ANL-76-48

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ANL-76-43

A+ALY^ , A CLD A r SHELL 4

ANALYSIS OF A CYLINDRICAL SHELL

RATING IN A CYLINDRICAL FLUID REGION

by

Ho Chiung, P. Turula, T. M. Mulcahy,

and J. A. Jendrzejczyk

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Distribution Category:

LMFBR Components

(UC-79k)

AN L-76-48

ARGONNE NATIONAL LABORATORY9700 South Cass AvenueArgonne, Illinois 60439

ANALYSIS OF A CYLINDRICAL SHELLVIBRATING IN A CYLINDRICAL FLUID REGION

by

Ho Chung, P. Turula, T. M. Mulcahy,and J. A. Jendrzejczyk

Components Technology Divisionne. ,*~e .

Woo " Vt e

August 1976

PREFACE

The work reported here was performed as part of thebase -technology activity under the Flow-Induced Vibration Pro-gram (189a No. CA054-A) sponsored by ERDA/ RRD. The over-all oLjective of the activity is to develop new and/or improvedanalytical methods and guidelines for designing LMFBR com-ponents to avoid detrimental flow-induced vibration.

The thermal liner of the Fast Flux Test F 3cility (FFTF)reactor is a relatively thin cylindrical shell separated fromthe main reactor vessel by a narrow, fluid-filled annulus. Themain function of the thermal liner is to shield the reactor ves-sel from excessive heat. Since the thermal liner is subjectedto an environment of sodium flow, the potential for flow-inducedvibrations must be assessed.

The present approach to the design of components sub-jected to flow includes separation o: the structural naturalfrequencies from excitation frequencies carried by the fluid.However, generally accepted techniques that can he used to de-termine the free vibration characteristics of a thin shell sur-rounded by a narrow, fluid-filled annulus are not presentlyavailable to the reactor design engineer. This report describesan analytical and experimental study of a model similar to thethermal liner. A previously issued technical memorandum'contained a summary presentation of some of the work reportedhere. The present report provides a more complete compi-lation of the basic data and theory, together with a discussionof other research work in the subject area.

3

TABLE OF CONTENTS

Page

P RE FAC E. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 2

NOMENCLATURE.......... ................................... 6

A BSTRAC T.................................. ...... 7

I. INTRODUCTION................................. . 7

II. VIBRATIONS OF SHELLS . . . . . . . . . . . . . . . . . . . . . . . . . 9

III. ANALYTICAL METHODS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

IV. EXPERIMENTAL INVESTIGATIONS . . . . . . . . . . . . . . . . . . . . 12

A. Therm al-liner Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

B. Test Proc.'dure. . . . . . . . . . . . . . . . . . . .. . .... .... ..13

V. DISCUSSION OF RESULTS. . . . . . . . . . . . . . . . . . . . . . . . . . . 14

A. Vibration in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1. Natural Frequencies.................................. 142. Frequency Response Analysis . . . . . . . . . . . . . . . . . . . . 173. Tirw History Analysis. . . . . . . . . . . . . . . . . . . . . . . . . 18

B. Effect of Inner Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

C. Effect of Fluid Annulus. . . . . . . . . . . . . . . . . . . . . . . . . 20

D. Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

VI. CONCLUSIONS . ... . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . 23

ACKNOWLEDGMENT ......................................... .23

REFERENCES........ . . . ........................... 24

4

LIST OF FIGURES

No. Title Page

1. Nodal Patterns of Clamped-Free Cylindrical Shell. . . . . . . . . . . 9

2. Schematic of Thermal-liner Model . . . . . . . . . . . . . . . . . . . . . 12

3. Electrom agnetic Exciter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3

4. Elastic -spring-supported Shell... . ........................... 15

5. Comparison of Experimental and NAS'. RAN-predicted NaturalFrequencies for Shell without Fluid. . . . . . . . . . . . . . . . . . . . . 16

6. Acceleration Mobility at Selected Points due to Harmonicoading at Point "" for a Clamped-Free Shell. . . . . . . . . . . . . 18

7. Acceleration Mobility at Selected Points due to HarmonicLoading at Point "a" for a Spring-supported-Free Shell . . . . . . . 18

8. Natural Frequencies for Shell Filled with Fluid. . . . . . . . . . . . . 19

9. Experimental and Predicted Natural Frequencies for Shell witha Fluid-filled Annulus . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 20

10. NASTRAN-predicted Natural Frequencies for Shell with aFluid-filled Annulus . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 21

11. Experimentally Determined Damping Ratios . . . . . . . . . . . . . . . 22

5

LIST OF TABLES

No. Title Page

I. Properties of Shell Model of Thermal Liner . . . . . . . . . . . . . . . 11

II. Comparison of Calculated Natural Frequencies for Shell inVacuum with Clamped-Free Boundary Conditions. . . . . . . . . . . . 14

III. Comparison of Calculated Natural Frequencies for Shell inVacuum with Experimental Values in Air for Various SupportConditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

