(THRu| i ,o.A?-(COZ · 1. 2 Basic Equations of Unsteady Motion 1.3 Initial and Boundary Conditions...

i,_: _ i _ °

i /

/V66 2494g

_ - (THRu|

(COZ

,o.A?- _Accession No.

t

A

S ID 66-46-I

STUDY OF LONGITUDINAL OSCILLATIONSOF PROPELLANTTANKS AND

WAVE PROPAGATIONS IN FEEDLINES

Part I--One-Dimensional Wave Propagationin a Feed Line

31March 1966 NAS8-11490

Prepared by

Henry Wing, Staff Investigator GPO PRICE $

Clement L. Tai, Principal InvestigatorCFSTI PRICE(S) $

II

Approved by Hard copy (HC)

.i---- /../.----,).. _ _ Microfiche (MF)

. _" If 653 July 65

F.C. Hung/Assistant Director, Structures and Dynamics

NORTH AMERICAN AVIATION, INC.SPACE and INFORMATION SYSTEMS DIVISION

https://ntrs.nasa.gov/search.jsp?R=19660015657 2020-07-22T20:25:02+00:00Z

/

4

0

Accession No.

A

31 March 1966

i S I D 66-46-I

STUDY OF LONGITUDINAL OSCILLATIONS

OF PROPELLANT TANKS AND

WAVE PROPAGATIONS IN FEED LINES

Part i--One-Dimensional Wave Propagation

in a Feed Line

NAS8-I1490

Prepared by

Henry Wing, Staff Investigator

Clement L. Tai, Principal Investigator

Approved by

F.C. Hung/Assistant Director, Structures and Dynamics

P NORTH AMERICAN AVIATION, INC.

SPACE and INFORMATION SYSTEMS DIVISION

. • TECHNICAL REPORT INDEX/ABSTRACT

D,

ACCESSION N UM B E RL__J_ DOCUMENT SECURITY CLASSIFICATIONUnclassified IITITLE OF DOCUMENT

"A Study of Longitudinal Oscillations of Propellant Tanks and

Wave Propagations in Feed Lines"

LIBRARY USE ONLY

C. L. Tai. M. M. H. Loh, S. A......... Fun H. Win S. Uchi am___aa

CODE ORIGINATING AGENCY AND /

OTHER SOURCES DOCUMENT NUMBER

, NAA ISID 66-46 (Five Volumes)

P"_'ET'CA"_ON DATE ] CONTRACT NUMBER

31 March 1966 1 NAS8-I1490

DESCRIPTIVE TERMS

fluid dynamics

pipeline dynamics

wave propagation

longitudinal oscillations

shell dynamics

ABSTRACT

The work done under this contract is being published in five separate parts:

(SID 66-46-i) Part I One-Dimensional Wave Propagation in a Feed Line

(by H. Wing and C. L. Tai)

(SID 66-46-2) Part II Wave Propagation in an Elastic Pipe Filled with

Incompressible Viscous Fluid

(by M. M. H. Loh and C. L. Tai)

(SID 66-46-3) Part III Wave Propagation in an Elastic Pipe Filled with

Incompressible Viscous Streaming Fluid

(by S. A. Fung and C. L. Tai)

(SID 66-46-4) Part IV Longitudinal Oscillation of a Propellmlt-Filled

Flexible Hemispherical Tank

(by H. Wing and C. L. Tai)

(SID 66-46-5) Part V Longitudinal Oscillation of a Propellm_t-Filled

Flexible Oblate Spheroidal Tank

(by S. Uchiyama and C. L. Tai)

FORM 131-V REV. 6-64

NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

m,

FOREWORD

This report was prepared by the Space and Information Systems

Division of North American Aviation, Inc., Downey, California, for the

George C. Marshall Space Flight Center, National Aeronautics and Space

Administration, Huntsville, Alabama, under Contract No. NAS8-11490,

"Study of Longitudinal Oscillations of Propellant Tanks and Wave Propaga-

tions in Feed Lines, " dated January 6, 1965. Dr. George F. McDonough

(Principal) and Mr. Robert S. Ryan (Alternate) of Aero-Astrodynamics

Laboratory, MSFC, are Contracting Officer Representatives. The work is

published in five separate parts:

Part I - One-Dimensional Wave Propagation in a Feed Line

Part II - Wave Propagation in an Elastic Pipe Filled With

Incompressible Viscous Fluid

Part HI - Wave Propagation in an Elastic Pipe Filled With

Incompressible Viscous Streaming Fluid

Part IV - Longitudinal Oscillation of a Propellant-Filled Flexible

Hemispherical Tank

Part V - Longitudinal Oscillation of a Propellant-Filled Flexible

Oblate Spheroidal Tank

The project was carried out by the Launch Vehicle Dynamics Group,

Structures and Dynamics Department of Research and Engineering Division,

S&ID. Dr. F.C. Hung was the Program Manager for North American

Aviation, Inc. The study was conducted by Dr. Clement L. Tai (Principal

Investigator), Dr. Michael M.H. Loh, Mr. Henry Wing, Dr. Sui-An Fung,

and Dr. Shoichi Uchiyama. Dr. James Sheng, who started the investigation

of Part IV, left in the middle of the program to teach at the University of

Wisconsin. The computer program was developed by Mr. 1%.A. Pollock,

Mr. F.W. Egeling, and Mr. S. Miyashiro.

o,°

- 111 -

SID 66-46-1

• NQRTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

b

ABSTRACT

In this report, the dynamics of elastic propellant /

feed lines with streaming fluid are analyzed using the

nonlinear one-dimensional unsteady compressible

fluid flow equations. The viscous effect of the fluid

is assumed to be expressible inthe form of a hydrau-

lic resistance corresponding to turbulent flow.

Transfer functions relating the pressures, velocities,

and the corresponding phase angles were derived

from the linearized equations for the perturbed state.

The governing equations were also converted into a

system of four first-order ordinary nonlinear differ-

ential equations by the method of characteristics.

These were then transposed into finite difference form

and solved numerically on a digital computer. Impulse,

step, and sinusoidal velocity disturbances were con-

sidered in the numerical solution for a feed line with

and without a control device. The physical param-

eters of the liquid hydrogen line on the Saturn S-II

engine are used for the numerical calculations to

illustrate the results.

- V -

SID 66-46- 1

NORTH AMERICAN AVIATION, INC. SPACE and INF_0RMATION SYSTEMS DIVISION

g

C ONT ENT S

NOMENC LAT URE

INTRODUCTION

SECTION i. EQUATIONS OF UNSTEADY FLOW IN PIPES

1. 1 Variation of Cross-Sectional Area of a Pipe as a Function

of Pressure

1. 2 Basic Equations of Unsteady Motion

1.3 Initial and Boundary Conditions

I. 3. 1 Air Chamber

1.3.2 Surge Tank

i. 3.3 Tank-Type Damper

SECTION 2. SOLUTION OF EQUATIONS OF UNSTEADY

MOTION BY LINEARIZATION

2. I Boundary Condition at X = _ With Pipe Vibration

2. 2 Physical Constants and Results of Linearized Analysis

SECTION 3.

UNST EADY

CHARACTERISTICS .

3.1

3.2

3.3

3.4

3.5

SOLUTION OF NONLINEAR EQUATIONS OF

MOTION BY THE METHOD OF

Specified Time Intervals

Selection of Mesh Ratio

Computational Problem

Interpolation Formulas

Finite Difference Approximation

SECTION 4.

4.1

4.2

4.3

COMPUTATIONAL PROCEDURE

Initial Conditions.

Boundary Condition at Left End of Line

Boundary Condition at Right End of Line

4. 3. l Velocity Impulse Disturbance

4. 3. 2 Velocity Step Disturbance

4. 3. 3 Sinusoidal Velocity Disturbance

4. 3.4 Feed Line With Pressure Regulating Device

4 ...... Modified Form of Boundary Condition

4.3. 5 Numerical Procedure

Page

ix

4

5

6

7

7

8

11

17

19

21

23

23

25

25

28

31

31

32

33

34

36

36

38

39

39

- vii -

SID 66 -46- 1

NORTH AMERICAN AVIAT!ON, INC.


SECTION 5. COMPUTATIONAL APPROACH TO SPECIFIC

PROBLEM .

SECTION 6. NUMERICAL RESULTS.

6. 1 Discussion of Results.

C ONC LUSIONS

AREAS FOR FURTHER STUDY

REFERENCES

APPENDIX A. TRANSFER FUNCTIONS

APPENDIX B. VELOCITY AND PRESSURE DATA

Page

41

43

43

47

49

51

53

67

- viii -

SID 66-46-I

NORTH AMERICAN AVIATION, INC. SPACE and 1NI_)RMATION SYSTEMS DIVISION

P,

4

a

a m

A

A d

AhA

S

A.pl

C

D

E

f

f(t)

g

g(t)

h

II , 12

J

k_, k t

K

K 1

L

NOMENC EAT URE

Velocity of pressure wave

Missile acceleration

Cross-sectional area of feed line

Cross-sectional area of tank damper

Cross-sectional area of feed line wall

Cross-sectional area of surge tank

Cross-sectional area of pump inlet

Feed line restriction coefficient

Feed line diameter

Young's modulus of line material

Darcy-Weisbach friction coefficient

Prescribed function for pressure at left end of line

Gravitational constant

Prescribed function for mass flow rate at right end of line

Wall thickness

Functions defined by Equation (48)

Square root of minus one

Laminar and turbulent resistance coefficient

Modified bulk modulus defined by Equation (13)

Bulk modulus of fluid

Length of feed line

- ix -

SID 66 -46 - I

NORTH AMERICAN AVIATION, INC.

I

SPACE and INFORMATION SYSTEMS DIVISICrN

m I , m 2

P

P

P

Q

Ql(t)

r

R

s

t

u

u

v(tl

V

x

z

Z

(1

A

e

Roots of Equation (46)

Fluid pressure

Reference pressure

Laplace transform of fluid pressure

Mass flow rate

Mass discharge rate

Internal radius of line

Reynold's number

Laplace transform variable

Time

Average fluid velocity over line cross section

Reference fluid velocity

Fluid discharge velocity

Volume

Spatial coordinate along axis of feed line

Axial displacement component

Laplace transform of axial displacement component

Density of line material

Incremental change

Damping coefficient of bellows

Liquid elevation in surge tank from equilibrium level

Angle between axisoffeed line and normal to flight path of

missile

4

- x -

SID 66-46-I

NO RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

KK,

I)

¢o

Subscripts

A, B, C,

R,S

Constant characterizing pressure regulating device

Viscosity of fluid

Poisson's ratio of line material

Density of fluid

Function defined by either Equations (3) or (4)

Frequency

Pertain to points according to Figure 2

Designate value at steady-state condition or value at x = 0

Designate value at x = L

Superscripts

o Initial guess value

i Value of first iteration

- xi -

SID 66-46-1


INT R OD UC T ION

Pressure wave propagation in a pipe filled with fluid has been investi-

gated extensively, from the early studies of surging phenomena to the recent

work on unsteady flow in blood vessels and hydraulic control systems

(References 1 to 17). Various simplifying assumptions were employed in

each particular problem to facilitate a mathematical solution. In general,

the formulation is not applicable to any problem with conditions other than

those appropriate to the particular problem.

The classical theory of surging phenomena or "water hammer," formu-

lated by Joukowsky (Reference l), is based on the linearized one-dimensional

Navier-Stokes equation for an inviscid fluid. It assumes that the pressure is

uniform across any section of the pipe and that the deflection of the pipe is

equal to the static deflection due to the instantaneous pressure in the fluid.

This theory has since been extended to two-dimensional space (References 12

and 13), with some of the simplifications made in the Joukowsky theory

relaxed (Reference 14). As far as the relations between fluid velocity,

pressure, and hoop stress are concerned, however, the improved theory

gave essentially the same results as Joukowsky's theory, which had been

c onfirmed experimentally.

Recently, the method of characteristics was applied successfully to

the numerical solutions of one-dimensional "water hammer" analysis

(References 16 and 17), including the nonlinear terms for the convective

acceleration and fluid friction. Similar nonlinear differential equations

describing the unsteady motion of a viscous compressible fluid in tubes in

terms of a hydraulic resistance were also applied to the self-induced oscil-

lations of heavy fluids (Reference 18). In this reference, the fluid density

term is assumed to be large, and the pressure gradient term is neglected,

thus making it possible to reduce the problem to the solution of first-order

quasilinear partial differential equations with nonperiodic boundary

c onditions.

The investigation of the pressure wave in the blood vessel was con-

cerned primarily with the influence of the elasticity of the walls on the

propagation of sound through the fluid medium in the vessel. Since the

physiologists considered the distensibility of the tube to be of far greater

importance than the compressibility of the fluid, the authors assumed an

incompressible and viscous fluid (References 6 to 8). The stresses and

shear are expressed in terms of the displacements, but the inertia terms

- 1 -

SID 66-46-I


are neglected. The significance of the effect of viscosity and internal

damping on the wave propagation is somewhat obscured in the authors' over-

simplification of the complicated solution.

The approaches used in the hydraulic and pneumatic control systems

are generally straightforward. Since pipe diameters are small, the

elasticity of the pipe may be neglected; and, because the flow rate is usually

low in the control system, the use of linearized equations with the laminar

friction loss (References 9 to ii) is justified. The results of the theoretical

frequency-response analysis compare quite well with the experimental data

from the test.

In the present investigation, it is assumed that the propellant line

connects the propellant tank and propellant pump. The diameter of the line

is much larger than that of hydraulic control lines or blood vessels, but

much smaller than the diameter of penstock in the lower plant. The source

of disturbance may be either the fuel sloshing in the propellant tank or the

variation of flow in the propellant pump. Another possibility is that the

changes in the natural frequency of the missile structure at different flight

times may also create unsteady flow in the fuel line.

The aim of this study is to determine analytically the general dynamic

behavior of the unsteady motion and to construct a reasonable model that

can be applied directly to the overall system analysis of longitudinal

oscillations of large missiles.

-2-

SID 66 -46-I

NbRTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

Pb-

SECTION i. EQUATIONS OF UNSTEADY FLOW IN PIPES

If we consider the motion of the fluid in a pipe as a whole and

introduce the _average values of velocity u, density p, and pressure p over

the cross-section A, the basic equations describing the unsteady motion of

a viscous compressible fluid in an elastic tube in terms of a hydraulic

resistance are the law of conservation of mass

a a

a--x(puA). = - -_ (pA) (i)

and the law of momentum

a a ap(puA) + _-x (puZA) = -A _x - A_(u) (Z)at

The function _ (u) is taken to be linear for laminar flow

_(u) =_u, for R< Z000 (3)

and quadratic for turbulent flow

f Z

_(u) =_--_ pu , for R > Z000 (4)

where _ is the viscosity, D the inside diameter of the pipe, R the Reynolds

number, and f the Darcy-Weisbach resistance coefficient.

To the above equations must be added the state equation for compres-

sible flow

dp = -_I dp (5)

where K 1 is the bulk modulus of the fluid which is defined as the change in

pressure per unit change in specific volume.

-3-

SID 66-46-I

NORTH AMERICAN AVIATION, INC. SPACE end INFORMATION SYSTEMS DIVISION

If the pressure variation is small, Hooke's law can be considered

valid for a liquid, i.e.,

p-P_KI ))P=Po +(6)

Po

dp = -K-_Idp(7)

where the subscript "o" designates the variables at steady state.

I. 1 VARIATION OF CROSS-SECTIONAL AREA OF A PIPE AS A FUNCTION

OF PRESSURE

It will be assumed that the pipe has a thin elastic wall and is made of

a material for which Hooke's law holds true. Let the initial pressure in the

pipe be p and the radius be r. Then the tangential and axial forces per unit

width are given by

Ne No

N o = 0-@h = pr

N =0-h=Cx x ipr

(8)

Figure i.

and the tangential strain by

1 (_8 pr i) (9)_0 =_E - VO-x) = Eh (1 - vC

where C I = 1/2 for a pipe fixed at one end and free to move throughout the

rest of its length, C 1 = v for a pipe anchored against longitudinal movement

along its length, and Cl = 0 for a pipe having expansion joints between

anchors throughout the length of the pipe.

-4-

SID 66 -46 - I


or

When the internal pressure is varied, the tangential strain varies by

5r 1

5_ 0 = m = N (1 - vC1) (pSr + rSp)r Eh

_r =

r25p (1 - vC1)

Eh - rp (1 - vC1)

Multiplying the above equation by Z_r and denoting C = 1 vC1, it becomes

5A- 2rACSpEh - Crp (10)

From Equation (9), Eh =(C/e0)Pr. Since c 0 << 1, the term Crp is

negligible in comparison with the term Eh, and hence

CDA

dA- Eh dp (11)

1.2 BASIC EQUATIONS OF UNSTEADY MOTION

Combining (i), (5) and (11) yields

[1 +CD] 8p Ou I_l+ CE] Dp (IZ)

Define

: v + = p (13)a

where a is the velocity of pressure wave propagation in a fluid medium con-

tained in an elastic pipe. Since the velocity of sound in the fluid is

a2 = dp/dp = K1/p, the term CD/Ehis a correction of K1, accounting for

the elasticity of the pipe wall. Equation (12) now becomes

From (I) and (2), we have

ap o___Ep = z O___u8--_+ u 8x - pa 8x

(14)

Ou 8u 1 @p 1 _(u)o-[ -;Ox-;

(15)

_

SID 66 -46 -1


Equations (14) and (15) are the basic equations of unsteady motion in

an elastic pipe. For small pressure variations, the value of p in the above

equations may be considered to be equal to Po of the steady state.

In the case where the effect of missile acceleration a m and the angle, O,

which the pipe makes with the horizontal are to be taken into consideration,

the term a m sin 0 has to be added to Equation (15); i.e. ,

0u 8u _ 1 8p 1 _(u) + a m sin e (15a)7_- + u Ox p Ox p

I. 3 INITIAL AND BOUNDARY CONDITIONS

It will be assumed, in this treatment of unsteady fluid motion in pipes,

that up to the moment t = 0 the fluid is in steady motion; i.e. ,

o) = u0

p(x, o) = Po

(16)

The boundary conditions depend on the nature of the disturbance of flow

at the boundaries. The following boundary conditions are expected to be

encountered in the present problem: (I) at one end of the pipe the pressure

is given as a function of time (as a particular case, this pressure may be

constant); (2) at the other end of the pipe a device is connected which varies

the flow rate of the fluid with time according to some known law--this may

be a pump, turbine, etc. This device may be connected either directly to

the pipe or through a chamber serving to regulate the flow rate or to

diminish the pressure oscillations. The boundary conditions in such cases

as these can be written functionally as

x = o p(o,t) = Po + f(t) (17a)

8x = £ K _Q(_,t) + Q(£,t) = g(t) (17b)

where f(t) and g(t) are given functions, with

f(t) = g(t) = 0 at t < 0

Q is the mass flow rate and K is a positive constant characterizing the type

of chamber in those cases where a chamber is present.

4

-6-

SID 66 -46-I


The boundary conditions of three cases at x = % when the pipe is

connected to a chamber designed to reduce the pressure oscillations will

be discussed in the following paragraphs.

I. 3. I Air Chamber

Let p andVbe the pressure and volume of the air in the vessel, and

let Po and Vo be their steady flow values. If the compression of the air in

the chamber is considered as isothermal, then

pV = PoVo (185

During unsteady flow the increase in the quantity of liquid in the vessel will

be

d(V - V)o

Po dt - (puASx= % - Ql(t) (19)

where (puA)x=_ = Q is the mass flow rate of the liquid out of the pipe and

into the air vessel, and Ql(t) is the mass discharge rate of the liquid from

the vessel.

For small pressure variations, it is permissible to use the linearized

continuity equation. From (18), (14), and (135 it follows:

d___y= _.P°V° 8p Vo K 8Q

dt 2 0t Po 8xp PoAo

(20)

The substitution of (20) into (195 gives the following boundary conditions

atx= _:

VKo 8Q

Po Ao OxQ = Ql(t) (21)

1.3.2 Surge Tank

If the same notation is used, then the rate of increase of the liquid

volume in the surge tank is

dV

Po-_ = Qx=_ - Q1 (t) (22)

-7 -

SID 66 -46 - i


where

V= qAs

rl = the increase of liquid level in the surge tank

A = cross-sectional area of the tanks

The increase in pressure of the liquid in the tank will be

V

P - Po = Po gq = Po gs

dvA AIK o __ _ s Op _ s 'Pdt pog at pog _ _-Ao x]o

Hence the boundary condition at x = £ will have the form

As K aQ

A pog axo

+ Q = Ql(t)(23)

1.3.3 Tank-Type Damper

The variation of the mass rate of the liquid contained in the damping

tank is

dpVd--t-- = (pAu)x=£ - Q1 (t) (24)

where V is the volume of the tank, Qx=# = (pAu)x=,_. is the mass of liquid

passing from the pipe into the tank, and Ql(t) is the mass discharge rate.

If the tank volume is V = AdL with the variation of tank length being

assumed small, then

dAd(25)dpV dd__td---_-=V + pL dt

-8-

SIX) 66 -46 - I

NORTH AMERICAN AVIAT!ON, INC. SPACE and INFORMATION SYSTEMS DIVISION

From (ZO),

then

(7), and (11)

0p_ K cgQ cgp Po 0p

at Po Ao 0x at K 1 at at

aAd CdDdAd 0p

Eh at

dpV -KL% 1 8Q

where Ad,

restraint constant of the damper, respectively.

D d and C d are the cross-sectional area, diameter, and the end

The boundary condition for x = £ will be

(26)

Ad /_ll OQCdDd_

KL_- ° + _-_dT_-x + Q: Ql(t)(Z7)

-9-

SID 66 -46-I

N'O RT H AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

SEC TION 2. SOLUTION OF EOUATIONS OF UNSTEADY

MOTION BY LINEARIZATION

For small pressure variations, the equations of unsteady motion in an

elastic pipe with turbulent friction losses from (14) and (15) are

Op ap z o____u (z8)8-_ + u _xx = -Po a 8x

8u 8u I 8p f 2

8-T"+ u -_x - Po 8x 2D u (29)

Let

P = Po +AP

u = u +AuO

or written simply as

or written simply as

P=Po+P

U=U +U

O

(30)

Equations (28) and (29) yield the following sets of equations for the steady

and the perturbed motions.

8Po 2 8UoU _" -o --_-x Po a Ox (31)

uo

8u fo I 8Po 2

-- = U

8x Po 8x 2D o(3Z)

_- + (u° + u)\ ox +_x = -_oa \_ +(33)

+ u) _ + - \axP o

+ u)2 (34)

-I1-

SID 66 -46 -I


Considering the steady-state values of Po and u o to be known and their

variations along the x-axis small, then the perturbed equations after neg-

lecting terms of higher order will be

ap ap Z au (35)a-l-+ u ....o 8x Po a ax

Ou 1 ap8u + u ...... ku (36)

at o ax Po ax

where

fuo

k =--=kD t

o

for turbulent friction loss and

32Vk - -----_ - k%

p Do o

for laminar friction loss

Using Laplace transform and the assumption of zero initial conditions

p(x,O) = O, u(x, O) = 0 (37)

Equations (35) and (36) become

ap 2 8U (38)sP+uo _x Po a 8x

8U 1 8P (39)(s + k)U + u .....o ax Po 8x

or upon differentiation,

82P aP 2 82U

u _ + s 8x Po a 8x 2°Sx

(4O)

uo

8U 1 8ZP

--+ (s + k) _xx = - p% 8x 2

(41)

- 12-

SID 66-46-i

N'O RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

After some algebraic manipulation between Equations (38) through (41), we

have two decoupled differential equations, one for U(x, s) and one for P(x, s).

