University of California, Los AngelesMechanical, Aerospace and Nuclear Engineering Department

LABORATORY SIMULATION OF THEEFFECT OF ROCKET THRUST ONA PRECESSING SPACE VEHICLE

FINAL REPORT

Participating Students:

Oscar AlvarezHenry BausleySam Cohen

Miguel Falcon-MartinGary Furumoto, Report EditorAsikin HorioDavid LevittAmy Walsh

Instructor:

R. X. Meyer, Adjunct Professor

Sponsored by NASA/USRA

(NASA-CR-186bPl) LABORATORY SIHULATI_N CFTHE EFFFCT OF KOCKET THRUST ON A PRFCESSINGSPACE VEHICLE Fin3l Report (CaliforniaUniv.) lib p CSCL 2IH

May 22,1990

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https://ntrs.nasa.gov/search.jsp?R=19900015866 2018-07-26T17:48:55+00:00Z

Abstract

Ground tests of solid propellant rocket motors have shown that metal-containing

propellantsproduce variousamounts of slag(primarilyaluminum oxide)which istrapped in the motor case, causing a loss of specifiC impulse. Although not yetdefinitely established, the presence of a liquid pool of slag also may contribute tonutational instabilities that have been observed with certain spin-stabilized, upper-

stage vehicles. Because of the rocket's axial acceleration-absent in the ground tests-estimates of in-flight slag mass have been very uncertain. Yet such estimates areneeded to determine the magnitude of the control authority of the systems requiredfor eliminating the instability. A test rig with an eccentrically mountedhemispherical bowl was designed and built which incorporates a ,follower" forcethat properly aligns the thrust vector along the axis of spin. A, program thatcomputes the motion of a point mass in the spinning and precessing bowl waswritten. Using various RPMs, friction factors, and initial starting conditions, plotswere generated showing the trace of the point mass around the inside of the fueltank. The apparatus will be used extensively during the 1990-1991 academic yearand incorporate future design features such as a variable nutation angle and a filmheight measuring instrument. Data obtained on the nutational instabilitycharacteristics will be used to determine order of magnitude estimates of control

authority needed to minimize the sloshing effect.

Introduction

Many rocket motor solid propellants in current use contain a significant amount ofaluminum which, when burned, produces a slag consisting of aluminum oxide andelemental aluminum. Most of this material is expelled throughout the rocketmotor nozzle and adds to the thrust, but some remains trapped in the motor case.

The melting point of the (z-form of AI20 3 is about 2050°C, below the temperature of

the combustion gas. The liquid slag, in the form of small droplets, is subject to acombination of forces that include the drag from the combustion gas, the inertialforce resulting from the axial acceleration of the rocket, and (for spin-stabilizedvehicles) the centrifugal force resulting from the vehicle spin.

The present analysis postulates that, because of the high level of turbulence in themotor, slag droplets entering the gas stream are ejected, and that trapped slag isformed primarily by liquid slag flowing along the surfaces toward the point ofminimum potential energy in the accelerating and spinning motor. Also, thepresent analysis concludes that slag will accumulate to some degree in all spinningor accelerating rocket motors with aluminum-contaning propellants and submergednozzles.

PRECEDING PAGE BLANK NOT FILMED

2

A number of spin-stabiifTed vehicles that use aluminized propellant have shown amarked tendency for a "coning" instability; i.e., a precession with steadily increasingnutation angle. These motors have a submerged nozzle geometry, resulting in adownstream annular pocket which is likely to favor slag retention. It has beensurmised, therefore, that the sloshing motion of a liquid slag pool may be acontributing cause of the observed flight instability. The effects of liquid slag on thestability of spinning vehicles is similar to the effects produced by fuel slosh inspacecraft. Slag retention also requires examination because of its potentially

deleterious effect on specific impulse.

Through installation of witness plates downstream of the nozzle, where some of the(now solid) slag particles are deposited, estimates of the size distribution and totalmass of the expelled particles have been made. Ground tests of this type, however,

take no account of the processing of the droplets in the nozzle.

This report consists of a mechanical design that simulates the motion of a spherical

fuel tank in a thrusting spacecraft. A true simulation of the thrust was thought to be

impossible due to the gravitational support forces present in the laboratory.However, through the means of an eccentrically mounted spacecraft model on thetop of a turntable, the simulation of thrust aligned with the vehicle axis is possible.The mechanical design was finished during the 1990 winter quarter and the test rigwas built in the spring. The comparison of the initial description (see figure 1) withthe design actually built (see figure 2) shows the evolution of the design concept.Qualitative analysis will be provided by photographs of fluid profiles at given time

intervals and quantitative analysis by correlation of film thickness from capacitancemeasurements between two platinum wires located in the bowl. This sensor will bedesigned, built, and incorporated into the test rig slip-ring assembly during the 1990-1990 academic year. From these data, nutational instability characteristics and order

of magnitude estimates of control authority needed to eliminate the instability willbe determined.

A computer program was written to simulate the shape of a fluid in a spinning and

precessing container with a nutation angle equal to zero. The fluid was assumed to

be in hydrostatic equilibrium. The fluid depth as a function of position along with

the shoreline of the fluid was determined. A more general code was written which

computes the motion of a point mass in a spinning and precessing hemispherical

container. Using various RPMs and friction factors, plots were generated to

compare the motion of the point mass and validate the theoretical model (see

figure 4).

3

W2

SLIPRING

SHAFT12

11

LEADPLATES

'FOLLOWERFORCE"T

4 3 c;U

BRUSHASSEMBLY

1 BEARING"'ASSEMBLY

SLIPRING

2

MOTOR_t00 RP

AC

1. generalbearing assembly2. AC motor (variable rpm)3. pulleyfor motor shaft4. main drive pulley5. secondary pulley (stationary)6. main shaft7. control arm

8. bowl pulley9. bowl bearing housing assembly

10. bowl mounting flange11. hemispherical bowl (lucite)12. bowl suport shaft13. idler guide

Figure 1: Appratus Diagram (Not to scale)

4

ORIGINAL PAGE ISOF POOR QUALITY

Figure 2: Completed Test Rig(a) top view showing liquid sloshing in bowl(b) front view showing dual motor assembly

5

Viscous Dissipation:

The degree of instability of a thrusting, spin stabilized spacecraft depends strongly on

the amount of internal energy dissipation. The dominant energy dissipationmechanism is thought to be caused by the sloshing of liquid slag at the bottom of thesolid motor casing which directly influences the body's motion. Oscillatory, andsometimes violent, motion of the fluid induce corresponding oscillations of the

body. Viscous effects in the fluid also influence the body causing the nutation angleto change thereby affecting the stability. It is, therefore, important to estimate the

energy losses in the fluid.

Once these energy losses are estimated, one can predict the body motion by reducing

its kinetic energy at the same rate. _ approach is known as the "energy sink"procedure. Due to the growing nutation angle from energy dissipation, thrustcorrections need to be fired to stabilize the craft. This requires more fuel to beincluded for stabilization during launch which ultimately increases launch mass.

Having to fire these correcting thrusters at the right time creates yet another

problem in the attitude dynamics and control of the spacecraft. IdeaLly, nutationalinstability characteristics and order of magnitude estimates of control authorityneeded to eliminate the instability would allow designers to provide the lightest

control system necessary to minimize this phenomenon.

Scale Model Principles

Many different'models have been developed to test sloshing and its effect onspacecraft. Most of these models, however, are made to simulate the sloshing of a

spacecraft in which thrust is absent. One of the recent problems is that an instabilityevidenced by a growing nutation angle has been observed during the firing of liquid

and solid perigee and apogee motors. A new model to simulate this motion wasneeded which properly aligns the "thrust" vector with the model axis.

A simple design of a spacecraft model mounted eccentrically on a turntable can be

used. This rig simulates the thrust as a "follower" force (see figure 3). Previousmodels were subjected to gravity forces acting at the center of mass. But the newmodel produces a resultant of gravity and inertial forces that remains aligned at alltimes with the vehicle axis. Hence, this thrust "follows" the model as is spins and

precesses around on the turntable.

6

z axis

marble

pointmass

"follower"force

R = turntable radius

= angula_velocity

g = 8rarity

(_2 R = centrffugaJ acceleration

Figure 3: "Follow Force Diagram"

0.15

10 Second Marble Trace40 RPM, friction factor = 0.5

0.10-

0.05"

E 0.00

-0.05-

-0.10-nutation angle = 15 deg rim at R = 0.15 m

I I tI l l

-0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15

x [m]

Figure 4: Computer Code Results

0.20

Because of space and cost constraints, it is necessary to have a model that is not full

scale. It must then be shown that the model behaves in the same way as the

spacecraft. Therefore, it is required for the model to have the same inertia ratio asthe spacecraft:.

odel pacecraft

It also follows that the ratio of the precession rate to the spin rate be the same inboth the model and the full scale model. To simulate the dynamics of the sloshingrequires that the Froude numbers of the model and spacecraft be the same:

Froude number"Ri(d(_/dt) 2"

5

g/cosO model

[.Rt(d_/dt)_

= [ TIM

Solving for (dO/dr)model:

(d_/dt)mode 1 = (d(_/dt) s ._pacecraft '_ (Rt)modelTCosO o

Using these equations, a good approximation to a thrusting spacecraft can be madein the laboratory.

