Linearized LongitudinalEquations of Motion

Robert Stengel, Aircraft Flight Dynamics

MAE 331, 2010

6th-order -> 4th-order -> hybrid equations

Dynamic stability derivatives

Phugoid mode

Short-period mode

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

6th-Order LongitudinalEquations of Motion

Symmetric aircraft

Motions in the vertical plane

Flat earth

x1

x2

x3

x4

x5

x6

!

"

########

$

%

&&&&&&&&

= xLon6=

u

w

x

z

q

'

!

"

#######

$

%

&&&&&&&

=

Axial Velocity

Vertical Velocity

Range

Altitude()

Pitch Rate

Pitch Angle

!

"

########

$

%

&&&&&&&&

!u = X / m ! gsin" ! qw

!w = Z / m + gcos" + qu

!xI= cos"( )u + sin"( )w

!zI= ! sin"( )u + cos"( )w

!q = M / Iyy

!" = q

State Vector, 6 componentsNonlinear Dynamic Equations

Fairchild-Republic A-10

4th-Order LongitudinalEquations of Motion

!u = f1= X / m ! gsin" ! qw

!w = f2= Z / m + gcos" + qu

!q = f3= M / Iyy

!" = f4= q

x1

x2

x3

x4

!

"

#####

$

%

&&&&&

= xLon4=

u

w

q

'

!

"

####

$

%

&&&&

=

Axial Velocity

Vertical Velocity

Pitch Rate

Pitch Angle

!

"

#####

$

%

&&&&&

State Vector, 4 componentsNonlinear Dynamic Equations,neglecting range and altitude

Caproni-Campini N1

Fourth-Order HybridEquations of Motion

Transform LongitudinalVelocity Components

!u = f1= X / m ! gsin" ! qw

!w = f2= Z / m + gcos" + qu

!q = f3= M / Iyy

!" = f4= q

!V = f1=1

mT cos ! + i( ) " D " mgsin#$% &'

!# = f2=1

mVT sin ! + i( ) + L " mgcos#$% &'

!q = f3= M / Iyy

!( = f4= q

x1

x2

x3

x4

!

"

#####

$

%

&&&&&

=

u

w

q

'

!

"

####

$

%

&&&&

=

Axial Velocity

Vertical Velocity

Pitch Rate

Pitch Angle

!

"

#####

$

%

&&&&&

x1

x2

x3

x4

!

"

#####

$

%

&&&&&

=

V

'

q

(

!

"

#####

$

%

&&&&&

=

Velocity

Flight Path Angle

Pitch Rate

Pitch Angle

!

"

#####

$

%

&&&&&

i = Incidence angle of the thrust vector

with respect to the centerline

Replace Cartesian body components of velocity by polar inertial components

Replace X and Z by T, D, and L

Hybrid LongitudinalEquations of Motion

!V = f1=1

mT cos ! + i( ) " D " mgsin#$% &'

!# = f2=1

mVT sin ! + i( ) + L " mgcos#$% &'

!q = f3= M / Iyy

!( = f4= q

!V = f1=1

mT cos ! + i( ) " D " mgsin#$% &'

!# = f2=1

mVT sin ! + i( ) + L " mgcos#$% &'

!q = f3= M / Iyy

!! = f4= !( " !# = q "

1

mVT sin ! + i( ) + L " mgcos#$% &'

x1

x2

x3

x4

!

"

#####

$

%

&&&&&

=

V

'

q

(

!

"

#####

$

%

&&&&&

=

Velocity

Flight Path Angle

Pitch Rate

Pitch Angle

!

"

#####

$

%

&&&&&

x1

x2

x3

x4

!

"

#####

$

%

&&&&&

=

V

'

q

(

!

"

#####

$

%

&&&&&

=

Velocity

Flight Path Angle

Pitch Rate

Angle of Attack

!

"

#####

$

%

&&&&&

Replace pitch angle by angle of attack ! = " # $

Why Transform Equations and

State Vector?

Phugoid (long-period) mode is primarily described byvelocity and flight path angle

Short-period mode is primarily described bypitch rate and angle of attack

x1

x2

x3

x4

!

