Aircraft Flight Dynamics Robert F. Stengel Lecture18 Advanced Longitudinal Dynamics

Advanced Problems of Longitudinal Dynamics Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012 " – Steady-state response to control" – Transfer functions" – Frequency response" –

Advanced Problems of

Longitudinal Dynamics


Robert Stengel, Aircraft Flight Dynamics

MAE 331, 2012 "

–   Steady-state response to control"

–   Transfer functions"

–   Frequency response"

–   Root locus analysis of parameter

variations "

flight condition"

Copyright 2012 by Robert Stengel All rights reserved For educational use only.!

http://www.princeton.edu/~stengel/MAE331.html !

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

Distinction Between

α = q

α ≠ 0 ≠ q; q = 0

!  With no vertical motion of the

c.m., pitch rate and angle-of-attack rate are the same "

!  With no pitching, vertical heaving (or plunging) motion of the c.m., produces angle-of-attack rate but

no pitch rate"

Vertical velocity distribution induced

by pitch rate!

Angle-of-Attack Rate

Has Two Effects "

!  Pressure variations at wing

convect downstream,

arriving at tailΔt sec later"

!  Lag of the downwash"

!  Delayed

tail-lift/pitch-moment effect"

!  Vertical force opposed by a

mass of air ( apparent

mass ) as well as airplane

mass"

!  Vertical acceleration

produces added lift and

moment "

Flight Dynamics, pp 204-206, 284-285"

Δ q = M q Δq + MαΔα + M δEΔδE+ MαΔ  α

Δ  α = 1−L q

V N

%

&

)

*Δq − Lα

V N

( )Δα − LδE

V N

( )ΔδE − Lα 

V N

( )Δ  α

Δ q − Mα  Δ  α= M q Δq + MαΔ α+ MδEΔ δE

Δ  α + Lα 

V N

( )Δ  α = 1−L q

V N

%

&

( )

*Δq − Lα

V N

( )Δ α − LδE

V N

( )Δ δE

1 −Mα 

0 1+ Lα 

V N

( )

#

$%

&

'(

#

$

%

%

%

&

'

( ( (

Δ q

Δ  α

#

$

%

%

&

'

( (=

1−L q

V N

* +

- / − Lα

V N

( )

#

$

%

%

%

&

'

( ( (

Δq

Δ

#

$

%

%

&

'

( (+

M δE

− L δE VN

#

$

%

%

%

&

'

( ( ( ΔδE

Angle-of-Attack-Rate Effects Principally

Affect the Short-Period Mode"

!  Lift and pitching moment proportional to angle-of-attack rate"

!  Bring effects to left side"

!  Vector-matrix form"

1 −Mα

0 1 + Lα

V N

( )

#

$%

&

'(

#

$

%

%

%

&

'

( ( (

−1

=

1 + Lα 

V N

( )

#

$%

&

'( Mα

#

$

%

%

%

&

'

( ( (

1 + Lα 

V N

( )

#

$%

&

'(

Δ q

Δ  α

#

$

%

%

&

'

(

(=

1 −Mα 

0 1+ Lα 

V N

( )

#

$%

&

'(

#

$

%

%

%

&

'

( ( (

−1

1−Lq V

N

* +

- / − Lα

V N

( )

#

$

%

%

%

&

'

( ( (

Δq

Δ

#

$

%

%

&

'

( (+

M δE

− L δE VN

#

$

%

%

%

&

'

( ( (

Δ δE

1 2 33 4

3 3

5 6 33 7

3 3

!  Inverse of the apparent mass matrix"

!  Pre-multiply both sides by inverse"

Δ q

Δ  α

#

$

% &

' (=

1+ Lα 

V N

( )

#

$%

&

'( Mα 

#

$

%

%

%

&

'

( ( ( 1+ Lα 

V N

( )

#

$%

&

'(

1−Lq VN

* +

- / −Lα

VN

#

$

%

%

%

&

'

( ( (

Δq

Δ

#

$

% &

' (+

0 1 22 3 2

4 5 22 6 2

Δ q

Δ  α

#

$

% &

'

( = 1 1+Lα

V N

( )

#

$%

&

'(

1+ Lα

V N

( )

#

$%

&

'(Mq+Mα *1−Lq VN +

-.

