Aircraft Equations of Motion - 2!

8/8/2019 Aircraft Equations of Motion - 2!

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Aircraft Equations of Motion - 2Robert Stengel, Aircraft Flight Dynamics, MAE 331,

2014

Copyright 2014 by Robert Stengel. All rights reserved. For educational use only.

http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html

• How is a rotating reference frame describedin an inertial reference frame?

• Is the transformation singular?

• What adjustments must be made toexpressions for forces and moments in anon-inertial frame?

• How are the 6-DOF equations implementedin a computer?

• Damping effects

Learning Objectives

Reading: Flight Dynamics

161-180

1

Euler Angle Rates

2


2/28

Euler-Angle Ratesand Body-Axis Rates

Body-axisangular ratevector(orthogonal)

! B =

! x

! y

! z

"

#

$$$$

%

&

'''' B

=

p

q

r

"

#

$$$

%

&

'''

Euler-anglerate vector

Form a non-orthogonal vectorof Euler angles

! =

"

#

$

%

&

'''

(

)

***

!! =!"

!#

!$

%

&

'''

(

)

***+

, x

, y

, z

%

&

''''

(

)

**** I

3

Relationship Between Euler-AngleRates and Body-Axis Rates

• is measured in the Inertial Frame

• is measured in Intermediate Frame #1

• is measured in Intermediate Frame #2

• Inverse transformation [(.)-1 # (.)T]

"̇

"̇

"̇ p

q

r

!

"

###

$

%

&&&= I

3

!'

0

0

!

"

###

$

%

&&&+H

2

B

0

!(

0

!

"

###

$

%

&&&+H

2

BH

1

2

0

0

!)

!

"

###

$

%

&&&

p

q

r

!

"

###

$

%

&&&=

1 0 'sin( 0 cos) sin) cos(

0 'sin) cos) cos(

!

"

###

$

%

&&&

!) !(

!*

!

"

###

$

%

&&&=L I

B !+

!!

!"

!#

$

%

&&&

'

(

)))=

1 sin! tan" cos! tan"

0 cos! *sin!

0 sin! sec" cos! sec"

$

%

&&&

'

(

)))

p

q

r

$

%

&&&

'

(

)))=L

B

I + B

Can the inversionbecome singular?

What does this mean?

• ... which is

4


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Euler-Angle Rates and Body-Axis Rates

5

Avoiding the Singularity at " = ±90°!

Don’t use Euler angles as primarydefinition of angular attitude

! Alternatives to Euler angles

- Direction cosine (rotation) matrix

- Quaternions

!

Propagation of rotation matrix(9 parameters) - From previous lecture

!H B

I h

B = "!

I H

B

I h

B

!H I B

t ( ) = ! "" B t ( )H I B

t ( ) = !

0 !r t ( ) q t ( )

r t ( ) 0 ! p t ( )

!q t ( ) p t ( ) 0 t ( )

#

$

%%%%

&

'

((((

B

H I B

t ( )

Consequently

6H I

B0( ) =H

I

B ! 0,"

0,#

0( )


4/28

Avoiding the Singularity at " = ±90°

! Propagation of quaternion vector

o see Flight Dynamics for details

e1

e2

e3

e4

!

"

#####

$

%

&&&&&

=

Rotation angle, rad x-component of rotation axis

y-component of rotation axis

z-component of rotation axis

!

"

#####

$

%

&&&&&

! Quaternion vector: single rotation from

inertial to body frame (4 parameters)

!

e t ( )=

!e1

t ( )

!e2

t ( )

!e3 t ( )

!e4

t ( )

!

"

##

####

$

%

&&

&&&&

=

0 'r t ( ) 'q t ( ) ' p t ( )

r t ( ) 0 ' p t ( ) q t ( )

q t ( ) p t ( ) 0 'r t ( )

p t ( ) 'q t ( ) r t ( ) 0

!

"

##

####

$

%

&&

&&&&

e1

t ( )

e2

t ( )

e3

t ( )

e4

t ( )

!

