GOODYEAR AEROSPACE

GOODYEAR AEROSPACE

~ U A S I - C B - 120261) USERS IIANUAL: DY YALIICS U74-29309 OF THO BODIES COUYECTSD dY A N ELASTIC r E T B E I , S I X EEGBAiLS OF F A i i Z D O n FOREBODY ,192 F I V E DEGREES OF (Goodyear Aerospace Unclas :ore.) 139 p Hc S10,UO

--

SSCL 20K 63/32 16556

CODE IDENT NO. 25500

GOODYEAR AEROSPACE CORPORATION

AKRON 15. OHIO

USERS MANUAL

DYNAMICS O F TWO BODIES CONNECTED

BY AN ELASTIC TETHER - SIX DEGREES OF FREEDOM FOREBODY

AND F I V E DEGREES OF FREEDOM DECELERATOR

(REF. NASA CONTRACT NAS~-29144 s / A ~ )

GEORGE R. DOYLE, JR. &

JAMES W. BURBICK

APRIL 1974

ABSTRACT

One important a s p e c t t o recover ing a body f a l l i n g through the atmosphere, is t o

d e c e l e r a t e and s t a b i l i z e i t . This is u s u a l l y accomplished by means o f a parachute .

The d e s i g n of t h e recovery system n e c e s s i t a t e s a knowledge of t h e dynamics and

loads dur ing parachute deployment and i n f l a t i o n . I n many c a s e s , a p i t c h p lane

a n a l y s i s w i l l p rovide adequate informat ion. However, i f t h e body is i n e g e n e r a l

tumbling motion, i t is necessary t o analyze i t s motion i n +Free dimensions.

T h i s r e p o r t c o n t a i n s t h e equa t ions of m t i o n and a computer program f o r t h e dy-

namics of a s i x degree of freedom body joined t o a f i v e degree of freedom body

by a q u a s i l i n e a r e l a s t i c t e t h e r . The forebody is assumed t o be a completely

g e n e r a l r i g i d body w i t h s i x degrees of freedom; the d e c e l e r a t o r is a l s o assumed t o

be r i g i d , bu t w i t h only f i v e degrees of freedom (symmetric about i t s l o n g i t u d i n a l

a x i s ) . The t e t h e r is represen ted by a s p r i n g and dashpot i n p a r a l l e l , where t h e

s p r i n g c o n s t a n t i s a f u n c t i o n of t e t h e r e longa t ion . Lagrange's equa t ion is used

t o d e r i v e t h e equa t ions of motion w i t h the Lagrange m u l t i p l i e r technique used t o

express the c o n s t r a i n t provided by the t e t h e r . A computer program is included

which provides a time h i s t o r y of the dynamics of both bodies and t h e t e n s i o n i n

t h e t e t h e r .

...

GER- 16047 .* 9 ,

TABLE OF CONTENTS

Abstract

L i s t of Figures

Naecnc la t u r e

Chapter

I Introduct ion

I I Derivation of Equations of Motion

1. Coordinate System

2. Euler Angle Transformation

3. Kinet ic Energy

4. Po t en t i a l Energy

5. Rayleigh's Diss ipa t ion Function

6. Lagrange ' s Equation

7. General Equations of Motion

8. Simplified Equations of Motion

9. Generalized Forces - Aerodynamics

I11 Computer Program

1. Features of Computer Program

2. Input

3. Output

4. Numerical Solut ion

5. P l o t t i n g Routine

6. English t o Metric Conversion

7. Conclusions and Recornendat ions

IV Program L i s t i ng and Sample Computer Run

Page

1

iii

References

>- .f

GER- 16047 k

- M

Figure No.

LIST OF FIGURES

T i t l e - Coordinate Systems

Euler Angle Rotations

Aerodynamics Coordinate System

Input Parameters f ~ r Parachute Added Air Mass

Parameters for Program Calculated Parachute Drag Area Time History

Page

The fol lowing is a l i s t of variables used i n the computer program and i n the

derivat ion of the equations a s discussed i n t h i s report. A br i e f descri , , ion

and associated units are included.

GER- 16047

METRIC ENCLISH FORTRAN STANDARD DESCRIPTION UNITS UNITS

A a Distance along t h e l o n g i t u d i n a l a x i s of -4

AALPDE (8)

AALPPE (16)

AALPME ( 16)

AALPPE (8)

AAM(8)

AAI@ (8)

AERATO

t h e forebody (5) from t h e i n t e r s e c t i o n

of t h e body axes t o t h e te ther- torebody

confluence p o i n t , p o s i t i v e towards the

nose (m) f t

Dummy v a r i a b l e s use3 t o express incre- (mlsec) f t l s e c

a e n t a l v e l o c i t i e s of t h e forebody i n

t h e Runge-Kutta i n t e g r a t i o n

An a r r a y of e i g h t va rLab les s i g n i f y i n g

angle-of-a t tack of t h e forebody used

wi th damping c o e f f i c i e n t s

An a r r a y of s i x t e e n - a r i a b l e s s i g n i f y i n g

angle-of-a t tack of t h e forebody used

wi th fo rce c o e f f i c i e n t s

An a r r a y of s i x t e e n v a r i a b l e s s i g n i f y i n g

angle-of - a t tack of the forebody used

wi th moment c o e f f i c i e n t s

An a r r a y of e i g h t v a r i a b l e s s i g n i f y i n g

angles of a t t a c k of t h e d e c e l e r a t o r


Mach number of t h e forebody used wi th

f o r c e and moment coe f f i c i e n t s


Mach number of t h e f o r c ~ o d y used wi th

damping coef f i d e n t s


Mach number of t h e d e c e l e r a t o r

Suspension Line AE Rat io

(AERATO = AE/AENy 1

GER- 16047 IY.

t . FORTRAN STANDARD DESCRIPTION - AIPHI

AIPHID

AJALPD

A JALPF

A JALPM

AKAM

ALPE 0(

Number of elements i n PPHIE a r r a y

( = 2 t o 8)

Number o f elements i n PPHIDE a r r a y

( = 2 t o 8)

Number of elements i n AALPDE a r r a y

( = 8)

Number of elements i n AALPFE a r r a y

( = 8 o r 16)

Number of elements i n AALPME a r r a y

( = 8 o r 16)

Number of elements I n AAM a r r a y

( = 2 t o 8)

Number of elements i n AAMD a r r a y

( = 2 t o 8)

Angle-of-attack of t h e forebody

ALPPE aP Angle-of - a t t a c k of the d e c e l e r a t o r

AM Mach number of t h e forebody

AMAX l Larger o r d i n a t e o f two points on the

I-- -0 M Y - AMAY 1

l o n g i t u d i n a l added mass versus

Do l o g log p l o t

METRIC ENGLISH UNITS UNITS -

kg s l u g

Smaller o r d i n a t e of two po in t s on the

1ongI tudinal added mass ve rsus

Do l o g log p l o t kg s lug

Larger o r d i n a t e of two po in t s on thc

l a t e r a l added mass versus

Do log log ~ l 0 t

-vi-

kg s lug

COODVEAR AEROSPACE C 0 I . 0 1 . 1 6 0 .

GER- 16074 - -. --

I .* r J3zmau STANDARD DESCRIPTION

AMAY 2 Smaller o r d i n a t e o f two p o i n t s on the


METRIC ENGLISH UNITS UNITS

AMP

kP

Do log l o g p l o t

Mach number of t h e d e c e l e r a t o r

Dis tance a long t h e l o n g i t u d i n a l a x i s

of t h e d e c e l e r a t o r (X ) from t h e c.g. pb

t o t h e t e t h f r - d e c e l e r a t o r con£ luence

po in t

Exponent of l o n g i t u d i n a l added mass

equa t ion

(MPAL = RHOOO * BX * DS ** AX)

Exponent of l a t e r a l added mass equa t ion

(WAS = RHOOO * BY * DS * AY)

Dis tance a long the l a t e r a l a x i s of t h e

forebody (Yb) from t h e i n t e r s e c t i o n of

t h e body axes t o t h e te ther- forebody

confluence p o i n t , p o s i t i v e towards t h e

l e f t wiqg

Dummy v a r i a b l e s used t o express inc re -

mental v e l o c i t i e s of tile d e c e l e r a t o r i n

t h t Runge-Kutta i n t e g r a t i o n

C o e f f i c i e n t of l o n g i t u d t n a l added mass

equa t ion (MPAL " RHOOO * BX * DS ** AX)

C o e f f i c i e n t of l a t e r ? ? added mass

eque t ion (MPAS = RHOOO * BY * DS *" AY)

Dis tance a long the v e r t i c a l a x i s of t h e

forebody (Z,,) from the i n t e r s e c t i o n of

t h e body axes t o t h e te ther- forebody

conf luence p o i n t , p o s i t i v e up

kg s l u g

GER- 1604 7 4.P.

$ I METRIC ENGLISH FORTRAN STANDARD DESCRIPTION UNITS UNITS - CA c~ Axial f o r c e c o e f f i c i e n t of forebody

CAP f 'AP

Axial f o r c e c o e f f i c i e n t of d e c e l e r a t o r

CC(3,3) C i j

Elements of t r ans fo rmat ion mat r ix from

i n e r t i a l coord ina tes t o body coord ina tes

of the f orebody

CCAP (8,8)

A t h r e e dimensional a r r a y of v a r i a b l e s

s i g n i f y i n g a x i a l f o r c e c o e f f i c i e n t s

of the forebody corresponding t o

AAM(8) , AALPFE ( 16) , and PPHIE (8)

A two dimensional a r r a y of 64 v a r i a b l e s

s i g n i f y i n g a x i a l f o r c e c o e f f i c i e n t s of t h e

d e c e l e r a t o r vFth r e s p e c t t o ang le of

a t t a c k corresponding t o AAMP(1) th ru

. .- AAMP (8)

B - . , CCHI c x c o s ( x

CCLL(8,16,8) A t h r e e dimensional a r r a y of v a r i a b l e s

s i g n i f y i n g r o l ' : ng moment c o e f f i c i e n t s

of the forebody c o r ~ e s p o n d i n g t o

AAM(8) , AALPME (16) , and PPHIE (8)

CCLLP (8,8,8) A t h r e e dimensional a r r a y of v a r i a b l e s

s i g n f f y i n g r o l l damping c o e f f i c i e n t s

of t h e forebody corresponding t o

AAMD(8) , AALPDE (8) , and PPHIDE (8)

A t h r e e dimensional a r r a y o f v a r i a b l e s

s i g n i f y i n g p i t c h i n g moment c o e f f i c i e n t s

of the f orebody corresponding t o

AAM(8) , AALPME (16) , and PPHIE (8)

