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English Pages 649 [325] Year 1973
TOPICS
IN
ADVANCED
Gordon George
K. J.
William
The
Cambridge,
MODEL
ROCKETRY
Mandell Caporaso
P.
Bengen
MIT
Press
Massachusetts,
and
London,
England
Copyright © 1973
by
The
To
Institute
Massachusetts
Technology
of
without
No part of this book may be All rights reserved. or electronic reproduced in any form or by any means, or by recording, including photocopying, mechanical, without system, any information storage and retrieval permission in writing from the publisher. This
book
and
bound
was in
printed Columbia
by
The
Colonial
in
the
United
Library
of
on
Milbank
Press, States
Congress
Fernwood
0262020963
model
begun;
and
of
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TL844 M36
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Mandell, Gordon K Topics in advanced
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not
FOREWORD
PUBLISHER'S
of
preparation
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J. K.
Cambridge,
May, 1971"
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the
of
Description
the
Flight
2.3
Drag and Side Force
3.
that
can
Aerodynamic Disturbances
3.2
Mechanical
Disturbances
this
and
presently
Massachusett
work
a foundation governing
Caporaso Mandell
°
Dynamics
Introduction 1.
8
Equations
of Motion
il
12
APPROACH
43
45 47
TO
Gordon
capability.
Mandell
CHAPTER A UNIFIED
1
34
technically
in
K.
Forces
3.1
models
PLIGHT
Caporaso
Descriptionof the Perturbing Forces
contained
ROCKET
and
Differential
written
the design of model rocket vehicles. George Gordon
of
31
pastime
find
field
Separation
Weight
altitude
will
1.2
2.2
a
designing
maximum
and RigidBody
high
information
the
PointMass
12
projects
just
J.
OF MODEL
6
1.1
Thrust
welleducated,
use
practical
a basis the
similar
and
this
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means
early
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challenging
proficient
you
show
to
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level,
of the Problem
2.1
late
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at
written
material
Definition
and
inclined
technically
By
you.
for
are
undergraduate
college
or early
hope
you
hand,
assimilating
of
volume
this
2.
other
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on
capable
school
larger
Company) .
Engineering If,
1.
1
DYNAMICS
Gordon
Centuri
the
and
Industries
TO THE
of
the
of
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through
INTRODUCTION
George
of
treatments
elementary
AN
oriented
Handbook
OF CONTEDTS CHAPTER
literature.
Stine's
Harry
G.
has
who
technically
less
TABLE
mathematics
and
anyone
its
or
rocketry
Estes
notably
(most
manufacturers
to
available
rocketry
in
topics
technical
or
excellent,
the
and
Rocketry
Model
average,
of
themselves
avail
to
modelers
science
experienced,
less
younger,
in
model
to
exposure
previous
schoolwork
above
consistently
been
advise
feel
anyone
to
age,
of
years
whose
sixteen
under
anyone
to
4t
recommend
not
do
We
everyone.
for
not
is
it
and
technical
highly
is
pdook
this
caution:
of
word
2
AERODYNAMIC K.
STABILITY
53
and Moment
of Inertia
Mandell
63
The Dynamical
67
Equations
1.1
Euler's
Angles
1.2
Angular Velocity
1.3
Applied Moments,
67 69 Angular
Accelerations,
74
xii
Fuler's
61
The Linearized Theory
Damping
and
2.1
Corrective
2.2
The Linearization
2.3
Coupled
2.4
Homogeneous,
Solutions
3.1
3.2
84
for
Equations
Particular
Generalized Homogeneous Response
3.1.2
Complete
Response
to Step
3.1.3
Complete
Response
to
3.1.4
Steady
Dynamical
Behavior
Input
3.2.3
Complete Response to Impulse Input
167
3.2.4
Steady
Forcing
3.2.5
Roll Stabilization
Rate
170
4.2
Normal
Method
Force
185
Coefficients
5.3
The
138
138 163
to Sinusoidal
Damping
at
the
Roll
and
Locating the Center of Gravity
Center
of
44
The Damping Vament Coefficient
5.5
The Longitudinal Moment of Inertia
Barrownan
201 205
241
249 and Criteria
Design Definition; Inertia 255
6.4.2
Static
6.4.3
Demping Ratio
6.4.4
Roll Rate
6.4.5
Construction and Testing
Stability
254
Center of Gravity Margin
and Moments
255
255
256
257
CHAPTER3 DRAG OF MODEL ROCKETS THE AERODYNAMIC William P. Bengen Introduction
2.
261
271
276
Basic Concepts Relating to the Study of Drag 2.1
203
235
Parameters
1. Basic Considerations
196
4.3 The Corrective Moment Coefficient
the
229
234
6.4.1
134
Pressure:
Coefficient
Moment
222
Coefficient
of Varying, the Parameters
178
fnalytical Determination of tne Dynamic Parameters 4.1
Tne Corrective Moment
130
Rate
Complete
Response
5.2
Design Procedures
3.2.2
State
217
6.4
Poll
213
Experiment
Moments
Rolling Rockets
Forcing,
Parameters
the TorsionWire
Inertia:
of
5.1
6.3 119
the
Zit
Model Rocket Design
6.
of
211
Experimental Determination of the Dynamic Parameters
109
Input
Input
Properties
Effects
Response
to Step
General
6.2
Generalized
Response
4.7
Radial Moment of Inertia
93
3.2.1
Homogeneous
Tne
Representative
Nonzero
at a Constant,
92
4,6
6.1
to Sinusoidal
Response
Interest
of
Cases
93
3.1.1
Impulse
90
Solutions
SteadyState
and
Dynamical Behavior at Zero Roll Rate
State
89
of Equations
Systems
Particular,
Dynamical
to the
81
Moment
Approximations
Decoupled
and
76
Equations
Dynamical
w
1.4
azili
260 Atmospheric Properties for Model Rocket Filgnt 2.1.1
Density
261
260
of
xiv
2.3
Dimensionless Coefficients and Quantities The Reynolds
2.2.2
The Drag Coefficient
2.2.3
The Coefficient
302
303
3.2
The
Distinction
3.3
The Laminar Boundary
3.4
The
3.5
Boundary
Turbulent
3.5.1
308
of Viscosity
Importance
3.6
5.2
299
Laminar
Between
and
Layer
Transition
on
308
Flow
Fluid
Flat
6.
3.6.1
Body Corrections
3.6.2
Fin Corrections
Pressure Drag 4.1 Introd
362
uction
.
36
to the
347
FlatPlate
4.2
BoundaryLayer Separation
Pressure Drag of the Forebody and Fins
4.3.3
Launch lug Drag
SkinFriction
7.
363
(Forebody) Pre
ee 387 Besse tee 4.3.2 3.2 Fin Pressure Drag 388
Increase
Drag
Rotation
to
Due
of the
The
United
at Angle
417
of Attack
of Simple Stability
Model
Rockets
ZeroLift Drag Coefficient of the Fins
471
6.1.2
ZeroLift
Body
436
Coefficient
of the
6.3
General
l
7.2
of the Datcam Method
Analysis
429
Datcam
Control
and
The Datcom Method Applied to the Javelin Rocket
Model
428
in Turbulent Flow
6.2
7
376
Drag
Air Force
Drag
414
420
ZeroLift
States
12
6.1.1
Method
453
The General Configuration Rocket (GCR)
6.3.2
Dependence of the Drag Coefficient General Configuration Rocket 460
6.3.3
Dependence of the Drag Force on Reynolds Number for the General Configuration Rocket 468
Rocket
Drag
at
App.
Limit
Ss on
Drag
Divergence
the
Transonic
Jicability
and Supersonic
of
454
on Reynolds
Speeds
Incampressible
Number
473
Analysis
473
475
Determination of Transonic and Supersonic Drag 7.3 Semiempirical Coefficients 478
56 8.
429
483
6.3.1
360
4.3
Nosecone
355
359
2
4.3.1
,
342
SkinFriction Drag of Boundary Layers With Transition Corrections
Total
Calculation 6.1
3.5.3
357
5.2.4
Drag Due to Surface Roughness
Effects
Coefficients
FinBody Interference Drag at Angle of Attack
5.4
3.5.2
‘ThreeDimensional
5.2.3
316
342
Roughness
Fin Drag at Angle of Attack
Plate
330
402
Body
5.2.2
Drag
Effects of Pressure Gradient and Reynolds Number of Surface
Drag at Angle of Attack
52.1
5.3
Plate
400
401
Drag at Small Angles of Attack
313
Flat
a Snooth,
400
Introduction
Flow
Turbulent
Layer on a Smooth,
Boundary
Layer
Real
in
391
Base Drag
Other Contributions to Model Rocket Drag 5.1
of Pressure
Drag
The
5.
292
Constituents of the Total Drag Coefficient
3.1
4.4
292
Number
2.2.1
Viscous (SkinFriction)
“ky
286
Viscosity and Kinematic Viscosity
2.1.2
2.2

Experimental Determination of Drag Coefficients
484
for the
a
kvii
xvi
8.1
Wind Tunnel and Balance
8.2
Vertical Wind Tunnel Drop Test
8.3
Vertical
8.4
Conclusion
The
485
System 489
490
George
The
J.
Caporaso
Differential
Equations
Representation of the Flight
Mathematical
1.2
Selection of the Coordinate System and Differential Equations of Motion 517
The NonOscillating Vertically 520
2.2
The
Specialized
Rocket:
Solutions
Differential
2.1.1
FehskensMalewicki
2.1.2
Caporaso—Bengen
2.1.3
CaporasoRiccati
for
The Differential Equations of Motion
3.2
Numerical Methods Trajectories 564 for the Digital Canputation of Nonvertical
3.3
Examples of Nonvertical Model Rocket Trajectories
Solution
Solution
The
4,2
A Numerical
4.3
Solutions for Standard Foreing Functions
2.2.2
Extended Fehskens—Malewicki
Numerical ceMethods Peto Sh
for a NonOscillating
of the
for
553
the
Vertical
for
520
Case
522
Methods
528
529
Solution
Rocket
568
4.4
Differential
Method
Performance
578
Equations of Motion With Perturbation Terms for
the
Digital
Camputation
in Cases of Oscillating Rockets
of
582
585
4.3.1
Homogeneous Response for General Initial Conditions
4.3.2
Step
4.3.3
Impulse
4,3.4
Response
The Effect
Response
590
to Sinusoidal Foreing
of Dynamic Oscillations
535
in the Coasting
t Computatio n
Compared
Phase
539
of Altitude to
Numerical
Solutions
5.1
Bengen's Maxima
5.2
Model
594
591
on the Altitude Performance
603
Design Optimization
606
5.2.1
Initial Design Definition
607
5.2.2
Drag Coefficient
5.2.3
Weight
5.2.4
Dynamic Stability Optimization
5.2.5
Reduction of Drag at Angle of Attack
5.2.6
Philosophy
Rocket
586
588
Response
Typical Model Kocket
579
Altitude
Recapitulation and Qualitative Features of the Analytical Results
Solution
the Digi tal
Approximate
by the Interval Method
the General
Launched
to Multistaged Vehicles
CaporasoBengen
2.4
of
525
Extended
Solutions
Formation
523
2.2.1
Validity
564
4.1
514
Forces
Vehicles
Equation
Solution
Extension of the Solutions
2.3
2.5
509
of Motion
1.1
2.1
ons for Vehicles Launched at Any Angle
3.1
505
General
Soluti
Coupling of Dynamic Oscillations to the Trajectory Equations
497
ELEMENTS OF TRAJECTORY ANALYSIS
1.
Rocket: “see
494
CHAPTER 4
Introduction
NonOs cillataing he Veren
607
Optimization
608
of Design and Flight
610 611
611
of a
602
xviil
APPENDIX CORRESPONDENCE
A AND ENGLISH
BETWEEN METRIC APPENDIX
PHYSICAL CONSTANTS
APPENDIX
APPENDIX
618
CHAPTER
1
C AND DECIMAL NOTATION
620
AN
D
A WORD ABOUT THE NATIONAL ASSOCIATION FIGURE
617
B
AND PARAMETERS
CORRESPONDENCE BETWEEN SCIENTIFIC
UNITS
CREDITS
623
OF ROCKETRY
INTRODUCTION
TO
THE
DYNAMICS
OF
MODEL
621 George
J.
Caporaso
and Gordon
K.
Mandell
ROCKET
FLIGHT
SYMBOLS Symbol
A
coefficient used in writing example of P(t)
Ar
reference
area
Cc
Constant
of
integration
Cp
drag coefficient
°Do
coefficient
Cy
normal
of drag
force
at zero
angle
of
attack
coefficient
drag Magnitude of thrust, whose direction is assumed to be forward along the vehicle centerline F
thrust
F(t)
thrust
Fay
average
Pi
thrust
by
of time
thrust values
approximate
used
in
Iy
total
L N
characteristic length : nomal force
R
radius measured
R 3
c
te)
method
impulse
total
imp ulse
specific
e
computing
summation
Isp
R
ee
,. 2Fg
1
as a function
‘
impulse
from
e
the
center
of
the
Earth
Reynolds number radius side
of
the
Earth
force
magnitude is assumed centerline
of exhaust to
be
velocity,
rearward
along
whose tie
direction
vehicle
.
ye
symbol =
Meant
exhaust
symbol
velocity
ac )
differential of ( )
d(
)/dat
derivative
f(
)
function
of
( ) with
of angle of
A(
of
respect
magnitude
z=
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specific
of
values
exnibit
have
the
providing
in
consumed
of using
engines
rocket
seconds.
problems
propellant
of
mass
103
of
Isp = 1+/me the
acceleration
highperformance,
th e
Strictly
efficiency.
propellant's
a
the
meters/sec.@).
grains
the
of
from
Specific
g is
Model
this
to
impulse
abl F e ‘ obtain
the
mathematical
(5)
Tsp
The
the
where
roc ket
the
answer
at?"
is,
of
u measure
thus the
&
called
aq uantity
that
of
is
powerful
“How
rockets
umong
rom
questions
first
the
of
types, > one
l: ble aila avai
many
;sounding
propellant
4
selecting
When
use.
in
currently
the
in
used
mixtures
field.
1s
the
force
A rocket
Of
exerted
mass
on
it
by
m will
have
the
Barth's
by Ww
as
rocket
before,
3
ng
zg ig
the
(vector)
acceleration
of
ro
ae
ee
aed
=32
field.
The
of the
gravity
field,
Barth  or “straight value
the
speaking,
th ®
of
vertical.
e
to th
My
(12)
Me = { oT20 ) at
where
mg
1
the
(13)
at the Barth's
Accordingly,
_
rocket
model
any
ever
apogee Ro.
The
book
The
of
be
This
mass
from
the
the
was
established
rocket
would
the
all
between
equation
does
it
thrust
(2).
mass
the
result
as
the
function
rocket in
is
ceases
burning
at
time
tp,
this
during
its
flight,
due
the
expulsion
to
thrust. rate
(2) of
that
the
mass
at
or,
of propellant
of
can time
In mass
now in
be the
the mass
of the
Section expulsion
used form
since
in
the
engine
before
of
the
rocket
burn
and
the
model
In
large
constitutes
at
ignition.
change
due
to
the
This
rocket
fact,
the
when of
for
all
a
model
the
point
showing
any
tendency
acts
in
by
flight
through
that
4,
away
matter
the
called
the
rotate
0.G.
about
object in
any the
downward
may
0.G., along
this
and tae
the
book
or balance
all
direction.
vertical.
of mass,
suspended
be
the
remaios
object,
of
of gravity,
relatively
vector
center
ceater
tne
performance.
«ay
scope
of
considerably
from
the
to
is
weignt
the
occur the
for
"tips"
veuicle,
fraction
altitude
tae
the
nowever,
Chapter
of
or
small
of
propellant
direction or
boosters
within
which to
the in
space mass
rockets,
of
burnout
purposes
identical
point:
a
rocket
point
a
liftoff
model
after
and
only
shown
rotates
from
the
continuously
lighter
model
rocket,
practical
as
a model
the
rocket
taken
be
be of
1s
has
In
will
thrust, the
of
burnout
decreases
Missiles
expenditure
as
originate
to
ut
calculation
tae
weight
always
most
had
appears
of
guided
it
invarient
may
Ne
a
Unlike
which

welght
engine
simplifies
The
My
such
mass
mass
=
ignition.
slight.
, cc
engine
is,
the
propellant
so
than
the
at
the
an
m =m,  ('2L) . the
than
g,.
is
Equation a
greater
a model
produces and
in
calculations
vary
variation as
of
magnitude
rocket
motor
model's
weight
In
weight
rocket
as
the
(10) When
a model
relationship
determine
on the
have
inflight
2.1
to
of
to
total
Mp
during
than
higher
much

meters
however,
negligible.
taken
weight
though.
confirmed),
of altitude
wholly
& will
greater than Ry and Even a 20,000foot
of R only one tenth of one percent
effect
therefore,
any radius
6,000
(about
flight
officially
value
Ris
of g at that radius.
g is the magnitude
= f zou)
ignition,
where R, is the radius of the Earth, g> 1s the magnitude of ¢ surface,
= My
relation
&80 = FyR 2
(9)
ty
(11)
Strictly
d increasing . altitude
with
according
surface
tue Barth's
above
1
loca
the
along
down"
of = decreases
cen ter
the
is
case
this
in
which
points
always
force
weight
gravity
3}
direction
the
in
without rotations
weight
local
vector
vertical.
TUE Foye
ae
tos
34
Drag and Side Force i nteresting» ly the most
2.3
Probab 1
atmosP here
is
sage,
Which
physicists
@ nd
which
it
the
moves,
write on
a model
of
magnitude
moving
v
given
is
Ay
is
a reference
crosssectional
the
rocket
parameter
it
area
of
which
and is
depends
also
(more
C.G.
with
betw
area
normal
its
drag
to
(15)
Re
= iv
f is
the
drag
where
the
total
drag
of
the
air,
v
is
is
the
length
of
is
also
a
at
f
wnich largest
the
is
usually
diameter
coefficient,
taken body
to
velocity
tube
be
the
used
in
a dimensionless
shape
of
the
model
generally)
on
the
Reynolds
and
its
number
usually denoted
by the
symbol
fen the instantaneous velocity vector respect t o the air and the Orientation of
of
the
«, the
denoted
by
R,
given
by
the
factor
scaling
aerodynamic
about
density
usually
is
physicist
English
the
for
negotiate.
to
airflow
named
(18421912),
Reynolds
dimensionless
a
number,
Reynolds
The
the
for
difficult
more
becomes
rocket
tne
of
shape
effective
the
and
drag,
the
calculating
in
used
A,
area
reference
the
than
larger
becomes
presents
it
which
airflow
tue
to

perpendicular
1s,
effective
tie
since
increases,
also
increases
rocket
model
a
of
attack
coefficient
that

facts
all,
of
enough
be
angle
at
operating.
The angle of attack, angle
on
zero
(again)
a
1s
and
relation
is model's
rocket's
so
in
and
the
air,
the
magnitude
of
the
velocity
the
rocket
viscosity,
object
other defined
of
vector
and
interest.
L
There
as
ye P that
sometimes
is
number
Reynolds
the
(17) The
of
or
viscosity
“#is the
density
kinematic
16)
area the
from
As
of
angle
the
6.
illustrated
is
attack
of
angle
of
concept
The
axis.
be
will
by
and Op is the drag
attack,
which
the
of
air
rocket
will
it
fundamental First
through
model
a
present
rockets.
Figure
jn
Osborne which
Ds 3 0ps,v?
where
eS
to model
rocket
(14)
For the
a few of the most
just
down
as it pertains
of
3.
in Chapter
presented
of
drag
the
influence
factors
these
all
surface
by
mechanisms
physical
the
of
discussions
Detailed
with
of attack.
angle
its
and
characteristics,
finish,
its
of
quality
medium
the
detailed
its
of
drag
velocity
the
shape,
and
size
its
moves,
it
which
through
and
density
of the
is a function
a body
pas
its
to
of
viscosity
moving
* body
on The
drag:
call
engineers
longitudinal
the most
resistance
aerodyn umic
the
the
i + ugh throug
tainly
acting
force
late,
calcu
to
ex @ na difficult
cer
and
as
written
Re = VL v
Chapter
(17) (a)
is
viscosity
of
concept
For
3.
can The
be
the
summed
flow
dynamically
about
be
will
more
the
present,
up in two two
similar
fully
importance
the
of
and
equations
statements:
geometrically
if
discussed
Reynolds
similar
numbers
oblects
for
explained
(15)
rene ge el
37
Sete
the
two
(bd)
axis
angle
6:
Figure is
deflecte a through
as sumes
velocity
respect
is referred
«
vector Ve
that
with
to
the
The
of attack.
angle x fro m the
the
to
as
the
is
no
horizontal
respe ct
to
the
tnere
ground.
air
is
rocket's
the
of
direction
"angle
of
wind,
so
that
same
as
This
the its
with
inertial
force
viscous
force
any
in
effects
of
numbers
less
means,
for
placed
in
the at
flight
is
the
same
some
or
apply
may
not
per
second.
Reynolds
varying
of
sealevel
at
room
The
In
number
is
temperature
unit
any
air
combination
case,
this
model
1.225
the
on
x
when
when drag
1073
is
if
it
is
airplane
about
maintained
is
that
so
a
in
used).
wind
tunnel
second
per
150
meters
of
effect
the
while
the
be
may
speed
coefficient.
is
tunnel
the
flying
the
that
so
considering
gram/cm.>,
It
an
of
used,
5 meters
of
testing
information
placed
rocket
model
(or
a lower
and
important
is
(b)
Statement
air
than
airspeed
same
that
to
a
an
generate
only
can
on
cases
two
fluid
model
prototype,
the
of
1.0.
testing.
airspeed
tunnel
the
tnan
aerodynamic
of
accurate
yield the
The
windtunnel
as
of
effects
greater
field
decreased
taken
data
Conversely,
other
inertial
the
speed for
is
viscosity
kinematic
if
prototype the
pressurized
which
will
ratio
Reynolds
quarterscale
a
that
at
numbers
such
used,
times vL
product
while
to
tne
situation.
predominate
1.0,
tunnel
a wind
expresses
Reynolds
instance,
fullscale four
than
are
and
fluidflow
important
models
which
given
at
is
(a)
Statement
to
viscosity
predominate
in
identical; number
diagram
rocket's
velocity
The
are
Reynolds
velocity
instantaneous
attack".
the
longituay
cases
The
density
its
viscosity
(18° ) is 1.827 x 107+ gran/(cm.sec.). gram/ (cm.sec.)
makes
the
value
is
also
of P/M in
known
CGS
as
units
the
equal
poise.
to
that
that
and
corresponds
A response
Shown
80
of
A,
1
seen
there A,
a
the
for the validity
square
and
«,
constants.
Ay {in equation
o«.,
and
of
the
root of a negative
be both a nonzero
writing for
tine
and
Ww (25)
Ry.
da,/dt,
we
by
and are
a set
setting
obtain
=
dy
FT) Ta Seno  T2
Tate. t TT Leo TT
kind

Tb
1s called overdumped;
Like
the
hazardous
its
eriticallydamped
response. condition.
also
it
has no overshoot;
a criticallydamped
extremely
Az
gives
Theo
13.
response
+
=
,
this
in Figure
Overdamped
than
=
=
o
Gi
large
solution
forn
=0
Vas
was
expressions
(26) T=
of tris
two different
condition
conditions.
A.
or
be
constants
Xxo
Solving
We
the
occur,
is
I Cc moe
the Validity
there
solutions
D.
in
for
requirement that the square root of w ceyutive nunber
occur,
In
ae
+CA,R involving
not
~4
2 cq oh >;
coudition
to be the
gives
equation
dynamical
the
in
tQA Those
 %
Az g + 2%
a
(25)
@
ga
Ae

&
Ss
@,
;
x
2
respons
the
of
constants
time
e
are
Aa
1
th
expressions
these
stituting A
AL
A 2 ra
°
Sub
called
formulae
derivative
The
are
These
With
in the flight path almost ae severe 4 s
overdamping
slowly
overdamping
make
large
resulting
ie
the
sore
decays
features
those
behavior
potion,
from
caanges neutral
slope=xo
o&, (rad)
Oxo
t (sec)
pa
A statically
0+
joefsictent gaaner
13:
Figure con
ditions
to
more of
in
prope rties
the
slowly infinite
{nitial &%. p)
the
Overdamped
e6
beloy,
poth
19:
pb;
Given
initial
of
A
=
Tae
Pigure
are
A,+A,+% \ 2 =
which
from
“T)
yosition =
Oro
t (sec)
and
that does not persist
in Section for all
time
_
120
of time t)as shown imagine
the
interval
shorter
and
shorter
becomes
greater
its
to
process
however,
case,
H,
and
the
a
second
response
criterion
transient
yaw
may
of
by
a
which
disturbances
be
defined
as
may
there
obtainable of
are from
singularity
the
the be
An
of
H.
offers
rocket
impulsive
(t
0
= H
rigorous socalled
definitions limiting
impulses),
such
an
understanding
disturbances
on
physical
of
systems.
is an idealization of physical encounter
Area
Strengtn«
F
=
Figure
20:
of
steps
Mg1
and
Mgl»
finite
H
still
in
such
the
duration. for
The
a
in
To
less
The
dyncmsec.
that
(c), the
"strength"
a
step
of
product
of
the
The
time
step
duration
a
to fe) t (sec)
t (sec)
persists
but
iA
Development of the concept of an impulse from a series
of
{mpulse
to
input
ie)
c t (sec)
Ty;
(b)
the
than
limiting
the
impulse
in
(a)
product
has T,
such
case
intensity

has
Moi?)
an intensity that
of
inteusity
infinite of
step
is
and
still
this but the
an ictensity 1s
Myo the
equal
to
greater
Mgoto
product
behavior
is
infinitesimal duration
than

is
the duration called
H dyncmsec.
0)
tue
of
among
formal
effects
Like
reality. disturbing
impulse
arguments
includes,
to
and
S
2
*
(t = 0)
(which
steps
=
>
H dyncmsec.
strength
input
the
=
functions
never
evaluated.
of
f,(t)
of
can
resistance
an
(t = 0)
necessary
rocket
such
= 0
study
a
to
f,(t)
more the
rocket
this
has by
of
P
at
follows:
O°f;(t) While
given
In
zero
impulse
an
called
2
:
AS} Ms»
thig
"type"
denote
hereafter
shall
is
kind
this
of
function
A forcing
I
which
Mgt,
value:
definite
a
with
infinity
"this"
of
product
the
since
infinity,
2
Y
Ket),
arrive
special
rather
a
considered
be
must
M,
carry
infinity.
is
M,
and
zero
is
t;
that
such
configuration
a
we
¢
5
;
F
6
€9
Input
step
ultimately
will
we
conclusion,
log {cal
If
becoming
product
the
constant.
remains
rectangle,
the
of
area
the
“action, Noy
persist sts e
step
that
greater in such & Way
and
f
Mgt).
1s
applied
the
of
magnitude
the
as
forcing
area
the !
whi
during
this
whose
which
ne
time
of
of
rectangle
a
forms
20,
in Figure
n
representatio
graphical
The
interya)
some
after
again
to zero
“steps down"
but ratner
0,
©
the
of
inputs the
other
things,
precision of
theory
is not
impulsive
step,
the
impulse
You know very well moments
of
that
infinite
Paed
t== 0,
©
DQ
>
we
122~ley,
and
duration,
however,
The
response
moment
M
of
wheel
the
flywheel
applied
is
to
of
for
a
input
our
of
Figure
5.
moment
the
of
I
to
of
angular
acceleratio,
that,
for
a
constant
which
angular
is
position
4s
thus
The
an
from
the
original
rotative
now,
that
the
t approaches
constant
at
wheel
the
by
conditions
the
zero
value
woment
remain
in H.
during constant
H I
such The tne at
M approaches a way
that
angular
tue
velocity
interval tne
infinity
value
t will,
and
product
Mt
imparted under
the
to these
the
of
angular
velocity
application
of
strength
H will
of
the
to vO
flywoeel
azpear
impul
icstantan
se.
zero
damping
cause
an angular velocity
H
I,
~
instantaneously,
to
will
be
moments
while
zero.
in
any
the
Does way
angular the
alter
displacement
presence the
at
of nonzero
state
of
time
corrective
the
rocxet
effect
due
at
t = 0? it
Moment
the of
And, the
with
is
stability,
occured;
zero.
remains
finite
of the yawing rocket is Precisely analozou:as, dk damping and corrective moments are both z ero:
ne the
arise
static
moment
reached
is
tae a
time
ax
yet time
the
Well, 2
Suppose,
4H
case
ca cause
to
impulse
is mM 42 a= iFt
since
decrease,
problem
provided
and displacement
to at
equal
angular
.
diacement will
s
eously
to
resulting
OWever
3T t
Limiting
velocity
Mt
the
_
impulge
the
w= and
h
,
11
“= the
of
by
yt is eiven
when
somewhat
types
Recall
resulting
the
ytually
other
to
of
t,
displacement
rocket
examoles
conce
discussion in
angular
liftorr,
during
consideration
a
inertial time
is
responses
the
illustrated
as
frictionless
taan
gne
im.ulsive.
virtually
{impulsive
the
of
the
from
are
staging
during
Ordep
high
a
direction
contact
launcher
to
to
residue
solid
of
facilitate
to
return
we
let
the
are
an
grasp
to
flywheel
the
to
order
In
imputs.
of
response
difficult
more
in
waich
functions
forcing
fluctuutions
encountered
disturbances
and
impulse
due
arising
moments
nozzle,
an
ejection
oblique
line,
thrust
a
Momentary
accuracy.
of
as
treated
ve
can
short
of
disturbance
strong
any
duration;
zero
{ntensity
is
hence
as
finite
there
does
arise
the
of
at t 20
moment
increment
therefore
any
be
cannot
displacement
corrective
velocity
and
there
angular
no
the
although
angular
that
clear
to
can produce
no
simultaneously this
{moulse,
the
has
is itself
moment
4 damping due
rocket
to
change
in eitner zero
amount
tb e angular velocity in a angular displacement or presence of aerodynamic It thus turns out that the time. ect
the initial eff Moments does not modify This effect produces an impulsive input.
upon
the
the
rocket
following
set
of
ewe
e
124initial
conditions:
Xxo
(37»)
2
=
O
a,
(37a)
(rad)
of
the
Unlike
no
xo
a
with
equations
response
equations
in
phase
angle @
=
=0
and
anrckan
Applying
(19).
through
(15)
the
rocket
initial
the
following
(37)
results
in
the
initial
amplitude:
by
given
is
conditions
values
for
(0)
A= Ty The given
characteristic
This
to
an
impulsive
disturbance
equations
(16)
is
by
(38) where
response
kx W
=7Re and
motion
The
equations
is
D are shown
Aim determined in
Figure
criticallydamped
(22)
and
Wk
(23).
by
impulse
In
and
(17).
21.
this
response
Case
we
igs
have
described
by
then
=
7]
Underdamped
gne
angular
jnitial
sinfarctan (sy
arctan()
figure
yaw angle
by
given
those
underdamped
an
of
impulseresponse
equations
the
@ homogeneous
a4
time.
of
values
tm
definition
its
from
see
:
the
speaking,
(37).
The
given
positive
conditions:
initial
of
set
special
can
you
actually
is
response
impulse
all
for
zero
is
as
for
H  Borctan(¥) Aum = Tage
it
with
associated
properly
More
t>0
zero,
has
input
impulse
for
is
itself
impulse
the
step,
response
particular
complete
ra
response
particular
the
=
attained,
res
velocity
;
Ponse to an impulse of st rength H in yaw. imparted
by the
impulse
’
the
maxinum
and the time at which it occurs are sho wn.
126
Ou
H
A,
where a
again
Dis
(24)
at
a
y
damped
to
Tesponse
an
impul
the initial yaw rate, the mazin
which
the
maximum
yaw angle
_
.
occu
se of yaw
um
strength H
an
and
“
Ts.
= a
Slope
results
the
cives
Critically
:
showing
time
conditions
initial
the
22
in yaw me
obeys
forcing
impulsive
pigure
22.
Figure
by
jllustrated
Applying
(26).
through
response
imoulse
The
(16)
from
resul tins
motion
equations
these
is
rocket
criticallydamped
eouations
equation
py
given
Overdamped
to
form
the
assumes
motion
— pt H L A T hx =
(39) of
=i
characteri stic
the
and
o, (rad)
A,= 0
Axm
H
=
A
C,
rocket
will
to
tne
being
value reduce
of
dependence of
Iy
the
It
equations
and
would
severity
of
the
(38)
initial
thus a
is of
inertia
of
moment
seem
rocket's
inversely rocket
the
through amplitude
that
a
impulse
velocity
angular
initial
the
disturbance
impulsive
an
disturbed,
that
shows
(37b)
from
23.
Figure
in
longitudinal
4nverse
t.
(40)
large
Iz
proportional
is
response.
oo
\T2
xm ~ 147)
on
the
desirable We
can
time
at
to
the
which
deflection
%T2
Overdamped
showing
angle
an
 TmAnlu/2)
nm”
yaw,
is
the
occurs
also
response
{nitial
critically
for the
is
which
reveal
factors
(e)**(@)™
= H%T2
23:
Figure
illustrated
resulting
a.
by
=
Equation
tm
overdamped
An
(25). described
impulseresponse
an
exhibit
equation
by
determined
are
Tz
thus
Xx
(40)
as
I, (%,%2) and
where
t ec)
£0 x rej
Tr
HT,
=
A,
3
(%\T2)
Ie
'
TT
XT
more
maximum sooner
damped
yaw
yaw and
to
rate,
angle
the
maximum
of strength H in yaw
is attained.
its value
responses
gradual.
an impulse
angle,
and
the
The maximum
is smaller than
is the
and the return to zero yaw
case
i ~129
128
. »~*dim
~
(41a)
Hg
the
,bt
H eo
criticallydamped
him Ax
an
Finally, equation
zero
way 28
O
that
.
motion
we
from
Sgt
gives
us
&
»~% , KHT
HEC
C
_
= you
(43b)
you
will
trigonometric
can
obtain,
after
>
“,
8 ome
7
expressions
D,
%,
into
make
the
following
motion
= 0
motion,
underdamped
For
criticallydamped
For
overdamped
an
of
I,
applicable
(20),
seem
at
first
are
desirable
only
in
the
not
is
in
the
which
whole
the
story,
damping
2,TL g= ~ 2Ne We see
that
an
Ta Particular,
qd amped
responses
increase
in
(41d),
Secti on (42b),
>
and
w x me
of
case
@
in
I; —— increases
underdamped
for
is
that
if we
given
lurge
x
xem °
values
xotion.
examine
equation
as
ly invari
ably
reduces
the
damping
ratio.
ad or criticallyIz, can cause overdampe have already been shown to be undesirable
increasin g
(which
in
I, decreases GeCreases
indicate
to
however,
ratio
’
=
derived
in
increase
might
results
manipulation
changes in Iz have no
otion,
motion,
same
ecguations
discoveries:
mot
P
on Xxkm
fF ron
.
equations
increase
an
expo ponential
in much the
(=)2
the
and
of
called
ec,
~
(~)
For
These
the
erse
algebraic
substitute T,
P als °
rithn 8,
 2
7
TCG)
Caritim of" "
inv
functions,
which
This overdamped
if
effect
= TL.D2 of
we
xem W
have
0
case
(40)
for
is
AH%2
and
of
;
lop°
atural
mathematical
the
are
»
"nat
i
and
the
:
the
Natural logari
parentheses.
tables
3.1.1
dim lanctan (¥)]
= +
in
27°
(43b)
H
(42b)
which
_
Wt
cot
for
junctions and may be found in tables arrangeded ase)
_
*
Ty
a (42a)
pt
DH =>
_
in
stands
enth
ote
so
—~Dancten(F)
‘
n
tos
Now
X com = iw
For
of
y
ocarithms
yaperta
is
case
(75)
m
(41b)
,>*
condition
the
imposing
underdamped
wh Tay t
phe quantity
of
obtain
we
which
from
+
Wh
value
{
:
equation.
resulting from
by
"ln" n”
notation
the
ee
DH _ Tw
the
on
velocity
angular
and
resultin g
equation
The
the
satisf. y
will
that
t
zero
to
smallest
the
determin jing
be
computed
wnere
ee ee
dx,/dt
they
wuic hb
These
a“
To
~
ae
—
setting
at
t
of
maxima,
+
_
(43)
Lr(Bx,
wet,
—
I
Se
nee
values
can
occur,
displacement
2
the
and
onse
jmpulseresp
of
case
eac h
with
associated
max imum
the
deriving
by
thoroughl y
more
angular
respon
the
of
severity
the
go verning
quantities
the
investigate
131
~130
overdamping
against
importance
significant
of
4n
which
In
the
the a
properties roll
this
of
of
input
an
to
rate
section
of
the
positive
static
to
going
an
I
of
response
a
be
the
situation
a
significant
of
is
alone
of
stability
creates
soon
interest. about
talking
fi(t
xit) theInto
for
yawing
form
the
_
4+
Ci Xx =
solution
.
Again
Wet
to this equation is imown to be
The
rocket
motion of
the
of
sim (wet
Ar
=
Xx
(44)
~
form
the
rocke t motor
of tue
the expression Given above ¢ for © us the differentia) equati on
AX SOR 2 dk
0,
4
phe particular
the
having
rocket
phenomena
1°)
of the
form
are
benavior
nowever,
time)
with
away
particular
the
for
respons e
particular
the
recainder
zero
dies
dampinz,
finite
nature,
(which,
response
charact2ristic and
periodic
and
prolonged
a
of
cases
In
motion.
resulting
the
of
moments
disturbing
the
where
I. aa?
taat
character
the
determining
in
> ddx
were
in
oscillatioc n
nature.
response
particular
tue
and
response
homogeneous
the
both

responses
such
was
cases
these
in
rocket
wits
discontinuous
or
transient
bas
model
a
of
behavior
The
of
functions
forcing
to
sections
previous
the
In
conce rned
were
we
7
Gives
(13)
quation
Forcing
Sinusoidal
to
R esponse
State
Steady
3.1.4
;
.
gubstituting
forcing.
{mpulsive
pepresentation
to
sensitivity
rocket's
ta e
reduce
to
and
"flutter" er
aerodynamic
baysical
suara
to
both
helps
inertia
of
mo ment
longitudinal
large
ne
responds
disturbance.
"leads"
response
sinusoidal is
frequency
The
amplitude
the
forcing
time
The
radians.
the
whose
own
its
to
+9) forcing
with
identical
to
is
different,
function
of the
derivatives
by
a
a sinusoidal the
frequency and
however,
phase
response
angle
thus
of
described
ee
a
Such
7
tarust
are
response
result
transient is
phenomena
identical
obtained
from
been
has
to
the
going
have
the on
died
based
away,
that
so
such
the
alone.
response
on
that
this
sufficiently
time
a
for
particular
an analysis
assuuption
an
lone
complete Tne
assumption
1s
=~
dhe _ dk?
—
Ar
We
~ Wt
;
The values of A, and *Xpressions
for
xX
(wet
cot
Ar @ and
,
am
+)
(wet +
are determi its
time
ned
) by
.
on
substituting
derivatives
the
int o the differential
nee
all
input
based
ee
that
be
will
LAx
ee
“sinusoidal”
sim Wek
+
analysis
Ag
arn
The
=
epee
fy (t)
Ce
as
design
desirable
a
is
; the
in
~
le
I,
instabilit y
to 8 Sinusoidal forcing,
Fesponge
=

c=
the
upper
expressions
for
the
+ eet
WwW

(Y+2
This
behavior. must are form
be to
which
determine is
carefully be
avoided.
a
these
of
rather
carried For
give
tricky
out the
2
+
step
case
Ne rae
———
2
ZX*w?
awe
x2 Wz? (Y+z S42) +z
jos.
us
the
se"4? )* +ZX
by
step
Xw,=0,
correct in
procedure if the
2
zw
LC (Y+z
must
22")
E GE 6 e/(rr 8 Get) Ll
We
root
of
of
algebra:
the
nuuber
this
a
positive
In
number.
a .4Uantity
thus obtained
is
squared,
is cacea, quantity
alwuys all
the and
the
caowiedce
is Levevosably
here,
example
our
V(Y+z)
limiting case
owe
assume
= .*
=
,
Now
}Y+z
=
discussing
to
the
If
we
case furtner have
(Y+Z)
if
staticallystable in
which
(Y+
Z)
rockets,
Y is negative
stipulate
that
the
roots
value
of
(¥+2Z)"
(Y+Z)0, but
In
always
“the
Y+zZ
and
the
= Y+z
‘
inconsistencies roots
when
bear in wind
of the oricinal
sign
algebraic
we must
complete
2
27
x*a)
rule
results
lost. fhe final result of such a sequence of operations is the absolute value of the original quantity, whica is by definition
in
sign
the
these
4
2u)2> [eee
we Vf
of
for
equation:
X*W
square
or
the
choose
to
obliged
Choosing
b@.
in
following
the
have
A=\
os
it
sign
are
we
simplifying
following
sign
a
chosen
In
=
upper
the
choose
can
Having
terus.
root
however,
we
that
in
ae,
quartic
the
of
roots
we
(r+z BOP ez we
in
root
have
we
(53b),
we
+ ZK
(ye e A)
a
here
square
the
for
in
z
&
option
an
is
there
Now
Stay
obtained
as
»?
for
2 wy? \2
+ Lb
Yt2
Awe
CE 
ata Solving
equation
quadratic
a
as
solved
be
give
to
a®
for
can
It
equation.
biquadratic
we is, be
are 0,
referring is
always
positive).
underdamped,
we
will
ee oe 2) ee
151
go
that
if
Wz70,
correctly
staticallystable zero.
For
the
in
opposite
the
limit
limit,
that
of of
W2z Ipa5We
be
seen
that
¥
4
2
W
4
Ie
as
°F terms,
collecting
~
Ir
7
relation
this
Iet We" 41,4
C2* Te> wat 41
be
can
inequality
the
express
can
his
*
Ig W2 Ty
we
relative
rocket
model
subside,
rollcoupled
for
stable
positively
41.
We
i
has
motions is
must
be
of
oscillations
the
do not.
to
wants
designer
or
2
2
relation
there
the
eyes
about
response
yawing
that
range
this
in
is
Co? Ip? War
initial
spinning
rocket
a model
nonzero to
to
response
homogeneous
Undamped
solving
by
explicit
It
rolling
which
motion
characteristic
Figure
is
condition
27.
conditions
other
rocket
the
I,
and
Cp,
under
The
general
Figure
in
positivelystable
coupled,
of
case
A representative
damping.
has
effectiveness
tae
reduce
to
serves
coupling
roll
that
so
elapsed,
most
tae
time
sufficient
a
after
a
As be
will
mode
slow
the
oscillation
the
oscillation.
decoupled
that
means
this
of
a
such
than
slowly
more
decays
8Ss
mode.
oscillation
4 decouvled
than
rapidly
more
decays
mode
fast
fast
the
than
mode
decaying
slowly
a more
be
tperefore

a
have
2
made
more
161
Slope = 2yo
ppon ——
fe)
a, (rad)
Kot
examining
(55),
= t (sec)
we
the
see
original
that
this
is
expression Just
the
for
“F
given
requirement
in equation
that
C,>O
cnaracteristic
motion,
whether
Fe 2
fe)
ay (rad)
4
27:
Characteristic
damping
and
yaw
and
pitch,
the
properties
oscillations and
true
nonzero
roll
showing of
the
eventually
response rate
the
to
decay
to
a
general
relation
response.
of
of
model
rocket
initial
the the
yawing
zero
and
the
finite
conditions
initial
Both
with
slow
mode
approaches
zero
in
conditions and
model
to
pitching
regains
straight
fast
0,
= O
so
flight.
as
just
stable,
remains
rocket
the
if
it
°F *
t
aztrz
approaches
mode
indefinitely;
deflected
0;
As
rolling.
not
Co*Ie
Wa
constunt
of
+
41,7
the
inverse
time
the
angular
deflection
flight
moment is
were
slow
At
Zero. it
neutrally
is
negative,
becomes
that
¥
of
the
of
decoupled
the
of
damping
the
twice
or
the
of
constant
time
from
positive,
is
Wz
if
the
of
the
of
tnat
while
inverse
The
0,/Iy»
that
but
oscillation,
above
from
moment
frequency
angular
the
— ins
of
negative.
approaches
mode
spinning
corrective
the
as
1s,
zero),
value
a
is
is
2
Ig we 4It
toward
approaches
Wz
222
(that
mode
if
rocket
2
Wz
fast
below Figure
2
Ie"
C2
“412
decreases
coefficient
t (sec)
2, x +
toward
decreases \
a
tL 2
the
As the value of
about its centerline. Slope = Qyo
or not
stable
positively
for
required
thus
1s
stability
static
Positive
increases
coefficient
rolling
There
or
of
with is
upper
the
integrand
D=
case
not to
in
are
and
enclosed
of
in
in
Table
Figure
a
17.
the
Ua.
whole to AB
mass
surface,
from
velocity
Assigning
control
the contributes positive
flowing
the
momentum
2. that
the
to
momentum
flux
through
total

far
segment
requires
the
space
contribution
to
rect
a
problems)
over
value
control
the
by
undisturbed
xdirection.
the
in
sufficiently
moreover,
negative
by as
is
their
symmetry, the
as
constant
consider
in
fluidflow
443,53,
region
in
conservation
limit
ek
h
drag
force the
D
be
control
(U.?utU2 ue)dy
of integration may is
expression
obtain
a
surface
of
plate,
the
expressed
p=$b(
(70)
such
the
drag
or
The
This
6 = 0.664
equal
surface,
in
flowing
imaginary
points,
momentum
region
thickness
surrounded
plate
analysis
to
lies
surface,
(69)
=
parallel
in
fluid
the
skinfriction
a flat
(an
corner
forces
exactly
y=o
in the
A,B),
wall
in
its
boundarylayer
the
consider
used
by
of the
@ to
surface
Momentum
§Un?@= $) u(Unu)dy 6 =
relate
Pressure
that
or
(68)
rate
a quantity
or (66)
due
potential
define
concept The
may
manner:
following
the order
we
1/8 (54).
equation
by
defined
identified
(15). In
approximately
Now
be exactly identical,
turbulent
of
cases
in
momentum
certainly
18
flat
distributions
velocity
the
zero,
to
equal
the
the
for
while
it),
through
flow
fluid
at
that
than
less
be
must
pipe
the
of
end
downstream
gnis Upstrean
zero
for y > h;
be changed
to
infinity,
as
hence
u (Uso > u) ay applies
to a plate wetted
on only
the drag for a plate wetted on both sides,
one
side;
we evaluate
to
TABLE
2 control
surface
Zi
Se
Rate
n
CrossSectio
LO
b
4 By
the
("(Ue, co u) dy
S 5 J U,,( U,u)ay
total
of
surface
control
turbulent
h
MELT
_—
Se
“ay
Sb
VIII)
LTT
5 u2ay
—x
h
————h
Eh Foy
0
———
'
a
%
Volume
net
flow
=
total
rate
= drag
0
moment
mm flay
—__
flow
surface
and
momentum
pictured
skinfriction
drag.
in
flux
Figure
accounting 17,
for
use
associateg with in
caleulaty aUhng
Figure
17:
due
a
to
angle
of
Oontrol turbulent
attack.
7
ZE
ulay)
nn) —
Po
Veo
= control
2:
j
—
h
b f) udy
BB)
Table
Fo), Ue iW
VITTITTTL
y
0
b { Udy
aay
sum
X Direction tn
ie)
h
AB
Flow
of
surface boundary
F— Us
for
calculating
layer
over
the
a flat
friction
plate
at
irag zero
3382p=Sb Jue
not
b is
merely
replaced
by
the
integral
may
Second,
on
case
in which
body,
leading
the
from
extending
region
the
on
circular wall
the
to
skin
friction
edge
to
that
particular
momentum
(72)
U26
This
expression
equations
(73) boundary
(66)
and
is
identical
and
be
recalled
(71);
to
hence
the
we
integral
o=)
that
appearing
return layer
velocity
»
now on
a
boundary
(6)
have
layers
2 ()
the
be of the
The
law"
the
pipe
flow
in the
form
we
the
turbulent
empirical
results
adopt
"1/7thpower
the
for
case,
can
results
form,
this along
and
represent
relationship a flat
thus them
of experiments
one
plate
curve
requires in
that
turbulent
plotted
all
flow
in dimensionless
all.
with
turbulent
boundary
the
shearing
stress
can
thickness
now
be
obtained
in terms
of
at
§ by direct
the
from equations
integration
$
te (I~ Gz) ay
© 5)" [IA ly (* (E)hay( (Bray = 2575
= 755 if
we
a7
pth
Combining
equation
(73)
with
respect
to
x
we
= tw = sult
equations
ys
Substituting
(79)
(76) and (77) then ylelds
Here gt (78)
Er
into
(75),
we noe
= 0.0225 (2)'
4n explicit expression for § as a function of x can now be obtained
layers
differentiate
obtain
(B)

profiles
same
coordinates
in
From
of
T.
laminar
velocity
plate.
consideration
y\%
Un
in
explicit
flat
distribution u
(74)
an
that
Then
in
Now
to
shown
(74):
u (U.u)dy
yro
(70)
turbulent
the
= §
it will
D= be U6 We
4s
°
(65)
also
relation
thickness
poundarylayer
u
equation
have
the
y 4 = 0.0225 (z) U.. $
Te
ou
station.
Now from
pipes
obeys
(75)
{The
due
drag
the
give
will
it
x
station
any
at
evaluated
be
the
body
cylinder,
the
of
circumference
the
cylindrical
a
of
case
the
in
plate;
flat
a
only
body,
here,
cylindrica,
symmetrical
any
to
made
be
should
integrals
is applicable
(70)
equation
First,
4n
these
concerning
remarks
Two
=u) dy
to
(63)
for
Mercer. B = 1700
(function
22 can be used directly
6.
MThreeDimensional Qorrections to the FlatPlate
Approximate
laminar 4)
to find
coefficients for use in the method described
SkinFriction
109,
Section
22, along with the pure
from equations
5X
for model
by Mark
(101)
2
conservative,
number
gathered
equation
@ in Pigure
pure
3:6
should
value
case,
rocket,
Since
matter the
no
a model
for
drag
a particular
Since
1700.
range
3 x 10° 8700
in
calculation
about
and
Section
for
for
of
(101)
known.
determined
average
1 x 106 3300
equation
must
corresponding
in
derived
B
5 x 109 1700
Ro,44
skinfriction skinfriction
of
(15):
to
of
1s plotted
B_
turbulent
order
rockets
 (Cé) iam  coefficient
(Ry)%
3 x 105 1050
quantity
6
 (Ce)iam_
skinfriction
(Ry) 1.328
below
Rerit :
The
(100)
0074
=
values
listed
assume
Letting
(86)
are
making
(99)
are
(C#)iam
the
or
we
(1020)
In

friction.
skinfriction
(Ceeun
layer
same is
_
(1028)
approximate
reduction
boundary
drag
decrease
from
introduces
the
turbulent
plate
assumed
way
if
the
be
region
the
Veo b Xcrit [ce eury ~ (Cian
2nd
change
laminar
just
for
The
(Of)turp
the
drag
can
AD= £
where of
edge.
all
flat
boundary
turbulent
the
point,
transition
the
behind
that,
the
Coefficients
methods have been developed for estimating
359
tne effects friction of
an
pe
used
for
of threedimensionality on the values of the skin
coefficients
approximate
to
method
correct
application
derived
for the flat
described
in
twodimensional
to
plate.
Reference
The 9,
results
which
can
skinfriction
coefficients
surfaces,
presented
threedimensional
are
pelow
3.6.1 Figure
with
Ronit
22:
boundary
Skin
friction
layer
= 5 X 105
coefficient
transition,
(corresponding
based
to
for
on
flow
the
B = 1740
over
a
assumption
in equation
flat
For
plate
of
(101)).
held
Body
laminar
with
its
Corrections
boundarylayer
axis
parallel
friction
coefficient
over
the
length
given
same
(103)
2
(0c+),,,
In
a
previous
of
0.00382
number
of
is
over
a
stream,
circular the
cylinder
increase
of a twodimensional
approximately
by
in
plate
skin
having
(9)
= aie
example,
we
determined
laminar
flow
1.206
x 109.
For
ratio
at
over
10
=
Ibex lo7*
skinfriction
the
a
a flat
a model
of
(Crom
adjusted
the
that
for
todiameter
The
to
flow
same
skinfriction
plate
rocket
body
Reynolds
coefficient
at
coefficient
a Reynolds with
number,
a lengthwe
obtain
is thus
= 328 . 2k ‘ a = (Cediom t UNC Ham = “aan * ARy = 003986 (Ce with of
equation
(103)
accounting
for
@ 4.4%
increase
in
the
value
Cr.
In the
coefficient
case
of turbulent
is found from (9)
flow,
the increase
in skinfriction
361
=360
02
(204)
(205) In
over found 10
coefficient
a flat to
and
plate
be
same
Then
the
at
a
will
completely
Assuming
be
that
we
of 1.206
have
for
the
boundary
skin.
layer
x 106 was
a lengthtodiameter the
ratio
skinfriction
coefficient
3
Now
Cy = 2 Ce the
to
the
increase
cylinder
equation
flatplate for
the
the
value

laminar
magnitudes
on
(105)
accounting
for
a bit
than
case.
will
be
less
In
used
Section
to
constantdiameter 3.6.2
Fin
A model
rocket
be
represented
as
of
the
airstream
468 higher
than
finite
thickness.
This
to
the
thickness
and
c
Then
for
fin
is
tube
the
percentage
corrections the
from
of
inorease
these
skinfriction
sections
of
the
a model
(108) For
drag
rocket.
a
is
seen
about of
fin
generally
a fin the
the
1s sometimes
The
with
not
average
no
thin
side
colloquially
flow,
force
enough
tangential
a symmetrical
undisturbed
produces
quite
D
(or
called)
at
Section
and
zero
‘The
angle
as
area
will
model
t denotes
coefficient,
effects,
rocket
fin
introduced
shall In
fin maximum
thickness
based
we
on
fin
planform
area,
obtain
having
a thickness
ratio
of
(108)
by the use of equation
have
that
occasion
section
to
also,
return
to
procedures
formulae
these for
converting
drag coefficients based on body tube lateral
planform be
where
1e proportional
10%.
6.
section
fin
a fin of
t
we
skinfriction
"lift", (9).
in
for the air to negotiate
= AC (1+ 2)
be
velocity
airfoil
even
to
drag
correction to
increase
in turn,
by
t
thickness
typical
the
0.05
fin
for
Cg
the
b
the
Again,
plate.
the
force
6,
determine
body
of attack when side
1/3
increase
t/o,
caused
1s proportional
chords
fin
denotes
pressure
increment,
ratio
Ac
Qorrections
a flat
that
1.3%
dynamic
required
is
= . 0045593
a
in
velocity
corrected with
in friction drag with thickness
increment
4n flow
of
107°
cA + aa Cia
(Ce eure =
recalled
turbulent
number
number,
=5.43x
adjusted
it
a Reynolds
Reynolds
eure
(0Ce
example, for
0.0045.
the
(106)
is
area,
reference
= ~ (Rg)
turbulentflow
friction
plate,
a flat
drag coefficient of a flat plate wetted on both sides, planform area (area of one side only) used ae the
friction with the
1.6x1073 (L/d)
(ACs)
our
Ce)turk for
+074/RyX6
=
(Or)turp
since
or,
(A/d)
(ACE) = or
area
presented.
to
coefficients
based
on maximum
area
frontal
i
ae
36}
362
4.
4.1
drag the
over
integral
forces
acting
motion;
that
(you
the
body
surface
of
directly
may
unit
opposite
Section
direction
of
2
as
the
pressure
of
components
the
the
in
the
rocket's
is
useful
integral which over
the
to
Figure divide
over
the
be
phenomenon
and
no
with
launch
boundary
layer
presence
of
inserted,
drag.
In
forebody,
against
the
the
event
airflow

is
will
of
the
into
as
the
base
it the
drag,
the
(pressure
the
intimately
In
general,
streamlined
fin
profile,
be
rocket,
result
separation
engine
casing
component
flow
of
separation of
drag
larger
unavoidable
the
large
pressure
considerably
blunt of
the
the
of due
to
must
pressure
occurs
on
surfaces
directed
forebody
can
begin
on
with
and
concerning
the
not or
is take
the
into
4.2
the
and
layer
remains
attached
This
study
phenomenon
important
question,
of pressure
in
Section
and
discusses
in
to
1s noted
that
this
exhaust
of
on
(if
Mark
Section
used
effects
of
launch lug
is
the
engine
sources
the 4.3
which
it
either
laminar
this
Section
although
3 for
prevention
to the
rockete:
base.
skinfriction
formed.
most
model
BoundaryLayer
The
and
pressure
two
first
of
of
drag.
the
account
influence
mechaniem
1s essential
expression
presented,
base
information
the
empiricallyderived
drag
the
the
there 1s very little quantitative
wellconstructed one)
times
a discussion
concerning
has
of
separation
Unfortunately,
available
several
understanding
4.2.
foredrag),
associated
an
we
integral
separation.
which
extensive

notation
first,
second,
fins
because
1s
a relatively that
and and
nose,
This
rear
opening
however
called
4.4;
the
perhaps
drag,
data
surface
analysis,
rocket,
body
drag
foredrag.
creates
the
base
the
of
parts:
boundarylayer
the
to
two
drag
streamlined
from
purposes
the
4.3.
pressure
of
a
lug,
pressure
the
of
Section
to
regard
into
the
rocket
Section
the
rocket
the
in
For
of
in
normal
memory
integral
area
of
in
everywhere
your
this
remainder
existence
a
large,
boundarylayer
model
again).
base
The
for
11
discussed
to be discussed with
n is
refresh
to
will
of
drag
vector
wish
consulting
the
previously
is,
by
the
defined
s
where
than
was
Dp = (f poos(n,V)ds
(109)
quite Since
Introduction
Pressure
be
pe
Pressure Drag
4.4
Mercer's
the
evaluate
base
formula
stabilizing
the
base
the
does
fins
drag.
Separation coefficients
turbulent
flow
to the
assumption
is
which
were
presuppose
solid
surface
generally
valid
derived
that
the
on which for
in
Section
boundary
it ie
a flat
plate,
as the pressure gradient along the plate's surface dp/dx is zero; but in regions of increasing pressure (dp/dx positive, &
this not
point
happens, applicable To
& blunt
it will
and
the
the
break away from
skinfriction
beyond
illustrate
body,
gradient)
boundary
layer
to follow the contour of the body surface
may be unable & certain
pressure
adverse
socalled
the
this
point
the
surface.
of
Section
coefficients of
phenomenon,
beyond When 3 are
separation. we
examine
the
flow
past
such ag the oircular cylinder shown in Pigure
364
from
Bernoulli's
equation,
between
A and
B and
thickness
of
dn
element
it
is
in
virtually
the
exterior
moves
from
the
increasing,
then
decreasing.
of
energy
that it
a
had
in
inviscid
fluid at
subjected
the
kinetic element
to
overcome
side
of
will
is
actually
25,
which
the
then
process shows
of
is
no
C with
the
which
not
have
pressure
at
the
the
some
point
influence
its
the
of
of
elements of
Hence
kinetic
the
energy
downstream
will
pressure
motion
much
B.
motion
the
its
exterior
consequences
velocity
its
velocity
consume
A to
on
predicts
fluid
sufficient
gradient
direction
against
from
first
fluids
same
forces
travelling
energy
dissipation
perfect
frictional
and
the
there
as
pressure
cylinder,
however,
under
how
at
the
layer,
will
and
of
theory
arrive
wall
reverse
rear
so
be
gradient, that
it
airstream.
are
profiles
depicted in
the
in
Figure
boundary
near
layer
the
Flow
Inviscid
cylinder
circular
a
about
theory
("poteatial")
transverse
held
predicts
tae
taat
the
to
flow
of the cylinder will be accelerated from point
surface
from
point
B and
decelerated
again
boundary
layer
develops
inflection
pressure
gradient
to
station.
kinetic
in
is,
taken
the
airstream.
the
same
of
boundary
positive
moving
the
in
cylinder,
the
Because
gained
the
Acting
4t
the
large
near
arrested.
This
to
energy
the
will
Within
are
to
flow,
element
A.
an
front
because change
continual
a
is
the
at
flow
flow,
and
23:
Figure
is
that
over
pressure
constant
exterior
the
in
undergoes
distribution, as
layer
pressure
the
to
equal
the
downstream
flow;
fluid
the
layer
boundary
the
in
station
any
at
the
distribution
0, = 1  4sin@?)
real
in
layer
boundary
the
upon
along
pressure
24 as the curve
(which appears in Figure impressed
increase
theoretical
This
Q.
B to
from
surface
a corresponding
pressure
static
in
decrease
4
is
there
with
accordance
In
C.
point
B to
point
decelerated
are
and
B,
point
A to
point
from
accelerated
are
elements
fluid
the
fluid,
"perfect"
a
of
flow
inviscid
the
In
23.
an
between
B and
0,
and
B to point
C.
In
due
separation
to
the
actuality, the
adverse
occurs
at
8.
A

Oe 367are
I
/
__
a ee
2
Cp=14sin’e
altered
near
nas
a
stable
of
the
of
increasing
to
be
full,
60°
8S
i)
flow
about
curve
obtained
from
potential
number
Reynolds
of
curves
average
with
6
x
(based
on
between
a line
drawn
23
and
a
periphery.
point
line
drawn
Point
C to p=
from from
Bin
180°.
is
data
taken
The
diameter). the
the
Figure
cylinder axis 23
to thus
at
the
Pis
angle
subcritical
forward
au)
(y
—
Figure
the
on
the =
90° t)
cylinder and
to and
the
general a
rapid e).
of the
c,
which
equations
boundarylayer
separation, conditions
and on
moves
well
into
particles
velocity
gradient
the wall
has
condition
upstream
the
at
a zero
zero
region
the
wall
develops at
consequently
of
normal
Mathematically
of
separation their
a
so
made
thick,
in
the
the
vortex
in the
boundarylayer
becomes were
point,
motion,
of fluid
(equations
layer"
a,
profile
the
reverse
layer
flow
the
expressed
accumulation
The
a "boundary the
of
the
point
layer
begin an
the
wall.
lost
all
velocity
as
0
increase
assumptions
layer
=
at
boundary
velocity
The is
The
the
nearest
a
wall
c,
has
wall
Downstream nearest
point
the
As
however, at
and
profile
c.
momentum.
at
(110)
angle
other
until,
of fluid
gradient
the A in
to
4ts
separation.
velocity point
point
layer
of
pressure,
retarded
The
number
point
corresponds
compared
and
to
axis
any
theoretical
Reynolds
supercritical
the
and
10
pressure
displayed
theory
experimental
cylinder
10°
3x
flow
of
The
cylinder.
circular
a
in
coefficient
variation
experimental
and
Theoretical
24:
Figure
point
separation
inflection 180°
120°
a
layers is
boundary
thickness in fact,
38 and 39)
of
apply
resort
had
be
the
the
only
to
of
the
series
of
photographs
from
and
@ original
boundary
and
to exist. to
the
experiment
rear
body
up
fluid
leads
are no longer valid
approximations
must
layer (points
as such can no longer be said
the
formed,
that
derivation
of
which
to the
Hence,
point
determine flow
has
Separated. The
4),
made
the
actual
cylinder.
remarkable
by Prandl
and
development In
Plate
4a,
Metjens of the
(Reference
separation flow
in
has
at
12),
the
just
Plates
rear
begun;
4a
through
illustrates of
a circular
the
boundary
layer is very thin, and conditions conform very closely to ideal,
Plate the
4: onset
gradient.
Flow of
around
boundary
the layer
rear
end
of
separation
a blunt due
to
body, an
illustrating
adverse
pressure
“
Jave Of of Javelin with airfoiled fins
 
of Javelin with square edged
—_— [>
BC70
[__1"——
BC78
0.41
'S
0.42
12
_—>
BC76
0.42
2.14
—
BC74
0.43
>———~  )
Hemisphere
ee
(0.70)
>

cone
0.59
45°
cone
0.61
Sligh lightly
 rounded Flat
—_§_
coefficient
nosecon one
shapes 3 p
a2 nd
aiagrams
of
nose
for the used by
of
tne
fins
the
shapes the
fins.
of ~ +,“9@
r Profiles
shapeg
each
Figures
Javelin mTy
8.
rst
tests
(designations
Centuri
Javelin
using
4 fin
yp
column
beg
Engineering
Cor
rocket).
nose
in
Snape,
Parentheses
were
obt
drag
coefficient
a
2.24
0.87 1.32
p
Drag
2.16 _
60°
:
21:
214
0.51
——_—
[>
(2.35
0.51
BC79
—
EE
[
2.10
BC72
[>
[
0.41
y
Figure


2.24

2.51 , ,
2.77
nw +4
 ;
Q\r
Nosecone sha pe
Figure
nose
32:
shape
Variation
with
in
fineness
the
ratio
L/d.
of
a paraboloidal
367
~386
negligible. varying
function
Mark shapes on
It appears from tests that (10)
Figure
29,
actual
model
kit produced to
various
greater
in
five
nosecones, are
are
his
for
numbers to
rocket
nosecones.
these
shapes
from
ratio
of
the
least
test
nosecone
first
of
what
The
2;
(b)
tested
and
body,
the
nosecone
being
its
base;
(c)
a smooth
boundary
the
Mercer's rocket
flow;
data
Stine
the
drag
slight
no
be
blunt
considered
of
only
The
as the to
between
its
surfaces an
which
distinguish
to
nosecone
the
tube
convex
side
facing
empirical
the
at
the
about
slightly.
differences
2.0;
quantitatively
L/d
drag
of
further
This
separation
of
as opposed
were
drag
31).
ratio
behavior
is
paraboloidal
extension
nosecone
in Figure
coefficient
the
in
32,
shape
of
the
nose
accounts
for
the
in drag among the five
taken
reduced
the
the
streamlined
is reduces
blunt
some
point
off,
separation
relatively
nosecones
to
Profile
0.02
edge
will
that 
If
occur
of
the
or
of the rounded
streamlined
rounded edges
streamlinedfin
respect
experienced

the
the
reattach
itself
the
trailing
edge
also,
rocket's to
section edge
(the
shape)

(or "profile")
leading
and
to
is
the
also
in
fin
a paperthin
with
structural
flow
fin
at
squared
a
“base
drag”
separation,
ts required.
accomplished
simply
twodimensional
a gentlysloping
in a sharp trailing edge (the socalled
compromise
in
body.
minimize
be
in
encountered
resulting
main
usually
increase
separated
will
there
leading
a 236%
commonly
or greater),
can
nose
at
configuration.
ratios
rather,
fin
streamlining
culminating Since
leading
prevent
a
than
trailing
downstream.
Streamlining
must
the
(usually
a
One
of
rather
thicknesstochord
rocketry
the
in this
in Cp from 0.70 to 2.35 when all the fin edges
from
to
edges.
fin edges,
used in his tests
that
Providing
of blunt
rocket
Javelin
at
To
model
the effect
The
tapered
model
that
from
and/or trailing
1s considerable
and
At
the leading
edges,
off,
over
to the flow result
streamlined
squared
analogous
to
flow from
left
edges
Drag
drag due to the fins must
indicate
to
an increase
pressure the
data
Adequate
lengthtodiameter
pressure
ratio
any
Mercer's
Fin Preseure
the fin surfaces are generally parallel
direction,
flow.
guide
tests.
Since
"typical"
a lengthtodiameter
tangent
with
BC74)
furthermore
considered
(a)
stock
and
29;
s
4.3.2
(Figure
particularly
BC=76
Figure
transition
curve
clearly
(Centuri
features
are
toward
streamlining.
(18).
up
be
generally
is demonstrated
significantly increased
(d)
thus
importance
streamlining from
may
nosecone The
and
drag
BO=72, in
trend
Note
Mercer'
the
which he modifieg is
31).
lowest
might
of
interest,
The
in
increaseq,
a commercially~availapie
BC78,
a smooth
of
Company,
shape
R is
seven
bluntness
essential
others
as
ones
(Figure the
1s a slowly.
all
purposes.
BC70,
the
representative
at
Engineering
exhibiting
similar
tested
Javelin,
measurements
shapes
decreasing
additional the
Centuri
model
toward
ae

increasing
catalog
roughly
they
some
rocket
with
number,
windtunnel
configurations
drag
the
has plus
by the
demonstrated that
Reynolds
Mercer
in
an
of
(r/h)crit
durability
by
analogy aftersurface
"knifeedge").
requirements
trailing edge 1s very easily damaged.
here,
The
of a wellstreamlined model rocket fin is illustrated
~389
388
drag
its
in
rockets
in
this
due
to
often
on
this
that
of
Mercer,
as
well
so
rockets
of as
the
effects
assessed.
lug
to
bluntfinned
the
drag
to
that
much
larger
for
is possible
the
parasitic
drag.
it
relates
influence
of
a
body
We
by
launch note,
about
lug
streamlinefinned
that
an
to suggest an average drag
the
fin
configurations, now
be of
increased
roughly
this
is
Cy
addition
Javelin
On
the
cannot
increment
version.
on
and
placement,
of Mercer's
0.21,
lug
diameters
rockets
model
to
basis
equal it
coefficient increment
the presence
of a launch lug on model
are
such that
the ratio
of body
of Mercer's
that
to
is identical
diameter
such that the lug placement is similar. develop
models
ratio
Javelin
a tentative in
to
which
the
formula
the
body
experiments,
To
do
a
this
launch
tube
for extending
lug
Mercer's
diameter
stands
than
the
diameter
by computing
while
diameter
1.93
of
will
be
the drag
was
in
case
a
in
coefficient
of
square
coefficient times
a large
the
greater
Denoting
area.
flow
of
separation,
area of the
the
lug may then
be written
on
ratio the
tube of
from
diameter the
A,,
rocket
entire
by
denote
outer
an
crosssectional
on the
we have
itself,
body tube in its vicinity.
coefficient
used
area
body
area
tube
The is

that the lug must from
drag
therefore
crosssectional
(Cp)rug = 5.75
but
of
is

Aug
(which,
cross
circular
or 4,/ALug = 23.0.
that indicates
only
drag
for the
of
the
on its own frontal
based
(Cp)rug
one
not
of
the
drag
shall
ratio,
based that
it by
to
we
which
lug
lug
has
area included within the
the
than
and
value,
the
equal
ratio
and
4.8,
diameter
the
of
is

launch
Javelin
the
The
computing
lug
standard
of
tube
then
is
to
the
the
size has a diameter of about 0.40
body
recalled,
tube), of
section
23
that
centimeters.
for
area
body
the
note
the
diameter
lug
the
the
we
of the Javelin's
centimeter,
4t
available
only
can +o
reference

boundary
the
from
The
launch
however,
version
and
to
drag
Pressure
protrude
as
of
javelin,
a model
separation
lug.
the
which
different
of
coefficient
as
effect
the
accurately
the
to
referred
important
data
values
rather
for
reported
of
face
objects
similar
and
lugs
blunt
the
from
case,
is
layer
agrees
boundarylayer
is
again,
once
culprit,
to
to
configurations lug
giameter
area.
(7).
The
whose
due
¢ne launch lug when in place based on its own included frontal
of
67%
or
0.28,
finding
This
or more
50%
of
rockets
the
the
rear
the
near
about
by
rocket
rocket.
a lugless
# 0.25
aifferent
streamlinedfin
Cp of the
the
test
Javelin
estimates
with
well
location:
for
value
the
(presumed
his
of
version
lug
small
that
showed
research
increased
of S°p
results
variations
large
Mercer's
tube)
body
the
of
5.
that
is
drag
coefficient.
a launch
of
addition
in
role
Section
in
produce
can
a body
of
shape
the
in
Profile
We
aerodynamic
of
aspect
remarkable
changes
the
Launch Lug Drag
4.3.3 A
of
an important
discussed
as
lift
to
due
drag
determining
edges
 does not substantially
play
it does
but
drag,
pressure
affect
lateral
or "airfoiled"
they are flat
whether
thi e
of
character
The
38.
in Figure
quite cause
a substantial
The general expression
increment due to a bodymounted
launch
391
390
==
(0¢n)\,,
(119)
and
drag
increment.
0.50
for
his
Skychute
XI
Reynolds
numbers
(Rp =
Mercer's
tests.
Since,
the
Cp
less
of
than
the
lug
0.15
the
Skychute
0.35,
the
located
=
rocket
2.5
x 10°)
XI
the
in
0p
of
bodymounted
lug
on a configuration
wind
Wichita
tunnel State
used
lower
Mercer,
so
is
results
directly.
it
this
following
due
to
(120)
however,
tentative
a launch
lug
the
formula
mounted
at
of
Section
the
presence
to
cannot
be
same
lowspeed
level
of
covering
variations
a problem
requiring
than
that
used
by
to
compare
of
presently
available
best
we
can
to
for
the
the
drag
finbody
is
empirical
in lug
configuration,
further
2
has
near
drag
and
research.
for
(A0D)1ug
will
of
Base only
the
base
axis
and
is
angle
the
placement
is
80
is
behind
resulting
this
and
established
lug
the
launch
in varying
a particularly
successful
the design
in
kits.
remains
area
on
the
Competition Model Rockets of
art
reiterate at
the
 of
professional
that
time
the
of
prediction
writing
"guesstimating",
rocketry.
of model
that
It
rocket
formulae
on a firmer
is
drag
for
to
hoped will
adopt
that be
predicting
analytical
almost
more launch
foundation.
Drag section
of
with,
I must
empirical
from
be
analysis
zero
drag
by which
or left
developed
lugs
investigated,
our
(lai)
launch
however,
launch
future
in
(at
lug
lug and has incorporated
of
of
the
pressure
generally of
attack),
second
term
as
expressions size,
to
colloquialism the
launch
Mechanisms
One firm,
launch
a matter
The
joint:
device.
after
Virginia,
due
wholly
reducing
experimented
conclusion,
4.4
increment
of
commerciallyavailable
drag
lug
to
been
popoff
thoroughly
a
present
coefficient
of
the
problem
also
soon
of success.
In of
launching
retracted
several
their
nature
do
form
in
at had
limited
accurate,
considerable
tunnel
have
a
AV ug/4r
have
(0Cv),., = 3.45 fuss = 3.45 jee
development
a
may
accurate
not
than
for
ratio
Kansas)
6,
of
more
Mercer
rod
means
the lug completely and launch from a tower or
Alexandria,
of
probably
much
by
of the
extremely
important
methods
due
Wichita,
necessarily the
those
is
(the
turbulence
Given
rocketeers,
the
The
not
concerning
model
air
near
as
lug
be
degrees
at
same
determined
by Malewicki
University
significantly
data
increase
lug
launch
joint
60%
a launch
the
the
a
only
The
the
to
without
finbody
with
about
according
increment
at
tested
can
lug
about
of
Cp
overall
an
determined
Malewicki
effective
closedbreech
model's
launch
the
decrease
substantially
can
body
its
fins
most
to eliminate
that the
between
joints
the
of
{The
indicate
(20)
Malewicki
J. one
in
lug
launch
the
placing
Aug = 5.75(dse) dp H,
5.75
by Douglas
taken
Data
2
Dy = GSSb pede
rocket
drag
is
now
remaining
the
base.
perpendicular base
Since
to
the
flow
pressures
act
along
in equation
(28) may
to
be
the
considered
plane
direction the
be written
drag
simply
392
(122)
Coy
TFs,
Theoretical
analysis
base
the
predict
flight.
time
drag
of
a model
from
boundary
layer
separates
equations
layer
separation
A
(c)
well for
to
researched
presented last
rocket
separates
known
and
as the
as there
is
the
when
fins
phases
of
sources; blunt
the
boundary
the the
beyond
hobby
empirical
consequently into
unable
expected
engine
is
not
converges
1s considerable
to
professional
take
a separation
be
"deadair"
the
expression
to
then
or
been
for
base
either
level drag
of
the
account.
essentially
flow
the
the
about
firing.
The
downstream,
region.
This
motion
of the
phenomenon.
the
base
boundary
enclosing
term
of
Mgure a model
layer a volume
igs actually
air in this
region
of
cause
not
have
rocketry,
to
and
reattaching
is in
decrease
further
a
poattail
a
and
region
base
the
into
drag
base
in
a misnomer,
region
the
boundary
then
low at
when
pressure
point
circulation
R

the
downstream
pattern
the
showa
a conical
boundarylayer
as
the
flow
base
flat
pressure" the
in
with
tne
"base of
rocket
§ is
thickens
layer
separates
and
angle
boattail
the
is
The
thickness.
flow,
bodies;
finless
for
€
a model
of
base
tne
about
Flow
33:
Figure
poattail.
the
disturbs
a decrease
in
peculiar
The
is
drag
depicts
the
cause
either
phenomena
Base 33
on
flow.
here
two
all
drag.
problems,
two
subsonic
of
exhausting
believed base
not
observed
that
jet
during
from
valid
are
resulting
generally from
accurately
point;
presence
The
(b)
can
previously,
as mentioned
and,
base
which
following
the
arise
The
last
rocket
Complications
(a)
The
theory
no
this
@iffious,
extremely
is
drag
base
the
of
at
is
there
fact,
in
D
=
rocket.
the
base
is
over
passes
forming
reached,
P)

