DASH, computer program for dynamic analysis of ship hulls

REPÖRT No. 159S

September 1971

(S 3f)

NEDERLANDS SCHEEPSSTUDIECENTRUM TNO

NETHERLANDS SHIP RESEARCH CENTRE TNO

SHIPBUILDING DEPARTMENT

LEEGHWATERSTRAAT 5 DELFT

DASH,

COMPUTER PROGRAM FOR

DYNAMIC ANALYSIS OF SHIP HULLS

(DASH

REKENPROGRAMMA VOOR DE DYNAMISCHE ANALYSE VAN DE SCHEEPSROMP)

by

IR S. HYLARIDES

(Netherlands Ship Model Basih)

VOOR WOO RD

In 1964 werd het rapport no. 58S ,,Nurnerieke berekening van de verticale. romptrillingen van schepen dOor discretisering van het trillingssysteem" door J. de Vries, gepubliceerd. Dit rapport beschreef een methode orn de -scheepstrillingen te berekenen, waarbij de romp in principe als een z.g. Timosheñkobalk werd beschouwd. Dit rekenprogramma wordt nog stecids gebruikt en levert bevredigende resultaten voor de trillingen van lagere orde in niet te complexe gevalleñ.

th recentere rapporten van het Scheepsstudiecentrum, ño. 144S bijvoOrbeeld, is gewëzen op de nadelen van de balkmethode en dit is ook in het hier gepresenteerde rapport herhaald.

Met de publikatie in 1967 van Hylarides' eeñte rapport over de analyse vän scheépstrillingen met behulp van dé eindige ele-meñten methode (rapport no. 107S) werd eennieuwe benadering geintÑduceerd, welke verder is uitgewerkt in het tweede rapport (no. 153 S) dat dit jaar werd uitgegeven.

Met de in het laatstgenoemde rapport beschreven techniek is

een rekenprograrnnia ontwikkeld eñ het is dit programma dat

het onderwerp is van dit nieuwe rapport.

Dit DASH (Dynamic Analysis of Ship Hulls) programma is zeer zeker geen eenvoudig programma en het is zeker nag vat

baar voor verbetering maar het voldoet veci beter aan de be hoefte aan een betrouwbare methode orn het complexe cine

dimensionale trillingsprobleem van de scheepsromp aan te vatten dan de prograrnma's die zijn gëbaseerd op de baikmethode.

StET NEDERLANDS SCHEEPSSTUDIECENTRUM TNO

PREFACE

In 1964 thç report no. 58S ,,Nurnerical calculation of vertical hull vibrations by discretizing the vibration system" by J. de

Vries was publishecL This report describéd a method to calculate

the hull vibrátions considering the hull fundamentally as a so

called Timoshenko beam. This computer program is still being used and prodüces satisfactory results for the lower-noded hull vibrations in ñot too compléx cäses.

In more tecent reports of thé Ship Research Centre, no. 144S for instance, the disadvantages of the bearn móthOd have been pointed out and this is also repeated in the present report.

With the publication in 1967 of Hylarides' flrt report on ship vibration analyses by finite element technique (report ño. 107S) a new approach was introduced, this has been developed fUrther in the second report (no. i 53S) issued this year.

With the technique described in the last rnentioned report, a

computer program has been developed and it is this prograrn

that is subject of the present report.

This DASH (Dynamic Analysis of Ship Hulls) program is by no means a simple program and it certainly is capable of

improvement, but it supplies the want for a reliable method to tackle the complex three dimensional vibration problem of the

ship hull much better than the programs based on the beam

rnethod

References

. .

. ...--

13

Appeñdh

- 14

CONTENTS

page

List of symbols

6

Summary

7

i

Introductión

7-2.

Slender beam method versus fiite element method

7

3

The fundamentals oÉ DASH

8

4

Short description of the. DASH computer program

10

5

Results of DASH

- IO

LIST OF SYMBOLS

c

Damping fOrce per unit length pet unit velocity

wig

Mass per unit length, including entrained

'atçr

f

Load vector

U

Displacemeñt vector

K

Stiffness matrix

M

Mass matrix

Circular frequency of the harmomc forces and vibrations

Natural circular frequency

DASH,

COMPUTER PRO3RAM FOR DYNAMIC ANALYSIS OF SHIP HULLS *

by

Ir. S. HYLARIDES

Summary

For detailed hull vibration analysis two calculations methods are available: the beam method and the fiñite element method.

These two methods are compared and it is shown that the finite element method is by far supenor

Based on the finite element method a computer program has been written, which is called DASH. A short description of DASH is given, eli.itidated with some results.

In the appendix a more detailed description is given.

1

Introduction

In modern ship design completely new concepts have

to be realised:

- extremely increased ship dimensions, accompanied

iby a correspondingly increased propulsive power;

- extremely increased propulsive power in order to

obtain high speeds;

-

combination of these demands.

In the earliet freighters, vibrations Were mostly of

secondary concern. For ships with extremely increased

dimensions or propulsive power the vibration probleth

becomes as important as, for example, the strength

problem or the hydrodynamic hull performance.

For the propeller-excited vibrations at manoeuvring

and service speed extrapolation techniques fail, so that

use has to be made of a detailed calculation techmque

At present two methOds are available:

- slender beam method;

- finite element method.

Until recently the calculation of hull vibrations was

based on the principle of a slender beam as formulated

by Bernoulli. This method shows to be inadequate for

the calculatión of complex hull vibrations, in spite of

several refinements applied to take account of

three-dimensional effects of the hull, which are neglected by

the simple beam method.

