fast time to derive an equation governing the slow dynamics. Section

3

is devoted to the analysis near

the 1:1 resonance. The multiple scales method is applied to the slow dynamics to derive an autonomous slow flow. Analysis of equilibria of this slow flow provides analytical approximations of the entrained amplitude-frequency response. A multiple scales method is performed in a second step on the slow flow to approximate quasiperiodic solution and its modulation domain. Analytical prediction of the variation of the hysteresis area as a function of the high frequency Ω as well as the influence of different parameters of the system on the hysteresis suppression is provided.

We perform numerical simulation and we compare with the analytical finding for validation. In Sect.

4,

we perform a similar analysis near the 3:1 resonance.

Section

5

concludes the work.

2 Slow dynamics

In this section, we use the method of direct partition of motion (DPM) [18] to derive the slow dynamics of system (1). We introduce two different time scales, a fast time T

0

= Ωt and a slow time T

1

= t , and we split up x(t ) into a slow part z(T

₁

) and a fast part φ (T

₀

, T

₁

) as follows:

x(t ) = z(T

₁

) + φ (T

₀

, T

₁

) (2) where z describes slow main motions at time scale of oscillations, φ stands for an overlay of the fast motions and indicates that φ is small compared to z.

Since Ω is considered as a large parameter, we choose ≡ Ω

⁻¹

, for convenience. The fast part φ and its derivatives are assumed to be 2π-periodic functions of fast time T

0

with zero mean value with respect to this time, so that x(t ) = z(T

₁

) where ≡

_2π¹ _2π

0

() dT

₀

defines time-averaging operator over one period of the fast excitation with the slow time T

₁

fixed.

Introducing D

_i^j

≡

^∂^j

∂T_i^j

yields

_dt^d

= ΩD

0

+ D

1

,

d²

dt²

= Ω

²

D

₀²

+ 2ΩD

0

D

1

+ D

²₁

, and substituting (2) into (1) gives

⁻¹

D

²₀

φ + 2D

0

D

1

φ + D

²₁

φ + D

²₁

z + z + φ

−

α − β(z + φ)

²

(D

0

φ + D

1

φ + D

1

z)

− γ (z + φ)

³

= h cos ωT

1

+

⁻¹

(aΩ) cos(z + φ) cos T

0

. (3) Averaging (3) leads to

D

²₁

z + z −

α − βz

²

D

1

z − γ z

³

= h cos ωT

₁

+

⁻¹

(aΩ)

cos(z + φ) cos T

₀

. (4)

(3)

Subtracting (4) from (3) yields

⁻¹

D

₀²

φ + 2D

0

D

1

φ + D

²₁

φ + φ

−

α − βz

²

(D

₀

φ + D

₁

φ) + β

2φ +

²

φ

²

(D

0

φ + D

1

φ)

− γ

3z

²

φ + 3

²

zφ

²

+

³

φ

³

=

⁻¹

(aΩ) cos(z + φ) cos T

0

−

⁻¹

(aΩ)

cos(z + φ) cos T

0

. (5)

Approximation of φ is obtained from (5) by consid- ering only the dominant terms of order

⁻¹

as D

²₀

φ = (aΩ) cos z cos T

₀

(6) where it is assumed that aΩ = O(

⁰

). The stationary solution to the first order for φ is written as

φ = −a cos z cos T

0

. (7)

The equation governing the slow motion is derived from (4). Inserting cos(z + φ) = cos z − φ sin z + O(

²

) into (4) and retaining the dominant terms of order

⁰

, we obtain

D

²₁

z + z −

α − βz

²

D

1

z − γ z

³

= h cos ωT

₁

− (aΩ)sin z φ cos T

₀

. (8) Inserting φ from (7) and using that cos

²

T

₀

= 1/2, we find the approximate equation for slow motions D

²₁

z + z −

α − βz

²

D

1

z − γ z

³

= h cos ωT

1

+ 1

2 (aΩ)

²

cos z sin z. (9) This equation is similar to the original equation (1) in which the non-autonomous term aΩ

²

cos x cos Ωt is replaced by the autonomous one

¹₂

(aΩ)

²

cos z sin z.