"

##

####

$

%

&&

&&&&

=

Q t ( )e t ( ); e 0( )=

e ( 0,) 0 ,* 0( )

7

Rigid-BodyEquations of Motion

8


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Point-MassDynamics

• Inertial rate of change of translational position

• Body-axis rate of change of translational velocity –

Identical to angular-momentum transformation

!r I = v

I = H

B

I v

B

!v I =

1

mF I

!v B

= H I

B!v I

! "" B

v B

=

1

mH

I

BF I

! "" B

v B

=

1

mF B ! ""

Bv B

F B

=

X

Y Z

!

"

#

##

$

%

&

&& B

=

C X qS

C Y

qS

C Z qS

!

"

###

$

%

&&&

v B =

u

v

w

!

"

###

$

%

&&&

9

!r I t ( ) =H B

I t ( )v B t ( )

!v B

t ( ) =1

m t ( )F

B t ( )+H I

Bt ( )g I ! "" B t ( )v B t ( )

!! I t ( ) =L B

I t ( )" B t ( )

!! B t ( ) = I B

"1t ( ) M B t ( )" "! B t ( ) I B t ( )! B t ( )#$ %&

• Rate of change ofTranslational Position

• Rate of change ofAngular Position

• Rate of change ofTranslational Velocity

• Rate of change ofAngular Velocity

r I =

x

y

z

!

"

###

$

%

&&& I

! I =

"

#

$

%

&

'''

(

)

***

I

v B =

u

v

w

!

"

###

$

%

&&& B

! B =

p

q

r

"

#

$$$

%

&

''' B

• TranslationalPosition

• AngularPosition

• TranslationalVelocity

• AngularVelocity

Rigid-Body Equations of Motion(Euler Angles)

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6/28

Aircraft CharacteristicsExpressed in Body Frame

of Reference

I B =

I xx ! I xy ! I xz

! I xy I yy ! I yz

! I xz ! I yz I zz

"

#

$$$$

%

&

'''' B

F B =

X aero + X thrust

Y aero +Y thrust

Z aero + Z thrust

!

"

###

$

%

&&& B

=

C X aero +C X thrust

C Y aero +C Y thrust

C Z aero +C Z thrust

!

"

####

$

%

&&&& B

1

2' V

2S =

C X

C Y

C Z

!

"

###

$

%

&&& B

q S Aerodynamicand thrustforce

Aerodynamic andthrust moment

Inertiamatrix

Reference Lengths

b = wing span

c = mean aerodynamic chord

M B =

Laero + Lthrust

M aero + M thrust

N aero + N thrust

!

"

###

$

%

&&& B

=

C laero +C lthrust ( )b

C maero +C mthrust ( )c

C naero

+C nthrust ( )b

!

"

#####

$

%

&&&&& B

1

2' V

2S =

C lb

C mc

C nb

!

"

###

$

%

&&& B

q S

11

Rigid-Body Equations of Motion:Position

! x I = cos! cos" ( )u + #cos$ sin" + sin$ sin! cos" ( )v + sin$ sin" + cos$ sin! cos" ( )w

! y I = cos! sin" ( )u + cos$ cos" + sin$ sin! sin" ( )v + #sin$ cos" + cos$ sin! sin" ( )w

! z I = # sin! ( )u + sin$ cos! ( )v + cos$ cos! ( )w

!! = p + qsin! + r cos! ( ) tan" !" = qcos! # r sin!

!$ = qsin! + r cos! ( )sec"

• Rate of change of Translational Position

• Rate of change of Angular Position

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Rigid-Body Equations of Motion:Rate

!u = X / m

!

gsin" + rv!

qw

!v =Y /m+ gsin# cos" ! ru+ pw

!w = Z /m+ gcos# cos" +qu! pv

! p = I zz L+ I xz N ! I xz I yy ! I xx ! I zz( ) p+ I xz2 + I zz I zz ! I yy( )"# $%r{ }q( ) I xx I zz ! I xz2( )

!q =1

I yy

M ! I xx ! I zz( ) pr ! I xz p2 ! r2( )"# $%

!r = I xz L+ I xx N ! I xz I yy ! I xx ! I zz( )r+ I xz2 + I xx I xx ! I yy( )"# $% p{ }q( ) I xx I zz ! I xz2( )

• Rate of change of Translational Velocity

• Rate of change of Angular Velocity

Mirror symmetry, I xz # 0 13

FLIGHT -Computer Program to

Solve the 6-DOFEquations of Motion

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http://www.princeton.edu/~stengel/FlightDynamics.html

