Solutions of a shallow arch under fast and slow excitations

FAST AND SLOW EXCIT A TIONS

Faouzi Lakrad

Institute B of Mechanics, University of Stuttgart, 70550 Stuttgart, Germany

lakrad@recherche-maroc.org

Mohamed Belhaq

Laboratory of Mechanics, Faculty of Sciences Ain Chock, BP 5366 Maarif, Casablanca, Morocco

Abstract: Quasi-Periodic (QP) and periodic burster solutions of a two degree of freedom shallow arch model, subjected to a very slow parametric excitation and a reso- nant external excitation, are investigated. A multiple-scales method is applied to have slowly modulated amplitudes and phases equations. Behavior charts and explicit analytical approximations to QP solutions are obtained and comparisons to numerical integration are provided.

Key words: Shallow arch, quasi-periodic solutions, slow manifold, bursters.

1. Introduction

Quasi-periodic (QP) excitations consisting of slow and fast periodic func- tions are very important sources of multiple scales phenomena. A wide range of these latter can be written as singularly perturbed systems or ODEs with slowly varying parameters. Several methods are available to analyze solutions of such systems. For instance, Bogoliubov and Mitropolskii [1] developed an averaging method considering the slowly varying parameters as constants dur- ing the averaging process. Belhaq and co-workers [2; 3] used the so-called double perturbation method to investigated non-linear QP Mathieu equations.

For other techniques and applications see [4].

The main goal of the present work is to approximate analytically QP solutions and periodic bursters of a two mode double hinged shallow arch model. It is

233 G. Rega and F. Vestroni (eds.),

IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, 233–240.

© 2005 Springer. Printed in Great Britain.

excited by two forcings, in the vicinity of a buckled symmetric state, under 1:2 internal resonance and the principal external resonance. The two excitations consist of a resonant external periodic excitation of the symmetric mode, and a very slow parametric periodic excitation of the two modes. This parametric excitation is due to the imposed motion of the arch support. Hence three time scales rule our system; the fast scale which is induced by the external excita- tion, the slow scale that is induced by the modulated amplitudes and phases, and the very slow scale caused by the parametric excitation.

This paper is organized as follows: in the first section we discuss the shal- low arch model. Then, a Multiple scales Method (MSM) is applied to deter- mine the slowly modulated amplitudes and phases equations. Finally, explicit conditions on the existence and stability of different dynamics are obtained.

Analytical approximations of QP solutions are computed and comparisons to numerical simulations are provided.

2. Shallow Arch Model

A double-hinged shallow arch is assumed to be subjected to a lateral load- ing consisting on a static loading and a periodic excitation. It is also subject to an imposed slow periodic motion of its support. The non-dimensional equa- tions of motion describing the evolution of the straightened amplitudes, of two sine functions taken as approximations of the two first modes of the shallow arch, can be written as

Q ¨ 1 + β 1 Q ˙ 1 + ( 1 + h cos (τ)) Q 1 + Q 1 ( Q ² ₁ − q ₀ ² + 4Q ² ₂ ) + q 0 − λ 0

+ρ cos (ν t ) = 0 (1)

Q ¨ 2 + β 2 Q ˙ 2 + 4 ( 4 + h cos (τ)) Q 2 + 4Q 2 ( Q ² ₁ − q ₀ ² + 4Q ² ₂ ) = 0 .

Here t is time and τ is a very slow time scale defined as τ = ε ⁿ t with the integer n ≥ 2. The variables Q 1 and Q 2 are the amplitudes of the symmetric and the asymmetric modes respectively. The parametric excitation h cos (τ) is due to the periodic motion of the end point of the arch. The parameters q 0 and λ 0

represents the non-dimensional initial rise and the static loading respectively.

The details of this derivation and the definition of non-dimensional variables and parameters can be found in [5], in which the externally excited version of equations (0) was studied.

Under the action of the static loading λ 0 alone, increasing the initial rise q ₀ , a

stable symmetric static solution bifurcates to three symmetric solutions. Two

among them are stable. In what follows we will deal only with the two stable

symmetric solutions. In all the numerical applications, q 0 = 2 . 5, λ 0 = 6 . 95 and

the damping coefficients β i = 0 . 03 ( i = 1 , 2 ) .

