Advanced topics in model rocketry 0262020963

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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

0-262-02096-3

model

begun;

and

of

world,

the

to

the

and

rocketry advanced

without

whom

G.

Harry

could model it

continue.

Vellum

America.

Cataloging model

in

Publication

629.47'52

Data

rocketry. Rockets George J.

72-10386

\

AB2484

Stine,

not

have

rocketeers

could

i

ISBN

whom

Carlisle

Opaque

1. Rockets (Aeronautics)--Models. 2. (Aeronautics)--Performance. I. Caporaso, II. Bengen, William P. III. Title.

TL844 M36

H.

Inc. of

Mandell, Gordon K Topics in advanced

Orville

not

FOREWORD

PUBLISHER'S

of

preparation

the

A

form.

book

after

a considerable

pense

of

to

act

of

detailed

is

from level

detail A

the

author's

delay

recognize

of

the

in

and

Perhaps

to

table

typeset of

been It

photographed

is

of

books

contents

is

consistency issued

a

to

edited

comprehensibility

and

standard

the in

has

book

print

may

inevitably

hobby

satisfactory

become

of

under

minor our

editorial

imprint.

included.

of

of

MIT

Press

rate

of

with

has

rocketry.

the

an

until

they

may

treatment

performance.

difficulty

of

fail

advanced

mathematics,

been,

model

and

complexity

physics,

the

and

designer's

recent

years,

the

ask,

should

model

one

examines

the

rocketry

and

communication

by

has

(of

projects

within,

industry; various

nothing

our

kits, the

this

which of,

supplies,

proliferation of

the

challenging

time

our

retention

improved

due

to

book

The

and

or

interesting

and

miscon-

widespread

that

Since

avocation.

Sections

a

measurably

content

technical

technically-oriented rocket

had

attention.

modelers

of

because

older,

its

of

state

twelve-

the

1969

In

most

losing

was

hobby

if years.

several

past

rocketry

their

merit

the

self-evident

participants

advanced

increasing

the

rocket

model

sponsored

of

this

people

and

sanctuary

Why,

becomes

over

programs

model

of,

stability

higher

that

these

value

intrusion

community,

technical?

to of

Perhaps

rocket

intuition

that

enough

the

modeling

the

that

hobby

ception

model

or

upon

better-educated The

rocket

analysis

answer

year-old

the

for,

associated

so

the

of

model

The

not

need

resent

engineering

directly

though

the

they

of

justification.

subjects

ex-

currency

members

require

to

that

it

may

unassisted

typescript.

completeness

present

detailed

this

of

text

necessarily

to

work

some

either

time

The

To

specialized

publication

composition

and so

or

though

in

affected.

The

of

editing

publication

prevent

content

text

publication

its

all.

at

not

or

delay

formal

to

transition

the

and

significant

of

number

large

make

manuscripts

work

a monographic

between

gap

time

the

close

to

is

format

this

of

aim

The

PREFACE

is

one)

improving

aimed the

introduction

literature of

National

annual

a

by

number

at

lines of

of

more

the

conventions

Association

e

-vilii-

"senior

rocketry's

projects,

making

by

,

our

college-level

model

rocketeers

our

hobby

a

of

limits

jas

been

chusetts

M.

Institute

in

part,

analytical

philosophy

state

frontiers

of

Though by

are of

different

unified model

dynamic

of

our

the

of

the

so

1965

work

of

analysis

drag,

analyses

in

and

of

flight

trajectory

Chapter

and

the

2 are

of

altitude

have of

and

logical

in

The

of

the

results

Chapter

4

to

of

provide

enable

the

drag

an

a working

as

of

Harry

Stine,

tended

to some

fully

sound

current

analyses

of

from

state

of

the

of

of

Chapters

experienced

knowledge

optimized

Mark

gaps

precisely

It

what

factor

2

modeler

algebra

standpoint

our

knowledge

before

due

to

present

jet

two.

