Purdue Aeroelasticity AAE 556 - Aeroelasticity Flutter-Lecture 21 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

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

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

Purdue Aeroelasticity Begin at the beginning We bet that f(t) = sinwt and we won our bet as long as we were pre-flutter We also bet that w was a real number but math told us that we were wrong if we had a frequency merging condition or post-flutter Purdue Aeroelasticity

What we should have done Talk about algebra! We would get two simultaneous matrix equations One with coswt The other with sinwt We would have two characteristic equations We would have two unknowns, b and w Purdue Aeroelasticity

What to do instead Math is our friend and suggests complex algebra Now we get a determinant whose solution leads to either pure imaginary numbers (sinusoidal response) or damped or undamped oscillatory response. Purdue Aeroelasticity

Purdue Aeroelasticity Our eigenvectors are developed under the assumption that s is a complex number so things are a little backwards real then motion is sinusoidal complex with real and imaginary parts then motion is either damped or exponentially increasing s is purely imaginary then motion is sinusoidal s has a real and imaginary part then motion roots are stable or unstable In either case, a complex mode shape means that motion is 90o out of phase and the system is unstable Purdue Aeroelasticity