The equations of motion for the pitching and heaving of a ship

CHALMERS UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF NAVAL ARCHITECTURE

AND MARINE ENGINEERING

GOTHENBURG - SWEDEN

THE EQUATIONS OF MOTION

FOR THE PITCHING AND HEAVING OF A SHIP

by

BENGT HALLBERG

TI EQUATIONS OF MOTION POR TI PITCH G

AND HEAVING OP A SHIP

A study of the pessjbjljtjea ef using

:8.

nen..linear system ef

diffjaJ. equations to simulate the pitching iand heaving

motioÉ of a ship on an n1og ceputer, for both natural

Oscillation in. calm water 84 in regular head or

ollcwing

waves.

by

Report Ne. 29 from the Division of Ship hydromechanies at

Chalmers University of Technology

Sponsored by the Swedish Technical Research Council

Pile numbers 2236 and 2691

Gothenburg, March l96

Curt Palkenió

CONTENTS

Page

Introduction ₃

i. The ship motions in -the longitudinal plane 6

2,. The equation of motion 8

Simulation of natural oscillation in calm _water 12

Simulation of forced oscillation in regular waves 16

S rnvy

₂₃

Appendix 1. Nomenclature 26

- _{- 2. References}

33

ft

-

_3.

Basic kinematic and d,namio definitions ₃₉

and relatiohips when considering ship

motions

Il

_-

_{4 Model tests}

48

- 11

-

5

Analog computer progmmming ₅₂

- 6. Theoretical calculation of the hydrostatIc ₅₃

force and moment

7. Theoretical calculation of the 'added mass ₅₅

and mass moment of inertia

- _{8. Theoretical calculation of the damping 60} - ti

-9. Theoretical calculation of the hydrodinam1c ₆₄

coupling

- 10. Theoretical calculation of the exöiting 66

-3-INTRODUCTION

During recent years the behaviour _{of ships in waves, including}

such problems as motions, speed loss, _{strength, stability,}

etc0, has been studied by using _{scientific methods on a}

large scale. _{The fact thát this, has not been done}

earlier depends on the great difficulties encountered iñ this connection.

References were made earlier to extremely simplifIed

con-siderations; maintenance of speed, for example, was estimated from model tests in calm water, bending moment by a study of the ship in a fixed wave and latera], stability by means of

hydrostatic calculation.

During 'recent years, however, _{developments within the fields}

of oceanography and statistics have resulted in much lmowledge

concering the nature of ocean waves at the same tine as

electronic computer techxiique ha made _{it possible to soie}

problems with an extensive volume of calculation.

The behaviour o± a ship in waves can be determined beforehand

either from model tests or theoretical calculations. Model

tests apply only to a certain hull form and a certain wave

system. It is not economically possible in the _preliminary

design work. to 'have model tests carried out in all the types

of waves likely to 'be encountered by the ship. _{In addition to}

this, it wöuld not 'appear to be possible either,to use systematic

tests to make a complete study of all hull _{forms and their}

properties at sea. _{A limited number of model tests, for example}

-4-ithknown factörs in the equations bf mot±on. _{These can then be}

used for a móre. general study. _{Solution of the equations must,}

however, be carried out numerically, for example by means of an analog computer.

The following is a study of the equations of _{shipmotion as such.}

Considering the extent of this problem it has been found

necessary to make the following restrictions: The waves are regular

The ship moves in head or following waves The ship has no forward speed.

The ai of this paper is:

to study the possibility of arriving at an equation of

motion that can be used. to simulate the actual ship

motion,

to study in practice a large analog computer and its

suitability for the solution of the equations of motion, to study the magnitude and frequency dependence of the

coefficients in the equation of motión,

to investigate the significance of the various functions

involved in

the equation of motion.

In the study of the equations of motion the external forces _are

divided into:

Hydrostatic forces

Hydrodynamic forces, independent of the waves

(e) _{Hydrodynamic forces, dependent on the waves}

(a) has been calculated from the hull form (Appendix 6), _{(b) has}

been studied with the help of natural oscillation _{testé (section}

5

The problem has been tackled by setting up a non-linear

differen-tial equation system oÍ' the second order (section 2) _{and comparing}

its solution with the model test results (Appendix 4).

The solution of the equations was obtained _{with the aid of the}

SAAB analog computer SEDA (of the type PACE, Appendix 5).

The differential equation system has been varied and comparisons made between the differeiit résulta.

The coefficients have been calculated according to the different

methods in the literature and hape been compaied wi±h the reSults from the analog computer (Appendices 7 - io).

A summary of the results is shown in _{section 5.}

The author should like to express his thanks to Professor Curt

Falkemo, who made this work possible, _{to Mr. Ike Rullgârd, Mr.}

Bengt Westeriund and others at SAAB who _{made up the programming}

for the analog computer and to all those who have aided in carrying

out the numerical calculations and in writing out this paper.

The help of the Swedish Technical Research _{Council (Statens} Tek-nïska Poskningsrâd) who sponsored the work, is gratefully acknow-ledged.

-6-1. SHIPMOTIQNS IN THE LONGITUDINAL PLANE

Prom a theoretical _{iewpoint, the motione of a ship in the}

longitudinal plane can be treated indepeidentiy from the other

modes of motion. In head Or following two-dimensional waves, the

external forces act in the longitudinal plane _{so that a symmetrical}

hull is not influenced by any forces laterally.

In order to simplify the treatment of this problem, the following assumptions are made:

The ship is considered as being a rigid body.

The relationships betWeen the ship and the model

ae

assumed to be governed only by gravity.

The motion is assumed to be so small that the distinc-tion between the moving and fixed coordinate system of

tie sflip can be neglected (see Appendix 3).

The waves are assumed to be regular and of a sinus forme

With these assumptions in mind, the following is a treatment of

the pitching and heaing motions when the shit has no forward

speed. The position of the ship relative to the position of

equilibrium in calm water is determined by three coordinates, two

modes of translational motion and one mode of rotational motion

(see Appendix

3).

Surging x

Heàving z

Pitching

x is defined as the horizontal displacement of the centre of

gravity,

z is defined as the vertical displacement of the centre of gravity,

is defined as the angle between the waterline of flotation and

z

CG

X

With the assumptions in section 1 and _{in accord&ioe with equation}

(26) in Appendix _{3, the equations of ship motion in te}

longitu-dinai plane can be

I (1) (2) (3) 2. _{EQUATION ÖF MOTION} where:

m = the mass oÍ the ship,

I = I = the moment of inertia f _{the ship around the}

T)

transverse axis,

R _{= external, horizontal force,}

X

R_z = -"- vertical _-"- and

moment round the transverse axis.