If

4,

optical

Malewicki

of

Chapter

theory

must

await

exhaust

that

lower

influence found

altitude

in

has

had

this

area.

preface

to

this

volume

of

and

would

gases some

the

is

be

and

can

on

by

as

the

as

much

as

as

some

magnitude

experimental

complete

suggest

great

become

relative

a

determined

its

reliability the

however,

arguments

jet

G.

their

have

estimating

little

and

been

coefficient

predictions

systems

in

and

for

accuracy

tracking

are,

performance

instance,

as

generally

design,

could

be

there

the

Institute

have

drag

for

in

such

of

rocket

the

3;

verified

conducted

Douglas

experimentation

No

Massachusetts

the

--

unfortunately,

deficiencies

at

coefficient

must

accurate

tests

and

yet,

the

drag

a method more

Chapter

effect

tunnel

researchers

flight not

experimentally

various

current

of

been

by

results

the

has

exhaust

of

taken

the

theory

have

in wind

Mercer,

in

2

Laboratory

Data

elimination.

the

author

confirm

still

Chapter

Projects

Technology.

suspect,

the

and

performance

chapter's

written

dynamic

intuition

the

oscillations

treatments

will

nearly

results

that

a

and

that

of

the

as

that

static

with

yawing

permits.

hobby.

analysis

and

regults

models

current

whole

pitching

The

criteria

rocket

thrusting-phase

been

of

combined

physical

the

rocket

purpose

viewpoints

used

the

Gerald resulting

The

model

volume

analysis.

Dr.

represents

the

a consistent from

--

establishes

this

research

Mercer.

design

and

of

structure

as to present

Mark

a

over

Barrowman,

S.

within

chapters the

and

1970

Massa-

the

of

Society

through

James

art

members

Rocket

rocket

designer's

various

authors,

from

model

by

Model

Malewicki,

J.

engineering

rockets

stability,

upon

Douglas

performed

Technology

of

extending

period

Gregorek,

current

from

taken

five-year founded,

research

good

design

of

material

the

of

Most

then

Aeronautical

for

opportunity

of

goldmine

contributions.

engineering

and

scientific

veritable

the

within

found

who

effects

performance.

are

to

by

existing

the

provide

The

several

by

written

were

herein

presented

treatments

The

3 to

1967.

in

racing

slot-car

of

hobby

sister

befell

that

cycle

fad

“poom-and-bust"

of

kind

the

our suffering

the

of

cross-section

broader

against

guard

helped

have

public,

4

with

to

interest

of

activity

an

rocketry

model

and

and

programs

These

hobby.

the

of

continuity

and

stability

and

problem",

of

altitude

Chapter

the

insure

help

to

by thereby

on

del mode

solve

help

to

part

their

done

have

all

--

exclusively

a

monthly

independent

first

estimates

ho bby

rocket

the

serving

publication

the

magazine,

Rocketry

Model

of

establishment

the

Rocketry;

of

-1x-

a

reality. verification

inherent paucity

without

in of

a

If book page less than 290: PDFPage = (BookPage/2) + 9 If book page greater than 290: PDFPage = (BookPage/2) + 10 -x-

has

not

had

no

We

we

for be

children

can

herein which It

is

will our

and

be

put

possess hope,

of

that

model

rocketry

is

not

but

an

activity

adolescents,

rewarding

even

for

will

find

that

You

adults. to

good

good

and

stability

moreover,

that

for further advanced

for

further

refinement

some

of

research of

those

in

you

in the

theories

J. K.

Cambridge,

May, 1971"

.*

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 Rigid-Body

high

information

the

Point-Mass

12

projects

just

J.

OF MODEL

6

1.1

Thrust

well-educated,

use

practical

a basis the

similar

and

this

have

means

early

and

challenging

proficient

you

show

to

we

level,

of the Problem

2.1

late

a

at

written

material

Definition

and

inclined

technically

By

you.

for

are

undergraduate

college

or early

hope

you

hand,

assimilating

of

volume

this

2.

other

the

on

capable

school

larger

Company) .-

Engineering If,

1.