"

#####

$

%

&&&&&

=

V

'

q

(

!

"

#####

$

%

&&&&&

=

Velocity

Flight Path Angle

Pitch Rate

Angle of Attack

!

"

#####

$

%

&&&&&

Why Transform Equations and

State Vector?

Hybrid linearized equations allow thetwo modes to be examined separately

FLon

=FPh

FSP

Ph

FPh

SPFSP

!

"

##

$

%

&&=

FPh

small

small FSP

!

"

##

$

%

&&'

FPh

0

0 FSP

!

"

##

$

%

&&

Effects of phugoid perturbations on

phugoid motion

Effects of phugoid perturbations on

short-period motion

Effects of short-period perturbations on

phugoid motion

Effects of short-period perturbations on

short-period motion

Nominal Equations of Motion inEquilibrium (Trimmed Condition)

!xN(t) = 0 = f[x

N(t),u

N(t),w

N(t),t]

!VN = 0 = f1 =1

mT cos !N + i( ) " D " mgsin# N$% &'

!# N = 0 = f2 =1

mVT sin !N + i( ) + L " mgcos# N$% &'

!qN = 0 = f3 =M

Iyy

!!N = 0 = f4 = q "1

mVT sin !N + i( ) + L " mgcos# N$% &'

T, D, and M contain x, u, and w effects

xN

T= V

N!N

0 "N

#$

%&T

= constant

LinearizedEquations of Motion

Sensitivity Matrices for LTI Model

!!x

Lon(t) = F

Lon!x

Lon(t) +G

Lon!u

Lon(t) + L

Lon!w

Lon(t)

F =

! f1

!V! f

1

!"! f

1

!q! f

1

!#

! f2

!V! f

2

!"! f

2

!q! f

2

!#

! f3

!V! f

3

!"! f

3

!q! f

3

!#

! f4

!V! f

4

!"! f

4

!q! f

4

!#

$

%

&&&&&&&&&&&

'

(

)))))))))))

G =

! f1

!"E

! f1

!"T

! f1

!"F

! f2

!"E

! f2

!"T

! f2

!"F

! f3

!"E

! f3

!"T

! f3

!"F

! f4

!"E

! f4

!"T

! f4

!"F

#

$

%%%%%%%%%%

&

'

(((((((((( L =

! f1

!Vwind

! f1

!"wind

! f2

!Vwind

! f2

!"wind

! f3

!Vwind

! f3

!"wind

! f4

!Vwind

! f4

!"wind

#

$

%%%%%%%%%%%

&

'

(((((((((((

Velocity Dynamics

!V = f1=1

mT cos! " D " mgsin#[ ] =

1

mCT cos!

$V 2

2S " CD

$V 2

2S " mgsin#

%

&'

(

)*

Nonlinear equation

Thrust incidence angle

neglected

First row of linearized dynamic equation

! !V (t) =" f

1

"V!V (t) +

" f1

"#!# (t) +

" f1

"q!q(t) +

" f1

"$!$(t)

%

&'

(

)*

+" f

1

"+E!+E(t) +

" f1

"+T!+T (t) +

" f1

"+F!+F(t)%

&'()*

+" f

1

"Vwind!Vwind +

" f1

"$wind!$wind

%

&'

(

)*

! f1

!V=1

mCTV cos"N # CDV( )

$VN2

2S + CTN cos"N # CDN( )$VNS

%

&'

(

)*

! f1

!+=#1m

mgcos+ N[ ] = #gcos+ N

! f1

!q=#1m

CDq$VN

2

2S

%

&'

(

)*

! f1

!"=#1m

CTN sin"N + CD"( )$VN

2

2S

%

&'

(

)*

Coefficients in first row of F

Sensitivity of Velocity Dynamics

to State Perturbations

CTV !"CT

"V

CDV !"CD

"V

CD# !"CD

"#

CDq !"CD

"q

Sensitivity of Velocity Dynamics toControl and Disturbance Perturbations

! f1

!"E=#1m

CD"E$VN

2

2S

%

&'

(

)*

! f1

!"T=1

mCT"T cos+N

$VN2

2S

%

&'

(

)*

! f1

!"F=#1m

CD"F$VN

2

2S

%

&'

(

)*

Coefficients in first rows of G and L

! f1

!Vwind= "

! f1

!V

! f1

!#wind

= "! f

1

!#CT!T

"#C

T

#!T

CD!E

"#C

D

#!E

CD!F

"#C

D

#!F

Flight Path Angle Dynamics

Linearized equation

!! = f2=1

mVT sin" + L # mgcos![ ] =

1

mVCT sin"

$V 2

2S + CL

$V 2

2S # mgcos!

