1. Purdue Aeroelasticity
AAE 556 Aeroelasticity
The P-k flutter solution method
(also known as the “British” method)
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2. The eigenvalue problem from the Lecture 33
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3. Genealogy of the V-g or “k” method
Equations of motion for harmonic response (next slide)
Forcing frequency and airspeeds are is known parameters
Reduced frequency k is determined from w and V
Equations are correct at all values of w and V.
Take away the harmonic applied forcing function
Equations are only true at the flutter point
We have an eigenvalue problem
Frequency and airspeed are unknowns, but we still need k to define the numbers to compute the elements of the eigenvalue problem
We invent ed Theodorsen’s method or V-g artificial damping to create an iterative approach to finding the flutter point
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4. Purdue Aeroelasticity
Go back to the original typical section equations of motion, restricted to steady-state harmonic response
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5. The coefficients for the EOM’s
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6. The eigenvalue problem
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7. Another version of the eigenvalue problem with different coefficents
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8. Purdue Aeroelasticity
Definitions of terms for alternative set-up of eigenvalue equations for “k-method”
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9. Return to the EOM’s before we assumed harmonic motion
Here is what we would like to have
Here is the first step in solving the stability problem
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10. Purdue Aeroelasticity
The p-k method will use the harmonic aero results to cast the stability problem in the following form
…but first, some preliminaries
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11. Purdue Aeroelasticity
Revisit the original, harmonic EOM’s where the aero forces were still on the right hand side of the EOM’s and we hadn’t yet nondimensionalized h P
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12. This lift expression looks strange; where is the dynamic pressure?
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13. Writing aero force in different notation - more term definitions
The Qij’s are complex numbers
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14. Aero force in terms of the Qij’s
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15. Purdue Aeroelasticity
Focus first on the term
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16. Purdue Aeroelasticity
The second term
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17. Purdue Aeroelasticity
Let’s adopt notation from the controls community to help with our conversion
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18. Continue working on the first term in the aero force expression
The expression for A11 reads
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19. Purdue Aeroelasticity
The term with the p in it looks like a damping term so let’s work on it
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20. Finally, the exact expressions for each term are as follows
Both terms are real numbers, there is no j here.
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21. Aerodynamic moment expression
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22. Purdue Aeroelasticity
The Qij’s
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23. Purdue Aeroelasticity
The p-k process
Step 1
Choose a value of k and compute all four complex aerodynamic coefficients
These are the complex Aij’s with the Theodorsen Circulation function in them
These will be a set of complex numbers, not algebraic expressions
Choose an air density (altitude) and airspeed (V)
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24. Perform this computation
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25. Compute the aerodynamic damping matrix, defined as
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26. Purdue Aeroelasticity
Take the results and insert them into an eigenvalue problem that reads as follows
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27. Purdue Aeroelasticity
Summary
Choose k=wb/V arbitrarily
Choose altitude (r), and airspeed (V)
Mach number is now known
Compute AIC’s from Theodorsen formulas or others
Compute aero matrices-B and Q matrices are real
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28. Purdue Aeroelasticity
Solving for the eigenvalues Convert the “p-k” equation to first-order state vector form
State vector =
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29. State vector elements are related
The equation of motion becomes
Solve for
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30. State vector eigenvalue equation – the “plant” matrix
Assume a solution
Result
Solve for eigenvalues (p) of the [Aij] matrix (the plant)
Plot results as a function of airspeed
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31. Purdue Aeroelasticity
1st order problem
Mass matrix is diagonal if we use modal approach – so too is structural stiffness matrix
Compute p roots
Roots are either real (positive or negative)
Complex conjugate pairs
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32. Purdue Aeroelasticity
Eigenvalue roots
wg=s is the estimated system damping
There are “m” computed values of w at the airspeed V
You chose a value of k=wb/V, was it correct?
“line up” the frequencies to make sure k, w and V are consistent
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33. p-k computation procedure
Input k and V
Compute eigenvalues
No, change k
Repeat process for each w
yes
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34. Purdue Aeroelasticity
What should we expect? w
Right half-plane s
Root locus plot
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35. Back-up slides for Problem 9.2
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36. A comparison between V-g and p-k
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37. A comparison between V-g and p-k
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38. A comparison between V-g and p-k
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39. A comparison between V-g and p-k
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40. Purdue Aeroelasticity
Flutter in action
Accident occurred APR at STEVENSON, AL Aircraft: WITTMAN O&O, registration: N41SW Injuries: 2 Fatal.
REPORTS FROM GROUND WITNESSES, NONE OF WHOM ACTUALLY SAW THE AIRPLANE, VARIED FROM HEARING A HIGH REVVING ENGINE TO AN EXPLOSION. EXAMINATION OF THE WRECKAGE REVEALED THAT THE AIRPLANE EXPERIENCED AN IN-FLIGHT BREAKUP. DAMAGE AND STRUCTURAL DEFORMATION WAS INDICATIVE OF AILERON-WING FLUTTER. WING FABRIC DOPE WAS DISTRESSED OR MISSING ON THE AFT INBOARD PORTION OF THE LEFT WING UPPER SURFACE AND ALONG THE ENTIRE LENGTH OF THE TOP OF THE MAIN SPAR. LARGE AREAS OF DOPE WERE ALSO MISSING FROM THE LEFT WING UNDERSURFACE. THE ENTIRE FABRIC COVERING ON THE UPPER AND LOWER SURFACES OF THE RIGHT WING HAD DELAMINATED FROM THE WING PLYWOOD SKIN. THE DOPED FINISH WAS SEVERELY DISTRESSED AND MOTTLED. THE FABRIC COVERING HAD NOT BEEN INSTALLED IN ACCORDANCE WITH THE POLY-FIBER COVERING AND PAINT MANUAL; THE PLYWOOD WAS NOT TREATED WITH THE POLY-BRUSH COMPOUND.
Probable Cause
AILERON-WING FLUTTER INDUCED BY SEPARATION AT THE TRAILING EDGE OF AN UNBONDED PORTION OF WING FABRIC AT AN AILERON WING STATION. THE DEBONDING OF THE WING FABRIC WAS A RESULT OF IMPROPER INSTALLATION.
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41. Purdue Aeroelasticity
Things you should know
Royal Aircraft Establishment
The RAE started as HM Balloon Factory. From it was called the Royal Aircraft Factory, but was changed its name to Royal Aircraft Establishment to avoid confusion with the newly established Royal Air Force.
Farnborough was known as a center of excellence for aircraft research. Major flutter research was conducted there. Famous R&M’s such as the “flutter bible” came from this facility. The RAE played a major role in both World Wars. So confident was Hitler that he could occupy England with relative ease that he spared the RAE from bombing in the hope of benefiting from its research.
Recently the RAE (now known as the Royal Aerospace Establishment) was absorbed into the DRA (Defence Research Agency), itself renamed as DERA (Defence Evaluation and Research Agency). The world famous initials are no more.
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