Aircraft Flight Dynamics Robert F. Stengel Lecture11 Longitudinal Dynamics

Linearized Longitudinal Robert Stengel, Aircraft Flight Dynamics • 6th-order -> 4th-order -> hybrid equations" • Dynamic stability derivatives " • Phugoid mode" • Short-period mode" Cop

Linearized Longitudinal

Robert Stengel, Aircraft Flight Dynamics 


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

•   Dynamic stability derivatives "

•   Phugoid mode"

•   Short-period mode"

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 !

•   Symmetric aircraft"

•   Motions in the vertical plane"

•   Flat earth "

x1

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x6

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= xLon6=

u w x z q

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Axial Velocity Vertical Velocity Range Altitude(–) Pitch Rate Pitch Angle

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u = X / m − gsinθ − qw

w = Z / m + g cosθ + qu

x I= cosθ( )u + sinθ( )w

z I= − sinθ( )u + cosθ( )w

q = M / I yy

θ = q

State Vector, 6 components!

Nonlinear Dynamic Equations!

Fairchild-Republic A-10!

u = f1= X / m − g sin θ − qw

w = f2 = Z / m + g cos θ + qu

q = f3 = M / Iyy



x1

x2

x3

x4

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= xLon4 =

u w q

θ

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Axial Velocity, m/s Vertical Velocity, m/s Pitch Rate, rad/s Pitch Angle, rad

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State Vector, 4 components!

Fourth-Order Hybrid Equations of Motion

Transform Longitudinal Velocity Components"

u = f1= X / m − g sinθ− qw

w = f2= Z / m + g cosθ+ qu

q = f3= M / I yy



θ= f4 = q



V = f1= T cos$% (α + i)− D − mgsinγ &' m

γ = f2= T sin$% (α + i)+ L − mg cosγ &' mV

q = f3= M / I yy



θ = f4= q x1

x2

x3

x4

!

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=

u

w

q

θ

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Axial Velocity Vertical Velocity Pitch Rate Pitch Angle

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x1 x2 x3 x4

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=

V

γ

q

θ

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Velocity Flight Path Angle Pitch Rate Pitch Angle

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i = Incidence angle of the thrust vector with respect to the centerline

•   Replace X and Z by T, D, and L"

Hybrid Longitudinal Equations of Motion"



V = f1= T cos$% (α + i)− D − mgsinγ &' m

γ = f2= T sin$% (α + i)+ L − mg cosγ &' mV

q = f3= M / I yy



θ = f4= q



V = f1= T cos$% (α + i)− D − mgsinγ &' m

γ = f2= T sin$% (α + i)+ L − mg cosγ &' mV

q = f3= M / I yy



α = f4 =  θ − γ = q − 1

mV$%T sin α + i( )+ L − mg cosγ&'

x1

x2

x3

x4

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=

V

γ

q

θ

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Velocity Flight Path Angle Pitch Rate Pitch Angle

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V

γ

q

α

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Velocity Flight Path Angle Pitch Rate Angle of Attack

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•  Replace pitch angle by angle of attack! α = θ − γ

θ = α +γ

Why Transform Equations and

State Vector?"

•  Phugoid (long-period) mode is primarily

described by velocity and flight path angle"

•  Short-period mode is primarily described

by pitch rate and angle of attack"

x1

x2

x3

x4

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V

γ

q

α

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Velocity Flight Path Angle Pitch Rate Angle of Attack

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Why Transform Equations and State Vector?"

