Torsional axial vibrations of a ships propulsion system Part II: Theoretical analysis of the axial, stiffness of the shaft support at the thrustblock location

REPORT No. 132 M

October 1969

NEDERLANDS SCHEEPSST UDIECENTRUM TNO

NETHERLANDS SHIP RESEARCH CENTRE TNO

ENGINEERING DEPARTMENT

LEEGHWATERSTRAAT 5, DELFT

*

TORSIONALAXIAL VIBRATIONS OF A SHIP'S

PROPULSION SYSTEM

PART II

THEORETICAL ANALYSIS OF THE AXIAL STIFFNESS OF THE SHAFT SUPPORT

AT THE THRUSTBLOCK LOCATION

(TORSIE-AXIAAL TRILLINGEN IN EEN SCHEEPSVOORTSTUWINGSSYSTEEM)

DEEL II

(THEORETISCHE ANALYSE VAN DE. STUFHEID IN AXIALE RICHTING VAN

DE ASONDERSTEUNING TER PLAATSE VAN HET STUWBLOK)

by

Jr. W. VAN GENT

and

Jr. S. HYLARIDES

Research scientists, Netherlands Ship Model Basin

Ir. C. DRAYER Ir. N. DIJKSHOORN

Drs. C. A. M. VAN DER LINDEN Prof. Dr. Ir. J. D. VAN MANEN Dr. Jr. R. WERELDSMA

Ir. A. DE MooY (ex officio)

De tendens naar snellere schepen, uitgerust met langzaamlopende hoofthnotor met een groot vermogen en de tendens naar de toe-passing van korte asleidingen leiden tot een trillingsgedrag van het assysteem, dat aanmerkelijk gecompliceerder is dan bij motor-installaties in een nog recent verleden.

Deze complicaties zijn voornamelijk het gevoig van het op-treden van a! of niet gekoppelde torsie- en axiale trillingen van het assysteem bij toerentallen in het draaigebied van de motor. Teneinde ontoelaatbare extra mechanische belastingen van het assysteem, die als gevoig hiervan kunnen optreden, te voor-komen is het noodzakelijk in het ontwerpstadium een betrouw-bare voorspelling van het trillingsgethag te kunnen doen.

Een berekeningsmethode, die, rekening houdend met de koppelingseffecten van de krukas, tot de voorspelling van de eigen frequenties van het assysteem leidt, werd beschfeven in rapport no. 39 M: ,,Crankshaft coupled free torsiOnal-axial vibrations of a ship's propulsion system".

Ter verificatie van deze berekeningsmethode en ter bepaling van nog niet met voldoende nauwkeurigheid bekende parameters van het stuwbiok en de schroef, die mede bepalend zijñ voor het tril]ingsgedrag, werd een nauwkeurige en uitvoerige trillings-meting uitgevoerd aan <leasteiding van het motorschip ,,Koude-kerk" (gepubliceerd in rapport no. 116 M).

Omdat bij de berekening van de eigenfrequenties van de axiale trillingen de stijtheid in axiale richting van de asondersteuning ter plaatse van het stuwbiok een belangrijke grootheid is, wordt in het onderhavige rapport deze parameter geanalyseerd.

Gebleken is dat de, volgens de in dit rapport vermelde me-thode, berekende stijtheid een goede overeenkomst vertoont met de experimented gevonden waarde.

Aanvullend onderzoek is echter gewenst ter verfijning van de opgestelde rekenmethode, waarbij o.a. het dynamisch gedrag van de dulibele bodem beschouwd dient te worden.

Tevens dient bet imaginaire gedeelte van de complexe smeer-filmstijtheid aan een nauwkeuriger onderzoek te worden onder-worpen.

NEDERLANDS SCHEEPSSTUDIECENTR!JM TNO

The tendency towards faster ships propelled by slow speed diesel engines of high outputs and the application of short shaftlines have led to a vibration pattern of the shafting that is consider-ably more complicated than that of propulsion plants used in relatively recent years.

These complications mainly arise when torsional and axial vibrations, either coupled or not, are to be expected in the shaft-ing. In order to avoid inadmissible extra mechanical stresses due to thó vibratory behaviour a reliable prediction concerning the vibrations that can possibly be expected has to be done in thedesign stage. A method of calculatingthe natural frequencies of the shafting, taking into account the coupling effects of the

crankshaft, is described in Report No. 39 M: ,,Crankshaft

coupled free torsional-axial vibrations of a ship's propulsion system".

In order to verify the calculation procedure, and to determine the parameters of thrustblock and propeller, which are not yet known with acceptable accuracy, extensive and accurate

vibra-tion measurements were carried out on the shafting of the

m.s. Koudekerk (as described in Report No. 116 M). Because a calculation of the vibratory behaviour of the shafting needs an accurate value of the stiffness of the shaft support in axial direc-tion at the thrustblock locadirec-tion this report deals with an analysis

of this parameter. This analysis was performed with some

assumptions simplifying the.prøblem.

It was found that the calculated stiffness agrees with the experimental measured value.

Further investigation is desired in order to refine the method of calculation now available, paying attention to the real struc-ture and the dynamic behaviour of the double bottom.

Also the imaginary part of the complex oil film rigidity, which can lead to complications, has to be studied in more detail.