GOODVEAR AEROSPACE CO.CO.4TI0U

GE R- 1604 7 *, . * I METRIC ENGLISH

FORTRAN STANDARD DESCRIPTION U N I T S U N I T S

C C W (8 , a , 8) A t h r e e dimensiona 1 a r r a y of v a r i a b l e s

t : s i g n i f y i n g p i t c h damping c o e f f i c i e n t s A,


CCLNR ( 8 , 5 , 8 )

CCNP (8,8)

AAMD (8) , AALPDE (8) , and PPHIDE f 8)


s i g n i f y i n g yawing moment c o e f f i c i e n t s


AAM(8) , AALPME (16) , and YPHIE (8)


s i g n i f y i n g yaw damping c o e f f i c i e n t s of

t h e foreoody corresponding t o

AAbi(8) , AALPDE (8) , and PPHIDE (8)

A t h r e e dimensional a r r a y o f v a r i a b l e s

s i g n i f y i n g normal f o r c e c o e f f i c i e n t s of

t h e forebody corresponding t o A A M ( ~ ) ,

AALPFE (16) , P P H I E (8)

A two dimensional a r r a y of 64 v a r i a b l e s

s i g n i f y i n g t h e p i t c h i n g moment c o e f f i c i e n t s

o f t h e d e c e l e r a t o r wi th r e s p e c t co ang le

of a t t a c k corresponding t o AAMP(1) t h r u

AAMP (8)

A two diinensjonal a r r a y of 64 var iaLles

s i g n i f y i n g t h e normal f o r c e c o e f f i c i e n t

of the d e c e l e r a t o r wi th r e s p e c t t o angle-

of a t t a c k corresponding t o AAMP(1) t h r u

AAMP (8)

Elements of t r ans fo rmat ion mat r ix from

i n e r t i a 1 coord ina tes t o body coord ina tes

of t h e d e c e l e r a t o r

GOODVEAR AEROSPACE CO.CO14110m

GER- 16047

METRIC ENGLISi FORTRAN STANDARD DESCRIPTION UNIT S UNITS - CCRIT c l c r = damping r a t i o :: 0.06

4

CCY(8,16,8) A t h r e e dimensional a r r a y of v a r i a b l e s

CDAP

CGAMP C 'fp

CHIE 7(

CHIPE X P

CLL C1l

CLLP

CLM C m

CLN 'n

CLNR

s i g n i f y i n g s i d e fo rce c o e f f i c i e n t s of

t h e forebody corresponding t o AAM(8),

AALPFE (16) , and PPHIE (8)

Drag a r e a o f d e c e l e r a t o r

Center of g r a v i t y

Cos ('6 )

cos ( '6

F l i g h t path angle o f forebody i n

h o r i z o n t a l p lane, measured from X

a x i s toward Y a x i s

F l i g h t path angle of d e c e l e r a t o r I n

h o r i z o n t a l p lane, measured from X

a x i s toward Y a x i s

Rol l ing moment c o e f f i c i e n t of the

f orebody

Rol l ing damping c o e f f i c i e n t of the

f orebody

P i t c h i n g moment c o e f f i c i e n t o f the

f orebody

P i t c h damping c o e f f i c i e n t of t h e

fo re body

Yawing moment c o e f f i c i e n t of the

f orebody

Yaw damping coefficient of t h e forebody

METRIC ENGLISH UNITS UNlTS - -I_--

FORTRAN STANDARD

CN C~

CNP C N P

COM(20)

DESCRIPTION - Normal f o r c e c o e f f i c i e n t of the forebody

Normal f o r c e c o e f f i c i e n t of t h e d e c e l e r a t o r

Input v a r i a b l e used t o d e f i n e computer

s imula t ion - up t o e i g h t y f i g u r e s

CPH I C b Cos (4)

CPHII cdi

CPHIPI Cbpi

CPSI c

CPSIP C y P

CS Cs

CSIGP

P

Damping c o e f f i c i e n t of t e t h e r

Cosine of one h a l f t h e apex angle of

t h e cone formed by the suspension l i n e s

CTHE CQ

CTHEP CQ P

CY CY D d

Cos (8 ) P

Side fo rce c o e f f i c i e n t o f forebody

Aerodynamic re fe rence l eng th of

forebody (m) f t

A two dimensional a r r a y of v a r i a b l e s

s i g n i f y i n g t h e c o e f f i c i e n t s of the (kg) s lug second d e r i v a t i v e s i n t h e equat ions of

01: , cnotions (kg- mL) s lug- f t

Z

-0 -a 7"' PI- -0 m u -

DDP (3,3) A two dimensional a r r a y of v a r i a b l e s (kg) s l u g s

s i g n i f y i n g t h e c o e f f i c i e n t s of the o r (kg-m2) s l u g - f t 2 second d e r i v a t i v e s i n t h e equat ions of

motion cf the d e c e l e r a t o r

T o t a l s t ~ s p e n s i o n l i n e d e f l . . t ion a r r a y m f t DELSX

GE R- 16047

HLTRIC ENGLISH FORTRAN STANDARD DESCRIPTION - UNES UNITS

DELTX Tc ta l t e t h e r l i n e de i l ec t i on a r r ay m f t

DLTO I n i t i a l elongat ion of t other beyond

DLTX

a ". 3 * I

DPR

DSP

'0 - Inw -

n -. DSY 1 -I. ' Lu I d E

unstretched length. DLTO is negative

i f the forebody and dece ldra tor con-

fluence points a r e c loser together

than LTO (m) f t

Tether de f l ec t i on component i n array

element (DLX(1) associated with load

PX(1) m f t

Ef fec t ive sp r ing def 1ectic.t a r r ay m f t

Rate of change of longi tudinal added

mass kglsec s luglsec

Aerodynamic reference length of

dece le ra tor (m) f t

Degress per radian - 57.2957795

Parachute diameter associated with SP m f t

Parachute projected diameter associated

witn DS m f t

Larger absc issa of two points cn the

longi tudinal added mass versus

Do log log p lo t m

Smaller abscissa o f two points on the

longi tudinal added mass versus

Do log log p lo t m

Larger abscissa of two points on the


D3 log log p lo t m

COOOVEAR AEROSWCE CO.CO..~IO.

GER- 16047

METRIC ENGLISH FORTUS I

STANDARD DESCRIPTION UNITS UNITS - DSY 2 Smaller absc issa of two points on the


Do log log p lo t

DT In t eg ra t i on increment

m1

DTT

DYPR q

DYPRP P

EE ( 6 )

EPL

EPS

EPSI

-I,. ' W W K EPT

Number of i n t eg ra t i ons betweek; da ta

output

Number of i n t eg ra t i ons between da ta

output when T b TDTC

f t

s e c

In t eg ra t i on increment when T < TDTC sec

Estimated parachute system period112 sec sec

Dynamic pressure of forebody (~lrn*) l b f / f t 2

Dynamic pressure of dece le ra tor (Il/m2) lb f 1 f t 2

An a r r ay s ign i fy ing the nonhomo- (Nlm) f t - l b f

geneous terms i n the s i x equations or

of motion of the forebody (N) Lb f

Suspension l i n e s t r a i n a r ray d m f t l f t

Number used t o check f o r incons,:;tent - 13

equations i n PIVERT Subroutine, !O

Number used t o check i f 8 i s approaching 2n+l

a s ingular point 8 = - 2 T. I f 0 is

approaching a s ingu la r po in t , t he

acce le ra t ions a r e kept fixed u n t i l t h i s

region is passed. EPSI = 0.00003bl

f reezes the acce le ra t ions i f 8 i s within

0 . 2 ~ of a s ingular point.

Tether l i ne s t r a i n a r r ay m/m f t l f t

COODVEAR comeonateom A- f i

GER- 16047 i

METRIC ENGLISH FORTRAN - STANDARD DESCRIPTION UNITS UNITS - ETA1 Number which con t ro l s D!I i f 0 i s near a L

s ingu la r point. ETAI = 0.00061 s e t s

DT = "'15, i f 0 is within 2 O of a

~. s i n g u l a r i t y

An a r r ay s ign i fy ing the acce le ra t ions (m/sec) f t / s e c o r

of the dece le ra tor rad/sec

Estimated psrachute sys tem frequer :y l / s e c I / sec $ 3 FREQP

Ultimate design f a c t o r of s a f e t y f o r

parachute

Generalized force on forebody i n

X d i r e c t i o n

Body force i n d i r e c t i o n o f 1 6 due t o

aerodynamics

FXP Generalized force on dece le ra tor i n

X d i r ec t i on

Body force i n d i r ec t i on of X pb


Y d i r e c t i o n

Body force i n d i r ec t i on of Y dce b

t o aerodynamics

FYB

FYP G a e r a l i zed force on dece le ra tor i n

Y d i r e c t i o n

FYPB

FZ

Body force i n d i r ec t i on of Y P"


Z d i r e c t i o n

FZB Body force i n d i r ec t i on of Zb due t o

aerodynamics

7- . . . , - . . .

.. .- .-I- f $?

,L f COOOVEAR AEROSPKE

C O I C O . 4 l l O I

GER- 16047 ;*. .

h fl METRIC ENGLISH FO-TRAN STANDARD DESCRIPTION UNITS UNITS

IXPB I Apparent moment o f i n e r t i a about xpb

X a x i s 2 (kg-m) s l u g - f t

2 pb

IXY B I Product of i n e r t i a a s s o c i a t e d w i t h xyb

X,, and Y axes (kg-m2) s lug- f t 2 b

IXZB 'xzb Product of i n e r t i a a s s o c i a t e d wi th

X,, and Z axes 2 (kg-m ) s lug-f t

2 b . ..

.R, k: IYB I Moment of i n e r t i a about Y a x i s 2

(kg-m ! s lug- f t 2

#- yb b

IYPB I Apparent moment of inertia about Y P ~

Y a x i s 2 (kg-m ) s l u g - f t 2

pb

IYZB I yzb

Product of i n e r t i a a s s o c i a t e d wi th t Yb and Z axes 2 a (kg-m j s lug- f t

2

ST b ir

IZB Moment of i n e r t i a about Zb a x i s 2 (kg-m) s l u g - f t 2

d I z b

KS

LS

LSCL

LTD

LTO

M

MP

MPAL

Ks Te ther s p r i n g cons tan t

Suspension l i n e l eng th

Dis tance a long parachute c e n t e r l i n e

between t h e confluence p o i n t and the

p ro jec ted diameter p lane

5 Tether l eng th - d i s t a n c e between

confluence po in t s

i Time r a t e o f change o f t e t h e r length

5 0 Unstretched t e t h e r length

m Mass of forebody

m Real mass of d e c e l e r a t o r P

Added mass o f t h e d e c e l e r a t o r a long X pb

axis

(m) f t

(mlsec) f t / s e c

(m) f t

(kg) s l u g s

(kg) s l u g s

GER- 16047

METRIC ENGLISH FORTRAN STANDARD DESCRIPTION UNITS UNITS - WAS Added mass of the d e c e l e r a t o r a long Y

~b o r Z a x i s

P" (kg) s l u g s

MPL m Apparent l o n g i t u d i n a l (X ) mass o f PI P"

d e c e l e r a t o r (kg) s l u g s

r MPS

-- . 3 ' . OHYBE

0mBE /

OPAM

OPDA

m Apparent s i d e (Y o r 2 ) mass o f P s pb pb

d e c e l e r a t o r (kg) s l u g s

Number of parachute suspension l i n e s

Nuruber o f t e t h e r l i n e s

Option v a r i a b l e : i f OMETRC = l., Inpu t

and Output a r e i n t h e m e t r i c system.

