1. Purdue Aeroelasticity
AAE Aeroelasticity
Flutter-Lecture 21
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2. 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
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3. Purdue Aeroelasticity
The prize
Remember what the bars mean.
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4. Calculate the determinant what do you hope to discover?2b
c.g.
shear center
aero center
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5. Quadratic equation for frequency squared
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6. 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.
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7. Purdue Aeroelasticity
The transition point between stability and instability for this idealization is frequency merging
Two solutions with the same frequencies
instability
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8. Transition to instability
?????
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9. Purdue Aeroelasticity
Two roots
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10. Purdue Aeroelasticity
Frequency merging
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11. Solution for frequency
When the airspeed is zero then these eigenvalues are real and distinct - they also depend on airspeed ...
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12. Purdue Aeroelasticity
When the frequencies are real and distinct then no energy is input or extracted over one cycle
Mode shapes are important
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13. 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
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14. Flutter has 90o out of phase motion phase motion
negative
positive
Complex eigenvalue results are an example of math trying to talk to us
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15. 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
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16. 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
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17. 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.
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18. 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
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