On problems in ship structural dynamics

LEF

Department

University

Torri ngto n

London

WC1

ON PROI3LEMS IN SHIP

_{STPUCTUPL DYNAMICS}

by

R

_{Eatock Taylor}

cf Mechanical Engineering

College Lonilon

Place

E 7J

Technische Hogschoo

T1,

Deift

UCL flep.IivArch. 3/12

Áì.igust 1972

1. INTRODUCTION

Part II of this report describes a study of the three-dimensional

fluid flow around a vibrating elastic body. The investigation was

intended as the first stage in development of numerical methods for

calculating hydrodynamic loads. These involve both forces associated

with oscillatory motions of marine vehicles, excited by machinery on board

or wave action, and the wave exciting forces themselves. Development of

these methods would lead to rational analyses of the structural dynamics

of such craft as floating platforms and srni-submersibles; accurate

calculations of complex interactions between main hull and local structural

vibrations of certain ship types; and a more sophisticated treatment of

wave induced response than could be outlined in Part I, under the limitations

of simple beam theory.

Existing work in this field having relevance to marine vehicles is

for the most part concerned with unsteady flow past ship hulls oscillating

as beams, The hull cross-section is assumed rigid, and the fluid forces

are calculated assuming the flow to be two-dimensional. A three-dimensional

correction factor is then applied, based on an assumed mode of motion of

a spheroid of comparable dimensions to the ship. (A by-product of the

research described here is an illustration of the dependence of such

correction factors on the assumed mode.) Wave excitation is conventionally

analysed using the assumptions of strip theory to derive hydrodynainic loads.

For heaving and pitching modes of an essentially rigid ship, experience has proved this to be a useful approach, though it is not rigorously based on

a rational hydrodynamic theory (Newman 1970) For distortional modes the

*

2

practical value of strip theory has unfortunately not been established :n

a satisfactory manner, either by model tests or by full scale measurements.

Numerical methods, based on a three-dimensional approach, could usefully be developed to give generalised hydrodynamic loads on ships and other 'rine vehicles, associated both with the lower modes and with highly

distortional modes at higher frequencies. The research reported here is

directed towards development of such methods.

NOTATION

a Wave amplitude; major semi-axis of ellipse

A,I Finite element sectional area and moment of inertia

b Half beam of ship; minor semi-axis of ellipse

d Draught

E Young's modulus

(x)

f Wave force per unit length at point x

{f} Vector of wave forces corresponding to {q}

{f0} Vector of wave forces corresponding to

{q0}

{f*} _{Vector of wave forces corresponding to}

{q*}

{F} Vector of total hydrodynamic forces corresponding to q}

{F0} Vector of total hydrodynamic forces corresponding to

{q0}

{}

Vector of total h3rdrodynamic forces corresponding to {}

F(x') Function defining body surface S

F(rs) Force corresponding to rt mode due to motion in 5th mode

g Acceleration due to gravity

g'3 Metric tensor

(i,j

= 1,2,3)

Three-dimensional correction factor for rth mode

k Wave number Element length

[] Added mass matrix (complex)

[]

High frequency added mass matrix

[]

Residual frequency dependent added mass matrix (complex)

_(x)

Strip theory added

[M], [C], [K] Mass,

[MJ, ['], ['!]

Mass,

M*],

[c*j,

K*] Mass,

mass per unit length at point x

damping and stiffness matrices corresponding to

{q0}

damping and stiffness diagonal matrices

corresponding to {}

damping and stiffness diagonal matrices

corresponding to {q*}

N Number of elements in finite element idealisation

p(r,t) Fluid pressure at time t at point defined by position vector r

P Number of terms in hydrodynaniic solution

q} Time independent part of vector {}

{q}

Vector of generalised coordinates in finite element representation

{}

Vector of principal coordinates of "dry" system

{q*} _{Vector of principal coordinates of "wet" system}

[Q] Matrix of modal columns corresponding to {q}

[Q]

Matrix of modal columns corresponding to {q0} t Time; element number

U Forward speed of body

Velocity vector in fluid (i 1,2,3)

u,w Longitudinal and transverse displacements

w,O Deflection and rotation of finite element node

x1 Coordinates in space (i = 1,2,3) xa _{Coordinates on surface S (a = 1,2)}

Displacement vector in mode i at point defined by position vector Dimensionless coordinates of ends of spheroidal element

Wave depression

K Parameter for ellipse ( /a - b2)

p,O,v Spheroidal coordinates

[p]

Diagonal matrix of damping factors Non-dimensional coordinate (

x/)

P,Pf Fluid density Body density

Velocity potentials

[]

Matrix of modal columns corresponding to {}

Natural frequencies

{+]

=

['-L_j + [']

n Normal vector to surface s

3, ANALYSIS OF THE MOTION DEPENDENT FLUID FORCES

3.1 The Boundary Value Problem

In order to find the high frequency approximation, obtained by ignoring the matrix [m., we require the high frequency motion dependent

forces mentioned in Section 2.6. These are developed in this section by

considering the double body problem in an infinite fluid. The theory

for a body of general shape is derived, for which it is convenient to use

tensor notation0 The derivation is similar to that given by Tirnrnan and

Newman (1962).

