Purdue Aeroelasticity AAE 556 Aeroelasticity The P-k flutter solution method (also known as the “British” method) Purdue Aeroelasticity

The eigenvalue problem from the Lecture 33 Purdue Aeroelasticity

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 Purdue Aeroelasticity

Purdue Aeroelasticity Go back to the original typical section equations of motion, restricted to steady-state harmonic response Purdue Aeroelasticity

The coefficients for the EOM’s Purdue Aeroelasticity

The eigenvalue problem Purdue Aeroelasticity

Another version of the eigenvalue problem with different coefficents Purdue Aeroelasticity

Purdue Aeroelasticity Definitions of terms for alternative set-up of eigenvalue equations for “k-method” Purdue Aeroelasticity

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 Purdue Aeroelasticity

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 Purdue Aeroelasticity

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 Purdue Aeroelasticity

This lift expression looks strange; where is the dynamic pressure? Purdue Aeroelasticity

Writing aero force in different notation - more term definitions The Qij’s are complex numbers Purdue Aeroelasticity

Aero force in terms of the Qij’s Purdue Aeroelasticity

Purdue Aeroelasticity Focus first on the term Purdue Aeroelasticity

Purdue Aeroelasticity The second term Purdue Aeroelasticity

Purdue Aeroelasticity Let’s adopt notation from the controls community to help with our conversion Purdue Aeroelasticity

Continue working on the first term in the aero force expression The expression for A11 reads Purdue Aeroelasticity

Purdue Aeroelasticity The term with the p in it looks like a damping term so let’s work on it Purdue Aeroelasticity

Finally, the exact expressions for each term are as follows Both terms are real numbers, there is no j here. Purdue Aeroelasticity

Aerodynamic moment expression Purdue Aeroelasticity

Purdue Aeroelasticity The Qij’s Purdue Aeroelasticity

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) Purdue Aeroelasticity

Perform this computation Purdue Aeroelasticity

Compute the aerodynamic damping matrix, defined as Purdue Aeroelasticity

Purdue Aeroelasticity Take the results and insert them into an eigenvalue problem that reads as follows Purdue Aeroelasticity

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 Purdue Aeroelasticity

Purdue Aeroelasticity Solving for the eigenvalues Convert the “p-k” equation to first-order state vector form State vector = Purdue Aeroelasticity

State vector elements are related The equation of motion becomes Solve for Purdue Aeroelasticity

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 Purdue Aeroelasticity

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 Purdue Aeroelasticity

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 Purdue Aeroelasticity

p-k computation procedure Input k and V Compute eigenvalues No, change k Repeat process for each w yes Purdue Aeroelasticity

Purdue Aeroelasticity What should we expect? w Right half-plane s Root locus plot Purdue Aeroelasticity

Back-up slides for Problem 9.2 Purdue Aeroelasticity

A comparison between V-g and p-k Purdue Aeroelasticity

A comparison between V-g and p-k Purdue Aeroelasticity

A comparison between V-g and p-k Purdue Aeroelasticity

A comparison between V-g and p-k Purdue Aeroelasticity

Purdue Aeroelasticity Flutter in action Accident occurred APR-27-95 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. Purdue Aeroelasticity

Purdue Aeroelasticity Things you should know Royal Aircraft Establishment The RAE started as HM Balloon Factory. From 1911-18 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. Purdue Aeroelasticity