Aircraft Flight Dynamics Robert F. Stengel Lecture9 Aircraft Equations of Motion 2

Aircraft Equations of Motion - 2 Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012 " • Rotating frames of reference" • Combined equations of motion" • FLIGHT 6-DOF simulation pr

Aircraft Equations of Motion - 2


Robert Stengel, Aircraft Flight Dynamics, MAE 331,

2012 "

•   Rotating frames of reference"

•   Combined equations of

motion"

•   FLIGHT 6-DOF simulation

program"

Copyright 2012 by Robert Stengel All rights reserved For educational use only.!

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

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

Euler Angle Rates

Euler-Angle Rates and Body-Axis Rates"

Body-axis

angular rate

vector

( orthogonal ) " ωB=

ωx

ωy

ωz

"

#

$

$

$

$

%

&

' ' ' 'B

=

p q r

"

#

$

$

$

%

&

' ' '

Euler-angle

rate vector "

non-orthogonal vector

φ θ ψ

%

&

' ' ' (

)

*

*

*

Θ =

φ

θ

 ψ

%

&

' ' ' (

)

*

*

*≠

ωx

ωy

ωz

%

&

' ' ' '

(

)

*

*

*

*I

Relationship Between Euler-Angle Rates 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 ] "

€

˙

φ

€

˙

θ

€

˙

q r

!

"

#

#

#

$

%

&

&

&

= I3

φ 0 0

!

"

#

#

#

$

%

&

&

&+ H2

0

 θ 0

!

"

#

#

#

$

%

&

&

&

+ H2H1

0 0

 ψ

!

"

#

#

#

$

%

&

&

&

p q r

!

"

#

#

#

$

%

&

&

&

=

0 cosφ sinφ cosθ

0 −sinφ cosφ cosθ

!

"

#

#

#

$

%

&

&

&

φ

 θ

 ψ

!

"

#

#

#

$

%

&

&

&

=LI

BΘ

 φ

 θ

 ψ

$

%

&

&

&

'

(

) ) )=

1 sinφtanθ cosφtanθ

0 cosφ −sinφ

0 sinφsecθ cosφsecθ

$

%

&

&

&

'

(

) ) )

p q r

$

%

&

&

&

'

(

) ) )=

LI B

ωB

"Can the inversion become singular?"

" What does this mean? "

•  which is "

Euler-Angle Rates and Body-Axis Rates" Avoiding the Singularity at θ = ±90°"

!  Don’t use Euler angles as primary

definition of angular attitude"

!  Alternatives to Euler angles"

-   Direction cosine (rotation) matrix"

-   Quaternions"

!  Propagation of rotation matrix

(9 parameters)"

-   From previous lecture"

HI B( )t = − ωB( )t HI B( )t = −

0 −r t( ) q t( )

−q t( ) p t( ) 0 t( )

#

$

%

%

%

%

&

'

( ( ( (

B

HI B( )t ; HI B( )0 = HI

B

φ0,θ0,ψ0

Consequently"

Avoiding the Singularity at θ = ±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

e t( )=

e1( )t

e2( )t

e3( )t

e4( )t

!

"

#

#

#

#

#

#

$

%

&

&

&

&

&

&

=

0 −r t( ) −q t( ) − p t( )

r t( ) 0 − p t( ) q t( )

p t( ) −q t( ) r t( ) 0

!

"

#

#

#

#

#

#

$

%

&

&

&

&

&

&

e1( )t

e2( )t

e3( )t

e4( )t

!

"

#

#

#

#

#

#

$

%

&

&

&

&

&

&

= Q t( )e t( ); e 0( )= e(φ0,θ0,ψ0)

Rigid-Body Equations of Motion

Point-Mass Dynamics"

•  Inertial rate of change of translational position"

•  Body-axis rate of change of translational velocity "

–  Identical to angular-momentum transformation "

rI = vI = HB IvB

vI = 1

m FI

vB = HI BvI −  ωBvB = 1

m HI

BFI −  ωBvB

= 1

m FB−  ωBvB

FB=

X Y Z

!

