On vibration of spherical shells interacting with fluid

ON

VIBRATION

OF

SPHERI-

CAL

SHELLS

INTERACTING

WITH

FLUID.

Institutt

for

Skipskonstruksjoner

Division

of

Ship

Structures

by

Tor

Vinje

Meddelelse

SK/M 23 TECHNSC! 1R!!IVERSiTET Laboratorium voor Scheepshydromechanlca Archief Mekelweg 2, 2628 CD Deift T 015.786873 Fax: 015.781833 March 1972 Trorujhei m

UNIVERSITETET

I

TRONDHEIM

Bibliotheek van de 'Qrafdeling derSche- .ouwkunde

Tce

Ho school, Deift DOCUMENTA j - S,4 DAT

TIER PUBLICATIONS IN ThIS SERIES SXB II/Ml Gerritsma, J. and van den Bosch, J.J.: "Ship Notions and Roll Stabilization, 1964. SF5 II/Nl Iversen, P.A.: 'En stivhet3:atrise for tetreader-elementer", 1964. 5KB II/M3 Kowalik, J.: "Iterative Methds for Large System of Linear Equations in Matrix Structural Anaysis", 1965. 5X3 II/M4 Kaviie, D., Kowalik, J. and Moe, J.: Structural Optimization by Means of Non- Linear Programming", 1965. XB II/MS Moe, J. og TØnnessen, A.: "Eksperimentell og teoretisk underskelse av spennings- forlpet i dekk ved skip med to lukerekker", 1966. 5KB II/M6 Fredriksen, K.E. og Moe, J.: "Styrkeundersøkelser av trefartØyer. Del I. Studier av en dei styrkeelementer p 53 fots fiskeb.t", 1967. 5KB II/M7 Moe, J. and Lund, S.: 'Cost and Weight Minimization of Structures with Special Emphasis on Longitudinal Strenght Members of Tankers', 1967. 5KB I/M3 Moses, F.: "Some Notes and Ideas on Mathematical Programming Methods for Structural Optimization", 1967, SKB II/M9 Fredriksen, K.E., Pedersen, G. and Moe, J.: "Strength of Wooden Ships. Part II. Full Scale Tests of Glued Laminated and Conventional Wooden Frames", 1967. 5KB II/Mio Ivorsen, P.A.: "Triangulare elementer med 12 frihetsgrader', 1968. 3KB Ii/Mil Hagen, E., Leegaard, F.0., Lund, S. og Hoe, J.: "Optimalisering av skrogkonstruk- sjoner", 1968. 3KB II/M13 Kapkowski, J.: "A Finite Element Study of Elastic-Plastic Stress Distributions in Notched Specimens under Tension', 1968. .K3 II/Ml4 Moe, J,: "Finite Element Techniques in Ship Structures Design, 1969. .L II/M15 Beyer, E., Gisvod, K.M. and Hansen, H.R.: "OUtline of a General User-oriented Computer Aided Design System, as applied to Ship Design - BOSS", 1969. 3 II/M16 Moe, J.: "Design of Ship Structures by Means of Non-Linear Programming Tec- nicues", 1969. 3FB II/:Ii7 Lund, S.: "Tanker Frame Optimization by Means of SUMT-Transformation and Behaviour Models, 1970. F3 II/M18 Gisvoid, K.M.: "A Method for Non-Linear Mixed Integer Programming and its Application to Design Problems", 1970. 3KB II/M:.9 Hagen, E. and Leegaard, F,O.: "Cost and Weight Optimization of Tanker Sections in Oil Tankers', 1970. XB II/M20 Hansen, H.R.: "Man-Machine Communication and Data-Storage Methods in Ship Structural Design', 1970.

an de ol, Dept ñ2.1 nderadI1fl L) CN Meddelelse S M23 ON VIBRATION OF SPHERICAL SHELLS INTERACTING WITH FLUID by Tor Vinje March 1972 TECHNISCHE (JNIVERSITEIT Laboratorium voor Scheepshydromechanlca Archlef Mekelweg 2, 2628 CD Deift Tel: 015- 786873.Fax 015.781838

Page (iv) r + _-._L-. + e e sin

Page 5 Equation (1.28) is read

Line number i from the top is read

w r

(Asin me + Acos rnO)F(cos)

(1.25) Jdrjfr,r2sin d dr (IV. 32) k:O

Equation (IV.31) is numbered (IV.32)

Equation (1.26) is read x r (Bsin mO + Bcos mO)P(cos) (1.26) is read - sin pr (CSsinmO+ C mk Ccos mO)Pk(co5c) mr o k:o

Line 3 from bottom of page 5

is read

the equations for each

m

are independent of the number

Line i from bottom is read frequencies, 5mk' Which are

independent of

m C

for each k).

Page 6, first line is read This means that for

mrO (axial symmetry) the Whole

Page 31 Line number 16 from the top is

read

compressibility, (IV.2l) may be rewritten

in the following Page 33 ERRATA Equation (1.27) is read D: h3 is read D: Eh3 Equation (1.28) is read 12(1 - y2) 12(1 -y2) VP(cos) r -k(k+l)P(cosq) Page (Vi) 2 r 12()2 is read l2() 32 1 a 2 3 1

13

V

r

(sin

h

+ r2sin24 + T (r -) is read 2 V r r 1 32 V1 r hsin4

h + $j2g

Page 3 Equation (1.13) is read

cotg

+ (1+2v)r)

Page 60 Equation (VIII.1O) is read O K(u1) AQU1 -Aiuo11e ?so4)OrR -C < ct (VIII. 10) (X0141+ >14)o)Ir_R >

Page 78 Equation (3) is read Page 79 Line number 8 from the top is read

P1(x) r

(P (x) r

n

Page 81 Line number 9

troni the top is read

d°

(x2

1)n

2n.n dxt

V2(pm(COS) 51°(rnO))r _n(n+1)pm(cos)sin(mO)I

n

cos

n

cos

Page 36

line number 7 from the top is

read

According to (V.11) and (V.12),

(V.5) is automatically

ful-Page 46 Equation (VII.5) is read f() r

V '0, Eh ' R nro (c2M+ 1)i4,

Equation (VII.7) is read

B P (cose) no

2n-1

n-1

P(x) r ---xP0_1(x) - --- P0_(x)

(3)

Line number 4 from the bottom is

read

Page 57 Line number

I

from the top is read

n

3

which is indicated in

Fig. (VII.6) and (VII.8).

Page 59 Equation (VIII.7) is read

2

A survey of recent works done on vibration of spherical shells interacting with fluid is given in this report.

The eigenfrequencies of completely filled, half

fluid-filled, fully submerged arid semisubmerged unconstrained sphe-rical shells are found.

The effects of compressibility of the fluid, the effect of the potential energy of the free surface of the fluid and the effect of prestressing of the shell are discussed.

In addition the eigenfrequencies of nearly half fluid-filled shells with different shell-thickness in the upper and in the

lower part are found.

ACKNOWLEDGEMENT

This study

is carried out

at

the

_{Department of Ship Structures} of the Technical University of Norway for the

lic.

techn.

degree.

The author will thank the University for covering _the

expenses of the study, and the staff of the Department for valuable suggestions and discussions during the work.

Special thanks go to Mrs. Irene Norvik for typing _{of the} manuscript, and to Mr. Gudmunn Johansen, who _{prepared the}

S UMMAR Y

ACKNOWLEDGEMENTS _i

CONTENTS _ii

NOTATIONS _iv

CHAPTER I THE GOVERNING EQUATIONS OF THIN

SPHERICAL SHELLS _i

CHAPTER II VIBRATION OF EMPTY SHELLS 7

CHAPTER III AXISYMMETRIC VIBRATIONS OF CLOSED

COMPLETELY FLUID-FILLED SPHERICAL SHELLS 13

CHAPTER IV THE EFFECT OF COMPRESSIBILITY OF THE

FLUID IN THE CASE OF COMPLETELY

FLUID-FILLED CLOSED SHELLS ₂₅

CHAPTER V

CONTENTS

AXISYMMETRIC VIBRATIONS OF CLOSED COMPLETELY FLUID-FILLED SPHERICAL MEMBRANE SHELLS WITH DIFFERENT SHELL-THICKNESS IN THE UPPER AND IN THE

LOWER PART

CHAPTER VI THE EFFECT OF PRESTRESSING OF THE

SHELL ₄₂

CHAPTER VII FREE AXISYMMETRIC VIBRATIONS OF

SPHERICAL SHELLS, HALF FLUID-FILLED. 45

CHAPTER VIII _{FREE VIBRATIONS OF NEARLY HALF}

FLUID-FILLED SHELLS ₅₈

Page

34

CHAPTER IX THE EFFECT OF THE POTENTIAL ENERGY

OF THE FREE SURFACE OF HALF

Page

CHAPTER X VIBRATIONS OF SUBMERGED SHELLS 66

CHAPTER XI THE TRANSIENT PROBLEMS 70

CHAPTER XII CONCLUSION 714

APPENDIX 0.A References 75

APPENDIX l.A Legendre functions ₇₈

APPENDIX II.A Assymptotic integration 83

APPENDIX III.A Short discussion of the

assumptions (111.1) - (111.3) 87

NOTATIONS

In addition to these there are some local definitions.

