Aircraft Flight Dynamics Robert F. Stengel Lecture20 Flying Qualities Criteria

Flying Qualities Criteria Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012 " Copyright 2012 by Robert Stengel.. • CAP, C*, and other longitudinal criteria" • ϕ/β , ωϕ/ω , and ot

Flying Qualities Criteria


Robert Stengel, Aircraft Flight Dynamics


MAE 331, 2012 "

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 !

•   CAP, C*, and other longitudinal

criteria"

•   ϕ/β , ωϕ/ω , and other

lateral-directional criteria"

•  Pilot-vehicle interactions"

•  Flight control system design"

•  Satisfy procurement requirement (e.g., Mil Standard)"

•  Satisfy test pilots (e.g., Cooper-Harper ratings)"

•  Avoid pilot-induced oscillations (PIO)"

•  Minimize time-delay effects"

•  Time- and frequency-domain criteria"

MIL-F-8785C Identifies Satisfactory, Acceptable,

Damping Ratio"

Step Response"

Frequency Response"

Short-period angle-of-attack response to elevator input!

Longitudinal Criteria

Long-Period Flying Qualities Criteria

•  Static speed stability "

–  No tendency for aperiodic divergence"

•  Phugoid oscillation -> 2 real roots, 1 that is unstable"

–  Stable control stick position and force gradients"

•  e.g., Increasing pull position and force with decreasing speed"

A   Non-terminal flight requiring rapid

maneuvering"

B   Non-terminal flight requiring gradual

maneuvering"

C   Terminal flight"

1   Clearly adequate for the mission"

2   Adequate to accomplish the mission, with some increase in workload"

3   Aircraft can be controlled safely, but workload is excessive"

Level of Performance!

Flight Phase!

Long-Period Flying Qualities Criteria

1   ( Δγ/ΔV)SS < 0.06 deg/kt"

2   ( Δγ/ΔV)SS < 0.15 deg/kt"

3   ( Δγ/ΔV)SS < 0.24 deg/kt "

ΔV SS = aΔδE SS+ ( ) 0 Δ δT SS + bΔδF SS

Δ γSS = cΔδE SS + dΔδT SS + eΔδF SS

• Lecture 19"

ΔγSS

c

• From 4 th -order model"

Long-Period Flying Qualities Criteria

•  Phugoid stability "

1   Damping ratio ≥ 0.04"

2   Damping ratio ≥ 0"

3   Time to double , T 2 ≥ 55 sec"

€

Time to Double!

Short Period Criteria"

•  Important parameters "

–  Short-period natural frequency"

–  Damping ratio"

–  Lift slope"

–  Step response"

•  Over-/under-shoot"

•  Rise time"

•  Settling time"

•  Pure time delay"

–  Pitch angle response"

–  Normal load factor response"

–  Flight path angle response (landing)"

Space Shuttle Pitch-Response Criterion"

Short-Period Approximation

Transfer Functions"

•   Elevator to pitch rate" ΔΔq(s)δE(s)= k q(s − z q)

s2 + 2 ζSPωn SP s +ωn SP

2 ≡

k q()s + 1Tθ2+,

s2 + 2 ζSPωn SP s +ωn SP

2

• Pure gain or phase change in feedback

control cannot produce instability"

Bode Plot!

Nichols Chart!

Root Locus!

Short-Period Approximation

Transfer Functions"

•  Elevator to pitch angle"

•  Integral of prior example"

Δ θ(s)

Δ δE(s)=

k q(s − z q)

s 2 + 2 ζSPωn SP s +ωn SP

2

•  Pure gain or phase change in feedback control cannot produce instability"

Bode Plot!

Nichols Chart!

Root Locus!

Normal Load Factor"

•   Therefore, with negligible L δE (aft tail/canard effect)"

Δn z =V N

g (Δ α − Δq) = −V N

g

Lα

V NΔα +

LδE

V N Δδ E

%

&'

( )*

1

%

&

)

g

%

&

)

∂ Δ δE(s)

positive down!

positive up!

Δα(s)

ΔδE(s)≈

kα

s2

+ 2ζSPωn SP s +ω n SP

2

•   Elevator to angle of attack (L δE = 0 )"

http://www.youtube.com/watch?v=xFemVFgsJAw!

