AAE556 Lecture04Divergence2011

AAE 556
Aeroelasticity
Lecture 4
Reading: 2.8-2.12

4-1

Purdue Aeroelasticity


Agenda

i

Review static stability

– Concept of perturbations
– Distinguish stability from response
i

Learn how to do a stability analysis

i

Find the divergence dynamic pressure using a “perturbation” analysis

4-2

Purdue Aeroelasticity

Perturbed airfoil
i

In flight this airfoil is in static equilibrium at the fixed angle θ but what happens
if we disturb (perturb) it?

∆L = qSCLα ( ∆θ )

lif t + perturbation lif t
∆θ
αo+θ

MS=KT(θ+∆θ)
torsion spring
KT

V

i

There are three possibilities

4-3

Purdue Aeroelasticity


Example

i

Perturb the airfoil when it is in static equilibrium

i

To be neutrally stable in this new perturbed position this equation
must be an true

(K

T

) (

− qSeCL θ + KT − qSeCL
α

α

) ( ∆θ ) = qSeC
4-4

Purdue Aeroelasticity

Lα

αo


The 1 DOF divergence condition

i

( KT − qSeCLα ) ( ∆θ ) = 0

Neutral stability

KT = qD SeCLα
i

KT
qD =
SeCLα

or

4-5

Purdue Aeroelasticity


Observations
i

The equation for neutral stability is simply the usual static equilibrium equation with
right-hand-side (the input angle αo) set to zero.

i

The neutral stability equation describes a special case

–
–

only deformation dependent external (aero) and internal (structural) loads are present
these loads are “self-equilibrating” without any other action being taken

4-6

Purdue Aeroelasticity


Stability investigation
i

Take a system that we know is in static equilibrium (forces and moments sum to zero)

K h
 0

i

 Kh
 0

i

i

0  h 

  − qSC Lα
K T  θ 

0
0


− 1  h 
− 1
0
  = qSC Lα α o   + qScCMAC  
e  θ 
e
1

Perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium)

0   h + ∆h 

 − qSC Lα
KT  θ + ∆θ 

0
0


−1  h + ∆h  (?)
 −1

 = qSCLα α o   + qScCMAC

e  θ + ∆θ 
e

Is this new, nearby state also a static equilibrium point?

 Kh

 0

0 
− qSCLα

KT 

0
0


−1   ∆h  (?) 0 
= 


e    ∆θ  0 

Static equilibrium equations for stability are those for a self-equilibrating system

7

Purdue Aeroelasticity

0 
 
1 


Neutral stability
i

Neutral stability is only possible if the system is “self-equilibrating.”

 Kh

 0

0 
− qSCLα

KT 

0
0


−1   ∆h  0 
= 


e   ∆θ  0 

i

The internal and external loads created by deformation just balance each other.

i

The system static stiffness is zero.

i

We’ll see that this requires that the system aeroelastic matrix become singular (the determinant is zero).

8

Purdue Aeroelasticity


The deformations at neutral stability are eigenvectors of the
problem

i

At neutral stability the deformation is not unique (∆θ is not zero - can be plus or
minus with indeterminate amplitude)

i

At neutral static stability the system has many choices (equilibrium states) near
its original equilibrium state.

– wing position is uncontrollable - it has no displacement preference when a

load is applied.

4-9

Purdue Aeroelasticity


For stability, only system stiffness is important. This graph shows where the equilibrium point for twist is located

M shear

center

M structure = KT θ
Structural

Aero overturning

resistance

M aero = qSeCLα ( α o + θ )
Slope depends on qSCLa
Equilibrium point

twist θ
4-10

Purdue Aeroelasticity

When we perturb the twist angle we move to a different position on the graph. One
of the moments will be larger than the other/

M structure = KT θ

M shear

∆θ

center

M aero = qSeCLα ( α o + θ )

Equilibrium point

twist θ
4-11

Purdue Aeroelasticity


The slope of the aero line is a function of dynamic pressure so the line rotates as speed increases.
This is a plot of the lines right at divergence.

M aero = qDiv SeCLα ( α o + θ )

M shear

Lines are parallel

M structure = KT θ

center
The equilibrium point lies at infinity

twist θ
4-12

Purdue Aeroelasticity


When the dynamic pressure is larger than the divergence dynamic pressure the crossing point is
negative. This is mathematics way of telling you that you are in trouble.

M shear

M aero = qSeCLα ( α o + θ )

center

M structure = KT θ

twist θ
4-13

Purdue Aeroelasticity


Let’s examine how aeroelastic stiffness changes with increased dynamic pressure

(K

T

)

− qSeCLα θ = Le + M AC = M SC
The standard definition of stiffness is as follows

∆M SC ∂ M SC
=
= K effective = K e
∆θ
∂θ

M sc

K effective = KT − qSeCL

α

twist θ

Aeroelastic stiffness decreases as q increases

4-14

Purdue Aeroelasticity

As we approach aeroelastic divergence we get twist amplification

i

Consider the single degree of freedom typical section and the expression for
twist angle with the initial load due to αo

i

neglect wing camber

qSeCL α o

qα o
θ=
=
KT ( 1 − q ) 1 − q
α

4-15

Purdue Aeroelasticity


Write this expression in terms of an infinite series

qαo
θ=
1− q
∞

 1 

2
3
n
qα o 
÷ = qα o 1 + q + q + q + ... = 1 + ∑ q ÷
n =1


 1− q 

4-16

Purdue Aeroelasticity


The first term is the uncorrected value of twist angle with no aeroelasticity

θ = qα o ( 1 + q + q + ...)
2

Plot the relative sizes of terms

1

with qbar=0.5
0.75

q bar = 0.5

0.5

the sum of the infinite series is 2

0.25
0
1

2

3

4

5

4-17

Purdue Aeroelasticity

6

7


Let’s take a look at the series and explain it as an aeroelastic feedback
process

θ = qα o ( 1 + q + q + ...)

2

θo is the twist angle with no aero
load/structural response "feedback"

θo =

qSeC L α o
α

KT
4-18

Purdue Aeroelasticity


Write the series slightly differently

θo =

qSeC L α o
α

KT

θ = θ o ( 1 + q + q + ...)
2

θ = θ o + qθ o + q θ o + ...
2

4-19

Purdue Aeroelasticity


The second term is the response to the first term

θ1 = q θ o =

qSeC L θ o
α

KT

This is the response to angle of attack θo instead of αo

…and, the third term

θ 2 = q θ o = q θ1
2

4-20

Purdue Aeroelasticity


Conclusion
Each term in the series represents a feedback "correction" to the twist created
by load interaction

∞

θ = θ 0 + ∑θ n
n =1


n
θ = θ o 1 + ∑ q 
 n =1 
∞

Series convergence

q <1

Series divergence

q ≥1
4-21

Purdue Aeroelasticity


Summary

i

Divergence condition is a neutral stability condition

i

Divergence condition can be found using the original equilibrium
conditions

i

Stability does not depend on the value of the applied loads

4-22

Purdue Aeroelasticity