We focus the analysis on small vibrations around the origin by expanding in Taylor’s series the terms sin z z − z

³

/6 and cosz 1 − z

²

/2. Keeping only terms up to order three in z, (9) becomes

D

²₁

z +

1 − 1 2 (aΩ)

²

z −

α − βz

²

D

1

z

−

γ − 1 3 (aΩ)

²

z

³

= h cos ωT

₁

. (10)

In (10), it appears that the effect of an FHE introduces additional apparent stiffness in the linear stiffness [9]

and in the nonlinear one. These effects have been ob- served in a spring-connected two-link mechanism at a vibrating support [12].

Hence, (10) can be written as

¨

z + ω

²₀

z −

α − βz

²

˙

z − ξ z

³

= h cos ωt (11) where ω

²₀

= 1 −

¹₂

(aΩ)

²

, ξ = γ −

¹₃

(aΩ)

²

and an overdot denotes differentiation with respect to time t .

3 The fundamental resonance 1:1

We express the 1:1 resonance condition by introducing a detuning parameter σ according to

ω

²₀

= ω

²

+ σ. (12)

Analytical investigation of periodic and quasiperiodic responses near the resonance requires the application of a double perturbation technique by introducing two small bookkeeping parameters, μ and η. To derive a slow flow of the slow dynamics (11), we use the parameter μ in the first perturbation, and for implementing the second perturbation on the slow flow, we introduce the other parameter η in a second step. This strategy of using a perturbation analysis on a slow flow in systems under quasiperiodic excitation has been proposed in [19] and applied successfully to obtain analytical approximations of quasiperiodic solutions [20] and analytical expressions of the stability chart for quasiperiodic Mathieu equations [21–23].

3.1 Slow flow and entrainment Rewrite (11) as

¨ z + ω

²

z

= μ − σ z +

α − βz

²

˙

z + ξ z

³

+ h cos ωt

. (13) Using the multiple scales technique [24], we seek a two-scale expansion of the solution in the form z(t ) = z

₀

(T

₀

, T

₁

) + μz

₁

(T

₀

, T

₁

) + O

μ

²

(14) where T

_i

= μ

ⁱ

t . In terms of the variables T

_i

, the time derivatives become

_dt^d

= D

0

+ μD

1

+ O(μ

²

) and

d²

dt²

= D

₀²

+2μD

0

D

1

+ O(μ

²

), where D

^j_i

=

^∂^j

∂T_i^j

. Sub-

(4)

Fig. 1 Amplitude-frequency response for different values ofΩ. Analytical approximation: solid (for stable) and dashed (for unstable).

Numerical simulation: circles

stituting (14) into (13) and equating coefficients of the same power of μ, we obtain

D

²₀

z

₀

(T

₀

, T

₁

) + ω

²

z

₀

(T

₀

, T

₁

) = 0, (15) D

²₀

z

₁

(T

₀

, T

₁

) + ω

²

z

₁

(T

₀

, T

₁

)

= −2D

0

D

₁

z

₀

− σ z

₀

+

α − βz

²₀

D

₀

z

₀

+ ξ z

₀³

+ h cos ωT

0

. (16) The solution to the first order is given by

z

0

(T

0

, T

1

) = r(T

1

) cos

ωT

0

+ θ (T

1

)

. (17)

Substituting (17) into (16), removing secular terms and using the expressions

^dr_dt

= μD

1

r + O(μ

²

) and

dθ

dt

= μD

₁

θ + O(μ

²

), we obtain the first-order autonomous slow flow modulation equations of amplitude and phase

dr

dt = Ar − Br

³

− H sin θ, r dθ

dt = Sr − Cr

³

− H cos θ,

(18)

where A =

^α₂

, B =

^β₈

, S =

_2ω^σ

, C =

_8ω^3ξ

and H =

_2ω^h

. Equilibrium points of the slow flow (18), corresponding to periodic solutions of (11), are determined by setting

