Forced vibration resonators and free vibration of the hull

FORCED VIBRATION RESONATORS AND FREE VIBRATION

OF THE HULL

by

1. General.

The increase in dimensions of the merchant ships and the outputs of their propulsive plants since the end of the second World War has put numerous technical problems to shipbuilders; among these problems it can be cited those con-cerning the ships's vibrations.

This increase in the size and output of propuls-ive plants of the ships has been at the origin of complex problems and especially those in

connec-tion with vibraconnec-tions. It can be said that their

solution

requires at present the adoption of

more scientific methods than those applied till now. The execution of simplified calculations, often deduced from empiric results, the extra-polation of experimental values previously obtained on less important ships very often leads

to aberrant results. Such situations result in important expenses caused not only by repair

works but also by a loss of time resulting from the long stoppings of huge tonnage modern ships

due to technical difficulties. Actually, the

technical problems are more and more com-plicated but fortunately the engineers have at

present at their disposal very powerful theoretical as well as experimental investigation means. The electronic computer facilitates very much these researches although even to-day some people consider them as too complicated or ex-pensive. But finally these researches are not at

all expensive if one considers the aim to be

obtained which is an efficiency and continuous exploitation of huge tonnage ships whose daily expenses are particularly important.

One of the problems, perhaps the less known

but particularly difficult to deal with, being at

present that of the vibrations, it is not surprising that very numerous searchers have tried to de-termine its numerous aspects in order to better

C)_{Manager of Bureau Veritas Maritime Services, Paris, France.}

") Head of Special Study and Research Section of Bureau Veritas

Maritime Services, Paris, France.

G. Bourcesu ' and G.C. Volcy

know the concerned phenomena.

The authors will only mention the papers of Pr. C.W. Prohaska and General Engineer Dieudonne [1] and [2] as it is they who have marked

important steps in the evolution of the knowledge

of this question. These papers have not only

enabled to better know the question of ship's girder

vibrations but have also given practical and quick methods and ways for the understanding and re-solution of numerous problems concerning them. However, the evolution of the technique always brings surprises and new problems to be solved and this is the case also with the part of vibra-tions being called generally 'ship's vibravibra-tions'.

It is quite evident that it is interesting to cal-culate and determine the values of natural fre-quencies which correspond to the various vibratory modes of hull girder but in most cases this study is ended with this knowledge.

As it is known, it is not practically possible to change considerably these values of natural

frequencies by acting on the scantling of the ship's girder; moreover, these values are not

constant; they vary within certain limits,

ac-cording to ship's loading conditions. Besides, it is always possible to find:

- either coincidence between the frequencies of

the various excitations, the origin of which

exists on the ship and the calculated frequen-cies which correspond to the different vibra-tory modes of the hull;

- or the presence of forces applied in the anti-nodes and/or couples applied to the anti-nodes of the various calculated vibratory mode-forms of the ship.

Too often one is to face a dilemma, what to do?

And yet, after the tests it is not rare to note

that in spite of the calculation results and the considerations showing the possibility of ap-pearance of various resonances, the expected

contrary, one notes the appearance of often un-expected forced vibrations which, after a closer

study, can finally be considered as due to the

excitation sources which were considered as not very important: the non rational alignment and

the whipping of the tail shaft, the transverse

vibrations of the main engines, the longitudinal

vibrations of the thrust-block, etc....

More-over, certain of these phenomena do not appear on a ship while they appear on a sister-ship. Of course, the differences between reality and cal-culation results could be attributed to the

imperfection of the last ones; we shall revert to this questionwhich presents a particular interest mainly for the frequency values of higher modes of free hull vibrations.

However, the stated differences and the sur-prises met owing to the appearance of other vibra-tions than those, which are the main subject of the classical theory of the calculations of the free vibrations of the ship s girder, have led to think of the possibility of appearance of other vibratory phenomena than those related to the free vibra-tions of the hull assembly.

Actually, in what follows, it would be question of forced vibrations of the hull and of local vibra-tions, either free or forced ones, of the various parts of the steel-work as well as the ship's pro-pulsive plant.

In fact, during the researches and

measure-ments carried out by Bureau Veritas and con-cerning ship's vibrations, it has been noted that the most severe forced vibrations of the hull steel-work are appearing in a very pronounced manner

in case of a resonance between the excitation

source(s) of any origin (coming either from the propeller or from the propulsive plant) and the

resonant response of a part of either the

steel-work or even the propulsive plant itself (the tail shaftorthe thrust-block for instance). It is also the case of local vibrations of the double-bottom and the propulsive apparatus for instance.

th all these cases, being in the presence of a resonance between the excitations and the re-sponse of the concerned local systems, it must occur a dynamical amplification of the efforts in presence which consequently increase the

importance of these excitations.

One seems to be in the presence of a vibration resonator.

In the case of the engine-room double-bottom

vibrations, the steel-work of which is embedded in the outside plating steel-work which forms

it-selfapartof the ship's girder assembly, the

in-duction and accumulation of vibratory energy seems to be sufficiently important to be able even to excite this ships's girder assembly:

- either in forced vibrations,

- or, being close to the natural frequency of one mode, in free vibrations.

Accordingtothe authors, for the huge tonnage ships representing very important virtual masses which practically can be excited in free vibrations only by sea conditions (slamming for instance) it is essential to try to find out such resonators of local vibrations. If these resonators are found out it is then necessary to detune the resonances found in order to eliminate the dynamical

am-plification of the excitation efforts which are

present onthe ship. If it is not the case, the ac-cumulation of vibratory energy could be felt on the ship as forced vibrations of one of her parts or even, in case of coincidence, with the natural frequency(ies) of one of the modes of the ship's

girder assembly, as free vibrations, having in

this case very pronounced amplitudes.

So, one could mention, as corresponding to a very important case, the necessity to be able to calculate the values of natural frequencies of the assembly constituted by the propulsive plant and the engine-roomdouble-bottom,and this in order to be able to study the repercussions of the vibra-tions of this assembly on the behaviour of the hull

girder and/or its superstructures.

This question of interdependence between the

vibrations of the superstructures (which very often vibrate much more than the propulsive

plants which excite them!) and the vibrations of

the double-bottom and the propulsive plant is

more and more of actual interest. One can

estimate that it is necessary, apart from the

calculation of the ship's girder frequencies, to

have means and methods of calculation of the

natural frequencies of the grillages of the

double-bottom and plating steel-work. It is also

very important to know their responses to the excitations being able to put them in forced

vibrations.

Of course, in such a case, it must not be for-gotten to mention the coupling effects between the different kinds of vibrations: their existence does

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can appear on the ship,

according

to their

character, the origin, of their coupling as well as the responses

of these

various elastic systems.