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

FLon = FPh FSP

Ph

FPh SP FSP

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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 period motion"

= FPh small

small FSP

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& ≈

FPh 0

0 FSP

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Nominal Equations of Motion in



VN= 0 = f1= T cos $% ( αN+ i ) − D − mgsinγN&' m

γN = 0 = f2= T sin $% ( αN+ i ) + L − mg cosγN&' mV

qN= 0 = f3= M Iyy



αN= 0 = f4= q − 1

mV $% T sin α ( N+ i ) + L − mg cosγN&'

xN

T

=# V N γN 0 αN

$ %&

T

Equations of Motion

Sensitivity Matrices for

Longitudinal LTI Model"

ΔxLon(t) = FLonΔxLon(t) + GLonΔuLon(t) + LLonΔwLon(t)

F =

∂ f1

∂V

∂ f1

∂γ

∂ f1

∂q

∂ f1

∂α

∂ f2

∂V

∂ f2

∂γ

∂ f2

∂q

∂ f2

∂α

∂ f3

∂V

∂ f3

∂γ

∂ f3

∂q

∂ f3

∂α

∂ f4

∂V

∂ f4

∂γ

∂ f4

∂q

∂ f4

∂α

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'

(

) ) ) ) ) ) ) ) ) ) )

G =

∂ f1

∂δ E

∂ f1

∂δT

∂ f1

∂δ F

∂ f2

∂δ E

∂ f2

∂δT

∂ f2

∂δ F

∂ f3

∂δ E

∂ f3

∂δT

∂ f3

∂δ F

∂ f4

∂δ E

∂ f4

∂δT

∂ f4

∂δ F

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( ( ( ( ( ( ( ( ( (

L =

∂ f1

∂V wind

∂ f1

∂αwind

∂ f2

∂V wind

∂ f2

∂αwind

∂ f3

∂V wind

∂ f3

∂αwind

∂ f4

∂V wind

∂ f4

∂αwind

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( ( ( ( ( ( ( ( ( ( (

Velocity Dynamics"



V = f1 =1

m[T cosα − D − mgsinγ]= 1

m C TcosαρV

2

2 S − C D

ρV2

2 S − mgsinγ

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&

( )

•   Nonlinear equation"

Thrust incidence angle neglected!

•   First row of linearized dynamic equation"

Δ  V (t) = ∂ f1

∂ V ΔV (t)+

∂ f1

∂γ Δ γ (t)+

∂ f1

∂ q Δq(t)+

∂ f1

∂α Δ (t)

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&

)

*

+ ∂ f1

∂δ E Δ δ E(t)+

∂ f1

∂δ T Δ δ T (t)+

∂ f1

∂δ F Δ δ F(t)

%

&'

( )*

+ ∂ f1

%

&

)

*

∂ f1

1

m ( CT V cosαN − CD V) ρVN

2

2 S + C ( T N cosαN − CD N) ρVNS

%

&

)

*

∂ f1

−1

m [ mg cosγN] = −g cosγN

∂ f1

−1

m CD q

ρVN2

%

&

)

*

∂ f1

−1

m ( CT NsinαN + CDα) ρVN

2

%

&

)

*

•   Coefficients in first row of F"

Sensitivity of Velocity Dynamics

to State Perturbations "

C TV≡∂C T

∂V

C DV≡∂C D

∂V

C Dq≡∂C D

∂ q

C D

α ≡∂C D

∂α



V = C( Tcosα− C D)ρV

2

2 S − mgsinγ

%

&

( )

* m

Sensitivity of Velocity Dynamics to Control and Disturbance Perturbations "

∂ f1

−1

m CD δ E

ρ VN2

%

&

)

*

∂ f1

1

m CT δT cos αN

ρ VN2

%

&

)

*

∂ f1

−1

m CD δ F

ρ VN2

%

&

)

*

•   Coefficients in first rows of G and L"

∂ f1

∂ f1

∂V

∂ f1

∂ f1

∂α

C T δT ≡∂C T

∂δT

C D δ E≡∂C D

∂δ E

C D δ F≡∂C D

∂δ F

∂f2

∂V =

1

mV N (C T VsinαN+C L V)ρV N

2

2 S + C( T NsinαN + C L N)ρV N S

$

%

' (

− 1

mV N2 (C T NsinαN + C L N)ρV N

2

2 S − mg cosγN

$

%

( )

∂f2

∂γ =

1

mV N[mg sinγN]= g sinγ N V N

∂f2

∂q=

1

mV N C L q

ρV N2

2 S

$

%

' (

∂f2

∂α =

1

mV N (C T NcosαN + C Lα)ρV N

2

2 S

$

%

( )

•   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!

