AAE556 Lectures p k Method

AAE556Lectures 34,35

The p-k method, a modern alternative to V-g

Purdue Aeroelasticity 1

Purdue Aeroelasticity

Genealogy of the V-g or “k” method

i Equations of motion for harmonic response (next slide)– Forcing frequency and airspeed are known parameters– Reduced frequency k is determined from w and V – Equations are correct at all values of w 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 invented V-g artificial damping to create an iterative approach to finding the flutter point

2

Equation #2, moment equilibrium

22 2 2 2 2 0hM M

h hx r r

b b

2

1 1

2 2h hM M a L M a L

1 1

2 2h hM a L

3Purdue Aeroelasticity

Divide by w2

2 22

2

10h

h hx r r M M

b b

Include structural damping

2 22

2

101 hig

h hx r r M M

b b

The eigenvalue problem

Purdue Aeroelasticity4

2

22

2

2 0 1

0

101

20

h

h h

h

h hxb b

x rr

hL L a Lb

M M

2

2

2

2

2 11 1

0

10

2 h

h

h hh hL L a Lx

b bx r

M Mr

Return to the EOM’s before we assumed harmonic motion


Here is what we would like to have

The first step in solving the general stability problem

12

2 32 0

ij j ij j ij j

ij j ij j

p M K A

p A p A

1 2 3 0ij j ij j ij j ij j ij jM K A A A

ptj j e p j

25-5

The p-k method casts the flutter problem in the following form

2 210

2ij ij ij ijp M p B K V A

pt pth

bt e e


…but first, some preliminaries

p j

6


Setting up an alternative solution scheme

hx Kh h P

b b m b mb

2 2 2ax I K Mh

b b mb mb mb

22 2

1 0

0

h

a

K Phh

mbmbbI

K Mmb

m

x

bx

mb

7


The expanded equations

22 2

22

4 2

2

1 0

0

1 0

0

1

2

1

2

h

a

h

h h

K Px hh

mbmbbI

Kx Mmb

mb mb

Kx hh

mbbI

Kxmb

mb

L L a Lb

mb

21 1 1 1

2 2 2 2h ha L a

hb

M L L a

8


Break into real and imaginary parts

3 2

2

3 2

1

2

1 1 1 1 1

2 2 2 2 2

1

2Real

1 1 1 1 1

2 2 2 2 2

h h

h h

h h

h h

L L a Lb

mba L M L a L a

L L a Lb

mba L M L a L a

2

3 2

2

1

2Imag

1 1 1 1 1

2 2 2 2 2

h h

h h

L L a Lb

jmb

a L M L a L a

9


Recognize the mass ratio

2

2

2

1

2Real

1 1 1 1 1

2 2 2 2 2

1

2Imag

1 1 1 1

2 2 2

h h

h h

h h

h

L L a L

a L M L a L a

L L a L

j

a L M L

21

2 2ha L a

10

Multiply and divide real part by dynamic pressure

Multiply imaginary part by p/jw

22

22

2

1

21 2Real

2 1 1 1 1 1

2 2 2 2 2

1

2Imag

1 1

2 2

h h

h h

h h

L L a Lk

Vb

a L M L a L a

L L a Lp

jj

2

1 1 1

2 2 2h ha L M L a L a

11Purdue Aeroelasticity

Multiply and divide imaginary part by Vb/Vb

22

22

1

21 2Real

2 1 1 1 1 1

2 2 2 2 2

1

2Imag

1 1

2 2

h h

h h

h h

L L a Lk

Vb

a L M L a L a

L L a LV k

pb

a

21 1 1

2 2 2h hL M L a L a

Define Aij and Bij matrices

2 2

22

1

2Real

1 1 1 1 1

2 2 2 2 2

1

2Imag

h h

ij

h h

h h

ij

L L a LV k

Ab

a L M L a L a

L L a LV k

Bb

2

1 1 1 1 1

2 2 2 2 2h ha L M L a L a

Place aero parts into EOM’sNote the minus signs

2 2

2

1

2Real

1 1 1 1 1

2 2 2 2 2

1

2Imag

h h

ij

h h

h

ij

L L a LV k

Ab

a L M L a L a

L L a LV k

Bb

21 1 1 1 1

2 2 2 2 2

h

h ha L M L a L a

2 0

0ij ij ij ij

hbp M p B K A

What are the features of the new EOM’s?

