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Bistability of a Pendulum in a Flow

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Abstract

Stochastic aspects of the transition between stable positions of a pendulum confronted with an air flow.
Bistability of a Pendulum in a Flow
Stochastic dynamics of transitions between stable fixed points
A. Gayout, N. Plihon, M. Bourgoin
Laboratoire de Physique `
a l’ ´
Ecole Normale Sup´
erieure de Lyon - UMR 5672
1 - Introduction
The dynamics of a simple pendulum in a wind tunnel flow, of mean velocity U, exhibit a complex hysteritic behavior [Obligado et al. 2013]. This
hysteretic behavior finds its origin in the dependence of the Lift (
L) and Drag (
D) coefficient of a flat obstacle (a disk of center D, diameter 4 cm) with
the angle θbetween the flow and the normal to the obstacle [Flachsbart 1932].
G
O
D
g
θ
U
N=
D+
L
L
D
J¨
θ=mgOGsin(θ)ksgn( ˙
θ)+1
2ρSCN(θ)U2OD
Aerodynamical coefficients for a disk pendulum at different angles.
Flachsbart, 1932.
Static hysteretic cycle done experimentally with attraction
bassins of upper and lower equilibrium positions.
2 - Double-potential Well
Etot =Ek(˙
θ)+Eptot(θ,U) = 1
2J˙
θ2mgOGcos(θ)+1
2ρSU2ODZπ/2
θCN(α)dα
There are 3 ways of escaping
one potential well to fall in the
other:
˙
θ2(from the kinetic energy),
U2(natural dispersion of U),
θ2(variations of θ).
4 - Arrhenius law: τ=τ0exp(∆E(θ)/kT )
Thermal noise on the pendulum can result from either U2or θ2, since
˙
θ2is coupled with θ2by the pendulum dynamics.
Characteristic time τ0is the period of the pendulum τ0= 2πrJ
mg = 0.94 s.
The relevant noise seems to be θ2, which leads to the conclusion that
the pendulum interaction with the flow is the most important part of the
phenomenon of spontaneous transition.
6 - Conclusion
When decorrelated from U2, linked to ambient turbulence,θ2is related
to vortex shedding. Even if θ2is relevant for the transition time, it cannot
alone explain all excursions, like type B excursions where the pendulum
relaxes with natural oscillations.
U
Ambient
turbulence
U
Vortex shedding
3 - Escape-time
If θ0is set close to the Stall angle, spontaneous transitions occur from the
Drag to the Lift branch. Statistics of escape-time are then systematically
measured. They follow an Exponential distribution, depending of θ0.
5 - Dynamics of a single transition
The spontaneous transitions cannot be explained by the static CN(θ)model.
In the waiting time before the transition, some excursions are observed.
Due to the limitations of the static model, reconstructed aerodynamical
torque Γaero =J¨
θ+mgOG sin(θ) + ksgn( ˙
θ)is employed to understand
these phenomena.
There are 2 types of excursions:
Excursion type A,t'0.10.2 s < τ0, with sharp V-signature on Γaero.
Excursion type B,t'0.51 s 'τ0, with less strong asymmetric Γaero.
M. Obligado, M. Puy, and M. Bourgoin, J. Fluid Mech.,728, R2 (2013).
O. Flachsbart, Messungen an ebenen und gew¨
olbten Platten (1932).
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