In the influence of the surging on the heaving and pitching

motions is neglected, -the above _{equations become:}

= R z =

-8-expressed as follows:

m(3+4i)

_=Rx

m( -k)

= R. = S(1) It is usual to divide

R and S into suitable partial functions

from the calculating point of view (this division is quite formal):

= Mz +N +

+B +P +

where

M

_Z,

_q)i the force (or moment due to the added màss or added

mass

mothent of inertia) q)

T,

damping force (or moment)

,

"

t! hydrodinamic coupling force

(or

oment)

B

q)" "

hydrostatic restoring force (or moment)

, q)

" _{hydrostatic coupling force (or moment)}

q)

" exciting force (or moment)

Now, these functions can be set out as follôws:

(4) . _.. =

-m

_Mq) =

-m

= n +

n//

= + = + ₌ Qq)ì + B

biz

+

b2z2

+

b3z3

Bq) = bq)1 + ₊

P1

+ P2q) + Pq)z Pq) ₌ Pq)1Z + _Pq)2Z +

Pq)zJ

=

ces (uit )

sin (t

+

where wis the frequency

_and

t the time. The coefficient in the

funotiòns M, N, Q, B and P with indices z and q) are assumed t be

constant in time and indepedent of the waves. The exciting

unctions and apply to the regular

_and

sinusoidal head or

following waves and the factors

' ,w ) are also

assumed tö be constant in time. _{The time is zero, when the top}

of wave is at L/2. _{(See also Apèndices 6 - io).}

Shifting M and to

the

left-hand sides in the eq. (3) and expressing: (5)

-

I +

a =m+m

Z Z 9

one obtaines: a =

n

+ n +

q1

+ +

b1z

₊

_b2z2

₊ z z]. z2

b3z3

_{+ +} + ₊ c°

(t +c)

where

(e)

applies to head. waves.

The function x can be assumed to be:

X =

i

sin (ùt. 10

-a nön-line-ar differenti-al equ-atiòn system of the seøond order with constant coefficien±s and sinusòidal exciting functions.

If it is assumed that the surging gives rise only to a phase

displacement in the exciting functions, this influence _{can be}

included in the equation system by expressing:

()

=

!

cos(t

+E + x)

F(1) = sin(wt

+E(, +

2ir

=x-

is the so-called wave factor and the plus sign

The system of equations cannot be solved mathematically _exactly,

but is suitable for solution by using _{an electronic analog}

compu-ter. _{The programme, of which a brief description is given in}

Appendix 5, has been made up in such a way that every coefficient in the differential equation system is, in principle, represented

by a variable potentiometer. _{In this way the coefficients can be}

a =

+ + + +

b1

₊

easily varied. The results are obtained in the for oÍ' recorded

curves, either on an x-yrcorderor, if several are

12

-3. SIMUlATION OF NATURAL OSCILLATION

_{IN CAL WATER}

If the exciting functions F, P _{are assumed to be zero9 a} homo-geneous differential equation system is obtained which corresponds

to the natural oscillation in calm water. TWo oscillation tests

for a ship model (model test i and 2, see Appendix 4) have been

simulated in this way and the coefficients in the equations of motion have been determined.

The hydrostatic forces and möments (B,

_{P, B, p,) have been}

accurately determined (see Appendix 6) and _{the coefficients}

according to (32) have been used in the following _{analysis. The}

other coefficients, for the virtual _{mass and the virtual moment}

of inertia (as, a), for the damping fbrce and moment

n1, n2) and for the hydrodinamjc coupling force _{and momênt} q2, qj1, q2), have then been 'varied and dète±mined by

bringing the solution of the differential equation system ïn

agreement with the results of the model tests (see Pigs.' 2, 3, 4 and 5).

By selecting the model tests _{so that each of the variables z and}

has a small initial value, it was possible not only to determine the hydrodynamjc couplïng (see Figs. 3 arid 4) but also to study the other coefficients. relatively independent of this coupling

(see Figs. 2 and 5). _{The determination was also facilitated by}

the fact that the virtual mass and the virtual moment of inertia

affect mainly the frequency and the damping affects mainly the amplitude.

13 -Results: (9) ...,. a. = 57 a = 46

zi

= -135

n1 = -210

225

n2 = 810

=

l0.o

q1

_-13.5

= -1.6

_{q2 = -0.5}

The results (apart from

_n1,

_{"f2' n1}

_{and n) are compared with}

the theoretical calculations in Pigs. 11,

₁₃

and 14).

Por the sake of comparison, simulation of the two model tests

has been made with the simp1ified diffei'ential _equation:

/

a = n + b z +

z z z z

1

_..

= n

+ b+ pz.

The values of the coefficients for the hydrostatic forces and

moments (b,

p

b, p

) used in this simulation were

approxi-mate values according to equation (33). _{In the same way as}

before. the coefficients a

, a ., n , n were varied and determined

Z()Z

(

(see Pigs. 6, 7, 8 and 9), resulting in,

(11) a

₌₅₇

a=50

= -105 n = -75

See also Figs 11 and

12.

The damping can also be studied with the help _{of the decrease in}

amplitude. A graph is made of the amplitude difference _between

meafl value of the same extreme values (z

) (see Fig. io).

mean mean

The graphs obtained in this way can be compared with the damping

coefficients obtained earlier if these are re-written as follows

for the non-linear damping

7t

4n,,

Z1

z

-

-2-w a

mean

₃ a

mean

zz

z

TEn1

4n2

2

5(j) 2 w a

mean -

3 a(,, (j)

mean

=57,

a

=46

and

TEn

(13)

8

z

-

14 --

-z

2wa

_mean

Many experimental determinations of the coefficients in the

equation of motion are. given in the literature. Determinations

are usually carried out by comparing model tests in which the ship models have been subjected. to a kiown sinusoidal force

and/or moment, with

_a

linear differential equat±on With lmown

harmonic exciting funòtions [2_1,

_[3],

[E],

[7]' [13], [32 ]

Particularly for the determination of the virtual mass end the

mass moent of inertia (a end at), another method based energy calculation in the case of impuls has been used [33]. Determination of the coefficients by a study of the natural

7E fl(4)

- 2wa()

_mean

= 57, a = 50

for the linear damping.

The natural frequencies have been determined by measurement

-

15

-ossillations has aisò been carried out [5], [32].

A compaBison is here made With the result of experiments with

a series of models carried, out by Motora [32]. He gives simple

diagrams for a, a,

_n añd n for the natural frequency as a

fwiction of' the length-breadth ratio, the depth-breadh ratio

and the block coefficient. The following values have been

determined from these diagiarns:

(15) ... a = 55 a = 40

= -91 n = -72

See also Figs. 11 and 12.

Pigs Il, 12, 13 and 14 show the results for the various

deter-minations of the coefficients in the. equation of motion: a,

_a,

0,03

z

nl

3

o

Model test result

sec.

Solution o± the equation of motion

o

ci:

I-b.

o

c)

I-J. c-o t-:J1

(D s

Ocf

H (D

UHg

(D

d

(I)

H

CD C-,.

-(D (D C-,. o

-0,02

3oÏution of the equation

of motion

LIodel test result

o

rn

o

f4. c+

o

iO

r-J. -.. c-1-i-'.

o o

c-1-f-I. CD 1-,J O

o

-Q-(D id

(Df-'

cf-(D c-F

-(D (D fJ. o

Solution of the equation of

motion

e

0,03

0,02

0,01

-0,01

-0,02

_-0,03

4,

rad.

