Equations of Motion With Different Axis

8/6/2019 Equations of Motion With Different Axis

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Aircraft Equations of MotionRobert Stengel, Aircraft Flight Dynamics

MAE 331, 2008

Angular kinematics

Euler angles

Rotation matrix

Angular momentum

Inertia matrix

Rotating frames of reference

Combined equations of motion

Copyright 2008 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

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

Nonlinear equations of motion

Compute exact flight paths andmotions

Simulate flight motions

Optimize flight paths

Predict performance

Provide basis for approximatesolutions

Linear equations of motion

Simplify computation offlight paths and solutions

Define modes of motion

Provide basis for controlsystem design and flyingqualities analysis

What Use are the Equations of Motion?

dx(t)

dt= f x(t),u(t),w(t),p(t),t[ ]

dx(t)

dt= Fx(t)+Gu(t) +Lw(t)

Cartesian Frames

of Reference

Two reference frames of interest I: Inertial frame (fixed to inertial space)

B: Body frame (fixed to body)

Common convention (zup) Aircraft convention (zdown)

Translation Relative linear positions of origins

Rotation Orientation of the body frame with

respect to the inertial frame

Measurement of Position in

Alternative Frames - 1

Two reference frames of interest I: Inertial frame (fixed to inertial

space)

B: Body frame (fixed to body)

However, differences in frameorientationsmust be taken intoaccount in adding vector components

r =

x

y

z

"

#

$$$

%

&

'''

rparticle = rorigin +(rw.r.t. origin

Inertial-axis view

Body-axis view


2/10

Euler Angles Measure the Orientation of

One Frame with Respect to the Other

Conventional sequence of rotations from inertial to body frame Each rotation is about a single axis

Right-hand rule

Yaw, then pitch, then roll

These are called Euler Angles

Yaw rotation (! ) about zI Pitch rotation (") about y1 Roll rotation (#) about x2 Other sequences of 3 rotations can be chosen; however, once

sequence is chosen, it must be retained

Effects of Rotation on

Vector TransformationYaw rotation (! ) about zI

Pitch rotation (") about y1

Roll rotation (#) about x2

x

y

z

"

#

$$

$

%

&

''

'1

=

cos( sin( 0

)sin( cos( 0

0 0 1

"

#

$$

$

%

&

''

'

x

y

z

"

#

$$

$

%

&

''

'I

=

xI

cos(+ yI

sin(

)xI

sin(+ yI

cos(

zI

"

#

$$

$

%

&

''

'

; r1=H

I

1rI

x

y

z

"

#

$$$

%

&

'''

2

=

cos( 0 )sin(

0 1 0

sin( 0 cos(

"

#

$$$

%

&

'''

x

y

z

"

#

$$$

%

&

'''

1

; r2=H

1

2r

1=H

1

2H

I

1rI=H

I

2rI

x

y

z

"

#

$$$

%

&

'''B

=

1 0 0

0 cos( sin(

0 )sin( cos(

"

#

$$$

%

&

'''

x

y

z

"

#

$$$

%

&

'''

2

; rB=H

2

Br

2=H

2

BH

I

2rI=H

I

BrI

The Rotation Matrix

HI

B(",#,$) =H2

B(")H

1

2(#)H

I

1($)

=

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

&

'

(((

)

*

+++

=

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#

&

'

(((

)

*

+++

The three-angle rotation matrix is theproductof 3 single-angle rotation matrices:

Properties of the Rotation Matrix

HI

B(",#,$) =H2

B(")H1

2(#)H

I

1($)

The rotation matrix produces an orthonormal transformation Angles are preserved

Lengths are preserved

rI= r

B; s

I= s

B

"(rI,s

I) ="(r

B,s

B) = xdeg

Inverse relationship

Because transformation is orthonormal, Inverse = transpose

Rotation matrix is always non-singular

rB=H

I

BrI

; rI= H

I

B( )"1

rB=H

B

IrB

HB

I= H

I

B( )"1

= HI

B( )T

=H1

IH

2

1H

B

2

HBI

HIB

=HI

B

HBI

=I


3/10

Measurement of Position in

Alternative Frames - 2

rparticleI= rorigin"BI

+ HBI#rB

Inertial-axis view

Body-axis view

rparticleB= rorigin"IB

+ HIB#rI

Angular Momentum

of a Particle Moment of linear momentum of differential

particles that make up the body Differential masses times components of their

velocity that are perpendicular to the momentarms from the center of rotation to the particles

