Introduction – Equations of motion G. Dimitriadis 07

Aeroelasticity
Lecture 7:
Three-Dimensional Wings
G. Dimitriadis
Introduction to Aeroelasticity


Wings are 3D
•! All the methods described until now
concern 2D wing sections
•! These results must now be extended to
3D wings because all wings are 3D
•! There are two methods for 3D wing
aeroelasticity:
–! Strip theory
–! Panel methods
Introduction to Aeroelasticity


Strip theory
•! Strip theory breaks
the wing into
spanwise small strips
•! The instantaneous lift
and moment acting on
each strip are given
by the 2D sectional lift
and moment theories
(quasi-steady,
unsteady etc)
Introduction to Aeroelasticity

dy
S
y


Panel methods

Introduction to Aeroelasticity

s

Wake Panels
i,j+1
i+1,j+1

y
i,j
i+1,j

z

•! The wing is
replaced by its
camber surface.
•! The surface
itself is replaced
by panels of
mathematical
singularities,

solutions of
Laplace’s
equation

0
x
c


Hancock Model
•! A simple 3D wing model is used to
introduce 3D aeroelasticity
!

1*+,-

A rigid flat plate of span s,
chord c and thickness t,
suspended through an axis
xf by two torsional springs,
one in roll (K!) and one in
pitch (K").

!!"$
!!"#

"
$

0

!"'

!

!"&

The wing has two degrees of
freedom, roll (!) and pitch
(").

)

.

(

!"%
!"#
!"#
!

!"$
!

/*+,!!"#

!!"$
!!"#
)*+,-

Introduction to Aeroelasticity


Hancock model assumptions
•! The plate thickness is very small
compared to its other dimensions
•! The wing is infinitely rigid (in other words it
does not flex or change shape)
•! The displacement angles ! and " are
always small
•! The z-position of any point on the wing is

(

)

z = y" + x # x f $
Introduction to Aeroelasticity


Equations of motion
•! As with the 2D pitch plunge wing, the
equations of motion are derived using energy
considerations.
•! The kinetic energy of a small mass element
dm of the wing is given by
1
1 2
dT = z˙ dm = dm y"˙ + x # x f $˙
2

2

(

(

))

2

•! The total kinetic energy of the wing is:

m
2 2
2
2 ˙2
˙
˙
˙
T=
2s " + 3s c # 2x f "$ + 2 c # 3x f c + 3x f $
12

(

Introduction to Aeroelasticity

(

)

(

) )


Structural equations
•! The potential energy of the wing is simply
1
1
2
V = K" " + K# # 2
2
2

•! The full structural equations of motion are
then:
$ I"
&I
% "#

I"# ' *"˙˙- $ K"
+ ˙˙. + &
)
I# ( ,# / % 0

(

)

0 ' *" - * M1 )+ . = + .
K# ( ,# / , M 2 /

(

)

I" = ms2 /3, I"# = m c $ 2x f s /4, I# = m c 2 $ 3x f c + 3x 2f /3
Introduction to Aeroelasticity


Strip theory
•! The quasi-steady or unsteady
approximations for the lift and moment
around the flexural axis are applied to
infinitesimal strips of wing
•! The lift and moment on these strips are
integrated over the entire span of the wing
•! The result is a quasi-steady pseudo-3D lift
and moment acting on the Hancock wing
s

M1 = " # yl( y )dy
0

s

M 2 = " # mx f ( y )dy
0

Introduction to Aeroelasticity


Quasi-steady strip theory
•! Denote, "=# and h=y!. Then
l

mxf

•! Carrying out the strip theory integrations
will yield the total moments around the y=0
and x=xf axes.
Introduction to Aeroelasticity


3D Quasi-steady equations
of motion
•! The full 3D quasi-steady equations of
motion are given by

•! They can be solved as usual
Introduction to Aeroelasticity


Natural frequencies and
damping ratios

Introduction to Aeroelasticity

Unsteady aerodynamics
•! Wagner function unsteady aerodynamics
can be implemented on the Hancock
model in exactly the same way.
•! The aerodynamic states need to be
redefined as

Introduction to Aeroelasticity


Incremental Lift
•! The incremental lift on each strip becomes
l

Introduction to Aeroelasticity


Incremental Moment
•! The incremental moment on each strip
becomes
mxf

Introduction to Aeroelasticity


Full equations of motion

Introduction to Aeroelasticity

Aerodynamic state
equations of motion

•! As in the 2D case, the unsteady equations
of motion need to be completed by four
extra equations
•! These are obtained from the aerodynamic
states

Introduction to Aeroelasticity


Natural frequencies and
damping ratios

Introduction to Aeroelasticity


Theodorsen function
aerodynamics

•! Again, Theodorsen function aerodynamics
can (unsteady frequency domain) can be
implemented directly using strip theory:
l

mxf

Introduction to Aeroelasticity

Flutter determinant
•! The flutter determinant for the Hancock
model is given by

•! And is solved in exactly the same was as
for the 2D pitch-plunge model.
Introduction to Aeroelasticity


p-k solution

Introduction to Aeroelasticity


Comparison of flutter speeds
Wagner and
Theodorsen
solutions are
identical.
Quasi-steady
solution is the
most conservative
Ignore the other
solutions

Introduction to Aeroelasticity


Exercise

•! How good is strip theory?
•! If it works for the steady case, it might work
for the quasi-steady or unsteady case.
•! Develop the strip theory solution for a simple
wing geometry under steady conditions.
•! Compare with the 3D lifting line solution.
•! How good is the strip theory approximation?

Introduction to Aeroelasticity


Exercise reminders
•! Reminder 1: 2D lift for flat plate:
•! Reminder 2: 3D lift for elliptical
wing:
•! Reminder 3: 3D lift for rectangular
wing
C L = C L! !
C L! = 0.995
E = 1+

cl!

E + cl! / " AR

2#
AR (1 + # )

Introduction to Aeroelasticity

cl = 2!"

2! AR
CL =
"
AR + 2

CL!=wing lift curve slope
cl!=sectional lift curve slope
E=Jone
s correction

"=taper ratio
AR=Aspect ratio


Vortex lattice aerodynamics
•! Strip theory is a very gross approximation
that is only exact when the wing’s aspect
ratio is infinite.
•! It is approximately correct when the aspect
ratio is very large
•! It becomes completely unsatisfactory at
moderate and small aspect ratios (less
than 10).
Introduction to Aeroelasticity