• Aucun résultat trouvé

16 Beyond the tip of the parametric instability tongue

N/A
N/A
Protected

Academic year: 2022

Partager "16 Beyond the tip of the parametric instability tongue"

Copied!
1
0
0

Texte intégral

(1)

rencontre du non-lin´eaire 2021 1

Beyond the tip of the parametric instability tongue

Alvaro A. Grandi1, Suzie Proti`ere1& Arnaud Lazarus1

Institut Jean Le Rond d’Alembert, CNRS UMR7190, Sorbonne Universit´e Paris, France alvaro.anzoleaga grandi@sorbonne-universite.fr

Parametric instabilities are dynamical instabilities which can arise when the mechanical state of a structure is periodically modulated in time. It is often described as a phenomenon one wishes to avoid:

for example the parametric rolling observed on sailing ships and is also exploited to study vibrating fluids (Faraday waves [1]) or Nano-Electro- Mechanical Systems [2]. One well-known limitation when fully exploiting classic parametric instabilities based on small periodic modulation of a mechanical state is that inherent friction forces rapidly cancel sub harmonic parametric resonances. To overcome this drawback, we propose to periodically vary the evolution function of a dynamical system to enhance and better control parametric instabilities.

Figure 1. Triggering extremely high orders of parametric resonances. a) Experimental Floquet oscillator: pen- dulum with a metallic marble symmetrically placed between two identical attracting electromagnets whose force depends on the imposed electrical currentI(see sketch in inset). The scalarω2(I), characterize the local evolution function of the electromagnetic pendulum. The electromagnets are periodically turned on and off in a square wave fashion. b) Extended stability diagram of the Meissner equation [3] showing the evolution of the Floquet exponent up to 0.4 in the parameter space. Grey dots (m, n) represent the discrete solutions using our geometrical relation. Green crosses represent our experimental results. Inset zooms on the classic first instability regions.

By combining theoretical models and desktop experiments, we have developed a geometrical relation to trigger high order parametric resonances (Grey dots in Fig 1.b)). As a proof of concept, we study the motion of an electromagnetic pendulum (Fig 1.a) and we are able to observe extremely high orders of parametric resonance which we can control, even in the presence of dissipation (Green crosses in Fig1.b)).

These concepts being universal, we can then think of new promising dynamic functionalities which could be applied to dynamical systems with a natural cycle which time scale could be periodically varied.

References

1. S. Douady, Experimental study of the Faraday instability,J. Fluid Mech,221, pp383-409 (1990).

2. Y. Jia, S. Du & A.A. Seshia, Twenty-Eight Orders of Parametric Resonance in a Microelectromechanical Device for Multi-band Vibration Energy Harvesting,Scientific Reports,6, 30167 (2016).

3. John A Richards, Analysis of time-varying systems,Springer Science and Business Media(2012).

c Non Lin´eaire Publications, Avenue de l’Universit´e, BP 12, 76801 Saint- ´Etienne du Rouvray cedex

Références

Documents relatifs

Road stiffness influence on rolling noise: Parametric study using a rolling tire

Three-mode parametric interactions occur in triply resonant optomechanical systems: Photons from an optical pump mode are coherently scattered to a high-order mode by mechanical

the analogy by comparing time and frequency domain measurements of the mechanical response in the non-linear regime and under the inuence of external, gate-controlled frequency

For a given parametric representation of a curve, PTOPO computes the special points on the curve, the characteristic box, the corresponding graph, and then it visualizes the

Im Bezug der Lernstrategien, und -techniken lässt sich die Meinung von Rampillon und Bimmel 2000 zitieren, die der Meinung sind, dass die Lernstrategien die wichtigste

In addition to the parameters usually considered, like the geometry of the mean flow [4, 5] or the magnetic boundary conditions [6, 7], the turbulent fluctuations of the flow seem

Pour cela, nous présenterons également, dans les deux premiers chapitres de cette thèse, la mise au point des réactions de condensation de Hantzsch et de Biginelli en décrivant

Le diagramme de bande énergétique tracé indique que le premier semiconducteur (l’InN) est en accumulation et le deuxième semiconducteur (l’InP) est en désertion. Nous avons