.. ,_

~

"

..,

2 - J. I ' ' 11'Sfl1'f1q1:1LL~~1 ~.ti. UVl 10 tlii'tM 1 "au11:1u 2530

' IJrlY

The Effect of Internal Flow on Marine Risers

Somchai Chucheepsakul

Dept. of Civil Engineering

King.Mongkut's Institute of Technology Thonburi

Abstract

The effect of internal flow on marine risers is presented.

The differential equation of. motion for a marine riser with flowing

fluid is developed and two additional terms are shown to occur.

The first term combined with the effective tension, thus reducing

the riser stiffness, and the second one produced the Coriolis

damping. The methods of analysis are presented using a series

solution and finite elements. The results indicated that the

effect could be significant with low effective tension and high

fluid velocity

3

INTRODUCTION

The dynamic behavior and stability of pipe conveying fluid has been studies extensively for more than a quarter of century. The vibration of both straight and curved tubes have been investigated by several authors l1-6l. Review of the related literature can be found in Refs. 17, SI. The effect of internal flow on marine risers has received relatively little attention. Until recently, Paidoussis and Luu 191 have studied the dynamics and stability of a cantilevered pipe conveying fluid, which might be used in oceaning mining. Irani and et al 110! have investigated the dynamic response of marine riser with internal flow and concluded that the effect could be significant.

The purpose of this study is to bring into attention and to elaborate on the effect of internal flow on marine risers. The differential equation of motion is developed by equilibrium of fluid/ riser pipe element, or alternatively by energy approach. For most of the typical riser problem, the solution can be found in Ref. 111 I. For this problem, a closed-form solution cannot be found, therefore, two approximate methods based on a series solution and finite element method are proposed. Although the methods presented here are quite well-known, however, they are conveniently applied. 'The method of series solution applied to similar problems has been used by Mote and Naguleswaran 1121 and Kirk et al j13I. Application of finite element method is found in Ref. 1141 and correction is pointed out in Refs. I 1 s, 16 I.

Numerical examples are given for a typical riser problem. For the case where no internal flow exists, two methods yields the very good results. For the case with internal flow, the finite element has been employed in order to give the reliable results.

Equilibrium of a Fluid,/);{iser Pipe Element

The riser with internal flow syst~m considered in this analysis shown in Fig. 1. In the present investigation, only two-dimentional case is studied. It is assumed that static equilibrium is nearly in the vertical position and vibration from equilibrium position is small.

4

The system consists of a pipe of length, L, flexural rigidity EI. and mass per unit length m, conveying a fluid of mass density pi with a constant velocity v. A large tension is applied at the top t9 avoid compression buckling at the bottom. In deriving the differential equation of motion of a riser. the transverse deflection x(z,t) is measured along the z-axis which is ass1.Dlled to be the centerline of W\disturbed riser pipe. Fluid and riser pipe elements of differential segmant dx with forces and moments acting on elements are shown in Fig.2. Consider only the element or fluid inside, there are pressureforce F per W\it length acted on fluid by the riser walls, axially pressure force P.A. and the transverse loading exerted by the fluid on the pipe

1 1a a 2 j21 PiAi Cat + v az) x per unit length.

By neglecting the effect of shear stress due to fluid friction acting along the length of .the pipe, s\.Ullfllation of forces in the x-direction gives

z

---- Riser pipe

L

z

x

Fig.1 A physical model of marine riser conveying fluid

'

5

P\.A~ +CJ (I> A~) di +de w

\

(b) Riser element

(a) Fluid element

Fig.2 Forces and moments acting on fluid and riser elements

For the riser pipe element, the internal forces consist of axial force T, shear force V, and bending moment M. The

02 x external forces per unit length are the inertia force mW- and

( 1 )

pressure force F. Similarly,.summation of forces in the x-direction gives

a2 x o v o [ T ... ddxz J -F - m~ - oz + 62:' = 0

Summation of moments at the center of pipe element gives

oM v - oz = 0

(2)

( 3)

From elementary beam theory the moment-curvature relation is

M EI a2 x azr (4 )

6

Combining Eqs. (1) and (2), and using Eqs. (3) and (4) yields

a2

[ a2

x ) - oz2 EI B"zr

( 5)

Including the effect of external c:?x

fluid pressure :z [ P eAe ~~] and added mass inertia force m -- , Eq. (5) becomes

aat2

a [r l ox] a2 x +;;- T-P.A.+PA ;;- -(m+m +p.A,)a:rt uZ l. l. e e J uZ a l. l.

= 0 ( 6)

or

02 [ 01 ) B?' EI i5"zr ( 7)

where the effective tension T is defined as

" T =T-P.A.+PA 11 ee

{8)

A

in which T is the true riser tension and P., P are the internal and i e

external pressures and A~, A are the internal and external cross J. e

sectional areas of the pipe. The total mass mtot is defined as

= m+ m' + p .A . a l. J.

where m is an added mass term written as a

( 9)

m = C p A ( 1 0 ) a a e e

where C -is the added mass coefficient. The second and third terms a

in Eq. ( 7) can be combined_, which gives the differential equation of motion of riser conveying fluid,

7

0

where T is the combined tension which corrected for pressure c

( 11 )

differences and internal velocity. Furthermore, the combined tension T also depends on the wet weight W of the riser,

c

( 12 )

in which T 8 is the combined bottom tension defined as

~ 2 TB = T8 -P.A. + P A - p .A .v 1 1 e e 1 1 ( 1 3)

and the wet weight of the riser w is

w = g(p -p)A + g(pi -p)Ai s s ( 14 )

where p are the densities of steel and sea water and A is the s

steel area.

The boundary conditions for the pinned ends riser are

x(O, t) = x (L, t) = 0 l J

( 1 5 ) 32 x ( 0) t ) 3

2 x (L, t) 0 azt' = Fr =

Hamilton's Principle

The differential equation of motion can also be derived from Hamilton's princi~le 111 in which the total kinetic and potential energy of the system are considered.

As the fluid element flows along the pipe with velocity v along the z-axis, the velocity component in the x-axis due to flowing fluid. is ( :~ + v :: ) .

Therefore, the total kinetic energy of the riser pipe and fluid is

( 16)

8

in which the total mass mtot is defined in Eq. (9).

The potential energy of the riser pipe can be expressed as

1 f L ( ai 12 1 J L ,.. ( a 12 f L 11 = 2 EI 0 z~ J dz + 2 T a: J dz - fxdz 0 ' 0 0

( 1 7)

The first term on the right of Eq. (17) is bending strain energy, the second term is strain energy due to tension, and the last term is the potential energy of the external force. For free vibration problem the last term may be disregarded.

Hamilton's principle stated that

t2 6 f (K - 11 ) dt = 0 t,

Substituting Eqs. (16) and (17) into Eq. (18) gives

6 {2 t1

( 18)

( 19)

If the first variation of Eq. (19) is performed, one obtains the differential equation of motion which is identical to Eq. (7).

Equation of Motion of a String

~or a very long and flexible riser, the effect of bending term can be neglected. Thus Eq. (11) reduces to

:z[Tc =~J- 2 PiAiv ::~z - mtot ~:: = 0 ( 20) Substituting of Eq. (12) into Eq. (20).yields.

I

9

= 0 ( 21 )

This is the lateral equation of motion of the string.

APPROXIMATE METHODS OF ANALYSIS

Due to the mixed derivative term in Eq. (21), .an exact solution of this equation is difficult to obtain. Therefore. the approximate solutions are proposed instead. It is the author's intention to present two methods of so~ution rather than a singl~ one. so that the numerical results can be compared. These two methods are well-known and suitable for the riser problem; the first one is series solution and the second' one is finite element mentod. Detail procedures of these analysis are given below.

Series.Solution

Let the solution of Eq. (21 ) be on this form

X (z, t) = ~ . 1Tz ( ) " Sl.nJy- qj t j=1

(22)

Substitution of x(z, t) and its derivatives into Eq. (21) gives

l[ . 11 2] l j l (TB + W z ) ( f ) .. 1lz W(j1l) .11Lz} (t) sin J L - . r cos J q j (23)

Equation (23) is multiplied throughout by sin m~z and integrated over the length L of the riser. By using the integrals given in Appendix. the result is

[TB [ mL V J2 f + w [ ~V ] 2 ~~] L +[~ w[ t'!-J2 E (m, j) - ~ w["] F (m, j)J qi

11 + l: 2 P;A, v ( ~) F (m, j) q-: = 0

Rearranging Eq. (24) in the matrix form yields

..

[M) {q } + [C] {q } + ( K] { q } = { 0} m m m

The entries of (M], [CJ and [KJ are given as

M . m, J

c m, j

K j m,

=

=

=

{

mtot L 2

0

m = j

m -:/:. .j

l 0 m = j . fl 2p.A. v ( L ) F Cm, j ) m # j l. l. WLJ L (~ >2 [TB +2 2

jfl 2 w ( L) E (m, j) - w ( t fl ) F (m. j),

m = j

m :I j

where coefficients E(m,j) and F(m,j) are given in Appendix.

Finite Element Method

10

(25)

( 26)

( 2 7)

( 28)

Fy the finite element method, the riser is discretized into finite number of beam elements. The riser length L is divided into N elements of length L. The displacement x is approximated by a cubic polynomial of z, which provides continuity of the displacements and slopes at nodes. Let the lcx::al degrees of freedom be

x (0)

x'co) {qe} = ( 29}

x ()

x'(l.)

..

'

' 11 if

Thus, the displacement in the element is approximated by

x = [N] {q } e

where (NJ is the shape function matrix [17].

(30)

The kinetic and potential energy for the entire riser structure expressed in Eqs. (16] and [17] are obtained by adding the contributions of all the finite elements, that is

N K t K

e=1 e r 31 >

N 11 = t 11

e=1 e ( 32)

Introducing Eqs. (31] and [~2] into Eq. [18] and after carrying out the variation procedure, one obtains the equations of motion for free vibration

(M]{ q} + (C] {q} +'[K-]{q}"" {O} (33) where [M] = ~ 1 [m J, [C] = ~ 1 ['c 1. [K] = t 1 [K ] and {q}, {q} and e= e e= e e= e {q} are global nodal accelerations, nodal velocities and nodal displa displacements respectively.

The element mass matrix [m ]:. element damping matrix [c 1 and e e

element stiffness [k ] are expressed as e

[m ] e

[c J

[k ] e -

f I. T mtot [N 1 [NJ dz 0

0 f I. II T .. . T (EI [N ] [N ] + T [N' J (N'] ) dz c

(34)

(35)

(36)

where prime denotes the derivatives with respect to z. If the bending term is neglected, then Eq. ( 36) reduces to

12

(k ] e

=

0 fl.Tc (N' ]T [N'] dz ( 3 7} Matrices form of [m J and [k ] can be found in standard text

e e book such as Ref. 1171. While matrix form of [c ] is lessfamilar and

e

given here

[c J = e

0 61. 30 -61. p.A. v

1 1 -61. 0 61. -l. 2 30

-30 -61. 0 61.

61. l. 2 -61. 0 J

Matrix (c ] is skew symmetric and known as the Gyroscopic e

( 38}

matrix or Coriolis matrix l1sl. The element matrices [m ]. [c ] and e e

[k J can be evaluated by Gaussian quadrature integration and then e

assembled to the global system.

Eigenvalue Analysis

If subscript min of Eq. (25) is disregarded, then the equations of motion has the same form as Eq. (33). Introducing the two dimensional state vector

{y(t}} = { l q (t)

q (t) } (39) Thus, Eq. (32) can be written in the form

{y} - (H] {y} = {O} (40)

where

- (HJ ( 41 )

t

13

in which [I] is a unit matrix.

The eigenvalue problem Eq. (40) can be solved by letting

{y} = (42)

where w is a complex eigenvalue and {0} is eigenvectors.

Substituting Eq. (42) into Eq. (40) yields

Jw[I)- [HJ!{} = { 0 } (43)

Equation (43) is solved by a reduction to upper. Hessenberg form and utilization of the QR algorithm l19j. If the number of degrees of freedom of Eq. (33) is n, therefore (HJ is a square matrix of order 2n which gives 2n eigenvalues occurring in pairs of complex conjugates. The motion characteristics depend on the nature of eigenvalues which generally contain real and imag~nary parts,

w =A. +iB r r r

r = 1 , 2, , 2n ( 44)

Three basic cases of the solutions are classified as follows

1. If A. = 0 i.e. eigenvalues are pure imaginary. This is r

the case of pure harmonic oscillation about the equilibrium point. the . /

motion is said to be stable.

2. If A. < 0 r

negative real parts. i.e. eigenvalues possess complex conjugates with

This case the solution approaches equilibrium point as time increases, the motion is said to be asymptotic stable.

3. If A > 0 i.e. eigenvalues possess positive real parts . r

This case the solution increases exponentially with time. the motion is unstable.

NUMER !CAL RESULTS

The basic data given in Table 1 is used to calculate the natural frequencies of the riser by the procedure described above. To simplify the problem, an unbuoyed riser of uniform section is considered in the

14

analysis. Introducing a dimensionless parameter a,

a = (45)

wnich is the ratio of effective bottom tension to riser weight. The value of a is equal to 0.1 for a given data in Table 1.

Table Example of data used in the analysis

Bottom tension Top tension Water Depth Outside diameter Inside diameter Density of steel Density of sea water Density of fluid Steel mass in air Riser wet weight Total riser mass Cross-sectional area Elastic modulus Added mass coefficient

43290.6 476197.63

300.00 0.26 0.20

7850.00 1025.00

998.00 170.16

1443.02 255. 94

0.02168 200 x 106

N N rt\

m

m

kg/m 3

kg/m 3

kg/m 3

kg/m N/m kg/m

2 m

!cN/m 2

In case where there is no internal flow, the third term in Eq. (21 ) is dropped out, then the closed form solution can be found and given in Ref. I 20 I. The natural frequency w . is written as

J

(IJ J

= [ ~~ J j = 1. 2, .. (46) in which mtot is the total, mass .of riser, T t and T 8 are the effective riser top and bottom tension respectively. Numerical results from the series solution with 20 terms, the finite element method with 20 elements, and Eq.. (46) are compared and given in Table 2. For the first ten modes, the results are in very good agreement. Therefore. either one of these two methods can be used for the analysis.

..

..

411

15

Tables 3, 4 and 5 give the numerical results of the natural frequencies for the first four modes for various values of a. Then the bottom tension is very low, the effect of internal flow could be significant as can be seen from Table 3. For the first mode, as flow velocity increases the natural frequencies will decrease and then change to the negative values at higher velocities in the range of

J 10 m/s to 30 m/s. That means there is no vibration arises, since the riser is displaced from its equilibrium position and asymptotically approaches its position as time increase. For the second mode, the riser loses stability when velocity is higher than 20 m/s. For the third and fourth modes, the riser is quite stable for the range of flow considered.

l

As the values of a increase, the effect of internal flow decrease as can be seen from Tables 4 and 5. There are the possibi-lities that static divergence or buckling can occur at the very high velocity. However, that range of velocity is out of practical interest.

Table 2

.

Comparison of numerical results of the first ten natural frequencies.

Mode series FEM Eq. (4. 2)

1 0.28911 0.28905 0. 29395

2 0.58524 0.58511 0. 58790

3 0.88018 0.87992 0.88185

4 1 .1 7480 1. 1 7433 1 . 1 7580

5 1 .46943 1 .46859 1 .46975 6 1. 76437 1. 76280 1. 76370

7 2.06006 2.05706 2.05765

8 2 .35745 2. 35146 2.3516 9- 2.65802 2.64612 2.64555

10 2. 96449 2.94120 2. 93950 '

Table 3 First four natural frequencies for o 0

v 11.11 w2 w3 w4

(m/sec) Real Im Real Im Real Im Real Im

0 I- 0. 1080 x 10 -5 0.2051 - 0. 1240 x 10 -5 0 .4 324 -0.4161 x10 -6 0. 6591 -0.2217x10 -5 0.8858

10 - 0. 2801 x 10 -5 0. 1 392 0. 1 788 x 10 -5 0.3594 - 0. 1080 x 1 0 -5 0.5810 - 0. 304 7 x 10 -5 0.8050

20 i-1.3877 0. - 0. 1341 x 1 0 -5 0.1360 0. 1192 x 10 -5 0. 3598 - 0. 8941 x 1 0 -7 0.5863

25 i-2.1115 o. 0. 5 743 0 . ::.. 0. 6631 x 10. 6 0. 2046 0.1490 x10 -6 0.4339 .

26 -2.2533 0. 0. 711 7 0. 0.5194x10 5 0. 1 789 - 0. 2831 x 1 0 -6 0 .4020

27 -2.3943 0. 0. 8499 0. 0. 9537 x 10 6 0. 1399 -0.1389x10 -5 0.3660

28 - 2. 5344 0. 0. 98836 0. - 0. 0205 0. - 0 .4 917x10 -6 0.3239

29 ~ 2. 6737 0. 1 . 1 266 0. - 0. 1 955 0. - 0. 7749 x 10 -6 0. 2796

30 - 2. 8123 0. 1 . 2642 0. -0.3254 o. 0. 365 8 x 1 0 .5 0.2414

O'\

.... ..........

. -

~ .,

... ,.

.,, .~

Table 4 First four natural trequencies for a

v (1)1 .(1.12

(m/sec) Real Im Real Im

0 - 0. 2222 x 10 -6 0. 261 9 - 0. 1 523 x 1 0 -5 0.5335

10 0.3874x10 -6 0.2567 0.6706x10 -6 0.5234

20 0. 2764 x 10 -5 0.2375 0.2026x10 -5 0 .4 868

25 - 0. 5364 x 10 -6 0.2139 -0.6959x10 -5 0.4427

26 0. 2980 x 10 -6 0.2044 - 0. 5960 x 10 -6 0.4258

27 - 0. 8345 x 10 -6 0. 1 883 -5 0.1039x10 0.3998

28 0. 1788x1 0 -5 0.1444 0. 9239 x 10 -6 0.3521

29 0.2032 0. 0 .44 70 x 10 -6 .o. 2756

30 0 .4 839 0. -0.2012 x10 -5 0 .2299

,,

0.05

(1)3

Real Im

- 0. 6975 x 10 -6 0. 8035

0.3949x10 -6 0.7885 .

0.3675 x10 -5 0. 734 3

0. 3619x1 0 -5 0.6699

- 0. 5960 x 10 -7 0. 6466

- 0. 35 76 x_10 -6 0.6143

- 0.2682x10 . 5 0. 5683.

0. 34 72 x 10 -5 0.5132

o.r73o x10- 5 0.4674

_,_

(1)4

Real

- 0. 1 306 x 1 0 -5

-0.5849x10 -6

';o -5

-0.2235 x10

- 0. 71 52 x 1 o-6

, 0 . 1 0 99 x 1 0 - 5

0. 2339x10 -6

0.1401 x10 -5

. -5 - 0. 3435 x 1 0

0.1237x10 -5

Im

.,

1 . 0 730

1 . 0531

0.9812

0.8978

0. 8698

0. 8345

0. 7914

0. 7449

0.7015

w-

....

-.J

....

ll '!

Table 5 First four natural frequencies for ~ = 0.1

v (I) 1 (1)2 w3 w4

(m/sec) Real Im Real Im Real Im Real Im .

0 - 0. 5879 x 10 -7 0. 2890 0. 3823 x 10 -6 0.5851 0. 7424 x 1 0 -6 0.8799 0.2341x10 -5 1. 1 743

10 0. 1371 x 1 0 -5 0 .2853 0. 24 74 x 10 -5 0.5777 0.1427x10 -5 0.8689 -0.2287x 10 -5 1. 160 I

20 0. 2414 x 10 -5 0.2730 - 0. 101 3 x 1 0 -6 0.5539 0.1967x10 -5 0.8333 ,;0.1401x10-5 1 . 11 23

30 0.3129x10 -5 0. 24 78 - 0. 201 9 x 1 0 -5 0. 5046 0.2727x10 -5 0. 7598 0. 1606 x 10 -5 1.0146

35 -0.4172x10 -6 0. 2223 - 0. 9239 x 10 -6 0.4552 0.2146x10 -5 0.6863 -0.6408x 10 -6 0. 91 78

36 0. 8345 x 10 -6 0.2134 -0.1460 x10 -5 0.4382 0. 8717x10 -6 0.6616 -0. 7749 x 10 -6 0.8868

37 0.1848x10 -5 0.2000 -6 0.149x10 . 0.4135 - 0. 1304 x 10 -7 0.6278 -0.2604 x 10 -5 0. 84 74 I

-6 -6 -6 -5 J . 38 - 0. 745Q x 10 0. 1 704 0. 5066 x 10 0. 3691 -0.4023x10 0. 5782 0. 1 382 x 1 0 0.7977

39 0. 1381 0. 0.4642x10 -5 o. 2818 - 0. 1892 x 10 -5 0.5124 -0.4917 x 10 -6 I o. 7412 40 0.4 702 o. . -5 -6 0.4553 .-6 I - 0. 24 74 x 10 0.2226 0.2012 x10 -0. 2831 x 10 I 0.6871

I I

' . I 1 i i I ....

00

-

... ~ ...

19

CONCLUSIONS

The effect of internal flow on marine risers has been investigated. The differential equation of motion can be obtained from the equilibrium of fluid/riser element or from Hamilton's principle. Due to the flowing fluid, two additional terms are produced.

ll!k One is apparently included in the effective tension, thus reducing the stiffness of the riser system, the other one is called Coriolis

I. ..

L.

damping. At present an analytical solution to the differential equation cannot be found, therefore two approximate methods are presented, namely series solution and finite element method.

Numerical results have been given for a typical riser system. The major effects of flow velocity. are to reduce the natural frequencies and to contribute to the damping. For deep water risers, and for flexible risers the tension can be very .low and with high internal flow velocities, the effect may be significant.

APPENDIX

INTEGRALS

The integrals used in deriving Eq. (23) are given below.

1 L m = j 2 I f L sin mtlz sin jflz d (4 7) L T z =

0 0 m I- j I

0 I m = j f L jflz d . mflz (4 8) sin -y;- cos = -r- z m+j

0 mL [1-(-1)]= F Cm, j) m I- j T 2 .2 . m -J

L2 m = j

4

j\ sin m z sin j z d (4 9) L -r- z L 2 [ 1 - ( -1 )m+j]

2 { - ) E{m. j), m I- j 1I (m+j)2

20

REFERENCES

1. Housner, G.W., "Vibrations of a Pipe Line Containing Flowing

Fluid", Journal of Applied Mechanics, Vol. 19, Trans. ASME, 1952, pp. 205-209.

2. Gregory, R.W., and Paidoussis, M.P., "Unstable Oscillation of Tubular Cantilevers Conveying Fluid, Part 1: Theory", Proceedings

of the Royal Society, London, Series A, Vol. 293, 1966, pp. 512-

527.

3. Chen, S.S., "Out-of-plane Vibration and Stability of Curved Tubes

Conveying Fluid", Journal of Applied Mechanics, Vol. 40, No. 2,

June 1973, pp. 362-368.

4. Hill, J.L., and Davis, C.G., "The Effect of Initial Forces on the Hydroelastic Vibration and Stability of Planar Curved Tubes",

Journal of Applied Mechanics, ASME, Vol. 41, No. 2, June 1974,

pp. 355-359.

5. Paidoussis, M.P., and Issid, N.T., "Dynamic Stability of Pipes Conveying Fluid", Journal of Sound and Vibration, Vol. 33, 1974, pp. 267-294.

6. Doll, R.W.,and Mote, C.D., "On the Dynamic Analysis of Curved and Twisted Cylinders Transporting Fluids", Journal of Pressure Vessel

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7. Blevins, R.D., Flow Indiced Vibrations, Van Nostrand Reinhold

Company, New. York, 1977.

8. Paidoussis, M.P., "Flow-Induced Vibrations in Nuclear Reactors and

Heat Exchangers: Practical Experiences and State of Knowl9fge in Practical Experiences with Flow-Induced Vibration", Naudasc~er, E., and Rockwell, D., eds.,Springer-Verlag, Berlin, 1980, pp. 1-81.

9. Paidoussis, M.P., and Luu, T.P., "Dynamics of a Pipe Aspirating Fluid Such as Might be Used in Ocean Mining", Journal of Energy

Resources Technology, ASME, Vol. 107, June 1985, pp. 250-255.

I

'

,

22

19. Martin, R.S., Peters, G., and Wilkinsom, J.H., "The QR Algorithm for Real Hessenberg Matrices", Handbook for Automatic Computation, Vol. II-Linear Algebra, 1970, pp. 219-231.

20. Engineering Procedure Guide Marine Engineering, Marine Riser Systems, Edited by Moe, G., Conoco, Inc., June 1986.

21

10. Irani, M.B., Modi, V.J., and Welt, F., "Riser Dynamics With

Interndl Flow and Nutation Damping", Proceeding ~f the Sixth

International Offshore Mechanics and Arctic Engineering Symposium,

Tx., 1987, Vol. 3, pp. 119-125.

11. Chakrabarti, S.K., and Frampton, R.E., "Review of Riser Analysis

Techniques", Applied Ocean Researches, Vol. 4, No. 2, 1982., pp.

73-90.

12. Mote, C.D., and Naguleswaran, S., "Theoretical and Experimental

Band Saw Vibrations", Journal of Engineering for Industry, ASME,

May 1966, pp. 151-156.

13. Kirk, C.L., Etok, E.U., and Cooper, M.T., "Dynamic and Static

Analysis of a Marine Riser", Progress in Engineering Sciences,

Vol. 1, Dynamic Analysis of Offshore Structures, Edited by Kirk,

C.L., C.M.L. Publications, Southampton, U.K., 1982.

14. Kohli, A.K., and Nakra, B.C., "Vibration Analysis of Straight and

Curved Tubes Conveying Fluid by means of Straight Beam Finites

Elements", Letters to the Editor, Journal of Sound and Vivration,

Vol. 93, No. 2, 1984, pp. 307-311.

15. Pramila, A., Comment on "Vibration Analysis of Straight and Curved

Tubes Conveying Fluid by means of Straight Beam Finite Elements",

Letters to the Editor, Journal of Sound and Vibration, Vol. 99,

No. 2, 1985, pp. 293-294.

16. To, C.W.S., and Healy, J.W., Further Comment on "Vibration Analysis

of Straight and Curved Tubes Conveying Fluid by means of Straight

Beam Finite Elements", Letters to the Editor, Journal of Sound and

Vibration, Vol. 105, No. 3, 1986, pp. 513-514.

17. Cook, R.D., Concepts and Applications of Finite Element Analysis,

2nd Edition, Wiley, 1981.

18. Meirovitch, L., Computational Methods in Structural Dynamics,

Sijthoff & Noordhoff: The Netherlands, 1980.

..

i