AE-430-3 (1).pdf
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AE 430 - Stability and Control ofAerospace Vehicles
Static/Dynamic Stability
Longitudinal StaticStability
We begin with the concept of Equilibrium (Trim).Equilibrium is a state of an object when it is at rest or insteady uniform motion, (i.e., with constant linear andangular momenta).
The resultant of all forces and moment about the CG mustboth be equal to zero.
Stability is defined as the ability of an aircraft to return to agiven equilibrium state after a disturbance (it is a property
of the equilibrium state) STATICALLY STABLE when
if it is disturbed from its equilibrium state by a smalldisplacement, then
the set of forces and moments so caused initially tend toreturn the aircraft to its original state
Static Stability
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Trimmed flight when all the forces and moments are balanced(trust = drag; lift = weight; pitching moment = 0; yawing moment = 0rolling moment = 0)
The steady flight condition may involve a steady acceleration e.g. acorrectly banked turn, or a steady dive or climb.
Pitch trim would be accomplished by deflecting the horizontalstabilizer, the elevator, or the elevator trim tab.
Trimmed state IS NOT NECESSARILY A STABLE STATE i.e. all the forces and moments may be balanced, but as soon
as the state is perturbed the aircraft departs from equilibrium.
Forces 0; Moments 0= =
0 for trimGM =
Trimmed Flight
(or steady unaccelerated flight)
Types of Stability
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Static Stability
Static stability of a body is anStatic stability of a body is an initialinitialtendency of that body totendency of that body to return toreturn to
its equilibrium stateits equilibrium stateafter aafter a
disturbance.disturbance.
Static longitudinal instabilityStatic longitudinal instabilityIn this case there is no tendency tono tendency to
return to equilibriumreturn to equilibrium
Any disturbance from equilibrium
leads to a larger disturbance, themotion is said to be divergent E
Neutral static stability is theNeutral static stability is the
boundary between stability andboundary between stability andinstability,instability, there is still no tendencyno tendency
to return to equilibriumto return to equilibrium, the motion is
therefore not stable
But, the motion does not diverge E
Energy is being dissipated
Positive damping
Energy is added to the systemNegative damping
Artificial damping is needed Stability Augmentation System SAS
Static Stability
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Dynamic Stability
DYNAMIC STABILITY characterizes the time history of motionDYNAMIC STABILITY characterizes the time history of motion
after a disturbance from equilibriumafter a disturbance from equilibriumAn aircraft is said to be dynamically stable if, after a
disturbance, it eventuallyreturns to its equilibrium state
and remains there
ABSOLUTE dynamic stability is not concerned withhow long this return takes
RELATIVE dynamic stability examines how long ittakes and what the behavior of that return motion is
To be dynamically stable, a system must first be staticallystable
A system can be dynamically unstable and be
statically stable -- but not vice versa
Dynamic Stability
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Dynamic Stability
Pilot-InducedOscillation
PIOs occur when a pilot over-controls an aircraft and a sustained oscillation results
Pilot-induced oscillations occur when the pilot of an aircraft inadvertently commands an often increasingseries of corrections in opposite directions, each an attempt to correct for the previous overcorrection withan overcorrection in the opposite direction. The physics of flight make such oscillations more probable forpilots than for automobile drivers. An attempt to cause the aircraft to climb, say by applying up elevator willalso result in a reduction in airspeed.
Another factor is the response rate of flight instruments in comparison to the response rate of the aircraftitself. An increase in power will not result in an immediate increase in airspeed. An increase in climb ratewill not show up immediately on the vertical speed indicator.
A pilot aiming for a 500 foot per minute descent, for example, may find himself descending too rapidly. Hebegins to apply up elevator until the vertical speed indicator shows 500 feet per minute. However, becausethe vertical speed indicator lags the actual vertical speed, he is actually descending at much less than 500feet per minute. He then begins applying down elevator until the vertical speed indicator reads 500 feet perminute, starting the cycle over. It's harder than it might seem to stabilize the vertical speed because theairspeed also constantly changes.
The most dangerous pilot-induced oscillations can occur during landing. A bit too much up elevator duringthe flare can result in the plane getting dangerously slow and threatening to stall. A natural reaction to thisis to push the nose down harder than one pulled it up, but then the pilot finds himself staring at the ground.An even larger amount of up elevator starts the cycle over again.
http://www.dfrc.nasa.gov/Gallery/Movie/F-8DFBW/HTML/EM-0044-01.html
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Statically Stable Response
noseup
nosedown
Balanced
positive pitchstiffness
(restoring moment)
Equilibriumpoint
Other necessarycondition to trim at
positive angle of attach,
m m Lm
L
dC dC dCC
d dC d
= = 0
m
L
dC
dC
Longitudinal Static Stability
Longitudinal static stability componentsLongitudinal static stability moments asa function of angle of attack. The curveis a composite of all the moment curvescaused by the different components ofthe airplane, (the wing, fuselage, tail,thrust, etc).
