Determination of the complete bifurcation behaviour of aeroelastic systems with freeplay

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The three degrees of freedom are constrained by three springs. The control surface spring contains a freeplay nonlinearity

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Kholodar and Dowell, AIAA Journal, 37 (5), 1999

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Conner et al, Journal of Fluids and Structures, vol 11, 89-109, 1997

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Inner system Outer system

VF1 flutter event V=6.85m/s VF2 flutter event V=13.11m/s VF3 Stabilization event V=25.91m/s VFlin Flutter event V=23.95m/s

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Describing function for freeplay Flutter speed variation with control surface stiffness

Equivalent linearization models the LCO of a nonlinear system by the flutter of a linear system. It also creates an equivalent linear stiffness depending on the response amplitude

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LCO branch 1

LCO branch 2

There are two LCO branches:

Branch 1: begins at VF1 as a subcritical Hopf, undergoes a fold and

asymptotes towards Vflin. Branch 2: begins at VF2 as a subcritical Hopf, undergoes two folds and ends at VF3 as a

supercritical Hopf.

Between 10m/s and 25m/s two LCOs are possible. LCOs occur at airspeeds less than VF1.

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LCO branch 1

LCO branch 2

There two LCO branches have completely different frequencies, which are delimited by the flutter frequencies of the

underlying linear systems. In the case of branch 1, the LCO frequency

asymptotes to the linear flutter frequency.

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Phase-plane diagrams of the system’s response. Dashed line: Equivalent linearization Solid line: Exact response from shooting

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Full bifurcation diagram Detail

Branch 1 Sub-branches

•!A shooting calculation is launched from every equivalent linearized LCO. Some calculations fail to converge, others tend to group around specific regions.

•!The shooting calculations have revealed the existence of sub-branches of branch 1, termed 1+ and 1-.

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Branch 1 Branch 2 Branch 1+ Branch 1- Area 1 Area 2 •!Blue lines denote stable LCOs, red lines unstable LCOs. •!In areas 1 and 2 there are many coexisting LCOs. •!In area 2 there are no stable low amplitude LCOs on branch 1.

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Branch 1 Branch 2 Branch 1- Branch 1+ Branch 2 Branch 1 Branch 1+ Branch 1- Area 1 Area 2 No stable LCOs

•!Sub-branches split from branch 1 in area 1, undergo many fold bifurcations and changes in stability and eventually join branch 1 again in area 2.

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•!The left plot was obtained by Kholodar and Dowell using straight numerical integration. The right plot was obtained using the present method.

•!The chaotic motion regions correspond to the area where no stable LCOs coexist or where many LCOs coexist.

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