Nonlinear vibrations of a shallow arch under a low frequency and a resonant harmonic excitations

N O N L I N E A R D Y N A M I C S , I D E N T I F I C A T I O N A N D M O N I T O R I N G O F S T R U C T U R E S

Nonlinear vibrations of a shallow arch under a low frequency and a resonant harmonic excitations

F. Lakrad . _{A. Chtouki} . _{M. Belhaq}

Received: 1 February 2016 / Accepted: 30 May 2016

Ó

Springer Science+Business Media Dordrecht 2016

Abstract In the present work we investigate analyt- ically and numerically nonlinear dynamics of a two degrees of freedom model of a shallow arch subject to a resonant external harmonic forcing and to a very slow harmonic imposed displacement of one of its supports. Charts of behaviors are determined, espe- cially the zones of existence of periodic bursters and chaos. Periodic bursters are found to exist in the boundaries of the instability regions. Various bursters involving fixed points, quasi-periodic and chaotic solutions are found. More importantly, it is shown that small amplitudes of the slow parametric excitation may suppress chaos from wide regions of control parameters.

Keywords Shallow arch Periodic bursters Suppression of chaos

1 Introduction

Arches have a wide range of uses in civil, mechanical and aerospace engineering [1]. For instance, they are used in MEMS and NEMS with electrical load

actuation [2, 3] and energy harvesting devices [4].

Arches are characterized, compared to the straight beams, by their initial curvature, strength and the bi- stability behavior.

Arches can be classified following their shallow- ness parameter, that is the sag to the span ratio, as shallow or non-shallow. The latter case was exten- sively studied analytically and experimentally by Benedettini and co-workers [5, 6]. In particular, they studied the dynamic instability of a double-hinged circular arch, excited by a sinusoidally varying force applied on the tip. The bifurcations and the regions of co-existence of mono and bi-modal solutions are highlighted, as well as the complex responses arising inside the instability region. Moreover, the spatial shapes visited in average by the experimental model, obtained by a Karhunen Loe´ve decomposition, put into evidence that two spatial shapes are enough to catch more than 90% of the signal power of oscilla- tions. This occurrence justifies the use of a two degree- of-freedom (d.o.f.) model both in regular and non- regular regimes.

Many authors studied the nonlinear dynamics of shallow arches under various types of loadings and internal resonances. Tien et al. [7, 8] investigated global bifurcations, using a Melnikov perturbation method. They determined the chaos occurrence in a 2-d.o.f. model of a shallow arch subject to a static and a harmonic loading under 1:2 and 1:1 internal resonances. For the same model, Bi and Dai [9]

studied numerically the period doubling cascades F. Lakrad (

&

) A. Chtouki M. Belhaq

Laboratory of Renewable Energy and Dynamics of Systems, Faculty of Sciences Ain Chock, University Hassan II of Casablanca, Casablanca, Morocco e-mail: f.lakrad@fsac.ac.ma

DOI 10.1007/s11012-016-0470-7

leading to chaos in the 1:2 internal resonance case. El- Bassiouny [10] examined the various responses, using the multiple scales method, under 1:3 internal resonance and two external harmonic resonant forcings. Lakrad and Schiehlen [11] reported on the effect of a slowly varying parametric excitation on a single degree of freedom shallow arch model.

Periodic bursters and chaos were observed and analyzed using a Poincare´ map and the Melnikov method. It is worth pointing out that from geomet- rical point of view, periodic bursters can be seen as generalized heteroclinic orbits. For detailed classifi- cation of bursters, see [12, 13]. It was shown in [14]

that a necessary condition for the occurrence of periodic bursters is that the slow excitation is of parametric type. Lakrad and Belhaq [15] investi- gated the behavior of a shallow arch subject to a very slow parametric excitation and a fast resonant excitation. Averaging over the fast dynamics was used to obtain the slow flow. Then, the chart of behaviors and the regions of existence of periodic bursters were derived in a special case.

The present work can be viewed as an extension of the previous one [15] reporting on new kinds of bursters and further parameter variations. It is shown that the slow parametric excitation can suppress chaos initially induced by the resonant forcing. Indeed, the slow frequency can transform the chaotic attractor to a periodic burster. In what follows a two d.o.f. model of a planar simply supported shallow arch subject to a harmonic resonant distributed load and a very slow harmonic displacement of one of its support is investigated. The nonlinear dynamics of this reduced order model is investigated under 2:1 internal reso- nance and 1:2 external resonance.

