Bifilar Pendulum Technique for Determining Mass Properties of ...

37
i'~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~.... ..... . .-. ' .... I I '-_a!i .iII AD-787 506 BIFILAR PENDULUM TECHNIQUE FOR DETER- MINING MASS PROPERTIES OF DISCOS PACKAGES R. A. Mattey Johns Hopkins University Prepared for: Naval Plant Representative Office July 1974 DISTRIBUTED BY: National Technical Information Service U.S. EPARTMENT OF COMMERCE I -aom i I I

Transcript of Bifilar Pendulum Technique for Determining Mass Properties of ...

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i'~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~.... ..... . .-. ' .... I I '-_a!i .iII

AD-787 506

BIFILAR PENDULUM TECHNIQUE FOR DETER-MINING MASS PROPERTIES OF DISCOSPACKAGES

R. A. Mattey

Johns Hopkins University

Prepared for:

Naval Plant Representative Office

July 1974

DISTRIBUTED BY:

National Technical Information ServiceU. S. EPARTMENT OF COMMERCE

I -aom i I I

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I nclassificd PLEASE FOLD BACK IF NOT NEEDEDSECURITY CLASSIFICATION OF THIS PAGE FOR BIBLIOGRAPHIC PURPOSES

REPORT DOCUMENTATION PAGE1 REPORTNIIMSER 2GOTACCESSION NO 3. RECIPIENT'S CATALOG NUMBER

'i12524, TIILE IOwdSubtotlet 5 TYPE OF REPORT & PERIOD COVEVIED

BIFILAR PENDULUM- TEC1HNIQUE FOR DETERMINING Technical M'vemorandumMASS PROPERTIES OF DISCOS PACKAGES ____________________________________

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR~s) 8, CONTPACT OR GRANT NUMBERI%)

R."A. Mattey NOO17-72-C-4401

9. PERFORMING ORGANIZATION NAME 5, ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

The Johns I-Iopkins University Applied Physics Laboratory AREA & WORK UNIT NUMBERS

8621 Georgia Ave.Silver Spring, Md 20010

11. CONTROLLING OFFICE NAME & ADDRESS 12 REPORTODATENaval Plant Representative Office July 19748621 Georgia Ave. 13. NUMBER OF PAGESSilver Spring, Md. 20910-43 31

14, MONITORING AGENCY NAME & ADDRESS 15. SECURITY CLA~SS. lof this report)Naval Plant Representative Office Unclassified8621 Georgia Ave.Silver Spring, Md. 20910 15a. DECLASSIFICATION/OOWNGRADING

SCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

*417. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, is ifent from Report)

18, SUPPLEMENTARY NOTES

19, KEY WORDS (Continue on reverse side ii necessarv and identify by block number)

Bifilar Pendulum, Moment of Inertia, Cross Products, Center of Mass

20 ABSTR ACT (Continue on reverse side if necessary and iden tii'y by block number)A bifilar pendulum was used to determine the mass properties of the Discos packages on

the Triad satellite. The pendulum design was unique in that it allowed the package to be rotatedinto six different angular orientations. Each angular position was located with respect to thecenter of mass of the package. Analysis and results are given that show that the inertia matrix

w , be car I I ... .. , ..-nhmc1_g k hifilns' t)nd~nnil .

RePirodisced by

NATIONAL TECHNICALINFORMATION SERVICEU S Ot'ixrtment of Comma.,ece

Spriu.p'feld VA 13/~

D D I1JAN73 1473 EUIvU nclassified

SUR vCLASIFtCATION OF THIS FAGE

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TG 1252JULY 1974

Technical Memorandum

I BIFILAR PENDULUM TECHNIQUE FORI DETERMINING MASS PROPERTIES

OF DISCO0S PACKAGESI by R.A. MATTEY

THE JOHNS HOPKINS UNIVERSITY * APPLIED PHYSICS LABORATORY8621 Georgia Avenue o Silver Spring, Maryland o 20910Operating under Contract NOD017-72-CA401 with the Department of the Navy

Approved for public release; distribution unlimitrd.