A rapid examination of this figure confirms the above mentioned opinion regarding the com-plexity of the concerned phenomena. So,

it is

quite evident that treating these problems in de-tail, in order to finally arrive at solutions which can bring remedies, is even more complicated. However, one must come to the conclusion that

for a rational and efficient study of vibratory phenomena, the most important question is to

understand the physical nature of this problem. Of course, the study of methods of calculation of natural frequencies is very important for the

un-derstanding of the nature of a given problem: how

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ever it does not assure the full understanding of the vibratory phenomena likely to appear on a ship. It is evident that the electronic computer, thanks to its calculation power and its rapidity,

has suppressed any discouragement as to the

execution of very complicated calculations. But it is perhaps because the calculations are no

longer the tiresome side of the researches that it is all the more important, and indispensable, in each case to understand the physical pheno-menon of all vibrations to be studied and their in-terdependence.

It is in such a spirit that the authors have

written this paper in order to give their point of view on the physical side of the various vibra-tory phenomena they met on ships.

Owing to the similarity between the mechanical

vibratory phenomena and the electric

oscilla-tions, electric analogies are also mentioned each time it has appeared useful to do it.

2. Some theoretical considerations.

2.1. Calculation of natural frequencies of the ship's girder considered as a Bernoulli's

beam and the reality.

It is not by accident that the increase of ship's tonnage is accompanied by the more frequent ap-pearance of vibratory phenomena. In fact, in con-nection with this general tendency in the present shipbuilding, there is:

- on one hand, a sensible increaseof the flexibili

-ty of the metallic steel-work leading to the de-crease of the natural frequencies of the various

structures and the hull assembly;

- on the other hand, an important increase of the stiffnesses of the line shafting of propulsive

plants as well as a very sensible increase of

the hydrodynamic as well as mechanical excita-ting efforts.

This questionhas been analysed in detail else-where [10 to 21].

One of the first questions which are put before the naval architects is the determination of the

values of the natural frequencies of the ship's

girder free vibrations according to various

modes. The calculation methods used till now are based on the hypothesis that a ship can be

con-sidered as a beam: thus it is thought that the stresses appearing in different structural ele-ments are proportional to the distance of this element to the neutral axis which is passing

through the gravity center of the surface of the working elements. This hypothesis, which is that of Bernoulli, leads to say that the transverse plane sections remain plane during the deformation of the assembly and do not suffer any local deforma-tions. This hypothesis is of course very doubtful when it is question of a structure as complex as thatofaship. So, for this reason one can expect

sensible differences between the real stresses

and those which are calculated according to the classical beam theory. This question has been studied by several searchers [3]: one has come to the conclusion that the differences between the

working conditions of the real steel-work and

those of the theoretical beam are leading to

ir-regularities in the distribution of stresses and

by a modification of the flexibility conditions; but this last one must then be considered for the

de-termination of the deformations and the study of the vibrations.

As the distribution of stresses is not that which should correspond to the Bernoulli hypothesis, it

results that the straight section deforms and it

is no longer possible to give a doubtless defini.-tionofthe flexibility. To overcome this obstacle it has been necessary to make certain approxi-mations which are expressed by the introduction of notion of efficiency of the working section as

well as the equivalent (fictive) inertia of this

section.

The introduction of these notions in the cal-culation of the natural frequencies of the hull has enabled the accordance between the results of calculation and the reality; while taking into

con-sideration the values of the moment of inertia

calculated according to the conventional method and comparing the results, for instance with the measurements executed on tankers having

similar dimensions, too high natural frequencies were obtained. The ratio of the fictive moment of inertia to the calculated one,which is general

-ly higher than 95 per cent for tankers, can

de-crease to 85 per cent and even below for cargo

ships; on this occasion it is worth noting that

also the position of the neutral axis can deviate considerably from the position calculated accor-ding to the classic beam theory.

The distribution of stresses in the transverse section can be even very different between several ships of a same series, and this due in particular to the fact that the steel plates have been more or less planed. When one studies the work of a plate, the result is extremely different if it works only in traction or if this plate, having an initial

de-flection, works in fact in flexion. This is

cer-tainly a further reason for the difference between the calculation and the results of the measure-ments and it is to be feared that it can never be eliminated. However, it is interesting to remark that generally the study of the differences between measured and calculated values of the natural fre -quencies of the ships shows the existence of a re-lation between this difference and the number of

nodes of the various vibratory modes studied. Now, the general tendency being to build bigger and bigger ships, the values of natural frequen-cies according to various modes are decreasing; however, the revolution number of the propulsive plants remaining quite the same and the number

of blades of the propeller having a tendency to

in-crease, there is a clear tendency of sliding

to-wards the possibility of resonances between the

resulting excitating frequencies and the fre-quencies of the higher modes of the hull. So,

one is in the frequency zone where the

differ-ences between the measured and calculated

values are the most marked. This phenomenon could be explained also by another hypothesis taken as a basis to establish the equations of the vibratory movement of a beam (namely if it is

question of a structure, or a body, where two dimensions, breadth and height for instance, are very small in respect to the third one, i.e. the length). For such a beam, to calculate the

flexion deformations, it is then used the formula established in 1744 by Bernoulli, which is:

d2 1(x) 2 dx df(x) 2 3/2 [1+( _dx ) I 2 T d f(x) M -+

_-

=+ 2

-E.J

dx

where: p curvature radius of the neutral axis

x abscissa along the beam

f(x) displacement from the equilibrium position

MT bending moment

E elasticity modulus

J moment of inertia of the considered

section with respect to the neutral

axis

This formula can be used as long as the distance

betweentwo points (i.e. in the case of vibrations

between two points where the curvature is

practically non-existent) is very high with respect to the other transverse dimensions, for instance. Now, in case of vibrations according to modes having an important number of nodes (three to four and even five) this distance becomes quite

similar to, and even smaller, for the modern

large tankers. than the transverse dimensions in particular.

It is one reason more to explain the differences stated between the values of natural frequencies measured on board and those calculated according

(1)

to the different methods existing at present and

based on the hypothesis which considers the steel

-work of a ship as a beam. The very numerous studies carried out starting from this hypothesis do not seem to have brought satisfying solutions

[24 to 301.

To better illustrate the difficulties met and to

try to draw more general conclusions, it has

been tried to establish the definition of the dif-ferences between the calculated and measured values of natural frequencies. In this respect a parameterR, which defines the ratio of the cal-culated frequencies F and those really measur-ed Fm, has been introducmeasur-ed:

(2)

When studying the values of this ratio con-cerning vertical vibrations, which have been

published by several authors [24 to 30] and which

are presented in Table I, interesting remarks can be made.

Table I

Summary of values obtained by various authors

F

24 to 30 of the ratio R = between the naturai,

Fm

calculated (Fe) and measured (Fm), frequencies.

Generally, the valueR is smaller than the unit for the fundamental natural frequencies. On the contrary, this value becomes greater and greater as the number of nodes increases.