The paper is organized as follows: in Sect. 2 we formulate the problem by presenting the model of a shallow hinged arch and the various internal reso- nances. In Sect. 3, the multiple scales method (MSM) is applied to derive slow flow equations. In Sect. 4, numerical and analytical results are presented and compared. A conclusion closes the work.

2 Mathematical model

We consider a double-hinged shallow arch subjected to a lateral sinusoidally distributed loading P(x, t) consisting of a static and a harmonic loadings and an

imposed horizontal slow harmonic motion of its support u(L, t); see Fig. 1. Under the assumptions [5] of small change of curvature, large extensional strain, negligible longitudinal inertia and constant elongation along the axis, the equation of motion governing the inplanar lateral deflection w(x, t) is given by

m wðx; € tÞ þ c wðx; _ tÞ þ EIw

⁰⁰⁰⁰

ðx; tÞ EA

L ðw

⁰⁰₀

ðxÞ þ w

⁰⁰

ðx; tÞÞ

uðL; tÞ þ 1 2

Z

L 0

ðw

⁰²

ðx; tÞ þ 2w

⁰₀

ðxÞw

⁰

ðx; tÞÞdx

¼ Pðx; tÞ

ð1Þ where the dot and the prime denote the derivatives with respect to time t and to the variation of the length x, respectively, E is Young’s modulus, I is the moment of inertia of the cross-section, m is the mass per unit length, L is the projected length of the arch and c is the viscous damping coefficient.

The model (1) of the shallow arch is, recently, widely used in the study of curved MEMS and NEMS, see for instance [2, 16]. Moreover, it was suggested by Alaggio and Benedettini [5] that the model (1) is adequate only for arches having a shallowness ratio of the order of 1 / 100.

w

0

ðxÞ ¼ q

0

sin px

L ð2Þ

Pðx; tÞ ¼ ðp

0

þ q cosðmtÞÞ sin px

L ð3Þ

uðL; tÞ ¼ H cosðXtÞ ð4Þ

Fig. 1

The shallow arch model

Here q

0

is the initial rise of the unloaded arch, p

0

is the static loading, q and m represent the amplitude and frequency of the resonant harmonic excitation, respec- tively. H and X are the amplitude and the frequency of the very slow imposed harmonic displacement, respectively. The double-hinged shallow arch is subjected to the following boundary conditions:

wðx; tÞ ¼ 0; w

⁰⁰

ðx; tÞ ¼ 0 at x ¼ 0; L ð5Þ The Galerkin method is used to reduce the equation of motion (1) to a set of ordinary differential equations by selecting appropriate shape functions. The transverse motion w(x, t) of the arch is approximated by the following expression:

wðx; tÞ ¼ q

₁

ðtÞ sin px

L þ q

₂

ðtÞ sin 2px L

: ð6Þ

We set t

¼ p

L

2

ffiffiffiffiffi EI m r

t; r ¼ ffiffiffi I A r

; k

0

¼ p

₀

2rEI L p

4

;

q

₀

¼ q

₀

2r ; h ¼ HL

r

²

p

²

; X

¼ XL

²

p

²

ffiffiffiffiffi m EI r

q

¼ q 2rEI

L p

4

; m

¼ mL

²

p

²

ffiffiffiffiffi

m EI r

b

_i

¼ cL

²

p

²

m ffiffiffiffiffi

p EI ; Q

1

¼ q

1

q

0

2r ; Q

2

¼ q

2

2r : ð7Þ where r is the radius of gyration of the cross section. In what follows the stars will be omitted for simplicity of notations. The non-dimensional equations of motion describing the evolution of the straightened ampli- tudes of the two fundamental modes read

Q €

₁

þ b

₁

Q _

₁

þ ð1 þ h cosðXtÞÞQ

1

þ Q

₁

ðQ

²₁

q

²₀

þ 4Q

²₂

Þ þ q

0

k

0

¼ q cos ðmtÞ;

ð8Þ Q €

₂

þ b

₂

Q _

₂

þ 4ð4 þ h cosðXtÞÞQ

2

þ 4Q

2

ðQ

²₁

q

²₀

þ 4Q

²₂

Þ ¼ 0; ð9Þ In the absence of the imposed displacement i.e., h ¼ 0, Eqs. (8) and (9) were studied by many authors, see for instance [7, 8].

The main non-dimensional parameters are: the viscous damping parameters for the first and second mode b

₁

and b

₂

, respectively, the static loading parameter k

0

, the initial rise parameter q

0

, the

amplitude h and the frequency X of the imposed displacement.