Mechanical Design:

A distinct design evolution was experienced in attempting to construct a test rig

which would adequately simulate the conditions present during the burn of a solidpropellant rocket motor. As a preliminary experiment it was primarily designed toprovide a qualitative analysis of fuel and slag sloshing and aid in the development

of future experimentation.

The design problem was to simulate rotation about the rocket's own axis and the

subsequent precession about an associated axis, both of which are effects of spin

stabilization. It was initially agreed that dual rotating shafts were best fitted to

produce the kinematics of the situation, and subsequently the design problem waslimited to developing a system that would drive the two shafts with correctdirection and rates of spin. In order to achieve this effect several proposals weremade, first of which entailed using a set of belts and pulleys driven by a single

electric motor. Succeeding designs included such elements as a planetary gearsystem, a set of rubber wheels, or a set of dual motors. In the end, the initial conceptof belts and pulleys was adopted for their availability and ease of use.

8

The rig is mounted on a half inch thick aluminum table, approximately one metersquare and held up by four nine inch long aluminum legs. The main shaft ispositioned verticallythrough the middle of the table,housed by a bearingassemblywhich is mounted to the under face of the table. This shaft is driven by a belt,

connected to a variable RPM electricmotor also mounted from beneath the table.

To the top of the main shaft is mounted a control arm made from an aluminum Tbeam. On one side of the control arm isthe fuel tank assembly and on the other,an

equal counter weight made of lead plates.

The hemispherical bowl, turned from a Lucite block, is mounted to a second shaftwhich rotateswithin the bearing housing mounted to the control arm. P6sitioned

on the main shaft and on the bottom of the second shaftare two pulleys. 'ITmpulley

on the main shaft is secured and remains stationary with respect to the table. The

other pulley is secured to the second shaft and produces the rotation of the bowlabout itsown axis. A crossing belt connects the two pulleys,and as the main shaft

rotatesat an average rate of forty RPM, the second shaft rotatestwice as fastin the

opposite direction. In order to keep an adequate tension in the belt,the bearing

assembly housing the shaft,can shifthorizontallyby ± 0.5 inch. In addition,an idler

isincluded on the controlarm to guide the beltand maintain itstension.

During the next academic year (1990-1991) a sensor will be designed which

determines the film thickness by measuring the capacitance between two platinum

wires. This will hopefully provide a means to quantify the force and momentum

produced by the rotatingHquid in the bowl atvarious RPMs. In order to incorporatethisinstrument, an electricconnection to the bowl is needed through a set of slip

rings in the rotating mechanism. Just below the bowl and above the bearing

assembly ismounted the firstslipring. And at the bottom of the main shaft below

the bearing assembly is mounted the second slip ring. To connect the wires from

the controlarm to the second ring,a hole is drilleddown the center and through the

entirelength of the shaft. Through thishole the wires are run to the slipring.

Computer Stmulafiom

A theoretical analysis which approximates the fluid in the bowl with a point masswas developed. The result was a system of two ordinary differential equationswhich can be solved numerically by Heun's method for initial value problems. A

code was generated which determines the x, y, and z coordinates of a "marble"rolling around inside the bowl given a friction factor, initial starting coordinates,bowl RPM, and nutation angle. The friction factor was varied to simulate the effectsof fluid viscosity and friction of the point mass. The larger the friction value, themore of a damping effect the marble exhibited. For smaller values, the marble took

longer to stabilize and rose higher in the bowl (see figures). When the actualexperiments begin this fall, the code can be properly validated with better estimatesof the friction factor, RPM, and nutation angles necessary to demonstrate a valid

theoretical model and test rig.

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TABLE OF CONTE_,FI'S

INTRODUCTI ON ............................................... I

LIST OF SYMBOLS ............................................

REFERENCE FRAMES ........................................... H

STABILIZATION PRINCIPLES ................................... _"

$SCALE MODEL PRINCIPLES .....................................

MECHANICAL DESIGN .......................................... 13

DESIGN SKETCHES ............................................ 16

COMPUTER ANALYSES - A. :SIMULATION OF --LUID SHAPE ........... _"

B. SIMULATION OF POINT MASS MOTI ON .... ./_,

HOT TUNGSTEN WIRE ANALYSIS ................................. /_'_

LIST 0F MATERIALS .......................................... _I

CONCLUS ION ................................................. _

PRELIMINARY DESIGN ............................. _"

CALCULATION FO FLUID ,SHAPE SIMULATION .......... _:_

POINT :_ASS MOTION HANDOUT ...................... g0

POINT ....IASS HOTION COMPUTER PROGRAM L:STING ..... _"-

UELOCITY CALCULATIONS FOR H0T tI!RE ............. G_

REFERENCES ................................................. lOS""

SIBLI 0GRAPHY ............................................... / 0_)

APPENDIX A:

APPENDIX B:

_mFENDI}," C:

_RPENDIX D:

APPENDIX E:

INTRODUCTION

ORIGINAL PAGE ISOF POOR QUALITY

In

permit any energy dissipation.

all spacecraft and rockets

including elastic structural

motion of fuel

Since Explorer I,

theory, the definition of a rigid body does not

However, it is known that

have some non-rigid entities

deflection and the liquid

n its tanks, otherwise known as slosh.

t has been known that these properties

can have a major effect on the motion, that is, there can be

instabilities that depend _argely on the internal energy

dissipation. In most spinning spacecraft and rockets, the

largest amount of energy dissipation comes from the liquid

s Iosh.

This report consists _f _ mec-_- r, ca_ _es.gn that

simulates the motion of a spherical fuel tank, in a thrusting

spacecraft. A true simulation of the thrust _4as t_,ought to

_e _mpossible ,_ue to the gr_vitaional support _orces _resent

n the ' abor atory. However,

eccentr i,=al ! ;," mounted spacecraft

turntable, i t _Jas discovered

through "he means

model :n the

in reference ! ,

-f &n

-_imu!atic, n ,_-,_ thrust &l igne._ ,Ji th the ,ehi,-!e

noeec _ossib!_. The mechanical design is -_=cheou'.eo to be

{n -he "orm _f pictures and -_ata c_llected ._=,bout the depth

of _Jater __t -ertain points. The re=_ul ts tJill then be

:ua] i tat ,,,el::_ presented .;Ji th .=onclusions drawn ._oout the

; I ,_n i r _ _.-r _.';e ,Je i . r-I _,aci,,,._roun_ ,:n :7";e ;zes i :_F_ B.r_O & ] St

:f mater :-]s r eedeo _;r t_:e _o_=ign &re nc!_=e_,

too _f a

-hat the

_XJS _Ja$

Also presented in this report is a theory on the

possibility of using a heated tungsten wire to measure the

depth of liquid in the model. By relating the heat loss of

the wire to the depth of fluid inside the cup, it _Jas

thought that an accurate reading of the depth could be

obtained. However, after many calculations it _Jas found

that there were too many uncertainties for the hot wire to

be an accurate device. Nevertheless, these calculations are

valuable and therefore included.

A computer program _,_as _vritten to simulate the shape of

a fluid n a spinning ancl precessing container v,Jith a

nutation angle equal to zero. The fluid was assumed to be

in hydrostatic equilibrium. The fluid depth as a function

of posi tion _Iong ,,vith the shorel ine :_f the _luid _ere

determined. A program that determined the motion -.f a point

_ass in a spinning and precessing hemispherical container

,,,,.as ._.Iso _,,_ri tten. L_si ng different condi t ions, , .e. ,

different RPrMs and friction f__.ctors, plots were generated to

.-_ompsre the motions of the point mass.

oRIGllxlAt- PAGE IS

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LIST OF SYMBOLS

(X,Y,Z)

(x,×,z)

0

M

m

r

Og

I

T

_e

h

D

_r

_..lu

Inertial reference _rame

Body-fixed frame

Precession rate

Spin rate

Nutation angle

Total mass including fuel

Fluid mass

Kinematic ,Jiscositi_

Radius

Centripetal .acceleration

Density

Acceleration of gravity

Moment of inertia

Thrust

Reyno!ds number

_roude number

Per_i nine to smacecrsEt

Pertsning to model

Height

Oressure

Angular .Jeloci ty _subscr i_s c_v_cus)

Prandt! number

_3r _o_ _umber

Nusselt number

3

ORIGINAL PAGE IS

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£zx

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Z

×

ORIGINAL PAGE IS

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q "3,, _f

STABILIZATION PRINCIPLES

There are primarily two

techniques currently in use. They

and three-axis stabilization,

advantages and disadvantages. Since

rockets and thrusting spacecraft, we

in spin stabilization.

Spin stabilization is based on the

produced due to the rotation of the

Dual-spin, which involves %_o types _f

precessing :f the

technique as opposed

rockets carry much

the motion of

the presence of

_rom itS _omir_&l

_iosning" .

--_ e

strong_i cn =_e

most _ominant

sloshing vJhich

different

ape

each

we are

will be

stabilization

spin .stabilization

of which has its

concerned with

most interested

rocket is the

to single-spin.

of their fuel

this fuel ,due to

thrust is of

position s

the

g r e ._t

_..vh a t

gyroscopic stiffness

rocket or spacecraft.

sp inn ing _nd al so

most common stabi! izaion

Since spacecraft and

in the form of a liquid,

spin stabilization and

interest. This motion

,s _:nown _s "! iquid

.smour. t -_ -_tern.._l ener_>' ,dissipat;,.-_,n, The

energy dissipation rr,ech&n i sm ;s the fuel

influences the boO>'"s moticn immediate]>" and

=ir-ect]/. OSCi _ '._.tor.>', _.nc; .-_ometimes ....._¢.,ient, motion of the

_l,Jid ncluce :orre-=.ponding

"i=cous.,_ e_c_:s..._ _n. the Z_uic_,

osci_',ations of the body.

also influence the body .:.._using

thereby &ffecting the

:moor. rant to est {rr.;,._tethe

the nutat i on .:-,ng! e to ,:ha.nge

_herefor-_sta.bi! ; t:/. "t , sj .

•: r_ e r "_ 7 : 5 _. -=

5ORIGINAL PAGE IS

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i

The ener'gy dissipation r.s. te ',s assumed to be ._. ;'Jnction

of the many spacecraft parameters as follows,

where,

M = total mass including fuel

m = fluid mass

= kinematic viscosity

R = radial distance of tank center from spin axis

r_ = radius of tank

r4 = radius to free __urfa.ce

a = centripetal acceleration, r ,." _ )

etc.

expanding in an infinite series representation,

_:Jher._ ei_ = exponents, --- !,2,, , ,!2,. i = !.,2 ....

Using the non-dimensional ization process using the

"Buc_:ingham P! Theorem" ':see reference 2) __.nct >Jriting the

,.esu;t in the original form ,.'_gher.e terms of like exponent=_.

have been _.r.ought together ind the exponents droppe,_),

some cf these terms can be " rzvertecl to =.how their more

= Reynolds number

= Frou_e r, urnber

= _:r ic:!cr_ ._i: "

ORIGINAL PAGE IS

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-)?,

Once these energy _osses are estimated, one can then

predict the body motion by reducing its Kinetic energy at

the same rate. This approach known as the "energy sink"

procedure is beyond the scope of the class. A more _etailed

discussion of it and energy dissipation can be found in

reference 3 and reference 4.

One of the results of this growing nutation angle from

energy dissipation is that c,orrecting thrusts will have to

be fired to restabilize

_uel and _horten the

these correcting _rusts

inother _ro=!em ;n the

The simplicity of spin

sloshing will still remain

the craft. This will use up more

life of the craft. Having to fire

__t the right time creates :/et

stabilization means that :iquid

a c.hallenging problem.

-7

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SCALE MODEL PRINCIPLES

Many different models have been developed to test

sloshing and its effect on spacecraft. Most of these

models, however, are made to simulate the sloshing of a

spacecraft in which thrust is absent. One of the recent

problems is that an instability evidenced by a growing

nutation

liquid perigee

Therefore, a

founO.

angle has been

and apogee

new model

observed during the firing of

rocket motors (reference I).

to simulate the motion had to be

As seen _n reference I, an intricate model need not be

made. Instead, a rather simple design of a spacecraft model

mounted eccentrically on a turntable can be used. This rig

simulates the thrust as a "follower force" (figure I). As

opposed to figure 2, where the model is subject to gravity

r-eact!ng force_ acting at its center =f mass, _igure 3 shows

the _esign _here the model i ies eccen_r icai l>- on a

turntable. This _ro_uces _ resultant of _ravi ty and

inertfal _orces that remains aligned it il! times _Jith the

vehicle ixis. Hence, this thrust is of the type _e_erred to

_S a "zol _ ower Zorce "

Because cf space _.no cost :snstrlznts, i t is necessary

to h._ve i model that is not full _cale. :t must then be

shown that the _o_el behaves in the same .va>" is the

spacecraft. ?herefore, it is required for _he model to have

the ._ame _nert!_ ratio as the _Dacecra÷t,

ORIGINAL PAGE IS

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!t also fol lows that the ratio of the precession r_te to the

spin

mode I .

that

rate De the same in both the moc_el and the full scale

To simulate the dynamics of the sloshing requires

the Froucle numbers of the model and spacecraft be the

same

Fr-om this Jt is seen that

| J ' _ ' ' I

Using these equations, a true simulationof s thrusting

spacecreaft _an be made.

ORIGINAL PAGE IS

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OR1GINAL PAGE ISOF POOR QUALITY

MECHANICAL DESIGN

A distinct design evolution was experienced in attempting to

construct a test rig_ which would adequately simulate the

conditions present during the burn of a liquid propellant rocket

engine. As a preliminary experiment it was primarily designed to

provide a qualitative analysis of fuel sloshing and aid in the

development of future experimentation.

The design problem was to simulate rotation about the rocket_

own axis and the subsequent precession about an associated axis,

both of which are effects of spin stabilation. It was initially

agreed that dual rotating shafts were best fitted to produce the

kinematics of the situation, and subsequently the design problem

was limit_l to developing a system that would drive the two

shafts with correct direction and rates of spin. In order to

achieve this effect several proposals were made, first of which

entailed using a set of belts and pulleys driven by a single

electric motor. Succeeding designs included such elements as a

planetary gear system, a set of rubber wheels, or a set of dual

motors. In the end, the initial concept of belts and pulleys was

adopted for their availability and ease of use. The details of

test rig's design are the following:

The rig is mounted on a half inch thick aluminum table,

approximately one meter square and held up by four nine inch long

aluminum legs. The main shaft is positioned vertically through

the middle of the table, housed by a bearing assembly which is

mounted to the under face of the table. This shaft is driven by

• m

a belt, connected to a variable RPM electric motor also mounted

from beneath the table. To the top of the main shaft is mounted a

control arm made from an aluminum T beam. On one side of the

control arm is the fuel tank assembly and on the other, an equal

counter weight made of lead plates.

The hemispherical fuel tank, turned from a lucite block, is

mounted to a second shaft which rotates within the bearing

housing mounted to the control arm. Positioned on the main shaft

and on the bottom of the second shaft are two pulleys. The pulley

on the main shaft is secured and remains stationary with respect

to the table. The other pulley is secured to the second shaft and

produces the rotation of the fuel tank about its own axis. A

crossing belt connects the two pulleys, and as the main shaft

rotates at an average rate of forty RPM, the second shaft rotates

twice as fast in the opposite direction. In order to keep an

adequate tension in the belt, the bearing assembly housing the

shaft, can shift horizontall_ by + or - half an inch. In addition,

an idler is included on the contol arm to guide the belt and

maintain its tension.

If a feasible means of determini_the depth of the fluid could

be produced, an electric c0_ce_ction to the fuel tank would be

necessary. For this purpose, incorporated into the design is a

set of slip rings to make possible electric connections to the

rotating mechdnisms. Just below the fuel tank and above the

bearing assembly is mounted the first slip ring. And at the

bottom of the main shaft below the bearing assembly is mounted

the second slip ring. To connect the wires from the control arm

to the second ring, a hole is drilled down the center and through

the entire length of the shaft. Through this hole the wires are

run to the slip ring.

The following are detailed drawings of the individual parts.

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Simulation of Fluid Shape

In order to gain insight into the instability of a

spacecraft due to liquid slag, a simple study of a liquid

under simulated static forces should be made. This study can

be made by applying Bernoulli's Equation (i) for both

spherical and _rtesian coordinate systems in 3 dimensions.

-(V-P + _-g-V-h) = _-a(I)

If applied properly a shoreline--the line where the

fluid meets the cup--can be graphed to obtain a top view of

the cup. Moreover a cross-section of the front view may also

be obtained. Due to the difficulty that the actual spacecraft

precession poses with its nutation angle, a simplified study

shall be made using a cup without a nutation angle.

Before the detailed study of the fluid shape under

static forces is made, it is useful to conduct a qualitative

examination of the fluid shape. Since the cup is spinning

about its own axis, as well as about a point located at a

fixed distance from the cup's center, one can assume that the

fluid's actual shape is the superposition of the shapes

caused by each motion separately. Figure i shows a picture of

the posed problem while figures 2,3 and _ are purely

qualitative drawings of the fluid_ shape before any

calculations were done.

