MAE331Lecture12.pdf

Linearized Longitudinal Equations of Motion!

Robert Stengel, Aircraft Flight Dynamics !MAE 331, 2014!

•! 6th-order -> 4th-order -> hybrid equations"•! Dynamic stability derivatives "•! Long-period (phugoid) mode"•! Short-period mode"

Copyright 2014 by Robert Stengel. All rights reserved. For educational use only.!http://www.princeton.edu/~stengel/MAE331.html!

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

Reading:!Flight Dynamics!

452-464, 482-486!Airplane Stability and Control!

Chapter 7!

1!

Learning Objectives!

The Jets at an Awkward Age!Chapter 7, Airplane Stability and Control,

Abzug and Larrabee!•! What are the principal subject and scope of the

chapter?"•! What technical ideas are needed to understand the

chapter?"•! During what time period did the events covered in

the chapter take place?"•! What are the three main "takeaway" points or

conclusions from the reading?"•! What are the three most surprising or remarkable

facts that you found in the reading?"

2!

Longitudinal LTI Dynamics “Wordle”!

3!

6th-Order Longitudinal Equations of Motion!

•! Symmetric aircraft"•! Motions in the vertical plane"•! Flat earth "

x1x2x3x4x5x6

!

"

########

$

%

&&&&&&&&

= xLon6

!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 components!Nonlinear Dynamic Equations!

Fairchild-Republic A-10!

4!

uwxzq!

"

#

$$$$$$$

%

&

'''''''

=

Axial VelocityVertical Velocity

RangeAltitude(–)Pitch RatePitch Angle

"

#

$$$$$$$$

%

&

''''''''

4th-Order Longitudinal Equations of Motion!

!u = f1 = X / m ! gsin" ! qw!w = f2 = Z / m + gcos" + qu!q = f3 = M / Iyy!" = f4 = q

x1x2x3x4

!

"

#####

$

%

&&&&&

= xLon4

State Vector, 4 components!

Nonlinear Dynamic Equations, neglecting range and altitude!

5!

uwq!

"

#

$$$$

%

&

''''

=

Axial Velocity, m/sVertical Velocity, m/sPitch Rate, rad/sPitch Angle, rad

"

#

$$$$$

%

&

'''''

Fourth-Order Hybrid Equations of Motion!

6!

Transform Longitudinal Velocity Components"

!u = f1 = X / m ! gsin" ! qw!w = f2 = Z / m + gcos" + qu!q = f3 = M / Iyy!" = f4 = q

x1x2x3x4

!

"

#####

$

%

&&&&&

=

uwq'

!

"

####

$

%

&&&&

=

Axial VelocityVertical Velocity

Pitch RatePitch Angle

!

"

#####

$

%

&&&&& 7!

Replace Cartesian body components of velocity by polar inertial components"Replace X and Z by T, D, and L"

Transform Longitudinal Velocity Components"

!V = f1 = T cos ! + i( )"D"mgsin#$% &' m

!# = f2 = T sin ! + i( )+ L "mgcos#$% &' mV!q = f3 =M / Iyy!( = f4 = q

x1x2x3x4

!

"

#####

$

%

&&&&&

=

V'q(

!

"

####

$

%

&&&&

=

VelocityFlight Path Angle


!

"

#####

$

%

&&&&&

i = Incidence angle of the thrust vectorwith respect to the centerline

Replace Cartesian body components of velocity by polar inertial components"Replace X and Z by T, D, and L"

8!

Hawker P1127 Kestral!

Hybrid Longitudinal Equations of Motion"

!V = f1 = T cos ! + i( )"D"mgsin#$% &' m

!# = f2 = T sin ! + i( )+ L "mgcos#$% &' mV!q = f3 =M / Iyy!( = f4 = q

x1x2x3x4

!

"

#####

$

%

&&&&&

=

V'q(

!

"

####

$

%

&&&&

=



!