^dr_dt

=

^dθ_dt

= 0. Using the relation cos

²

θ + sin

²

θ = 1, we obtain the amplitude-frequency response equation

A

3

r

⁶

+ A

2

r

⁴

+ A

1

r

²

+ A

0

= 0 (19)

where A

₃

= B

²

+ C

²

, A

₂

= −2(AB + SC), A

₁

= A

²

+ S

²

and A

₀

= − H

²

. The discriminant of (19) is given by

= P

³

27 + Q

²

4 (20)

where P =

^A_A¹₃

−

_3A^A²²2 3

and Q =

₂₇²

(

^A_A²

3

)

³

−

^A_3A²^A2¹ 3

+

^A_A⁰₃

. Equation (19) has three real positive roots if is neg- ative. Furthermore, (19) has only one positive root if is positive. In what follows, we fix the parameters α = 0.01, β = 0.05, γ = 0.1, h = 0.1 and a = 0.02. In Fig.

1(a), we show the frequency-response curve, as

given by (19), for Ω = 0. The solid lines denote stable branches and the dashed lines denote unstable ones.

The effect of the frequency Ω on the backbone curve is illustrated in Fig.

1(b), (c) for the values

Ω = 25 and Ω = 40, respectively. The plots show that as Ω increases, the backbone curve shifts left and its characteristic switches from softening to hardening. For validation, analytical approximations are compared to numerical integration (circles) using a Runge–Kutta method.

3.2 Quasiperiodic modulation and hysteresis suppression

We transform the polar form (18) using the variable

change u = r cos θ, v = − r sin θ , and we implement a

second perturbation analysis by introducing the book-

keeping parameter η in damping and in nonlinearity

(5)

components. We obtain the Cartesian system du

dt = Sv + η Au − (Bu + Cv)

u

²

+ v

²

, dv

dt = − Su + H

+ η Av − (Bv − Cu)

u

²

+ v

²

.

(21)

Approximations of periodic solutions of the slow flow (21), corresponding to quasiperiodic motion of the slow dynamics (11), can be obtained by using a multiple scales technique [19,

21]. We expand solutions as

u(t ) = u

0

(T

0

, T

1

) + ηu

1

(T

0

, T

1

) + O η

²

, v(t ) = v

0

(T

0

, T

1

) + ηv

1

(T

0

, T

1

) + O

η

²

,

(22)

where T

_i

= η

ⁱ

t. Introducing D

_i

=

_∂T^∂_i

yields

_dt^d

= D

0

+ ηD

1

+ O(η

²

), substituting (22) into (21) and col- lecting terms, we get:

– Order η

⁰

:

D

²₀

u

₀

+ S

²

u

₀

= SH,

Sv

0

= D

0

u

0

; (23)

– Order η

¹

: D

₀²

u

₁

+ S

²

u

₁

= S

− D

₁

v

₀

+ Av

₀

− (Bv

₀

− Cu

₀

)

u

²₀

+ v

²₀

− D

0

D

1

u

0

+ AD

0

u

0

− D

0

Bu

0

+ Cv

0

u

²₀

+ v

₀²

, Sv

₁

= D

₀

u

₁

+ D

₁

u

₀

− Au

₀

+ (Bu

₀

+ Cv

₀

)

u

²₀

+ v

₀²

.

(24)

The solution to the first-order system (23) is given by u

₀

(T

₀

, T

₁

) = H

S + R(T

₁

) cos

ST

₀

+ ϕ(T

₁

) , v

0

(T

0

, T

1

) = − R(T

1

) sin

ST

0

+ ϕ(T

1

) .

(25)

Substituting (25) into (24) and removing secular terms gives the following autonomous slow slow-flow system on R and ϕ:

dR dt =

A − 2BH

²

S

²

R − BR

³

, dϕ

dt = − CR

²

− 2CH

²

S

²

.