Legeridre coefficients A n B n C n D n M0 N0

N0

c Velocity of sound in fluid

c1 z c2

zl+v

c3 h3 - 12(1 -

y2)

Young's modulus h z Shell-thickness 1T O I . I .

J P (dos)P.(cos)sin

d JP(x)Pi(x)dx ni, n,i 11 -1 1 J J .

JP(cos)Pi(cos).sin

d fP(x)P.(x)dx ni, n,i o o

J(x)

z Besselfunction of order p. K z Linear differensial-operator ----h s = Bending moments Membrane forces D E

p k P (x) n P (x) m Q Qe R r r' t t ti t2 U u, u. j-u Ue V V w, w. F(x) Radial loading

z Isotropic pressure in fluid

Legendre polynomials

z Shear forces

z Mean radius of shell

z Radius (spherical coordinates)

z Displacement vector of fluid

z Time (in Chapter I and II)

z Shell thickness (in Chapter V)

z Displacement potential of the shell z Shell displacements vector

z {u,u0,w} referenced to spherical coordinates

(, O, r)

z Shell displacements

z Shear force - potential of the shell

z Fluid velocity vector z Radial shell displacement z Gammafunction

.Yn z -n(n + 1) + c3

z-+

cotg.-z A small quantity with different meanings in

nfl z -n(n + O Spherical coordinate X A , X', X Perturbation coefficients of A n n n V Poisson's ratio

l2()2

PS z Density of the shell

Density of the fluid

z Spherical coordinate

Fluid velocity potential

Fluid displacement potential

W, W z Eigenfrequencies Q2

Q, Q

_n V V < > < > T < > < > a.E: ,

0

-t a wR V'E / p s 1 1

(sin

h

i

2 d z

_T

_rsin

_;; cÎI 1 . ) z _--(sin - sin2

z Scalar inner products, defined in Chapter VIII

The coma means differensiation with respect to the variable succeeding

- da

Fig. (1.1) The spherical shell

The shell treated is supposed to be a closed spherical shell, with mean radius R and with homogeneous thickness h. The shell

is made of a linear elastic material of density p5, and the ratio (h/R) is supposed to be so small that the thin-shell

theory may be used. The governing equations of the shell

are taken from Shan, Ramkrishnan and Datta /1/ (here

re-written in Kraus' /2/ notation). The effect of rotational

inertia and shear deformations are taken into account. The

equations are as follows:

i

N + (N - N )cotg

+ N90

ir +

e

where

N0

+ N00

+ 2N0 cotg _{+ Q0}

1

+ M + M 0,0 (M - M0)cotg - RQ p h3 S ( 12 C2u + Rk_{r q,tt}J i +

+ M00

2M0 cotg - RQ0 p h3 12 (C2u0 tt + Rk

rO

1 + M _{+ MOrO} sin

Mr cotg

- (M + M0) - RN , r 12 C2w,tt + Rk

rr

2 k2 -1

C22,

k1 -1+..,

k (1 ± )2 _{(+ is used if external pressure, - if}

internal).

is the radial pressure introduced by the author to take into account the effect of interaction with

fluid. (1.2)

hR(kìu0tt + Rk20)

s + Qe,0 + Q cotg - (N0 + N)

Phkiw,tt + Rk2

- Rk p

_pr

(1.3) (1.5)

The relations between the forces and stress-couples and the displacements and rotations ( - s) are as follows:

N Eh (u + (1-2)R vucotgq + (1 + ')w + J +

u00

r Eh (\)11 N -e

+ ucotg

+ (1 + 0,0 + VRr) (1.7) (1.8) N r -Eh

{(u

+ ucotg

(l-2)R + 2w + u ) +

+ ßcotg

(1.9) e,e sin4: i ) +

+2

+ r 0,0 sin N 0 Eh -

+u

,e (1.10) u0cotg

2(l+)R0,

Q Eh - u + (1.11)

2(l+)Rk,

Qe Eh i U ₊

R)

(I. 12)

2(l+)Rk5,

Sjfl

+ v + 0cotg ₊

(i+2))

(1.13) 8,0 sincI Me + cotg + si

+ (i+2)J

(I.1L) D(l-v) + (I. lb) (s8, - bcotg 2R

where and M D(l-v) - 2Rk5 M D(1-v) Or 2Rk 8r,O sine s Eh3

Dz

12(1-v2) k

s5

-(1.16) (1.17)

(which is shown in /1/ to be a good approximation)

Supposing free, harmonic vibration, with frequency w,

intro-ducing U, V, x and A in the following way:

u z u - Xsin (1.18) i z _{8 sin (1.19)} z V - Asin (1.20) , z V 1 _(1.21) e

0 sin

Inserting these into the equations (1.1) - (1.17) and

separating w and X, one gets:

(8v

+ E6V

+ eV

+ 2V ₊ z + 6oJPr (1.22)

(v

+ y2V + z Q (1.23) re 2 1 V1 z _{---(sin .)} + sincP (1.214.)

The reader, interested in details about the

6'S

and y's, will find them in reference /2/.

The (S-s are given by:

::_

Eh ₆₆

R i

where 66 is taken from /1/.

When dealing with closed spherical shells, w,X and

r may be

written formally in the following way:

w (A5ksin mO + Ackcos m8).Pk(cos)

m o

ko

X (BS_mksin mO + BC cos mO).Pk(cos)

mk m

ko

mz o f s . CC mO)Pk(cos) '..0 sin mO + cos m mk mk m o kzo

where the dependence on t is omitted.

In the following, relation (1.28) is of great importance.

This is taken from Appendix (I.A).

VPk(cos)

_m(m+l)Pk(cos) _(1.28)

When inserting (1.25) - (1.28) into (1.22) and (1.23), there is found that these equations are separating in O, and that the equations for each k are independent of the number k.

(1.22) and (1.23) now turn into secular equations in the

frequencies, _Wmk which are independent of k (for each m).

(1.25)

(1.28)

This means that for n. O (axial symmetry) the whole spectrum

of w-s is found. (According to Appendix (I.A) this is the

only value of m which gives the whole spectrum.)

This conclusion is based on the assumption that the Cks may be found as linear combinations of the Amk_5 and Bmk_S which is the case when

r is due only to the dynamic pressure

II. VIBRATION OF EMPTY SHELLS

When the shell is closed and empty (that means

r O), the

calculation of the elgenfrequencies is quite simple. This

calculation is done in the way shown in Chapter I, with the

simplification that Mr M0r _r and Nr are put equal to

zero. (According to the argumentation in Chapter I, only

axi-symmetric vibration is assumed.) The details of the

calcula-tion and the numerical results are taken from Long /3/. Instead of (1.18) - (1.21) it is, in this case, suitable to

introduce

u z

Q V

The equations, connecting U, V and w then become:

+ (l-2)(k1 + 2k2)Q2 + (1-))u + (i + (l+)k -) (1_v2) s + ((i + ) -

2k2Q2(lz))w

o (1 + (2k3 + kr)(l_V2)2 + (i))u + 2(l-)k ( + (1-2)k Q2

(1)

(l+v) VR s r 2k s - (

+ k(1-2)2 + (1-))w

û (II.)

- (l-)() +

(2

- k1(1-)2)w

o (11.5) (11.3)

(12

_{+ y2A + 10)u0} z Q _(11.8)

where B

lB

A -

+ cotg-

S --(sin

h

C2

k3- -,

w2p R s E E Youngts modulus. w eigenfrequency.

The equations (11.3) - (11.5) are solved in the way explained in Chapter I and the results are given in Fig. (11.1) - (11.3). Fig. (11.1) and Fig. (11.2) show the two lowest eigenfrequencies

as function of n for z l.2lO and for

l.2lO.

