tema 5 mecánica de vuelo: Stability & Control

powerpoint sobre mecánica de vuelo. exactamente sobre la estabilidad y el control de vuelo aéreo.

1. Stability & Control
Aerospace Science & Technology
F. Mellibovsky
Escola d'Enginyeria de Telecomunicacio i Aeronautica de Castelldefels
Universitat Politecnica de Catalunya
March 2, 2011

2. Summary
Basic Concepts
Longitudinal Flight
Longitudinal Equations
Longitudinal Equilibrium
Longitudinal Balance
Longitudinal Stability
Static Stability
Dynamic Stability
Lateral Flight
Lateral Equations
Lateral Equilibria
Lateral Stability
Static Stability
Dynamic Stability

3. Outline
Basic Concepts
Longitudinal Flight
Longitudinal Equations
Longitudinal Equilibrium
Longitudinal Balance
Longitudinal Stability
Static Stability
Dynamic Stability
Lateral Flight
Lateral Equations
Lateral Equilibria
Lateral Stability
Static Stability
Dynamic Stability

4. Denitions
IFlight Equilibria: All forces and moments are balanced
IFlight Parameters: Speed, ascent/descent rate, turn radius...
IAircraft Control / Piloting: Ability to prescribe ight parameter
values and modify equilibria
IPerformances: Equilibrium optimisation
IFlight Dynamics: Response to unbalanced force/moment
ITime-evolution of ight parameters
IStability
IStability conditions: Unstable, Neutral, Stable
IStatic Stability: Initial response to perturbation
Restoring Force
IDynamic Stability: Dynamical behaviour
Restoring Force
Damping Force
IHandling & Flying Qualities: Dynamical response optimisation
IManoeuvrability: Response celerity to command input
IStability Augmentation: Automatic Control

5. Body Axes
Angular velocities:
Ix:p(roll)
Iy:q(pitch)
Iz:r(yaw)
Aerodynamic forces:
Ixa:D(drag)
Iy:FY(lateral force)
Iza:L(lift)
Aerodynamic Moments &
Commands:
Ix:Ml,l
Iy:Mm,m
Iz:Mn,n symmetry
plane
"wings"
plane

6. Primary Control Surfaces & Commands
Axis
Primary
Control Surface Command
Roll (p)
Longitudinal
x-axis
Ailerons
Stick / Yoke
(left-right)
l!Cl
Pitch (q)
Lateral
y-axis
Elevator
Stick / Yoke
(forward-backward)
m!Cm
Yaw (r)
Vertical
z-axis
Rudder Rudder pedals
n!Cn
x-axis Power Setting Throttle
x

7. Secondary Control Surfaces & Control Systems
Secondary Control Surfaces
IHigh-Lift Devices:
Iaps, slats, slots
ITrim tabs:
Ielevator trim
Irudder and aileron trims
(large aircraft)
ISpoiler / Air Brakes
IOthers: aperons, stabilators...
Flight Control Systems:
IMechanical
IHydro-mechanical
Iy-by-wire

8. Other Surfaces
Empennage:
ITailplane:
IHorizontal Stabiliser
IElevator
IElevator trim tab
IFin:
IVertical Stabiliser
IRudder
ICombined: V-tail (with ruddervators)
Canards
ILifting-canard
IControl-canard
Winglets

9. Outline
Basic Concepts
Longitudinal Flight
Longitudinal Equations
Longitudinal Equilibrium
Longitudinal Balance
Longitudinal Stability
Static Stability
Dynamic Stability
Lateral Flight
Lateral Equations
Lateral Equilibria
Lateral Stability
Static Stability
Dynamic Stability

10. Longitudinal Equations
Hypothesis:
IAircraft reexion-symmetry plane is vertical!= 0
INo roll or yaw rotation velocities!p=r= 0
IAerodynamic Forces within the symmetry plane!= 0
IPropulsion Forces parallel to the symmetry plane
IAerodynamic Moment orthogonal to the symmetry plane
IPropulsion Moments orthogonal to the symmetry plane
Consequences:
ILongitudinal Equations decoupled
IFlight occurs in the vertical plane