FLIGHT - MATLAB Program

15

http://www.princeton.edu/~stengel/FlightDynamics.html

FLIGHT - MATLAB Program

16


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Examples from FLIGHT

17

Longitudinal TransientResponse to Initial Pitch Rate

Bizjet, M = 0.3, Altitude = 3,052 m

• For a symmetric aircraft, longitudinalperturbations do not induce lateral-directional motions

18


10/28

Transient Responseto Initial Roll Rate

Lateral-Directional Response Longitudinal Response

Bizjet, M = 0.3, Altitude = 3,052 m • For a symmetric aircraft, lateral-directional perturbations do induce longitudinal motions

19

Transient Response toInitial Yaw Rate

Lateral-Directional Response Longitudinal Response


20


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Crossplot of TransientResponse to Initial Yaw Rate


Longitudinal-Lateral-Directional Coupling

21

Aerodynamic Damping

22


12/28

Pitching Moment due to Pitch Rate

M B = C

mq Sc ! C

mo+C

mqq +C

m"

" ( )q Sc

! C mo +#C

m

#qq +C

m"

" $

% &

'

( ) q Sc

23

Angle of Attack DistributionDue to Pitch Rate

• Aircraft pitching at a constant rate, q rad/s, produces anormal velocity distribution along x

!w = "q! x

!" =!w

V =

#q! x

V

• Corresponding angle of attack distribution

!" ht =qlht

V

• Angle of attack perturbation at tail center of pressure

lht = horizontal tail distance from c.m.

24


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Horizontal Tail LiftDue to Pitch Rate

• Incremental tail lift due to pitch rate, referenced to tail area, S ht

• Lift coefficient sensitivity to pitch rate referenced to wing area

! Lht = !C

Lht ( )

ht

1

2" V

2S ht

C Lqht !" #C Lht ( )aircraft

" q=

" C Lht

"$

% & '

( ) *

aircraft

lht

V

% & '

( ) *

!C Lht ( )aircraft = !C Lht ( )ht S ht

S

" # $

% & ' =

( C Lht ()

" # $

% & '

aircraft

!) *

+,,

-

.//=

( C Lht ()

" # $

% & '

aircraft

qlht

V

" # $

% & '

• Incremental tail lift coefficient due to pitch rate,referenced to wing area, S

25

Moment CoefficientSensitivity to Pitch Rate of

the Horizontal Tail

• Differential pitch moment due to pitch rate

! " M ht ! q

= C mqht

1

2# V 2Sc = $C Lqht

lht

V

% & '

( ) * 1

2# V 2Sc

= $ ! C Lht

!+

% & '

( ) * aircraft

lht

V

% & '

( ) *

,

-..

/

011

lht

c

% & '

( ) * 1

2# V 2Sc

C mqht = !

" C Lht

"#

lht

V

$ % &

' ( ) lht

c

$ % &

' ( ) = !

" C Lht

"#

lht

c

$ % &

' ( ) 2

c

V

$ % & ' ( )

• Coefficient derivative with respect to pitch rate

• Coefficient derivative with respect to normalized pitch rate isinsensitive to velocity

C m q̂ht =

! C mht

! q̂=

! C mht

! qc

2V ( ) = "2

! C Lht

!#

lht

c

$ % &

' ( )

2

26


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15/28

Roll Damping Dueto Roll Rate

•

Tapered vertical tail

• Tapered horizontal tail

ˆ p = pb

2V

C l ˆ p( )ht =! "C l( )ht

! ˆ p= #

C L$ ht

12

S ht

S

%

&'

(

)* 1+ 3+

1++

%

&'

(

)*

•

pb/2V describes helixangle for a steady roll

C l ˆ p( )vt =! "C l( )vt

! ˆ p= #

C Y $ vt

12

S vt

S

%

&'

(

)* 1+ 3+

1++

%

&'

(

)*

29

Yaw Damping Due to Yaw Rate

C nr

! V 2

2

" # $

% & ' Sb = C

n r̂

b

2V

" # $

% & '

! V 2

2

" # $

% & ' Sb

= C n r̂

! V

4

"