%

&'

(

)*

Nonlinear equation

! !" (t) =# f

2

#V!V (t) +

# f2

#"!" (t) +

# f2

#q!q(t) +

# f2

#$!$(t)

%

&'

(

)*

+# f

2

#+E!+E(t) +

# f2

#+T!+T (t) +

# f2

#+F!+F(t)%

&'()*

+# f

2

#Vwind!Vwind +

# f2

#$wind!$wind

%

&'

(

)*

! f2

!V=

1

mVNCTV sin"N + CLV( )

#VN2

2S + CTN sin"N + CLN( )#VNS

$

%&

'

()

*1

mVN2

CTN sin"N + CLN( )#VN

2

2S * mgcos+ N

$

%&

'

()

! f2

!+=

1

mVNmgsin+ N[ ] = gsin+ N VN

! f2

!q=

1

mVNCLq

#VN2

2S

$

%&

'

()

! f2

!"=

1

mVNCTN cos"N + CL"( )

#VN2

2S

$

%&

'

()

Coefficients in second row of F

Sensitivity of Flight Path Angle

Dynamics to State Perturbations

Coefficients in second row of G and L in Supplemental Slide

CTV !"CT

"V

CLV !"CL

"V

CLq !"CL

"q

CL# !"CL

"#

Pitch Rate Dynamics

!q = f3=M

Iyy=Cm !V

22( )Sc

Iyy

Cm may include thrust as well

as aerodynamic effects

Nonlinear equation

Linearized equation

! !q(t) =" f

3

"V!V (t) +

" f3

"#!# (t) +

" f3

"q!q(t) +

" f3

"$!$(t)

%

&'

(

)*

+" f

3

"+E!+E(t) +

" f3

"+T!+T (t) +

" f3

"+F!+F(t)%

&'()*

+" f

3

"Vwind!Vwind +

" f3

"$wind!$wind

%

&'

(

)*

! f3

!V=1

IyyCmV

"VN2

2Sc + CmN "VNSc

#

$%

&

'(

! f3

!)= 0

! f3

!q=1

IyyCmq

"VN2

2Sc

#

$%

&

'(

! f3

!*=1

IyyCm*

"VN2

2Sc

#

$%

&

'(

Coefficients in third row of F

Sensitivity of Pitch Rate

Dynamics to State Perturbations

CmV !"Cm

"V

Cmq !"Cm

"q

Cm# !"Cm

"#

Angle of Attack Dynamics

!! = f4= !" # !$ = q #

1

mVT sin! + L # mgcos$[ ]

Nonlinear equation

Linearized equation

! !"(t) =# f

4

#V!V (t) +

# f4

#$!$ (t) +

# f4

#q!q(t) +

# f4

#"!"(t)

%

&'

(

)*

+# f

4

#+E!+E(t) +

# f4

#+T!+T (t) +

# f4

#+F!+F(t)%

&'()*

+# f

4

#Vwind!Vwind +

# f4

#"wind!"wind

%

&'

(

)*

! f4

!V= "

! f2

!V

! f4

!#= "

! f2

!#

! f4

!q= 1"

! f2

!q

! f4

!$= "

! f2

!$

Coefficients in fourth row of F

Sensitivity of Angle of Attack

Dynamics to State Perturbations

Longitudinal Stabilityand Control Derivatives

Dimensional Stability-DerivativeNotation

Dimensional stability derivatives portray acceleration sensitivities tostate perturbations

Redefine force and moment symbols as acceleration symbols

! f1

!V" #DV =

1

mCTV cos$N # CDV( )