I f OMETRC = 0.0 Inpu t and Output ar.2 i n

t h e Engl ish system.

Wxb Angular v e l o c i t y about Xb a x i s

y ~ b Angular v e l o c i t y about Yb a x i s

Wzb Angular v e l o c i t y about Zb a x i s

Option v a r i a b l e : i f OPAM = l . , added

mass of t h e d e c e l e r a t o r f 0; i f

OPAM = O., added mass of d e c e l e r a t o r

= 0

Option v a r i a b l e : i f OPDA = l . , damping

moment c o e f f i c i e n t s of t h e forebody a r e

read i n a s a r r a y s ; i f OPDA = U, damping

moment c o e f f i c i e n t s a r e read i n a s

cons tan t s

Option v a r i a b l e : i f OPOS = l., a t l e a s t

one of t h e c.g. o f f s e t s o r products o f

i n e r t i a o f t h e forebody f 0.; i f OPOS = O.,

a l l c.g. o f f s e t s and products of i n e r t i a = 0.

deg l sec

deg l sec

deg l sec

WOOVEACl A E R O W CO.CO.A1IOm

GER- 16047

METRIC ENGLISH UNITS UNITS - FORTRAN STANDARD DESCRIPTION

Option v a r i a b l e : i f 0PPZI)T = I . , a p l o t

t ape can be made; i f 0PPII)T =@., no p l o t

t ape i s made.

OPPRIN Option v a r i a b l e : i f OPPRIN = l., a l l aero-

dynamic c o e f f i c i e n t a r r a y s a r e p r i n t e d ou t ;

i f OPPRIN = Q, no aerodynamic c o e f f i c i e n t

a r r a y s a r e p r i n t e d o u t

OPDT Option f o r automat ic DT determinat ion

(OPDT = 1)

Option f o r automatic parachute a r e a

c a l c u l a t i o n s (OPSP = lJ

OP SP

OPSYM Option v a r i a b l e : i f OPSYM = l., t h e f o r e -

body is aerodynamically symmetric such t h a t

Cy% =Cm=O; i f OPSm = 0, t h e forebody i s

no t symmetric

Parachute o v e r i n f l a t i o n a t r ee fed

s t a g e ( I ) . (percent/100)

PCTO 1 Parachute over in f l a t i o n a t reefed

s t a g e 1. (percent/100)

Parachute o v e r i n f l a t ion a t r ee fed

s t a g e 2. (percent/100)

Parachute over inf l a t i o n a t reefed

s t a g e 3. (percent / 100)

PHIAE Aerodynamic r o l l ang le of forebody,

0 S PHIAE \( 180' . a

PHIDDE 0 Angular a c c e l e r a t i o n about Xb a x i s . PHIDE 8 Angular v e l o c i t y about X a x i s b

GER- 16047 - - ~~

METRIC ENGLISH FORTRAN STANDARD DESCRIPTION - UNITS UNITS

PHIE 0 Euler angle ro t a t i on about % ax i s deg

PHIIE Ii Aerodynamic r o l l angle o f forebody,

-180' * tii ,( 180" deg

PHIPI 8 Aerodynamic r o l l angle of dece le ra tor , o i

POROS Parachute porosity. Use POROS = 0.15

PPHIDE (8 ) An a r r ay of e igh t var iab les s ign i fy ing

forebody r o l l angle used with damping

coe f f i c i en t s deg

PPHIE (8) An ar ray of e igh t var iab les s ign i fy ing

forebody r o l l angle used with fc.~*ce

and moment coe f f i c i en t s de8

PSIDDE Angular acce l e r a t i on of forebody

about -2 a x i s deglsec 2

PSIDE Angr?!sr -:c?eclty of fo i~bodj i about

-Z ax i s deglsec

PSIE Euler angle r o t a t i o n of forebody about

-Z a x i s de8

PSIPDE Angula r ve loc i ty of dece le ra tor about

-Z axis

Angular r o t a t i o n of dece le ra tor about

-2 ax i s

PSIPE

Angular acce l e r a t i on of dece le ra tor about

-2 axis deg/sec 2

PSPDDE

PULAN Angle between t e t h e r and forebody

cen t e r l i ne del3 de8

Suspension l i n e load a r r ay N l b f

GER- 16047 A. -4 - '* *

FORTRAN STANDARD DESCRIPTION METRIC ENGLISH UNITS UNITS -

I PSX

m

Tota l suspension l i n e load a r r ay N lb f

Tether l i n e load a r r ay N lb f

PTX N d

Tota l t e t h e r l i n e a r ray lb f

x a Ef f cc t i ve spr ing load array N

lbf

Parachute load due t o r a t e of change

of mass of the parachute t i m e s the

r e l a t i v e ve loc i ty , XPBDR N lb f

QPHI Q0

Generalized force about X,, a x i s (m-N) f t - l b f

QPSI Q Y Generalized force of forebody about

-2 ax i s (m-N) f t - l b f

QPSIP Q ~ P

Generalized f o r of dece le ra tor about

-2 a x i s fm/N) f t / l b f

QTHE Q~ Generalized force of ,orebody about

negative l i n e of modes (m/N) f t / l b f

QTHEp Generalized force of dece le ra tor about Qe, negative l i n e of modes

RATIO Nondimensional length used i n the

dece le ra tor ' s body torque expressions

(m) f t Radius of e a r t h - 20,926,435. :A

RHO Density of atmosphere a t 2 (1962

Standard)

A i r dens i ty r a t i o (RHO/RHOO)

Aerodynamic reference a r ea of forebody

sin ( X

s i n ( Xp)

s i n ( X 7 - SGAM

GER- 16047 :I ! a ! f - METRIC ENGLISH 4

\ / FORTRAN STANDARD DESCRIPTION UNITS UNITS 3 - 2

SP

SPD

Parachute drag a r ea

Time r a t e of change of parachute

drag area 2 2 3

m I s ec f t I s ec

SPHI S0 s i n (0)

SPHII sOi

SPHIPI SQpi

s i n (gi)

s i n (0 .) P l

I n i t i a l parachute area m 2

f t 2

F i r s t reefed s tage parachute drag a rea 2 f t 2

Second reefed s tage parachute drag a rea m 2 f t 2

Third reefed s tage parachute drag a rea m 2 f t 2

SPRQ

SPRl

SPRL Parachute drag a rea associated with

reefed s t age (1-1) m 2

f t 2

SPRU Parachute drag area associated with

reefed s t age ( I ) m 2 f t 2

SPSI s 'I SPSIP st',

SSP (16) An ar ray of s ix t een var iab les s ign i fy ing

aerodynamic reference a rea of the

decelerator corresponding t o TTIP (16) (2) f t 2

STHE SO s i n (0) - 0 - 40

7" STHEP SO so m w

P s i n (0 )

P - 0 .. r T F l igh t time - LL

s e c

lbf

I W UP

TENS Tension i n t e t h e r

-c THEDDE '0' Angular acce l e r a t i on of forebody about

negative l i n e of nodes

COOWEAR A€- COmCOm.TIO*

GER- 16047

METRIC ENGLISH UNITS UNITS - FORTRILV STANDARD DESCRIPTION

THEDE Angular v e l o c i t y of forebody about

negat ive l i n e of nodes deglsec 2

THEE Euler angle r o t a t i o n of forebody about

negat ive l i n e o f nodes deg

deglsec

deg

deglsec 2

Angular v e l o c i t y of d e c e l e r a t o r about

nega t ive l i n e of nodes

THEPDE

THEPE Euler angle r o t a t i o n of d e c e l e r a t o r about

negat ive l i n e of nodes

Angular a c c e l e r a t i o n of d e c e l e r a t o r about

r ~ e g a t i v e l i n e of nodes

TPPDDE

TFI Parachute i n f l ~ t i o n time from Stage ( I )

t o Stage ( I + 1) s e c

TINF Time when i n f l a t e d a rea f i r s t

equa l s SPRU s e c s e c

Tin.? a t s t a r t of i t f l a t i o n of

I cefed s t a g e ( I )

T o t a l Time spen t i n region where

(1-ABS,(SIN(THE)). LT .EYSI) s e c

TNINY

s e c

s e c I n i t i a l t ime s e c

One ' a ? f t h e time spen t i n t h e o w r -

i n f l a t i o n of s t a g e I scc s e c

Time a t s t a r t of i n f l a t i o n o f f i r s t

reefed s t a g e

TOTRO

s e c s e c

s e c s e c

s e c s e c

TOTRl Time a t s t a r t o f i n f l a t i o n of

second reefed s t a g e

Time a t s t a r t of i n f l a t i o n of

t h i r d reefed s t a g e

TPD T o t a l t e t h e r load ( t e n s i o n + damping) N lb,

GOOUVEAR AERO- C O n C O n A I I O I

GER- 16347

TPDRB

1

TPDXB

TPDYB

TPDZS

" ". m

I -, METRIC ENGLISH 1

FORTRAN STANDARD DESCRIPTION - UNITS UNITS ! 1

TOTR3 Time a t end . t h i r d ree fed s t a g e s e c s e c , I ,IC

TRO

TTIP (16)

TXB Txb

TTT

TYB

TYPB

TZB

Component of t e t h e r load normal t o

f orebody c e n t e r l i n e N l b f

Tether load component along f orebody

XI3 a x i s N l b f

Tether load component a long forebody N lb f

YB a x i s

Te ther lozd component along forebody

ZB a x i s N lb f

Time from "TO" t o s t a r t of i n £ l a t i o n

of f i r s t s t a g e s e c s e c

Time from "TO" t o s t a r t of i n f l a t i o n

of second s t a g e s e c s e c

Time from "TO" t o s t a r t of i n f l a t i o n

of t h i r d s t a g e s e c s e c

Time from "TO" t o end of t h i r d

s t a g e (TR3 ) TTT) s e c s e c

An a r r a y of s i x t e e n v a r i a b l e s s i g n i f y i n g

i n £ l a t i o n time even t s s e c

T o r q u e a b o u t X a x i s d u e toaerodynamics (m-N) i t - l b b f

F l i g h t time a t which s imula t ion i s ended s e c

Torque about Yb a x i s due t o aerodynamics (m-N) f t - l b f

Torque about X a x i s due t o aero- pb

dynamics (m-N) i t - l b f

Torque about Zb a x i s due t o aerodynamics (M-N) f t - l b f

GER- 16047 . -

t t r $ 8

FORTRAN STANDARD DESCRIPTION METRIC ENGLISH UNITS UNITS --

4 4 i TZPB T Torque about Z a x i s due t o aero-

zpb P" dynami cs (m-N) f t - l b f

v

VP

VS

WT

WTC

WTCM

WTL

WTLM

WTP

X

Tota l ve loc i ty of forebody (mlsec) f t l s e c

f t l s e ~

f t l s e c

l b

l b

s lug

l b

s lug

l b

Tota l v e l c c i t y of dece le ra tor (mlsec)