(26)

Then, the approximate periodic solution of the slow flow (21) is given by

u(t ) = H

S + R cos νt, v(t ) = −R sin νt,

(27)

where the amplitude R is obtained by setting

^dR_dt

= 0 and given by

R =

A B − 2H

²

S

²

, (28)

and the frequency ν (frequency of the slow-flow limit cycle) is given by

ν = S − CR

²

− 2CH

²

S

²

, (29)

and hence the approximate quasiperiodic response of the slow dynamics (11) reads

z(t ) = H

S cos ωt + R cos(ω + ν)t. (30) On the other hand, using u(t ) = r(t ) cos θ (t ), v(t ) =

− r(t )sin θ (t ), the modulated amplitude r(t ) of the quasiperiodic oscillations is approximated by

r(t ) =

A B − H

²

S

²

+ 2R H

S cos νt . (31) The envelope of this modulated amplitude is delimited by r

_min

and r

_max

given by

r_min=min

A

B −H² S² +2RH

S,

A B−H²

S² −2RH S

,

(32)

r_max=max

A

B−H² S² +2RH

S,

A B−H²

S² −2RH S

.

(33)

In Fig.

2

we draw, for Ω = 25, the modulated qua-

siperiodic area given by (32), (33) (solid lines) and

obtained numerically using Runge–Kutta method (cir-

cles). The comparison between the two results shows

a good agreement and clearly confirms the accuracy

of the analytical approach used here. In contrast, the

approach used in [17], based on the invariance of the

slow flow under the transformation θ → − θ +

^π₂

,

(6)

σ → − σ and ξ → − ξ , presented a slight discrepancy between the theory and the simulation, and fails in the vicinity of jumps phenomena, especially. Figure

2

shows that outside the synchronization area, quasiperiodic behavior with two predominant frequencies takes place. Moving away from the synchronization area, the depth of modulation amplitude decreases. When approaching the entrainment area, the lower band of the modulation amplitude domain drops to zero, and then increases to collide with the upper limit of the modulation amplitude, exactly on the loci where the frequency-locked response takes place. Note that this dynamics has not been captured analytically using the invariance approach [17].

Fig. 2 Modulation amplitude vibration forΩ=25. Analytical approximation: solid lines. Numerical simulation: circles. QP:

modulation area of quasiperiodic response

In Fig.

3(a), (b) we show a global picture including

the quasiperiodic modulation area and the amplitude- frequency response for the values Ω = 0 and Ω = 40, respectively. Figure

3(c) illustrates the frequency re-

sponse for the critical case Ω

_c

corresponding to the vanishing of the nonlinear stiffness component ξ of the slow dynamics (11). This threshold is given by Ω

_c

=

√ 3γ

a . (34)

By analyzing the sign of the discriminant given by (20), we obtain an analytical approximation of the hysteresis width as illustrated in Fig.

3(a), (b). In Fig.4,

we plot this hysteresis area as a function of the fast excitation frequency Ω . It can be seen from Fig.

4

that a complete elimination of the hysteresis is achieved in a certain range of the frequency Ω around the threshold Ω

c

27.4 located exactly in the middle of the suppression domain. In this domain, jump phenomenon is completely eliminated and hence a smooth transition between the quasiperiodic response and the frequency- locked motion takes place. This analytical prediction illustrated in Fig.

4

is confirmed by comparison to numerical simulations provided in Fig.

1.

In Fig.

5, we plot the effect of different system pa-

rameters on hysteresis. When varying a parameter, the others are kept fixed with the values as given above.

Figure

5(a) shows the effect of varying the amplitude

of the external forcing h on the hysteresis area and on the hysteresis suppression domain. It can be seen that for increasing values of h, the hysteresis area increases, whereas the suppression domain is still un- changed. This indicates that the external forcing am-

Fig. 3 Effect of the high frequencyΩon the hysteresis width. H: hysteresis area; E: entrainment area

(7)

plitude h has no effect on the hysteresis suppression area. Figure

5(b) shows the effect of varying the linear

damping α on the hysteresis. This subfigure indicates that for the given parameters, the linear damping has

Fig. 4 Variation of the area of the hysteresis loop versus the frequency excitationΩ. Analytical result from (20)

no effect on the hysteresis suppression domain and a small influence on the hysteresis area. Figure

5(c) in-

dicates that increasing nonlinear damping β , the hysteresis area decreases and the suppression domain increases significantly. It can be seen from Fig.