(az -uZ) --aZU-uo(ZS +k)_%U- s(s +k) U = 0o 8x 2

(42)

(az - U Z) 82P 8P

o 8x Z Uo(ZS +k)_x - s(s +k) P = 0 (43)

Let

U(x, s) = A(s) emx

mxP(x, s) = B(s) e

where m is determined by the characteristic equation

(44)

(a2 2- u 2)m - u (2s + k)m - s(s + k) = 0 (45)o o

/22Uo(ZS +k) +VUo k + 4aZs(s +k)

m -- (46)2(a 2 - u 2)

O

Hence

= m2xU A 1 em Ix + A 2 e

P = B 1 emlx + B 2 em2 x

(47)

The relations between A and B can be determined by putting (47) into

(38) and (39)

I I xIA2 oa2 Im°xAlPoaZml+Bluoml+Bl s eml + mz+Bzuomz+B2 s e _ --0

x m,_ xluoml+_ml+Al(S+k) e mt + 2Uom2+_m2+A2(s+k e -

Po Po=0

13-

SID 66 -46 -I


Since

mlxe #0,m 2 x

e #0

we have

p°aZml 1

B I =-AI Luoml +sj

[Po(Uoml + s +k)]B 1 = _ A 1[ ml

Hence

2

po a m

u m+so

- PoUo

s+k

+ Po m

Poa2m2 ]

BZ = -AZ [Uomz + sj

p oluomz + s + kJB 2 = _ A 2 m 2

_[I1,1 for m = m 1

[i2, for rn m 2(48)

-4

and

U(x, s) = A 1 e mix + A 2 e m2x [

!mlx m2xt

P(x, s) = -AII 1 e A212 e J

(49)

Let the boundary condition at x = 0 be

U(o, s) = Uo (s) P(o, s) = Po (s)

When applied to (49), it yields

U =A I +A 2o

P = -Allo 1

A 1 =

P + Uol 2o

12 - II

P + Uol Io

A 2 = ii I2

14 -

SlID 66 -46 - 1


Hence

Atx=£

U(x,s) =

P(x,s) =

P + Uol 2 P + Uol 1o mlx oe

I2 - I1 IZ - I1

rn2xe

II(P +UoI 2) I2(P + UoI I)o mix oe +

12 - I 1 12 - I 1

e m2x }(5O)

[i2 ii i [ 1U_ U _ eml£ __ em2_ 1 (eml£= o I2 I1 - Iz - I1 + Po I2 - I1 - em2"_)

[iii2 [i2 iI ]P£ = Uo 12 -T 1 (em2_- eml + Po 12 -I 1 em2 - 12 - I1 eml _

Let G(s) be the transfer matrix defined by

(51)

G(s) =

/ I2 mi£ I1 m2£e e

Iz - I 1 12 I 1

I 112

IZ - II

(em2"6 _ eml

then (50) can be written as

I (eml £ _ m2_ )

12 _ ii e

Iz em2_ Ii eml

Iz - II Iz - II

(5Z)

U£

P£

= G(s)

U o

Po

(53)

which represents the dynamic characteristics of the unsteady friction flowin an elastic line.

-15

SID 66 -46- I


From (53), we have the following relations:

Uo(S ) I2 + 1 em2_ - I1 + 1

(54)

P_(s)

Po(S)

(Iz - Ii) e (ml+m2)i

ml_ ZII + I2 e - I2 + II e m2

(55)

U£ e(ml+m2) _

Uz(s) _ (I2 - Ii)

)Uo(S) UZ UZI2 + 1 em2'Z - I1 + e ml

(56)

where m I, m 2, II and I2 are functions of s as defined in Equations (46)

and (48).

Substituting s = jco in (54) to (56), we can compute the magnitude and

phase angle ofU (jo_)

0

(a) the ratio of velocity to pressure deviations, p (j¢0)' at x = 0,versus frequency o

(b) the ratio of pressure deviation at x = £ to pressure deviation at

P_(jo_)

x = 0 p (jco)' versus frequency.O

(c) the ratio of velocity deviation at x = £ to velocity deviation at

u_(j_)

x = 0, U (J_• ), versus frequency.O

-16-

SID 66-46-1

NORTH AMERICAN AVIAT!ON, INC. SPACE and INFORMATION SYSTEMS DIVIS.ION

The boundary condition at x = % is

a a 7.(_, t)AIE _xZ(£, t) + _ _-_ = p£(A-Api ) (62)

where A is the cross-sectional area of pipe, AI, the cross-sectional area of

pipe wall, Api the cross-sectional area of pump inlet and _ the coefficient of

damping of the bellow• The Laplace transform of (6Z) is

a

AIE _x Z(x, s) Ix=£ + _sZ(x, s) Ix=£ = p_(s)[A-Api ] (63)

Sub stituting

Z(x, s)Ix= _ = B(s) sinh_-_sxlx=_ (64)

into the above equation and solving for B(s), we have

B(s) :P_(s) [A - Api ]

V_ v_-_s s_nn_- sAlE s cosh £ +_ s " -%/'_-S

(65)

Hence, (64) becomes

z(t,s)=P_(s) [A - Api ]

S [AIE_//-_"_ cot h%/_ s_ + _]

(66)

The transfer function relating the velocity at x=% to the pressure at

x=¢, is

A - As Z(_,s) _ pi

P£(s) AIE%/_ cot h%/'_ s£ +{

This factor is to be added to the term U£/P_, in Equations (54), (55), and

(56). The deviations in magnitude and phase which occur near the natural

frequency of the longitudinal vibrations of the pipe can be seen from the

computational results shown in Appendix A.

(67)

- 18-

SID 66-46-i

NO=RT H AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

2.Z PHYSICAL CONSTANTS AND RESULTS OF LINEARIZED ANALYSIS

The following physical constants are based mostly on information

applicable to Saturn S-II Stage.

a

Constant

Table I

internal radius of pipe (D = 2r)

length of pipe between tank and

P

K1

h

E

(1

A

A h

Api

C

v

V

uo

Values for LH 2 Line

0. 333 ft.

20.83 ft. (outboard)

pump

velocity of pressure wave

1a =

cD+ E---/-

density of fluid at steady state

bulk modulus of fluid

thickness of pipe wall

Young' s modulus of pipe wall

density of pipe wall

cross-sectional area of pipe

31.9Z ft. (center

3700 ft/sec

0. 136 Ib-secZ/ft Z

2. 15 x 106 ib/ft Z

-31.83 x l0 ft

4.32 x l09 ib/ft 2

14.8 ib-secZ/ft 4

0. 348 ftZ

cross-sectional area of pipe wall

cross-sectional area of pump inlet

restriction factor of pipe

Poisson' s ratio of pipe wall

damping coefficient of bellow

viscosity of fluid

steady-state fluid velocity

0. 00383 ft2

0. Z56 ft2

0.85

0.30

0 (assumed)

-50. 029 x 10

51.5 ft/sec

Ib - se c /ftg

-19-

SID 66-46-I

NO RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISIQN

U_

P

k t

Constant

Table I (Gont)

steady-state velocity-pressure

ratio (determined from the steady-

state pressure versus flow curve)

laminar resistance coefficient

kf = 3Z_/po D2

Darcy-Weisback resistance coeff.

Values for LH 2 Line

-41.55 x 10 per sec

-41.55 x 10 per sec

turbulent resistance coefficient

fu0

kt - D

O. 0775 (outboard}

O. 059 (center}

5.98 per sec (outboard)

4.54 per sec (center)

With the above information the frequency response equations (54),

(55), and (56) were programmed on the IBM 7094 for both LOX and LH 2 lines.

The numerical results for the outboard LH 2 line of S-If stage are illustrated

in Appendix A, Figures A-I through A-12. The magnitude and phase angle

of Uo(jco)/Po(j_) versus 0_ are shown in Figures A-I through A-4 according to

the type of hydraulic resistances and with or without the effect of line

vibrations. Similarly, the magnitude and phase angle of P_(jc0)/Po(j_0)

versus _0 are shown in Figures A-5 through A-8. The magnitude and phase

angle of U_(j_)/Uo(j¢o ) versus ¢0 are shown in Figures A-9 through A-I2.

20-

SID 66 -46-I


SECTION 3. SOLUTION OF NONLINEAR EQUATIONS OF UNSTEADY

MOTION BY THE METHOD OF CHARACTERISTICS

The transient pressure of unsteady flow with a turbulent resistance in

an elastic pipe undergoing an acceleration am requires the solution of the two

nonlinear partial differentialEquations (14) and (15a}.

310 + 3p 3uat U_x = -paZ a-_" (68)

au + ua___u= lop_ f_L.u 2 + am sin8 {69)3t 3x p 3x 2D

The governing equations cannot be solved analytically. However, these two

equations can be transformed into a set of ordinary differential equations by

the method of characteristics and then solved numerically.

Define a characteristic curve of Equations (68) and (69) to be a curve

along which a solution of Equations (68) and (69) may have discontinuities in

the derivatives of the velocity and pressure. Suppose that such a curve is

given by

x = x(s), t = t(s) (70)

and that u and p are prescribed at each point of this curve. Then u and p

become functions of s and we have a system of four equations in four unknown

partial derivatives Ux, ut' px and Pt" These four equations are

3__u+ uOU + 1 3p _ f u 2 + a m sin O8 t 3x p 3x 2D

ZOU _p+ 3ppa _xx + at U_xx : 0

(continued on next page)

-21 -

SID 66 -46-1


8u dt + 8ua-i- = du

8_p_pdt + 8pat _x dx = dp

(7l)

The discontinuity relation requires that there be at least two distinct

sets of values of these derivatives which satisfy Equations (71). The con-

ditions under which Equations (T1)will admit of more than one solution

require that

1

i u o ?-

0 pa 2 1 u

dt dx 0 0

0 0 dt dx

=0 (72.)

and

1 u 0 - fu2 + am si'neZD

0 pa z 1 0

dt dx 0 du

0 0 dt dp

= 0 (73)

Expanding these determinants (72) gives

dx-- - U =-l-adt

(74)

and (73) gives

du f u2 1 (dx _tp)d--i-+ _-_ -amsinO +----_ _-- upa

=0 (75)

- 22 -

SID 66 -46-1

NORTH AMERICANf

AVIAT!ON, INC. SPACE and INFORMATION SYSTEMS DIVISION

Putting the relation (74) into (75), it becomes

du _D i dpd--t-+ ug - a m sin O ±_-_ d-_ = 0 (76)

In these equations where the ± sign appears, the plus sign is associated with

the characteristic C + and the minus sign with the characteristic C-.

Equations (74) define the slope of the characteristics C + and C- as

a function of velocity of flow, u, and the velocity of pressure wave, a.

Equations (76) are the compatibility equation relating the velocity u and

pressure p along the characteristic lines C + and C-.

Equations (76) are two separate, total differential equations with t as

independent variable and u and p as dependent variables. If u = u(x,t) and

p = p(x,t) satisfies Equations (68) and (69) then Equations (74) determine

two families of characteristic curves C + and C" in the x,t plane belonging

to the solution u(x,t) and p(x,t). These four Equations (74) and (76) can be

solved by finite difference approximations with appropriate initial and bound-

ary conditions.

3. 1 SPECIFIED TIME INTERVALS

There are two ways in which the characteristic curves C + and C- may

be used to obtain an approximate numerical solution to the original partial

differential equations. One method involves the use of a grid of character-

istics and the other makes use of specified time intervals. In using the

former method, it would be difficult to arrange the computations so that the

points of intersection of the characteristics occurred at values of x and t

on a rectangular grid. The use of specified time intervals has one main

advantage over the use of a grid of characteristics in that it provides results

of the flow field at different times. Hence, from a programming viewpoint,

the ordering of the computations would appear to be easier by this method

than by the use of a characteristic grid, since the values of the dependent

variables are known at predetermined points and the additional information

of the corresponding x and t coordinates does not have to be recorded.

For this reason, this is the approach which will be used in the present

analy s is.

3. Z SELECTING MESH RATIO

The first step in using the method of specified time intervals requires

selecting a mesh ratio_t/Ax for the finite difference approximation. The

- 23 -

• SID 66-46-1


selection of an "admissible" mesh ratio plays a vital part in the success of

the numerical process. It insures that the difference equations actually do

approximate the differential equations so that anumerical solution of the

former should be a good approximation to the exact solution.

The region bounded by the two characteristics C + and C-, and a third

non-characteristic curve such as PRCS shown in Figure g is a region in

which a unique solution to the original partial differential equations would

exist provided conditions along RCS are prescribed. Thus, it is only logical

to select a mesh ratio such that the solution will lie within this "zone of

influence. " This condition will be satisfied if the mesh ratio is less than the

slope of the characteristic curves as given by Equation {74), since geometri-

cally the mesh ratio plays the role of "finite difference characteristics. "

Under this conditon, the finite difference scheme should be a valid represen-

tation of the differential Equations (14) and (15a).

If the variation in the velocity of wave propagation is assumed small

so that it may be taken equal to its steady-state value a o, then for

At i-<- (77)

Ax u±a0

we would expect the solution of the difference equations to tend to that of the

differential equations as At --- 0. Geometrically, this means that R must

lie within AC and S within BC in Figure 2. The condition given by Equa-

tion (77) also insures the stability of the numerical process. Since the

_Lh

C

\C ÷

x

Figure 2.

- 24 -

SID 66-46-I


velocity u < < a o, the criteria used in the present study for selecting the

mesh ratio was

At 1

Ax a0

The accuracy of the finite approximations deteriorates if the mesh ratio is

much less than 1/a o although their solution will be stable. Optimum accuracy

is obtained when the finite difference characteristics coincide with the con-

tinuous characteristics of the governing differential equations. For a system

whose true characteristics are curved such as in the present case, the

optimum accuracy cannot be obtained using a fixed rectangular grid since

the finite difference characteristics must necessarily have smaller slopes

than the true characteristics to ensure stability of the numerical process.

3.3 COMPUTATIONAL PROBLEM

The computational problem which must be dealt with is illustrated in

Figure Z. On the rectangular grid in the x,t plane as shown here, the

velocity and pressure are known at each mesh point, such as A, B, and C

for time tl, either as given initial conditions or as a result of a previous

stage of the calculations. The problem is to determine the conditions

(velocity and pressure) at a typical point P corresponding to time ti + I.

The coordinates of point P in the x,t plane are therefore known. Through

this known point, we pass the two characteristic curves C + and C-, inter-

secting the line t = ti at points R and S respectively. Both the velocity

and pressure are unknown at the three points P, R, and S, and the space

variable is unknown at the latter two points, giving a total of eight unknowns.

3.4 INTERPOLATION FORMULAS

In the analysis to follow, the various variables associated with a given

point will have that point's designation used as a subscript, such as uA,

PA, XA ..... etc.

Since the characteristics C + and C- pass through points R and S

respectively, expressions are needed for the velocity and pressure at these

two points in order to compute the condition at P using these two curves.

This is done by fitting a parabola through the known values at A, C, and B,

and then interpolating. For example, let us consider the case of calculating

the velocity uR at point R. (See Figure 3.}

-25 -

SID 66-46-1


uC

uR

u A

//

XA x R x C

Figure 3.

Using the second order equation

u(x) --a + b(x - Xc) + c(x - Xc)2 (78)

and applying it to the three points x A, x B, and x C where the velocities are

known, we obtain three algebraic equations for determining the coefficients

a, b, and c. Applying Equation (78)to x = x C shows immediately that

a=u C(79)

Application to the remaining two points yields

uA = a + b(x A - Xc) + c(x A - xc) Z (80)

u B = a + b(x B - Xc) + c(x B - xc)Z (81)

-26 -

SID 66 -46-I

NdRTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

Since successive mesh points along the x axis are _x units apart, we can

express xA and xB in terms of x C as

x A = x C -_x

x B = x C + Ax

On substituting these two relations into Equations (80) and (81), they become

uA = a -b • _x + c(_x) z (82_)

uB = a + b • Ax + c(_x) Z (83)

Making use of the result given by Equation (79) and solving Equations (8Z)

and (83) simultaneously, we find that

1

b - aax (UB - UA) (84)

C - 2 (UA + UB - ZUc)(85)

With the three coefficients a, b, and c determined, we apply Equation (78)

to x --x R and obtain

]

UR : Uc + Z_x (UB - UA)(XR - Xc) "_z(m×)

Z (UA + UB - 2Uc)(xR - Xc)2

(86)

The same functional form given by Equation (78) is also assumed for

the pressure variation between the three adjacent points xA, XB, and x C.

Under this assumption, the coefficients given by Equations {79), (84), and

(85) are valid provided that the velocities uA, UB, and u C are replaced

-27-

SID 66-46-I

NORTH AMERICAN AVIATION INC. SPACE and INFORMATION SYSTEMS DIVISION

respectively by the corresponding pressures PA' PB' and PC" Thus, the

following quadratic interpolation relationships can be written down

immediately:

i i

PR = PC + 2-_--_x(PB - PA){XR - xC) +2(a×)

Z {PA + PB - ZPc)(XR - xc )Z(87)

1 1Us --uc + z-N-£x(uB " UA)(xs - xc) +

Z(Ax)Z (UA + uB- ZUc)(X S - xc)Z

(88)

I

PS = PC + _ {PB - pA)(X S - Xc) + 1 (PA + PB - 2Pc)(Xs - Xc )22(Ax)z (89)

These four equations form a necessary part in the computational scheme.

The other four equations needed to obtain a solution are given by the charac-

teristic Equations (74) and (76).

3.5 FINITE DIFFERENCE APPROXIMATION

Let us confine our immediate attention to the two characteristic equa-

tions associated with the characteristic C+; i.e., we consider the plus sign

in Equations {74) and (76). These two equations are transposed into finite

difference form by replacing the differential coefficient by a divided difference

over an interval and replacing the other variables by the arithmetic mean

over the interval. This finite difference approximation is one of second

order, and hence it is compatible with the interpolation formulas given in

Equations {86) through {89).

On applying the foregoing procedure to Equations (74) and {76), and

transposing, we obtain

uR + Up o)XR = Xc - 2 "+ a At (90)

f Up)g PP - PRUp - UR+-_-_(UR+ _t-a sin 8.At + = 0 (91)m Po ao

- 28 -

SID 66-46-1


It is assumed that the friction coefficient is a constant, and furthermore,

that the variations in density and velocity of wave propagation are sufficiently

small so that they may be taken equal to their respective value at steady-

state conditions. The subscript "o" refers to steady-state values.

Similarly, the corresponding equations associated with the character-istic C- are

_(u S + Up o)XS = XC 2 a At (92)

PP - PSUp) 2Up - u S + (uS + At- a sin O • At - O (93)m Po ao

The above four equations, together with the four interpolation formulas,

Equations (86) through (89), form a set of eight nonlinear simultaneous

algebraic equations for the determination of the eight unknowns XR, UR,

PR' xs, us, PS, up and pp. The problem now is to organize a computational

tional procedure suitable for solving these eight equations on a digital

computer. Without resorting to a simultaneous solution of these eight equa-

tions, the most practical way would be by iteration.

- 29 -

SID 66 -46-I


SECTION 4. COMPUTATIONAL PROCEDURE

One method of obtaining the conditions at point P by iteration using the

nonlinear algebraic equations (86) through (93) is the following:

1. Make an estimate of the conditions at R, S, and P by assuming

that they are the same as those existing at C. In other words,

o o o

u R = uS = Up = u C

o o o

PR = PS = PP = PC

The superscript "o" refers to the initial guess, or zeroth

approximation.

Z. Using these estimates, calculate x_ and x ° from Equations (90)

and (92), respectively.

1 1 13. Making use of the results in step 2, calculate u ' PR' Us' and PSfrom Equations (86) through (89) inclusive, respectively.

4. With the values obtained thus far for the first iteration, solve

Equations (91) and (93) simultaneously for u 1 and p_.

The foregoing process is continued until successive values for both

velocity and pressure at point P are within the prescribed tolerance.

4. 1 INITIAL CONDITIONS

In order to start the computational procedure, as outlined above, we

must have the initial conditions corresponding to time t = 0. The initial

conditions in the propellant line are taken as those corresponding to the

steady-state condition. Due to friction losses along the length of the pro-

pellant line, the pressure at each mesh point on the x-axis must necessarily

be different from one mesh point to another. The friction coefficient can be

calculated from the experimental data that gives pipe resistance versus flow

rate. It is tacitly assumed that this friction coefficient obtained under

steady-state conditions is applicable to transient phenomena and also remains

a constant throughout the line.

-31 -

SID 66-46- 1

NORTH AMERICAN AVIATI. O N, INC. SPACE and INFORMATION SYSTEMS DIVISION

Within the framework of one-dimensional flow, the velocity u in the

line must be a constant in the case of steady flow. This is immediately

apparent from the continuity Equation (1) when one invokes the condition of

steady flow whereby the flow field is independent of time, and hence, reducing

it to

_-x-x(pUa ) = 0

Since we are considering propellant lines of constant cross-sections and

ignoring any variation in p, it is readily seen that u must be a constant.

Under this condition, the equation of motion given by Equation (15a) reduces

to

Po

2U

dp + _..f. o-- -a sin Odx D g m

= 0

On integrating this equation and applying the result to two consecutive mesh

points, we have

Pl

U o

- P2 = Po _ _--a sinm(94)

where the subscript "1" designates the upstream mesh point and the subscript

"2" the downstream mesh point. In the absence of the acceleration term

a m sin e, Equation {94) is equivalent to the well-known formula used in

hydraulics to calculate the head loss in a pipe.

4. Z BOUNDARY CONDITION AT LEFT END OF LINE

For the boundary condition at the left end of the line, we assume that

the pressure in the tank remains a constant. This condition is very nearly

fulfilled since the behavior of the system is of interest only during the first

few seconds following the initiation of the disturbance. The value of this

constant pressure is taken equal to the steady-state value just prior to the

introduction of the disturbance in the line.

- 32 -

SID 66-46-I


A C B

Figure 4.

To calculate the velocity corresponding to this given pressure, we may

proceed in the following manner (refer to Figure 4):

1. Assume initially that the velocities Up and u S are equal to u A ,

i.e., u_ = u_ = uA .

2. Using these assumed values, calculate x_ from Equation (92) with

x C replaced by xA = 0 .

1 13. For this value of

x S, compute u_ and p_ from Equations (88) and(89), respectively.

4. Solve for uI from Equation (93) using the results of Step 3.

This process is repeated until two consecutive values of up agree within the

specified accuracy.

4. 3 BOUNDARY CONDITION AT RIGHT END OF LINE

It is assumed that the disturbance in the system originates at the right

end of the line where the pump is located. Some of the various boundary

conditions which may be considered have been already discussed in Section

1. 3. Before considering these boundary conditions where some sort of

regulating device is present in this preliminary investigation, we will merely

specify a particular disturbance. Since any sort of disturbance will affect

the flow rate and hence the fluid velocity, we will consider the disturbance

as a change in velocity. The disturbance is taken in such a manner that it

causes a decrease in velocity at the point of origin, and hence in turn pro-

duces an increase in pressure.

- 33 -

SID 66-46- I


4.3. 1 Velocity Impulse Disturbance

Art

Figure 5.

If the righthand end of the feed line is provided with a valve and this

valve is suddenly closed a certain amount and then reopened instantaneously,

we would have an impulse disturbance. The magnitude of this disturbance

will depend upon the amount of closure before reopening takes place. The

maximum impulse will occur, of course, when the valve is closed completely

before it is reopened. (See Figure 5.)

For a velocity change that takes the form of an impulse, the conditions

at the right end must be recalculated before proceeding with the solution in

the x,t plane. The corresponding pressure change Ap due to a Au change

in velocity can be found from momentum considerations. The result is

Ap = Po ao Au (95)

Therefore, the conditions which we have at the right end at time t = 0 due

to the impulse are

U ----UO

P = PO

(96)

where u and Po are the steady-state velocity and pressure, respectively,• O

prtor to the introduction of the impulse.

- 34 -

SID 66 -46-I


A C B

C +/

Figure 6.

In calculating the two unknowns at point P when t > 0, another relation-

ship is required besides the equation for the characteristic curve C +. This

additional relation is obtained from the condition that the discharge at anytime must obey the orifice law. Hence,

1/2Ip_\

Up = u (-_) (97)

where the quantities with a bar over them represent reference values.

They may be thought of as the "equilibrium conditions" about which the

transients caused by the disturbance oscillate. In the case of an impulse,the steady-state values are used as reference.

The calculation of Up and pp may proceed as follows:

1. Assume that initially the velocities at R and P are equal to that

at B (refer to Figure 6); i. e. ,

o o

u R = Up = u B

2. Calculate x_ from Equation (90) with x C replaced by x B = • .

- 35 -

SID 66 -46 - 1


1

4.

ICompute uI and PR from Equations (86) and (87), respectively.

1 1

Solve Equations (91) and (97) simultaneously for Up and pp .

As before, this procedure is repeated until two successive values of both

Up and pp are within the specified tolerance.

4.3.2 Velocity Step Disturbance

We define a step disturbance as that caused by an instantaneous partial

closure of the valve with the valve held at this new position. Thus, the step

disturbance in this sense does not mean that the velocity is maintained at

this new value for all time, but the position of the valve itself represents

a step function. The position of the valve creates a new "equilibrium

position" which the flow will eventually attain after the transients due to the

disturbance have subsided.

As in the case of a velocity impulse, we must first calculate the new

conditions at the right end due to the step disturbance before proceeding with

the solution. These new conditions are computed in exactly the same manner

as was done in the case of the impulse. The pressure change is given by

Equation (95), and the new conditions by Equation (96). The only difference

between the two cases is that for the step disturbance we use the new con-

ditions given by Equation (96) for reference values in Equation (97) instead

of the steady-state values. With this one change, the computational proce-

dure for calculating the velocity and pressure at point B on the boundary

for t > 0 is identical to that outlined for the case of an impulse.

4.3.3 Sinusoidal Velocity Disturbance

The procedure given in Section 4.3. Z for handling a step disturbance

may be extended to handle any other type of disturbance or pulse shapes

simply by replacing the given disturbance by an equivalent series of step

functions. By equivalent, it is meant that both the composite step function

representation of the given disturbance and the disturbance itself have the

same area under the curve.

