AAE556 Lecture_3435_pk_flutter

Genealogy of the V-g or “k” methodi Equations of motion for harmonic response next slide – Forcing frequency and airspeeds are is known parameters – Reduced frequency k is determined fro

AAE 556

Aeroelasticity

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

The eigenvalue problem from the Lecture 33

2

2

2 2

2 2

0

1

0 1

i

h

b M

g

M

θ θ

2

h h

Genealogy of the V-g or “k” method

i Equations of motion for harmonic response (next slide)

– Forcing frequency and airspeeds are is known parameters

– Reduced frequency k is determined from ω and V

– Equations are correct at all values of ω and V.

i 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

Go back to the original typical section equations of motion, restricted to steady-state

The coefficients for the EOM’s

The eigenvalue problem

Another version of the eigenvalue problem with different

h

B = µ x θ + L α − L   + a  ÷

Definitions of terms for alternative set-up of eigenvalue equations for

D = µ x θ + M − L   + a  ÷

0 0

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

( ) { } ( ) { } { }

1 2

The p-k method will use the harmonic aero results to cast the stability

problem in the following form

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

ba

V

shear center airfoil chordline

P

h

This lift expression looks strange; where is the dynamic

h

2

3 2

2

e L

a 2

1 L

b

h L

b V

2

2 2

2

e L

a 2

1 L

b

h L

V

b 2

b 2

Writing aero force in different notation

- more term definitions

1 2

Focus first on the term Q 11

The second term

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

Continue working on the first term in the aero force expression

p

Q Q

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

2 2

ω

1

2 11,

Finally, the exact expressions for each term are as follows

Aerodynamic moment expression

4 2

2

1 1 2

2

2

4 2 22

– These will be a set of complex numbers, not algebraic expressions

i Choose an air density (altitude) and airspeed (V)

Perform this computation

Compute the aerodynamic damping matrix, defined as

,

1 2

ij imaginary ij

Take the results and insert them into an eigenvalue problem that reads

ij imaginary ij

i Choose k= ω b/V arbitrarily

i Choose altitude ( ρ) , and airspeed (V)

i Mach number is now known

i Compute AIC’s from Theodorsen formulas or others

i Compute aero matrices-B and Q matrices are real

p M     η − p B     η +       K − ρ V   Q     η =

Solving for the eigenvalues

Convert the “p-k” equation to first-order state vector form

State vector elements are related

State vector eigenvalue equation – the “plant” matrix

Solve for eigenvalues (p) of the [Aij] matrix (the plant)

Plot results as a function of airspeed

1st order problem

i Mass matrix is diagonal if we

use modal approach – so too

is structural stiffness matrix

ij

I A

Eigenvalue roots

i ωγ=σ is the estimated system damping

i There are “m” computed values of ω at the airspeed V

i You chose a value of k= ω b/V, was it correct?

– “line up” the frequencies to make sure k, ω and V are consistent

p-k computation procedure

Input k and V

Compute eigenvalues p i = ω γ i ( i ± j )

i i

b k

What should we expect?

Back-up slides for Problem 9.2

A comparison between V-g and p-k

Purdue Aeroelasticity

36

( )

22

22

22

0

1

0 1

h

b M

g

M

θθ

θ θθ

α

θ θθ

θ

ω ω

ω ω

A comparison between V-g and p-k

0

2 2

A comparison between V-g and p-k

0

2 2

A comparison between V-g and p-k

α θ

h b

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

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