Cross Product

r " v =

i j k

x y z

vx vy vz

= yvz #zvy( )i + zvx # xvz( )j+ xvy # yvx( )k

dh = r " dmv( ) = r " vm( )dm

= r " vo+ # " r( )( )dm

" =

"x

"y

"z

#$%%%

&'(((

Angular Momentum

of the Aircraft

Integrate moment of linear momentum of differential particles over the body

h = r " vo +# " r( )( )dmBody

$ = r " v( )%(x,y,z)dx dy dzzmin

zmax

$ymin

ymax

$xmin

xmax

$ =hxhy

hz

&

'

(((

)

*

+++

%(x,y,z) = Density of the body

h = r " vo( )dmBody

# + r " $ " r( )( )dmBody

#

= 0% r " r "$( )( )dmBody

# = % r " r( )dm "$Body

#

& % rr( )dm$Body

#

Choosing the center of mass as the rotational center

Cross-Product-

Equivalent Matrix

r " v =

i j k

x y z

vx vy vz

= yvz # zvy( )i + zvx # xvz( )j+ xvy # yvx( )k

=

0 #z y

z 0 #x

#y x 0

$

%

&&&

'

(

)))

vx

vy

vz

$

%

&&&

'

(

)))

=

yvz # zvy( )zvx # xvz( )xvy # yvx( )

$

%

&&&

'

(

)))= rv

Cross-product-equivalent matrix

r =

0 "z y

z 0 "x

"y x 0

#

$

%%

%

&

'

((

(


4/10

Location of the Center of Mass

rcm = 1m

rdmBody" = r#(x,y,z)dx dy dz

zmin

zmax

"ymin

ymax

"xmin

xmax

" =xcm

ycm

zcm

$

%

&&&

'

(

)))

The Inertia Matrix

where

h = " r r # dmBody

$ = " r r dmBody

$ # = I#

Inertia matrix derives fromequal effect ofangular rateon all particles of the aircraft

I= " r r dmBody

# = "0 "z y

z 0 "x

"y x 0

$

%

&&&

'

(

)))

0 "z y

z 0 "x

"y x 0

$

%

&&&

'

(

)))

dmBody

#

=

(y2 + z2) "xy "xz

"xy (x 2 + z2) "yz

"xz "yz (x2 + y2)

$

%

&&&

'

(

)))

dmBody

#

" =

"x

"y

"z

#$%%%

&'(((

Moments and

Products of Inertia

Inertia matrix

I=

(y2

+ z2

) "xy "xz"xy (x2 + z2) "yz

"xz "yz (x2 + y2)

#

$

%%%

&

'

(((

dmBody

) =Ixx "Ixy "Ixz"Ixy Iyy "Iyz"Ixz "Iyz Izz

#

$

%%%

&

'

(((

Moments of inertia on the diagonal

Products of inertia off the diagonal

If products of inertia are zero, (x, y, z) are principal axes --->

Newton!s 2nd Law, applied to rotational motion (in inertial frame) Rate of change of angular momentum = applied moment (or torque), M

dh

dt=

dI"( )dt

=I" +I =M =

mx

my

mz

#

$

%%

%

&

'

((

(

Ixx 0 0

0 Iyy 0

0 0 Izz

"#$$$

%&'''

Inertia Matrix of an Aircraft

with Mirror Symmetry

I=

(y2+ z

2) 0 "xz

0 (x2 + z2 ) 0

"xz 0 (x 2 + y 2 )

#

$

%%%

&

'

(((dm

Body)=

Ixx 0 "Ixz

0 Iyy 0

"Ixz 0 Izz

#

$%

%%

&

'(

((

Nose high/low product of inertia, Ixz


5/10

Inertia matrix,

with constant density, $

Moments

and

Products of

Inertia(Bedford & Fowler)

I= "

(y2+ z

2) #xy #xz

#xy (x2 + z2 ) #yz

#xz #yz (x2 + y2)

$

%

&&&

'

(

)))

dxdydzBody

*

Can construct aircraftmoments of inertia fromcomponents usingparallel-axis theorem

Angular Momentum

and Rate

Angularmomentum andrate vectors arenot necessarilyaligned

h = I"

How Do We Get Rid of dI/dtin the

Angular Momentum Equation?