Control Anticipation Parameter, CAP"

•  I nner ear senses angular acceleration about 3 axes"

€

Δ ˙ q (0) = M δE− Mα

V N + LαL δE

&

' ( )

*

+ ΔδE SS

Δn SS=V N

g Δq SS= −

V N g

&

'

)

*

M δE Lα

V N − Mα

L δE

V N

&

'

)

*

M q Lα

V N + Mα

&

'

)

* ΔδESS

Δn SS

=

V N + LαL δE

%

&

)

* M q Lα

V N + Mα

LαM δE − L δE Mα

•  Inner ear cue should aid pilot in anticipating commanded normal acceleration"

Initial Angular Acceleration!

Desired Normal Load Factor!

Control Anticipation Factor!

MIL-F-8785C

Short-Period

Flying

Qualities

Criterion"

CAP =

Lα

V N + Mα

ωn2SP

n z/ α

€

ωn SP vs. n z

α

1   Clearly adequate for the mission"

2   Adequate to accomplish the mission, with some increase in workload"

3   Aircraft can be controlled safely, but workload is excessive"

Level of Performance!

with L δE = 0!

•  CAP = constant

along Level

Boundaries "

CAP!

Control Anticipation Parameter vs Short-Period Damping Ratio "

− M q Lα

V N + Mα

Lα g

≈ ωn SP

2

n z / α

!  Below V crossover, Δq is pilot s primary control objective"

!  Above V crossover, Δn pilot is the primary control objective"

C*=Δn pilot+V crossover

g Δq

= (l pilot Δ q + Δn cm) +V crossover

g Δq

= l pilot Δ q + V N

g (Δq − Δ α )

$

%

( )+V crossover g Δq

Fighter Aircraft: V crossover ≈ 125 m / s

•   Hypothesis "

Gibson Dropback Criterion

•  Step response of pitch rate should have overshoot for satisfactory pitch and flight path angle response !

Δq(s)

ΔδE(s)=

k q s + 1

Tθ

2

$

%

&& ' ( ))

s2

+ 2ζSPωn SP s +ω n2SP

=

k q s +ωn SP

ζSP

$

%

( )

s2

+ 2ζSPωn SP s +ω n2SP

z q  − 1

Tθ

2

ζSP

%

&

)

*

•  Criterion is satisfied when !

Gibson, 1997!

Lateral-Directional Criteria

Lateral-Directional Flying Qualities Parameters"

Lateral Control Divergence

•  Aileron deflection produces yawing as well as rolling moment"

–  Favorable yaw aids the turn command"

–  Adverse yaw opposes it "

•  Equilibrium response to constant aileron input "

ΔφS

ΔδAS =

Nβ+ N r

Yβ

V N

%

&

)

* L δA − Lβ+ L r

Yβ

V N

%

&

)

* N δA

g

V N(LβN r − L r Nβ)

•  Large-enough NδA effect can reverse the sign of the response"

–  Can occur at high angle of attack "

–  Can cause departure from controlled flight "

LCDP ≡ C n

β −C nδA

C l

δA

C l

β

Nβ

( )LδA − L( )β NδA

LδA = Nβ−

NδA

LδA Lβ

•  Aileron-to-roll-angle transfer function "

Δφ(s)

kφ(s2+ 2ζφωφs +ωφ2)

s − λ S

( ) (s − λ R) s2+ 2ζDRωn DR s +ω n DR

2

  ωϕ is the natural frequency of the complex zeros"

  ωd = ωnDR is the natural frequency of the Dutch roll mode "

• Conditional instability may occur with closed-loop control of roll angle, even with a perfect pilot"

ωϕ/ω Effect"

Δφ(s) ΔδA(s)=

kφ s2

s −λS

( ) (s −λR)s2

+ 2 ζDRωn DR s +ωn2DR

ϕ/β Effect"

•   ϕ/β measures the degree of rolling response in the Dutch roll mode"

–   Large ϕ/β: Dutch roll is primarily a rolling motion"

–   Small ϕ/β: Dutch roll is primarily a yawing motion "

of the state component in the i th mode of motion"

λiI − F

Eigenvectors!