C T V≡ C T

∂V

C L V≡ C L

∂V

C L q≡ C L

∂

C Lα ≡ C L

∂α



γ= C( Tsinα+ C L)ρ

2

2 S − mg cosγ

%

&

)

* mV

∂ f3

1

Iyy Cm V

ρVN

2

#

$

' (

∂ f3

∂ f3

1

Iyy Cm q

ρVN

2

#

$

' (

∂ f3

1

Iyy Cmα

ρVN

2

#

$

' (

•   Coefficients in third row of F"

Sensitivity of Pitch Rate Dynamics to State Perturbations "

C m V ≡∂C m

∂V

C m q≡∂C m

∂q

C m

α ≡∂C m

∂α

q = Cm ρ V2

2

( ) Sc Iyy

∂ f 4

∂V

∂γ

•   Coefficients in fourth row of F"

Sensitivity of Angle of Attack

Dynamics to State Perturbations "

∂ q

∂α

 α =  θ −  γ = q −  γ

Dimensional Stability and Control Derivatives

Dimensional Stability-Derivative

Notation"

!  Dimensional stability derivatives portray acceleration

sensitivities to state perturbations"

Drag mass (m) ⇒ D ∝ V 

Lift mass ⇒ L ∝ V γ

Moment moment of inertia (Iyy) ⇒ M ∝ q

Dimensional Stability-Derivative

Notation"

∂ f1

m ( CT Vcos αN− CD V) ρ VN

2

2 S + C ( T Ncos αN− CD N) ρ VNS

&

'

* +

∂ f2

Lα

mVN ( CT NcosαN+ CLα) ρVN

2

%

&

)

*

∂ f3

Iyy Cmα

ρVN2

%

&

)

*

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

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

Longitudinal Stability Matrix"

FLon = FPh FSP

Ph

FPh SP

FSP

!

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& =

LV

VN

g

VNsinγN

Lq

− LV

g

VNsinγN 1− Lq

VN

* +

-.

VN

!

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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 Longitudinal Stability

Derivatives"

D V−1

m C T V − C D

V

( )ρV N

2

2 S + C T N − C D

N

( )ρV N S

#

$

' (

Assuming γN  αN  0

L V

V N 1

mV N C L V

ρV N

2

2 S + C L N ρV N S

"

#

%

&

' −mV1

N

2 C L N

ρV N

2

2 S − mg

"

#

%

&

M q= 1

I yy C m q

ρV N2

2 Sc

"

#

%

&

Mα= 1

I yy C mα

ρV N

2

2 Sc

#

$

&

'

Lα

mV N (C T N + C Lα)ρV N

2

2 S

#

$

' (

Origins of Stability Effects

Velocity-Dependent Derivative

Definitions"

•   Air compressibility effects are a principal source of velocity dependence"

CD

∂ ( V / a ) = a

C D V ≡∂C D

∂V =

1

a

#

$

&

'

(C D M

C L V ≡∂C L

∂V =

1

a

#

$

&

'

(C L M

C m V≡∂C m

∂V =

1

a

#

$

&

'

(C m M

C D M ≈ 0

C D M > 0 C D M < 0

a = Speed of Sound

M = Mach number = V a

Pitch-Moment Coefficient

Sensitivity to Angle of Attack"

MB= Cmq Sc ≈ C ( m o+ Cm qq + Cmαα ) q Sc

Cm

αnet ( hcm − hcp net)

αnet

xcm − xcp net

c

$

%

(

αht

Pitch-Rate Derivative Definitions"

C m

q=∂C m

∂ q =

c

2V N

"

#

&

'C m ˆq

C m ˆq=∂C m

∂ ˆq =

∂C m

∂ qc 2V( N)=

2V N

c

"

#

%

&

'C mq

ˆq = qc 2VN

M q=∂M

∂ = C m q( ρV N2 2)Sc = C m ˆq c

2V N

#

$

% &

' ( ρV N 2 2

#

$

% &

'