5

Mode of vertical vibrations

_/

1/2 / 2/3/ 3/4/ 4/5/ s// 6/7

/

Number of nodes Sibliography 1.200 0,865 1,011 1.114 1,230 Method integral 0,894 1,006 1,071 1,113 0,739 1,118 1,256 24 Method HOLTZER' 1,111 1,296 1,433 'MYKLESTAD' 1,045 1,128 1,415 25 1,037 1,116 1,309 26 0,912 0,968 1,111 1,242 1,356 1,345 0,884 0,963 1,025 1,183 1,156 1,036 27 0,955 1,011 1,078 1,233 1,215 1,115 28 0,960 0,978 1,004 1,022 29 0,891 0,955 0,983 1,056 1,036 30 0,959 0,950 0,963 0,970 (Average of 14 ships)

Thus, one has the confirmation of the remarks presented before and concerning the calculation hypothesis considering the ship as a Bernoulli beam. However, there are still other phenomena which can influence the results of the natural fre-quency values, among which:

The quantity of water being able to participate inthe vibrations of the ship and called 'virtual

mass',

The distribution of loading of the beam, The beam stiffness (this has already been dealt with at the occasion of the analysis of the

ef-ficiency of the working section and the moment

of fictive inertia to the flexion of the ship's girder),

The influence of the shearing forces on the deformation of the ship's girder,

The coupling between the horizontal and

torsional vibrations,

The effect of inertiaof the rotation of the sec-tions on the oscillatory movements.

The coupling between the vibrations of the ship's girder assembly and local vibrations

(mainly those of the double-bottom steel-work, together with either the propulsive plant or the ship's loading).

The importance of the quantity of water (a) which participates to the vibratory phenomena has already been recognized at the beginning of this century. The numerous researches carried out have enabled to assess this question; how-ever, the obtaining of concrete values related to this quantity of water (besides often questioned), is always long and difficult to determine. More recent evaluations lead to admit that this virtual mass diminishes with the number of nodes of the corresponding vibration mode. Regarding the values of natural frequencies mentioned in Table

I, it is to be pointed out that this question has

been taken into consideration for the execution of the corresponding calculations.

As to the influence of loading distribution (b)

for the execution of the calculation of natural

frequencies (which is interesting to know before

building and which must be carried out at the

project stage), when real values are not available one is obliged to take estimated values as a basis, values introduced in the calculations and so

dif-ferent from real values. Moreover, even the

evaluation of the weight of a given vessel presents

so many uncertainties that one can be sure not to be able to have, even for a known ship, actual values to introduce them in the calculation.

For the values of stiffness (c) the same remarks as those made for (b) can be presented in addition to those mentioned during the discussion of the Bernoulli beam. However, in addition to the ef-ficiency of the working sections, there is also, and this already for the execution of the

calcula-tion of moments of inertia to the flexion, the

prob-lem of the effective breadth of the flanges of the

partial beams forming the

hull steel-work.

Numerous researches carried out in this respect are giving values which are not always coherent: they influence of course the calculated values of the frequencies of the concerned beam. It can be

said that generally for beams, where ratio length - height is near 12 to 14, the value of the real natural frequency, calculated with the value of the effective breadth normally used, is close

to the calculated value. This remark is similar

to that concerning the influence of the shearing forces on the beam deformation and

consequent-ly on the value of the vibrations natural

fre-quency.

The influence of the shearing stresses on the deformation of the beam can be defined by the

equation;

df(x)K

T

dx G.A

wherexandf(x)havethe same meanings as those indicated in the formula (1).

K constant (as a function of the section shape)

T shearing force

G modulus of shearing elasticity

A surface of the section working in shearing Of course, in order to calculate the values of the natural frequencies it is necessary to take ac-count of the deformation resulting from both types of solicitations, i. e. those due to flexion and those due to shearing.

Taking a mode of vertical vibrations one notes that the second differential of the displacement is almost zero at the nodes. Consequently,from the equation of flexion (1)

it results that the

cor-responding bending moment is also practically zero; so at these nodes there are only shearing forces which are acting on the sections. Mainly the vertical elements of the ship's steel-work i.e.

the hull outside plating and the longitudinal bulk-heads for tankers, withstand these efforts.

Onthe contrary, as it is known, to the bending moment are resisting mainly the horizontal ele-ments of the hull girder located most farther from the neutral axis i. e. the steel-work of the double-bottom and the resistance upper deck.

However, the shearing forces acting at the

nodes are already provoking in their vicinity the appearance of the bending moment and, conse-quently, the corresponding solicitations mainly in the double-bottom and the resistance upper

decks. So this transmission of solicitations is

possible thanks to the stresses due to shearing which appear in the horizontal elements.

The logical consequence of this evolution, and of this transmission of the efforts, is that not all the elements of the double-bottom and decks are obliged to withstand, and on their whole length, the solicitations coming from the bending mo-ments. So one revert again to the question of the effective breadth which was treated above.

A conclusion can be drawn on the subject of the effective stiffness which withstands the flexion and especially, that it diminishes when the

num-ber of nodes increases, i.e. when the ship's

girder begins to vibrate according to higher

modes. This is of course valid for vertical and horizontal vibrations. The calculations carried out with these hypothesis (the results of which correspond to the average on forteen ships which is mentioned at the last line of Table 1) seem to give rather satisfying results. However, the re-sults of these calculations, when considered

separately,

are less satisfactory and show a

rather important dispersion. For these last cal-culations, it is also necessary to precise that it has not been taken account of the virtual mass; it diminishes with the increase of number of nodes of the vibratory mode-form. Consequently, and eveninthis case, the problem is still not solved

as regards the differences between the results

of the calculations and the experimentation. Owingtothe absence of symmetry plane of the hull, it also occurs a coupling between the hori-zontal and torsional vibrations (e). This question is often omitted simply because it is very long and difficult to determine correctly the moment of inertia and the torsional stiffness of the hull. As to the effect of inertia of the rotation of the sections on the oscillation movements (f), it is

relatively less important than that due to the

shearingforces. Moreover, itdoes not intervene in the determination of hull stiffness.

However, another coupling can occur between the vibrations of the hull and the vibrations of the cargo or the propulsive plant, which act on the double -bottom (g). It is quite evident that the last one is an elastic steel-work: it deforms perpen-dicularly

to its surface and,

besides, it is elastically connected to the ship's girder. Con-sequently, it is to be considered that the cargo,

representing the largest part of the vibrating

mass is fixed elastically to the ship. So it must be taken account of the location of these masses i. e. either in the nodes or in the antinodes of the vibratory mode-forms. It has been noted that the fundamental frequency of the system constituted by the elastic steel-work of the double-bottom and the cargo or the propulsive plant is located

in the zone of natural frequencies of the hull girder

according to rather high modes. Consequently, one can expect the interference of these

vibra-tions with vibravibra-tions of the hull assembly ac-cording to

higher modes. This question is

developed more in detail in the following text. The question of the influence of the damping of ship's girder vibrations due to the type of cargo carried and, toacertainextent, of the surround-ingwater, gives rise to parameters worth more

developing. Although the influence of the damping

on the value of the natural frequency of a system

is less important, it provokes nevertheless a

sliding of the value of this frequency to smaller values; it introduces new unknown random ele-ments in the determination of the frequency cal-culation of the hull considered as a beam.

It can be considered that these comments tend to show that the methods of calculation of natural frequencies of the hull free vibrations generally

applied, based on the theory of Bernoulli beam, which do not take account in a satisfactory manner of the real stiffness of the hull girder and of the influence of the masses lying on the elastic parts of the steel-work (such as the double-bottom for

instance), cannot give satisfactory results for the values of natural frequencies according to

higher modes of the hull.

