Supervised by Dr David A. W. BartonDepartment of Engineering Mathematics

Objectives and Motivation

Aerofoil Model

Analysis

Control-based Continuation

Results and Conclusions

Jiří Klein

To model flutter - the nonlinear and potentially destructive behaviour of an aerofoil.To determine the onset of flutter, diminish the effects of it, and to manage it with appropriate control design. The control of flutter is increasingly important with state-of-the-art aircraft with high aspect ratio wings, such as the Boeing SUGAR Volt Concept - while this design has a decreased induced drag, it is more susceptible to flutter.

The model is designed as a set of 2 DOF equations (pitch and heave), similar to a mass-spring-damper system with external forces. Our extended model has a third DOF in the form of a trailing edge flap, acting as a control surface.Our model can be numerically fitted with appropriate parameters and used to predict the behaviour of an aerofoil in a wind tunnel experiment.

Control-based Continuation (CBC) is a technique for tracking the solutions and bifurcations of nonlinear experiments. The idea is to apply the method of numerical continuation to a feedback-controlled physical experiment, such that the control becomes non-invasive. The PID controller is extended to enable behaviour in the unstable regions of the system. This eventually allows us to show the existence of, for example, Unstable Periodic Orbits of a subcritical Hopf bifurcation in a wind tunnel experiment. This is achieved by replacing the d0 parameter with an appropriate function of time.

To find the point of linear instability - flutter speed Uf.To investigate what bifurcations occur at and beyond Uf.Achieved by both eigenvalue analysis and by numerical continuation tools.

The model is able to fully reproduce experimentally verified behaviour, undergoing a Hopf bifurcation at Uf.The analysis revealed which parameters change the quality of the behaviour, such as the criticality of bifurcations.The controlled system succesfully eradicates early onset of flutter.Finally, the model is capable of simulating Control-based continuation.

The study suggests methods for an innovative wind tunnel experiment - one that is able to physically show e.g. Unstable Periodic Orbits

Control StrategiesThe 3 DOF system is reduced to 2 DOF (removing 1 EOM) to allow for control of the trailing edge flap with a well-designed PID controller. While very sensitive to tuning, this method is successful at diminishingthe effects of flutter.

Boeing SUGAR Volt Concept (High AR wings)

-6 -4 -2 0 2 4 6Heave [m] #10-3

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

Pit

ch [

rad]

Phase Portrait of the Controlled behaviour, kp

= 1

Simulation Run of 50 SecondsOne Period of the Unstable PO

-5 0 5Heave [m] #10-3

-0.06

-0.04

-0.02

0

0.02

Pit

ch [

rad]

Phase Portrait of the Controlled behaviour, kp

= 5

Simulation Run of 50 SecondsOne Period of the Unstable PO

CBC DesignThe behaviour of the aerofoil is illustrated with the following figures - a phase portrait and a numerical continuation plot

θ(t)=kp×(d0−α(t))

Aerofoil Expetiment

Sensors

Actuators

Fourier Series Estimator

P Control Noise Filter

Real-time Controller

x(t)

kp

w(t), p(t)ϕ(t) -

--c = 2b

b

hahb

α

Elastic axis

Displaced aerofoil

Mean position of the aerofoil

q1 = h

q2 = α

0 5 10 15 20 25 30U

-0.3

-0.2

-0.1

0

0.1

0.2

Pit

ch [

rad]

Numerical continuation in ζ2, U

f

Stable RegionUnstable Region

q3 = θ

M q̈(t)+Cq̇(t)+Kq(t)+N (q(t))=Cq(t)

-0.05 0 0.05Heave [m]

-0.15

-0.1

-0.05

0

0.05

0.1

Pit

ch [

rad]

Phase Portrait of the Unforced Behaviour, U = 15

Tracking the Onset of Flutter in an Aerofoil