Speed of sound a t Z (mlsec)

Weight of forebody (N)

Parachute canopy weight i;,o*m/sec 2

Mass of parachute canopy kg

Parachute suspension l i ne s weight kg*rr./sec 2

Mass of parachute suspension l i n e s kg

Weight of dece le ra tor (N)

Down range i n e r t i a l ax i s or d i sp l acemnt

of forebody (m)

Longitudinal body ax i s o r displace-

ment of forebody (m)

c.g. o f f s e t alocg 5 a x i s (m)

Xb body ax i s ve loc i ty (mlsec)

XBAR

XBD

Down ranbz ve loc i ty of forebody (mlsec; X,

XDD

XP

Down range acce le ra t ion of forebody 2

(mlsec )

Down range displacement of

dece le ra tor

X body ax i s v e l o r 4 t y of dece le ra tor (mlsec) P"

XPBD

XPBDR Velocity of a i r en t e r ing or ex i t i ng

the parachute r e l a t i v e t o the

parachute ve loc i ty rnlsec

COOWEAR AEROSPACE C O . C O ~ . l I O I

GER- 16067

METRIC ENGLISB UNITS - UNITS STANDARD DESGRIPTION

XPBDI Para-hute ve loc i ty , XPBD, a t T TINT.

It i s used t o ca lcu la te f i l l time m/sec f t /sec

XPD 4 Down range veloci ty of dece lera tor (m/sec) P

f t / s e c . . XPDD X Down range acc , l e r a t i o n of decelerqtor (m/sec ) f t l s e c 2 2

P

Cross range i n e r t i a l ax is o r displace-

ment of forebody (m)

Letera l body ax i s or displacement of

f orebody (m)

I - YBAR Y

5 v

c.g. o f f s e t along Yb ax is (m)

1' YBD Yb

t ! YD ; . . YDD Y

Yb body ax is ve loc i ty (m/sec)

Cross range ve loc i ty of forebody (m/sec) n

Cross range acce lera t ion of forebody (m/sec )

Cross range i n e r t i a l displacement of

decelerator (m)

YPBD Y body ax is veloci ty of decelerator (m/sec) P b

YPD 4 P

Cross range ve loc i ty of decelerator (m/sec)

Z Cross range acce lera t ion of dece lera tor ( m / ~ ? c ) YPDD

z Vert ica l i n e r t i a l ax is o r displacement

of forebody (m)

Ver t ica l body ax i s or displacement of

forebody (m)

GER- 16047

METRIC ENGLISH FORTRAN STANDARi) DESCRIPTION - 'UNITS UNITS -

i ZBAR 2 c.g. o f f se t along Zb axis (m) f t $ 4

!

ZBD Zb Zb body axis veloci ty

Vert ical ~ ~ 1 0 1 : i t y i ~ f forebody (mlsec) f t / sec 0. 2 2

ZDD 2 Vertical accelerat ion of forebody (m/sec ) ft./sec

Z P 2 Vert ical i n e r t i a l displacement of P

decelerator (m) f t

ZPBD 2 Z body axis veloci ty of decelerator (mlsec) f t / s e c pb P"

t P D Z Vert ical veloci ty of decelerator (m/sec) f t / sec . P

2 2

ZPDD Vert ical accelerat ion of decelerator (m/sec ) f t l s e c 2

P

CHAPTER I - INTRODUCTION

The system t o be simulated is two r i g i d bodies joined by an e l a s t i c t e the r . The

forebody may have a completely general shape and mass charac ter is t ics , and w i l l be

free t o move with s i x degrees of freedom (three t rans la t ional , three ro ta t ional ) .

The decelerator is assumed t o be symmetric i n shape and mass cha rac te r i s t i c s about

its longitxdina: ( r o l l ) sx f s , and sill be f r e e t o move with ffve degrees of freedom

(three trscslat iona ' l , two ro ta t ional ) . A f r i c t ion less swipe1 is assumed a t the

decelerator-tether confluence poict . Thus the r o l l motions of t h e forebody w i l l not

couple with the decel.erator. The t e the r i s simulated by a spring and dashpot i n

para l le l . Damping coefficfents fo r t e the r l i n e s are d i f f f c u i t t o obtain; but spring

constants f o r a te ther cqn be f o ~ i d fro^ experimental s t r e s s s t r a i n curves. Conse-

quently; the damping coeTficient is a s s u e d constant, while the spring constant is

assumed t o be a functfon of clangs-Lion i n tile computer program, thereby introducing

2 qunsilinear spring.

CHAPTER I1 DERIVATION OF EQUATIONS OF MOTION

SECTION 1 - COOPJ)INATE SYSTEM

Figure 1 shows t h e d i f f e r e n t coordinate s y s t e r s used t o de r ive t h e

equations of motion. X y Z is an i n e r t i a l orthogonal coordinate

system at tached t o a f l a t non-rot;.ting ea r th . XYZ and X y 2 a r e P P P

orthogonal axes f i xed t o t he forebody and dece l e ra t e r a t "On and

"0 " respect ively . Coardinate systems XYZ and X Y 2 t r a n s l a t e P P P P

with the bodies but do not r o t a t e , always remaining p a r a l l e l t o

corresponding i n e r t i a l axes. The displacements X, Y , 2 , Xp, Y P'

and 2 a s measured Prom t h e o r i g i n of HY 2 , a r e t h e s i x P'

t r a n s l a t i o n a l degrees of freedom of t h e two bodies. The reference

forebody body axes, longi tud ina l (Xb) , l a t e r a l ( Y b ) , and v e r t i c a l

(2 ) , i n t e r s e c t a t "On, the o r i g i n of t he aerodynamics load system b of t h e forebody. The re fe rence dece le ra tor body axes, longi tud ina l

(X ) , l a t e r a l (Ypb) , and v e r t i c a l ( 2 ) intersect a t '0 ", t he pb pb - P

c.g. of t h e dece le ra tor , The va r i ab l e s X , P, a r e t he d i s tances

frozr "a" tu the c o g . of t h e forebody measured p o s i t i v e l y i n t h e

d i r e c t i o n of the p o s i t i v e body axes Xb8 Yb, Zb r espec t ive ly . For

o r i e n t a t i o n purposes, t h e reader should pos i t i on himself a s a

p i l o t i n an a i rplane. In t h i s pos i t i on , Xb is p o s i t i v e toward t h e

nose, Yb is p o s i t i v e toward the l e f t wing and Zb is p o s i t i v e up. +

is the vector d i s tance from t h e i n t e r s e c t i o n of t h e longi tud ina l ,

l a t e r a l , and v e r t i c a l axes of the Lorebody ('On) t o t he t e t h e r +

confluence point of the forebody. r2 i s t h e vector d i s tance from

t h e c.4. of t he dece le ra tor ("0 " 1 t o the t e t h e r confluence po in t -+ P

of the dece le ra tor ; r2 lies aloqg X pb'

F I G U N i 1 - COORDINATE SYSTEMS

8 p

COOOVEAR AEROSPWE CO.~O.A~lO.

GER-16047 --.

f 3 . , SECTION 2 - EULER ANGLE TRANSFORMATION

I n order t o spec i fy t he angular o r i e n t a t i o n of a body w i t h r e fe rence

t o a non-rotat ing coordinate system (X, Y , Z ) , t h r e e succesr'-r-

r o t a t i o n s a r e made a s shown i n Figure 2. The f i r s t r o t a t i o n i s . i n

t h e d i r ec t ion , -02, such t h a t OX and OY a r e r o t a t e d tnrouy. an angle

$ i n t o Oa and ON respec-ively. The second r o t a t i o n is i n t h e I d i r e c t i o n , -ON, such t h a t Oa and OZ are ~ o t a t e d t h ~ o u g h an angle 0

i n t o OXb and Ob respec t ive ly . The finl: r o t a t i o n i s about O X , i : 3 .

such t h a t ON and Ob a r e ro t a t ed through an angle 4 i n t o OYb and 3 # -

i- s OZb respect ively . The t h r e e angular ro ta t io l i s ($, 0 , 4 ) spec i fy [: \

t h e o r i e n t a t i o n of t h e body axes (Xb , Yb, Z b ) with respec t t o t h e e F 5 -

i n e r t i a l axes ( X , y , ). Again, from a p i l o t s viewpoint, a 1 'r

[. t p o s i t i v e is a nose t o t he r i g h t yaw; a p o s i t i v e 0 is a nose up p i t ch ; ?' [ and a p o s i t i v e @ is a r i g h t wing down r o l l . g ,--- I

. i

The transformation matrix between the body axes and i n e r t i a l axes

is now found by considering one r o t a t i o n a t a time and then combining. ,~ -

The f i r s t r o t a t i o n i s given by:

where S$ = s i n $ and C$ = cos$.

' -0 The second r o t a t i o n is: - Q

, I.0

FIGURE 2 - EULER ANGLE ROTATIONS

a

3

The f i n a l r o t a t i o n is: a---

By s u b s t i t u t i n g Equation (1) i n t o ( 2 ) and (2 ) i n t o ( 3 ) , t h e

transformation matrix [Cl is formed

- Since [CI i s a l i n e a r orthogonal ( 2 Cij Cik - 6 jk: j , k = 1.2.3) i=l

transformation, i t s inverse i s equal t o i t s transpose. . Therefore . cqce -c$ses$+s+c$

s$ses$+c$c$

ces4 cec4

For t h e dece le ra tor , t h e r e i s no r o t a t i o n about tho longi tud ina l

axis. Consequer.tly, t h e transformation matr ix i n ( 4 ) i s s imp l i f i ed

by 1e;lting $ = 0, The r e s u l t i s [C 1. P

The t o t a l angular v e l o c i t y of the forebody i s given by: + .+

( 7 )

1 From the inverse of ( 3 ) : i

t i

From the inverse o f (2'

i; = se?,+ cet,

subst i tut ing ( 8 ) , (91 , and (10) i n t o (7) :

The components of angular v e l o c i t y for the forebody are:

Likewise, f o r the decelerator , the angular v e l o c i t i e s are:

SECTION 3 - KINETIC ENERGY

The k i n e t i c energy of t h e system is due t o t h e t r a n s l a t i o n a l and

r o t a t i o n a l v e l o c i t i e s of t he forebody and t h e dece le ra tor . The

forebody is completely general i n shape, and products of i n e r t i a

and c.g. o f f s e t s w i l l e f f e c t t h e k i n e t i c energy. On t h e o t h e r hand, the dece le ra tor is assumed t o be symmetric about t h e longi tud ina l

a x i s and t h e aerodynamic loads a r e referenced t o t h e cog. Therefore,

a l l products of i n e r t i a and cog. o f f s e t s a r e zero. The expression

f o r k i n e t i c energy is: ( 2 )

I n Equation (18) , mp fi and m include d i r e c t i o n a l m a s s terms due i PS >

i ,

ts t he a i r enclosed i n t h e canopy, Ixpb, I ypb, and 1 zpb are apparent mass moments of i n e r t i a .