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSI.VCR SPING. MANYLANO

CONT11'-NTIS

List of Illustrations . .

'1 1. Introduction . . .7

2. Analysis .. 9

3. De(velopment of the -MeasuringTechnique . . . .15

j . Final Pendulum Design . 19

5. Electronics Instrumentation . 27

6. Test Results 29

1 7. Concluding Remarks . . .33

Bibliography 35.

Appendix A 3

Moment of Inertia Equations for Bodiesthe Shape of the Discos Packages

34

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THiE J0HNS HOPKINS UNIVERITY

APPLIED PHYSICS LABORATORYSLvar SPRINO. MARYLAND

ILLUSTRATIONS

1 Disturbance Compensation System(Discos) 8

2 Line with Respect to a Mass

Particle 10

3 System Measurement Axes . 10

4 Line with Reference Designations . 12

5 Prototype Moment of Inertia Design . 17

6 Prototype Setup, Closeup View . 18

7 Package on Bifilar Pendulum withArrows Showing AdjustmentMethods 20

8 Package on Bifilar Pendulum in Ote ofthe Orthogonal Axis Positions . 21

j 9 Package in One of the Skew Positions. 22

10 Center of Mass Measurement 23

11 Complete System Layout . . 25

12 Homogeneous Package Printout . 30

13 Final Program Printout . 31

14 Package Mass Properties . . 34

Preceding page blank

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ITHE 1OHNS HOPKINS UNIVEICKITY

APPLIF.D PHYSICS LABORATORY

1 1. INTRODUCTION

lI

The accuracy of the Navy navigation satellite, Trar-Isit, is dependent on precise orbit determination and orbitprediction. Improvements in orbit prediction have been

limited by uncertai~ities in orbit disturbance from solarpressure and atmospheric drag. The Disturbance Compen-sation System (Discos) was designed at Stanford Universityunder subcontract to the Applied Physics Laboratory of TheJohns Hopkins University. It was flown on the Triad satel-lite, and it has opened possibilities for improvements inaccuracy and operational convenience for the Transit sys-tem by freeing Triad's orbit of disturbances larger than5 X 10-12 g.

To eliminate the effect of forces on the Discos sys-tem because of the various satellite components, each mainelectronics package (shown in Fig. 1) had its mass proper-ties measured. To accomplish this a bifilar pendulum wasdesigned that could rotate a body into six distinct positionswith respect to a coordinate system that had its origin fixedIL at the body's center of mass. With this rotational capability,six independent moment of inertia measurements were made.The data were then reduced with the aid of a computer, and

[ the complete inertia matrix with respect to the package cen-ter of mass was determined.

A

, p

II

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THE JOHNS HOPKINS UNIVERSITY

APPLIED) PHYSICS LABORATORYSILVIR SPRING MARVLAND I

lb

UPPER BOOM

UPPER 4~.TANK .- -. '

PRSSR 2VALVE DISCOS PROPULSIONTRANSDUCER (I OF 6) SUBSYSIEM (DPS)

-UPPER SUPPORT

LOW PRESSURETRANSDUCER -RGUAO

PIKF LOWER ,,PROOF MASS

PICOFFPIC KOF FELECTRONICS HOUSING

S CAGINGA MECHANISM

II

ELCRNC ROTATED 1800(INSIDE BOTTOMj

BOTTOM'SUPPORT TUBE)SUPPO;i VTUBE

AtA

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYj SILVIR SPRIiNG MARYLAND

I 2. ANALYSIS

I

Figure 2 shows a line with respect to a mass parti-cle mj. The moment of inertia of the particle with respectto the line L (X, A, y) is

I 'L 2 2xx +A 2 yy + 2zz - 2)1Ixy - 2Xylxz - 2AvIyz. (1)

I X, t, and a are the direction cosines of a line L in spacethat passes through the origin of a fixed Cartesian system

j of coordinates.