0 2

3

Lα 

V N

( )

#

$%

&

'(Mα −Mα 

Lα VN

( )

0 2

3 5 1−Lq VN

* +

α

VN

#

$

%

%

%

%

%

&

'

( ( ( ( (

Δq

Δα

#

$

% & ' (

+ 1+ Lα

V N

( )

#

$%

&

'(M δE−Mα 

L δE

VN

− δE

VN

#

$

%

%

%

%

&

'

( ( ( (

ΔδE

0

1

7 7 7 7 7

2

7 7 7 7 7

3

4

7 7 7 7 7

5

7 7 7 7 7

!  Multiply matrices"

!  Substitute"

Simplification of

Angle-of-Attack-Rate Effects"

Δ q

Δ  α

#

$

%

%

&

'

(

( 

M q+Mα 

{ } Mα−Mα 

Lα

V N

( )

V N

( )

#

$

%

%

%

%

&

'

( ( ( (

Δq

Δ

#

$

%

%

&

'

( (+

M δ E−Mα 

L δ E

V N

− L δ E

V N

#

$

%

%

%

%

&

'

( ( ( (

Δδ E

!  Typically" L q and Lα have small effects for large aircraft*

M q and Mα are same order of magnitude and have more significant effects

!  Neglecting" L q and Lα

* but not for small aircraft, e.g., R/C models and micro-UAVs"

2 nd -Degree Characteristic Polynomial with"

!  Short-period characteristic polynomial"

!  Damping is increased"

!  Natural frequency is unaffected"

Δ s( ) =

s − M( q+Mα)

α −Mα Lα

V N

( )

$

%(

&

')

V N

( )

$

%(

&

')

= s − M$ ( q+Mα) &' s + Lα

V N

( )

$

%(

&

')− Mα −Mα Lα

V N

( )

$

%(

& ')

Δ s( )= s2

+ Lα

V N

( )− M( q+Mα)

$

%&

' ()s + Mα− M q Lα

V N

( )

$

%&

' ()

* + ,

- /

= s2 + 2ζωn s +ω n

2

= 0

L q and Lα  0

= s2

+ Lα

V N

( )− M( q+Mα)

#

$%

&

'(s + Mα− M( q+Mα)Lα

V N

( )

#

$%

&

'(+Mα

Lα

V N

( )

) +

,

Linear, Time-Invariant Fourth-Order

Longitudinal Model "

Δ V (t)

Δ γ (t)

Δ q(t)

Δ α(t)

$

%

&

&

&

&

&

'

(

)

)

)

)

)

=

L V

V N 0 0 LαV N

V N 0 1 − αV N

$

%

&

&

&

&

&

&

&

'

(

) ) ) ) ) ) )

ΔV (t)

Δγ (t)

Δq(t)

Δα(t)

$

%

&

&

&

&

&

'

(

) ) ) ) ) +

$

%

&

&

&

&

&

'

(

) ) ) ) )

ΔδE(t) ΔδT (t) ΔδF(t)

$

%

&

&

&

'

(

) ) )

•  Stability and control derivatives are defined at a

trimmed (equilibrium) flight condition "

Perturbations to the Trimmed Condition"

•   Initial pitch rate [Δq(0)] = 0.1 rad/s" •  Elevator step input [ΔδE(0)] = 1 deg"

•  Small linear and nonlinear perturbations are virtually identical "

Steady-State Response

LTI Longitudinal Model"

ΔVSS

ΔγSS ΔqSS

Δ SS

$

%

&

&

&

&

&

'

(

) ) ) ) )

= −

LV

− V

$

%

&

&

&

&

&

&

&

'

(

) ) ) ) ) ) )

