A e 5362 Ac Modeling

Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA

EQUATIONS OF MOTION FOR A CONVENTIONAL

AIRCRAFT


PROBLEM DEFINITION

Roskam, J., Airplane Flight Dynamics and Automatic Flight Controls - Part I, Design, Analysis and Research Corporation, Lawrence, KS, 1995.


Note: These equations are only valid in Inertial Space

Newtons Law of Linear Motion

ForcesAppliedMomentumLinear

dsdVdVdtd

dtd

SVA

VA +=

Fgr

Eulers Law of Angular Motion

Moments AppliedMomentumAngular

+=

SA

VA

VdSdVdV

dtd

dtd Frgrrr

EQUATIONS OF MOTION general


+++=

++=

dmdtddm

dtddm

dtddm

dtd

dtdLHS

dmdtd

dtdLHS

V

P

VVP

P

VP

PV

'P

rrrrrrrr

rrrr

''

''

' )()(

dVr Ap =+= dm define also and :using gEliminatin rrr

Moments AppliedMomentumAngular

+=

SA

VA

VdSdVdV

dtd

dtd Frgrrr

ANGULAR MOTION EQUATION


Assumption 1: P is the Vehicle Center of Mass and rP is a constant for V. Assumption 2: Mass is a constant, i.e.,

0= V

dmr

0=dtdm

+=

+++=

+++=

dmdtddm

dtd

dtdLHS

dmdtd

dtddmdm

dtddm

dtd

dtdLHS

dmdtddm

dtddm

dtddm

dtd

dtdLHS

Vv

PP

V

P

VVP

v

PP

V

P

VVP

P

VP

rrrr

rrrrrrrr

rrrrrrrr

''

''

''

''

''

0 0



MASS DISTRIBUTION aircraft and spacecraft

! A constant mass distribution infers that the center of gravity remains constant during the period of interest. ? For stability and control purposes, aircraft responses are typically evaluated

for a duration of 30 to 60 seconds. ? For orbital spacecraft, the period of interest can range from several minutes

to several hours. ! Phenomena which violate the constant mass/moment of inertia

distribution include: ? Fuel sloshing ? Shifting payloads

i wandering passengers on commercial aircraft i solar panel deployment and communication antenna reorientation on spacecraft

? Expendable payloads i weapons i reconnaissance pods


CONSTANT MASS aircraft and rockets

17.2 19,300 - 100,000 112,000 Delta

27.7 1,800,000 - 4,500,000 6,500,000 Saturn 5

0.05 3.3 200 1,020 6,800 GA Twin

0.18 99 5,940 17,600 54,000 Fighter

0.22 1,500 90,000 291,000 675,000 SST

Mass Change % Takeoff weight

Mass Change (lbs)

Cruise Fuel Consumption

(lbs/hr)

Maximum Fuel Weight

(lbs)

Takeoff Weight (lbs)

Type

t = 60 seconds

Reference 2-1

! If the mass change is within 5% after 60 seconds, the constant mass assumption is considered acceptable. ? Reasonable for airplanes, but not for rockets.


+=

++=

dmdtd

dtddm

dtdLHS

dmdtd

dtddm

dtddm

dtd

dtdLHS

VV

PP

VV

PP

v

PP

rrrr

rrrrrr

2

'2'

2

'2'

''0

Special Identity:

0

0

0)(

0

0

2

2

2

2

2

2

=

=

+

=

=

V Spp

V

pp

V SVpp

V SVp

V SVp

dsdmdmdt

d

dsdmdmdtd

dsdmdmdtd

dsdmdmdtd

dtd

Frgrr

r

Fgrrr

Fgrr

Fgrr



Returning to the LHS:

++=

+=

dmdtd

dtddSdmLHS

dmdtd

dtddm

dtdLHS

VSP

VP

VV

PP

rrFrgr

rrrr

''

2

'2'

Now Looking at the RHS:

+++=

+++=

SSP

VVP

SP

VP

dSdSdmdmRHS

dSdmRHS

FrFrgrgr

Frrgrr''

'' ])[(])[(

0 Vover constant is V

== Vrggrg:3 Assumption dmdm



GRAVITATIONAL FORCES aircraft and spacecraft

! The Flat Earth Assumption implies that the gravitational vector is oriented along the positive Z axis. ? Aircraft

i not reasonable for intercontinental flight ? Orbital spacecraft

i not reasonable even for small vehicles ! Assuming constant gravitational acceleration over the entire volume

means that each mass element is subject to the same gravitational acceleration. ? Aircraft

i applies in virtually all cases ? Orbital spacecraft

i does not apply


Forming the Total Equation:

++=

++

SSP

VP

VSP

VP dSdSdmdmdt

ddtddSdm FrFrgrrrFrgr ''''

Therefore: =

SVdSdm

dtd

dtd Frrr

Under the assumptions: 1. P is the Center of Mass 2. Mass is constant 3. g is constant over V Note: The Rotational Motion Equations are only a function of r.



frame. rotating Z)Y, (X,

in theobserver an by seen as of change of rate theis t

:Where

tdtd

:is frame coordinate fixed )Z,Y ,X( in theobserver an by seen as of change

of rate then theframe, coordinate fixed )Z,Y ,X( the torelative , rate,angular an at rotating is that frame coordinate a is Z)Y,(X, If

resides.observer hein which t framecoordinate on the depend will vector theof change of rate observed The

Z)Y,(X,

Z)Y,(X,)Z,Y ,X(

AA

AAA

A

A

+=

VECTOR DERIVATIVES


Eliminating r using: dmdVAP =+= and' rrr'

+=

+

SVVP dsFdmgdmrrdt

ddtd )( '

Again assuming: Mass is a constant, i.e., and g is constant over V. 0=dtdm

+=+SVV

P dsmdmdtddm

dtd

dtd Fgrr ][][ 2

2'

0

ForcesAppliedMomentumLinear

dsdVdVdtd

dtd

SVA

VA +=

Fgr

LINEAR MOTION EQUATION


LINEAR MOTION EQUATION

TAPP mm FFgVV ++=+

][

0

+=+SVV

P dsmdmdtddm

dtd

dtd Fgrr ][][ 2

2'

TAV

P mdmdtd FFgV ++= ][

:gives over constant are and , that Noting VPP VV

Notes: ! The Linear Motion Equations are only concerned with the motion of the Center of Mass

Defining new variables: t

ds P P T A S

'

and ; r V F F F = + = 2 '

2p

P t

=r

V


Returning to the Angular Motion Equation

=

SVdSdm

dtd

dtd Frrr

Expanding using the Chain Rule

=

+

SV

dSdmdtd

dtd

dtd Frrrrr 2

2

)(

0

Introducing the notation: TS

AdS MMFr +=

TAV

dmdtd MMrr += 2

2



Now accounting for the rotation of the coordinate frames:

rrrrr

rrr

+++=

+=

ttdtd

andtdt

d

2

2

2

2

Therefore:

TAVV

dmdmdtd

dtd MMrrrrr +=+=

)([22

Assumption 4: The body is rigid. i.e., 022

==tt

rr



ELASTIC STRUCTURES aircraft and spacecraft

! The Rigid Body Assumption implies that all mass elements maintain their relative distance to each other, except for mass elements which are a part of rotating machinery.

! Volume and surface integrals can only be evaluated if the external geometry is known. ? Aircraft

i geometry is known if the aircraft is rigid i if highly elastic (the usual case), then aeroelastic equilibrium must be

established before the external shape of the aircraft can be determined i an assumption which is nearly always made, but which is never actually valid

? Spacecraft i lightweight, elastic structures, often significantly more elastic than aircraft i validity varies from spacecraft to spacecraft i antennas and solar arrays large contributors to elasticity


i, j, and k are unit vectors along the (X, Y, Z) body fixed axes.

kjigkjiMkjiF

kjirkjiVkji

P

zyx

zyx

gggNMLFFF

zyxWVURQP

++=

++=

++=

++=

++=

++=

BODY AXIS unit vectors and components


BODY AXIS

Note: Positive Signs Shown

component definitions

Linear and Angular Velocities Aerodynamic and Thrust Moments

Aerodynamic and Thrust Forces Acceleration of Gravity

Ref. Roskams book


INERTIAS body axis moments and products

Y

X

Z

Convair B-36D Peacemaker

=

=

=

+=

+=

+=

Vyz

Vxz

Vxy

Vzz

Vyy

Vxx

dmyzI

dmxzI

dmxyI

dmyxI

dmzxI

dmzyI

)(

)(

)(

22

22

22

By Symmetry: Ixy = Iyz = 0


INERTIAS highly asymmetric aircraft

Blohm & Voss BV 141 B-0

Oblique Flying Wing


EQUATION SUMMARY Linear Motion

"")(

"")(

"")(

EquationLiftFFmgVPUQWm

EquationSideforceFFmgWPURVm

EquationDragFFmgWQVRUm

zTzAz

yTyAy

xTxAx

++=+

++=+

++=+

Drag Equation Sideforce Equation Lift Equation

Angular Motion Assuming the x-z plane is plane of symmetry, i.e., 0== yzxy II

TAxzxxyyxzzz

TAxzzzxxyy

TAyyzzxzxzxx

NNQRIPQIIPIRI

MMRPIPRIIQI

LLRQIIPQIRIPI

+=+++

+=++

+=+

)(

)()(

)(

22

Rolling Moment Equation Pitching Moment Equation Yawing Moment Equation


! Most airplanes are equipped with spinning rotors (propellers/turbine engines) that exert gyroscopic moments on the airframe.