IV. Finite-element Size Sensitivity of Natural Frequency for Shellwith Fluid-filled Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Symbol

fn

hi, R, I

KR, K 0 , KZ

m

n

Z, a

B

E

p

P

NOMENCLATURE

Desc ription

Natural frequency

Thickness, radius, and length of shell

Elaatic spring constant in radial, circumferential,and axial directions

Number of axial nodes

Number of circumferential waves

Axial and circumferential coordinates

Bulk modulus of fluid

Young's modulus

Viscous damping factor

Poisson's ratio

Mass density of shell

Mass density of fluid

6

7

ANALYSIS OF A CYLINDRICAL SHELLVIBRATING IN A CYLINDRICAL. FLUID REGION

by

Ho Chung, 13. Tarula, T. M. Mulcahy,and J. A. Jendrzejczyk

ABSTRACT

Analytical and experimental methods are presented forevaluating the vibration characteristics of cylindrical shells

such as the thermal liner of the Fast Flux Test Facility (FFTF)reactor vessel. The NASTRAN computer program is used tocalculate the natural frequencies, mode shapes, and responseto a harmonic loading of a thin, circular cylindrical shell situ-ated inside a fluid-filled rigid circular cylinder. Solutions ina vacuum are verified with an exact solution method and the

SAP IV computer code. Comparisons between analysis and ex-periment are made, and the accuracy an-i utility of the fluid-

solid interaction package of NASTRAN is assessed.

1. IN'I PRODUCT ION

The reactor vessel of the Fast Flux Test Facility (FFTF) is a heavysteel i ylinder which holds the coolant and reactor internal structures. Thethermal liner is a thin shell inside and concentric with the upper plenum ofthe reactor vessel. Liquid sodium from the primary coolant loops enters thereactor vessel near the lower end. Some inlet coolant is passed through thenarrow annual region between the thermal liner and the reactor vessel tolimit the probability of developing an excessive temperature gradient. Sincethe thermal liner is subje, ted to sodium flow, the potential for flow-inducedvibrations must be assessed.

The Hydraulic Core Mockup (11CM) is a 0.286-scale model of the FFTFreactor vessel, including some reactor internals. The 11CM was built to eval-uate the hydraulic and vibrational characteristics of the FFTF reactor coreand attendant components. Water is used to model sodium coolant because ithas similar physical properties such as density and bulk modulus. The resultsof 11CM analysis may be used to characterize the prototype FFTF throughproper scaling laws and extrapolation.

The present report is an analysis of the vibration characteristics ofa model of the thermal liner of HCM, FFTF. The analysis includes calculations

S

and experimental evaluations of natural frequencies, mode shapes, twm)dal

damping, and response to harmonic forcing functions. The the rmal liner \ asmodeled as a thin, circular cylindrical shell with a fluid-filied annulus. Theouter cylinder (reactor vessel) was assumed to be rigid, in t onsideration ofits relative stiffness compared to that of the inner shell (thermal liner).

Sinc . the early work of Arnold and Warburton, the vibration problems

of thin shells have been of interest to many structural engineers. ' eissa'collected many results on vibrations of shells from published literature andsummarized them in a unified manner. His report presents extensive fre-

qiency predictions fo:- various boundary conditions and other complicating

effects such as a surrounding fluid medium. Many authors 4-10 have investigatedsome natural-vibration characteristics for a cylindrical shell filled with fluidfor various boundary conditions. The cylindrical shell with an annular fluidgap was first analyzed by Mnevi'' and recently 6y Levin and Milan,13Krajcinovic,14 and Chen and Rosenberg.'5 However, these analyses were re-

strictud to shells with simply supported boundary conditions and no axial

constraint at the ends. This is 0h, most amenable boundary condition in the

analytical method designed to obtain a closed-form solution. The study pre-

sented here was undertaken to provide information for the cylindrical shellwith a fluid-filled annulus rigidly clamped at its base and free at the top. This

may be considered as a better representation of the boundary conditions of thethermal liner than simply supported conditions at both ends.

The following steps of analytical and experimental work were performed

in order to characterize t:ie dynamic behavior of the thermal-liner model:

1. Free-vibration analysis in vacuum (analytical and experimentalstudies).

2. Frequency response analysis in vacuum (analytical and experi-mental studies).

3. Time history analysis in vacuum (analytical study).

4. Free-vibration analysis for the thermal liner filled with water

(analytical study).

5. Free-vibration analysis with a water-filled annulus (analyticaland experimental studies).

In the analytical studies, case I was performed by using an exact so-lution method.1 and two general-purpose finite-element computer programs:NASTRAN and SAP IV. Cases 2, 4, and 5 were carried out by using theNASTRAN computer program." However, case ; was done by the use of theSAP IV program'

9

II. VIBRATIONS OF SHELLS

The modes of vibration of thin cylindrical shells are characterized bythe number of circumferential waves (n) and axial nodes 'r), as shown in

Fig. 1. For each nodal configuration,CIRCUMFERENTIAL NODAL PATTERN three natural frequencies exist. The

essential difference among these vibra-

tions is the relative amplitudes of the

motions in the axial, circumferential,

So Mal_ n- and radial directions. Only the lowestfrequency for each configuration is usu-

ally of practical importance because the

other two are normally much higher invalue.

AxIAL NODAL PATTERN For the modes having low n val----- -- ues, the dynamic behavior of a cylindrical

j~~ ~L-.shell is largely dependent upon the mem-T. T"? -s brane stiffness of the shell. As the cir-

NOA, AGcumferential mode number increases,

.5.m.4 the bending stiffness also becomes im-portant in determining the dynamic be-havior of a cylindrical shell. For the

i -.kcuWELENTAL NODE nodes having higher n values, the mem-

AM-A, NODE brane stiffness is negligible and thebending stiffness is of prime importance.

Fig. 1. Nodal Pattermn of (I.jnlpLed- This phenomenon is generally true forFrre (ylandrical ShLlt any values of axial-node number (m).

The lowest natural frequency is associatedwith the mode of the lowest total strain energy of the shell. Arnold andWarburton] first showed that the lowest natural frequency frequently corre-sponds to a mode having a higher n value rather than the modes having lown values such as n 0, 1.

Forsberg" investigated the influence of boundary conditions on themodal chEracteristics of thin cylindrical shells for a wide variety of boundaryconditions. However, a reasonable analytical representation of actual boundaryconditions still remains in question. There a-e a number of efforts20,U1 toclarify the frequent discrepancies between analytical predictinns and ex;ri-mental observations of shell natural frequencies, especially for the modesassociated with large membrane stiffness of the shell. These discrepanciesare often attributed to imperfect boundary support,'' nonlinear behavior of theshell,1o and different levels of modal dampings. As described in this report,an effort was made to bracket the experimentally observed natural frequenciesby analytical predictions for a few boundary conditions that could reasonablysimulate the experimental setup. In addition, frequency response analysis wasperformed to compare the level of output for different modes of vibration inorder to investigate the difficulty in finding the low-n-value modes in the

experiment.

10

III. ANALYTICAL METHODS

Free-vibration problems for cylindrical shells in a vacuum have beensolved by many contributors through many different ways. Exact solutionswere obtained by Forsberg, 9 Warburton,2 2 and Chung.1 6 Utilizing the finite-element method, many shell problems with sophisticated geometries andphysical properties have been also treated.1' 3' 5 During the past few years,a number of general-purpose finite-element analysis programs have beendeveloped and widely used in industry. In this report the NASTRAN finite-element analysis program 7 was used for analytical computations. The SAP IVfinite-element codelp and an exact solution procedure developed by Chungi6were alt o used as corroborating solutions for the shell in vacuum.

NASTRAN is a general-purpose digital-computer program designed toanalyze the behavior of elastic structures subjected to a wide range of loadingconditions. Considerable analytical versatility has been built into NASTRAN,including determination of real and complex eigenvalues "or use in vibrationanalysis, and evaluation of dynamic response to transient loads, steady-stateharmonic lcads, and random excitations. NASTRAN is designed to treat largeproblems with many degrees of freedom; however, limits are imposed bypractical c-nsiderations of running time and the capacity of secondary devicesused b; the computer. The numerical method used in NASTRAN takes advan-tage of matrix sparsity and bandwidth to reduce analysis time.

The structu-al model used in NASTRAN consists of grid points definingdegrees of freedom, structural elements connected between grid points. andloads applied to grid pr:nts. NASTRAN has a wide variety of elements andconstraint between degrees of freedom. One of the unique features of NASTRANis the fluid-elastic analysis capability. The program can perform free-vibration analysis, frequency response, and transient analysis for an axisym-,r.etric (geometrically) structure containing flid. The compressibility andgravity effect of a free surface can be included in the analysis. The motionsof the fluid are assumed to he small compared to the size of the container.

The SAP IV finite-element code has similar capabilities, but its anal-ysis versatility is limited compared to NASTRAN. However, the SAP JV pro-gram was found to be more econornica: for problems analyzed here in termsof computer run-time.

An exact solution methodt" also was used to evaluate natural frequenciesof the thernial liner in vacuum. The solution was obtained from a direct-solution procedure of Sanders' shell equations in which the modal displacementfunctions were constructed from simple. Fourier-series expressions.

The uimensions of the shell considered in this report were scaled fromth'.e thermal liner of the FFTF reactor. Several fluid-annulus sizes were con-sidered. The scale factor is about 1/14, giving approximate model dimensions

11

of 20.5-in. (52.1-cm) height, 17-in. (43.2-cm) diameter, and 0.039-in.

(0.098-cm) thickness. The geometrical, as well as material, properties cifthe scaled shell model of the thermal liner are given in Table I.

TABLE I. Properties of Shell Model of Thermal Liner

Inner Shell

Thickness

Radius

Length

Material density

Young's modulus

Poisson's ratio

h = 0.058 in. (0.15 cm)

R = 8.510 in. (21.62 cm)

= 20.125 in. (51.12 cm)

p = 701 x 10-6 lb-sec2 /in. 4 (7492 kg/m3 )

E = 26.5 x 106 psi (1.83 x 1011 Pa)

v = 0.3

Water Annulus

Radial-gap size

Fluid density

Bulk moduliar

0.151 in. (0.38 cm)

0.253 in. (0.64 cm)

0.538 in. (1.37 cm)

1.033 in. (2.62 cm)

2.94 in. (7.47 cm)

p = 93.6 x 10-6 lb-secZ/in.; (1000 kg/m')

B - 300,000 psi (2.07 x 10' Pa)

Outer Shell

Assumed rigid

12

IV. EXPERIMENTAL INVESTIGATIONS

A. Thermal-liner Model

The experimental model of the thermal liner was fabricated by rollinga steel plate, seam-welding it. and soldering it to a 0.5-in.-thick (1.27-cm)

brass plate which was mounted to a 1-in.-thick (2.54-cm) steel plate. This

steel plate was then bolted to a heavy steel base block. To test the thermal-

liner model with different water-gap sizes, several outer cylinde, s were

made. To obtain a uniform gap, in spite of slight out-of-roundness or imp-"r-

fection. , these cylinders were formed by casting relatively thick (-2 in.)

(5.1 cm) concrete around the st-el test cylinder using hard durometer neoprene

spacers to obtain the desired spacing.The concrete was water-proofed and

EM attached to the steel base block asXCITER STEEL CYLINDER (LINER MODEL) shown in Fig. 2.

The average diameter of theWATER GAP (VAPABLE) fabricated shell was determined by

CONCRETE CINDER taking the ave4 age of 36 measurements.The deviation of these measurements

BASE PLATEwas less than 0.5%. The annular gapsize was determined by measuring the

ig. 2. schem c of ThcrniI- nL riodei volume of water necessary to fill the

ANL Neg. No. 1!:3-5732 Re. 1. annular region. For any of the gapsizes, the maximum deviation from the

average values was about 10%. (The geometric accuracy of the model may

exceed the uniformity expected in a prototype thermal liner.) The density andelastic modulus of the thermal-liner model were determined by weighing andby performing cantilevered-beam frequency tests of strips of steel plate usedin fabricating the shell. The dimensions and material properties of thethermal-liner model have been given in Table I.

The inner steel cylinder was excited by an electromagnet positionedinside the shell (see Fig. 3). The exciter coil (280 turns of No. 16 enamel-covered wire) was mounted o a heavy steel support column, and had adriving area of about 26 in.(0.017 mZ). The gap between magnet and steelcylinder was about 3/16 in. (0.48 cm). The magnet drive current was pro-vided from an audioamplifier (McIntosh Model MI-200A13, Z00 W/ controlled bysignal generators (Hewlett-Packard Model 203A Variable Phase Generator,and Model HOI-3722A Noise Generator). Both sinusoidal and wide-band ran-dom signals were generated.

Motion of the top of the steel cylinder was monitored by seven miniaturepiezoelectric acceleiometers (Endevco Picomin No. 22) cemented to the insidetop surface of the test cylinder every 30", with 0 defined to be opposite thecenter of the magnet. In addition, three accelerometers were mounted along

1 3

the 0 longitudinal line on the cylinder

r to examine the longitudinal-mode shape.

A n.ovable accelerometer mounted on

the magnet was used to search for cir-

cumferential nodal points fir the modes

having higher n values. These accel-

erometers were connected through

harge amplifiers to a patch panel for

ease in m~nomtoring the signals.

13. Test Procedure

The test procedure consisted

of three ph a es. First, natural fre-

quencies were determined by excitingthe shell with a wide-bard random

- force and inspecting power-spectral-density plots produced by a r'ourieranalyzer Aewlett-Packard Mode!

A) from the tim'--history signals

- f st vcral Uccel."rormeters. Second,- - -the shell was excited mgain with a

sinusoidal current supplied to the coil,

Using a range of frequency, in the

- vicinity of .-ach of the natural vibra-

ti(on frequencies detected by random

excitation. For each natural frequency"

Vfn, the accelerometer signal in a nar--ow band about this frequency was

1):ro" sSdt th - wrier anal',-tr 1t( provide a more accurate value of the

1 .I t:-t'tl n v and to establish th. rms acceleration at each plc celerometer

position. This into rnat ion was plotted to identify mode shapes correspndini

to the natural Irtqutm iles. In the bird phase of testirg, the Lransfer functionbetween the rmts displacetent and peak coil current was plotted at discretepoints io .I na Irow band about cachIi nature al frequency so that the equivalentvi-t o s dallping ratio could be calculated by the half-power-point bandwidthmitthb d.

14

V. DISCUSSION OF RESULTS

A. Vibration in Vacuum

Before considering the shell vibrating in a fluid, we studied the shell

in vacuum to establish the significance of variations in the boundary conditionsand the degree of correspondence to be expected between analytical and exper-imental results.

1. Natural Frequencies

The natural frequencies for the thermal liner in vacuum werecalculated by an exact method, NAST1RAN. and SAP IV. The finite-elementmodel used in the computer codes was the same and consisted of 10 divisionsvertically and 9 divisions over a quarter of the shell circumferentially. Theseanalytical predictions are compared in Table II for clamped-free boundaryconditions. NASTRAN gave frequencies up to 5% higher than those obtainedby the exact-solution method. The solutions of the SAP IV code are much

closer to the exact solutions (less than 2% deviation).

TABLE II. Comparison of Calculated NaturalFrequencies (in liz) for Shell in Vacuum with

Clamped- Free Boundary Conditions

Exact Method

855.10403.72223.34171.77199.16268.86361.92472.54599.03

928.28644.48

494.69142.00464.59539.45648.34

SAP IV

405.1225.6174.3201.7272.036(.2478.4

505.8454. 3176.7

NAST R AN

856.3410.1232.2180.5206.2275.5370.1483.5614.0

943.2671.4529.3478.0496.9567.267S.2

Although the experimental setup was intended to simulate a fixedcondition at the base, complete rigidity of axial and rotational restraint wasnot achieved. Thus the base boundary conditions should be properly

n

2222222

15

modeled in the analytical method to correlate with the experimental results.

Five different cases of bottom boundary conditions were considered:

(1) Clamped (C)

(2) Simply supported (SS)

(3) Spring supported (Sp): KR = K = 106, KZ 6

(4) Spring supported (Sp): KR = K 9 = 106, KZ = 101

(5) Spring supported (Sp): KR = Ke = 106, KZ 10.

Here the unit of spring restraint is lb/in, per 1.4844 in. (4.645 kN/m per m)of shell circumference. In the clamped condition, all displacements and ro-tations were restrained; in the simply supported condition, only displacementswere restrained. In the NAST RAN model, the base boundary condition wasfurther relaxed by allowing movement in axial, circumferential, and radial

directions against elastic springs at the grid points (see Fig. 4).

r

-n

1rig. 4

Elastic-spring-supported Shell

K

p.L

Table III and Fig. 5 compare the experimental results for the shellvibrating in air (the effect of air is often neglected) to the analytical predic-tions of the NASTRAN program for different bottom boundary conditions. Incomparison with the analytical predictions for the clamped-free shell, theexperimental results are smaller for the modes having lower n values andlarger for those having higher n values. This may be due to some degree ofeccentricity' in the experimental model and errors in evaluating its physicalproperties. Although not distinguishable in Fig. 5, the natural frequenciesare higher (< 1%) for the clamped base condition than for the simply supportedbase condition. Thus it can be concluded that rotational restraint at the basedoes not significantly affect the overall results and is primarily related to alocal flexure condition.

16

TABLE III. Comparison of Calculated Natural Frequencies (in Hz) for Shellin Vacuum with Experimental Values in Air for Various Support Conditions

NASTRAN PredicitionsSp-F Experiment

n m C-Fa SS-Fb Kz = 10' Az = i0s Kz = 10 in Air

01234S6789

10

34567P9

10

856.3410.1232.2180.5206.2275.5370.1483.5614.0760.6

943.2671.4529.3478.0496.9567.2673.2804.9

856.2406.7228.8180.4204.9275.5366.6483.5613.7760.6

670.9527.7477.3496.2566.7673.2804.5

782.5357.5198.1162.6199.3271.8365.6482.3613.2760.0

895.2628.5492.6449.3476.3552.9663.1797.4

525.8204.7117.4127.6185.4266.6365.5480.9612.3759.4

860.9585.9455.6422.4458.854 2.0655.9792.7

14.5

21.9

112.5

265.0

480.5

759.3

573.1

415.7

539.6

152202.2292.7399520.8665815

461437488.9580.6710820

b -F: lamped-Free.SS-F: Simply supported-Free.

cSp-F: Spring-supported-Free iKR

LAPPEDD FRit 4_D SMPLI SUPOIT(D-FREE

CJ/E(sPm'WG Mu'itiD-fmif

CAV( I It 106

4 or Q

S, op0

: 2.

C t 4 a I '0OOl %0 (Al

K_ 106 at the base).

Fig. 5

Comparison of Experimental and

NASTRAN-predicted Natural Fre-quencies for Shell without Fluid.ANL Neg. No. 113-5728 Rev. 1.

In contrast, the influence of axial constraint (Kz) is evident andsignificant throughout most of the region of interest. The natural frequenciesbecome much smaller as the axial constraint (Kz) is relaxed. The lowest

C-

I.

C,

17

natural frequencies for shell-type vibration (n z 2) occur at n = 2 for Kz = 10,in contrast to n = 4 for Kz = 106. For higher axial half-waves (m = 2), thedifferences are much smaller throughout the entire region of circumferentialwave number n. The experimental results are somewhat closer to the case of

spring-supported base condition of Kz = 106 for higher circumferential wavenumbers (n z 4).

All experimental efforts failed to detect the response of modes

associated with lower circumferential wave numbers (n < 4). In addition toelectromagnetic excitation, these efforts included acoustic excitation at

various amplitude levels by a loud speaker. Fourier analysis of the acceler-ation response to a random excitation did show a response at 140 Hz, whichprobably corresponds to the n 3, m = 1 mode, but there was no corre-sponding response when a single harmonic excitation at this frequency wasgenerated.

Other investigator s220-t2 have also reported difficulty in obtainingnatural frequencies for the low-n-value modes. However, the explanations

posed here and elsewhere are not totally satisfactory. The frequencies ofthe low-n-value modes are highly dependent on the stiffness Kz, as shown inFig. 5, so that the relatively undeterminable and possibly nonlinear nature ofthis restraint may greatly reduce the sharpness of the associated response.However, for the modes that gave a clear response, increasing the drivingforce by a factor of 4 lowered the peak frequency by at most 0.5%; hence,nonlinearity of support appears to be negligible.

One explanation is in the strain-energy consideration:20 2 2 Thelow-n-value modes associated with larger membrane energy may be difficultto excite, compared to the higher-n-value modes with larger bending energy.Also, the difficulty may be the masking effect of the higher-n-value modes atfrequencies close to that of a low-n-value mode. This is particularly suspectedin the tests using electromagnetic excitation; when a pure harmonic currentis applied to the device, the resulting force function carried higher harmonicfrequencies of an amplitude up to 20% of the fundamental harmonic. Finally,geometrical imperfections in the as-fabricated shell are a factor that shouldnot be ruled out of consideration.

2. Frequency Response Analysis

The steady-state response to a radial harmonic force was predictedby NASTRAN for a thermal-liner model with two different boundary conditions:(1) clamped-free, and (2) spring-supported (Kz 106)-free. (In the previoussection, the experimentally determined natural frequencies were found toapproximate the case of spring-supported-free shell.) The latter boundaryconditions were simulated in an effort to identify the difficulty in excitingthe low-n-value modes. In operation, the driving force was applied at onepoint at the top of the shell and the responses were calculated at selected

18

points on the shell surface. The particular response of interest was the

acceleration due to a sinusoidal force with 1 lb (4.45 N) maximum amplitude,

the so-called acceleration mobility. The analysis was based on the normal-mode method, with the lowest 35 normal modes and an assumption of 2%structural damping.

Figures 6 and 7 show the spectrum of acceleration mobility forthree selected points to a loading at the top of the shell: (a) the same pointas the load point, (b) a point 900 from the load point, and (c) a point diametri-cally opposite the load point. As evidenced by the figures, the accelerationresponses corresponding to modes (n = 2, 3) with low circumferential "vave

number are smaller than those with higher n for both cases of shell boundaryconditions. Also, the modes (n, m = 8, 1; 6, 2 in Fig. 6; and n, m = 8, 1;5, 2; 7, 2 in Fig. 7), having nearly equal natural frequencies, were superim-posed, and the response is much larger than the others.difficult to isolate the component natural frequencies.

'280' a .5

91

200 >.*cp ;4%

Fig. . Acceleration Mobility at Selected I'.ints ducto Harmonic Loading at Point "a" for aClamped-Free Shell

Consequently, it is

ft1 7,2 t124'' 3/1 5 121, i 6,1 a

o 9

.-4.

I, 1

?" s.

IkQ(%- M

1

Fig. - Accv Iration Mobility at Selected Pointsdue toHarmonic Loading at Point "a" for a Spring-upported (Kl KV KZ 106)-Free shell

Figures 6 and 7 also show that the response at point "b" does notexperience the resonance of the modes with odd n values (n = 3, 5, 7, ... )because this point falls on a node line.

3. Time History Analysis

When the natural frequencies for different shell-vibration modesapproximate each other, it may be difficult to excite one particular mode bya sinusoidal force. This may produce a situation in which these modes arecoupled together, with consequent difficulties in identifying the mrade s.apefor a particular resonance. An analytical study was conducted, using theSAP IV computer program, to demonstrate this aspect of shell-vibrationphenomena.

0

t3

c

eP

19

The time history analysis was performed for a clamped-free shelldue to a sinusoidal force. The force was applied in radial direction on a smallarea (rather than at a point) near the top of the shell. First, the shell wasexcited with the natural frequency corresponding to the n = 3, m = I mode.

Examination of the resulting time-history data showed the shell was vibratingwith the n = 3, m = : mode. In Fig. 6, frequency response analysis showedthis mode was distinctly identified and not close to other modes. However,

when the shell was excited with the natural frequency of the n - 8, m = 1 mode.

the resulting shell motion was a coupled one of the n 8, m = 1 mode and the

n = 6, m = 2 mode. Therefore, in all experiments an extremely careful dis-crimination is essential to accurately identify the mode shapes and not to missany natural frequencies.

B. Effect of Inner Fluid

Vibrations of a cylindrical shell containing a liquid have been studiedextensively. 3"' Since the fluid has a considerable inertial effect 3n the shell,the natural frequencies are reduced significantly.

Free vibrations of the thermal-liner model filled with fluid were ana-

lyzed, using the NASTRAN program. The support condition for this thermalliner was assumed as clamped-free. The finite-element model consisted of

10 vertical divisions and 15 circum-E -ferential divisions over a quarter of

the shell, and 10 divisions radiallythrough the inside fluid. First,

S ANASTRAN solutions for other shellgeometries were evaluated by com-

4DC paring them to the available analyti-cal closed-form solutions. 5 -' 10-12

For a variety of shell geometries andtwo cases of boundary conditions

(clamped-free and simply supportedwithout axial constraints), NASTRAN

SELLL WiTh a yielded satisfactory predictions offLU'D ANNULUS

natural frequencies.

Figure 8 shows the natural fre-

quencies for the thermal-liner model

filled with fluid. Also shown are nat-0 1c (a[T.. ODr e 9 o rural frequencies for the thermal-liner

model in vacuum and with a fluid-filled

Fig. 8. Natural Frequencies for shell annulus only. For modes n = 2-7 andFilled with fluid (mn l, m = 1, the inside fluid of the shell

lowered the natural frequencies by thefactor of 1.4-2.3; however, with a fluid-filled annulus of 0.151 in. (0.38 cm),the modes were lowered by a factor of 3.6-Q.5. These factors become smalleras the circumferential wave number n increases for both cases of the inside

fluid and the fluid-filled annulus. Natural frequencies for the shell with dif-ferent size of fluid-filled annulus are discussed in Sec. C below.

C. Effect of Fluid Annulus

The natural frequencies of the shell %%ith a fluid-filled annulus werestudied by three different ways:

300 '- LEGEND

-0 EXPERIMENTAL DATA 4

73 EXTRAPOLATION FROMIN-AIR DATA i

25O- /250 -- NASTRAN

PREDICTIONS

Sz200 --

U /, n r

'50-(6 2)

72

6 - '

00-- a -

C '

0 0 GA 4 sE. -0

Fag. 9

Experimental and Pr.cted Natural Frequen-cies for Shell with .I Fluid-filled Annulus.

ANL Neg. Nil. 113-7,730 Re. 1.

an experimental investigation, a simple addedmass correction1 5 to the solutions in air, and

a NASTRAN program analysis. Figure 9summarizes the results.

The added mass corrections due to thefluid annulus were made with the experimen-

tal data for the shell vibrating in air. Theanalytical evaluation of the added mass fac-tors was made by assuming the shell wasinfinitely lor;. A comparison between the

experimental results and these extrapolations

shows this method appears to give frequencypredictions that are somewhat low, but gen-

erally within 10% of the experimental results.

The finite-element model for the

NASTRAN analysis consisted of five divisionsvertically, six divisions (15 segment) over aquarter of the shell circumferentially, andfive divisions radially through the fluid, i.e.,a 15, 6, 5) rodel. To check the sensitivityof the frequency to element size, refinedmodels with finer meshes were analyzed.First the number of divisions radially throughthe fluid was varied (see Table IV), but thisdid not affect the results significantly, indi-

eating that the five-division model can be considered fine enough for accurateresults. Then the meshes for the shell were refined up to 10 divisions vertically

1 A1 .J I\V. ftnlto -- o le nwlnl 11-, . S n ttlvltys 11 \aturrs F er .bir Sh4 11 tth F lid-fb llvd Annal I'si

\,r, 'I l)nv I ilins i,,r Shell 1,-. r h Not. if I .,,ti..I 1..t ratl ( I I iflw.

era . aI (.Irt -* ,.1-ren,,.l I)IvsI - uton Ir % .1,-r Frieg. no .s. lit .-

I,

5

S

I. 11+

77 07 7 .'.%

-; 7, M

17

.471'1

'.I

5 I.Mi4lb t i, ., 1 710 '1 41 t,7 10 '71

IA 1 S t .4 MN i',

' rlan -el- rr ,1. I th 1 fill-in. 11 ti s qI J ... r oil I I

tr AS111A " I@'% I Ii4.1 -. ra r an ,m 1100t S 170-Ito .16 t '' r.stl. .r+ se.nt t~ , %al ast. .111.froq (I n to11

and 15 divisions circumferentially (6 segment). This (10, 15, 5) model re-

sulted in a 20% lower natural frequency. However, the cost of running a se-

ries of fluid-solid interaction problems with NASTRAN, particularly with amuch finer mesh than used here, would be prohibitive. To find one natural

frequency, the runs involving l5, 6, 5) mesh required about I min of IBM

S, 370-195 CPU time; the (10, 15, 5)run using the NASTRAN level 15.1.

700 r

600

'500

z

W. 300

SMELL 'N vACuO,

GAP

0

' n

3

200' / , 0253

/,/0 151

/L'

'00' \ .. 7

2 4 6 aCiRCUMFERENTIAL MODE NUMBER Wn)

Fg . NASTRAN-prdicted Natural

Frequencies fir Shell with aI luId-filled Annulus (m 11.ANL Nei. No. 113-5731.

frequency-response curve was f

case required about 8 min. Both were

Figure 10 shows the natural fre-

quencies predicted by the NASTRAN pro-

gram for the clamped-free shell with fourdifferent fluid gaps. Natural frequenciesfor the shell in vacuum are also shown

for comparison. As the gap size decreases,the frequency becomes smaller for theentire region of circumferential wave

number n. These NASTRAN-predictedfrequencies are also compared to otherresults in Fig. 9. The NASTRAN predic-tions are 20-30% higher than the experi-mental results. This difference may be

narrowed by using the finite-elementmodel with finer meshes. However, one

may still expect the uncertain shell-

boundary condition (see Sec. A above) tocause some discrepancies between ana-

0lytical predictions and experimentalresults.

D. Damping

For purposes of determining 'heequivalent viscous damping ratios, the

irst determined in the 0.1-0.5-g accelerationrange and then redetermined at the highest acceleration level compatible withthe available equipment. The ratio of acceleration levels was at least twoand usually five to ten. The damping-ratio variation with amplitude level wasin the 10-40% range. Figure II shows selected average values of damping forthe shell vibrating with various water gaps and in air. Clearly, the smallerfluid gaps are associated with higher damping ratios than those measured inair, typically twice as large. Generally, larger damping is associated withsmaller water gaps. For those few cases that do not follow this later trend,the frequency-response curves probably were distorted (broadened) by su-perposition of the response corresponding to an adjacent natural frequencywhich could not be accounted for in the half-power bandwidth computation.

-

22

I I I I I

mn

0.02 2 6 .27%

2680.01 -8 -

0

0 04 -1 4

0 03 -

0.02

1 60.01-

00 0.2 04 06 08 10

WATER GAP, in AIR

Fig. 11. Experimentally Determined Damping Ratios.ANL Neg. No. 113-5809 Rev. 1.

VI. CONCLUSIONS

Vibration-analysis methods for shells, such as the thermal liner of

the FFTF reactor, are illustrated and comparisons were made with experi-

mental results irom a scale model of the thermal liner. Correspondence of

experimental and analytical results is within acceptable limits for designpurposes once the shell-boundary conditions are properly modeled. The ex-

perimental model was designed to be a fixed-free shell; however, its modal

characteristics were similar to the shell with elastic spring support at thebase. Flexibility of the base support in the axial direction of the shell wasshown to considerably lower the natural frequencies, especially for the

small-n-value modes. Natural frequencies were shown to be rather insen-

sitive to flexibility of the base support in the radial and tangential directions.