5(c) that

for higher values of β , hysteresis can be suppressed from lower values of Ω . Finally, Fig.

5(d) illustrates

the effect of the amplitude a of the FHE on the hysteresis. The plots reveal that as a is increased, the suppression domain decreases and shifts left toward smaller values of Ω. This suggests that adjustment of the suppression area to a desired frequency range of Ω may be achieved by acting on the amplitude of the FHE.

4 The subharmonic resonance 3:1

In this section, we analyze the effect of an FHE on frequency locking and on hysteresis near the subharmonic resonance 3:1. We express this resonance condition by introducing the detuning parameter σ ac-

Fig. 5 (Color online)

Effect of system parameters on hysteresis area and on hysteresis suppression domain. (a) Effect of the external excitationh, (b) effect of the linear dampingα, (c) effect of the nonlinear dampingβ, (d) effect of the amplitudea of the FHE

(8)

cording to ω

²₀

=

ω 3

2

+ σ. (35)

Analytical treatment of periodic and quasiperiodic solutions is carried out as in the previous case of resonance 1:1.

4.1 Slow flow and entrainment

The first perturbation step to derive a slow flow is im- plemented using the parameter μ. According to (35), rewrite (11) as

¨ z +

ω 3

2

z

= h cos ωt + μ − σ z +

α − βz

²

˙ z + ξ z

³

. (36) We seek a two-scale expansion of the solution in the form

z(t ) = z

0

(T

0

, T

1

) + μz

1

(T

0

, T

1

) + O μ

²

(37) where T

_i

= μ

ⁱ

t . Substituting (37) into (36) and equating coefficients of the same power of μ, we obtain a set of linear partial differential equations

D

₀²

z

0

(T

0

, T

1

) + ω

3

2

z

0

(T

0

, T

1

) = h cos ωT

0

, (38) D

₀²

z

₁

(T

₀

, T

₁

) +

ω 3

2

z

₁

(T

₀

, T

₁

)

= −2D

0

D

1

z

0

− σ z

0

+

α − βz

²₀

D

0

z

0

+ ξ z

³₀

. (39) The solution to the first order is given by

z

0

(T

0

, T

1

)

= r(T

1

)cos ω

3 T

0

+ θ (T

1

)

+ F cos ωT

0

. (40) Substituting (40) into (39), removing secular terms and using the expressions

^dr_dt

= μD

₁

r + O(μ

²

) and

dθ

dt

= μD

1

θ + O(μ

²

), we obtain to the first order the autonomous slow-flow modulation equations of amplitude and phase:

dr

dt = Ar − Br

³

− (H

₁

sin 3θ + H

₂

cos 3θ )r

²

, dθ

dt = S − Cr

²

− (H

₁

cos 3θ − H

₂

sin 3θ )r,

(41)

where A =

^α₂

−

^βF₄²

, B =

^β₈

, C =

^9ξ_8ω

, S =

^3σ_2ω

−

^{9ξ F}_4ω²

, H

₁

=

^{9ξ F}_8ω

, H

₂

=

^βF₈

and F = −

_8ω^9h2

.

Equilibria of the slow flow (41), corresponding to periodic oscillations of (11), are determined by setting

dr

dt

=

^dθ_dt

= 0. We obtain the amplitude-frequency response equation

A

₂

r

⁴

+ A

₁

r

²

+ A

₀

= 0 (42) and the phase-frequency response relation

tan 3θ = (A − Br

²

)H

₁

− (S − Cr

²

)H

₂

(A − Br

²

)H

₂

+ (S − Cr

²

)H

₁

(43) where A

2

= B

²

+ C

²

, A

1

= − (2AB + 2SC + H

₁²

+ H

₂²

) and A

0

= A

²

+ S

²

.

Figure

6(a), (b) shows the variation of

z(t )-amplitude-frequency response, as given by (40), (42) and (43), for Ω = 0 and Ω = 50, respectively. The other parameters are fixed as follows: α = 0.01, β = 0.05, γ = 0.1, h = 1 and a = 0.02. The solid line denotes the stable branch and the dashed line denotes the unstable one. For validation, analytical approximations are compared to numerical integration (circles) using a Runge-Kutta method. The effect of Ω on the amplitude-frequency response is illustrated in Fig.

