This paper deals with the problem of the aeroelastic stability of a typical aerofoil section with two degrees of freedom induced by the unsteady aerodynamic loads. A method is presented to model the unsteady lift and pitching moment acting on a two-dimensional typical aerofoil section, operating under attached flow conditions in an incompressible flow. Starting from suitable generalisations and approximations to aerodynamic indicial functions, the unsteady loads due to an arbitrary forcing are represented in a state-space form. From the resulting equations of motion, the flutter speed is computed through stability analysis of a linear state-space system.
Flutter is the dynamic aeroelasticity phenomenon whereby the inertia forces can modify the behaviour of a flexible system so that energy is extracted from the incoming flow. The flutter or critical speed
Theodorsen [
The main objective of this paper is to investigate the aeroelastic stability of a typical aerofoil section with two degrees of freedom induced by the unsteady aerodynamic loads defined by the Leishman’s state-space model.
The mechanical model under investigation is a two-dimensional typical aerofoil section in a horizontal flow of undisturbed speed
A typical aerofoil section with two degrees of freedom.
The equations of motion for the typical aerofoil section have been derived in many textbooks of aeroelasticity and can be expressed in nondimensional form as
The first term in (
The indicial response method is the response of the aerodynamic flowfield to a step change in a set of defined boundary conditions such as a step change in aerofoil angle of attack, in pitch rate about some axis, or in a control surface deflection (such as a tab of flap). If the indicial aerodynamic responses can be determined, then the unsteady aerodynamic loads due to arbitrary changes in angle of attack can be obtained through the superposition of indicial aerodynamic responses using the Duhamel’s integral.
Assuming two-dimensional incompressible potential flow over a thin aerofoil, the circulatory terms in (
One exponential approximation is given by Jones [
The state-space equations describing the unsteady aerodynamics of the typical aerofoil section with two degrees of freedom can be obtained by direct application of Laplace transforms to the indicial response as
The main benefit of the state-space formulation is that the equations can be appended to the equations of motion directly, very useful in aeroservoelastic analysis. Furthermore, it permits the straightforward addition of more features to the model, such as gust response and compressibility.
The indicial approach and the state-space formulation lead to a dynamic matrix that governs the behaviour of the system and enables future prediction. The analysis of flutter in this case is straightforward and it can be performed in the frequency domain, since the eigenvalues of the dynamic matrix directly determine the stability of the system. If, for a given velocity, any of the eigenvalues has a zero real part, the system is neutrally stable, that is, it defines the flutter onset.
In this section, the stability analysis of the state-space aeroelastic equation is presented. The results have been validated against published and experimental results.
Theodorsen and Garrick [
Aeroelastic parameters for the validation.
Case |
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|
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a | 0.2 | 1/3 | −0.4 | 0.25 |
b | 0.2 | 1/4 | −0.2 | 0.25 |
c | 0 | 1/5 | −0.3 | 0.25 |
d | 0.1 | 1/10 | −0.4 | 0.25 |
Figure
Comparison of flutter boundaries from Theodorsen and Garrick [
This discrepancy is probably due to numerical inaccuracies in the curves presented in the original work. Zeiler [
Figure
Comparisons of flutter boundaries from Zeiler [
An experiment on flutter speed was performed at 5 × 4 Donald Campbell wind tunnels. Pitch and plunge are provided by a set of eight linear springs. Airspeed was gradually increased until the onset of flutter. The parameter values used in the experimental study are
The nondimensional flutter speed resulting from the present computation flutter analysis is
A model to determine the flutter onset of a two-dimensional typical aerofoil section has been implemented and then validated. A traditional aerodynamic analysis, based on Theodorsen’s theory and Leishman’s state-space model was used. The validation was performed, firstly, by solving Theodorsen and Garrick’s problem for the flexure-torsion flutter of a two-dimensional typical aerofoil section. The stability curves obtained are in close agreement with the results reported by more recent solutions of the same problem, whereas the original figures from Theodorsen and Garrick are found to be biased, as was previously reported by Zeiler. Secondly, validation with experimental data was conducted and the results showed a fairly close agreement.