In

addition the results, when using the classical theory (putting

k2 k3 _kr k5 O and k1 1) is plotted. Fig. (11.3)

shows the highest mode as function of n. This mode does not

occur when using the classical theory.

The solution of equation (11.6) is taken from Wilkinson and }(alnins //, and the results are shown on Fig. (II.'4). In this case the high-frequent mode is not found by classical

theory.

Remark that the solution O for n = O is found both

for the tortional mode and the bending mode. These modes are

due to ridged body rotation and translation respectively. When comparing the results, found by the classical and the improved theories, one finds out that for n reasonably small

and for > l.2lO ( < -) the results show good accordance.

When dealing with spherical domes, the introduction of the boundary conditions somewhere at the shell (usually at the

circle z ÎS complicating the calculations. In the

case of the closed shell, the solution was to be found in

z 1.210'

.ú1)

Improved theory MemLrane theory

'f

/

J. 4 5 7 n z 1.21U ( z 0.1) Membrane theory -.

-.-'laosical bending theory _{Improved theory}

4 t

L

-n L Fig. (11.1) Elgenfrequencies of closed, Fig. (11.2) Eigenfrequencies of closed, empty shells empty shells 9 8 7 6 '4 3 2 i

140 3 1. 22 20 IB 12 lo i.7.IO (4 Membrne and improved theory

I

Fig. (11.3)

Upper branch in the case of

Fig. (II.)

Eigenfrequencies for tortional

imiroved theory

vibration ¿f closed, empty shells.

Improved tflecr only n i 3 14 6 7 8 10 ii 12 13 14 15 US) =

terms of series of Legendre polynomials of integer order; in the general case, however, the order of the

Legendre-func-tions used may be of complex order, and this number may be a function of the unknown eigenfrequencies. To solve the

seculare equation, which occur when inserting the solution

of the equation into the boundary condition, is then a

rather cumbersome task.

To simplify the calculation, Langert s /5/ method of

assympto-tic integration may be used. This method is described in

Appendix (II.A). The simplification is that one gets the

solution in terms of Bessel-functions of order zero and of

order one. These functions are tabulated and described in

detail in the literature, which is not the case for the Legendre

functions of not-integer order. As mentioned in Appendix

(II.A), the equation, found by this method, coincides with the shallow-shell equation at the apex ( z O) and with the

Geckler approximation near equator ( z Here the results

for spherical domes are presented as they are found in Kraus /2/, Naghdi and Kamins /6/, Ross /7/ and Ross and Matthews /8/. These results are due to the classical theory. Fig. (11.5) shows the eigenfrequencies of hemisspherical caps with different boundary conditions (at equator). The upper branch

is due to typical longitudinal modes, the lower is due to

bending modes. Fig. (11.6) is taken directly from Kraus /2/,

and shows the variation of the lowest frequency of clamped

r 0.3

r 30

r 1 2 3 rrp ly u1urt. 5 £ 7 8 9 L1ìntped nd with free îg Simply supported 10 lI 12 13 1 iS n Fig.(II.5)

Eigenfrequencies of hemispherical caps

a

Fig.(II.6)

The lowest eigenfrequencies of clamped spherical segments with c ïr/2.

o 0 2 0 L4 0.6 0.8

III. AXISYMMETRIC VIBRATIONS OF CLOSED, COMPLETELY FLUID-FILLED SPHERICAL SHELLS

When the vibrating shell is interacting with fluid, some

additional problems arise. In the first place one has to

choose the model, describing the fluid and its velocity field. In the second place, one has to introduce a fluid free-surface condition when the shell is not completely filled. These

problems are discussed in Appendix (III.A). According to

this discussion, the fluid is supposed to be homogeneous, incompressible and ideal and the velocity field is supposed

to be irrotational. In addition the displacements (and

velocities) are assumed small and the effect of the gravity

is assumed neglectible. This means that the velocity is given

by:

V z (111.1)

where fulfils the Laplace equation:

0 _(111.2)

In addition, the isotropic pressure, p, isgiven by the

Eulerian equation:

Because the displacements are assumed small, the

displace-ments vector, r' , is given by:

Dr'

- - z

-

(11I.'4)

dt

when neglecting v.Vr' (which is a small quantity).

one gets: r' where v2p o and p

The kinematic condition at the interface between the shell

and the fluid then becomes:

w-at r R.

in Equation (1.22), is then given by:

(111.10)

at r R.

When comparing the results given by the classical shell theory and by the improved theory in Chapter II, there was found

that for moderately small n-s (n 7, say) and () <

the effect of the improvements (rotational inertia and shear deformations) on the eigenfrequencies was neglectable. When

the shell is interacting with fluid, the total kinetic energy is raised, while the potential energy is unaffected by the

existence of the fluid. This indicates that the effect of

in a wider range of n and F, because of the kinetic energy, due to this improvements of the model, will be a smaller part of the total kinetic energy than in the case of the empty

shell. This is the reason that the classical shell theory

is used in the rest of this chapter.

According to the discussion in Chapter I, only the axisymmetric vibrations of the completely filled shell are going to be

regarded. Since the tortional vibrational modes are unaffected by the existence of fluid, u0 O is going to be assumed.

As in Chapter II, the following is introduced:

where

and assuming harmonic vibrations the guiding equations

become: RV

- (l-v)A() +

(2 -

(l-)Ç2)w

( + (l-v))U +

(1_2)

+ (l+v)w + (1-v2)22U o VR (L + (l-v))(U--w) - (l-v2) 0 (111.15)

vR

p s and E s - 2M+ 1-v -02 ₀ - _{+ cotg} (III. 13)

in addition

r rR

(111.16)

and

v2p o _(111.17)

The solution of (111.17), which is analytic at r O is

rn

D RF (cos)()

_(111.18)

n n

nl

(See Withaker and Watson

/l4/).

According to (1.25) - (1.27) one may write:

and U A P (cos) nzl n n Eh - P (cos) nzl n n w C P (cos) nfl n=l (111.19) (111.20) (111.21)

Remark that n O is omitted in (111.21) (and of course iri

(111.19), because hPo(cos) E 0). This is due to the fact

that the fluid is supposed incompressible, and that Po(cos) is the only mode which causes a change of the fluid volume. The time dependence is taken care of in (111.13) - (111.15)

and is therefore omitted in (111.18) - (111.21).

Inserting the series-expansions of U, V, w and i4i into

(111.13) - (111.16) and using the orthogonality conditions of the Legendre polynomials, the following equations occur:

A - C3fl B + 2C _-

2c3(Cn + MD) z

o nfl

nn

n y A + c1B + c2C + 22c1A O

nn

n n n y A - c1B - y C O nfl n nfl o z nD n n for n 1, 2, 3, ¿4 Here 02 z 1+\), 03 = l-v, _Ci z l-v2z

_C2C3

-n(n+l) and y z _n + 03 n

Eleminating B and Dn one gets:

n (1 )

n _C2 nnyn (2+ C2

i

o _Q2 + 0

03(1+)

n n

This algebraic eigenvalue problem is solved by calculating the roots of the second order equation which occurs when putting the determinant (of the matrix) equal to zero. The results are given in Figs. (111.1) - (111.5). Fig. (111.1) indicates that the lowest eigenvalues show small

variation with , but a greater variation with M. When

raising n, the variation with becomes greater. The same

tendency is shown in Figs. (111.3) - (111.5), where is

plotted for i z 2,3, as function of M+ and for various .

Fig. (111.2) shows that there is nearly no variation of the

zU

C n'

eigenvalues of the upper branch, when varying . What

is not shown at Fig. (111.2) is that the variation with is small, which is due to the fact that these modes

cause nearly tangential displacements.

Fig. (111.3) shows that there is a remarkable small

varia-tion of when varying E, and that for most practical

purposes the membrane theory is well fit for calculation

of this eigenfrequericies, which are the lowest ones.

9

Fig. (111.1) The lowest branch of the eigenfrequencies

for closed, fluid-filled spherical shells

M:1O,

Mi3, Çb1O

M:iO,

s 2 i ¶ =Û.j M:iO -

,s

-5. fi rJ' lys. i 2 3 6 7 8

Fig. (111.2) The upper branch of the elgenfrequencies

for closed, fluid-filled spherical

shells

Fig. (111.6) just shows M as function of for various

(p

Some of the mode shapes of axisymmetric vibration are shown

in Fig. (111.7) and Fig. (111.8). Fig. (111.7) shows shapes

of the bending modes, Fig. (111.8) the shapes of the

0.b o» 0.2 0.1 o Lb 0.0 0.3 0.2 X2 = 0.3 M 3, Mr O F.1 M7103

Fig. (111.3) X2

as a function of

for various

Fig. (III.)

À3 as a function of

for various

10 5.102 M

13 lU 010

5 13_I

2.102 102 101 . 10° O . a 2 0'

-4zH,

1.87.10 4. M u, = LO3 ,3'4.1O trtipZ.i .

Fig. (111.5)

X14

as a function of M

for various

5.101 102 s iu

r

T (TTT \

f

nf

fnr various (o

1o)

- z

r -T io 5. bc 1O 2.i0 5.i0 106 2.10' 102 5.i_,_. _O i0 2.1O 5.1Q

Fig. (111.7) The mode shapes of the modes of the lower branch (n2,3,,5)

If a further simplification of the calculation of the

elgenfrequencies is wanted, this may be done in some special

cases. That means when the variation of A2 with is

neg-lected (that means putting equal to infinity) and in

addi-tion one of the following assumpaddi-tions is done:

Neglecting the value of 2' in the secular equation

given by putting the determinant of equation (111.26)

equal to zero.

Neglecting the kinetic energy given by u.

Both assumptions lead to a equation of first degree with

the following solutions:

a) b) n(n-'-l) - 2

1-

+ (l+)(n(n+1) - (1-u)) + 2(l+) X? n(n+l) - 2

(l+)(n(n+l) -

(l-u)) n

Here An is the estimate of n

For n constant and - both A? and _{x? will}

assympto-tically approach

n(n+l) - 2

3_

(n(n+l) - (l-u))

According to the assumptions made, the following inequality

is valid: 2 x? < n (111.27) (111.28) (III. 29) (111.30)

Fig. (111.9) and for n 2 and n 3 and for

various M+. 0.3.

In Fig. (111.9) the ;alues of the n_ and are shown for

0.3, n 2 and n 3, and for various values of M+.

is calculated for 3.106. n

n= 3

M X O 0.507 0.540 0.755 0.718 0.757 0.885 1 0.379 0.400 0.503 0.567 0.590 0.661 10 0.116 0.119 0.126 0.194 0.197 0.204 50 0.0285 0.0286 0.0290 0.0301 0.0494 0.0497 0.049g 0.0532 100 0.0147 0.0147 0.0148 0.0151 0.0257 0.0257 0.0257 0.0266

IV. THE EFFECT OF COMPRESSIBILITY OF THE FLUID IN THE CASE OF COMPLETELY FLUID FILLED CLOSED SHELLS

When supposing the fluid to be compressible, its governing equation is, according to Appendix (III.A):

z c2V2

This equation has been treated in detail in a lot of

text-books on mathematical physics. The solution takes a special,

simple form when the problem is given in spherical coordinates:

jut pm( SJfl m .j

1(r)

e _cose) (mO)(A r n cos n n+ c mz o nzo w + Bmr_.J_(fll)(r)) n

(See for instance Sneddon /11/).

Here J(x) is the Bessel function of order p.

J (x) - as x - O _(IV.3)

p _2F(p+l)

This implies that the solution, that is finite at r = O,

is given by:

lut _{Pm(cos) sin}

)ArJ 1(r)

4) e (me cos n n+ c or m= o

no

i J

() -

i< r1' c n

when ()

O (IV.2)

L

i wt _{sin n} - e

AmF(COS)

(mO)r nfl cos nZo

mo

which is the way to expand i if incompressible fluid is supposed.

According to the series expansion of the Bessel functions

xt if 8( 2n+l)( 2n+3) or in our case If

()2

« 1

2 (2n+ 3) x 1 i x2

2F(n+) j

_1(x)

-

2(2n+3) n+

the fluid may be assumed incompressible.

This is the extreme value, when putting r R. The mean

value over the fluid volume is found to be 3/10 of this

maximum.

In the case when c 1500 m/s, R 15 m, n 2

11

and

(IÚR)2

c _{- 1.8.10_2}

which is much smaller than 1.

This indicates that for even great R-s the effect of

com-pressibility may be neglected.

To examine the effect of raising R when n is kept constant

may be done in the following way: 2(2n+3)

« 1

(IV. 6)

as n is constant and M+ -

,

_or 2 h (

K

-n R3 as n is constant and M+ According to this

(fl)2

c h 2(2n+3) r\

when n is constant and M+

In other words: The effect of compressibility is shown to

be neglectible when raising F., and keeping n constant.

According to equation (111.26) - K1n2 as

is kept constant. This gives that

(WR)2 c

_-K

2(2n+3)

fl-const.

R>5

h n-p. n K2 (IV.9) (IV.11) + and M (IV. 12)

In other words: To assume the fluid to be incompressible

is only valid for small n.

One may give an estimate for the upper bond of the n, for which one may calculate the eigenvalues by means of

the incompressible fluid model when is kept constant.

Here this calculation is restricted to typical membrane

shells. I.e.:

n

l.l5/7T

-In this case one may use the fact that Q2< X', where X"

is given by Eq. (111.28). According to this:

wR

2 C < 1.15 2(2n+3)

-In other words: If the eigenfrequencies may be found by

using the membrane theory,and in addition ()2/2(2n+3)

_{« 1}

for n _{2, the eigenvalues may be calculated by means of}

the incompressible fluid model.

Remark that (__)2/2(2n+3) may be small, even if (IV.13)

is not fulfilled. The following will ensure this.

The exact solution, using the compressible fluid model, is

given by Advani and Lee /12/. They have been using the

assumption (IV.4) together with the conditions (111.8) and

(111.9). Instead of (111.26) they have got an equation in the following form (when assuming axisymmetric vibrations):

2(2.2 +3)

(RJ1)

p2(Q2;n)

(Q2;n)

1(rJ

(-.) dr n+ c for each n. Here _n -n(n+1) and

is a polynomial in Q2 of degree m and with

coefficients which are functions of n Advani and Lee /12/ have calculated the solution for an

idealized brain/skull system. In this case the following

constants are assumed:

z 0.2

c l'-60 rn/s

/E/p

2500 rn/s

z _0.1

J_n+21(x) is oscillating in

x,

so that there exist ari infinite

number of roots of Eq. (IV.15).

The results of Advarii and Lee /12/ are plotted in Fig. (IV.l).

Only the three lowest roots are regarded and compared to the incompressible fluid solution and with the empty shell

solution. There is a remarkable good agreement between the solution using the incompressible fluid model and the

solu-tion for m z 1 when using the compressible fluid model.

The curvature of the line for m z justifies the solution

n z 3 of (IV.13).

z 47

and the solution becomes

z 0.46 (lowest, for n z 2) In this case: (LR)2

4.5.lO_2<

1

_«

_(IV.

16)

2(2ri3)

_nz2

o

0.56 y 0.2 .7 l.2l0

The compressible fluid-modeL

3 L _{5 6}

mr

mr i

Fig. (IV.l) Results, using the compressible

fluid-model, compared to the results, using the incompressible fluid-model

When

» 1

and K2

E/p5c2

is of order 1, one is able to calculate the effect of compressibility (when assumed to

be relatively small). As found before, in most cases the

eigenfrequencies for moderately small n and great are

mainly varying as 1/Mt In this case one may assume:

Q2 X X1 + s2X2 +

1

where

« 1.

X The empty shell

The incompressible fluid-mode:

7-and in addition:

1) z + + E21P2 +

and

u z _u0 + su1 + s2U2 + _{. (IV.l9)}

where U is the displacement vector of the shell.

In this case (IV.1) becomes:

-K2X1,U z _{R2V2iL (IV. 20)}

The governing equation of the shell may be written formally

in the following way:

K(u) -

Au _Alper (IV. 21)

where the operator K is defined according to Chapter III.

Remark that Au at the left hand side of Eq. (IV.18) is of

order s and produces a correction in X of the same order.

(This is the correction which is neglected when putting .) When one is only interested in the effect of compressibility, (IV.18) may be rewritten in the following

way:

K(u) z

Xe

_{(IV. 22)}

Separating (IV.20) and (IV.22) in powers of s (remark that (IV.17) - (IV.19) are assumed valid for all s which are small enough to ensure convergency) one gets:

Power 1.

K(u0)

z

Power 2:

K(u1) z

(X11 + X20)e

-K2X11)0

and so on.

In addition one has got the kinematic condition at r

which in separated form is written:

rzR

for all powers (i) of .

The solution of (IV.22), (IV.24) and (IV.27) is the in-compressible solution (except for X1U in Eq. (IV.2l)). The operator, K, is selfadjoint, <u1 K(u0)>

where

R2Jf()sin d

arc therefore:

z X1<iwo> + X2<iowo> (IV. 28)

According to (IV.27)

<wio> - <wo> =

< o> - _{(IV. 29)}

Using the integral theorem of Gauss on Eqs. (IV.2L1.) and

(IV.26) one gets:

° where K2X -

<10>

- R <PO1PO>T (IV.27) (IV. 30)

JdrJf(r,)r2sin

d dr

Introducing (IV.30) into (IV.28) one gets:

2>

T

X2 z _K2X

<lj) oWo>

or by using Gauss' theorem on Eq. (IV.32)

2> T

X2 z _K2X (IV. 33)

As a numerical example the following is given:

2 z _50,

z 3l0,

Qz

X1 z 2.8710 (nz2) K2 11.2 (water/steel) and u0 0.97P2(cos)e

02dP2(cos)

r d e z

0.85.RP2(cos)()2

Inserting this into (IV.32) X2 becomes:

X2 z -1.80

or

Q2

Q(l-2.l02),

Q

Q0l-1.2102

The conclusion of the discussion

in

this chapter is that for most practical purposes, the incompressible model is welifit when only calculating the lowest eigenvalues. The higher

order solutions may be affected by the compressibility. The

t2

+ ti

and

Introducing t and E as variables one gets:

t z + E

t

z 1 - £

The governing equatiun of the shell is written in the

follow-ing way:

+1

K(u) - Xu - X _er z EX

e

2

-i

(V.1)

where U is the shell displacement vector,

z ç2 _and

K _{is a operator defined according to Chapter III.}

z

PS t

The boundary conditions given at z _are:

z _u2 z _(V.2)

V. AXISYMMETRIC VIE ATIONS OF CLOSED COMPLETELY FLUID-FILLED SPHERICAL MEMBRANE SHELLS WITH DIFFERENT SHELL-THICKNESSES IN THE UPPER AND IN THE LOWER PART

The thicknesses of the shell are given as

hzt1

for Tr and

hzt2

for Tr

The following definitions are introduced:

2t1t2

- t2+ti

-where the index 1 means

(um

) and the index

11m

2 (

Expressing (V»-i) by u and using (V.2), one gets:

and (w )i (w _,q )2 w

(N)1

(N)2

t1(u)i

t2(u)2 + (1+v)(t2-t1)w

at By Introducing w _uSer w P (cos)

n1

n n where u U U'P (cos)

nl

n n U U2P (cos)

nl

n n Tt lT (V.5) (V.8) (V.9) (V. 10)

)"RP(cos)

(V.7) nzl

where harmonic oscillations are supposed, the kinematic

condition

w at rR

(V.8)

is fulfilled in addition to Eq. (V.3) and the r-component

of Eq (V.2).

The Legendre polynom ls fulfil the following conditions and d2 P (cos) n -

P(cos)

O for and n odd P (cos) O for -d n -and n even

According to (V.11) and (V.12), (V.6) is automatically

ful-filled for n odd, whilst the _{-component of (V.2) automatically}

is fulfilled for n even. By putting:

and

U' U2 U

n n n

the -component of (V.2) is fulfilled for all n-s.

Introducing: U1

(l+)U

EW n n n(n+l) n 2 l+v U _{= (1-)U} + EW n n n(ri+l) ri for n odd (V.13) (V.14) (V.15)

then (V.5) is fulfilled for all n-s. The assumptions (V.6)

-(V.15) now ensure that the boundary conditions (V.2) _{- (V.5)}

are fulfilled.

Using Galerkin's principle on (V.1); multiplying by

(sin P1(cos)) and integrating by from = _{O to =}

n, oe

where (-1(1+1) + (1-v))U1 + (1+)w. - (1-2)Q2U._i 21+1 + 2 (-n(n+1) + (1-v) - c22)(J .- I .). ni ni 1+') (U -n n(n+1) n + (i(i+1))U. + 2w1

-w)

21+1 _{-n(n+1)(J .- I} .)(U n(n+1 n ni ni n + 2 nz2,,6,8. w - (1_v)M+Ç2

Jni - I) }

o n1,3,5,7. rr/2 J - r P (cos)P1(cos)sin d

ni

J n o and TÍ I . P (cos)P1(cos)sin d ni _J n Tr/2

In most cases c is relatively small, so that perturbation

(V. 18) Because P I . ni in the more (A -where (-x) n 22B)x u}

(-l)'J

compact (_1)flp (x) n ni matrix form +

(C - 22D)x

(V.16) 0 and (V.17) (V.19) become: (V.20) (V.21)

technique may be usec. Suppose that e is that small that

the following expansions are valid

Xo + eX1 + e2X + _(V.22)

and

x X, + ex1 + e2x2 _(V.23)

Introducing this into (V.21) and separating powers of e,

one gets: Power e: (A - X0B)x, O _(V.214) Power e': (A - X0B)x1 - À1Bx0 + (C - X,D)x0 0 (V.25) Power e2: (A - X,B)x2- À,Bx, - X2Bx,4- (C - X0D)x1- X1Dx0 0 _(V.26) and so on.

Suppose that (X0,x0) is one solution of (V.214), calculated according to Chapter III.

Because xA Ax for x chosen freely, it is also necessary

to introduce the solution (X0,x) of the problem

*

x,(A - A,B) 0

(V.27)

where is normalized by:

*

xoBxo

1 _(V.28)

The solution (X1,x,) of (V.25) exists in the form, supposed,

x(A

-X0B)x1- X1xBx0

+ x(C - X0D)x0 0 (V.29) or by use of (V.27) and (V.28) * X1 -x0(C - X00)x0 * x0(C - X0D)x0 O because J

-I

0 nn nfl This makes X1 0.

X1 is then given by:

(A - À0B)x1 -(C - X0D)x0 _(V.33)

The solution of this equation exists according to (V.29)

and might be found by standard methods.

The solution

(À2,x2)

of (V.26) then exists if and only if:

* * 14

x0(A -

X0B)x2 - X1x0Bx1 - X2x0Bx0

+ x(C - X0D)x1 -

A1xDx0

O (V.3L)

Because (J . - I .) O if (n+i) is even, the solution

ni ni

of (V.33) only has got odd components if Xo consists of one

even component and vice versa. This means that

xBx1

O

because B is diagonal. According to (V.32)

xDx0

z _0.

This gives that:

(V.30)

In Chapter III there was found that the solutions of (V.2k) separate in index n and the same is the case, of course, for

(V.27). According to this:

If (X0,x0) is due to typical bending mode, then

(Uom)2 «(worn)2. In the normalized form Worn is equal to i

and therefore:

Wom

(l-u) (1+)

rn

according to (V.28).

Equation (111.28) gives approximately:

X o

-rn(m-1-l) - 2

(i+)(m(m+l) - (l-u))

m

for

M+»

m.

Because (X0,M) then will be of order (M)°, X1, given by

+ _o

(V.33) will be of order (N ) , and therefore X2 will be,

ac:cord-ing to (V.35) of order (Mf)'.

2

In this case À

X0(l +

)

and _X2/À0 is of order

(M+)0, _{which means that the relative correction on X0 is of}

order 2

One complete calculation is performed in the, following case:

z _{18 m}

z 37.L mm

lo

87 (Al-shell filled with LNG)

X0 is found to be

X0 1.7.10_2 for m z 2 and _À2/À0 -0.76

This gives that

X

X0(l -

0.76 £2) X0(l 1.3.10_2)

(V. 36)

arid

Qo(l - 0.65.10_2)

In fact, the complete spherical shell, filled with fluid is supported and influenced by the gravity field, and therefore

prestressed. To calculate the effect of this prestressing

on the eigenfrequencies, it is necessary to take into account non-linear (static) effects, which is rather difficult. To

get an estimate of this effect one may suppose that the shell is acted on by the mean inner static pressure

P,gR (VI .1)

(g acceleration of gravity)

throughout the whole sphere.

In this case, one may follow Flügge /10/ (Chapter 7.3), introduce the vibrational terms and get instead of (111.22)

- (111.25)

A

- c3rB

+ 2C - 2c3(C + MD )

nn

n n n n

c3R2Pg

This leads to:

(î

A +

(ii +

L.)C ₎ O 2Eh

nn

n n

y A

+ c13

+ c2C + n nfl n n

ciR2Pg

C -A)zO

- 2Eh n n y A - c1B - y C O

nn

n

nn

(VI. 2) (VI. 3) (VI. L4) C_n nD (VI. 5) n

By assuming:

-(i

+ n) (l(C2

-

H) Ci (n

(l-b)

-n c2 c3nnll) c3(fl+ )ll - H2 i o o

c3(l+)

n where 2Eh M' (;.)

v"g1V(E/P5) is the ratio between a velocity, characterizing

the gravity field, and the velocity of sound

Th the shell.

Jut of (VI.6) one gets the following secular equation:

c3(l+)H

+ ( n y

_nn

n c2 + c3(n+)ll +

((l+) +

ll))H2 yn

((cz

Ci c - cll) C2 0311) 0 (VI.7)

Putting R = 15 m, v'E/p3 5000 m/s, RI h 100 and

= 1/8, 11 is given as

which indicates that 11 is a small quantity.

o (VI.6)

A n

s

or

ç2 o

+ HAÌ +

* +

introducing chis into (VI.7) and separating in powers of II, one gets for

**

* 2C3(l +

)X0X1

+ c3(X + 5

+ )X0

n n + (-2 +

c3(l +

i

* c2 n c - 2). (1 n

---)0

- * Y *

c3(î

+ 5 + -)X0 - 2X (l-e) n n n nfl

2c3(l+)X+(

n Y n 2 _c2

+c3(i

n _Ci *

Assuming

«1, n

2 and (111.28) for Xo, one gets:

0.18

(VI. 8)

(VI. 9)

(VI. 10

(VI .11)

This means that the effect is neglectible, not only because

that II is small, but also because X is small.

According to (VI.11)

.*rI gR

2 _(F/p

s

which is small.

For the parameters supposeU:

Xll 0.09

5.610

VII. FREE AXISYMMETF1C VIBRATIONS OF SPHERICAL SHELLS, HALF FLUID-FILLED

The free axisymmetric vibration problem of spherical shells containing fluid has been treated during the latest yes by

Rand and Di Maggio /18/, Advani and Lee /12/ and IKumar /19/,

all handling the problem of a shell, completely filled with

fluid. In addition to that, Taj and Uchiyama /20/ have been treating the problem of free vibrations of hemispherical caps,

filled with fluid.

As far as it is brought to the author's knowledge, no work

has yet been done on the vibration problem of complete spherical

shells, partly filled with fluid. Generally formulated this

problem is untractable, because of mathematical difficulties, but limitting the treatment to the half-filled shell, with the highfrequency assumption made in the free surface condi-tion of the fluid, the problem is solvable, using the Galerkin

principle in a some unusual way.

There might be shown (by comparing the equations found in // with the equations in /3/ and taking advantage to the fluid pressure) that the same eigenvalues will be found by

non-symmetric vibrational modes as by symmetric. The tortional

vibration is not affected by the existence of fluid inside the shell, so these modes are of no special interest here. So the loss in generality is small when only regarding the axisymmetric, non-tortional vibrations of the shell.

The governing equations are given, according to Chapter I

and u and Q are expressed, according to Eqs. (11.1) and

(11.2). Remark that

r is put equal to zero for O < <

The boundary condition at the free surface, , is given,

according to Eq. (16) in Appendix (III.A).

O at

The governing equatio.s then become: where where c1R

(A+c3)U+c2w+_V+c1Ç121J

O (A + c3)(U - w) c1R

VO

Eh V) 2w - o3Q2w

f()

A -+ cotg 02 1 + \); 03 1 -ci

c2c3;

+ v R M l2()2 and ç2 2

following expressions are introduced:

U A P (cos)

_nn

nzo co - c3R B P (cose)

nn

no

O,

f(4)

(VII. 5)

22M'

-- , < (VII. 2) (VII. 3) (VII.L.) (VII. 6) (VII. 7)

w = C P (cos) nfl

no

2n+1 RD P

(ccs)()

n 2n+1

no

according to Chapter I. Separating the coefficients, one gets

from Eqs.(VII.2) and (VII.3)

y A + c1B + C2C + c1Q2A z _Q n n ri n n y A - c1B - y C O

nn

n

nn

for n z 0, 1, 2, Here -n(n+l) + c3 y n

According to the Galerkin principle, using the weighting

function, sine, Eq. (VII.4) turns into:

A - c3fl B + 2C - c3Q2C n n

nn

nn

-

M+c32

2n+l _D

2i+ln

z Q

2.

_izo 0, 1, .. . , n -n(n+l) and

I,tz

(See Appendix (VII.A).

(VII. 8)

(VII. 9)

(VII. 12) (VII. 10)

D n

0jTj,2n+l

n 2n

J o

n 0, 1, 2,...,

Eleminating the B and the D between

(VII.10)- (VII.1L),

n n

one gets after some algebra the following equation in the

matrix form: (r - 22A)x O (VII.15) where (A.) and c (c.) J J , I O and A F3 F O E where Y. E ..

-(l + )..

11J Ci 13 1 Y1

12ij

(Cz

-F . .

î-.(l ).

313 1 13

By using, what Collatz /21/ calls "Boundary methods" the

equation

r w at

rR,

(VII. 13)

is multiplied by sin.P21(cos)

and integrated between

and 'rr. This gives:

(VII. 1L)

a

x=

C

I.. z 1J JJ 21-1-1 E..

c3(.

+ 2 2n+1

'j,2n+12n+1

,1 1J 1] nzo for i 0,1,... , arid j

Here is the Kroenecker-ó, given by

O

ij

z

'J _i

izj

By taking i equal to 1, the root Q2z O is found. The

dis-placement vector for this mode is given by: u z -sine, w cos4, which corresponds to a vertical rigid-body

trans-lat ion.

By putting i equal to zero, one gets

D30 z

Because of this, the mode (U z P0(cos), w z 0) separates

with the eigenvalue

Q2

Ci 02 ç

Q is less than zero, which seems curious. But by taking

a look at the eigenvector, one observes that the displacement

vector u z {u,w} is identically zero.

To get the solution of (VII.15) one has to limit oneself to a finite number of unknowns, hoping that the convergency

Then the problem is rdud to find the eigenvalues and eigen-vectors of the following equation

(G - XL)y o (vII.16)

where G and L are finite submatrices of r and A.

If G' or L' exists, the problem is simplified to find the eigenvalues and eigenvectors of one of the following equations:

(L'G

- XI)y O

(VII. 17)

(GL -

I)y O

where I means the identity matrix.

This is a well-defined algebraic eigenvalue problem treated

in every textbook on numerical analysis.

In this case L' exists and the solution of (VII.16) is found

by standard methods.

The convergency of the lowest dimension of y is found to be

10 and

5l0

(which

when putting the dimension of

48 are shown in Fig. (VII.l).

Q3

eigenvalues when raising the

quite good. The results for

means a typical membrane shell)

y equal to 12, 18, 24, 36 and Q₆ 12 0 0,'432 0.558 0.702 0.896 1.043 18 0 0.431 0.548 0.620 0.684 0.844 24 0 0.431 0.547 0.619 0.668 0.705 36 0 0.431 0.546 0.619 0.667 0.705 48 0 0.431 0.546 0.619 0.666 0.705 Fig. (VII.1)

1.2 1.0 0.8 0.6 0. 0.2 O

The calculations show That the eigenvalues are given by 3

branches: The upmost nearly coinciding the upper branch for

the empty and completely filled shells, as shown in Fig. (VII.3), the lowest lying near to the lowest branch for the completely filled shell and the middle lying near to the lowest branch

for the empty shell. The two last ones are shown in Fig. (VII.2),

together with the eigerivalues for the empty and the completely

filled shell. The n-values for these coincide with the index

of the Legendre polynomials.

D

o Completely fluid-filled shell

Half fluid-filled shell

D Empty bel1

r

3 4 6 7 8 9

Fig. (VII.2) The lowest branch and the middle branch of the

eigenfrequencies of half fluid-filled spherical shells, compared to the eigenfrequencies of completely fluid-filled

iO y 0.3 510'

O ('ornpletely fluid-filie

x Half fluid-filled shel

o Empty shells

n

Fig. (VII.3) The upper branch of the eigenfrequencies

of half fluid-filled spherical shells, compared to the eigenfrequencies of completely fluid-filled and empty

shells.

The mode shapes for the three lowest eigenvalues for M+ 10,

5lO

and 0.3 are shown in Fig. (VII.). These

are characterized by large deformations in the fluid-filled

part of the shell, On Fig. (VII.5) the mode shape for the

same shell is shown for a typical member of the middle branch.

This one has got large deformations in the empty part of the shell, and the numbers of nodes coincides with the ones for

Fig. (VII.) The mode shapes of the modes with the eigen-frequencies in the lowest branch

12

1.0 0.8 0.6 Qt4 0.2 o

Fig. (VII.5) The mode shape of the

mode with the eigenfrequency given

by n in the middle branch

10 y 0.3

5lO

'ianiped .emispherica1 Cap n 6 7 P

Fig. (VII.6) The eigenfrequencies of the two lowest branches

for

510

and for various M+

n L on Fig. (VII.Li.

On Fig. (VII.6) the eigenfrequencies are shown for M varying

from 5 to 50 whilst is kept constant, equal to It

shows that the eigenfrequencies of the lower branch are varying much with M. The eigenfrequencies of the middle

branch show a small variation with M+. This is in good agree-ment with the mode shapes, shown on Fig. (VII.'-) and Fig.

(VII. 5).

On Fig. (VII.7) the results are shown for M constant, equal

to 10, whilst is varying from 5l0

to 5lO.

As for the

empty shell, the variation with in the lower branch is small

for small n-s and great for greater n-s. The variation with

in the middle branch is relatively great for all n-s.

On Fig. (VII. 8) the eigenfrequencies of the middle branch

are compared to the eigenfrequencies of hemispherical caps clamped at equator and hinged at equator when

- 310.

The agreement is shown to be good. The curves for the

hemi-spherical cap are plotted, according to equation (kl) of

Ross and Matthews /7/. This work is based on Ross /8/, where

the theory of assymptotic integration is used. These calcula-tions are based on the following assumpcalcula-tions (which are not

mentioned in /7/ and /8/):

(x2 (f22 -

l) » 1

when rAki is of order (Q2-. and

when 2j 2_

l

is of order °.

Both these lead to the restriction:

2

lj

is of order

Fig.(VII.7) The elgenfrequencies of the two lowestbranches

for MlO and for various

1.6 1. L4 1.2 0.8 0.6 0. 0.2-I' 'I 1 2 3 5 6 7 so

x Half fluid-filled shells

y 0.3 3-10h Hemi Clamj phericai1 cap. ed at ecLuator C1od .er Hemisprieric Simply uppo at equator 1 cap. ted il ri 1 2 3 4 5 6 7 9

Fig.(VII.8) The eigenfrequencies of the middle branch, compared

to the eigenfrequencies of hemispherical caps, simply supported and clamped at equator, and the closed empty shell

1»4 1.2 1.0 0.8 0.6 0. L4 0.2 o

In /7/ the solution 13 given as:

2

(2n)

-for 106 the restriction on 2_ 1 is fulfilled for

n 5 which is indicated at Fies. (VII.6) and (VII.8).

The conclusion of this chapter is:

The bending modes of the half-filled shells are separating

into two different groupes of modes. One low-frequent group

with the main part of the kinetic energy due to fluid motion and with nearly no deformation of the shell in the part which

is not interacting with the fluid. The eigenfrequencies of

these modes are slightly greater than the eigenfrequencies

for completely fluid-filled shells.

The other group is high-frequent, with nearly no interaction

with the fluid. The eigenfrequencies of these modes are in

good agreement with eigenfrequencies for hemispherical caps,

supported at equator.

The eigenfrequencies of the longitudinal modes are nearly

not affected by the fluid, and are nearly ttie same as for the empty shell.

VIII. FREE VIBRATION OF NEARLY HALF FLUID-FILLED SHELLS

Fig. (VIII.l) The shell and the perturbed

free surface.

Suppose that the shell is nearly half-filled, which means that the free surface is at a level R above as shown

in Fig. (VIII.l). The problem is then formulated in the

following way:

TI

o

K(u) - Q2u

ç2 (VIII. 1)

E:

rR

O beneath (VIII. 2)

atOF

(VIII. 3) TI

>E

-W

at (VIII. '4) R

The operator K is th same as the one used in Chapter IV. u is the shell displacement vector.

Supposing E is small, the following expansion is introduces:

1)o + E1U + +

U Uo + EU1 + E2U2 +

Xo+ EX1 + E2X +

K(u0) -

X0u Xa

V2o z

Q

O at O

z R

which is the problem formulated and solved in Chapter VII. There might at once be mentioned that 1)Q _{is, according to a}

Taylor-series expansion,of order C when z <O,ER>.

Introducing (VIII.9) and (VIII.5) (VIII.7) into (VIII.l)

-(VIII.L.) and (VIII.8) and separating in powers of E, one

gets for power

The surface condition (VIII.2) is transformed to the level

O in the following way:

a4) (cR)22

zQ

'Tr at 2 O O le rzR (VIII. 8)

Here

(u0,

_0, X0) is the solution of the following problem:

(VIII. 9)

z M4

K(u1) - X0u1 - X1u0

lT at

4z_

V21 z

Q -- W1 at

rzR

-- wo) at -R

According to the series expansion of ((o/r) - wo) around

z _ir/2, ((Dio/r) - w0) will beof order °. Because

the area of the shell between z - _and z _is

of order , the effect on X1 of integration over this area

will be of order too, and therefore neglected.

Multiplying (VIII.lO) by (sin u0) and integrating between

z _and z _{ir, one gets:}

<K(u1)uo> - X0<u1.u0> - X1<u0.u0> - X1 0W0 >0

n M4 Xo [<owo>o +

<1Wo>0fl]

where

<f()>

Jf.sin d for r z R z _Jf.si d for r z R rr/2 and

<f()>o

z _Jf.sin d for r z R

Ir o

XoioIR

2 - (VIII.lO) (Xoi1+ X10)I _{rzR >}

rz R

(VIII. 11) (VIII. 12) ( VIII. 13 ) (VIII.lLf)

The operator K is th same as the one used in Chapter IV.

u is the shell displacement vector.

Supposing is small, the following expansion is introduces:

+ _El1)i + + u

u0 +

Eui + E2U2 + Q X+ EX1 + + M

K(u0) -

X0u A0 o + i1)o O at () O R

which is the problem formulated and solved in Chapter VII. There might at once be mentioned that 11.o is, according to a

Taylor-series expansion,of order E when z <O,ER>.

Introducing (VIII.9) and (VIII.5) (VIII.7) into (VIII.l) -(VIII.L) and (VIII.8) and separating in powers of E, one

gets for power

The surface condition (VIII.2) is transformed to the level

O in the following way:

14) (ER)2211)

14) + ER - +

2 + . .

- O at (VIII.8)

Here

(u0,

14), X0) is the solution of the following problem:

O

rR

(VIII. 9)

+ R O O 3r Wi at r R - w1 - WO) -e 3r 7F at at

According to the series expansion of ((o/r) - wo) around

7F/2,

((o/r) - w0)

will be of order e°. Because

the area of the shell between c - e and is

of order e, the effect on X1 of integration over this area will be of order e too, and therefore neglected.

Multiplying (VIII.lO) by (sinc u0) and integrating between

O and rr, one gets:

<K(u1)uo> -

X0<u1.u0> -

X1<u0.u0> - X1

X0 + _<iWo>0n] where d for r R ff.sin d for r R 7172 and rr/2 d for r R - _{e < <}

rR

(VIII. il) (VIII. 12) (VIII.13 ) (VIII.l) O lT

K(u1) - X0u1 - X1u0 M - e < < (VIII.lO)

R

(Xo1+ X1i40)[

_rR

Because ijo is of order c for - c < < and < >

is of order E,

<Owo>O

is of order

E2,

and therefore

eg1ect-ible compared to

The operator K is selfadjoint, and therefore

<(K(u1) - X0u1)u0>

<(K(u0) -

X0u0)u1>

O

R n

according to Eq. (VIII.9).

Using Green's theorem, one gets:

R I r )o _dr <l)1wO>n - j Tr o

when the fact that V2ì40 O is used.

Introducing this into Eq. (VIII.lq), one gets:

R + M+ J

(0)2

AO

where (VIII.11) is used.