Write the dynamic equation in a body-referenced frame Inertial properties of a constant-mass, rigid body are

unchanging in a body frame of reference

... but a body-referenced frame is non-Newtonian or non-inertial

Therefore, dynamic equation must be modified to account fora rotating frame

dI"( )dt

=I" +I

Chain Rule ... and in an inertial frame

I" 0

Angular Momentum Expressed

in Different Frames of Reference

Angular momentum and rate are vectors

They can be expressed in either the inertial or body frame The 2 frames are related by the rotation matrix

hB=H

I

Bh

I; h

I=H

B

Ih

B

"B=H

I

B"I

; "I=H

B

I"B


6/10

Vector Derivative Expressed

in a Rotating Frame

Chain Rule

Consequently

hI

=HB

Ih

B

+ HB

Ih

B

Effect ofbody-frame rotation

Rate of changeexpressed in body frame

Alternatively

hI=H

B

Ih

B+ "

I#h

I=H

B

Ih

B+

Ih

I

=

0 #"z

"y

"z

0 #"x

#"y

"x

0

$

%

&&&

'

(

)))

... where the cross-product-equivalent matrix of angular rate is

HB

Ih

B=

Ih

I=

IH

B

Ih

B

External Moment Causes

Change in Angular Rate

hB=H

I

Bh

I+ H

I

Bh

I=H

I

Bh

I"#

B$ h

B

=HI

Bh

I"

BhB=H

I

BM

I"

BIB#

B

=MB"

BIB#

B

In the body frame of reference, the angular momentum change is

Positive rotation ofFrame B w.r.t.Frame A is anegative rotation ofFrame A w.r.t.Frame B

MI =

mx

my

mz

"

#

$$$

%

&

'''I

; MB =HIBMI =

mx

my

mz

"

#

$$$

%

&

'''B

Moment = torque = force x moment arm

Rate of Change of Body-Referenced

Angular Rate due to External Moment

For constant body-axis inertia matrix

hB=H

I

Bh

I+ H

I

Bh

I=H

I

Bh

I"#

B$ h

B

=HI

Bh

I"

BhB=H

I

BM

I"

BIB#

B

=MB"

BIB#

B

In the body frame of reference, the angular momentum change is

B= I

B

#1M

B#

BI

B"

B( )

Consequently, the differential equation for angular rate of change is

hB= I

B

B=M

B#

BI

B"

B

Euler-Angle Rates and

Body-Axis Rates

Body-axis angular ratevector (orthogonal)

" B ="x

"y

"z

#

$

%%%

&

'

(((B

=

p

q

r

#

$

%%%

&

'

(((

Euler-angle rate vector

Form a non-orthogonal vectorof Euler angles

" =#

$

%

&

'

(((

)

*

+++

" =#$%

&

'

(((

)

*

+++

,

-x

-y

-z

&

'

(((

)

*

+++I


7/10

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

... which is

Inverse transformation [(.)-1" (.)T]

""" pq

r

"

#

$$$

%

&

'''

=

(0

0

"

#

$$$

%

&

'''

+H2

B

0

)0

"

#

$$$

%

&

'''

+H2

BH

1

2

0

0

*"

#

$$$

%

&

'''

p

q

r

"

#

$$$

%

&

'''

=

1 0 (sin)

0 cos* sin*cos)

0 (sin* cos*cos)

"

#

$$$

%

&

'''

*)+"

#

$$$

%

&

'''

=LIB, "#$%

&

'''

(

)

***

=

1 sin"tan# cos"tan#

0 cos" +sin"

0 sin"sec# cos"sec#

%

&

'''

(

)

***

p

q

r

%

&

'''

(

)

***

=LBI , B

Can the inversionbecome singular?

What does this mean?

Point-Mass

Equations of Motion

Rate of change of the center of mass!s translational position

Express translational dynamics of the centerof mass in the body frame of reference

rI= v

I=H

B

Iv

B

vI=

1

mFI

vB=H

I

BvI"

BvB=

1

mH

I

BFI"

BvB

=

1

mFB"

BvB

FB =

fx

fy

fz

"

#

$$$

%

&

'''B

; vB =

u

v

w

"

#

$$$

%

&

'''

rI =HBI

vB

vB=

1

mFB+H

I

BgI"

BvB

" =LB

I #B

B= I

B

#1

MB#

BIB"

B( )

Rate of change of

Translational Position

Rate of change ofAngular Position

Rate of change ofTranslational Velocity

Rate of change ofAngular Velocity

rI=

x

y

z

"