"

#####

$

%

&&&&& 9!

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

Hybrid Longitudinal Equations of Motion"

!V = f1 = T cos ! + i( )" D "mgsin#$% &' m

!# = f2 = T sin ! + i( ) + L "mgcos#$% &' mV!q = f3 = M / Iyy

!! = !( " !# = f4 = q "1mV

T sin ! + i( ) + L "mgcos#$% &'

x1x2x3x4

!

"

#####

$

%

&&&&&

=

V'q(

!

"

####

$

%

&&&&

=


Pitch RateAngle of Attack

!

"

#####

$

%

&&&&&

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

! =" +# 10!

Why Transform Equations and State Vector?"

•! Velocity and flight path angle typically have slow variations "

•! Pitch rate and angle of attack typically have quicker variations"

x1x2x3x4

!

"

#####

$

%

&&&&&

=

V'

q(

!

"

####

$

%

&&&&

=


Pitch RateAngle of Attack

!

"

#####

$

%

&&&&&

11!

Small Perturbations from Steady Path Approximated by Linear Equations"

!x(t) = !xN (t)+ !!x(t)" f[xN (t),uN (t),wN (t),t]+ F t( )!x(t)+G t( )!u(t)+L t( )!w(t)

12!

!x(t) = 0 + !!x(t)" f[xN (t),uN (t),wN (t),t]+ F!x(t)+G!u(t)+L!w(t)

Steady, Level Flight!

Rates of change are “small”"

Nominal Equations of Motion in Equilibrium (Trimmed Condition)"

!xN (t) = 0 = f[xN (t),uN (t),wN (t),t]

!VN = 0 = f1 = T cos !N + i( )" D "mgsin# N$% &' m

!# N = 0 = f2 = T sin !N + i( ) + L "mgcos# N$% &' mVN!qN = 0 = f3 = M Iyy

!!N = 0 = f4 = 0( )" 1mVN

T sin !N + i( ) + L "mgcos# N$% &'

T, D, L, and M contain state, control, and disturbance effects"

xNT = VN ! N 0 "N

#$

%&

T= constant

13!(See Supplemental Material for trimmed solution)!

Small Perturbations from Steady Path Approximated by Linear Equations"

14!

Linearized Equations of Motion!

!!xLon =

! !V! !"! !q! !#

$

%

&&&&&

'

(

)))))

= FLon

!V!"!q!#

$

%

&&&&

'

(

))))

+GLon

!*T!*E"

$

%

&&&

'

(

)))+"

Linearized !Equations of Motion!

15!

Phugoid (Long-Period) Motion!

Short-Period Motion!

Approximate Decoupling of Fast and Slow Modes of Motion"

Hybrid linearized equations allow the two modes to be examined separately"

FLon =FPh FSP

Ph

FPhSP 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"

=FPh smallsmall FSP

!

"##

$

%&&'

FPh 00 FSP

!

"##

$

%&& 16!

Sensitivity Matrices for Longitudinal LTI Model"

!!xLon (t) = FLon!xLon (t) +GLon!uLon (t) + LLon!wLon (t)

FLon =

! f1!V

! f1!"

! f1!q

! f1!#

! f2!V

! f2!"

! f2!q

! f2!#

! f3!V

! f3!"

! f3!q

! f3!#

! f4!V

! f4!"

! f4!q

! f4!#

$

%

&&&&&&&&&&&

'

(

)))))))))))

GLon =

! f1!"E

! f1!"T

! f1!"F

! f2!"E

! f2!"T

! f2!"F

! f3!"E

! f3!"T

! f3!"F

! f4!"E

! f4!"T

! f4!"F

#

$

%%%%%%%%%%

&

'

((((((((((

LLon =

! f1!Vwind

! f1!"wind

! f2!Vwind

! f2!"wind

! f3!Vwind

! f3!"wind

! f4!Vwind

! f4!"wind

#

$

%%%%%%%%%%%

&

'

(((((((((((17!