Suppose that in the absence of propagation effects, the disturbance

under consideration took the form of a sine function OAB as shown in

Figure 7. We first divide the half-time period tA into suitable time incre-

ments At. From the computational standpoint, it is convenient to choose

AT equal to either At, the time interval used in the difference equations, or

a multiple thereof. Within each interval, the ordinate for the "step" is

selected by either visual inspection or other means such that the area under

the "step" is equal to that under the actual curve. In this way we replace

- 36 -

SID 66 -46-I


4

A B

Figure 7.

the sine curve by the "staircase"approximation 01234 A...B, and this

function is the one which willbe dealt with in our subsequent discussion.

Corresponding to the instantaneous velocity change AU a for the

first step "012" of the staircase approximation, the pressure change is

obtained from Equation (95) as before. Since it is assumed that the dis-

turbance takes place only after steady-state conditions have prevailed in

the line, the new conditions at the righthand end after the disturbance has

taken place is given by Equation (96). The calculation for the pressure

and velocity at the right end for 0 < t < AT now proceed in the manner

given in Section 4.3. l where the new conditions existing there are used

as reference values. This iteration process is continued for as many

steps as necessary until the specified tolerance on the dependent variables

have been met. We now arrive at point "2" on our staircase approxi-

mation so that step "234" comes under consideration. The response of

the system due to this step input during the time AT < t < 2 AT is

handled in the identical fashion as for the previous step in which

the reference values to be used now in Equation (97) are furnished by the

- 37 -

SID 66 -46 - I


conditions existing at time t = AT with the effects of AU b included. This

same computational procedure is continued for each succeeding "step. "

4.3.4 Feed Line With Pressure Regulating Device

In Section i. 3, the boundary condition at the right end was derived for

the case of a pressure regulating device, such as a surge tank, air chamber,

or tank damper, being present in the line. In each of these three cases, it

was shown that the boundary condition can be expressed as a first order

partial differential equation in the form

OQ +K-_- Q g(t) at x = L (98)

where Q is the mass flow rate entering the regulating device, and g(t), a

prescribed function, is the mass flow rate discharging from the device. The

constant g is dependent upon the type of regulating device under considera-

tion, and the value it assumes for the various devices is given in Section 1.3.

It would appear that one approach of incorporating Equation (98) as a

boundary condition into the righthand end of the feed line would be to inte-

grate it and obtain a relation between Q(t) and g(t) . For any prescribed

disturbance function g(t) then, Q(t) will be known and hence also the velocity

u(g,t) and the discharge velocity are known. Therefore the computational

procedure given in Sections 4.3. I, 4.3.2, or 4.3.3 may be employed,

depending on the type of disturbance under consideration. Since Equation(98)

is a boundary condition, what we are seeking is a particular solution.

For any given disturbance, there are a number of particular solutions which

would satisfy Equation (98). Without knowing the general form of the closed

form solution for the pressure and velocity along the feed line itself, it is

impossible to tell which one of the particular solutions is the correct one.

Therefore the only other alternative is to express this equation in finite

difference form and solve it simultaneously with the other appropriate

equations at the right end, numerically.

The derivative 8Q/Ox appearing in Equation (98) can be expressed by

a divided difference over an interval Ax. However this "average" slope of

Q over the interval may be quite different from the slope 8Q/0x at x = g.

Although this difficulty may be alleviated somewhat by dividing the last inter-

val into smaller intervals, the numerical procedure would be complicated.

The reason for our difficulty in obtaining a representative finite

approximation for Equation (98) is that although it dictates the condition

which must prevail for all values of time t , it involves the derivative aQ/ax

- 38 -

SID 66 -46-I


which in the x, t plane is in a direction normal to the t axis. Thus if the

boundary condition can be rewritten so that any derivatives present are

with respect to time instead of the spatial variable x , then it can be put in

finite difference form solely in terms of the variables at x = g. Having the

equation in this form should permit a more accurate numerical treatment

of the boundary condition at the right end of the feed line than would be

pos sible otherwis e.

4.3.4. 1 Modified Form of Boundary Condition

The boundary condition given by Equation (17b) stems from the con-

tinuity in mass flow which must exist at the junction of the device and the

feed line. For each of the three pressure regulating devices considered,

this boundary condition can be rewritten in the form

A K @--_-P+ Q = g(t) (99)o at

by utilizing the relation between 0Q/0x and 0p/at given in Section I. 3.

The constant K characterizes the damping device and is equal to K divided

by the modified bulk modulus K defined in Equation (13), and A o is thecross-sectional area of the feed line.

If our attention is focussed on feed lines of constant cross sections

such as in the present case, and the variation in the fluid density is assumed

sufficiently small so that it is essentially equal to its steady-state value, it

is advantageous from a computational point of view to rewrite the quantity Q

in Equation (99) in terms of the fluid velocity u. Since Q = 9oAou ,

Equation (99) takes the form

_P

_T + u = v(t) (100)

The quantity v(t) will be equal to the discharge velocity, provided the dis-

charge cross-sectional area is the same as that of the feed line; otherwise

it will be simply proportional to it. The proportionality constant in this

case is equal to the ratio of the discharge cross-sectional area to the feed

line cross-sectional area.

4. 3. 5 Numerical Procedure

Equation (I00) may be transposed into a finite difference equation

using the technique given in Section 3.5. Employing the same notation

- 39 -

SID 66-46-I

NORTH AMERICAN AVIATI. ON, INC.SPACE and INFORMATION SYSTEMS DIVISION

as shown in Figure 6, the resulting equation takes the form

PP - PB 1 1

B) = (Vp+V B) (101)

The three quantities with the subscript B are known from the calculation for

the previous time interval, and since v is a prescribed function, Vp is also

known. Thus pp and Up are the two unknowns in Equation (101). The com-

putational procedure to determine these two unknowns may proceed in the

following manner.

I. Assume initially that u_ = u_ = u B

O

2. Calculate xR from Equation (90) with x C replaced by L .

3. Compute u_ and p_ from Equations (86) and (87) respectively.

4. Solve Equations (91) and (101) simultaneously for up and p_ .

This process is continued until successive values for both Up and pp are

in agreement within the specified limits.

- 40 -

SID 66 -46-1


SECTION 5. COMPUTATIONAL APPROACH TO SPECIFIC PROBLEM

We are now in a position to handle a specific problem under the bound-

ary conditions described previously. Referring to Figure 8, suppose that we

have decided to divide the propellant line into five equal incremental lengths

so that the left and right boundaries coincide with A0 and A5 respectively.

One way in which the computation can be carried out would be the following.

From the given steady-state condition at A5, the velocity at A0 through A5 isknown and the corresponding pressure at these mesh points can be calculated

from Equation(94). If the disturbance under consideration involves an instan-

taneous velocity change, such as a step function, the conditions at mesh point

point A 5 must include the effect of the velocity change before we proceed with

the calculations. The incremental pressure change corresponding to this

incremental velocity change is computed from Equation(95). Superimposing

these incremental changes in velocity and pressure according to Equation (96)

onto the steady-state values at A5, we obtain the new conditions which mustprevail there because of the disturbance.

In the case where the feed line contains a regulating device and the

disturbance is a sinusoidal function, the foregoing calculation is not neces-

sary since the disturbance was not approximated by a series of step functions

and hence did not involve any instantaneous velocity changes.

t

t 2

t IBo 81 8 2 8 3 8 4

t=O

A0 AI A2 A3 A4 A5

Figure 8

85

X

- 41 -

SID 66 -46-I


Having the conditions for all mesh points at t=0, we now "march out" to

time tI, and proceed to calculate the conditions at these mesh points. For

the prescribed boundary condition of constant pressure at B o, the velocity

may be determined by the procedure outlined in Section 4. 2. The conditions

at mesh points B l through B 4 are found by the procedure given in Section 4. 0.

At mesh point B 5 where both the velocity u and pressure p are unknown, we

proceed in the manner described in Section 4. 3. I, 4. 3. 2, 4. 3. 3, or 4. 3.4,

depending on the type of velocity disturbance and whether a regulating device

is present or not. As long as suitable boundary conditions are prescribed at

both ends of the feed line, the solution can be "marched out" as far as

desired.

- 42 -

SID 66 -46-I

NO:RTH AMERICAN AVIATION, INC.SPACE and INFORMATION SYSTEMS DIVISION

SECTION 6. NUMERICAL RESULTS

Some numerical results obtained from the application of the foregoing

numerical procedure to the center liquid hydrogen feed line on the SaturnS-II

engine are shown in AppendixB, Figures B-1 through B-48. The numerical

values used for the various parameters in the analysis are given in Table I

(Section Z. Z). Calculations were carried out for the feed line with and without

an air chamber when subjected to all three types of velocity disturbances

discussed previously. The air chamber was selected over the other two

damping devices discussed since it combines the space economy provided by

the tank damper and, to a lesser degree, the effectivenesswithwhichthe surge

tank alleviates pressure changes. Thus it represents a device which might

well be incorporated into a propellant feed line.

Two amplitudes were considered for each disturbance expressed as a

percent of the steady-state velocity. The amplitudes chosen were 2 and 5 per-

cent since they correspond approximately to the minimum and maximum

variation expected in the pump flow of the S-II respectively. Four frequen-cies were selected for consideration in connection with the sinusoidal dis-

turbance. They were 7. 20, 9. 70, 11. 70, and 21. 11 cps, corresponding

respectively to the first four structural modes of the spring mass "POGO"

model of the Saturn S-II engine at 95 percent burntime.

6. 1 DISCUSSION OF RESULTS

Pressure and velocity time histories were obtained both at the right

end and at the middle of the line for the case when the feed line does not con-

tain a regulating device. The results corresponding to the impulse disturb-

ance and the step disturbance for amplitudes of 2 and 5 percent of steady-

state velocity are shown in Figure B-1 through B-8. In all cases, since only

partial reflections take place at the right end, both the pressure and velocity

in the line regain steady-state conditions in a very short time, approxin_ately

0. 10 second, as shown by a decrease in amplitude of each succeeding "spike"

or "step" following the initial disturbance. This is to be expected due to the

stabilizing influence of the unidirectional flow. With the exception of the

numerical values obtained, the general behavior of the feed line subjected to

either a 2 or 5 percent impulse disturbance is the same. This also applies

to the step disturbance.

The results for the sine disturbance are given in Figure B-9 to B-24

inclusive. The sine disturbance introduces an additional parameter, input

43-

SID 66 -46-1


frequency, which does not appear in either the impulse or sine disturbance.

For the liquid hydrogen line under consideration, it takes the pressure wave

caused by a disturbance at the right end 0. 03455 second to complete one

cycle, corresponding to a frequency of 28. 96 cps. The initial reflected pres-

sure wave of a sine disturbance with this input frequency will arrive at the

right end just when the disturbance is zero. Under this condition, the maxi-

mum fluid velocity is equal to the initial steady-state velocity and the trend

of velocity variation is toward this value. In the present four cases, the

input frequencies were allbelow 28. 98 cps so that the disturbance and the

reflected waves are out of phase with one another, causing the general veloc-

ity trend to increase with time. This was true both at the right end and at

the midpoint of the feed line. The pressure tends to decrease with time

but at a much lower rate than does the velocity. If the plots were carried out

for a sufficient length of time, the velocity in all four cases tends to oscillate

about some equilibrium value, as does the pressure. After the initial tran-

sients have subsided in the first cycle, the peak-to-peak variation of each

consecutive cycle for both the velocity and pressure is essentially zero.

Hence the effect of pipe friction is very small.

Figure B-25 through B-48 contain the results for all three disturbances

when an air chamber is located at the right end of the feed line. It was

assumed that the air chamber had a cross-sectional area of 3 sq ft (diameter

of approximately 2 ft) with the air occupying a 2-foot length of the chamber

initially, giving a volume V o of 6 ft3 under steady state flow. Since Po is

4,320 psf, this yield aVvalue of 4 x 10 -3 ft3/Ib. This value of _ was used

throughout the study whenever an air chamber was considered as present in

the feed line.

The results obtained for the feed line with and without an air chan_ber

cannot be compared directly to ascertain exactly how effective the air cham-

ber is in alleviating pressure surges, since in the former case the discharge

velocity is prescribed for all time while in the latter it obeys the orifice law.

This is obvious since the time history in the two cases will differ consider-

ably from one another whether the air chamber is present or not. However,

they can be compared with each other from a qualitative point of view. This

is particularly true for the pressure variation since, broadly speaking, the

general trend and maximum peak amplitudes are approximately the san_e

whether the discharge velocity is "fixed" or "free" as in the case where the

orifice law applies.

The effect of the air chamber on velocity and pressure at the right end

is illustrated in Figure B-29 and B-31 for a 2 and 5 percent step disturbance

respectively. It is seen that the pressure fluctuations which occur in the

feed line without a pressure regulating device are completely eliminated by

the air chamber and the overpressure decreases very gradually fronl its

- 44 -

SID 66 -46-I

NORTH AMERICANAVIAT!ON, INC. _,_


initial value caused by the disturbance. The pressure fluctuations actually

are much larger than for a "free" boundary since the damping influence of

the streaming fluid is much less in the fixed case. This is shown quite

clearly in Figure B-30 and B-32 by the larger pressure fluctuations in the

middle of the line. From this observation then, we see that the air chamber

is quite an effective damping device.

The results for a prescribed impulse disturbance with an air chamber

in the feed line are shown in Figure B-Z5 through B-Z8. One feature of these

results is that they are very similar to those for the step disturbance and

differ only from a quantitative point of view. This is not too surprising when

it is seen that the only difference between the two cases is that v(t); the

prescribed discharge velocity, assumes a slightly different fixed value after

the initiation of the disturbance. Since both cases have the same boundary

condition at the right end which differ only by a constant, the two results

must necessarily be similar.

Just how effective the particular air chamber is in alleviating pressure

surges is illustrated in Figures B-33 to B-48, which are the results for the

fixed sine disturbance. Considering the 5 percent sine disturbance, we

noticed that in no case does the maximum peak-to-peak pressure variation

exceed 15 psf, which is about 1.5 percent of the overpressure for the cor-

responding "undamped" case. Naturally for the Z percent case, the peak-to-

peak pressure variation is proportionately less. The air chamber also

suppresses the velocity so that in all cases it did not vary more than I/2 fps

from the steady-state velocity of 51.5 fps. The high frequency oscillations,

which appear in the velocity variation such as in Figure B-33 and in the

pressure variation as shown in Figure B-34, have no physical significance

but are mathematical in nature. As was mentioned earlier in Section 3.2, if

the time interval selected is such that the solution obtained coincides with

the true characteristics, then these oscillations will be eliminated.

- 45 -

SID 66-46- 1


CONCLUSIONS

The principal objective of the present investigation was to develop

some mathematical models for the unsteady fluid motion in a propellant

feed line which may be incorporated in a closed loop dynamic system for the

analysis of longitudinal oscillations of a large liquid rocket booster. With

this in mind, the dynamic behavior of the feed line was looked at both from

a linear and a nonlinear viewpoint. The analysis of the linear mathematical

model resulted in the transfer functions given in Appendix A. The similarityof results for both the frictionless line and one with friction taken into

account indicate that friction loss for this particular line is very small.

Since the cross-sectional area of the line is relatively large, this result is

to be expected. The inclusion of the longitudinal natural frequency of the line

resulted in a modification of the frequency response curve which is similar

to those given in Reference 10 and 11.

Some results based on the nonlinear differential equations governing

unsteady compressible fluid flow are shown in Appendix B. Although these

results cannot be compared directly with any known published data, they do

correspond to a modification of the water hammer problem encountered in

hydraulics in which the valve is only partially closed and is a function of

time. By neglecting the friction term and setting the velocity to zero at the

right end, the solution obtained agrees with the classical waterhammer solu-

tion. Hence the method provides a useful tool for inclusion in a system

dynamic study. Various disturbances were considered under two different

boundary conditions to illustrate the flexibility of the method.

- 47 -

SID 66-46-1


AREAS FOR FURTHER STUDY

The propellant feed line considered in the present analysis was straight

and of constant cross-section, which is a very special case. In actual prac-

tice, feed lines may involve a step change in cross-sectional area, branch

lines, bends, or any combination of these. Therefore, in order to obtain a

more realistic mathematical model of a propellant feed line, further study

should be undertaken to incorporate these additional parameters. The method

developed for the nonlinear model can be extended to handle the step change

in cross-sectional area or a feed line with branch lines quite readily by

using it as a "building block. " What these two cases require is a modifica-

tion of the boundary conditions at either end of the line so that it can accom-

modate the continuity condition for both mass flow and pressure at the point

of discontinuity in cross-section or junction of the branch lines.

Another area for further investigation in connection with the present

problem is a parametric study of the various pressure regulating devices to

determine how each parameter influences the dynamic response. For

example, the volume of air at steady-state flow may be varied by the con-

stant_ that characterizes the air chamber. In the case where two or more

parameters are involved, such as for the tank damper, such a study would

indicate which parameter has the greatest influence on the effectiveness of

any given device to dampen pressure surges. Going a step further, this

data can then be used to make a trade-off study, say between weight and

allowable space (total volun_e) versus damping effectiveness of a particular

device. This will provide information which can be used directly to "tailor"

a damping device for a particular feed line system and a particular n_issile.

Besides relieving the overpressure by mechanical devices, there are

other means based on the high compressibility of a gas which nlerit con-

sideration. One such method is injecting a gas, say heliun_, into the feed

line. The gas bubbles "soften" the fluid column and also act as an absorp-

tion device whenever a disturbance causes a propagating pressure wave in

the line. The other idea is to insert a gas-filled plastic bag in the annulus

between two concentric lines. The inner line, which carries the propellant,

contains a series of holes, allowing contact between the propellant and the

plastic bag. This arrangement has the effect of making the feed line highly

compliant to any pressure surges.

- 49 -

SID 66-46-I


RE FERENCES

.

.

4.

5.

6.

.

.

.

10.

11.

12.

13.

14.

Joukowski, N. Water Hammer. Proc. of Am. Water Works Asso.

(1904)

Allievi, L. Allgemeine Theorie {iber die veranderliche Bewegung des

Wassers in Leitungen. Julius Springer, Berlin, Germany (1909)

Symposium on Water Hammer, The A. S. M. E. (1933)

Parmakian, J. Waterhammer Analysis. Dover (1963)

Rich, G. R. Hydraulic Transients. McGraw-Hill (1951}

Morgan, G. W. "On The Steady Laminar Flow of a Viscous Uncom-

pressible Fluid in an Elastic Tube," Bul. Math. Bioph.,Vol. 14 (195Z)

Karreman, G. "Some Contributions to the Mathematical Biology of

Blood Circulation Reflection of Pressure Waves in the Arterial System, "

Bul. Math. Bioph., Vol. 14 (1952)

Morgan, G. W. and W. R. Ferrante. "Wave Propagation in Elastic

Tubes Filled with Streaming Liquid," J. Acou. So. of Am. (July 1955)

Blackburn, Reethof, and Shearer. Fluid Power Control. John Wiley

(1960)

D'Sonza, A. F. and R. Oldenburger.

Lines," J. Basic Eng. (Sept. 1964)

"Dynamic Response of Fluid

Regetz, Jr., J. D. An Experimental Determination of the Dynamic

Response of a Long Hydraulic Line. NASA TND-576 (Dec. 1961)

Jacobi, W. J. "Propagation of Sound Waves Along Liquid Cylinders,"

J. Acou. So. of Am., Vol. Z] (1949), pp. 120-127

Thomson, W. T. "Transmission of Pressure Waves in Liquid Filled

Tubes," Proc. of 1st U.S. Nat. Congress of App. Mech A.S.M.S. (1951)

Skalak, R. "An Extension of the Theory of Water Hammer,"

Trans, A.S.M.E. (Jan. 1956), pp. 105-116

- 51 -

SID 66-46- 1


15.

16.

17.

18.

19.

Z0.

P1.

Sabersky, R. H. Effect of Wave Propagation in Feed Lines on Low-

Frequency Rocket Instability. Jet Propulsion (May - June 1951)

Streeter, V. L. and C. Lai. "Waterhammer Analysis Including Fluid

Friction," ASCE Trans., Vol. IZ8 (1963)

Contractor, D. N. The Reflection of Waterhammer Pressure Waves

from Minor Losses. A.S.M.E. Paper No. 64-WA/FE-16

Kochina, N. N. "On Self-lnduced Oscillations of Heavy Fluids in

Tubes," PMM, Vol 27, No. 4 (1963), pp. 609-617

Lister, M. "The Numerical Solution of Hyperbolic Partial Differential

Equations by Characteristics, " Mathematical Methods for Digital

Computers, Edited by A. Ralston and H. S. Will. New York:

John Wiley & Sons, Inc. (1960)

Hartree, D. R. Some Practical Methods of Using Characteristics in

the Calculation of Non-Steady Compressible Flow. Atomic Energy

Commission Rept. AECU Z713 (195Z)

Crandall, S. H. En_ineerin_ Analysis. New York: McGraw-Hill

(1956), pp. 355-368

- 52 -

SID 66 -46-I

NdRTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

APPENDIX A. TRANSFER FUNCTIONS

Transfer functions for the outboard liquid hydrogen line on

the Saturn S-II engine

- 53 -

SID 66-46-I


.OilI111111111111II lllll III IIIlllllll lllt II

•oi411 I I I I I I I I I I IlllIlllllllllI I II I I I I IIII III I I I I I I I IIIII I I

-OSll I I I I I I I II Ill I II I I I I [ I I I! III I II I I I I I I I I!III I II I II I I I I IIII III I I I I I I I IIII I I

lllllllllllll IIIIIIIIIi Ill Ill III II Ill Illlllllllllllllll lllllllllI I I I I I I I I I I I I 1.I I I I I I 1 II I I I I I I I I I I I I IAI I I I I I I II I I l l l l l l l l l l Ill | l l l , l lt II I l I 1 III I I I I, III I I I I IIII III III I III HIll I I I 1 II 11 I I1 I I1 I ill IIII1 III II1 I I 1 I 1 I I 1 I I I I I III I I 1 I I II I 1 I I I I I I I I I 1 I III I I ! I I II I t I I I I I 1 1 I I 1 I III I I I I I I.or|

I? I I I I I I I 1 I 1 II 1 1 I I I I I I I I I l I I I II III I I I I I II i i i I I I i I I k I I I I I I I I I I I J I I 111 III I I I I

I I II I I I II II II I II I II I II I III/ I• -OO_l I I I I I I I I I I I I I I I I I I I I I I I I III I u I I I I I I

• I I I I I I I III I I I I I I I I I I I I I I I I Ill I I I I I I I I

I I I11 I Ill I I I I I 1 I I I I I I III I I I I I I I I I•o_1 I I I I I I Ill I III I I I I I I 1 I I I I I I I I t III I I I 1 I !

, I/l I _I I I I I Il I I I I ] i/l I I I I I I I I I I I I I I I l III I I l_l I I I I I

-00411 IIIII/11111111111111tt111/1 I [ I i I l t I I !

i I I ] I [ I#l I I I IM I I I_ Ill l_l i I [I II t i i I lil I I I i I I\I I l IIII II I Y _

-00Z I +_ ; i IA J I I J II]l I_;il_l l_fl;JlIil ; i I i iN I ; ]I ', I I Yl [ I , I lI It It i I, il , ] I i I_1 , ,

I T i.,FI I I J I I I I 1 t F_,,L I i I I IJl I I I ! bd II ;_FI l f I I I I t I I I I I'_,_11 L,_ l I I I I I 11 11 ] I P'_k f

.0 I._T ] I I I I f I I 1 ] I I ] I I _P-,,.i'_ I I I ! I I I ] 1 ] ; I I I I'r -I! 20 40 I0 80 lO0 120 140 IQO llO

FREQUENCY (CPS)

8O

• ll ill el OO I|O 180 tllO Ill le_' lllO Ill ttl "_ Ill tel III

FREQUENCY (CPS)

Figure A-1. The Ratio of Velocity to Pressure Deviation at x = 0 Versus

Frequency for Outboard LHg Line, Frictionless and Without Pipe Vibration

55-

SID 66-46-I

NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVI6ION

.oil

I _ [ I I IA it ] i IJI Ili II t II1_1

•ol4[ [ I ] I llill

l{',f { '""II1|1I

__ I 1 I I1[1[

• I i|i

• I I I i I I

1 i I i fit I] I i 11 IIIr ; _ /lilt;[ i i i ] 111 l_

+ J i [i ]"! I I

'_:i illi [ I I [ I

i .____ I I ] tI I 1

i , t I i i! ] ] I I I

;+.i,+J- it l.o I!I0 ZO 40

I 1 II 1 Ill I :II I llll IIIIll|l

I l 1 I I 111 I

II i ll II I llll1 I 1 I I IlnlI I I I I I 1 I It II I I 1.1 I 1 I nlII 1 1 111 llllI I [ I l I t I III

" DHt| "'It " itlI I [ I I It I11I I i [ I 111 I|1I I I I I I Ill IU[ I I I I I III l iIt I I II II[llI I I I I I [I r iI i ] i I I l/ ] I IL1 I ] I I II I [I I_IlL iL I1| 111_

"' "[I/l IfI ii _ II! I : I I I Ill II I .