Besides, and owing to the high flexibility of the modern ship's steel-work as well as the existence of more and more important hydrodynamic and mechanical oscillating efforts, it is indispensable

to be able to calculate the response of this

as-sembly in forced vibrations. Now, in this re-spect, it is necessary to mention that the

cal-culation methods of the hull vibrations are not at

all easy or practical for the execution of such

vibrations.

Finally, there is a conclusion which comes out from this preliminary analysis: that the present methods do not seem to be able to take account of the aspect of the space three -dimensionality of the hull steel-work.

It is to provide against the inconveniences which

may be implied from this fact that a method is being developed [22, 23] to enable to consider the hull has a steel-work having three dimensions.

Namely it is question of representing the ship as a structure constituted by plates and bars: it is the method called of finite elements. It could be expected that the application of this methodwill

be more developed and will lead in the near

future, to the most satisfactory results.

Buttill now, it can be estimated that the main concern of vibration studies should be the prob-lem of forced vibrations and the research of the possibilities of resonances which could appear between the excitation frequencies existing on ships and the natural frequencies of the different elastic systems of the ship's steel-work and her propulsive plant.

In the following text, it has been tried to deal with this question; it

is tried in particular to

search the presence of resonators of forced vibra-tions, coming from any source of excitations and likely to put in forced or free vibrations the various parts in presence as well as the assembly of the ship's girder.

Moreover, atthe end of this paper. the question of the repercussion of the free vibrations, excited mainly by the state of the sea on the hull steel-work, is also dealt with.

2.2. Treatment of the mechanical vibrations by means of the electric analogy.

The existence of an analogy between the

pheno-mena relating to vibrations, respectively mechanical, optical, acoustical. hvdrodynamic and electric ones, is a well-known fact.

Since several years. one has given his atten-tiontothe treatment of mechanical vibrations by electric analogy. To this aim panels of electrical circuits have been realized for the execution of analogue calculation. On them were analyzed electrical phenomena similar to the mechanical phenomena to be studied, and then, by means of suitable transformations, it has been endeavoured to pass to the mechanical phenomena in question.

However, this method does not seem to have

brought the expected results; this can be explain -edby the difficulties inherent in the choice of the electrical elements as well as the realization of the circuits leading, unless particular realization conditions, to an excessive bulkiness of the last

ones; the result is that there is a lack of

flexi-bility indispensable to assure the efficiency and the rapidity of the treatment of these problems appearing every day.

However, the use of the electronic computer seems to open also in this field a wide range of application. A computer can be a very convenient and powerful instrument to solve numerous

mechanical problems relating to various questions concerning ship's vibrations; in

particular the association of digital and analogue computers enables to use simulation assemblies which present no inconvenience inherent to the one and the other type of material for such studies. The first results which have been obtained in

Bureau Veritas for such a treatment of the

questions of mechanical vibrations are most

promising:we have thought that it could be useful to show the possibilities of particularly interest -ing future developments.

It has been deemed useful to remind some es-sential notions concerning mechanical and elec-trical vibrations and then to present the existing analogies.

A real mechanical system put in forced vibra-tions can be exposed to four types of different ef-forts which correspond respectively:

- to the elasticity of the system (characterized

by the stiffness k);

- to the inertia of the mass of the system (func-tion of M) and the accelera(func-tion ;

- to the damping (which depends on the speed inamedium presenting a resistance value r to the movement in question);

- to the outside excitation of sinusoidal variation: F0 sin t.

0,

The values of the vectors are

indicated if)square brackets []

r

x

a) Elastic system exposed

to sinusoidal excitations.

[ri...,. x0]

kx

c) Vectorial representation of the equatica [4]

rx k.x= F0sn....t J

ri

R

F

D

e, Determination of the impedance z and iy osa function of

Mechanical impedance. z Impedance of the electrical circuit.

Figure 2. Analogy between the vibrations of mechanical andelectrical systems.

Then one obtains an equation of movement

which can be written:

M5+r.,+k.x=F0sinc.t

(4)

On Figure 2 is presented schematically such a vibrating system (see sketch a).

To exert an influence on the vibratory behaviour

of such a system one can act on both essential elements i.e.

- on the excitation;

- on the response of the vibratory system. a. Regarding the action on the excitation, it can

be exerted:

b) Electrical circuit exposed to forced oscillations.

E0

d)flnother method of vectojial represeolation of the equation [4] the lues of the vectars

expressed as function of v0 I

LQ

f) Electrical analogy of figure d) giving th vectorial representation of the equation L5]

LQR+*E,st 1

either on the absolute value of this force i. e.

on F0;

or on the pulsation by changing the excita-ting frequency;

or on the phase (which it will be told about

further).

b. Regarding the possibility of acting on the

re-sponse of the system, there are still three

possible actions, namely:

- change of the elasticity efforts; one proceeds to the modification of the constant k (action on the stiffness of the system);

- change of inertia forces;

one modifies the

9

mass(es) participating in the vibratory move-ments;

- change of the damping; this can be realized either by the installation of a damper, or by

the modifications of the characteristics of the already used damper.

However, it must not be forgotten that there is an interdependence between the effects of the possibilities offered. For instance, by increasing the stiffness one increases the natural frequency; but this increase of the stiffness leads to the in-crease of the mass leading to the diminution of the frequency: finally, it can happen that instead of improving the situation it can be even worse.

Let us leave, for a moment, the mechanical

vibratory phenomena to consider the electrical

one.

For an electric circuit: - with a self-inductance L; - a capacity C;

- a resistance R.

to which is applied a tension E E0 sin t the movement of electrical charges Q as a function of the time t satisfies to the equation:

LQ+RQ-1-

= E0sint

(5)

On the sketch b of Figure 2, such an electrical circuit is schematically presented.

dQ is the charge per unit of time i.e.

(6)

- dt the intensity

di is the differential of the intensity

'dt in respect to time

It is also reminded:

The tension to the self-induction terminals L is obtained from:

EL= L LQ (8)

The tension at the terminals of an ohmic resist-ance R is:

ER = Ri = RQ

The tension at the terminals of a condenser:

E

-C _c

It follows from the second Kirchoff law that:

EL + ER + Ec = E

from which one arrives directly to the equation (5) As both equations (4) and (5) are of the same form, it can be easily established a parallel

between the electrical and mechanical

combina-tions (6): so the mechanical problems can be transposed on electric problems, the study of

which is nowaday the subject of well elaborated and powerful methods.

However, there are two methods to deal with the mechanical vibratory phenomena by analogy with the electrical vibratory phenomena: one can called 'old' thatwhich is the best known, the new one being that called of Firestone. Although it is the second methodwhich is the best adaptedtothe

solution of complex systems (instead of

re-presenting an equilibrium of forces such as in the 'old' method, it used the equation concerning the speeds of various elements, equation made by analogy with the equation of tension in an electric circuit); to simplify the reasoning we have restricted ourselves to establish analogies between the various mechanical vibratory

pheno-mena met on ships and the electrical vibrations by using the analogy method called 'old'.