6(c)

for different values of Ω. The plots in this figure indicate that as Ω increases, the backbone curve shifts left and changes its bending from softening to hardening. Near the threshold Ω

_c

given by (34), the backbone curve becomes smaller, the frequency locking and hysteresis are reduced and jumps are attenuated.

Hysteresis is completely eliminated at the threshold Ω

c

corresponding to the vanishing of the nonlinear characteristic stiffness of the slow dynamics (11).

In Fig.

7, we show the

z-amplitude-frequency re-

sponse near the two resonances, 1:1 and 3:1. This fig-

ure indicates that for Ω = 0, the 3:1 resonance area is

in a narrow range comparing to the 1:1 resonance area

(Fig.

7(a)). In contrast, for

Ω = 50 the 3:1 resonance

curve becomes larger (Fig.

7(b)) indicating that in the

softening case the resonance 3:1 can be considered as

negligible, whereas in the hardening case the 3:1 reso-

nance is significant in terms of magnitude and width.

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Fig. 6 Amplitude-frequency response of the slow dynamicsz(t ). Analytical approximation: solid (for stable) and dashed (for unsta- ble). Numerical simulation: circles. (a)Ω=0, (b)Ω=50, (c) effect of different values ofΩon the backbone curve

Fig. 7

Amplitude-frequency response near 1:1 and 3:1 resonances

4.2 Quasiperiodic modulation and hysteresis suppression

Following the same analysis as for the fundamental resonance case, we construct analytical approximation of the limit cycle of the slow flow (41) corresponding to quasiperiodic motion of the slow dynamics (11) near the 3:1 subharmonic resonance. The Cartesian system corresponding to the polar form (41) is written as

du

dt = Sv + η Au − (Bu + Cv)

u

²

+ v

²

+ 2H

1

uv − H

2

u

²

− v

²

, dv

dt = − Su + η Av − (Bv − Cu)

u

²

+ v

²

+ H

1

u

²

− v

²

+ 2H

2

uv ,

(44)

where the bookkeeping parameter η is introduced in damping and nonlinearity terms. Using the multiple scales method, the solution to the first order of system (44) is given by

u

₀

(T

₀

, T

₁

) = R(T

₁

) cos

ST

₀

+ ϕ(T

₁

) , v

0

(T

0

, T

1

) = − R(T

1

) sin

ST

0

+ ϕ(T

1

)

. (45)

The amplitude R and the phase ϕ vary with time according to the following slow-flow system:

dR

dt = AR − BR

³

, dϕ

dt = −CR

²

.

(46)

(10)

Fig. 8 Entrainment area (E), hysteresis (H) and modulation amplitude of slow-flow limit cycle (QP)

The first-order approximate periodic solution of the slow flow (44) is then given by

u(t ) = R cos νt,

v(t ) = − R sin νt, (47)

where the amplitude R and the frequency ν are given by

R = A

B , (48)

ν = S − CR

²

, (49)

and finally, the approximate quasiperiodic response of the slow dynamics (11) reads

z(t ) = R cos ω

3 + ν

t + F cos ωt. (50) In Fig.

8(a), (b), we show the analytical entrainment

zone along with the numerical quasiperiodic modulation domain of the slow-flow limit cycle for Ω = 0 and Ω = 50, respectively. The numerical quasiperiodic modulation domain is marked with double circles connected with a vertical line.

Figure

8(a) shows the existence of two hysteresis

loops near this subharmonic resonance. One hysteresis loop (right) is found to be very small, whereas the other one (left) is quite large and may produce a significant jump phenomenon from higher to lower amplitude when the forcing frequency is swept backward.

In the hardening case (Fig.

8(b)), only one similar

jump phenomenon occurs when the forcing frequency is swept forward.

In Fig.

9, we present examples of time histories of

the slow dynamics z(t ) obtained by numerical simulation for some values of ω picked from Fig.

8(a). As

predicted, away from the resonance, a slow-flow limit cycle exists and attracts all initial conditions. The related time trace is shown in subfigure (a) corresponding to ω = 2.72. In subfigures (b) and (c) we plot the time traces corresponding to ω = 2.8. These figures indicate the coexistence of periodic (stable slow-flow equilibrium) and quasiperiodic (stable slow-flow limit cycle) responses. As the forcing frequency increases, a transition from quasiperiodic vibration to a response at the 3:1 subharmonic frequency occurs. The region between ω = 2.862 and ω = 2.939 is associated with this 3:1 frequency-locked motion, in which the response of the system follows the 3:1 subharmonic frequency.

The corresponding time trace is shown in subfigure (d) corresponding to ω = 2.9. It is worth noting that in the region corresponding to the small hysteresis, coexistence phenomenon can also occur. Moreover, saddle- node and heteroclinic bifurcations involving the slow- flow limit cycle and the cycles of order 3 can take place in this small region. However, this problem is beyond the scope of the current paper.

In Fig.

10, we show the variation of hysteresis area

(obtained from (42)) as a function of the frequency Ω for h = 1. This plot indicates that in the softening region (Ω < Ω

c

) the hysteresis area is relatively small, whereas in the hardening region (Ω > Ω

c

) it becomes large. It can be seen from this figure that the hysteresis is attenuated around the threshold Ω

_c

and completely eliminated at this threshold.

In Fig.

11, we plot the effect of different system

parameters on the attenuation domain of hysteresis.

Figure

11(a), (b) shows that varying the amplitude of

(11)

Fig. 9 Examples of time histories of the slow dynamicsz(t )

Fig. 10 Variation of the hysteresis area versus the frequencyΩ near the 3:1 subharmonic resonance

the external forcing h and the linear damping α has small effect on the attenuation area of hysteresis. Fig- ure

11(c) indicates that increasing the nonlinear damp-

ing β , the hysteresis area decreases and the attenuation

zone of hysteresis increases. Figure

11(d) illustrates

the effect of the amplitude a of the FHE on the hysteresis. As it was seen in the primary resonance case, the plots in subfigure (d) indicate that increasing the amplitude a causes the attenuation domain of hysteresis to shift left toward smaller values of Ω .

5 Conclusion

In this paper, we have investigated the effect of an FHE on hysteresis and on frequency locking in a forced van der Pol–Duffing oscillator near the primary and the subharmonic 3:1 resonances. Averaging procedure and multiple scales techniques are performed near each resonance to derive a slow dynamics and its slow flow. Analysis of equilibria of each slow flow provides analytical expressions of the corresponding backbone curve. The results showed that adding the FHE causes the backbone curves to shift left and the nonlinear characteristic to change from softening to hardening.

Analysis of the influence of different system parame-

ters on the hysteresis near the resonances reveals that

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Fig. 11 (Color online) Effect of system parameters on frequency area.

(a) Effect of the external excitationh, (b) effect of the linear dampingα, (c) effect of the nonlinear dampingβ, (d) effect of the amplitudeaof the FHE

increasing the amplitude a of the FHE decreases the suppression domain and shifts it toward lower values of the fast frequency Ω . Moreover, adding the nonlinear damping β increases the hysteresis suppression area.

Concerning the 3:1 subharmonic resonance, it is shown that in the softening situation (Ω < Ω

c

), the 3:1 resonance may occur in a small range of the forcing frequency, whereas in the hardening situation (Ω > Ω

c

), the 3:1 resonance domain is large (Fig.

7).

In addition, near this subharmonic resonance, two hysteresis loops may exist, one of them is large (Fig.

8)

and can produce a significant jump that cannot be ig- nored in practical applications.

The results suggest that the high frequency Ω can be chosen so as to completely eliminate the hysteresis loop for a significant range of the frequency near the 1:1 resonance and in a small range near the 3:1 resonance. This suppression offers a smooth transition between the quasiperiodic responses and the entrained motions, preventing jumps near the resonances.

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