(Sc = C m ˆq

ρV N Sc2 4

#

$

% &

' (

M B = C m q Sc ≈ C m

o+C m

≈ C m o+∂C m

∂q q + C mαα

$

%

(

)q Sc

•  Pitch acceleration sensitivity to pitch rate "

Pitch-Rate Derivative Definitions"

in terms of a normalized pitch rate "

c

2VN

"

#

$ %

&

'Cm ˆq

C m ˆq=∂C m

∂ˆq =

∂C m

∂ qc 2V

N

2V N

c

"

#

$ %

&

'C m q

ˆq = qc 2VN

M B = C m q Sc ≈ C m

o+C m

≈ C m o+∂C m

∂q q + C mαα

$

%

(

)q Sc

•  Then"

•  But dynamic equations require ∂C m /∂q "

Angle of Attack Distribution

Due to Pitch Rate"

•  Aircraft pitching at a constant rate, q rad/s, produces a normal velocity distribution along x"

Δw = −qΔx

Δ =Δw

V N

=−qΔx

V N

•  Corresponding angle of attack distribution"

Δ ht=ql ht

V N

•  Angle of attack perturbation at tail center of pressure"

€

l ht = horizontal tail distance from c.m.

Horizontal Tail Lift

Due to Pitch Rate"

•  Incremental tail lift due to pitch rate, referenced to tail area, S ht"

•  Lift coefficient sensitivity to pitch rate referenced to wing area"

ΔLht = ΔC ( L ht)ht

1

2 ρ VN

2

Sht

C L qht ≡∂ ΔC( L ht)aircraft

∂C L ht

∂α

%

&

)

*

aircraft

l ht

V N

%

&

' ( )

*

ΔC L ht

( )aircraft = ΔC( L ht)ht

S ht

S

"

#

%

&

' = ∂C L ht

∂α

"

#

&

'

aircraft

Δ

* +

, ,

-

/ /=

∂C L ht

∂α

"

#

&

'

aircraft

ql ht

V N

"

#

$ %

&

'

•  Incremental tail lift coefficient due to pitch rate, referenced to

wing area, S"

Moment Coefficient Sensitivity to Pitch Rate of

the Horizontal Tail"

∂ ΔM ht

∂ q = C m qht

1

2ρV N

2Sc = −C L qht

l ht

V N

%

&

( )

1

2ρV N

2Sc

= − ∂C L ht

∂α

%

&

( )aircraft

l ht

V N

%

&

( ) ,

0

1l ht

c

%

&

( ) 1

2ρV N

2

Sc

C m qht= −∂C L ht

∂α

l ht

V N

$

%

& ' ( ) l ht

c

$

%

' ( ) = −∂C L ht

∂α

l ht

c

$

%

' (

2

c

V N

$

%

& ' ( )

•   Coefficient derivative with respect to normalized pitch rate is insensitive to velocity"

Cm ˆqht = ∂Cm ht

∂ ˆq =

∂Cm ht

∂ qc 2V ( N) = −2

∂CL ht

∂α

lht c

$

%

& ' ( )

2

Comparison of Fourth-

and Second-Order

Dynamic Models

•   0 - 100 sec "

4th-Order Initial-Condition Responses of Business Jet

at Two Time Scales"

•   0 - 6 sec "

•   Plotted over different periods of time"

–  4 initial conditions "

Second-Order Models of

Longitudinal Motion"

•  2 nd -Order Approximate Phugoid Equation"

ΔxPh= Δ V

Δ γ

#

$

%

%

&

'

(

(≈

−D V −g cosγ N

L V

V N

g

V NsinγN

#

$

%

%

%

&

'

( ( (

ΔV

Δγ

#

$

%

%

&

'

( (+

T δT

L δT

V N

#

$

%

%

%

&

'

( ( ( ΔδT +

−D V

L V

V N

#

$

%

%

%

&

'

( ( (

ΔV wind

ΔxSP= Δ q

Δ α

#

$

%

%

&

'

(

(≈

M q Mα

1 −L q

V N

+

,- /0 −

Lα

V N

#

$

%

%

%

&

'

( ( (

Δq

Δ

#

$

%

%

&

'

( (+

M δ E

−L δ E

V N

#

$

%

%

%

&

'

( ( (

Δδ E +

Mα

−Lα

V N

#

$

%

%

%

&

'

( ( (

Δ wind

•  2 nd -Order Approximate Short-Period Equation"

!

