LR 257φ Theoretical Manual - Calculation of ship responses in regular waves by strip theory

PEF: R.& T.A. REPORT NO. 5103

TITLE: LR 257Ø - THEORETICAL MANUAL

CALCULATION OF SHIP RESPONSES IN REGULAR WAVES BY STRIP THEORY

EDITED BY: A. Blixell, Civ.ing., C.Eng.

DEPARTMENT: Structures Section, Research and Technical Advthsory Services

CONTENTS

ACKNOWLE DGE MENT S

INTRODUCTION THEORY NOMENCLATURE REFERENCES PAGE i 2 4 21 25

The work described in this report substantially follows that given in Ref. (1), and forms the basis of a computer program module which was

written under contract by Oceanics Inc., Plainview, New York, for Lloyd's Register of Shipping. The program, which links with Lloyd's Register's Ship and marine structures/ocean-environment simulation system (2) , constitutes one of a number of modules for calculating ship motions

LLOYD'S REGISTER OF SHIPPING

1.

INTRODUCTION

1.1 This report describes the application of strip theory to the calculation of bending moments, shear forces, motions and accelerations etc.

in regular waves, je. the calculation of transfer functions. This basic data is required before proceeding to calculate ship behaviour in

irregular waves (2).

1.2 The vessel, which is assumed to behave as a rigid body, is taken to be composed of a number of transverse strips or segments to facilitate the determination of the coefficients and exciting forces, and also the integration along the ship. Then the equations of motion are solved to give the ship responses and loads.

1.3 The strip theory approathto the determination of ship responses is thus relatively complex, and

This Certificate is issued upon the ternis of the Rules and Regulations of the Society, which provide

that:-"The Committees of the Society use their best endeavours to ensure that the functions of the Society are properly executed, but it is to be understood that neither the Society nor any Member of any of its Committees nor any of its Officers, Servants or Surveyors is under any cirurestances whatever to be held responsible or liable for any inaccuracy in any report or certificate issued by the Society or its Surveyors, or in anyentry in the Register Book or other publication of the Society, or for any act or omission, default or negligence of any of its Committees or any Memberthereof, or of the Surveyors, or other Officers, Servants or Agents of the Society".

N (Rpt iOc.Loa.) 2.12,69 (RADI AND PRDI1 IN INOLAND)

71,

Fenchurch Street, London, E.C.3

April, 1972.

LR 257Ø - THEORETICAL MANUAL

CALCULATION OF SHIP RESPONSES IN

REGULÄR WAVES BY STRIP THEORY

limited use can be made of the theory.

1.4 A detailed explanation of the theoretical background is given. For information on the

associated computer program reference should be made to LR 257Ø - Userst Manual (3).

2.0

THEORY

2.1

CO-ORDINATE SYSTEMS

The two co-ordinate systems used, one for the wave system and one for the ship, are shown in Fig. 1.

EJrection of shp speed V

X

4-Wave direction of propagation at speed c.

z and z1 pOsitive downwards

Following sea Q0, head sea 18O°

FIG. i

Wave

trough Wavecrest

Wave trough

I1

Axis fixed

The wave propagation, at a velocity c, is fixed in space. The ship moves at a speed, V, at some angle, , with respect to the wave direction.

2.2

WAVE VELOCITY POTENTIAL

The wave velocity potential for waves of short period (independent of water depth) is expressed

as:

=_ace cos k(xt+ct)

where a = wave amplitude c = wave velocity =

g = acceleration of gravity 2iî

k = wave number =

-À = wave length

z'= vertical co-ordinate from the undisturbed water suif ace (with positive downwards)

X=

axis fixed in space t = time

The x-y axes have their origin at the C.G. and the centre of gravity of the ship moves with the ship. The xt co-ordinate of a point in the x-y plane can be expressed as:

xt =-(x+Vt)cos + ysin (2)

The surface wave elevation,n , positive upwards, can be expressed as:

Z:

(1)

2ir

In x-y co-ordinates:

r= asin k [-xcos +ysin +(c-Vcos )t] (4)

D ₌ 12

- 4_J

(x,t)

n= akccosk[ -xcos S+ysin 8 +(c-Vcos )t] (5)

and

r= = -akgsin k [-xcos +ysin +(c-Vcos )t] (6)

The results of the equations of motion will be referred to the wave elevation,, at the origin of the x-y axes:

r= asin k(c-Vcos )t or r= asin Wet where 2î

We T\Tc0s ;)

We is the frequency of encounter.