II, I'I]Attll

ili I i/ll I illl_fll IIII I [ _11 I i I I I I

• t [ I I 1 11 I LliO O0 tO0 lZO 140

FREQUENCY (CPS)

II'lillllJll,l_, [ !li1 I I I I I I I I I [ I I IIII I I I I I i I I I i I I lli [ ]il 11

l I 1 I [ l I I I I i

i] Ill I1 I llllll II I 1 I I I I I l I I, , I I lI I I I I I I In I i i ] I 1

I II I I l I I I [ II I I III I I I I !I I I I 11 I I ] I • I 111 II 1I I 1 1 I I I | I 1 I 1 II1 IIllll t I Ill Ili l I I11I

I I I_ i I IiI I I I Ill il 1 I i !l I I I I I I III II [ 1I I

I I l l ] I II I [li I ! ] l11 i i I I 1 !111t[ I [ I I I [ I I I I

['lllllllllllll [ ][I111111111111/11111111[i i] i 11 l Ill ] ] I 1 11

I I I I 1 1 Ill I I 'l I ] I I[11111111111#11 1 X I I IIIIII 1111 II1111 Iil 1 I II[ I I i I I I I 1111 1 1 I I ' I 1I ] l IlVillll -.J I I

<,,III ]-,\PJ I I I 11 I I/11 11 I I [I '_ 1 1I l",d 1 b,4" I I 1 "_1 1

I I t t ! It , >.,i i !r-_160 t$O I_O0 $1_0 1_4O LqiO

f

illo

iiI ii

I Ii i i

I 11I Ir I !

I Ii 1 I

I Ii I I

i II I1 I

; ]iI iI II |

I I' !t I

111i rl

]1 i,'i i

i I, ] I

$oo

eo

QO _ "+'--+--'_+

:::lli,,l +

, -+--+--+--+

Ill i+O + 14--.+---+--i

_ • _ __

Ill -iO _ _

-410 _

II I0 4o lO I0 I00

E

rio 140 riO tlO I00 llO 149

FREQUENCY (CPS)

Fp.-

F

+-

F

Ill

+-__

_- -I-- -+-+-

i;-4-

..L|11 llllll


Frequency for Outboard LH 2 Line, With Turbulent Resistance and

Without Pipe Vibration

- 56 -

SID 66-46-I

NORTHt

AMERICAN AVIATION, INC.__ SPACE and INFORMATION SYSTEMS DIVISION

3

0

D

.ore

.ois " i :

.014

.ore _- i _

.olo iii

.000

• 006 : :

°01_ " " :

•" IIi i , .

: [ I] :, .: ::. : :

:_ ]J ii iJ ! lilt :: ' l/It Ii l .n i ]

ill! J " 1

1 1111 I Ifill 1 i

I I !II I I II I'. 1 I II 1 "_ l " I

I I I I I I i !I I l " I I

l

i ! ii_ ;_ ! ! !: i !! i :: : : i: : : : : : ! ! !

!

i

" ; : I Itll N i.oo_

o 2o 4o 6o 8o

I 1 I I I I '' [ il : " i

] I I J I i

t 1 I] ! :I 1 I I . :11 I1 I :I i il ! :

I f I 1 : :I ] IN1 t Ill i :1 I lilt

' I¢'i i11 It ;I 1I ] _ll III :

J l i i] ] /I I I. I,I ] II t :

III I I

l_l_ i ,I 1 J I I !1 i :

I I I I I :II / l]: .I

i iI! _ll]I" iil _|_" ]1 •

sT! II :: :_'- I I ! ] . i "-

tO0 IZO 140 160 180

FREQUENCY (CPS)

I I L! I i; I

I

III

II

1!!

: 1!

i ': I: !: !

I

I

I • LI

I

!!!!!!_

!!giiii

iiiiiii

I I I I

IJlllll

"]I]"[I II

][ IIII il]Illl,,i]II]''..... !!

i !I!/i ii_ t! [_ I

IIIIXIIi ! I _! [ I

-FI !, I_0 _0 _

8O

' _ I i i

•o---'(--- ' % .... i :

Z .... " ''

" _ . , ,

n - , ; _ I i., ! !

"'0 ....:TTr [{i!I!_. ,...... _:_! !!

-soo ......... _:: ':lo _o *o IO

1 1

tl l

I

i 11

iIl

II

]I

i

|00

i:

!;i:

I :

i :I :

] :• ] :

! : :

] : :

: J- /!

: I

tQO

i !r

i :1 :I

I :

J Ii !I

! 'I1IIIIIt

i

FREQUENCY (CPS)

:]

:1

1]I!

]!]

\i]

t!00 Ill Ill

/l::i]

;i i I I _11 i

illlilt

: : | _ , |

[iii'' 1I I I 1 _ _ ]

!!!_111;!!'_ t1I I ; i ,Il 1 1 1 1 : 1

L !!!!!!!

; _i1!!_1I10 IlleO 111111

Figure A-3. The Ratio of Velocity to Pressure Deivation at

Frequency for Outboard LH 2 Line, Frictionless and

Including Pipe Vibration

x = 0 Versus

- 57-

SID 66-46-i

NORTH AMERICAN AVIATION, IN(:3. SPACE and INFORMATION SYSTEMS DIVISI'ON

°-,_ o

0

.080

0 20 40 QO 80 100 120 t40 t60 tSO 200 220 240 2QO £10 $00

FREQUENCY (CPS)

Io

SO

40

(_ - £0

<

-SO0

_- - eo

I fO 40 60 lO tO0 ItO t40 tOO rio tO0 llO t40 till tiQ lilt

FREQUENCY (CPS)


Frequency for Outboard LH Z Line, With Turbulent Resistance and


- 58 -

SID 66-46-i

oNORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

3}o'"_ o

,,,(Q.

I

[1Jli!!_III1

y

:!

•-}i;ili}

'''li!iI}llll!

.... :i!

• ,ii ill j, . [ .

3 ' :

''l 7!2

-- lJ " :

20 40 SO

II IIII!ii _i _: : i i

i! iJi i '

ii ::

• !

8O

II "" III

I I I.I

+ + ii _m :,i i i

Ii illJ i

] I I I _I , , ,

: i I I It ' '

I I I I i _ i] I I i +

II | I

I', I11i iii!I 1

'' t iJl i- i '.

+ I I ! ! i_ :" _ li+ ;

I 11 ;+ : II! ' ! "+ _ '

:+4+u+__7- I + _,

100 120 140 160

FREQUENCY

[ I

I 1, +

i'

, i !

+ I

: ii

180

(CPS)

t

iiiI l 1

i

/:

1"00

ii, !i I ;_;I

ii++<' t

3++ ++I + i I_i: ,! ..... j

.... i I;ill ++ +,, +++I: =: : ; [

i '\ ! : : I ' ' _ - .

1'1,0 1,40 _0 1,00 $00

- _0

-I00

l.l,J

-I$0

I,Ll

--,I -zoo

Z

Ill10 -+o,

"i"t'l -it,

-400

:::-:.++ -+.::4"++ --:-+4:: ••t .... -3:+ ::---t:::t----Z::---,.., i 3 t.,,3 '+:.i 3 i it' i i .......................... i i + '.......... , , =- _='-;t :-,uP,-+-" +-'I-.... -_+ - - _ - • -_ - - - _ - - +..... _ I : ; :II ; ; ; I +]_++ _ " "-]+ +_ +--.... ' ! ............ , _.:,.::1:'.'.11'..:1__ __ __1 ..t..+j

- ,. a , _ _ +_= ---_'_x_+:l_,,:,:+_, _:,:::,_:: :::l;_--l_=:_b_,- ¥.-d:--_::7-]-:_-=+ +7 ; 1- II ; 1 i _ ; 1 ; I ; I ; ; i : I + II 1 -- ._+ + L +_ .... J ....', ; ; I i 7 I i)._ ! i ; i ! i : ! _ , r ; : + ....... + : : :

i i + + + : ! i -='_-+---L: ', : r : : : i :.; i i '+ l _ ! ; ; i I i , : .:-L_--+_.-Z.Z_ ....; i ; ! ; i t i ; ; I 1 7%_,,I '+ ! ! + + ! ' ' " ;iJ ' ; : 1 ' + + I '+'| ++_+++ _

......... ::': "_xi;i !137 :ii[ i i+ ...... '+' +

....... _-' : : i l ' I " ; I I i I I++,I+''I ....: : + : ', t 1 'A! ] I i I I I ill i i ] I ] ' " + + '

................ i li 1 .... ---_i " + ;_ _'i " + ".... ', '. _ ill i i _ " i i :II ; _ : I ++ : ! I ...... I "+I" " "I i i ' ' ' I I I I ] l I I 1 l I I I I • I I 1 I : ! 1 t l i ', _ ' _'-_-++ + ++ + • _*+ " + +: ! _ i 7 +, : i I I 1 ] I 1 i I z i ! It I + ! ',-Ii + ',+ .... 111,3:TI:.+'.l_I"3 '. i I J L i I i ; I l l [ , i ; ! ! 11

+ I .......... ._+ + __ + • .............

.... ;i: ;i+i+liiiltfilflitiN_+-+ _:::t:::] I ,-1...t..-;_L_._;_42_.7+C-i- i ___.i ._.+_l_.l_4--i-'-.+4-1--bJ+-d-+.i-u-I _ _k-..,+,,,i_++ + + .+ ; . I ; • ; 1 • . '. I i '. '. i . '. '.-+ +_++-,--i+++-+--+-+---+-+-.i--+-++--,.-_-+-+--l-_+-+-+ -_ _ + *-+--b + + .... +I • " " I • • " 'i • • I ' " "..... -I" "++-'I-+ +-+_-++-+--i+ .... ++_+ ' ++--+-- + ++ _ ...... 1 " "" I "" "I " " I " • "-.+ --+-I -+-+- : ? +.+--_-_ ,-_-'. -+ +--t+'+-l'_ + 7-f-l-..-+ +7-P + " " : : + : ....... ; 't " " "I I'0 lO I0 I0 I00 IIII llO llO II0 lOO llO lifo II0 1,10 ill

FREQUENCY (CPS)

Figure A-5. The Ratio of Pressure Deviation at x = ,! to Pressure Deviation

at x = 0 for Outboard LH2 Line, Frictionless and Without

Pipe Vibration

- 59 -

SID 66-46-i

NORTH AMERICAN AV IATI. O N, INC. SPACE and INFORMATION SYSTEMS DIVISION

{!!ttli II!!i!ill I I _ili,ii_ ;i

• • :;,I!!] lu

iti

__ ; ];

- L ' IJ , ,!

0 zo 40

FREQUENCY (CPS)

(9LIJEl

LI,,IJ(9Z

LI.I

,<.1-{:L

0

"'°° ,I ; ' I ........ I ' , .

' * ' [ ' ' _ [ I [ [ j [ ''''1 ..... I .... _ -- ._L.

-ZOO , , , , , , , I , , : : ; : ' __ -

, i , , _ i _ l ; ! _ ! ' : : :

'- ' ' ,-F ...... { .............

i_ _- "_ _" :::::::::::::::::::::::-,,, :'_ :_ -r __LL: : : _. : : :_ : : H : _t:-, ,, :t +:;:- i _*_ _ :! ._'+.t i i ::_i : :_ : : :i : : : i : : :

i::t:: .... it'"1- ........ - .................. :::_:::_:::_:::!II ZO 40 tO IO |00 110 140 |lO ||0 I00 Ill |&O lllO llll 1Oil

FREQUENCY (CPS)

Figure A-6. The Ratio of Pressure Deviation at x : g to Pressure Deviation

at x = 0 for Outboard LH2 Line, With Turbulent Resistance and

Without Pipe V_brations

- 60 -

SID 66-46-1

N(_RTH AMERICAN AVIATION, INC.SPACE and INFORMATION SYSTEMS DIVISION

I

_: Ill ]ll Ill Ill 11 III

_i:l]]]]:]illJ' T , ]

I

...... !l!!ti!!l!!!l!!!,!_il! _ _ ......

_ i ! I[i[ Itllll!

• ' [ I " ] [i:llil'E]:'' I Jill TJI i;] [Ji I lillil] 1:1

t :ill :;11; ]::: : : _I h; ' i i i i _ i i i i ill i _ i ] t J _ i I I

i, Ill II!I] I]I I_:I I I [ i [ : : I : ] : !

l i i : : ...... ! ; ] :] _ - : : ............

_! ] : : I : : : I : : I

:ll .........I! ....

3

[ i ] i : i i , . : . , :

'I _iJ'i:iil;;i:;II1 ! ! I I ] , i : I ] : .

Ill ' ' ' " ' " - " "w, _i ii : ;:] i

iiI I ] I : : : I : I : : ,

I _! T ] , ] I 1 , I _ } I _! i i ! _ !i ! , } I' ' I tI !_!!_I !: ! !i i ! 4 ; ! • - _ _ _ i : ' : : " " : - : : :• 1 : : I : ] I : . : '. ] . I . . .

! 7/: : : '.t_ I ] i ', I ', : : 1 ' I I • I l I 11 I I I I T l I I ', ! I ] ', 1 ] lit I T : ' 1 " ' " I : ' : •

_----- : : ............ [ i : i :-. : : : , , , . o .......... . : • -

0 Z rl 40 60 80 tO0 IZO 140 160 18Q _00 220 _140 _0 _lllO $00

FREQUENCY (CPS)

_ _r-_.-K ................... ..... _ =____- so .......... _r:_ iT-- :

.......... I _'n . i,, ........

:::_-:-- _,_ i;': : .... " " , 1 1;::I: I I *. : : 1 I * I- ? x " 7"._7 :

hl ....

• -J -zoo

(.9 ................... - { :,-Y_-T i ----- +-Z .... _ -- _ ?--r_-'l_ "-F_ I _ ', I_! i l 1 l i ] i i t i _ i ] i i ] i i ; i : i ; , _ "

. . --

{,/') -!I00 - : ; ; ......

"1-

o. -,,o:: : : _t: : : l: : ,-_- ::_: :_ : ::q _:,_:7l-: ;:,_ :;::1 :: _: :: 1: i :-I :::I :::.... . . . * . * • _ .... 4 . . . 4 . . . 4 * +4--,I-.+ _'+r-- _4--" d'4--_4 " * _4 " * " 't" " * " 4 " * * & . . •

I ZO 40 I0 IO IOO tie 140 lillO |lO IO0 IlO 140 IlO leo lot

FREQUENCY (CPS)

-1OO

-ISO

Figure A-7. The Ratio of Pressure Deviation at x = _ to Pressure Deviationat x = 0 for Outboard LHz Line, Frictionless and Including

Pipe Vibration

-61 -

SID 66 -46-i

NORTH AMERICAN AVIATION ' INC. SPACE and INFORMATION SYSTEMS DIVISI'ON

t0

|20 140 160 100

FREQUENCY (CPS)

.,.=...

C3+tO0

ill

.-I -.o(.9Z

-ZOO

1.1.1{/_ -ZgO

,<"I-CL -_oo

-390

-4tlO

ZO 40 I0 l| |00 Alto liO |I| II0 I0| |tO |lO II0 ll| ll)O

FREQUENCY (CPS)

Figure A-8. The Ratio of Pressure Deviation at x = i_ to Pressure Deviation

at x = 0 for Outboard LH 2 Line, WLth Turbulent Resistance and


- 62 -

SID 66-46-!

N() RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

So0

0 20 40 60 80 tO0 120 140 tSO t80 tOO ttO tdO tQO ISO 500

FREQUENCY (CPS)

• 81 LI i| i| tO0 181 liO lii |l| tn ttl t61 RQ! tn Joe

FREQUENCY (CPS)

Figure A-9. The Ratio of Velocity Deviation at x = _ to Velocity Deviation at

x = 0 for Outboard LH2 Line, Frictionless and Without Pipe Vibration

- 63 -

SID 66-46-1

NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVIS'ION

t.4

t.|

t.0

0.4

0.2

• L0

r- _-

E "E

20 40 I0

¢--

VV

L80 tOO t20 140 rio

FREQUENCY

tiO

(CPS)

200 PJ_O 240 _lO £10 SO0

.J ,,,, ii ]It[ [ [ 1 I II|] I I I [

o 5o1| | i I I I I _ I i i !IIJ ! l J J / ! ]

ITL I [ I J 1 i

-too [ I I I I ! i-- L I

l, I,,, iI i

1 ] [ ] t I

Z_ -tso | I I I i L i I [I I 1 I ] [ [ [I I I I i

I_ | l I l : E I

i J-I_I ] J I I J

I II I I I

I I I I i i I I

I I I [ I I

-1901 I I I i I rI I I I

I I I 1 i I I -

l'l II lI,[ I i

I I I [ i [ I

-III 1 [ I 1 [ !

I [ [ I iI | I I i

J J J J [ I l-III I I [ I I 1 I

I 1O 40 1O il 1O0 Ill |40 Ill Ill

FREQUENCY (CPS)

Ill III III l|l Ill I|1

Figure A-10. The Ratio of Velocity Deviation at x = I) to Velocity Deviation

at x = 0 for Outboard LH 2 Line, With Turbulent Resistance and

Without Pipe Vibration

- 64 -

SID 66 -46-1


t.O

0.5

0

i

I

I

i

4O

II

I1

II

!I

i

i

I

I I I

ii.i[ • !

I'-I

i ii iI II I

! :!!

i '::

• .-::

: .-::

i iii

i| |

i i i-.

iiitOO |1_0 $40 tl_ 180 _0 _lO ILNkO )00

FREQUENCY (CPS)

° 5O

ILl -loo

-150

LI.I/

(.9 --ooZ<[

-£90

LIJIt}

-see"t-O.

-I!*1

°4U

[i[

|| 1| I)O I0 I00 I10 I|O IlO ||l I|1 141 Ill

FREQUENCY (CPS)

: : : : -

: : : _=

iiii!!!!Ni;i_{ii lli[I I IIII 1 IiiI I libI I lib

i i llllI I libI I I IIIi i i uiI I IllI I lUl1 I 111I I IIII I III

if:::• . ." =:

!!!!!iiiiiiii

I|O

i

i i

; I

: i

i,

I I

IIIII IINI

Figure A-ll. The Ratio of Velocity Deviation at x = _ to Velocity Deviation

at x = 0 for Outboard LH2 Line, Frictionless and Including

Pipe Vibration

-65-

SID 66-46-I


4.11

l.l

i.0

I.l

1.0

O.S

I

I

ii

::I

iI1

i0 tO 4O 8O ttO 140 NO tOO

: ;

!

iIi

i

41k ]I

SO0

- 5O

(9I_1 -ira

IJl.J(9-_mZ

-Illll

hi(/)<I: -el"l-n

-111

• !!%

i

i

ii!i

il ',, I !

I II 41 II

!i

.i i

! i

: iii Ii

I !

i :

; iI :

i

: +

Ii i

ll I I,

: I1 I1t

, !!

IIIl

]

iIll ill

FREQUENCY (CPS)

I I ILI

ii]: : :

i : : :

i .iii11 •., III • II,IU

IIII III11 ]

IlII I

IIII I

I Ill II III I! HI I

I M]I MI II

IIll I

; ::: :; : 2: :I

iiiitO0 Ill -ill till

: i

i

I .

: il I

: I

: I

-r"i :

i i

!1I I

Illlll lilll

Figure A-12. The Ratio of Velocity Deviation at x = I to Velocity Deviation

at x = 0 for Outboard LH 2 Line, With Turbulent Resistance and


- 66 -

SID 66-46-I

_O'RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

APPENDIX B. VELOCITY AND PRESSURE DATA

Velocity and pressure time histories for the center liquid hydrogen

line on the Saturn S-II engine

-67 -

SID 66 -46-1

NORTH AMERICAN AVIATION, INC. SPACE and INFOI_MATION SYSTEMS DIVISION

St.l

S1.4

50.G iI

.D1

I I t I I l1 I I I I 1J I [ I l f[ t I I I I ]

I 1 I I i 111I 1 I I I ]11

l lt t lflHI ] I II,ill ....I I I I I IUI I I ' I II

I',IIII'I t I ' J Jll t I I I III I I 1 | 1"T i I I ) I

',',IIIII I I ! I II I I , I I

IIIIIII I t ' 1 II I I I l Ii i i i I lI I I I I I

IIIIIIl l I I l I

i I I ',,,,11i I I l I I I

.02 .03 .O4

I I I I I I I I I I I I IIII , I I I I I I I I I I I

I',lq-M-l-l-l-l-4,''',,',

,J l, ,,f- -_r- --1-Ii II J l ' IIII I I I]l I I I I !I I I I • I I I l'Tl I I I

-I-H-I--J+-H-IiI ,''',,,i1 I 1 ' r t I I ' 1 1 ' I I II I I , I l I I , l ] I I II I I I I I I I I I I I I II I I I I I I I I I I I , II I I I I I I I I I I I I Ii I I , I I I I I I I I I II I I I i I I I I I I I I II I I I I I I I I I I I I II I I I I I I I I I I I I II [ [ I I I I I I l I , I II I i I I J I I I I I I _ l

I I I I I I I I I I I I I II I I I I I I I I

I i J I l I I I I I l l , lI I Il l I i i i,,lil,llllll l I I I I I I I , I I I II I I I I I I I I I I I I II ] I I I I I I I I l I I II I I I I I I I I I I I I II I I I I I I I I I I

I I I I I l l 1 I I I

1I I I ' I I ' 1 I I I I II I I I I I 1 I I I I I I I I

.OS .GG .O?

TIME (SEC)

II,II

Ill - i

it,,

I !!HI It-I-II'11till I idi lllll I !II 11411 " t|

IIIII II1|11

IIIII 1I IIIII

lllll II IIIIII ""' : II

IIIII II ,,,,j II

,_ ,_ .lO ,11

9000

4BOO

4400

Q..

4ZOO

4OOO

.0

]-t

fix.01 .O2 .O:5

-T

I.D4 • 05 .Oi .DT .08

TIME (SEC).10

-+--4

-+-,-4

-+--I

-+-+ -+-i

_.___+ _+_

4--1

_-_ _-,

_[_[_'

Figure B-I. Velocity and Pressure Time History at Right End of Feed Line,

Impulse Disturbance 2 Percent of Steady-State Velocity

- 69 -

SID 66-46-i


St .I

"" r51,4

•--_ 51.2

U

50.61

.0 .01

]

I

+J_TUY

.02 .03 .04 .05 .{)6 .07 .08

TIME (SEC)

,1{

1 tI I1 I1 11 11 1

I I1 (I I

.Og .I0 .11

52OO

5OO0

0. 44oo

4200

4000

.0 .01 .OZ .03 .04 .D5 .06 .I)7 .08 .09 .10 . t I

TIM E (SEC)

Figure B-Z. Velocity and Pressure Time History at Midpoint of Feed Line,


- 70 -

SID 66-46- 1

. N6RTH AM ERICAN AVIAT!ON, INC. _ _ SPACE and ][NFORMAT,OI_ SYSTEMS DIVISION

! llllll

=.mill I f {11 I I I I ! I I I I t 1 I I IIii I I ! III I I I1 I I I I I t I I I I I

,,,, !l!!! ,,, ,, ,,,,,i, :,,nil! illll ill -n _- llllltl= i'_l, _llll It IIIIIII I

Itll ] Illil Ill- L iii,_iI IIIII III II " IIIII I I I II [I II I II

•,,I,I [ ,ll,l Ill ,l ,,,,,,, [II II , IIIII III II III I I II

II II ] i Ill II III II Ill II ,l I

I l l l l l I I I -H_L i I I I l l l Illll lllll lllllll,711 I I I i I I I I I I I 11 I I I I l

.0 .01 .02 .03 .04 °05 .06 .07 ,1_ ,09 • lO . 11

TIM E (SEC)

.DI .02 .OT

k.03

_ --j-

F- :- 'L

.08 ,og

i

I

:- i

-t- .---_----1

- i- -----_-....

_:. : -:L7 ....

I_ iTh -

_ lf [2.I0 .|I

Figure B- 3. Velocity and Pressure Time History at Right End of Feed Eine,


- 71 -

SID 66-46-1


oUJO_

F-U.

$2.0

SI.S

51.0

50.5

50.0

49.5

49.0

.0 .01 .02 .03 .04 .05 .OI .07 .Oil .09 .10 . 11

TIME (SEC)

I"h

IXl_1

9OOO

I_. 40O0

i I1 lt 1I t

55O0 i II !I |I I] ]

5ooo I [

I I

_! L /I I t, ,= .__k

4500 1 "i _ ll' I_- ", !