SECTION 4 - POTENTIAL ENERGY

The p o t e n t i a i energy of t h e system i s due t o t h e g r a v i t a t i o n a l

p o t e n t i a l of both bodies and t h e e l a s t i c p o t e n t i a l o f t h e t e t h e r .

L ~ o is t h e unst re tched length of t h e t e t h e r ; and LT i s the

s t r e t ched length of t h e t e t h e r a s given by t h e geometry of t h e

system. Referring t o Figure 1:

Sl and d2 a r e t h e vec tors from t h e i n e r t i a l coordinate system

( X 7 2 ) t o t he confluence po in t s of t h e forebody and

dece le ra tor respect ively . For t h e forebody,

a, b, and c a r e measured along p o s i t i v e body axes Xb8 Yb8

and Zb r espec t ive ly . Using t h e coordinate transformation

matrix ( 4 ) :

Similarly f o r the decelerator:

Substituting (25) i n t o (24) and using matrix Equation ( 6 ) , I

1 j

Define the variables Z, E, and F such that: i

Further on i n the derivat ion it w i l l be necessary t o know 7

the t o t a l t i m e der ivat ive o f L, and the p a r t i a l der ivat ives 5 3

.L

o f x, g, and F with respect t o the generalized coordinates. 1

SECTION 5 - RALEI'3I'S DISSIPATION FUNCTION

If the viscous damping force is proportional to the velocity

of the particle at which the force acts, an expression analogous

to the potential energy of a spring may be used. This fufiction,

F, 2s known as Rayleigh's di~sipation function, and is defined as (1 1

For this problem, Rayleigh's dampirt : r :maidered only in

the tether.

SECTION 6 - LAGRANGE'S EQUATION L he Lagrange equation for a non-conservative (aerodynamic forces)

system with a holonomic (can be expressed as an algebraic

c% expression), scleronomous (independent cf time) constraint and b- -0 S W

Rayleigh's dissipation function (dampir-g in the elastic tether) I 0 .. -LL can be written as:

(1 1 , ;Y

I n Equation (54), t h e t e r m X 2: expresses t h e general ized A

force exer ted by t he t e t h e r on t h e ith degree of freedom.

The zons t r a in t equation is:

Qi is t n e general ized fo rce due t o t h e aerodynamics.

is t h e fo rce due t o damping i n t h e tether. Ti Tne Lagrangian is equal t o *.e t o t a l k i n e t i c energy of t h e

system (Equation (18)) ini3us t h e t o t a l p o t e n t i a l energy of

t h e system (Equation (19)) . With s u b s t i t u t i o n s from Equations

( 4 ) , (6) and (12) t o (17), t h e Lagrangian is:

-{I,,, r - i tces4)-i ( ~ $ 1 I r-j, (ceco)+i ( ~ 4 ) l+rXzb 1-i (se1+611-$ (c~c+)+; (s+)I

f . ~ .. Note: I Y P ~

= I due t o decelerator symmetry. '% zpb

SECTION 7 - GENERAL EQUATIONS OF MOTION ".

I . Equation ( 5 6 ) displays a l l of the generalized coordinates - i

e x p l i c i t l y except those appearing r? IT. The terms t o be

1 subst i tuted i n t o Equation (54 ) are now developed, 5 *

X Equation

3

i d a~ .. aF = x{rnl+${m [P(-s~ce)+P(slysesg+clycg~+Z(slysec~-c~s~ I 1

I ax

Y Equation

t t

i 2 Equation

i

Z- Equation

y Equation

\ . 9 Equation

@ Equation

+" ''xb (-ce) + [ I ~ ~ ~ (S~P +I,,, (cm 1 (-s e 1 1

+$* #-

IIXyb (CQ)-lxzb (SO) I (Ce) }

. . LT Equation

k z Substituting Equations (101) t o (104) i r t o (54) y i e lds :

? = -[Ks($-LTO)+csiT~ (105)

f

i e The value o f A i n Equation (54) i s now defined and is expressed r !- i n "srms o f the e leven generalized coordinator and the i r t i m e f

5 der. datives. The eleven simultaneous, nonlinear,

1 coupled d i f f e r e n t i a l equations o f motion are then writ ten as:

$ 1. X Equation

I

2. Y Equation

Y {m)+T{-m [ Z C ~ ~ + % ? C ~ ~ + T C ~ ~ I 1

+'B'{m [X(s$se)+Y(s$ces~) +~(s)cecg) ] 1+${n [ F c ~ ~ - Z ~ ~ ~ 1 1

= $4 {-2m[X[c+se)+Y(c+ceso) +T(c$cecm) ] ~ + $ ~ { ~ : ~ [ ! T C ~ ~ - ' Z C ~ ~ ] 1

+ B ~ I - ~ ~ [ P ( s $ c ~ c ~ ) -H(sqces@) I ~+$2~m[~~c,2+~~22+'Zcc,,~ 1

+B2{m [~c~~+~(s~ses~)+T(s)secg) 1 1

3. Z Equation

. ~ { m } + i i { m [y(ce)+~(-seso+~(-secg) I ~ + i ' ~ m I ~ c ~ ~ - % ~ ~ I

= 8){2m[Y(~ec~)-T(sesg) 1 I + B ~ { ~ [ X C ~ ~ + ! T C ~ ~ + Z C ~ ~ I 1

4. Xn Equation

x s.

5. Y Equation

i

7. IJ Equation

8. 0 Equation f

Y{m [F(-c$s~) +F(-c$c~s$) +Z(-C~CCIC~) 1 I

- Q 9 . I$ Equation -

P;" tz Inw - P { ~ ~ ~ Y E ~ ~ - R ~ ~ ] }+f {m [ ! F c ~ ~ - Z C ~ ~ ] }+i'{m [iiC jr-Z~23 ] 1 0 ; td

W K . . +${-1xb~13+~xyb~23+~xzb Cj3}+B {Ixyb (CQ1-I (SO) 1+;{1&}

-C xzb

, ,-

-33-

6 F

-0 - m *el

PI- -0 Y)w - 0 .- -I. ' W

w a - . .

SECTION 8 - SIMPLIFIED EQUATIONS Or MOTION

muations (106) to (116) are for the most general situation

possible, and as a result, are quite lengthy. Under some

circumstances, these equations can be simplified. If this can.

be accomplished, a significant decrease in computer time will

be realized. The first simplification occurs if the forebody's

aerodynamic (body) reference axes are principal axes. In this

case = P = 'i = 0 and I - xl'b

= I - Ixzb yzb = 0. Equations (1061,

(107), (1081, (1121, (1131, and (114) become:

The above s i x equations of motion have not only been shortened,

bu t they a l s o have been uncoupled i n t he t r a n s l a t i o n a l

acce le ra t ions making them e a s i e r t o solve. The second

s imp l i f i ca t ion involves t h e dece l e ra to r degrees of freedom.

I f t he added masses of t he dece l e ra t c r a r e ignored (mpe= mps

= m p ) , E q u a t i ~ n s (109). ( 1 1 0 ) . (ill), (115). and (116) become:

Like t h e s i m p l i f i e d equat ion f o r t h e forebody, t h e d e c e l e r a t o r

equat ions have a l s o shortened. Furthermore, they have

completely uncoupled i n t h e second d e r i v a t i v e s making numerical

i n t e g r a t i o n easy.

SECTION 9 - GENERALIZED FORCES - AERODYNAMICS

The nonconservat ive f o r c e s a c t i n g on t h e forebody a r e due t o

aerodynamics. The aerodynamics and t h e convention used i n t h i s

r e p o r t apply t o t h e Space S h u t t l e S o l i d Rocket Booster (S.R.B..).

If a d i f f e r e n t body i s t o be s imula ted , t h e aerodynamic c o e f f i c i e 3 t s

and p o s s i b l y t h e convention used t o d e f i n e them, would change.

For t h e S.R.B., t h e aerodynamics a r e a f u n c t i o n o f r o l l angle ,

angle-of-at tack, and Mach number. The angle-of-a t tack i s

measured from t h e t o t a l v e l o c i t y v e c t o r t o t h e p o s i t i v e l o n g i t u d i n a l

a x i s (Xb) a s shown i n F igure 3 .

The normal f o r c e c o e f f i c i e n t , CN, i s i n t h e p lane formed by

t h e v e l o c i t y v e c t o r and t h e l o n g i t u d i n a l a x i s , and is perpend icu la r

t o t h e l o n g i t u d i n a l a x i s ( X b ) . The r o l l ang le , i s then

measured from t h e normal f o r c e c o e f f i c i e n t t o t h e Zbbody a x i s .

The a x i a l f o r c e c o e f f i c i e n t i s de f ined a s u s ~ . a l , p o s i t i v e i n

t h e nega t ive Xb d i r e c t i c n . F i n a l l y , t h e s i d e f o r c e c o e f f i c i e n t

i s perpendicular t o t h e Xb body a x i s and t o t h e normal f o r c e ,

such t h a t t h e d i r e c t i o n s o f CA, CN, Cy form a right-handed

or thogonal coord ina te system. Mathematically, t h e aercdynamics

r o l l ang le i s given by:

-1 Oi = t a n [-Pb/-ib]

COOWEAR AEROSPACE CO.COI.TlO*

FIGURE 3 - AERODYNAMIC COORDINATE SYSTEM

i

GOODVEAR AEROSPACE CO.COn.l IO~

GER-16047

The positive directions of the moment coefficients are shown

in Figure 3 as double arrows. Dampirig . z n t coefficients

are about the body axes (Xb, Yb, Zb). Aerodynamic body axes

forces are given as:

The body axes forccs are converted to inertial a;?s force

using the elzments of LC], Equation 4.

-S are: Body axes torqu-

The body axis torques are transformed to generalized

torques using Equations (5) and (1) and iAotlng s i gn

conventions.

GOOWEAR AEROSPACE C O I . O I . l I C .