According to (VIII.9)

<Owo>n

_J

Jr2sin(Vo)2dr

d

o Tr/2

i

E _RT

Assuming M+

» 1,

one gets:

R

f(0)2

r dr J z

zo

xl -AO

<O

o

r dr

zO

(VIII. 15) (VIII. 16) (VIII. 17) (VIII. 18) (VIII .19)

In the case of 0, iû and n 2, one gets:

Xl - QL35 lo (VIII.20)

or

X X(l - O»435 E) (VIII.2l)

When assuming (VIII.21) to be valid for O < E < , one gets

that the frequency of the completely fluid-filled shell is

ç2 x0 + sA1 + +

Assuming s to be small (which is discussed in Chapter VI),

one may put:

and the shell displacement vector:

U Uo + SU1

+ E2U2 +

. (IX.5)

where

(u0,0,X0)

is the solution of the problem defined in

Chapter VII.

As in Chapter VII the governing equations are:

K(u) -

M

(IX. 3)

IX. THE EFFECT OF TFE POTENTIAL ENERGY OF THE FREE SURFACE OF HALF-FILLED SHELLS

The free surface condition in Chapter VII is correctly written:

o at

r g

Defining

(E7p ) (IX. 2)

(IX.l) is transformed to:

er

rR

2

and

o

K(u1) - X0u1 - X1u0

R Po

()5.

+ Xoi1i z _{Q at}

In the same way as in Chapter VIII this equation gives rise to the following equation (paralell to Eq. (VIII.l)):

+ -

À0<u1.u0>

-M X1

- <ip1w0>o

n for < > -at (IX. 9)

r zR

4

Introducing (IX») - (IX.6) into (IX.7 and (IX.3) and

separat-ing in powers of c, one gets for the power

£1:

lt

2

According to Eq. (VIII.15) , Eq.(IX.12) is transformed into:

X1(<uo.uo>

X0 (<owo>o - (IX. 13)

Introducing Eq. (IX.11) into Eqs. (VIII.1E) and (IX.13),

one gets:

(IX. 10)

and

and

Xo O.O485

A1 QL.35

In the case of a steel shell of radius 18 m

E 6-10

-6

-5

5. .10 AO

-or A

Ao(l + 5.-l0)

This correction is said to be neglectable.

M J R O)2 r(1 dr 3een used. then is given dr ATr > O has by: and n 2 (IX. 1) (IX.15)

where Eq. (VIII.17)

Assuming M+ o

» 1

X R O)2 o A1

<°>T

In the case of M 50,

510e

jT

M v1J)o1VPo>T) +

X. VIBRATION OF SUBÎRGED SHELLS

Vibrational problems of submerged shells are handled in the same way as described in the preceeding chapters. As with

the filled spherical shell, the simplest problem to handle,

is the one of the fully submerged shell. This subject has

been treated by Sonstegad /16/ and Huang and Chen /17/

amongst others.

Sonstegard is dealing with both compressible and incompressible

fluid. In the case of the compressible fluid model there is of some sense to introduce a energy absorbing boundary at

r - , because of the possibility of travelling acoustical

waves in this medium. This boundary condition only affects

the damping coefficient and not the real part of the

eigen-value (i.e. the eigenfrequency). This problem, of course,

does not occur when the fluid is incompressible, because the incompressibility ensures that any signal is known anywhere in the fluid instantaneously, and therefore no travelling

wave solutions are possible. In general this problem is a

boundary value problem, and is handled in the same way as

described in Chapter IV. Sonstegard concludes that when

calculating the lowest eigenfrequencies the effect of

compressibility is of no practical importande for reasonable

great M+ and c//E/p5, which is corresponding to the conclusions

of Chapter IV of this work. The results of /16/ are given

in Fig. (X.l).

When comparing the results of fully submerged shells and completely fluid-filled shells, the presentation of the data done by Huang and Chen /17/ is better fit than that of

Sonstegard /16/. /17/ is only dealing with incompressible

fluid. The method of /17/ is exactly the same as the one used in Chapter III, except for a minus-sign at the right hand

side of Eqs. (111.13) and (111.16) and that the following

610

[4. 3 l0 2 .l0 10&

3l0

2.10

l.510

10

5.jQ2

O z _{D RP (ccs)} r -(n+l) n n nzo which is analytic at r

-The result of this is that () in Eq. (111.28) is replaced

n

by

TT'

The results for

1.33l0,

M z _{14.35 and z 0.3 are}

given in /17/. In Fig. (X.2) and Fig. (X.3) these are compared

to the eigenfrequencies of the same shell, filled with the same fluid.

710

w (rad/sec) / // 2 3 [4 R=0,5 m w (rad/sec) / , , / 1 2 3 '4 3. 10' R=12,5 m (X.1) vIE/p z- _{-z0.128 vzO.3} c s 3 o s

/

Fig.(X.1) Results given in /16/.

23

4 o zl. 2l0 R=2,5 m Empty shell Compressible fluid-model Incompressible fluid-model

1.2 1.0 0.'4 0.2 o _D i .35 vrD.3 1.3310k o 2 o

Fig.(X.2) The eigenfrequencies of fully submerged shells,

compared to the eigenfrequencies of completely

fluid-filled shells (the lower branch).

35 vr0.3 :1.33.10

I

i 2 3

Fig.(X.3) The eigenfrequencies

of

fully submerged shells,

compared to the eigenfrequencies of completely

fluid-filled shells (the upper branch).

D F1A :ubrrierged .nelis

omçiee1y fluid-filled

sheìi

ru: submerged sheli5

o mp1etely fluid-filled

1.0

0.8

o

Ql

Remark that for the :ubmerged shell, the possibility n O

occur, which is due to the possibility of displacing fluid

at r

-As expected the eigenfrequencies for the submerged shells are greater than for the fluid-filled shell.

When regarding the semi-submerged _{shell, the calculation is} the same as for the half-filled shell, with _{the same exceptions} mentioned for the fully _{ubmerged shell. This gives that}

instead of the expression of E1 _{in Eq. (VII.15) one has got}

to use:

With this modification, the _{same computer program is used,}

and the results for M+ 10, _{and 0.3 are given}

in Fig. (X.). It shows that for the semisubmerged _{shell the} eigenfrequencies are greater than for the _{half-filled shell.}

U.0 2i+l + _Ln+3 E..

c3(.

. + M

2n+l 1j,2n+l2n+l

.) L

,1

no

D j D X X v0. 3 5 6 7 (X. 2)

:i ells

Fig. (X.Li.) _{Comparison between the eigenfrequencies of}

fluid-filled and semisubmerged shells. _{(The two lowest branches.)}

-ñ--- D D D

j

o

XI. THE TRANSIENT P1-DBLEMS

In the preceeding chapters one has been dealing with the pure eigenvalue problem of the spherical shell, interacting

with fluid. There is of some interest to look at the transient

problem too. There are two types of actual external loadings

in the case of the spherical shell.

External pressure acting on the shell:

z pr,e,t)

Prescribed timedependent displacements (or forces)

intro-duced somewhere at the shell.

The first case, a), is discussed in some detail by Baker,

Hu and Jackson /22/. They have used the common series-expansion

technique, which is going to be described here.

At first: One knows that the homogeneous eigenvalue problem has got a solution (given in Chapter III or VII) and that the eigenfunctions are orthogonal (according to some rule) and form a closed set (which is possible to prove, because

of the type of problem treated).

These eigenfunctions, in normalized form, ae given as:

for each i (XI.l)

u.

j-The variables are then supposed expanded in the following

way:

u

z _a.(t) (XI. 2)

+

r - aM --(Ea.i.)e

ji

r

i

where a and K are some constants given in Chapter I.

Eleminai-ing by w and remembering that

K(u.)

i

-

i

i

Q

where H is some matrix, given by Chapter III or Chapter VII,

one gets

+

ä.Hu.

Kpe

(XI. 5)

Using the orthogonality condition, one gets:

(. +

i

ii

a.)<U.

Hu.>

K<p w.>

i i

ri

where < > is the scalar product. Eq. (XI.6) becomes:

ä.

i

+

ii

a. p.(t)

i

t a1(t)

fP1(T)

sin(2.(t-T))dT i i o

In the special case when

r

Pr0)5t the solution

becomes:

(XI.

(XI. 7)

where t is supposed dimensionless according to R and (E/p5)

and

p(t) = K<p

w.>/<u..Hu.>

r i i i

The solution of (XI.7), supposing a1(0) O is

given as: