Aero Elasticity

Introduction to Aeroelasticity

Lecture 4: Theodorsen for non-sinusoidal

motion

G. Dimitriadis

Aeroelasticity


Time domain responses

@ Theodorsen analysis requires that the equations of motion are only valid at zero airspeed or at the flutter condition. @ They are also valid in the case of forced

sinusoidal excitation. @ We can calculate the response of an

aeroelastic system with Theodorsen aerodynamics to any excitation force


Frequency Response Function

@ Imagine that we excite the pitch-plunge airfoil at the leading edge with a force F0expjt. @ The equations of motion become

@ This equation is of the form H()q0=F, where H-1() is the Frequency Response Function.

1x f

# $ %

& ' ( F0


FRF for pitch-plunge system

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FRF of h The two modes are clearly present FRF of The first mode is present as an anti-resonance


Working with the FRF @ If the force is non-sinusoidal, F0=F0(). @ The systems response to such a force is

obtained as q0()=H()-1F(). @ If F()=1 then the inverse Fourier

Transform of q0() is the systems impulse response. @ The impulse response can also be used to

perform stability analysis.


Impulse response of pitch-plunge airfoil

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Damped sinusoidal motion @ The previous discussion shows that:

Theodorsen aerodynamics are only valid for sinusoidal motion

Yet Theodorsen aerodynamics can be used to calculate damped impulse responses

@ Stability analysis is slow and and can be less accurate when performed on impulse responses @ We need a method for calculating the

damping at all airspeeds directly from the equations of motion


The p-k Method

@ The p-k method is the most popular technique for obtaining aeroelastic solutions @ It was started in the 80s and since then

has become the industrial standard @ Virtually all aircraft flying today have

been designed using the p-k method


Basics @ The p-k method uses the structural

equations of motion in the standard form

@ Coupled with Theodorsen aerodynamic forces of the form

With k=b/U


Basics (2)

@ Remember that this is only correct if the response is sinusoidal, since the Theodorsen lift is equal to

@ The p-k method mixes h(t), which is a general function, with h0expjt.


Basics (3)

@ Therefore, the equations contain terms that depend on frequency @ The basis of the p-k method is to define

@ Then, the equations of motion become

@ Where q=[h ]T.

p2Ms +K s 12 U

2Q p( )$ %

& ' q = 0


Using p

@ Using the p notation, the Q(p) matrix

becomes:

@ i.e. it is a polynomial function of p (or p/U).

Q p( ) =

2cC k( ) pU 2b2 pU

$ %

& '

22cC k( ) 2b2 pU 2cC k( )

34 c x f

$ %

& ' pU 2b

2 pU

$ %

& '

2

2ec 2C k( ) pU + 2 x f c2

$ %

& ' b2 pU$ %

& '

2 2ec 2C k( ) 234 c x f

$ %

& ' b2 pU +

2ec 2C k( ) 34 c x f$ %

& ' pU 2b

2 x f c2

$ %

& '

2 pU

$ %

& '

2

b44

pU

$ %

& '

2

(

)

* * * * * * *

+

,

- - - - - - -


The p-method

@ The p-method consists of solving this eigenvalue problem for p.

@ Its a nonlinear eigenvalue problem but polynomial so it can be solved. @ The p values will generally be complex. @ There is no guarantee that the real parts of

the p values will have the correct value

p2Ms +K s 12 U

2Q p( )$ %

& ' q = 0


The p-k method @ The p-k method is more sophisticated than

the p-method in that it performs frequency matching @ The equations solved are

@ Since it is known that the aerodynamic matrix is only a function of frequency (not of damping) @ Again, k=b/U

(2)

p2Ms +K s 12 U

2Q jk( )$ %

& ' q = 0


Application to 2-dof model

@ The p-k equations for the 2-dof model are:

@ Notice that the Q matrix depends only on k, not on flight condition

m SS I

#

$ %

&

' ( p2 +

Kk 00 K

#

$ %

&

' (

12 U

2

4C k( ) jk + 2k 2 2cC k( ) 2bjk 4C k( ) 34 c x f# $

& ' jk + 2b2k 2

4ecC k( ) jk 2 x f c2

# $

& ' k 2

2ec 2C k( ) 2 34 c x f# $

& ' bjk +

4ecC k( ) 34 c x f# $

& ' jk + 2 x f

c2

# $

& '

2k 2 + b

2

4 k2

,

-

.

.

.

.

.

.

.

/

0

1 1 1 1 1 1 1

#

$

% % % % % %

&

'

( ( ( ( ( (

h

2 3 4

5 6 7

= 0


The p-k solution

@ The solution of these equations is iterative. @ We guess a value for the frequency (and

hence k) and then we calculate p from the resulting eigenvalue problem. @ The norm of p should be equal to . @ If it is not, we change the value of until the

scheme converges @ This is called frequency matching


Frequency matching


p-k method characteristics

@ Converges very quickly to the correct eigenvalue @ Suitable for large computational

problems @ Calculates sub-critical damping ratios @ Flutter speeds are very similar to the k-

method results


Results


Rogers Approximation @ Another way to transform the p-k equations to

the time domain is using Rogers Approximation. @ The frequency-dependent part of equations

(2), Q(jk), is approximated as:

@ Where nl is the number of aerodynamic lags and n are aerodynamic lag coefficients.

Q jk( ) = A 0 +A1 jk +A 2 jk( )2

+ A 2+njk

jk + nn=1

nl


Rogers EOMs @ The equations of motion of the complete

aeroelastic system then become:

@ Where

@ Usually:

q =

M 1C M 1K M 1A 3 M 1A nl +2I 0 0 00 I V 1 /bI 0 0 I 0 V nl /bI

$

%

& & & & & &

'

(

) ) ) ) ) )

q

M = Ms 12 b

2A 2, C = Cs 12 UbA1, K = K s

12 U

2A 0, A j = 12 U

2A j

nl = 4, n = 1.7kmaxn

nl +1( )2 , kmax = maximum k of interest


Practical Aeroelasticity @ For an aircraft, the matrix Q(jk) is obtained using a

panel method-based aerodynamic model. @ The modelling is usually performed by means of

commercial packages, such as MSC.Nastran or Z-Aero. @ For a chosen set of k values, e.g. k1, k2, , km, the

corresponding Q matrices are returned. @ The Q matrices are then used in conjunction with

the p-k method to obtain the flutter solution or time-domain responses. @ The values of Q at intermediate k values are

obtained by interpolation.

Aero Elasticity

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