<

-X

Z

I _ .... jl Ir=up 0 r=u_

< >

r_m

Figure 1.

>

+X

ii

Fig.2 rotation about

the cups center.

!

|

Fig.3 rotation about a

fixed point.

Summing figures 1 and 2 figure 3 is obtained.

• " " r

!I

Fig.3 superposition ofshapes in Fig. 2 and 3.

Now that a qualitative understanding of the situation

$_ Om_rNALPAGEISOF POOR OUALITY

has been made, a theoretical solution should be constructed

using equation (i) applied to our posed problem of fig. I.

Applying Bernoulli's equation in the x direction, eqn.

(I) reduces to eqn. (2):

d

-[p'dx (2)

where: -r=u_ _ x <- r__u_

> acceleration due to the cup's

rotation about a fixed point.

acceleration due to rotation

about the cup's axis.

After integrating and assuming zero gauge pressure

outside of the cup, equation (3) is obtained in the xy-

plane:

W=L,D 2 + Wa.-,. 2 W_,-,,1 2 *r_r-m

z(x) - *x 2 + .x + z(0) (3)

2"g g

Similarly applying eqn. (i) in the y direction, the

following is obtained (%):

d

_[p+ :0a dy \

where a_ is :

ORIGINAL PAGE IS

OF POOR QUALITY

(%)

ay = w_p2-y where: -r_p _< y _ r_p

Note that there is only acceleration due to the cup's

rotation about its own axis in the yz-plane at the instant

time we apply eqn. (I). Following the same integrating

procedure used to obtain equation (3), equation (5) can

easily be shown:

W_up 2

z(y) = .y2 + z(0) (5)

2-g

Combining equations (3) and (5) we get a general three-

dimensional expression for the height of the fluid in

Cartesian and spherical coordinates (6a) & (6b). See appendix

for the definition of spherical coordinates used.

z(x,y) = ...... x 2 + °x +

2"g g

*__ u p 2

+

2-g

•y2 + z(0,0) (6a)

z(R,¢,e) =

W,-up 2 + Wa,-,.2

2"E

•R-cos(¢)cos(8) + w.,-m2"r.,m

"R'cos(¢)cos(@) + z(0,0) (6b)

In the xz-plane (front view, z(x,y=0) ), we obtain a

ORIGINAL PAGE iS

OF POOR QUALITY

o

parabolic equation. By adjusting the constant z(0,0) one can

determine how high the fluid may be at the middle of the cup

before it would spill out. A quantitative plot was made and

the initial height of the fluid determined to be 0.91cm using

MathSofts's MathCad program. The following graph, constructed

for the xz-plane, indicates the qualitative drawing for the

final fluid shape in the xz-plane was indeed correct

r,, iiIII ,¢' I

.- .'"¢ ., ?'

,.*'

" ... "..%",.., ..."

To obtain a shoreline of the fluid (xy-plane,top view)

one must combine the equations for the height of the cup

with the equation obtained for the height of the fluid shape

(6b).

The since the cup is a sphere, the equation of the cup

may be described by equation (7a) or (7b).

ORICINAL PAGE IS

OF FOGR QUALITY

Z2 + X2 + y2 = R2 (7a)

Z = J R 2 - x 2 - y2 (7b)

_Combining equations (7b) and (6b) then letting MathCad

handle the algebra and plotting, we get the following graph.

. .....-'*"--" -'--, ."'°°..

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., ...

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If we examine equation (6a) we see that the form of the

equation for the xy-plane is that of an ellipse displaced

some distance from the origin (8).

A.x z + B.x + C-y 2 = D

or

A'.(x + b)2 + C.yZ = D

where A,A',B,C,D are constants.

ORIGINAL PAGE ISOF POOR QUALITY

(8)

This form of our equation certainly agrees with the plot

of the shoreline performed on MathCad as one can easily see

by inspection. Thus we may conclude that the plot agrees with

the equation obtained for the fluid shape.

One may attempt to dispute the validity of the

equations: however, recall that the problem of the actual fluid

sloshinE was not solved but rather a simplification of the

problem. The solution thouEh is still useful in EaininE

insight to the degree of fluid sloshing under the simulated

conditions that have been chosen.

COMPUTERSIMULATION OF A POINT PIASS

A theoretical analysis representing _ fluid by

displaced mass point has been developed by reference 5.

Using this knowledge to validate our efforts, it was desired

to track the motion of a point mass in a spinning and

precessing hemispherical container with a nutation angle

equal to zero• This was done in reference 6, (see appendix

_), resulting in the following _overning e0uations:

Using Heun's method for -olving ordinary differential

equation=_., as in reference 6, and noting that the q-antities

_, _, and _ _re near!'/ constant: "he posit!_n -_f the F-oin_.

mass _s _ ._,_'nc'.!._n,of time ,.,.J._s_.__unc]• A computer -.rogr_m

,,Jas vJritter, -o gener.__te the _at_ neecled t_ develop -. plot _f

".he mot ions .:see e.ppen_i." C?,.

'Sever_m! nitial conditi_-,ns _,,ere '_=ec_ " •e. _0, 40, end

30 RPMs, -end friction ._ac_rs -_ '°'.50, v.,5, i _0) ano plots

.._ere maoe ;or e.sch ,of the n!ne -onoi t _ons. " t ..._as ;oun,_

that no matter ',_hat valu_ the f_ictic.n _:._ct__r had, the point

,mass _!ew c,ut _f the cup .._.t 'SO RPM. In _ct, _irel>.

differ-ence --..motion v_as ibser.'.,eo .'.:=_ee_!gures "-_;.

However, __.t 40 RPM, the s_z÷ :f the _;cti:zn _ac_or

nf!.-enced the mo_ion in two distir:c*. '.Jays, see =:._ur.es 4-_x,.

ORIGINAL PAGE IS

OF POOR QUALITY!I r

At a lower value, the motion rose higher in the cup as

expected. Also as expected because of the greater motion,

the point mass in the case of lower friction factor took

much longer to stabilize and have a smooth motion.

In the case of spinning at 20 RPM, see figures 7-9,

much less is obvious. Since the spin rate is so low, the

point mass stays much :_ser _o the bottom _f the cup. The

most interesting observation is that again the friction

factor plays a large role in compactness of the motion.

This theoretical analysis, though it is valid, will not

have much of an influence in the report except that it shows

that a stable motion can be achieved. Applying this to our

problem we can hope that an equilibrium of the fluid

motion can be found.

ORfGINAL PAGE IS

OF POOR QUALITY

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ORIGINAL PAGE 18

OF POOR QUALITY

t_-+

HOT WIRE

In the process of designing the experimental fluid

sloshing machine, the question of how to measure the

depth of fluid in the sloshing container turned out to

be a very challenging project. Professor Meyer proposed

the use of a heated tungsten wire inside of the fluid

container (cup). The heat los& b_ the wire would in

turn be related to the depth of fluid inside of the cup.

Other proposals, llke pressure tabs and simple capillary

tubes, were also considered and thoroughly discused.

Since the hot wire proposal seemed to be the most

promising path, several calculations were done to relate

the heat los& o% the tungsten wire to the actual depth

of fluid in the cup.

The hot-wire was intended to be an application of the

Hot-Wire Anemometer commonly used in measuring the speed

of fluids. The hot wire anemometer is basically a

thermal transducer. An electric current is passed

through a fine filament which is exposed to a cross

flow. (This fine wire is actually one of the

resistances in a Wheatstone bridge circuit). As the

flow rate varies, the heat transfer from the wire to the

flowing fluid varies (increases with increasing velocity

and decreases with decreasing velocity). This variation

occurs because the electrical resistivity of the wire is

"Q

a very strong function of temperature. Hence, when the

wire loses heat (it cools) its electric resistivity goes

down (true for metals).

There are basically two techniques to monitor flow

conditions with the hot wire: constant temperature and

constant current flowing through the wire. When the

current in the wire is kept constant, the changes in its

electrical resistivity unbalance the Wheastone Bridge.

This is recorded as a voltage drop across the bridge.

On the other hand, when the temperature of the wire is

kept constant, then a feed-back control will have to be

part of the Wheatstone bridge. This feed-back control

will sense the increase or decrease of heat transferred

by the wire to its surroundings, and will adjust the

amount of current flowing through the wire in order to

keep the temperature constant.

The problem with the hot wire proposal is not one but

many. For example, it is necessary to determine whether

the wire is losing heat through a free convection

process or a forced convection process. Moreover, it is

critical to determine how much heat is lost to the

sloshing fluid, and how much is lost to the surrounding

air. This is important because if the difference

between the heat lost to air and to water is not very

significant then the recorded datum would be misleading.

_ nat

-- 4V, 8r+d._e

FIg.A Block diagram of a constant temperature

anemometer. The hot wire is the probe

acting as one of the resistors in the

Wheatstone Bridge circuit. The feed back

control adjusts the current to keep the

bridge balanced.