It is worth pointing out that the slow imposed displacement is acting as a parametric excitation and the initial curvature q

0

softens the first and the second modes and introduces a quadratic nonlinearity beside the static load.

The static equilibria of the arch are computed from Eqs. (8–9) by dropping the time derivatives and excitations. In Fig. 2 are shown the number of static equilibria in the plane ðq

0

; k

0

Þ. The zone I corresponds to Q

2

¼ 0 and one stable static solution of Q

1

. The zone II corresponds to Q

₂

¼ 0 and three static solutions of Q

1

. Two of the three equilibria are stable.

In zone III, there is coexistence of the zone II equilibria and two unstable equilibria corresponding to ðQ

1

¼ ðq

0

k

0

Þ=3; Q

2

¼

¹₂

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

²₀

4 Q

²₁

p Þ.

In what follows, we work around the unbuckled stable static equilibrium corresponding to a zero second mode Q

₂

¼ 0 and a first mode Q

₁

¼ g

₀

. This latter is solution of the following algebraic equation

Q

³₁

þ Q

1

ð1 q

²₀

Þ þ q

0

k

0

¼ 0 ð10Þ

3 Perturbation analysis

We perturb the variables ðQ

1

; Q

₂

Þ in Eqs. (8) and (9) around the stable unbuckled static solution corre- sponding to ðQ

1

¼ g

₀

; Q

2

¼ 0Þ. Thus,

0 0.5 1 1.5 2 2.5 3

−6

−3 0 3 6 9

Initial rise q

0

Static load λ 0

Zone I Zone III

Zone II

Fig. 2

Number of static equilibria in the plane

ðq0;k0Þ. In zone

I: one static solution with

Q2¼

0. In Zone II: three solutions

with

Q2¼

0. In zone III: coexistence of equilibria of zone II and

two unstable static solutions with

Q26¼

0

Q

1

¼ g

₀

þ ex

1

ðtÞ ; Q

2

¼ ex

2

ðtÞ ð11Þ where e is a small positive parameter. Let q ¼ e

²

q; ~ h ¼ e h ~ and b

_i

¼ e b ~

_i

with i ¼ 1; 2. Equations (8-9) become up to the order Oðe

²

Þ

€

x

1

þ x

²₁

x

1

¼ hg ~

₀

cosðXtÞ þ eð b ~

₁

x _

1

3g

₀

x

²₁

4g

₀

x

²₂

q ~ cosðmtÞ h ~ cosðXtÞx

1

Þ

ð12Þ

€

x

₂

þ x

²₂

x

₂

¼eð b ~

₂

x _

₂

8g

₀

x

₁

x

₂

4 h ~ cosðXtÞx

2

Þ ð13Þ The slow frequency X ¼ Oðe

ⁿ

Þ with n 2. The linearized frequencies corresponding to the first and second modes are represented by x

1

and x

2

, respec- tively. They are given by

x

1

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3g

²₀

þ 1 q

²₀

q

; x

2

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 4q

²₀

þ 4g

²₀

q

ð14Þ The initial rise q

0

and the static load parameter k

0

can be used to tune the natural frequencies x

1

and x

2

in order to realize various internal resonances. Indeed, in the absence of the static load, i.e., k

0

¼ 0, g

₀

¼ q

0

and x

1

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2q

²₀

p and x

2

¼ 4. This result is in agreement with [17] where it was shown that, in the case of a double-hinged shallow arch, the only natural

frequency that is affected by the initial rise q

0

is the first one.

In Fig. 3 are shown various zones of internal resonances in the plane ðq

0

; k

0

Þ. These regions are computed by setting

p 0:1 x

1

x

2

p þ 0:1; with p ¼ 1 2 ; 1; 2:

ð15Þ In Fig. 3, the black zone corresponds to 2:1 internal resonance. The strong and light grey zones correspond to 1:1 and 1:2 internal resonances, respectively.

