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Introduction to Aeroelasticity
Lecture 4: Theodorsen for non-sinusoidal
motion
G. Dimitriadis
Aeroelasticity
Introduction to 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
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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.
Introduction to Aeroelasticity
Impulse response of pitch-plunge airfoil
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Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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.
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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
(
)
* * * * * * *
+
,
- - - - - - -
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
Frequency matching
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
Results
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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
Introduction to Aeroelasticity
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.