CO_$

_uT re. %Sourc_

Fig. 8 The block diagram of a constant current

anemometer. The probe is one on the

resistors in the Wheatstone Bridge circuit.

The voltage across the bridge is shaped and

amplified before being recorded.

,_BL

The following calculations will show that, in view of

all the "forced" assumptions to idealize the process,

the hot wire is not the most reliable method to

determine the depth of fluid in the cup.

I

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: 7.Sx|o m.

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"]'4 : _uid's Temp.

: 2.1 .II"C

: Ai," Te-_e.

= 3.Sl xlo "4 _/'K

Cp = 4.174 r/g_ "Cu_/m "C

_._:= o. 6 30

To he +,e=_'ed Qs Q vet+i c-a(

.¢cess_.y cond'l +',on "ls: [1 ]_'_,

tO-_ < _rPr< IO_Z, _o,

cy {{_der_ -t-k¢

Pr: 4.53

= 3_, _141

Aad :

&_ Pr - 40Z, go4 /

valid _r %_e #cf,l,_T,,_pera_ure, T#- I-.o* v._2.

• * _ree Conv¢c41otq RcaT%r_t_er _s _st_ed.

%ee gppeadlr E

Ii

= O.8Z_ ÷_o.-+8"_(r=,.P_,)'_

']+(_ _qX_78/.7-,t Pr_ / J

I

Kf

To ._e'l-"f't,,_he..afLo _ff"J

%_ _ A.#.T+-7.)(I-0)

Q**,_: 5".q27 x ,o "t vv'

......... ..l--...................... "....

° •

NOW if the same wire were exposed only to air, the air

could not be assumed stagnant with respect to the wire

since the cup is rotating about a point ten [nches away.

(see fig.C)

,,Qfm _l,

' ,-ew _¢e.

,\

Flg. C Wlre rotates about point "o"

Hence, the velocity of the air hitting the vertical wire

is approximately given by the tangential velocity of the

wire:

vQr= R.uJ, (I-A)

However, this velocity indicated by eq. ( ) does not

take into account the fact that the wire is also

rotating about its own axis. So, the absolute velocity

of a point on the surface of the wire is the sum of V

given by eq. ( ) and the velocity of the point with

respect to the center of the wire:

Ve_= r. w z (_'_

5o: - C .C)V'B,AE$. = _'" u_. + P, _J,

,5"7. o_-L

-.. t

l.o &qO MAec.

T_- _.l. tl _ T_- _- _l°°_J%, So.oo

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VeD? i

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_-u- _g_ t._z_5,D

K_And

-_ 5_co '*'/°K, _'

30; Fro_ e$ (_I-D)-7.

_.: g_o..= 2.11 x_o wi

(,:_-e)

-. -

The ratio of Qair to Qwater when the wire is in air only

and water only respectively is:

_wQ%zr= lq. _I

(_ O,t f"

This ratio is very likely to go down when more of the

wire is exposed to air as it would be the case during an

actual experiment. So, @

L,= Le,t_i'%o_. w,,'e _ _4zO" o.oozS ,.,.,// o.oo)o,,,

® _ qOQA._r -----

Q _fer - 304--ll --

In view of all the assumptions and of the given sample

ratios, this proposal of the hot wire presents too many

uncertainties. For this reason no experimental work was

carried on and the proposal had to be turned down.

However, there is another proposal that is likely to

give more precise results in a simpler manner. This

proposal, given by James M_olesco (TA), will be

explored further this Spring.

o

0

§I I I I 0

o o 6o _ _ _0 0

(£--30L x) 'M 'J.S01 .L_H

The purpose of these calculations is to demonstrate that

the heat loss of tunosten wire is oreater in water than in

air and hence the voltage drop is also, greater in water.

However, a very simplified version of calculatino the 17eat

transfer c,f the hot-wire probe is as follows:

A tunosten wire of length , L : 7.5mm and diameter, D = 1.27x

1{[3mm is used. There are two, ca__es where this tunosten wire

is either exposed to the air or water in the r¢,tati_no cup

with no nutation angle. Several assumptions are made due to

the complexity of these problems_, such as forced cc,nvection

heat transfer of wire, the wire is entirely exposed to the

air or _ubmeraed in water, the temperature of _Jire is below

the boiling-point of water, and both wire and r_up have equal

anoular velocity (or con__tant .-elocitv) on the r__tatic,nary

arm.

The sample calculations of these two cases are as

fol lows:

I. To find the velocit./ of water in the cup using Navier-

Stokes equation. "-

m ¢ I._

e: . -

Poo ou uvv

"_ j,'{

If. For'f-_d convection of wire in ai_-.

.4,,- F,_F',_ :t'd._,_t_ .r

, "%: 7_ :z_c : 2qg/<T_,_, + -I%

r+ _=l_._l x,o-_ _/_, _'_=_7o_-, _: _.,_6_/,_

E7ORtGINAL PAGE IS

OF POOR QUALITYAf.

#. W: o._,l[s,

r&_: 3.44q

_: -_A [ T_,- T.,] o4._ _c'__aM , "f_ 3.1_xIo -_ Voa_

III. Fc, i-cEd cr.mvectic, rl of _Jtre _ _ate_-.

": 62-6k"

il .4522

-5/b,a 7-: 641 x_o _t_I+

{'ft ORIGINAL PAGE i$

OF POOR QUALITY

I - 517 ,4[

We cain observe that the heat loss of _ire i_n the air is

much less compared to the heat los= of wire in the _ater. It

can therefore be argued that the heat lo=s in air is quite

neol iolble.

After performing these sample ralculations, we still

need to examine the effects of heat loss and also the voltaoe

difference of tungsten wire having a different height

submeroed in water. Let us make several assumptions fc,r this

case by :

i. neglecting heat los=_, of tungsten wire in the air ,:as

shown earlier where heat loss in air is mucln smaller than

in water ).

2. havino the temperature of wire below the boilino-point of

water.

3. having constant velocity (at Ve= C,.63m/s).

We have the same conditions for wire and water

temperatures, except that the height of wire submerged in

water i_ 5.0mm. This sample calculation follows:

Forced cc,nvectic,_ of wire in water.

ul O#I_|N.&L PAGE IS

OF FOOR QUALITY

l

E,:aminino these three cases, we can argue that for

smaller ,ealues of tungsten wire height being submerged in

water, we will obtain lower heat transfer coefficients_, lower

values of heat loss and thus, lower voltage differences. So,

based on these "cough" assumptir, ns that as the convectio_

heat transfer coefficient decreases, so will the voltage

different.e=_, and we can try to correlate these to give _ =

"_,(v).

ORIGINAL PAGE ISOF POOR QUALITY

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ORIGINAL PAGE iS

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.i FI

CONCLUSI ON

A1 though

effects, the

sloshing still

character ist ics

full scale tests.

results wil!

Two different

liquid slosh

three-axis stabiTization presen_ few sloshing

ease of spin stabilization means that liquid

remains a problem. Nutational stability

have generally been established by costly,

By use of the "follower force" method,

be more easily obtained and just as accurate.

simplified methods for the observation of

will be used in this design. Pictures will be

taken fc- qualitative

various points in

analysis. It

results wil_ be

problem of liquid

analysis anc_ the depth of fluid at

the cu_ wil_ be used for 0uanti tative

is hoped that after the model is built, the

helpful in gaining an insight into the

slosh ing.

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F'arameters of the modeled system

ra "= 15'cm rc := !5cm (,)c := 80'rpm r'a 1= 40'rpm

"The equation representing the fluids' sr_ape:"

I 2 2 ]z(x) := • )a ra + ""'cm _,'cm + CONST2

"The equation representing the __hape of the sphere:"

f(:_,) := -, (rc) - (::'cm) + 15cm

30 •cm

z(;:) ,f(:')

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ORIGINAL PAGE IS

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7 q i

Parameters of the modele_ system

0 := 0, .08 ..2_ ra := 15"cm rc := 15"cm (Oc := 80'rpm

CONST .'= .91"cm _ := 0 R -= rc

ca :-- AO'rpm

"The equation representing the fluids' shape:"

2 2

J'.(Ja+ (,:ic

'ra +

2

R'co_(@) 'cos((-])

+ (R'cc,_(#) 'sin(C)) )

22 _c

'-- + CONST

2'g

• R • cos ( _, .I • cos ( ('_

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F','R,_,A) := R

"Sphererical Coordinates:"

•:(R,_,O) = R'cos(#)-cos(os)

v(R,_,(_)) - R'cos(_)"_in(O)

z(R,_),O) _ R'sin(#)

15 "cm

v(L(R,_,O) ,_ ,0 ) ,y(F'(R,_,O) ,_,0 )

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TOP VIEW

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ORIGINAL PAGE I$

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d_mer,_,,on u_(I_b l" d,i_y,<,_o.",_(lO,,_'l , bill'>..)