For this reason the Netherlands Ship Model Basin,

stimulated and supported by the Netherlands Ship

Research Centre TNO, developed a more suitable

calculation method. Ïhig method is based on the con

cept of breaking down a structure into finite elements.

In this way the three-dimensional character is

main-tained.

In the following chapter these methods are

compar-ed, from which the justification of the use of the finite

element in this field of problems will be shown.

* NSMB publication no. 367.

2

Slender beam method versus finite element method

In vibration analyses two aspects have to be considered

-

the natural frequencies and corresponding vibration

modes;

- the level of the forced vibrations.

For the calculation ofthe natural frequencies of the

lower-noded vertical hull vibrations (2-, 3- and

4-noded modes) rather a good approximation is to be

expected by means of the sleñder beam method.

How-ever, for the higher-noded hull vibrations the inaccuracy

of this method becomes unacceptable, which is a

serious drawback as at service speed especiàlly the

higher-noded vibration modes are important for

modern ships, suffering from propeller-excited hull

vibrations.

For what happens:

- with increased ship dimensions the natural frequen.

cies of the hull vibrations decrease and those of the

higher-noded vibrations come into the range of the

blade frequency at manoeuvring and service speed as

the propeller rpm remain of the same order or increase

slightly;

- for small ships the propulsive power increases

signif-icantly, accompanied with a considerable increase of

the propeller rpm, so that the higher-noded vibrations

come into the range of the blade freqûency.

In the calculation of the level Of the forced

vibra-tiöns by mèans of the beam method, use has to be

made of the natural frequencies and corresponding

modes in the frequency ìànge of interest. Any

inaccu-racy in these data will be found again in the calulated

level of the forced vibrations

That the beam method fails is caused by the neglect

of certain mechanical properties of the hull [1, -2],

which become important at the higher-noded

vibra-tions due to the increased complexity of the vibration

mode. These properties :are:

8

- shear lag;

- sprung masses;

- structural discontinuities;

- three-dimensional thass distribution.

However, if the finite element technique is used, it is

possible to take account of these properties directly

without special efforts in the preparation of the input

for the computer program [1, 3, 4].

To calculate the horizontal vibrations by means of

the slender beam method the same type of equations

are used as for the vertical vibrations. This means that

the coupling with the torsional vibrations is neglected,

an unacceptable approximation, especially if the ship

is provided with large hatch openings.

At the higher frequencies the natural frequencies of

the vertical, horizontal and torsional hull vibrations

are so close to each other, that coupling between the

several vibration modes always occurs, so that it is

requisite to consider all types of vibrations

simulta-neously in the calculations. A hard job to realize by

means of the beam method, but automatically done

by the finite element techniquç.

In the fluite element techmque a structure is broken

down into a large number of elements [1, 3]. In this

way the three-dimensional character of the structure s

maintained, a fundameñtal difference with the beam

method.

For the preparation of the input and for the

calcula-tion of the solucalcula-tion much time is needed for the finite

element method. However, if the beam method is

extended in such a way that it has a comparable

accuracy, a similar amount of time is needed for the

preparations and calculations. But then the finite

le,

ment method has the advantage of being

fundamn-tally bettet, so that unknown effects can already be

noticed at the evaluation of the calculated results,

whereas the beam method does not show these effects,

because they are not explicitly formulated in the cal

culatioñ concept

The computer program DASH (Dynamic Analyses

of Ship Hulls)

is therefore based on the finite

le

mént method. A description of this program is given

on the following pages..

3

The fúndamèntalg

f DASH

Ïn this chapter an overall description of the

coni.puta-tion technique, as used in DASH, is given. For more

detailed information reference is made to the appendix

and the bibliography.

In the finite eleieñt technique a structure is broken

down into simple elements (triangular or rectangular

plate and bar elements) which are connected to each

other in a limited number of points (the corners of

plate elements) The elements are chosen of such a

simplicity that it is possible to express rather accurately

the stresses in these elements as a function of the

dis-placements of the connection points or joints

Then the stresses thus found at the sides of an

ele-ment are replaced by an equivalent force system

apply-ing at the joints. Also the external forces are

concen-trated in the joints. Requiring equilibrium betweeh the

internal and external forces at each joint in each of

the co-ordinate directions gives as many linear

equa-tions as there are (yet unknown) joint displacements.

In each of the co-ordinate directions these equations of

equilibrium are set up. In this way a solvable set of

equations has been obtained [1, 3, 5, 6].

A ship hull

is a three-dimensional

structure of

plates and bars. Therefore the bending stiffness of

these plates and bars plays second fiddle in the overall

hull response to dynamic forces, so that in the

calcula-tions only the displacements of the joints are considered

In vibration analyses also the masses have to be

taken into account. This is done by introducing Inertia

forces which are concentrated in the joints That means

that the mass is lumped and placed at the joints in

such a way that the overall mass distribution does not

change An impression of a possible choice of finite

eléments and their joints for a ship hull is given in

Fig.

1. This is a very coarse network and is on1

suited for the calculations of the lower noded vibration

modes. (Fig. 2 and Table I) [4].

Table 1. Natural frequencies (Hz) of vertical hull vibrations.

(given in Fig. 2). calculâtiòñs

number of finite element beam method

nodes technique measùrements

2 1.20 1.24 (not measured)

{

* at two resonances à four-node vertical hull vibration has been measured, this is caused by the resonance of the dOuble bottom of a hold [21.

A much finer network of elements is given in Fig. 3.

Maintaining this network over the entire hull would

lead to an enormous set of equations to be solved.

