AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2 Lecture 2: Dynamic Aeroelasticity G. Dimitriadis Aeroelasticity 1
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Lecture 2: Dynamic Aeroelasticity
G. Dimitriadis
Aeroelasticity
1
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Aeroelastic EOM
In the previous lecture we developed the aeroelastic equations of motion for a pitching and plunging flat plate:
2
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
2nd Order ODEs The equations are 2nd order linear ODEs of the form
where A = m S
S Iα
!
"##
$
%&&, B = πb2
1 c2− x f
!
"#
$
%&
c2− x f
!
"#
$
%&
c2− x f
!
"#
$
%&
2
+b2
8
!
"
#####
$
%
&&&&&
, C = 0 00 0
!
"#
$
%&, q = h
α
!
"#
$
%&
D = cπ1 3c
4− x f
!
"#
$
%&+
c4
−ec c2− x f
!
"#
$
%&
2
+3c4− x f
!
"#
$
%&c4
!
"
#####
$
%
&&&&&
, E =Kh 00 Kα
!
"
##
$
%
&&, F = cπ 0 1
0 −ec
!
"#
$
%&
A+ ρB( ) !!q+ C+ ρUD( ) !q+ E+ ρU 2F( )q = 0
3
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
First order form The second order equations can be easily written in first order form:
where M=A+ρB The first order ODEs are of the form
where
!!q!q
!
"##
$
%&&=
−M−1 C+ ρUD( ) −M−1 E+ ρU 2F( )I 0
!
"
##
$
%
&&
!
"##
$
%&&
!z =Qz
z =!qq
!
"##
$
%&&, Q =
−M−1 C+ ρUD( ) −M−1 E+ ρU 2F( )I 0
!
"
##
$
%
&&
4
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Analytical solution Recall from last year’s Flight Mechanics course that first order linear ODEs have an analytical solution:
or, after decomposing the matrix exponential:
where c=V-1z(0), n is the number of states, V is the eigenvector matrix of Q and λ are the eigenvalues of Q.
z t( ) = eQtz 0( )
z t( ) = vieλitci
i=1
n
∑
5
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Frequency and Damping The absolute values of the eigenvalues are
the natural frequencies, ωn=|λ| The damping ratios are defined as: ζ=-Re(λ)/ωn The damping ratios are measures of the
amount of damping present in each mode of vibration It must be kept in mind that both natural
frequencies and damping ratios are functions of airspeed and air density because the matrix Q is a function of these two quantities.
6
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Variation with airspeed
As the airspeed increases, the two natural frequencies approach each other. One of the damping ratios increases while the other first increases and then decreases. The critical damping ratio becomes zero and then negative. Instability ensues. This phenomenon is called flutter and the zero damping speed is the flutter speed.
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Subcritical System response Solve the equations of motion for the time responses of the system from initial conditions (α(0)=5o).
Time responses for U=30m/s. Both pitch and plunge decay with time.
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Critical System Response Solve the equations of motion for the time responses of the system from initial conditions (α(0)=5o).
Time responses for U=35.9m/s. Both pitch and plunge oscillation amplitudes remain constant.
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Supercritical Responses Solve the equations of motion for the time responses of the system from initial conditions (α(0)=5o).
Time responses for U=38m/s. Both pitch and plunge oscillation amplitudes increase with time.
10
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Stability criteria The stability of the system can be estimated directly from the eigenvalues of the system matrix: – If all eigenvalues have negative real parts, the
system is stable – If at least one real eigenvalue is positive, the
system has undergone static divergence – If at least one pair of complex conjugate
eigenvalues has positive real part, the system has undergone flutter.
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Determining the flutter speed The flutter speed can be determined by trial and error: – Choose an air density (i.e. flight speed) – Calculate the system eigenvalues for a
starting airspeed – Keep increasing the airspeed until at least one
pair of complex eigenvalues has positive real part
– Continue to try different airspeeds until the real part is almost zero.
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Routh-Hurwitz (1)
The static divergence and flutter speeds can also be obtained directly from the characteristic polynomial This can be achieved using the Routh-Hurwitz stability criterion. The criterion applies to a polynomial of the form
a4λ4 + a3λ
3 + a2λ2 + a1λ + a0 = 0
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Routh-Hurwitz (2) The system is unstable if
– any of the coefficients ai is zero or negative while at least one is positive
– There is at least one sign change in the first column of the matrix H
The matrix H is given by
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Routh-Hurwitz (3) The condition a0<0 gives the static divergence
condition, Kα<ρU2ec2π The condition c1<0 yields
Which, when expanded, yields a 4th order polynomial in U. Two of the solutions are U=+0 and U=-0 The other two solutions are U=+UF and U=- U=-UF
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Numerical searches Routh-Hurwitz can be easily applied to a 2-DOF system. Aircraft aeroelastic models can have more than 100 DOFs. Routh-Hurwitz is totally impractical for such large systems. Numerical methods can be used instead. These are generally divided into two categories – Directed searches, e.g. Newton-Raphson – Indirect searches, e.g. trial and error
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Newton-Raphson Newton-Raphson is a very widely used method for solving nonlinear problems. Suppose we need to solve the nonlinear equation f(U)=0. We start with a first guess Ui. This is a guess so f(Ui)=0. However, we want to calculate a correction ΔU, such that f(Ui+ΔU)=0. We expand f(Ui+ΔU) in a Taylor series around Ui:
f Ui +ΔU( ) = f Ui( )+ dfdU Ui
ΔU = 0
17
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Solving for ΔU we get:
Now we can calculate a better approximation for the solution of f(U)=0, which is Ui+1=Ui+ΔU.
This value is still not exact. We need to re-apply the procedure in order to calculate Ui+2, which will be an even better approximation.
We keep iterating until |ΔU|<ε, where ε is the required tolerance.
Newton-Raphson
ΔU = −dfdU Ui
#
$%%
&
'((
−1
f Ui( )
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Flutter test functions For flutter determination we need to define a
suitable function f(U)=0. Several different test functions work well. The
simplest is:
Where n is the number of states. This test function is equal to 0 when the real part
of any of the eigenvalues is equal to 0. If we want to detect only flutter and not static
divergence, then we can choose to include only the complex eigenvalues in the product.
f U( ) = ℜ λ j U( )( )j=1
n
∏
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Flutter derivative As the calculation of the eigenvalues is numerical, it is not possible to evaluate the derivative analytically. We can use a forward difference scheme to calculate the derivative numerically:
Where δU is a very small user-defined speed increment.
dfdU Ui
=f Ui +δU( )− f Ui( )
δU
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Starting guess The starting guess for the flutter speed should not be close to 0. Aeroelastic systems without structural damping flutter at U=0. Aeroelastic systems with structural damping can flutter at negative airspeeds. Choose an airspeed within the flight envelope but far from 0. Some aeroelastic systems may have many flutter airspeeds. Only the lowest flutter airspeed is of interest.
21
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Effect of flexural axis The position of the flexural axis has a significant effect on both flutter and static divergence.
For this aeroelastic system the flutter speed is always lower than the static divergence speed, unless xf/c>0.75.
Also note that placing the flexural axis in front of the aerodynamic center is bad for the flutter speed!
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Unsteady Aerodynamics
As mentioned in the first lecture, quasi-steady aerodynamics ignores the effect of the wake on the flow around the airfoil The effect of the wake can be quite significant It effectively reduces the magnitude of the
aerodynamic forces acting on the airfoil This reduction can have a significant effect on
the values of the flutter
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Starting Vortex (1) The simplest unsteady flow is a flat plate at 0o angle
of attack in a steady flow of airspeed U. At a particular instance in time, t0, the angle of
attack is increased impulsively to, say, 5o. This impulsive change causes the shedding of a
strong vortex, known as the starting vortex. The starting vortex induces a significant amount of
local velocity around the airfoil. However, it travels downstream because of the steady flow U. As the starting vortex distances itself from the wing,
its effect decreases After a while it has no effect at all and the flow
becomes steady
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Starting Vortex (2)
Wake shape of an airfoil whose angle of attack was impulsively increased to 5o.
The starting vortex is clearly seen
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Effect on lift
cl t( ) /cl ∞( )
Initially the angle of attack is zero. As the airfoil is symmetrical, its lift coefficient is also zero. When the change in angle of attack occurs, the lift jumps to half its steady-state value for the new angle of attack. The unsteady lift then asymptotes towards its steady-state value
26
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Wagner Function (1) The effect of the starting vortex on the
aerodynamic forces around the airfoil can be modeled by the Wagner function The Wagner function states that the
instantaneous lift at the start of the motion is equal to half the value of the steady lift (i.e. the value of the lift if the flow had been steady) The instantaneous lift then slowly increases
to reach its steady value as time tends to infinity
27
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Wagner Function (2) The Wagner function is equal to 0.5 when t=0. It increases asymptotically to 1.
