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Lec 2 Point Mass Dynamics
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Transcript of Lec 2 Point Mass Dynamics
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Point-Mass Dynamics and
ero ynam c rus orces
Properties of the Atmosphere
Frames of reference
Velocit and momentum
Newtons laws Introduction to Lift, Drag, and Thrust
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The Atmosphere
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Air Densit D namic
Pressure, and Mach Number
=Air density,function of heightz
=
= 1.225 kg / m3; = 1 / 9,042m
levelsea
levelsea
=Airspeed[ ] [ ] 2/12/1222 vv Tzyx vvvV =++=
Dynamic pressure = =q2
2V
ac num er = ; a = spee o soun , m sa
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Air density and pressure decay exponentially with altitude Air tem erature and s eed of sound are linear functions ofaltitude
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Zero wind at Earths surface = Inertially rotating air mass Wind measured with res ect to Earths rotat in surface
Airspeed = Airplanes speed with respect to ai r mass Inertial velocity = Wind velocity + Airplane veloci ty
n e oc y ro esvaryover me yp ca e s ream e oc y
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Contours of Constant
Dynamic Pressure, q In steady, cruising flight, SqCSVCLiftWeight LL ===
2
2
to maintain constant dynamic pressure
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for a Point Mass
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Newtonian Frame of
Reference
Newtonian (Inertial) Frame of
Reference
Unaccelerated Cartesian framewhose ori in is referenced to an
inertial (non-moving) frame
Right-hand rule Origin can translate at constant
linear velocit
Frame cannot be rotating withrespect to inertial orig in
Position: 3 dimensions
x
a s a non-mov ng rame
z
rans a on c anges e pos on o an o ec
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Velocit of a article
xvxdx
&
&
&
=
===
z
y
v
v
z
ydt
&
xv
Linear momentum of a particle
v
v
vmm
z
y
== vp
particleofmassmwhere =
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Newtons Laws of Motion:
First Law Ifno force acts on a particle, it remains at rest or
continues to move in a strai ht line at constant
velocity, as observed in an inertial referenceframe -- Momentum is conserved
( )21
0tt
mmm
dt
dvvv ==
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Newtons Laws of Motion:
Second Law A particle of fixed mass acted upon by a force
changes velocity with an acceleration
proportional to and in the direction of the force,
as o serve n an ner a re erence rame;
The ratio of force to acceleration is the mass ofthe particle: F = ma
( )
=== y
x
f
f
F
dt
dmm
dt
dF
vv ;
=== y
x
f
f
m
m
m
mmdt
d
/100
0/10
00/111
FIFv
3
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Newtons Laws of Motion:
Third Law ,
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Equations of Motion for a Point
Mass: Position and Velocity
&
Rate of change
of position
==
==
z
y
v
v
z
y
dt
dvr
r
&
&&
fmv 00/1&
Rate of changeof velocity
==
==z
y
z
y
ff
mmm
vvdt
/1000/10Fv
v
&&&
Vector of
xf
combined forces Ithrustcsaerodynamigravity
z
yI
f
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Equations of Motion for a
Point Mass
Written as a sin le e uat ion
[ ]Fxfx ),()()( ttdxt ==&
With
y
x
=
=
xv
z
Velocity
Position
v
rx
z
y
v
v
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Scalar Dynamic Equations for
a Point Mass
x xvx 000001000
&
&
+
=
=
y
x
z
y
f
fzvz 000100000
&
&
zyyy
x
fmvmfv 0/10000000/
&
&
zz
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Gravitational Force:Flat-Earth Approximation
Flat earth reference is an inertialframe, e.g.,
North East Down
mg is gravitational force
Independent of position
Range, Crossrange, Alt itude () z measured down
( ) ( )
=== 0mm fE
gravityI
gravity gFF
0g
'2 .0
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Force
Inertial Frame Body-Axis Frame Velocity-Axis Frame
CX X
21 CX C
CZIZ
Y
I
I
2 SqC
C
BZ
YB
=F SqCYV =F
Sq
C
C
Z
Y
=
Referenced to theEarth not the aircraft
Aligned with theaircraft axes
Aligned with andperpendicular to
the direction of
motion
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Non-Dimensional
Aerodynamic Coefficientso y- x s rame e oc y- x s rame
tcoefficienforceaxialCX tcoefficiendragC
D
=
tcoefficiencenormal for
tcoe c enorces e
CBZ
Y
=
tcoefficienLift
tcoe c enorces e
CL
Y
Functions of flight condition, control settings, and disturbances, e.g.,CL = CL(, M, E)
Non-dimensional coefficients allow application of sub-scale model-
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SVCThrust T2
2
Non-dimensional thrust
,
CT is a function of power/throttlesetting, fuel flow rate, blade angle,
, ...