! Gyroscopic moments are often negligible due to opposite rotations: ? Counter-rotating propellers ? Twin spool turbines

SPINNING ROTORS gyroscopic moments

Boeing V-22 Osprey Tiltrotor

iR

Assume one or more spinning rotors with total angular momentum

where each rotor has moment of inertia and spins with angular velocity

x y zh h h= + +h i j k

1

n

i ii=

= h h

1i i

n

i R Ri

I =

= h

iRI


( )( ) ( )

( )

2 2

XX XZ XZ XX YY z y A T

YY XX ZZ x z A T

ZZ XZ YY XX XZ y x A T

I P I R I PQ I I RQ Qh Rh L L

I Q I I PR IXZ P R Rh Ph M M

I R I P I I PQ I QR Ph Qh N N

+ + = +

+ + + = +

+ + + = +

A A TV

d d dr dvdt dt dt

+ = +r h M M

SPINNING ROTORS gyroscopic moments

The angular momentum equation can then be written

Taking the vector derivative of and assuming , 0iR

=ddth

Note that the gyroscopic moments appear with coupling.


The orientation of the body frame relative to the fixed frame can be specified using Euler Angles.

RELATIVE ORIENTATION inertial and body coordinate frames

Reference 2-2


EULER ATTITUDE ANGLES definition

Reference 2-2

yaw attitude anglepitch attitude angleroll attitude angle


=

RQP

seccossecsin0sincos0

tancostansin1

Notes: 1. This is a set of non-linear differential equations 2. In general, 3. These equation are not defined at i.e., there is a singularity.

RandQP

,.,deg90=

KINEMATIC EQUATIONS Euler angle and body axis rates

1 0 sin0 cos cos sin0 sin cos cos

PQR

=


EQUATIONS OF MOTION observations

! The Equations of Motion are a set of 9 differential equations: ? First order ? Nonlinear ? Coupled ? Ordinary

! The Variables are: ?

! The aerodynamic and thrust Forcing Functions:

?

are functions of: ? Velocity ? Angle-of-attack ? Sideslip angle

TATATATATATA NandNMMLLFFFFFF zzyyxx ,,,,,,,,,,

and , R, Q, P, W,V, U,

? Time ? Altitude ? Configuration


Angle-of-Attack

Sideslip Angle

Grumman F11F-1 Tiger

AERODYNAMIC ANGLES

XI

XB

VP

VP XB

definitions

Note: All Angles Shown Are Positive


Angle-of-Attack

Sideslip Angle

AERODYNAMIC ANGLES approximations


v U1

u

1

1 1

sin v vU U

=

u

U1

w

1

1 1

sin w wU U

=


FORCES AND MOMENTS longitudinal perturbations

! With

! There are separate equations for the aerodynamic and thrust forces and moments.

1 1

and ; controls and :E Fw w

U U = =

X X X X X XX E F

E F

Z Z Z Z Z ZZ E F

E F

E FE F

F F F F F Ff u qu q

F F F F F Ff u qu q

M M M M M Mm u qu q

= + + + + +

= + + + + +

= + + + + +


FORCES AND MOMENTS lateral / directional perturbations

! With

! There are separate equations for the aerodynamic and thrust forces and moments.

RR

AA

RR

AA

RR

YA

A

YYYYYY

NNrrNp

pNNNn

LLrrLp

pLLLl

FFrr

FppFFFf

+

+

+

+

+

=

+

+

+

+

+

=

+

+

+

+

+

=

1 1

and ; controls and :A Rv v

U U = =


FORCES AND MOMENTS shorthand notation

! Dividing by mass, each term becomes a longitudinal linear or angular acceleration.

! Letting , dimensional derivatives are the linear or angular acceleration per change in the associated motion variable. ? is the pitch angular acceleration imparted to the airplane as the result

of a unit change in angle-of-attack.

! There will be separate equations for the aerodynamic and thrust forces and moments.

FEquYY

FEquZ

FEquX

FE

FE

FE

MMqMMMuMmI

ZZqZZZuZfm

XXqXXXuXfm

+++++=

+++++=

+++++=

1

1

1

M

variablemoment 1 momentvariable m

=


FORCES AND MOMENTS shorthand notation

! Dividing by mass, each term becomes a longitudinal linear or angular acceleration.