It can be useful to recall briefly the question of mechanical impedance to pass then to its elec-tric homologue i. e. the impedance of elecelec-trical circuit 8).

Let us take again the equation (4) of the move-ment of a damped oscillator which is then sub-mitted to a sinusoidal excitation force or couple

of period T = -. For the constant speed this

equation has a solution which can be written under the form:

X = X0 sin (c t + (12)

The above equation will contain a sum of

sinusoidal functions having the same pulsation; so it can be possible to determine X0 and by

using the Fresnel construction (see Figure 2,

sketch c).

if one represents the function kx by a vector

of origin 0 with a length kx0 and an arbitrary

direction, rxis representedby a vector of length r c x0 with a quarter period in advance and Mx

by a vector of length M2 x0 in opposition of

phase to the first one.

vector F of amplitude F0 it makes an angle p with

the vector kx. One sees that the out-of--phase

angles can be still supposed situated between

and +Tr.

p is in the present case:

2 - situated between -iT and - - if M co > k - situated between - and 0 if M k

X is always lagging in phase behind F. Besides one has (0 A F triangle):

cor tgp -k -M co (13) and F02=X02 [r2co2+(k_Mco2)2] (14) F0 from what X0 2 2 2 2 (15) co +(k-Mco

The same vectors presented on sketch c of Figure 2 can be presented in another way see

sketch d, _{their length being expressed with}

respect to the amplitude v0 of the speed:

v = =v0sin(cot+p)

(16) = p

F0

Itcanbe seen that the ratiowhich is called

V0

mechanical impedance of the considered system has a value:

F0

z=-=

_M2)2

(17)

Co

and that one has: k - -M. co CO tg p =

r

r

(18) cos p = -sinp =

-the choice of -the signs resulting from:

To enable a rapid discussion of the variations

of p, p, Z, and x0 when 1 . M, k and r being constant it is supposed that co varies from 0 to the sketch e presented on Figure 2 could be de-duced from the sketch d.

F0

The extremity D of the vectorof lengthZ0

then described from bottom to top the straight line D, and the D'of the opposite vector, of length

F

1 °

-

=-

describes the curve which is deduced from

Zv

₀

by inversion; it is the circle C of diameter

F0

z0 is minimum and v0 maximum and equal to

-whenD is in D0, i.e.. whenM. = or Co=Co 0

ThenonehasLp=0 and

_{p = -;}

these twoangles decrease constantly when co increases. Table II summarizes this discussion.

Table II

Forced sinusoidal oscillations. Influence of the imposed pulsation.

V0

As to the amplitude x₀

= -

it does not pass

CO

exactly by its maximum at the same time as v0. The minimum of the denominator of the equation (15) takes place for:

d _{[r2co2+M2(co02_co2)2]}

=0

d(co2)

where r2 -2 M2(co 2 _2) = 0

0

The solution of this equation

II (A) 0 ₀ +00

4\

o\-0 \

¶

-3-

\

\

-7r z +00

r

/

x0 x0

/

X \

0 Tr -iT <p <0 and - < 2

1

-

C) 2 r2

0

-2M2

is called pulsation of resonance.

It is seen that it is always lower than the natural pulsation

but it is not very different if r is

small i. e. if the damping is weak, and they are

equal for r = 0. The corresponding value of the

maximum amplitude is: F0

-

rV2

-(20)

M2

It is high if r is small and would be infinite if r was zero.

For =Othe elastic system takes the deviation:

F0 F0

x = =

0

9k

(21)

M.

then one has a static phenomenon and the dis-placement of the system results of the equilibrium between the constant force F0 applied and the response force kx0.

After this analysis of the sinusoidal forced

vibrations applied to the mechanical systems, it would be easy to pass to the notion of impedance of an electrical circuit (see equation (5)).

The intensity i=Qof a current is sinusoidal and

in advance in respect with the difference of

potential E of an angel

(0) such as:

1

CCD

-L.

(19)

Table III

Correspondences between the vibrations of

mechanical and electrical systems (according the

old analogy).

3. Various resonators of forced vibrations appearing on the ship.

After the short analysis of the methods of cal -culation of the natural frequencies of the ship's

girder as well as that relating to the theory of

forcedvibrations, it is time now to deal with the solution which can be contemplated for the pract-ical treatment of the vibratory phenomena, ap-pearing more and more often on modern ships.

As it has already been said, one can tell that ships of big and very large tonnage in service are

only very seldom excited in free vibrations: it

could even be said that the present big ships do not vibrate !On the contrary, when it is said that

a large tonnage ship vibrates, it is question of either forced or free vibrations of the various parts of the hull steel-work as well as its

ap-pendices.

In other words, itcanbe said that the very im-portant masses, represented by such a ship and its cargo, cannot be put into free vibrations, and in addition be sustained, but only by sea-swell (slamming for instance).

Linear mechanical system _I Electrical circuit

jFrls;nct C

L

EE0sinut

R

MirxkxFsnwt

LQRQ#E0sinW

Displacement x Electrical charge of the_condenser Velocity V=X intensity of the current I f=D

Acceleration

r=x

Moss M Inductance L

Stiffness k In verse capacitance

Damping coefficient r Resistance R

External force F sin ut External voltage EuE0 sin ut

2

Potential energy kx Electrostatic_energy (&l 2 Kinetic energy --Mv2 Electromagnetic_{energy 2}

Mechanical impedance z=4 Impedance of z=L

the circuit

Z== R

(

L.c)

E0 \J 2 2 '0

C.(

(22)

By replacing in this expression R by r, by k

and L by M one finds the mechanical impedance (see equation (17)).

In Table III are indicated the correspondences

between the electrical and elastic vibrations,

according to the 'old' electric analogy.

-R

For the graphic presentation see sketch f of Fig-ure 2.

The quotient, called electrical impedance, has for expression:

In this category of excitation sources can also be included the one which is provoked by the en-counter of a ship with a heavy swell, the excita-tion frequency of which, resulting from its

propagation speed combined with the ship's speed, would fall in resonance with one of the natural frequencies of one of the vibratory modes of the

ship.

However, as it has been already mentioned, the heavy vibrations of the hull-girder are felt, es-pecially in case of resonant response of a part either of the steel -work or of the propulsive plant

to the excitations coming from either the

pro-peller or the propulsive plant itself.

So it may be deemed reasonable, at present, not to deal only with the question of the calcula-tionof the natural frequencies of the free vibra-tions of the hull but to consider essentially the following questions:

- the response to the exciting forced oscillations of the various elastic systems existing on the

ship; Radio installation e=F#F,5in (J1t + F; F; cm + Ship Source of excitation The efforts engendered, either by the

prcpulsor or by the propulsive apparatus. etc....

Emitter Exciter e e3

As

Amplifier

L

_e3 ;e=12.3 LR CR Resonator Resonator

Elastic system in resonance with the source of excitatiónfresonant response) hence dynamic amplification of the excitation forces.

