Vibration

Aeroelasticity2DOF FlutterDarin HaudrichDarin.p.haudrich@boeing.com3/23/2015BackgroundWe previously seen that system phase and amplitude are dependent on the velocity, forcing frequency, and system damping and frequency.A linear aeroelastic system will change damping and frequency with airspeed when the system damping becomes unstable, flutter occurs. All systems typically have a flutter speed (and a divergence speed).

2DOF AirfoilRigid AirfoilP reference pointC Center of MassQ Aerodynamic centerT chord pointb chorde and a go from -1 to 1X = e-a

EOMUsing Lagranges EquationsPotential EnergyKinetic Energy Velocity is Needed as center of mass, C:Inertial Velocity at P:

Going to the global coordinates K:Generalized Forces

EOMLagranges Equation:

EOMs:

Lift slope is 2

Flutter FormulationUncoupled, natural frequencies at zero airspeed:

Substituting Lift and frequencies into EOM

Note the units are consistent in the above equationsAssuming

Flutter FormulationFlutter EOM:

Although the above equation can be solved introducing some dimensionless parameters can help:r is radius of gyration, is a frequency ratio, is a mass ratio, and V is velocity ratio

Resulting in:

Divergence can be obtained when s=0

Resultsa = -1/5, e = -1/10, = 20, r2 =6/25, and =2/5

s= ( i)/