Severe vibration modes corresponding to solutions with low n values

eluded experimental detection. Efforts to resolve this experimental diffi-

culty included a frequency-response analysis and a time history analysisusing NASTRAN and SAP IV. The efforts revealed that acceleration mobilitiesfor the membrane energy dominant modes (small-n-values modes) were lower

than those for the other modes. However, further investigation is required to

clarify the significance of these modes in design and the reasons for the dif-

ficulty in detecting them experimentally.

The feasibility of using coupled fluid-elastic finite-element analysisin NASTRAN to solve vibration problems involving shells in fluid mediumhas been demonstrated; however, the computer time costs incurred prohibitextensive application.

ACKNOW LEDGME'T

We wish to express our gratitude to Dr. M. W. Wambeganse for hir

comments.

REFERENCES

1. T. M. Mulcahy, P. Turula, H. Chung, and J. A. Jendrzejczyk, Analyticaland Experimental Study of Two Concentric Cylinders Coupled by a FluidGap, ANL Technical Memorandum, ANL-CT-75-36 (Apr 1975).

2. R. N. Arnold and G. B. Warburton, The Flexural Vibrations of the Walla ofThir Cylindrical Shells having Freely Supported Ends, Proc. Roy. Soc.(London) A197, 238-256 (1949).

s. A. W. Leissa, Vibration of Shells, NASA SP-288 (1973).

4. L. Rayleigh, On the Vibrations of a Cylindrical Veesel Containing Liquid,Phil. Mag. 15, 385-389 (1883).

5. M. L. Baron and R. Skalak, Free Vibrations of Fluid-Filled CylindricalShells, J. Eng. Mech. Div., ASCE 88 (EM-3), 17-43 (June 1962).

6. U. S. Lindholm, D. D. Kana, and H. N. Abramson, Breathing Vibrations of aCircular Cylindrical Shell with an Internal Liquid, J. Aerospace Sci. 29,1052-1059 (Sept 1962).

7. R. Kumar, Flexural Vibrations of Fluid-Filled Circular Czylindrical Shells,Acustica 24, 137-146 (1971).

8. A. S. Arya, S. K. Thakkar, and A. C. Goyal, Vibration Analysis of ThinCylindrical Containerc, J. Eng. Mech. Div. ASCE 9? (EM-3), 317-331(Apr 1971).

9. A. A. Lakis and M. P. Paidoussis, Free Vibration of Cylindrical 'helloPartially Filled oith Liquid, J. Sound Vib. 19(1), 1-15 (1971).

10. W. E. Stillman, Free Vzbration of Cylinders Containing Liquid, J. SoundVib. 30(4), 509-524 (1973).

11. Ye. N. Mnev, "Vibrations of a Circilar Cylindrical Shell Submerged in aClosed Cavity Filled with an Ideal Compressible Liquid, Second All-UnionConf. on the Thecre. of Plates and See ola, Lvov, Sept 15-21, 1961, Kiev,lzd-vo AN USSR, 284-288 (1962).

12. Ye. N. Mnev and A. K. Pertsev, iydroelacticity of Shells, English Trans-lation, Foreign Technology Division, U. S. Air Force, FTD-MT-24-119-71(1971).

13. L. Levin and D. Milan, "Coupled Breathing Vibrations of Two ThinCylindrical Coaxial Shells in Fluid," Int. .s podium on VibrationProblem in Industry, Keswick (1973).

14. D. Krajcinovic, Vibrations of Two CO.zxial Cylindrical Shella ContainingFluid, Nucl. Eng. Des. 30, 242-248 (1974).

15. S. S. Chen and G. S. Rosenberg, fynice of a Coupled Shell/Fluid Syatom,Nucl. Eng. Des. 32, 302-310 (1975).

16. Ho Chung, Free Vibrations of Circular Cylindrical Sello, ANL TechnicalMemorandum, ANL-CT-76-9 (Aug 1975).

17. C. W. McCormick, Ed., The NASTRAX Uaer's Nanual (Leel 16), NASA-SP222(01) (1972).

18. K-J. Bathe, E. L. Wilson, and F. E. Peterson, SAP IV--StructuralAnalyaia Progran for Static and Pynamio Neoponse of Linear Syotea,EERC 73-11 (June 1973).

19. K. Forsberg, Influence of Boundar:- Conditions on the Modal Characteristicsof Thin Czylindrical Shells, AIAA J. 2(i2), 2150-2157 (Dec 1964).

20. J. G. A. Croll, Coupled Vibration Modes, J. Sound Vib. 38(1), 27-37 (1975).

21. V. I. Weingarten, Free Vibrati,)nsq of Thin Cylindrical Sheila, AIAAJ. 2(4), 717-722 (1964).

22. G. B. Warburton, Vibration of Thin Cy4indrical Shells, J. Nech. Eng.Sci. 7(4), 399-407 (1965).

23. S. K. Sen and P. L. Gould, Free Vibration of Sheile of Revolution UsingFinite Element Method, J. Eng. Mech. Div., ASCE 100(EM2), 283-303(Apr 1974).

24. S. Ghosh and E. Wilson, Dynamic Stress Analysis of Axisymnetric Structuresunder Arbitrar. Loading, EERC 69-10, University of California, Berkeley(Sept 1969).

25. J. H. Argyris, Matrix Displacement Analysis of Plates and S:'ells,Ing.-Arch. 35, 102-142 (1966).