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CR 112068

- 3/

Final Report on Study of

Feedline Dynamic Effects onShuttle POGO Stability

5 My 7972

Contract NAS1:1"0836 SD72-SH-0039

FOR U.S. GOVERNMENT AGENCIES AND THEIR CONTRACTORS ONLY

Space DivisionNorth American Rockwell

https://ntrs.nasa.gov/search.jsp?R=19720024234 2020-07-18T22:01:29+00:00Z

CR 112068

Final Report on Study of

Feedline Dynamic Effects onShuttle POGO Stability

Prepared by O.D. DiMAGGloT. NISHIMOTO

R.H. LASSENStudy Manager

5 May 1972

Contract NAS1-10836 SD 72-SH-0039



FOREWORD .

This report is a description of work performed

under Contract NAS1-10836, Study of Feedline

Dynamic Effects on Shuttle POGO Stability, by the Space

Division (SD) of North American Rockwell Corporation

(NR) for the National Aeronautics and Space Administra-

tion, Langley Research Center, Hampton, Virginia,

during the period of 25 June 1971 to 2^ April 1972.

This work was conducted at NR/SD by 0. DiMaggio

and T. Nishimoto. Study Manager was R. H. Lassen of

NR/SD and the Technical Monitor was L. D. Pinson of

NASA Langley Research Center.

- 111 -SD 72-SH-0039


SUMMARY

The mathematical representation of a feedline is an important element in a

POGO analysis. The dynamic characteristics of a feedline have been determined

and the results presented in the form of transmission parameters. These

transmission parameters include the effects of wall deformation, distributed

damping, convective acceleration terms and the bulk modulus of the liquid.

In order to reduce the characteristic equation associated with the system

equations to a polynomial, it is necessary to approximate the transcendental

terms appearing in the transmission parameters by appropriate polynomial

expressions. The transmission parameters have been approximated using both

power and product series expansions. The feedline transfer functions of a

Shuttle Orbiter feedline configuration have been obtained using power and

product series approximations of 60th, 120th, 180th, and 2Uoth order. Bode

plots using the above polynomial approximations have been obtained and the

results compared with the "exact" solution. The "exact" solution to the

feedline transfer function have been obtained by using the transcendental terms

appearing in the transmission parameters.

The results show that the Shuttle Orbiter feedline may be modeled adequately

by using polynomial approximations for the transcendental functions appearing

-in—the—transmission—parameters-.—T-he—power—series—approach-has—been-shown—to

be preferable to the product series method.

- v -SD 72-SH-0039


CONTENTS

Section PaSe

I. INTRODUCTION - • • 1

II. SYSTEM EQUATIONS . . . • • . • • 3

n i . FEEDLINE DYNAMICS . . . . . . . 51 . Transmission Parameters . . . . . 62. Feedline Transfer Function for Straight

Feedline With Simple Compliance Pump . 183. Feedline Transfer Function of Straight

Feedline With Double Compliance Pump . 27

IV. , SHUTTLE OR BITER FEEDLINE . . . , . . 571. Modeling . . . • • • • • • 5?2. Numerical Results . . • .... • • 64

V. EIGENVALUES AND THE SOLUTION VECTOR . . 85

VI. CONCLUSIONS AND RECOMMENDATIONS . . 95'i

VII. ' REFERENCES . . • • • • • 97

APPENDICES . . • • • . . . • 99A. DETERMINATION OF TRANSMISSION

PARAMETERS . . . . . . . 9 9B . MERMORPHIC FUNCTIONS . . . . 1 0 5C. PRODUCT SERIES EXPANSION OF THE

FEEDLINE TRANSFER FUNCTION . . 107

D. LIST OF SYMBOLS U1

- vn -,SD 72-SH-0039


ILLUSTRATIONS

Figure Page

1 Engine Free-Body Diagram . . . . . . . . 42 Phase Shift Due to Convective Acceleration Term g

3-8 Power and Product Expansions - Hyperbolic Cosine(4th, 6th, 8th, 10th, 12th, and 14th Orders) . . . .12

9 Simplified Feedline . . . . . . . . . . i g10 Straight Line With Single Compliance Pump - "Exact" Solution • 2111 Straight Line With Single Compliance Pump - Damping

Lumped (One Segment) . . . . . . . .2212 Straight Line With Single Compliance Pump -

(Five Segments) . . . . . . . . . . 2 313 Peak Gain Vs. Number of Segments at w = 21.75 rad/sec . . 2414 Peak Gain Vs. Number of Segments at w = 215 rad/sec . • 2515 Peak Gain Vs . Nondimensional Frequency . . . . . 26

16-17 Straight Line With Double Compliance Pump- "Exact"Solution . . . . . . . . . . . -30

18-29 Straight Line With Double Compliance Pump - Power Series(4th, 8th, 12th, 16th, 20th, and 40th Order) . . . - 3 . 2

30-41 Straight Line With Double Compliance Pump - ProductSeries (4th, 8th, 12th, 16th, 20th, and 40th Order) ... 44

42 Shuttle Orbiter LOX Feedline 5843 Shuttle Orbiter LOX Feedline - Forward End . . . .5944 Shuttle Orbiter LOX Feedline - Aft End 6045 Feedline Geometry . . . . . . . . . • 61

46-47 Shuttle Orbiter Feedline - Exact Solution (Gain and Phase) . 6548-55 Shuttle Orbiter Feedliiie~ - Power Series (60thy -120th,--