[30OO

.0 .01 .O2 .05 .04 .05 .Oil .07 .011 .09 .10 .1 i

TIME (SEC)

Figure B-4. Velocity and Pressure Time History at Midpoint of Feed Line,


- 72 -

SID 66 -46-I


S! .5

i I II [ I

a I !| I I

SI.OI I !i I• I i• IB i '

_-51L__

"' _,.ol I

IIU. 4o.s

! I

! I49.01 I

I II iI !I t

48.SI II 1I !| ]I 1

40.01 I

.O

,,,:,",l"'I"," t-HU-""''"'IH-'' , _]_I*!1-'__i" ii!t,,I I III I ] -- _

I I I 11 I 1J I 1 II I Ii ll!lr'' '"'' H4-H+4-,, ,.,, _I I Ill t I 1 I l I I 1 I

I I III I 1 I I I II I J I( I I I I

I I I b I I I I I I I I II I I| I 1 I I I I 1 I II I I I I I I I I I 1 I II I Il I I I ,

I I I 1 I I 1 I { 11 J I I1 I I I I I I I I III I II I I I I 1 I I I II I I1 I I 1 I I 1 1 1 III I II I I I I I I I I IIJ I I

'' ''" III"'II I I 1 I 1 i I I l

I I I I I I J I I I I I I II I I I I I I I I I I II I i I I I I I I I I I I I I

.01 .02 .03 .O4

TIME

!-T-

1 I II I II 1 I 1 I I 1 i

I 1 I I I 1 I t111 llll!llllllll III '

,"",, iiiiiiii!!!ii ,_:J,, ,I111 II11111111111

IIII IIIIIII111111 !I I I I I11 I11 I II I II_[n _uiiniln_J[ n_ illll IJJJJllllllll

_ j _ _ ! ! !4- I i t i i i i i i i , , i ] n n J _i i i i

,,,, ;l_--ffSH-Hddqll ;'"'llll .*

.05 ._ .07 .08 ,_ .10 .11

(SEC)

4900

• _a. 45°°l_

4400_

4300

.0 .01 .02 .03 .04 .OS .Olt .07 .00 .09 .10 . Z Z

TIME (SEC)

Figure B-5. Velocity and Pressure Time History at Right End of Feed Line,

Step Disturbance 2 Percent of Steady-State Velocity

73-

SID 66-46-i


./

_ SPACE and INFORMATION SYSTEMS DIVISION

51 .S

51 .C

50.5

(..)I_1 so.o(/)

_ 49.s

49.0

48.5

48.0

.0

m

p--

7-m

.01 .02

m

i iE

.03 .De,

-- _- F--

E,05 ,06

TIME (SEC)

_÷÷

i -i.07 .08 ._ .10 .11

52OO

5100

5_

N

4_

Jv

4 TO0

4600 --

.O .01 .02

I I Ill I I i

l I l ) II

t Iit-t-I 11

ltl ,'I 11 ' 1

i ' I I I I

I I I I i

I I I I I

I | I I I

I III I

I I I 1---I I [ I

, '111 ''

.03 .04 .05

II

1

III

1

I

I

I1I

I

L--

I

.D6

TI M E (SEe)

Illll Ill[illI I 11 i I I 1_1 I I I ( -

IlIIIIl]_lllli 1

_3All l I [ l I I l I I ] I_F_i--[IIIlll ii [ I

I]111[ II 'III1[11! : -_

lllll Illlllli i I

J l 1 1 I I I I I I I l ] ' " I

lllll I I1 I II l [ i I

lllllllll[ll _--fl

l I I I I I l I-I_ I I ] _

I I I l I l I I_ __l I l I Ill l .

I I I+-H4-F-_:II .

Iii.07 .08 .09 .tO .11


Step ]Disturbance Z Percent of Steady-State Velocity

- 74 -

SID 66- 46- l


P-

S8

5O

48

46

44

42

.0 .01 .02 .05 .04 .O5 .06 .O? .06 .05 o10 .11

TIME (SEC)

I,-

M_

m

.J

v

5800

56_

54OO

52O0

5OOO

48GG

46OO

44OO

42OO

.O .Ol .OZ .05 .D4 .05 .06 .07 °06 ° D9 .10 .1 I

TIM E (SEC)

Figure B-7. Velocity and Pressure Time History at Right End of Feed Line,

Step Distrubance 5 Percent of Steady-State Velocity

- 75-

SID 66 -46-I


SZ

4Z.O .01 .(Y'_ .03 .04 .05 .06 .07 .08 .09 , 10 . 11

TIME (SEC)

5_

5_

LU

.J

41_o

4500.0 .01 .09. .0_ .D4 .05 .06 .07 .08 .09 . tO .11

TIME (SEC)


Step Disturbance 5 Percent of Steady-State Velocity

76-

SID 66-46-I

_ORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

N

$S

S4

U sstU¢/)

MI-LL.

ss

::3

so

49

I I I I I

iiiii: : ". : :: ." : : :

: : : : :

iiiii: ." : : :

iiiii: .- .- : :; : : : :

!!!!!: : .= : =

iiiiiI||111JIJIBI

[ I Jm I

I | |111I i f I ]

I |R I III/I I I

LJl III

r!!!!: : : : :; : : : :

iiiii: .- : .- :

i_iii: = : : :

iiiil.06 .to

lall

i!!!iiii

_ I I I"

iiiiiiii

!!!!

iiii

iiii

fi!!!!!!iiiiiiii.12

41

• 0 .O_ .04 .OS .14 .16 .22 .24 _ .?JI .10 .sup .]4

TIME

._!!!!

: : : :: : : :

: : : :

iiiii: : : : :: : : : :

: : : : :: : ; :

ii!iiiiiii: : : : ;: : : : :

i!!!: : : : :

iiii!: : : : ;

ii!iiiii]I I I Lf

"k.J J, SI i

iii!: : : ."

iiiil.to .20

(SEC)

NI--M.

In._1

a.

soot}IIII!!!!

liiiIII

!!!!

;ill

iiii!!!!iiii!!!!

i!!!

!!!!!iiiiiiiiiiiiiii: : : : :

ii!!!: : : : :

iiiii! ! !L_!iii_ilirllIi/11 i}.v!! !i01 i i i

.v! !! !a.!!!!

iii!iiiiii: ;., : :

.tO • ".t _'" "

3400

.O .i_ .04 .14 .16 .22 J4 .Lq .20 .30 ._ ,$4

TIME

iiiiiiiiii!!

,iiiii_l I I

i;; ;.

! ! ! !_,

iiii

i!!!iiii

.tll .20

(SEC)

Figure B-9. Velocity and Pressure Time History at Right End of Feed Line

Under Sinusoidal Disturbance, Amplitude Z Percent of Steady-

State Velocity, Frequency 7.20 cps

- 77 -

SID 66 -46-I

NORTH AMERICAN AVIATION. INC.SPACE and INFORMATION SYSTEMS DIVISION

N

$S

41.0 .Ir_ .l_, .el .Oil .10 .12 .14 .16 .18 .20

TIME (SEC)

4_0

Illlllllllllllllllllllllllllllllllllt IIIIII II II

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIILLIIIIIIIIII II1141OOll II I I I I III I I I111 I I I II I I I I III I _ IN I I 1 III I I ii i i I I II

II1_11/1%1111111111111111111[ll _ I11N II Illll Illlll IIIlYl]_II,IIIIIIIIIIIIIIIlilIIMIIIII MIIIIII Illlllll

,.. Illllll IMII Illlill II III I III1¥111 I1111111111 II I l I1 II

_1-- Ilillll I,II IIIllll II III 111 I1_ III III I _11[111 II11 I1 II47_IIAIlIIII_IIIIIIIIIIIIIIIIlIIIIIlIIII_IIIII IIIII111

i_ IIIIIIll I I1_ II Ill III III I I I I IIIll II II III I%1111 i III l I I IIIIIII III I Ilkll III1111111 I I lYlII II IIIII1_111 I Ill I I I II

lYll I I I I II1%11 III I III III I I I_lllllllll I1 _111 I I111 II I1[111111111111111111111111 VIIIIIII Ill,Ill I111 II I14*oo111]1111111 _IIIIIIIIIIIIIIIIIIIIIIIIIIIMII IIIIIIII

Illl III I III1_11111 I III I I Vl II1111111Jllll_ I1 I111 I I IIi-I IIIIIIIIIIII N IIIIllllll/lllllllllllllll_l] IIII II II

III111 I I III I1_111 II1111111 I IIII II II III I II1_ I !! ! ! I III.liJJlJiilJltliMllllllllllllllllllllllllllM III/

45_llillillllllllillllllll/lllllllllllllllllll_ !!!!IIVIIIII1111 IIIIIIIN II II IIAI I I IIIIII III I I I III I _ IIAIIIII11111111111N IIIlYlllllllllllllllllllll'.iiiiI#llII11111111111111N IIYlIIIIIIIIIIIIIIIIIIIII %111YlllIIIIIIIIIIIIIIIIlr"c.flltllllllllllllllllllll 1"1...1-4.'1111

i,_lllllllllllllllllllllllllllllllllllllllllll I1111111.0 ,_ ._ ._ ,01 .10 ,1l .14 .11 .ll ,10 ,ll .14 .11

TIME (SEC)

iiii iiiiiiiIIII IllllilI111 I111111Ill/r .11111 IIIVII _kllllllMII I,Illll1111 I1%11111t!!!! Ilkllll.... !!!_.!!!.... !!!_!!!: : : : : : : : : : :

: : : ; : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

: : : : : : : : : : :

.i," " "._o" ".il"" "._

Figure B-10. Velocity and Pressure Time History at Midpoint of Feed Line

Uncler Sinusoidal Disturbance, Amplitude 2 Percent of Steady-State

Velocity, Frequency 7.20 cps

- 78 -

SID 66-46-i

NO'RT H AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

INi!!!!!!

i!iiii

iii!!!

!!!!!!

iiiii!

iiiiii

|1|111

lillyIIIII

I I I IIII I I WI

1111!IliA!

_ I I fl JI IA I I

; : I Yl I II_ I I I

_.6,drllll

.&. - =

!!!! J: : : : :

: : : : :

Nii i!.r,i i _iiM iI l IM

!ii_, !iiii i: : : : :

iiiii: : : : :

iiii: : : : : :

: : : : i

: : : : I

:::: ;

iiii !: : : : :

iiii !: : : : :

4000

46OO

I'-- 44OOU.

131

41OOO

118OO

36o0

.O .02 .30

....... illlillllllllitilllllllllllilittlii!!!!!!!

lllliliIil_llJillllllIijlllillJll,....... i,,,IIlilri_tli,,lil,IiilJl IllxJ_i, .......iiiiiii IJilllllw II,llililllllliilii#lil_ I...... llll ,I,lll allillllllliillil_l_llliiiiiii: :: : : : II I I I I Iill I II I1 III I I I II I I III I I/I I1 I I%1 : : : : : : :

_1 I' I I1 It 11 II I 1 I I I IIJ II I I I I I I I III I/I III I I ILiJi J

,,,,II ,I,,,s,,,,I, I,,,ll,,,,,,,,," I""'ll Ilitit illlltllli illllltliiiilliillllllillllll I

iiii i i i It I I I Iil II I I I I IIII I I I I I I I IIiil I II1 I I Ikll Itill illtlitlililliilll I II Ilii IlUl i ilillllUlill I

' "II ""'""'"'"'"'""'"'"'""'"'"IIII I%1I I I I I I I'll I I 11 I I I tllll I I I I I I I Ill I I I I I I I I I!1 I

'"'II ,,,,,,,,,,,,l,,II1%1 II I I I11 II I I II I I Illl III I I I I1¢1 I I Itl I I I I Ill '

,,a, ll iiii iiiii11111111111 i11111111 i i Iiiiiiii!1Illl ii i i!lll i i11 I I IIIII II I I _ I 1/11 I I III I i I I IIII I I I

:111%11 IIII/llll Illl II,lllllll III/ll I I IIII IIIllill II

:I IIl I IJ , ig, 111 I I I' ' ' tit lit I I ' ' I!'' ' i ' it I ' ' ' lilt '], I'ii itl#l,llllillllilltlllill#llllllllllil Ili]l]i

iI ,,,, Ilill IIIlll I1tllllllllll VIII IIIII IIII111%1I1I1_1 ilViiiiliiilllllll IMIlilIIIIIIIIIIII ]l'ltli II,, IIAlllllllllllllllllllVlllllllltllll II1%11I I I1" M'I I I III I I It I1111 111_LL4 I Ill I I Ill I I II'11 I1_1

11 it, ilJlllllllll It,ltlllllliiltl tilllllll IIIITi Ill iltiilll,lll Illitl Illlllll,l ,lllll,lllllll.04 .I)l .Oil .tO .ll .14 ,11 °ll .tO .ll .14 .ZS .ll

TIME (SEC)

!!!!!: : : : :

...ii1/11 I I.v!! ! !

II I I I I

'::::: ::

: : : : :

: : : : :

!!!!!::

: : : : : : :

: : ; : : : :

iiiiiiii!!iiii::: :: II,.

!!iiii!: : : : : : :

• ".it" ' ".;_


Under Sinusoidal Disturbance, Amplitude g Percent of Steady-State

Velocity, Frequency 9. 70 cps

79-

SID 66-46- I


M

SS

U S_

LLJ

__. szU-

SO

41

.0

IIIIIIIIIIII IIIIIIIlilI IIIIIIIIIIl]lllllllllllllllllllll IIIIIIIIIAI kllll

iIIIIIIl]lllllllllJllllllllllllllllillllllllrll_lMIll

IIII|IIIIIII[I_I_IIIIIIIIIIIIIIIIIAII I1_111I I I 111 IIIIII IIIVl I_11 I IIII ] I I I I II I Vii I_ Ill 1]

I IIII [11111111_ I IIN I I III II I I I 111 Ill l I I I! [ I1 _1 I I1[I Ilill I IIVIIII NIl I It1[ |11 I II1_ I|111 I [11

i _ IIIIIIIII]IIIA II_llllll]lllli_llll IlllllIV • IIl[lllillllll_ I]llillll[ll]ll[lll]l[l:|llll

[A _ IIIIIlilllllWlillll_]llillllllllAIIIll[ll|lI • Ililllllll[IJlllll _ IIIIIIIIIl/lllll III RI

1 • _IIIIIIIIII[II/IIIIIIIIII[IIIIIIIIF]IIIllI]II-[

_ IIII111 III_IIIIIIIIRIIIIIIIIIIIlIlll I]llIIIIIIII1111 • llllllllllllllllllllllllllllllllllllllll:lllll

I IIII!1111111_111111 II_IIIIIIIIIIlIIIIIIIVIIIIII:IIIIII IIIIAIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIRIIIIIIIIIIIIIII IIIII

IIIIIIIlllllllllllllllllllllllllllllllllllllll_llllll ! 11111l I V Iilllll 1 11111111111111111111/1111111111111

IIII • I llllll I IIIIIIIIIIIIIKIIIIIIIlIIIIII:IIIIIk _ llll V IIIIIIIllllllll/lllll]l IIIII

I IIII11111 Ii111 6 IIIIIIIIIIIItlllll/llllllllllllllI _ I If lllllllllllllllllllllllllll_llllIIIIIIIII I1111

ll,I J Illlll Vlllllllllllllll'lll_ IIIIIIIIllllll1 RI I I IIIIklllllJ]llllllllllllll _ I_ Illllllll IIIII1 _ IVlIIIIIIIIIIIIIMIIII/IIIIIIIIIIIIIIIIII]IIIIIIIIII IIIII

Ik iJlllllllllllllllllllll[llllllllllllllllllllllllllllllll

Ill I _ V [ilillllllllllllllllllllllll:lllllII1 • FiiiiiiiIiiiiiiiil-_P.1111111111111111111111111111111 lll]l111 _ A I IIIII IIII II1111111111]11111111[1111 |1111

I _IIIVlII I IIIII 1 Illllllllllllllllllllll[ll:lllll

I1111 _]IIAII[I illllll[llllllllllllllllllllil]ll IlllllllllIIIIIIIlI.,.L._IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

TIME (SEC)

4_00

400O

,,",. 471_

N

LI.

_O

..J

CL 4see

4400

43oo

.o • , , o ° o o . . °

TIM E (SEC)

,$1 .14


Under Sinusoidal Disturbance, Amplitude 2 Percent of Steady-State


- 80 -

SID 66-46- 1


ll

$4

iIi

/

i

iI[I

.14 .is

(SEC)

so°°l I i I I I I I I I I I I I I I t I I i I I I I I I I!1 I II I II I Ill till II III I I I I I I I I I I/F-I_I I I I I I I I ,,'L I I I i

4ad)Ol 11 I I I I I I | I I ,/' _,1 i I I I I I I f % i I i III ii _ I I I i I I]tl I I I I I I ' I I I II I III %1 I I I I I I I I I I I I I/I I I I_1 I I I I I I / _1 I I I

I_1 I I

IA i i i ill i i I i I I I I I I I t I I I lU _ I I I I I_i t I ,

d_ 4gOOl I I I_

$4 .0 .Oe .04 .041, .011 .10 .t2 .t4 .It .rib .20 .Lq_

_ TIME (SEC)



Velocity, Frequency ii. 70 cps

- 81 -

SID 66-46- 1

NORTH AMERICAN AVlAT I.O N, INC. SPACE and INFORMATION SYSTEMS DIVISIQN

I!

.02 .04 ,06 .D$ .10 .12 .14 .15

TIME (SEC)

.18 ._ ._ .P.4 oM ,21

4_

41QJ

4400

43O0.0 .OZ

470_

I-i,

46oDrn..J

45OO_L

• o ° • ° • °

TIME (SEC)

.24 .tl

4

oll


Under Sinusoidal Disturbance, Amplitude Z Percent of Steady-

State Velocity, Frequency 11. 70 cps

- 82 -

SID 66-46-I

N O_R T H AMERICAN AVIAT!ON, INC. SPACE and INFOI:tMATION SYSTEMS DIVISION

oILl

IS .0

(/1

5|.5

U.

_S

SO.S

!!!

!!!

iii

iiii

iiiiiiii

iiiiii!!

;;il;;;;

iiii°_ ._ .OS ._ .09

TIM E (SEC)

!!!!!!!:::::: ::::::: :

iii!!!!::::.-: :

iiiiiiii_iiiiiiIklllllI1,1111II1_1111IIIMIIIII1,11III I P_IIIIIII1 _lllllll!!!!!!!

iiii iii.... !!!:::::: :

iiiii!i:::::::

iiiiiiiiiii[l[

.IO .11 .11

!!!!!

:::::

iii_II_lIIAIII#111

iiii!iiiii:::::

iiiii!i!i!:::::

iiiii:::::

!!iii::::;

iiiii:::::

.ii " " .t,

!

!'

i!!

i

!

ii

ols

! 1I

i'II

iIIII

!

.ill

l-U.

m

.J

Q.

11.5_111t_ I M I!11111

_lllfll

IIIR

::::::::::::::::::

iiiiii::::::

iiiii!iiiiii_

iiiiii

iiii!!

iiii!!M" '.U

slo0.O .DI .Or .05 .04 .05 .OI ,01 .Oil .09 .12

TIME (SEC)

IIII '"'"_",,,,,,,,IIII '"'" 11IIIIVIII

IIII '""'""Illllllll tlIIII IIIVllliII;;;; illi_!!!!

Ililll

III .'_ 1IIII '"'" ",.,,,,IIIIIIII '"'""'_'11111""

IIII .,,,,,IIII..... rlllllll III..... II11111111

I,l," ,,,,,,IIIIII llllll

11/1 1111111111Iil t11111''1]_wl ll_ll_ii_l

'hl,"1 I I Illlllllll.iS .14 .iS .ll

Figure B-I5. Velocity and Pressure Time History at Right End of Feed Line

Under Sinusoidal Disturbance, Amplitude g Percent of Steady-State


- 83 -

SID 66-46- i

NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVIS_ION

O

I.,J

07

S3.©

52.!

$2.0

Sl .$

SO.S

SO.O

4i.5

.0 .01

_ e-r...r-.-4--4-

-4- -4-4---4- -4-4--

_ .+--*- _,--+-

_ I--4--4-

I--4-'4-t--.4-4-t.--f.-4-

*05

,--w-e--e"

=====

.06 .0$ .09 .10

TIME (SEC)

.11 .12 .13 .14 .15 .II

SlO0

SOOO

4gO0

I-

_ 471111

El

-J

_ 46111:1

r'l

450o

4400

43(_

.04 .OS .OG.11 .1 l' .15 .14 .SS .11,

Figure B-16. Velocity ancl Pressure Time History at Midpoint of Feed Line

Uncle:" Sinusoidal Disturbance, Amplitude 2 Percent of Steady-

State Velocity, Frequency 21. II cps

84-

SID 66-46-I

NO_RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

$500III1 I IIII IIIIllll 1I I I | • ,I I I I I I 11 I I I I I -- 1

_ Ill .. _ IIII I1111111 • • 1

N II f _ IIII Jlllllll _ I1 I_11 _ [111 IIIIII1_ 1f: Illl fill ]lllllll _ 1i I I kl t i i I ii I I i I Vi 1

I IIIN % IIII lllllI/l I If II Ik II II III I II_I _ I

4_ I I I11 _ I I I I I I I I Iil I • I

j I I II I II I I III llJll IIII I I I _ I I I I I I I I III I I I IiiII _ _ IIII IIIIVlII I I

t I I I I I I I I I Ill Ii I I I 1 [_. 4000 I I I I _ I I I I I I I I_ I i I 1 II_ 11 I I % I I I I ti I _11 I I I tj Illl % I I111 IIlIIIII I I

I I I I %11 I I I I I_ I I I I i IIIII N I11 IIVI I III I 1

3500 _ I I_ I I I III I I l I I I lIIII _ I_I I I_I III II I II I I I / I I_I I 14 1 1 1 [ I I I IIIII \ I111 _IIIIIII I II I I I • _ I I I\I II I I I I I I I I

II II II I_ IIII II III I +I I II "" III I'_ / II I I I III II I I I I I I I I _'-_ II I I I I I I IIIII I IIII IIIIIIII I I

l$_ IIII I IIII IIIIIIII I I

.o ._ ._ ._ .os .to jz .t4 .Is .. ._ .== ._ ._ .-- ._ -_ -_

TIME (SEC)

Figure B-17. Velocity and Pressure Time History at Right F.nd of Feed Line



- 85 -

SID 66-46- 1


$$

---, l0

t_W

O_

b-b.

so

4s

4DJ

!|ill[1111Illl]illll;IJll!l!lI!|Ill[1111IIII

1111it11]

llliIlli

|ll|IIIIIllllilllill

_ I I INilI1_1

III1 TM

I111IIIIIIIIIII1IIIIIlllIIII

o_

[ f I' i I fi I f f f [ I I I I I 1'1 III I I i i I I I I i t I |i f i I i i I I i I I i I [ I I 1 I I I I t I I | "Lit Illlllltlllll Illltl IIIIIIIIIIIlilI]IILIIIIIIIIIIIIIII]IllI1 1. IIII1 Jill I III llllll J I I I II i I till II I| I ] ]llil I l|lJ IJL I I 11 II I 11 I I I I I I I II I I11 II I| I [ I I I I I II [ IIII I I I I I J_[I I I I I I i I I I 1 i I I[ t I i i t t t I I I I t I t I I I I 1 1 I 1 I I I I I I I 1 I I I I I I 1 I I 1 I IJrl 1-_1 I 1 I 1 1 1 1 I I I 1 iI I I Illll II11 Illl IZllllltZll IIZ _ II III I 111 II1 Vii 1 I IINilII [] IIIILII I 1:1 III I I I I I I I I I I 11 I 11 I I I !11 1 I I I I 1 [ I I I I 11 I_I11 I 1 I I 1M I I I I LJ_LLLJ[ll_ltililll[lll IJl /1 i 1 I ! I I I 1 t I I I I t t I I t I,AI'I It [ I 1 ]% I..J_LL_E]Z]_Z_IL I I : 1 I I I f | I I I i I I I_ I rl I I I I l I I I I I I I I I I 1 I 1 Yl I I i i i i i i i_l L L I I I I I I[ I I 1 I I I I I I i 1 I I,"fl -"1_ t I i I 1 I I I I I I l I I I 1 I I lil i I I I I I I ] f IN |] I I I I i I[ l I ] I I J l J l I I I l I I I/1 I J I J I_1 I l I I J l l I i ; J J I | lJl I I I J I i i | I I 1 I_ I I J I ) ) I_ 1 I I t I I 1 I I I I [ I I I/i I I 1 1 I rim I I I i I I I I I I i i I i i vi i 1 i i 1 i 1 11 i 1 i i_1 i i i I i i: I I, ] I II I I I I I I 1_1 I I 1 |IM I 1 I I I I I I I I 1 I IIAI I I I I I I I 1 I I1 i 1 l_]l I I I I: | Ir | I I I I I I 1 I I¥1 1 I I I I 1 II I1kl 1 I I I I I I I I I 1 I¥1 I 1 I I 1 I I 1]_1_1 I I I/

1 1 : 1 I I I 1 I I I I I/I I I I 1 I rl I I_ I I I I I [ i I I I I vI I I I I I I I I 1 I I I I I I I I l'_J [ I II I : I I I I I 1 I I Ill I I I I1 I | I I i I IM I I I I I I I I I i_f I I I I I I I I I I I I I I I I 1 1 I I I-I I I[ I i 1 I [ I I I I I !/I I I I I III I I f I I 1 kl I I I I I I I lyl 1 1 1 I I I I I I I I I I I I [ 1 I I 1i III 1 I I I I I IJl I I I I i I I I 1 I 1 1 I 1 1%l I I I I I ly* 1 1 1 I I 1 1 l I I I I 1 II | I I I I I I I lYl ] 1 i I I I ] I I I I I 1 I I J_l I ] I I y'l I I I j I I I l l I ] I ] I I I ] I I J ] II I I I I I I 1 1 1/I I I I I 1 | I I I I I I I I I I I I'I_LI L.J_I I I I I I I I 1 I I I I I I I [ I I I 1 1 I 1 I I [ IIII I I I i I IA i i I I I 1 1 III I I I I I 1 I 1 [ I I [1 I I I I I I 1 1 I I I i I I I I I I I 1 I i i I I i 1 I I1111111111111111 lllllllllllllllllllllllllll 111 lllllllllllllll I

_1 I t I [ I IV'I I 1 I I I I I I I I I I I I I 1 I I I I I I 1 [ l I I I I I 1 I I I I I 1 I I I I | I I ! I I I I I [ 1 I_LII I I IJ"] I 1 I I 1 I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I I I i III 1 1 I I I 1 I 1 I I 1 I I I

I'1,,J IJ_FI I I | f I i I I I I I I I I I I I I I I | I I i I i I I 1 ] I I I I I I I I I I 1 I I I 1 I 1 i i I I i I I I1 II I llll IIII I III |lllllllll I[I I I Ill Ill Illl I I1 III I ] Ill II Illl I I I IJlllllll IIIlllil IIIllllllllllll]lJJlllllll]llllllllJllllllllJ II Ill IIIl II [i IIIl I Illlll Ill I 11 | 1 II II I I Illl Illll I 1 II I I I1 [II Ill I II Ill Illl II II III I I I1[11 IIll III li II Illl Ill| I Illl [II 1 Illl ill I II 1 IIlllllllllllllll IIIIIlllllllllllllllJlllllllll IIIIllllllllllll

.04 .06 .08 ,tO .12 .14 ,16 .18 .20 .22 .24 ._ ,Z8 .:_10 .3Z .34

TIME (SEC)

4 8GO __+..4_ _

.j _

42_ _

4000 _.0 ._ .04 .04_ .08 ,10 .12 .14 .1S .lid .20 .Z2 .24 .2S .l_8 .$0 .$_ .$4

TIME (SEC) --


Under Sinusoidal Disturbance, Amplitude 5 Percent of Steady-


86-

SID 66-46-I

N_RTH AMERICAN AVIATION, INC,

/


OI¢1

hv

uIllllIIllUU]

U;;i;;;;;;;I I!!!!!!!!??