1

DYNAMICS

Gordon

Centuri

the

and

Industries

TO THE

of

the

of

some

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 Torsion-Wire

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

Steady-State

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

Flat-Plate

4.2

Boundary-Layer Separation

Pressure Drag of the Forebody and Fins

4.3.3

Launch lug Drag

Skin-Friction

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

Zero-Lift Drag Coefficient of the Fins

471

6.1.2

Zero-Lift

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

Zero-Lift

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

Skin-Friction 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

Fin-Body Interference Drag at Angle of Attack

5.4

3.5.2

‘Three-Dimensional

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

5-2.1

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 (Skin-Friction)

“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 Non-Oscillating Vertically 520

2.2

The

Specialized

Rocket:

Solutions

Differential

2.1.1

Fehskens-Malewicki

2.1.2

Caporaso—Bengen

2.1.3

Caporaso-Riccati

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 Non-Oscillating

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

Caporaso-Bengen

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

Non-Os 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

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physicist

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16)

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jn

Osborne which

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3.

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than

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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

critically-damped

(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

impulse-response

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

critically-damped

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 ~ 14-7)

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

impulse-response

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

critically-damped

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

critically-damped

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

jmpulse-resp

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

statically-stable 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

statically-stable 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,

roll-coupled

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

positively-stable

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

x-direction.

the

in

sufficiently

moreover,

negative

by as

is

their

symmetry, the

as

constant

consider

in

fluid-flow

443,53,

region

in

conservation

limit

ek

h

drag

force the

D

be

control

(U.?-ut-U2- 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

skin-friction

a flat

(an

corner

forces

exactly

y=o

in the

A,B),

wall

in

its

boundary-layer

the

consider

used

by

of the

@ to

surface

Momentum

§Un?@= $) u(Un-u)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

Cross-Sectio

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

skin-friction

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

"1/7th-power

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)" [I-A ly (* (E)hay-( (Bray = 25-75

= 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

poundary-layer

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.

MThree-Dimensional Qorrections to the Flat-Plate

Approximate

laminar 4)

to find

coefficients for use in the method described

Skin-Friction

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

skin-friction skin-friction

of

(15):

to

of

1s plotted

B_

turbulent

order

rockets

- (Cé) iam | coefficient

(Ry)%

3 x 105 1050

quantity

6

- (Ce)iam_|

skin-friction

(Ry) 1.328

below

Rerit :

The

(100)

0-074

=

values

listed

assume

Letting

(86)

are

making

(99)

are

(C#)iam

the

or

we

(1020)

In

|

friction.

skin-friction

(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 three-dimensionality on the values of the skin-

coefficients

approximate

to

method

correct

application

derived

for the flat

described

in

two-dimensional

to

plate.

Reference

The 9,

results

which

can

skin-friction

coefficients

surfaces,

presented

three-dimensional

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

boundary-layer

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 two-dimensional

approximately

by

in

plate

skin-

having

(9)

= aie

example,

we

determined

laminar

flow

1.206

x 109.

For

ratio

at

over

10

=

Ibex lo7*

skin-friction

the

a

a flat

a model

of

(Crom

adjusted

the

that

for

to-diameter

The

to

flow

same

skin-friction

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 skin-friction

-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 length-to-diameter the

ratio

skin-friction

coefficient

-3

Now

Cy = 2 Ce the

to

the

increase

cylinder

equation

flat-plate for

the

the

value

--

laminar

magnitudes

on

(105)

accounting

for

a bit

than

case.

will

be

less

In

used

Section

to

constant-diameter 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

skin-friction

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

skin-friction

"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)

turbulent-flow

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.

skin-friction

formed.

most

model

Boundary-Layer

The

and

pressure

two

first

of

of

drag.

the

account

influence

mechaniem

1s essential

expression

presented,

base

information

the

empirically-derived

drag

the

the

there 1s very little quantitative

well-constructed 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:

boundary-layer

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,

boundary-layer

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

skin-friction

beyond

illustrate

body,

gradient)

boundary

layer

to follow the contour of the body surface

may be unable & certain

pressure

adverse

so-called

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=1-4sin’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

boundary-layer

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

boundary-layer

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

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of Javelin with square edged

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nose

32:

shape

Variation

with

in

fineness

the

ratio

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of

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-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

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the

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differences

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31).

ratio

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nosecone

in Figure

coefficient

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32,

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in drag among the five

taken

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ts required.

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compromise

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culminating Since

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prevent

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than

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a

One

of

rather

thickness-to-chord

rocketry

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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

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tapered

model

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and/or trailing

1s considerable

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off,

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to the flow result

streamlined

squared

analogous

to

flow from

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edges

Drag

drag due to the fins must

indicate

to

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pressure the

data

Adequate

length-to-diameter

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ratio

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Fin Preseure

the fin surfaces are generally parallel

direction,

flow.

guide

tests.

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"typical"

a length-to-diameter

tangent

with

BC-74)

furthermore

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(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,

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in

increaseq,

a commercially~availapie

BC-78,

a smooth

of

Company,

shape

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seven

bluntness

essential

others

as

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(Figure the

1s a slowly.

all

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BC-70,

the

representative

at

Engineering

exhibiting

similar

tested

Javelin,

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shapes

decreasing

additional the

Centuri

model

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ae

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catalog

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rocket

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number,

wind-tunnel

configurations

drag

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by the

demonstrated that

Reynolds

Mercer

in

an

of

(r/h)crit

durability

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analogy aftersurface

"knife-edge").

requirements

trailing edge 1s very easily damaged.

here,

The

of a well-streamlined model rocket fin is illustrated

~389-

-388-

drag

its

in

rockets

in

this

due

to

often

on

this

that

of

Mercer,

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rockets

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lug

to

blunt-finned

the

drag

to

that

much

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the

parasitic

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body

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by

launch note,

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lug

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to suggest an average drag

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Cy

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the

cannot

increment

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of Mercer's

0.21,

lug

diameters

rockets

model

to

basis

equal it

coefficient increment

the presence

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are

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of body

of Mercer's

that

to

is identical

diameter

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models

ratio

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to

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formula

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body

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this

launch

tube

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than

the

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1.93

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in

coefficient

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Denoting

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that indicates

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of

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launch

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the

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to

the

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size has a diameter of about 0.40

body

recalled,

tube), of

section

23

that

centimeters.

for

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note

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4t

available

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reference

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boundary

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from

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launch

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version

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to

drag

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protrude

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the

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the

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important

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values

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from

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layer

agrees

boundary-layer

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again,

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culprit,

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to

configurations lug

giameter

area.

(7).

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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

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near

about

by

rocket

rocket.

a lugless

# 0.25

aifferent

streamlined-fin

Cp of the

the

test

Javelin

estimates

with

well

location:

for

value

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(presumed

his

of

version

lug

small

that

showed

research

increased

of S°p

results

variations

large

Mercer's

tube)

body

the

of

5.

that

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drag

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a launch

of

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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

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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 body-mounted

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

body-mounted

lug

on a configuration

wind

Wichita

tunnel State

used

lower

Mercer,

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results

directly.

it

this

following

due

to

(120)

however,

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lug

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formula

mounted

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of

Section

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to

cannot

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low-speed

level

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than

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to

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the

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2

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angle

the

placement

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80

is

behind

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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

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of

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rocketry.

of model

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lug

lug and has incorporated

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attack),

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expressions size,

to

colloquialism the

launch

Mechanisms

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launch

a matter

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joint:

device.

after

Virginia,

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wholly

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4.4

increment

of

commercially-available

drag

lug

to

been

pop-off

thoroughly

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present

coefficient

of

the

problem

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soon

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In of

launching

retracted

several

their

nature

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in

at had

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have

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AV ug/4r-

have

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development

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may

accurate

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than

for

ratio

Kansas)

6,

of

more

Mercer

rod

means

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Alexandria,

of

probably

much

by

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extremely

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methods

due

Wichita,

necessarily the

those

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(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

fin-body

with

about

according

increment

at

tested

can

lug

about

of

Cp

overall

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determined

Malewicki

effective

closed-breech

model's

launch

the

decrease

substantially

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body

its

fins

most

to eliminate

that the

between

joints

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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

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now

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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

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from

boundary

layer

separates

equations

layer

separation

A

(c)

well for

to

researched

presented last

rocket

separates

known

and

as the

as there

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when

fins

phases

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the the

beyond

hobby

empirical

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unable

expected

engine

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to

professional

take

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"dead-air"

the

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level drag

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about

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igs actually

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decrease

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R

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pressure" the

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with

tne

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finless

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€

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Flow

33:

Figure

poattail.-

the

disturbs

a decrease

in

peculiar

The

is

drag

depicts

the

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phenomena

Base 33

on

flow.