−1

0 0 L δF / VN

0 0 −L δF / VN

$

%

&

&

&

&

&

'

(

) ) ) ) )

ΔδESS ΔδTSS ΔδFSS

$

%

&

&

&

'

(

) ) )

ΔxSS = −F−1G ΔuSS

•  How do we calculate the equilibrium response to control? "

Δx(t) = FΔx(t) + GΔu(t)

•  For the longitudinal model "

Algebraic Equation for

Equilibrium Response "

ΔV SS

ΔγSS

Δq SS

Δ SS

$

%

&

&

&

&

&

'

(

)

)

)

)

)

=

−gMδE Lα

V N

$

%&

'

D V Lα

V N − Dα

L V

V N

$

%&

' () M V

Lα

V N − Mα

L V

V N

$

%&

' () $% (DαM V − D V Mα)LδF / V N'(

−gMδE

L V

V N

$

%&

'

$

%

&

&

&

&

&

&

&

'

(

) ) ) ) ) ) )

g M V Lα

V N − Mα

L V

V N

ΔδESS

ΔδTSS ΔδFSS

$

%

&

&

&

'

(

) ) )

ΔV SS

Δ γSS

Δq SS

Δ SS

$

%

&

&

&

&

&

'

(

) ) ) ) )

=

0 0 0

$

%

&

&

&

&

'

(

) ) ) )

Δ δE SS

Δ δTSS

Δ δFSS

$

%

&

&

&

'

(

) ) )

• Roles of stability and control

derivatives identified"

• Result is a simple equation relating

input and output "

Be Counterintuitive"

ΔV SS = aΔδ E SS+ ( ) 0 ΔδTSS + bΔδ F SS

ΔγSS = cΔδ E SS + dΔδT SS + eΔδ F SS

Δq SS= ( ) 0 ΔE SS+ ( ) 0 ΔδTSS+ ( ) 0 Δδ FSS

ΔαSS = f Δδ E SS+ ( ) 0 ΔδTSS + gΔδ F SS

•  Observations"

–   Thrust command"

–   Elevator and flap commands"

–   Steady-state pitch rate is zero!

Δ θSS = Δ γSS + Δ αSS = c + f( )Δ δE SS + dΔδT SS + e + g( )Δ δF SS

•  Steady-state pitch angle "

Effects of Stability Derivative

Longitudinal Modes

Primary and Coupling Blocks of the Fourth-Order Longitudinal Model"

FLon=

−DV −g 0 −Dα

L V

− V

#

$

%

%

%

%

%

%

%

&

'

( ( ( ( ( ( (

= FPh FSP

FPh FSP

#

$

%

%

&

'

( (

•  Some stability derivatives appear only in primary blocks (D V , M q , Mα)"

–   Effects are well-described by 2 nd -order models "

•  Some stability derivatives appear only in coupling blocks (M V , Dα)"

–   Effects are ignored by 2 nd -order models "

•  Some stability derivatives appear in both (L V , Lα)"

–   Require 4 th -order modeling"

ΔMα Effect on 4 th -Order Roots !

Short

Period!

Short

Period!

Phugoid!

Phugoid!

ΔLon(s) = s 4

+ DV+Lα

V N − Mq

+ g − D( α)L V

V N + D V

Lα

V N − M q

( )− M q Lα

V N − Mαo

$

%&

' ()

2

+ Mq(Dα− g)L V

V N − DV Lα

V N

$

%&

' ()+ DαM V − DV Mαo

+ g MV Lα

V N − MαoL V

V N

( ) −ΔMαs2+ DV s + g L V

V N

≡ d(s)+ kn(s)

•  Primary effect: The same as in the approximate short-period model"

•  Numerator zeros "

–   The same as the approximate phugoid mode characteristic polynomial "

–   Effect of Mα variation on phugoid mode is small#

Short Period!

Short Period!

Phugoid!

Phugoid!