Solution of the equation of motion

Model test result

ora

O H

cf Ç

I-cf

o

i'

r-'

.- cl-I-J.

o o

c+ I-'. CD

see.

H O

Ot

o

cl-i

CD (fl

H

cf (D

cl-r\) 0

CD c-I- O J

z

Model test result

sec.

Solution of the equation of motion

o'

C/

Ha,.

o

&

o

i-i.

rio

(D pI Il ,-J.

Oc

H 'W

(Dd

(1]

1'

c1 t-J.

I

o

- Solution o

the equation of' motion.

lYlodel test result

Solution of the equatiòn

of motion

°

Solution of the e:quation of

motion

_ôz l0

'lo

P±tchj,

Çompar.isön of decrease in amplitude

according to model test and to

simulated damping funktions.

Heaving

_{on-linear damping}

acc. to (9) and (12)

Linear damping acc. to (ii)

_{and (i 3)}

)< odei test I

10

20

30

40

₅₀

_ZmeanslO

l'O

2b

_3b

Nön-linear damping acc. to

(9) and (12)

X model test 2

40

5Ó

Pigure lo

Linear damping ace.

to (ii) and (13)

3

X theoretical calculation

_{ace, to Grim.}

O theoretical

_{calculation ace, to}

Korvin-65

IrOukovsky,

from 'simulation with

_comlete

_equation,

from simulation with

_{simplified equation,}

O

+ from diagram

_{by Motora.}

60

X

55.

+

50

aoefficients for virtual

mass and

masS'moment of

inertia (a

_{and ad,)}

Summery o± results.

025

0,25

,

0,69

7

i ,35

0,69

1 35

w-.

2g

J 55

Pitching

X

504

o

45

C

C

o

o

40f

'

.-150

1 00

loo:

50

1 50 50

-&-nz

0oefficien-s

_{or linear damping}

(n

_{and n). Sumery of results.}

3'

0,2.5 X

0,25

x theoretical calculation acc. to Grim.

O

_{theoretical calculation}

_{acc. -to}

órvin-Kroukovsky

w from simulation with simplified equation

o

+ rom diagram by Motora

0,69.

7 w 1 ,35 0,69

i 35

Figure 12

o Heavii X

10

-Coe±'f.iciens for hydrodynarnic

coupling in the first deriva;es

and q4,1

). Suery of results.

0,25

3

0, 6G

i ,35

C

Pitching

Pigure 13

x theoretical calculation acc. to Grim.

from simulation with complete ecuation.

X X

Heaving

i

20

1.5 X

Coeffiients for hydrodynamic

coupling in the second derivates.

and q2 ). Summery of results.

X

o

X theoretical calculation acc. to Grim.

O theoretical. calculation acc. to

Korvin-'o1ikovs1r.

from simulation with conaplete equation.

Heaving

o

Pigre 14

3 5

w

0,25

0,69

1 35

2g

X

Pit oiling

16

-4. SIMULATION OF FORCED OSCILLATION IN REGULAR WAVES

Since the coefficients in the homogeneous part of the differential

equation system are determined both by simulation of the natural

oscillation tests and by theOretical, calculations9 _{the exciting}

functions remain to be studied.. _{This has been done by simulating}

four tests in regular waves (model tests _{3, 4, 5 and 6, see}

Appendix 4). _{The coefficients determined earlier are considered}

to apply here. _{The. factors in the force and the moment functions}

(,

,

E,

c) have been varied and determined by _{bringing the}

solution of the equation of motion into agreement with the

results of the. model tests. _{The frequency (w) during model}

tests in regular waves when the ship speed is zero is equal to the frequency of the wave.

The intent-Ion of dividing up the study of the equation of motion

in the way as was done in sections 3 and _{4 is to decrease the.}

number of unlmowns in the simulations and thereby _{improve the}

result. _{The determination of the coefficients as done in section}

3, however, is not completely satisfactory _{as a basis of an}

analysis concerning the forcing functions since result could only

be obtained for the natural frequency

_of.

_{the model.}

All the coefficients with the exception of the hydrostatic force arid rnoent are dependent on the frequency (see Figs. li -14).' This cannot be neglected without causing _{excessively large}

errors. _{It is, however, possible to carry out extrapolation}

to the correct frequency with the a-id _{of the theoretical}

calcula-tions. _{This has actually been done but still implies a little}

17

-of the coefficients for the non-linear damping

_(r1i,

"2'

i'

since no theoretical calculating methods Thr these

coefficients are yet known. _{No extrapolation of the coefficients}

for the hydrodynam!c coupling (q1,

_{q2, q1, q2) has been done}

either, since the degree of uncertainty _{would appear to be too}

high.

Simulation of the wave tests has therefore _{been carried: out with}

the simplified equation of motion:

(16) ... a = n

_{+ b1z + b2z2 +}

_b3z3

_{+ P1(j) +} z2 +

+ pz +

cos (uit +

1

2 _{3 2} a(,cI) np + b(1,1 + +

13(I)

+

P1Z

+ + P(,ZZ(1' + eos(wt

+ ea,)

The values of the coefficients for the hydrostatic force and

moment (b and p with index) are calculated _{in Appendix 6(32).}

The coefficients

_{a, a,}

r1,

_{%have been extrapolated in the way}

mentioned above to the correct frequency by _{using Figs. 11, 12,}

13 and 14. _{The result of the extrapolation is shown in Tab. 1}

which even shows the result of the determination of the applied

force and moment compared with theoretical _{calculations.}

A comparison between the solution of the equation of motiòn, which

was traced on a multi-channel recorder to determine _{the phase}

whatso 18 whatso

-ever since the coefficients

'

and can always be

chosen so that there is full agreement in amplitude and phase

displacement. No special study has been _{made òf the form of the}

curves, i.e. their deviatin from the _{sinus form, since}

19

-Table 1. _{Comparison between exciting forces and moments by}

simulation and those by theoretical calculation.

model

test

no.

Coeff. thrgh simulating

alm water test

Coeff. thrgh simulating

wave tet

a Z a nZ n FZ P_(I) E.Z C 3 58 48 120 80 -12 15 -15° 0 4 60 50 130 85 12 26 0 0 5 62 52 140 85 34 68 27° 0 6 60 50 130 85 16 -48 75° 450

model Coeff. thrgh _{Coeff. ace.}

test hydrostatic to

Korvin-no. calculation _Kroukovsky

F C E 3 -.8 27 0 0 -17 38 20° 22° 4 8 69 0 0 15 55 _2l0 -13° 5 ₂₃ 55 0 0 38 61 40 l9 6 8 -69 0 0 model test no. wave direction

-z max aftei wave-crest

-(JJ max aftêr wavecrest 3 head 0,7 4,66 0,015 550 0,021 24° 4 11 1,0 3,95 0,018 89° 0,048 300 5 It 1,3 3,45 0,033 114° 0,056 36° 6 following

0

_3,95 0,012 84° 0,044 65°

20

-As a supplement to the simulation of the wave tests, a study has

been made of the effect of 'the non-linearity and couplings on

the solution of the equation and also the effect of surge on

heaving and pitching. This investigation was carried out without

any comparison with model tests.