D4:

program ballC*************** MAIN PROGRAM

C***************

implicit double precision (a-z)parameter (pi = 3.141592654,neqns=4,eps = I.OD-06)

common wr,wthet,wpsi,wthett,wpsit,gthet,gpsi,athet,apsi

dimension U(IO)

integer n,nsteps,maxit,i,j,npi

open(15,fil

open(20,fil

open(l&,fil

read(l&,*)read(16,*)

read(16,*)read(16,*)

read(16,*)

read(16,*)

read(16,*)

read(16,*)read(16,*)

read(16,*) u

read(16,*) u

read(16,*) tread(16,*) t

print *, "in

read *, maxi

• = "xy.tgp')• = "xyz.out')

"input.dat')• _m

rpmrl

rrO

r2

r3

nuu(1)

u(2)(3)(4)a

bput maxit and

t,h

h: "

nsteps = abs((tb-ta)/h)wl=rpm*2.0*pi/60.O

do 10,n=O,nstepst:tb*n/nsteps

print*, "n,u(1),u(3): ",n,u(1),u(3)

call Par ams(wl,r,rO,rl,r2,r3,U,t)

call Heun(neqns,maxit,h,eps,t,U)

the = U(I)

psi = U(3)

if (dabs(the).gt.2.0*pi) thennpi = dint(the/2.0,/_i_the = the-npi*2.0*pi

endif

if (dabs(psi).gt.2.0*pi> thennpi = dint(psi/2.0/pi)psi = psi-npi*2.0*pi

endifU(1) = theU(3) = psi

I000

1002

i0

x = r*sin(U(1))*cos(U(3))

y = r*sin(U(1))*sin(U(3))z = r-cos(U(1)>

if (MOD(n,50).eq.O) then

write (15,1000) x,y

write (20,1002) n,t,x,y,zformat (e15.8,2x,e15.8)

format (i6,2x,elO.5,2x,e15.8,2x,elS.8,2x,e15.8)

endif

if (U(1).gt.(pi/2.0)) then

stopendif

if (U(1).It.(-pi/2.0)) then

stopendif

continue

end

C*********** THE VECTOR FUNCTION

C***********

double precision function f(i,U,t)

implicit double precision (a-z)common wr,wthet,wpsi,wthett,wpsit,gthet,gpsi,athet,apsi

integer i,npidimension U(IO)

print *, "func',U(1),U(3)

if (i.eq.1)f = U(2)

else if (i.

f = U(4)

% nu*(

else if ( i.f = U(4)

else if (i .

f = (-2.

Y.

endif

end

then

eq.2) then*U(4)*sin(U(1))*cos(U(1)) + (gthet - athet)/r2) - wr*(wthet - 2.0*U(4)*sin(U(1))) - wthett

eq.3) then

eq.4) thenO*U(2)*U(4)*cos(U(1)) + (gpsi - apsi)/r

nu*U(4)*sin(U(1)) - wr*(wpsi + 2.0-U(2))+ wthett)/sin(U(1))

C********** THE FORHULAS FOR THE PARAHETERS

C__

subroutine Params(wl,r,rO,rl,r2,r3,U,t)

implicit double precision (a-z)common wr,wthet,wpsi,wthett,wpsit,gthet,gpsi,athet,_psi

integer npidimension U(IO)

_'(0 il'- "JL

print *, "param',U(1),U(3)

g = 9.81hat = atan(wl*wl*rl/g)wx = wl*sin(U(1))*cos(r2*wl*t/r3)

wy = wl*sin(U(1))*sin(r2_l*t/r3)wz = -wl*(r2/r3+cos(U(1)))

wr = Wx*sin(U(1))*cos(U(3)) + wy.sin(U(1))*sin(U(3))

%

wthet

X

wpsiwxt

wz*cos(U(1))

+

= wx*cos(U(1))*cos(U(3)) + wy*cos(U(1))*sin(U(3))

wz*sin(U(1))

= -wx*sin(U(3)) + wy*cos(U(3))= -wl*wl*r2./r3*sin(hat)*sin(r2*wl*t/r3)

wyt =wzt =

wthett =

X

wpsit =

gx =

gy =

gz =gthet =

%

gpsi =ax =

ay

az

athet =

%

apsi =end

wl*wl*r2/r3*$i n (hat)*cos(r2*wl*t/r3)

0.0

wxt*cos(U(1))*cos(U(3)) + wyt*cos(U(1))*sin(U(3))

wz t*s in(U( I))

-wxt*sin(U(3)) + wyt*cos(U(3))

-g*si n(hat)*cos(r2*wl*t/r 3)

-g*si n(hat)*si n (r2*wl*t/r3)

g-cos(hat)gx*cos(U(1))*cos(U(3)) + gy.cos(U(1))*sin(U(3))

gz*si n(U( I))

-gx*sin(U(3)) + gy*cos(U(3))-wl*w1*r0*cos(U(1))*cos(r2*w1*t/r3)

-w1*w1*r0*cos(U(1 ))*si n(r2*w1*t/r3)

-w1*w1*r0*si n(U(1))ax*cos(U(1))*cos(U(3)) + ay.cos(U(1))*sin(U(3))

-az*si n(U(1 ))

-ax*sin(U(3)) + ay*cos(U(3))

Cw__

C**_*******

SUBROUTINE JACOBN

subroutine Jacobn(J,U,t)

common wr,wthet,wpsi,wthett,wpsit,gthet,gpsi,athet,apsi

dimension J(lO,10)dimension U(IO)

implicit double precision (a-z)

integer npi

print *, 'jaco', U(1),U(3)

J(1,1) = 0

J(l,2) = 0

J(l,3) = 0

J(l,4) = 0J(2,1) = U(4).U(4)*(cos(U(1))*cos(U(i))-sin(U(1)>*sin(U(1)))

J(2,1) = J(2,1)+wr*(2*U(4)*cos(u(1)))

J(2,2) = -NUJ(2,3) = 0

J(2,4) = 2.W(4>*SIN(U(l>)*COS(U(1)>+2*wr*_in(U(!>>

J(3,1) = 0,T(3 _ = 0

J(3,3)J(3,4)J(4,1)J(4,1)J(4,1)J(4,1)J(4,1)J(4,2)J(4,3) =J(4,4) =RETURN

END

=0

=I

= -2.U(2)*U(4)*cos(U(1))+((gpsi-apsi)/r)

= J(4,1)-nu.U(4)*sin(U(1))_r*(wpsi+2*U(2))+wthett

= J(4,1)*cos(U(1))=sin(U(1)).(2.U(2)*U(4)*sin(U(1))-nu*U(4)*cos(U(1)))-J(4,1>

= J(4,1)/(sin(U(1))*sin(U(1)))= -2*U(4)*COS(U(1))/SIN(U(1))-(2*WR/sin(U(1)))

0

(-2*U(2)*COS(U(1))/SIN(U(1)))-NU

r

C

C

c MARBLE • finds the position of a marble

C _m_m

C

c James Marcolesco

c Winter 1990

c MANE 199 Prof. Meyer

c

c This routine drives the Heun subroutine

C

p Iaced in the TEACUP

C

C

c maxit = maximum number of iterations

c h = step size

c eps = convergence tolerance

c neqns = number of equations to be solved in

c F = function vector

c J = Jacobian matrix of function vector

c du = vector to add to solution vector

c U = solution vector to be solved in HEUN

c t = current time at call to HEUN

C

system

c

c224_7

program main

implicit doubl

integer maxit,

common r,nu,gt

dimension u(10

e precision <a-z)

neqns,nsteps, i ,j,n,npi ,rad,count,l imit

he,gph i ,athe,aphi ,w3r,wthe,wph i ,wphi dt,wthedt

)

C

C • • • • •

C

open (15, file = "xy.tgp" , status = "unknown')

.open (16, file = "x.out" , status = "unknown')

open (17, file = "×.out" , status = _unknown')

open (18, file = "z.out" , status = "unknown')

open <19, file = "xyz.out" , status = "unknown')

open (20, file = "input.dat', status = "old')

................................. set

neqns = 4

g = 9.80665d0

pi = 3.141592654d0

pi2 = 2.dO * pi

pigO = pi / 2.dO

rtd = 180.dO / pi

control parameters

s_ri re(*, 1002)

print*, "Enter

print*, "

write(*,lO01)

print*, "max_t

.4rite(*,1001)

max iter&tions:"

= I is explicit

_, 1 is implicit

Heun method"

trapezoidal methoa"

write(*,lO02)

print*, "inputwrite(*,lO01)

read*, eps

convergence tol er ance, eps:"

write (19,1002)

if (maxit.eq.l)

if (maxi_.gt.i)

write (19,1002)

write(19,*) "**

wrlte(_9,*) "**

Explicit

Implicit

Heun-s Method **"