! Letting , dimensional derivatives are the linear or angular acceleration per change in the associated control variable. ? is the sideforce linear acceleration imparted to the airplane as the result

of a unit change in rudder deflection.

! There will be separate equations for the aerodynamic and thrust forces and moments.

controlforce 1 force

control m =

RY

RArpZZ

RArpXX

RArpY

RA

RA

RA

NNrNpNNNnI

LLrLpLLLlI

YYrYpYYYfm

+++++=

+++++=

+++++=

1

1

1


LIFT AND DRAG FORCES definitions


Lift

U1

Drag

XB

ZB


In body axes,

1

211 12

21 11 1 1 1 1 12 2

1

cos cos

cos cos (2 )

X

X

A D

A DD

F D U SC

F CU S SC UU U

= =

=

1 1

1

211 1 1 112

1 1 11

1

1111

2 21cos cos

cos 2

X

X

u

A D DD D

AD D

F C CC Cu U S u q S uU U U U UU

U

F uu q S C CU U

= =

=

Then

Or

NON-DIMENSIONAL DERIVATIVES

UCU

UUCC DDDu

=

= 1

1

Define the derivative:


In body axes:

1

211 12

21 11 1 1 1 1 12 2

1

cos cos

cos cos (2 )

X

X

A D

A DD

F D U SC

F CU S SC UU U

= =

=

uUC

UUC

USqu

UC

UCSUu

UF DDDDAX

=

=

1

1

11

1

2112

1

1

11212Then:

Or [ ]1

1

11

2UuCCSqu

UF

DDA

u

X =

NON-DIMENSIONAL DERIVATIVES

UCU

UUCC DDDu

=

= 1

1

Define the derivative:


DIMENSIONAL DERIVATIVES longitudinal linear accelerations

)sec(

)sec(2

)sec(2

)sec()2(

)(sec)2(

)sec()sec()2(

)(sec)2(

)(sec)2(

21

1

1

11

1

1

211

1

1

2121

1

1

11

1

1

11

1

11

=

=

=

+

=+

=

=+

=

+=

+=

ftmSCq

Z

ftmU

CcSqZft

mU

CcSqZ

ftm

CCSqZ

mUCCSq

Z

ftmSCq

Xftm

CCSqX

mU

CCSqX

mUCCSq

X

E

E

q

u

E

E

XuXu

L

L

q

L

DLLLu

DLD

TT

uTDD

u

Note: All Quantities are Linear Acceleration per Unit of Applied Variable


)(sec

)(sec2

)(sec

)sec()2(

)sec()2(

21

1

1

2

1

21

11

1

1

11

1

1

1

1

=

=

=

+

=

+

=

YY

M

YY

M

YY

M

YY

TT

uT

YY

MMu

ICcSq

M

UI

CcSqM

ICcSq

M

ftUI

CCcSqM

ftUI

CCcSqM

E

E

MuM

u

DIMENSIONAL DERIVATIVES longitudinal angular accelerations

Note: All Quantities are Angular Acceleration per Unit of Applied Variable


)(sec)(sec

)(sec2

)(sec2

)(sec)sec(

)sec()sec(2

)sec(2

)sec(

2121

1

1

211

1

21

2121

211

1

1

1

1

121

==

==

==

==

=

=

XX

L

XX

L

XX

Lr

XX

L

p

XX

LY

YYr

Y

p

Y

ICbSq

LI

CbSqL

UICbSq

LUICbSq

L

ICbSq

LftmSCq

Y

ftmSCq

YftmU

CbSqY

ftmU

CbSqYft

mSCq

Y

R

R

A

A

rp

R

R

A

A

r

p

DIMENSIONAL DERIVATIVES lateral linear accelerations

Note: All Quantities are Linear Acceleration per Unit of Applied Variable


)(sec)(sec

)(sec2

)(sec2

)(sec)(sec

2121

1

1

211

1

21

2121

==

==

==

ZZ

N

ZZ

N

ZZ

Nr

ZZ

N

p

ZZ

N

TZZ

N

ICbSq

NI

CbSqN

UICbSq

NUICbSq

N

I

CbSqN

ICbSq

N

R

R

A

A

rp

T

DIMENSIONAL DERIVATIVES lateral angular accelerations

Note: All Quantities are Angular Acceleration per Unit of Applied Variable


STATIC STABILITY CRITERIA

r q p w v u Forces and Moments wU1

=vU1 =

xx TAF F+

yy TAF F+

zz TAF F+

TAL L+

TAM M+

TAN N+

Du 0C >

y 0C

l 0C nr 0C

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A e 5362 Ac Modeling