0 1 2 3 4 5 6

0 1 2 3 4 5 6 7 8 9 10

Initial rise q₀ Static load λ0

2:1

Fig. 3

Various internal resonances, relating

x1

and

x2

, in the plane

ðq0;k0Þ. The black zone corresponds to 2:1 resonance; the

strong grey zone to 1:1 resonance and the light grey zone to 1:2 resonance

Fig. 4

Behavior chart of Eqs (17) in the external excitation parameters

ðm;qÞ

plane, in the absence of the base displacement

h¼

0,

q0¼

2:5 and

k0¼

6:95. Zone A, only the single mode consisting of the first mode exists and is stable. Zone B, coexistence of the single and coupled modes. Region C, only the coupled mode is stable. Region D, both modes are unstable

1 1.175 D1 D2 1.373 1,5

−0.06

−0.04

−0.02 0 0.02 0.04 0.06

ν Q2 [n 2π/ν]

Fig. 5

Bifurcation diagram of

Q2

versus

m

for

q0¼

2:5;

k0¼

6:95 and

q¼

0:002

In what follows, we shall perform the analysis in the black region i.e., 2:1 internal resonance and near the principal external resonance

x

1

¼ 2x

2

þ e~ r

1

; m ¼ 2x

2

þ e~ r

2

ð16Þ where r

i

¼ e r ~

i

with i ¼ 1; 2 are detuning parameters.

Using the MSM [18] one can eliminate the fast time scale dependence. This method ultimately results in the following modulation equations of amplitudes a

1

; a

2

and the phases of the first and second modes, respectively

a

⁰₁

¼ b ~

₁

2 a

₁

q ~ 2x

1

sinðc

₁

Þ g

₀

x

1

a

²₂

sinðc

₁

2c

₂

Þ a

1

c

⁰₁

¼ðr

2

r

1

Þa

1

g

₀

x

1

a

²₂

cosðc

1

2c

₂

Þ q ~

2x

1

cosðc

1

Þ hX ~ 2x

1

a

1

cosðXtÞ a

⁰₂

¼

b ~

₂

2 a

2

þ 2g

₀

x

2

a

1

a

2

sinðc

1

2c

₂

Þ c

⁰₂

¼ r

2

2 2g

₀

x

2

a

1

cosðc

1

2c

₂

Þ 2 hY ~ x

2

cosðXtÞ ð17Þ

−0.04 −0.02 0 0.02 0.04

−5 0 5

x 10⁻³

Q2

dQ 2/dt

(a) ν = 1.2971

−0.04 −0.02 0 0.02 0.04

−8

−4 0 4 8

x 10⁻³

Q2

dQ 2/dt

(b) ν = 1.2975

−0.04 −0.02 0 0.02 0.04

−8

−4 0 4 8

x 10⁻³

Q2

dQ 2/dt

(c) ν = 1.29792

−0.04 −0.02 0 0.02 0.04

−8

−4 0 4 8

x 10⁻³

Q2

dQ 2/dt

(d) ν = 1.298

Fig. 6

Poincare´ sections of

Q₂

for various values of

m

and for

q₀¼

2:5;

k0¼

6:95 and

q¼

0:002.

am¼

1:2971,

bm¼

1:2975,

c m¼

1:29792,

dm¼

1:298

where X ¼ 1 ð6g

²₀

=x

²₁

Þ and Y ¼ 1 ð2g

²₀

=x

²₁

Þ. The prime denotes the derivative with respect to the slow time scale et. Here we have considered the slowly varying parametric excitation as constant during the averaging process.

The solution of Eqs. (8–9) are approximated up to OðeÞ as follows

Q

1

ðtÞ ¼ g

₀

þ a

1

ðsÞ cosðmt c

₁

ðsÞÞ hg

₀

x

²₁

cosðXtÞ

ð18Þ Q

2

ðtÞ ¼ a

2

ðsÞ cos m

2 t c

₂

ðsÞ

þ OðeÞ ð19Þ

where s ¼ Xt is a very slow time scale.

4 Results and discussions

In this section we will discuss the effects of the low frequency harmonic parametric excitation on the local dynamics of the shallow arch under the resonant forcing and the 2:1 internal resonance. Thus, we first present and recall results corresponding to the case of the absence of the low frequency excitation i.e., h ¼ 0.

Fig. 7

Behavior chart of Eq. (17) in the external excitation parameters

ðm;qÞ

plane, in the absence of the base displacement

h¼

0,

q0¼

5 and

k0¼

5. Zone A, only the single mode consisting of the first mode exists and is stable. Zone B, coexistence of the single and coupled modes. Region C, only the coupled mode is stable

6.8 6.85 6.9 6.95 7 7.05

−0.03

−0.02

−0.01 0 0.01 0.02 0.03

ν Q2 [n 2π/ν]

Fig. 8

Bifurcation diagram of

Q2

versus

m

for

q0¼

5;

k0¼

5 and

q¼

0:008

(a)

h= 0.001

(b)

h= 0.002

(c)

h= 0.004

Fig. 9

Evolution of the behavior chart of Eq. (17) in the

external excitation parameters

ðm;qÞ

plane, for various base

amplitude displacement

h. Theblue lines

correspond to the case

h¼

0.

