Lec 2 Point Mass Dynamics

7/30/2019 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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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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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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Lec 2 Point Mass Dynamics

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Transcript of Lec 2 Point Mass Dynamics