Direct Thrust Effect on Speed

Stability, T V"

∂T

∂V =

< 0, for propeller aircraft

≈ 0, for turbojet aircraft

> 0, for ramjet aircraft

#

$

%

&

%

∂T

∂V −

∂D

∂V > 0

T N − D N = C T N 1

2ρV N

2

S − C D N 1

2ρV N

2

S = 0

•   Effect of velocity change "

•   Small velocity perturbation grows if "

•  Therefore"

–   propeller is stabilizing for velocity change"

–   turbojet has neutral effect"

–   ramjet is destabilizing"

•  Thrust line above or below center of mass induces a pitching moment"

•  Aerodynamic and thrust pitching moments sensitive to velocity perturbation"

Martin XB-51!

McDonnell Douglas MD-11!

Consolidated PBY!

Fairchild-Republic A-10!

•   Negative ∂M/∂V (Pitch-down effect) tends to increase velocity"

•   Positive ∂M/∂V (Pitch-up effect) tends to decrease velocity"

decreases thrust, producing a pitch-up moment"

–   Up: Lake Amphibian, MD-11"

–   Down: F6F, F8F, AD-1 " ∂ M thrust

∂T

Douglas AD-1!

Lake Amphibian!

Grumman F8F!

McDonnell Douglas !

MD-11!

Pitching Moment Due to Thrust, M V!

Roots"

•  Large positive value produces

oscillatory phugoid instability "

•  Large negative value produces

real phugoid divergence "

ΔLon (s) = s4

+ D V+ α

V N − M q

+ g − D( α )L V

V N + D V

Lα

V N − M q

( )− M q

Lα

V N − Mα

$

%&

' () 2

+ M q(Dα− g)L V

V N − D V Lα

V N

$

%&

' ()+ DαM V − D V Mα

gMαL V

V N+M V Dαs + g Lα

V N

€

Dα = 0

Short Period!

Phugoid!

frequency of the phugoid"

short-period"

•  Lα /V N: Increased damping

of the short-period "

•  Small effect on the phugoid

mode"

Short

Period!

Phugoid!

Pitch and Thrust Control Effects

Longitudinal Model Transfer

Function Matrix (H x = I, H u = 0)"

H Lon (s) =

n δ E V

(s) n δT V

(s) n δ F V

(s)

n δ Eγ

(s) n δTγ

(s) n δ Fγ

(s)

n δ E q

(s) n δT q

(s) n δ F q

(s)

n δ Eα

(s) n δTα

(s) n δ Fα

(s)

$

%

&

&

&

&

&

&

'

(

) ) ) ) ) )

s2

+ 2ζPωn P s + ω n P

2

+ 2 ζSPωn SP s + ω n SP

2

ΔV (s)

Δγ (s)

Δq(s)

Δα(s)

$

%

&

&

&

&

&

'

(

) ) ) ) )

= H xA s( )G

ΔδE(s) ΔδT (s) ΔδF(s)

$

%

&

&

&

'

(

) ) )

=H Lon (s)

ΔδE(s) ΔδT (s) ΔδF(s)

$

%

&

&

&

'

(

) ) )

A Little More About Output Matrices"

•   With H x = I and H u = 0!

Δy = Δx = H x Δx; then H x = I4

and

Δy1

Δy2

Δy3

Δy4

"

#

$

$

$

$

$

%

&

' ' ' ' '

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

"

#

$

$

$

$

%

&

' ' ' '

Δx1

Δx2

Δx3

Δx4

"

#

$

$

$

$

$

%

&

' ' ' ' '



ΔV

Δγ

Δq

Δα

"

#

$

$

$

$

$

%

&

' ' ' ' '

•   Only output is ΔV !

Δy = ΔV= "# 1 0 0 0 $%

ΔV

Δ γ

Δq

Δ α

"

#

( ( ( ( (

$

%

) ) ) ) )

•   ΔV and Δ α are measured"

Δy2

"

#

$

$

%

&

' '=

ΔV

Δ α

"

#

&

' =" 1 0 0 00 0 0 1

#

& '

ΔV

Δ γ

Δq

Δ α

"

#

$

$

$

$

$

%

&

' ' ' ' '

A Little More About Output Matrices"

•   Output (measurement) of body-axis velocity and pitch rate and

angle"

•   Transformation from [ΔV, Δγ, Δq, Δθ] to [Δu, Δw, Δq, Δα]"

Δu Δw Δq

Δθ

#

$

%

%

%

%

&

'

( ( ( (

=

cosαN 0 0 −V NsinαN

sinαN 0 0 V NcosαN

#

$

%

%

%

%

%

&

'

( ( ( ( (

ΔV

Δγ

Δq

Δα

#

$

%

%

%

%

%

&

'

( ( ( ( (

•   Separate measurement

of state and control

perturbations!

Δy = Δx

Δu

"

#

%

&

' = H x Δx + H u Δu

Δy1

Δy2

Δy3

Δy4

Δy5

Δy6

"

#

$

$

$

$

$

$

$

$

%

&

' ' ' ' ' ' ' '

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

"

#

$

$

$

$

$

$

$

%

&

' ' ' ' ' ' '

ΔV

Δγ

Δq

Δα

"

#

$

$

$

$

$

%

&

' ' ' ' ' +

0 0

0 0

0 0

0 0

1 0

0 1

"

#

$

$

$

$

$

$

$

%

&

' ' ' ' ' ' '

Δδ E ΔδT

"

#

&

'

Elevator-to-Normal-Velocity Numerator"

H xAdj sI − F( Lon)G =# sinαN 0 0 V NcosαN %

n V (s) nγV (s) n q V (s) nαV (s)

n Vγ(s) nγ(s) n qγ(s) nαγ(s)

n V q (s) nγq (s) n q (s) nαq (s)

n Vα(s) nγα(s) n qα(s) nα(s)

#

$

( ( ( ( ( (

%

&

) ) ) ) ) )

0 0

M δ E

0

#

$

( ( ( (

%

&

) ) ) )

= n δ E w

(s)

•  Transform though αNback to body axes "

n δE w (s)= # sinαN 0 0 V NcosαN

n qγ(s)

#

$

( ( ( ( ( (

%

&

) ) ) ) ) )

= M δE ( sinαN)n q V

(s)+ V( NcosαN)n qα(s)

•   Scalar transfer function numerator "

Elevator-to-Normal-Velocity

Transfer Function"

Δw(s)

Δδ E(s)=

n δ E w (s)

ΔLon (s) =

M δ E(s2 + 2ζωn s + ω n2)Approx Ph(s − z3)

s2+ 2ζωn s + ω n2

( )Ph(s2 + 2ζωn s + ω n2)SP

Elevator-to-

Response"

Δw(s)

ΔδE(s)=

n δ E w (s)

ΔLon(s)≈

M δ E s2 + 2ζωns + ωn

s2 + 2ζωns + ωn

( )Ph s2

+ 2ζωns + ωn

( )SP

0 dB/dec!

+40 dB/dec!

0 dB/dec!

–40 dB/dec!

–20 dB/dec!

•  (n – q) = 1"

almost (but not quite) cancels phugoid

Elevator-to- Pitch-Rate "

Numerator and Transfer Function"

H xAdj sI − FLon( )G ="# 0 0 1 0 $%

n V (s) nγV (s) n q (s) nα(s)

n Vγ(s) nγ(s) n qγ(s) nαγ(s)

n V (s) nγq (s) n q (s) nα(s)

n V (s) n V (s) n V (s) n V (s)

"

#

( ( ( ( ( (

$

%

) ) ) ) ) )

0 0

M δ E

0

"

#

( ( ( (

$

%

) ) ) )

= n δ E q

(s)

Δq(s)

ΔδE(s)=

ΔLon (s)≈

MδE s(s − z1)(s − z2)

s2 + 2ζωn s + ω n2

+ 2ζωn s + ω n2

•  Free s in numerator differentiates

pitch angle transfer function "

Elevator-to-Pitch-Rate Frequency Response"

+20 dB/dec!