.iiiiiiiiiii i::::::::::::::::;::::::;

_iiii!!iii!!!;iiii!!iiiil

_!!!!!!!iii!i_iiiiiilliii

,lllllll:llIlll

_l''llllzl I,.?! ....

-iiiiiiiiJiiiIIIl|l|||lll

_iii iiiillll iIIlllllllIll

NI I I I !...... IIIIII,_I I_ l I I I I I I I i

III_IIIIIII_

III _ IIIIIIi

4s|l,,_,,,,,,ill_::llllIllll_|llllFI 11 I I _11 I 1VII I I I I I _11 _ !

_iiiiiiii_ii

I I ]

iii

;ii.i i iI Illi IT ii A i111 I

111

! ,

ill

.I0 olZ .Z2 .Z4 .L'S ._I .liO .34

iliii!!!

iiii:1;:

iiiii!!!iiii!!!!

iiii

iiiiiiiiiiii

!!!!

iiii.14 .16 .18 ._

TIME (SEC)

: a

hi!'!

iiII

! I

: I

I !

°_

l--h

..1

??!!!!::::::

::::::

::::::

iiiiii::::;:

iii[ii::::::

i[ii[i::::::

::::::

iiiiii::::::

::::::

ii!!!!::::::

iiiiii/11111, ::::::

:::::

_ ii::_!!?

?

.I

I

[[!!

!iii.

iiiiI I Ill

?!_Y!iiiiI III I

I I/I I

!._!!

iJiiIII I IIII I I

!!!?,_;;i

!?!!!

!!!!

!!!!All .ZO

(SEC).02 .04 .10 .lZ .14 .1S .2Z .Z4 .Z$ .Zll .$0 .34

TIME




-87-

SID 66-46- 1

NORTH AMERICAN AVIATION, INC. SPACE and INFORMAT]ON SYSTEMS DIVIS.T.ON

9O

...... ,,,,,,,,,,,,,,III,,,,,,,,,,,,,,,,,,,,,,,,, iiii i'''''''''''''''"'""""'"III

till lliIllilllilll llllllllililllilliilllll% i]fill JJJJJlJlJlJJJJJ_-NJJJ]Jt_JttJtJmJJll]l!! !-:::: llliltllllilllJlll N llilli|lllilAllillil:: :_

IIl]llllllillYiil _ IIItlllVlll Ill[

_r%JI IIIIIIIIllllilll _ 1 IlYlIIlilIII

II N IIIIIIIIIIIhlil[[llllN IIIlll M llllllllli,,_,i[1,11[[I,_[[i , _ i_fi,Iir[II,,fl_]l Illll'LlliillllllVlli I lll_lillVllilllill ilIJll _IIIIIIiIi''IIIIIIIIlilI_LII]/Illll `llllllll

I,,i _,,,,,I,,vII,,,ll rT,,,_,,II ,

IIII itll'1.+,rlllllllll] iiiiiiiiiiiiiiiiiiiiii_ll, ,ll,,,l,lllll_ i,,,lllzlil,l_l,l,lt,,,l] ]

Itll i,,I,I[i,I,,1,,llLI,i11,,ll,,,I'I'llil'''' I!!!I il tllil[lllllll lllllllllilllllllllllliiliilil[[liIIi iliiIIlillllilIi liillll "

iiii llllillilillllil111illllllllllllllllllllll.I0 ,IZ ,14 .IS .18 ,_ ._ ._ ._ ._ "_ "_

TIM E (SEC)

I I i_l I I I I1 I I I I I I I I I,,l,_ I I I II I illNi I I II I I I i i I I I I I II I/'1 rM I II

lm_l I II I_l_l I II I I I I 11 I I ...... I I I lkl I It I li t'[ki il t I t i t I i t i i i ill 1 1 il, I i 1lilt lill_l Illllll I III III/11111111I Ill I II 111 If I I I I I I I I I 11 Vl I I I I I_1 1

I till Iti i_li i I I I I I i I i i i/] i i I I Illkl,-.. llll I II 1 111l I I I I I I 11 I 1 I111 I I I I I I_1

ll/ll III l|ll III l I I I I III I I III II_I4soor-111 iiII Ill I111 llli;_ii i III IIIi

14. I I IJ III I I%11 I I I I I I 11 1111 11 I I I I I II 11 I 1111 Ill 11 I I I I I I I/I I I 11 I I I I I

l:l) I III III lll_lllIllllllll III IIIII44_Ol l J I I I 1 l I III I I t I I I I I!1 ! I I I I I ) )

--I II IIIII I ll_lllll IIIIIII i

1 ,,,,,,,I I 11 I I I 1 il 11_11 I I I 1111 I I I I I I III I1111 1 li11_1111/1 lJ I I11 IIIII llllll ill IIN IlYl I IIIII I I1111

4o0_! I I I 11 I I I I I I I IM VII I I I I I I I I II I I I I1 i III I I 11 n i i i i I Ii li 1 i [ i i i i

llllllllllllllllllllllllllllllIIIIIIIIIIIIIIIIIIII IIIIIII_OOlI IIIII Illllllll IIII III I lllJlJ

,0 .Of ,04 .04 .08 .10 .lZ .14

TIME

i!!!:t::

i!!!

iiii

iiii

iiib

III

lIAI111

Ji/lYliI_11

.II .18" " ",ZO

(SEC)

llllllll/llllll)llllllIIlllllll]llJ111111111 I

l,_l I Ill I I I 1111 I I III 11 I111M I I I11 I 11 I11 I I 1 11 IZlll_lllillllll IIIIIIVI IIII I_IIIIII I III I III II#II "

II1111'11111 II Illllyll iI I I I I_1 111 11 I I11 I I 11III I I_IIII I I l III III I/11 I :III II!11 I11 Ill 11 II I/1 I) !III I I1_111111 III lllll I I :II Illl_llll I II11111/1111

II I I I I' I I 11 III I I I I111 ' I iII 11 111_111 I I 111 I1'1111II11111_111 I II11111 11

III I ' ' 11 I I 11 I i I 1 Ill II I Ililill I1_111 111 lll/I III1111 11_111 I I11 LII II I1 "III III I1_/ I I Ill I I/ll 11 iIll I Illl Ikl I I II Itllll] l[ "

l l I I I I I l'l I III I 11 1 l_-_-_J I-Ill l' , 1/1111 .I I I I I I II I I_I I II I/II I I I I

I I l 11 I 11 I I_LI I I_11 I I l I " iIIIIIlilIINlYlIIII1 iIllllllllili_l Illlll l Ili l, lill I, 1 i II Jill 'J..Illlllllll I III IllllI +_L

.22 .24 .Zll .II ,$0 .St .14

Figure B-Z0. Velocity and Pressure Time History at Midpoint of Feed Line


Velocity, Frequency 9,70 cps

- 88 -

SID 66-46-i

NC)RTH AMERICAN AVIATION, INC._) SPACE and INFORMATION SYSTEMS DIVISION

so

.fit .04 .06 .08 .10 .12 .14 .16

TIME (SEC)

.18

i

II

.2 .IN, .1_

; i

; i! !

; iII

! |

i i

II! !i !

°21J

S!_ll

I-

I)D_1

zsl_

o@ .18.16

TIME (SEC)

,-*-4--

.-t.- 4-

:t::1:

-.t- 4-

.20 .22 .t4 .t8.Lqr_



State Velocity, Frequency ii. 70 cps

- 89 -

SID 66-46-I

NORTH AMERICAN AVIATION, INC SPACE and INFORMATION SYSTEMS DIVISION

TOI

' 1

15

fl0 J

,,, ," ,. /

u. \ /

:_ L: /

4$

40.I) .t_ ._1 .1_, .1_ .ID .lZ .14 o1(

TIME (SEC)

_1 t'l

% f

1,

/f

.ill .ZO .lit .14

I!1 _

"111i1l1

I!Il

I [1 1i !

I1I1I!lI

.H ollO

I I II [ II I 1

- I IIi _ | 1

• i, \ /I \ l! ItI_. _ I I

"" • I _ Iii II I

I" '1 I, / I II I: I 1_ I _ 1 1It. i / [ t

% I I l [/1'1 460O I t _ I l I

I _ I 1 I--I i / [ tI / l l

4400 I I I

a. x x x i iI I I 1

I I [ [

I _ _ 1 1/ j l 1

_ k I k J |4000 % / _ /! 1

_ J • • k tJ I- ,, .r I

13100 I

.0 ,OZ .04 .06 .Oltl .10 .IZ .14 ,ll .111 .tO .It ,Z4 .Z41 .!1

TIM E (SEC)



State Velocity, Frequency ll. 70 cps

-90-

NO'RTH AMERICAN AVIATION, INC SPACE and INFORMATION SYSTEMS DIVISION

U

0w

F-u.

t1111111111111111111111IIIII111111111111111111IIIIIllllllllllllllllll

K IIIIIIIIIIlllllllllllllIIIIIIIIIIIIIIIIIIIIIIIIIIIllllillllllllllllllIIIIit1111111111111111111111111111111111111111IIIIIIIIIIit11111111111IIIIIIIIIIIIIIIIIIIIIIL

IIlIlIIIIIIIIIIIIIIII,HIIltllllllllllllllll/rltl

MIIIIIIIIIIIIIIIIYIIIIIIIIIItlIIIIIIIVIIII[I_11111111111111_ IIIlilIklllll[lllllllilllllll11_11111111111111111111III_[IIXPLIII)'IIIIIIIIIIIIII[IIIIIT]IIIIIIIIII

. IIIIIIIIIIIIIIIIIIIIIit

IIII!111111111111111IIIIIIIIIIIIIIIIIIIIIIIIIIIIIllllll]lllllllllllllll[llllllllIit1111111111111111!IIIlillllllllllllllltlIIIIIIIIIIIIIIIIA_Tllllllllllllllll_ liIIIIIIIIIIIIIIIMIIIIIIIIIIJllllll_llll

1MIItlIIIIIII _ IIIIII_111111111 H IIIIIIIII]_llilllJ/llllilllIIIIrH-I-H_IIItlIIIIIIIIIIIIlllllllltlllIIIIIIIIIIIIIIIIIIIIIllllllllllllllllllltlllllllllllllllllllIIIIIlllli[lllllllllIIIIIIIIIIIIIIIIII1[

TIME (SEC)

IIII III1 1111 1 IIIII Illl 1111 I II111 I111 _ I IIII1 IIII _11t I IIIII I IIII / III1 • I1111 I1.11 III1 I_k

• IIII I IIII II11 1 1_LIII IIIIJ IIII I II_I IIH IIII I Ill]'h L,MI llll I IIIII _ 7111 IIII I IIlll IIII IIit I IIIII IIII IIII I IIIII IIII IIII I IIIII IIII Iltl I IIIII IIII II11 I IIIII IIII IIII I II111 IIII IIII I IIIII IIII IIII I IIIII IIII IIII I I

.10 o11 .12 .15 .14 .15 .11

Iooo

5o0o

45O0

I-.-I,i.

40O0IZl_J

Q.

z$oo

Lq)oo

.O .It .12 .13 .14 .15 .|1

lllllll[lllI]lll I] ll]llllll,,,,,,,,,,,.....I IIIIIII IIIIIIIIII'IIII['llJ.Jl llll = : 2IIIIIIIlilllllllll II IYI _ llll I

,,,, ............ l,,Itlllll,_,,.,,,.lllllIl,i,lllll ,._""_'I I I lfl_l I II I I II I II Ir VI I 11%11;I I

ll_lllll_l{lltll ] llllll'"''"' ...... "l"lllll"'llll]'I'llllll 1IIIII II IfillIE I

........ '"+''"'I"lllll""llll ,,,,, llllllllIllll ....11111111!1111111 II I ! ] ....

IIIIIIIIIIII II III il,,,,,...._,II,,IIIIII,,,_llllllll,,llllllllllll"iiii,,I ........ ,,II IIIIIIII ]IIIIIII III111111111_ IIIIJ I III Ill ....

IIIIIIIIII llllll III IIII,,,,,,,,,,I,,,,,,Ill,,,llllllll,,::]:lllllll"]i£....iiii,,,,_ .... ,,,,,,,,_,,,,,,,,,,,,,,!!!!,,!!!!!!!!!!,,, iiii

[ III1]j I]]llllljl I]llliljlkflIIIIIIIlllllllll II IIII ....

........ ' " IIIIIIII IIIIIIIII I iiiilllllllllll_lll II lm,,,,,,,,.....,,,II,III.,IIIIIIIIIIII_'IIIIIIII"'I,,,,, .IIII...illlllllll} l| III _I lllltl II....,,,,,_,I_,,II,,,I,,IIIIIIIII,,,,,llltlll,_II ...iiii.IIII|IIIIII _II I£,,,,,,,,,,,I,,,,II111 .., lllllllllll"l_]llllll"'lljl ,, ....iiiiT II I I ]1 I ] II ]_ I I ill Ill ]_l ]j

I II VIllllll[ll]I .......... i I IIII[ ikl i ii_111 1 1 III I IIIIII l I Ill I I _II I II ....

ill l]t|ll][lillll il]l]].... II ......... II::

IIIIIlI[lltllllllll II IIIIllllllllllllllllll lillll

Illllll ..II ..[lll]lll[lllllllllllllllllllllllllllllllllllllll .-.m ._ .o= .m os ._ .o_ o8 .o<+ .io

TI M E (SEC)



State Velocity, Frequency 21. ii cps

-91 -

SID 66-46- 1

NORTH AMERICAN AVIAT!ON. SPACE arid INFORMATION SYSTEMS DIVISI.ON

U.

io

a

.0 .0!

lllll+llllllllllll IIII 1111

lll]"'"'" '"' "",_ll,_,,i]l]l ,iJ, ,,i,

,,,,,,,,,,,, ,,,, ,i,, ,,,,II11 Illllll .... IIII llll

illlllllili_],,Jll liliIlIIIllliilillli !1_1 INI Illl

II1 iiill Illl IIII

lllllJll'II[lllll,'llli I1[I illllIIII IllllllVIlil] IIII XIII

IIVI illll IIII I_11

11111111111 III1 IIII III.IIIIIIIIVIII !llii Ilil Illl"iliiAlll Illll I1'' IlllIMIIIII#I|II III1 II11 IIIIIMIIlYlI,II !!![ II11 ....

'""'"" JJJi,l,iIIIlllllllll IIII IIII

TIM E

111111

IIIIIIIIII1[I]1]1]llllllllllllIIIIIIIIIIIIIIIIIIIIIIIIiillllIllil_IIIII1iillVIIIIMI

JilllullM

IIII[11111

llllllllllI

IlllllII11,1IIII11

IlIllllllllllllll[lllli I, [lllllllllllllllll|llll

Illllllll]lllllllll MIillll Ilillilllllllllllllill

1 1 '.d"'.l.I I I I I I [ I i I 1 I I I 1 I 1 1,'1[ I

'Nil ii_iiiiiii,:,_iii,iiJjIIIM IILllllllllll/llll/

!I]II ,,,,,,,1,,,,,,,,,,,,,,I III51 II I II 1111 I VI 111 I ,

,,,,,,,,,,,,,,,,,,,,,,I Illlltdl1111Allllll I I

I I I I I I I I'I'_LJ,_I I I I f I I I I I

IIII ,,,,,i,,,,,,,,,,,,,,,,.... I|111111111111111|1111IIIIllllllllllllllllll

ilii "'"'""'""'"""IIIIllllllllllllllllll.... III1111111111111111111iiii '"'"'"'"'"'"""JJllllllllllll|lllllll

IIII ,,,,,,,,,,,,,,,,,l,,,,IIIIIl,lllllllllllllll,OI ,lID ,II ,IZ ,I3 ,I4

(SEC)

IIIIIIII

-'I_ I Il I'X.III1_1! ! ! .n.

!!ii

III 1III Iiil,iiii!!!!

iJJJiiii

°I5

!

J

iI

i

.11

A

IMI--

.Jv

Q.

15dllIlll II/1 Ill I1111%1 IMI Ill IIJJl IIllilI I I llll

': : : :',::::::

iiiik." ::: 1::::: ::::t:::::;;

ii!!!:::::::::::::::

iiiii!!iii::::::::::

ii+!i:::::

iiiii

iiiii.hi .1_

lllIl• I

J_!!!

lllll

!!!!!iiiiiI|111

!!!!!iJiiiIIIII

!!!!!:::::

iiiiiiiiii!!!!!J1!!!_

iiiiiIll III11 ffl

,,IIIII I

!!iJ!!_,.,.Rill

.O! ._ .ot

[Ill llil.... IIII

IIII I111J.,,'t_ I IIII

rl I N.I IIII

III_ ',I ',1tllll II)1IIII1 IIII1111 IIII

ll'i ":il1,1, _,,,'"' "]I.... Illiiii lull.... lllll::: : IIII

iiii l_ll.... 111

iiii li,il.... IIJiii [l_[1111 illllIIII llill

IIII II" II

LII] IIIIIIII Illl

IIII IIIIIll[ Jillllll IIII

°01 .01

TIME

IIII '"'1'"' " ""

IIII ,,,,,,,1, ,, iiiiiiiii _I'11 llll I

Illl II1£1111%1 11 IIill !!

,,,,,,,,_, ,,,,, ,, ,,i iil_ 11

.........:::: I III 111 1 II! 1 II I I II I I I1 [I 11 ff(I III111

|1.... _1111ii 1 II ,_IIilii li.lllllllllii'llllllli I[

,,,,ll III I ,,1"'" ,I IIIII I/I]

:;:; II111111111_11111111111_IIII II

II '_-- II]I"IIIIITMI, _1,,I ,,,1,,"""I'"'I11_,,,"IIlli_ lllllllllllllllllLIIIlllll]!! iti1111 iiilll llllllllllllllll .... II H

1141 Illll] IlIIIIINIIIIIIill

1711 IIIlllillllllllllllllllllll I IIIA I I I II III I I1_1111 _ 11

lllltlllllllllll IllYlII I111111 llll Illll

"1111 IlIllllIlIllll]l%JJ"11[II lii_Illl iillll lilllIllllllllllllll'

.00 ._ .10 .It .I, .ll .14 .I$ .if

(SEC)



State Velocity, Frequency 21. iI cps

9Z-

SID 66-46-i

NOR'TH AMERICAN AVIATI. ON, INC. SPACE and INFORMATION SYSTEMS DIVISION

$2

i i ; i

i I :! _ i . !

i i 'l

ii_i _il I

,i i i, •

I 'l p_--

! ! ,

I iI i I

i i •

i ! ' i[ i :_ '

.03 .04

: IIIIIIIi III IIIIIIIIII

I III IIIIII III iIII III

ill ill 1Ill Ill ,, Ill Ill i

llJilll I,

, Iltllll iill I_l iII _111 IIllt_-f-t'T'--

IIIiiii IIll JII i

IIlilll II IIIII iIll Ill Illll]il I

TIME (SEC)

'I[

I

ilII

IIUIrIIIo0

i {, i

i i= II

I

i !i i

i !i

i.--t

JI II I

IIIllJI"II]

!'JIill

ill IIII.

llJIIII

III.lliIll!

.I.-I_!ill.ill

.I0

J

II

ii

i

I

].II

41KX)I

i:II I

411OO11I!I1

IIII! I

4"1_} II I

=li45_1 I

¢'1 II! I

44(_: ,

.0

! !

i I :

i i

i.113

I I

i l: I

I

i!il' I

i:

IIII I II I IIII

II I!III I II I III I

I I II I I

i II II I II I II I II I |111

I I IIIII I |I I !111

Itl111I I |li I

.04 .05

TIME

llll {llIIII ,IIIIII III

_ IIII JII_ III I I ' ['T"I'III;,,,I IIIIIII III' IIIII

_i I _lll1111.II1 IIIII1" II_iIIII IIIIII III

lll! 'llIII II

III IIIII II

i11 IIIII II

Ill ", |1IIII II

illl IIfill II]Ill II

._ .o'

(SEC)

, Ii

i

P

.011

' I'

I

' III

• I

III I1II I

IIIII

I |.ol

I

I

FII

1i ii ii !!

i !I

I iI ]i !

°10 .11


With Air Chamber, Impulse Disturbance 2 Percent of Steady-State

Velocity

-93 -

SID 66 -46-i

NORTH AMERICAN AVIAT(ON, INC.SPACE and INI_ORMATION SYSTEM8 DIVISION

$1

I1_11

II,li

_-. !1o 4.1["' I[(/)

"ItP

II= II

I1-,11

I1I1II

-ll.0

I!1I.01

I .....

!I I

i i

i[ ..--

I ! i ;

i

i !

I I

i iI !

.o3

IIIIIIIIII

I IIIIII1111IIIIIII111

i IIIIII I1111hlllll

' W'_I I_ I IILL-' IIIII

IIIII: IIII/

IIII1IIIIIIIII1Illll 'I1111

.04 .OS .0t_

TIME (SEC)

I iI

I iI i

III I

_Z

I.08

Ii]J.

I!.0t)

!t Ji l

II

I III:li

I i' II .I

I'

1 [I_.li II

[I

,10 ,11

I i i1

sloo; i i.

Ii I III]

--- !!!

I :I :

41oo [i.o .01

l: III il

[ [:: ilII II

II I

• ii t

ii t.OZ .03

:**: [iiI ! [ I I I I I I

l_I,,,,:,,,I

I '_"' ItlllllI[ I I I I 1 IIfii,_.l.--.L_ 1 I I ] 1 I

Ill I II1 I I I I I I I I] , I I lii I i [ I , ! a l

II I ] Ill I I I I II I i II 1

• 1 I |1 I II IJ I I II I II I

I I I I I II I " I I I1 1 11 1I i ] Ill I i I ill i i J|[

I I I I l] ] I I II I l II I

I 1 I I II I l I1[ 1 Ill !' I III I I11tllll_lr,,,,, ,,,

i_ltlitl'"' '='''I Ill I Ill

ti I 1 I I I I III 1 III iII 1 I I I I I IIII III I

III l I I I I i 1 1ll I IIIII I I t I I I 111 I II1

III U """ '"I II1 I III

Itl I_Li "' '_'-I'"" tlltl "tlllllll[,,, ,,I I I I l I I

,04 .05 .OI .07

TIME (SEC)

,,illI,,III [Illl'

1III I ,, _lll ;IIIII,l

II1" jIII,ll

II, j11,111,{|111II1III1]1

II|1,111

I1,! III|itt.i

.o$

I! I_I

i :: i. I I

,1 Ir II ,

IIi II : i I

[ II I : III I,i_ i i

i i,[ ! ,

" J"ITi |l ,II !

i II "

! : , i iI I

1 I 1 [ ]1 L I,Ot ,I0 ,It

Figure 15-26. Velocity and Pressure Time History at Midpoint of Feed Line


Velocity

- 94-

SID 66-46-i

NOR_TH AMERICAN AVIATION, INC.SPACE and INFORMATION SYSTEMS DIVISION

]

I

IL __

i[Ir

.1o .li

46oo

44oo

420O.0 .01 .04 .05 .06 .07 .Oil

TIME (SEC).11



Velocity

-95-

SID 66-46-I

NORTH

/

A+E,,,CANAV,AT,ON.,NO.< SPACE and INFORMATION SYSTEMS DIVISION

S$

.0!

i

iLi

.03

I I !! I

IIII]1I1II

J_

I iIIIIII II

iI

I

I iI II II II I iI [ II I1 I i

.04 .05

TIME

ii lili II I I

I I i II I II ' i I

11] I tI I I

I I

I I _: II II I

I I1 _ I I

-- I I,j.

