Solve for the three dynamic pressures that create neutral stability

AAE556 2.1 Three identical uncambered wing segments are connected to each other by torsional springs and to the wind tunnel walls by additional torsion springs, as indicated in the figur

Homework set #2 Due February 4, 2011

A-1

AAE556 2.1 Three identical uncambered wing segments are connected to each other by torsional springs and to the wind tunnel walls by additional torsion springs, as indicated in the figure The four torsion springs have the same spring constant KT The wing segments are mounted on bearings

on a spindle attached to the tunnel walls

(a) Develop the equations of static equilibrium at neutral stability The three degrees of freedom are θ1, θ2 and

θ3

(b) Write the expression for the strain energy stored in this configuration as a function of the torsional displacements Use the energy method to derive the system stiffness matrix in terms of the three torsional deflections Compare this result to that found in part (a) They should be identical

(c) Solve for the three dynamic pressures that create neutral stability

(d) Find the mode shapes at all neutral stability points Describe the mode shapes (how do the surfaces deflect relative to each other?).

(e) Which of the three dynamic pressures in part (c) is the divergence dynamic pressure?

Problem 2.1

Problem 2.2

Two wing sections are mounted on shafts attached to each other and to a wind tunnel wall, as indicated Note that the two torsion spring stiffnesses are equal, but are offset an amount d This configuration is similar to, but not identical to, the example in Section 2.18 When the sections are placed in the wind tunnel at an angle of attack αo, the two springs deform, as indicated in the diagram Lift on the two, identical, uncambered wing sections is

L1 = qSCLα( αo + θ1)

L2 = qSCLα( αo + θ2)

(a) Place an aileron on the outer (right-hand) section This aileron has aerodynamic coefficients

L MAC

C and C

Develop the static equilibrium equations for the system when there is no initial angle of attack, but the control surface is deflected downward an amount δo

(b) Specialize the result in part (a) by making the aileron flap-to-chord ratio E = 0.15 with d/e = 1 and e/c = 0.10 Solve for the rolling moment generated by the aileron as a function of dynamic pressure, q Use Section 2.18

as an example of how this is done

(c) Solve for the reversal speed

Part (a): The static equilibrium equations for the system when there is no initial angle of attack, but the control

surface is deflected downward an amount δo

1 1

2

0 1

L L

d

δ α

δ

The aileron coefficients are computed to be:

and

With d/e = 1 and c/e = 10 I get:

δ

A-3

Problem 2.2

( ) ( ( ) )

1

2 2

0.4754 0.4703 0.956 0.4805

q

q

θ

−

The lift on each section is:

1

2 2

0.4754 0.4703

0.956 0.4805

q

q

 

2

2

1.021 3.011 1

2 4 1 1.00 2.011 1

2 4 1

L o

L o

δ

δ

δ δ

=

=

Roll

   

 

=     =       = +

     

2 2

L o Roll

M

=

Problem 2.3

The wing idealization is shown in the figure will be tested in the wind tunnel at several different airspeeds and in several different configurations All testing is to be done at sea level conditions and at such low speed that the flow field is incompressible Changes in the position of the wing box change the wing shear center position and the offset distance between the shear center and the aerodynamic center For instance, when the shear center is at the mid-chord, when the shear center is at the wing mid-chord the

offset distance e is equal to c/4 Preliminary wind tunnel testing shows that the wing divergence dynamic

pressure is

100 lb / ft2.

The team aerodynamicist predicts that the ratio C MAC /

CLα

is -0.075 and the lift curve slope is 4.0; the

reference planform area is

S = 10 ft.2

The section angle of attack

αo

for all tests will be pre-set to 3o

Problem

a) Develop the expression for the lift force when the shear center is located at the mid-chord Find the

twist angle θ

as a function of wind tunnel airspeed

b) Consider the same wing with different wing shear center locations The offset distance e o = c/4 when

the shear center is located at the mid-chord (part (a)) is to be used as the reference and

ei

is the offset

when the shear center is moved Develop the equations for wing lift L

and twist θ as functions of the ratio

ei

eo

Write these expressions in nondimensional form, but use the divergence q for the reference

wing as the reference in all expressions, because the divergence speed changes with

ei

eo

(c) Plot wing lift and twist angle vs airspeed for four

ei

eo

values of 0, 0.25, 0.50 and 0.75 Use an airspeed range between zero and 150 ft/sec

A-5

wing box lift

shear center range

Wind tunnel wing segment model cross-section

Hints for Problem 2.3 in the text

A rigid wing is attached to a wind-tunnel wall by a linear torsion spring This wing also has a small wing attached to its tip The idealized tip surface produces lift

LS

according to the assumed simple relationship

LS = qStipCLαSαS

where

αS

is the streamwise angle of attack of this small surface;

CLαS

is the tip surface lift curve slope, while

Stip

is the tip surface area

The tip device is connected to the wing tip by a torsion spring with stiffness k in-lb/radian When the tip device

rotates an angle β with respect to the wing tip a restoring torque kβ is generated to oppose this rotation

The idealized wing lift is

L = qSCLαα

with

α = αo+ θ

Problem statement

a) Write the equations of torsional equilibrium for this system when the entire system is given an initial angle

of attack Express these equations in terms of the following non-dimensional parameters;

L T

qSeC q

K

α

=

;

tip L s R

L

S C e

S

SC e

αδ α

=

;

R T

k k K

=

Derive the characteristic equation for aeroelastic divergence

D

q

in terms of these parameters

b) Plot the divergence dynamic pressure parameter

D

q

as a function of SR when

10

R

and

1

R

Partial answers

tip s L o

T tip s L tip s L

L

S e C K

SeC

α

α

αδ

α

β

 

V

St

gap accentuated for illustration

torsional spring

KT

shear center aerodynamic center

A

A

e s

view A-A enlarged view of cross-sectional geometry

V

β

αo+θ

Problem 2.4 - Wing with flexible tip support

2

D

q

=

with

L T

qSeC

q

K

α

=

tip L s R

L

S C e S

SC e

αδ α

=

and

R T

k k K

=

The ratio

R

is the ratio of divergence q’s for the tip device alone and the wing alone

When kr = 10 and SR = 0.1 then

0.9083

L T

qSeC q

K

α

When kr = 10 and SR = 0.5 then

0.6592

L T

qSeC q

K

α

The larger the size of the tip device, the lower the value of divergence dynamic pressure

The divergence dynamic pressure is plotted against the ratio of tip device area beginning with SR = 0.1 and ending with SR = 1

A-7

System divergence q vs SR when kR = 0.1