3. Perturbation Analysis

We perturb the variables (Q 1 , Q 2 ) in equations (0) around one of the two stable buckled symmetric solutions (η 0 , 0 )

Q 1 = η 0 + ε q 1 Q 2 = ε q 2 (2)

where ε is a small positive parameter. Let

ρ = ε ² ρ, ˜ h = ε h ˜ , and β i = ε β ˜ i , i = 1 , 2 . (3) Inserting (1) and (2) into equations (0) leads to the following nonlinear varia- tional equations up to O(ε ² )

¨

q 1 + ω ₁ ² q 1 = − ˜ hη 0 cos (τ) + ε(− ˜ β 1 q ˙ 1 − 3η 0 q ₁ ² − 4η 0 q ₂ ² − ˜ ρ cos (νt)

− ˜ h cos (τ) q 1 ), (4)

¨

q 2 + ω 2 ² q 2 = ε(− ˜ β 2 q ˙ 2 − 8 η 0 q 1 q 2 − 4 h ˜ cos (τ) q 2 ),

where ω ² 1 = 3 η ² 0 + 1 − q ₀ ² and ω ² 2 = 16 − 4q ₀ ² + 4 η 0 ² are the linearized frequencies corresponding to the first and second modes, respectively. We will restrict our analysis to 1 : 2 internal resonance and to the principal external resonance

ω 1 = 2 ω 2 + εσ 1 , ν = 2 ω 2 + εσ 2 , (5) where σ 1 and σ 2 are the detuning parameters. For an exhaustive review of the recent works on two-degree-of-freedom quadratic systems, under external or parametric excitations, we refer the reader to [6].

Reduced System

The MSM (see [7]) is applied to (3) to eliminate the fast time scale depen- dence. This process ultimately results in the following modulation equations of amplitudes a 1 , a 2 and phases of the first and second modes respectively

a ₁ = − β ˜ 1

2 a 1 − ρ ˜ 2 ω 1

sin (γ 1 ) − η 0

ω 1

a ₂ ² sin (γ 1 − 2γ 2 ) a 1 γ 1 = (σ 2 − σ 1 ) a 1 − η 0

ω 1

a ² ₂ cos (γ 1 − 2 γ 2 ) − ρ ˜ 2 ω 1

cos (γ 1 )

− h X ˜ 2 ω 1

a 1 cos (τ) a ₂ = − β ˜ 2

2 a 2 + 2 η 0

ω 2

a 1 a 2 sin (γ 1 − 2 γ 2 ) (6) γ 2 = σ 2

2 − 2 η 0

ω 2

a 1 cos (γ 1 − 2 γ 2 ) − 2 hY ˜ ω 2

cos (τ),

where X =1 −( 6 η ² 0 /ω ² 1 ) and Y =1 −( 2 η ² 0 /ω ² 1 ) . The prime denotes the derivative with respect to ε t. Here we have considered the slowly varying parametric excitation as constant during the averaging process.

The solutions of equations (3) are approximated up to O(ε) as follows

q 1 ( t ) = a 1 (τ) cos (ν t − γ 1 (τ)) − h ˜ η 0

ω ² 1

cos (τ) + O(ε), (7) q 2 ( t ) = a 2 (τ) cos ( ν

2 t − γ 2 (τ)) + O(ε). (8)

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

0 0.5 1 1.5 2 2.5

3 x 10 ⁻³

ν

ρ

A

B C D B

Figure 1. Boundaries of dy- namic instability of the mod- ulation equations (5) in the external excitation parameters plane (ν, ρ) for h = 0. Re- gion A, only single symmet- ric mode exists and is stable.

Region B, coexistence of cou- pled and single modes. Re- gion C, only coupled mode is stable. Region D both modes are unstable.

In figure 1., we show the stability chart of system (5) in the absence of the parametric excitation i.e., h = 0. In region A, only the symmetric mode ( a 1 , a 2 = 0 ) is excited and is stable. In region B, coexistence of the stable symmetric mode ( a 1 , 0 ) and two coupled modes ( a 1 , a 2 = 0 ) , one of them is stable. In region C, coexistence of the destabilized single mode and the stable coupled mode. In region D, destabilization of the coupled mode through a Hopf bifurcation. For more results about this case see Tien et. all [8].

In figure 2, we consider a point in the region A with a static displacement of the support of the arch i.e., = 0 and h = 0. For low and high values of h only the stable semi-trivial solution exists and for h ∈ [0 . 00118 , 0 . 005] there is coupling between the existing modes.

One interesting contribution of the slow parametric excitation i.e., h = 0 and

= 0 is that it enables the crossing of the borders between the different re-

gions in figure 1 during one period of τ . This effect is the main cause for the

occurrence of periodic bursters. See figure 3 for a burster solution Q 2 ( t ) of the

original system (0). It consists of a heteroclinic cycle between a fixed point

and a QP solution.

0 1 2 3 4 5 6 7 8 x 10 ⁻³ 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

h

a 1 , a 2

a2

a1

Figure 2. Stationary responses of (5) for ρ = 0.0007, ν = 1.29, = 0. Continuous lines refer to stable solutions and dashed lines to unstable ones.

1 1.5 2 2.5 3 3.5

x 10 ⁴

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03 0.04

t Q 2

Figure 3. Periodic burster solution Q ₂ ( t ) of the original equation (0) for ρ = 0 . 002 , ν = 1 . 25 , = 0 . 001 and h = 0 . 01.

QP solutions and bursters

Equations of modulations (5) can be writen as a slow-fast system

ε z ˙ = f ( z , τ), τ ˙ = 1 , (9)

where the vector z = ( a 1 , γ 1 , a 2 , γ 2 ) and the dot means the derivatives with respect to the very slow scale of time τ . In the limit ε → 0 one can compute the slow manifold given by M = { ( z , τ) : f ( z , τ) = 0 }. It is composed of two types of solutions

The semi-trivial solution

a 1 (τ) = ρ ˜

2 ω 1

[ ( ^{β ˜} ₂ ¹ ) ² + (σ 1 − σ 2 + ₂ ^{h X ˜} _{ω 1} cos (τ)) ² ]

, (10)

a 2 (τ) = 0 . (11)

The non-trivial solutions

a 1 (τ) = | ω 2

2 η 0 |

( β ˜ 2

2 ) ² + ( σ 2

2 − 2 hY ˜ ω 2

cos (τ)) ² . (12)

The two nontrivial amplitudes of the anti-symmetric mode a 2 (τ) are obtained

by solving a fourth order algebraic equation.

It is known through the singular perturbation theory that when the slow man- ifold M is stable, there exists a solution of the frozen system (8) tracking M at a distance of order ε and admitting asymptotic series in ε . Near bifurcating equilibrium branches the center manifold theorem can be applied to reduce the dimension of the system to the number of bifurcating eigenvalues, see for instance Bajaj and co-workers [9] and Berglund [10]. In figure 4., we show the stability chart of system (5) for h = 0 . 001 in the plane of external excita- tion parameters. The stability behaviors of the solutions in the zones A , B , C and D are the same as in figure 1. However, instead of fixed points we deal now with periodic solutions. In the grey zones these periodic solutions are changing their nature or stability during one period of τ . These zones are the zones of existence of periodic bursters solutions.

Using equations (1), (6) and (7) we conclude that away from the boundaries of dynamic instability, i.e., in regions A , B and C, the stable solutions of the initial system (0) are mainly QP. The amplitude of the symmetric mode Q 1 ( t ) is a QP solution with basic frequencies ν and . The amplitude of asymmetric mode Q 2 ( t ) is a trivial solution in the zones A and B . It is a QP solution in the zones B and C with basic frequencies ν/ 2 and . In region D, 3-period QP solution and non regular behaviors can be observed. In the grey zones, periodic bursters solutions exist and they alternate between the dynamics of zones delimiting them. As an example see figure 3.

In figure 5, we show for different values of parameters of the excitations, the comparisons between the results of MSM and the numerical simulations of equation (0). In every case, excellent agreements are shown. For the behavior of the solutions near the resonances boundaries (grey zones in figure 4.) and in the zone D, a more detailed investigation is in progress, and results will be reported elsewhere.

Figure 4. Behavior chart of equations (5) for h = 0.001.

The grey areas correspond to zones of existence of bursters.

The qualitative behaviors of

the lettered zones are the same

as in figure 1.