a h¼

0:001,

b h¼

0:002,

c h¼

0:004. (Color

figure online)

4.1 Absence of the low frequency excitation i.e., h ¼ 0

In Fig. 4, we show the behavior chart of the modu- lation equations (17) in the absence of the base displacement i.e., h ¼ 0. In region A, a single mode consisting of the first mode ða

1

; a

2

¼ 0Þ is excited and is stable. In region B, coexistence of the previous stable single mode and two coupled modes ða

1

; a

2

6¼ 0Þ, one of them is stable. The coupled modes appear through a saddle-node bifurcation. In region C, destabilization of the single mode without a change in the coupled mode. In region D, destabiliza- tion of the coupled mode through a Hopf bifurcation.

In Fig. 5 is shown the bifurcation diagram of Q

2

versus the frequency m of the external load, for

q ¼ 0:002, obtained by numerically integrating Eqs (8–9) with initial conditions ð1:523; 0; 0:001; 0Þ.

For m 2 ½1; 1:175½, the regions A and B, the single mode corresponding to Q

2

¼ 0 is stable. In region C i.e., m 2 ½1:175; D1 ¼ 1:287½ the coupled mode is stable and corresponds to Q

₂

6¼ 0. In the region D corresponding to m 2 D1; D2½ with D2 ¼ 1:312, the dynamics are non-periodic. In the region D, the system undergoes a torus doubling sequence as a route to chaos; see Fig. 6.

In Fig. 7 the chart of behaviors of Eq. (17) are shown for q

₀

¼ k

0

¼ 5. The main differences with Fig. 4 are: (i) the absence of the zone D that

1.28 1.285 1.29 1.295 1.3 1.305 1.31 1.315 1.32

−0,02 0 0,015

ν

Maximum Lyapunov Exponent

h=0.01 h=0

h=0.004

Fig. 10

Maximum Lyapunov exponent versus

m

for various values of

h

and for

q¼

0:002

0 5 10 15 20

x 10⁻³

−0.02 0 0.01

Amplitude of the low frequency h

Maximum Lyapunov exponent

Fig. 11

Maximum Lyapunov exponent versus

h

for

m¼

1:3,

q¼

0:002 and

X¼

0:001

(a) h = 0

(b) _{h = 0.004}

Fig. 12

Snap-through boundaries in the plane of the external

forcing parameters

ðm;qÞ, forh¼

0 and

h¼

0:004.

X¼

0:001

for the same parameters as Fig.

4. Theblack zone

corresponds to

the buckled state and the

white zone

to the unbuckled state.

a h¼

0,

bh¼

0:004

corresponds to the nonregular dynamics; (ii) the increase of the threshold of various instability zones in terms of the forcing amplitude q; and (iii) the v- shaped instabilities zones instead of a w-shaped one in Fig. 4. To confirm the chart of behaviors given in Figs. 7, 8 shows the bifurcation diagram of Q

2

versus the external load frequency m, for q ¼ 0:008, obtained by numerically integrating Eqss (8–9) with initial conditions ð4:8989; 0; 0:001; 0Þ. Furthermore, the region of nonregular dynamics is reduced to a very thin region near m ¼ 6:876.

4.2 Effects of the low frequency excitation

In the presence of the low frequency parametric harmonic excitation, the equations of modulations (17) can be written as a slow-fast system

e z _ ¼ fðz; sÞ ; s _ ¼ 1 ð20Þ

where the state vector z ¼ ða

1

; c

₁

; a

₂

; c

₂

Þ and the dot is the derivative with respect to the very slow time scale s ¼ Xt, with X ¼ Oðe

²

Þ. In the limit e ! 0 one can compute the slow manifold given by M ¼ fðz; sÞ : fðz; sÞ ¼ 0g. It is composed of two types of solutions:

– The single mode solution a

1

ðsÞ ¼ q ~

2x

1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b~1

2 2

þ r

1

r

2

þ

_2x^hX~

1

cosðsÞ

2

r

ð21Þ

a

2

ðsÞ ¼ 0: ð22Þ

– The coupled modes solutions

(a) ρ = 0.002 (b) ρ = 0.005

(c) ρ = 0.0055

Fig. 13

Basins of

attraction, in the plane

ðQ1ð0Þ;Q⁰₁ð0ÞÞ, of the

unbuckled state in

white. for

various values of

q, and for h¼

0:004,

X¼

0:001 and

m¼

1:4. The unbuckled

static equilibrium is

represented in

blue.a q¼

0:002,

bq¼

0:005,

c q¼

0:0055. (Color

figure online)

a

1

ðsÞ ¼ x

2

2g

₀

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ~

₂

2

!