#

$$

$

%

&

''

'I

" =#

$

%

&

'

(((

)

*

+++

vB=

u

v

w

"

#

$$$

%

&

'''

" B =p

q

r

#

$

%%%

&

'

(((

TranslationalPosition

AngularPosition

TranslationalVelocity

AngularVelocity

Rigid-Body Equations of Motion Aircraft Characteristics Expressedin Body Frame of Reference

MB =

L

M

N

"

#

$$$

%

&

'''B

=

Clb

Cmc

Cnb

"

#

$$$

%

&

'''B

1

2(V2S=

Clb

Cmc

Cnb

"

#

$$$

%

&

'''B

q S

IB=

Ixx

"Ixy

"Ixz

"Ixy

Iyy

"Iyz

"Ixz

"Iyz

Izz

#

$

%%%

&

'

(((B

FB =

X

Y

Z

"

#

$$$

%

&

'''B

=

CX

CY

CZ

"

#

$$$

%

&

'''B

12

(V2S=

CX

CY

CZ

"

#

$$$

%

&

'''B

q S

Aerodynamic andthrust force

Aerodynamic andthrust moment

Inertia matrix

b = wing span

c = mean aerodynamic chord


8/10

Rigid-Body Equations of

Motion (Scalar Notation)

u =X/m " gsin#+ rv "qw

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

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

p = IzzL + I

xzN" I

xzIyy"I

xx"I

zz( )p+ Ixz2+ I

zzIzz"I

yy( )[ ]r{ }q( ) IxxIzz "Ixz2( )q = M" I

xx"I

zz( )pr "Ixz p2" r

2( )[ ]Iyyr = I

xzL + I

xxN" I

xzIyy"I

xx"I

zz( )r + Ixz2 + Ixx Ixx "Iyy( )[ ]p{ }q( )

IxxIzz"I

xz

2( )

xI = cos"cos#( )u+ $cos%sin#+ sin%sin"cos#( )v + sin%sin#+ cos%sin"cos#( )w

yI = cos"sin#( )u+ cos%cos#+ sin%sin"sin#( )v + $sin%cos#+ cos%sin"sin#( )w

zI = $sin"( )u+ sin%cos"( )v + cos%cos"( )w

"= p+ qsin"+ rcos"( )tan##= qcos"$ rsin"%= qsin"+ rcos"( )sec#

Rate of change of Translational Position

Rate of change of Angular Position ( Ixyand Iyz= 0)

Rate of change of Translational Velocity

Rate of change of Angular Velocity

Aircraft with

mirror symmetry,Ixz! 0

Longitudinal Transient

Response to Initial Pitch Rate

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

Transient Response to

Initial Roll Rate

Lateral-Directional Response Longitudinal Response


Transient Response to

Initial Yaw Rate

Lateral-Directional Response Longitudinal Response



9/10

Crossplot of Transient

Response to Initial Yaw Rate


Relationship of Inertial

Axes to Body Axes

Independent ofvelocity vector

Represented by

Euler angles

Rotation matrix

u

v

w

"

#

$$$

%

&

'''

=HIB

vx

vy

vz

"

#

$$$

%

&

'''

vx

vy

vz

"

#

$$$

%

&

'''

=HBI

u

v

w

"

#

$$$

%

&

'''

Relationship of Body

Axes to Wind Axes

No reference to theinertial frame

u

v

w

"

#

$$$

%

&

'''

=

Vcos(cos)

Vsin)

Vsin(cos)

"

#

$$$

%

&

'''

V

"

#

$

%

&&&

'

(

)))

=

u2+ v

2+ w

2

sin*1

v /V( )tan

*1w /u( )

$

%

&&&

'

(

)))

Relationship of Inertial

Axes to Velocity Axes

No reference to the

body frame Bank angle is roll

angle about thevelocity vector

vx

vy

vz

"

#

$$$

%

&

'''I

=

Vcos(cos)

Vcos(sin)

*Vsin(

"

#

$$$

%

&

'''

V

"

#

$

%

&&&

'

(

)))

=

vx2

+ vy2

+ vz2

sin*1

vy/ v

x

2

+ vy2( )

1/ 2$%&

'()

sin*1 *vz /V( )

$

%

&&&&

'

(

))))


10/10

Alternative Frames of Reference

Orthonormaltransformationsconnect allreference

frames

Next Time:Effects of Stability, Control,and Power on Design

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Equations of Motion With Different Axis