GER-16047

The aerodynamics of 3 dece l e ra to r (parachute) a r e not veil

known because a parachute i s not a r i g i d body, ana does not

lend i t s e l f t o e a s i l y obta inable t e s t daLa, espec iaJ ly

under dyna;.ic condit ions. Consequently the aerodynamics

of a symmetric c 'ocelerator tend tc be re la t ive ly simple

due t o a lack of bet'er understanding r a t h e r than the

i n a b i l i t y t o use ava i l ab ' t ineormation. I f b e t t e r

aerodynamic da t a i s a t t a i n a b l e , it i s a simple matter t c j

a l t e r the body forces and torques appropr ia te ly .

For t h i s r epo r t , t he decelerator body forces and t ~ r q ~ e e are :

The general ized iorces f o r the dece lerator are:

F xp

= F + F C xpb 'pll ypb p21 + Fzpb C p31

F YP

= F xpb 'pl2 + F ypb $22 + Fzpb $32

F z P

= F xpb Cp13 + 'zpb 'p33 - mpg

- Q * ~ - - Tzpb 'p33

= -T Y P ~

GER- 16047

CHAPTER 111 COMPUTER PROGRAM

SECTION 1 - FEATURES OF THE COMPFCER PRIGRAM

The computer program con ta ins t h e fo l lowing f e a t u r e s .

1. The progran has many opt ions which s i m p l i f y t h e inpu t of d a t a o r dec rease the

program run time. Use o f the op t ions a r e conta ined i n the f i s t i n g of t h e

program a s comment ca rds . These op t ions a r e :

a. Opt i )ns a r e included which change t h e dimensions of t h e aerodynamic

c o e f f i c i e n t a r r a y s a s d i c t a t e d by inpu t requirements.

b. An op t ion is provided (OPDT = 1.0) which au tomat ica l ly determines t h e

magnitude of t h e i n t e g r a t i o n t i m e i n t e r v a l , DT.

c. An op t ion is provided (OPSP = 1.0) which c a l c u l a t e s t h e parachute d rag

a r e a (SP) time h i s t o r y . I f t h i s op t ion is r o t used the d rag a r e a versus

time i s inpu t i n t o the program i n t h e form of look-up a r r a y s .

d. An o p t i o n f o r including l o n g i t u d i n a l and l a t e r a l added a i r mass e f f e c t s

on t h e parachute (OPAM = 1.0) i s included i n t h e program.

e . A p rov i s ion i s made t o use s i m p l i f i e d equa t ions of motion (OPOS = 0.0) t o

reduce run t ime, i f a l l t h e forebody products of i n e r t i a and c e n t e r 3f

mass o f f s e t s a r e equal t o zero.

f . An o p t i n ( o P P L ~ = 1.0) f o r making a p l o t tape is a v a i l a b l e .

g. English o r me t r i c systems may be used f o r d a t a inpu t and out by equat ing

OMETRC t o 0.0 o r 1.0 r e s p e c t i v e l y .

2. A l l aerodynamic c o e f f i c i e n t s a r e read i n t o the program a s func t ions of angle

of a t t a c k , r o l l ang le , and mach number i n t h e form of t h r e e dimensional look-up

a r rays .

-EAR A- CO.CO..IIOrn

GER- 16047

3. The i n i t i a l s t a r t cond i t ions f o r t h e forebody and a f t body a r e completely

genera 1.

4. The s t a c k i n g of des ign c a s e s is poss ib le .

5. The at tachment l o c a t i o n of t h e t e t h e r t o t h e forebody is completely genera l .

6. The t e t h e r load and t h e angle i t makes wi th t h e c e n t e r l i n e of t h e forebody a r e

program ou tpu t s .

7. -411 load and t r a j e c t o r y d a t a a r e output a t p re - se lec ted t imes.

8. Termination of a des ign case occurs et a predetermined time o r a l t i t u d e .

9. The program c a l c u l a t e s t h e e f f e c t i v e system s p r i n g constant .

10. The program c a l c u l a t e s t h e parachute phys ica l p r o p e r t i e s a s t h e perachute

i n f l a t e s a s a func t ion of time.

11. The parachute may have t h r e e s t a g e s of r e e f i n g , i f t h e automat ic d r a g a r e a

versus t ime op t ion is chosen.

12. As t h e parachute i n f l a t e s , t h e d rag a r e a ve r sus time follows a second degree 2

curve (y = ax ) .

SECTION 2 - INPm

GER- 16047

Except f o r t h e v a r i a b l e COM, a l l i n p u t s a r e r e a d i n under t h e format s t a t e m e n t

8F10.0. COH i s a n 80 column h e a d e r ca rd . A l l of the fo l lowing v a r i a b l e s a r e

d e f i n e d i n t h e nomenclature.

INPUT ITEM

a

VARIABLE

AIPHI, AIPHID, AJALPF, AJALPM, AKAM, AKAMD,

OPSYM, OPDA

PPHIE

AALPFE

AALPME

AAM

CCA

CCN

CCLM

CCY

CCLL

CCLN

CLLP, CLMQ, CLNR

PPHIDE

AALPDE

AAMD

CCLLP

CCLMQ

CCLNR

AALPPE

AAMP

CCAP

CCNP

CCMP

NUMBER OF CARDS

1 c a r d

1 c a r d

1 o r 2 c a r d s

1 o r 2 c a r d s

1 c a r d

4 t o 128 c a r d s

4 t o 128 c a r d s

4 t o 128 c a r d s

0 o r 4 t o 128 c a r d s

0 o r 4 t o 128 c a r d s

0 o r 4 t o 128 c a r d s

0 o r 1 c a r d

0 o r 1 c a r d

0 o r 1 c a r d

0 o r 1 ca rd

0 o r i t o 64 c a r d s

3 o r 4 t o 64 c a r d s

0 o r 4 t o 64 c a r d s 1

1 c a r d

1 ca rd

2 t o 8 c a r d s

2 t o 8 c a r d s

2 t o 8 c a r d s

W E A R AEROSPKE CO.CO.4llOm

GER- 1604 7

INPUT ITEM

x)

Y)

2)

aa)

bb)

VARIABLE NUMBER OF CARDS

PS, PT, EPL, EPT 1 c a r d

X, Y , X, V , GAME, CHIE, EPSI, ETA1 1 c a r d

WT, IXB, I Y B , IZB, IXYB, IXZB, IYZB 1 c a r d

S , D, XBAR, YBXR, ZBAR, OPPRIN, OPPIIOT, OPOS 1 c a r d

PSIE, THEE, PHIE, OMXBE, OMYBE, OMZBE 1 ca rd

A , B, C, OPAM, OMETRC 1 c a r d

PSIPE, THEPE, PSIPDE, THEPDE, VP, GAMPE, CHIPE 1 ca rd

LS, LTO, DLTO, NS, KT, DP, CCRIT 1 c a r d

AMAX1, AMAX2, DSX1, DSX2, AMAY1, AMAY2, DSYl, DSY2 1 ca rd

WTC, WTL, OPSP, OPDT

I F (OPSP .EQ.O .O) GO TO ITEM kk)

TRO , TR1, TR2, TR3 1 ca rd

SPRO, SPR1, SPR2, SPR3, PCTOl, PCTO2, PCT03, POROS 1 ca rd

T r I P , SSP 4 c a r d s

I F (OPDT.EQ.O.0) GO TO ITEM m)

GWAD, FSULT, AERATO, TO, DTP1, TTT, HHH 1 ca rd

DT1, TO, DTP1, TTT, HHH 1 c a r d

COM 1 ca rd

The va lues of t h e v a r i a b l e s r ead i n i n p u t i t e m "a" de termine , i n p a r t , t h e s i z e s

of t h e aerodynamic a r r a y s . The a x i a l and m m e n t c o e f f i c i e n t s have t h e o p t i o n o f

u s i n g e i t h e r e i g h t o r s i x t e e n a n g l e s - o f - a t t a c k (one o r two ca rds ) . I f , f o r example,

f i v e o r e l e v e n a n g l e s - o f - a t t a c k a r e needed, one o r two c a r d s a r e needed r e spec -

t i v e l y . The r o l l and Mach number a r r a y s may va ry from two t o e i g h t . A s a n

example cons ide r t h e a r r a y CCA where t h e va lue of CA depends on f i v e r o l l a n g l e s f

-0 - (D (di), e l e v e n a n g l e s - o f - a t t a c k (q) a l d seven Mach numbers (AM). The a r r a y

I-m

C 6 PPHIE would be r e a d i n on one ca rd c o n t a i n i n g f i v e d i s t i n c t r o l l a n g l e s , t h e l a s t m w - D .- t h r e e f i e l d s o f t e n d i g i t s would he b lanks . The a r r a y AALPFE would be r ead i n on -LL ' W i 4 nz two cards . The f i r s t c a r d ~ o u l d c o n t a i n e i g h t d i s t i n c t a n g l e s - o f - a t t a c k , and t h e

, 5 second cerd would c o n t a i n t h r e e d i s t i n c t a n g l e s - o f - a t t a c k snd f i v e b lank f i e l d s c f i: - r: t e n d i g i t s . The a r r a y AAM would be read i n on one c a r d c o n t a i n i n g seven d i s t i n c t

t Mach numbers and one b lank f i e l d of t e n d i g i t s . The f i r s t e lement i n each o f t h e

above a r r a y s shou ld s t a r t a t z e r o and i n c r e a s e numer i ca l ly u n t i l t r h i g h e s t

GER- 16047

. i r poss ib le va lue expected t o be encountered i s s p e c i f i e d . I n t h i s p a r t i c u l a r example,

t h e a r r a y s i z e used w i l l be CCA(5, 16, 7). The proper read sequence i s t o f i r s t

read two c a r d s c o n t a i n i n g t h e va lues of C a t e l e v e n a n g l ~ - o f - a t t a c k , t h e i n i t i a l A r o l l a n g l e (zero) and t h e i n i t i a l Mach number (zero). These ca rds a r e followed by

two ca rds con ta in ing values o f C a t e l e v e n ang les -o f -a t t ack , t h e second r o l l ang le A and t h e i n i t i a l Mach number. T h i s i s continued f o r f i v e r o l l a n g l e s a t t h e i . n i t i a l

Mach number. A f t e r these t e n c a r d s , t h e same procedure i s followed f o r t h e s :ond

Mach number, and t h e t h i r d , e t c . up t o seven s e t s o f t e n cards.

A l l t h e aerodynamic c o e f f i c i e n t a r r a y s a r e read s i m i l a r l y . However, n o t i c e t h a t

t h e angle-of-a t tack a r r a y a s s o c i a t e d wi th t h e moment c o e f f i c i e n t s i s d i f f e r e n t than

t h a t a s s o c i a t e d wi th the f o r c e c o e f f i c i e n t s . Also, t h e damping moment c o e f f i c i e n t

a r r a y s ( input i tems "p", "q" , and "r") may no t be read i n a t a l l , depending on the

v a l u e o f OPDA. I n s t e a d , inpu t i t e a "1" can be used i f the damping c o e f f i c i e n t s a r e

cons tan t . F i n a l l y , t h e damping c o e f f i c i e n t s correspovd t o the a r r a y s read i n inpu t items llm1l, " 11 n , and "0".