- the research of what can be called resonators of forced vibrations; in fact under this name is to be understood an elastic system which, having the value of one of the natural

frequen-cies according to various vibratory modes,

very near or equal to one of the excitating fre-quencies existing on board ship, responds to

this excitating frequency by a resonance

pheno-menon which produces a dynamic amplification of the efforts coming from this source of ex-citation. The diagram of such a resonator is presented on Figure 3.

When proceeding like that, the electric analogy being very useful for such purpose, one can di-vide the vibratory phenomena of a ship, which are presented on Figure 1, in various parts.

In what follows, the analysis of the excitation sources and the response of the elastic systems in question, using the result of experimental

re-searches, is treated in such a manner that an

electric analogy can be established between the studiedvibratory phenomena and the oscillations

e, -AamJ, 5*')CJet

Receiver

A5= f(Aompt1.CJe)

Hull steel- work

Respondinç either as forced vibrations or even as free vibrations to the

excitation-foltes amplified by the resonator

Figure 3. Scheme of a vibration resonator and of transmission of the forces.

'3 donc at( 1 4 1 VLR.CR donc e2-AsesinQbt

of an equivalent electrical circuit. The

trans-position from the equations describing the elec-trical phenomena to those related to elastic os-cillations can be effected by the intermediary of the electronic computer enabling the complete study of the phenomenon by the suitable variation of the parameters in play.

It can be even specified that the task of the

person who is in charge of investigating the troub -les likely to be provoked by the mechanical

vibra-tions is contrary to that of a radio-operator. In fact, to obtain correctly a radiomessage the last

one is obliged to tune the frequency of the response

of the installation of his receiving-set to that of

the transmitter. On the contrary, as regards

mechanical vibrations, at first it is question of detecting, determining, and then proceeding to a

detuning between the excitation frequency (ie s) and

that of the response of the concerned elastic sys-tem(s).

Now, let us analyse the various types of such

resonators and the mechanism of the

trans-mission [31] of the corresponding efforts and

vibrations of the hull by using the notion of

mechanical impedance and the electric analogy.

In what follows are presented the essential

types of resonators which may be encountered on modern vessels. They are treated according to the diagram shown on Figure 1, and according

to their character they can be divided in two

categories: active and passive ones.

The first category concerns these ones which the elastic system itself canbe excited into reso-nance by its proper forces during its own func-tioning. The second category concerns those when the elastic system is able to respond into

reso-nance only under the influence of excitations

originating elsewhere.

The first category of resonators is treated in

paragraphs 3.1 to 3.4 and the second one in

paragraph 3. 5.

Anyhow their detailed description and analysis

being too long it has been deemed useful to present

hereafter only their essential points and refer the reader to:

- either the original publication issued in Nou-veautés Techniques Maritimes 1969;

or the different items of the bibliography deal-ing with corresponddeal-ing phenomena.

3.1. Action of the propeller being at the origin of the exitation of the hull due to

hydrodynamie phenomena.

At first let us tell about the effects of the in-fluence of the propeller action on the steel-work beingtransmitted to it by the intermediary of the hydrodynamic effects.

It is during its functioning in the non-uniform wake field and each time a propeller blade passes in the neighbourhood of the stern (or the hull) that it occurs a variation of hydrodynamic pressure. This variation which is propagated at the speed of the sound through the surrounding element, i.e. through water, gives rise to a variable press -ure field. The corresponding impulses are ex-erted on the aft part of the underwater hull i. e. the outside plating as well as the rudder, located

of course behind the rotating propeller.

In addition to the above described effects, the action of the propeller is exerted on the hull by mechanical effects which are due to forces and torques induced by the propeller into the tail shaft

and consequently into the propulsive plant assem-bly. This type of action, and excitation, acting in vertical, horizontal and axial directions have

a static as well as dynamic character.

To better characterize these two types of fects. i. e. the hydrodynamic and mechanical

ef-fects, on Figure 4 are presented all the efforts

acting onthe propeller and the tail shaft while on Figure 5 is presented the block diagram

concern-Surface of action of the transverse forces.

TJ Qo Wght r. Torque FoThru=t Transverse force e Eccentrc,ty of thrust ml Rake Genera trix 'a

Figure 4. Forces acting on the propeller and the

Mechanical characteiist,cs of the sfeol-eack

2

In the axial directto,,

Hydrodynarnic _j

I

[ On the hull 1 On the ,iidder

Transverse hydra dynamic

cat ficient,

Vibration measurements of the hull structure

For the right-handed propeller

6 components

EXCITATIONS COMING FROM THE PROPELLERS

Hyoradynamlc cot ft/dents , , 2 lateral components 2 axial components Equation of movements of the rudder

EQUATION OF THE MOVEMENTS OF THE SHIPS GIRDER STEEL -WORK

I

V,brat,an measurements Measurements of the Measurements of axial iifrations of the line-shafting, the forces occuHn,n the of the propulsive ,nglal/atice, aid propeller and the rudder suppor I lateral vibrations of the steel-veu*

Legend 0 couplings

Figure 5. Block-diagram of the study related to the exciting efforts of the propeller and of the response to their

action of the elastic systems of the ship.

Mechanical characteristics

Interact,or, between a propeller blootâ and the L%ronJW hut.

In the tcegentiol direction

Mechanical on the line- shafting

Lateral 4 Ia erol components Lateral hydra dynamic coefficients 2 axial components 0 Mechanical characttrist,c dde Ire -shoftag and tb supports C P Axial hydrnidynamic coetfcents V V V Mechanical characteristics of the line-she /1mg

and the thrust-bind,

IEqua ion of movements of the line- Equation a/axial movements of shaftngondthe propeiler'ieiherling) the propulsive installation

Equation of vertical and transverse movements of the outside-plating

and the double- bottom

Parts of ship's ,s,derwo$e lou/I eohio't -e afff t3t The

pressure pulsations provoked by the prop&"°"

Result for the two directions

Regions where the loading of the blades of a right-handed propeller are increased due its presence in the field of walle,

For th. left-handed propeller

Eccentricities of thrust and lateral forces

Figure 6. Interaction between the propeller and the ship's underwater parts and the resulting effects.

L000tis, of the pressure

hsieer septa 0 5.

Abiaded piopeii.r

fart cai paeihe,

5/a blaSt

transducer B

Var,atcns at hydmdyflatriic pressu,-es

measured 0th. upper port 0/the

aperture at the stern Figure 7. Variations of the hydrodynamic pressures

provoked by the action of the propeller.

ing the exciting efforts and the response of the vibrating elastic systems.

OnFigure6 are presented the variable efforts imposed on the aft part of the hull by the action of the rotating propeller.

The detailed treatment of these phenomena is presented in [15. 32, 35. 36 and4Ol.Theresults 4bladedpropellei: foctor,sl dipgrst,t sf he eccentricity of thrust

j

Vectorial pogrom of the transverse forces

'en.

so.

-tO 0 . 0 c5 sf0

V. a/propel/er rothuc V. of average thrust

Madotcat,a, n est.r flare pra.ie.ed by change of ruOr angie

Figure 8. Increase of the variations of the

hydro-dynamic pressures due to the modification in the condi-tions of water flow provoked by change of rudder angle.

of the experimental researches concerning this problem are published in [40] and shown on Fig-ures 7 and 8.

The analogies between the oscillations of these systems and the electrical oscillations are pre-sented on Figure 9.

a

-Vertical transverse force

5 bladed propellec

t/Sofarevolution of the propeller Vertical eccentricity of thrust

tie. 0 -to 0 atO V. of prspe er radius Rodier angie p.34Pf S. tots ra/an 4 bled.d propeller no tOf 5 Troaa&o.r A ?rjoolut,aa Transducer S

Ilenotions at nydrsdynsmc preoscres m.oesad

in the spear port of tOO oparture of the atari,

o Vectorial diagram of the eccentricity of thrust Vectoriol diagram of he transverse forces 24 0 .t OP V. of average thrust

Figure 10. The dynamic phenomena occuring on 4 and 5 bladed propeller during runing in wake-field.

Hor,zoo tat traitSperSe o 5-a 5-B a 72' ear Var,at,05 of thrust 0 72' 72' Variation of resisting torque

tAd. Horizontal transverse :1 72' farce Harira stat eccefltncity \ at thrust

Figure 9. Electrical analogy of the excitations due to hydrodynamic effects provoked by the propeller.

a a 0 70 - 9-8-. 7- 6- 5- 4- 2-'0

'A,

5F ' '-"\,_'' '-' - 0 0 180 270 350 Thrust variation .".I

Figure 11. Comparison of the measured fluctuating efforts generated by the three types of propellers of a tanker-model. iIlIr 11

-A Caeffc,ent at dynance 342 450 430 amplitcatsan F .1630ue/,, Natural frequency calculated

V(7

-YnaMol 74

zoo

0 90 180 270 360

Transverse bending moment

Rudder angle.

a.-natcrai Resonant condifioos

S Number of blades. 5 bladedpropeller f) 704 rpm line -shoft,ng Fear 5x104 520 aec/mn of th order N,5

- O3ffd.._-_. iln ff45 calculated. natural frequency calculated

09' - 7074 recorded. natural frequency recorded

4 "natuml 4.50 - AX rnear,cnl scheme af Ito l,r,e.shaft,ng, Actual pasitmorm at lmae.shafting

Figure 12. Effect of conditions of liaison between the tail-shaft and its forward bush on the dynamic behaviour of the line-shafting.

VV Vertical vibrations VT Transverse vibrations

The ship with the same draught aft.

VT

® Before rational shaft alignment ® After rational shaft alignment

Figure 13. Forced hull vibrations generated by the propeller-shaft vibrations of a 58000 T.D.W. tanker (5 bladed propeller).

3.2. Exitation of vibrations induced into the steel-work by the intermediary of the tail

shaft supports.

The second category of excitations coming from

the propeller, acting on the hull steel-work,

whichwe are studying, are the repercussions on this steel-work of the lateral forces and torques induced into the tail shaft due also to the

function-ing of the propeller in the non-uniform wake

field.

It is in fact the above mentioned efforts which provoke in the tail shaft solicitations which can

excite in flexural vibrations the tail shaft and

even an important part of the line shafting. More -es.

n =97rp.m._

VT

VV Sn,pk ,mpViV.on

Measurement of the propeller - shaft vibrations.

over, all these effects are at the origin of the

reaction variations which act on the hull through the line shafting supports.

Let us come back to the efforts and couples originated by the propeller action during its rota -tion: they mainly concern the aft extremity of the tail shaft.

This problem has been studied by numerous searchers. Recently, it

was the subject of

particularly rational researches done by

R. Wereldsma from Netherlands Ship Model

Basin at Wageningen [37, 38, 391.

These researches concerned mainly the excita-tion forces and couples. On Figures 10 and 11 are shown the essential results of these

Vor,at,onu of the hydrodynarn,c pressures provoked by the action of the propeller

V

Exc,tatng forced due to the sea- cond,fiotts

Loadnq of the ta,'- shaft supports

A

Exc,f,ng forces of the I

propulsive apparatus

Vibratory movements of the

hull

Dynamic characteristics of transfer the hull

Searing rear t,on

Bear ng supports

I Transverse dynamic chsuacter,st,cs of the propeiler, the tail - shaft and

f's supports

Vibratory < movement of the

bearings

Figure 14. Block-diagram of lateral dynamic behaviour of the tail-shaft in conjunction with the elasticity and the vibrations of the hull steel-work, as well as the lateral coefficients of the propeller.

On the contrary the question of the response of the elastic systems constituted by the propeller. the tail - and intermediate shafts as well as the

hull steel-work, to these excitations has been treated in [11, 15. 16, 17, 19 and 30J. In fact, this response is the function of shafting

align-F1

F2

ment conditions and some dynamic phenomena

connected to the real behaviour of stern gear,

shafting alignment and forced vibrations, some examples are presented on Figures 12 and 13.

Figures 14 and 15 are showing some hints

con-'Tx

F0

Figure 15. System of reference coordinates and a summary of the excitations and the elasticities in the lateral

direction, of the propeller shaft-supports and the after part of the underwater hull.

I c, F _{- C,}F, F, t, F, £, F, y F, 'F, F, 'p, F, x F, P2 F, 'r F, y F F7 t, F1 5, F2 2 F2 11 F1 'F, F1 'F, F , F7 P F2 'F F T,

r

T, T, 1, 1, ty 1 T, ;; T, Px 1, T, ; 1, y T2 T1 t0 i,, T P0 I! '2, P ii 'Ft '22 Excitations Of ffF0 propel/er- shaft F,, F Excitotions of the after part ofthe hull-body

FF'T o

Hull steel-work

Transve, so locdrog of the Transverse osOvllot,ng forces and the tail- shaft, the wh,rI,ng of eccentricity of propeler thrust

e7 e= 5,r,(Jt

(()

k, k k k, t(M;) f(M,b) f(M,) f(R) M23 f(Mç) f(M,5) S M5

k'

Whirling(precession and nutation)

6= (cos(S'+)+j sin()J+._(cos(-f)-jsin(

-)J

k, k,

-4

R)

Figure 16. Electrical analogy of lateral vibrations of the line-shafting.

T6)

z Couplings e_ f(e)

ef(eT

19/ V1812i671' f= 21 M, R f(M)

cerning the theoretical approach of treatment of this type of resonators.

Figure 16 shows the analogy between the mechanical and electric systems corresponding to the question which has just been treated.

3.3. Longitudinal vibrations of the thrust bearing due to the thrust variations and

longitudinal vibrations of the main engine crankshaft.

The essential role of the action of a propeller is to transform in thrust the torque coming from the propulsive apparatus. Now, this transforma-tion is also shown by the appearance of thrust

variations due essentially to the presence of a

limited number of blades of the propeller and to

the fact that it works in the non-uniform wake field.

The thrust variations are at the origin of the axial vibrations of the line shafting, the thrust-block and even the propulsive apparatus, i. e.

either the crankshaft of a Diesel engine, or the shaft and wheel of the second reduction of the main gearing of a turbine driven propulsive plant. So one is in the presence of a new resonator of

forced vibrations coming from the propulsive

plant and likely to induce forced vibration exci-tations in the hull steel-work.