and
Viscous
pressure
the
then effects
region.
eee ~394
39
“pump"
it away.
the
(59),
coefficient
(123) where
Crp
Cg, =
as
the
basis
(124) The
nature
Cy of
2
wetted
area of the
coefficient
a
smaller
base
a forebody
>
bricks
the
and
base, above
by
Ce
=
of
area
determined
of
proportional
is
(54),
drag
friction
the
O¢'
Sy
the
is
of
forebody Section
base,
the
of
skinfriction 3,
we
relationship

that
generally
limited
employed
is,
the
form
of
base
drag.
on models
which
in
diameter
than
is
referred
to
Figure the
expect
flow
the
33)
angle not
1s
reduces
diameter
of
small
separate
boattail
base
to
casing;
the
enough
is
the
engine
to
there
a rocket
exhaust.
be
a
rear
geometry
the
the
base
(about
effect
to
tube
section
fit"
to
the
body,
area.
If
to
the
degrees)
the
surface,
drag.
The
technique
which,
for
some
reason,
must
use
in
diameter
than
the
significantly is,
just
of
course,
to enable
greater no
sense
it to
be
recovery
and
the
models
pressure
engine,
section
rocket
lateral
greater
due
base
to
are
technique,
5 or 10
boattail
the
by varying
“glove
of
trading
with
pressure
those
section
not
rocketeers
body
a reduced
from
does
A graduallytapered
€
in
body
added
a main
would
to
increase
a@ main
which
downstream
the
usefulness
require
boattailing.
as
flow
will
that
stated,
widelyused
most
The
base
model
rocket
in
variations
in
drag
base
(which
a decrease
experimentation
finish,
surface
drag
rocket
or
fins
reduced
for
viscous
previously
as
some
been
drag
to
body
produces
of
effects
a model's
in
in
that
= f (Cy) this
of
boattail
exclusive
body
discussion)
roughness
guide
$
methods
the
friction
the
however,
has
there
in
roughness)
of
the
increased
Dt 9 Se
increase
account
into
=
(125),
Equation
Although
(see
‘2s
an
is
(124)
0.024 Vn,
effects
the
reductions
drag.
equations
compare
example,
of
that
7 En is
the
8S, is
Sp is the (on
such
_
implies
If we now define
(60)).
and
reduction
(for
drag
skinfriction
smaller
thickness
boundary
in
increase
a
boundarylayer
the
that
seen
have
We
effective
in
results turn
in
which
pressure,
base
the
take
=
Do
expected,
arag
effect
pump
jet
the
and An
reduced.
is
pump
the
therefore
thickness
layer
to
of
pressure
dynamic
separation),
we
4ncludes
boundary
The
121).
after
between
region,
deadair
the
and
flow
outer
the
sheet
insulating
an
as
acts
however,
layer
shear
free
the
becomes
(which
layer
As
to
consequently
is
base
the
at
equation
(see
results
drag
and
air"
"dead
the
pressure
static
The
base
and
reduced,
of
a
flow,
tries
The empirical function that has been
equation
for
C
(125)
34.
in
along
is
limited
enlarging
built
with
the
a
boattail! Smee
as
with
mixes
"Jet",
acting
(9)
the external
is somewhat like a jet pump:
suggests,
as in Figure
agetermined
Hoerner
as
4 whole,
as
flow
the
of
character
The
(16).
point
at the convergence
from flow reversal
data,
and

stream
free
the
and
volume
deadair
the
between
region
experimental
of
a plot
from
determined
be
can
f 
function
ee
resulting from mixing along the free shear layer  the boundary
—
YT.

0.30
397
An
0.204 Cop calculated using base area 0.10
0.05

0
06
0.4
0.2
[e)
08
Cro
Figure
revolution and
Variation
34:
with
of
forebody
base
drag
coefficient
friction
drag
coefficient.
the
shown
experimental
data.
represent
a
"best
a
of
body
can
maximum
frontal
d,,
then
(126)
SE
=
to
the
base
is
of
a
denoted
§,,
boattailed
and
its
body
the
If
(125).
config
associated
body
dy \*
S,,
(Z)
drag
coefficient
frontal
area
is
Cy, Dg =
drag
equation
from
friction
(127)
a large
area
diameter
pody
curve
The fit"
of
for
developed
be
uration
The
function
semiempirical
collection
of
expression
of
the
forebody
based
on
maximum
D orebod —forgbede 4 Sn
Now
= (Dszststs )( 42)  O.(¢
= a a“
Ce
(126)
Furthermore,
where
(Cp),
frontal
drag
2
db
 (
simplifying
1s
(Cpp),
apparent
drag of the as
the
that
ratio
body.
engine
on maximum
body
algebraically, 3
the
based
0.0249
db (Cy), = (+)
decreasing
the
coefficient
Lin /
Cy
Ve. (LIS 0.024
m
It
tail
base
Then
(Cp)
a3)
of
the
area.
(130) or,
is
[de a (=) ~
De “a Sy
_ ~
D 43
(Co.),=
(129)
boattail
dp/d,
The
itself.
VC
cam be reduced,
or by increasing
first The
og
technique
danger
angle €
of
is
the
by
skinfriction
limited
separation
(as defined
either
by
the
diameter
from
the
boat
in Figure
33)
is
398
will
not
apply.
of
section
the
precisely,
longer
body
tube
with
boattail
that
the
rocket
it
would
have
than
Differentiation
it
cylindrical
to
be
not
should
Tasers
must
care
denote
differentiation
denote
diameter 1s
dp/dg
that
skinfriction
we
unrelated
drag
require A(Cpp)m
~
nose
2.
find
from
(170)
ARy = 46.1
= 4b dm
OGIVE
is
OSive
Sm
body
area
1.055
)
Ss
(
CYL
Sm
ogival
crosssectional
to
area
cylindrical
()Sm
obtain
we
= 1645
wetted
Sim.
is
= +.0135+ .04l = +.0025 dh) dm
(LvLA/dm)® of
ratio
_
3175
43
=
[\+ The
*
_
Q,
8
3.94
84.
66.2
779 x 1074
6.18
1.75 x 10° =
1.57 x 107
x 1079
eee.;
:
1.94 x 1073
2 x 10?
8 x 10° 9 x 109 1 x 106
and finally
(209)
x 10
Da (1)
;
3.33 x 107?
5
3 x 10? 4 x 109 5 x 10°
2
C
1.16
10%
7 x 10% 1.5
, we write
Rot v2
x
1.03 x 107
1 x 10°
>m
D=aS\oge
5
De (i)
04
1xil
coeffioy,
drag
the
a
Rocket
Re? »* “hy?
U2 = 2 =
Then
Cy
S, and U,
To eliminate
and
20.5
terms
in
ressed
} as
tone
Gres
a
Configuration
General
for the
the
assumption
28.0 27.2
10.1
21.0
15.7
19.6
drag
force
variable with
of
the
in
drag
newtons
on
the
coefficient
approximate
(0p))pp = 0.473
drag
at various
470
1 x 10 through 1 x 107. 1 x 109, the
the
drag
force
0.25 newton at liftoff) flight
regime
that
in
force
predicted
constant
Cp
be
dependence
of
behavior
of
values
for
large
the
drag
Rg
coefficient
is
inversely
(Cpo)pp
on
decreases.
as
»
that,
of
the
data drag
the
of
Figure
of a
assumption
for
the
Reynolds
Since
the
proportional
number
to
the
drives
skinfriction
laminar
square
camnot
increase
rapidly
enough
to
Reynolds
number
increases
into
the
climbs
rapidly,
root
offset
of
this
effect.
zone,
the
the
drag
force
experiencing
a tenfold
increase between Rg = 6 x 10° and Rp = 2x 10. with
the
“exact"
function
D.
function
for
the
drag
derived
assuming
the
drag
Reynolds
number
this
constant
wee
Phase
as
2 flight,
(Cpo)pp
Da,
in
coefficient
The
range.
is the or
= 0.473,
value Rg =5
so
Figure
to
value of
from
be
52
constant
I have
From
an
equation
Plotted along approximating (210)
over
elected
(Cpo)pp at
x10,
is
the
Table
transition
body
by
the
to
entire
choose
initiation 6 we
read
for
of this
meeps
As
52:
Variation
GCRx drag
approximate
quadratic
the
the
variable
9) pp is (Cp
although
10°,
than
less
Rg
force
D downwards
Rp
is
in this
seen.
for this
reason
The very
can
by
encountereg
so minute
the
and
than
less
values
Dp (N)
either
commonly
is
between
(210)
by equation
th
representation
graphical
distinction
no
52,
Figure
shown
in the
levels
force
drag
The
engines.
rocket
in model
or the thrust
with
weigh
rockets
model
few
(very
weight
rocket
compared
small
is very
up to about
numbers
For Reynolds
range
number
Reynolds
the
over
(210)
equation
from
obtained
rocket.
of
Deg
coefficient drag
obtained
drag
is
force
the
newtons
“exact”
computed by
in
taking
by
the
drag
with
obtained
Datcom
(Cpo)pp
Reynolds
as
method;
0473.
using Dg
number
the is
the
472
(211)
D=
approximates
Reynolds
numbers
large
values
in
drag
force
the
of
is a _
not
the
or
vehicle
than
these At
drag is
are “may
be
for
all
because
value 5 hood
know of
both
a
whether newton
three
or
the
when four
drag
performance.
the
force
thrust
orders
of
Reynolds
numbers
at
in
Rp
constant
51,
1/1000
of
and
weight
of
model the
in
the
of
10% up to
Rg
51,
greater
to the
chosen
as
the
the
approximate,
first,
because
Rp 25x
onset
of body
transition
of
the
7 and
configuration;
drag
Reynolds
is
obtainable
numbers
converted
be
given
Rocket
Drag
71
Limits
on
As and
The
Table
the
number
Model
than
coefficient
zone.
a value
106
in
equal
flight,
to
requiring
assumption
the
attained
that
performance
of
calculations
transformation
actual
drag
x
when
used
assumption
the
at
Rp
in
closed
of
=
5 x
10>
a
axis
of
velocity
to
a
in
equation
at
Figure
52,
axis
like
that
applying
by
of
Pigure coordinate
the
(207).
Transonic
and
Supersonic
Speeds
greater
on which
regardless
magnitude
is,
transition
was
reasons:
to represent
of Op to within
Figure
body
calculations
rockets,
exact
saw
that
3
estimates
Reynolds
can
7.
about
exceed
to
Cp.
The
It
is
magnitude
approximation
= 5 x 109
for two major
considered
a good
as we
constant

acceptable
insig
are
deviations
of
Cp with
altitude
form
magnitude
number
constant
a
yields
the
attains
Op
absolute
the
rocket
model
of
(as
Reynolds
of
numbers
although
10°
4x
these
that
small
so
Cp for the
based
=
calculation),
to
(Cpo)pp
constant
Rp
exact
because,
nearly
of
Furthermore,
a
expected
not
rockets
model
singlestaged
for
rule:
deviating
flight,
Reynolds
for
function
the
Dz, represents
force
value
rocket
of
range
semiempirical
a
of
form
the
in
stated
be
can
result
This
values.

very
model
below
2/1000
higher
5 x 109
over
this
that
the
closely in
rockets.
seen
be
108.
calculation
are
may
large
necessary
newton
x
2.2
is
the
in
nificant
it
is
error
percentage
7,
exact
the
and
10>
x
4
between
newtons
model
moderatesized
to
small
of
flight
the
during
encountered
in
quite
Dg
2
Rp
force
interest
of
from
10%
than
°
Table
of
function less
drag
inspection
From
.
x 10
approximate
the
gives
1.58
473
Figure
and
10°
of 2.2 x 106 » very nearly the practical limit of Ry
this
only of
in
chapter
if
the
from
the model
this
slight" 0.316,
Section
2.1.1,
treatments
can
be
compression is
Applicability
relatively
discussion
compression
that
the
the
in
accurate
The
the
due
Analysis
analyses
first
an a
reader
analytical
corresponds
of
the
on
atmosphere
slight.
the
Incompressible
results
contained
assumed of
of
six
priori
to
the
may
criterion
sections
basis airspeed
also
of
recall
“sufficiently
to a Mach number M of less
than
where
second,
from
in the
52
of
stated
semiempirical
the
this
neighbor
and
air:
oc is
the
the
speed
socalled
Strictly
with
which
“speed
speaking,
the
of
sound
waves
travel
through
the
sound".
Mach
number
associated
with
a given
474
475
varies
varies
with
you may
variation
of
sound
atmosphere.
is
by
(213) Tyzq
is
the
in
question
and
at
the
in
time
measured It
is
the
speed 
over
c
this
about
the
phenomena.
at
the
in
as
the
density.
rocket
of
the
air
The
becomes
were
of
at for
the
by
few
of
it
its
sealevel
standard
calculations airspeed
theory
can
of
vicinity
and
the
within
the
boundary
layer
leading
to
be
the
the
stagnation
edges
layer
thickens
,
of
the
increases
anda
its
the
a number
fins,
we
our
have
only
around
as
is
the
not
exhibit
not
are
numerically
the
short,
of
a
the
model
effects
rocket
are
in
bit
of
more
complex
incompressible
the
the
vast
fluid
theory
the
above
M
of
M
Mach that
such
ana
the
(.2
Drag
As a model
Sound

regions
from
on
value
below
as flow
to
make
theory
they
be
used
calculations.
the
airspeeds
the
various
thus
performance
at
rockets
therefore,
may
of
drag
a manner
= 0.316,
= 0.9
is
numbers
incompressible
compressibility
negligible
at
flow
as 2 model
deviation
designer in
such
up
drag
coefficient
to
approximately
meters/second.
as
literature
overall
revolution,
a a treatment
compressible
that
these
such
model
closedform
of
fill
highperformance fact
to
of
What
interact
up
a
treat
appreciable
to
each
of
present
to
of
A
would
quite
aspect
case.
rocketeers.
flow
valid
to
portion
results
accurate
are
able
of
any
gift
modification
In
body
phenomena
analytically
model
model
of
incompressible
Although
the
study
to
designers
Nature's
of
purpose
their
the
effects
small
interest
if
the
been
a
as
of
involved
fimned
are
306
drag
not
of a
by
true.
points
noticeably
velocity
us
does
without
meters/second.
through
of
to
is
behave
geometrically
experimentallyobserved
It
this
results
known
the
to
compressibility
of
on
appreciable
O.9
assume
considered
107
of
predicted
present
which
up
is
overall
It
grown
rocket,
practical
interest
compressibility
Presence
nose
is
has
coefficient
at
value
which
approximately
its
air in the
at
of
is
the
mathematics
by
fins
description
Furthermore,
interest
that
altitudes
reasonable
all
be
percent
which
range
must
the
The
than,which
itself.
those
that is
upon
as
flow.
altituae
altitude
however,
a
for
the
Kelvin.
therefore
influence
makes

or
is
maximum
boundary
(213),
temperatures
the
that
Rankine
slightly
over
at
at
tha
analytical
phenomena book
adh
lower
precise,
States
temperature
temperatures
equation
is therefore
speed
The
local
ratio
Shows
United
temperature
the
incompressibleflow
Above
tip
The
valid
which
the
¢ upon
scales,
only
constant
of
well
varies
meters/second
where
4 and
remains
from
for
temperature
absolute
It
analytically
flow
the
and
achieve.
rocketeers.
obtained
actual
can
model
of
atmospheric
and
340
altitude
4,
profile
this
here,
rockets
about
Figure
In
than
the
range
model
models
that
the
itself
Ta
Figure
sound
c
temperature.
consult
with
since
Tr
question,
of
to
and
dependence
standard
from of
launch
day
The
T is
one
found
most
to
on
wish speed
C= Cs
where
conditions,
composition
again
standard given
atmospheric
atmospheric
connection the
with
reba
airspeed
Divergence
rocket
form
approaches
near
the
nose
"Mach
tip
one"
and