Trapezoidal Method

I001 format (/)

1002 format (//)

I continue

count = 0

write(*,lO02)

print*, "input time step,write(*,lO01)

read*, h

write(*,lO02)

print*, "for plotwrite(*,1001)

read*, limit

write(*,lO02)

h:"

files, skip how many points?"

read(20,*) RPM

read(20,*) rl

wl = RPM * 0.I0472d0

THETA = datan(wl * wl * rl / g)

read(20,*) r

read(20,*) ROread(20,*) R2

read(20,*) R3

read(20,*) nuc

c ...................................... read

C

do 2, i = 1,4

read(20,*) u(i)2 continue

in initial conditions

read<20,*) ta

read(20,*) tb

the = u(1)phi = u(3)

,_rlte <I,9,'000> h,THETA*r, td,wl,r,rl,RO,R2,R3,nu,the,phi

i000 format _:x, step size, h = ' _:7,4, ---ec _, :

%ORiGiNAL PAGE IS

OF POOR QUALITY

• Ix,'THETA = ",f7.4," deg',2x,'wl = ",f7.4," raci/s',

• 2x,'r =',f6.4," m',/,'rl =',f6.4," m',

• 2x,'RO =',f6.4," m',Ix,'R2 =',f6.4," m',• 2x,'R3 =',f6.4," m',2x,/,'nu =',f&.4,

• 2x1_the(O) =',f6.4," rad',2x,'phi(O) =',f6.4," rad',//,

. 3x,'time step',5x,'time',gx,'x',12x,'y',14x,'z')

cc ...................................... (I) calc nsteps

c

nsteps = dabs(tb-ta) / h + It = ta - h

C

c ...................................... (2) begin time steppingc include t = ta

c Calculate coefficients..•

c

do I00, n = I, nsteps

t = t + h

msin = dsin(the)mcos = dcos(the>

psin = dsin(phi)

pcos = dcos(phi)

tsin = dsIn(THETA)

tcos = dcos(THETA)

rsin = dsin(R2 / R3 * wl * t)

rcos = dcos_R2 / R3 * wl * t)

w3x = wl * tsin * rcos

w3× = wl * tsin * rsin

w3z = - w l * (R2 / R3 + tcos)

w3r = w3x * msin * pcos + w3y * rosin * psin + w3z * tacos

w3the = w3x * mcos * pcos + w3y * mcos _ psin - ,;J3z _ msin

w3phi = - ,_3x * psin + w3y * pcos

w3xdt = - ,,Jl* wl * R2 /" R3 _ tsin * rsin

w3ydt = wl * wl * R2 / R3 * tsin * rcos

w3zdt = O.dO

wthedt = w3xdt*mcos*pcos + w3ydt*mcos*ps n -w3zdt*msin

wphidt = - w3xdt * psi n + w3ydt * pcos

gx = - g * tsin * rcosgy = - g * tsin * rsin

gz = g * tcos

gthe = gx * mcos * pcos ÷ gy * mcos * ps n - gz _ msin

gphi = - gx ÷ psin + .__y ÷ pcos

-_.Ox ---- - ','J1 - ,.41 * "._" _ .'.C.,.,'-S _ r-rOS

ORIGINAL PAGE IS

OF POOR QUALITY

aOy - - wl * wl * rO * tcos * rsin

aOz = - wl * wl * rO * tsin

aOthe = aOx * tacos * pcos + aOy * mcos * psin - aOz * msin

aOphi = - aOx * psi n + aOy * pcos

cc ......................................... call solving subroutine

ccall heun(neqns,maxit,h,eps,t,U)

the = u(1)

phi = u(3)

C

ClmiJwwliwwwwwumwmm.ewwwmwlwwwwlwwwmlwlwwww

C

if (dabs(the).gt.pi2) then

npi = dint(the/pi2)the = the - npi * pi2

endif

reduce to Srder<2 pi)

if (dabs(phi).gt.pi2) then

npi = dint(phi/pi2)phi = phi - npi * pi2

endif.C

CilIIIIIlIIIIIlIIIIIIIIIIIIIIIIIIIIIIIIIII!

C

x = r * dsin(the) * dcos(phi)

y = r * dsin(the) * dsin(phi)z = r * dcos(the)

cc ...................................... (II)

C

find x,×,z coords

print out results

if (cour, t.gt.limit) then

write(15,1102) x,y

write(!6,1101) x

write(17,1101) y

write(IS,1101) z

write(19,1105)count = 0

endif

n_t,x,y,zORIGINAL PAGE ISOF POOR QUALITY

stop if out of teacup

1101 format(Ix,el5.8)

1102 format (Ix,e 15.8,2x,e15.8)

.'105 format(4x,i5,7x,f6. _,S(2x,e12.4))

C

CJlliillJIJliJeIJJIIllIIlliIliIJIlliliIIIll

if (dabs(the).gt.pigO) then

write(*,!O00)

#Pint*, mar_ie #Jew cut .s_ teacup: -HETA = ,the_rtd," _eg

q2_

stopendif

I00

u(1) = the

u(3) = phi

continue

close(15)

close(l&)

close(17)

close(18)close(19)

close(20)

encl

functions

C _mm

double precision function f(i,U,t)

implicit real*8(a-z)

integer icommon r,nu,gthe,gphi,athe,aphi,w3r,wthe,wphi,wphidt,wthedtdimension u(lO)

cl = dcos(u(1))

c2 = cl * clsl = dsin(u(!))

s2 = sl * sl

I0

2O

goto (I0,20,30,40), i

f = u(2)

return

f = u(4) * u(4) * cl * sl + (gthe - athe) / r - nu * u(2)- w3r * (wthe - 2.dO * u(4) * sl) - wthedt

return

3O

4O

f = u(4)

return

f = (-2.c10 _ u(2) * u<4) * cl + ,_gphi-aphi)/r - nu * u(4) * sl-w3r * (wphi + 2.dO * u(2)) + wthedt) / sl

return

end ORIGINAL PAGE IS

OF POOR QUALITYC

c .]acobian subroutineC"

sul_routine ]acobn(J,U,t)

impl icit real*8(a-z>

zc.mmon -,r,u,._he,gpr_ ,a_he,aprJi ,_43r.,:,.v_rze,_,Jpn! ._Jpnict,_herJt

q?

dimension J(10,10),u(10)

cl = dcos(u(1))

c2 = cl * cl

sl = dsin(u(1))

s2 = sl * sl

J(l,l) = 0.d0

J(l,2) = l.d0

J(l,3) = 0.d0

J(l,4) = 0.d0J(2,1) = u(4) * u(4) * (c2 - s2) + 2.d0 * u(4) * w3r * cl

J(2,2) = - nuJ(2,3) = O.dO

J(2,4) = 2.dO * u(4) * cl * sl + 2.dO * w3r * sl

J(3,1) = O.dO

J(3,2) = O.dO

J(3,3) = O.dO

J(3,4) = 1.dOJ(4,1) = (2.dO * u(2) * u(4) * sl - nu * u(4) * ci) / sl

- cl/s2 * (-2.dO * u(2) * u(4) * ci ÷ (gphi - aphi) / r- nu * u(4) * sl - w3r * (wphi + 2.d0 * u(2))

+ wthedt)

J(4,2) = (-2.d0 * u(4) * cl - 2.d0 * w3r) / sl

J(4,3) = 0.d0J(4,4) = (-2.dO * u(2) * cl - nu * sl ) / sl

return

end

HUEN's METHOD

James Marcolesco

Fall 1989

MANE 192C Prof. McDonough

This routine solves Initial Value Pr-ob!ems for ODEs

C

C

C

C

c

C

C

C

c

m = number of final iterations

maxit = maximum number of iterations

h = step size

a,b = time ,domain

F

Jdu

LJ

= function vector

= ]acobian matrix of _unction vector

= vector to add to solution vector

i _olution vector to be solved in NEWTON

..'_4_7OR!G!NAL PAGE IS

OF POOR QUALITY

subroutine heun(neqns,maxit,h,eps,t,U)

implicit double precision (a-h,j,o-z)

dimension Jf(10,10),JFF(10,10),FF(10),u(10),uold(10),du(10),

• ustar(10),g(10)

double precision maxclif

integer delta,i,j,m,maxit,n,neqns,nsteps

external f,cleltacc ...................................... (3)

c3 tol d = t - h

m = 0

begin Newton iterations

do 80, m = I, maxit

if (m.gt.l) goto 6cc ...................................... (4) Evaluate

4 do 20, i = I, neqns

g(i) = f(i,u,told)ustar(i) = u(i) + h * g(i)

20 con t inue

C

c ...................................... (5)

5 do 30, i = !, neqnsuold(i) = u(i)u(i) = u(i> + 0.5

30 continue* h * (g(i)