"

#

#

#

$

%

&

&

&B

=

CXqS

CYqS

CZqS

!

"

#

#

#

$

%

&

&

&

vB=

u v w

!

"

#

#

#

$

%

&

&

vB( )t = 1

m t( )FB( )t + HI B( )t gI− ωB( )t vB( )t

ΘI( ) t = LI B( ) t ωB( ) t

ωB( )t = I B

−1( )t #$MB( )t − ωB( )t I B( )t ωB( )t%&

•  Rate of change of Translational Position "

•  Rate of change of Angular Position "

•  Rate of change of Translational Velocity "

•  Rate of change of Angular Velocity "

rI=

x y z

!

"

#

#

#

$

%

&

&

&

I

ΘI=

φ θ ψ

%

&

' ' ' (

)

*

*

*

I

vB=

u v w

!

"

#

#

#

$

%

&

&

&B

ωB=

p q r

"

#

$

$

%

&

' '

B

Position "

Position "

Velocity"

Velocity "

Rigid-Body Equations of Motion"

(Euler Angles)"

Aircraft Characteristics Expressed 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

"

#

$

$

$

$

%

&

' ' ' '

FB=

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

2ρV

2

S =

C X

C Y

C Z

!

"

#

#

#

$

%

&

&

&

B

q S

Aerodynamic

and thrust

force "

Aerodynamic and

thrust moment "

Inertia

matrix "

Reference Lengths

b = wing span

c = mean aerodynamic chord

MB=

L aero + L thrust

M aero + M thrust

N aero + N thrust

!

"

#

#

#

$

%

&

&

&

B

=

!

"

#

#

#

#

#

$

%

&

&

&

&

&

B

1

2ρV

2

S =

C l b

C m c

C n b

!

"

#

#

#

$

%

&

&

&

B

q S

Rigid-Body Equations of Motion:

Position "

x I= cos( θcosψ )u+ − cos( φsinψ+ sinφsinθcosψ )v+ sin( φsinψ+ cosφsinθcosψ )w

y I= cos( θsinψ )u+ cos( φcosψ+ sinφsinθsinψ )v+ − sin( φcosψ+ cosφsinθsinψ )w

z I= − sin( θ )u+ sin( φcosθ )v+ cos( φcosθ )w

φ = p + ( q sinφ + r cosφ ) tanθ

θ = q cosφ − r sinφ



ψ = ( q sinφ + r cosφ ) secθ

Rigid-Body Equations of Motion:

Rate "

u = X / m − gsinθ + rv − qw

v = Y / m + gsinφ cosθ − ru + pw

w = Z / m + g cosφ cosθ + qu − pv

p = I ( zzL + IxzN − I { xz( Iyy− Ixx− Izz) p + I " xz2 + Izz( Izz− Iyy) $ } q ) IxxIzz− Ixz2

q = 1

Iyy M − Ixx ( − Izz ) pr − Ixz p2

− r2

r = I ( xzL + IxxN − I { xz( Iyy− Ixx− Izz) r + I " xz2 + Ixx( Ixx− Iyy) $ p } q ) ( IxxIzz− Ixz2)

Mirror symmetry, I xz ≠ 0#

FLIGHT - Computer Program to Solve the 6-DOF Equations of Motion

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

FLIGHT - MATLAB Program"

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

FLIGHT - MATLAB Program"

Examples from FLIGHT

Longitudinal Transient

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

perturbations do not induce lateral-directional motions "

Transient Response

Lateral-Directional Response" Longitudinal Response"

Bizjet, M = 0.3, Altitude = 3,052 m! •  For a symmetric aircraft,

induce longitudinal motions "

Transient Response to

Lateral-Directional Response" Longitudinal Response"

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

Crossplot of Transient

Response to Initial Yaw Rate"

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

Longitudinal-Lateral-Directional Coupling"

Alternative Reference

Frames

Velocity Orientation in an Inertial

Frame of Reference"

Polar Coordinates" Projected on a Sphere"

Body Orientation with Respect

to an Inertial Frame"

Relationship of Inertial Axes

independent of

velocity vector"

–  Euler angles"

–  Rotation matrix "

vx

vy

vz

!