Reference area, S, may be aircraft
wing area, propeller disk area, orjet exhaust area
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1
rusam noNNTN 2
=
(.)N=Nominal(or reference) value
Turbojet thrust is independent of airspeed over awide ran e
Ifthrust is independent of velocity (= constant)
T
VTNT
C
SVCSVVV 2
0 2
+
==
NT
V
N=
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SVCVelocityThrustPPower 31==
Ifpower is independent of velocity (= constant)
NTT SVCSV
V
C
V
PN 2
3
2
10 23 +
==
NTT VC
V
CN
/3=
Velocity-independent power is typical of propeller-driven propulsion (reciprocating or turbine engine,
with constant RPM or variable-pitch prop)
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=- (with wingtips level)
u(t ) : axial velocity along vehicle centerlinew : norma ve oc y
V (t ) : velocity magnitude
(t ) : angle of attack
along net direction of flight angle between centerline and direction of f light
angle between direction of flight and local(t ) : flight path angle
(t ) : pitch angle
horizontal
angle between centerline and local horizontal
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-=
(t ) : sideslip angle angle between centerline and direction of f light
an le between centerline and local horizontal
(t ) : heading angle
(t ) : roll angle
angle between direction of flight and compassreference (e.g., north)
angle between true vertical and body z axis
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Lift and Dra
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Lift and Drag are Orientedto the Velocity Vector
SVCCSVCLi t L 22 11 +=22 0
Lift components sum to produce total lift Pressure differential between upper and lower surfaces ng
Fuselage Horizontal tail
[ ] SVCCSVCDrag LDD 22222 0 +=
Dra com onents sum to roduce total dra Skin friction Base pressure dif ferential Shock-induced pressure dif ferential (M > 1)
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CL 222 111 = LLLLL
htfw 222 0
Streamlines
Chord Section
Fast flow over top + slow flow over bottom =
Mean flow + Circulation Speed difference proportional to angle of attack Kutta condition (stagnation points at leading and
trailing edges)
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InwardOutwardFlow .
TipVortices
IdenticalChordSections
Infinitevs.
Finite
Span
Inward flow over upper surface Outward flow over lower surface Bound vorticit of win roduces ti
vortices
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SVCCSVCCCSVCDrag LDDDDD wip 222 0
+++=
Dra com onents Parasite drag (friction, interference, base pressure
differential)
Induced drag (drag due to li ft generation) ave rag s oc - n uce pressure eren a
In steady, subsonic fl ight Parasite (form) drag increases
as V2
Induced drag proportional to2
Total drag minimized at oneparticular airspeed
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2-D Equations of Motion
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2-D E uations of
Motion for a Point Mass Restrict motions to a vertical plane (i.e., motions in
ydirection = 0)
x xvx 000100
&
&
+== z
x
xx
z
x fmv
z
mf
v
v
z
0/10000/
&
&
zzz mvmv
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from Cartesian to
o ar oor na es
+=
+=
=
=
v
vv
zzxVVvx
z
zxx
1
22
1
22
cos
&
&&
&
&
VVz
+
=
+
=
vvdt
d
v
vvdV zxzx
2222
&
&
VdtVz1sins n
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Longitudinal Point-Mass
Equations of Motion Assuming thrust is aligned with the velocity vector
sin1 2 tmStVCC
1
2s n)(
2
mm
tmgragrusttV
=
=&
)(2)(
)(cos)( tmV
mg
tmV
tmgLiftt
L
=
=
&&
&
)(cos)()()(
s n
ttVvtxtr
ttvtzt
x
z
===
===
&&
When airplane is in steady, level flight,CT= CD
V = velocity
=flight path angle
h = height (altitude)
=
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Aircraft Equations
of Motion - 1
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