AAE556 lecture 05 MDOF response

Mach number depends on altitude and airspeed so two expressions must be 0 M q SeC... Summary i Lift curve slope is one strong factor that determines divergence dynamic pressure – depend

AAE 556 Aeroelasticity

Lecture 5 – 1) Compressibility;

2) Multi-DOF systems

Reading: Sections 2-13 to 2-15

Homework for Monday?

– Uncambered (symmetrical sections) MAC = 0

– Lift acts at aero center (AC) a distance e ahead to the shear center

i Problem 2.3 – wait to hand in next Friday

Aeroelasticity matters Reflections on the feedback process

Topic 1 - Flow compressibility (Mach number) has an effect on divergence because the

lift-curve slope depends on Mach number

L

T D

SeC

K q

SeC

K q

0

M

q SeC

But wait! – there’s more!

Mach number depends on altitude and airspeed so two expressions must be

0

M

q SeC

The divergence equation which contains Mach number must be consistent with the “physics” equation

2 1

2 2

2. Find the speed of sound

3. Square both sides of the above equation and solve for

2

1

a

q = ρ

q M

1.00 0.75

0.50 0.25

0.00 0 50 100 150 200 250

2

1 2

1

M a V

q

q q

a

a atmosphere

ρ

ρ =

=

=

If we want to increase the divergence Mach number we must increase

stiffness (and weight) to move the math line upward

1.00 0.75

0.50 0.25

20,000 ft.

40,000 ft.

Summary

i Lift curve slope is one strong factor that determines divergence dynamic

pressure

– depends on Mach number

i Critical Mach number solution for divergence dynamic pressure must be added

to the solution process

Topic 2 – Multi-degree-of-freedom (MDOF) systems

i Develop process for analyzing MDOF systems

i Define theoretical stability conditions for MDOF systems

i Reading - Multi-degree-of-freedom systems – Section 2.14

e

b/2 b/2

shear centers

aero centers

Torsional degrees of freedom

Here is a 2 DOF, segmented, aeroelastic finite wing model - two discrete aerodynamic surfaces with flexible

connections used to represent a finite span wing (page 57)

Introduce “strip theory” aerodynamic modeling to represent twist dependent airloads

i Strip theory assumes that lift depends only on local angle of attack of the strip of aero

The two twist angles are unknowns - we have to construct two free body diagrams to

develop equations to find them

Wing tip

Wing root

Internal shear forces are present, but not drawn

Double arrow vectors are torques

Structural restoring torques depend on the difference between elastic twist angles

This is the eventual lift re-distribution equation due to aeroelasticity – let’s

see how we find it

Torsional static equilibrium is

a special case of dynamic equilibrium

0

0

1 2

2

2 5

2

1 2

1

o L

θ

α α

i The equilibrium equations are written in terms of unknown displacements and known applied

loads due to initial angles of attack These lead to matrix equations

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

0

0

1 2

2

2 5

2

1 2

1

o L

θ

α α

The aeroelastic stiffness matrix is     K T

Combine structural and aero stiffness matrices on the left hand side

0

0

1 2

2

2 5

2

1 2

1

o L L

θ

θ θ

θ

α α

The solution for the θ ’s requires inverting the aeroelastic stiffness matrix

8 6

4 2

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

STABLEUNSTABLE

Dynamic pressure parameter

Plot the aeroelastic stiffness determinant D against

dynamic pressure (parameter)

The determinant of the stiffness matrix is always positive until the air is turned on

α

= ( 1 q ) ( 6 q )

b/2 b/2

shear centers

aero centers panel 1

3K T 2K T

panel 2V

54

32

10

-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)

More algebra - Flexible system lift