An airfoil subjected to two-dimensional incompressible inviscid flow is considered. The airfoil is supported via a translational and a torsional springs. The aeroelastic integro-differential equations of motion for the airfoil are reformulated into a system of six first-order autonomous ordinary differential equations. These are the simplest and least number of ODEs that can present this aeroelastic system. The differential equations are then used for the bifurcation analysis of an airfoil with a structural nonlinearity in the pitch direction. Sample bifurcation diagrams showing both stable and unstable limit cycle oscillation are presented. The types of bifurcations are assessed by evaluating the Floquet multipliers. For a specific case, a period doubling route to chaos was detected, and mildly chaotic behavior in a narrow range of velocity was confirmed via the calculation of the Lyapunov exponents.

A great deal of qualitative information can be obtained about wing flutter by studying the aeroelasticity of a simple two degree-of-freedom-system (DOF). The system, shown in Figure

Schematic of the airfoil.

The model is an extreme simplification of a wing. Despite this simplification, the underlying dynamics of the model could still be quite complicated if a structural nonlinearity is taken into account. The nonlinear dynamic behavior of this model has been investigated by several researchers, and limit cycle and chaotic oscillations have been detected for velocities below the linear flutter speed [

In the present analysis, the integro-differential equations are further simplified to a set of six autonomous first-order ODEs. The new set of equations are much simpler than the previously presented equations [

The equation of motion for the two-dimensional airfoil shown schematically in Figure

For incompressible flow the aerodynamic force and moment may be obtained for any arbitrary motion of the airfoil [

Previously [

The above set of ODEs are either analyzed using collocation method via AUTO software package [

The linear flutter velocity

The real part of the eigenvalues as a function of

The nonlinear set of ODEs are used for bifurcation analysis of the system taking into account a cubic nonlinearity of the form

This yields

(a) Bifurcation diagrams showing both stable and unstable solutions for pitch motion, (b) expanded view of partial region from (a).

Another unstable periodic branch starts at point 2 and changes its direction and becomes stable at a limit point 6. This branch becomes unstable after a bifurcation to an invariant torus at point 7, reaches the limit point 4, turns back to a stable solution via another bifurcation into a torus at point 8. Passing the limit point 9, this branch becomes unstable and returns to point 2 where it started.

As explained above, a pitchfork bifurcation and two bifurcations into torus were detected for this case. The variations of Floquet multipliers around these bifurcation points are shown in Figure

The variation of Floquet multipliers around two points in Figure

Figure

Comparison of the bifurcation diagrams obtained via the defect-controlled method and AUTO.

A bifurcation diagram was also constructed using AUTO for the same airfoil but with

(a) Bifurcation diagrams showing both stable and unstable solutions for pitch motion, (b) expanded view of partial region from (a);

The period doubling cascade leading to chaos was also predicted in the defect-controlled results. To compare the results obtained using two different methods, both the stable results obtained using AUTO and the results obtained using the defect-controlled method are presented in Figure

Comparison of the bifurcation diagrams obtained via the defect-controlled method and AUTO;

Lyapunov spectra were also calculated for the same airfoil and nonlinearity as Figure

(a) Time evolution of the Largest Lyapunov exponents, (b) variation of the largest Lyapunov exponent with

Assuming the approximate exponential expressions for the Wagner function, the aeroelastic equations of motion for a two-DOF airfoil are conveniently transformed into a set of

Nondimensional distance measured from airfoil mid-chord to elastic axis

Semichord of airfoil

Nonlinear structural restoring force

Plunge motion of the airfoil

Mass moment of inertia about elastic axis

Linear structural stiffness in heave

Linear structural stiffness in pitch

Aerodynamic lift force

Mass of airfoil per unit span

Nonlinear structural restoring moment

Aerodynamic pitching moment about elastic axis

Nondimensional aerodynamic force

Nondimensional aerodynamic moment

Nondimensional radius of gyration about elastic axis

Nondimensional free stream velocity,

Nondimensional linear flutter velocity

Free stream velocity

Nondimensional distance from elastic axis to center of mass

Pitch rotation of the airfoil

Viscous damping ratio in pitch

Viscous damping ratio in heave

Airfoil-air mass ratio,

Nondimensional heave displacement,

Nondimensional time,

Air density

Wagner's function

Uncoupled frequency in pitch,

Uncoupled frequency in heave,

Frequency ratio,

The authors would like to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada.