02 dynamic aeroelasticity - ltas-aea ::welcome · 2014-10-02 · aero0032-1, aeroelasticity and...

AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 2

Lecture 2: Dynamic Aeroelasticity

G. Dimitriadis

Aeroelasticity

1


Aeroelastic EOM

  In the previous lecture we developed the aeroelastic equations of motion for a pitching and plunging flat plate:

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


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

!

"

##

$

%

&&

!qq

!

"##

$

%&&

!z =Qz

z =!qq

!

"##

$

%&&, Q =

−M−1 C+ ρUD( ) −M−1 E+ ρU 2F( )I 0

!

"

##

$

%

&&

4


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


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


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.

7


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.

8


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.

9


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


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.

11


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.

12


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

13


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

14


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

15


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

16


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


  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( )

18


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

∏

19


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

20


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


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!

22


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

23


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

24


Starting Vortex (2)

Wake shape of an airfoil whose angle of attack was impulsively increased to 5o.

The starting vortex is clearly seen

25


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


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


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

28


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


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

30


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( )

31


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

32


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


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


Unsteady equations of motion

This type of equation is known as integro-differential since it contains both integral and differential terms.

35


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


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



  The integral in the lift equation can be expanded by parts. Then, substituting for the aerodynamic states we obtain

38


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( )

"

#

$$$

%

&

'''



  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:

40


Leibniz Integral Rule

  E.g for w1(t):

  For all wi(t): (2)

41


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


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

43


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


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


Natural Frequencies and damping ratios

46


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


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


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

02 dynamic aeroelasticity - ltas-aea ::welcome · 2014-10-02 · aero0032-1, aeroelasticity and...

Documents