11. Longitudinal Equations
Equations (aerodynamic axes projection):
IPropulsion equation (P):m
_
V=
1
2
SV
2
CD+Tcosmgsin
ILift equation (L):mV_=
1
2
SV
2
CLTsin+mgcos
IPitching Moment equation (Q):Iy_q=
1
2
SlV
2
Cm+MT
IPitch Velocity equation (QV): _+ _=q
IAltitude equation (H):
_
h=Vsin
Simplifying Hypothesis:
I~
Tand
~
Vparallel!PandLdecouple regardingT
IMT= 0!PandQdecouple regardingT
ISmall(cos'1, sin')!PandLdecouple regarding

12. Longitudinal Equilibrium
Equations:
IPropulsion equation (P):T=
1
2
SV
2
CD+mg
ILift equation (L):mg=
1
2
SV
2
CL
IPitching Moment equation (Q):Cm= 0
IPitch Velocity equation (QV):q= 0
IAltitude equation (H):
_
h=Vsin= 0
Consequences:
ILongitudinal balanced ight
I!straight (q= 0,=ct)
I!horizontal (= 0)
IRectilinear ascent / descent!pseudoequilibria ( _'0)
Controls:
IElevators (stick or yoke)m()()V(())
IPower Setting (thrust lever / throttle)x()

13. Longitudinal Balance
Centre of Lift (xL):
IPlace where Lift is Applied
INo Pitching Moment
Cm(xL) = 0
IChanges with
Aerodynamic Centre (xAC):
IWhere Lift change is applied
IConstant Pitching Moment
(@Cm=@)(xAC) = 0
IFairly independent of
Pitching Moment Coecient (with
respect toxG):
Cm=Cm
(0) =Cm0
+Cm

Pitching Moment Gradient Model:
Cm=
xACxG
l
CL
but the aircraft is yet to be balanced...

14. Longitudinal Balance
IElevators (stick or yoke command):m
IModify lift (small eect):CL=CL(0) +CL
m
m
IModify pitching moment:Cm=Cm0+Cm+Cm
m
m
IQ:Cm=Cm0
+Cm
+Cmm
m= 0
Im$:m=(Cm0+Cm)=Cm
m
IDepends on: centering (xGxAC)
IIndependent of:h,V,,m
ITo sustain: elevator trim (secondary control surface).
IPitching moment due to engine axis neglected.

15. Longitudinal Balance
TailPlane:
ICompensating nose-down
pitching moment
IDownforce: subtracts from
wing lift
IPerturbed ow on tailplane
Canard:
ICompensating nose-down
pitching moment
IUpforce: adds to wing lift
IStalls before wing
IPerturbed ow on wing

16. Longitudinal Balance
IL:mgcos=
1
2
SV
2
CL()!V
2
(0) =ct
I$V
IDepends on:h,m
IIndependent of: centering (xGxAC)
IP:T=
1
2
SV
2
CD() +mg!T(x) =mg
Ix$
IDepends on:h,m,V
IIndependent of: centering (xGxAC)
IP=L:T=mg(1=E+)
IV$
IDepends on:h,m,V, Regimes
IIndependent of: centering (xGxAC)

17. Longitudinal Balance
IPower setting:x$
IElevator deection:
m$$V$
Flight Regimes:TD=mg
IR1: V <0
IR2: V >0
Drag (Required Thrust) vs. Speed:
D=DP+Di=ADV
2
+BD=V
2
AD=
1
2
SCD0BD=
kW
2
1
2
S
Power vs. Speed:
P=PP+Pi=ADV
3
+BD=V
Available Thrust:
T(x) =kTV
T
x

18. Static Stability
IPositive Stability
IG ahead of Neutral Point
INose-up!Nose-down
restoring moment
INeutral Stability
IG on Neutral Point
INose-up!No net moment
INegative Stability (Instability)
IG aft from Neutral Point
INose-up!Nose-up moment
Stability tunining:
ISize / position of horizontal
stabiliser (tailplane or canard)
IWeight distribution

19. Dynamic Stability
IFive longitudinal equations
IThree longitudinal modes:
IPitching Oscillation: oscillatory, involving
I: angle of attack
Iq: pitch angular velocity
IPhugoide: oscillatory, involving
IV: velocity
I: slope
IReturn to straight & level ight: aperiodic, involving
Ih: altitude

