Example of a non-smooth Hopf bifurcation in an aero-elastic system

TL;DR

This paper studies how non-smooth aerodynamic loads in aero-elastic systems can cause non-smooth Hopf bifurcations, leading to complex flutter behaviors, using a combination of numerical integration and dynamical systems analysis.

Contribution

It demonstrates the occurrence of non-smooth Hopf bifurcations due to non-linear, non-smooth aerodynamic loads in aero-elastic models, a novel insight into flutter dynamics.

Findings

Non-smooth bifurcation can generate limit cycle attractors.

Dynamic stall model's non-smoothness influences flutter behavior.

Bifurcation diagrams reveal complex oscillation patterns.

Abstract

We investigate a typical aerofoil section under dynamic stall conditions, the structural model is linear and the aerodynamic loading is represented by the Leishman-Beddoes semi-empirical dynamic stall model. The loads given by this model are non-linear and non-smooth, therefore we have integrated the equation of motion using a Runge-Kutta-Fehlberg algorithm equipped with event detection. The main focus of the paper is on the interaction between the Hopf bifurcation typical of aero-elastic systems, which causes flutter oscillations, and the discontinuous definition of the stall model. The paper shows how the non-smooth definition of the dynamic stall model can generate a non-smooth Hopf bifurcation. The mechanisms for the appearance of limit cycle attractors are described by using standard tools of the theory of dynamical systems such as phase plots and bifurcation diagrams.