%VN2

2S + CTN cos$N # CDN( )%VNS

&

'(

)

*+

! f2

!"# L"

VN=

1

mVNCTN cos"N + CL"( )

$VN2

2S

%

&'

(

)*

! f3

!"# M" =

1

IyyCm"

$VN2

2Sc

%

&'

(

)*

Drag

mass (m)! D;

Lift

mass! L;

Moment

moment of inertia (Iyy )! M

Thrust and drag effects

are combined and

represented by one

symbol

Thrust and lift effects are

combined and

represented by one

symbol

Elements of the Stability Matrix

! f1

!V" #DV ;

! f1

!$= #gcos$ N ;

! f1

!q" #Dq;

! f1

!%" #D%

! f2

!V"LVVN;

! f2

!#=g

VNsin# N ;

! f2

!q"LqVN;

! f2

!$"L$VN

! f3

!V" MV ;

! f3

!#= 0;

! f3

!q" Mq;

! f3

!$" M$

! f4

!V" #

LVVN;

! f4

!$= #

g

VNsin$ N ;

! f4

!q" 1#

LqVN;

! f4

!%" #

L%VN

Stability derivatives portray acceleration sensitivitiesto state perturbations

Longitudinal Stability Matrix

FLon =FPh FSP

Ph

FPhSP

FSP

!

"

##

$

%

&&=

'DV 'gcos( N 'Dq 'D)

LVVN

g

VNsin( N

LqVN

L)VN

MV 0 Mq M)

' LVVN

'g

VNsin( N 1'

LqVN

*+,

-./

' L)VN

!

"

########

$

%

&&&&&&&&

Effects of phugoid

perturbations on phugoid

motion

Effects of phugoid

perturbations on short-

period motion

Effects of short-period

perturbations on phugoid

motion

Effects of short-period

perturbations on short-period

motion

Primary LongitudinalStability Derivatives

DV!

!1m

CTV

! CDV

( )"V

N

2

2S + C

TN! C

DN( )"VNS

#

$%

&

'(

Assuming !

N! "

N! 0

LVVN

!1

mVNCLV

!VN2

2S + CLN !VNS

"

#$

%

&' (

1

mVN2CLN

!VN2

2S ( mg

"

#$

%

&'

Mq =1

IyyCmq

!VN2

2Sc

"

#$

%

&'

M! =1

IyyCm!

"VN2

2Sc

#

$%

&

'(

L!VN

!1

mVN

CTN

+ CL!( )

"VN

2

2S

#

$%

&

'(

Origins of Stability Effects

Velocity-DependentDerivative Definitions

Air compressibility effects are a principal source ofvelocity dependence

CDM

!"C

D

"M=

"CD

" V / a( )= a

"CD

"V

CDV

!"C

D

"V=1

a

#$%

&'(CDM

CLV

!"C

L

"V=1

a

#$%

&'(CLM

CmV

!"C

m

"V=1

a

#$%

&'(CmM

CDM

! 0

CDM

> 0 CDM

< 0

a = Speed of Sound

M = Mach number =V

a

Pitch-Moment Coefficient

Sensitivity to Angle of Attack

MB = Cmq Sc ! Cmo + Cmq q + Cm""( )q Sc

For small angle of attack and no control deflection

Cm! " #CN!net hcm # hcpnet( ) " #CL!net hcm # hcpnet( ) = #CL!netxcm # xcpnet

c

$%&

'()

= #CL!wingxcm # xcpwing

c

$

%&'

()# CL!ht

xcm # xcphtc

$%&

'()= #CL!wing

lwing

c

$%&

'()# CL!ht

lht

c

$%&

'()

= Cm!wing+ Cm!ht

referenced to wing area, S, and m.a.c., c

Pitch-Rate Derivative Definitions

Pitch rate derivatives are often expressed in terms of anormalized pitch rate

Cmq =!Cm!q

=c

2VN

"

#$%

&'Cmq

Cmq =!Cm!q

=!Cm

! qc2VN( )