Velocity Dynamics"

!V = f1 =1mT cos! " D "mgsin#[ ]

= 1m

CT cos!$V 2

2S "CD

$V 2

2S "mgsin#%

&'

(

)*

Nonlinear equation"

Thrust along xB!

First row of linearized dynamic equation"

! !V (t) = " f1"V

!V (t)+ " f1"#

!# (t)+ " f1"q

!q(t)+ " f1"$

!$ (t)%

&'

(

)*

+ " f1"+E

!+E(t)+ " f1"+T

!+T (t)+ " f1"+F

!+F(t)%&'

()*

+ " f1"Vwind

!Vwind +" f1

"$wind

!$wind%

&'

(

)*

18!

! f1!V

= 1m

CTVcos"N #CDV( ) $NVN

2

2S + CTN

cos"N #CDN( )$NVNS%

&'

(

)*

Coefficients in first row of F"

Sensitivity of Velocity Dynamics to State Perturbations "

CTV!"CT

"V

CDV!"CD

"V

CDq!"CD

"q

CD#!"CD

"#

!V = CT cos! "CD( ) #V

2

2S "mgsin$

%

&'

(

)* m

19!

! f1!"

= #1m

mgcos" N[ ] = #gcos" N

! f1!q

= "1m

CDq

#NVN2

2S$

%&

'

()

! f1!"

= #1m

CTNsin"N +CD"( ) $NVN

2

2S%

&'

(

)*

Sensitivity of Velocity Dynamics to Control and Disturbance Perturbations "

! f1!"E

= #1m

CD"E

$NVN2

2S%

&'

(

)*

! f1!"T

= 1m

CT"Tcos+N

$NVN2

2S%

&'

(

)*

! f1!"F

= #1m

CD"F

$NVN2

2S%

&'

(

)*

Coefficients in first rows of G and L"

! f1!Vwind

= "! f1!V

! f1!#wind

= "! f1!#

CT!T"#CT

#!T

CD!E"#CD

#!E

CD!F"#CD

#!F 20!

Flight Path Angle Dynamics"

Second row of linearized equation"

!! = f2 =1mV

T sin" + L #mgcos![ ]

= 1mV

CT sin"$V 2

2S +CL

$V 2

2S #mgcos!%

&'

(

)*

Nonlinear equation"

! !" (t) = # f2#V

!V (t)+ # f2#"

!" (t)+ # f2#q

!q(t)+ # f2#$

!$ (t)%

&'

(

)*

+ # f2#+E

!+E(t)+ # f2#+T

!+T (t)+ # f2#+F

!+F(t)%&'

()*

+ # f2#Vwind

!Vwind +# f2#$wind

!$wind%

&'

(

)*

21!

! f2!V

= 1mVN

CTVsin"N +CLV( ) #NVN

2

2S + CTN

sin"N +CLN( )#NVNS$

%&

'

()

* 1mVN

2 CTNsin"N +CLN( ) #NVN

2

2S *mgcos+ N

$

%&

'

()

Coefficients in second row of F"

Sensitivity of Flight Path Angle Dynamics to State Perturbations "

CTV!"CT

"V

CLV!"CL

"V

CLq!"CL

"q

CL#!"CL

"#

!! = CT sin" +CL( ) #V

2

2S $mgcos!

%

&'

(

)* mV

22!

! f2!"

= 1mVN

mgsin" N[ ] = gsin" N VN

! f2!q

= 1mVN

CLq

"NVN2

2S#

$%

&

'(

! f2!"

