Lectures 22, 23 Typical dynamic instability problems and test review

AAE 556Aeroelasticity Lectures 22, 23 Typical dynamic instability problems and test review 22-1 Purdue Aeroelasticity... Stability equation is original equilibrium equation with R.H.S.=0

AAE 556

Aeroelasticity

Lectures 22, 23 Typical dynamic instability problems and test review

22-1

Purdue Aeroelasticity

How to recognize a flutter problem in the making

2 2

2

2 2

Q

ω ω

ω ω

Given: a 2 DOF system with a parameter Q that creates loads on the system that are linear functions of the displacements

The modified determinant

2 1

2 2

2

2

2 1

Q

ω ω

ω ω

2 1

2 2

2 2

2 1

2 2

2 1

If flutter occurs two frequencies must merge

2 1

2 2

2 2

2 1

2 2

2 1

FLUTTER – Increasing Q must cause the term under the radical

sign to become zero

If one of the frequencies can be driven

to zero then we have divergence

2 2

2 2

2 1

Q

ω ω

ω ω

23-5

Purdue Aeroelasticity

Aero/structural interaction model

TYPICAL SECTIONWhat did we learn?

Perturbations & Euler’s Test

KT ( ) ∆ θ > ∆ ( ) L e

K T ( ) ∆ θ = ∆ ( ) L e

result - stable - returns -no static equilibrium in perturbed state

result - unstable -no static equilibrium - motion away from equilibrium state

result - neutrally stable - system stays - new static equilibrium point

Stability equation is original equilibrium equation with R.H.S.=0.

Only loads that are deformation dependent are included

The neutrally stable state is called self-equilibrating

23-9

Purdue Aeroelasticity

Multi-degree of freedom systems

e

shear centers

aero centers

From linear algebra,

we know that there is

a solution to the homogeneous equation only if the determinant of the aeroelastic stiffness matrix is zero

23-10

Purdue Aeroelasticity

MDOF stability

System is stable if the aeroelastic stiffness matrix determinant is positive Then the system can absorb energy in a static deformation mode If the stability determinant is negative then the static system, when perturbed, cannot absorb all of the energy due to work done

by aeroelastic forces and must become dynamic

Mode shapes? Eigenvectors and eigenvalues

K T

[ ] { } ∆ θi = { } 0

23-11

Purdue Aeroelasticity

Three different definitions of roll

effectiveness

• Generation of lift – unusual but the only game in town

for the typical section

• Generation of rolling moment –

• contrived for the typical section – reduces to lift

generation

• Multi-dof systems – this is the way to do it

• Generation of steady-state rolling rate or velocity-this

is the information we really want for airplane performance

• Reversal speed is the same no materr which way you

do it

23-12

Purdue Aeroelasticity

Purdue Aeroelasticity

Steady-state rolling motion

0 0

0 5 1.0 1.5 2.0

nondimensional divergence dynamic pressure vs wing sweep angle

sweep angle (degrees)

23-16

Purdue Aeroelasticity

Lift effectiveness

350

30 0 250

20 0 150

10 0 50

0

0 0

0 5 1.0 1.5

2.0

lift effectiveness vs.

15 degrees sweep

unswept wing

23-17

Purdue Aeroelasticity

Purdue Aeroelasticity

How to recognize a flutter problem in the making

2 2

2

2 2

Q

ω ω

ω ω

Given: a 2 DOF system with a parameter Q that creates loads on the system that are linear functions of the displacements

If flutter occurs two frequencies must merge

2 1

2 2

2 2

2 1

2 2

2 1

If one of the frequencies is driven to

zero then we have divergence

Fuel line flutter

A hollow, uniform-thickness, flexible tube has a mass per unit length of m slugs/ft and carries liquid fuel with density ρ to a rocket engine The fuel flow rate is U ft/sec through a pipe cross-section of A The tube is straight and has supports a distance L apart, the tube bending displacement is approximated to be

φ2 Unknown amplitudes of vibrational motion

The free vibration frequencies when the fluid is not flowing are:

Fluid flow creates system coupling, but through

the velocity, not the displacement

2 2

2

2 2

2

8

03

03

1 Find the divergence speed

2 Estimate the flow speed that flutter occurs, if it occurs

23-23

Purdue Aeroelasticity

Divergence is found by computing the determinant of the aeroelastic stiffness matrix

0

2 2

2 2

2

2 2

AU L

m

AU

o o

aesm

π

ρ ω

π

ρ ω

2 2

2

2 2

2

8

0 3

0 3

Assume that coupling leads to flutter and find an estimate of the merging point

2 2

2

2 2

The frequencies are approximated

2 2

2 2

2 1

AU L

m

AU

o

F o

ρ ω

L

L A

ρ

ω ω

2 2

2 2

1

2 2