In order to solve for the unknown cross products inEq. (1), it is necessary that the body shown in Fig. 2 bemoved to six different angular positions and that the momentof inertia is measured at each of those positions. Equation

1(1) was derived so that only three of the angular positionsI can be mutually perpendicular. Figure 3 shows three of the

mutually perpendicular axes x', y', and z'. Each of theseaxes could be placed on the axis of rotation. Also, the threeskew lines are shown in Fig. 3; their angular rotations andline numbers are indicated below:

I Line Rotation

2 -50,+45

3 +450

To use the moment of inertia data obtained from the1six different measurements, the equations are set up as

shown in Eq. (2). Equation (1) is rewritten where the primerefers to the moment of inertia of the axis being measured:

I!I

I -9-

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THE JOHNS HOPKINS5 UNIVERSITY

APPLIED PHYSICS LABORATORY

SILVtft SPRtING. MARYLAND

/lLl

0

XI/

A

102

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVER SPRING MARYLAND

NXIxy + XyIxz + MVIyz 2 X2 Ixx + + y2 I.z - IL . (2)

Now, setting up a matrix of the left-hand side of Eq. (2) inconsideration of Fig. 4 where Sc = sin (Y an! Cu = cos C(also for three skew lines) we obtain the following:

-$51SICxcSo I C ISxICc S ISc!C&1 -lcy-

S 2 Sac2 CA 2Sa 2 CA2Sxa2 Ca 2 S9 2 Sa 2 Ca 2 Ikz . (3)

L/ 3 S 3 C33S 3 C03 SX3 Ca 3 S3 3 Sa 3 Ca 3 Iz

In setting up the right-hand side in a matrix form, we ob-tain:

22 2I -(C81Scx)21cx~ SlSI2,~ 2 , ,W a I W SN 1 Ca Izz -I'

1 1 IYcIZL

(CA 2 Sue2 ) Ixx (8 2Sc 2 )2 I y 2 Izz I L (4)

a2 1/ 2 ,,, //- I 'll(CA S01 ) lXX (SO3 S 3 ) lyy Ca 3 Izz -

Using matrix notation on the left- and right-hand side,

FIXY[Cij] Ixz = [Di - ILi] (-) (5)

l IyzL5therefore letting Ixz Eij. Solving for the cross products

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• -o.,,- - iI I

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THE JOHNS HOPKINS UNIVERSITY

APFLIED PHYSICS LABORATORYSILVER SPRING, MARYLAND

z

COSOSIN a, SINOISIN a, COS a

a

Lt

y

11

.1

J

-12- 3

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSII.VER SPRING. MARYLAND

we have

E Cij]-I [Dij -lij] (6)[Eij2 (6)

Knowing all the cross products we can set up the followingmatrix equation to solve for the principal axis:

r

(Ixx - I) Ixy Ixz

Iyx (Iyy - I) Iyz =0. (7)

LIzx Izy (Izz - I)]

I If we let I = X we can solve the above as follows:

2 2 2(Ixxlyylzz - Izylxx - Izxyy - Iyxlzz + 2lyzlzxlyx)

+ )(IxxIyy + IyyIzz + Ixxlzz - Izy + Izx + Iyx) (8)

S4 X 2(Ixx + Iyy + Izz) - )p = 0 .

The three roots of the above will be the moments ofinertia about the principal axis of the package.

velc ISince for rotation about a principal axis the angularvelocity vector coincides with this axis, the set of numbers

Sx , wy, Wz I, which satisfies the following corresponding to**?' I I 1 of Eq. (8), consists of the direction numbers for axis I1 .

(Ix - ll)W x(1) - IxyO (1) - IxzW (i) 0j Ix y z

(1) (1) (1

-lyxWO(I + (Iyy - Il) 1 )c - IyzWa(I 0(9

-Izx x (1) _Izywy(1) + (Izz -I I) (1)

Solving Eq. (9) will result in the ratios -Wx(1). ( ) : zI and the axis associated with I1 is thereby define3 relative

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILV9R SPRING. MARYLAND

to the original coordinate system. To solve the ratios, let ione direction number equal 1. 0. Repeat this procedure forthe other two roots of Eq. (8) to define the other principalaxes with respect to the original coordinate system.

By using the above analysis and by knowing the mo- -.

ment of inertia of a package fo:: three perpendicular axesas well as three skew axes, the inertia matrix can be com-pleted. Once this inertia matrix is completed, the princi- -

pal moments of inertia of the body can be determined aswell as the orientation of the principal axes with respect tothe coordinat.: axis system useQ for the original measure-mernts.

3

-J

I

U

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVIIR SPNING MARYLAND

3. DEVELOPMENT OF THE MEASURING TECHNIQUE

The design goals for the measurement of the massproperties were as follows:

1. Error in the moment of inertia +0. 5% of the sumof the three inertias about any set of three mutually perpen- fdicular axes;

2. Mass center location determined within +0. 001inch of mounting surfaces and hole patterns; and

3. Mass measurements of all packages to within

±0. 050 gram.

Of all the reqiv.rements, the third was obviously theone that would be the easiest to meet. To determine theinertia matrix and center of mass it was apparent that ex-

jtraordinary precision would have to be obtained.

There were two basic proposals put forth by APL toIobtain the complete inertia matrix:

1. Obtain a large A-frame and suspend a torsionalpendulum at least 6 feet long from it, and

2. Develop a pendulum similar to a trifilar pendu-

lum.

The second proposal was selected for the following

reasons:

1. Package orientation and subsequent dynamic un-balance would have a minimal effect on the system.

2. A preliminary calculation of the error allowedon the smallest package showed it to be on the order of2. 5 x 10-5 slug ft 2 . It was apparent, therefore, that air

I K U-15-

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THEr

JOHNS HOPKINS UNIVERSITY

APPLIED PHYS:CS LABORATORYSILVER SPAI'. MARTIAND

drag on any system used would affect the period measure-ment and should be eliminated if possible. By using thesecond proposal the entire system could be put efficientlyinto a bell jar.

3. Other considerations would be the weight and, -i

therefore, the configuration of the basic system. A bifilarpendulum lends itself to efficient mass distribution. Mostof the mass can be concentrated at the center of oscillation,thereby eliminating large Mr 2 terms.

Figures 5 and 6 show the preliminary bifilar pendu-lum arrangement. Part of the testing on this bifilar pendu-lum concerned the problem of determining the best suspen-sion system wire diameter and material. The figures showthe pendulum supported by two 0. 007 inch BeCu wires. Sub-sequent to the use of the 0.007 inch BeCu wire, music wireand hypodermic needle tubing suspension systems were -tested. The final suspension system chosen was stainlesssteel hypodermic needle tubing with a 0.042 inch 0. D. anda 0.006 inch wall. -,

The basic equation for determining the moment ofinertia on a bifilar pendulum (knowing the period) is

S-T2WD2 (10)

16ff2 L

where I = moment of inertia, T = period, D = distance be-

tween the supporting wires, W = weight on the pendulum,and L = wire length.

Figure 6 shows the-pendulum witha test mass in .place and small pins that could be selectively removed andrelocated to produce a change in its moment of inertia.This test mass proved that a moment of inertia change ofapproximately ±0. 5% of the sum of three perpendicularaxes could be measueed. After it was proven that the bifi- -

lar pendulum would give the accuracy desired, the designof a more sophisticated system was begun.

16 -

p

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THE JOHNS HOPKINS UNIVERSITYA'PPLIED PHYSICS LAE30RATORY

SI.t~i~ SPRINr MANYLAND

I Fig. 5 PROTOTYPE MOMENT OF INERTIA DESIGN

-17-

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APTHE JONS HOPKINS UNIVERSITYAPLIED PHYSICS LAB3ORATORY

SLytR SPING NMrKLANO

L'F

LL.

18J

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVER SPRING MARYLANO

II

I 4. FINAL PENDULUM DESIGN

Based on the analysis and the preliminary experi-Iments, a platform was designed that could rotate to threeperpendicular axes and tihat could also be offset to locate

* I the package into three skew positions. Figures 7 and 8Ishow a package in two of the orthogonal ayis positions, and

Fig. 9 shows a package in one of the skew positions. Acounterweight was mounted beneath the package on the plat-form frame to compensate for the unbalanced momentcaused by the location of the package in its various posi-

I tions.

Inherent in the platform design was the necessity todesign it so that the package, when attached to the platform,could be positioned at its center of mass. In Fig. 7 thearrows show the adjustment methods as indicated below:

I, Arrow Function (see Fig. 7)

1 This shaft was adjusted by a screw onthe bottom that allowed the package tobe adjusted in the vertical direction.

S2 This shaft was adjustable to the right*• or left.

3 This screw adjusted the platform

toward or away from the viewer.

All of the above functions were adjusted in a specially de-signed fixture using precision measuring devices that mea-sured to 0. 0001 inch.

IFigures 10c and 10d show the fixture being usedwith )iscos package No. 5. The dial indicator (used forthe measurement of the vertical motion of the platform) aswel as the height gauge (used to locate the package counter-weight) are shown. Figure 10 also shows the micrometers

1 used for the lateral positioning of the package.

-19-

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THE JOHNS HOPKINS UNIVERSMTAPPLIED PHYSICS LABORATORY

aB.VER SPRING. MARns.AND

zI

0

z

< 0

- 20 -

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY

IL

0

z

0

z0w

D0.

LI

IL

-21-

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILALN SPRtING MAPLANO

z0I--

9, CI)0

0

0

0

IL

-22-

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORY

IL

IL

LU

F-

-23-

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THC JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSI.VER SPRING MARyLANU

All of the adjustments were made after the center ofmass was measured (see Figs. 10a and 10b). The beamused for the center of mass determination was designed sothat the moment of inertia platform was registered in aknown location on the beam. This beam was supported bya 0. 375 inch diameter ball in the center of the weighing panon a precision balance and two 0. 375 inch diameter balls atthe olner end.

The advantage of measuring the center of mass inthis manner was that the package (once installed on theinertia platform) was not removed until all of its mass prop-erties were determined.

To complete the system, the pendulum was sus-pended in a bell jar (Fig. 11), which rested on a collar.The collar had two ports that could be used for viewing,and when required the port cover could be disassembledfor system adjustment without disturbing the bell jar.

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'I'

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYIJILVCR SPRING MA IVLANO

A'A

1 -3

LU

C-,

ad

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVtRf SPRING MARYLAND

II

I5. ELECTRONICS INSTRUMENTATION

It was obvious in the development of the measuringtechnique that the bifilar pendulum was going to have to becarefully instrumented to be successful. Figure 11 showsall of the instruments external to the be!l jar, and Fig. 8is a view that shows the instrumentation inside the bell jar.

A shaft was attached to the bottom of the moment ofinertia platform. It was guided by a nylon bearing approxi-mately 0. 010 inch larger than its 0. D. To start the pendu-lum, a timing motor was actuated that rotated a cam-operated wire that "locked" onto the pendulum shaft. Afterthe pendulum was rotated 200, the wire was abruptly re-leased from the inertia pendulum shaft, allowing the pendu-1 lum to swing freely.

A fan-shaped disk with a small hole in it was locatedI directly above the pendulum shaft. The "fan" rotated be-

tween a light and an N-P-N planar silicon light sensor.When the hole passed over the light sensor, the light illumi-nating the sensor triggered the external electronics.

The following is a list of the external electronicsi and their use:

Instriment Function

' IOscilloscope To determine if the light wascentered over the hole in the"fan. "

Sanborn recorder To check for pendulum damp-

ing.

I Printer To record the period of thependulum.

Counter To check if the average of 10periods was remaining con-stant.

Preceding page blank - 27 -

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVER SPRING MARYLANDI I,

16. TEST RESULTS

I 'I

On 23 November 1971 the final calibration tests andpackage testing began on the bifilar pendulum. During allof the tests a vacuum of 30 inches of mercury was main-

I tained.

To account for any measurement variation owing tothe weight of the package being measured, a calibrationweight was made for each package. Also, to determine ifthe measured moments of inertia were reasonable, a com-puter program was written that calculated the moment ofinertia of homogeneous packages the same size as Discos(see Appendix A). Figure 12 shows the program printout.

I, To do the final calculations a computer program waswritten. This program calculated the test mass momentsof inertia as well as the final package moment of inertia.In this way the program was cont;nually checked for accu-racy by a known mass. Figure 13 is a sample of the pro-gram printout. The printout notation is as follows:

I..P = Final package moment of inertia and cross1 J product,

IL(ij) = Moment of inertia of skew line,m Ii Principal moment of inertia,

i J, 2, 3, and

I IPRO(ij) = Final moment of inertia matrix check.

P!

Preceding[ page blank -2

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THE OHNS HOPKCINS UNIVERSGITY

APPLIED PHYSICS LABORATORYSILVERt S#INGM. MARYLND

.0 0 . -I D

0' -4 .lt AN)

I- 0 A -00 N- *I IA 00.0f0C Ila SIN ,,a'4

.'J MMA- 00

0-r 441 on fnjON O N 0.47

4r 44 4 4

N~ If 'AI

NUN1 N-A NI% "Ift NIA1N!.J NJn.jN.N

a- r-

no no no no n

NS 61 NA C 0

'oS 00 AC ONQ.4Id I AN -

Sf- 00 SN.10 0100

On~~I Ow - 44NIt 04 04 .4f 4 0

44 zIL NL Nu N ube iN In K , 0 on on ft

'--4 01 . tI

'1 )w riJ. Uf y L'A It a nn 4- N0 W3 04 -

NM40A9- 0 -AlA V> t-A4 r v 4 le4. InW IWi

4- 'a ol S 4 n .;O 0 MC 0 fl 0 . C 0 tA u

4-4 3n N'?-1 w

0.00 4 0P 4-c ntu 4 A4 coNN .

00 NO -1Itt *N* 1'00411 s 4A .44

.4.4. i &444 SN AC.. i V14. MN j0 4I 04.1

-30-

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THE JIOHNS HOPK(INS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVER SPRINC4., MARYLAND

0 -Q a III

0 0 0 -1~ly 0 n I -r w

,J W 0 -A 0 I W J ..

~ 0 4 Z1 0 In 00 . i f

II '). N *- 0 ri 0 Nc - - -

V.~ A 0 ) N~I----- ---- ---- ----

-C1 10In In-

a, Cn 0.

0 I I 0.m A CIN N, In In 0. W 5' 5,

0. 'C N 0 0 1 N' C 0 001- :Q! i I0' £ .5 .0 0 0 AS < - ~ C 55 -

C S S' 0 O4 0 5 0 .4 to a, 5- cc

'C 5' C 0 V 0 '.5 0 . 0 N 00'0 A '

In . 000 c0

0~ 'It?" 04.

P' <-C B C O C C 5 , C 5t -5 C 5 C 52 O C C- C 54U 2 t tC tS 5 - 0 C0 '' S '

A.5, C N 0 CS UC~L. , N 'S C C C.

- C C C 5 A''U .SU' US US 5, C C .d

14 ' .5 4 ' 'CS OC CU~ C 5 0 0' *. N'

'C 'S C 'C ' 'C~ 0N''S C C N 'C 5 C

w ww . WNO c 4W I~UPON 0 -% -0 W- I. Ia.n n r 0.

IN . a BC R, 0C InS55K .1% M . 0 J S

10 m 4"Z o 20 z1IL~ 31 -"0 tAI 0 =

WI cJ ni

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ITH JOHNS HOPKINS UNIVERSITY I

SILVZR SPAING. MARYLAND

7. CONCLUDING REMARKS

IOn 11 April 1972 Figs. 14a and 14b were sent to

Stanford University, thereby completing the package massproperty documentation for the Discos electronic packages.

After insertion into orbit, Discos operated success-

fully for 1 year until it was commanded off. It exceeded itsdesign requirements for orbital perturbations of 10-11 g.

IIII

- I

IIi

|Preceding page blank -33..-

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THE JOHNS HMIKINS UNIVERSITYAPPLIED PHYSICS LABORATORY

SILVER SPRING. MARYLAND

0 D

+

(n

0a-C,,U,

(n

ILD

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APPLIED PHYSICS LABORATORYSII.VtRt SPAI14G MARYLAND

BI13LIOGRAPHY

1. D. B. Debra, "Disturbance Compensation SystemDesign, " APL Technical Digest, Vol. 12, No. 2,

April-June 1973.

2. H. Goldstein, Classical Mechanics, Addison-WesleyI Pub. Co., Reading, Mass., 1965.

3. C. M. Harris and C. E. Crede, Shock and VibrationHandbook, VoL. 1: Basic Theory and Measurements,McGraw-Hill Book Co., Inc., N.Y., 1961.

I4. S. W. McCuskey, Introduction to Advanced Dynamics,

Addison-Wesley Pub. Co., Reading, Mass., 1962.

35I

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IU

THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVtR SPft140. MARtYLAND

IIAppendix A

MOMENT OF INERTIA EQUATIONS FOR BODIES THESHAPE OF THE DISCOS PACKAGES

i To determine if the moments of inertia calculated onthe bifilar pendulum were reasonable, the equations for themoment of inertia of the three principal axis of the Discospackage were determined as follows:

I

j RR

4 d

d Ixx

x0

,

x/ R Cos 0

, pI

I Preceding page blank - 37-

Page 33: Bifilar Pendulum Technique for Determining Mass Properties of ...

THE JOHNS HOPKINS UNIVERSITY

APFLIED PHYSICS LABORATORYSILVItR SPRING, MARYLAND

The radius to the element of mass is2 2 21

D=[z2 +2 cos (A-1)

The element of mass is

dm = Pr d~drdz . (A-2)

Therefore, the M of Ixx is

Ixx= p ff D2 r d~drdz , (A-3)

H R

Ixx = J (z 2 + r cos20) r 2dOdrdz , (A-4)

2

and, finally,

HpcxCR - R4I Hp sin2c [R 4- RI4lxx = + 2 1 (A-5)/4 8

H3ai [R 2 R2

+ 2 112

- 38 -

Page 34: Bifilar Pendulum Technique for Determining Mass Properties of ...

APPLIED PH-YSICS LABORATORYSILVE SPRING MARYLAND

Iyy

Ix x '

Iydz

04f dy]

1 The element of mass is

dm =dy dz(b-a) p .(A-6)

'1 The radius to the element is

1 2 2r = y + z 1 (A-7)

ITherefore, the M of Iyy is

Iyy r 2 rdm (A-8)

IyyPf f (b- a)(y2 +z 2)dydz. (A-9)

1 -39-

Page 35: Bifilar Pendulum Technique for Determining Mass Properties of ...

THE JOHNS HOPKINS UNIVER.IMY

APPLIED PHYSICS LABORATORYSILVER SPIRNG MARYLAMP

And, finally,

h0R 4 2hpR 4sin 4 a hoR 4sin 4 PR 2h 3I 4 -16 2 tan O 12

(A-10)

+2R 2h 3sin PR 2h 3 in24 12 tanal

Izzz

Y

x

The element of mass is

dmn rd~drdzp P (A-11)

-40-

Page 36: Bifilar Pendulum Technique for Determining Mass Properties of ...

THE JOHNS HOPK(INS UNIVERSMT

APPLIED PHYSICS LABORATORY

SILVER SPRING MARYLAND

The radius of the element is R; therefore lzz is

H R1 O

2

And, finally,

Izz 20 2 1 H(A -13)L74

The above equations were programmed. and the

mass properties of homogeneous packages of the same

weight as the actual Discos packages were determinled.

*41

Page 37: Bifilar Pendulum Technique for Determining Mass Properties of ...

THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYSILVER S mING MARYLANDI

ACKNOWLEDGMENT

I

The author wishes to acknowledge the technicalassistance of P. Fuechsel, J. Smola, G. Sweitzer, and

W. Radford.

III

I,II

II

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--I