Chaos theory has spectacularly evolved since the pioneering work by E. Lorenz on chaotic motion in a simple, deterministic system. Since then, the chaotic behavior of many other deterministic, low-dimensional systems in a large variety of ﬁelds has been developed. In the particular ﬁeld of aeroelasticity of aircraft structures several reports of chaos have been documented. However, we are unaware of any report of chaotic systems of civil (non-aeronautical) use induced by an aeroelastic phenomenon. In this paper a well deﬁned civil, aeroelastic system, susceptible to exhibit chaotic behavior is presented. The system consists of a buckled beam from which a second beam is suspended. This last beam (hereafter, galloping beam) has a square cross- section and can undergo transverse galloping. The system is subjected to an uniform wind ﬂow and, as it will be shown in the paper, for wind velocities larger than a threshold value, the galloping beam begins to oscillate and induces, for a determined set of parameter values, a chaotic motion in the buckled bea

Alonso Rodrigo, Gustavo

Barrero Gil, Antonio

English

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

BBAA VI International Colloquium on:
Bluff Bodies Aerodynamics & Applications
Milano, Italy, July, 20–24 2008
CHAOTIC VIBRATIONS IN A BUCKLED BEAM INDUCED BY A
GALLOPING PHENOMENON
Barrero-Gil, A.?, Alonso, G.
?IDR/UPM, ETSI Aeronauticos, Universidad Politecnica Madrid, Pza. Cardenal Cisneros s/n,
28040 Madrid, Spain
e-mail: antonio.barrero@upm.es
Keywords: Aeroelasticity, Transverse galloping, Chaotic behavior
ABSTRACT: Chaos theory has spectacularly evolved since the pioneering work by E. Lorenz
on chaotic motion in a simple, deterministic system. Since then, the chaotic behavior of many
other deterministic, low-dimensional systems in a large variety of fields has been developed. In
the particular field of aeroelasticity of aircraft structures several reports of chaos have been do-
cumented. However, we are unaware of any report of chaotic systems of civil (non-aeronautical)
use induced by an aeroelastic phenomenon. In this paper a well defined civil, aeroelastic sys-
tem, susceptible to exhibit chaotic behavior is presented. The system consists of a buckled beam
from which a second beam is suspended. This last beam (hereafter, galloping beam) has a
square cross- section and can undergo transverse galloping. The system is subjected to an uni-
form wind flow and, as it will be shown in the paper, for wind velocities larger than a threshold
value, the galloping beam begins to oscillate and induces, for a determined set of parameter
values, a chaotic motion in the buckled beam.
Mathematical model
Let us consider a square cross-section beam (susceptible to transverse galloping) suspended
from a buckled beam (it is not difficult to imagine a practical situation where, for example, the
origin of the compression load could be due to thermal effects) under the action of an airstream
with uniform velocity. Both beams are restricted to oscillate perpendicular to the airstream di-
rection (Fig. 1). For the sake of simplicity, we assume that the vibrations of the system are
mainly dominated by the lowest mode according to each degree of freedom. Thus, the system
is analysed using a simplified model with two degrees of freedom that represents the balance
of inertia, damping, restoring and aerodynamic forces. Quasi-steady conditions are assumed to
model aerodynamic forces where we retain only the first nonlinear aerodynamic term. Further-
more, a simplified model of a buckled beam has been used [1]. In a non-dimensional form, the
dynamics of the system is represented by the following non-linear differential system:
η¨1 + c11η˙1 − c12(1 + c13)η1 + c14η31 + c12c13η2 = 0, (1a)
η¨2 + c15η˙2 + c16(1 + c17)η2 − c16c17η1 = c18(Ura1η˙2 + a3
Ur
η˙2
3), (1b)
1
Barrero-Gil, A. Alonso, G.
where,
η = y/D; τ = ω2t; c11 = c1m1ω2 −
ρU2Da
2m1ω2
; c12 = − k1m1ω22 ; c13 =
k12
k1
; c14 = k13D
2
m1ω22
;
c15 =
c2
m2ω2
; c16 = k2m2ω22 = 1; c17 = k12/k2; c18 =
ρD2
2m2
; Ur = U/ω2D; (·) = d/dτ .
and a1, a3 are, respectively, the first and third order coefficients of aeroelastic force (a1 = 2.7
and a3 = −168 for a square section [2]).
Figure 1: Graphical representation of the physical system and schematic model as a coupled oscillator.
The set (1) consists of two coupled non-linear and ordinary autonomous differential equa-
tions. Neither the system properties nor the forces action on the beams (we do not consider
turbulence in the airstream) vary randomly with time and, consequently, the system is fully
deterministic. If the nonlinearities appearing in the equations are strong, then a scenario of de-
terministic chaotic motion could appear as a consequence of the parameter values in the system.
Direct inspection of equations (1) does not lead to an easy, qualitative prediction on whether
the system can turn chaotic. However, for values of the coupling parameter c17 small compa-
red with the stiffness of the galloping beam, c16, equations 1a and 1b decouples. Under this
assumption, η2 in Eq. 1a becomes a forcing term with a known time dependence. This Eq. 1a