=2VN

c

"#$

%&'Cmq

q =qc

2VN

Mq =!M!q

= Cmq "VN22( )Sc = Cmq

c

2VN

#

$%&

'("VN

2

2

#$%

&'(Sc = Cmq

"VNSc2

4

#$%

&'(

often tabulated used in pitch-rate equation

MB = Cmq Sc ! Cmo + Cmq q + Cm""( )q Sc

! Cmo +#Cm#q

q + Cm""$%&

'()q Sc

Pitch acceleration sensitivity to pitch rate

Angle of Attack Distribution

Due to Pitch Rate

Aircraft pitching at a constant rate, q rad/s, produces a normalvelocity distribution along x

!w = "q!x

!" =!w

VN

=#q!x

VN

Corresponding angle of attack distribution

!"ht =qlht

VN

Angle of attack perturbation at tail center of pressure

lht = horizontal tail distance from c.m.

Horizontal Tail Lift

Due to Pitch Rate

Incremental tail lift due to pitch rate, referenced to tail area, Sht

Lift coefficient sensitivity to pitch rate referenced to wing area

!Lht= !C

Lht( )

ht

1

2"V

N

2Sht

CLqht!" #CLht( )aircraft

"q=

"CLht"$

%&'

()*aircraft

lht

VN

%

&'(

)*

!CLht( )aircraft = !CLht( )htSht

S

"#$

%&'=

(CLht()

"#$

%&'aircraft

!)*

+,,

-

.//=

(CLht()

"#$

%&'aircraft

qlht

VN

"

#$%

&'

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

Moment CoefficientSensitivity to Pitch Rate of

the Horizontal Tail

Differential pitch moment due to pitch rate

!"Mht!q

= Cmqht

1

2#VN

2Sc = $CLqht

lht

VN

%

&'(

)*1

2#VN

2Sc

= $!CLht!+

%&'

()*aircraft

lht

VN

%

&'(

)*,

-..

/

011

lht

c

%&'

()*1

2#VN

2Sc

Cmqht= !

"CLht"#

lht

VN

$

%&'

()lht

c

$%&

'()= !

"CLht"#

lht

c

$%&

'()2

c

VN

$

%&'

()

Coefficient derivative with respect to pitch rate

Coefficient derivative with respect to normalized pitch rate is insensitive tovelocity

Cmqht=!Cmht!q

=!Cmht

! qc2VN( )

= "2!CLht!#

lht

c

$%&

'()2

Comparison of Fourth-and Second-OrderDynamic Models

0 - 100 sec

Reveals Phugoid Mode

Initial-Condition Responsesof Business Jet at Two TimeScales

0 - 6 sec

Reveals Short-Period Mode

LTI 4th-order responses viewed over different periods of time

4 initial conditions

Second-Order Models of

Longitudinal Motion

Approximate Phugoid Equation

!!xPh =! !V! !"

#

$%%

&

'(()

*DV *gcos" NLVVN

g

VNsin" N

#

$

%%%

&

'

(((

!V!"

#

$%%

&

'((+

T+T

L+TVN

#

$

%%%

&

'

(((!+T +

*DVLVVN

#

$

%%%

&

'

(((!Vwind

!!xSP =! !q! !"

#

$%%

&

'(()

Mq M"

1*LqVN

+,-

./0

* L"VN

#

$

%%%

&

'

(((

!q!"

#

$%%

&

'((+

M1E

*L1EVN

#

$

%%%

&

'

(((!1E +

M"

*L"VN

#

$

%%%

&

'

(((!"wind

Approximate Short-Period Equation

Assume off-diagonal blocks of (4 x 4)stability matrix are negligible FLon =

FPh

~ 0

~ 0 FSP

!

"

##

$

%

&&

Phugoid Time Scale Short-Period Time Scale

Full and approximate linear models

Comparison of Bizjet Fourth- andSecond-Order Model Responses

FourthOrder

SecondOrder

Approximate Phugoid Roots

Approximate Phugoid Equation (!N = 0)

!!xPh =! !V! !"

#

$%%

&

'(()

*DV *g

LVVN

0

#

$

%%%

&

'

(((

!V!"