Because it is reasonably straightforward to include in the derivation

the influence of the body's forward velocity on these motion dependent forces,

we shall adopt this procedure here. However to apply the results of this

section to the preceding analysis, we must eventually iose the limitation

that the forward velocity is zero0 It is only included here for completeness,

snd as a basis for further work0 When the forward velocity is zero, the

forces obtained here are simply "added inertia" forces corresponding to

added masses in Eq(2.11)0

Let x (i = 1, 2, 3) be a general coordinate system mong through

the fluid with constant velocity U, and let x1 be a coordinate system

oscillating relative to x such that

= xi

-o

The x' coordinates are assumed fixed to the body, whose surface is

defined by the equatìon

2L

F(x')

0 (3.2)

are components of the disturbance vector, considered small, which will

depend on x1. We represent the velocity of the fluid by the vector

= v

+ etg13,.

(i = 1, 2, 3; = 1, 2, 3)

(33)

where y1 is the steady velocity field due to the uniform forward motion of the body at speed U, and

4(x1)

is the potential of the oscillatory

velocity vector, satisfying Laplace's equation, g13 is the metric tensor.

The boundary condition on the body is

DF o = ₌ BF

vX

k F k k iwt -

L.

iwe

+ v -g- (5, - . e -- x1 1 1 k k where .

Is the Kronecker delta and x represents the covariant derivative

of the vector k.

Now this condition holds on the actual surface S of the body. Using

Taylor's theorem we relate the velocity vector y1 on S to the vector

v on the mean surface S° (i.e. the mean position of S during oscillation).

Thus

1

li

it

'2

y

y +v

.e +Oçc.

o

oj

Therefore substituting Eq.(3.3) and Eq.(3.5) in Eq.(3.4), and neglecting

second, order quantitïes, we obtain

i 3F

iwtEj

i 3F i 3F i k 3F ij 3F

O =

y .+e

av ,--iuja ---va +g

3x1 _L j 3x' 3x1 i 3xk 'j

3x-The first term is the steady state boundary condition for

v--- =

0 _(3.6)

We are left with the disturbance boundary conditions, which may be

written

ij 3F

ji

i i 3F

g =

(iwa +Va

-av

J 3x' J _3x'

These terms are now all of the same order and it is no longer necessary

to distinguish between and x coordinate systems, or the mean and actual

positions of the body We may drop the subscript o and write the boundary

condition on the oscillatory potential :

g = (iwa1 + va1

-av

ji

)fl.

j

on S (3.7)

where n. is the covariant component of the normal to the body surface.

32 Choïce of Coordinate System

Up to this point we have made no assumption concerning the general

coordinate system. We now postulate that we may define a non-singular

26

to S. This implies that S is well-behaved and continuous. The

system x1 is taken to be a normal coordinate system consisting of a two

parameter surface

x (

= 1, 2) on S and a coordinate x3 normal to the

surface. We therefore have on S nl = n2 = O als o ga3 = _{O for} a 1, 2

Thus Eq.(3.7) becomes

g,3n3 =

(iwa3 + v3aI . - a3v3J .)n3 on S (3.8)

J J

Also, from Eq.(3.6),

V3 = O

and the incompressibility equation is

= O

Reduction of Eq.(3.8) and substitution of Eq.(3.9) and. Eq.(3.lO) yields

g33,3

= iwa3

(a3Vv)H

a

where a = IL, 2 and 1 denotes the two-dimensional covariant derivative.

In particular, if x3 measures distance normal to S such that g3 is a

unit vector, we have g33 = g33 = 1, and Eq.(3.11) may be written

= 1wa3 +

(a3v)H

a (3.9) (3.10) on S (3.11) ori S (3.12)

This may also be wrïtten - iwa3 1 ---(3V'jv2) on S (a3v'v1) +

-ax3

-

_{Ç ax'}

ax

where g = g11g22 - g122

We note, parenthetically, that although this section is concerned

with a body submerged in an infinite fluid, the boundary condition represented by Eq,(3.12) would also hold on the submerged surface of a body in the

presence of a free surface.