Table IV

Axial hydrodynamic coefficients of the propeller. I..'

05

ox

a. (is's)

MiOS,re*nfs.ft,ct,dn.op ton point 5, Crf& .0.015, *obi,-boIlonl 7 L -C 6 -o LO Q_- 4 o '.. c_0.0 0 .' 00 CL .2

U

8 5

éHHlIJlIIIIII

'0

Figure 17. Comparison of the vertical vibrations, due

to the longitudinal vibrations of the thrust-block, of the double-bottom steel-work of two ships, the steel-work of one being very flexible and the other very stiff.

N= Order of vibrafions

14(5,0! o.broPlr. op. 0t7 igid dsib4.5of,,,, iIiIint4

H I

III (III II

i!sb,oif's,. 0/0 itt fInib!.

00004.-bottin, Of,.! -0.0(0

40 50 60 70 80 ) 700 10 120

Thrust bearing type "MICHELLAR 1080 Stiffness of foundations k= 1260 tonnes/cm Figure 18. Dynamic magnification of the efforts due to thrustvariations provoked by the longitudinal vibrations

of the thrust-block of two similar tankers (of about

40000 T.D.W.) fitted with 4 and 5 bladed propellers

respectively.

Meanings Symbols Non dimensional_coefficients

Mass of additional virtual water Axial damping F2 C2 pnP3 Moment oi inertia of the additional mass Torsional mass !.pz pnR5

Coupling between the inertia and the torque

Tz

C pR4

Coupling between the inertia and thrust F

Coupling between the

axial speed and the torsionul torque

T

-z pnRd

Coupling between the axial speed and the

thrust F pnR 70 60 50 40 30 20 : N=5/

7

0 40 50 6O2t 80 700 710 120

The Nr4 and Nx&' order variations are due to a 4 bladed propeller

The N=S and N=10 order variations are due

too 5 bladed propeller

3

From the already developed considerations, the mechanism of the creation of the excitations by

the propeller seems to be already clearly

ex-posed.

The interaction of the thrust variations with other types of excitations also generated by the propeller, and especially those due to the lateral vibrations of the propeller and the tail shaft, is presented on Figures 1 to 5. However, and as it is noted onthese figures, there is also a coupling between the longitudinal and torsional vibrations and this is quite logical; the repercussion of this last type of vibrations on the ship's steel-work can also be noted and this question will be dealt with more in detail in the following text.

To come back to the question of the resonator due to the longitudinal vibrations excited by the

Absolute vertical vibra f/ohs measured in the steering-gear compartment.

Relative vertical vibrations of the tail- shaft.

Absolute vertical vibrations

of the thrust-block

Relative vertical vibrations

of the thrust-shaft.

propeller, we shall

still turn to the already

mentioned works of Mr. Wereldsma [37, 38,

39]

On Table IV are indicated the hydrodynamic coefficients describing the interaction between the propeller and the surrounding water in the axial direction.

Different theoretical and experimental

re-searches concerning the longitudinal vibrations

of the shafting have been undertaken and published

[32, 41, 42, 43 and 441.

OnFiguresl7, l8andl9 are presented the

es-sential and most interesting results showing the

response of the elastic system constituted by the shafting, the thrust-block and the

engine-room steel-work to the axial excitations coming from the propeller.

t"flfl"flflfl'fl'flh"fl'fl'flfl'flh"1

U U I

UU

jUl

UIMUIIDIIUIUI11NNIIl!MwHUIIIxIaUDaNMNIIIUma

IUUUURuURUUUUUu5UUmRUUnnRRRUuUui

IflflflIflflflflRflIliflflIflflflflflRRRflI

N=5 N=5

Figure 19. Vertical vibrations of the ship's structure of a 100000 T.D.W. tanker excited by the vertical

vibra-tions of the engineroom double-bottom being excited by the longitudinal vibravibra-tions of the thrust-block.

23

Ballast conditions N= 5

Absolute vertical vibrations

of the thrust-block

UURURRRIRRUUUUUkUUUUUUUSfltWUUUMUUUU

_Aflflflfli

Relative vertical vibrations

of the thrust-shaft.

I

I

d iUU ii

_UI

5 bladed propeller

Ballast conditions N= Order of vibrations

-d e.

''

P4, P4,, F;.E. (a,. '.)+(-). (-). ) f(P4) f(M,) f(M,J P(M,,) P(M) , ((P4,) ((P4,) ) J'P(*) r 'i T'

a) V,of*,g syP4'a' 04 lP0ff0Pg -(,ne (aa Porb,,a dpon pPOp,d$,l plant

On Figure 20 are presented the real systems

of the line shafting of twotypes of propulsive plant

respectively with turbines and with Diesel engine propulsive apparatus as well as the equivalent mechanical diagrams.

One notes in particular the presence of two

stiffnesses: one is the stiffness of the thrust-block opposing the action of the thrust and its

varia-tions, the other is the stiffness of the

double-bottom steel -work which is exposed to the action

of the excitations,

in the vertical direction,

coming from the longitudinal vibrations of the thrust -block.

On the same figure are presented the electric analogies of both mechanical systems used for the

treatment, by the electronic computer, of the dynamic axial behaviour of these mechanical

systems.

3.4. Vibrations of the double-bottom

steel-work due to exitations originating in the

main engine.

After the analysis of the vibrational excitations

originated by the propeller and the resonators

that they encounter to provoke the resonant

response of the elastic systems comprising the hull steel-work, the analysis of another source of excitation must be considered.

This concerns at first the vibratory phenomena whose origin can be attributed to the operation of the propulsion engine.

4P-op.(,-and ,,,,-P,,Ol aOP.

,

Th,ant- b,k C,onknl,0(t

LI

Mo, Lon a-n ha (hon p

M,¼ P4, iP, P4,, I,,,

-u .

Figure 20. Electrical analogy of the longitudinal vibrations of the line-shafting for both turbine and Diesel-engine driven propulsive plants.

Certain phenomena appear during operation:

they can be considered as essential sources of

vibration excitation coming from the main engine. Essentially, they concern the:

Longitudinal vibrations of the crankshaft: their action is exerted by the intermediary of the thrust-block;

Vertical andtransverseforces due to the bal-ancing of the moving masses of the engine: they exert their action on the foundations in-corporated in the engine room double -bottom steel -work;

Torsional vibrations of the line shafting as-sembly of the Diesel engine propulsion plant: they provoke vertical vibrations of the engine structure, of its foundations and consequently of the engine room double -bottom steel-work; Lateral orprecessionvibrations (whirling) of the crankshaft: they excite in vertical vibra-tions, by the intermediary of its bearings, the

engine structure, its foundations and the

engine -room double -bottom steel -work; Transverse vibrations of the engine structure due to the transverse components of the forces occurring during combustion, their action

being transmitted to the engine framework by means of the cross-head guides; this action is due to the reaction of the resistant torque

of the propeller acting on the cross-heads

through the crankshaft and connecting-rods. The effects of these vibrations are felt on the

2,th,,rt,o.,

TM n...hafP,ng

e2- F;',t M01,MPp M,0, a,