NETHERLANDS SHIP RSSEARCH CENTRE TNO

page

Summary 5

I Introduction 5

2 Hydrodynamicál rigidity of the fluid film in the thrust bearing 5

2.1 Introductory remarks 5

2.2 Force and moment in case of quasi-stationary motion 6 2.3 Dynamical behaviour for small variations round equilibrium 6

2.4 Hydrodynamical rigidity 7

2.5 Remarks on a three-dimensional description of the flow 8

3 Rigidity of structural support 11

3.1

Introductory remarks ...

11

3.2 Stiffness of the thrustblock on infinitely stiff bottom 12

3.3 Stiffness of the double bottom 12

3.4 Stiffness of the collar on the shaft 15

3.5 Stiffness of block on flexible bottom . . 16

4 Critical considerations . 16 5 Conclusions 17 6 Future work 17 7 Acknowledgement 17 References 17 Appendix 18

TORSIONAL-AXIAL VIBRATIONS OF A SHIP'S PROPULSION SYSTEM

PART II

THEORETICAL ANALYSIS OF THE AXIAL STiFFNESS OF THE SHAFT SUPPORT

AT THE THRUSTBLOCK LOCATION

by

Jr. W. VAN GENT

and

Jr. S. HYLARIDES

Summary

The effective dynamic stiffness experienced in axial direction by the propeller shaft in the thrustblock has been analysed.

The hydrodynamical rigidity of the oil film is derived from the dynamical behaviour of the thrustblock for small variations round equilibrium which derivation is based on the pressure distribution equation for the viscous flow in a narrow slot

The structural stiffness has been obtained by means of the flute element technique; only static calculations have been performed. In conclusion a qualitative picture is built up of the composition of the effective stiffness. The calculated value comes close to the experimentally obtained value.

1 IntrodUction

A detailed investigation has been performed in the

torsional-axial vibrations of the whole shafting of a

dry cargo motorship [1]. For the calculation of the

natural frequencies of the axial vibrations the dynamic

stiffness or rigidity of the axial support of the shaft

appears to be an important factor.

t i

-iLDD

tO.O835m D1.O.6731 m i.2065m B O,2857m

0 diameter at which thrust

resultant applies

U flDn velocity at

dIameter 0

slider surface aring s&rface

Fig. 1. Definition of symbols and bearing configuration. Sectorial segment

Cross-section of slider along line of constant diameter

C. Slot geometry

Therefore it was decided to carry out an additional

investigation into the effective rigidity of the axial

support of the shaft. This study consists of two

in-dependent parts:

the h'drodynamic rigidity of the oil ifim in the

thrustblock

the rigidity of the structural parts transiriitting the

thrust to the hull, i.e.:

- the collar on the shaft

- the thrustblock

- the double bottom.

The main scope of this. investigation is to obtain an

estimate of the magnitude of each of these rigidities

and to know which possibly dominates. Therefore,

in both parts of the study the problems are kept as

simple as could be justified.

For that reason the hydrodynamical considerations

are kept two-dimensionally and the calculations of

the structural rigidity only refer to static loadings on

the collar, thrustblock and double bottom.

2

Hydrodynamical rigidity of the fluid film in the

thiust bearing

2.1

Introductory remarks

The hydrodynamical theory of the lubricating

proper-ties of the visëous flow in a narrow slot is given in

several textbooks. See for instance the references

[2, 3]. In this study of the motion of the fluid film

be-tween the ring-shaped bearing surface and the

sec-5

tonal segmented slider

surface a two-dimensional

approximation is used. The underlying assumption

that the radial velocities are negligible is correct over

a large part of the radial dimension of the slider, as

the side leakage is restricted.

The bearing configuration considered and the

no-tatión used are shown in Figure 1. The main direction

M' = B $ (p - p0)xdx

of the fluid motion is along a line of constant radius,

0

but the curvature of this line

is neglected in the

calculations.

2.2

Force and moment in case of quasi-stationary

motion

The before mentioned theory results in an equation

relating the pressure p in the fluid film, the width H

of the slot and the relative velocities of the adjacent

surfaces. For the two-dimensional case this so-called

Reynolds equation is [2]:

4__(H3 -R-)

= 6p(Uj - U2)

- 12u( V1 - V2) (1)

In this form eq. (1) applies to the special case of

con-stant dynamic viscosity and concon-stant density.

A general motion of the bearing consists of a

transla-tion of the bearing surface (y = 0) with velocities in

x-and y-direction respectively U1 x-and V1 x-and a rotation of

the slider around hinge point S with angular velocity

c = a0 (Fig. 1).. The velocities of the slider surface

(y = H) corresponding with this

rotation are

in

x-direction U2

ct and in y-direction V2 =s(sx).

After substitution of these expressions into eq. (1) this

equatiOn can be integrated once. For a second

inte-gration it is necessary to know the slot width H as a

function of the distance x. The thrust bearing under

study has plane slider surfaces, thus the following

relation holds (Fig. ic):

In the Appendix these integrations are performed and

with use of the boundary conditions p(0) = p(l) = Pa

(pressure outside the slot) the resulting pressure

distri-bution is:

PPa

L[(U10

±2V}f1

+cl{2 (_L9f1 +f2}](3)

The functions f1 and 12 depend on xli and a/i. By

further integration over the distance x and

multipli-cation with the breadth B of the slider the force F' and

its moment M' with respect to the point x = 0 can be

calculated. This is also performed in the Appendix and

the results are:

F' = BJ(pp0)dx =

=

B[{UlO+2vl}kl +Cl{2(

s)k

_{± k2}]}

₍₄₎

The functions k1, k2, m1 and m2 only depend on a/I.

(Appendix).