= 1mVN

CTNcos"N +CL"( ) #NVN

2

2S$

%&

'

()

Pitch Rate Dynamics"

!q = f3 =

MIyy

=Cm !V 2 2( )Sc

Iyy

Nonlinear equation"

Third row of linearized equation"

! !q(t) = " f3"V

!V (t)+ " f3"#

!# (t)+ " f3"q

!q(t)+ " f3"$

!$ (t)%

&'

(

)*

+ " f3"+E

!+E(t)+ " f3"+T

!+T (t)+ " f3"+F

!+F(t)%&'

()*

+ " f3"Vwind

!Vwind +" f3

"$wind

!$wind%

&'

(

)*

23!

! f3!V

= 1Iyy

CmV

"NVN2

2Sc +CmN

"NVNSc#

$%

&

'(

Coefficients in third row of F"

Sensitivity of Pitch Rate Dynamics to State Perturbations "

CmV!"Cm

"V

Cmq!"Cm

"q

Cm#!"Cm

"#

!q =Cm !V 2 2( )Sc Iyy

24!

! f3!"

= 0

! f3!q

= 1Iyy

Cmq

"NVN2

2Sc#

$%

&

'(

! f3!"

= 1Iyy

Cm"

#NVN2

2Sc$

%&

'

()

Angle of Attack Dynamics"

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

1mV

T sin! + L # mgcos$[ ]

Nonlinear equation"

Fourth row of linearized equation"

! !"(t) = # f4#V

!V (t) + # f4#$

!$ (t) + # f4#q

!q(t) + # f4#"

!"(t)%

&'

(

)*

+# f4#+E

!+E(t) + # f4#+T

!+T (t) + # f4#+F

!+F(t)%

&'(

)*

+# f4#Vwind

!Vwind +# f4#"wind

!"wind%

&'

(

)*

25!

! f4!V

= " ! f2!V

Coefficients in fourth row of F"

Sensitivity of Angle of Attack Dynamics to State Perturbations "

! f4!q

= 1" ! f2!q

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

26!

! f4!"

= # ! f2!"

! f4!"

= # ! f2!"

Alternative Approach: "Numerical Calculation of the Sensitivity

Matrices (“1st Differences”)!

! f1!V

t( ) "

f1

V + #V( )$q%

&

'

(((((

)

*

+++++

, f1

V , #V( )$q%

&

'

(((((

)

*

+++++ x=xN (t )u=uN (t )w=wN (t )

2#V; ! f1

!$t( ) "

f1

V$ + #$( )q%

&

'

(((((

)

*

+++++

, f1

V$ , #$( )q%

&

'

(((((

)

*

+++++ x=xN (t )u=uN (t )w=wN (t )

2#$

27!

Remaining elements of F(t), G(t), and L(t) calculated accordingly"

! f2!V

t( ) "

f2

V + #V( )$q%

&

'

(((((

)

*

+++++

, f2

V , #V( )$q%

&

'

(((((

)

*

+++++ x=xN (t )u=uN (t )w=wN (t )

2#V; ! f2

!$t( ) "

f2

V$ + #$( )q%

&

'

(((((

)

*

+++++

, f2

V$ , #$( )q%

&

'

(((((

)

*

+++++ x=xN (t )u=uN (t )w=wN (t )

2#$

SpaceShipTwo Accident!October 31, 2014"

28!

Flight Profile of SpaceShipOne" (Precursor to

SpaceShipTwo)!

29!

SpaceShipTwo Cutaway Diagram"

30!

SpaceShipTwo Accident!October 31, 2014"

Chris Hart, ‘68, *70!Acting Chairman, NTSB!

Probable Cause, as of 11/3/14!Premature Feathering at M = 1!

31!

SpaceShipOne"Ansari X Prize, December 17, 2003"

Brian Binnie, *78!Pilot, Astronaut"

32!

33!

SpaceShipOne Altitude vs. Range!MAE 331 Assignment #4, 2010"

34!

SpaceShipOne State Histories"

35!

SpaceShipOne Dynamic Pressure and Mach Number Histories"

36!

Dimensional Stability and Control Derivatives!

37!

Dimensional Stability-Derivative Notation"

!! Redefine force and moment symbols as acceleration symbols"

!! Dimensional stability derivatives portray acceleration sensitivities to state perturbations"

Dragmass (m)

! D " !V

Liftmass

! L "V !#

Momentmoment of inertia (Iyy )

! M " !q

38!

Dimensional Stability-Derivative Notation"

! f1!V

" #DV !1m

CTVcos$N #CDV( ) %NVN

2

2S + CTN

cos$N #CDN( )%NVNS&

'(

)

*+

! f2!"

# L"VN!

1mVN

CTNcos"N +CL"( ) $NVN

2

2S%

&'

(

)*

! f3!"

# M" !1Iyy

Cm"

$NVN2

2Sc%

&'

(

)*

Thrust and drag effects are combined and represented by one symbol!

Thrust and lift effects are combined and represented by one symbol!

39!

Longitudinal Stability Matrix"

FLon =FPh FSP

Ph

FPhSP FSP

!

"

##

$

%

&&=

'DV 'gcos(N 'Dq 'D)

LVVN

gVNsin(N

LqVN

L)VN

MV 0 Mq M)

'LV VN'gVNsin(N 1' Lq VN

*

+,

-

./ '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"40!

Comparison of Fourth- and Second-Order

Dynamic Models!

41!

0 - 100 sec"•! Reveals Phugoid Mode"

4th-Order Initial-Condition Responses of Business Jet at Two Time Scales"

0 - 6 sec"•! Reveals Short-Period Mode"

Plotted over different periods of time"4 initial conditions [V(0), #(0), q(0), $(0)]"

42!

2nd-Order Models of Longitudinal Motion"

Approximate Phugoid Equation"

!!xPh =

! !V! !"

#

$%%

&

'(()

*DV *gcos" N

LVVN

gVNsin" N

#

$

%%%

&

'

(((

!V!"

#

$%%

&

'((

+T+TL+T

VN

#

$

%%%

&

'

(((!+T +

*DV

LVVN

#

$

%%%

&

'

(((!Vwind

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

FLon =FPh ~ 0~ 0 FSP

!

"##

$

%&&

43!

2nd-Order Models of Longitudinal Motion"

!!xSP =

! !q! !"

#

$%%

&

'(()

Mq M"

1* Lq VN+,-

./0 * L"

VN

#

$

%%%

&

'

(((

!q!"

#

$%%

&

'((

+M1E

*L1EVN

#

$

%%%

&

'

(((!1E +

M"

*L"VN

#

$

%%%

&

'

(((!"wind

Approximate Short-Period Equation"

FLon =FPh ~ 0~ 0 FSP

!

"##

$

%&&

44!

Comparison of Bizjet 4th- and 2nd-Order Model Responses"

45!

4th Order,"4 initial conditions

[V(0), #(0), q(0), $(0)]"

2nd Order,"2 initial conditions

[V(0), #(0)]""

Phugoid Time Scale, ~100 s"

Short-Period Time Scale, ~10 s"

Comparison of Bizjet 4th- and 2nd-Order Model Responses"

46!

4th Order,"4 initial conditions

[V(0), #(0), q(0), $(0)]"

2nd Order,"2 initial conditions

[q(0), $(0)]""

Approximate Phugoid Response to a 10% Thrust Increase "

What is the steady-state response?"47!

Approximate Short-Period Response to a 0.1-Rad Pitch Control Step Input "

Pitch Rate, rad/s! Angle of Attack, rad!

48!

What is the steady-state response?"

Normal Load Factor Response to a 0.1-Rad Pitch Control Step Input "

Normal Load Factor, g s at c.m.!Aft Pitch Control (Elevator)!

Normal Load Factor, g s at c.m.!Forward Pitch Control (Canard)!

nz =

VNg

! !" # !q( ) = VNg

L"VN

!" +L$EVN

!$E%

&'(

)*

•! Normal load factor at the center of mass"

•! Pilot focuses on normal load factor during rapid maneuvering"

Grumman X-29!