#

$%%

&

'((+

T+T

L+TVN

#

$

%%%

&

'

(((!+T

Characteristic polynomial

sI ! FPh = det sI ! FPh( ) " #(s) = s2+ DVs + gLV /VN

= s2+ 2$%ns +%n

2

!n= gL

V/V

N

" =DV

2 gLV/V

N

Natural frequency anddamping ratio

!n" 2 g

VN

; T = 2# /!n

$ "1

2 L / D( )N

Neglectingcompressibility effects

Effect of Airspeed and L/D on Approximate

Phugoid Natural Frequency, Period, andDamping Ratio

!n " 2gVN

" 13.87 /VN (m / s); Period, T = 2# /!n " 0.45VN sec

$ "1

2 L / D( )N

Velocity

Natural

Frequency Period L/D

Damping

Ratiom/s rad/s sec50 0.28 23 5 0.14100 0.14 45 10 0.07200 0.07 90 20 0.035400 0.035 180 40 0.018

Neglecting

compressibility effects

Approximate Phugoid Response to

a 10% Thrust Increase

What is the steady-state response?

Approximate Short-Period Roots

Approximate Short-Period Equation (Lq = 0)

Characteristic polynomial

Natural frequency and damping ratio

!!xSP =! !q

! !"

#

$%%

&

'(()

Mq M"

1 *L"VN

#

$

%%%

&

'

(((

!q

!"

#

$%%

&

'((+

M+E

*L+EVN

#

$

%%%

&

'

(((

!+E

!(s) = s2 +L"

VN

# Mq

$

%&'

()s # M" + Mq

L"

VN

$

%&'

()

= s2+ 2*+

ns ++

n

2

!n= " M# + Mq

L#

VN

$

%&'

(); * =

L#

VN

" Mq

$%&

'()

2 " M# + MqL#

VN

$%&

'()

Approximate Short-PeriodResponse to a 0.1-Rad Pitch

Control Step InputPitch Rate, rad/s Angle of Attack, rad

Normal Load Factor Response to a

0.1-Rad Pitch Control Step Input

Normal Load Factor, g!s at c.m.

Aft Pitch Control

Normal Load Factor, g!s at c.m.

Forward Pitch Control

nz=VN

g! !" # !q( ) =

VN

g

L"

VN

!" +L$E

VN

!$E%

&'(

)*

Normal load factor at the center of mass

Pilot focuses on normal load factor during rapid maneuvering

Grumman X-29

Next Time:Lateral-Directional Dynamics

SupplementalMaterial

! f2

!"E=

1

mVNCL"E

#VN2

2S

$

%&

'

()

! f2

!"T=

1

mVNCT"T sin*N

#VN2

2S

$

%&

'

()

! f2

!"F=

1

mVNCL"F

#VN2

2S

$

%&

'

()

! f2

!Vwind= "

! f2

!V

! f2

!#wind

= "! f

2

!#

Control and Disturbance Sensitivities inFlight Path Angle, Pitch Rate, and Angle-of-

Attack Dynamics

! f3

!"E=1

IyyCm"E

#VN2

2Sc

$

%&

'

()

! f3

!"T=1

IyyCm"T

#VN2

2Sc

$

%&

'

()

! f3

!"F=1

IyyCm"F

#VN2

2Sc

$

%&

'

()

! f3

!Vwind= "

! f3

!V

! f3

!#wind

= "! f

3

!#

! f4

!"E= #

! f2

!"E

! f4

!"T= #

! f2

!"T

! f4

!"F= #

! f2

!"F

! f4

!Vwind=! f

2

!V

! f3

!"wind

=! f

2

!"