20. Pitch Oscillation
Pitching oscillation ($q) with respect to a point ahead ofG
IDecoupling Hypothesis:V;h'ct
ISpring / Restoring eect:Cm
(pitching moment due to)
IDamping eect:Cmq
(horizontal stabiliser),CL
(wing)
IProperties
IShort period:
!n=V
q
1
2
Sl=Iy
p
Cm= (1=x)
p
g(xGxAC)
p
CL=CL
IStable while periodic, then aperiodic, then unstable
IStatic marginms= (xACxG)=l:Gforward!!", constant
damping
IAttention to Pilot-Induced Oscillations! Do not try to pilot it!

21. Phugoid
Oscillatory exchange betweenVand
IDecoupling Hypothesis:'ct
ISpring / Restoring eect: Weight component on speed axis
IDamping eect: Thrust-Drag imbalance
IProperties
ILong period:=
p
2=gV
IUsually stable
IPoorly damped:= (2T)=(2
p
2(E+ tan))
IIn the abscence of damping: Potential-Kinetic energy exchange
h+V
2
=(2g) =ct(V$h)

22. Return to Straight & Level Flight
Straight & level ight recovery associated to balanced ascent
IDecoupling Hypothesis: Only_h=Vout of equilibrium
IProperties
IAperiodic
IExtremely long period:=E(V=g+ 66T=V)=(2T)
IStable
Assume=ct, xed by the pitching moment equilibrium
=ct!CL=ct!

L:V
2
=ct
CD=ct

!D=ct
!T: T=mg
But thrust decreases with altitude:
T=kT(V
2
)V
T2
=kT(V
2
)
T=2

1T=2

23. Outline
Basic Concepts
Longitudinal Flight
Longitudinal Equations
Longitudinal Equilibrium
Longitudinal Balance
Longitudinal Stability
Static Stability
Dynamic Stability
Lateral Flight
Lateral Equations
Lateral Equilibria
Lateral Stability
Static Stability
Dynamic Stability

24. Lateral Equations
Hypothesis:
IConstant velocity and angle of attack!V; =ct
INo pitch rotation velocity!q= 0
ISmall sideslip angle!cos'1
Consequences:
ILateral Equations decoupled

25. Lateral Equations
Equations (body axes projection):
IRoll Moment equation (R):
Ix_p(IyIz)qer+Ixz(_r+pqe)
1
2
SlV
2
Cl
IYaw Moment equation (Y):
Iz_r(IxIy)qep+Ixz( _p+rqe)
1
2
SlV
2
Cn
ILateral Force equation (LF):
mV(
_
+rcosepsine) =mgsincose+
1
2
SV
2
CY
IRoll Velocity equation (RV):_=p+ tane(rcos+qesin)
Simplifying Hypothesis:
Ix
b
taken as principal inertia axis!Ixz= 0
INo roll velocity!qe= 0
ISmall angle of attack and pitch angle!e; e'0

26. Aerodynamic Coecients
CY=CY
+CYp
pl
V
+CYr
rl
V
+CY
l
l+CYn
n
Cl=Cl
+Clp
pl
V
+Clr
rl
V
+Cl
l
l+Cln
n
Cn=Cn
+Cnp
pl
V
+Cnr
rl
V
+Cn
l
l+Cnn
n

27. Aerodynamic Coecients
CY=CY
+CYp
pl
V
+CYr
rl
V
+CY
l
l+CYn
n
Cl=Cl
+Clp
pl
V
+Clr
rl
V
+Cl
l
l+Cln
n
Cn=Cn
+Cnp
pl
V
+Cnr
rl
V
+Cn
l
l+Cnn
n
Sideslip

