Purdue Aeroelasticity AAE 556 - Aeroelasticity Flutter-Lecture 20 Reading: 4.1-4.3; 4.5.1-4.5.3; 4.6.1-4.6.4 Purdue Aeroelasticity

Quasi-steady flutter with a typical section vibration idealization Flutter is a self-excited, dynamic, oscillatory instability requiring the of motion and interaction between two different modes an external energy supply Quasi-steady aerodynamic loads capture some dynamic effects of the lift force, but ignore lags between motion and developing forces and moments Assumed harmonic motion Purdue Aeroelasticity

Purdue Aeroelasticity The prize Remember what the bars mean. Purdue Aeroelasticity

Calculate the determinant what do you hope to discover? 2b c.g. shear center aero center Purdue Aeroelasticity

Quadratic equation for frequency squared Purdue Aeroelasticity

The A, B terms are airspeed dependent Purdue Aeroelasticity

Divergence is a special case set then Purdue Aeroelasticity

Purdue Aeroelasticity Divergence Purdue Aeroelasticity

Solution for natural frequencies When the airspeed is zero then these eigenvalues are real and distinct. They stay that way as airspeed increases. That means our original assumption of harmonic (sinusoidal) motion is correct. Purdue Aeroelasticity

Purdue Aeroelasticity The transition point between stability and instability for this idealization is frequency merging Two solutions with the same frequencies instability Purdue Aeroelasticity

Frequency depends on airspeed Purdue Aeroelasticity

Transition to instability ????? Purdue Aeroelasticity

Purdue Aeroelasticity Two roots Purdue Aeroelasticity

Purdue Aeroelasticity Frequency merging Purdue Aeroelasticity

Solution for frequency When the airspeed is zero then these eigenvalues are real and distinct - they also depend on airspeed ... Purdue Aeroelasticity

Purdue Aeroelasticity When the frequencies are real and distinct then no energy is input or extracted over one cycle Mode shapes are important Purdue Aeroelasticity

Purdue Aeroelasticity Free vibration is usually either “in-phase” or “180 degrees out of phase” negative work positive work In phase motion Sinusoidal motion assumption does not permit anything else Out of phase motion positive work negative work Purdue Aeroelasticity

Flutter has 90o out of phase motion phase motion negative positive Complex eigenvalue results are an example of math trying to talk to us Purdue Aeroelasticity