# $

%

& ' Sb

2

< 0 for stability

30

r̂ =rb

2V


16/28

Yaw Damping Dueto Yaw Rate

C n r̂ ! C n r̂( )Vertical Tail + C n r̂( )Wing

C n r̂( )Wing = k 0C L2

+ k 1C DParasite,Wing

k 0 and k 1 are functions ofaspect ratio and sweep angle

• Wing contribution

• Vertical tail contribution

! C n( )Vertical Tail = " C n# ( )

Vertical Tail

rlvt

V ( ) = " C n# ( )Vertical Taill

vt

b

$ % &

' ( )

b

V

$ % & ' ( )

r

r̂ =rb

2V

C n r̂

( )vt

=

! " C n( )Vertical Tail

! rb2V ( )

=

! " C n( )Vertical Tail! r̂

= #2 C n$ ( )

Vertical Tail

lvt

b

% & '

( ) *

NACA-TR-1098, 1952 NACA-TR-1052, 1951 31

Next Time: Aircraft Control Devices and

Systems

Reading: Flight Dynamics

214-234

32


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Supplemental

Material

33

Airplane Angular Attitude (Position)

• Euler angles• 3 angles that relate one

Cartesian coordinate frame toanother

• defined by sequence of 3rotations about individual axes

•

intuitive description of angularattitude

• Euler angle rates have anonlinear relationship to body-axis angular rate vector

• Transformation of rates issingular at 2 orientations, ±90°

! ," ,# ( )

34


18/28


H I

B(! ," ,# ) =

h11

h12

h13

h21 h22 h23

h31 h32 h33

$

%

&&&

'

(

))) I

B

H I

B(! ," ,# ) = H

2

B(! )H

1

2(" )H

I

1(# )

H I

B(! ," ,# )$% '(

*1= H

I

B(! ," ,# )$% '(

T

= H B

I (# ," ,! )

• Rotation matrix

• orthonormal transformation

• inverse = transpose

• linear propagation from one attitude toanother, based on body-axis rate vector

•

9 parameters, 9 equations to solve

• solution for Euler angles from parametersis intricate

35


H I

B(! ," ,# ) = H

2

B(! )H

1

2(" )H

I

1(# )

H I

B(! ," ,# ) =

1 0 0

0 cos! sin!

0 $sin! cos!

%

&

'''

(

)

***

cos" 0 $sin"

0 1 0

sin" 0 cos"

%

&

'''

(

)

***

cos# sin# 0

$sin# cos# 0

0 0 1

%

&

'''

(

)

***

H I

B(! ," ,# ) =

cos" cos# cos" sin# $sin"

$cos! sin# + sin! sin" cos# cos! cos# + sin! sin" sin# sin! cos"

sin! sin# + cos! sin" cos# $sin! cos# + cos! sin" sin# cos! cos"

%

&

'''

(

)

***

Rotation Matrix

H I

BH

B

I = I for all (! ," ,# ), i.e., No Singularities 36


19/28

Angles betweeneach I axis and each

B axis


Rotation Matrix = Direction Cosine Matrix

H I

B=

cos! 11

cos! 21

cos! 31

cos! 12

cos! 22

cos! 32

cos! 13

cos! 23

cos! 33

"

#

$$

$

%

&

''

'

37


e1

e2

e3

e4

!

"

#####

$

%

&&&&&

=

Rotation angle, rad

x-component of rotation axis

y-component of rotation axis

z-component of rotation axis

!

"

#####

$

%

&&&&&

!

Quaternion vector!

single rotation from inertial to body frame

!

non-singular, linear propagation of attitude based on body-axis rate vector

!

4 parameters; more compact than the rotation matrix

!

solution for rotation matrix and Euler angles fromparameters is intricate

38


20/28


Rotation Matrix from Quaternion Vector

H I

B=

e1

2 ! e2

2 ! e3

2+ e

4

22 e

1e2 + e

3e4( ) 2 e2e4 ! e1e3( )

2 e3e4 ! e

1e2( ) e1

2 ! e2

2+ e

3

2+ e

4

22 e

2e3 + e

1e4( )

2 e1e3 + e

2e4( ) 2 e2e3 ! e1e4( ) e1

2+ e

2

2 ! e3

2+ e

4

2

"

#

$$$$

%

&

''''

Euler Angles from Quaternion:

see p. 186, Flight Dynamics 39

Euler Angle Dynamics

!! =

!"