Further it has to be realised that in fãt for each

fre-quency the calculation of the vibrational response to

a certain excitation has to be repeated, because of the

fact that with the frequency the response changes.

FINITE ELEMENT TECHNIOUE SLENDER BEAM METHOD MEASUREMENT OF ACTUAL SHIP

Fig. 2. Càlculated, 2-, 3- and 4-noded vertical hull vibration

modes compared with the measured modes [4].

Fig. 3. Aftership with a rather fine break-down into finite ele-merits, as used in DASH.

io

Despite the high-speed automatic computers available

nowadays, the computation time would be

unaccept-able. Therefore. the following reduction technique is

used.

A part of the hull structure, fôr example a

bulk-head, or the double bottom of a hold, or the afterbody

shown in Fig. 3, is represented by a large number of

basic elements, leading to a large number ofjoints, each

with three unknown, displacements. From these joints

a limited number is chosen, sufficient to assure the

connection with the surrounding construction. The

other joints are expressed as a function of the final

ones so that a complex element remains with a small

number of joints Which represent implicitly the

elimi-nated joints. By this process the elastic characteristic

of the original break-down into simple elements is

maintained [4}.

The complete hull is broken down into a number of

complex (three-dimensional) elements composed of

basic elements. These complex elements or

substruc-tures together form the total hull structure

From elementary mechanics it has been learned that

for the mass distribution less accuracy is required than

for the representation of the elasticity of the structure

Therefore it suffices to distribute the mass over the

final joints. this gives the second condition to be

ful-ifiled by the final joints: an adequate mass distributión.

In this way the time needed for the final

(macro-scopic) calculations of the hull vibrations is sharply

reduced.

4

Short description of the DASH computer program

For the description of the calculation use is made of

matrix notation.

From the basic elements via the composed elements

the stiffness matrix K of the hull is obtained. The mass

of the ship, the cargo and the added mass of water is

distributed over the final jomts in such a way that the

lumped masses have the saine distribution as the

orig-inal mass. These mass-lumps are assembled in the mass

matrix M. The displacements of the joints are grouped

in a matrix of one columfi ¿r vector, which is called u.

The dynamic forces, concentrated in the joints in a

statically equivalent manner, are presented by the

vector f.

The derivation of the stiffness matrix, the mass

matrix and the load vector is described in more detail

in the Appendix.,

For undamped vibrations the matrix equation of

vibration write.

Assuming harmonic vibrations this leads to a set of

linear equations

[Kw2M]u =f

in which co is the circular frequency ofthe vibratiòn.

For the derivation of the solution several computer

techniques are

available. Due to the mechanical

characteristics of the hull, the solution is. very stable,

rounding-errors do not play an important ròle. For

these reasons the order of the system is to be chosen

nearly free. However, the computer determines the

limitation, because of the fact that the solution has to

be calculated for several values of the frequency co in

order to obtain the dynamic response in the

corre-sponding frequency range. In this way for several load

vectors the vibration mode u is obtained as a function

of the frequency. In fact u represents the. hull response

to the excitation forces given in the load vector.

The calculation refers to an undamped system

or

the lower modes this is rather an accurate approach

[4] but for the higher noded modes damping is very

important.

McGoldrick performed measurements and

calcula-tions for several ships, from which the influence of

damping can be deduced [7]. More information is

given in the Appendix. Being summarized, the entre

process consists of the following steps:

- choice of the final joints and the corresponding

sùb-.structures;

- calculation of the overall stiffness matrix based on

the final joints;

determination of the ma.ss matrix;

- determination of the load vector from the given

excitation;

solution of the equation [K - co M] U = f fOr several

values of the circular frequency co;

- estimate of the itifluence of damping.

In Fig. 4 a flow diagram of the entire process has

been given, indicating the preparations of the input

for the computer, the computer calculations and the

evaluation of the computer results.

For the preparation the following items are needed:

- structural. drawings;

- mass distribution,

- ship lines and Bnjean-curves;

- hull and propeller excitation.

An outline of the presentation of the calculated

results as well as a discussion of the consequences is

given in the following chapter.

.

5

Results of DASH

For each of the finaijoints the vibrational amplitude

which is generated by a certain excitation, applied

some-Determination of excitation forces Reading excitation forces

t

Calculation of force vector Móre excitation distribu-tions? yes no

Formulation of final joints

and sub-stñscturès from main scantlings

f-Derivitlön of input data for

each sub-structure from constructional drawings

Reading data of next sub-structure

t

Calculation of stiffness matrix of sub-structure

t

Reduction of stiffness matrix of su b-structure to final joints

Coupling with the stiffness

matrices of preceding sub-structure

Evaluatiòn of calculated modes u intO graphs and tables.

Estimating, influence of damping. Judgement of the results.

Formulation Pf input 4aa

of mass distribution of ship, cargo, etc.

Reading data of mass

distribûtion

t

Distribution of mass lumps over final joints

and additiôn to the

mass matrix

Derivation of the equation of motion (Kw2t1]u_{= f}

and its solution

Ou thratmn mode u

(u

Calculation of the added mass for each of the expectedvibration modes

arid Input preparation

L.

t

Distribution as mass lumps Över final

joints, but storing In different memory

places

L

Combination of the added mass from

expected vibration mode with the

structural and cargo mass

Fig. 4. Flow diagram fOr a complete hull vibration analysis as used for DASH.