the
speed
the
fin
leading
of
476
edges
in which
short
distance.
fin
trailing
transmit
through
fluid
of it
move
smoothly
and
become
tip
the
fin
and
transonic compressed volume,
nose of
fluid
thus
At
a Mach
the
decreasing
oblique from
called this
the
as
the
nose
the cone
Mach may
socalled
a
model to
the
shock.
As
conical,
intersect
boom.
the
of
ground,
In
do
in
on the
by
that
nose
its
air
original
original
that
would
M
increases
with a
the
the
cone
body
any
above
1.0
halfangle
of
shock
revolution
observers
sound
trailing
airplanes to
hear
produced
the body itself can only be heard within the Mach cone:
is
,
by
to an
again
be
=
one
Op
and drag
reason
before
1947,
the
and
the
a
as
rocket than
its
the
is
Mach
divergence.
that
it was some
Bell
to
to
and
is
increases
the
shock/expansion
quantity
Mach
The
drag
supersonic
flight
approached,
than
one
Cp
declines
subsonic
2.0.
The
between
occurs,
is
divergence
is
"the
barrier
first
Drag
and the
regime,
drag
airplane
thereafter.
the
toward
peak
of
to
transferred
velocity,
greater
call
expansion
The
be
drag
coefficient.
1.0
transonic
Xl1
and
by.
above
sonic
which
rocket
conditions.
Mach
expansion
completely
drag
existence
common years
and
the
the
at
The
rocketpowered
for
as
number
in
number
momentum
somewhat
Mach
gone
transonic
above
that
has
subsonic
rapidly
model
shock
subsonic
in
to
returned
over
under
experienced
the
of
airstream
the
not
rocket
amount
slightly
value
is
the
increases
at
phenomena
through
model
once
greater
increase
0.9
these
exhibiting
density,
PrandtlMeyer
the
as
known
passes
in
supersonic
in
rocket
ahead
the
of
expansion
decrease
rapid
of
a model
a peak
called
in
of
coefficient
rapid
behind
transferred
toward
drag
model
certain to
therefore
reaches
M
a
model
Typically,
axis
threedimensional
oblique
causing
is
the
a
the
state
causes
the
at
which
surrounding
from
The
importance
the
system
the
the
the
compression
horizontallyflying
addition,
is
pattern
by,
their
subsonic
one
entering
to
Now
elements
longitudinal
increases:
bounded
cases
fan.
aircraft.
pattern
flow
characteristic
the
very
a
with
occurs
coefficient
of
to
supersonicallyflying
in
that
passed
return
region
normal
number
has
a
expansion.
the
becomes
surface
cone;
sonic
1.0
a normal
Mach
The of
rapid
surface
region
shock.
the
endeavor
called
the
directly
they
against rocket
fluid
fluid
up"
"pile
a model
elements
of
(as
compressed,
Once
a
pass
flight
"lead"
look
(i.e.,
sound
the
must
region
the
into
atmosphere
travelling
upstream
upstream
body
of
the
further
therefore
or
a body
velocity
the
for
the
edges
As
fluid
regime.
a thin
compression
let
crushed,
number
rocket,
the
elements
undergoing
becomes the
to
leading
flight
sonic
time
in
aside
another
the
"inform"
upstream
The
flight).
to one another.
approaching
is
body
elements
see
to
order
in
it)
of
ground
the
on
observer
an
and
overhead,
passed
already
has
it
after
heard
be
only
the
that
fact
the
air
rocket's
the
by
with
the
can
airplane
supersonic
a
reason
this
For
silence.
perfect
in
flying
be
to
appears
vehicle
the
cone
the
outside
observer
the
near
form
in which
left
void
a relatively
is the speed with which fluid
approaches
cannot
base
associated
is
behavior
over
regions
tube
partial
information
the
in front
to
body
the
fill
to
the
of sound in a fluid
speed
the
and
compressed
manner,
similar
a
In
This
passage.
air is highly
edges
again
expands
can
the
477
sound
exceeded
divergence
it
also
478
one.
Only
our
very
highestperformance
this
feat,
and
then
only
by
the
F100,
F67,

+.__Half round
> 7
ot
r
0.474
3
Oo
02
04
06
——_
   
08
10
of
compressible
fluid
is
too
permit
great
to
supersonic the
drag
case
for
obtain
of
values (9)
finbody
(ogival
or
rounded
nose
Cys)
at
Ms
16
18
20
conical) shapes.
Cp
rises
affected
1.2
and
on
use
drag
in
in
noses, For
to
1.7
and
then
2.0.
The
by
falls
the
in
to
53a,
rocket
test
containing one
comprised
its
as
far
number and
models
subsonic to
the
value
about
configuration its aS
Cp
2.00p,
peaks for
into
sharp having
drag
(denoted
1.27
Cp,
is more at M
1.2
J
2.170ps approaching
rs
[X
,
/
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0.8
——

; 1
04
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t
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04
06
}
08
10
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12

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14
16

{


18
20
™
of
fall
having
of
Cos
/
1.6
calculations.
supersonic
results
rockets
again
roundnosed
a
rockets
sharpnosed
declines
for
transonic the
order
performance
data
c.
principles,
data
times
only
first
experimental
Compressibility: back
transonic
to
at
we
from
of
recourse
tested
and
calculation
have
coefficient
one
revolution
must
model
Figure
of
T
2.07
analysis
the
with body
directly
flight,
based
for
finned
practical
categories:
as M approaches at
the
shown
M = 1.05
severely
a
about
combinations
As
distinct
coefficient
Cp
presents
velocities.
, two
flow
subsonic
formulae
small
associated
coefficients
semiempirical
Hoerner
complexity
mathematical
The
was
14
)
247—
Coefficients
Drag
Supersonic
and
Transonic
of
Determination
Semiempirical
7.3
As
12
”
staging.
and
+—4
ee
0.877
and

enne
4.944
multiple
to
had
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often
must
Co
Cos
highestthrust
Bl4,
types

resort
then
even
and

on
So
powered
available
engines
rocket
model
when
accomplish
can
designs
T
16,—
Mach
exceed
to
rockets,
model
as
such
impulse,
total
limited
]
204—}+—
of
vehicles
small
for
difficult
extremely
is
it
that
means
24
Figure
53:
finned
bodies
behavior both
Variation
of
ogive
described
of the
and in
experimental
of
©
drag
coefficient
revolution. drag
coefficient
halfround
equations behavior
(a):
noses.
(214) to
and
within
with
Mach
Experimentally of
the
(bd): (215) 10%.
rocket
for
determined
pictured,
Analytical
that
number
using
functions
approximate
the
480
2.0.
The
classes
the
difference is
due
shock a
rounded
in
supersonic
cone
to on
be
the
other
front
surface
the
difference
and
also
the
of
of
nose
greater
momentum
urations
have
higher
the
off"
itself.
two
body
revolution.
transfer
the
coefficients
at
in
is
mathematically
drag
coefficients
53a
to
very
not
really Figure
Mach
1.5
high
worth 53a
are
the
are
of model
If
takes
the
with
the
transonic
rocket
of exceeding
construct
precision.
igs
config
and
since
(in
(which in itself Mach
one),
curves
data
the
has
assuming
super
model
rockets it
for Figure is
presented
curves
above
applicability not
been
to
established:
a highperformance
is justified
however,
of
procedure
the
their
configurations
of initially
a
fact,
and
formulae
the
Such
though,
only the highestperformance
capable
there
roundednose
represent
approximate
liberty
configuration
to
on extrapolation)
range
rocket
of
will
trouble,
only
based
possible
which
order
a wide
one
and
neighborhood
As
are
should
by the fact in fact prove
54:
Shock
bluntnosed
.
the
a
Figure
of
patterns, the
velocities. It
ahead
illustrates
associated
shock,
a
forward
shock
observed
like
a rounded
slightly
pattern
of
to
54
noge
shock
nose
its
Figure the
sharp
the
due
at
generateg
a
the
shock
"stands
attached
drag
of
rounded
between
a bluntbased
with
that
the
shock/expansion
base
itself
it
structure
than
in
is
The
by
i.e.,
point
between
shock
produced
the
configuration
nature
the
shock;
point.
is detached:
detached
sonic
from
in
and
shock
is an attached
hand,
in
nose
oblique
a pencil
the two
difference
a sharp
directly
on
the
appreciably
The
shed
and
the
with
flight
extremity
of
fundamental
nose.
suspended
nose,
a
associated
by
appears
to
in behavior between
possible
fans;
bodies
compression
wavy
produces
than
the
of
expansion
dotted
delineate
an attached
detached
patterns
about
sharpnosed
Solid
lines
indicate
revolution.
waves;
lines
nose
and
shock
lines
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expansion
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results of
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or
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482
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8.
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the
in
Section
in
flight
Under
the
assumption
experience reduce
of
a
model
respectively,
ana
the
model
rotations,
does
during
not
its
oscillate,
flight
of
or
(16)
and
(17)
Numerical
Methods
Nonvertical
In
Section
1 closedform, were
2 of
In the
case
the
Digital
Computation
of
Trajectories
this
approximate
sufficiently
for
chapter
we
solutions accurate,
of equations
were
to
the
simple,
(90)
and
able
to
determine
equations and
(91)
general
we
shall
of
motion
for
not
practical
be
coarser
The
generalization to
24
horizontal
of
several
approximate,
iteration,
in
the
machinery.
One
necessarily
accuracy
computation
the
nonvertical
flight
is
a relatively
the
rocket
of
following
Aye.
Ate
of
this
Once
has
its
into
done,
been
form:
at lect) > — mag]
~
= at lew
= fin
concomitant
m (t)
nigh
$]
.
(are?
(94)
Vt
(95)
Drag =
(96)
bya
= aefrcerd  meer
(97)
die
=
kv
.
kur]
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schemes
simple vertical @
theoreticalvelocityincrement
the
a con=
intervals.
time
of
motion
can
either
accept
either
the
intended
course,
of
hand,
in
degradation
the
or
2.4,
by
calculations
be besed
Section
of
those
computing
so must
interval
therefore
will
by
work
to
easy
of
components.
or
(93)
work
of
use
separating
(92)
do
the
Section
part
computer
over
section to
automatic
to
wish
of
this
of
iterative
such
amount
advantage
analogous
with
use
who
those
with
solutions
out
siderable
cosa —mitdg [keefulvy
The
for
carry
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techniques
numerical
put
Fit)©  kvx
Vx
m(t) Tt 22
.
components
equations o¢
m(t)dv;FY = F(t) y  metg  kvy
(91)
se.
differential
to
(90)
u
horizontal
are,
that
rigidbody
general
the
sufficleatly
still
and
purposes
a
offer
primarily
% cose ~[k+ef(«)] vx
F(t)
m(t) ays
the
and
mt) $= Fit)
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that
1,
Motion
of
on
closedform,
any
reveal
to
failed
hobby
techniques.
Equations
investigation
solution to (90) and (91) which is sufficiently accurate
for to
nand
Extensive
has
researchers
on
vertical
motion
f Vehicles Launched for
Solutions
Vertical
Differential
The
As
Rocket: :
the
from
Angle
3.1
fortunate.
(even)
NonOscillating
at Any
go
analytical
rockets.
The
3.
shall
we
ag
accurate
designing
of
purpose
the
for
need
ever
that are ag
results
in Section 21 produce
developed
solutions
analytical
closedform,
the
that
conclude
may
we
scatter,
of
matter and
single method
“loop”, appears
566
v= Vik)*+ Cy?
yryrday
(113)
y=yray
(101)
xX =X+4X
(114)
XxX
(102)
y= yraye
(115)
t
(103)
x =X +dX3
(104)
v = Moe
ground
(105)
t
different
At,
=
Ay,
= actual
actual
variables
Likewise,
the
of
Section
2.4
according
to
the
of
(106)
p=
(107)
vertical as
in
velocity
velocity Section
dragfrompriorvelocity
technique
set
are
horizontal
velocity

generalizes
equations
to
presented
OtlFie) > ~mitrg kvy]
y= _ me) ny = ACL y kvil m(t>
(108 )
ay = atly+ 5]
(109)
ax = at[x+5%
(110)
y=ytty
velocity
increment
equal
for
set
to
use
first of
the
meter
to or
equations
initial
angle
method
the
zero
velocity
presence
so of motion
by
that
a fixed
of
assumes
launch
with
the
(116)
OV;
=
(117)
Drag
(118)
0V3
(119)
Ay
=
(120)
ax
= atlv+ 42 ]sin
(121)
vy =V+AV9
(222)
y = y+ ay
(123)
x
(le4)
t=t+at
atl
of
simulate
from
®,
theoretical
of
using
Such
velocity
2.4.
method

nonvertical
below:
the
second
flight
=
=
the
launch
a slightly trajectory
a
scheme
angle
suitable
increments
F(t) — m(t)gcos 8,lo] a g
increment
increment
the
on
is
ee
= dragfree vertical
horizontal
the
necessary
calculations
on
Oy,
increment
"theoretical"
for
is
begin
oe
= dragfree
it
to
ewe
Ox,
order
k(v+ ave)
atl F(t) — mit)+ gees Oo=  k(vedve) 2
at [vs Ml] cos 8
=X+AX
eee
other
= t+at
=t+At
ee
the
rod
X+AX%
a
and
In
=
a
where
x
Se
ee
(100)
(112)
:
XK + ax
——
ax = at[x+
H=
ee em e.
(99)
(111)
OT 8 ee re wee a en =
ay = atly+ S22]
—
(98)
568
while
the
launch
phase
using
the
method
of drag
a nonvertical
from
prior
Aves
(126)
oy = atlv+ SY ]cosQe
(127)
AX
=
(128)
v=
v+Av
(129)
y=
t=t+at
schemes to
the
for
second.
The
above do
not,
2:3 The
with
the
nonvertical
herein
can
phase
be
of
requires
thrust
.001 and
and
F7
are
30°
at
smaller
0.1
or
is
same
as
at
to
Model
paths.
the
interval may
be
used
used use
with in
discussion
Rocket
Due
every
to
vast
to account
parameter
were
considered;
the
restricted
interest
i.e.,
quantity
for a full
relevant to
to
of
the
range
cases
B14,
data
BA,
that
would
of variation
nonvertical
certain
or illustrative
decision
to
under
the
launch
of
same
in
of
flight,
deemed
in
the
to
be
valae in practical
model
within
Three
cases
2s
the
most
the
rules
were
of
then
rocket,
and
one
with either
Section here.
Section
engine
the of 2.4;
equal
m °
to
corresponding
a body
with
second, third,
»
mended
tube
of
model
a value
Code
launch
the
angle
deviation
of
the
of
initial
condition
the
Mational laws
angle
@,
governing
of
for
each
engine:
one
first
case,
engine
alone
and
of
about
liftoff
weight
Case
of
the grams
FLOO
the
parameter
k
in
sealevel
air
used.
and a value
manufacturer's
engine,
(0.453
Kg),
woere
tne
this
being
the
Por
for
40
the
recon
maximum
of k approximately
case were used.
mass
initial
drag
0.3
to the engine were
considered
rocket
an a
repre
a typical
one representing
the
to
30°
of nonverticality
worstperfoming
coefficient
equal
from
hobby.
rocket,
the
glovefit
of my
of the first
453
a drag
burningphase
both quantities were slightly more tnan doubled;
that as
of
all
initial
by most
The
computed
For
engine.
mass
the
to
as
the
representing
that
for
well
severe
the
possible
an
Safety
rocketry.
bestperforming
the
of
by the
the
with
to perform
this being the greatest
Rocketry,
of
made
assumption
senting
representative
five
and
the
permitted
represents
possible
first
vertical,
practice
thus
was
Again,
Trajectories
discussed
‘the
than
for
Computing
vehicles
thrust
be
described
a number
the
burnout.
second
further
methods
calculate
flight
those
any
or
recommended
functions
both
multistaged
altering
time
0.01
numerical
rocket
by
to
staging
require
to
the
solutions,
extended
second
the
of the
employed
2.5 were
generated
particular
flight
a
mass
therefore,
model
each
Section
calculations
vertical
Examples of Nonvertical been
be
Association
while
second
to
in and
vertical
accuracy;
methods
have
of
appropriately
good
FLOO,
The
case
method
method,
has
p4,
used
rocketry
coasting
latter
types
calculations
presented
functions
first
they
as
=X +AX
the
former
the
rePresenteg
"
(131)
the
be
y+hy
x
mass
velocity
may
at [v+ 4¥] sin®,
(130)
and
flight
m(t)gcos@okv2] th
at[ F(t)
(125)
As was
32
of
56
times
An exception was made in the maximum the
value
maximum
of
Mm, m,
legal
was
taken
liftoff
570
through
14.
on
severe
burn
flight
considerable
burningphase
burning
of
a@ very or
risky
the
in fact,
ground
rockets;
results
in
You
should
note,
that
trajectories
sd
have
sobw
the
predicted,
recovery been
given
time
the
ground
applies
to
The
burning safety
as
some
engines
The the is
and
no
severe
of
in
into
called
10
with
ejection
from to
vertical
models
trajectory
to
a term which describes
Xp
is
23 m,
burnout
30°
launch
my = +140
14,
time
the
is
0.35
predicted
or
kg,
activation,
Kk = .002 ke/m; second.
impact in
on
The
the
seconds,
altitude
x» =3m,
Yp
18 40 m.
yys5m.
time
of
flight
ground
if
there
are
marked
sodels
m) = .020 kg, k = .00005 kg/m;
my = +050 kg, k = 00012 kg/m; x, = 9m, yy 215m.
“Dower
angle;
on
apex,
were the
Curve
Curve (c)s The engine
and
no
(b):
the
recovery
time
burning of
system
curves.
require
rule or
strike
which not.
long
of
its
angle
by models
known
range
Ourve (a):
for
for any
not
using
because
purnout
trajectories
predict
does a
Nonvertical
launch
activation