In case of head waves having the same frequency as the natural

frequency ( w = 5,4, see (14) ), then the values of the coeffic±en-ts

in the homogeneous part of the differential equation system as

determined in sectIon 3 can be used. As already mentioned, the

values according to (32) have been used for the hydrostatic

force and moment. The exiting force and moment have been esti..

iated by hydrostatic calculation to be:

(17) ...P_

= -10 cost

F(1) = -15 sin wt

(the negative sign depends on the high frequency, see [52, Pigs.

8]. and 83 ]). With these values 'the amplitudes obtained were::

0.025, ₁₎ c

0.030

br comparison reference was made of the same equation of

motion in the case of thè simulation of model tests in waves, equation (16), wich regard to the exciting terms as given in

equation (17). The values of the other coefficients are

shown in Table 2, line a. The solution of this equation can

be seen as an solid line in Pigé. 15-19.

By putting the coefficients: b2,

b3,

z2' zd1'

b2, b3,

= O and giving the coefficients bei, b and

21

-(see Table 2, line b), a study is made of the difference

between an exact function and a simple linearized one

con-cerning the hydröstatic force an4 moment.

The comparison is

shown in Pig. 15.

e. By also putting the coefficients:

p

= O (Table 2, me

c)

a study is made of the effect of the hydrostatic coupling.

This comparison is shown in Pig. 16.

à. Returning to the equation according to point a, q1, q1 are

put = 10 and

q2 = 1 (rounded-off values fiom the.

deter-mination in section 3) (see Table 2, line à).- Te sigiiific.nce

of the hydrodmRmic coupling can thereby be studied.

This

comparison is shown in Fig. 17.

e. On the basis of the equation according to point a, the linear

damping is replaced by the non-linear damping (see Table 2,

1ine e), whereby the differenbe can be studied.

This

com-parison is shown in Pig. 18.

f. On the basis of the equation according to point a, the

exciting functions are altered in accordance with equations

(7) and (8) to:

(18)

F

= -10 cos(wt +x)

-15 sin(t +x)

X

=xSinut and

where

x

= =

- 2.97 and

= 0.1 (largest probable amplitude of surging).

This gives some idea' of the significance of the phase

displace-ment caused by the surging motion.

This comparison is shown

in

22

-Table 2. Coefficients of miscellaneous terms of the equations

of motiöns. a a z 57 n z -105 n zi -n z2 -q zi 0 q z2 0 b zi -1685 b z2 625 b z3 -12500 p z].

-1512550

p_z2 p z 990 b - 0 0 -1697 0 0 -151 0 0 C

"

-

-

o o

"

o o o o o d - - 10 1 -1685 625 -12500 -151 2550 990 e " - -135 225 0 0 ti it ti tt it f -.105 - O O it it it ti it a n

n1

_"2

q1

q2

b1

b2

b3

_{i d2}

_(Z

a 46 -75 - - 0 0 -1554 2131 -6667 -139 681 4910 b - - O 0 -1524 0 0 ¿-151 o o C

"

-

-

o o

"

o o o o o d " " - - 10 1 -1524 2131

-6667 -19 681

4910 e - -210 810 0 0 " " " " " ii _{-75 - - O O} it t? ti it it

m

rd

O.Q4

z

Comparison between

3o1ut-±onof dIfferent

ecuations

of motion.

(-acc. to point a,

- - - - aco. to point b.)

m

_Ge

Oompaiison between solutions of different. eqoetions of motion

(

ace, to point a, - - - - ace, to point e)..

I I t

t

¡ _I

t

I

t

I

t.

I

'I

¡

t

Comparison between

soltjo

of different

euations of

otio C

ncc. to

oint a,

-ace. to point 'd)

Compari$on between

Soiutjon

of different

equations. of motion

(

ace0 to point a,

- - - - acc, to po±nt e).

Ooinparisori between solutions of different eqations

of motion

Pige 19

23

-5. SUIvfl&ARY OP TRE RESULTS

The results of the investigation can be siimmvized as follows: It is quite possible to use the proposed equation system to

represent the pitching and heaving of a ship concerning

amplitude, frequency and phase displacement both in the case of natural oscillation (see Pigs. 2-5) and also in the case of

regular waves.

The analog computer is the best aid for such a study. The

differential equation can naturally be solved by using an approximate numerical method, for example by using a digital computer, but the extensive variations caused by both the

equation of motion itself arid also by the component coefficients

cannot be handled in practice. An extra advantage in the use

of the analog computer is that the solution is obtained in a convenient form as records which can be compared directly with model test results.

A study of simple differential equation systems and of the

significance of the partial functions gave the following

results:

A simple equation system without non-linear damping,

hydrodynnmic coupling and non-linearity in the hydrostatic force and moment is considerably worse for a simulation of natural oscillation (compare Pigs. 6-9 with 2-5). (A

cor-responding comparison concerning the motions in waves

cannot be carried out since eventual deviations are here

compensated for by the exiting fOrce and. moment.

24

-(see Pigs. 10 and 18). _{A certain influence of the wall}

effect in the model tests cannot, however, be excluded in this case.

e. The hdrodynamic coupling must not apparently be neglected

(see point a above and Fig. 17)o

The influence of non-linearity in the hydrostatic force and

moment when considering motions in regular Waves is slight (see Fig. 15).

The hydrostatic coupling (non-linear or linear) is of 'the greatest significance for pitching and heaving, (see Fig.

16).

The coupling effect from a sinusoidal surging is negligible

during pitching and heaving (see Pig. 19).

4. A study of the component coefficients in the equation of motion gave the following results:

Por the magnitude of the motions concerned here, an exact

non-linear function (according to equations (31) and (32) )

for the hydrodyiiamic force end moment can be replaced by a 'linear approximation (as shown in equatIon (33)), this being

based on the assumption of vertical sides of the hull.

All the coefficients concerned with the effect of the

'hydrodynamic force vary considerably with frequency (see Pigs. li-14),

e. The theoretical calculation of the coefficients for the

- 25

hydrodynarnic coupling agrees on the whOle, but several

serious deviations are shown (see Pigs. 11-14). The

calciiating method suggested by Korvin-Kroukorsky is based on factors that are excess±vely inaccurate if a satisfactory

result is to be obtained. Better values re obtained

according to the same principle ii' calculations. are carried

out in the way suggested by Grim. (see igire

_37-39

and

11

rad'. /grade

ni coordinate system

fixed in the ship

ni (appendix 3)

C

m

NOMENCLATURE

As basic uni-bare adapted;

meter (ni) length

kilopond (kp) force second (s) time 26 -Appendix i X translational components

y

m z m

rad./grade rotational components

t?