3.3 A General Expression for the Fluid. Forces

So far we have postulated the existence of a velocity potential ,

satisfying Laplaces equation and the above boundary condition on S, which is to represent the effect of a harmonic perturbation of the body from a

steady forward velocity through the fluïd Next we derive ïn terms of

an expression for the fluid force on the body resulting from this oscillation.

The pressure in the fluid is, from Bernoulli's equation:

ac i

p =

ç. iwt iwt i

= piwe

+e

g .v. + y

y.

ji

2

neglecting terms of second order in the oscillatory potential . Now

28

p must be evaluated on the osculating surface S, so we must use the

Taylor expansion (v'v.)

2(vv.) +

et 1(1)J

+ Thus in Eci.(3.l3)

iwt.

ij

.)

-

=

1vv

+ e (iw + g o

01J

p

2001

The amplitude of the oscillatory pressure, the term of interest to us

here, is given by:

p = - p(iw + 1 +

i

3)

(3.15)

where we have again dropped the subscript o, as in this equation the

difference between mean arid actual position of the body is a second order

effect. We represent this oscillatory pressure as the sinn of two

components p =

p +p

whe re = -p(iw + J = _-

_{p, n.dS}

r) r) i S

Now p, and hence is linearly related to , and hence to c in

(3 i4)

We consider first the fluid forces acting on S due to p. We assume that any oscillatory motion of the body may be written as a sum of mode

shapes

r)'

where subscript r denotes the rth mode shape, and we define

the generalised fluid force corresponding to the rt mode as:

(3.17)

i (3.16)

Eq(3.T), since satisfies Laplace's equation and Eq.(3.T). But a1 we

have assumed as a sum of mode shapes, and we therefore obtain a matrix of force coefficients Ft), where

SI

0

=

_-

p, n.dS

1=1,2,3

(3.18)

(rs) r) i

is the force corresponding to the rth mode due to motion in the 5th mode.

Furth errno re

s)

P(iW$()

+ (s)'i

Vi)

Considering as before a coordinate system giving n1 _{= n2 = O and}

wìth n3 = /g33, we have

O

a

=

_{-p J}

[iw + y ]a, Vg33dS

(rs) (s) (s),ct r)

s

Consider now the identity

J

(a3/g33v)HdS+f

_a3g33vadS

=

J

(a/g33v)

Ii dS a

s

s

s

=

i

V(a/g33v")dS

= C

rfte line integral is taken around the closed curve bounding the submerged

surface S For a body in an infinite fluid this term is zero. Hence

f=

-

J

,a3/gg3vas

S S

Thus in Eq.(3,19) we obtain finally

a cpa vg33v dC

30

0

=

J

-

(3.20)

(rs)

s

where satisfies the boundary condition on the body given by

=

Ls)V833

(5)Vg33v)/g33

Ofl S (3.11)

We note the striking similarity between Eq.(3.11) and Eq.(3.20). In

fact we may write

0

=

f

(r),3

(s) /g33 dS (rs)

s

where is the potential corresponding to the rth mode but with the

direction of forward motion reversed (changing sign of y). This corresponds

to the Newman (1965) result, for slender or deeply submerged bodies.

3.L _{An Explicit Formulation in Terms of the}

Solution of the Boundary Value Problem

In

order to proceed further in obtaining an explicit formulation of the hydrodynamic forces in terms of the body motions, we write the

disturbance potential in the form

= amfm(x1,x2)hm(x3)

pp

p

mp

where the unknown coefficients am are chosen to satisfy the boundary

condition given by Eq.(3.11), and each solution

fhm

satisfies Laplace's

equation: In other words hm(X3) rGg f(x1,x2)1 p axaL ax i + (x1,x2) [Gg33

ah(x)

ax = o e re a G = [g33(g11g22 - g122)]2

Since c _{satisfies Laplace's equation we may use Green's theorem}

f(fmhm)

_pp

-- (f%°)ds = f

(f%°) -p- (hm)ds

an

qq

j

qq

an

pp

S S Therefore

h°'

I fmfn/53

3dS =

h%m

_f fmfn/g33

pq

j

pq

qp

j

pq:

S S where h° denotes q dhn(X3) q dx3

This leads to the conditions

?f'/gS =

0

pq

S = bm p n q j m = n

p =q

(3.22) g± cH₁₃

=0

Substituting the assumed solution Eq.(3.21) in the boundary

condition given by Eq.(3.11), we obtain

= [ig33 + (a3

)/g33v)J/g33

on S

mp

(s

We now multiply by fI/g33, integrate over body surface S, and use Eq.(3.22)

to obtain on S

am?bm

=

J

wa5)Ig33

(afl/g33va)lIJfmdS

pp p

S where 32

m

= 1(D + PS S where = I

_s)33ds

PS _J S E =

J

(a)Ig33vl f

ap

s Therefore on S (s) is given by hm +

Emlfm

_{evaluated on S}

mpbmhm

L P

PJ

Inserting this expression for

(s) in Eq.(3.20), and using Eq.(3.2L):

m

h

0

= p

[1m

m

[WDm

+ Em

(rs)

_mp

_L pr Epr

bhmt

L ps psj

pp

or, more concisely,

O- '

mHmm

(rs)

_mp

p Ps hrn Hm = p p p bmhm

PP

(3.23) (3.25)

may write

[F]

W2[Dm'][Hm]{Dm] - ÍW([Dm'][Hm][Em]

-

[Em'][Hm]{Dm]) + [Em'irHi[Em]

}

(3.26)

where the th row and rth column of [Dm] is

r etc.

Recalling Eq.(3.16), we see that there is in addition a fluid force

on the body resulting from the pressure component p°. As for we define:

JOE'

FOE

-

p, OE (rs) s) (r) s J

aflVg3

OE = _p

VV

OE J

a)v'g33

OE dS + p y OE dS

a.1,2

OElI

(s)

OE13 (s) s s

The total oscillatory force on the submerged body, induced by its motion,

is obtained from the matrix of force coefficients

[F] =

[F] +

[FOE] _(3.21)

Now the cnly frequency dependence of these terms is indicated explicitly

by the appearance of w2 and w in Eq.(3.26). Therefore we may write

[F]

= w2[] -

W[] -

_[cD]

and the matrices

L], []

and [] are independent of w. They clearly correspond to added mass, damping and stiffness. In fact

34

[LI

= [Dm']_Hm] [Dtm] (3.29)

[]

= ([Dm'1[H][Ern]

- [E']Fj[Dmi)

(3.30)

nl

and a corresponding expression may be written for [Ç].

3.5 Discussion of the Resulting Matrices

We shall not require [Ç], though it should be noted that all its terms are proportional to U2, the square of the steady forward velocity.

For bodies of approximately shiplike proportions at practical speeds,

this matrix is negligible compared with the elastic stiffness matrix [g].

The added damping matrix is proportional to U, and from Eq.(3.30)

it is seen to be antisymmetric:

rs

- 0sr

rs

(3:.31)

and 0rr

= O

Physically this implies that if the body has no steady forward speed, I

there is no added damping for this deeply submerged or high fquency case. And when the body has forward speed, there is no dissipation of ener-through hydrodynamic damping, merely a transfer of enerr between modes. The phenomenon has been called "dynamic damping", rather then dissipative

damping, but the conclusion that there is no dissipation has often

vibrations since it may be shown that in the _{case when the free surface}

influence is explicitly developed in the equation for the vertical vibration

of a ship's girder, that part of the hydrodynamic damping _{matrix involving}

the forward speed U also satisfies the relations in Eq.(3.31) (Newman _1965).

Hence neglect of all other hydrodynamic damping, of which wave generation by the body oscillating in the free surface is the most _{important, implies}

that the only sources of energy dissipation _{are structural damping and}

sources internal to the body. _{The only influence of the added damping is}

then to transfer energy between modes, and hence to modify the natural

vibration frequencies _{These considerations have been overlooked in the}

calculations by Goodman (1911) of wave excited vibration _{amplitudes, and}

his assumptïons lead to a non-zero term on the diagonal of the matrix [].

The amount of coupling between the modes, arising _{from the}

off-diagonal terms of the matrix {] defined in Eq.(3.30), _{is generally small,}

but not neglïgible in the lowest modes. _{This is a matter that should be}

investigated further.

For deeply submerged bodies with zero forward _{speed, the only}

motion dependent forces are represented by the matrix

_{[] .}

_{This is}

directly related to the matrix [j required _{for the method of analysis}

inicattd in the previous section. _{The high frequency motion dependent}

forces on a body moving vertically in _{a free surface have values one half}

of the corresponding values for a double body, formed by the submerged

body surface and its reflection in the free surface, _{moving vertically}

in an infinite fluïd. Hence _{is the high frequency added mass}

matrix

{J

_{for the body undergoing symmetric motions on a free surface,}

36

applying to any configuration of elastic body (providing the surface in contact with the fluid is well behaved, in the sense defined when deriving

Eq.(3.8)), subject to a vertical harmonic disturbing motion. We see

immediately that [] is symmetric, but to calculate this matrix we require an explicit form of the solution expressed by Eq.(3.21) and evaluation of

the integrals in Eq.(3.2l) . For certain bodies this is possible, but for

the general case numerical methods must be adopted to approximate these