It can be equally used to describe an impulsive change in angle of attack at constant airspeed
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Wagner Function (3) An approximate expression for the Wagner
function is given by
Where Ψ1=0.165, Ψ2=0.335, ε1=0.0455, and ε2=0.3. The lift coefficient variation with time after a
step change in incidence is given by
So that the lift force variation becomes
Φ t( ) = 1−Ψ1e−ε 1Ut / b −Ψ2e
−ε 2Ut / b
cl t( ) = 2παΦ t( )
l t( ) = ρπU 2cαΦ t( ) = ρπUcwΦ t( )
w=Uα is the downwash velocity 29
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Unsteady Motion Unsteady motion can be modeled as a
superposition of many small impulsive changes in angle of attack The increment in lift due to a small change in
pitch angle at time t0
So that the lift variation at all times can be obtained by integrating from time -∞ to time t, i.e.
l t( ) = ρπUc Φ t − t0( ) dw t0( )dt0-∞
t
∫ dt0
dl t( ) = ρπUcΦ t − t0( )dw t0( ) = ρπUcΦ t − t0( ) dw t0( )dt0
dt0
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Unsteady Motion (2) Using the thin airfoil theory result obtained in the first lecture, the downwash velocity can be written as
For a motion starting at t=0, w=0 for t<0 and w=w(0) at t=0. The lift generated at negative times is given by
w t( ) = Uα tot t( ) = Uα t( ) + h t( ) +34
c − x f$ %
& ' α t( )
l t( )t0 <0
= ρπUc Φ t − t0( ) dw t0( )dt0-∞
0
∫ dt0 = ρπUcΦ t( )dw 0( ) = ρπUcΦ t( )w 0( )
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Unsteady Motion (3) Then, the lift at all times is
Use integration by parts to get rid of acceleration terms inside the integral (these are very difficult to deal with) Result is:
l t( ) = ρπUc Uα 0( )+ !h 0( )+ 34c− x f
"
#$
%
&' !α 0( )
"
#$
%
&'Φ t( )+
ρπUc Φ t − t0( )0
t∫ U !α t0( )+ !!h t0( )+ 3
4c− x f
"
#$
%
&' !!α t0( )
"
#$
%
&'dt0
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Unsteady Motion (4)
This equation is the basis of Wagner function aerodynamics It includes the effect of the entire motion
history of the system in the calculation of the current lift force
l t( ) = ρπUc Uα t( ) + h t( ) +34
c − x f& '
( ) α t( )&
' *
( ) + Φ 0( ) −
ρπUc∂Φ t − t0( )
∂t00
t
∫ Uα t0( ) + h t0( ) +34
c − x f& '
( ) α t0( )&
' *
( ) + dt0
33
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Moment
The aerodynamic moment around the flexural axis due to the unsteady lift force is simply mxf(t)=ec l(t) However, for a complete representation of the
aerodynamic force and moment, the added mass effects must be superimposed, exactly as was done in the quasi-steady case. The complete equations of motion become
34
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Unsteady equations of motion
This type of equation is known as integro-differential since it contains both integral and differential terms.
35
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Integro-differential equations
Integro-differential equations cannot be readily solved in the manner of Ordinary Differential Equations. A numerical solution can be applied, based
on finite differences, e.g. Houbolt’s Method However, numerical solutions are not very
good for conducting stability analysis The equations must be transformed to ODEs
in order to perform stability analysis
36
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Transform to ODEs (1)
Use the following substitutions:
The wi variables are known as the aerodynamic states. They arise from the substitution of the approximate form of the Wagner function, Φ, in the equations of motion.
