Aircraft Equations of Motion - 1Robert Stengel, Aircraft Flight Dynamics, MAE 331,

2010

6 degrees of freedom

Angular kinematics

Euler angles

Rotation matrix

Angular momentum

Inertia matrix

Copyright 2010 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

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

Lockheed F-104

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)

Translational Position

Cartesian Frames of Reference

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

B: Body frame (fixed to body)

Common convention (z up) Aircraft convention (z down)

Translation Relative linear positions of origins

Rotation

Orientation of the body frame withrespect 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)

Differences in frame orientations must

be taken into account in adding vector

components

r =

x

y

z

!

"

###

$

%

&&&

rparticle = rorigin + 'rw.r .t . origin

Inertial-axis view

Body-axis view

Euler Angles Measure the Orientation ofOne 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 onVector Transformation

Yaw 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

=

xIcos' + y

Isin'

(xIsin' + y

Icos'

zI

!

"

###

$

%

&&&; r

1= H

I

1rI

x

y

z

!

"

###

$

%

&&&2

=

cos' 0 ( sin'

0 1 0

sin' 0 cos'

!

"

###

$

%

&&&

x

y

z

!

"

###

$

%

&&&1

; r2= H

1

2r1= H

1

2H

I

1!" $%rI = H I2rI

x

y

z

!

"

###

$

%

&&&B

=

1 0 0

0 cos' sin'

0 ( sin' cos'

!

"

###

$

%

&&&

x

y

z

!

"

###

$

%

&&&2

; rB= H

2

Br2= H

2

BH

I

2!" $%rI = H IBrI

The Rotation Matrix

HI

B(!,",# ) = H

2

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 theproduct of 3 single-angle rotation matrices:

Properties of the Rotation Matrix

HI

B(!,",# ) = H

2

B(!)H

1

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

Properties of the Rotation Matrix

Inverse relationship

Because transformation is orthonormal,

Inverse = transpose

Rotation matrix is always non-singular

rB= H

I

BrI; r

I= H

I

B( )!1

rB= H

B

IrB

HB

I= H

I

B( )!1

= HI

B( )T

= H1

IH2

1H

B

2

HB

IH

I

B= H

I

BH

B

I= I

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

Angular Momentumof a Particle

Moment of linear momentum of differentialparticles that make up the body

Differential masses times components of theirvelocity that are perpendicular to the momentarms from the center of rotation to the particles

Cross Product: Evaluation of a determinant with unit vectors (i, j, k)along axes, (x, y, z) and (vx, vy, vz) projections on to axes

r ! v =

i j k

x y z

vx

vy

vz

= yvz" zv

y( ) 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)dxdydzzmin

zmax

#ymin

ymax

#xmin

xmax

# =hx

hy

hz

%

&

''''

(

)

****

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

h = r ! vo( )dmBody

" + r ! # ! r( )( )dmBody

"

= 0 $ r ! r ! #( )( )dmBody

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

"

% $ !r!r( )dm#Body

"

Choosing the center of mass as the rotational center

Supermarine Spitfire

Cross-Product-

Equivalent Matrix

r ! v =

i j k

x y z

vx

vy

vz

= yvz" zv

y( ) i + zvx " xvz( ) j + xvy " yvx( )k

=

yvz" zv

y( )zv

x" xv

z( )

xvy" yv

x( )

#

$

%%%%%

&

'

(((((

= !rv =

0 "z y

z 0 "x

"y x 0

#

$

%%%

&

'

(((

vx

vy

vz

#

$

%%%%

&

'

((((

Cross-product-equivalent matrix

!r =

0 !z y

z 0 !x

!y x 0

"

#

$$$

%

&

'''

Location of the Center of Mass

rcm =1

mrdm

Body

! = r"(x, y, z)dxdydzzmin

zmax

!ymin

ymax

!xmin

xmax

! =xcm

ycm

zcm

#

$

%%%

&

'

(((

The Inertia Matrix

The Inertia Matrix

where

h = ! !r !r " dmBody

# = ! !r !r dmBody

# " = I"

Inertia matrix derives from equal effect ofangular rate on 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

#

$

%%%

&

'

(((dm

Body

"

=

(y2+ z

2) !xy !xz

!xy (x2 + z2 ) !yz

!xz !yz (x2 + y2 )

#

$

%%%%

&

'

((((

dmBody

"

! =

!x

!y

!z

"

#

$$$$

%

&

''''

Moments and

Products of Inertia Inertia matrix

I =

(y2+ z

2) !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

Ixx

0 0

0 Iyy

0

0 0 Izz

!

"

####

$

%

&&&&

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

All rigid bodies have a set of principalaxes

Stearman PT-17

Inertia Matrix of an Aircraft

with Mirror Symmetry

I =

(y2+ z

2) 0 !xz

0 (x2+ z

2) 0

!xz 0 (x2 + y2 )

"

#

$$$$

%

&

''''

dmBody

( =Ixx 0 !Ixz

0 Iyy 0

!Ixz 0 Izz

"

#

$$$$

%

&

''''

Nose high/low product of inertia, Ixz

North American XB-70

Moments and productsof inertia tabulated forgeometric shapes withuniform density

Moments andProducts of

Inertia(Bedford & Fowler)

Construct aircraftmoments and products ofinertia from componentsusing parallel-axistheorem

Model in Pro/ENGINEER,etc.

Rate of Change ofAngular Momentum

Newton!s 2nd Law, Applied

to Rotational Motion

In inertial frame, rate of change of angularmomentum = applied moment (or torque), M

dh

dt=d I!( )

dt=dI

dt! + I

d!

dt

= !I! + I !! =M =

mx

my

mz

"

#

$$$$

%

&

''''

Angular Momentumand Rate

Angularmomentum andrate vectors arenot necessarilyaligned

h = I!

Rate of Change ofAngular Momentum

How Do We Get Rid of dI/dt in 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

d I!( )

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

Even though they are changing over time, they are relatedalgebraically

hBt( ) = H

I

Bt( )h

It( ); h

It( ) = H

B

It( )h

Bt( )

!Bt( ) = H

I

Bt( )!

It( ); !

It( ) = H

B

It( )!

Bt( )

Vector Derivative Expressed ina Rotating Frame

Chain Rule

Consequently, the 2nd term is

!hI= H

B

I !hB+ !H

B

IhB

Effect of body-frame rotation

Rate of change

expressed in body frame Alternatively

!hI= H

B

I !hB+!

I" h

I= H

B

I !hB+ "!

IhI

!! =

0 "!z

!y

!z

0 "!x

"!y

!x

0

#

$

%%%%

&

'

((((

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

!HB

IhB= "!

IhI= "!

IH

B

IhB

External Moment Causes

Change in Angular Rate

!hB= H

I

B !hI+ !H

I

BhI= H

I

B !hI! "

B# h

B

= HI

B !hI! ""

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= H

I

BM

I=

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

B !hI+ !H

I

BhI= H

I

B !hI! "

B# h

B

= HI

B !hI! ""

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" "!

BIB!

B( )

Consequently, the differential equation for angular rate of change is

!hB= I

B!!B=M

B" "!

BIB!

B

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of Motion - 2