AAE556 lecture 06 MDOF stability

Purdue Aeroelasticity1 AAE 556 Aeroelasticity Lecture 6 – Multi-DOF systems Reading: Sections 2-13 to 2-15... Homework for Friday?Purdue Aeroelasticity 2... Purdue Aeroelasticity10 The s

Purdue Aeroelasticity

1

AAE 556 Aeroelasticity

Lecture 6 – Multi-DOF systems

Reading: Sections 2-13 to 2-15

Homework for Friday?

Purdue Aeroelasticity

2

aero centers

Purdue Aeroelasticity

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Torsional static equilibrium is

a special case of dynamic equilibrium

0

0

1 2

2

2

o L

θ

α α

L1 = qSCLα ( α o + θ 1)

i Matrix equation order, sign convention and ordering of unknown displacements (torsion angles) is important

0

0

12

2

25

2

1 2

1

o L

θ

α α

Combine structural and aero stiffness matrices on the left hand side

0

0

1 2

2

2 5

θ

θ θ

θ

α α

Purdue Aeroelasticity

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The solution for the θ’s requires inverting

the aeroelastic stiffness matrix

When dynamic pressure increases, the determinant

∆ tends to zero – what happens to the system then?

Purdue Aeroelasticity

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8 6

4 2

0 -8 -6 -4 -2 0 2 4 6

STABLE UNSTABLE

Dynamic pressure parameter

The determinant of the stiffness matrix is always

positive turning the air on reduces its size

Purdue Aeroelasticity

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Solve for the twist angles created by

an input angle of attack α o

1 2

4 7

aero centers panel 1

3KT 2KT

panel 2

V

A A

5 4

3 2

1 0

-10 -8 -6 -4 -2 0 2 4 6 8 10

outboard panel

inboard panel

inboard panel

outboard panel

unstable region

dynamic pressure parameter, q

Unstable q region

Outboard panel (2)

Purdue Aeroelasticity

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Lift re-distribution due to aeroelasticity

(originally presented on slide 13)

MDOF Divergence

i In general we have matrix relationships developed from

EOM’s

i These can be converted into perturbation relationships

1 2

0 0 0

When dynamic pressure increases, the determinant

∆ tends to zero – divergence occurs

4 2

0 -8 -6 -4 -2 0 2 4 6

STABLE UNSTABLE

Dynamic pressure parameter

This nth order determinant is called the stability

determinant or the characteristic equation

qSeC K

aero centers panel 1

3KT 2KT

panel 2

V

A A

5 4

3 2

1 0

-10 -8 -6 -4 -2 0 2 4 6 8 10

outboard panel

inboard panel

inboard panel

outboard panel

unstable region

dynamic pressure parameter, q

Unstable q region

Outboard panel (2)