Introduction – Equations of motion G. Dimitriadis 03

the source distribution normal to the wing’s surface must be equal to the velocity due to the wing’s motion and the free stream, i.e.. Introduction to Aeroelasticity Wing motion plunge

Unsteady Aerodynamics

• ! As mentioned in the first lecture, quasi-steady aerodynamics ignores the effect of the wake

on the flow around the airfoil

• ! The effect of the wake can be quite significant

• ! It effectively reduces the magnitude of the

aerodynamic forces acting on the airfoil

• ! This reduction can have a significant effect on the values of the flutter

• ! In other words, the wake contains a significant amount

of circulation, which balances the changes in

circulation over the airfoil

• ! It follows that the wake cannot be ignored in the

calculation of the forces acting on the airfoil

Kelvin’s Theorem

Pitching and Heaving

simulation results (top)

and flow visualization in

a water tunnel (bottom)

by Jones and Platzer

How to model this?

concerning starting vortices for example)

is changed, a new simulation must be performed

exist They were developed in the 1920s and

1930s

Simplifications

simplifications are assumed:

amplitude is small

fact Theodorsen worked on a flat plate with a control surface (3 d.o.f.s), so asymmetric wings can also be handled

flat wake assumption has little influence on the results

Basis of the model

solutions of the Laplace equation:

– !The free stream:

– !The source and the sink:

&

' (

)

* +

Circle

circle that can be mapped onto a flat plate through a conformal transformation:

z a = x a + iy a = z + R z2

and surface of the flat plate, balanced by

sources of strength -2! on the bottom surface

– !A pattern of vortices +"! on the flat plate

balanced by identical but opposite -"!

vortices in the wake

Complete flowfield – flat

The wing’s chord should have been 4b but this has been divided by 2 because we are

using sources of strength 2!

About the wing and wake

distribution that changes in time

not cancel each other out

vorticity that changes both in space and in time

•! The +"! and -"! vorticity contributions do not cancel each other out

Wing and wake are slits

parts of the wing

Circle lower surface

Boundary conditions

problems there are two boundary

Boundary conditions 2

the source and sink distribution

vortex distribution

because for every vortex +"! there is a countervortex -"! Therefore, the total

change in vorticity is always zero

Impermeability

a solid surface is equal to zero

the source distribution normal to the wing’s surface must be equal to the velocity due to the wing’s motion and the free stream, i.e

and w is the external upwash

!"

!n = #w

Impermeablity (2)

object the source strength is given by

flow is constant)

distribution is defined by the wing’s

motion

#n

Introduction to Aeroelasticity

Wing motion

plunge degrees of freedom

to

and goes from -1 to +1

Introduction to Aeroelasticity

Potential induced by sources

•! The potential induced by a source at x1 , y1 is given

by

sink at x1 , -y1 is given by

use non-dimensional coordinates

( )

( )

*

*

where x = x b , y = 1! x 2

Total source potential

and sinks is given by

( )

( )

*

*

$ 1 1

Introduction to Aeroelasticity

After the integrations

sessions has been censored Such scenes

and middle-aged engineering professors

! ( )x ,y = b U( " + ˙ h # x f" ˙ ) 1# x 2 + b2" ˙

2 (x + 2) 1# x 2

! ( )x ,y lower = " ! ( )x ,y

(1)

Introduction to Aeroelasticity

Pressure on the surface

•! Where p is the static pressure, " the air density

and q the local air velocity

apply this equation to the wing’s surface

the surface As the wing lies on the x-axis:

q = U cos! + u = U cos! + "#

"x $ U +

"#

"x

Introduction to Aeroelasticity

Pressure difference

Introduction to Aeroelasticity

Non-circulatory lift

•! Substituting for "p we get

•! Because we set that "(1)="(-1)=0

Carrying out the integrations we get

Introduction to Aeroelasticity

Non-circulatory moment

by:

the integrations we get:

Circulatory forces

impermeability condition

using the vortex distribution

Introduction to Aeroelasticity

Potential induced by vortices

Introduction to Aeroelasticity

Pressure difference

potential is, as before

propagate downstream at the free stream velocity Then

Pressure difference (2)

calculus we obtain

difference at one point on the flat plate by only one vortex In order to obtain the full circulatory aerodynamic loads we need to integrate for all vortices over all the wing

Introduction to Aeroelasticity

Lift - Integrate over wing

Introduction to Aeroelasticity

Lift - Integrate over the wake

trailing edge and extends to infinity

Introduction to Aeroelasticity

Circulatory Moment

flexural axis becomes

•! After substituting for "p, the integrals

become much more complicated

1 )

*

The nature of V

strength as they travel downstream, V is a

function of space

reference system that travels with the fluid

of both time and space, i.e

V = f Ut ! x( 0)

Kutta condition

V

local velocity at the trailing edge must be finite’

component since the wing lies on the x-axis

the sources and vortices

!"tot

!x x =1 = finite

+ b

2 %

x 02 # 1

1# x 2 (x # x 0)Vdx 0 1

&

'

Introduction to Aeroelasticity

Kutta condition (3)

denominator becomes zero there

numerator must also become zero at the trailing edge Therefore:

Kutta condition (4)

the necessary vortex strength for the Kutta condition to be satisfied

Introduction to Aeroelasticity

Circulatory lift

Total lift

non-circulatory is easily obtained by adding the two contributions:

the added mass terms

% )

&

' * (5)

% )

& ' *

Discussion

(5) and (6) for the full lift and moment acting

on the airfoil

– ! Attached flow everywhere

– ! The wake is flat

– ! The wake vorticity travels at the free stream

airspeed

value of C?

Prescribed motion

define C we need to know V

useful

and then determine what the resulting

value of V will be

Sinusoidal motion

motion is sinusoidal motion

Slowly pitching and

More about sinusoidal motion

near the airfoil

The vortex strength of the wake behind a pitching and plunging airfoil can

have any spatial and temporal distribution, V(x 0 ,t)

There are two special motions for which Theodorsen’s function can be evaluated: steady motion and sinusoidal motion

For sinusoidal motion: ! = !0e j"t

functions of the first

and second kind

A much more practical,

approximate,

estimation is:

With k="c/U

or k="b/U

Usage of Theodorsen

• ! Theodorsen’s lift force is now given by

• ! Theodorsen’s function can be seen as an

analog filter It attenuates the lift force by an amount that depends on the frequency of

oscillation

• ! Theoretically, Theodorsen’s function can only

be applied in the case where the response of the system is exactly sinusoidal

l c = !"UcC k( )% U# + ˙ h + 3c% & 4 $ x f ' ( ˙ #

& )

' ( *

Lift and moment

moment around the flexural axis using Theodorsen are:

Aeroelastic equations

*

+ ,

*

+ ,

- =

.l t( )

m t( )

( )

*

+ ,

h0

#0

* + ,

- / e j"t = !l t( )

m t( )

* + ,

- /

Equations of motion?

• ! As the system is assumed to respond

sinusoidaly there is no sense in writing out complete equations of motion

• ! Combining the lift and moment with the

structural forces gives

Validity of this equation

when the wing is performing sinusoidal

oscillations

only possible when:

damping – free sinusoidal oscillations

forced sinusoidal oscillations

self-excited sinusoidal oscillations

critical flutter condition

Flutter Determinant

matrix must be equal to zero, i.e D=0, where

•! D is called the flutter determinant and must be solved

Two equations with two unknowns.

Solution

obtained from

natural frequencies

Effect of Flexural Axis

Since sinusoidal

motion is assumed the

Theodorsen equations

are only valid at the

flutter point (or when

Theodorsen are less

conservative than the

quasi-steady results