AAE556Lecture03Stability in Aircraft

Lecture 3 – summaryi Aeroelasticity is concerned with interactions between aerodynamic forces and structural deformation i Develop simple static aeroelastic model with pitch torsion and

Lecture 3 – summary

i Aeroelasticity is concerned with interactions between aerodynamic forces

and structural deformation

i Develop simple static aeroelastic model with pitch (torsion) and plunge

(bending)

– Section 2.4

Reading topics

i 2.6 Lifting generation-flexible surface

i 2.7 Example problem – work it through by hand

i 2.8 Using simple results

i 2.9 Load factor

i 2.10 Simple model – 1 degree-of-freedom-emphasis on stiffness, not strength

i 2.10.2 – Stability definition – essential

i 2.11 Example problem using perturbation concept

i 2.12 Analysis example showing when stability is obvious and when it is not

i 2.13 Compressibility

Aero/structural interaction model

Lift and the aeroelastic parameter

MAC o

L

K qSeC K

qScC qSC

L

α α

α

1

Lift equation with wing flexibility

C q

L

1

0

Two degree of freedom aeroelastic model (Section 2.4)

Displacement, h, plunge at the shear center

Airspeed, V

twist , θ

Plunge is resisted by

spring, Kh Twist is resisted by spring, KT

Goal - add bending deformation (plunge) to the simple 1 dof model

+h

M

L

h K

0

0

K

θ

Write the aerodynamic loads in terms of h & θ

We use matrix methods – that’s our theme

h qSC

Le M

M SC = AC +

Twisting moment, at wing shear

center, positive nose-up

Idealized wing section lift

Aeroelastic static equilibrium equationIntroducing the aeroelastic stiffness matrix constructed out of thin air

1

0 0

0

MAC o

L L

h

qScC e

qSC

h e

qSC

h K

K

α θ

1

0 0

0

MAC o

L L

T

e qSC

h e

qSC

h K

K

α θ

Solution for wing deflections, h & θ

1

MAC T

o L

T L

T

L T

h

K

qScC e

K

qSC h

K qSeC

K

qSC K

K

α

α α

Divide by KT to get nondimensional terms

11

1

1

T L h L h

T T

MAC T

L h L h

T T

o L

K qSeC

K

qSC

K K

K

qScC e

K qSeC

K

qSC

K K

K

qSC

α α

α

ααθ

Invert 2x2

matrix

Get BHM

h L

T MAC

T L

h

o L

K qSeC

K

qSC K

qScC K

qSeC

K

qSC h

MAC L

T

o L

qSeC K

qScC qSeC

K qSeC

α α

αα

θ

1

1 1

plunge

twist

+h

New goals

i Define structural static stability

i Learn how to do stability analysis

i Find the wing divergence dynamic pressure using a “perturbation”

analysis

Math Summary

i Static equilibrium plays an essential role in aeroelastic analysis (surprise, surprise…)

– Static equilibrium equations are statically indeterminate (equilibrium depends on knowledge of force/deflection relationship)

– Multi-degree-of-freedom systems have as many equations of equilibrium as degrees of freedom

i Systems of simultaneous equations can be written (and solved) in matrix form.

i Static equilibrium aeroelastic equations yield two important matrices

– Structural stiffness matrix – symmetrical if you do it right

– Aerodynamic stiffness matrix – aero people will not recognize this term

– These matrices are added together to form the aeroelastic stiffness matrix

0

1 0

0

0

MAC o

L L

T

h

qScC e

qSC

h e

qSC

h K

K

α θ

Euler’s static stability criterion

i "A system in static equilibrium is neutrally (statically) stable if there

exist nearby static equilibrium states in addition to the original

static equilibrium state.”

i Stability - the tendency of a system (structural configuration) to

return to its original equilibrium state when subjected to a small

1707-1783 Advisor-Bernoulli

The perturbed structure

i Static stability analysis considers what happens to a flexible

system that is in static equilibrium and is then disturbed

– If the system tends to come back to its original, undisturbed position, it

is stable - if not - it is unstable.

i We need to apply these above words to equations so that we

can put the aeroelastic system to a mathematical test

Stability investigation

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

i Add a disturbance to perturb the system to move it to a different, nearby position (that may or may not

be in static equilibrium)

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

i Write static equilibrium equations and see if forces and moments balance

0

1 0

0

0

MAC o

L L

T

h

qScC e

qSC

h e

qSC

h K

K

α θ

torsion spring

KTV

αo+θ

∆θ MS=KT( θ + ∆θ )

lif t + perturbation lif t

Perturbed 1 dof airfoil

i In flight this airfoil is in static equilibrium at the fixed angle θ but what happens

if we disturb (perturb) it?

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 qSeC L ) ( K T qSeC L ) ( ) qSeC L o

Perturbation possibilities

i KT(∆θ )>( ∆ L)e

– statically stable because it tends to return

– no static equilibrium in the perturbed state

i KT(∆θ )<( ∆ L)e

– statically unstable

– motion away from original position

i KT(∆θ )=( ∆ L)e

– system stays in perturbed position

– new static equilibrium point ∆θ

– Euler test has found a neutral stability condition

( K T qSeC L ) ( ) ?

α θ

Static stability investigation is “stiffness based”

so

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

h

L T