Control- and Disturbance-Effect Matrices

Control-effect derivatives portray accelerationsensitivities to control input perturbations

Disturbance-effect derivatives portray acceleration sensitivitiesto disturbance input perturbations

GLon

=

!D"E T"T !D"F

L"E /VN L"T /VN L"F /VN

M"E M"T M"F

!L"E /VN !L"T /VN !L"F /VN

#

$

%%%%%

&

'

(((((

LLon

=

!DVwind

!D"wind

LVwind

/VN

L"wind/V

N

MVwind

M"wind

!LVwind

/VN

!L"wind /VN

#

$

%%%%%%

&

'

((((((

Primary LongitudinalControl Derivatives

D!T !"1m

CT!T#VN

2

2S

$

%&

'

()

L!FVN

!1

mVNCL!F

#VN2

2S

$

%&

'

()

M!E =1

IyyCm!E

#VN2

2Sc

$

%&

'

()

Horizontal Tail Lift Sensitivity

to Angle of Attack

CL!ht( )aircraft =Vtail

VN

"

#$%

&'

2

1()*)!

"#$

%&'+elas

Sht

S

"#$

%&'CL!ht( )

" = Wing downwash angle at the tailVTail = Airspeed at vertical tail;

scrubbing lowers VTail, propeller

slipstream increases VTail#elas = Aeroelastic correction factor

Wing Lift and Moment CoefficientSensitivity to Pitch Rate

Straight-wing incompressible flow estimate (Etkin)

CLqwing= !2CL"wing

hcm ! 0.75( )

Cmqwing= !2CL"wing

hcm ! 0.5( )2

Straight-wing supersonic flow estimate (Etkin)

CLqwing= !2CL"wing

hcm ! 0.5( )

Cmqwing= !

2

3 M2!1

! 2CL"winghcm ! 0.5( )

2

Triangular-wing estimate (Bryson, Nielsen)

CLqwing= !

2"

3CL#wing

Cmqwing= !

"

3AR Approximations are very close to 4th-order values because natural

frequencies are widely separated

Comparison of Bizjet Fourth- andSecond-Order Models and Eigenvalues

Fourth-Order ModelF = G = Eigenvalue Damping Freq. (rad/s)

-0.0185 -9.8067 0 0 0 4.6645 -8.43e-03 + 1.24e-01j 6.78E-02 1.24E-010.0019 0 0 1.2709 0 0 -8.43e-03 - 1.24e-01j 6.78E-02 1.24E-01

0 0 -1.2794 -7.9856 -9.069 0 -1.28e+00 + 2.83e+00j 4.11E-01 3.10E+00-0.0019 0 1 -1.2709 0 0 -1.28e+00 - 2.83e+00j 4.11E-01 3.10E+00

Phugoid ApproximationF = G = Eigenvalue Damping Freq. (rad/s)

-0.0185 -9.8067 4.6645 -9.25e-03 + 1.36e-01j 6.78E-02 1.37E-010.0019 0 0 -9.25e-03 - 1.36e-01j 6.78E-02 1.37E-01

Short-Period ApproximationF = G = Eigenvalue Damping Freq. (rad/s)

-1.2794 -7.9856 -9.069 -1.28e+00 + 2.83e+00j 4.11E-01 3.10E+00

1 -1.2709 0 -1.28e+00 - 2.83e+00j 4.11E-01 3.10E+00

Effects of Airspeed, Altitude, Mass, and Moment of

Inertia on Fighter Aircraft Short Period

Airspeed Altitude

Natural

Frequency Period

Damping

Ratiom/s m rad/s sec -122 2235 2.36 2.67 0.39152 6095 2.3 2.74 0.31213 11915 2.24 2.8 0.23274 16260 2.18 2.88 0.18

Mass

Variation

Natural

Frequency Period

Damping

Ratio% rad/s sec --50 2.4 2.62 0.440 2.3 2.74 0.3150 2.26 2.78 0.26

Moment of

Inertia

Variation

Natural

Frequency Period

Damping

Ratio% rad/s sec --50 3.25 1.94 0.330 2.3 2.74 0.3150 1.87 3.35 0.31

Airspeed

Dynamic

Pressure

Angle of

Attack

Natural

Frequency Period

Damping

Ratiom/s P deg rad/s sec -91 2540 14.6 1.34 4.7 0.3152 7040 5.8 2.3 2.74 0.31213 13790 3.2 3.21 1.96 0.3274 22790 2.2 3.84 1.64 0.3

Airspeed variation at constant altitude

Altitude variation with constant dynamic pressure

Mass variation at constant altitude

Moment of inertia variation at constant altitude