37
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Transform to ODEs (2)
The integral in the lift equation can be expanded by parts. Then, substituting for the aerodynamic states we obtain
38
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Transform to ODEs (3) The integrals have been absorbed by the
aerodynamic states. The full equations of motion are
(1)
M
C
K
W
39
m+ ρπb2 S − ρπb2 x f − c / 2( )S − ρπb2 x f − c / 2( ) Iα + ρπb
2 x f − c / 2( )2+ b2 / 8( )
"
#
$$$
%
&
'''
!!h!!α
"
#$$
%
&''+
πρUcΦ 0( ) c / 4+Φ 0( ) 3c / 4− x f( )
−ecΦ 0( ) 3c / 4− x f( ) c / 4− ecΦ 0( )( )
"
#
$$$
%
&
'''
!h!α
"
#$$
%
&''+
Kh +πρUc !Φ 0( ) πρUc UΦ 0( )+ 3c / 4− xf( ) !Φ 0( )( )−πρUec2 !Φ 0( ) Kα −πρUec
2 UΦ 0( )+ 3c / 4− xf( ) !Φ 0( )( )
"
#
$$$
%
&
'''
hα
"
#$
%
&'+
2πρU 3−Ψ1ε1
2 / b −Ψ2ε22 / b Ψ1ε1 1−ε1 1− 2e( )( ) Ψ2ε2 1−ε2 1− 2e( )( )
ecΨ1ε12 / b ecΨ2ε2
2 / b −ecΨ1ε1 1−ε1 1− 2e( )( ) −ecΨ2ε2 1−ε2 1− 2e( )( )
"
#
$$$
%
&
'''
w1w2w3w4
"
#
$$$$$
%
&
'''''
=πρUc !Φ t( ) h 0( )+ 3c / 4− xf( )α 0( )( )+ p t( )−πρUec2 !Φ t( ) h 0( )+ 3c / 4− xf( )α 0( )( )+ r t( )
"
#
$$$
%
&
'''
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Transform to ODEs (4)
There are two equations with 6 unknowns; 4 more equations are needed. These can be obtained by noting that the definitions of wi are of the form
Differentiating this equation with time:
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Leibniz Integral Rule
E.g for w1(t):
For all wi(t): (2)
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Complete Equations Equations (1) and (2) make up the complete aeroelastic system of equations. Equations (1) are 2nd order Ordinary Differential Equations (ODEs). They describe the dynamics of the system states. Equations (2) are 1st order ODEs. They describe the dynamics of the aerodynamic states.
42
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Complete Equations (2)
Q =
−M−1C −M−1K −M−1WI 0 00 W0
#
$
% % %
&
'
( ( ( , u =
h α
hα
w1
w2
w3
w4
#
$
% % % % % % % % % %
&
'
( ( ( ( ( ( ( ( ( (
W0 =
1 0 −ε1U /b 0 0 01 0 0 −ε 2U /b 0 00 1 0 0 −ε1U /b 00 1 0 0 0 −ε 2U /b
#
$
% % % %
&
'
( ( ( (
Here is the form of the complete equations
where
u = Qu
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AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Aerodynamic States The aerodynamic states are mathematical constructs that are used to represent history effects. As already mentioned several times, the aerodynamic forces depend not only the current state of the system but also on the history of the motion. This history is stored in the aerodynamic states. After all they are integrals.
44
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Solution of the ODEs
Now the unsteady aeroelastic equations are in complete ODE form (6 equations with 6 unknowns) and can be solved as usual, by injecting a harmonic component
A 8th order characteristic polynomial is obtained of the form
u = u0eλt
45
a8λ8 + a7λ
7 + a6λ6 + a5λ
5 + a4λ4 + a3λ
3 + a2λ2 + a1λ + a0 = 0
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Natural Frequencies and damping ratios
46
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Hard flutter There are significant differences between the unsteady and quasi-steady natural frequencies and damping ratios. The unsteady flutter speed is much higher than the quasisteady one: 50.7 m/s instead of 35.9 m/s. The bad news is that the unsteady flutter mechanism is much more abrupt: the damping drops very quickly to zero. This phenomenon is known as hard flutter and can be very dangerous.
47
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Effect of Flexural Axis The divergence speed is the same as in the quasi-steady case.
The flutter speeds obtained from Wagner’s method is always higher than that obtained from quasi-steady calculations.
48
AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2
Discussion Wagner function aerodynamics leads directly to time domain equations of motion. The application of this approach has been mostly limited to simple systems, such as the pitch-plunge airfoil or the pitch-plunge-control airfoil. Commercial aeroelastic packages calculate the aerodynamic forces in the frequency domain and then transform to the time domain, if needed.
49