Determination of the free

quencles of Interest and esti-mating the corresponding vibration modes 11 Reading 'equency and expected vibration mode Reading data of added mass

12 z O 10_1 w -J w u u 4

.6

> 4 ID w

I

I-O 4 C-Q) 4 w JI, Q) 'Ji G-X w w I--J a.

46

z, o 4 w -J Ji u u 4 e 4 10-2 B 4 X

I

x 0Ô4 INCONVENIENT PERCEPTIBLE BLADE FREQUENCY

where at the hull is calculated. Despite the applied

reduction techmque these calculations give yet a large

amount of iiiförmation, which, however, is not all of

the same importance. Only for the most essential places

at the hull a graph is made as represented by Fig. 5.

From the calculated results, which refer to an

undamp-ed system, an estimate of the influence of damping is

made.

If the calculations have been carried out for

propel-lers with different numbers of blades, then these results

have to be compared. mutually as is indicated

dia-grammetically by Fig. 6 This comparison makes it

Fig. 6.

Maximum vibration level ofthé hull at the

accommodation for different ñuthbers of propeller blades, calculated by means of DASH, using the network of Fig. 3. The

total hull excitation is supposed to be equal for the various propellers.

10-1 C, z O i02 4 w w u I-) 4

possible to choose at design state the propeller With

the lowest vibration leveL

Of course the set of graphs as indicated by Figs. 5

and 6 has to be completed with the results of the

anal-ySis of the vibrations of the shafting, also a vely

decisive item in the complete vibration analysis.

The calculations further lead to the lower noded

natural frequencies and corresponding modes as sho'n

by Fig. 7. From this result the effect of springing (wa'e

excited lower noded hull vibration resonance) can be

judged. Fröm Fig. 7 it follows clearly that not only

the vertical hull vibrations have to be considered in

lu

u

PE RC PTIBLE ANNOYING INCONVENIENT BLADE FREQUENCY Fig. 5.

The calculated effect of damping near resonance, the calculations

are based on a network of e1e

ments as given.in Fig. 3,

employ-ing DASH. UN DAMPE D VIBRATION DAMPED VIBRATION cg ww -0.02

APP. F.PP

0.65 Hz

2 NODE VERTICAL ViBRATION

0.7Hz

COUPLED HORIZONTAL-TORSIONAL VIBRATION DECK VIBRATION BOTTOM VIBRATION

Fig. 7. Example of lowest vCrtical and horizontal-torsiònal

resonance vibration modes denved by means of DASH

this question, but also the horizontal vibrations, which

are strongly coupled with the torsional vibrations.

6 Outline of a vibration analysis

A complete vibration analysis is given by the two

following items:

- the analysis of the excitation forces

- the calculation of the hull response to these forces.

As soon as the hull performance, the propeller, the

main scantlings and the weight distribution are known,

the complete analysis can be performed,

The determination of the propeller forces and hull

forces is very expensive. For an analysis, in which

propellers with dierent numbers of blädes are

con-sidered, it is therefore recommended to calculate first

the hull response to estated huÎI and propeller

[

excitation. As soon as the final propeller has been

chosen, based on hull vibrations as well as propulsive

or cavitative performance, the propeller and hulÏ forces

have to be determined, so that the forced vibration

level on board of the. ship can be calculated for the

rpm's of interest;

Of course such a complete analysis is very complex

and expensive. Generally speaking the time involved

athounts to 4-6 moúths, and the costs run from

Dfl. 50,000. up to Dfi. 200.000.. A very important

factor is the desired detail of the analysis

Due to these large costs such an analysis has. only

to be performed for complete new ship designs.

There-fore it is not possible to describe any input procedure

for the computer, because the choice of the

mathemati-cal model depends fully on the structural arrangement.

References

i. HYLARIDES, S., Finite element technique in ship vibration auia1/sis. mt. Shipb. Progress, Vol. 15, no. 169, September

1968:

Hyi1uDEs, S., Critical consideration of present hull

vibra-lion analysis Netherlands Ship Research Centre TNO

Report ño. 144 S, December 1970.

HYLARIDES, S., The finite element method in ship deSign. Part from: Design and ecoñomicál considerations on ship-building and shipping. Report of the post graduate course,

May 1969, at Deift. H. Veenman en Zonen N.y.,

Wage-ningen, Holland.

HvLths,. S., Ship vibration analysis by finite element

technique Part two Vibration analysis Netherlands Ship

Research Centre TNO, Report no. 153 S, May 1971. .5. HYLARiDES, S., Ship vibration analysis by finite element

technique Part one: General review and àpplicatiòn to simple structures, statiscally loaded. Netherlànds Ship

Research Centre TNO, Report no. 107 S, December 1967.

ZIENK1EWÎCZ, O. C. and Y. K. CHEUNG, The finite element

method in structural and continuum mechanics McGraw

Hill, 1968.

MCGOLDRICK, R-.. T., Comparison between theoretically and

experimentally determined natural frequencies and modes of vibration of ships. DTMB Report 9O6 August 1954.

JoosEN, W. P. A. and J. A. SPARBNBERG, On the longitudinal

reduction factor for the added mass of vibrating ships with rectangular cross section Netherlands Ship Research Centre TNO, Report no. 40 S, April 1961.

LEWIS, F. M., The inertia of the water surrounding a vibrat-ing ship. Trañs SNAME, Vol. 37, 1929.

HYLARIDES, S., Recent developments in hull an shaft

vibration analysis mt Shipb Progress Vol 17 no 190

June 1970.

14

Appendix

General

As desctibed fOr undamped, haEmónic hUll vibrations

the dynamiô hit!! response is given by the matrix

equatiòn

[Kw2 M]u =f

For the derivation of the mass matrix M, the

stiff-ness matrix K and the load vector f, computer pro

grams have been made Together with the solution

programs they are grouped under the name DASH.

The following items, forming together the analysis,

are treated separately

- choice of the final joints;

stiffness matrix K.;

-mass matrix M;

- load vector f;

- influence of damping.

Choice of the finaijoints

Because of the fact that the figal calculatidns will iefer

to a considerably reduced number ofjoints, over which

the total mass has to be distributed, the analysis of the

hull vibrations has to be started with the choice of these

joints. They have to satisfy the following conditions:

- connection

between

adjacent

sub-structures

is

assured;

their distribution over the hull is dense enough to

represent adequately the kmetic energy, when the

total ship mass is distributèd as lumps over these

joints;

Where the excitation forces or moments apply the

joints have t

be chosen in such a way that the

effec-tiveness of the forces ïs not changed.

- there have to be enough joints to qualify the

cal-culated vibration level in the regions at which special

attention has to be paid to vibrations.

After the determination of these final joints the

anal-ysis can be started by distributing the mass as lumps

over these joints and by formulating the sub-structures.

Stiffness matrix K

For the derivation öf the stiffness matrix the complete

drawings of the structure are needed as far asthey

con-cern the principal strength members. Constructional

members of only local mportance do not cOntribute

to the overall hull response. This means that at rather

an early stage of the hull design vibration calculations

can be performed.

For each sub-structure a local system of axes XY

ZK is chosen (Fig. Al). By means of its

direction-z*

21

Fig. Al. Substructure with basic elements and local co-ordinate frame.

cosines with the main system of axes XYZ for the

entire hull, the stiffness matrix of the sub-structure will

be transformed.

The basic elements are rectangular or triangular

plates and bars (Fig. A2). Slightly curved plates will

be considered to be flat. Not rectangular quadrangular

plates can be approximated by rectangular elements as

long as the deviation is small, if not, they can be

represented by two triangular plate elements.

When the network of elements of a sub-structure

has beeñ defined, the co-ordinates of all joints wih

'3

respect to the local system of axes are derived and fed

into the computer. Next, per element, its type, of a

plate element its effective stiffness or of a bar element

its effêctive sectional area is fed into the computer and,

finally, the names of the j oints at which the element is

connected. Then the computer deduces from the

co-ordinates of the joints the length and breadth

dimen-sions of the element as well as its spacial location with

refèrence to the local system of axes. With this

informa-tion the effect of the element in the stiffness matrix

can be calculated.

After the calculation of the stiffness matrix of a

sub-structure, the computer calculates the reduced stiffness

matrix based on the final joints that hold for this

sub-structure. Then this stiffness matrix is added to the

final stiffness matrix, holding for the complete

struc-ture (Fig. 4).

Mass Matrix M

The continuous mass. distributiOn has to be replaced

by a system of mass lumps, concentrated in the final

joints.. In this breakdown the redistribution has to be

carried out in such a way that the new system is

equiv-alent to the original one.

Until now it has been normal practice to give only

the longitudinal mass distribution of a ship. Therefore,

fòr the timê being, in DASH it is assumed that the

mass is distributed homogenously over the breadth

and the depth. Ifa superstructure is given separately its

mass lumps are concentrated in the upper joints.

The mass distribution of hull, superstructure, cargo

and fUel can be given separately or combined. The

added mass of water is concentrated in the lower joints,

the vertical added mass applies only to the vertical

displacements of the joint whereas simultaneously the

horizontal added mass applies to the horizontal

trans-verse displacement. For longitudinal vibrations no

added mass is taken into account, because this field is

yet unknown. However, this approach may be incorrect

as considerable longitudinal bottom vibrations have

been found from calculations.

Because of the fact that the added mass changes with

the vibration mode [8, 9], the added mass is treated

separately. Depending on the expected vibration mode

the corresponding distribution Of added mass is used

in the calculations. Comparing the calculated mode

with the supposed one gives an indication of the

accuracy.

The calculation of the added mass is based on the

ship lines and Bonjean-curves.

Fig. A3 gives an example of a mass distribution. The

input of DASH consists of the ordinate and

corre-sponding mass per unit length at each discontinuity in

the distribution.

Fr

xi

loo

Mass of the ship

Fig. À3. Distribution of ship mass and the input formulation

for the computer.

From these data the compUter distributes the mass

over the joints, maintaining the original distribution.

The results per joint of the several mass distributions

aré added to obtain the complete distribution.

Load vector f

The complete fluctuating load, generating hull

vibra-tions, is given by the entire propulsion unit. For

high-powered units very often use is made of turbine plants.

These are well balanced so that their iñfiüence is

neglected. Nowadays large diesel engines are so well

balanced that their free excitation (forces or moments)

may be neglected in comparison with the fluctuating

forces generated by the propeller. Therefore in

vibra-tion analysis the attenvibra-tion is focused on the propeller

forces.

The dynamic forces generated by the propeller can

be split up in the pressure fluctuations at the afterbody

and forces and moments applying at the propeller.

At the NS MB, special measuring devices have been

developed for measurements ön models of ship hulls

and propellers. With these measurements rather

reli-able estimates of the excitation forces can be obtained.

The results are given in harmonic components each

with its phase relation to the angular propeller position.

Only blade frequency is considered in the

calcula-tions, because the corresponding harmonics are

dom-inant. If desired, also mUltiples of the blade frequency

can be taken into account.

The forces and moments thus obtained are discretis-.

15

1.0

-E

0.5-116

ed in such a way that the concentrated forces are

statically equivalent to the onginal load system

These concentrated forces as well as their phase and

pomt of application (the joint in which they are

con-centrated) are fed iñto the computer, which calculates

the final load vector ffrom this information.

Influence of damping

The influence Of damping is especially of importance

around a natural frequency. In this region the

corre-sponding vibration mode will dominate For this reason

the hull can be ôonsidered a simple

mass-spring-system. with only one degree of freedom and a natural

frequency that equals that of the hull resonance

frequency. In this

case the relation between the

amplitude of the damped vibration and the amplitude

without damping writes:

Xdampcd =

(w)2}2

±

()

undamped

in which co

= circular frequency óf the excitation,

co,,

= natural circular frequency,

¿

= damping force per unit length per unii

velocity,

wig

mass per unit length, including en

trained water.

By means of vibratiOn tests excited by a generator for

several ships, McGoldrick [7] deduced that on thé

average for vertical vibrations cg/wco

0.034 añd for

horizontal vibrations cg/ww

0.041.

Based on these assumptions the curves of amped

vibrations in Fig 4 have been obtained

PUBLICATIONS OF THE NETHERLANDS SHIP RESEARCH CENTRE TNO

PUBLISHED AFTER 1963 (LIST OF EARLIER PUBLICATIONS AVAILABLE ON REQUEST)

PRICE PER COPY DFL. IO.- (POSTAGE NOT INCLUDED)

M = engineering department S = shipbuilding department C = corrosion and antifouling department

Reports

57 M Determination of the diamc properties and propeller excited

vibratiOns of à special ship stern arrangement. R. Wereldsma,

1964.

58 S Numerical calculation of vertical hull vibrations of ships by

discretizing the vibration system, i. de Vries, 1964.

59 M Controllable pitch propellers, their suitäbility and economy for largesea-gôing ships propelled by conventional, directly coupled engines. C. Kapsenberg, I 964.

60 S Natural frequencis of free vertical ship vibrations. C. B.

Vreug-deiThil, 1964.

61 S The distribution of the hydrodynarnic forces on a heaving and

pitching shipmodel in still water. J. Gerritsma and W.

Beukel-man, 1964.

62 C The mode of action of antifouling paints : Interaction between anti-fouling paints and sea water. A. M. van Londen, 1964.

63 M Corrosion in exhaust driven tthbochargers on marine diesel

engines using heavy fuels. R. W. Stuart Mitchell and V. A. Ogale,

1965.

64 C Barnacle fouling on agedanti-fouling paints; a survey of pertinent literature and sorne recent observatiOns. P. de Wolf, 1964. 65 S The lateral damping and added mass ola horizontally oscillating

shipmodel. G. van Leeuwen, 1964.

66 S Investigations into the strength of ships' derricks. Part I. F. X.

P. Soejadi, 1965.

67 S Heat-transfer in cargotanks of a 50,000 DWT tanker. D. J. van der Heeden and L.. L. Mulder, 1965.

68 M Guide to the application of method for calculation of cylinder liner temperatures in diesel engines. H. W. van Tijen, 1965. 69 M Stress measurements on a propeller model for a 42,000 DWT

tanker. R. Wereldsma, 1965.

70 M Experiments on vibrating propeUer models. R. Wereldsma, 1965. 71 S Research on bulbous bow ships. Part II. A. Still water

perfor-mance of a 24,000 DWT bulkcarrier with a large bulbous bow. W. P. A van Lammeren and J. J. Muntjewerf, 1965.

72 S Research on bulbous bow ships. Part II. B. Bchaviôur of a

24,000 DWT bulkcarrier with a large bulbous bow in a seaway. W. P. A. van Lammeren-and F. V. A. Pangalila, 1965.

73 S Stress and strain distribution hi a verticalty-córrugated bulkhead. H. E. Jaeger and P. A. van Katwijk, 1965

74 5 Research on bulbous bow ships. Part I. A. Still water investiga-tions into bulbous bow forms for a fast cargo liner. W. P. A. van Lammeren and R. Wahab, 1965.

75 S Hull vibrations of the cargo-passenger motor ship "Oranje

Nassau", W. van Horssen, 1965.

76 S Research on bulbous bow ships. Part I. B. The beháviour of a fast cargo liner with a conventional and with a bulbous bow in a sea-way. R. Wahab, 1965.

77 M Comparative shipboard measurements of surface temperatures

and surface corrosion in air cooled and water cooled turbine outlet casings of exhaust driven marine diesêl engine

turbo-chargers. R. W. Stuart Mitchell and V. A. Ogale, 1965. 78 M Stern tube vibration measurements of a cargo ship with special

afterbody. R. Wereldsma, 1965.

79 C The pre-treatment of ship plates: A comparative investigation

on some pre-treatment methods in use in the shipbuilding

industry. A. M. van Londen, 1965.

80 C The pre-treatment of ship plates: A practia1 investigation into

the influence of different working procedures in over-coating

zinc rich epoxy-resin based pre-construction primers. A. M. van Londen and W. Mulder, 1965.

81 S The performance of U-tanks as a passive anti-rolling device.

C..Stigter, 1966.

82S Low-cycle fatigue of steel structures. J. J. W. Nibbering and

J. van Lint, 1966.

83 S Roll damping by free surface tanks. J. J. van den Bosch and

J. H. Vugts, 1966.

84 S Behaviour of a ship in a seaway. J. Gerritsma, 1966.

85 S Brittle fracture of füll scale structures damaged by fatigue.

J. J. W. Nibbering, J. van Lint and R. T. van Leeuwen, 1966. 86 M Theoretical evaluation of heat transfer in dry cargo ship's tanks

using thermal oil as a heat transfer medium. D. J. van der

Heeden, 1966.