IT

kpin s -'t-q s r,v -"-kp/kpm N

ZJ)

Q i, B _tt P Z9 ( F _!'_ - 27

-vectorial velocity components of the system fixed in the ship

(appendix 3)

vectorial rotational components of the system fixed in the shit

(appendix 3)

directional cosines (eq 21)

kp m-1 s2 mass

I

kpms2

I

I

-"-

moment o± inertia

(eq.. 19)

R -

kp

R. external forces (eq. 19 a.nd 26) t' external moments (eq. 19 and 26)

functions for external forces and moments

(eq. 3) = 1,2,3) -1 u Ins V t, w p rad. s q t' r

-it-flqj1

kpms

-2 2 rZ2 kprn s 2

kprns2

kps

the 28

%2

b i

rad1

b

b3

bJj3 P(J)1 z2 P(1)2 P kp

m2

kp m rad

kpm

kp m rad kp

raf1

kp

rad2

kp

kp m' rad1

kp raC1

kp kpiri rad./grade rad./grade

coeff. for external diff. equa-tiori (eq. 4,5) forces in

m

-1 kpni i

kpm

2

s added mass coeff.

a

2 virtual mass coeff.

a z -1

kpm

s2 2 kp s z1 kp s

KG m ni b

pz

P(1)

kp

w LOG

n

_z

_Içpin

-1

s

kp

s

-i

kprn

kp m rad

-1

kp rad

kp

weight

m

distance of C.G. froni L/2

positive

aft L/2

'n 2 A rn

n

29

-coeff. for external forces in the

simplified linear diff. equational

system

(eq. io)

distance of O.G. from baseiine

dispiacenient

longitudinal inetacen-tric radius

length betw. perpendiculars

1ongitudinaicoordinate in general

max. breadth of waterline

draught

draught at A.P.

draught at F.P.

blockcoeff.

area of flöat ion waterline

hal-f-breadth of waterline

30

-GP m distance of centre of floatatjon

from C.G., positive aft CG.

A midship cti..area

B/2 d

height of waofile above nieaniine

n wave amplitude

wave length

wave nunther (= -)

kp wave aiergy per length and time unit

(eq. 42,44) t ie period of oscillation s natural period

s_L. _fl_.

rad

frequency

natural frequency, heaving

pitching dimensionless g n accelerät±on of gravity ( 9,81) h T (A) z

31

-added mass coeff., two dimensional

(eq. 37)

coeff. ace. to Korvin-Kroukovsky

(eq. 38)

n-daI±ng coef±.9 two dimensional

(eq. 45)

kp m s2 density oÍ' watêr

kp specific weight of water

amplitude difference betw. two extrerie values

rad. /grade

2_mean

m

mean amplitude betw. two extreme values

Inea rad. /grade

C z

k4

A

s

32

-notations used for the aa1og

-

33

-Appendix 2.

REFNCES.

[i] Proude,W.,: "On the rolling of ships". _{TIN4, 1861}

(The papers of William Proude, London

1955, p. 40-75)

Gerritsma,J.: "An experimental, analysis of shipmo-tions

in longitudinal regular waves." TiO

Re-port nr 30S, deò.

_1958.

Gerritsrna,J.: _{"Experimental determinations of damping,}

added mass and added mass moment of

inertia of a shipniodel." TNO Report nr

258, okt 1957.

Gerritsma,J.: 'tSoe notes on the calculation of pitching

and heaving in longitudinal waves." TNO

report nr

22S,

dec

_1955.

Golova-to9P.: "A study of the transient pitching

oscillations of a ship." J. Ship

Research,

_12,

4, mars

_{1959, p 22-29.}

Golovato,P.: _{'The forces and moments on a heaving}

surface ship." Journ. of Ship Research,

I, 1957, p 19-26.

Goodrich9G.J.: "On the forced pitching of _{a ship in calm}

water." Proc. Symp. on the Behaviour of

Ships in a Seaway,

_1957,

_{Wagenïnge±i,}

_1959,

,Grjja,o.:

-

34

-"Berechnung der durch Schwingungen eines Schiffekörpers erzeugten hydrodynamischen

Kräfte." J.S.T.G-.,

47, 1953,

_p

_277-299.

"Die Schwingungen von schwimmenden,

zweidimensionalen Körpern."

_Haburgische

Schiffbau-Versuchansta1t Bericht nr

_.1171,

sept

_1959.

[iój Grim,O.: _{"Durch Wellen an einen Schiffskörper}

er-reg-de It'äfte." _{Proc. Symposium orì the}

Behavitzr of Shs in a Seaway,

_1951,

Wageiién,

1959,

p

232-265.

[ii] Haskind,M.D.: "Difrakciya vom vokrtig dvizuscegosya

cil.indriceskogo sudna." (Diffraótjon

of waves around a moving cylindrical ship)

Pr.ikl. Math. I. Meh.,

17, 1953, p 431-442

Haskind,M.D.: _{"Kolebaniya plavayusöego kontura ná}

poverbnosti tyàzeloi zidkosti."

(Oscillations of two.dimensjona1 sections in the free sura..ce of a heavy fluid).

Prikl. Match. I. Meli.,

17, 1953.

Hàskind, M.D.: _{"Metod opredeieniya harakteristik kacki}

Riman,I.S.: _{korablya." (Method of determiring the}

pitching and heaving characteristics of

ships). _{Izvestiya 4kad. Nauk. SSSR, Otd.}

Tehn. Nauk.,

_1946,

_p

_1373-1383.

Haskind,M.D..: _{"Metody gidrodinamiki y problemah}

morehod-fosti korablya na voinenii."

(Hydro-dynamic methods in the problem of the

behaviour of ships in waves). Trudy CAGI

603, 1947..

Haskind,M.D.: "Priblizennye metody opredeleniya

gidro-djnamjceskjh hàrakterjstjk kacki."

(Approximate methods of determination of

hydrodynamic characteristics of ship

oscillations). Izvestiya Akad. Nauk SSSR

Otd. Tehn. Nauk,

1954, p 66-86.

35

-Haskind,M.D.: "Two papers on the hydrodynamic theory

of heaving and pitching of a ship",

(Translation). Technical and Research

Bulletin No 1-12, SNAL _1953.

Haskind,M.D.,: _{"Vozmuscayuscie sily i zalivaexnost cudov}

na volnenji," _{(Exciting forces and the}

influence of te ship on the waves).

zvestiya Akad. Nauk. SSSR, Otd. Tehn. 1957, p 65=79.

Havelock,T.H.: "Damping of the heaving and pitching

motion of a ship." Phil. Mag. 33,

p 666-673.

Havelock,T.H.: "The damping of heave and pitch: _a

com-parison of two-dimensional and

three-dimensional calculations." _{TINA, 98,}

1956, p 464-468.

Havelcjck9T.H.: "Waves due to a floating sphere making

periodic heaving oscillations." _PRS

A23l, 1955, p 1-7.

Holsteìn,H.: _{"ber die Verwendung des Energisatzes zur}

Lösung von Oberflächenwellenproblemen."

Ing. Arch. 8, 1937, _{p 103-111.}

Imloy9F.H.: _{"The coplete expressions for added mass}

of a rigid body moving in an ideal fluid.

DTL report 1528, juli 1961.

[23]' Iyarsson9A.: "Modêllförsök för bestämn-îng av böjande

moment i ett ordin't lastfartyg under

gâng . regelbundna vâgor." Insto för

skeppsbyggnadsteknjk, CTH, juni l961

[24] Jacobs,W.R0 "Guide to computational procedure for

and others:

analytical evoluatïon f

shipbending-moments in regular waves." _Davidson

36

-[25]

Joosen9W.P.A.,: "On the longitudinal reduction factor Sparenberg,J.A.,: for the added mass of vibrating ships

with rectangular cross-sections."

ISP 8, 80, april

_{1961, p 143-149.}

Jinnaka,T.:

Kaplan,P.: Hu9P.N.

"Some experiments on the exciting forces of waves acting on the fixed ship model."

J.Zosen Kiokai,

103, 1958, p 47-57.

"Three-dimensionál stripwise damping coefficients for heave and pitch of a

submerged slender spheroid." J. Ship

Research, 4, :L, juni

1960, p 1-7.

motiöns." J. Zosen Kiokai, 105,

_1959,

p 83-92.

Krylov,A.N.: _{"A General Theory of the Oscillations}

of a Ship on Waves." TINA,

40, 1898,

p 135-196.

[29]

Landweber,L.: "Added mass of a three-parameter family

Macagno9M.C.de: of twodiniensional forces oscillating

in a free surface.". _{J.. Ship Research,}

4 mars

1959, p 36-48.

[30]

Landweber,L.: "Added mass of two-dimensional forms

Mac.agno,M.C.de: oscillating in a free surface." J.

Ship Research, 1,

3, 1957, p 20-30.

[31] LeWis9F.M.: _{"The inertia of the water surrounding}

a vibrating ship." Tr. SNAJ,

_37,

1929, p 1-20.

[32]

Motora,S.: "On the measurement of added mass and

added moment of inertia for ship motions."

J. Zosen Kiokal,

107, juli 1960, p 83-96.

[33]'

Motora,S.: "On the measurement. of added mass and added mass moment of inertia for ship

-

37

-f34]

Newman,J.N.: _{"A linearized theory for the motion of}

a thin ship in regular waves." J.Ship

Research,

5,

_].,

juni 1961, p '34-55.

[3.5]

NeWman,J.N.: _{"The damping and wave resistance of a}

pitching and heaving ship." J. of

Ship Research,

3, 1959, p 1-19

[36]

Newinan,J.N.: _{"A note on the stripwise damping of a}

submerged spheroid." J. Ship Research,

4, 1, juni 1960, p 8-11.

[37]

Newman,J.N,: "The damping of an oscillating, ellipsoid

near a free surface." J. Ship Researôh

5, 3,

dec

1961, p 44-58.

[38]

"Nomenclature for Treating the Motion

of a Submerged Body Through a Fluid",

SNAIV _{Technical and Research Bulletin}

nr

1-5, april 1952.

[39]

011endorff,P.: _{"Die Welt der Vektoren", Wien}

₁₉₅₀

[40] P.avlenko, G.: _{"Oscillations of. ships." Leningrad,}

19350

[41] Peters,A.S.: _{"The motion of a ship as a floating rigid}

Stokèr,J.J.: body in

a

seaway." Comm. Pure and Appi.

Math., 10,

_{1957, p 399-490.}

[42]

ProhaskaC.W,: _{"Vibrations verticales du navire." ATM}

46, 1947, p 171-219.

[43]

Radosavijeyjc,L1.:"On the Smith effect." ISP,

4, 1957,

p 478-490.

[44] Srnith,W,E.,: "Hogging and sagging strains in _{a seaWay}

as influenced by wave structure." _TiNA,

2.4, 1883, p 135-153,

[45]

St.Denis9M.:

"On sustained sea speed." _{Tr. SNAJV,}

_59,

[53]

Watson9T.O.:

-

38

-ships in waves, part

_2:

ship motions."

ISP, 6 & 7, 63-70,

nov

159-juni 1960.

"Experimental investigätion of the

vertical forces acting on prolate

spheroids in sinusoidal heave motionÇ"

University of Cal. report,

82, 18,

april

_1961.

[46]

Streeter,V,L. "Handbook of Fluid DynriIcs", New

and others:

York

_1961.

[ii.'y] Tasäj,P.: _{"Ori the damping force and added mass}

of ships heaving and pit.hing."

J.Zosen Kiokal,

_{105, 1959, p 47-56.}

[48]

Ursefl,F.: _{"On the heaving motion of a circular}

cylinder on the surí'ace of a fluid."

QJMAM,

_{1949, p 218-231.}

[49]

Ursell,P.,: _{"On the virtual mass and damping} of

floating bodies at zero speed ahead." Proc.. Symp. on the Behaviour of Ships

in a Seaway,

₁₉₅₇

_Wageningen,

_1959,

p 374-387.

[50]

Ursell,F.:. _{"Short surface waves due to an}

oscillating immersed body." PRS A

220, 1953, p 90-103..

[51]

tlrsell9P.: _{"Water waves generated, by oscillating}

bodies." QJMA1,

7, 1954, p 427-437.

[52]

Vossers,G.: _{"Fundamentals of the behaviour of}

[54]

Wejnblu.za9 G.. _{"On the notions of ships at sea."}

St.Denjs,M.:

Tr. SNA,

_{58, 1950, p 184-248.}

[] Wendel9K.:

"Hydrodynamische Massen und

hydro-dynamische Massenträgheitsmomente.

JSTG.,

44, 1950, p 207-255.

Two co-ordinate systems9 one reference system (relative to the

forward

motiön of the ship in calm water)

_ana

one ovab1e co-ordinate system - fixed in the ship - are defined.

The reference

O,

origin

x axis along the mean course of the ship and horizontal, positive ahead,

y axis at right-angles to the mean course of the ship and

horizontal, positive to port,

z axis vertical, positive upwards.

The co-ordinate system (Q-r-system) fixed in the ship:

,

origin

axis

i

axis

axis

39

-system (o-xyz -system):

moving wïth constant speed

(=

the mean speed of

the ship) along a horizontal straight path ( _the

mean course of the ship) through the location of'

the centre of gravity when moving forward at the same speed in calm water,

Appendix 3.

BASIC KINEMATIC AND DYNAMIC DEFINITIONS AND THE RELATIONSHIP WHEN CONS IDERING SHIPMOTIONS

at the centre of gravity,

coincident with the longitudinal main inertia axis for the ship, posiiive ahead,

coincident with the transverse main inertia axis for the ship, positive to port,

Fig. 20. _{The movable co-ordinate system - fixed in the ship}

system).

The position of a ship relative tö the reference system, is

determined by six coordinates, three modes of _{trnslational and}

rotational components as shown below:

Desjnat ion Surging _x Swaying _y Heaving _z RoIling _p Pitching Yawing

The traslationa1 coponents (x, y, z) are identical with the

co-ordinates for in the O-xyz system.

The rotational components (,

, .) can be defined in different

ways, see for example [38], [39], [2e] and [56]. According t

[38] and [46], an abitrary Ñtation of a ship can be described in 4G

41

-the following way (see Fig. 21):

Assumé first that the axes , r, coinôide with the axes

y, z,

then turn the ship round the z axis, angle _{.3., whereby the}

t t

and the r axes reach the new positions 0 and, 01)

then turn round the r' axis, angle d whereby the 0 axis

and the axis reaóh the posïtións 0

_{and O,}

finally turn round the axis, angle p , whereby the axes 01)

and 0 become Or) and 0

In accordance with this, the following definit±ons _{are obtained:}

the angle between the z-p1ane (vertical) and the ¿

-plane (a rotation round the _{axis), positive för}

heeling to starbOard,

the angle between the xy-plane (horizontal) and the axis, positive for the bows downwards,

the angle between the xz-plane (vertical) and the z-plane (vertical) (a rotation round the z axis), positive for the bows to port.

42

-Fig. 21. _{Orientation of the 2-y-system relative to the}

reference yste by means of the a!lgles

p,

and.

The dynamic consideration is simplified 1±' the motion is regarded

and

-

43

-u velocity component in the direction at Q

y =

It _

ti Ii lt

-

it ii

Q

il t! it ti It ti

Q

p = tIe rOEtational velocity component aroufld the axis

q lt ti t, _ti I! -it _!

li lt ti ti ri r! ti

'r

FIg. 22. _{Velocity components th the QT)C system.}

If the external forces

_R,

_{influence the ship in the}

directions and the external mOments S , S , S around same

p q r

axes, then the equatiöns of shipmotion _{are (see for example [38 ]}

44. -= m(û - rv + qw) R. m(' - pw + ru)

(9)

= m( - qu + pv)

s

=

i+ (i

- I )qr Sq =

14 + (I

i

)pr =. I + (i - )pq, where

m = the mass of the ship,

= the mass moment of inertia of the ship

around the axis,

= the mass moment of inertia of the ship

around the r axis9

= the mass moment of inertia ôÍ' the ship

around the axis

(Note that the added mass and the mass moment of inertia are

here included in the external, forces and mome±its.)

Usuälly the motion Íor example in the _{case of model tests, is}

described in the reference system, henöe _{it is desirable to have}

the equation of motion in this sysiem. _{The transfòrmat.jon of the}

equation (1g) to the O-xyz system. is, hoWévei, _{complicated, so}

that the distinction betweên the two co-ordinate systems is

usually neglected. _{The approximation is good if the rotations}

are small, this being exemplified by the derivation _{on the}

45

-According to the laws for the turning of the co-ordinate system

then:

u = i a11 + r a12 + a13

v=ka21+ra22+a23

w = x a31 + Y a32 +.z a33,

where a. . are the directional cosines showing the direction of

-

-the arid _{axes relative to the 0-xyz system, arid are}

defined as:

a11 = cos(), a12 = cos(y),

a13 = cos(z _),

a21 = cos(x), a22 = cos(yr)), a23 = cos(zfl),

a31 = cos(x), a32 =cos(y),

a33 = cos(z _).

There are six relationships between the nine directional cosines,

hence the turning is completely determined by three angles, _for

example the rotational components _{p, and8 . with the ai of}

spherical trigonometry [39], the follo*ing is obtained (see

_a10

[38jand [46]).

... a11 = cos cos .3.

a12 = COS(l)

a3 = -

Sjfl

Sill

a21 = smp sin

cose - coscp

_Sifl&

a22 = sincp

a23

smp

sin cos

sïn.3. + coscp cos.3.

a31 coscp _Sin(IJ

005.3. + Siflcp COSi

a32 = coap a33 = coscp sin (j) cosd Sin , - Sin p

005

46

-If thé expressions for the directional cosines in equation (20) are inser±ed, then:

(23)

u

= COS 4,

COS & +

COS 4, sine _{- sin4,}

v=± (sin cpsin4,

_{cose - coscp sin}

) +

+ _{(sinq sin4, sin + coscp cos '&}

) +

+ Sin ( COS

w = k

(cos _{ç 4, cos 8 + sin p sin 8 )} +

+ _{r (cos p sin}

4, sin 8 - sin p ces 8 ) +

+ cos p cos 4,

In a similar way, the following is _obtained:

(24) _{p =}

-q = . sin p cos 4,

+ 4, cos p

r = _{cos p}

cos4, -

_sincp.

Therefore, for small values of

, 4, ,

8:

(25)

Uj*

z

)

r

If i-t is assijmed that the djs%jnctïn between the movable and the fixed co-ordinate systerns can be neglected, ihen the

47 -[R = m(f - ₊ 4) R = m( _ + k)

-4) ± +

(26)

..

sP = i°c+ (i

-

) S() _{= + (I} -) s&= I'4; + 'r) - I; ) ) where R , R

, R are the external forces in the x, _y z

direc-x _y z

tions and S , S S _{aré the external moments around the}

same

X

y

z axes.

48

MODEL TESTS

Appendix 4.

Model tests were carried out on a 4-metre long wooden model

of M/S _{CanadatT (see Table 3 for data, Pig. 23 för bOdy pian)}

at the Swedish State Shipbuilding Experimental Tank _{in 1960}

[23].

Heaving and pitching were measured and recorded on a two-channel

recorder by measuring the vertical motion

at

two points located

in the centre-line at equal distances fore and _{aft the centre}

of gravity. _{The model was restrained for lateral motion. During}

the tests the model was loaded with ballast _{so that it floated}

without trim. _{Its weight and longitudinal moment of inertia}

were determiied.

A series of tests in regular _{waves with various speeds,}

wave-lengths and wave heights as weil _{as some natural oscillation}

tests

in

calm water were carried out.

_Prom

_{these tests, two}

natural oscillation tests (model tests i and 2) and four wave tests (model tests 3, 4, 5 and 6) have been selected for this

paper. _{The results are shown in Pigs. 24 - 27 and Table}

4.

From the records of the wave tests, _{the amplitude of the motions}

as a mean value of about thirty extreme values _{and phase}

49

-Sonie supplementary model tests - mainly concerning natural oscillations iii calm water - have also been carTied out at the

Table 3. Data for model9 ni/s "Canada".

Model scale

Length betw. perpendiculars

Largest breadth

Draught

Weight

Mass

Longitudinal moment of inertia

Natu±a1 period, heaving

t, t,

-

50

-Vertical centre of gravity above ba seliné B ni ni I T z 1:36 3,936 ni 0,540 ni 0,226 in 313 kp 31,9 kps2ni1 27,8 2. 1,16 sek pitching T 1,16 sek

Longitudinal centre of gravity LÖG 0,044 ni

abaf t L/2

KG 0,226 n

Block coefficient _0,65

Area of floatation waterline _{1,697 ra2}

Waterlj.

ea coeff. 0, 8

Centre of floatation aft C.G. GP 0,089 ni

51

-Table 4. Resült of niodeltest in harmonic Waves.

no. wave X _T

--

_z

max (il'mmax

direction _{pp after} wave-top 'r a'ter wave-top 3 head _{2,84 0,7 4,66 1,35 0,071} 0,015 55 0,021 24 4 I! 3,94 1,0 3,95 1,59 0,066 0,018 89 0,048 300 5 5,17 1,3 3,45 1,82 0,066 0,033 114° 0,056 36° 6 following 3,94 1,0 3,95 1,59 0,065 0,012 84° 0,Ô44 65°

Body plan of rn/s Canada0

o

o

c»J

o

o

o

o

Model test i

.

Fiire 24

Measuring of heaving (z).

o

o

o

c,'J

o

o

o

o

o

Model test i.

_{Figure 25}

Measuring of pitching (i,)

('J

o

MOdel test 2

_{Figure 26}

Measuring of heaving (z).

Model test 2.

Measuring of pitching (4i)

CJ Q O O O O O 0 o, O O

O O

O

igure 27

O O

52

-Appendix 5

ANALOG COIVTUTER PROGRAMMING

In an analog computer the physical magnitudes are represented by

electric völtages. Ail the variables, including the derivatives,

must be suitably scaled so that they correspond to voltages of

betwêen lOO V, A time scale can also be introduced which then

influences the amplitude scale. In this case the computer time

was selected 5 times as slow as the actual time..

After the scaling had been carried out,, the original equations

were re-written in thè form of equations (see Pigs.

₂₈

_and

₂₉₎

which formed the basis for the setting up of a wiring diagram

(Pigs. 30 and 31). The computer components cöns±st in principle

of integrators ( - ), su.mmators (--j ) which can also be

used as sign reversers (small triangles), multipliôators ( -®- ),

inverters ( -®--- ) and potentiometers (small circles). The

potentiometers are used for multiplication by a constant, for

example a öoefficiént in the differential equation system.

The accuracy of the analog machine is stfficiently great in

this case. With the wiring used, the error is probably less

Pigure 28.

-

[2oJ

:

_L200]

_f

0.i

+

[oo

iii]

.ó.o

--

[200 q,] _'

-0.,

-t-

[o.ry}

_

;o

+

[ioo .]

[200

_{] =}

[20 ï]

2

[ioco ]

=

[200 ]

o

= -

[c

]

[200

J ''

.

,

+

_

[ooi.]

-- a,

[Q

[2oo}= [2o]..2

[î000

_{] =}

[200

_]

Scaled eq.uations (Scale

_{of tine:}

_t=5t)

>'a]

40

± Eioô

[0.5-

_{y: J}

=

-

[i000

]

+

{i4]

20

-

froVj

oo

-

[000

_{y j}

{,o4

2]

_;?2oo

₊

{i4

_p

y]

2oò

{a5Y2

figure

29. Scaled.

_{equations.. (cont)}

[,ÔOOv]

f

₂₀₀₀₀

'2

...1

L"

[/Cc V,]

[ioo

(,.t

_)]

co5(,w%)-(wt#e)Jsit(siz ti#)

[t000J

+

[1o4z2J

+[/o4y}

[oo

J

[too

3;/? (wt t

*

±

{,'oa

(w * e) $j

_¿v$)

1

''P3

J 200000

r

I d

r

L

6

\:20Z 2

- 1oo

J

-2o

Â

t too

/

OOo tíÒ4ZT

[Ñ4

LN]i

M!

t

Recorder (x)

LO

o

'

Wiring diagraza

_or

exoiting termS.

1ø4 .Ç'# O

Pigure 31

k,'

Mz

R2

Ri;

6>

t

53

-Appendix 6.

TEEORETICAL CALCULATION OF TI HYDROSTATIC FORCE AND TH MOMENT

The hydröstatic force and moment, which _{only vary with .z and}

_4',

are. obtained from the hull f rm by detemiriing the change _in

displacement which can be written:

... _Z,d)z,d1 f(LCB, ) .

g(d,d)

= h(z,4' ).

Tbø draught fore end aft as a function Of the motion is obtained as follows:

... d = d - z - (L/2 - LCG.)tg4' -(d-)

,co:(I)

-1

= d - z + (L/2 + LCG)tg4' -(d-

005(I)

-Since d = KG and _4' tg _4' then:

.. d = d - z - (L/2 = LG.)4'

d=d-z+(L/2+LCG)4'.

The displacement and th _{center of buöyancy for vaious draughts}

fore and aft have been worked out on an electronic computer

---b

Displacement (mm3) for mode].

0±

We

_{Canada as Thnct ion of draught}

(=) at AP and PP.

Figure 32

40

o

o

AP

=

Distance. () from 1/2 of center

of buóyancy (neg.=abaft

_{"p /2}

as function of draught

at

AP and PP.

O0 + 4.00

loo

200

300

400

Pigure 33

PP

I

(30)

(33).

-

54

-With knowledge of the above and ince:

B + P difference in

z . z

B +

¿

difference in

Then thô .iydrostatic force and moment can be determined for

various values of z and J , Figs. 34 and

35.

By using the least square method and the polynöms:

(31)

...

B + P =

b1z

+

b2z2

+

b3z3

+

p1

2 + PZ(l)Z4

B P

b1

+

b22

+

b3

one gets:

=bz= YVLz1697z

= =

-y

A 4 = -151 d1 B(1, = b

_-yA

_{L(1) =}

_-1524 P

_{_,}

A1

_{z = -151 z} (32)

_bi

₌ -1685 b _{= -1554}

b2

625 b(12

= 2131

b3

= -12500

_b3

_-6667 = -151

_{P(,1 = -139}

= 2550

p2

= 681 990 ₌ 4910

If the hull is assumed to be vertical in the vicinity of the

waterline of flotation, then the following linear approximation can be obtained:

Upward vertical hydrostatic force

_{Pigure 34}

in kp as function of Z and

'.

Por ce

kp

cI,=-0, 04

C1'=O,

02

4=0,02 and 0904

50

50

In

0,04

0,02

0,02

gydrostatic mömen.t (Ba, +P,) (kpm)

55

-Appendix 7

THEORETICAL CALCULATION OP THE ADDED MASS AND THE ADDED MASS

=

---=============---====---MONT OP IHERTIA

The virtual mass and moment of inertia. (a) consist partly of the

mass (m) and the mass moment of inertia"(I) of the _{ship itself,}

and partly of the added mass and the added mass moment of inertia

(m and respectively) according to equation (5).

Roughly it can be set according to [52]:

(34) ... m

m

32

14.

A three dimensional calculation of n arid for the ahips hull

cannot be carried out without certain _{simplified assumptions [16 j,}

[20],

t

_{22], [25], [34] and [41].}

The calculation is even then very complicated.

A so-called "strip method" is usually used for the deterttination of

m

and m . _{A calculation, is thereby made, on the assumption}

of

two-dimensional flow, first of the added _{mass and moment of}

inertia per unit of length (m) for the various cross-sections

and then these are integrated longitudinally:

(35)

... m

= L la' dl