I Figure 4 he lps io c l a r i f y t h e meaning of t h e inpu t parameters a s soc ia ted wi th t h e

added a i r mass on t h e parachute (Ref. Inpu t I tem f f ) .

Po

3

n Po 1 r(

cn V

(I) D

S $4 -4 u. TI a w P 4:

Parachute Reference Diameter ( f t ) ( log)

Figure 4. Input Parameters f o r Parachute Added A i r Mass

Figure 5 he lps t o c l a r i f y the meaning o f the paramters associated with OPSO = 1.0

which d i r e c t s the program t o ca lcu la te the parachute area time h i s tory as time

ad lances (Reference Input Items hh and j j) .

GER- 16047

SECTION 3 - OUTPUT

A l l o u t p u t v a r i a b l e s a r e d e f i n e d i n t h e nomenclature. Before beg inn ing t h e simu-

l a t i o n , t h e fo l lowing v a r i a b l e s , s p e c i f y i n g t h e c h a r a c t e r i s t i c s o f t h e r i g i d body

and i n i t i a l pa rame te r s , a r e p r i n t e d ou t .

L ine 1. COM

Line 2. IXB, IXYB, XBAR, S, CLIP, OPPRIN, OPSYM, AIPHI, AIPHID, DT1 EPSI

Line 3. IYB, JXZB, YBAR, D, CLMQ, OPPLOT, OPDA, AJALPF, AJALPM, AJALPD,

TTT, ETA1

Line 4. IZB, I Y Z B , ZBAR, WT, CLNR, OPOS, OMETRC, AKAM, AKAMD, HHH

I f CLLP, CLMQ, and CLNR a r e c o n s t a n t s f o r t h e s i m u l a t i o n , t h e i r va lues a r e p r i n t e d

o u t i n t h e a p p r o p r i a t e p l ace . I f t h e damping c o e f f i c i e n t s a r e found from i n t e r p o -

l a t i o n o f t h r e e d imensional a r r a y s , CLLP, CLMQ, and CLNR a r e set e q u a l t o z e r o f o r

t h i s p r i n t o u t only . S e v e r a l v a r i a b l e s d e a l i n g w i t h t h e d e c e l e r a t o r a r e t h e n

p r i n t e d o u t .

Line 5. A , LTO, LS, AMAX1, AMAX2, AMAY1, AMAY2, AP, GWAD, FREQP, OPAM,

PCTO 1

Line 6. B, NT, NS, DSK1, DSK2, DSY1, DSY2, CHIPE, FSULT, POROS, OPDT,

PCT02

Line 7. C, DLTO, DP, LTC, WTL, WTP, CCRIT, VP, AERATO, TO OPSO, PCT03

Line 8. TRO, TRl, TR2, TR3, SPRO, SPR1, SPR2, SPR3

The parachute suspens ion l i n e load and s t r a i n a r r a y s a r e p r i n t e d o u t nex t on L ines 9

anr! 10.

L ine 9. PS (1)

Line 10. EPL(1)

The t e t h e r l i n e load and s t r a i n a r r a y s a r e p r i n t e d o u t next on L ines 11 and 12.

Line 11. PT (1)

Line 12. EPT (1)

GOOOVEAR AEROSPACE C O I . O I A l l O m

GE'-.- 16047

The parachute i n f l a t i o n time h i s t o r y a r r a y and drag a rea a r r a y a r e x i n t e d ou t

next . I f t h e op t ion (OPSP = 1.0) t h e a r r a y s a r e s c t equa l t o ze ro LCI:ause they

a r e not known before i n i t i a l time TO.

Line 13. TTIP ( I )

Line l!. SSP ( I )

The computer program then checks t h e op t ion v a r i a b l e OPPRIN. I f OPPRIN = l.,

a l l the aerodynamic d a t a i s l i s t e d a s follows:

PPHIE ( I )

AALPFZ (J)

AALPME (J )

AAM (K)

I f OPSYM = O . , t h e fo l lowing iierodynamic d a t a i s l i s t e d

CCY(I,J,K)

CCLL (1, J , K)

CCLN(I,J,K)

I n t h e above aerodynamic c o e f f i c i e n t a r r a y s , AALPFE(J) i s a s s o c i a t e d with CCA,

CCN, and CCY; AALPME(Jj i s a s s o c i a t e d with CCLM, CCLL, and CCLN.

I f OPDA = I., the damping aerodynamics i s l i s t e d .

PPHIDE ( I )

AALPDE (J )

AAMD (K)

CCLLP ( I , J ,K)

CCmQ(Iy JyK)

CCLNR ( I , J , K)

GOODVEAR AEROSPACE C O I C O . A I I O I

GER- 16047

The aerodynamic a r r a y s a s s o c i a t e d w i t h t h e d e c e l e r a t o r t h e n fo l l ow .

AALPPE (I)

(1)

CCAP (I)

CCNP ( I )

ccm (I)

A f t e r 'he l i s t i n g of t h e i n p u t d a t a , t h e computer program b e g i n s n u m e r i c a l l y i n t e -

g r a t i n g . A t T = T and a t p r ede t e rmined t ime i n c r e m e n t s , t h e f o l l o w i n g d a t a i s 0

p r i n t e d o u t .

L i n e 1. T , X, XD, XDD, FX, CAY V , TENS, XP, XPQ, XPED, F a , CDAP, CMP

L i n e 2. TXB, Y , YD, YDD, FY, CN, AM, LT, YP, YPD, YPDD, FYP, CNP, AMP

L i n e 3. TYB, Z , Z D , ZDD, FZ, CY, DYPR, TPD, ZP, ZYD, ZPDD, FZP, TYPB, DYPRP

L i n e 4. TZB, PSIE, PSIDE, PSIDDE, QPSI, CLN, ALPE, O W E , PSIPE, PSIPDE,

PSPDJIE , QPSIP , TZPB , ALPPE

L i n e 5. GAME . THEE, THEDE, THEDDE, QTHE, CLM, PHIIE , OMYBE, THEPE , THEPDE, THPDDE, QTHEP, TPDXB, GAMPE

L i n e 6. CHIE, PHIE, PHIDE, PHdDE, QPHI, CLL, PHIAE, OMZBE, KS, CLLP,

CLMQ, CLNR, TYPDRB, PULAN

L i n e 7. MPAL, WAS, DMD, QMAXPB, IXPB, IYPB, SPD, SPY SPRU, SPRL, TINT,

TNINY, T F I , XPBDI

Whel-I t h e s i m u l a t i o n r e a c h e s HHH o r TTT, t h e computer w i l l write o u t "RUN ENDED

BY CONSTRAINTS". It w i l l t h e n a t t e m p t t o r e ad i n more d a t a c a r d s , t o i n i t i a l i z e

f o r a n o t h e r s i m u l a t i o n , s t a r t i n g w i t h i n p u t item "y". I f t h e r e a r e no d a t a

c a r d s a v a i l a b l e , t h e program w i 11 CALL EXIT.

GOOWEAR C O ~ P O ~ ~ T I O ~ AEROSPACE

SECTION 4 - NUMERICAL SOLUTION

For the most general type r i g i d body, t he re a r e s i x second order d i f f e r e n t i a 1

equations, coupled i n the acce lera t ion terms. These s i x equations can be wr i t t en

and solved simultaneously using the PIVEP,T subroutine. PIVERT uses Gauss elimina-

t i o n with complete pivot ing t o obtain the l a r g e s t diagonal elements. After solving . . f o r the acce lera t ions (ui i n equation (153)), the r e s u l t s a r e numerically i n t e -

(3) grated using Runge-Kutta , fou r th order techniques . If the forebody has the proper t ies t h a t H = P = Z = Ixyb = Ixzb - - I Y Z ~ = O*, the

equations of motion g rea t ly s implify f o r the forebody. I n t h e case of i n t eg ra t ing

t h e Euler angles , three equations remain coupled i n t h e acce lera t ion terms, end a r e

separated using PIVERT. The th ree t r a n s l a t i o n a l acce lera t ions a r e already i n a

su i t ab l e form t o in tegra te immediately. A simpler s i t u a t i o n occurs i f the added

Tssses of the dece lera tor a r e neglected. A l l f i v e equations of motion a r e un-

coupled i n the second de r iva t ive and a r e e a s i l y in tegra ted by 4th order Runge-

Kutta .

GOODVEAR AEROSPACE C O . C O ~ . l I O Y

k t

GER- 16047

P i

SECTION 5 - PLOTTING ROUTINE

i I f OPPIllT 1. , e leven v a r i a b l e s a r e saved i n a r r a y s . A t t h e end of t h e simula-

t i o n , any o r a l l of t h e s e v a r i a b l e s ;re p l o t t e d by c a l l i n g PLTRAJ and s e t t i n g the

a p p r o p r i a t e arguments. PLTRAJ was o r i g i n a l l y w r i t t e n f o r use on a CALCOMP 563 i t plo ' ter and 750 t ape d r i v e . It has been modified f o r use a t M.S.F.C. where 8

SC 4020 p l o t t e r i s t h e p r e f e r r e d p l o t t e r . The o r i g i n a l PLTRAJ w i l l p l o t up t o

4 v a r i a b l e s ve r sus time on one graph f o r each c a l l t o PLTRAJ. The modified PLTRAJ

f o r t h e SC 4020 p l o t t e r p l o t s only one v a r i a b l e versus time per p l o t ; t n e r e f o r e

f o u r p l o t s w i l l be made i n s t e a d of one f o r each c a l l t o PLTL4J. Two hundred d s t a

p o i n t s a r e p l o t t e d on each graph p e r v a r i a b l e .

GOODVEAR AEROSPACE C O I C O . . l I O ~

SECTION 6 - ENGLISH TO METRIC C O I ~ S l O N

The computer program operates i n e i t h e r Fnglish or Metric un i t s . Fhe program input

and output i s i n Englisk un i t s unless t h e op t icn paraae te r , OME;TRC, i s s e t equal t c ,

1. I f OMETRC = 1. the input and o ~ t p u t i s i n t h e Metric Systom. A conversion

t ab l e from Engiish t o Metric !.s given b e l a ! f o r comonlg used engineering parameters.