49!

Next Time:!Lateral-Directional Dynamics!

Reading:!Flight Dynamics!

574-591!

50!

SSuupppplleemmeennttaall MMaatteerriiaall

51!

Trimmed Solution of the Equations of Motion!

52!

Flight Conditions for Steady, Level Flight"

!V = f1 =1m

T cos ! + i( ) " D " mgsin#$% &'

!# = f2 =1mV

T sin ! + i( ) + L " mgcos#$% &'

!q = f3 = M / Iyy

!! = f4 = !( " !# = q "1mV

T sin ! + i( ) + L " mgcos#$% &'

Nonlinear longitudinal model"

Nonlinear longitudinal model in equilibrium"

0 = f1 =1m

T cos ! + i( ) " D " mgsin#$% &'

0 = f2 =1mV

T sin ! + i( ) + L " mgcos#$% &'

0 = f3 = M / Iyy

0 = f4 = !( " !# = q "1mV

T sin ! + i( ) + L " mgcos#$% &' 53!

Numerical Solution for Level Flight Trimmed Condition"

•!Specify desired altitude and airspeed, hN and VN!•!Guess starting values for the trim parameters, !!T0, !!E0, and ""0##•!Calculate starting values of f1, f2, and f3"

•! f1, f2, and f3 = 0 in equilibrium, but not for arbitrary !!T0, !!E0, and ""0"•!Define a scalar, positive-definite trim error cost function, e.g., "

f1 =1mT !T ,!E," ,h,V( )cos # + i( )$D !T ,!E," ,h,V( )%& '(

f2 =1

mVNT !T ,!E," ,h,V( )sin # + i( )+ L !T ,!E," ,h,V( )$mg%& '(

f3 =M !T ,!E," ,h,V( ) / Iyy

J !T ,!E,"( ) = a f12( )+b f2

2( )+ c f32( )

54!

Minimize the Cost Function with Respect to the Trim Parameters"

Cost is minimized at bottom of bowl, i.e., when "

! J!"T

! J!"E

! J!#

$

%&

'

()= 0

Error cost is bowl-shaped "

Search to find the minimum value of J "

J !T ,!E,"( ) = a f12( )+b f2

2( )+ c f32( )

55!

Example of Search for Trimmed Condition (Fig. 3.6-9, Flight Dynamics)"

In MATLAB, use fminsearch to find trim settings"

!T*,!E*," *( ) = fminsearch J, !T ,!E,"( )#$ %&56!

Elements of the Stability Matrix"

! f1!V

" #DV ;! f1!$

= #gcos$ N ;! f1!q

" #Dq;! f1!%

" #D%

! f2!V

" LV VN; ! f2

!#=gVNsin# N ;

! f2!q

"LqVN; ! f2

!$" L$ VN

! f3!V

" MV ;! f3!#

= 0; ! f3!q

" Mq;! f3!$

" M$

! f4!V

" # LV VN; ! f4

!$= #

gVNsin$ N ;

! f4!q

" 1# Lq VN; ! f4

!%" # L% VN

Stability derivatives portray acceleration sensitivities to state perturbations"

57!

! f2!"E

=1

mVNCL"E

#VN2

2S

$

%&

'

()

! f2!"T

=1

mVNCT"T

sin*N#VN

2

2S

$

%&

'

()

! f2!"F

=1

mVNCL"F

#VN2

2S

$

%&

'

()

! f2!Vwind

= "! f2!V

! f2!#wind

= "! f2!#

Control and Disturbance Sensitivities in Flight Path Angle, Pitch Rate, and

Angle-of-Attack Dynamics"! f3!"E

=1Iyy

Cm"E

#VN2

2Sc

$

%&

'

()

! f3!"T

=1Iyy

Cm"T

#VN2

2Sc

$

%&

'

()

! f3!"F

=1Iyy

Cm"F

#VN2

2Sc

$

%&

'

()

! f3!Vwind

= "! f3!V

! f3!#wind

= "! f3!#

! f4!"E

= #! f2!"E

! f4!"T

= #! f2!"T

! f4!"F

= #! f2!"F

! f4!Vwind

=! f2!V

! f3!"wind

=! f2!" 58!

