Introduction – Equations of motion G. Dimitriadis 07

Wings are 3D concern 2D wing sections 3D wings because all wings are 3D aeroelasticity: – !Strip theory – !Panel methods... Strip theory the wing into spanwise small strips and momen

Lecture 7:

Three-Dimensional Wings

G Dimitriadis

Aeroelasticity

Wings are 3D

concern 2D wing sections

3D wings because all wings are 3D

aeroelasticity:

– !Strip theory

– !Panel methods

Strip theory

the wing into

spanwise small strips

and moment acting on

each strip are given

by the 2D sectional lift

and moment theories

i,j+1

i+1,j i,j

1*+,-A rigid flat plate of span s,

chord c and thickness t,

suspended through an axis

x f by two torsional springs,

one in roll (K !) and one in

pitch (K ")

The wing has two degrees of

freedom, roll (!) and pitch

(")

Introduction to Aeroelasticity

Hancock model assumptions

compared to its other dimensions

• ! The wing is infinitely rigid (in other words it does not flex or change shape)

always small

z = y" + (x # x f )$

Introduction to Aeroelasticity

Equations of motion

equations of motion are derived using energy considerations

dm of the wing is given by

dT = 12 ˙ z 2dm = 12 dm y ˙ ( " + (x # x f )$ ˙ )2

T = m12 (2s2 " ˙ 2 + 3s c # 2x( f )" ˙ $ ˙ + 2 c( 2 # 3x f c + 3x2f )$ ˙ 2)

Structural equations

- / +

- / =

M1

M2

* + ,

- /

I" = ms2 /3, I"# = m c $ 2x( f )s/4, I# = m c( 2 $ 3x f c + 3x2f )/3

Introduction to Aeroelasticity

Strip theory

approximations for the lift and moment

around the flexural axis are applied to

infinitesimal strips of wing

integrated over the entire span of the wing

and moment acting on the Hancock wing

M1 = "#0s yl y( )dy

M2 = "#0s m x f ( )y dy

Quasi-steady strip theory

will yield the total moments around the y=0

l

m xf

3D Quasi-steady equations

of motion

motion are given by

Natural frequencies and

damping ratios

Unsteady aerodynamics

can be implemented on the Hancock

model in exactly the same way

redefined as

Incremental Lift

l

Full equations of motion

Aerodynamic state equations of motion

of motion need to be completed by four

extra equations

states

Natural frequencies and

damping ratios

Theodorsen function

aerodynamics

can (unsteady frequency domain) can be implemented directly using strip theory:

l

m xf

Flutter determinant

model is given by

for the 2D pitch-plunge model

p-k solution

Comparison of flutter speeds

Exercise

for the quasi-steady or unsteady case

wing geometry under steady conditions

Exercise reminders

E=Jone
s correction

Vortex lattice aerodynamics

that is only exact when the wing’s aspect

ratio is infinite

ratio is very large

moderate and small aspect ratios (less

than 10)

Comparison of 3D and strip

theory, static case

The 3D lift distribution

is completely different

to the strip theory

result!

Vortex lattice method

• ! The basis of the

v

i,j P w

ri,j

x y

Characteristics of a

vortex ring

ring and positioned at its midpoint (the

intersection of the two diagonals - also

termed the collocation point)

Panelling up and solving

panels

values of the vorticities ! on each wing panel

at each instant in time

change in time Only the vorticities on the

wing panels are unknowns

Calculating forces

known, the lift and moment acting on the wing can be calculated

difference acting on each panel

entire wing yields the total forces and

moments

Panels for static wing

Even if the wing is

not moving, the

wake must still be

Unsteady wake panels

Wake shapes

Wake shape behind a

rectangular wing that

underwent an

impulsive start from

rest

The aspect ratio of the

wing is 4 and the

angle of attack 5

degrees

Effect of Aspect Ratio on lift

coefficient

Lift coefficient variation

with time for an

impulsively started

rectangular wing of

varying Aspect Ratio It

can be clearly seen that

the 3D results approach

Wagner’s function (2D

result) as the Aspect

Ratio increases

The aspect ratio of the

wing is 4 and the roll

The aspect ratio of the

wing is 4 and the pitch

The aspect ratio of the

wing is 4, the pitch

All planforms can be treated

Wake shape behind a

Unsteady lift and drag

Lift coefficient

Unsteady lift and drag

coefficients for a

bird-like wing performing

roll oscillations at

10Hz with amplitude 2

degrees

Industrial use

to calculate

with unsteady vorticity, just like Theodorsen’s method

airspeed and in the free stream direction

few chord-lengths)

Aerodynamic influence

coefficient matrices

• ! 3D aerodynamic calculations can be further speeded up

by calculating everything in terms of the mode shapes of the structure

• ! This treatment allows the expression of the aerodynamic forces as modal aerodynamic forces, written in terms of aerodynamic influence coefficients matrices

• ! These are square matrices with dimensions equal to the number of retained modes They also depend on

response frequency

• ! Therefore, the complete aeroelastic system can be written

as a set of linear ODEs with frequency-dependent

matrices, to be solved using the p-k method

Commercial packages

that can calculate 3D unsteady

aerodynamics using panel methods:

– !MSC.Nastran

– !ZAERO (ZONA Technology)

although idealized geometries

ZAERO AFA example

an Advanced Fighter Aircraft model

Structural model

Aerodynamic model