i We still need k defined before we can evaluate the matrices

i Airspeed, V, appears.i The EOM is no longer complexi We can calculate the eigenvalue, p, to

determine stability

2 0

0ij ij ij ij

hbp M p B K A

The p-k problem solution

i Choose k=wb/V arbitrarilyi Choose altitude ( )r , and airspeed (V)i Mach number is now known (when appropriate)i Compute AIC’s from Theodorsen formulas or othersi Compute aero matrices-Bij and Aij matrices are reali Convert “p-k” equation to first-order state vector form

2

2

0

0

0

0ij

ij ij ij ij

ij ij K

hbp M p B K A

hbp M p B

A state vector contains displacement and velocity “states”

j jvelocity vector v x

{ } j

jj

xz

v

ì üï ïï ï=í ýï ïï ïî þState vector =

jdisplacement vector x



Relationship between state vector elements

{ } { }j jx v=

{ } { } { } { }0ij j ij j ij jM v B v K xé ù é ù é ù- + =ê ú ê ú ê úë û ë û ë û

{ } { } { }

{ }

1 1

1 1

j ij ij j ij ij j

j

j ij i ij ij jj

v M K x M B v

xv M K M B

v

- -

--

é ù é ù é ù é ù=- +ê ú ê ú ê ú ê úë û ë û ë û ë ûì üï ïé ùé é ùé ù é ùê úê ú ê úë û ë ûë û

ù ï ïé ù é ù= - í ýê úê úê ú ê úë û ë û ï ïë ûë ûï ïî þ

An equation of motion with damping becomes

Use an identify relationship for the other equations

Purdue Aeroelasticity19

{ } [ ][ ]{ }0 0 1 0

00 0 0 1

j j

j ij j

x xz I z

v v

ì ü ì üé ùï ï ï ïï ï ï ï é ùê ú= = =í ý í ý ë ûê úï ï ï ïë ûï ï ï ïî þ î þ

State vector eigenvalue equation

{ }[ ] [ ]

{ }1 1

0j j

j ij jj j

Ix xz Q z

v vM K M B

- -

é ùì ü ì üï ï ï ïê úï ï ï ï é ù= = =í ý í ý ê úê ú ë ûé ùï ï ï ï-ê úï ï ï ïê úî þ î þë ûë û

z(t) z estAssume a solution

Result

Solve for eigenvalues (p) of the [Q] matrix (the plant)Plot results as a function of airspeed

{ } { } { }j j ij jz p z Q z é ù= =ê úë û



1st order problem

i Mass matrix is diagonal if we use modal approach – so too is structural stiffness matrix

i Compute p roots– Roots are

either real (positive or negative)

– Complex conjugate pairs

1 1

0ij

ij ij ij

IQ

M K M B

K K A

{ } { } { }j j ij jz p z Q z é ù= =ê úë û

Eigenvalue roots

i =wg s is the estimated system dampingi There are “m” computed values of w at the

airspeed Vi You chose a value of k=wb/V, was it correct?

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

real imaginaryp p jp

p j


Procedure

Input k and VCompute eigenvalues

i i ip j

ii

bk

V

?i inputk k

yes real i i

imaginary i

p

p

Repeat process for each w

No, change k


P-k advantages

i Lining up frequencies eliminates need for matching flutter speed to Mach number and altitude

i p-k approach generates an approximation to the actual system aerodynamic damping near flutter

i p-k approach finds flutter speeds of configurations with rigid body modes


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AAE556 Lectures p k Method