• Eigenvectors , i, are solutions to the equation"

λiI − F

or

can be found (within an arbitrary constant) from"

Adj(λiI − F)=( a1ei a2ei … a nei ), i = 1,n

MATLAB

V,D

V: Modal Matrix (i.e., Matrix of Eigenvectors) D: Diagonal Matrix of Corresponding Eigenvalues

the corresponding eigenvector is"

eDR+=

e r

eβ

e p

eφ

#

$

%

%

%

%

%

&

'

( ( ( ( (

DR+

=

σ + jω

σ + jω

σ + jω

σ + jω

#

$

%

%

%

%

%

%

&

'

( ( ( ( ( (

DR+

=

AR e jφ

( )r

AR e jφ

( ) β

AR e jφ

( )p

AR e jφ

( ) φ

#

$

%

%

%

%

%

%

%

&

'

( ( ( ( ( ( (

DR+

•   ϕ/β is the magnitude of the ratio of the ϕ andβ eigenvectors"

φ

AR

AR

g

#

$

% &

'

V N +

Lβ

L r

#

$

' (

2

2

,

- .

/ 0

1 1

1

ϕ/β Effect for the Business

Jet Example!

eDR+=

e r

eβ

e p

eφ

#

$

%

%

%

%

%

%

%

&

'

(

(

(

(

(

(

(DR+

=

0.525 0.416 0.603 0.433

#

$

%

%

%

%

&

'

( ( ( (

DR+

φ

β = 1.04

Roll/Sideslip Angle ratio in the Dutch roll mode!

Early Lateral-Directional

T1 = 0.693 / ζ ωn

v = V Nβ

O Hara, via Etkin!

Ashkenas, via Etkin!

Time to Half!

Criteria for Lateral-Directional

Maximum

Roll-Mode Time

Constant"

Minimum

Spiral-Mode

Time to Double"

Minimum Dutch Roll Natural

Pilot-Vehicle Interactions

Pilot-Induced Roll Oscillation"

Δφ(s)

Δδ A(s)pilot in loop

= K p / T p

s + 1 / T p

$

%

&

' ( )

kφ s2

+ 2ζφωφs + ωφ2

s − λ S

( ) (s − λ R) s2

+ 2ζDRωn DR s + ω n2DR

/

0 0

1 2

3 3

Aileron-to-Roll Angle

Pilot Transfer Function ! Aircraft Transfer Function !

YF-16!

YF-17 Landing

•  Original design"

frequency"

Elevator-to-pitch angle Nichols

80°

Phase Margin!

Gibson, 1997!

•  Revised DFCS design"

frequency"

–   CHR = 2 " Input frequency, rad/s"

13 dB Gain Margin!

•   Alternative pilot transfer function:

gain plus pure time delay "

H jω( )pilot = K P e − jωτ

•   Gain = constant"

•   Phase angle linear in frequency"

•   As input frequency increases, ϕ(ω)

eventually > –180°!

But Stability Margins Were Large

K P e − jωτ = K P  K P e − jωτ

H s( )pilot=Δu s( )

Δ ε ( )s = K P e

− τ

Inverse

Problem of

is the control history that

actions "

interconnect (ARI)

Grumman F-14 Tomcat!

Yaw Angle" Roll Angle" Command"

Angle of attack (α) =

10 deg; ARI off"

α = 30 deg; ARI off"

α = 30 deg; ARI on"

Stengel, Broussard, 1978!

Flight Control System

Design

Control System

–  State observer"

–  Kalman filter (optimal estimator)"

•   Assume Gaussian errors"

•   Combine withLQregulator "

•   LQGregulator "

–  Robustness"

–  Gain scheduling"

–  Adaptive control"

Proportional Stability Augmentation

Δu t( )= C F ΔyC( )t − C BΔx t( )

Section 4.7, Flight Dynamics"

dim Δu t"# ( )$% = m × 1; dim Δx t"# ( )$% = n × 1

dim ΔyC"# ( )t $% = r × 1, r ≤ m

dim CF[ ]= m × r; dim CB[ ]= m × n

•  Full state feedback"

ΔyC(t), than independent control inputs, Δ u(t)"

Proportional Stability Augmentation

with Command Input !

Δx t( )= FΔx t( )+ GΔu t( )

Δy t( )= H xΔx t( ); H u  0

Δu t( )= C F ΔyC( )t − C BΔx t( )

Δx t( )= FΔx t( )+ G C#$ F ΔyC( )t − C BΔx t( )%&

= F − GC#$ BΔx t( )%& Δx t( )+ GC F ΔyC( )t

= FCL Δx t( )+ GCLΔyC( )t

gains of the closed-loop command/stability

augmentation system"

Section 4.7, Flight Dynamics"

•  Eigenvalues"

•  Root loci"

•  Transfer functions"

•  Bode plots"

•  Nichols charts"

•  "

Next Time:

Maneuvering and Aeroelasticity

Reading

Supplemental

Material

Large Aircraft Flying Qualities"

•   High wing loading, W/S"

•   Distance from pilot to rotational center"

•   Slosh susceptibility of large tanks"

•   High wing span -> short relative tail length"

–   Higher trim drag"

–   Increased yaw due to roll, need for rudder coordination"

–   Reduced rudder effect "

•   Altitude response during approach"

–   Increased non-minimum-phase delay in response to elevator"

–   Potential improvement from canard "

•   Longitudinal dynamics"

–   Phugoid/short-period resonance "

•   Rolling response (e.g., time to bank)"

•   Reduced static stability"

•   Off-axis passenger comfort in BWB turns"

Criteria for Oscillations and Excursions

Proportional-Integral Command and

Δu t( )= C F ΔyC( )t − C I∫#$Δy t( )− ΔyC( )t %&dt− C BΔx t( )

Section 4.7, Flight Dynamics"

• Full state feedback"

• Command = desired output"

–  r (≤ m) components"

Proportional-Integral Command and

Stability Augmentation !

Δx t( )= FΔx t( )+ GΔu t( )

Δy t( )= HxΔx t( ); H u  0

Δu t( )= C F ΔyC( )t − C I∫#$Δy t( )− ΔyC( )t %&dt− C BΔx t( )

Δx t( )= FΔx t( )+ G C{ F ΔyC( )t + C I∫#$Δy t( )− ΔyC( )t %&dt− C BΔx t( ) }

= F − GCB[ ]Δx t( )+ G CFΔy{ C( )t + CI∫#$ΔyC( )t − HxΔx t( )%&dt}

Section 4.7, Flight Dynamics"

Dynamics and Control!

Substitute Control in Dynamic Equation!

Proportional-Integral 


Command and Stability

Δy t( )= H xΔx t( ); H u  0

Δξ t( )  ∫$%ΔyC( )t − Δy t( )&'dt

= ∫$%ΔyC( )t − HxΔx t( )&'dt

Δξ t( ) ΔyC( )t − H xΔx t( )

Δx t( )= FCL Δx t( )+ GC F ΔyC( )t + GC IΔξ t( )

Δx t( )

Δξ t( )

#

$

%

%

&

'

( (=

−H x 0

#

$

%

%

&

'

( (

Δx t( )

Δξ t( )

#

$

%

%

&

'

( (+

GCL I

#

$

%

%

&

'

( (ΔyC

t

( )

Section 4.7, Flight Dynamics"

• Define integral state, Δξ(t)"

• dim[Δξ(t)] = dim[Δy(t)]"

• Augmented dynamic equation"

Proportional-Integral 
 Command and Stability

closed-loop command/stability augmentation system"

•  Eigenvalues"

•  Root loci"

•  Transfer functions"

•  Bode plots"

•  Nichols charts"

•  "

Δχ t( )  Δx t( )

Δξ t( )

$

%

&

&

'

(

) ); dim Δχ t$% ( )'( = n + r( ) × 1

Δχ t( )= FCL ' Δχ t( )+ GCL' ΔyC( )t

Proportional-Filter Stability

Δu t( ) = +∫#$C F ΔyC( )t − C BΔx t( )− −C IΔu t( )%&dt

Section 4.7, Flight Dynamics"

http://www.youtube.com/watch?v=GXdJxjvQZW4 !

http://www.youtube.com/watch?v=t6DdlPoPOE4 !

http://www.youtube.com/watch?v=j85jlc1Zfk4 !