"

#

#

#

#

$

%

&

&

&

&

= HB I

u v w

!

"

#

#

#

$

%

&

&

&

u v w

!

"

#

#

#

$

%

&

&

&

vx

vy

vz

!

"

#

#

#

#

$

%

&

&

&

&

Velocity-Vector Components

of an Aircraft"

Velocity Orientation with Respect

to the Body Frame"

Polar Coordinates" Projected on a Sphere"

frame"

about the velocity vector "

V

ξ γ

#

$

%

%

%

&

'

( ( (

=

v x + v y + v z

sin−1 v y / v( x + v y)1/2

#

sin−1

−v z / V

#

$

%

%

%

%

%

&

'

( ( ( ( (

vx

vy

vz

!

"

#

#

#

#

$

%

&

&

&

&I

=

V cosγ cosξ

V cosγ sinξ

−V sinγ

!

"

#

#

#

$

%

&

&

&

Relationship of Inertial Axes

Relationship of Body Axes

the inertial frame "

u

v

w

!

"

#

#

#

$

%

&

&

&

=

V cosα cos β

V sin β

V sinα cos β

!

"

#

#

#

$

%

&

&

&

V

β α

#

$

%

%

%

&

'

( ( (

=

u2

+ v2

+ w2

sin−1

v / V

( )

tan−1(w / u)

#

$

%

%

%

%

&

'

( ( ( (

Angles Projected

on the Unit Sphere"

€

α : angle of attack

β : sideslip angle

γ : vertical flight path angle

ξ : horizontal flight path angle

ψ : yaw angle

θ : pitch angle

φ : roll angle (about body x − axis)

µ : bank angle (about velocity vector)

•   Origin is airplane s center

of mass"

Alternative Frames of Reference"

•  Orthonormal transformations connect all reference frames" Next Time:

Linearization and Modes of

Motion

Reading Flight Dynamics, 234-242, 255-266, 274-297, 321-330 Virtual Textbook, Part 10

Supplemental Material

r I = H B I v B

vB= 1

m FB+ HI

BgI −  ωBvB

ωB = IB−1( MB−  ωBIBωB)

•  Rate of change of Translational Position "

•  Rate of change of Rotation Matrix "

•  Rate of change of Translational Velocity "

•  Rate of change of Angular Velocity "

rI=

x y z

!

"

#

#

#

$

%

&

&

&

I

ΘI = fcn H I B

( )

vB=

u v w

!

"

#

#

#

$

%

&

&

&B

ωB=

p q r

"

#

$

$

%

&

' '

B

Position "

Position "

Velocity"

Velocity "

Rigid-Body Equations of Motion" (Attitude from Rotation Matrix)"

HI B

= −  ωBHI B

r I = H B I v B

vB= 1

m FB+ HI

BgI −  ωBvB

ωB = IB−1( MB−  ωBIBωB)

•  Rate of change of

Translational Position "

•  Rate of change of

Rotation Matrix "

•  Rate of change of

Translational Velocity "

•  Rate of change of

Angular Velocity "

rI=

x y z

!

"

#

#

#

$

%

&

&

&

I

ΘI = fcn H I

B( )e

"# $%

vB=

u v w

!

"

#

#

#

$

%

&

&

&B

ωB=

p q r

"

#

$

$

%

&

' '

B

Position "

Position "

Velocity"

Velocity "

Rigid-Body Equations of Motion"

(Attitude from Quaternion Vector)"

e = Qe