becomes the extensively studied Duffing-Holmes equation that exhibits, as it is well known, a
chaotic behavior within a certain parameter range.
Numerical results, bifurcation diagram and chaotic behavior
In equations 1a and 1b there are several parameters characterizing the dynamic behavior. The
high dimension of space parameter leads to the practical impossibility of complete parameter
study. Thus, the study must restrict to a certain set of the most important physical variables in
the problem. The most interesting one is to keep the system characteristics (e.g, mass, structural
damping and stiffness) fixed and take the external actions (velocity of airstream and compressive
axial load) as control parameters. To make the analysis as transparent as possible is analysed
first the dynamic of the system with velocity of airstream as the only control parameter. Finally,
a study with the two control parameters is presented.
Bifurcation diagram with Ur as control parameter
The system under study is in a slowly evolving environment, so its coefficient Ur experiments
a gradual change. In nonlinear systems, behavior can change very suddenly and, depending of
2
Barrero-Gil, A. Alonso, G.
the velocity of the wind, the buckled beam can be found in a state of static equilibrium, periodic
oscillations around the equilibrium state or chaotic oscillations. Based on the precedent dis-
cussion, the original system (equations 1a and 1b) has been integrated with fixed values of the
parameters. The parameters taken are c11 = 0.31, c12 = 1.5, c13 = −2/3, c14 = 0.5, c15 = 0.01,
c16 = 1, c17 = 0.001, c18 = 0.001 and the wind parameter, Ur, is taken as the control parameter
and bifurcation diagram has been computed in order to achieve the possible behaviors as the
reduced velocity varies. In figure 2 the result obtained with a standard numerical Runge-Kutta
algorithm is plotted. Four distinct regions according to the value of reduced velocity can be
discern: (i) Ur < 3.7 the instability of galloping has not begun yet and the buckled beam re-
mains in the static equilibrium position, (ii) 3.7 < Ur < 4.4 the galloping beam is oscillating
and induces a periodic vibration around the static equilibrium position in the buckled beam, (iii)
4.4 < Ur < 6.6 the oscillations in the buckled beam turn erratic and chaotic and (iv) Ur > 6.6
vibrations in the buckled beam recover a periodic characteristic.
2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
U
r
d
Figure 2: Bifurcation diagram for system with Ur as a control parameter and d = (η21 + η˙
2
1)
1/2
From figure 2 it is clear that region (iii) can be considered as chaotic, although for values of
reduced velocity between 4.8 and 5.4 the motion is not totally chaotic (numerical simulations
in this region shows periods of chaos coexisting with periods with periodic motion). In figure
3 time series generated by numerical simulation after a considerable number of transient cycles
and the corresponding phase projection for Ur = 6 are shown. The top part of this figure shows
the response of the buckled beam and the lower part the response of the galloping beam. From
the figure can be observed that the buckled beam is oscillating in a somewhat erratic manner
about, and between, the two static equilibrium positions, located at η1 = ±1.
Analysis of chaotic regions with the wind velocity, Ur, and the compressive load, c12, as control
parameters
To investigate the possibility of chaotic motion a numerical study for the parameter of compres-
sive load ranging from 0 to 5 and Ur varying from 0 to 12 has been carried out. Principal lower
1We consider that this parameter is dominated by structural damping and aerodynamic damping is a second
order effect.
3
Barrero-Gil, A. Alonso, G.
1
h
t
2
h
2
h
&
t
2
h
1
h
&
1
h
Figure 3: Numerical time series describing the position of the buckled beam, η1, and the galloping beam, η2 for
Ur = 6.
and upper Ur chaos boundaries were found for a discrete series of c12. All results shown are for
simulations started from the initial values of η1 = 1, η˙1 = 0, η2 = 0.03, η˙2 = 0, t = 0. From
the figure can be observed that for no airflow or for no compressive load chaos does not oc-
cur. However, with both compressive load and airflow chaotic motion may occur as determined
from numerical simulations. The higher the applied compressive load the greater wind velocity
is necessary to the occurrence of chaos, and this takes place in a narrower range of reduced
velocity. By contrast, when the compressive load is negative but moderate chaos would occur
more easily.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
4
5
6
7
8
9
10
11
12
c12
U
r
 
 
CHAOS
Figure 4: Sketch of chaotic region in the (c12, Ur) parameter space.
REFERENCES
[1] Tseng, W. Y. & Dugundji, J. Nonlinear vibrations of a buckled beam under harmonic vi-
brations. Translactions of the American Society of Mechanical Engineering Journal of Applied
Mechanics, 38:467-476, 1971.
[2] Blevins, R. D. Flow-Induced Vibration, Krieger publishing company, New york, 1990.
4


Chaotic Vibrations in a Buckled Beam Induced by a Galloping Phenomenon

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