10:
using Type Bl4 engine.
the
a
through
impact
rocket
flight
to @ phenomenon
"gravity turn",
a dive
considerations
discussion
of
Figure
reader
whether
change
of
the
the
Figure
case
seconds
system
9secong
mediun
in
activates
flight
phase
until
safety
system

fe)
launch of
parachute
enable
that
and
cases
weight
Figures
Times
further
in
100
to develop
thrust
colloquially
were
such
seen
200
n>
engines,
a nonvertical
continued
flights,
burning due
and
say,
in nonvertical
low
behavior
recovery
to
time
rocket
bears
makes
drag
to
recovery
model
during
rocketry as the
at
The
E
performance
are
particularly
there
Needless
the
paths
examining
been
rocket
implications.
long burning
when
if
a delay
behavior
experienced
engine
high
Figureg
shortburningtime
flight
commonly
have
the
the
before
engines
altitude
curves
of
all
and
on
delay.
selection
behavior
activation.
of
the
of a nonvertd oa)
system
marked
position
influence
catastrophic
prang".
other
the
the
more
is
expect,
indeed,
power,
ground
in
the
F7
under
all
displayed
curvature.
type
proposition
heavyweight
14,
are
involving
increases,
3001
States.
calculations
for cases
time
time
United
As one would
subsequent
is least As the
of the
the
using
professional
the following
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results
in
wo
The launch
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te
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400
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Xp = 55 Ws Vp
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The
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Yp
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m,
(b):
Curve
m.
(c):
mg
engine
=
burning
kg,
time
= +00012 kg/n;
k
kg,
050
k = is
.002
l. 20
kg/m;
sec.
13:
using
Nonvertical
Type
FLOO
trajectories
engine.
Curve
Xp = Ol mM, Yp = 103m.
(a):
Yp = 47m.
Curve (c):
Xp
Yp
The
13m,
=
Ql
m.
engine
for
30°
launch
angle;
models
kg/m;
my = .230 kg, k = 0003 kg/m; my = .453 kg, k = 0045 kg/m; burning
time
is
0.50
sec.
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Nonvertical
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trajectories
Ourve
Oo
800
(a):
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for
=
30°
032
kg,
Figure
14:
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using
Type
angle;
launch
k =
fe)
models
800
=
236
m,
yp
=
342
m.
Curve
(b):
My
=
065
kg,
k
= .00017 kg/m
Xp
Xp
= 133
m,
yp
= 160
m.
Ourve
(c):
My
=
+125
ke,
k
= .0027 ke/m;
Xp
=
burning
time
is
2.90
Xp = 696 m, yp = 3ll m.
57M,
Yp
= 38m.
The
engine
sec.
the
1050
model
liftoff.
m,
yp
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(m)
4403 > 2000 1600
= 1079
. 1200
trajectories for 30° launch angle; Curve
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=
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Nonvertical F7
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(a):
on the ground
engine
pe
E
4
00
NE
4
Oe
@ _
er
1200
m, = .110 xg, x = .00012
Curve (bd):
xp = 28M, =
1000
ee
pigure
800
ee em
modelg
= 00005 kg/m;
k
kg,
100
angle;
launch
30°
021
=
Wo
(a):
Ourve
m.
for
trajectories
600 (m)
a
= 131
x
ae
yp
400
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200
RON eR REME
engine.
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ere
B4
600
500
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Mo
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230
kg,
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models kg/m;
0003
kg/m;
Mg = +300 kg, k = .0045 «g/m;
under power 6.84 is 9.00
sec.
seconds
after
wena.
Type
using Xp
x
400
300
Nonvertical
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longburning
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and
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thus
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continue.
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ee
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same is true of
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577
576
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rockets
model
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Sporting
N.A.R.
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dangers
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free
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those
portions
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burnout
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familiar
calculation
the
for
equetions
are
who
readers
Those
curves
points
zero.
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and
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Figures
those
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trajectory
of
difference
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through
lying
10
would
parabola
 which
igs to
14
result
a dragfree,
neglecting the Earth's curvature,
of an inverted
drag
under
motion
projectile
of
physics
elementary
the
with
will say
if
k
were
constantmass
be in the
that,
if there
were no drag, the portions of the trajectories above the burnout points would be symmetrical about a verti cal line drawn through
the flight apex.
is
lost;
apex,
the
giving
In cases of nonzero drag, however,
trajectory
falls
the
path
flight
orf
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more
steeply
@ppearance
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flight.
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or
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over",
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later
stages
of
A trajectory computed for finite drag will algo lie
entirely below (or inside") the zerodrag flight path.
thereby

launcher
gravityturning
and
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to
the
reader
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turn.
gravity
disastrous
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a light
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@,
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creating
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as
soon
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upwind,
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engines
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flight
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intended
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is attempted,
launch
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stability
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often
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Te
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general
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precisely given
slight are
the
wind
Impulsive
and
propulsive
to
uncertainties
vehicle
forcing
due
precise
nature
in flight,
a detailed
method
the maximum
altitude
attained
general
forcing
functions
for
as
with
any
for
would
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will]
to
such
a
of
of
modeler
to
4.1
solidparticle
The
to
of
motion
inereases
in
drag
we
shall
and
however,
to
ejection, Because
of
the
vertical
important
a far
more
of
this
forcing set
with
Perturbation
of
which
of vertical
motion;
Chanter
2 4s valid
Blven
dynamic
analysis
equation
(16)
due
section
to
the
1s not the
formation
Jt is exact
it is an approximation. only
provided
that
does not move laterally tea much duvine any
C.G.
disturbance.
an oscillating
of
led
to
wee
First,
(16) is in order.
of motion
eauation
this
in
equations.
these
of
assumptions
the
of
Later
vehicle.
due
thrust
effective
of
reduction
solution
terms
perturbation
included
the
differential
general
the
down
In
actuality,
the
momentary
ee
for
Se
for
encountered
Motion
eauation
the model's
is
of
differential
of completel¥
Equations
that
precise
Tt
basis
combination
remember
The
useless.
some
to
encountered
to
the
and
the
review
a
which
yawing
and
attempt
of the disturbances subjected
disturbances
from
wrote
we
1.2
equations
any
calculation
the
out
which is to say that any possible
Differential
Section
In
in
due
form
carried
Terms
pitching
forcing
which
be
disturbances.
standard
to
of
will
be
me
synthesized
know
departure
precision.
can
of this
a
different
encounter
the
function
of
solutions
rotative
flight,
be
solutions
functions
rocket
of
moment
forcing
rigidbody
section will
the
jn model
infinite
oscillations
from
possible
completely
have
standard
the
SRA ee
calculation
of
of
disturbances.
See
to
launch
by a rocket be

virtually
cause
change
his
in the nature
the
altitudes
possible
predict
of
may
moreover
malfunctions
impossible
of these
the
conditions
different
not
profile
detailed
because
which
which
therefore
effects,
also
at
directions,
flight.
staging
functions
winds
the
under
disturbances
The
is
for
performance
forcing
vehicle.
speeds
equations
impossible
set
all
in
capability.
wellnigh
a
Derivations
affected by inflicnt
ei
altitude
and
to
that
desioy
Se,
present

of
vurvose
the
Eh
rocket's
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altitude
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discussed
motion
directed.
assumption
a
this
facilitating
aes
to
be
of
in
to
least
contribute
Ee
attempt
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now
1s
are
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and
co
would
are
It
which
structural
vehicle
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ek
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we
rigidbody
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of
effects
attack,
the
in
the
in
t,)
for
aiscussion
nearly the
of
to
the
except for relatively short times  on the order of a few tenths
on
about
throughout
important
representation
The
Then
or moments,
the step forcing is due
to slight aerodynamic asymmetries, the strength of the perturbing moments Will vary a8 the square of the airspeed. Where the step
produces
torques,
rockets where
_
tn cases
lowering
My the
j 
arises from ® uniform horizontal wind the magnitudes of M, and
forcing
step
to
a model
into
t,
time
some
at
of
response
the
compute
the
that
assume
axig
yaw
the
and
Response
arises
suddenly
that
Step to
order
In
Veloot ty
conditions.
initial
setting
pitch
the
both
account
into
taken
I have
that
angular
of completeness
sake
for the
it is merely
one axis only;
about
and
consider
to
reason
horizontal despite
wi the
it
is
nd
effects
admitted
. fa > nee
assume
you
dis placement
angular
initial
nonzero
if
interpreted
easily
more
be
will
behavior
response
the
and
~589
590
response
be
be
zero;
to
considered
not
or
whether
true
is
this
disturbing
the
of
one
if
lost
will
igs
rocket
the
quiescent
prior
the
of
in

results
equations any
will
overall
and fy(t) are both effectively infinite at the
the
Again,
of
Ay
(160b)
Wx
and
Hy
pitch
t,
after state
impulse,
as
a homogeneous
liftoff specified
discussed
response
specialized
we
set
in
for
of
by
to
impulsive
again
assume
equations
initial
be
the (154).
the
hypothesized model
wees
Hy
already
been
taken
into
and
Hy

including
zero
for
as long as these values do not to
exceed
will
then
to
greater
than
to
performance
conditions:
0.2
radian
during
either
cause
the
the
subsequent
a wide
and
Hy
axes,
and
I, the
components
of
the
impulse
about
the
yaw
respectively,
of the
unique
nature
idealization
disturbing moment
of impulsive
involving
applied
the
forcing assumption
for an infinitesimal
rockets,
high
designing
specialized
two
rocket about Sinusoidal forcing of a nonrolling one
are
in
by analyzing
obtained
be
can
could
relating
information
the
all
interest
of
is
disturbances
and nonrolling
rolling
that
find
that
forcing
sinusoidal
of both
cases will
of
Poreing
casesé
=
=
variety
for
models
to Sinusoidal
ee
Response
rocketeer
sinusoidal
2,
Chapter
time
Although
forcing
(a)
a mathematical
infinite
have
(the
impulse
of an
time period)»
only,
axis
its
varies
in which as
frequency
the
square
varies
of
amplitude
the
of
en ee
response
=
wy
Because is
for
attack
forcing
where
this
;
acceptable
4.3.4
a model's
rotational
following
(160a)
(160c)
of
(160).
values
be
angle
Response
instant
the
produce the
fx(t) but
by setting the yaw and pitch
tine:
rem nen
at
effect
Impulse
to determine
occurring
with
to»
for ai}
=0
Fy (t)
=
be computed
zero
ee ee
In order
to
time
must
response. 4.3.3
be
f(t)
one
is
torques
spinning.
The
(161)
to
ee ewer
the
response
functions
regarding
information
no
that
concluded
be
may
it
4.3.1
Section
impulse
account
of
that
to
similar
reasoning
By
guaranteed.
be
to
is
analyses
dynami ¢
2's
an
forcing
actually,
bounds,
Chapter
of
validity
the
if
unavoidable
is
work
the
but
smal)
attack
of
angle
upper
such
establish
to
experimentation
numerical
some
be
the
the airspeed
directly
the
with
airspeed;
and
(b)
ling rocket in which Sinusoidal forcing of @ rol
the roll
exist
sinusoidal rate
about
nature
itself,
the
so
pitch
of that
and
the
1s
forcing
axes
to
the
noments
sinusoidal
yaw
due
of
angular om
0.2
of
they
It admittedly take,
response.
the
during
time
any
at
radian
that
an
exceed
not
will
rocket
the
sure
make
to
enough
only
subject
arbitrarily,
My
restri ection
the
to
My ang
choose
you may
calculations
response
step
In making
591
~—an
~~
592
fy (t) = Ag Sin wet
(62a) (162b)
fy (t) =
We
frequency
rocket's
the
to
yaw
(1632)
Ag = Aov2
(163)
We
per
of
as
the
perturbation
constants
values
of
of
Ap
proportionality
and
Oe at
an
which
airspeed
may
of
be
one
meter
the
second
type
exhibits
forcing
functions
(164)
fy (t) = Ag cos wat in model
means,
to velocity
tionality
of
since
the
1s proportional
the
inertial
to
strength of the
square
perturbation
the
of
aero
the
air
1s proportional
to
directly with
which in turn varies
rate,
roll
of the
square Both
types
most
=
Wx
started
arise as
the
immediately
behavior
actual
of cases
a
such
of interest.
at t = O with
rotationally
conditions:
Xx = Ky =O
(166b)
be
to
flight,
the
vast majority
therefore
should
assumed
represents
accurately
in the
forcing
are
throughout
persist
to
initial
(1662)
perturbations
model
sinusoidal
quiescent
of
and
liftoff
should
It
of (t) = Ag sin wzt
dynamic
that
causes,
t
=0
= (@)
Wy
2
of
inertial
airspeed.
form
spin
and
Calculations
(64a)
Since
aynamic
to
mathematical
second.
the
or
of
are
A disturbance of
and
applies whether the disturbance 1s due to aero
aynamic
upon
= Wov a,
(2008)
the
rockets roll
for most
scheme
used
is
rate
differ
and
in
3.2.4
sinusoidal
almost
may
somewhat
be
invariably
considered
cases
of interest.
for
the nonrolling
The
case
induced
linearly
same
can
sort
then
by
aero
proportional
of propor
be
used
be
noted
from
that
forcing
the
that
the
analyses
calculations of
we are considering
Chapter
presented 2's
the complete
here
Sections
TREY
thought
and
gquation
3.1.4
response
to
rather than the steady~state response only;
— eee
A,
WaoV
=
our numerical
calculations will therefore pick up socalled he discussion transients which are not considered in t
starting of
Chapter
however,
2.
The
be greatly
basic
character
of the
response
will
not,
altered and it 1s very nearly correct to