"

#

#

$

%

&

&

•  Full and approximate linear models"

Comparison of Bizjet Fourth- and Second-Order Model Responses"

•   Fourth Order"

•   Second Order"

because natural frequencies are widely separated"

Comparison of Bizjet Fourth- and

Second-Order Models and Eigenvalues"

Fourth-Order Model

F = 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-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 Approximation

F = G = Eigenvalue Damping Freq (rad/s)

Short-Period Approximation

F = G = Eigenvalue Damping Freq (rad/s)

Approximate Phugoid Roots "

•  Approximate Phugoid Equation (!N = 0)"

ΔxPh= Δ V

Δ  γ

#

$

%

%

&

'

( (≈

−D V −g

L V

V N 0

#

$

%

%

%

&

'

( ( (

ΔV

Δγ

#

$

%

%

&

'

( (+

T δT

L δT

V N

#

$

%

%

%

&

'

( ( ( ΔδT

•  Characteristic polynomial"

sI − FPh = det sI − F ( Ph) ≡ Δ(s) = s2+ DVs + gLV/ VN

= s2+ 2ζωns + ωn

2

•  Natural frequency and damping ratio"

2 L / D ( )N

compressibility effects"

Effect of Airspeed and L/D on Approximate

Phugoid Natural Frequency, Period, and

Damping Ratio "

ζ ≈ 1

2 ( L / D )N

Velocity

Natural

Damping Ratio

Neglecting compressibility effects"

Period, T = 2 π / ωn

≈ 0.45VN sec

Approximate Phugoid Response to

a 10% Thrust Increase "

•  What is the steady-state response?"

Approximate Short-Period Roots "

•  Approximate Short-Period Equation (L q = 0 )"

•  Characteristic polynomial"

ΔxSP= Δ q

Δ  α

#

$

%

%

&

'

( (

≈

M q Mα

V N

#

$

%

%

%

&

'

( ( (

Δq

Δα

#

$

%

%

&

'

( ( +

M δ E

−L δ E

V N

#

$

%

%

%

&

'

( ( (

Δδ E

Δ(s) = s2+ Lα

V N − M q

$

%

& '

(

)s − Mα + M q

Lα

V N

$

%

& '

( )

= s2+ 2ζωn s +ω n2

ωn = − Mα+ M q

Lα

V N

$

%

& '

( ); ζ =

Lα

V N − M q

$

%

' (

2 − Mα+ M q

Lα

V N

$

%

& '

( )

Generally,

Lα> 0

Mα< 0

M q< 0

Approximate Short-Period Response

to a 0.1-Rad Pitch Control Step Input "

Pitch 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 (Elevator)!

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

Forward Pitch Control (Canard)!

n z=V N

g (Δ α − Δq)=V N

g

Lα

V N

Δα +L δ E

V N

Δδ 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

Reading

Flight Dynamics, 96-101, 574-582, 587-591 Virtual Textbook , Part 12

Supplemental Material

Flight Path Angle Dynamics"

•   Second row of linearized equation"

γ = f2 = 1

mV[T sinα + L − mg cosγ]= 1

mV C TsinαρV

2

2 S + C L

ρV2

2 S − mg cosγ

%

&

( )

•   Nonlinear equation"

Δ  γ (t) = ∂ f2

∂ V ΔV (t) +

∂ f2

∂γ Δ γ (t) +

∂ f2

∂ f2

∂α Δ α (t)

%

&

)

*

+ ∂ f2

∂δ E Δ δ E(t) +

∂ f2

∂δ T Δ δ T (t) +

∂ f2

∂δ F Δ δ F(t)

%

&'

( )*

+ ∂ f2

∂ VwindΔVwind+

∂ f2

∂αwind

Δ αwind

%

&

)

*