Model Reduction of Nonlinear Aeroelastic Systems Experiencing Hopf Bifurcation

Abdessattar Abdelkefi,Mehdi Ghommem

doi:10.7726/jmsic.2013.1005

Abdessattar Abdelkefi, Mehdi Ghommem

Open Access

https://doi.org/10.7726/jmsic.2013.1005

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Abstract

In this paper, we employ the normal form to derive a reduced-order model that reproduces nonlinear dynamical behavior of aeroelastic systems that undergo Hopf bifurcation. As an example, we consider a rigid two-dimensional airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. We apply the center manifold theorem on the governing equations to derive its normal form that constitutes a simplified representation of the aeroelastic system near flutter onset (manifestation of Hopf bifurcation). Then, we use the normal form to identify a self-excited oscillator governed by a time-delay ordinary differential equation that approximates the dynamical behavior while reducing the dimension of the original system. Results obtained from this oscillator show a great capability to predict properly limit cycle oscillations that take place beyond and above flutter as compared with the original aeroelastic system.

Highlights

The need to model, capture and predict properly the nonlinear dynamics, such as limit cycle oscillations, bifurcations, chaos, ... (Dowell et al, 2003; Ghommem et al, 2010a; Ghommem et al, 2012; Abdelkefi et al, 2012a,b; Vasconcellos et al, 2012) associated with aeroelastic systems may lead to large and complex models (Beran and Silva, 2004; Beran et al, 2004)
proper orthogonal decomposition (POD) constitutes a common technique for extracting the coherent structures from a linear or nonlinear dynamical process
We use this form to come up with a simplified representation based on time-delay differential equation to reproduce its nonlinear dynamical behavior

Summary

Introduction

The need to model, capture and predict properly the nonlinear dynamics, such as limit cycle oscillations, bifurcations, chaos, ... (Dowell et al, 2003; Ghommem et al, 2010a; Ghommem et al, 2012; Abdelkefi et al, 2012a,b; Vasconcellos et al, 2012) associated with aeroelastic systems may lead to large and complex models (Beran and Silva, 2004; Beran et al, 2004). We follow a different approach based on observing a particular behavior or phenomenon from data obtained from experiments or numerical simulations and describing them with existing models such as self-excited oscillators Towards this end, we employ perturbations techniques to derive the normal form of the Hopf bifurcation of these aeroelastic systems. We consider a rigid two-dimensional airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads and apply the center manifold theorem on the governing equations to derive its normal form We use this form to come up with a simplified representation based on time-delay differential equation to reproduce its nonlinear dynamical behavior. The emphasis of our study is to predict properly the type of dynamic instability (Hopf bifurcation) and reproduce limit cycle oscillations (LCOs) that take place beyond flutter onset

Modeling of the aeroelastic system

Normal form of the Hopf bifurcation of the aeroelastic system

Time-delay representation

10 N m 15 N m 15 N m 30 N m μ

Conclusions