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all

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problems,

two

subsonic

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believed base

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during

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valid

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accurately

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presence

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(b)

can

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arise

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last

rocket

Complications

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The

theory

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in

D

=

rocket.

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and

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pressure

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then effects

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eee ~394-

-39-

“pump"

it away.

the

(59),

coefficient

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Cg, =

as

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nature

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relationship

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than

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Figure the

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33)

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the

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than

the

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recovery

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the

models

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would

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roughness

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121).

after

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dead-air

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As

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equation

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34.

in

along

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built

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a

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(9)

the external

is somewhat like a jet pump:

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agetermined

Hoerner

as

4 whole,

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of

character

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(16).

point

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dead-air

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f --

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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

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Figure

revolution and

Variation

34:

with

of

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friction

drag

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the

shown

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represent

a

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(126)

SE

=

to

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denoted

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config-

associated

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(127)

a large

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curve

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uration

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D orebod —forgbede 4 Sn

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= a a“

Ce

(126)

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db

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simplifying

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engine

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the

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db (Cy), = (+)

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Cy

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m

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tail

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a3)

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area.

(130) or,

is

[de a (=) ~

De “a Sy

_ ~

D 43

(Co.),=

(129)

boattail

dp/d,

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itself.

VC

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first The

og

technique

danger

angle €

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skin-friction

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the

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in Figure

33)

is

-398-

will

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of

section

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precisely,

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the

rocket

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would

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than

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~

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2.

find

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(170)

ARy = 46.1

= 4b dm

OGIVE

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OSive

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body

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)

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CYL

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Figure

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In

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horizontal

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the

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the

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Solutions

Vertical

Differential

The

As

Rocket: :

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The

3.

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v= Vik)*+ Cy?

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actual

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of

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2.4

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(107)

vertical as

in

velocity

velocity Section

drag-from-prior-velocity

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meter

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Oy,

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oe

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schemes to

the

for

second.

The

above do

not,

2:3 The

with

the

nonvertical

herein

can

phase

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of

requires

thrust

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restricted

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of

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BA,

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nonvertical

certain

or illustrative

decision

to

under

the

launch

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same

in

of

flight,

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in

the

to

be

valae in practical

model

within

Three

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the

most

the

rules

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then

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Section here.

Section

engine

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weight

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parameter

k

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engine,

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Por

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the

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case were used.

mass

initial

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to the engine were

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rocket

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the

to

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worst-perfoming

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both quantities were slightly more tnan doubled;

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to

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by the

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Again,

Trajectories

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than

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Computing

vehicles

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those

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functions

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rocket

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calculations

vertical

Examples of Nonvertical been

be

Association

while

second

to

in and

vertical

accuracy;

methods

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good

FLOO,

The

case

method

method,

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rocketry-

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latter

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As was

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value

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You

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14,

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the

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Non-vertical

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considerations

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Figure

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whether

change

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rocketry as the

at

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Those

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zero.

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those

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the flight apex.

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C.G.

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in

equations.

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Later

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any

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Section

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to

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winds

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is

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on

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produces

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tn cases

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j |

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4.3.1

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Chapter

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In making

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fy (t) = Ag Sin wet

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fy (t) =

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(163)

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second

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tionality

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since

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air-

1s proportional

to

directly with

which in turn varies

rate,

roll

of the

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types

most

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Wx

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arise as

the

immediately

behavior

actual

of cases

a

such

of interest.

at t = O with

rotationally

conditions:

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(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

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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

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rate

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in

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sinusoidal

almost

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be

invariably

considered

cases

of interest.

for

the non-rolling

The

case

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linearly

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can

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then

by

aero-

proportional

of propor-

be

used

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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 so-called 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