180th, and 240th Order) 6756-63 Shuttle Orbiter Feedline - Product Series (60th, 120th,

180th, and 240th Order) 7564-65 Shuttle Orbiter Feedline - Product Series - 60th Order

(Gain a n d Phase) . . . . . . . . . . 9 366 Tubular Element . . . . . . . . . . • 99

—67--6S—Straight-Line-^Wlith-Single_C_ompliance Pump (Gain and Phase) . Qg

- ix -

SD 72-SH-0039


TABLES

Table Page

1 Damping Data 622 Feedline Data 633 Shuttle Orbiter Feedline Peak Gains . . . . . 834 Closed-Loop Roots • • 905 Undamped Natural Frequency a n d Damping . . . . 9 16 Solution Vector for First Root 92

- xi -

SD 72-SH-0039


I. INTRODUCTION

In order to perform a POGO stability analysis on the Shuttle Orbiter vehicle

an accurate model of the propellant feedline is required. The model of the

feedline must be made compatible with the stability program. The current

stability program at NR can handle up to UO fourth order differential equations

with constant coefficients. It determines the characteristic polynomials

associated with the stability equation and extracts the roots from these

polynomials. Experience has shown that the program gives good numerical

results for systems up to about 80th order. The transmission parameters for

an elemental section of a feedline have been obtained by solving the linearized

wave equation in the LaPlace domain. The transmission parameters include thel

effect of the convective acceleration terms and distributed damping. The

"exact" transmission parameters have been used for obtaining results in the

frequency domain and are useful in checking the numerical accuracy of the

polynomial approximations required for inclusion into the stability program.

Bode plots of the feedline transfer functions have been obtained for a

simplified feedline and a Shuttle Orbiter feedline configuration, using both

a power series and product series approximation to the transmission parameters.

The results have been compared with the "exact" solution.

The stability of the system and the solution vector were determined using

both the power and product series approximations for the transmission

parameters and the results compared.

- 1 -SD 72-SH-0039


II . SYSTEM EQUATIONS

The POGO stability equations deal with the interaction of the elements of the

propulsion system and the structural system. Current Shuttle Orbiter designs

use bipropellant (l /LI ) rocket engines. For the present study only the LC>2

system will be considered and the hydrogen system will be neglected. Although

experience gained in the S-II stage of the Apollo Saturn V launch vehicle

showed this approximation valid, it may or may not be true for the Shuttle

Orbiter.

Since the present study. is concerned primarily with modeling of the feedline

and not with demonstrating the stability of the Shuttle, the hydrogen system

has been neglected. The stability analysis to be performed will be meaningful

only in that it may be used to compare the results for different approximations

for the feedlines. The stability results themselves are not necessarily

indicative of actual POGO tendencies. The system equations for the POGO,

stability analysis are given below:

X0H

Pt = A60 P* + BO)

- 3 -SD 72-SH-0039


In an attempt to account for the dynamic effects of the flow through the

bulkhead and the feedline, the effective thrust ~Te. may contain terms

which include the pump inlet pressure PS and the tank bottom pressure PT.

The most common procedure is to draw a free body diagram of the engine system

as shown in Figure 1,

/A~~Figure 1. Engine Free Body

Diagram

and obtain l«ft. as the resultant force on the system.

T«. — T - PS, A

It should be noted that the factor k appearing with the effective thrust

results from assuming that the feedlines to each engine are identical. This

was the case for the four engine orbiter configuration used in this study.

- 4 -

SD 72-SH-0039


. . . III. FEEDLINE DYNAMICS • - '

Feedline Dynamics concerns itself with wave propagation in the propellant

lines and the dynamic characteristics of the pump. The approximate wave

equations for the lines are developed in Appendix A. The solutions of these

equations have been obtained and the results presented in the form of trans-

mission parameters. The effect of the convective acceleration terms and

distributed damping have been investigated using a simplified feedline and

the "exact" or transcendental form of the transmission parameters. The

transmission parameters have been approximated by polynomial expressions

using both the power series and product series approach. The feedline transfer

functions for a simplified feedline and a Shuttle Orbiter configuration have

been determined using these polynomial approximations and the results compared

with the "exact" solution.

- 5 -

SD 72-SH-0039

Space DivisionNorth Americarr'Rockwell

1. Transmission Parameters For a Line Segment

The equations of motion of a compressible liquid in an elastic pipe are

shown in Appendix A. The effect of distributed damping and the convective

acceleration terms have been included in analysis. .The results are presented

in terms of transmission parameters and are given by the following equations:

N

CO) DCs)

where P^, PB, U^, UB are the pressure and fluid velocity at the upstream

and downstream end, respectively, and A, B, C and.D are the transmission

parameters given by the following expression: - ' ' ••

CCs) =

DCs)

a. Convective Acceleration Terms •

The linearized equations of motion for a compressible fluid in an

elastic pipe are given by the following equations:

- 6 -

SD 72-SH-0039


= 0

__ £P + C U = Oe ax

3"where C = 3" ° _ = damping parameter

( e — equivalent bulk modulus

LA0 •=. nominal fluid velocity

y p> — perturbed velocity and pressure

The effect of the convective acceleration terms oliJ-. and

^Xcan be determined from the transmission parameters A, B,

C and D given in the previous section. The effect of these terms is-TTs

to introduce a lag term S into the transmission parameters.

The effect of Mach No. — - and nondimensional frequency 1 on phaseCL. a.shift due to the convective acceleration term is given in Figure 2.

b. Power Series Representation

Neglecting the convective acceleration terms and the damping, the

transmission equation .for a segment of the feedline is given by:

**_ sha.

\ *i$ \ S @sh _ ch ±±.

The simplest procedure for reducing the hyperbolic functions to

polynomials is to use a Taylor series.

chil = | +-a. ' ai «• Ml

sh M - s* +^ CL - CL

- 7 -

SD 72-SH-0039


SNvmvy

UJ

I_UOO

Ouu

OO

O

tCO

UJto

11°

CD

0

II

e

- 8 -

SD 72-SH-0039


In order to better understand the errors associated with using a

Taylor series, Bode plots of cK — - have been obtained using

polynomials, of different order and are shown in Fig. 3-8. It

becomes clear that increasing the order of the polynomial approxima-

tion is not a very satisfactory method of expressing the hyperbolic

functions. The results indicate reasonably good approximations for• .

-— <. I , but inaccurate results for larger values of the argument. It

should be noted that when using this procedure some of the roots fail

to appear. If a single Taylor series for the feedline were used, it

would be possible to introduce instabilities due to the inaccuracies

of the roots of polynomial approximation. This problem can be

alleviated by using a segmented line and using terms no higher than

the fourth order.

The transmission equations for a segmented line without damping will

then be given by the following equations :

R - M 4- -L (Mf• • * ~ .<*-. /3i\o.;

A B

C D

A t 3k~

C, D,

FAa Ba-

_Cu D._

"A, 6,1

C^ D/v,

c. Product Series Representation

The hyperbolic sines and cosines may be expanded into product

series. (Ref. 3)

- 9 -

SD 72-SH-0039


Kz = TT i -t-

sK It

M 2*\1- t

^'Z'

Letting

s

Ss-t-c

^ cK

sh

sh

Letting 4- t') TT CL

(a.

we obtain the following matrix equation for the transmission equations.

A B

C D

A, 6k

C, P,

- 10 -SD 72-SH-0039


where

A, - TT

C, =

•>\-o

oO

ITw

It is clear from looking at the damping terms«

and 5,*, —

I (C.A" ~ ~

\)'

-"/ tha* the damPinS coefficient is inversely propor-

tional to the mode number.

One should, therefore, expect that the peaks which occur in the feed-

line transfer functions should be sharper at the higher frequencies.

This in fact, is what happens for this Shuttle Orbiter feedline

configuration (Figure U6) .

Bode plots of the hyperbolic cosine have been obtained using from 2

to 7 roots and are shown in Figures 3-8. The product expansions

by their very nature maintain the roots exactly, but the error in

gain increases with increasing frequency.

- 11 -SD 72-SH-0039


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- 12 -SD 72-SH-0039


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- 14 -SD 72-SH-0039


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- 17 -SD 72-SH-0039


2. Feedline Transfer Function For a Straight Line With Single CompliancePump

The dynamic characteristics of a feedline are most readily summarized by

means of the feedline transfer function. This transfer function is usually

defined as the pump inlet pressure (Ps) divided by the tank bottom pressure

(FT). In order to investigate relative merits of lumped vs distributed

damping, use will be made of a simplified feedline shown in Figure y.

Straight Feedline Pump

Figure 9. Simplified Feedline

The relationship between Pg and the Pip are given by the following equations:

PT = ACs) R, -*• &(*£-

-I-L = GO) =Ws

where A C . ) - ex {- £ - fA '(*)]•

•W A u,

Using the above relationship, the feedline transfer function becomes;

chwhere

r -- 18 -

SD 72-SH-0039


When using the single compliance model the following constants were used.

Ji. , o-'' "a, •r± - .,006a.

= C,:A.G8

M/ (30.H8K.)

, ooo JH.- ( 5-07 J :Sec, V sec

The transmission parameters previously considered had the damping distributed

along the line segment. An alternate procedure is to lump the damping at the

ends of elemental sections of the feedline. The pressure loss due to damping

in a section of the feedline is given by the following expression:

p = - $ Aa P_L _ a>rt ' a

The linear perturbation equations may be obtained by removing the steady state

terms. The resulting equations for the perturbed values of pressure and

velocity are given below:

AP = - C/

c =D

In order to compare lumped with distributed damping, Bode plots of the feedline

transfer function were obtained for an n-segmented line with lumped damping.

These results were compared with the "exact" solution using distributed

- 19 -SD 72-SH-0039


damping. Two typical runs using lumped damping for n = 1 and 5 are shown in

Figures 11 and 12 and may be compared with the case of distributed damping

shown in Figure 10. The results are seen to be identical except for the peak

gains.

The effects of the length of elemental sections on the error have been shown

graphically for the peak gains at the first and fifth peaks in Figures 13 and

. lU. A more informative curve may be obtained by plotting peak gains rs/p*(j O

vs nondimensional frequency 6 where © = —-. This curve (Figure 15) shows•O'V CC

that in the frequency range of interest, the error is acceptable provided ©<

It should be noted that ©-O corresponds to distributed damping.