Ii [I I! I! I

' I[ iI I

.06

(SEC)

! IIIIIII

i II1IIIIIIIIII

-- -..I.IIIIIII!I

! ' II

iIIII

I II 1

II

' I]IIIIIII

I

II i

i I L! 'I I

i!

i

!i

]!

Jl

C

: i

i.IO

l ; I[ : ,,| : :II

] : :I

I ili iI

I

I :1

Ii i1 t-i !1

.ill

: i I

! • I

i i i

; I

i • i

' I I; " I

.OZ .03

I I I [ I t I :iIi II i II I I " "I 1 " ; : "I I - i_i _ i *-.I 1 I].1 i :I I I!: I

,, ii ilI I " " _ *ii 1 lj ,I Ii I : { i I :

lli _i _I.

1 1 " i :: ' ;] L I I :

li ! I!I i I ,Ii i 'tI I I Ji i t l lt i ! t ;I I i ,! I i , I i

]

III [1 I I I I

,04 .05 .0_

TIME (SEC)

I I

I •

II ,II III III !II,ll!IIiiiII

IIIIII

[!IllI|1

IIIII111I11I I,.I rI II II I

if !II *ii iII, i i....,I I !] 1I ; i

t! !I :

li iI .

II :ifli ! I,i i i 1Ii ii III I ! II 1I I 11 II! ii IIII , I ! illI L II, 1i I i! i,¢:I 1 [ . rlI I I I I II I L I 1 I

,0? ,Of

I ;I II II I

_l!11 ,IIII i

! 1

I •I 1! I! I

J

i

i

I.io

I I . .

'l !!1

J_ iJl : +I . ,

I :_+

l .. :]

1i j_-_

-+._.

L _OH



Velocity

- 96 -

SID 66-46-I


m

1I

go

tt.

44

J

iIII F

.0i .02 .03 .04 .05

TIMEJ_

(SEC)

II I

IIII

IIIII

I JI ....... I

I.0e .0g .to .11

4=oo

44oo

U}-M.

41oo

..J

I_. 45oo

44o0

I II I1 I

I

I7 !

II1III

tI!

I I I _ • I

tIIII

I 1I

.o .m .oe .o3 .o4 .os .os .o7 .o, .oo . to .1,

TI M E (SEC)


With Air Chamber, Step Disturbance Z Percent of Steady-State

Velocity

- 97-

SID 66 -46-I


/

\---


R| . ,

l 1 1

50

: !!

iii

la.l¢.f) :;1'i

,- ill'_ 4s i

; : : t44

i:: ii :: I

I.,li_t,0 .01

I111I111 I

ttll i IJillIIII

i II I_._ IIliI r_llll L,Ill IIJlJ t I'I111 tllll _ i:

, Jill [i IiIIII I I.

, Illil I Ii I lJI , ! I j t[ I III , I i I ,

.02 .05

i i

i

i ii : ii

! !

i

!

i

I

I

i i

t

.04 .05

TIM

I]

I

I

iJI

E (SEC)

i : I

i i iI' l IIIIi

: I I1il

I Ii: I If

I iJi i iiiI II

i I II; !t il--'-t,

..-_1 I Ii; i _ _L_1i ]1

ii ilI.of .(_ ._

I

i

]

I

,AO

i

iI

,11

"®: i i

_,. [!!Ihl I : :

i, i!!

[ : :[ : :

"" lii• r •

i :: i

I i4 _1_'| •

o0

! I: I

i il

i i

i i

i i: I

: :{ :

i: I

i

" ii _

: ii ,: I

.02

.!

I I I I I1 !I J i I I If " • " i, ',I IIi_ i

! _l iii :I I 1 I I i

Iili II[' _;I " " ' !

i II ,,,,ii li II II ,,,

I_ II !: '!ii iI I !_I II i i, i ':' . !r i

1 , II! 'I ] i I i i ;

" J I :

,04 .05 .01

TIM E (SEC)

' II

i ii !

I

r W

o0

IIII III I | [ I I[ I 1 [ I! I I ' I l

1 II I 1 I I I

ill:I_ lli!II' IIII '_III III

II IIIII III

,tl 111

I t I' i t I; I 1 I |

i I_ ii ,.08 ,Og

]

: i

i

i i

i -- -1

! q

ii

-1

.lo .;_


With Air Chamber, Step Disturbance 2 Percent of Steady-State

Velocity

98-

SID 66-46-i

Nd'RTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISION

SS

4S

OWfJ_

U.

SS

3o

I$

.01 ._ .03 .04 .05 ._

TIME (SEC)

.i 1 1

r--It

I

,01 ,1_

I I:ll

tJ II I

I |I II II II II 1I II II II II II II 1I II II II I

- L,,,.,,_ ''w I II II II II I

,tO ,It

04I--It

m--I

Q.

.01 .02 .05 .04 .05 .041 ,OT .Oil ,0tl . tO . I t

TIME (SEC)



Velocity

- 99 -

SID 66-46-1

/

NORTH AMERICAN AVIATION, INC. ( _ SPACE and INFORMATION SYSTEMS DIVI$_ON

A

OtU(/1

I$

SO

45

4O

3S

3o

£$.0

:\tj

,01 ,03

i1II1II

I1

1fIIII

t ' I

, I

1

i iI t

I i I

i t !

1!IIIIIIIlIII

_,,

II

: I

1I[

1I

I I1

, i i l, I I I

I I 1I

I I I-I I 'I

_-I,I I1 ! 1 . lI ' III T _l

l i Ti II I 1I I |I 1 I1 1 I1 1 1I I I

i I i I I.05 .OS .O

TIME (SEC)

I' I

lIIII

tI1

i II I

!

!I

I ! :!I

, !

!, Ii i i! !J I i

' ,,114,• ' I I! i i

i I1-t: [!t

i J: ,:J

i ;'i: : |

: I

I

.-i i t: : ]• . i I

,OI .10 .11

II

Ioo0

Sloe

51lop

IM

I-J,_ 5ZoO%fZ)J s_o

a.480G

44_00

44OO.0

IE.Ol .oZ .03 ._

EE

.05 .Oil

TIME ($EC)

m

,OT

m

,08 .Ot

--t--_ --t--+_-_

,10 ,li



Velocity

- 100-

SID 66-46-I

NO_RTH AMERICAN AVIATION INC. SPACE and INFORMATION SYSTEMS DIVISION

illllllllJllll li II

SZ.UlIIIIIIII]IIIt II IIIIl[lllllllllllltlilllllIlilllllllllllllilllllllVi_ IIIIIII1111111111111

sl._lllllllllllllllllll[lllll

Ill lLIIIIIIIlilIIIIIIIIIIII MUlIIJlIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

s1.qllllllllllllllllllllllll

I1 VlllllllllllllllllliIIIIIl|lllllllllllllllllI1 IIIIIIIIIIIIIIIIIIIII

sl._llllllllllllllllllllllllIllllll!qlllllll _1111!I I IIIIIIIl_i_lll!II I I I I1 I RLII I1 Irl IIVII I

sl._l I I I IIII IVi LI/I%/I I III IllI1 i I I I I I I ItJWI i I i II I II1_1ITrlIIIIliFITIIIIIIIIILIIIIIIIIIIIIIIIIIIIIIIr

sl.elllllllllllllllllllllll.0 .lllJ .IO .1S .20

Illtlllllll}lllllllllllillllllll'W ]llilllllllllilill IIIIIIIIIIIIit111111111111 IIIIIIVI]I_[II IIIIIIIM[IIIIIIII IIIlllllllllllllllllllllll"[lllN IIIIIIIL./R_IIIliIIIIIIIIIIItltlIIIIIIIIIIIIIIIAIIIIIMIIIIIIAIItNIII iIIIII1111111111111 LWkLIIIIIII#IIIIIIIMIII L/IIIII_IIII IIIIIIIIIIt II I I I_] N_II II IFII I III11 IILIIIilIII I IIM 1II III II IIIIII II I IVI II IHIIIIM I1 I IIII I I IkT'TIIII I111111IIIIIIIIIIIIIIII _ Illllll,&?llllllllllllllllllllllll

IllililillllllllJlllllllllTIIIIIllllllllllllllllllll pIIIIIIIit1111111/1111111111111111111111111111111111

Ilililllilllllllfiiilliliilllllllllliliillllllliiiill IItlillllllillltrt tllllllllllliIllllIIlillllillllliI

i1111LtililA IltllJ I Iltll II IIIIIIII IIIIII IIIIIIIIIII till iitllli'rl'llil.lllffl IIIIII IIII III III I11111 I II111 IIII111

ill iI/I It II'II/Vlli It I IIIIllll IIII I Illll I IIIll IItllllll iLiii/lllllllrllllllltlllllllllllllllllllllllllllllll

IIIIIIItlllllllllllllllilllllllllllllllllllllllllllll IIllillllltllllllllllllllllllltlllllllllllllllllllllIIIrllllllllllilllllllllllllllllllllllllllllllllltll!!!II/1111111111111111111111111111t11111111111111111111

k/llllllllllllllllllllllllllllllllllllllllllllllllliiiIIIIIIIIIIIIIIIIIIIIIIIIIIIII1111111111111111111111 11

IlllllllllllltlllllllllllllllllllllllllllilllllllllllllIIIIIIllllllllll[lillllllllllllllllllllllllllllllll itoZ5 .30 .35 .40 .45 .SO .55 .r_ .6S .?0 .?S

TIME (SEC)

, !

: I: !

b'

IllIIIllII

Ill

iiii I1IIIII

.80 .b5

I11 111L,,.LI I I I I I I I I 1"11I I I IIIII11/11'%111111111111111II111111 i%11111IIIIII1111

4:_11111111 I I I1111 I III I I III I II I I III I I I II I I I I I I I I ll"_.l IIII11111 I11111111 I1711%1II I VII I I ttil I I tlt I I#1II I%1

"3z211 I1 I I I I I 1Ill I I I I 11111 I tlII Iit II II i Iltl I I 11 I If tll III

I'-" Illlllllllllllllllll IIIIPU. Ilblllllllllllllllli IIIII

43zoL/I I I I I I I t III I I I I#1 IllllG) Illllllllllllklllll,.,,,j I1111111t11111.1#11: i..,,. I11 I I I I I I I I I I r'l"l I I I

43.11i11111111111111111,,,, [illi,iilii,iliillilllll

I111111111111t11111

'r"illllllll IIIIIllll llllll IIIIIIIII11111111111I11111 IIIIIIIIIIIIIiill111111 I IIIIIIIIII

l'llll I I I I I I IIIIIIIlllllllll.o .os .lo .is .zo .:,5

t11111111i1111t11111111111111 II II II I IIllllltlllt]llll ,,,,,,,,,,,,,::,,I::,,IIII,,IIIIIII •,,' ' "III:III I!!"IIII,, l ii

II II 1111"IIII,,II .... 1 •ItllllltJltlltllllllltllllJll..ll ,.., : ,,_,IltltllllltllltllllllllllllllIIIIIIItll t111111111111111 It IIIII L.llll III IIII1[11_11 : ::

. I I I I IJ,,,.MI I I I I I I I 111111 I I I I I I I I I I 1#'11_11I I 11 I 11I I#11I ,ll.!

tilt rli%iJ li f t il t It,,FN t lii t itiillillktt ill t ili/tiil_I I IIA II I%11I I I I I I Itll I Ill I I I I I I 1t VI I I rll I I i III I.!1I1 I

Illl'lilltllllllili/lllillllillllllllllUlilllllilllll i IIlillllllliU ililllllillltlli li lil/ll Iliillil IliilllltiIll/lllllltilltlltl/ltllll]llltllilllltllllllll#1111111 ; IIiilillllllllllltl/lltlltllllllillllillllllliifillllli " 'II#ill III1_11111t/tlltllttll II IIIIIllllll'ill Illll IIIIII IiiIrltilt]iillliili#llilli;itlitllliilliliil_iil_lllillil illy Ii#llll Ilillilllllllllil iikl lil/lllii I III rll Illlililii _li

lililiiitli_illt,4111iiilit_ltl!!!!!!!!!!..N.J..'!!!!!!!!! i .:"1111111tll IIIIl/llllllltllklM

iii iiiiiiiiiiiiil i111111111111i3111111111111111 il II11 11 II lIIIIIIIIIIIIIit11111111111111 ......................

,,,,,,,,liiii,,,,,,,,,,,,,,,,iiiiiiiiiiiiiirl-HimIIIIIIIllllllllllllllllllllll II Illlllllll.30 .35 .40 .45 ,50 ,55 ,SO .15 .TO .l$ .80 .85

TI M E (SEC)


With Air Chamber Under Sinusoidal Disturbance, Amplitude 2 Percent

of Steady-State Velocity, Frequency 7.20 cps

- I01 -

SID 66-46-1

NORTH AMERICAN AV IAT I.O N, INC.SPACE and INFORMATION SYSTEMS DIVISION

¢.)t_J0_

U.

St .S4

51 .S2

St .SO

S1.4|

S1.4l

S 1.44

$1.4Z

51.4_

.0

TIME (SEC)

8S

O.

.0 .05 .tO .15 .ZO .Z5 ,30 .35 .40 .45 ,50 .55 .60 .iS

TIME (SEC)


With Air Chamber Under Sinusoidal Disturbance Amplitude 2 Percent

of Steady-State Velocity, Frequency 7.20 cps

- IOZ -

SID 66-46-I

N O'RT H AMERICAN AVIATION, INC, SPACE and INFORMATION SYSTEMS DIVISION

$1.$2

$1 .Sl

S! .SO

!! !iiiii+....II t] i_tl

g ..... t:I I I_ . .: M I _ : :

: iiiiiikP I I I lit II

(.) st'49i I ! ! !_!'

I Ill:t:

s_'+°i iiiiii...: : : : : : :

II I I1[:::

: iiiiii_ 51o4T .....

: : : : : :

st.,, illiii::::::

iiii!!SI.41 _::::

iiiiii!:::::

.DS .Oe

I I I I iI I I I I

iiiiii

',I111,]!!I . . -

!!!!

iiii:!!!

iiii

ii_dii i ! 1

I/1 _1 +

I/I I_|/

_.!!!

!!!!'. : : :

i[[iiIIIII

.10 .12 .14 .16 .18

TIME (SEC)

!!!!1!::::::

iiiii_:::::

iiiiiiiiii:::::

iiiiiii!!i!!!ii:::::

!!!!!

iiiii

iii_I I 111_I l Ill I-

_1111q i_Lfl I I

:::::

:::::

::::::::::

:::::

iiiii

!!!!!

iiii

iii!!!!!iiii

iiii

ii!!

iiii

[!!!I .,,L I

[_ib.iiii

ii!!

.24 .21

![[![![!: : : : : : : :

i!iiii[!: : : : : : : :

: : : : : : ; :

iiiiii[-i!!!!!!!!!!!!iiiiii!!!!!iiiiiii[i: : : : : : : :: : : : : : : :

!!!!!i!!

!!!!iiiiiii_iiiiI,l_l"ll[l.,

rl _Wl i I i i i

: : : : : : ,I .I

ii!!i;ii

iiii ,[[[

.18 .30 .31

4_4

_ 4322

l-b.

\ 4321

m.J,,...* 4320

4_|9

4318 i

I4117 |

.0

] !! ''If::::::

! ,, ....+llllll_

i illlll_

I t I I I I ?I

: ill;ill: !!t/t!!l: ii;iiii

i1_[]11

I [WIIIII

lllIlll

+ ,,,II!!I.: r:ii!!i

i II .....,liiiiit II,,iiiii• i_iiiii

II,,[[[11II,,[[]11

"'I[II[

!!!!!

!!iii: : : : :

iiii!: : : : :'. _ : : :

iiiii: : : : ;

iiiiiiiiii.!!Ill/

i i i

I lJl I ]

/_. III

iiiii: : : : :

: : : : :

[[i[i: : : : :: : : : :

iiiiiiiiii

.08 .10 .12 .14 .15 .18

TIM E (SEC)

,_iiii IIIII

!iiiii Jiiii

: : : : : : i i _.

+++i++::::::

Villi

:::::: :::::_::::: :::::

ii iiii.... _iii_

+++++A iiiii

Jii_i Jill][._-r. !!! iiiii:::::: :::::

: :: ::: i iiii

ttitii +++14.20 ._ .Z4 .21

I:tlI :

i+:!I _

I :

I :

:+I :I ..

_J

.21 ._,0 .$II


With Air Chamber Under Sinusoidal Disturbance, Amplitude

Z Percent of Steady-State Velocity, Frequency 9. 70 cps

- 103 -

SID 66-46- I


t_

h

IIIIIIIIIIIIIII IIIIIII

Ilo41 I I I I_ l l l I l l I I lI I_I I I I I I I I

I I I I I I I[I rllllllll llltlll

II.41 IIIIIII lllllll

IIIIIIII % IIIIIIIli.4711111111 IIIIIII

I:II:III ]II_"_ "I|1

"'" IIIIIII I''"",,,,,,llllllll IIIIIII

II;;III[ :I:''",,,,ll._ll]lllll I111111

IIIIIIII IIIIIIIlllllllt IIIIIII

:IIII:]: "'""IIIIIIIS|.43[1111lll IIIIIII

o0 ,_ ,_ ,_ ,_

I I I I I I I

I I I I I I II II II III I I I I I II I I I I I II I I I I I I

I II II IIi I J I I I II I I I I I II II II III I I I I I II I I I I I IF I I II III I I I I III I I I I III I I I I I II I IJ,..l I II I III _k I I

'SVI,1','I"_]

1111111I I I I I I II I I I I I [I I I 1 [ I II I I I IliI I I I I [ lI I I I I I II II II III I I I I I II I I I I I II II li II

.I0 .12 o14

I I I I I I IIII I I III I I I I I II I I I I I I

IIIIIIII I I I I I II I I I I I II I I I I I III I I I II

] I I I I I II I I I I I IIIII illI I I I I I II I I I I I III I I I II

I I I I I I II I I I I I I

II il I IIII I I I III I I I I I II I I I I I tII II IIII I I I I I II I I I I I I

%_I I I I I I I_LI I I I I IIM I I I I III I I I I III_II III

] ]11_1I11 IITII I I I 11 1I I I I I I I

l I I I I I I

.16 .18

TIM E (SEC)

IIIII IIIIII I11111111IIIII ill[ll I[lll[llllllll IIIlll IIIIIIII1III]1 IIII11 ]llllil]lIIIII IIIIII IIIIl[lil

]:I:: :II:II ,,,,,,,,,"'"'"I,IIII IIIIII IIIIIIIIIIIIII IIIIII IIIIIIIIIIIIII IIIIII IIIIIIIIIIIIII IIIIII IIIIIIIIIIIIII IIIIII IIIIIIIII

IIIII llllllIIIII IIIIII IIII|IIIIIIIIII IIIIII IIIIIIIIIIIIIII IIIIII IIIIIIIIIIIIIII IIIIII IIIIIIIIIIIIIII III]II IIIIIIIIIIIIIII IIIIII IIII|IIIIIIIIII II_III IIIIIIIIIIIIIIV_I'II IIIIIIIIII

,,'_II: :I:::: lllT]l:llli_ IIII IIIIII llllllllll!

I

lllll I,IIIIIIIII IIIIII IIIIIIIIIII

!!!!!

.I0 .IZ

I I [ I l l I l I I I I I I I I I I I I I I I I i I I I I I I I I I I I I I

.14 .IS .ll .lO .IID .14 .II .II .I0 .31

TIM E (SEC)


With Air Chamber _ Under Sinusoidal Disturbance, Amplitude

Z Percent of Steady-State Velocity, Frequency 9.70 cps

- 104 -

SID 66- 46- I


Q

¢J

¢fJ

b.

!!!! !!! ! !!! !

iii iii :: i::::::

iii iii !!!!!: : : : : : : : ; : ;

iii i ii i ii; i: : : ; : ......

iii iii : ::: :LI=_ --- iiiii

.z. iiiiiU/I I I I I .....V,=l--- i iiii,,._iii .....

,,,,,,, i[iiiI I I ' I/RII I I I Will l]llll I II I I ,4 III I/ IM I jr_

" " " _1 It j_ I! I/lliii I III I RIll

: :: ::: _r! !!v!

iii iii iiiii: : : : : : : : : : ;

iii !!i ii!ii: : : : : : : : : : :: : : : : : ; : : : :

iii ii! iii!!::: ::: ii iii i._ _ .o6 .IO

!!!!

ii!ii!ii

i!!i

!i!i

i!!iiiiiiiii: : .- :

iiiiiiii

_.111:3: : :

I I I/II NI/I

!]_.!

iiii.- : =. :

.12 " ".1.4"

TIME

!!!!

iiii

iiii

iiii

iiiiI I : :

iiii.- : : :

iiii

i,_i i irJll I1

II I%111

r! !7!

!!!!: : : .=

iiii

.t6 .i8" " ".2O

(SEC)

!! !!!! !!!!

iii!ii iiii!!ii!! !!ii: : : : : : ; ; ; :: : : : : : : : : :

!iii!! iiii: : : : : : ; : : :

iiiiii iiii: : : : : : : : : :: : : : : : : : : :

iiiiii iiiiiiiiii ii!!: : : : : : : : : .- :

:: :::: Jill-: : : : : :I I l Ir

: : : ; : : I I IJl: : : : : : I I IIII I : : : : )_l Vl

• 1 i i i i #l'1,.J iI1%1 1 IJ I I1 l I

r 11 I It'Ll [ I I I II ll_ J_a_ ! ! ! .! !V! !Y :: ::

iiiiii iiii: ; : : ; : : : : :

._., " .=, " " L_

431_!!! !!!

ill',lllrl I I _1 I I

!!! illilli ii i%1i

III I!1Ill III! !! I lll

Ill::: Iiiiii i1_:.-: !!!tiii i!!iii i!i: ." : : : :

iii iii: ." ". : .- ;

i!! !ii; ; ; : : :

iii ii!III iii

!!!

iiiIII

ii/!.i#

7i

,i I '_'!.

ii:: !

: :

ii.01 .10 .12 .14

TIME

!!!

!!!

iii• . . j

!!!,ii!/.= : _-

: :#:

iiiIII I¥11

,iS ,iS" " ._0

(SEC)

!!!]!! !!!!iiiiii i!!!!!!!!!_!!!!

:::::::::::

iiiiiiiiiii"_.J I I 1 I : : : : :....

ii!_iii iiiiii ii_J ii i!!!!

i I I I III I : : : :

i I II I_ I : : : :

;;;;;_;_::::

iiiiii iiiii

!!!!!!!_ ........... iiiiii

::::;: I_11![i!!iii N_v__::::::::::

'-._=--'.=,'--..-


With Air Chamber Under Sinusoidal Disturbances, Amplitude

2 Percent of Steady-State Velocity, Frequency II. 70 cps

- 105-

SID 66-46- 1

NORTH AMERICAN AVIATION, INC. SPACE and INFORMATION SYSTEMS DIVISI.ON

(JLIJ01

I"h

St .SO]_ I.LIII I1J/l'_ I I I

I1-1 III11

,1.4, t [ I 1'1 'IIX[_I

IlkIIIIIII1111111

$t.4_ I I I 1 I I I11111111111111IIIIIIIlIIIIll

51.4T I I I f I I IIIIIIII

ltl:lllIIIIIII

S|.46 I [ I I II I11111[I

11111

-.-111111,,11111I111illI1 III11

Sl.44r I 11 I II I.0 .r_ .(_1

IIIIIIII1IIIIlllllllllllltl111111111II1111111tlllllill111111111IIIIIIIII111111111IIIIIIIIIIIIIIIIIIIIIIIIIIL111111111II1111111IJ I I,_kl I I 1

_L _ _ IX1 I I I_ !11 111_111

IIIlll_lr_l

IIIIIII1_IIIIIII1_IIIIII111_II III II IISIIIIIIIIIII1111111IIIIIIIII111111111II1111111IIIIIIIIII I I.I I I I I IIIIIIIIII

._ ._ .10 .lZ

111111I1 II I1

Ill ii i1,i1111111111 I

llllll

t IIIII11 IIC !t,,i,I I I I II I1 II II

I', ',II tt ,i,ii,IIII III II11 11 IIII III I I II II II II I1II IIIII II II IIII1 II II II I1 I1Ill III1 IIII I1I111111 I111 III 11 IIII II11 III I I I I I I I I/l_l I I

] J ] 1_'

-_[ I I I I I/I I 1 I [ I II_1 IA.J Yl I 1 1 I I II_LYl r"l I I 1 I I I I

I I1 I ii [ill

III IIII I111 11ii]1111 I111 III II III1 II II II

.14 .16 .18 .20

TIME (SEC)

llllll l J11t111 IIIIIII 1

iIII,1 111 I )I ]tlIlll ,

LIIIII ]Illlll IIIIIII III1111 i111111 I111111 |Illtll I 1111111 !IIII11 _ I Il I I I I I iJ _m 1 1IIII11

[1111_" _i11111/I I111111 Illll_l IllJ.Jrll 1

_k I[1111 IYlIIII 1

1-111111 l111111 1llllll I

IIII1

0

I-U.

co.J

Q.

.o2 .04 .OG .08 .I0 .IZ .14 .2o .22 .Z4

TIME

!!!

1 Ill

:2:

i|1Ill

III ItIII II

L Ill IIk III III III II

|1 I

_ii

L,; : :

.16' ' ".i8

(SEC)

I I i i i,,tl,,,IlllllI I 1 I I I I I I 1 I I I

I I I I I I I I I I I 1 II [ 1 I I I I I I I I 1 I

llllll''lllIl,,| I I I | I I 1 I I I I I

i

1 i i i i 1 II I | I | 11 I | I I I II I I I I I I I¢1 I 1 [ II I 1 I I 1 I III ] l 1 II I I L I I I I I I J II I J I I_1 I i.LL_._

IIIll I I il I 1 II II I1 I I I | [,_tl ......... -HIll I Ill I I1 I | I 1Ui I 111 I III I I I 1 I

I Ill I1 I Ill I ] I 1I|l 111 |Ill I l I l iI 111 II | ]1111 I I I l I

III I [ II_LLJ_I i ii i i i_.111 I I 1_I I I

i_1 i1| i i Iii i ill I ] i i 1 I I I iii i !11 i i i 1 i 1 I IIi i iul I I I I I 1 I lii 1 i i i 1 1 i 1 i i 1

, I 1 | | ]L J_ _[_}1_m. _I I I I I

.Z(, .20 ._O .a='



2 Percent of Steady-State Velocity, Frequency Ii. 70 cps

I06 -

SID 66-46- 1

NORTH AMERICAN AVIATION INC.

/

,o-

11r

i!

/

k

|I

x

i

+ID .tZ .14 .16 .18 ._P

TIME (SEC)

fLr +1 J

!1 •

I'

4_.0

437.1.9

4321.0

I--

mJ

o.

43tg.§

A_

_L f

I !A

t IA I

I

..... --4 __.... r--- -_- .... "'_

A

I xI x f_

AtI l I III _ t

--il -- i--iz_,,q

I A

45iS.©

.0 .02 ' ._1 .06 .O8 .tO .rE .t4 .iS .ill .ZO .Z?.

TIME (SEC)

I II Iu 1

tI1I1

._4 ._



Z Percent of Steady-State \_elocity, Frequency 21. II cps

107 -

SID 66-46-i


hi

IL

It .SO0

91.4l$

$1.411C

S t .41g

51.480

91.475

51.4TO

$1 .d5

.0

IIIIIIIIIII '''''_'''_l',,,,I,_'llllll]llllllllllllllll,,,, 'IIIII"III'II,'IIIIIIIIII,,,,,,,,,,,,,,, II,,zl,,,,,, I , I I I L III I I I, , , , I , I , I I1 ' ' ' ' Illll ''_ 'lllJlllll'''''* ''IZll,,, lllllllllll,,,,[lilllIlllIIIIIl[]l|llllll III111111

illllllllIlll[]llllllllllllli[l|llllllll]ll III

__llllllllllllllllll l iii_i!!lll!!!l

,_ .08 .JO .12 .14 .16 .18 *_ ._ ._

TIME (SEC)

!iLi

.6.

I-Ll.

El--I

Q.

41141'

l I ) I I I [ I I

] I I I I

4141 ' [ |l I I II I[A lI l II I I

ell I II ] I II l I [I III I II 1 LI 11 l II /_

III1 I I1 I II II I II I I _ l[Ill I I I II II I II I I I I t I

4MGIIll I I I ]l II I I t ! Ii | till I III 11 I I ] I1 II ] I I I ! II t 1II III I l i II I I I I | I | I I II 1IIIII I I I I I I I I I I I I

I_ III R I I I I It I I I I4415g Ill II I I I I I! I I I

lltl "" '"' ' ' _ *n 1 I l 1 II I I

,,, ,,,, "'" ' I! i ; ]I '_ "'" " TM +_ _ ' _:Ill it II IllI , 11,|1 ,|lit I , ,,i ,,,,i -,,_ , , i I ,

,|,I, ,|,,, l _ , l, t _L__l+I I ,,,1. .1.. , .465? l i l II11 I Ill I I I " ii ', 1

,. ,i,,, I *,t'I ,, ,l 11 I UI lI II t IMI I I ] l/ I _

I I 1 I ----, ll+J I I I LJ I I l1131 I I "I I I l I U I I I

I 1 I I 1 I I I I

41135 I I I . l I I

.0 .O'Z .04 .06 .08 . lO .12 ,14 .16 .18 .20

TIME (SEC)

t-t..........22 .24 .II



2 Percent of Steady-State Velocity, Frequency gl. ii cps

- 108 -

SID 66-46-i


.05 .tO .iS _ .ZS .30 .35 .40 .45 .SO .SS .gO .115 .SO .IS

TIME (SEC)

41B$

IIIIII

IIIllJII]Ill

I Iqi

4310

I-Ll.

GO 4szo l/i.J ill

Ill

UJI_. 431!111J

|11

451© | | I

1ll

||1

III|11

4_ ! IJ

oD

____J____________________________________________________________i_________________

IlllN II]111111111 Ill II IiIiItllllllllllllllllllllllllllllllllllllllllllllllllllllllll IIlVllillllll IIIllllll II lllllllllIIIlllllllllllllllllllllllllllllllllllllllllllll

Ilill II.lllllllllllllll 11 I II11111111111111.1111111111111111 !1111111111111111111111t111

I|111 Ill Illl illllllLllllllltlllllllllitlilllillllltllll[lll[lllll[llllll

III11 IIUIlIIIIIIIII RIIIi I IIIIIIIIItlIlIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIILIIIIIIIIIIIIIII Illlllllllllllil II11111111111111111111111t111111111111111111111111/1"_11111111

IIIIII I llllll Illi/lllll 11 llltllllllllllllllllltlll IIII IIII IIIJ*kll IIII IIIIIi1111111111

ill I 1 I 11_1 III II Ill I III I I I I llllllJ,-l, lll t I I I I III 11 I I I I I I 1 t I I I I 1/11111 I I 1 III ll/llllll I IIt I I

111111 Illlllllllllllllill IIIIIIl]lll_llllllllllllrlMIIllllllllillllllllllllllllllllllll

rlllll __/__/__V_i__/__j__Itl II I t1|11 Illrlll I I II I I I II I I1/1 I Itil I I111 Illll I RI I I I II II I II I Illl I I I I I I Ii111111_111111

IIIIIIIIl_llllllllllllllllltllllllltltllllll_llll|lllllltl_lltl_llll Illlllllllltll|l

IIIll I I liN I 11!/I I I I I II Ill I 1|1 I Ii I 11 I Ill I I I I I I Illtl I Ikt I I I II I Ill tiiJl_] li I | IIi II I I I Ill I I I I

I II I I I III |1 I I!1 I1 I I I I III i I I I I I/lI 1 I llll I I I 11 t lJ Jl I I Ill I I II t I /I I I I1111 I I I I I/I I 11 Illlll I I I I11111 III11_11/1111111111i11111111111111t_1111111111_1111111111111 llllllllllllllllllllll

IIII1 IIIIIM/IIIIIIIIIIIIIIIIIIIlIIil Illtllt|lllllll III1_ IIIIII1_111 II#lilllllllllll

I I III I II1 III Ill I 1 I I I I I I_1 I 1]111tl I I Illll I I I1 Ii 11 III I Ill I I11 II1 ! I II11 IIIl I111 I I I I I11 I _1 I 1 I

11111 IIIllllllllll III Ill I11 |llllllllllllll IIIII111111 I I1111 111 Illl IIII II/ll]llltllll 111

I1111 IIIII I11111 II I I I | III 1 I/ Illlllllll I11 IIIltllllil IJ I_ I II IIIII Ill I11 IIII II I II _)_'J

111 1 11 I I111 I II II I I I I I 11 I_11]/l II11 I Ill Ill I 1111111 I I II RI 111/I I Ill 111111_11/I I t I II I I I II I1"II]111] lllllllllllllllllll1_J_lllllllllll Ill/llllllllll_[llllli|lll[[ll?'qlllll]lllllllll

Illll IIIIItlllll II111111illlllllllllll_llrll]lllllillMV[lllllllll Illll[lllllllllllll21111 illlllllllltllll IIIIIllllllll]lll_VIIIIIlllllilrlllllllll[llll Illlliilllllllll

Illll IIIllill[llillli IIIll]lltillll|ll FIItlIIIIIIIII Illllllilllllll|lllllllllllllll

Illli III]1111111111111111111111111il IIIllllllllllllllllllllllllllllllll]lll[]illill

IIII1 llllllllllllllll IIIIllltlllllllllllllllllllllllllllllllllllllllllllllllJllllll

IIII1 II111111111 11111 IIIII1 IIIllllli I Illllll[lllllil II11[ IIIllllli] I IIIIliI|lilI1-LI

III11 III1111111111111 IIII 11111111111 lllillllllllllll Illillllllllilllillllli[lllllll

.O_l, .lO .IS .Lq) .L*S .30 ._15 .40 .45 .50 .$5 .1_0 .IS .70 .?S .10 ._5

TIME (SEC)



5 Percent of Steady-State Velocity, Frequency 7.20 cps

- 109 -


St .SO

$1.55

51.5C

uJf/) si .4s

u.51.401

S1.35

illiiiii

_1111

I_lll

'IIIP_!!!!!

1:::::

Jiill:::11

iiiii'iiiii

1::::

!!1!!!!!!i1::::

IIIlllllllllll IIIII,Yll N lllll ....._t__ iii_i-_IIIIIIII11111! I VIII

II_ IIIII_III: Mill[lllllllll_lli IIIIII _ IIIIIIII_ II llllll_lllllilti_i lllll_IILIIIIIIIP IIIII[lllllllllllll .....

:::::::::::::: :::::

iiiiiiiiiii[ii :iiiii...!! ., !!!!!..!! ...

[iiiii!!!!!!!!.....iiiiiiiiiiiiil iiiii

!!!!i!!!!!!!! !!!!!!!!!i!!!iiiiii iii!!::::::::::1::; :::::

_iiill llllllliiiiii: _i_i_iii.0 .05 .g) .15 .20 .5_5 .30 .35 .40 .45 .50 .55 .60 .65 -_

TIME (SEC)

!!llll+i!!i!F

NIIIIJ *Ikllll I

iiii' { '::::::

[![![!:!iiiii

:::::::

!!!!!!i!iiiii

iiiiiis .8o .Is

4t,

414q

" " III III" '""'"lllllR illlllllllll IIIII I IIII IIIIII IIIII IIIIIIII " i::::':

III III ::II :::::_II4M. lllJlllllllllllllIlllllll llIIllllll IIIII Illlllll II ::::::::IIIIUlIII_IIIIIIIII IIIII I IIII IIIIIIIIIII IIIIIIII II II IIIl ll_llllllllli l, ili il.iAIl ill i , ii iii ill il ll, iilii III IIiIi , III I I liltit ....... iiiiiiii''"'""_"' "'""""'"''''"'""'''"'"'IIIIII" IIII" iii iiiii ........I LIIIIIIII_IIIIIII II II I IIIIIII I IlI I II II IIII II II II II I I II II "'' ' .... ::::::::

lllllllllllllllllnilllllilll1111111illllllllllllllllilliilllll I lillil iii iiiii _iiiiiiiii iiiiIii i,i uiiin i 1 i i i ill i iiii111 i i i i i i i i i LIII i i i i i i i 1 i i ..........................,llllllllllFJlllllllilllillllBllllllllllllllllllillllllllllliifllll i ilii. Lii iiiii _,,il]ili

.... ,= ;.-/ .....

llllllllllllllUlllllllllllllllilillillllllllllil_illlllllllAIIII.il I tll_ _tll I .... iiiiiiiiIIllillI II I IIIIIllI II [lltlIII III.III I II IIlilll_lI II I II II Il_Wiil_lI 1111IIt_llillilillllllllllllVllllllllllllllllllOVllflftlllllllli_liVtl..!!!!!.Vi Ill iiiiL ii_ii]iI011111 I II1 Ill IIlt111111 111111111111111 I I'I III III I I I IIll I {I I ........ II'I {!!!_- : ........lllllllllllllllllllllllllllnllllllllllllllllLlllllll_l II IIIIIIIII'I _'

III [ i i I I I lliillllli'li I i i I I iq i i I I IJ I Fi I I i illli I i i i i I' i i 1 I IYI.I I i i iil i iii i iii/i .........IIIIIIIIIIIIUIilllllllll[ll IIIEIIIIIIIIIII/IIIIIIIIIIIIII]IIIIIII|II 1 Ill i!_L.[_ _BI I 1 I I I ] I I I! Illlll I I I I I I I I IIII !,111 II III Ill I I I I II I i 11 I I II11 I I I.lll I III IIP_I 'II[I[I II l lII] llllllI lI ll lI I II|ISIIRII II II III LI I I/illlI I[ I IIII I IYVI I ' :;;,; iiiiii!,i_IlllililliliUlillllllllllllillllllli'.lllllllilliilllllllll IliU._iillli ill _/Iril_- _-_.._I I 1111 I I I II I III I I I I 111 I I 11"111111111I I I I I I I VIllA !1"11 I I 11 } t I"111_11t 11II II II I IIIIIII III I II I II IIIII1111111111 II l'lllllllI II II I IIII|I_ III "'" I_ii_ ____._._ _flllllllllllllllllllllllllUlqllllllllllll_/llllllllllllllllrllll lli I:::: :::::: :llllll 1 ! _1 11111]11111111 "_i i ........

lllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll ::: _ ........lllllll IIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIII :

4,_ Illlllllllllllllllllllllllllllllllllllllllllll II11 iii iiiii _lTllTl-i.O .05 .IO .15 .20 .P'5 .,L_O .3_ .40 .4_ .50 .55 .I+0 .I_ .?O .?S .lid *85

TIME ($EC)

4ibl;

I'_ 4ibll

u.

..J 4s_8

46_4




- ii0-

SID 66-46- 1


b

Sl .S8

$1 .SO

51.48

0 sl.,_U.IC_

51.44

U.

S1.40

51 .!14

SI.M

.0 n. .04 .06 .06 .10 .12

!iJi

i!!i.,: : : :

iii!iiii

iiii!!!!iii!

i!!iilii_ i,,r_.

Jl'li_'tlj!!!..L I ! I I

.14 .16 .18 .21)

TIME (SEC)

.OZ .OS .10 .12

--P-+-t--*_

.14 .1Q .18 .20

TIME (SEC).31


With Air Chamber Under Sinusoidal DisLurbance, Amplitude


- 111-

SID 66-46-1


I1 .SO

I1 .M

[!!i lILII

iiii iiiiiIIIII

.... !!!!111 N iiiii

!!!_ ii!i!!!! ii!!ii!i JiJi!!!! ,_,,iiii ]!_J'-:::: ::::

iii! !!!i!iii i!!!!!!! !iii!!!! i([i-._- ._...._ .10

!!!!iiii

iiiiiii!

!!!ii!i!i LA._L,_1 ! !"

!iii

iiiiiiiiiiiiiii!" ".IZ .14

TIME.1(1 .18

(SEC)

4

4e_ll

I_, 4ii31

4131

fill ....III ::::.... III ....

ILl_i_i li. !!!!IlliIlllIli,.liii,i_iii!llll 1111 in i m

II111 IllII_t_l, iiiiiII I II I It II Ill I!ll I IHI I I1 II Ill IIIII I I'1 II111 III 1!IIII I I1'_1111 IIIII1 I II Ill IIrill I II IIllIIII II Ill!!!!i_t !!p!

Ill

ill'"III ....

III'" iiiiI]1 ....tiitll iiii

.OI_ .O,I .O_i .Oe

If{ Ill_lllll{llll_llll{l[ll_lll[ Ill... lllllllllIllIIIIIIIIIIIIIIIIIIIIIII!IIIIIIIIIIIIIIIIIIIIIIIIIIII

ilililllilliiillilllilliillillil!ii II+III,II Ill II I;lllllIllllll III III

lil_l flu il I II II tl ]I] I II LII l_i II I II

I I II11IIIIIIA II lll_I I lll/I11l IIIllIl

...... _I I I I I I_Yl _I IJ I I I I i ll_i II I_I 1_ I/I_I I

_III,IIIIIBI_I Nlll_lillili M l|illlili llli llliil_ll I_I Ill II II_I I Ii I liiltii_I/lll lllllillll_II_llll_lli[llllllilll|l

!_111 1 1 I I I l J I I IYl 111IIII_lllI I I I I I I I I I I I_lII I I I I I I II II IIVIIUII l II I 11 I I II I I_J

II III IIIIlIIIIIII_IIIIIIIIIIIIIl!!. I IMIIIIIlIlIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1

iiii IIIIIIIIIIIII_IIIIIIIIIIIIIIIIII.1_ .14 .11 ,18 .20 .5_Z ._4 .1_1 .t8

TIME (SEC)

l

: II

.-f_

i

li,

.$0 ' .311




- I12 -

SID 66-46-i

NO I_TH AMERICAN AVIATION, INC./_) SPACE and INFORMATION SYSTEMS DIVISION

4=

5! .40

i!!!!: !!!!!!_..... !!!!iiiiiiii iiiiii _i!!!! !iiiii::::: ::::::

iiiii !iiiJi::::: ::::::

iiiii ::::ii::::: ::::::

!!!!! ii!!!!

_iiii ......,m,"'" iiiii I I I I_I I I I I II

III I_ I I/?'l"iIIIII _ IIIII

..... YII_!!iiiii ill:::iiiii .......12 .. '.;6"'.io

TIME (SEC)

.20 _Z .Z4 .H

'ILliii i: : : l : [ l

iii ] :ll::.. ! i.::: I : I I::: I : II

iii ' III :[I

' [Iiii I ;ll::: I : l,._. I II,rr_J I ill

g!_-r It;:: I ill::: I I II

iii ' _[_... _ _L=::: I , II

!!!' [ I'I . II

iii I [ III II• "' I IIiii I Ill

.211 °30 JR

4330

4314

.0 .02 .04 .OS .08 .10

--,f--4F-,-_-I-.-(-- --4--(.--4_.-(.-._

® .--F-4---t--_--+,-4--

---4-.4--_--+--*--4--

.IZ .14 .16 .18

TIME (SEC)

.LYO ._' .24 .21 .28 .10 .32

qp



5 Percent of Steady-State Velocity, Frequency 11. 70 cps

113 -

SID 66 -46-I

NORTH AMERICAN AVIAT!ON, INC.SPACE and INFORMATION SYSTEMS DIVISION

/

A

t_UJ0_

u.

IIt.lO

$1.40

51 ._1

.0 .rip

IIII!!!l%111I_llIIMII Ill I

ii_i111_1!!!k

I

Elf',!!!.iiiii l I 1

• • °(_j" ,O6

,'"",,,,,,,,,IIII'", ,,,,'"' ,,,,,,'""'II11

III III Jill =lllll..... III .... Illl illlll

iiiLL '" ill"" ""'I I I i Ill ,11111Illll II I " " lllIl..... III :::: IIII IIIIIiJJJi III JJii Jill lllIl..... III .... llIl lllIl

!!!! JJJ'"' ,""'i ._l ;;;; Llll lllli....._I I IFF l 1_ J i I I

,,,'J!!!,,,,, !!!!!iIII _lli' fill .....

::::: III I_II, IIII IZII'_::::: III I_III IIII _II ii::::: I II I111! IIII IIIII

..... Ill ..!!!,IIL,.G I IIIII::::: III :::::_-FII ,;;;;]iiiii lli ..... lIIi lllill

._ .10 .12 .14 .1Q .18 ._

TIME (SEC)

j I IIIIIIIII IIII J"Jll !

IIIIIIIIIIIIIIIII_lllIllil II IIIIIIII

I IIIIIIIIIIIIIIIIIIIIIII1 Ilil IIIII111

I It1111111111111111llllllllll III IIIII

I IIIII1111111111111llllllll II III II III

III1111111 IIIIIII1IIIIII 111111111111! III111111111111111 "

III IIIIIIIII..LIIIII ii IIIIIIIIII _ IT'I_LII

I III IIIIII _ IIIII_I J• IIIIII II _ IIIIIIII I, III IIIIIA I IIIIIIII '

I lllIllI/llllllllll i[ II1111 VIII llllllll

lllll _ III IIIII III 1• M_II IYI IIII IIIIIIII• II_IIIIIIIIIIIIII !I IIIIIIIIII11111111I IIllllllllllllllll i

4,

414t

4ql_

4541

I-LL

4640

rn--I

I_ •465e

41131

4534

.0

: : : ; i: : : : :

: ::: ;

i iiiiI IIll

I VI|III I III

i i f, liIi111III II IIIIIII1Ill I III!!! ! !v!

: : : : :

: : : : :

: : : : :

: : : : i: : : : :

: : : : :

: : : : :

.Op " ".l_4" *

::::::

::::::

;:;:;;

::::::

::::::

:::;:;

::::::

:::::;

::::::

J_ iiiJV|II_"llll_'I III II111I|1 III'i III 11[lill IIIIH i I

iii_iilllVlll::::::

::::::

JJJJJJ• _ .08 .10

IIIII"',,IIIII",,I III"III_11111IIIIII1,11II1_ ( 1_ I|llllllI1111111I I _111 _1A I1_1 I|1III IV I II1"ltllllllIIIIIIII,

Illill_IIIIIIIIlillIIIIIII

ii!!l,!.1_ .14

TIME

.....i i i i i

: : : : :

I iiiii

I i i Illll• II I_1

/_ ,i, I1!

ki:::..ii ii

: i i i i

: : : : :

l : : : : :

.15 .18 .Z0

(SEC)

' i II I III IIII IIIIIIII;' III lllllllllllllll IJ' II I IIIIIII IIIIIIII IIIIIIIIIIIIIII11[II I

: llllllilllllllllli Iii II1111111111111111,llllllllllllllllll_ IllllllltlllIlllll I

!_ IIIIIll,,,i,llllll IIIIIIIIIIIIIIIIMI

II/'tlllllllllllllll i,11111111 I1 III Ill PIII11 tl I I Ill11111 II

' Illl111 1 I II/1'II I "

iII IilI/IE I I,,I I Ill IM'I I I '

I II11111111 V_I IIII111 illli/lllllllillllll, IIIIMIIIJIII/IIIIIII :÷.

!lltl_,_l_,avl_I'l'l iII III l lUll l III Ill II: i III llll_'l[llllllli

]lllllllll]llllilll !J IIIIIIIIIIIIIIIIII ..22 .24 .I| .|8 .$0 .lid



5 Percent of Steady-State Velocity, Frequency ll. 70 cps

41

- I14-

SID 66-46-I


!I

St.St

51.45

i=I ,i

i'm .(34 .06 .GO .10 .12 .14 .16 ,18 .5D9 .24

TIME (SEC)

I .

i:

i

I-U,.

G)_1

.0_ .04 .0_ .08 .10 .IZ .14 .iS .22

TIME (SEC)

tFigure B-47. Velocity and Pressure Time History at Right End of Feed Line


5 Percent of Steady-State Velocity, Frequency 21. Ii cps

- 115-

SID 66-46- 1


II.90

$1.49

S1.48

U

LLJ $1.4T

C/)

i- si.,*U.

S1.4S

$1.44

S1.43

.0

.-I--I--4--

.10

TIM

-4--t---t--t-

.12 .14 .16 .18 .20 .l_ .?.4

E (SEC)

.8

414'*

414Z

4141

4MO

F-h

4139

m

--I_'* 4Q38

Q.

4137

4131

4135

.0 .04 .04 .01

--_-i-- +-

-.4--4--t--

.20 .22 .24

---i

.21

Figure B-48. Velocity and Pressure Time History at Midpoint of Feed LineWith Air Chamber Under Sinusoidal Disturbance, Amplitude

5 Percent of Steady-State Velocity, Frequency 21. II cps

- 116-

SID 66-46- l

(THRu| i ,o.A?-(COZ · 1. 2 Basic Equations of Unsteady Motion 1.3 Initial and Boundary Conditions...

Documents

Transcript of (THRu| i ,o.A?-(COZ · 1. 2 Basic Equations of Unsteady Motion 1.3 Initial and Boundary Conditions...