2.3 VERTICAL PLANE EQUATIONS

The coupled'.equations of motion for heave, z (positive downwards), and pitch, O(positive bow-up) are:

Xb

mz=

f

dx+Z

X5 dx

w

Xb

'ye=_

f

xdx+M

dx w 6

where ¡n = mass of ship

= local sectional vertical hydrodynamic force on ship

x,

x1 = co-ordinates of stern and bow ends of ship

z = wave excitation force on ship w

I = mass-moment of inertia of ship about y-axis

= wave excitation moment on ship

The general vertical force is:

dZ

A'

(-x+V8)

H

N' (_x+VO)_pgB*(_xe)

(11)

2

_Dtl

_{33 z}

where A'33 = local sectional vertical added mass local sectional vertical damping force coefficient

density of water local waterline beam

-2

f

1-3

pg2A 1w

= ratio of generated wave amplitude

and heave amplitude for vertical motion-induced wave

Expanding the derivative:

dA' I

=

-A' (-x+2V)- I

N'-V

(-x+VO)- pgB*(z_xO

) (13)

dx 33 I z dx

The equations of motion, (9)

and (lo),

are then transformed into the following familiar form:

a'z +bz +c'z - dO - eO -

_g'O

= (14) AO

Be + ce -

Dz - E

-

G'z = (15) with N' = p = B* = N' z =

dZ

[PgB*n _{+ IN'}

dA3

.11 -kE

dx

t z

dx

J fl+A3flj (19) where a' = m

fA3dx

b = JN'dx - V fd(A3)

C' =

pg

fB*dx

d = D =

fA3xdx

e = JN'xdx - 2V fA3dx - V

fxd(A3)

(16) g' = pg fB*xdx - Vb A = I +

fA3x2dx

B = fN'x2dx

- 2V fA3xdx - V !xd(A3)

C

pgfB*x2dx

- VE E =

fNxdx - V fxd(A3)

G' ₌ pg JB*xdx

where all the indicated integrations are over the length of the ship.

The right hand sides of equs. _{(14) and (15), which} represent the wave excitation, are given by:

Xb

=1

dx ₍₁₇₎ dx Xb dz

f

W M

=-

-

xdx

₍₁₈₎

w dx xs

The local sectional vertical wave force acting on the ship section is represented as:

dz

-= -

aeh

dx

where =mean sectional draught.

Substituting the expressions for _n, i and from

(4), (5) and (6) with y = O and applying the approximate factor for short wave lengths gives:

[(PgB*.A3kg) _{sin (-kxcos)}

+kc1N'

_zV

dA3\

dx _J

cos(-kxcos

cos w te

-dA' +{(pgB*_A3kg)cos _(-kxcos ) - kc (N' V dx ITrB*

S1fl(--- Sifl

sin (-kxcos ) sin

X

The value of is approximated by:

ii= HC

s

where H = local sectional draught

Cs = local sectional area coe-fficient

The steady state solution of the equations of motion are obtained by conventional methods for second order ordinary differential equations,using complex notation. The solutions are expressed as:

z = z sin (wt+d)

o

e = O0sin (wt+c)

where the zero subscript denotes the amplitudes

andand tare the phase angles,i.e. leads with respect to the wave elevation in eq. (7).

(22)

lo

-The vertical loading is given by:

dZ

dZ w

= 5m(zxO) + +

-dx dx

where m = local mass per unit length.

Eq. (23) is the summation of inertial, hydrodynamic, hydrostatic and wave excitation forces. The latter terms are given in eqs. (13) and (20).

The vertical bending moment at location x0 is then given by:

BM(x)=

f

or!

[ x _Xb z o Xb df (x-x )

-- dx

o ax

and expressed in a form similar to the motions:

BM_z = BM_zo sin (w t+)_e (25)

2.4 LATERAL PLANE EQUATIONS

The coupled equations of motion for sway, y (positive to starboard) , _{yaw, p(positive bow-starboard),}

and roll, (positive starboard-down) are given as:

Xb

my=f

_dx

dx+Y

w dY

p-i

= f

xdx+N

z _{xz dx w} xs (23) (24) df z dx

+ 5

-

(M ₃ + ₃ Dt s s dK D

3-N

_sc

_(+x3-Vi)

-

_{(M5,6)- 6N5}

-(29)

where I = mass-moment of inertia of ship about z-axis

= mass-moment of inertia of ship about

x -axis

= mass-product of inertia of ship in xz- plane

dY

= local sectional lateral hydrodynamic force on ship.

dK

local sectional hydrodynamic rolling moment on ship

YW,NWIKW = wave excitation force and moments on ship

= initial metacentric height of ship (hydrostatic)

The hydrodynamic force and moments are:

dY D -

[M

+x-V) F

-

rs

-N5(+x-Vp) +N

3 ] rs

-N3+N

(+x-Vp)

(30}

12

-where 0G = distance of the ship's CG from the waterline (positive up)

M5 = sectional lateral added mass N = sectional 1aeral damning force

coefficient

M = sectional added mass-moment of inertia due to lateral motion

N5 = sectional damping moment coefficient due to lateral motion

sectional added mass-moment of inertia sectional damping moment coefficient sectional lateral added mass due to roll motion

sectional lateral damping force coeff-icient due to roll motion

and the sectional added mass-moments and damping moment coefficients are taken with respect to an axis at the waterline.

The additional roll damping moment to account for viscous and bilge keel effects is taken as a particular fraction of the critical roll damping:

*

N _{= Cc/L_Nr(Wq)}

r

where N* _{= sectional damping moment coefficient due}

to viscous and bilge keel effects

fraction of critical roll damping (empirical data)

C _{= critical roll damping}

(31) I r = Nr = F rs = N =

mgGM

Ix +

L = ship length (L ₌

_xbxs)

= natural roll frequency

Nr(W)

= value of Nr at frequency

The critical roll damping is expressed in terms of the natural roll frequency:

C = 2 mg GM/w with

dM\

dxJ

-

N - N*'c3 + 'M + M )

(+xj-2V)

r rj s s /dM

dM'

+ + _N

_{-v I}

-

_I L _\

dx

dx,]

(32)

(33)

(34)

¡dF

-'N

_rs

+N

_s

+5N)+V( rssJ

_s

dx

where the integral is taken over the ship length.

Exp an din g the derivatives:

1dM

= -

i (+x-2v)

+

1v-a

N )

(+x-vp)

dx s

(dx

s ldF dN + (F + M + N +

-

rs

S rs s) rs s dx dx dK

r

IdI dM =

_-11+

(

s

rs dx dx / M + F + M

+ Iv i +

14

-The equations of motion (26) , (27) and (28) are then transformed:

a11y + a12y + a14' + a15i4 + a16

+ a17+ a18

=

a21v + a22y + a24 + a25i + a26 + a + a = N

27 28 w (35)

a31y + a32 + a34 + a35iJ.J + a36p + a37

+ a38+a39

= K

The coefficients on the left-hand sides are defined by: a11

= m+fMdx

a12 = fN dx-V fd(M5) a,4 = 1M xdx s a15 = IN xdx-2V 1M dx-Vfxd(M s s s a16

= -V12

a17

= -1F dx-G 1M dx

_rs s

a18 = -IN dx+OGV f d(M )-OG IN dx + Vf d(F

rs s s rs

a21 = fM5xdx a = IN xdx-Vfxd(M 22 s s = I +1M x2dx a24 s a = IN x2dx-2VIM xdx-V!x2d(M 25 s s s a26 = -Va22 a = -I -IF xdx-OGJM xdx 27 xz rs s

a28 ₌

_Nrsxdx+Ohj

_{(M5) -OGIN5xdx+Vfxd (Frs)}

a IM dx-OGJM dx

31 s s

a32 = -fN dx-OGfN dx+Vfd(M )+VOGJd(M)

s s s

a34

_{= -I-fM5xdx_OGfM5xdx}

a35 _{= _fNspXdX_OGJNsXdX+V!Xd(Mscp) +VIxd (M5) -2Va31} a36 = -Va32 a

= I +11 dx+fM dx+OGIF dx+2fM dx

37 x r s rs s a38

= 1(N +N*)dx+5fN dx+

IN

dx+52fN dx

r r s4 rs s _V[Id(Ir)+5

fd(M5)+

_{!d(Frs)+2fd(Ms)}}

a39 =mgGM

where all the integrations are over the ship's length.

(37)

The right hand sides of egus. _{(35), which represent} the wave excitations, are given by:

Xb _dY

=1

w (39) X s Xb _dY N

xdx

(40) X dK K

=1 dx

w dx s

The local sectional lateral force and rotational moment due to the waves acting on the ship are

represented as: W ₌

(pS+M)5.

V

V+NV+k (

s19 sin X rr ß * X sin 5 113* sin 5 s) -

16

-dI(

r

_B*3 DV N V i w

ID

-Ì w dx

_{L_(M5}

Sz) _{Dt s wj} dY 0G dx (43) V w

where V = lateral orbital wave velocity w

S = local section area

= local sectional centre of buoyancy, from waterline

The lateral wave orbital velocity is obtained from:

Vw = w

V

= -akc esin sin k [-xcos

+ ysin -1-(c-Vcos)t] (44)

and

DV

-w

-kh.

- akge sin ecos k [_xcos e-i- ysin +(c-Vcos)t] (45)

Substituting these expressions and expanding terms:

dY

= Tcos Wet+T2

_Wet (46)

where T1 = T3[gT4cos T6+cT5sinT6] T2 = T3[_gT4sin T6+cT5cosT6] sir' sin

-kh.

'A T3 = -ake sin Dt T4 = pS+M5-kM5 dM dM T = N

- V.

+kV-s dx dx T6 = -kxcos 'rrB*

- sin

X

18 -and dK = T cos w t+T sin w t x 7 e 8 e where T7 = T3 [gT9cos T6+cT10sin T6] T8 = T3 [_gTgsinT6+cT10cos T6] B*3 -

-T9 =p - Sz _M5q)_OG T4 dM

T =N +V

lo sq) dx 5

The steady-state solution of the equations of motion are expressed as:

y = y0sin (Wt+K) (48)

=

sin (wt+)

(49)

q) = q)0sin (wet+v) ₍₅₀₎

where the zero-subscriped quantities are the amplitudes

and ,ct and V are phase angles.

The local lateral and rotational loadings are given by: dY -

= - &n(+x-) +

₊ dx 2» B'3

- = -m

dx dK + w dx dx

-s-s0

-g&n ?;q) (47) (51) (52)

where

= local centre of gravity (relative to ship CG, positive down)

y = local mass-radius of gyration in roll

arid the hydrodynamic and wave excitation terms are given in Eqs. (33), (34), (46) and (47).

The lateral bending and torsional moments at location are then: X _Xb df

BM(x)

= or (x-x0) dx (53)

IX

X I o

;1X

bdm

L

TM(x) =1

or dx X O

LS

0

and are again expressed as:

BMy = BMyosin(wet +'r) TM_x = TM_xosin(w t +p)

e

The requirement on the local vertical mass centre

is:

Xb

m

dx = 0 ₍₅₆₎

X

s

similarly, the requirement on the local roll radius of gyration is:

2

my dx

Shear Moment

Acceleration

Relative motion

20

-The product of inertia in the x-z plane is defined

by: Xb

xz

=

f

6mxdx

(58) X s NON-DII'NSIONAL RESULTS

The non-dimesional form of the motions and forces are:

Linear motion (heave, sway)

ç motion amplitude a shear force pgB*La BM (or BM of TM z y X pgB*L4a Acc aw2 RN a I. Mathewson, rincipal Surveyor to

Lloyd's Register of Shipping. Angular motion (pitch, yaw, motion amplifude

-1

c = Wave velocity (LT

= Local sectional hydrodynamic rolling moment on ship (MLT2)

df

Z

-Local vertical load (MT2)

= Local sectional lateral hydrodynamic force on ship (MLT2) dK dx dZ dx 3.0 NOMENCLATURE

A = Ratio of generated wave amplitude and heave amplitude for vertical motion-induced wave

A3 =

Local sectional vertical added mass (ML)

a = Wave amplitude (L)

B* = Local waterline beam (L)

Vertical bending moment (ML2T2)

2 -2 Lateral bending moment (ML T

Critical roll damping (ML2T)

Local sectional area coefficient(-)

= _{Local sectional vertical hydrodynamic}

force on ship

(-2)

Frs = Sectional lateral added mass due to roll motion (M)

GM = Initial metacentric height of ship (hydrostatic) (L) g = _{Acceleration of gravity (LT2)} BM z BM = y Cc Cs

L M

s

22

-H = Local sectional draught (L)

h = Mean sectional draught (L)

I = Sectional added mass-moment of inertia

r

(ML)

= Mass-moment of inertia of ship about

2 x-axis (ML

= Mass-product of inertia of ship in xz- plane (ML2)

I = Mass-moment of inertia of ship y 2 about y-axis (ML 2'ir -1 =

Wave number = (L

X

= Wave excitation morflent on ship (ML2T2)

= Ship length (L)

= Sectional lateral added mass (ML) = Sectional added mass moment of inertia

due to lateral motion (M)

= Wave excitation moment on ship (ML2T2) m = Mass of ship (M)

Nr = Sectional damping moment coefficient

(MLT)

N = Sectional damping moment coefficient due to viscous and bilge keel effects

(MLT-1)

Nrs = Sectional lateral damping force coefficient

N

s

N(ô) =

Value of Nr at frequency w(MLT)

= Sectional lateral damping force

co-efficient (N1T)

N = _{Sectional damping moment coefficient}

s

-1 due to lateral motion (MT

N = Local sectional vertical damping force

coefficient (MLT1)

N = Wave excitation momenton ship (ML2T2) 0G = Distance of the ship's CG from the

waterline (positive up) (L)

S = Local section area (L2)

2 -2

TM = Torsional moment (ML T

x

t = Time (T)

V =

Ship speed (LT)

V = Lateral orbital wave velocity (LT)

x,y,z = Ship coordinates (L)

Y Sway (L)

X,

Xb = Coordinates of stern and bow ends of ship (L)

x' = Axis fixed in space (L)

= Wave excitation force on ship (MLT2)

Z = _{Wave excitation force on ship (MLT2)}

z _{Heave (L)}

z' = Vertical coordinate from the undisturbed water surface (positive downwards) (L)

n

e

'p

24

-z = Local sectional centre of bouyancy, from waterline (L)

= Heading ()

y = Local mass..radius of gyration in roll (L)

= Phase angles ()

Local mass per unit length (ML)

Local centre of gravity (relative to ship CG, positive down) (L)

Fraction of critical roll damping (empirical data)

Surface wave elevation (L)

Roll angle ()

Pitch angle ()

Yaw angle ()

2 Wave velocity potential (L /T)

Wave length (L)

Density of water (ML3)

Frequency of encounter (T)

4. REFERENCES

KAPLAN,P. and RAFF, A.I. 'Evaluation and Verification of Computer Calculations of Wave-Induced Ship

Structural Loads', Oceanics, Inc., Tech. Rpt. No. 70-77, November, 1970.

'Simulation by Computer of Motions and Sea Loads for the Design of Ships and Offshore Structures', Environmental Loads and Motions Group, R.& T.A.S. Report No. 5105, Lloyd's Register of Shipping, 1972.

BLIXELL, A. 'LR 257Ø - Users' Manual. Calculation of Ship Responses in Regular Waves by Strip Theory',

R. & T.A.S. Report No. 5116, Lloyd's Register of Shipping, 1972.