- 20 -

SD 72-SH-0039


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- 21 -SD 72-SH-0039


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Figure 11 Straight Line with Single Compliance PumpDamping Lumped (One Segment)

- 22 -SD 72-SH-0039


TTT TTT TTT TTT

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Figure 12 Straight Line with Single Compliance PumpDamping Lumped (Five Segments)

- 23 -SD 72-SH-0039


>S/Pt

49 db-

48 db

UNIFORM DAMPING (C

1 2 3 4 5 N O O F SEGMENTS

Figure 13 Peak Gain Vs No. of Segments at w = 21.75 Rad/Sec

- 24 -

SD 72-SH-0039


41

40>

't 39

38

37

36

35

\DISTRIBUTED

DAMPING

I I I I I

1 . 2 3 4 5 6 7 8 9 1 0NO. OF SEGMENTS

Figure 14 Peak Gain Vs No. of Segments at o> = 215 Rad/Sec

- 25 -SD 72-SH-0039


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15

- 26 -

SD 72-SH-0039


3. Feedline Transfer Function of Straight Feedline with Double Compliance Pump

Prior to the analysis of the Shuttle Orbiter feedline, a simplified feed-

line consisting of a straight 1200 in. (30.U8 m) line with a double compliance

pump will be considered. Because of the lack of bends in the line, the only

damping is that due to the shear stresses already considered when determining

the transmission parameters. Current state of the art is to obtain the dynamic

pump characteristics experimentally in terms of the pump termination impedance

or the pump inlet pressure (P ) divided by the flow rate (W ). For use inS S

this simplified line as well as the Shuttle Orbiter feedline, use will be made

of the pump termination impedance obtained from tests on the J-2 engine in lieu

of more precise information on the Shuttle engines.

The relationship between the pump inlet pressure (P_) and the tank bottomS

pressure (Pm) are given by the following equations:

RT = AO) PS -t-^ 5Cs)VJs

%L - GOOVvswhere

G = K K(s)

a T s s^

5, - S.64 Hz,

3.58 sEc/5.H4«io 5 =IN* V

- 27 -SD 72-SH-0039


The feedline transfer function may now be obtained in terms of the feedline

transmission parameters and the pump characteristics.

where Q- - O.C5S

K A 3-M, ( 30,48

A = a50 \H? (.161 M?

i-l = O.OOGCL

These numbers have been chosen so that they are representative of the Shuttle

Orbiter. .

In order to evaluate the method of expanding the transmission parameters by

power and product series, it was necessary to obtain an "exact" Bode plot by

using the transcendental functions which appear in the transmission parameters.

These results are shown in Figures 16 and IT. The Bode plots using the power

and product series representation are shown in Figures 18 - 29 and Figures

30 - Ul, respectively. Results have been obtained using polynomial approxima-

tions correct to the Itth, 8th, 12th, l6th, 20th, and Uoth order in '

The results show that the power series approximations are in fairly good

agreement provided the nondimensional frequency < where — is the length* f\

of the elemental segment. For frequencies above this value, the gains and to

- 28 -

SD 72-SH-0039


a larger extent the phase, begin to vary from the "exact" solution. The

product series solutions of the same order have about the same accuracy as the

power series results up to the 8th order. The higher order solutions, however,

are not as satisfactory. The gain curve tends to slope downward. This can be

explained by looking at the product series expansions of the hyperbolic cosine

curves given in Figures 3-8. The product expansion of the hyperbolic cosine

(as well as the hyperbolic sine, not shown) have errors which increase the

gain at the higher frequencies. Since these terms appear in the denominator

of the transfer function PS/PIJI they have the effect of reducing the gain at

higher frequencies.

\

- 29 -SD 72-SH-0039


22

18

16

14

12

to10

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Figure 16 Straight Line with Double Compliance Pump"Exact" Solution

- 30 -SD 72-SH-0039


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- 31 -SD 72-SH-0039


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- 32 -SD 72-SH-0039


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- 33 -

SD 72-SH-0039


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- 34 -

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- 40 -

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- 44 -

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- 45 -

SD 72-SH-0039


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- 47 -SD 72-SH-0039


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- 49 -

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- 50 -SD 72-SH-0039


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- 51 -SD 72-SH-0039


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- 52 -SD 72-SH-0039


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- 53 -SD 72-SH-0039


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- 54 -

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- 55 -SD 72-SH-0039


IV. SHUTTLE ORBITER FEEDLINE

1. Modeling

The Orbiter L02 System (Figures 1*2 - UU) has been modeled using power

series, product series and the "exact" transcendental equations for the

transmission parameters. In each of these methods, the damping due to bends,

changes in cross-section, inlet losses, etc., were lumped at their respective

locations. The damping due to these causes is lumped at locations A through

I shown in Figure 1*5- The losses due to a bend in the line, entrance losses,

etc., as given by the following formula.

*p = i K>UIRemoving the steady state terms, we obtain the following expression for the

losses due to the perturbed flow.

A p - K e u*aIn terms of flow rate, this equation becomes

A p = R W

where R = 3. P0/W0

P = steady state head loss

The losses which occur at sections A - I are given in Table 1. The feedline

may be conveniently subdivided into three main sections which have different

properties. Additional information required for determining the feedline

transfer function is summarized in Table 2.

The damping due to the shear stresses distributed along the line are limned

at the ends of the elemental sections when using .the power series. This

damping is handled directly when using the product series expansion.

- 57'-.,.

SD 72-SH-0039


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- 58 -SD 72-SH-0039


LOX TANK

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(.457 METERS)

Figure 43 Shuttle Orbiter - LOX FeedHne - Forward End

- 59 -SD 72-SH-0039


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- 63 - 72-:


2. Numerical Results

Bode plots of the feedline transfer function have been obtained using the

"exact" transcendental functions for the transmission parameters. The results

are given in Figures U6 and Vf. This has been used as a basis for comparing

the numerical results when using the power and product series expansions.

Results have been obtained for 60th, 120th, l80th, and 2UOth order expressions

for the orbiter feedline and are given in Figures 1*8 through 63. A

comparison chart showing the peak gains for each of these approximations is

shown in Table 3.

The NR stability program will allow the use of 60th order approximation to

the line. The higher order results show the improvement to be expected as

the order is increased. For the Shuttle Orbiter configuration, it is clear

that the power series representation is preferable to the product series

approximation for the transmission parameters. This is due in part to having

to subdivide the line for the inclusion of lumped damping due to bends and

changes in the cross-section of the line.

- 64 -SD 72-SH-0039


-1020 HO 60 BO 100 120 140 160 160 800 SSO 3HO 260 280 300

FREQUENCY RADIANS PER SECOND .

Figure 46 Shuttle Orbiter Feedline"Exact" Solution

- 65 -SD 72-SH-0039


6.3

6.0

20 HO . 60 80 100 ISO 140 160 ISO 200 320 3tO £60 £60 300


Figure 47 Shuttle Orbiter FeedHne"Exact" Solution

- 66 -SD 72-SH-0039


300


Figure 40 Shuttle Orbiter FeedlinePower (60th Order)

- 67 -SD 72-SH-0039


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- 68 -SD 72-SH-0039


800 220 210 260 260 300


Figure 50 Shuttle Orb'Uer FeedTlnePower Series (120th Order)

- 69 -SD 72-SH-0039


6.5

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Figure 5] Shuttle Orbiter FeedlinePower Series (120th Order)

- 70 -SD 72-SH-0039


-e80 HO 60 80 100 120 1HO 160 180 £00 £50 840 £60 880 300


Figure 52 Shuttle Orbiter FeedHnePower Series (180th Order)

- 71 -SD 72-SH-0039


6.5

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HO 60 80 100 180 140 IPO 100 800 820 840 8S!> 880 300


Figure 53 Shuttle Orbiter FeedlinePower Series 080th Order)

- 12 -SD 72-SH-0039


-8MO 60 8 100 ISO 1HO 160 180 S03 Z20 3HO 250 SBO 300


Figure 54 Shuttle Orbiter Feedllne' . . ' . ' Power Series C240th Order)

- 73 -SD.72-SH-0039


6.5

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SO HO 60 60 100 ISO 140 "150 I BO £00 280 240 2GO 280 300


Figure 55 Shuttle Orbiter FeedlinePower Series (240th Order)

- 74 -SD 72-SH-0039



Figure 56 Shuttle Orbiter FeedlineProduct (60th Order)

- 75 -

SD 72-SH-0039


to

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Figure 57 Shuttle Orbiter FeedlineProduct (60th Order)

- 76 -SD 72-SH-0039


-16HO 60 BO 100 120 ItO 160 160 200 220 2HO 260 280 300


Figure 58 Shuttle Orbiter FeedlineProduct Series (120th Order)

- 77 -

SD 72-SH-0039


6.5

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- 78 -SD 72-SH-0039


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- 79 -SD 72-SH-0039


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- 81 -SD 72-SH-0039


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- 82 -SD 72-SH-0039


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- 83 -SD 72-SH-0039

Page Intentionally Left Blank


V. EIGENVALUES AND THE SOLUTION VECTOR

The stability equations have "been reduced to a system of linear homogeneous

equations with constant coefficients. They are represented "by the following

equation:

i c o) "M r owhere y represents a vector whose components are the dependent variables

appearing in the stability equation and is a matrix whose elements <

\tare polynomials in the differential operator D. Letting ^J ~ ~E Q.

we obtain -. c / \ \ „J-CA) '-M - O

For a non-trivial solution, it is required that | j(A) = 0. The order of the

characteristic equation 5C )| = 0 determines the number of arbitrary

constants in the solution.

The eigenvalues A^ of the characteristic equation are obtained using a program

currently available at NR and has successfully been used for POGO stability

studies. The analysis need only consider the case in which all the eigenvalues

are distinct. The system of equations J(O)M.= 0 can be reduced to a system of

linear differential equations with constant coefficients such as - = Axclt,

such that one could then obtain the eigenvectors associated with the matrix A.

For the purpose of the present problem, it is more meaningful to obtain the

solution vector y for each of the characteristic values X« obtained from

the solution of the characteristic equation. Letting F(XK) be the adjoint of

) we then have: (Ref. U) .

- 85 -SD 72-SH-0039


F(x) = A(X)

for X - XK where A(^Xi<) - C

KXO FOO = O

The rank of F (An) is 1 so that the matrix F ( AK) has only one independent

column vector. It is clear then that the solution vector y is proportional to

any non-null column of the adjoint matrix F(Aj<). (For the case of multiple

degeneracy, the adjoint matrix is null. This case, however, is only of

academic interest for the POGO stability problem and will hot be considered).

For the system of equations given for the Shuttle stability analysis, the

solution vector y can most easily be obtained by simply choosing /H,(Ai<)"1T=

and directly solving for •>u^(AK) where /\n(Ai<) represents the remaining

dependent variables PS , Prp, etc.

Performing the indicated operations one obtains:

>?(T-

-v

PT =

H(s)= C hcs) ; C = 15-0 iw*

- 86 -SD 72-SH-0039


where —h C£> =

(l-i- 5/7<?xio~*s)(|-t--F.,q6xtd3s-H.f

The gain and phase measurements obtained from flight data occur at the

characteristic frequency of the system. Comparison of these results with the

solution vector should give a check on the accuracy of the POGO model.

The structural data used in the stability analysis is summarized below;J3

generalized mass *>Vtj. = -aa4 * lO^St-UGS (3.a»*U

natural frequency it), S.fe'H

damping ^ ~ O.OO-5T

modal displacement at gimbal 0' 9

modal displacement at sump r i C T) ~

liquid L02 level \\ - 3OO \tt. (7'<*'a. K.)

modal pressure

The eigenvalues were obtained by solving the system equations with the currently

available stability program. Because of the size limitations of the program,

power and product series approximations to the 60th order were utilized. The

stability analysis was performed using only one structural mode in which the

bulkhead was treated using a spring-mass analogy. For a more accurate stability

analysis , the bulkhead must be modeled using a multi degree of freedom hydro-

elastic model.

For the purpose of the present study which is concerned primarily with numerical

methods in treating the propellant line, the simplified treatment of the bulk-

head was deemed adequate. It should be emphasized the results will be applicable

only in the frequency bandwidth where the feedline transfer function obtained

from either the power or product series approximation is valid.

- 87 -SD 72-SH-0039


The closed loop roots and the associated solution vectors have been obtained

with and without the convective acceleration terms and are given in Tables h

and 6. The undamped natural frequency and the damping are given in Table 5.

The results indicate the convective acceleration terms were of minor importance.

Since only one structural mode was used it should be emphasized that only the

first root has significance. For this reason the solution vector has been

tabulated for only the first characteristic value.

In order to perform the stability analysis using the product series expansion

for the transmission parameter it was necessary to alter the model from that

used to obtain the Bode plot shown in Figure 56 and 57. The damping due to the

bend at B (see Figure U5) was included with the entrance losses at A. This was

done to reduce the number of equations in order to make the feedline model

acceptable to the stability program. The order of the product series approxima-

tion, however, was kept at 6dth order. The effect of this model change can be

seen by comparing the original Bode plot (Figure 56) with the amended one with

the bend damping lumped at the entrance given in Figures 6k and 65.

The largest effect appears to occur at the first mode where the decrease in

gain has the effect of increasing the damping from 2.8 percent to 3.3 percent.

Although the remaining results are not too meaningful, since only one structural

mode was used in the analysis, the first five closed loop roots obtained from

the power and product series methods are in substantial agreement. In order

to get agreement in a higher frequency range, higher order approximations are

required.

The computer time required to obtain the closed loop roots was from 3-^ CPU

minutes when the feedline was modeled with either power or product series

- 88 -

SD 72-SH-0039


expansion. Bode plots of the feedline transfer function of the Shuttle Orbiter

configurations for each of the polynomial approximations utilized about 0.3

CPU minutes. The computations were performed/on an IBM Model 165 computer."'%•.

•)K

Nyquist diagrams were not presented because the numerical procedures used are

not meaningful beyond a given frequency. An instability at 100 Hz for example

would result in an instability in the Nyquist diagram. It should be again

emphasized that the stability results are only valid in the range where the

Bode plots of the approximate feedline transfer functions are in agreement

with the "exact" results.

- 89 -SD 72-SH-0039


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too aoo 300

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- 93 -SD 72-SH-0039


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Figure 65 Shuttle Orblter Feedllne-Product Series (60th Order)

- 94 -SD 72-SH-0039


VI. CONCLUSIONS AND RECOMMENDATIONS

The results of this: study show that the Shuttle Orbiter feedline may be modeled

adequately by using polynomial approximations .to .transcendental functions

appearing in the transmission parameters. The . adequacy of the approximations

may be determined by a comparison of the Bode plots of the feedline transfer

function obtained from either the power or product series approximations with

the "exact" solution obtained using the transcendental functions. For complex

feedline with multiple bends this study indicates the power series method is

to be preferred over the product series . This study has also shown that the

line damping may be adequately approximated by lumped damping provided the

elemental sections are made sufficiently small. When using the power series

method this is automatically accomplished by the requirement that the nondimen-

sional frequency ty A of the largest elemental section be less than one.

Procedures have been developed for including the effect of the convective

acceleration terms in the feedline representation. It has been shown that

the sole effect of this term is to cause a phase shift of - in the feedline<x a.

transfer function. For the Shuttle Orbiter configuration this is quite small

and may be neglected. This term, however, was included in the feedline transfer;

function and the stability analysis.

The Shuttle Orbiter line was modeled using polynomial approximation of up to

the 2UOth order. Using the NR stability program the closed loop roots were

obtained using a 6oth order polynomial approximation for the line.

- 95 -SD 72-SH-0039

Space Division .North American Rockwell

The system equations used for obtaining these results did not include the

hydrogen system and only one LOg bulkhead mode was used in the analysis. The

stability results therefore, as well as the solution vector, should only be

used for comparing the results of different numerical procedures.

The computer time required to obtain the closed loop roots was from 3-U CPU

minutes when the feedline was modeled with either power or product series expan-

sion. Bode plots of the feedline transfers function of the Shuttle Orbiter

configurations for each of the polynomial approximations utilized about 0.3

CPU minutes. . .

The procedures developed in this study for obtaining the approximate and the

"exact" Bode plots of the feedline transfer functions are completely general,

although they were used for a specific feedline geometry. These procedures should

be automated in a digital program so as to handle an arbitrary feedline with

branches, bends, pumps, etc., to obtain the required dynamic feedline character-

istics. The dynamic characteristics may then be obtained using a Taylor series

approximation for the elemental sections and the results checked with the "exact"

solution. Having modeled the feedline to the required degree of accuracy

necessary one can then proceed to determine the closed loop roots and the

associated solution vector by incorporating the feedline model into the system

equations. . . .

- 96 -

SD 72-SH-0039


VII. REFERENCES

1. Titchmarsh, E. C. The Theory of Functions. Oxford University Press (1939)

2. LeVine, Dickey Root Finding Techniques for a Transcendental Function.M.S. Thesis, Purdue"University (1968).

3. Jolley, L. B. W. Summation of Series, Dover Publications, Inc. New York.

U. Frazer, R. A., Duncan, W. J., Collar, A. R. Elementary Matrices,Cambridge at the University Press (i960).

- 97 -SD 72-SH-0039


APPENDIX A

DETERMINATION OF TRANSMISSION PARAMETERS

The equations of motion of a compressible fluid in an elastic pipe may

be obtained from the following three equations:

1. Continuity Equation

2. Momentum Equation

_fL I f»-i> cL\t

3. Equation of State

e 3-e-where 5 is the surface of V,-tr->v. is the component of -v- along the outward

normal to 5> and - denotes differentiation with V held fixed.2x

The differential form of the equations of motion may be obtained by using

a tubular element as shown in Figure 66.

x+'c(_x Fig. 66 Tubular Element

Since we are concerned with turbulent flow (N = lO^) the shear stress isK

given by the following formula: f . .z

••r = ec 8

where f is the Darcy Weisbach constant.

- 99 -SD 72-SH-0039


Evaluating the integral equations (l) and (2) for the elemental volume

shown in Figure 66 results in the following differential equations:

5- Continuity Equation -x

6. Momentum Equation .s 0

The pressure area relationship required to further reduce the above

equation depends on the end conditions for the pipe. The area of the pipe

is given by: ^ R<L _ ^^ | '

where £, is the hoop strain.

Case I _ . . ' • . . . , . . - . '

Pipe with end closed such that hoop stress is twice axial stress.

«i - P RA e _ PP / . -« v -•«*« PR/at ' '" ^ y

Case II - • . , .

Pipe constrained from having axial motion

-•^*t °T - O

2EtCase III

Pipe with one end free such that

< r a , o

Using these three expressions for hoop strain one obtains the following

„expression for

Case I Â _ A ( I -''•&.)

- 100 -SD72-SH-0039


Case II Â _ AD

^P " E-t

Case III Â _ ./ VZA D

^P ^S ' Êt

Substituting the above into the continuity equation one obtains

(8) where J_ _ J_ . J_&* ~ O A

For Case I which will be used in this study, one obtains for the effective

bulk modulus :

The momentum equation becomes:

do) AiL + Ji. If + q .1 3* +f ' +

Combining equation (?) and (10) one obtains

(ID ' *£ û.^L^'AL 4.1 : =0^t ^x. e 3x ^DEquations (7) and (ll) constitute two nonlinear equations of motion which

govern the motion of compressible turbulent flow in a flexible pipe. There

are methods of handling nonlinear equations in the time domain, such as the

method of characteristics, but we are here concerned only with linear methods

which may be used for POGO stability analysis. Linear perturbation equations

may be obtained by removing the steady state terms .

LLÛO+S: ; p = p 0 + p

Substituting into equations (7) and (ll) we obtain the following equations

for the perturbed values. (The perturbed values 'u", "p", have now been redefined

as u and p) .

- 101 -SD 72-SH-0039

Space Division-North American Rockwell

+ .. + = 05X ^X 3t

(13) . . . .* u 4. ix0 Su + J...LP 4 cu. = a

e

where c = J

DOTaking the Laplace Transform of (12) and (13), one can arrive at the

following expressions: ;

— - ' f \. s \. . .*• v

U. =-/u°?l û. /UcV as+c'N sCsvc')^~' J ."" ^ VÂ ~^~) ce

-z. °

(16) &e

Letting u = e^x one obtains:

Letting R^ and R2, respectively, be the roots associated with positive

and negative signs in front of the radical, one obtains the following

expressions for the pressure P and the velocity U.

p = Av e*'x + Aa-.«

where

B, S4W-0R,

Dr

- 102 -SD 72-SHr0039


Letting

one obtains

(20)

(21)

p&UB

p a. UA =

P

U

P

U

-RL

A-r X = O

x = o

AT

P - a.Ua

. rt

B"

D/• . ,_

f Ps"/ p«.UB^ » «

By neglecting second order terms in -^- and — we obtain the following

simplified version of equations (20) and (21).

(20) I "*

A =

B =

C =

D =

ex p. i - rf

ex P. f u_o f as+cVJ.

exp. I.Uo

The above expressions may be further simplified by eliminating product

~ - - " " ~terms in— andCL

The exponential term thereby reduces to exp

UThe second expression in A and B behaves as — 2. . ±. ascJ-*O. For large values

3.0. cuof the expression — « | and the product o . . can be neglected. The

- 103 -SD 72-SH-0039


resulting simplified transmission parameters are:

A = exp 1 _ u-c> „( o*,

B = exp / _ Up f_

C = exp { __ U.o

D = exp J _ U0

If one is interested in the frequency range &)> I Hz and notes that for

the Shuttle feedlines C — .1 rad/sec., the following approximations hold:

= exp J - Zl - iila . a .

B(s) •=• exp A _

C ( s ) = exp J-:l£u%ii

D ( s ) = exp J-J±S. ^1 y u i'

- 104 -

SD 72-SH-0039


• APPENDIX B

MERMORPHIC FUNCTIONS

THE MITTAG - LEFFLER'S EXPANSION THEOREM

Consider a class of functions $Cx) whose only singularities in the

finite part of the plane are simple poles aj_, o.^ a ......... where ja,

aol^laoj.. • Let b^_, bg, bo be the residues at these poles and let

it be possible to choose a sequence of circles Cm (the radius of Cm being

with center 0, not passing through any poles, such that | Jw| is bounded on Cm

Then it can be shown that[Reference (l)J

INFINITE PRODUCT EXPANSION

The expansion theorem of integral functions into a product series

follows directly from the Mittag-Leffler's Expansion Theorem. Let £(z) be

an integral function with simple poles at a]_ , a^ ...... '.. . and suppose that

is a function satisfying all the conditions of the Mittag-

Lef fler Theorem. It can be shown that :

Applying the above theorem to the functions SIN 2 ^ COSZ.

and chi one obtains:

- 105 -SD 72-SH-0039


The exponential terms are missing because (l) the roots appear in pairs

+_ ais + a2 such that "TT S ** =€?- I » and (2) . = 0 so thate *efr = 1.

Letting f(a) = A ch z + BZ sh Z. the same situation arises. The roots

to this equation are given by the following formula: (Reference 2).

where Z = X 4-

The roots of the above equation occur in pairs. Denoting these roots

by+ y , + y , and noting that ^ . = 0, we obtain the following product- i - a. > (o)

expansion for Ach Z + BZ sh Z :

Ach2 4- BZ sh-2. = TTA\ - I

- 106 -

SD 72-SH-0039


APPENDIX C

PRODUCT SERIES EXPANSION OF THE FEEDLINE TRANSFER FUNCTION

The feedline transfer function for a straight feedline with a simple

compliance pump has already been obtained and is given by the following

expression:o f ** sJttPs _ ex p. I-5:' 5TJ

Letting Z = yj SCS+c) (£) • !±f - o

Because of the special nature of the roots of the feedline transfer function,

it may be expanded into a product series.

where cot y = I y

The solution of this transcendental equation for 1= .208 has been

obtained and a few roots are given below.

y^ = 1.308

y2 = **.023

y3 = 6.898

yu = 9.883

A Bode plot of Ps/Pt has been .obtained using the first 20 roots of yn '

and is shown in Figs. 6? and 68. This is a similar curve to that shown in

Fig. 10. Because of the plotting techniques used the peak gains should not

- 107 -SD 72-SH-0039


be read off the curve shown in Pig. 6U. It is clear though that the gain

drops down with increasing frequency in the product expansion. This is the

same phenomena that occurred in the more complex Shuttle Orbiter feedline

when product expansions of the transmission parameters were used.

- 108 -

SD 72-SH-0039


80 HO 60 80 100 ISO 140 160 180 800 230 240 ESO 280 300


Figure 67 Straight Line with Single Compliance PumpPs- '

nn=l

- 109 -SD 72-SH-0039


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- 110 -SD 72-SH-0039


APPENDIX D

LIST OF SYMBOLS

A Area of feedlineD DiameterM Mass of vehicleP PressureR Damping constantT ThrustA(s) , B(s), Transmission parametersC(s), D(s)G(s) Pump termination impedance PS/WNR Reynolds numberPs Pump section pressurePt Tank bottom pressurePs/Pt Feedline transfer functionTeff Effective thrustU0 Nominal flow velocityWs Flow rate*a Acoustic velocityc Distributed damping constant /UO/D/ Darcy Weisbach constantg Acceleration of gravityh Height of liquid in LO2 tank1 Lengths Complex variable in Laplace transformqi Generalized coordinateXg Displacement of engine gimbal pointx0 Displacement of orbiter c. g.p Bulk moduluspe Effective bulk modulus<f>i Modal displacement

Mass densityYJL Modal pressure\i Characteristic valuep FrequencyU)l

Nondimensional frequencyct

H-J Generalized mass

*WS is weight flow in English units and mass flow in Metric System.

Ws D'pAUsg ENGLISH SYSTEM

Ws D^pAUs METRIC SYSTEM

This causes some constants to have different units in each system (e. g. ,damping R is sec/in.^ in English units and 1 /meter sec in the Metric System).

- Ill -SD 72-SH-0039

Documents

Final Report on Study of Feedline Dynamic Effects on ... › archive › nasa › casi.ntrs.nasa... · This report is a description of work performed under Contract NAS1-10836, Study