scholarly journals Control of Limit Cycle Oscillations of a Two-Dimensional Aeroelastic System

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
M. Ghommem ◽  
A. H. Nayfeh ◽  
M. R. Hajj
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycle Oscillations ◽  
Feedback Controls ◽  
Aeroelastic System ◽  
Freestream Velocity ◽  
Nonlinear Springs ◽  
Subcritical Hopf Bifurcation ◽  
Static Feedback ◽  
Nonlinear Static

Linear and nonlinear static feedback controls are implemented on a nonlinear aeroelastic system that consists of a rigid airfoil supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. The normal form is used to investigate the Hopf bifurcation that occurs as the freestream velocity is increased and to analytically predict the amplitude and frequency of the ensuing limit cycle oscillations (LCO). It is shown that linear control can be used to delay the flutter onset and reduce the LCO amplitude. Yet, its required gains remain a function of the speed. On the other hand, nonlinear control can be effciently implemented to convert any subcritical Hopf bifurcation into a supercritical one and to significantly reduce the LCO amplitude.

scholarly journals Rendering a subcritical Hopf bifurcation supercritical

1996 ◽  
Vol 317 ◽  
pp. 91-109 ◽  
Author(s):  
Po Ki Yuen ◽  
Haim H. Bau
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Thermal Convection ◽  
Control Strategy ◽  
Chaotic Motion ◽  
Feedback Controller ◽  
Nonlinear Feedback ◽  
Naturally Occurring ◽  
Subcritical Hopf Bifurcation ◽  
Stable Periodic

It is demonstrated experimentally and theoretically that through the use of a nonlinear feedback controller, one can render a subcritical Hopf bifurcation supercritical and thus dramatically modify the nature of the flow in a thermal convection loop heated from below and cooled from above. In particular, we show that the controller can replace the naturally occurring chaotic motion with a stable, periodic limit cycle. The control strategy consists of sensing the deviation of fluid temperatures from desired values at a number of locations inside the loop and then altering the wall heating to counteract such deviations.

scholarly journals Uncertainty Quantification and Bifurcation Analysis of an Airfoil with Multiple Nonlinearities

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Haitao Liao
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance Method ◽  
Inequality Constraints ◽  
Maximization Problem ◽  
Limit Cycle Oscillation ◽  
Structural Parameter ◽  
Limit Cycle Oscillations ◽  
Aeroelastic System ◽  
Equality And Inequality Constraints ◽  
The Stability

In order to calculate the limit cycle oscillations and bifurcations of nonlinear aeroelastic system, the problem of finding periodic solutions with maximum vibration amplitude is transformed into a nonlinear optimization problem. An algebraic system of equations obtained by the harmonic balance method and the stability condition derived from the Floquet theory are used to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the proposed approach is validated, and the effects of structural parameter uncertainty on the limit cycle oscillations and bifurcations of an airfoil with multiple nonlinearities are studied. Numerical examples show that the coexistence of multiple nonlinearities may lead to low amplitude limit cycle oscillation.

scholarly journals Limit Cycles in a Model of Olfactory Sensory Neurons

2019 ◽  
Vol 29 (03) ◽  
pp. 1950038 ◽  
Author(s):  
Yonghui Xia ◽  
Mateja Grašič ◽  
Wentao Huang ◽  
Valery G. Romanovski
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Sensory Neurons ◽  
Limit Cycles ◽  
Center Manifold ◽  
Calcium Oscillations ◽  
Olfactory Sensory Neurons ◽  
Generalized Hopf Bifurcation ◽  
Subcritical Hopf Bifurcation ◽  
Unstable Cycle

We propose an approach to study small limit cycle bifurcations on a center manifold in analytic or smooth systems depending on parameters. We then apply it to the investigation of limit cycle bifurcations in a model of calcium oscillations in the cilia of olfactory sensory neurons and show that it can have two limit cycles: a stable cycle appearing after a Bautin (generalized Hopf) bifurcation and an unstable cycle appearing after a subcritical Hopf bifurcation.

scholarly journals Wiener–Askey and Wiener–Haar Expansions for the Analysis and Prediction of Limit Cycle Oscillations in Uncertain Nonlinear Dynamic Friction Systems

2013 ◽  
Vol 9 (2) ◽  
Author(s):  
Lyes Nechak ◽  
Sébastien Berger ◽  
Evelyne Aubry
Keyword(s):  
Hopf Bifurcation ◽  
Friction Coefficient ◽  
Limit Cycle ◽  
Bifurcation Point ◽  
Nonlinear Dynamic ◽  
Polynomial Chaos ◽  
Dynamic Friction ◽  
Limit Cycle Oscillations ◽  
Hopf Bifurcation Point ◽  
Modeling And Prediction

This paper is devoted to the robust modeling and prediction of limit cycle oscillations in nonlinear dynamic friction systems with a random friction coefficient. In recent studies, the Wiener–Askey and Wiener–Haar expansions have been proposed to deal with these problems with great efficiency. In these studies, the random dispersion of the friction coefficient is always considered within intervals near the Hopf bifurcation point. However, it is well known that friction induced vibrations—with respect to the distance of the friction dispersion interval to the Hopf bifurcation point—have different properties in terms of tansient, frequency and amplitudes. So, the main objective of this study is to analyze the capabilities of the Wiener–Askey (general polynomial chaos, multielement generalized polynomial chaos) and Wiener–Haar expansions to be efficient in the modeling and prediction of limit cycle oscillations independently of the location of the instability zone with respect to the Hopf bifurcation point.

An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system

2012 ◽  
Vol 71 (1-2) ◽  
pp. 159-173 ◽  
Author(s):  
Abdessattar Abdelkefi ◽  
Rui Vasconcellos ◽  
Ali H. Nayfeh ◽  
Muhammad R. Hajj
Keyword(s):  
Experimental Investigation ◽  
Limit Cycle ◽  
Limit Cycle Oscillations ◽  
Aeroelastic System

scholarly journals Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems

2013 ◽  
Vol 371 (1999) ◽  
pp. 20120467 ◽  
Author(s):  
C. M. Postlethwaite ◽  
G. Brown ◽  
M. Silber
Keyword(s):  
Hopf Bifurcation ◽  
Unstable Periodic Orbits ◽  
Feedback Controls ◽  
Bifurcation Theorem ◽  
Non Invasive ◽  
Subcritical Hopf Bifurcation ◽  
Bifurcation Problems ◽  
Spatio Temporal ◽  
The Stability ◽  
Equivariant Hopf Bifurcation

Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.

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