2nd Semester, 2011

by

Dr.Chinnapat Thipyopas

Department of Aerospace Engineering Faculty of Engineering, Kasetsart University

(Flight Dynamics: Stability and Control)

This Eq.,

- we consider only internal force from aircraft configuration

(Matrix “A”) + internal forces due to the control of aircraft

(Matrix “B”; control surface + throttle).

- In addition, flight condition is uniform. Derivatives of forces &

moments are induced by change of the motion of aircraft itself.

BηAxx

BηAxx

r

a

rp

rp

rp

ra

ra

r

NN

LL

u

Y

r

p

NNN

LLL

u

g

u

Y

u

Y

u

Y

r

p

00

0

0010

0

0

cos)1(

00

0

000

T

T

Zw

M

T

MZw

MMT

ZZT

XX

q

w

u

gw

Muw

Mq

Mw

Zw

Mw

Mu

Zw

Mu

M

guw

Zu

Z

gw

Xu

X

q

w

u

00

0100

0sin

0

0sin

0

0cos0

BηAxx

Control Input

Characteristic of

Aircraft

Last 2 classes, only the free response of aircraft is examined.

η = 0

BηAxx

Control Input

Characteristic of

Aircraft

To control AC (change speed/altitude/turn/etc..),

the control input (η) are applied

Pure Pitching Motion eq eMMMM )(

solution for a step change e in the elevator angle

)1sin(1

1)( 2

2

te

t n

t

trim

n

η ≠ 0

input

outputnctionTransferFu

inputnctionTransferFutoInputOutput )(

Longitudinal Equations of Motion [M.V.Cook]

Apply Laplace Transforms

and

Then

Writing in Matrix format

Now Cramer’s rule can be applied to obtain the response transfer

function to an elevator’s input ( η ), and a throttle input ( τ );

Here we exhibit only response to elevator, by assume thrust

remains constant (τe), so τ(s)=0

Dividing through eq. 5.9 by η(s)

Apply Cramer’s rule to find the elevator response transfer function

The numerator polynomials are given by the determinants of

and

Where the common denominator polynomial is the determinant:

The response for lateral motion can be achieve via the similar

concept

for Ex. Roll rate response to aileron

Response to Control

The aircraft response to controls are readily obtained by finding

the inverse Laplace Transform

The most commonly used inputs are impulse and step functions

Therefore, for example roll rate response to an aileron step

input of magnitude k is

In Real Flight

It is rarely calm but usually windy, gusts + turbulence

Wind gust created by movement of atmospheric air mass (driven by solar heating, earth’s rotation, …)

can degrade the performance & flying qualities of aircraft

G.Norman photo © 1996

Local variation (speed and direction) of the wind vector (measured in vertical and horizontal direction).

Produced by boundary layer, thunderstorms, two different air masses (Fig.-above), etc

http://www.youtube.com/watch?v

=DhR-fivekCY&feature=related

http://www.youtube.com/watch?v

=IxB5-r32s9s&feature=related

Force / Moment depend on the relative motion of AC to atmosphere

Rewrite velocity as term of inertial and gust

gagaga

gagaga

rrrqqqppp

wwwvvvuuu

Force / Moment are modified

Produced by variation of

ggg rqp ,,

ggg wvu ,,

x

wq

y

wp

g

g

g

g

Previously (in uniform flow; ch 3), we have:

For ex. Force in X-axis

With gust or any nonuniform flow

......

wXuXw

w

Xu

u

XX wu

...)()(

...)()(

gwgu

gg

wwXuuX

www

Xuu

u

XX

can be expressed as

gq

gw

gu

qM

wM

uM

wZ

uZ

wX

uX

T

T

Zw

M

T

MZw

MMT

ZZT

XX

q

w

u

gw

Muw

Mq

Mw

Zw

Mw

Mu

Zw

Mu

M

guw

Zu

Z

gw

Xu

X

q

w

u

000

0

0

00

0100

0sin

0

0sin

0

0cos0

Longitudinal Equation

g

g

g

rpv

rpv

v

r

a

rp

rp

rp

r

p

v

NNN

LLL

Y

NN

LL

u

Y

r

p

NNN

LLL

u

g

u

Y

u

Y

u

Y

r

p

ra

ra

r

000

00

00

0

0010

0

0

cos)1(

0

0

0

000

Lateral Equation

CBηAxx

Characteristic of

Aircraft Control Input

Atmospheric

Input

See example 6.2

Open Loop

Stability Augmentation (Control)

To improve stability of aircraft, FCS (Flight Control System) is used, then

air data measurements and motion sensors must be added

The detail will be discussed next classes by

Dr.Chaiwat

About 2nd Examination

Date: