U. AN EXPEKIMtNIAL bIULY OF AXISYMMETRIC MODES IN … · struct simple structural components, basically thin shells of revolution, whose natural frequencies and responses to forced

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(NASA-C ) ~ EXPERIr1ENTAL STUDY OFAXISYMMEZ MODES IN VARIOUS PROPELLANTTANKS CO:fNING LIQUIV Final Report D.D.Kana, et, (Southwest Research Inst.) May1971 63 ,0 (NASACRORTMX UKAuu"Wm,... CSCL 13D CU.

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)AN EXPEKIMtNIAL bIULY OF AXISYMMETRIC

MODES IN VARIOUS PROPELLANT TANKSCONTAINING LIQUID

by

Daniel D. KanaAndrew Nagy .-I

FINAL REPORTContract No. NAS8-30167

Control No. DCN 1-9-53-20030SwRI Project No. 02-2583

Prepared for

National Aeronautics and Space AdministrationGeorge C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama

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https://ntrs.nasa.gov/search.jsp?R=19720007132 2020-03-23T14:40:49+00:00Z

SOUTHWEST RESEARCH INSTITUTE

Post Office Drawer 28510, 8500 Culebra Road

San Antonio, Texas 78228

AN EXPERIMENTAL STUDY OF AXISYMMETRICMODES IN VARIOUS PROPELLANT TANKS

CONTAINING LIQUID

by

Daniel D. Kana

Andrew Nagy

FINAL REPORT

Contract No. NAS8-30167Control No. DCN 1-9-53-20030

SwRI Project No. 02-2583

Prepared for

National Aeronautics and Space Administration

George C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama

May 1971

Approved:

H. Norman Abramson, DirectorDepartment of Mechanical Sciences

TABLE OF CONTENTS

LIST OF ILLUSTRATIONS

INTRODUCTION

DESCRIPTION OF APPARATUS AND PROCEDURES

EXPERIMENTAL RESULTS

ACKNOWLEDGEMENTS

REFERENCES

PRECEDING PAGE BLjKW NOT FILMDEiii

Page

iv

1

2

6

11

12

LIST OF ILLUSTRATIONSPage

1. Cylindrical Tank with Hemispherical Bulkheadon Eight-Column Support 14

2. Spherical Tank Mounted at Upper Half-Radiusin Massive Support Structure 15

3. Wall Thickness Distribution for HemisphericalBulkheads 16

a. Number 1 and 2 Aluminumb. Number 3 Aluminumc. Lucite Plastic

4. Wall Thickness Distribution for EllipsoidalBulkheads 17

a. Number 1 and 2 (Small)b. Number 3 and 4 (Large)

5. Aluminum Hemispherical Bulkheads and Mode Shapes 18

6. Natural Frequencies for Aluminum HemisphericalBulkheads 19

7. Plastic Hemispherical Bulkhead and Mode Shapes 20

8. Natural Frequencies for Plastic HemisphericalBulkhead 21

9. Pressure Profiles for Aluminum HemisphericalBulkhead 22

10. Pressure Profiles for Plastic Hemispherical Bulkhead 23

11. Spherical Tank Supported at Center Flange and ModeShapes for Configuration 1 24

12. Natural Frequencies of Spherical Tank Supported atCenter Flange for Configuration 1 25

13. Spherical Tank Supported at Center Flange and ModeShapes for Configuration 2 26

14. Natural Frequencies of Spherical Tank Supported atCenter Flange for Configuration 2 27

iv

LIST OF ILLUSTRATIONS (Cont'd.),Page

15. Small Ellipsoidal Bulkheads and Mode Shapes 28

16. Natural Frequencies for Small Ellipsoidal Bulkheads 29

17. Pressure Profiles for Small Ellipsoidal Bulkhead 31

18. Small Ellipsoidal Tank and Mode Shapes 32

19. Natural Frequencies for Small Ellipsoidal Tank 33

20. Large Ellipsoidal Tank 34

21. Natural Frequencies for Large Ellipsoidal Tank 35

22. Mode Shapes for Large Ellipsoidal Tank 36

23. Relative Pressure Response for Large Ellipsoidal Tank 37

a. h/d = 0.329b. h/d = 0.50c. h/d = 0.614d. h/d = 0.843

24. Comparison of Natural Frequencies for Large andSmall Ellipsoidal Tanks 39

25. Spherical Tank and Mode Shapes for Support atLower Half-Radius

26. Natural Frequencies for Spherical Tank with Supportat Lower Half-Radius 41

27. Spherical Tank and Mode Shapes for Support at UpperHalf- Radius 42

28. Natural Frequencies for Spherical Tank with Support

at Upper Half-Radius 43

29. Cylinder and Sphere Tandem Configuration 44

30. Natural Frequencies for Cylinder and Sphere TandemConfiguration 45

a. Sphere Liquid Depth, hs/D = 0b. Sphere Liquid Depth, hs/D = 0.5c. Sphere Liquid Depth, hs/D = 0.95

v

LIST OF ILLUSTRATIONS (Cont'd.)Page

31. Cylindrical Tank With Hemispherical Bulkhead -Stiffener Rings Configuration 46

32. Natural Frequencies for Cylindrical Tank WithHemispherical Bulkhead - Stiffener Rings Configuration 47

33. Cylindrical Tank With Hemispherical Bulkhead -Stiffener Rings, Stringers, and Baffles Configuration 48

34. Natural Frequencies for Cylindrical Tank withHemispherical Bulkhead - Stiffener Rings, Stringers,and Baffles Configuration 49

35. Cylindrical Tanks with Inverted Hemispherical Bulkhead 50

36. Natural Frequencies for Cylindrical Tank with InvertedHemispherical Bulkhead 51

37. Cylindrical Tanks with Inverted Ellipsoidal Bulkhead 52

38. Natural Frequencies for Cylindrical Tanks with InvertedEllipsoidal Bulkhead 53

39. Cylinder with Flat Rigid Bottom and Top Support 54

40. Natural Frequencies for Cylinder with Flat RigidBottom and Top Support 55

41. Cylinder with Hemispherical Bulkhead and Top Support 56

42. Natural Frequencies for Hemispherical Bulkhead andTop Support 57

vi

INTRODUCTION

Recent advancements in space technology and the continuedoccurrence of POGO oscillations in launch vehicles make it verydesirable to develop further a more accurate spring-mass modelwhich can be used to analyze the longitudinal dynamics of suchsystems. The limited experiments which have been performed onsome practical shaped tanks verify that much useful informationcan be obtained by studying simplified models. A basic model forsuch representation was developed by Glaser . To determine thevalidity of this model, it became desirable to provide experimentaldata on simple, simulated vehicle structural systems for comparison.Some of the results from these experiments were presented in the

Interim Report 2 of the present contract. The data provided in thisReport are the results obtained from additional configurations whichcomplement the experimental data reported in Reference 2.

* Superscript numbers refer to References at the end of text.

2

DESCRIPTION OF APPARATUS AND PROCEDURES

The experimental procedure utilized in this phase is similar tothat used in the first phase of the program. It was desirable to con-struct simple structural components, basically thin shells of revolution,whose natural frequencies and responses to forced vibration may becompared with results obtained by using theoretical models. Figures1 and 2 show two of the models used for the current work and are a wayto show the difference between the two supporting systems used in thetests.

Figure 1 shows a cylindrical tank, having a hemispherical bulk-head,supported on columns. This was a standard approach for support-ing cylindrical tanks with bulkheads, other than flat-rigid, and also forsupporting spherical or ellipsoidal tanks at their center flanges. Thetank shown in Figure 1 is a thin-walled cylinder which was describedin detail in the Interim Report2 and was designated throughout thatreport as the "Lower Tank". With the exception that, in the presentcase, the bulkhead is an aluminum hemispherical bulkhead rather thanellipsoidal bulkhead, the model is identical in every respect to thatshown in Figure 32 in the Interim Report2 .

This supporting technique worked satisfactorily for all configu-rations where tanks with their coupling flanges were mounted on thecolumns directly, as shown in Figure 1. Despite the fact that the flangeshad some effect on the systems, since they were not infinitely stiff aswould be desirable, their influences were readily identified during thetests. For example, a problem arose when this column-type systemwas attempted to be used for supporting spherical tanks at half radiuson their lower or upper halves. A 3/4-inch-thick steel ring was in-stalled on top of the columns and was intended to be used for supportingthe spherical tank. When the system was excited, it was found that,even though the input was controlled by the acceleration at one point onthe mounting ring, the supporting system was not sufficiently rigid forthe intended purpose. Bending responses occurred in both the mountingring and the columns, and the affect of these responses became sostrongly coupled with the response of the tank that it was impossible toidentify the natural longitudinal modes. As a result, the massive sup-porting structure used for these configurations was fabricated and isshown in Figure 2.

The structure consists of a cast aluminum, heavy wall cylinderand two end plates which are bolted to this cylinder. Coupling to theelectrodynamic shaker was accomplished by the lower end plate, while

3

the upper ring provided the means of supporting the models to be tested.Figure 2 shows the aluminum spherical tank supported at its upper halfradius inside of the massive support structure. Windows cut on thesupporting cylinder provide access for obtaining bulkhead mode shapeson the models being tested. This massive support structure has dimen-sions such that it is capable of supporting the "Upper Tank" 2 at its top.This is shown later in Figures 39 and 41.

Model components, namely the Upper and Lower Tanks, Skirt,Plastic Hemispherical Bulkhead, Ellipsoidal Bulkheads, and Flat-RigidBulkheads were described in detail in the Interim Report 2 and will not,therefore, be discussed again here.

In addition to the above-mentioned components, three aluminumhemispherical bulkheads were also used in the program. They werefabricated from 6061-0 aluminum sheets by a spinning process, as werethe ellipsoidal bulkheads. Flanges, identical to those on the, smallellipsoidal bulkheads2, were provided during the spinning process toassure interchangeability. One of these hemispherical bulkheads,designated as No. 3, was then modified by the addition of an extraflange at its mid-radius. This modification was accomplished bymachining a 1/4-inch-thick aluminum ring to the contour of the bulkheadat the desired location and electron-beam welding it into position. Thefinal step in the fabricating process, was subjecting all three bulkheadsto heat treatment to obtain a minimum yield strength of 30, 000 psi.

The spinning process for fabricating the above-described bulk-heads, in essence, consists of stretching the sheet metal to conform toa pre-machined shape which, in this particular instance, is a hemisphere.This process provides a relatively uniform thickness around a givenlatitude circle, but an unfortunately significant variation in thicknessfrom the outer circle to the bottom center (i. e., along a meridian).The question then arises as to what thickness should be used when theo-retical calculations of natural frequencies are made for such bulkheads.An equal-weight average of thickness is the simplest answer. However,some additional consideration should be made before a final judgementcan be effected on this matter. In this regard, it was called to ourattention that Reference 3 presents a membrane analysis of a sphericalshell with liquid. Results are given as a plot of the nondimensionalnumber

4

In this parameter, p is the density of the contained liquid, w is thenatural frequency of the system, R and t are the radius and thick-ness, respectively, and E is the modulus of elasticity for the materialof the shell. For a depth of r = h/2R = 0. 5, which is the equivalentof a hemispherical shell filled with fluid, the lowest value of the non-dimensional number 2 is given as 1. 20. This is the theoretical valueobtained for the first natural longitudinal mode and should prove to beclose to that procurable by experiments.

Use of the above-given definition for Q2, with values applicableto the present aluminum models of the hemispherical shells, resultedat first in a number equal to 0. 966, which is considerably lower thanthat given above. This value was obtained by using an average wallthickness of 0. 027 inch for the bulkhead in question. Upon furtherreflection, it was realized that all parameters in the expression for 2were well defined with the exception of the wall thickness, which variesalong the meridian of the shell. By substituting the smallest valuemeasured for the wall thickness, a larger value of the frequency para-meter, namely S2 = 1. 10 was obtained, which is in closer agreementwith the referenced article. These results indicate that proper choiceof an effective wall thickness is very essential. Therefore, to assistin the selection of an appropriate average wall thickness for use intheoretical calculations, a graph of the measured wall thickness vari-ation for each hemispherical and ellipsoidal bulkhead is presented inFigures 3 and 4, respectively. Some suitable scheme for developingan effective average thickness remains to be determined. However,from the physical nature of the geometry, it is believed that the thick-ness near the flange probably has the most influence on symmetricvibrational modes, and a straightforward use of that thickness there-fore would be appropriate. As a result, the above-given value of

= 1.10 is probably a reasonable value for the case described.

Two sizes of ellipsoidal bulkheads were utilized in the tests asindicated by the graphs in Figure 4. The small bulkheads were espe-cially fabricated for use in this program and were designed to fit withthe cylindrical components described in the Interim Report2 . The largebulkheads were made for a previous program; however, since theywere a scaled-up version of the small shells, it seemed useful to alsoutilize these bulkheads to obtain natural frequency data for comparisonbetween the two sizes.

Liquid propellant was simulated by using distilled water in allmodels, as was done in the first part of the program. A 12. 0-inch-diameter, 0. 125-inch-thick aluminum cover plate formed a closure forthe hemispherical and semi-ellipsoidal tanks. It will be shown later

5

that this plate also added rigidity to the support. Ullage pressure wasthen provided to preclude the possibility of buckling of the invertedbulkheads or the cylinder supporting the liquid-filled spherical tank and,in all cases, to increase the natural frequencies of nonsymmetric modesabove the frequency range of interest. As was noted in the InterimReport2 , ullage pressure had only a negligible effect on the symmetricmodes.

Excitation of the models was accomplished with the electro-dynamic shaker which was used throughout the preceding phase, and theinput was controlled by the accelerometer which is denoted as A1 onthe schematics supplied in this report. These schematics also indicatethe additional accelerometers and pressure transducers, where appro-priate, and their respective locations on the structures. Amplitudesand relative phase angles between these transducers were employedfor determining the natural frequencies of the structures.

6

EXPERIMENTAL RESULTS

During the first year, attention was concentrated on severalmodel configurations which consisted of one or more cylindrical shellsof revolution. These models were outfitted with flat rigid or elasticbulkheads and, in all cases, were supported at their lower-most flanges.Experimental data obtained with these models are presented inReference 2. During the current work, the following additional con-figurations were tested:

a) Hemispherical and spherical tanks supported atvarious points on the tanks

b) Semi-ellipsoidal and ellipsoidal tanks supported attheir flanges (which is at the major diameter on theellipsoidal tank)

c) Cylindrical and spherical tanks in tandem configuration

d) Cylindrical tanks with hemispherical and invertedhemispherical bulkheads supported at their loweststructural points

e) A cylindrical tank with inverted ellipsoidal.bulkheadsupported at its lowest point

f) A cylindrical tank with flat rigid and elastic bulkheadsupported at its top.

Experimental results obtained on these models, as in the pre-ceding Interim Report 2 , are presented by plotting the natural frequenciesagainst liquid depth. The plotted data for each configuration are accom-panied by a schematic depicting the model in question which is suppliedto facilitate visualization of the model structure for which these dataare presented. Geometrical and material property data essential fortheoretical calculations are included with the schematics in a tabulatedform. In most configurations for which the elastic bulkheads were

accessible, mode shapes were noted for each natural frequency and arepresented in a sectional line drawing form along with the schematics.They are identified by corresponding mode number on the frequency data.

It should be mentioned here that, as before, results are pre-sented only for those modes which were identified to be axisymmetricabout the longitudinal axis. Some bending modes which were excited

7

through eccentricities in the structures were identified in the courseof the testing program, but were omitted, since these were not primarylinear responses to longitudinal excitation.

Data resulting from experiments conducted on two of the aluminumhemispherical tanks, made from Bulkheads No. 1 and No. 2, are shownin Figures 5 and 6. Natural frequencies for these bulkheads are pre-sented on a common plot to demonstrate their identical responses undersimilar test conditions. Corresponding bulkhead mode shapes for thesefrequencies are shown in Figure 5. When counting the flange for one nodalcircle and neglecting the nodal circle appearing very close to the flangeon the first mode, the number of the nodal circles corresponds to themode shape numbers. The second nodal circle observed for the firstmode probably resulted from bending at the flange and coupling betweenthe shell and the supporting structure. This same behavior was demon-strated by the plastic hemispherical bulkhead for which the experimentalresults are shown in Figures 7 and 8. Comparison of the data for thealuminum and plastic bulkheads verifies the fact that the elastic propertiesof the bulkhead will affect the natural frequencies, lowering them in thecase of the plastic bulkhead. Pressure profiles along the center linesand meridians of the two aluminum and one plastic hemispherical bulk-heads are shown in Figures 9 and 10. It should be pointed out that inorder to introduce a pressure transducer into the liquid, the cover plateused during the acquisition of natural frequency data had to be removed.The removal of this plate softened the support systems to some degree,which is indicated by the lower natural frequencies shown in Figures 9and 10. This effect was quite pronounced for the first mode on thealuminum bulkhead. It must be emphasized again that this was a resultof a softened boundary condition, rather than any effects from changedullage pressure.

For the next step in the program, the No. 1 and No. 2 aluminumhemispherical bulkheads were joined to form a spherical tank supportedat its center flange. Two configurations were constructed: Configuration 1,in which bulkhead No. 1 formed the bottom, and Configuration 2, in whichbulkhead No. 2 formed the bottom. Natural frequency data along withbulkhead mode shapes for these two conditions are shown in Figures 11-14.General mode shapes for the two configurations are identical, while thenatural frequencies deviate from each other to a small degree and only atthe liquid levels approaching full condition. If the stiffening effect of theupper half of the tank and the cover plate for the bulkhead were equal(i. e., if the flange boundary conditions were essentially rigid in both cases),then data for the spherical tank at the liquid depth ratio h/D = 0. 5 shouldbe identical to that for the hemispherical tank filled with liquid. Com-parison of the natural frequencies for those two conditions are in closeagreement and seem to support the preceding statement.

8

Results for the small ellipsoidal bulkheads are shown inFigures 15-17, and the natural frequencies for the two bulkheads areagain presented on a common frequency versus liquid depth plot. Thefigure clearly exhibits a difference in natural frequencies for corre-sponding modes in the two different bulkheads. This frequency differ-ence is the direct result of the thickness difference between the bulk-heads as indicated by Figure 4a. The thickness variation is particu-larly apparent in the near vicinity of the supporting flange and the wallthickness is considerably lower at this location for Bulkhead No. 1.This suggests a decrease in stiffness resulting in a lower frequencylevel and is in agreement with Figure 16. Pressure profile data pre-sented in Figure 17 are similar to the general mode shapes of thehemispherical bulkheads. The only exception is the second mode alongthe center line shown in Figure 17a. Examination of Figures 9a, 10a,and 17a seems to indicate that the pressure distribution along the centerline is strongly influenced by both the elasticity and geometry of thebulkhead, particularly for higher modes.

Ellipsoidal bulkheads No. 1 and No. 2 were joined to form anellipsoidal tank for which data are presented in Figures 18 and 19. Again,as it was applied to the spherical tank, a comparison may be drawn be-tween the ellipsoidal bulkhead filled with liquid and the ellipsoidal tankat a liquid depth ratio of h/d = 0. 5. Frequencies for these conditionsare also in good agreement.

Examination of the ellipsoidal tank mode shapes may raise aquestion concerning the validity of the first and second modes, sincethey seem to indicate identical conditions. The difference resulted fromthe fact that while the first mode appeared to either lead or lag the con-trol accelerometer at resonance, the second mode consistently main-tained the same phase relationship when resonance was reached.

Results for the large ellipsoidal tank supported at its centerflange are shown in Figures 20-23. Natural frequencies of this tankwere considerably lower than those for the small tank and became verydifficult to identify above 600 Hz when liquid depth in the tank surpassedthe mid-level. To help identify the natural frequencies, transfer func-tions indicating relative pressure response were run for various liquiddepths. Representative figures for liquid depth ratios h/d = 0. 329,0.5, 0.614, and 0. 843 are presented in Figure 23. These figures alsovery vividly show the change in the amplification ratio with increasingliquid depth in the tank, along with the presence of split modes whichmost likely would not be predicted by a linear vibration theory.

9

Figure 24 shows some final data for both the large and smallellipsoidal tanks. The natural frequency parameter described earlierfor hemispherical tanks is applied to the ellipsoidal tank data for thefirst two modes. The correlation appears to be quite good when thethickness is taken as the minimum measured value which occurs nearthe flanges. These results again seem to indicate that this minimumvalue is the significant thickness of the tanks.

In the next two configurations, for which data are shown inFigures 25-28, the spherical tank is supported first at its half-radiuson the lower hemisphere, and then at its half-radius on its upper hemi-sphere. Natural frequencies for the latter case are appreciably lowerthan for the sphere supported on its lower half, which emphasizes thedifference in effective stiffness between the two conditions.

The next model is somewhat analogous to the lower tank withflat rigid bulkhead and top mass shown in Figures 3 and 5 in theInterim Report 2 . The difference between these models and the cylinderand sphere tandem configuration shown in Figure 29 of this Report isthat, in the present case, the top mass is replaced in part by an effec-tive liquid mass which is frequency dependent. Results for this modelare presented in Figures 30a,b, and c, which give the natural frequenciesof the structure for varying liquid depths in the cylinder and for fixed liquiddepth ratios hs/D = 0, 0. 5, and 0. 95, respectively, in the sphere. Thestructural mode appearing at 515 Hz for the empty condition of the sphereis consecutively lowered with increasing liquid depth in the sphere. Thismode was identified as the first structural mode for the cylinder. Basedon the available data, the structural mode shown at 450 Hz in Figure 30cwas judged to be the same mode which is located at 970 Hz for the half-full sphere in Figure 30b; however, it was not strong enough to be ob-served for the complete liquid level range in the cylinder when the spherewas empty. It should also be pointed out that above 800 Hz the naturalmodes became increasingly weaker and highly coupled for the case ofh/D = 0. 95 liquid level in the sphere, making identification very difficult.

Results for the cylindrical tank with aluminum hemisphericalbulkhead and stiffener rings are given in Figures 31 and 32. Two externalmodes, namely column and support flange resonances, were detected andare indicated in Figure 32. These modes coupled with the liquid modesof the cylinder, thereby causing some distortion. The expected fre-quencies of the liquid modes in the absence of these coupling effects arealso indicated.

10

Following this configuration is the model just mentioned withstringers and baffles added onto the cylinder for which the results areshown in Figures 33 and 34. Natural frequencies obtained for thebulkhead alone (data from Figure 6) are indicated on the frequency plotfor the model in question. They show very close agreement with thefrequencies between liquid levels h/1 = -0. 379 and 0 in the cylinder.A comparison could also be attempted between Figures 46 and 47 inthe Interim Report 2 and the figures just discussed, to thereby showthe combined effect of the rings, stringers, and baffles and a lowerbulkhead elasticity on the natural frequencies of the structure.

Data for cylindrical tank with inverted hemispherical andellipsoidal bulkheads are presented in Figures 35-38. These modelsshow an effect of higher structural stiffness, resulting from the geo-metrical shape of the bulkheads. That is, corresponding liquid modesare higher in frequency for the ellipsoidal configuration. Also, notethat the structural mode occurs near 325 Hz for both cases.

Finally, data for the cylindrical tank, designated in the InterimReport2 as the upper tank, with flat rigid and hemispherical bulkheadsand supported at its top, are presented in Figures 39-42. These datashow not only the effect of bulkhead elasticity between the two condi-tions, but also the change in system response when the structure issupported and excited at its new location as compared to Figures 30and 31. When supported at its top, the model exhibited a decrease instructural stiffness resulting in lower frequencies, particularly forlower order modes.

It should be emphasized that, for the experimental resultsthroughout this report, curves through the data points have been drawnby best judgement, based on experimental observations, and may re-quire some alterations when compared with results obtained throughthe use of the theoretical models.

ACKNOWLEDGEMENTS

The authors are most grateful to Messrs. George W.

Downey, Jr. and Dennis C. Scheidt for conducting the experi-

mental tests, and to Mr. Victoriano J. Hernandez for prepar-

ing the illustrations.

12

REFERENCES

1. Glaser, R. F., "Longitudinal Mass-Spring Modeling of LaunchVehicles," NASA TN D-537, Washington, D. C., August 1969.

2. Kana, D. D., and Nagy, A., "An Experimental Determinationof the Longitudinal Modes of a Simulated Launch VehicleDynamic Model, " Interim Report, Contract No. DCN 1-9-53-20030,Southwest Research Institute, March 1970.

3. Balabukh, L. I., and Molchanov, A. G., "Axisymmetric Oscilla-tions of a Spherical Shell Partially Filled with Fluid, "Inzhenernyi Zhurnal-Mekhanika Tverdogo Tela, Sept. -Oct.,1967, pp. 56-61.

APPENDIX

Illustrations

13

Figure 1. Cylindrical Tank with Hemispherical Bulkhead on Eight-Column Support

14

NOT REPRODUCIBLE

Figure 2. Spherical Tank Mounted at Upper Half-Radius in Massive Support Structure

15

0.022

O. OZ4 \1 =2

0.030 2

( i Number I and 2 Aluminum

WALL THICKNESS ( in.) WALL THICKNESS ( in. 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030 0.040 0.050 0.060 0.07

PRESSUREI~ ~ ~ ~ .0 PRESSURE CELL

MOUNTL~VMMJON

45 0.060

0.0600

I b) Number 3 Aluminum Ic ) Lucite Plastic

Figure 3. Wall Thickness Distribution for Hemispherical Bulkheads

16

70

WALL THICKNESS ( in. )0.020 0.024

BULKHEAD NO. 1 --- O---BULKHEAD NO. 2 0--

3328

(a) Number 1 and 2 (Small )

WALL THICKNESS ( in. )0.032 0.036 0.040 FLANGE

BULKHEAD NO. 3

BULKHEAD NO. 4 --- 0---

3329

-PRESSURE CELL ( b ) Number 3 and 4 ( Large)MOUNT

Figure 4. Wall Thickness Distribution for Ellipsoidal Bulkheads

17

C

, 0.020vA

· 0.024-j

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. 0.032

v)

z 0.036

I-

0.3: 0.040

SUPPORTCOLUMN (8)<

HEM I SPHERICAL-BULKHEAD

/P

1

3

;M

2

4

7

2 LIQUIDDEPTH " h "

3330

Figure 5. Aluminum Hemispherical Bulkheads and -Mode Shapes

18

Structu ra I Equation Inside Wall Material EElement of Dia. Thickness Density x 10l

Surface (in. ) (in.) ( #/in ) psi

Bulkhead 2 +y2 = 52I D 10. 0 Fig. 3 0.098 10

A,--

e ^ r~~

Z< 1200

1000

ULLAGE PRESSURE = 3 psig-- BULKHEAD NO. 1

---- BULKHEAD NO. 2600 MODE SHAPE 1 0

203 4

400 l0 0.1 0.2 0.3 0.4 0.5

LIQUID DEPTH, h/D3331

Figure 6. Natural Frequencies for Aluminum Hemispherical Bulkheads

19

Al- I

SUPPORTCOLUMN ( 8 )

HEMISPHERICALBULKHEAD

P

D/2

DEPTH "h"

1

3

3332

Figure 7. Plastic Hemispherical Bulkhead and Mode Shapes

zo

ra Equation Inside Wall Material EStructurl of Dia. Thickness Density x 106

Surface ( in. ) ( in. ) ( #in? ) psi

Bulkhead x2+y2=4.932 D=9.86 Fig. 3 0.043 0.4

D4\_ &)04w)A= ---r7Jv7f f

I .////////x////////,d

X ,

//

\

ULLAGE PRESSUREMODE SHAPE 1

234

l l- ~0.1 0.2

LIQUID

= 3 psigo

0

0.3DEPTH, h/D

hp-_

0.4

Figure 8. Natural Frequencies for Plastic Hemispherical Bulkhead

21

1600

140a

1200

1000N

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U-

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C

600

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z Z0.8 0.8

0.9 ' 1 0.9 0

i.0 1.0-1.0 -0.4 0 0.4 1.0 -1.0 -0.4

NORMALIZED PRESSURE, Px/P NORMALIZED F(a ) Along Centerline 333s4 ( b ) Alon

Figure 9. Pressure Profiles for Aluminum Hemispherical Bulkhead

0 0.4 1.0PRESSURE, Px/ Pig Meridian 3335

22

0..

U

I!

0.3

0.4

0.5 __

0.6

0.7-

0.8

0.9

IL

!U.0\

-0.4 0 0.4NORMALIZED PRESSURE, Px/P

(a ) Along Centerline

-- 1 0 IULLAGE PRESSURE - 0 psig

TOTAL LIQUID DEPTH - 4.93 in.0 347 Hza 464 Hz

0.

LIJ

"

0.:

Q 0..

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z 0.'

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(b) Along Meridian

Figure 10. Pressure Prdiles Plastic Hemispherical Bulkhead

23

CA

L:

-

-

N

1.0

3337

1 I I J jn2 %

0

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HEMISPHERICAL-BULKHEAD NO. 2

A,-

SUPPORTCOLUMN (8)-


I

I 4 l

3338

Figure 11. Spherical Tank Supported at Center Flange andMode Shapes for Configu ration 1

24

LIQUIDDEPTH " h "

Structu ral Equation Inside Wall Material Eof Dia. Thickness Density x 106

Surface (in.) ( in.) ( #/in? ) psi

Bulkheads x2+y2 52 D 10.0 Fig. 3 0.098 10

2000

1800

1600

1400

1200

1000

800

600

ULLAGE PR

MODE S400

200

0 0.2

Figure 12. Natural

0.4 0.6 0.8 1.0LIQUID DEPTH, h/D 3339

Frequencies of Spherical Tank Supported atCenter Flange for Configu ration 1

25

Nr-

cr)

z

wLA-

-- ~ ~ -

HEMISPHERICALBULKHEAD NO. 1

SUPPORTCOLUMN ( 8 )


j,

`,P

D

/ _

/~~~~

2 , '

__

I 4

3340

Figure 13. Spherical Tank Supported at Center Flange andMode Shapes for Configuration 2

26

I LIQUIDDEPTH "h"

Structural Equation Inside Wall Material EElements of Dia. Thickness Density x 106

Surface ( in. ) ( in. ) ( #in ) psi

Bulkheads x2+y - 52D -ID10.0I Fig. 3 0.098 10

RPA I FI+AA/Jlrz41111 I

II///////////////// /

/r

N

C,

CYLa

L-

Figure 14. Natural

0.4 0.6 0.8 1.0LIQUID DEPTH, h/D 3341

Frequencies of Spherical Tank Supported atCenter Flange for Configuration 2

27

- I

A,-- ,

SUPPORTCOLUMN ( 8) -

ELLIPSOIDALALRBUL KHFAD

D -

r i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~d1d/2

ILIQUIDDEPTH " h "

1

4 5

3342

Figure 15. Small Ellipsoidal Bulkheads and Mode Shapes

28

2~''_ ~P

Structu rail Equation Inside Wall Material Eof Dia. Thickness Density x 106

Surface (in. ) ( in.) ( #/in? ) psi

Bulkheads 1= Fig. 4 0.098 101 5 ~~d - 7.07 I

FF I 1 , + , 1 I/ 7 1 7 7 7 7 7 7 7I J l 1 !F

////I//////.////////////i

'j-P/

I."w,,,.

2200

z 1200

Li

LL

1000

800

ULLAGE PRESSURE = 3 psig---- BULKHEAD NO. 1

600 --- BULKHEAD NO. 2

MODE SHAPE 1 o20

400 - 34A5 O

200 I I I0 0.1 0.2 0.3 0.4 0.5

LIQUID DEPTH, hid 3343

Figure 16. Natural Frequencies for Small Ellipsoidal Bulkheads

29

3

4

5

6

7

9

1.-1.0I:

k


(a) Along Centerline

I I I V I IULLAGE PRESSURE 0 psig

TOTAL LIQUID DEPTH * 3.53 in.O 695 Hza 1175 Hz

LI;

r-

0

U-

oz

I-

-J

0

n u.e U.'

0.3 -

0.4

0.5 -

0.6

0.7 -

0.8

0.9

1.0 I II1.0 -1.0

3344

kw

-0.4 0 0.4NORMALIZED PRESSURE, PxIP

( b ) Along Meridian

Figure 17. Pressure Profiles for Small Ellipsoidal Bulkhead

'P!,i' iB~i x IK OT ?azTN

31

0

0.1

0..-

x

LI.

W.

z 0.,

X-

0

L0.

N 0.iI.E

0.7

1.0

3345

I

I -

r

&

I%

7. 7

ELLIPSOIDALBULKHEAD NO. 1

A,

SUPPORTCOLUMN (8)-

ELLI PSOI DALBULKHEAD NO. 2

d

ILIQUIDDEPTH " h "

5

\ 6/

334_ \ -

3346

Small Ellipsoidal Tank and Mode Shapes

32

/L - D -

Equation Inside Wall Material EI of Dia. Thickness Density x 106

Element Surface ( in. ) ( in. ) ( #1in3 ) psi

Bulkheads x2 I D-10. Fig. 4 0.098 1052 d = 7. 07

Figu re 18.

.. o. in. O ......mPLx

~%%%%%%%%%%%%%%%%%%%\\\\\\\

0.2 0.4 0.6 0.8 1.0LIQUID DEPTH, h/d 3347

Figure 19. Natural Frequencies for Small Ellipsoidal Tank

2000

1800

N

LaI

LL

'"r

>.~zlwj

1600

1400

1200

1000

800

600

400

200 -

OO0

ALUMINUM FLANGE

LUCITE CAP

ALUMINUM RIM

ELLI PSOI DALBULKHEAD NO. 4

A,

ALUMINUM SUPPORTTAN K ( 0.060" WALL )


LIQUIDDEPTH " h"

SYMBOLS USED FOR INDICATINGMODE SHAPES ON LARGE ELLIPSOI

GENERALDAL TANK

3348Figure 20. Large Ellipsoidal Tank

34

Equation Inside Wall Material EcElement uof Dia. Thickness Density x 10

Surface (in. ) (in. ) ( #lin?) psi

Bu lkheads - D 24.71 Fig. 4 0.098 1012.3752 d = 17.5

MODE SYMBOL

1 ·2 o3 04 ·5 O67 8 Z9 &

MODE SYMBOL

10 o1112 a13 A14 a15 O16 O17 018 _

MODE|SYMBOL

19 ·20 A21 O22. Q23 ·24 025 I26 a27 ·

1400

1300

1200

1100

U

wi1

35

i0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0I0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

LIQUID DEPTH, h/d 3s4,

Figure 21. Natural Frequencies for Large Ellipsoidal Tank

26/~~ ~ "

1~~~~~~~

/~~I

0 '~~~~~~~~~~~~~~~~~~~~~'i· "~"

1F5 6 7 Sae 8

I~~~~~~~~~~~~~~

9 10 11 12

13 14 1 L t 16

-- '- N ~ /--·

I ~ ~~~~~ / /

17 "4 18 19

I' N~~~~~~~~~~

N~ I --

I- N -~~~

20 21 22

\n~~~~~~~~~~~~~~~ I-- -

A~~ /

23 24 25

-- r --- -' ~-. -- , -' N ' - -; -. -

26 27

Figure 22. Mode Shapes for Large Ellipsoidal Tank

36

ULLAGE PRESSURE · 3 psig

100

FREQUENCY, Hz(a) hld-0.329

1000

ULLAGE PRESSURE - 3 psig

100

0.

l - ' 'l

0 200 400

Figure 23.

600 800FREQUENCY, Hz(c) hld-0.614

1200

1000

ULLAGE PRESSURE 3 psig

100

5 IC

00 2 00 600 800 1000 1200 1400

.ia:

*5

I

w

14003353

FREQUENCY, Hz(b) hd -0.50

3352


100

10 - Ill .

I0.1 D {II I

' 0 200 400 600 800 IO) 1200 1400FREQUENCY, Hz(d) hd -0.843

0

Relative Pressure Response for Large Ellipsoidal. Tank

37

o..E'-FS

w

t-

0.

!5"q

-i

V

A

V

mI

mm

.9 InFr)

CD

yr~

~~

~~

~~

~~

~~

~~

~~

0C

D~~~~~~~~~~~~~~~~~~~~~~~~.-

C

C L

-

II

# o.

Ln

0~

C

A

(

m~~~~

oo ~

~~

~

~~I I

Cu

_'

_ E

IJ_

_

CY

dC

o~

~~

~~

~~

~~

~~

t'4

-

, -

U

Cu~

~~

~~

~~

C

_j ~

.. c

Er"

'

Cl

-1

1.1

~~

~~

~~

~~

~

*6

0 C

D

Cn

CD

I~C

lJ 0

' '

0 o

G-P

A~

~~

~~

~~

~~

~-

39~ ~~

~~

~~

~~

~~

V

tI ,,

o

(~ 0~~~~~

L.-0 ' M

t33

WV

l~d

XD

Nn

b3

l_-

/ /~

(A

0~

~~

~~

~L

-U

-

0~

~~

~~

~'I-

~ ~ ~

~ v

CO

0

\ 00

0

N

~CUJ C

.J -

-

~3'tI]13WLA

VI~d A3N

~flO]lU

HEMI SPHERICALBULKHEAD NO. 2-

A -

HEMISPHERICALBULKHEAD NO. 3-

SUPPORTSTRUCTU RE

4

'- -

Figu re 25.

D

LIQUIDDEPTH"h"

5I

3356

Spherical Tank and Mode Shapes for Support at Lower Half-Radius

40

Equation Inside Wall Material Eof Dia. Thickness Density x 106

Element Surface (in. ) (in. ) ( #in?) psi

Bulkheadsx2 +y2, 52 D 10.0 Fig. 3 0.098 10

2000

1800

1600

1400

1200

Z 1000

800

600ULLAGE PRESSURE 3 psig

MODE SHAPE 1 020

3 050

200

0 0.2 0.4 0.6 0.8 1.0LIQUID DEPTH, h/D 3357

Figure 26. Natural Frequencies for Spherical Tank with Supportat Lower Half-Radius

41


Al

HEMI SPHERICALBULKHEAD NO. 2

SUPPORTSTRUCTURE

3358

Figure 27. Spherical Tank and Mode Shapes for Support at Upper Half-Radius

42

StructuraI Equation Inside Wall Material Eof Dia. Thickness Density x 106

Element jSurface ( in. ) (in.) ( #/in?) psi

Bulkheads I2 2 +y2 = 52 1 D = 10.01 Fig. 3 0.098 10

2000

r 1000

800

0

400

ULLAGE PRESSURE = 3 psigMODE SHAPE 1 0

200- 2 r

3 ©

00 0.2 0.4 0.6 0.8 1.0

LIQUID DEPTH, h/D 3359

Figure 28. Natural Frequencies for Spherical Tank with Supportat Upper Half-Radius

43


A3


SKIRT -

A2

TAN K

FLAT RIGID -

BULKHEAD

II

-I[

D

P2

P

M////

hs

"/A 3.72#

1LIQUIDDEPTH "h"

I I

3360

Figure 29. Cylinder and Sphere Tandem Configuration

44

Effective Inside Wall Material ES cElement u Length (A ) Dia. Thickness Density x 106

( in.) ( in.) (in.) (#in?) psi

Tank 14.5 10.0 0. 005 0. 29 29S ki rt 7.5 10.3 0. 025 0. 098 10

Bulkheads D - 10.0 Fig. 3 0.098 10

I/ ///, //

1�

0

k

-i i ',44.52#

A,

0 0.2 0.4 0.6 0.8LIQUID DEPTH, h/A

( a ) Sphere Liquid Depth hs / D - 0

1400

1200

1000ooo

N

C4

,z,LUOaLUU-

800

600

0.2 0.4 0.6 0.8 1.0LIQUID DEPTH, h/l 3362

(b) Sphere Liquid Depth hs/D - 0.5


I \·s.L _, I I/ D IQ o

A m~~~C n i

0 0.2 0.4 0.6 0.8 1.0LIQUID DEPTH, hil 3363

(c ) Sphere Liquid Depth hs/D - 0.95

Figure 30. Natural Frequencies for Cylinder and Sphere

Tandem Configuration

45

I I IULLAGE PRESSURE - 3 psig

(lo

600 o oi

1200

400 __ IL

800

1400

N

I.

1.03561

1400

N

k

Cy

r- X S

cfC~~~~k ~> C>0-0

D OC m~11

--

P-0-0�.W -<>

uru-"11 -

400

200

0

A2-

STI FFENERRING ( 11 )

TANK

A,

SUPPORTCOLUMN (8)


'"' 20.4#

A

LIQUIDE DEPTH"h"

5.35#

P

3364

Figure 31. Cylindrical Tank with Hemispherical Bulkhead-Stiffener Rings Configuration

46

Effective Inside Wall Material EStructura I Length ( ) Dia. Thickness Density x 106

Element (in.) (in.) (in.) (#lin? ) psi

Tank 14.5 10.0 0.005 0. 29 29

Bulkhead D - 10.0 Fig. 3 0.098 10

Number Material E Dimensions SpacingStutu Used On Density x 106 (in.) (in.)

Tank (#lin.3) psi

Stiffener 10.0 I.D. SymmetricalRing 11 0.098 10 10.5 O.D. about 1. 24

0.032 Thick midspan

T;

Support 1600Flange Resonance

SupportColumn Resonance@ 1224 Hz

1200

0.2 0.4 0.6 0.8 1.0LIQUID DEPTH, h/A 3365

Figure 32. Natural Frequencies for Cylindrical Tank with HemisphericalBulkhead-Stiffener Rings Configu ration

47

STI FFENERRING ( 11 )

STRINGER ( 12 )

BAFFLE ( 11 )

Al

SUPPORTCOLUMN (8)


LIQUIDDEPTH "h"

5.35#

Structural Effective Inside Wall Material EElement Length ( ) Dia. Thickness Density x 106

(in.) I in.) (in.) ( #/lin ) psi

Tank 14.5 10.0 0.005 0.29 29

Bulkhead D=-10.0 Fig. 3 0.098 10

Structural Number Material E Dimensions Location SpacingUsed On Density x 106 (in.) (in.)

Tank (#lin?) psi

Stiffener 10.0 I.D. SymmetricalRing 11 0.098 10 10.5 0.D. about 1.24

0.032 Thick midspan

0.125 x Symmetrical EquallyStringer 12 0.098 10 0. 125 x about spaced on

14.0 midspan inner cir-cu mference

8.25 I.D. SymmetricalBaffle 11 0.306 16 9.68 O.D. about 1.25

0.0125 Thick midspan

3366

Figure 33. Cylindrical Tank with Hemispherical Bulkhead-Stiffener Rings,Stringers, and Baffles Configuration

48

=z 1000

e8

800

600 in -

200

0-0.379 -0.2 0 0.2 0.4 0.6 0.8 1.0

LIQUID DEPTH, h/1 3367

Figure 34. Natural Frequencies for Cylindrical Tank with Hemispherical Bulkhead-Stiffener Rings, Stringers, and Baffles Configuration

49

\\\XX \XXN 20.4 #

TAN K

SKI RT


A,

FLAT RIGID iiSBULKHEAD

SUPPORTSTRUCTURE

P

....... r / / . . . I./A////////VY////// // /

cP'

D/2

A

3.72#

LIQUIDDEPTH "h"

J 2IrL1 ',',','', , . , 7 I

3368

Figure 35. Cylindrical Tanks with Inverted Hemispherical Bulkhead

50

Effective I nside Wall Material ELength (A ) Dia. Thickness Density x 10

Ein.( ) (in. ) (in .) (# in?) psi

Tank 14.5 10.0 0.005 0.29 29Skirt 7.5 10,3 0.025 0.098 10

Bulkhead D = 10.0 Fig. 3 0.098 10

"tp r�T4-aA'- P-A-10hi

7~~/_/i/L/_///X

%

)Ii

I

1L

I

W

I

Wm

A3 - ,I\, SS\XEE\

I L

1400

TOP MASS: 20.4#ULLAGE PRESSURE = 3 psig

1200

1000

200

00 0.4 0.8 1.2 1.6 2.0 2.4

LIQUID DEPTH, h/D 3369

Figure 36. Natural Frequencies for Cylindrical Tankwith InvertedHemispherical Bulkhead

51

A3 '

L N'"

TAN K

SKI RT


A,FLAT RIGIDBULKHEADSUPPORTSTRUCTURE

N

\\\\\\\'

P ,\

'. X ' ~\\\l\ l

///////1///////I Ill_1II '1 ///7 //11--- D

'/1

d/2EXI

m--

20.4#

3.72#

IA

ILIQUIDDEPTH"h "

3370

Figure 37. Cylindrical Tanks with52

Inverted Ellipsoidal Bulkhead

Effective Inside Wall Material ESrElucmtut Length ( ) Dia. Thickness Density x 106Element( (in.) (in.) ( in. ) (#/in?) psi

Tank 14.5 10.0 0.005 0. 29 29Ski rt 7.5 10.3 0.025 0.098 10

Bulkhead D 10.0 Fig. 4 0.098 10d -7.07

I" I m ILL....... I .. .

I/VI

A?

5LAIrA

r-

rI

I

I

I

IPP

\,N

E=r-

;J II

71

1600

1400 ULLAGE PRESSURE 2 3 psig

1200I \

1000

z 800

OYI.,I

600

400

00 0.4 0.8 1.2 1.6 2.0 2.4

LIQUID DEPTH, hiD 3371

Figure 38. Natural Frequencies for Cylindrical Tankswith Inverted Ellipsoidal Bulkhead

53

Al

TANK

SUPPORTFRAME

A2

FLAT RIGIDBULKHEAD

BASE PLATEATTACHED TOSHAKER

hi

/;=/

/

=/J//J

7

A

cu

l-7 IT

LIQUIDDEPTH " h "

9.01 #

3372

Figure 39. Cylinder with Flat Rigid Bottom and Top Support

54

r///777]772[

P/

t .ucal ..I Effective Inside Wall Material EElement Length ( ) Dia. Thickness Density x 106

emenin.) (in.) (in.) (#in.3 ) psi

Tank | 14.5 10.0 O 0.005 0.29 29

L\\\\\\

1600

1400

1200 -

1000

800

00 0.2 0.4 0.6 0.8 1.0

LIQUID DEPTH, h/l 3373

Figure 40. Natural Frequencies for Cylinder with Flat RigidBottom and Top Support

55

A,

BASE PLATE '--ATTACHED TO SHAKER

LIQUIDDEPTH " h "

35 #

1

3374

Figure 41. Cylinder with Hemispherical Bulkhead and Top Support56

Effective I nside Wall Material ES cElement Length ( ) Dia. Thickness Density x 106

( in.) ( in.) ( in.) (#/in?) psi

Tank 14.5 10.0 0.005 0.29 29

Bu I kh ead D - 10.0 Fig. 3 0.098 10

1400 _

1200

1000

: = 800

Lo

US 600

400

ULLAGE PRESSURE = 3 psig200 MODE SHAPE 1 0

2 [3 <4 A

- 0.379 -0.2 0 0.2 0.4 0.6 0.8 1.0LIQUID DEPTH, h/l 3375

Figure 42. Natural Frequencies for Hemispherical Bulkhead and Top Support

57

Documents

U. AN EXPEKIMtNIAL bIULY OF AXISYMMETRIC MODES IN … · struct simple structural components, basically thin shells of revolution, whose natural frequencies and responses to forced