Roll Speed
p
Yaw Speed
r
Ailerons
l
Rudder
n
CY
Lateral Force
CY

<0
Lateral Lift
Yaw damping
CYp
'0 CYr
'0 CY

l
'0 CY
n
'0
Cl
Roll Moment
Cl

<0
Dihedral Eect
Spiral Stability
Clp
<0
Roll Damping
Clr
>0
Roll-Yaw Coupling
Cl

l
<0 Cl
n
&0
Cn
Yaw Moment
Cn

>0
Vertical Stabiliser
Directional
Stability
Cnp
'0
Roll-Yaw Coupling
Cnr
<0
Yaw Damping
Cn

l
&0
Adverse Yaw
Cn
n
<0

28. Lateral Equilibria
Equations (body axes projection):
IRoll Moment equation (R):
Cl=Cl
+Clp
pl
V
+Clr
rl
V
+Cl
l
l+Cln
n= 0
IYaw Moment equation (Y):
Cn=Cn
+Cnp
pl
V
+Cnr
rl
V
+Cn
l
l+Cnn
n= 0
ILateral Force equation (LF):
mV(rcosepsine) =mgsincose+
1
2
SV
2
(CY
+CYn
n)
IRoll Velocity equation (RV):p=rtanecos
Additional Equations:
ICrosswind Straight Flight:r= 0,=e
ICrosswind Landing
IEngine Failure
IBanked Turn:r=
ICoordinated Turn:= 0
ISkidding (Operational) Turn:n= 0

29. Lateral Equilibria
ICrosswind Straight Flight:r= 0,=e>0
Il=(Cl

=Cl

l
)0
In=(Cn

=Cn
n
)0
Isin=(CY

=CL)0
IBanked Turn:r= >0
ICoordinated Turn:= 0
Il=(Clr
=Cl

l
)(l=V)>0
In=(Cnr
=Cn
n
)(l=V)<0
I=V
2
=(Rg)>0
ISkidding (Operational) Turn:n= 0
Il= (Cl

CnrClr
Cn

)=(Cn

Cl

l
)(l=V)
?0
I=(Cnr=Cn

)(l=V)>0
I= (mV+
1
2
SV
2
CY

(Cnr
=Cn

)(l=V))(=(mg))
> CT(to compensate for lateral force)
Crosswind landing

30. Directional & Lateral Static Stability
Directional Stability
ITendency of the airplane to recover
from a Yaw perturbation
ICoecientCn
>0 for stability
IContributions:
IWing sweep: Stabilising for backward
sweep
IFuselage: Usually destabilising
IVertical Stabiliser (n)
IEngine failure
Lateral Stability
ITendency of the airplane to recover
from a Roll perturbation
IStrongly coupled to Yaw
IRecovery is mostly dynamical

31. Dynamic Stability
IFour lateral equations
IThree lateral modes:
IRoll Subsidence: aperiodic, involving
Ip: roll angular velocity
IDutch Roll: oscillatory, involving
I: sideslip angle
Ir: yaw angular velocity
ISpiral mode: aperiodic, involving
I: bank angle

32. Roll Subsidence
Damping of Rolling motion (p)
IDecoupling Hypothesis:g=V;Cnp
'0
INo spring eect
IDamping eect:
Clp
(wing span / aspect ratio)
IProperties
IAperiodic
ITime constant:=2Ix=(SVl
2
Clp)
IStable

33. Dutch Roll
Yaw oscillation ($r) with roll coupling
IAnalogous to Pitch oscillation in a
horizontal plane...
I... but asymmetry induces roll.
IDecoupling Hypothesis:g=V;Cnp
'0
ISpring eect:Cn
(vertical stabiliser)
IDamping eect:Cnr
(vertical stabiliser)
IProperties
IShort period:!n=V
p
LS=(2Iz)
p
Cn

IUsually not very well damped
ISize and distance of vertical stabiliser
aect damping
IFrequency decreases with aircraft size

34. Spiral Mode
Return to Straight & Level Flight (from banked turn)
Slow aperiodic change in bank angle
IProperties
IAperiodic, very slow
ITime constant:
=
V
g
ClpCn

Cl

Cnp
ClrCn

Cl

Cnr
IStable if
ClrCn

|{z}
engaging
Cl

Cnr
|{z}
disengaging
<0
IIt can be left unstable.
IClr
>0,Cn
>0,Cnr
<0
IOnlyCl
can change sign. Positive dihedral (Cl
<0) is required for
stability (sweep also can inuence).