−1.536 −1.534 −1.532 −1.53 −1.528 −1.526 −1.524

−5 0 5

x 10 ⁻³

Q ₁ d Q 1 / d t

−1.536 −1.534 −1.532 −1.53 −1.528 −1.526 −1.524

−5 0 5

x 10 ⁻³

Q ₁ dQ 1 /dt

−0.015 −0.01 −0.005 0 0.005 0.01 0.015

−0.01

−0.005 0 0.005 0.01

Q ₂ d Q 2 / d t

−0.015 −0.01 −0.005 0 0.005 0.01 0.015

−0.01

−0.005 0 0.005 0.01

Q ₂ dQ 2 /dt

−1.538 −1.536 −1.534 −1.532 −1.53 −1.528 −1.526 −1.524

−5 0 5

x 10 ⁻³

Q ₁

−1.538 −1.536 −1.534 −1.532 −1.53 −1.528 −1.526 −1.524

−5 0 5

x 10 ⁻³

Q ₁ dQ 1 /dt

−0.02 −0.01 0 0.01 0.02

−0.01

−0.005 0 0.005 0.01

Q ₂ dQ 2 /dt

−0.02 −0.01 0 0.01 0.02

−0.01

−0.005 0 0.005 0.01

Q ₂ dQ 2 /dt

dQ 1 /dt

(a)

(b)

(c)

(d)

(a’)

(b’)

(c’)

(d’)

Figure 5. Comparisons of the approximated solutions and numerics of equation (0), for ν =1 . 25 and = 0 . 06. The right figures refer to the computer generated solutions and the left ones to the analytical solutions. The plots (a) (resp. (a’)) and (c) (resp. (c’)) correspond to a point in the zone A with ρ = 0 . 00065 and h = 0 and h = 0 . 001 respectively. The plots (b) (resp. (b’)) and (d) (resp. (d’)) correspond to a point in region C for ρ = 0.0015 and h = 0 and h = 0.001 respectively.

excitation. The dynamic on the slow manifold is studied after a reduction through the MSM. Behavior charts and explicit analytical approximations of solutions are computed and comparisons to numerical integration are pro- vided.

It is worth noting that for high amplitudes of the parametric excitation h the dynamics of the arch leaves the neighbourhood of the considered symmetric buckled position to oscillate around the other one. A global analysis is re- quired to understand this jump.

4. Summary

In this paper, we have investigated QP solutions and periodic bursters of a

shallow arch subject to a very slow parametric excitation and a fast external

Acknowledgments

The first author (F.L) would like to acknowledge that this research is fi- nanced by the Alexander von Humboldt Foundation. He also acknowledges the hospitality of Institut polytechnique priv´e de Casablanca.

References

[1] N.N. Bogoliubov and Y.A. Mitropolskii, Asymptotic Methods in the Theory of Non- linear Oscillations. Delhi: Hindustan Publ. Corp, 1961.

[2] K. Guennoun, M. Houssni and M. Belhaq, “Quasi-periodic solutions and stability for a weakly damped nonlinear quasi-periodic Mathieu equation,” Nonlinear Dynamics 27, 211-236, 2002.

[3] M. Belhaq, K. Guennoun and M. Houssni, “Asymptotic solutions for a damped non- linear quasi-periodic Mathieu equation,” Int. J. of Non-linear Mechanics 37, 445- 460, 2002.

[4] J. Kevorkian and J.D. Cole, Multiple Scale and Singular Perturbation Methods.

Berlin: Springer-Verlag, 1996.

[5] W.M. Tien, Chaotic and stochastic dynamics of nonlinear structural systems. Phd thesis in University of Illinois at Urbana-Champaign, 1993.

[6] A.H. Nayfeh, Nonlinear Interactions, Wiley , New York, 2000.

[7] A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973.

[8] W-M. Tien, N.S. Namachchivaya and A.K. Bajaj, “Nonlinear dynamics of a shal- low arch under periodic excitation-I. 1:2 internal resonance,” Int. J. of Non-linear Mechanics 29(3), 349-366, 1994.

[9] A.K. Bajaj, B. Banerjee and P. Davies, “Nonstationary Responses in two degree-of- freedom nonlinear systems with 1-to-2 internal resonance,” Proceedings of IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics in Mechanics, 13-22, Kluwer Academic, 1999.

[10] N. Berglund, “Dynamic bifurcations: hysteresis, scaling laws and feedback control,”

Proc. Theor. Phys. Suppl. 139, 325-336, 2000.