!#

!$

%

&

'''

(

)

***=

1 sin" tan# cos" tan#

0 cos" +sin"

0 sin" sec# cos" sec#

%

&

'''

(

)

***

p

q

r

%

&

'''

(

)

***

= L B I , B

L B I is not orthonormal

L B

I is singular when ! = ± 90°

40


21/28


22/28

!r I

= H B

I v

B

!v B =

1

mF B + H

I

Bg I ! ""

Bv B

!! B

= I B

"1M

B

" "! B

I B

! B( )

Rigid-Body Equations of Motion

(Attitude from 9-Element Rotation Matrix)

!H I

B= ! ""

BH

I

B

No need for Euler angles to solve the dynamic equations 43

Quaternion Vector Dynamics

!e t ( ) =

!e1

t ( )

!e2

t ( )

!e3

t ( )

!e4

t ( )

!

"

######

$

%

&&&&&&

=Q t ( )e t ( )

=

0 'r t ( ) 'q t ( ) ' p t ( )

r t ( ) 0 ' p t ( ) q t ( )

q t ( ) p t ( ) 0 'r t ( )

p t ( ) 'q t ( ) r t ( ) 0

!

"

######

$

%

&&&&&&

e1

t ( )

e2

t ( )

e3

t ( )

e4

t ( )

!

"

######

$

%

&&&&&&

44


23/28

!r I = H

B

I v

B

!v B =

1

mF B + H

I

Bg I ! ""

Bv B

!! B = I B"1M

B " "! B I B! B( )

Rigid-Body Equations of Motion(Attitude from 4-Element Quaternion Vector)

!e =Qe H B

I , H

I

B are functions of e

45

Alternative ReferenceFrames

46


24/28

Velocity Orientation in an InertialFrame of Reference

Polar Coordinates Projected on a Sphere

47

Body Orientation with Respectto an Inertial Frame

48


25/28

Relationship of Inertial Axesto Body Axes

•

Transformation isindependent ofvelocity vector

• Represented by

– Euler angles

– Rotation matrix

v x

v y

v z

!

"

##

##

$

%

&&

&&

= H B I

u

v

w

!

"

###

$

%

&&&

u

v

w

!

"

###

$

%

&&&

= H I B

v x

v y

v z

!

"

####

$

%

&&&&

49

Velocity-Vector Componentsof an Aircraft

V , ! , "

V , ! ,"

50


26/28

Velocity Orientation with Respectto the Body Frame

Polar Coordinates Projected on a Sphere

51

• No reference to the bodyframe

• Bank angle, % , is roll angleabout the velocity vector

V

!

"

#

$

%%%

&

'

(((

=

v x2+ v y

2+ v z

2

sin)1

v y / v

x

2+ v

y

2( )1/2#

$&'

sin)1 )v

z /V ( )

#

$

%%%%%

&

'

(((((

v x

v y

v z

!

"

#

###

$

%

&

&&& I

=

V cos' cos(

V cos' sin(

)V sin'

!

"

#

##

$

%

&

&&

Relationship of Inertial Axesto Velocity Axes

52


27/28

Relationship of Body Axes to Wind Axes

• No reference tothe inertial frame

u

v

w

!

"

#

##

$

%

&

&&

=

V cos' cos(

V sin(

V sin' cos(

!

"

#

##

$

%

&

&&

V

! "

#

$

%

%%

&

'

(

((

=

u2+ v

2+ w

2

sin)1

v /V

( )tan

)1w / u( )

#

$

%%%%

&

'

((((

53

Angles Projectedon the Unit Sphere

" : angle of attack

# : sideslip angle

$ : vertical flight path angle

% : horizontal flight path angle

& : yaw angle

' : pitch angle

( : roll angle (about body x ) axis)

µ : bank angle (about velocity vector)

• Origin isairplanes center

of mass

54


28/28

Alternative Frames of Reference• Orthonormal transformations connect all reference frames

55

Aircraft Equations of Motion - 2!

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Transcript of Aircraft Equations of Motion - 2!