2

þ r

2

2 2 hY ~ x

2

cosðsÞ

²

v u u t

ð23Þ The two nontrivial amplitudes of the anti-symmet- ric mode a

2

ðsÞ are obtained by solving a fourth order algebraic equation.

In Fig. 9, we show the evolution of the chart be- haviors of the modulation Eq. (17) for various amplitudes of the base motion h in the plane of external harmonic excitation parameters. The behav- iors of the solutions in the zones A, B, C and D are the same as in Fig. 4. However, instead of fixed points we

deal now with periodic solutions of the modulation equations. Thus, in regions A, B and C the solutions of the initial Eqs. (8) and (9) are quasi-periodic. In the grey zones these periodic solutions are changing their nature and/or stability during one period of the slow time scale s. These zones are the zones of existence of periodic bursters. The areas of these zones are increasing by increasing the amplitude of the base motion h. Moreover, the zone D disappears leading the system to evolve periodically rather than aperiodi- cally; see Fig. 9c.

Figure 10 shows the maximum Lyapunov exponent versus the frequency m for various h. It can be observed that for h ¼ 0 and h ¼ 0:004 the dynamics is chaotic for m 2 1:295; 1:312½ and for m 2 1:295; 1:305½, respectively. For h ¼ 0:01 the dynamics is regular.

Figure 11 confirms the suppression of chaos when h is increasing.

It is worth pointing out that all the presented charts of behaviors are based on the study of local dynamics near the stable unbuckled static equilibrium. Conse- quently, this study is no more valid when the snap- through occurs. In Fig. 12 are shown Snap-through zones, in black, in the plane of the external forcing parameters ðm; qÞ, for h ¼ 0 and h ¼ 0:004. Indeed, the arch is considered to be undergoing snap-through if Q

1

ðtÞ 0 at any time during the numerical integra- tion of Eqs. (8) and (9). Figure 12 shows that the snap- through is occurring for q [ 0:0035. Thus, the local analysis and the presented chart behaviors are valid since q is taken, in our case, below the threshold of this escape phenomenon.

In Fig. 13 are shown basins of attraction phase portraits, in the plane ðQ

1

ð0Þ; Q

⁰₁

ð0ÞÞ, of the unbuckled configuration for various values of the external forcing amplitude q. Numerical integrations of Eqs. (8) and (9) are performed using ðg

₀

; 0; 0; 0:001Þ as an initial condition. It is observed that increasing q decreases the basin of attraction of the unbuckled position (the white zone), till vanishing for q [ 0:006.

In order to validate the perturbation method, Fig. 14 shows comparisons between the analytical solutions (18-19) and the numerical solutions of Eqs. (8) and (9). Figure 14(a) shows a quasi-periodic solution and Fig. 14(b) shows a periodic burster relating a station- ary solution and a quasi-periodic solution. We observe the delay in the loss of stability of the trivial solution, for more informations about this phenomenon see (a) _{h = 0.001}

(b) _{h = 0.01}

Fig. 14

Comparisons of time histories of Eqs. (8–9), analyt-

ically (18–19) in

grey

and numerically in

black, for m¼

1:25;

q¼

0:002 and

X¼

0:001.

ah¼

0:001,

bh¼

0:01

[11]. In the case q

0

¼ 2:5 and k

0

¼ 6:95, besides the burster involving a trivial solution and a quasi- periodic (QP) attractor shown in Fig. 14b, various bursters can be found. Thus, in Fig. 15a is shown a

burster relating two QP solutions. In Fig. 15b a periodic burster involving two QP solutions and a trivial solution is shown. Figure 15c shows a burster involving a QP solution and a chaotic attractor.

5 Conclusion

The multiple scales method is used to determine charts of behaviors of a two degree of freedom model of a shallow arch subject to a resonant external harmonic forcing and to a very slow harmonic imposed displacement of one of its supports. The zones of various periodic bursters and chaotic dynamics are determined. Bursting solutions were found to involve a fixed point and/or quasi-periodic solutions and/or chaotic solutions. It is shown that the slow parametric excitation may suppress chaos from wide regions of control parameters space even for small amplitudes.

The effects of the low frequency excitation near other resonances are to be studied in order to gain more insight on the interaction between various time scales dynamics.

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