87 S Model experiments on sound transmision from engineroom to

accommodation in motorships. J. H. Janssen, 1966.

88 S Pitch and heave with fixed and controlled bow fins. J. H. Vugts,

1966.

89 S Estimation of the natural frequencies of a ship's double bottom by means of a sandwich theory. S. Hylarides, 1967.

90 S Computation ofpitch and-heaveinotions for arbitrary ship forms. W. E. Smith, 1967.

91 M Corrosion in exhaust driven turbochargers on marine diesel

engines using heavy fuels. R. W. Stuart Mitchell, A. J. M. S. van Montfoort and V. A. Ogale, 1967.

92 M Residual fuel treatment on board ship. Part H. Comparative

cylinder wear measurements on a laboratory diesel engine ¿ising filtered or centrifuged residual fuel. A. de Mooy, M. Verwoest and G. G. van der Meulen, 1967.

93 C Cost relations of the treatments of ship hulls and the fuel 'con-sumption of ships. H. J. Lageveen-van Kuijk, 1967.

94 C Optimum conditions for blast cleaning of steel plate. J.

Rem-melts, 1967.

95 M Residual fuel treatment on board ship. Part I. The effect of cen-trifuging, filtering and homogenizing on the unsolubles in residual fuel. M. Verwoest and F. J. Colon, 1967.

96 S Analysis ¿f the modified strip theory for the calculation of ship motions and wave bending moments. J. Gerritsrria and W. Beu-kelman, 1967.

97 S On the efficacy of two different roll-damping tanks. J. Bootsma and J. J. van den Bosch, 1967.

98 S Equation of motion coefficients for a pitching and heavingides-troyer model. W. E. Smith, l967

99 S The manoeuvrability of ships on a straight course. J. P. Hooft,

1967.

100 S Amidships forces and moments on a CB = 0.80 "Series 60"

model in waves from various directions. R. Wahab, I 967. 101 C Optimum conditions for blast cleaning ofsteel plate. Conclusion.

J. Remmelts, 1967.

102 M The axial stiffness of marine diesel engine crankshafts Part I.

Comparison between the results of full scale measurementsand

those of calculations according to published formulae. N. J.

Visser, 1967.

103 M The axial stiffness of marine diesel engine crankshafts. Pai-t II. Theory and results of scale model measurements and comparison with published formulae. C. A. M. van der Linden, 1967. 104 M Marine diesel engine exhaust noise. Part I. A mathematical model.

J. H. Janssen, 1967.

105 M Marine diesel engine exhaust noise. Part II. Scale models of

exhaust systems. J. Búiten and J. H. Janssen, 1968.

106 M Marine dièsel engine exhaust nOise. Part 1H. Exhaust sund

criteria for bridge wings. J. H. Janssen en J. Buiten, 1967.

107 S Ship vibration analysis by finite element technique. Part I.

General review and application to simple structures, statically loaded. S. Hylarides, 1967.

108 M Marine refrigeration engineering. Part L Testing of a

decentraI-ised refrigerating installation. J. A. Knobbout and R. W. J.

Kouffeld, 1967.

109 S A comparative study on four different passive roll damping tanks. Part I. J. H. Vugts, 1968.

110 S Strain, stress and flexure of two corrugated and one plane

bulk-head subjected to a lateral, distributed load. H. E. Jaeger and

P. A. van Katwijk, 1968.

IIi M Experimental evaluation of heat transfer in a dry-cargo ships'

tank, using thermal oil as a heat transfer medium. D. J. van der Heeden, 1968.

112 S The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. J. H. Vugts, 1968.

113 M Marine refrigeration engineering. Part II. Some results of testing a decentralised marine refrigerating unit with R 502. J. A. Knob-bout and C. B. Colenbrander, 1968.

114 S The steering of a ship during the stopping manoeuvre. J.. P.

Hooft, 1969..

I 16 M Torsional-axial vibrations of a ship's propulsion system. Part I. Comparative investigation of calculated and measured

torsional-axial vibrations in the shafting of a dry cargo motorship.

C. A. M. van der Linden, H. H. 't Hart and E. R. Doffin, 1968.

I 17 S A cömparative study on four different passive roll damping

tanks. Part II. J. H. Vugts, 1969.

i 18 M Stern gear arrangement and electric power generation in ships propelled by conüo11able pitch propellers. C. Kapsenberg, 1968. i 19 M Marine diesel engine exhaust noise. Part IV. Transferdâmping

data of 40 modelvariants of a compound resonator silencer.

J. Buiten, M. J. A. M. de Regt and W. P. H. Hanen, 1968. 120 C Durabilitytests with prefabrication primers in use of steel plates.

A. M. vari Londen and W. Mulder, 1970.

121 S Proposal for the testing of weld metal from the viewpoint of

brittle fracture initiation. W. P. van den Blink and J. J. W.

Nib-bering, 1968.

122 M The corrosion behaviour of cunifer 10 alloys in seawaterpipiñg-systems on board ship. Part I. W. J. J. Goetzee añd F. J. Kievits,

1968.

123 M Marine refrigeration engineering. Ptht III. Proposal for

aspecifi-cation of a marine refrigerating unit and test procedures. J. A.

Knobbout and R. W. J. Kouffeld, 1968.

124S The design of U-tanks for roll damping of ships. J. D. van den

Btint, 1969.