Ustar for use inHuen's method

Calculate inital guess for

trapezoidal rule from Huen'smethod

+ #(i,ustar,t))

if (maxit.eq.1) return

C

...................................... (6)

C

6 call Jacobn(Jf,U,t)

C

Cmwwlewememlmmmlwlmmwewmmmmmwma•mmmmmle_7_

40

50

C

Evaluate FFI,'Um) and J(F.'.,

do 50, i = I, neqnsFF(i) = u(i) - 0.5*h*f(i,u,t) - (uold(i) +

do 40, j = I, neqns,JFF(i,j) = delta(i,j) - 0.5 * h . Jf(i,j)

continuecontinue

C , • • , , • • • • , • , • • • • • • • • • • .... , , , • • • • , , ° , • (`8)

C

8 call GAUSS(neqns,JFF,du,FF>

_-...................................... (,9)

ORIG!NAL PAGE IS

OF POOR QUALITY fl_"

0.5*h*g(i))

Solue for du using Gauss

C_lculate max norm of

.£-.F_C, !F_cFeme_- _ _i >

du<i)

9 dumax = 0.d0

do 60, i = I, neqns

if (dabs(du(i)).gt.dumax)

u(i) = u(i) - du(i)

60 continue

c

c ...................................... (10)

C

C

10 if (dumax.lt,eps) return

dumax = dabs(du(i>)

Test convergence of Newton

iterations

80 continue

print*, "Newton iterations failed to converge at time step n + I"

return

end

C

c ...................................... define

c

function delta(i,j)

implicit integer(a-z)

Kroniker

if (i.eq.j) delta = I

if (i.ne.j) delta = 0

return

end

delta function

C

C

E

C

C

C

C

C

C

G A U S S E L I M I NA T I ON

James Marcolesco

MANE 192C

Fall 1989 Prof. McDonough

S U 8 R 0 UT " H E

C

c234567

subroutine gauss(n,A,X,B)

implicit real*8 (a-h,m,o-z)

dimension A(10,10),×(10),B(10),M(10,10)

C

c .......... forward elimination

C

do !00, k = !, n-!

c ............ row pivoting

c

imax = K

amax = dabs(A(k_k))

%

I0

do 10, i = k + I, n

if (dabs(A(i ,k)).gt.amax) thenamax = dabs(A(i,k)>imax = i

endif

con t inue

2O

if (imax.eq.k) go to 30

do 20, j = k, n

atemp = A(k,j)

A(K,j) = A(imax,j)

A(imax,j) -- atempcon t inue

btemp = b(k)b(k) = b(imax)

b(imax> = btemp

3O do 60, i = k + I, n

N(i,k> = A(i,k> / A(k,k)b(i) = b(i) - N(i,k) * b(k)

4O

do 40, j = k + I, n

A(i.jj) = a(i,j) - M(i,k) * A(K,j)continue

60 continue

I00 continue

c

c .......... back substitution (solve stage)C

x(n) = b(n) / A(n,n)

do 200, i = n - I, I, -ix(i) = 0.d0

210

do 210, j : i + i, n

x(i) = x(i) + A(i_j) * x(j)con t inue

200x(i) = (b(i) - x(i)) / A(i,i)

continue

return

end

APPENDIX

Calculation of velocity of sloshing fluid around wire

Assume two concentric cylinders. The inner cylinder is

fixed to the outer one, and both of them are rotating

with the same angular velocity about their common axis

of symmetry. Moreover, assume that the velocity of the

fluid around the inner cylinder is only a function of

the distance r from the center of this cylinder

(wire). For simplicity no flow in the direction of the

axis of symmetry is assumed (this is no_a very good

assumption), and the cylinders are very long in their

axial direction. Also, as it will be shown, the main

problem with this analysis is the Boundary Conditions.

%

r !

}{ l

I ' II

{ ' II

g ' '{

I{ i

{llq_

I

No v,er-Sto_e's e_.:

Ul)

C_0 ,o

or"

_'v:+ ,'v:-v_-o (B)

Infe_ ra _on 0 t

Ve = _"Y _to

Cx.(x-_,)r_-')y._

_.; v c_ : (s,.,_#i+,_+-;,,,_)

_)

-_ F X _×-'- 1"x =o

OY'"

r+Cx=-t)-o .'. ,+--+---.i

J

Ve-- C,F + C_,I_ )

_-oy"m _ _ve.._:

_9 oP,L

j TuJo _o_a_y

B.C. i: Ve (._=R,) = uo R_

?fS.C. 2."

Ve_- w_z.= c, Rz I" C.

F<,- V._

_4ouo _v_r (F._) ;s

&_. _eye &e _-,ned +o be 4;ff.e,"e,4"."_r_ t_ O T ck (_e u _ c_a g y

_-L_

_o"o

c,+_ .I

C,= ua- .---C'- (D-t)R;

and ;,

c c ml

Conu1_no_'t;on of e,1_.[D-O

C_. __-uJ R, I"4,' R_,- r_,_- _I.,

Sul_s'l-i-lu-l-ion o[ eels. _Iq

C_t UC_..•

OF/

E valuQfinq (FJ e_t c_ lime..

w_re is Fully _brner_

I0/ o_L

R m7._

y- =

W

_5 c_

Rz.

L.O0 cm

40 rp_

_ORPH, , _ ._ -_-_.

t._l"_ XIO-& m/sP_.c.

10% ++(-

o_

.p.._

o

.i--,4

©

©

.i,--i

-P--i

r"!!

q_

I

• ,,,--4

©

• ,--,,i

" t-'l ;_mili I

-,-'-4r-,--4

/

!I

iI

IiiI

uIII

! ..................................

REFERENCES

I. Meyer, R.X., "Stability Tests of Spin-Stabilized

Spacecraft in the O_escence of Thrust, _ Journal ofGuidance and Controls, August 1990.

2. Personal communication between N. Furumoto and J.P.

Vanyo.

B Vanyo, J.P. and P.W. Likins, "Measurement of Energy

Dissipation in a Liquid-Filled, Precessing, Spherical

Cavity,H Transactions of the ASME, September 1971

1 Garg, S.C., N. Furumoto, and J.P. Vanyo, "SpacecraftNutational Instability Prediction by Energy-Dissipation

Measurements, _ Journal of Guidance, Control, and

Dynamics, May-June 1986.

o • . . "Nutational Stability of aMingori , D L and Y Yam,Spinning Spacecraft with Internal Mass Motion and Axial

Thrust," AIAA Paper 86-2271, 1986.

6. Handout by James Marcolesco.

BIBLIOGRAPHY

Agr iwal , Brij N., Desi 9n of Geosynchronous Spacecraft,

Prentice-Hall, Inc., New Jersey, 1986.

Brown, R.C., P. Andreussi, and S. Zanelli, "The Use of Wire

Probe for the Measurement of Liquid Film Thickness in

Annular Gas-Liquid Flows," The Canadian Journal of

Chemical Engineering, Dec. 1978.

G ar g, S.C., N. Furumoto, and J.P. Vanyo, "Measurement of

Energy Dissipation in Forced Precession Compared to

Flight Data," AIAA Guidance and Control Conference,

August 1984.

Garg, S.C., N. Furumoto, and J.P. Vanyo, "SpacecraftNutational Instability Prediction by

Energy-Dissipation Measurements," Journal of

Guidance, Control and Dynamics, May-June 1986.

Harrison, J.A., S.C. Garg, N.

Technique to Measure

Applications," AAS/AIAA

Conference, August !983.

Furumoto, "A Free-Fall

Nutation Divergence, and

Astrodynamics Specialist

Incropera, F.D., and D.P. DeWitt, Fundamentals of Heat and

Mas# Transfer," John Wiley and Sons, 1985.

Kaplan, Marshall H., Modern Spacecraft Dynamics and Control,

John Wiley and Sons, 1976.

Lomas, C.G., Fundame_&l_ _f Hot Wip_ Anemometryl Cambridge

University Press, 1986.

Martin, Ernesto R., "Fuel Slosh and Dynamic Stabil ity of

Intelsat IV," AIAA Guidance, Control and F1 ight

Mechanics Conference, August 1971.

Meyer, Rudolf X., "Stability _ests of Spin-Stabilized

Spacecraft in the Presence of Thrust," AIAA Journal

of Guidance and Control, August 1990.

_unn_ _.H., !ntermedi&t? Clui_ Mechanics:

Publishing Corporation, 1989.

Hemisphere

Perry, A.E.,1982.

Not-Wire Anemometry_ Clarendon Press, Oxford,

[Jan YO, J.P.and P.W. Likins, "Measurement of Energy

Dissipation in a Liquid-Filled, Precessing, Spherical

Cavity," Transactions of the ASME, Septemoer 1971.

lOG ,l:

Vanyo, J.P., "An Energy Assessment for

Precessing Spherical Cavity,"

Mechanics, December 1973.

Liquids

Journal

in a

of

Filled

Applied