+20 dB/dec! +40 dB/dec!

0 dB/dec!

–20 dB/dec!

Δq = 0 ( ) ΔδE + 0 ( ) ΔδT + 0 ( ) ΔδF

•  (n – q) = 1"

•  Negligible low-frequency response, except at phugoid natural frequency"

•  High-frequency response well predicted by 2 nd -order model "

Δq(s)

Δδ E(s)=

n δ E q (s)

ΔLon (s)≈

M δ E s s − z( 1 ) (s − z2 )

s2

+ 2ζωn s + ω n

+ 2ζωn s + ω n

Transfer Functions of Elevator Input

to Angle Output*"

Δθ(s)

ΔδE(s)=

n δEθ (s)

ΔLon (s); n δE

θ (s) = M δE$%&s + 1Tθ1'

(

2

$

%

( )

Δα(s)

ΔδE(s)=

n δEα(s)

ΔLon (s); n δE

α (s) = M δE s2 + 2ζωn s +ω n2

Δγ(s)

ΔδE(s)=

n δEγ (s)

ΔLon (s); n δE

V N%&'s + 1Tγ1(

)

*

•  Elevator-to-Flight Path Angle transfer function "

•  Elevator-to-Pitch Angle transfer function "

* Flying qualities notation for zero time constants"

Frequency Response of Angles to Elevator Input"

•  Pitch angle frequency response (Δ θ = Δ γ + Δα)"

–  Similar to flight path angle near phugoid natural frequency"

–  Similar to angle of attack near short-period natural frequency"

ΔγSS = cΔδ E SS

Δ SS = f Δδ E SS

ΔθSS = c − f( ) Δδ ESS

Transfer Functions of Thrust

Input to Angle Output"

Δθ(s)

ΔδT (s)=

nδθT (s)

ΔLon (s); nδT

θ

(s) = TδT$%&s + 1TθT'()

Δα(s)

ΔδT (s)=

n δTα

(s)

ΔLon (s); n δT

α (s) = T δT s s + 1Tα

T

$

%&

' ()

Δγ (s)

ΔδT (s)=

n δTγ

(s)

ΔLon (s); n δT

γ (s) = T δT L V

V N s

2

+ 2ζωn s + ω n2

•  Thrust-to-Flight Path Angle transfer function "

•  Thrust-to-Pitch Angle transfer function "

Frequency Response of Angles

to Thrust Input"

•  Primarily effects flight path angle and low-frequency pitch angle "

Gain and Phase Margins:

The Nichols Chart

Nichols Chart: 


Gain vs Phase Angle"

•  Bode Plot"

frequency not shown"

20 log 10 AR(jω) "

and PM "

H blue ( jω ) = 10

jω + 10

"

#

%

&

100 2

jω

( ) 2 + 2 0.1 ( ) ( 100 ) ( )jω + 100 2

"

#

&

'

H green ( jω ) = 10

2

jω

( ) 2 + 2 0.1 ( ) ( ) 10 ( )jω + 10 2

"

#

&

jω + 100

"

#

%

&

Gain and Phase Margins in

Elevator-to-Pitch-Angle

Bode Plot"

Elevator-to-Pitch-Angle Nichols Chart"

•   Gain Margin: Amplitude ratio below 0 dB

when phase angle = 180°"

•   Phase Margin: Phase angle above –180°

when amplitude ratio = 0 dB"

Pilot-Vehicle Interactions

Pilot Inputs to Control"

* p 421-425, Flight Dynamics"

Effect of Pilot Dynamics on

Pilot Transfer Function = Δu s( )

Δ ε ( )s = K P

1 / T P

s +1 / T = K P

1 / 0.25

s +1 / 0.25

•   Pilot introduces neuromuscular lag while closing the control loop"

•  Example "

–   Model the lag by a 1 st -order time constant, T P, of 0.25 s"

–   Pilot s gain, K P, is either 1 or 2 "