the free vibration of a cylindrical shell on an elastic foundation

D. N. Paliwal Professor.

Rajesh K. Pandey Graduate Student.

Department of Applied Mechanics, Motiial Nefiru Regional

Engineering College, Allafiabad-211004, India

The Free Vibration of a Cylindrical Shell on an Elastic Foundation The frequency equation for a thin circular cylindrical shell resting on an elastic foundation is developed by using the first order shell theory of Sanders and eigenfrequencies are calculated. These eigenfrequencies are plotted against the axial wave parameter. Effects of the axial wave parameter, circumferential wave number, non-dimensional thickness and foundation parameters on eigenfrequencies are investigated. It is found that the foundation modulus chiefly affects the radial mode eigenfrequency and has no effect on torsional and longitudinal modes. On the otherhand, shear modulus does have influence on radial as well as tangential modes of vibrations. Though the effect on radial mode frequency is more pronounced.

Introduction The thin circular cylindrical shells used in process equip

ments and pipings, missiles and rockets, submarines, underground and under-sea pipelines are often subjected to dynamic loads. Therefore the study of the vibrations of cylindrical shells assumes importance. Rayleigh (1881), Love (1888), Arnold and Warburton (1949), Warburton (1965) and Forsberg (1964, 1965) contributed significantly to the vibrations of cylindrical shells. Dym (1973) studied the influences of the wave numbers and the thickness-ratios on the frequencies and the modal amplitude ratios. Markus (1988) provided excellent description of the free vibrations of the cylindrical shells.

Herein, the authors intended to study the free vibrations of the cylindrical shells that are continuously in contact with an elastic medium either at the outer or the inner surface. The cylindrical shells are assumed to be resting on the winkler and Pasternak foundations.

The primary contribution of this work is the presentation of the results, using the "bes t" first-order shell theory of Sanders. The effects of axial wave parameter, circumferential wave number, and thickness-to-radius ratio on the vibration frequencies have been extensively studied. Besides this, the most distinguishing feature is the investigation of the effect of foundation parameters on the eigenfrequencies. The lowest eigenfrequency, predominantly the radial vibration mode, being the most vital and sensitive component of the eigenfrequency cluster, is studied more deeply.

Formulation of tlie Problem

The following equations of vibration of cylindrical shell on a Pasternak foundation have been derived as follows (Markus, 1988; Leissa, 1973)

„ , 5'M 1 , , , d^u R ,^ , d'-v

dx

^( . - .v f .(»)

f" + u) d'u ^ R'

dxd4> 2

C! 1-2

^ ( 1 E

dw

^)R^^, lib) at

n=2

V=0-3

I |J=0.5,G=0.55

Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March Fig. 1 The dependence of SIH (the lowest eigenfrequency) on the ratio 1995; revised Feb. 1996. Associate Technical Editor: L. A. Bergman. lilR, \ , p and G for n = 2

Journal of Vibration and Acoustics Copyright © 1998 by ASIVIE JANUARY 1998, Vol. 1 2 0 / 6 3

Downloaded From: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 04/25/2014 Terms of Use: http://asme.org/terms

^du dv [I + KR\l - v^)] vR \ w

dx d4> Eh

d" + ~K,\R ^ + 2 « ' ^ ^ , ^ , ^ ^ .

G( l - i^')/?' /d^w 1 d^w

"" Eh \ dx^ " R^ dd>^

P_ E

= ^ ( 1 - ^ . ^ ) / ? ^ ^ 1(c) 5V

where fc, = {\in){hlRf The factor, ki, determines the effect of bending. If Ki = 0,

the bending effect vanishes. Equation (1) can be written in operator form as

E (2)

where, u, is the column vector of the displacements u, D, and w i.e.

u = COI(KJ)

where

a 20

n = l,V=0-3

-^=OOO2,O-06,O-1

0=O.5,i5=O-5S

P= 0-5,6=0

M, = {u,v, w) and L = —--at

Fig. 2 The dependence of Up (the lowest eigenfrequency) on the ratio hIR, foundation parameters ji and Q and axial wave parameter X.

LoM is the matrix of the differential operators of Donnell-Mushtari's theory of thin cylindrical shells, and is expressed in a matrix form as follows

All the equations of motion of the thin shells, due to various theories can be written with the help of Donnell-Mushtari's basic differential operator, LQM, and a modification operator, EMOD,

EDM —

r,2 5 ' 1 , , , 9 ' R ,, , d^ -vR-

dx

uR^-^, -dx dcj)

R

2 ^^ " '''' 8x8(1) 2 ' R \ , , 8'

(1 - 1^)-:-^ +

1 + KR\l - v^)

Eh kAR

dx"- d(l>

a^

2 '

dx^ d<f>' Eh

(3)

N o m e n c l a t u r e

C, (;' = 1, 2, 3) = vibration amplitude constants in axial, tangential and radial directions

D = flexural rigidity E = Young's modulus of

elasticity G = Shear Modulus of the

foundation G - nondimensional shear

modulus = G( 1 — v'')IEh

h = shell thickness K = foundation modulus

L = length of the cylindrical shell m, n = number of axial half-waves and

circumferential half waves, respectively

R = mean radius of the shell t = time coordinate

u, V = displacement components tangential to the middle surface of the shell

w = displacement component normal to the middle surface of the shell; positive inwards

x, (j), z = shell coordinates (axial, angular and radial, respectively)

X, y, z = rectangular coordinates with z normal to a planar boundary

X.* = a parameter (mir/L) \ = an axial wave parameter {rmrRI

L) V = Poisson's ratio p = mass density of shell material

n^ = dimensionless frequency parameter = (,plE) (1 - u^)uj^R^

uj = angular frequency (natural) Jl = nondimensional foundation

modulus = KR^d - v^)IEh

64 / Vol. 120, JANUARY 1998 Transactions of the ASME


10 80

60

AO

20

10 S

6

Ci 1 0 -

0-B-

0 6 -

OA-

0-2

0-1 008 006

OOA

002

001

n=1

n=0

n = 0 —

n:1 —

h R

1

= 0-1,V =

:0,G=0

0'3

10

Fig. 3 The change of the frequency parameter n ' with respect to axial wave parameter (X) and circumferential wave number (n); for lowest value of fl; for highest value of ft

10 80;

60-

^0

20

10 8

" G I . O

0-8

0-6

0 2

0-1 008 006

004

001 (fl

-• "

-

-

-~ -~ ~

• "

-

-

-

-

5 -

4 -

3-

2 -

1 -

n = 0 -

5 -

4 -

3 -

2 -

1 -

n = 0 -

1

^=0-1,V=0-3

|i=0.5,G=0

_ — -

-

• ~ \ ' ' ^

\ . — — —- "^ ^ " X - ' ' ''

X -^^ ''-,

/ /! -- \ ' - ' ' "' -

\ - - / /

^ / ^ / /

- i — /

-\....-"' i ^ ^ ^ \ _ _ — • — - — ' i / j / f ~ -

~\y— "i^^

/

1

A Fig. 4 The change of the frequency parameter ft' with respect to axial wave parameter ( \ ) and circumferential wave number (n); for lowest value of fJ; for highest value of il

(hoM + KILMOD)U = ^ (1 - v^)R^Lu E

(4)

The solution of the Donnell-Mushtari's equations of motion for a cylindrical shell of finite length, simply supported at the both ends (Markus, 1988) yields the following three algebraic equations;

C,(Q^ - / / , ) + — ( 1 + v) Cj - i/VCj = 0

C, — (1 + i ) + C2(n ' - //j) + Can = 0

Journal of Vibration and Acoustics

-C,p\ + Cin + Q(n^ - //a) = 0

with the notations

C, = unknown constants, i = 1, 2, 3

U^ (1 - u )oj R , a frequency parameter.

\ = \*R = mirR

(5)

and m = 1, 2, 3 . . . is the number of half waves in the axial

JANUARY 1998, Vol. 1 2 0 / 6 5


10 80 60

40

20

10 8

6

CM

C! 1.0 0'8 0-6

0-2

0-1 008 0 06

0-04

0 0 2 -

001 10'

~ = 0-1,V--0-3

A

40

20

10 8

6

4|-

1 0 -0-8 •

CM 0 - 6 -

0-4

0-2-

0-1 7 0 0 8 ;

0 0 6 -

0 . 0 4 -

002

001

n :0

^=0-1,V=0-3

ii = 0 .G = 0

10- X

Fig. 5 The change of the frequency parameter U" with respect to axiai wave parameter ( \ ) and circumferential wave number (n); for lowest value of O; for highest value of U

direction of the cylindrical shell. Other notations are defined as follows

Fig. 6 The change of the frequency parameter VF with respect to X and n; graphs for the third value of O for foundation parameters; p. = 0,

e = o

G = nondimensionai shear modulus. Eh

For the nontrivial solution of Eq. (5), the determinant of the coefficients of the constants, C,, must be zero. This yields the following bi-cubic frequency equation.

n^ - n\Hy + H2 + Hj) + n H1H2 "+ / /2^3

Hi = k^ + — {1 ~ ly)

H, = - i \ - v ) + n^

Hi = [\ + Ki(\^ + n^y + ]l + G(k^ + n^)]

+ H,Hi ~ u\' -n' - — (\ + ^y 4

n A (1 + t^yn,

Eh nondimensionai foundation modulus.

= 0 (6)

where n stands for fi^.



40

20

10 8

6

C!

10 0-8

0-6

0-4

0-2

0-1 -0 - 0 8 -

0 0 6 -

0 0 4 -

0-02

0-01 10

n=1

n = 0

R •0>1,V=0-3

M = 0-5,G = 0

A

40

20

10 8

6

4

10 <N 0-8 •

0-6h

0-4-

0 2

0-1 008

0 06

0-041-

0-02-

0-01 10 r 1

n=1

n = 0

^=0-1,V=0-3

M = 0-5,G = 0-55

Fig. 7 The change of the frequency parameter il' with respect to X and F'9- 8 The change of the frequency parameter il' with respect to X and n; graphs for the third value of SI for foundation parameters; p. = 0.6, 6 " : graphs for the third value of fJ for foundation parameters; p = 0.5, G = 0 = 0-55

The following bi-cubic frequency equation is obtained for Sanders theory, using Eq. (4).

+ H^H\ — S\ ~ S2 " S^\ -\- YS\H^ + S2H2

where

+ SlH[ + 2S,S2S, - H[H{H',] = 0 (7)

(1 +u)-~K,(l -ly) 4

KM (I - ly) - vk

S, = n {^(3-.)H-n^} \+k,{j(3

H[ = \' + -{1

«s4(. ly) + n^ + AT, 9\'

(1

-u)

v) + n^

//5 = [1 + K.ik'- + n'-Y + p + G(\" + n')]

Results and Discussions The influence of the h/R ratio and the foundation parameters

p and G on the frequency characteristics is shown in Fig. 1. Figure 2 shows that the sensitivity of the lowest frequency increases visibly with increase in n. For a shell on a Winkler foundation, divergence of the frequency curve is more pronounced as the circumferential mode number (n) increases, whereas for the shell on a Pasternak foundation, the effect is negligible. It may be noticed that with increasing n, the fre-

Journal of Vibration and Acoustics JANUARY 1998, Vol. 1 2 0 / 6 7


20

10 8

6

10

0 8

0-6

0-4

0-2

G 0-1 0 08 0-06

OOA

0 0 2

0 01

0 0 0 8

0 006

0004

0 002

0001

^ = 0002,V=0-3

ja= 0 0 , 5 = 0

n=0

n=1

10

Fig. 9 Curves for the lowest value of ll^for the shell (radial vibration mode) for foundation parameters / i = 0, G = 0

u

0 8

U

-

-

1

| - = 0'002,V=0'3

jj = 0-5,G=0 -

-

vo 0-8

0 6

0-4

0-2

<N

a 0-1 0-08

0 06

0-04

0 02

0 01 0-008

0006

0 00«

0002 -

0001 10'

n = 1

n=0

X Fig. 10 Curves for the lowest value of ff for the shell (radial vibration mode) for foundation parameters p = O.S, G = 0

quency increases. Figure 3 indicates that for a given L/R ratio of a shell, a fundamental frequency is lowest at about \ = 2. The highest eigenfrequency goes on increasing with increasing circumferential wave number. Fig. 4 shows that the Winkler foundation modulus, jQ, affects the lowest eigenfrequency which represents the radial mode, the effect being more pronounced as - increases. Figure 5 reveals that for a shell resting on a Pasternak foundation, the frequency increases with increasing circumferential mode number. However, for very short shells, the frequencies for all circumferential modes approach asymptotically the frequency for « = 0. A saddle is present for n = 0 in Figs. 6 and 7. This means curve represents longitudinal

mode for \ < 1.5 and torsional mode for X. > 1.5. It is evident from Fig. 8 that the shear modulus G affects the middle frequency considerably. Comparison of Pigs. 9, 10, and 11 for h/ R = 0.002 with Figs. 6, 7 and 8 for hJR = 0.1, reveals that for the short shell the frequency decreases with the reduction in the value of h/R. Figure 10 shows the effect of the foundation modulus, p on the frequency. We also witness in Fig. 11, a large influence of G on the frequency. The results shown in Figs. 12, 13 and 14 indicate that for simply supported shells having X. > 0.3, the minimum eigenfrequency occurs in the domain \ > 2. However, for very long shells, i.e., X < 0.3, the minimum frequency always occurs at n = 1. In Figs. 15, 16



/u

10

e 6

u

2

10 0 8 0.6

0-^

0-2

G 0-1

0 0 8 0 06

004

0-02

0 01 0 008

0 006

0 0 0 4

0 002

-

5 v ^

4 ^

3

2 ^

n=1 .

n = o s^^

1

- j ^ = 0002,V=0-3

j j : 0 .5,5=0-55

1

>/-

/ / '

-

-

:

-

—

-

-

-

-

-

10' A

Fig. 11 Curves for the lowest value of il' for the shell (radial vibration mode) for foundation parameters ji = 0.5, G - 0.55

and 17, the effect of hIR on the radial frequency is accounted for by an increase in the hIR ratio and an increase in \ . The radial frequency curves tend to approach the longitudinal frequency curves.

Conclusions The following is a summary of the main conclusions:

(1) Radial vibrational mode frequency is considerably influenced by the foundation modulus. Longitudinal and torsional mode frequencies remain unaffected.

Fig. 12 The change of the lowest eigenfrequency UR with respect to the number of circumferential waves n and axial wave parameter X

40

2 0 -

0-2

CJ

0.1 0-08

006

004

0.02.

30 ^0

h R V:

= 0'02

:0-3 0 5 G: :0

n

Fig. 13 The change of the lowest eigenfrequency OR with respect to the number of circumferential waves and axial wave parameter ( \ ) for nondimensional foundation parameter p. = 0.5

(2) Shear modulus (G) affects all the three vibrational frequencies; however, its influence on the radial vibrational frequency is pronounced.



10' 8 6

a 1.0 08

06

0.2

0.1

:

- n=0,V=0-3 ii=0-5,G--0

—~[

-

1

Fig. 14 The change of the lowest eigenfrequency OR with respect to the number of circumferential waves and axial wave parameter ( \ ) for nondimensional foundation parameters p. = 0.5 and G = 0.55

10

Fig. 16 The behavior of the eigenfrequency for rotationally symmetric vibrations of the cylindrical shell with respect to the parameter X for foundation parameters /oi = 0.5, S = 0

10 8 6

C! 10 0 8 0-6

0-2

01 10

n:0,V=0-3

ii=aG=o

•^=0-002-

A

10 8

G1-0 08

0-6

OA

0-2

0-1

n=0,V=0-3 i i : 0-5,5'0-55

Fig. 15 The behavior of the eigenfrequency for rotationally symmetric vibrations of the cylindrical shell with respect to the parameter X for foundation parameters p, = 0, G = 0, Q = 0

^o'

Fig. 17 The behavior of the eigenfrequency for rotationally symmetric vibrations of the cylindrical shell with respect to the parameter \ for foundation parameters fi = 0.5, S = 0.55.

(3) The effect of h/R ratio is more visible for higher values of circumferential wave number.

(4) For circumferential wave number, n = 0, the effect of

h/R ratio is significant for higher values of axial wave number \. As the value of h/R ratio increases, the resulting curve tends towards the longitudinal frequency curve.



References Leissa, A. , vibration of Shells, NASA SP-288, Washington, D.C., 1973, pp.

, ^ '™"^ ' ^- ^••. ""''.^arburton, O^B. 1949, "Flexural Vibrations of the Walls ^^^ove, A. E. H., 1888, "On the Small Free Vibrations and Deformations of Thin of Thin Cylindrical Shell Having Freely Supported Ends, Proceedings of the c. »• ou n .. m -i / • i-r .• />,i D J f „ tj j c • , „ r „ , , Aim .no Tc? Elastic Shells, Philosophical Transactions of the Royal Society of London, ?:ene?, Royal Society of London, A197, pp. 2iS-256. 401 s/fi

Dym, C. L., 1973, "Some New Results for the Vibrations of Circular Cyliii- \ , , i^^'cZ'. ,^ . . , „., ,. x ^ ,• ^ • , c „ c • ders," Journal of Sound and Vibration, Vol. 29, No. 2, pp. 189-205. ' ^" ' ' "^ ' S" '^^^' ^'"^ Mechanics of Vibrations of Cylindrical Shell, Elsevier

Forsberg, K., 1964, ' 'Influence of Boundary Conditions on the Modal Characteris- Science Publishers, The Netherlands, pp. 1 -101 . tics of Thin Cylindrical Shells," AIAA Journal, Vol. 2, No. 12, pp. 2150-2157. Rayleigh, Lord, 1881, "On the Infinitesimal Bending of Surfaces of Revolu-

Forsberg, K., 1965, "Review of Analytical Methods Used to Determine the lion." Proceedings of the London Mathematical Society, Vol. 13, pp. 4-16. Modal Characteristics of Cylindrical Shells," Lockheed Missiles and Space Com- Warburton, G. B., 1965, "Vibration of Thin Cylindrical Shells," Journal of pany Technical Report No. 6-75-65-25. Mechanical Engineering Science, Vol. 7, No. 4, pp. 399-406.



the free vibration of a cylindrical shell on an elastic foundation

Documents