2.3

Dynamical behaviour for small variations round

equilibrium

For small, low-frequency vibrations of the thrust

bearing the dynamical behaviour of the lubrication

film can be derived from the equations for

quasi-stationary motion of the foregoing section.

The equilibrium conditions are characterized by:

U1 = U; V1 = 0 and c = 0. The force and moment

follow from eq. (4) and (5):

F = 6jiUBk1/cz

(6

M = 6pUB1m1/a

(7

In the operating thtust bearing the slider will adjut

itself in such a way that the force vector is pointed

through the hinge point S or, in other words, th

equilibrium condition:

M = sF

(8)

must be satisfied. Substitution of eq. (6) and (7) into (8)

results in the equation:

With the expressions for k1 and m1, given in the Ap

pendiic, eq. (9) represents a relation between the ratis

a/i and s/i. As for a certain thrust bearing the value

of s/i is fixed, the ratio a/l can be determined .and with

that the coefficients k1, k2 m1 and m2 are also fixed.

The deviations from equilibrium are characterized by:

- the elevation of the bearing surface h

-

(AH)L5

and V1 = v = Ii;

- the deviation of the slope of the

slider surface

cc = Acc0 and c = a.

The additional force corresponding With these dóvia

tions is, by definition, expressed in the following way:

H = c0(ax)

(2)

m1/k1 = s/I

(9)

6uBl[tu

+2Vi}mi+cl{2(i _s)

_{+ m2}]}

(5)

F'F = F[hfh±ccf2+vj+efj

In view of eq. (6) and the relation

= --EH

(as)

ico cc0

which is derived from eq. (2), the coefficients f,, and f

are:

lf'dFA

1 1 dk1

fh-1 (dF\

1 [

as I

dk1

f=i--i =--12+----

_F\dcc0Jh=o

_ccoL 1

k1 da

In view of eq. (4) the coefficients f,, andf are:

1(12MB \.2

jv

_{- _;i}

iki ,

-F0

cc0

J

cc0U

1[6.uB1{2(a_s)k+k}]1[O]O

_(lOd)

which follows from eq. (9), and the relations for k1, k2

and m1 of the Appendix.

In a similar way the additional moment is

ex-pressed by:

M'M = M[hmh+ccm+vmO+smJ

(12)

where:

1 (dM\

1 1 dm1

m= ---I--- I

=

M\dHJ=o

cL0lm1 da

(12a)

2_()

(12b)

M0 dcc0

h=0 cc0

1112/2B

\

2

m=I

m1

I=-M\ cc

I

1 r6Bl2 I fas\

₌

L1

ccoULmi

k1 I

m1 da

(10)

The quantities h and v are dependent upon external

conditions, that is to say they are interactions of the

vibrating shaft. The quantities cc and

only play a role

in the internal performance of the thrust bearing and

depend on h and v. The relation between cc and s on

one side and h and v on the other side follows from the

equilibrium condition:

(lOc)

When a coefficient C is introduced, defined by:

(12d)

M' = sF'

(13)

Substitution of eq. (10), (12) and (8) into (13) results in:

hfh+ccf+vfV = hmh+ccm8+vmv+eme

or after rearranging and with.e =

k2 [1 dk1

:

_{dmil /[m2}

k2

2k[k1 da

m

da ]/ [m1

k

This equation is a first order differential equation for

the rotation of the slider under influence of the axial

elevation h. The numerical values of C and k1/k2

depend on the bearing configuration and the value

of U/I, in addition, on the number of shaft revolutions

per second.

2.4

Hydrodynamical rigidity

For the thrust bearing under study s/i = 0.5943 (See

Fig. 1) and the corresponding solution of eq. (9) is

a/i

1.625. With this value the various coefficients

introduced before are (See Appendix):

k1 = 0.0666

mi = 0.0395

ldk1/da = 0.1945

m2 = 0.0794 ldm1/da = 0.1218

k2 = 0.1379

(12c) (15) (16)

C = 2.8

Further: U/I = nirD/(irD/1 3.6) = 1 3.6n (See Fig. 1).

For harmonical vibrations:

h =

exp (2irvti),

v = 2itvhi

cc = exp (2irvti),

a = 2irvcci

and eq. (lOa, b, c, d), (12a, b, c, d) and (11) are

sub-In eq. (lOd) use is made of the relation

stituted into eq. (14) this equation becomes:

CU

CU2k1fh

Tcc=

_T1Z1T

- ₍₁₄₎

as

k2

Substitution of these expressions and numerical values

into eq. (16) gives, after rearranging of the terms:

2k1 /k2 Ii

0.97

h

and a stiffness:

I (2irnl/CU)(v/n)i

1

1 +0.165(v/n)i

1

-0.97

[h

0357V

1±0.027(v/n)2 L 1

U

With the numerical values from the beginning of this

section the coefficients in eq.. (10) become:

(lOa) fh =

;

(lOb) f =

(lOc)

and it has already been found in eq. (lOd) that:

fe=0

-Substitution of eq. (17) into eq. (10) then leads tO:

F'F =

[.{2.92

0.97

1+0.027(v/n)2J

( 0.346 ) i

1.

1 + 0.027(v/n)2j U

As the thrust bearing has 12 sliders the total

axial-vibration force is:

f

12(F'F0')

while the total axial static force is:

F0 = 12F0'

From eq. (6) follows:

I

-

[

"

1

[ F0

In Fig. 3 modulus and argument of the rigidity are

- [6iUBkj

3.88

given for some values of n as a function of the

exci--

tating frequency v. It can be concluded that in the

fr-With these expressions and eq. (18)f finally becomes

quency range of interest

KI > 12 x iO Nm' ana

60° <

<900.

f= h[F2.17+0.08)l

+

[( 1+0.027(v/n)

J

± 2ir

.JO.192+O.0045(v/n)211[

'

1

n 1 +0.027(v/n)2

_{i] [npBD3J}

The hydrodynamical rigidity is defined by;

K =

= K±2irviR

and according to eq. (19) it is composed of a damping:

52.17±0.088(v/n)21

[_±

1 (17) 1.

1+0.027(v/n)2

5

[nuBD]

R

0.192+0.0?45(v/n)21

[ F

it I. 1 + 0.027(v/n)2 J

LniBD3

(20a) (2Gb)

In these formulas D is the average diameter of the

bearing.

A review of this section learns that the freedom

of the slider to rotate decreases the stiffness and

in-creases the damping if compared with the case of a

ftxed slider; The stiffness and damping in the latter

case follow from eq. (20a, b) fOr

v/n -+

It must be remarked that the results of this. section

hitherto apply to geometrically similar thrustblock

configurations with 12 sliders and s/I = 0.5943.

Application:

FOr the thrustblock considered in [1]

the relation is given in Fig. 2 between the axial load F0

and the number of shaft revolutions per minute n

The viscosity of the lubricating oil is

= 0063

Nsrn2 at a temperature of 45 °C and D = 0.94 m,

B

0.27 m (see Fig. 1).

For convenience the hydrodynamical rigidity

is

written as:

K

.

IKI {cosq + isinq}

where, according to eq. (20):

IKI = [K2+(2irvR)2j4

= arctan [2icvR/K]

2.5

Remarks on a three-dimensional description

of the flow

With aid of numerical methods it is possible to solve

the Reynolds equation also for the three dimensional

case. In [2] solutions are given, in graphical form, fOr

rectangular and sectorial segmented sliders. Although

this description takes into account the end effects and

the curvature of the streamlines,

it

overestimates

presumably the influence of the side leakage. This

out-flbwis less than predicted from viscous flow theory in

0

85 90 95 100

consequence of the formation of a meniscus or a

"vena contractã" [2].

These three-dimensional solutions can be used to

describe the dynamical behaviour for the case of a

fixed slider. In Table I the valUes of the various

quan-tities describing this dynamical behaviour are

com-pared with the values of the two-dimensidnal

des-cription.

The values for the three-dimensional cases are

found by interpolation in the graphs of [2]. Apart from

the unknown influence of the slider rotation, the

three-dimensional description predicts higher values of the

damping R and the stiffness K. Presumably the slider

rotation has a reducing effect on the hydrodynamic

rigidity, but just like in the two-dimensional

descrip-105

n = NUMBER OF SHAFT REVOLUTIONS [RPM]

Fig. 2. Relation between thrust and number of shaft revolutions per minute.

110 115

Table I. Comparison of results of various calculation methods for the case of a fixed slider.

eq. (6):

Fcc/z UB

eq. (10):

f U

eq. (10): JhO1 C4. (20a):

R [F/pBD3n3]

eq. (20b):

K[F/pBD3n]

3-dimensional sectorial 2-dimensional rectangular segmented

9 120 100

90

-u, 80

z

0

I-.

-J

I-. U, 70 I-U Ij.0 60 50 40 110 MEAN DRAFT= 8.14m 0.4 0.138 0.131 2.0 2.0 2.0 2.92 2.56 2.30 0.164 0.279 0.286 3.25 4.85 4.47

80

60

48

40

32 I-.

0

'2 20 IL.

0

16 U) -J

0

o

12

JL10

8 6 90 60 30 0

U1. FREQUENCY OF AXIAL SHAFT VIBRATION

[i.ta]

Fig 3. Hydrodynamical rigidity of lubricating oil film in thrustblock.

x109

A

I

/

n

[rPV/B

th 100

90

4th ORDE' VIBRATION 12th

8th

VIBRATION

---4th ORDER

______________

go

1''

120 4rpm]

I

2 6 8 1O 12 16 20 24 3 16 32 2 3 4 6 8 10 12

= FREQUENCY OF AXIAL SHAFT VIBRATION

[Hz]

tion this effect may besmall. Thus the two-dimensional

calculations of this chapter give a lower bound for the

hydrodynamic rigidity.

3

Rigidity of structural support

3.1

Introductory remarks

The total structural rigidity of the shaft support in

axial direction is given by the collar on the shaft, the

thrustblock and the double bottom. This rigidity

problem can be solved by means of the finite element

technique [4].

The main scope of this investigation is the

deter-mination of the dominant parameters. Therefore in

the calculations only static loadings are considered.

For that reason it is advisable to speak in this section

of ,,stiffness" instead of ,,rigidity", because this latter

expression is generally reserved for dynamic problems.

In particular the following problems have been

in-vestigated:

the stiffness of the thrustblock;

the support of the thrustblock: is the block

sup-ported by the double bottom only or is the engine

also involved?

the stiffness of the double bottom, taking account

of the engine;

the stiffness of the collar on the shaft.

The dimensions of thrustblock and bottom, needed in

the calculations, have been obtained from the

tech-nical drawingsoftli m.s. ,,Koudekerk", which were

made available by the engine- and shipbuilder.

Fig. 4. Schematical representation of the construction of the thrustblock as used for the finite element calculations.

3.2

StWness of the thrustblock on infinitely st

For this case th calculations give an axial stiffness of

bottom

the block: K

3.35x i0 Nm1.

The thrustblock is a heavy and solidly built structure,

hence, it can be stated that its lowest natural freqUency

will surpass considerably thç frequencies of interest in

this investigation Therefore an investigation after its

statical behaviour only sUffices.

In Fig. 4 an outline of the construction is given. For

the finite element calculations it had to be divided into

elements (simple plates arid bars) interconnected at

nodal points. These nodal points are indicated in

Fig. 5.

First it has been assumed that the thrustblock was

also supported at the engine by its side plating. Double

bottom and engine are supposed to be infimtely stiff

Next it has been assumed that the block is not

supported at the engine by its side plating thus only

at the bottom. Then the stiffness of the thrustblock

becomes: K = 3.00 x i0 Nm'. Hence,

the

side

plating is riot a decisive factor in the problem, but has

to be taken into account man accurate calculation.

The results of these calculations are summarized in

Table II, together with the results of the calculations

of the bottom stiffness. The calculated systems are

schematically indicated in that Table.

3.3

Stiffness of the dOüblè bottOm

In the investigation of the influence of the double

Table II. Review of the several configurations considered in the calculation of the stiffness of the thrustblock, statically loaded.

/

\

/

$ ENGINE

Double bottom bided b foundation K = 3.00x l0°Nm-'

DOUBLE BOITOM forces of thrustbbock. SIDE PLATING

/THRUSTBLOCK

ENGINE DOUBLE BOTTOM

Thrustblock supported at the double bottom and by its side plating at the

engine.

Double bottom and engine inter-connected by side plating thrustbJock. Load corresponds to foundation forces of thrustbiock.

K 3.35 x l0°Nm'

Thrustbbock supported at the double K = 3.00 x 10°Nrn' bottom only

K = 6.30x10°Nm'

13

THRUSTBLOCK WITH Twodimensional presentatiOn of the K 2.09 x 1O9Nrn'

SIDE PLATING total problem.

'I /j

ENGINE

Fig. 6. The influence of the side plating, interconnecting double bottom and engine in the overall bottom stiffness. I I

r- -

-

---I

-I: I I I __ I

b.

L.L

_I.

I I

L

----bottom on the whole stiffness of the axial support of

the shaft a static loading has been considered, although

previous calculations have shown that the lowest

natural frequency of the double bottom in the engine

room of the investigated ship is 5.9

cs_i

[5]. It is,

however, accurate to do so because also this particular

investigation refers to the determination of the

im-portance of the underlying parameter in the whole

problem. If it would appear, for example, that the

static stiffness of the double bottom is much higher

than the static stiffness of the thrustblock, its influence

would also be small in dynamic considerations. If the

reverse holds true then only the double bottom has to

be considered in the determination of the. thrustblock

stiffness.

The stiffness of the double bottom has also been

determined by means of the finite element technique. In

agreement with the introductory character of the

investigation the double bottom has been represented

by a beam clamped at its ends. (Fig. 6 and Table II).

The centre girder and the side girders have been taken

as the web of this beam. The support is taken at the

front and aft bulkhead of the engitie room. Deflections

of the thrustblock will first intro4uce important local

deformations of the double bottom in the

neighbour-hood of the block. Then the remaining part of the

bot-tom will be deformed. With increasing distance from

the thrustblock a larger width of the bottom plating

will be involved. This has been taken into account by

supposing a linear increase of the effective stiffness of

the beam.

An estimate has been made of effective influence, of

the engine on the bottom stiffness. Due to the solid

structure of an engine a rough estimate is enough.

The loadings of the double bottom are the.

foun-dation forces exerted .by the 'thrustblock due to the

unit of thrust. The stiffness of the double bottom is

defined as the inverse of the extra displacement, which

the centre of the thrustblock obtains due to the bottom

deformations.

First the thrustblock was omitted. Then the bottom

stiffness appears to be 100 x iø Nm. In Fig. 6a the

deflection of the double bottom has been given. It can

be stated that this deflection is mainly caused by the

shear deformation of the centre and side girders under

the thrustblock. Therefore the influence of the side

plating of the thrustblóck was investigatecL Taking it

into account the double.bottom stiffness increases from

3.00x iO

Nm1 to 6.30x i0 Nm

(Table II, 'Fig.

6b).

From this result and that of the foregoing section it

can be concluded that the side plating, although not of

significant importance for the block stiffness,

sub-stantiálly influences the bottom stiffness.

A more significant conclusion of this section is,

however, that such a static consideration of the double

bottom is not acceptable. The lowest natural frequency

of the out of plane vibrations of the bottom is of the

order of 6 cs, i e, much lower than the lowest

natural frequency of the axial vibrations of the shaft.

As,. however,, the higher natural bottom frequencies can

have a significant influence, a further investigation is

needed.

3.4

Stiffness of the collar on the shaft

To obtain an approximation of the axial stiffness of the

collar, use has been made of the formulae given by

Roark [6]. The-total-- deflection is composed of two

parts:

due to bending

ccF0a2 Yb Et3

due to shear

O.375F0a (

Ii-2irtG

_'

where

= a constant depending on

/3

=a/b;

a = mean diameter at which the thrust applies;

b = inner diameter of the collar;

F0

total thrust;

t

= thickness of collar;

E

modulus of elasticity;

G modulus

of rigidity

0'

-4=. ___&__

-.;'___

2 In_/3 15

Fig. 7. Collar on shaft. (Dimensions in in).

0.1 525

0

Cd Cd

With the dimensions given in Fig. 7 the stiffness of

the collar, obtained from the inverse of the total

deflection for F0

IN equals K

48.5x i0 Nth'.

3.5

Stiffness of block on flexible bottom

To conclude these structural considerations the

inter-action between thrustblock and double bottom has

been investigated. Therefore a two-dimensional model

of the thrustblock has been derived with equal rigidity

as found for the three-dimensional model.

For the combination of thrustblock and double

bottom the calculation leads to the stiffness K

2 09 x

x io Nm'. From the results given in Table II it

follows that for the actual structure the interaction is

small.

Despite the close conformity with the experimentally

obtained value, K = 2.0 x iO Nm

which suggests

a high reliability of this two-dimensional approach,

this way of calculation cannot be used in general,

because several difficulties have been encountered in

representing the thrustblock two-dimensionally.

4

Cntical considerations

The effective axial rigidity, which is experienced by

the shaft in its thrüstblock, is composed of the various

rigidities of the structural parts which carry the thrust

from shaft to hull.

The calculation and measurement of the natural

frequencies of the propulsion system [1] mutually

agree, when for the thrustblock rigidity a value of

2 0 x l0 Nm

is assumed Strictly speaking it is not

correct to call it thrustblock rigidity as also other

structural parts contribute to the total rigidity. In this

study a picture is built up of The ng4dity of the whole

support. Figure 8 gives a schematic representation

and a review of the calculated values From left to

right the components are:

AAAAA

N V V V V V

collar

the collar on the shaft with rigidity = 48.5 x l0

Nm;

the oil film with complex rigidity, which depends

on the frequency. See section 2.4 and Fig. 3;

c

the thrustblock with rigidity = 3 35 x iO Nm

thedoublebottomwithrigidity = 6.30x iO Nm1

the remaining part of the ship, which has not been

considered in this investigation.

The reciprocals of the rigidities must be added to

obtain the reciprocal of the total rigidity. Thus the

greatest rigidities give the smallest contribution. In

view of this the influence of the collar on the shaft is

small. AlsO in dynamical considerations it may be

expected that its influence remains small due to its

sOlid construction.

The cOntribution of the oil film is

L=_l!

!

Ri Ri

Ri

In the frequency range of practiôal interest (see section

2.4 and Fig. 3) holds:

cosço

-<

0.47

=mN

10

-IKI

12x109

25

and

sin4,.

1 .

iO-IKI

12x iO-

12

whereas the contributions of the sevefal constructive

parts are of the order 10

9/3 mN'. Therefore it

can

be stated that the influence of the oil film is small for

this rough calculation.

In an accurate description of4he dynamic behaviour

of the whole support the influence of the oil ifim must

be taken into account and likely, also the influence of

the collar deformation on the behaviour of the oil film

An interesting phenomenon to signalize

is

the

generation of axial vibrating forces by torsional vi

brating motions. It is easily seen from the fOrmulae

in sebtion 2 that SF/F

i.n/n. This effect cannot be

overlOoked in a complete description of the dynamic

behaviOur of the axial support.

The imaginary part of the complex film rigidity can

lead to complications which are not studied in this

investigation Because of the di.fficulties met in the

comparison between the calculated and the measured

axial and torsional vibrations of the shafting, a

further investigation in this aspect is needed.

The main parameters in the overall rigidity of the

axial support of the shaft are thus the thrustblock

it-self and the double bottom, the latter in combination

with the engine and the side plating of the thrustblock.

Complex

46.5x109 _section 35x109 6.30 x 10

2.4)

Fig 8. Schematic representation of axial support Of the shaft.

harmonical thrust rigidity [Nm1J Hull dynamics

NW

oil film ihrustblock double bottom remaining part

It has been assumed that the lowest natural

fre-quency of the thrustblock will considerably

ed the

frequency range of iriterest. In that case the statical

considerations lead to a good approximation of the

rigidity =

3.35,x io

Nm1. To complete the

investi-gation the dynamical behaviour has to be investigated

to be sure of the assumption made.

FrOm statical considerations the bottom rigidity

appears to be 6.30 x iO Nm '.. Thus its importance

in the whole problem is of the same order as that of the

thrustblock. Calculation of the static stiffness of the

combination of block and double bottom gives an

overall stiffness K = 2.09x iO Nm'.

However, it has to be noted that the calculations are

based on a

statical consideration of the bottom,

whereas theoretically it has already been shown that

the lowest natural frequency of the double bottom is

5.9 cs1. This means that for an accurate calculation,

the dynamic stiffness or rigidity of the double bottom

has to be taken into account.

The natural frequencies of the shaft system in axial

vibrations are 11.17 cs

1 and 15.25 cs_i [1]. At these

frequencies it can be expected that the double bottom

will vibrate mainly in its vertical vibration modes.

With a rather rough division into elements the dynamic

stiffness or rigidity of the double bottom can be

cal-culated by means of the finite element technique.

Further the two-dimensional representation of the

double bottom, in whiáh local effects have beèñ taken

into account by means of a linear increase, of the

effective stiffness with growing distance to the

thrust-block, has to be kept in mind. A more accurate

esti-mate of the effective stiffness has to be calculated from

the real structure. Also for this reason a second

in-vestigation into the behaviour of the double bottom

is needed.

Despite the nice agreement between the calculated

and experimentally obtained values of the overall

rigidity of the axial support of the shaft it can only be

concluded that the proposed calculation technique

has led to satisfactory results in the static approach

of the problem and that it

will

also lead to good results

in the dynamic and more detailed approach.

It will be noted that for the oil film the damping

has been taken into account. This is not done in the

structural considerations because there, damping plays

a very small role. Only in the neighbouthood of

res-onance it has to be taken into account. But at these

frequencies the rigidity will almost disappear. Strictly

speaking this effect has also to be investigated

5 Conclusions

The rigidity or dynamic stiffness of the axial support

of the shafting depends mainly on:

the hydrodynamical rigidity of the oil film in the

thrustblock;

the structural rigidity of the thrustblock;

the Structural rigidity of the double bottom.

The calculated value agrees well with the

experimental-ly obtained value for the

rigidity.

However, the

dynamical behaviour of the double bottom has yet to

be investigated to complete the analysis.

The hydrodynamical rigidity

is composed of a

stiffness and a damping term, both depending on the

frequency of the shaft vibration and on the mean

thrust.

For the structural analysis the finite element

tech-nique has proved to be an accurate and handsome

means of calculation.

6

Future work

In order to calculate the rigidity of the shaft support

in axial direction the dynamic behaviour of the double

bottom has to be investigated.

Also the imaginary part of the complex oil film

rigidity, which possibly can lead to complications, must

be studied in more detail.

7 Acknowledgement

The Netherlands Ship Research Centre TNO

acknowl-edges the cooperation of Stork N.Y., Hengelo and

Van der Giessen - De Noord N.Y., Krimpen aan de

IJssel, which greatly facilitated the work reported herein

References

I VAN DER LINDEN, C. A. M., H. H. 't HART and E. R. DOLFIN,

Torsional axial vibrations of a ship s propulsion system

Neth. Ship Research Centre TNO, report no. 116 M.

Dec. 1968.

2 PINKUS, 0. and B. STERNLiCHT, Theory of hydrodynarnic lubrication. McGrawhill, 1961.

3 RADZIMOVSKY, E. I., Lubrication of bearings. Ronald Press,

New York, 1959.

4 HYLARIDES, S, Ship vibration analysis by finite element

tech-nique. Part I. General review and application to simple

structures, statically loaded. Neth. Ship Research Centre TNO, report no. 107 S. December 1967.

5 HYLARIDES, S.-, Estimation of the natural frequencies of a

ship's double bottom by means of a sandwich theory. Neth. Ship Resarch centre TNO, report no. 89S, April 1967.

6 ROARK, R. W., Formulas for stress and strain. University of

Wisconsin.

PPo =

[{U1L0+2V1}f1 +e{tc0+2(a -s)}f1 + elf2]

ic0

eq. (1) this equation reads:

(e)

with:

After substitution of U2 = et and V2 = e(s -. x) into

_(H3F)

6u(U1 +et)

- 12t(V1 -es+cx) (a)

Integration and division by H3 results in:

ss)

61L6x2

_C

(b)

dx

112 _{H3 H3 H3}

In eq. (e) the term t0 can be neglected as compared

Introduction of eq.(2): H = cc0(a - x) makes further

with 2(a -s). In the calculation of the force and its

moment the following integrals appear:

p(x)

64u(U1 +st)11(x)- l2ji(V1-es)12(x)-

_k1

-4_Jf1 dx

= ln---1

_2a-i

- 6,cI3(x) + C14(x) +p(0)

(c)

with:

k2=--Jf2dx=

lnai

integration possible:

11(X) = .1

=

1

o H

xxdx

1 x2

12(x) =

B3 - 2

a(a - x)2

x2dx

1

a

x(2a -3x)

13(x)

_{= 5}

=

L

a-x

2(a-x)2r

dx

1

x(2a-x)

14(X) =

= 2

a2(a_x)2

xl(l-x)

1

(a-x)2(2a-l)

x(2a-x) (a-i)2

_{In -- -ln-_}

l(2a-l)(a-x)

a-i

a-x

1

a(3a-21)

a

6a-1

m1

_{-J xf1dx}

= i(2a-i)

I

2(2a-i)

1 2 2

i

aa-i)

2

a

m2 - j xf2dx = -

in

+

12o

_l3(2a-i)

a-i

2a+l

12(2a-l)

a-i

41

Also of interest are the following relations:

1dk1

da

a(a-l)(2a-l)2

The constant C follows from: p(l) = p(0)

_{= Po}

or:

o = 6z(U1 +st)I1(1)

12(V1 -es)12(I)-

_1d1

2(3a2 -3a1+ i2)

a

l(6a2 -9a1±4i2)

- 6iI(l) + C14(l)

(d)

da

(2a

2

a -1

(a -1) (2a

APPENDIX

Elimination Of C from eq. (c) and (d) gives:

x

a-x

PUBliCATIONS OF THE NETHERLANDS SHIP RESEARCH CENTRE TNO

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

PRICE PER COPY DFL

10,-M = engineering department S = shipbuilding department C = corrosion and antifouling department

Reports

57 M Determination of the dynamic properties and propeller excited vibrations of a special ship stern arrangement. R. Wereldsma,

1964.

58 S Numerical calculation of vertical hull vibrations of ships by discretizing the vibration system, J. de Vries, 1964.

59 M Controllable pitch propellers, their suitability and economy for large sea-going ships propelled by conventional, directiy coupled engines. C. Kapsenberg, 1964.

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

Vreug-denhil, 1964.

61 S The distribution of the hydrodynamic forces on a heaving and pitching shipmodel in still water. J. Gerritsma and W. BeukCl-man, 1964.

62 C The mode of action of anti-fouling paints: Interaction between anti-fouling paints and sea water A. M. van Londen, 1964. 63 M Corrosion in exhaust driven turbochargers on marine diesel

engines using heavy fUels.. R. W. Stuart Michell and V. A. Ogale,

1965.

64 C Barnacle fouling on aged anti-fouling paints; a survey of perti-nent literature and some recent observations.. P. de Wolf, 1964. 65 S The lateral damping and added mass of a horizontally oscillating

shipmodel. G. van Leeuwen, 1964.

66S 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 vibratingpropeller models. R. Wóreldsrna, 1965. 71 S Research on bulbous bow ships. Part H. A. Still water

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

72 S Research on bulboU bow ships. Part. H. B. Behaviour of a 24,000 DWT bulkearrier with a large bulbous bow in a seaway. W. P. A. van Lammeren.and F. V. A. Pangalila, 1965. 73 S Stress andstrain disfributionin a vertically corrugated bulkhead.

H. E. Jaeger and P. A. vàiTKatwijk, 1965.

74 S Research on bulbcvus.,boiv 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 ILorssen, 1965.

76 S Research on bulbOus bow, ships. Part I. B. The behaviour of a fast cargo liner with aconventional 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 diesel 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. WCreldsma, 1965.

79 C The pre-treatment' of ship plates: A comparative investigation on some pre-treatment methodin use in the shipbuilding indus-try. A. M. van LOnden, 19.-.)

80 C The pre-treatmentpf ship.plàs: A practical investigation into the influence of'different: *king procedures in over-coating zinc rich epoxy-resin based fe-construction primers. A. M. van Londen and W.Mulder;I965.

81 5 The performanô of Utanks as a passive anti-rolling device.

C. Stigter, 1966.

82 S Low-cycle fatigUe of st'eel structures. J. J. W. Nibbering and I. van Lint, 1966.

83 S Roll dampingby fr surface tanks. J. J. van den Bosch and J. H.

Vugts, 1966.

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

85 S Brittle fragtirof full 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 transmission 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 of pitch and heave motions for arbitrary ship forms. W. E. Smith, 1967.

91 M Corrosion in exhaust driven turbochargers on marine diesel en-gines 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 II. Comparative cylinder wear measurements on a laboratory diesel engine using ifitered 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. Remmelts,

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 of the modified strip theory for the calculation of ship ttnotions and wave bending moments. J. Gerritsma 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 heaving des-troyer model. W. E. Smith, 1967.

99 5 r The manoeuvrability of ships on a straight course. J. P. Hooft,

1967.

1005 Amidships, forces and moments on a GB = 0.80 "Series 60" model in waves from various directions.. R. Wahab, 1967. 101 OrOptimum conditionsfor blast cleaning of steel plate. Conclusion,

:-j. Remmelt, 1967.

102 MThe axial stiffness of marine diesel engine crankshafts. Part I. 'Comparison between the results of full scale measurements and those of calculations according to published formulae. N J. :-Visser, 1967.

103 M.The axial stiffness of marine diesel engine crankshafts. Part 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 modJ.

J. H. Janssen, 1967.

105 M Marine diesel engine exhaust noise. Part II. Scale models of exhaust systems. J. Buiten and J. H. Janssen, 1968.

106 M Marine diesel engine exhaust noise. Part. Ill. Exhaust sound 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 I. Testing of a decentral-ised refrigerating installation. J. A. Knobbout and R. W.. J.

Koufi'eld, 1967.

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

110 S Strain, stress and fiexure of two corrugated and one plane bulk-head subjected to a lateral, distributed load. H. E. Jaeger and P. A. van Katwijk, 1968.

111 M Experimental evaluation of heat transfer in a dry-cargo ships' tank, using thermal oil as a heat transfer medium. D. J. van der

Hóeden, 1968.

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

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

I Ni'S The steering of a ship during the stopping manoeuvre. J. P. Hooft, 1969.

115S Cylinder motions in beam waves. J. H. Vugts, 1968.

116 M Torsional-axial vibrations of a ship's propulsion system. Part 1. Comparative investigation of calculated and measured

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

118 M Stern gear arrangement and electric power generation in ships propelled by controllable pitch propellers. C. Kapsenberg, 1968. 119 M Marine diesel engine exhaust noise. Part IV. Transfer damping

data of 40 modelvariants of a compound resonatorsilencer. J. Buiten, M. J. A. M. de Regt and W. P. H. Hanen, 1968. 120 C Durability tests with prefabrication primers in use of steel plates.

A. M. van Londen and W. Mulder, 1969.

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.

Nibbering, 1968.

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

1968.

123 M Marine refrigeration engineering. Part III. Proposal for a specifi-cation of a marine refrigerating unit and test procedures. 1. A. Knobbout and R. W. J. Kouffeld. 1968.

124 S The design of U-tanks for roll damping of ships. J. D. van den Bunt, 1969.

125 S A proposal on noise criteria for sea-going ships. J. Buiten, 1969. 126 S A proposal for standardized measurements and annoyance rating of simultaneous noise and vibration in ships. J. H. Janssen, 1969. 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 III. A. de Mooy, P. J. Brandenburg and G. G. van der Meulen, 1969.

130 M Marine diesel engine exhaust noise. Part V. Investigation of a double resonatorsilencer. J. Buiten, 1969.

131 S Model and full scale motions of a twin-bull vesseL

M. F. van Sluijs, 1969.

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

scarcely saponifiable vehicles (Dutch). A. M. van Londen and P. de Wolf, 1964.

12 C The pre-treatmentofship 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 prevent 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. Hamm. 1968.

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

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