Velocity-Dependent Derivative Definitions"

Air compressibility effects are a principal source of velocity dependence"

CDM!"CD

"M=

"CD

" V / a( )= a "CD

"V

CDV!"CD

"V=1a#

$%&

'(CDM

CLV!"CL

"V=1a#

$%&

'(CLM

CmV!"Cm

"V=1a#

$%&

'(CmM

CDM! 0

CDM> 0 CDM

< 0

a = Speed of Sound

M = Mach number = V a

59!

Wing Lift and Moment Coefficient Sensitivity to Pitch Rate"

Straight-wing incompressible flow estimate (Etkin)"CLq̂wing

= !2CL"winghcm ! 0.75( )

Cmq̂wing= !2CL"wing

hcm ! 0.5( )2

Straight-wing supersonic flow estimate (Etkin)"CLq̂wing

= !2CL"winghcm ! 0.5( )

Cmq̂wing= ! 2

3 M 2 !1! 2CL"wing

hcm ! 0.5( )2

Triangular-wing estimate (Bryson, Nielsen)"

CLq̂wing= ! 2"

3CL#wing

Cmq̂wing= ! "

3AR60!

Control- and Disturbance-Effect Matrices"

•! Control-effect derivatives portray acceleration sensitivities to control input perturbations"

•! Disturbance-effect derivatives portray acceleration sensitivities to disturbance input perturbations"

GLon =

!D"E T"T !D"F

L"E /VN L"T /VN L"F /VNM"E M"T M"F

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

#

$

%%%%%

&

'

(((((

LLon =

!DVwind!D"wind

LVwind /VN L"wind /VNMVwind

M"wind

!LVwind /VN !L"wind /VN

#

$

%%%%%%

&

'

(((((( 61!

Primary Longitudinal Stability Derivatives"

DV !

!1m

CTV!CDV( ) "VN

2

2S+ CTN

!CDN( )"VNS#

$%

&

'(

Small angle assumptions

LVVN!

1mVN

CLV

!VN2

2S + CLN

!VNS"

#$

%

&' (

1mVN

2 CLN

!VN2

2S ( mg

"

#$

%

&'

Mq =1Iyy

Cmq

!VN2

2Sc

"

#$

%

&' M! =

1Iyy

Cm!

"VN2

2Sc

#

$%

&

'(

L!VN!

1mVN

CTN+ CL!( ) "VN

2

2S

#

$%

&

'(

62!

Primary Phugoid Control Derivatives"

D!T !"1m

CT!T

#VN2

2S$

%&

'

()

L!FVN!

1mVN

CL!F

#VN2

2S$

%&

'

()

63!

Primary Short-Period Control Derivatives"

M!E = Cm!E

"NVN2

2Iyy

#

$%&

'(Sc

L!EV = CL!E

"NVN2

2m#$%

&'(S

64!

Flight Motions "

Dornier Do-128 Short-Period Demonstration"http://www.youtube.com/watch?v=3hdLXE0rc9Q"

Simulator Demonstration of "Short-Period Response to Elevator Deflection"

http://www.youtube.com/watch?v=1O7ZqBS0_B8"

Dornier Do-128D!

Dornier Do-128 Phugoid Demonstration"http://www.youtube.com/watch?v=jzxtpQ30nLg&feature=related"

65!

Simulator Demonstration of Phugoid Response !http://www.youtube.com/watch?v=DEOGM_9NGTI!

MAE331Lecture12.pdf

Documents

Transcript of MAE331Lecture12.pdf