nose up (+)
nose down (-)
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Stable, neutral, and unstable static stability
DC-9. Note the contributions from the variouscomponents and the highly nonlinear post-stallcharacteristics
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There are different
degrees of stability
Some aircraft tend toreturn to equilibrium
faster
An aircraft can bestable at lower angles
of attack but may beunstable at higher
angles of attack
Wing Contribution
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WingContribution
( ) ( )
( ) ( )
cos sin
sin cos
w
w
w
cg
cg w w w cg ac w w w cg ac
w w w cg w w w cg ac
Moments M
M L i x x D i x x
L i z D i z M
=
= +
+ +
( ) ( )
( ) ( )
212
Dividing for :
cos sin
sin cos
w
w
cg cgac ac
m L w w D w ww wcg
cg cg
L w w D w w mw w ac
V Sc
x xx xC C i C i
c c c c
z zC i C i C
c c
= +
+ +
( ) ( )
( )
( )
( )0
cos 1; sin ;
negligible
;
w w
ww w w w
w w w w w w
L Dw w
cg cgacm L L w w mw wcg ac
cg
L w ww
cg cgac acm m L m m L L wwcg ac cg ac w
i i i
C C
x zxC C C i C
c c c
zC i
c
x xx xC C C C C C C
c c c c
= + +
= + = + +
0wL L L ww wC C C
= + Lift Coefficient
Well designed aircraft
Normal flight operation
Wing Contribution
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Nonlinear contributions
( )w w
cg cgac acm m L D w ww wcg ac
x xx xC C C C i
c c c c
= + +
2w
w w
L
D d
CC C
eAR= +
( ) cg
Lw w w Dw
zC i C
c
+
0wL L L ww wC C C
= + 2
w
Wind drag turn
Wing Contribution
0 0
0
w
w
cg acm m Lacw w
m m m wcg w wcg ac
m Lw w
x xC C C
c cC C C
x xC C
c c
= +
= +
=
To have a wing alone statically stable 0m wC < cg acx x
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Wing Contribution
Positive camber give nose-down pitching moment
Negatively cambered airfoil gives nose-up pitching moment and
cancels nose-down moment caused by lift and weight vectors
For straight-winged, tailless airplane, negative camber satisfies
conditions for stable, balanced flight
Not in general use
Dynamic characteristics poor
Drag and Clmax poor
Swept back wing with twisted tips
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Conventional and forward tail
arrangement
Tailless Aircraft
One example of
a tailless aircraftthat trims using a
positive Cm0airfoil section: theAeroVironment
Pathfinder, solar-powered aircraft
on a flight to over50,000 ft (15.2
km).
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Example # 1
For a given wing-body combination, the aerodynamic center lies 0.03
chord length ahead of the center of gravity. The moment coefficient
about the center of gravity is 0.0050, and the lift coefficient is 0.50.
Calculate the moment coefficient about the aerodynamic center.
, ,
, ,
,
,
,
( )
( )
0.005 0.5(0.03) 0.01
cg w ac w
ac w cg w
ac w
cg ac wM M Lw
cg ac w
M M Lw
M
x xC C C
c c
x xC C C
c cC
= +
=
= =
Example # 2
Consider a model of a wing-body shape mounted in a wind tunnel. The
flow conditions in the test section are standard sea-level propertieswith a velocity of 100 m/s. The wing area and chord are 1.5 m2 and
0.45 m, respectively.
Using the wind tunnel force and moment-measuring balance, the
moment about the center of gravity when the lift is zero is found to be
-12.4 N m.
When the model is pitched to another angle of attack, the lift and
moment about the center of gravity are measured to be 3675 N and20.67 N m, respectively.
Calculate the value of the moment coefficient about the aerodynamiccenter and the location of the aerodynamic center.
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Example # 2 Contd
,
, ,
2 2 2
,
1 10.225*100 6125 /
2 2
12.40.003
6125*1.5*0.45
0.003 at zero lift
cg w
cg w ac w
cg w
m
m m
q V N m
MC
q Sc
C C
= = =
= = =
= =
Example # 2 Contd
,
, ,
, ,
,
,
,
,
36750.4
6125*1.5
20.670.005
6125*1.5*0.45
( )
0.005 ( 0.003)
0.4
0.02
cg w
cg w ac w
cg w ac w
Lw
cg w
m
cg ac wm m Lw
m mcg ac w
Lw
cg ac w
LC
q S
MC
q Sc
x xC C C
c c
C Cx x
c c C
x x
c c
= = =
= = =
= +
= =
=