ENGLISH TO METRIC CCINVERSION

REFERENCE NASA SP 7012

*EXACT

FJRCE

LENGTH

MASS

SPEED

PRESSURE

volume

ffim

ACCELERATION

INERTIA

TORQUE

DENSITY

v i scos i t y

SPRING CONSTAFT

(LR) X 4.4482216152605* = (1 ) MEWTON

(FT) X .304802~0 * = (1) METER

(SLUG) x 14.5939029 = (1) KILOGRAM

(FT/SEC) X ,3048 = ( 1 ) METERS SEC

(r.a/$ ) x 4: .882e58 = ( 1 ) N E " V I T O N / ~ ~ E R ~

( F T ~ ) x .02831684659* = METERS^ (F?) X .09290304* = MFTERS~

(FT/SEC~) x .3048+ = METER/SEC 2

( SL~G-I?? ) X 1.35581794 5= KILOGRAM- METER^ (FT-LB) x 1.355817948 = METER - NEWTON

(SLIG/FJ!~) X 515.379 = KILC)GRAM/METJB~

(SI~UG/FP-SEC) X 47.880258 = NLurON SEC/ME'TET~~

(LB/FT ) X 14.59390293 = NEWTON/METER

* ESract Numbers - No rolmd o f f s

mjsec

a/m2 01 3 2

m

m/sec 2 2

kg-m

GOODVEAR AEFICSPACE c O n P O n . l l O l

CER- 1604 7

SECTION 7 - CONCLUSIONS AND RECOMMEN'DATIONS

The i o l l o w i n g conc lus ions and recommendations a r e made f o r u s e of t h e 6+3 P.0.F.

computer progr3m.

1. The (6+5,\ DOF l oads a s se s smen t computer program s h o u l d be used p r i m a r i l y t o

a n a l y z e t h e loads induced on a wobbling o r s p i n n i n g body when t h e body i s

s t a b i l i z e d by t h e deployment o f a drogue pa rachu te .

2. A f t e r t h e body h a s been s t a b i l i z e d by t h e drogue , f u r t h e r pa rachu te deployments

(main c h u t e s ) shou ld bc ana lyzed u s i n g tt p l a n e r (3+3) DOF computer program.

The (3+3; program shou ld be used because o f t h e f o l l o w i n g r ea sons :

a . The (3+3) program i s f a s t e r and e a s i e r t o u s e t han t h e (6+5) program. 0

b. Terminal d e s c e n t w i t h ,he forebody p i t c h a i ~ g l e e q u a l t o 290. p r e s e n t s

no mathemat ica l s o l u t i n n problent u s i n g t h e (3-t3) WF program.

3. It should be noted h e r e t h a t t h e (6+5) DOF p r o g r a s h a s a mathemat ica l s i n g u l a r i t y

p o i n t a t a forebody p i t c h a n g l e o f 'go0. To permi t passage through t h i s p o i n t

t h e s i x forebody a c c e l e r a t i o n s a r e f r o z e n a t t h e i r l a s t va lu2 when t h e p i t c h

a n g l e i s i n t h e r e g i o n o f 89.8O ( < 90.2"' T h i s ' ~ d u c e s some e r r o r i n t h e

t r a n s l a t i o n a l c o o r d i n a t e s and a t t i t u d e of t h e forebody, b u t i t has been shown

t o be s m a l l f o r normal v e l o c i t y pas se s througn t h i s p o i n t . A t ime count.:

(TNINY) f o r t h e t ime s p e n t i n t h i s r e g i o n i s a program ~ ' ~ t p u t .

CHAPTER I V - PRNRAM LISTING AND SAMPLE COMPUTER RUN

The fol lowing prc:ram l i s t i n g i s f o r t h e Univac 1108 a t M.S.F.C. and adapted from

che IBM 360 l i s t i n g used by Goodyear Aerospace Corporation.

The sample problem, SRB S t a b i l i z a t i o n by 54' Drogue Palachute , r e p r e s e n t s t h e

depJoyment of a 54' drogue f r o r a SRB which i s wobbling and flyiklg broadside

t o t h e wino vector. The t r a j e c t o r y of t h e SBB is near ly v e r t i c a l . The drogue

is s t a r t i n g t o i n f l a t e and i s s t r e t c h e d ou t normal t o the SRB c e n t e r l i n e . The

drogue has one s t s g e o f r e e f i n g (0.82 of f u l l open a rea ) .

Some of tile xiore important i n i t i a l cond i t ions a r e given i n t h e t a b l e below.

- d - 4) Pl7 The output from the samp1.n problem s t a r t s on Page 57, and s e l e c t e d por t ions of

I A t l t i t u d e F t

Vel city i:t/Sec

A n g l e o f A t t s c k Deg.

F l i g h t Path Angle 3eg.

Body Axis Rates

P i t c h DegISec

Yaw Deg/Se c

Rol l DegISec

r-- -0 Ins

t h e t r a j e c t o r y a r e found s t a r t i n g on page 137. -

19,000.

553.

90.

- 85.

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a a a a a a a a a o o a a a o o

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c r c c r -u a a L z o o x a a a a 0 -

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m u rL - a - Q - - - * -- x - 0 - t * J a % - a - 4 - - - J o r , o * - . , x - w - * - W L O Y ) -L - u r n = N - - ~ m - r m x - - a -u - K O - w a - 3 a r w e - r - % C * - m c - = - I & - L d U & U a - - 3 - -0 - 3- - J a * 3 z r - _ I L U C u a W a 3 o o x o w X , O o - = a + - , .=a -. - - 3 - N O - c u o ~ a - a - ~ - m - - m a - A - - - x - - - - a & - - - a * a a o w m * - - L O O - n o a c n o a = - - * - 0 - - --- - 0 0 - a * O - a * O C ~ L ~ * u m L m o - c n a c - C - N W -- 3 J J O 3 1 - a - u - c - a x u L

m a - - w x - 0 - - N U - - 6 - r - c n - - N - -*t 0 - X C A - C z L - W - - Q x - O L - P C - - - 0- c - 2 0 3 ~ - P L Q L X U - - m a - 3 O I 3 N O - 0 u a - * * w a - I O U - N O - - - x -r - u o a - x a 0 L r A " J d a - m x 0..

n r u - - ~ I U - - z - a n - - + a - U C - a e m a - u r n - 1 0 - II)-A - r a x o r - u - - x r a ~ X L - P a -* I L Q _ U O I X L - - Q - 0 - - e s w - L O - - - r - H A * - a - - - a m - - 3 P W O 0 - N O = * l l - 6 - 4 Z O C - -::?

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- , 0 - 0 0 - -. - - 0 - 0 - - - - 0 " . . . . . . . . o o . . - - 0 0 . . - - * O O - . * - O O . r e r e O C ) . e # - e O

a 1 n 0 . 0 ~ 0 0 n 0 - 0 - 0 0 ~ o - O r 0 0 O O n b * 0 0 rn o n o r c r o * 0 r o a o o i

- u a p r mm t o o 9 1 m n o o N @ * 0 0 0 r I + w o o o ~n P ~ L I I , f t ~ t Q O a t h 0 W e 0 C N O * c l * C - 0 0 b O z ? g : @ : z ? g

i ' - O N O Q 0 0 * o n o o o o t o o o m o o * o m o a 0 0

. O D O . . . D N O . 1 1 ; . . 0 h - . . Q 0 r e . . . ,

1 '

z a @ 9 r o y r t 8 8 I @ @

i . . . O C e . . . . . . . .o m 0 , O n N - o t o a - - o r r c o r n ~ . 0 r h o t o * * - h . . 0 0 - h . . o m (

m t n r a t n ~ I ! - c u m - h e 4 0

n - n - 1 ' 1

8 , I i . o . 0 * 0 n . t . O t O h

W W * ; C I - O C O N 0 Q - C C e l . cn D Q C 0 ~ 4 0 0 0 0 r P ~ O O C * . . . . . . . . . . . . . . . . . . . W C O L r L a * 8 h C O O * - o 9 r a c o o o n a - m w o o.-o w 0

- 4 - r -

.> m o m o o n o a0 0001.1 w u N - - a o o . h - = m o o -

mu-( e r n - n o o t . J ) - Q 3 o . r &&-...a 0 .* .. e m o . . . . . mrn

S * A X X O t o m @ o n o m @ o > * O ( L I H m - o 9 m - o 9

n * M l o o o n m * t o o o n n - 0 0 0 0 . m - * t o o .

m - m o o t m . e o o . r , O . . . . . . . . . . . , v 0 0 9 8 o * * I ) , m - o 4 m -n 9

W W Y O m t n o r o n a o r n o s o r 0 0 0 0 O O W V h O . , C I Q O O O . O D O R n o - o m o n r o w e + on

* - r r r 0 -

h on o n c - * O t a a - 0 r r e - 0 0 0 0 I.-n n - N * r C 0

0

r r r o r 0 0 - 0 0 0 n 0 0 o a t t o e - N * r m

r r r r r O O 0 0 * 0 * 0 0 m a - t o o * - 0 - 0 n t o o - C m - a - a

. r r r 0 0 O O O I G O B - - 0 0

r r r 0 0 0 C oa O L O O * 1 - - 0 0

r . . O h - 0 r

II) O t 0- t - n 0 I m o n * r 0 - C . . O * -

a t n r - c u m w -

Y Y U 'V. c - 0 x o m * - u c * x t x L C A * o o o m

= a * a 0 0 0 - - * n 0 0 r . U - C I O O * r * r r r n - * a II) -*

. - 0 o r .- - o* on0 r -ma a ; * m * * * h . ' n u

C I . ' I

N 0-0-00 ' t on oh o r , * o m o - 0 . 0 . o r . - I * - - n , * ' i , i

8

* or on o o 0 o* 00 00 a o t on o r , . - 0 . . . n 0 1 0 1 t n t

II) I . . l l o m O t

0 0.. 0. 0 . 0 * 0. Odl * .. r

e t n a - -.

U Y U I O O O I 030. .

0 o* 0 -00 t O* OC 0 0 O 0-0111 0 0 * o m o r o o r o* 0 - 0 0 m 0 9 0 - 0 0 ~ o r , o o o . * o o o m o r a o * o o o - r r r r . .. 0 . . . r r r r - t J 8 l 1 1 I C) * # a @ * + m a t a n t a n

a a f . . m

a

r . r O* or) m on o* 0 - n w e e o r ,

1 .. r m a - a -. -

I

l - o r on a 0- olm 0 . n 9 o o o m

a 0 . 0

0 r r 0 - 0 4 O O h 0 0 0 . - * a n o n

O r r - r m n

'dl I.

REFERENCES

1 ) Goldstein, H . , Classical Mechanics, Addison-Wesley Publishing Company

Inc . , Seventh Printing, 1960.

2) Wells,DaveA.,LagrangianDynamics,SchaumPublishingCompany,

N.Y., N.Y., 1967.

3 Korn, G . A., Mathematical Handbook for Sc i en t i s t s and Engineers,

McGraw-Hill Inc . , 1961.

4 Doyle, G. R . , Jr . , Three Dimensional Dynamics of Two Bodies Connected bv

an E las t i c Tether - Six Degrees of Freedom Forebody and Five Degrees o f

P-;ecZz= !?n==lnr=r21r~ CER- 15957, July 1973.

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