1 S A proposal on noise criteria for sea-going ships. J. Buiten, 1969.

126 S A proposal for standardizedmeasurenients and annoyance rating of simultaneous noise and vibration in ships. J. H. Janssen, I 969. 127 S The braking of large vessels II. H. E. Jaeger in collaboration with

M. Jourdain, 1969.

128 M Guide for the calculation of heating capacity and heating coils for dOuble bottom fuel oil tanks in dry cargo ships. D. -J. van der Heeden, 1969.

129 M Residual fuel treatment on board ship. Part ifi. A. de Mooy,

P. J. Brandenburg and G. G. van der Meù]en, 1969.

l30 M Marine diesel engine exhaust noise. Part V. Investigation of a

double resonatorsilencer. J. Búiten, 1969.

131 S Model and full scäle motions of a twin-hull vessel. M. F. van

Slùijs, 1969.

132 M Torsional-axial vibrations of a ship's propulsion system. Part 11. W. van Gent and S. Hylarides, 1969.

133 S A model study on the noise redúction effect of damping layers aboard ships. F. H. van ToI, 1970.

134 M The corrosion behaviour of cunifer-lO alloys in

seawaterpiping-systems on board ship. Part H. P. J. Berg and R. G. de Lange,

1969.

¡35 S Boundary layer control on a ship's rudder. J. H. G. Verhagen,

1970.

137 M Torsional-axial vibrations of a ship's propulsion system. Part 1H. C. A. M. van der Linden, 1969.

138 S The manoeuvrability of ships at low speed. J. P. Hooft and

M. W. C. Oosterveld, 1970.

139 5 Prevention of noise and vibration -annoyance aboard a sea-going

passenger and carferry equipped with diesel engines. Part 1:

Line of thoughts and predictions. J. Buiten, J. H Janssen,

H. F. Steenhoek and L. A. S. Hagernan, 1971.

140 S Prevention of noise and vibration annoyance aboard asea-going

passenger and carferry equipped with diesel engines. Part II: measures applied and comparison of computed values with

measurements. J. Buiten, 1971.

141 S Resistance and propulsion of a high-speed single-screw cargo

liner design-. J. J. Muntjewerf, 1970.

142 S Optimal meteorological ship routeing. C. de Wit, 1970.

143 S Hull vibrations of the cargo-liner "Koudókerk". H. H. 't Hart,

1970.

144 S Critical consideration of present hull vibration analysis S Hyla

rides, 1970.

146 M Marine refrigeration engineering. Part IV. A Comparative study on single and two stage compression. A. H. van der Tak, 1970. 147 M Fire detection in machinery spaces. P. J. Brandenburg, 1971. 148 S- A reduced method for the calculation of the shear stiffness of a

ship hull. W. van Horssen, 1971.

149 M Maritime transportation of containerized cargo. Part II. Experi-mental investigation concerning the carriage of green coffee from Colombia to Europe in sealed containers. J. A. Knobbout, 1971. 151 M Maritime transportation of containerized cargo. Part I.

Theoretical and experimental evaluation of the-condensation risk

when transporting containers loaded with tins in cardboard

boxes. L A. Knobboút, 1971.

152 5 Acoustical investigations of asphaltic floating floors applied on a steel deck. J. Buiten, 1971.

153 S Ship vibration analysis by finite element technique. Part II. Vibra-tion analysis. S. Hylarides, 1971.

156 S The behaviour of a five-column floating drilling unit in waves. J. P. Hooft, 1971.

159 S DASH computer program for Dynamic Analysis of Ship Hulls

-S. Hylarides, 1971.

Communications

Il C Investigations into the- use of some shipbòttom paints, based on

scarcely sapoñiflable vehicles (Dùtch). A. M. van Londen and

P. de Wolf, 1964.

-12 C The pre-treatmnt of ship plates: The treatment of welded joints

- prior to painting (Dutch). A. M. van Londen and W. Mulder,

1965.

13 C Corrosion, ship bottom paints (Dutch). H. C. Ekama, 1966.

14 S Human reaction to shipboard vibration, a study of existing

literature (Dutch). W. ten Cate, 1966.

15 M Refrigerated containerized transport (Dutch). J. A. Knobbout,

1967.

16 S Measures to ptevent sound and vibration annoyance aboard a seagoing passenger and carferry, fitted out with dieselengines

(Dutch). J. Buiten, J. H. Janssen, H. F. Steenhoek and L. A. S. Hageman, 1968.

17 S Guide for the specification, testing and inspection of glass reinforced polyester structures

in shipbuilding (Dutch). G.

Hanim, 1968.

18 5 An experimental simulator for the manoeuvring of surface ships. J. B. van dén Brug and W. A. Wagenaar, 1969.

19 S The computer prOgrammes system and the NALS language for numerical control for shipbuilding. H. le Grand, 1969.

20 S A case study on networkplanning in shipbuilding (Dutch). J. S. Folkers, H. J. de Ruiter, A. W. Ruys, 1970.

21 S The effect of a contracted time-scale on the learning ability for manoeuvring of large ships (Dutch). C. L. Truijens, W. A. Wage-naar, W. R. van Wijk, 1970.

22 M An improved stern gear arrangement. C. Kapsenberg, 1970. 23 M Marine refrigeration engineering. Part V (Dutch). A. H. van der

Talc, 1970.

24 M Marine refrigeration engineering. Part VI (Dutch). P. J. G. Goris and A. H. van der Tak, 1970.

26 S On optimum propellers with a duct of finite length. Part II.