REPORT No. 102 M
August 1967
NEDERLANDS SCHEEPSSTUDIECENTRUM TNO
NETHERLANDS SHIP RESEARCH CENTRE TNOENGINEERING DEPARTMENT LEEGHWATERSTRAAT 5, DELFT
*
THE 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
(DE AXIALE STIJFHEID VAN SCHEEPSMOTORKRUKASSEN)
DEEL i
(VERGELIJKING VAN DE RESULTATEN VAN WARE-GROOTTEMETINGEN EN BEREKENINGEN VOLGENS GEPUBLICEERDE FORMULES)
by
N. J. VISSER
IR. C. DRAYER IR. N. DJJKSHOORN
DRS. C. A. M. VAN DER LINDEN PROF. DR. IR. J. D. VAN MANEN DR. IR. R. WERELDSMA
IR. A. DE
Moor (ex officio)
VOORWOORD
De snelle ontwikkeling van de langzaamlopende scheeps-hoofdrnotor naar eenheden met groot vermogen en de ten-dens naar de toepassing van korte asleidingen in moderne schepen, hebben geleid tot cen trillingsgedrag van het
as-systeem, dat aanmerkelijk ingewikkelder is dan bij motor-installaties in een betrekkelijk recent verleden het geval was.
Deze complicatie is voornamelijk bet gevolg van het op-treden van al of niet gekoppelde torsie- en axiale trillingen van bei assysteem bij toerentallen in het draaigebied van de motor.
Teneinde ontoelaatbare extra mechanische belastingen van het assysteem, die als gevoig hiervan kunnen optreden te voorkomen, moeten in bet ontwerpstadium doelmatige maatregelen worden genornen. In verband hiermede is bet noodzakelijk een betrouwbare voorspelling van het trillings-gedrag te kunnen doen.
Na uitvoerig theoretisch- en modelexperimenteel onder-zoek, werd een rekerimethode ontwikkeld, die rekening hou-dend met de koppelingseffecteri van de krukas, tot de voor-spelling van de eigenfrequenties van bet assysteem leidt.
Deze methode werd beschreven in rapport No. 39 M:
"Crankshaft coupled free torsional-axial vibrations of a ship's propulsion system".Een nog niet afdoende opgelost vraagstuk, dat zieh hierbij voordoet, is het elastisch gedrag van de krukas.
Door verschillende onderzoekers zijn theoretisch en/of experimented afgeleide formules gepubliceerd, die dit elas-tisch gedrag beschrijven.
Teneinde deze formules op bun bruikbaarheid te toetsen werden aan een aantal verschilende krukassen metingen verricht. De resultaten van dit experimented onderzoek wer-den vergeleken met de uitkomsten van een aantal in de lite-ratuur versehenen formules. Tevens werden de uitkomsten van de formules onderling vergeleken.
Uit het vergelijkend onderzoek moet worden
geconclu-deerd dat de verschillen tussen de uitkomsten van de formules te groot zijn orn deze als betrouwbaar te heschouwen. Tevens
is gebleken dat geen der gebruikte formules een goede over-eenkomst met de resultaten van bet experimented onderzoek vertoont.
Voor cen aanvaardbare oplossing van bet probleem zal het in dit rapport voorgestelde verder onderzoek noodzake-lijk zijn.
NEDERLANDS SCI-SEEPSSTUDIECENTRUM TNO
PREFACE
The rapid development of the low speed marine diesel engine
to highly-powered units, and the tendency towards applica-tion of short shaftlines in modern ships, have led to a vibra-tion pattern of the shaft system which is more complicated than that of propulsion systems used in relatively recent
years.
At speeds in the running range of the modern engine, the torsional and axial vibrations of the shaft system will either be coupled or not and, therefore the pattern of vibrations is complicated accordingly.
In order to avoid inadmissible extra mechanical loads on the shafts that will be due to this phenomenon, suitable pre-cautions should be taken in the design stage.
In this connection, a reliable prediction of the vibratory behaviour of the shaft system is required.
Extensive theoretical and model experimental investiga-tions have resulted in a calculation method which takes into account the coupling effects of the crankshaft and thus the natural frequencies of crankshaft coupled torsional-axial vibrations of the propulsion system can be predicted. The
method is published in Report No. 39 M: "Crankshaft
coupled free torsional-axial vibrations of a ship's propulsionsystem".
A problem which has not yet been solved sufficiently, however, is the elastic behaviour of the crankshaft, although several authors have published theoretically and/or experi-mentally deduced formulae for this behaviour.
In order to verify the suitability of some of these formulae,
measurements were carried out on a number of different crankshafts. This report compares the results of the experi-ments with those of the formulae.
It also compares the results of the formulae. In this
manner, an indication is obtained that the formulae used are not reliable enough to justify their practical application for crankshafts of different shapes and sizes.In the author's opinion, further investigations as suggested in this report will have to be carried out in order to solve the problem of elastic behaviour of crankshafts.
page
Summary
7I
Introduction
72
The experimental work
72.1 Crankshaft 1 9 2.1.1 Results g 2.2 Crankshaft 2 9 2.2.1 Results g 2.3 Crankshaft 3 io 2.3.1 Results 10 2.4 Crankshaft 4 10 2.4.1 Results io
3
Discussion of the results
114
Theoretical work
Ii4.1 Calculation of crank elasticity Il
4.2 Formulae used Il
4.2.1 Formula of Dorey 12
4.2.2 Formula of Draminsky and Warning 12
4.2.3 Formula of Andersson, Olsson, Gustavsson and Brämberg 13
4.2.4 Formula of Guglielmotti and Maciotta 13
4.2.5 Formula of Johnson and Mc.Climont 13
4.3 Comparison of the formulae 13
5
Comparison of calculated and measured elasticities
145.1 Crankshafts 14 5.2 Cranks 18 5.2.1 Crankshaft 1 18 5.2.2 Crankshaft 2 18 5.2.3 Crankshaft 3 19 5.2.4 Crankshaft 4 19
5.3 Comparison of calculated elasticities 19
6
Results of calculations and measurements of cranks
207 Conclusions 20
8
Future work
209 Acknowledgement
20LIST OF SYMBOLS
a angle between adjacent cranks (0 < a < 1800)
a- a between crank i and 1+ 1
aa,,
(a - a,) /2
y crank elasticity
y,
yofcranki
à
av a,,,,
rotation about the axis perpendicular to the crank plane
AA penetration factor according to ref. 3
penetration factor according to ref. 4
a journal length
A area of crankspin section A,,, area of web section
(=ol,2,3) A,,. of a section at 0.5 R, 0.75 R and 1.0 R of shaft centre line
B a/El,,
B,, h/3E1,, B,, p/El,,
B,,, R/EI,,,,
d diameter of journal and crankpin
E Young's modulus
G modulus of rigidity
h web thickness at 0.75 R of shaft centre line
I,, moment of inertia of journal
l
moment of inertia of crankpinI,(j=l,2,3) moment of inertia of crank web at 0.5 R, 0.75 R and
1.0 R of shaft centre lineIc1 (1< i g) dimensionless factor dependent on a
n number of cranks p length of crankpin
p,, p dimensionless factors independent on a
P,, axial force
THE 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
by
N. J. VISSER
Summary
The measurement of the axial stiffnesses of the cranks of foui full scale crankshafts, supported by the engine main bearings, is described. The results are discussed and compared with those of the calculations using six published formulae.
The relationship between the coefficients and the angles of adjacent cranks, or the "mean" crank angle, is investigated.
The experimental results are verified with the theoretically determined shear force moments for axially loaded adjacent cranks.
The experimentally and theoretically determined stiffnesses are compared and finally proposals are made for future work.
i
Introduction
Practical experience has shown that the increasing
main engine outputs required for the propulsion of
modern motorships, attended with the decreasing
number of cylinders of low speed directly coupled
engines, as well as the tendency towards
applica-tion of a short length shafting between the engine
and the propeller, may result in the occurrence of
complicated vibration phenomena in the shafting
system.
The dynamic properties of the shafting, the
elastic behaviour of the bearings and the vibratory
components of the engine and propeller forces may
cause inadmissible torsional and axial vibrations
to occur at speeds in the manoeuvring range of the
engine.
Due to the heavy weights of reciprocating parts
of large bore engines, the consequences of the
dependence of the moment of inertia of these parts,
relative to the crankshaft centre line, on the
rota-tion angle, as described by DRAMINSKY, is becoming
more important.
In Report No. 39 M: "Crankshaft coupled free
torsional-axial vibrations of a ship's propulsion
system" [7], the results of a theoretical- and model
experimental investigation of crankshaft coupled
free torsional-axial vibrations are described. In the
said report, a method is developed to predict the
natural frequencies and modes of these coupled
vibrations in the design stage.
Other authors also developed calculation
meth-ods for the natural frequencies of axial vibrations.
The calculation procedure of these natural
fre-quencies is practically analogous to the classical
calculation of torsional vibration frequencies,
ex-cept for an important difference due to the
rela-tionship of the axial stiffness of a crank and the
angle between the crank and other, especially the
adjacent, cranks.
The determination of the axial stiffness of a
crank from the axial stiffness of the crankshaft, and
vice versa, cannot be effected in a manner similar
to that for torsional stiffnesses.
This is due to the fact that the axial stiffness is
not only dependent on the axial deformation caused
by an axial force for a clamped crank, but that it
is also affected by other influence factors. These
influence factors are related to the angle between
the crank under consideration and the adjacent
cranks, and are also affected by the bearing
stiff-nesses.
To predict the dynamic behaviour of the shafting
system, accurate data of the elastic properties of
the engine crankshaft are required. In order to
attain better understanding of the axial stiffness of
a crankshaft, experimental investigations were
car-ried out on four marine diesel engine crankshafts.
The results were compared with those calculated
with theoretically and/or experimentally developed
formulae given in refs. i to 5.
The formula of POOLE, given in ref. 6, was not
used because this formula only applies to
crank-shafts with angles between adjacent cranks of l8O.
2
The experimental work
Measurements were carried out on the four
crank-shafts shown in Figs. 1, 2, 3 and 4. Dimensions and
8
ahead
For all shafts: E = 2.1 x 10 kgf cm-2 G = 0.835x 106 kgf cm-2
p/2
ahead
Item Crankshaft i Crankshaft 2 Crankshaft 3 Crankshaft 4 Dimension
a 40.5 37 50 40.5 cm 1 2380 2780 3020 2380 cm2 .1,,,, 3040 3610 4050 3020 cm2 'w2 2600 3610 3650 2330 cm2 Ba 42.9 29.6 32.8 42.9 l0-'2/kgf cm B5 11.1 7.7 8.9 10.4
l0'2/kgfcm
B 45.6 29.6 28.0 42.9 1012/kgfcm B6.1 133 188 67 142l0'2/kgfcm
B2
194 188 74 218 10 12/kgfcm B6,3 235 188 242 270 lO-22/kgfcm d 55 59 62 55 cm h 31.5 29 40.5 29.5 cm 'a 449 595 72.5 44.9 10 cm4 44.9 59.5 72,5 44.9 10 cm4 It,,, 25.1 17.2 20.3 20.3 55.4 49.8 26.0 16.9 l0 cm4 10 cm4 'Wa 14.2 20.3 15.2 13.6 10 cm4 n 10 9 9 6 p 43 37 42.6 40.5 cm R 70 80 77.5 77.5 cmFig. i Crankshaft I Fig. 3 Crankshaft 3
h p/2 ahead
i ig. 2 Crankshaft 2 Fig. 4 Crankshaft 4
The results mentioned below are those from
mea-surements at maximum load, as the inaccuracy of
these measurements is relatively small.
It is realized that the experimental determination
of the axial stiffness on a non-rotating crankshaft
does not fully represent the elastic behaviour of a
rotating crankshaft, since rotation implies a
con-siderable difference of bearing conditions. Although
the influence of the bearings on the elastic
prop-erties of the crankshaft is known, the available
number of theoretical and experimental data of
bearing properties is not sufficient to evaluate this
influence.
2.1
Crankshaft I (Fig. I)
Measuring was done on board the m.s. Scheide
Liqyd. The axial loads, applied at the free end by
means of a hydraulic jack, pressed the crankshaft
against the astern pads ofthe thrust block. Expressed
in tons, the successive axial loads were: 1, 5, 1, 10,
1, 15, 1. The total axial deformation of the shaft
was calculated from three measured displacements,
namely: free end with respect to engine bed plate,
bed plate with respect to the ship's structure, and
the ship's structure with respect to rear end of
crankshaft.
The axial deformation of cranks 4, 5, 7 and 10
was determined by measuring the decrease of the
web span at the centre line. The measurements
were repeated.
2.1.1 Results
The results are given in Tables 3 and 4, and shown
graphically in Fig. 6.
Cranks 4, 5 and 7, with aav = 108°, and crank
10, with aav = 180°, were measured individually.
From the total elasticity, y = 62. 10-e mm/ton,
follows the mean value of crank elasticity: 6.2. l0
mm/ton.
The investigated cranks show elasticities that
equate respectively with only 56%, 57%, 60% and
83% of the mean value.
It is considered to be impossible that the other
cranks would have any much greater elasticity. It
has not been possible to decide either that the
measured values of the individual cranks are too
low or that the measured total elasticity of the
crankshaft is too high.
The inaccurate results may be due to bad
ac-cessibility of the crankcase.
It will be shown later that a somewhat better
correspondence exists between the measured and
the calculated value of the total elasticity than that
between those values for one crank.
2.2
Crankshaft 2 (Fig. 2)
These measurements were carried out in the
work-shop on an engine with dismantled reciprocating
parts.
The axial load was applied by a hydraulic jack
and transmitted to the shaft by a rod in the bores
of journal and thrustshaft. The axial load was
raised to 8 tons, and reduced to zero, in three
un-equal steps.
The load was measured by means of a strain
gauge dynamometer.
The axial displacement with respect to the floor
was measured on both sides of the crankshaft.
Two different types of instruments were used to
determine the axial deformation of the cranks,
namely: dial gauges with an accuracy of 1 0
mm,
with which the reduction of the web span of a
crank was measured, and HOTTINGER inductive
displacement pick-ups, with which the reduction
of the distance between the webs of adjacent cranks
was measured. On one crank, both methods were
used. All measurements were carried out for three
different crankshaft positions.
2.2.1 Results
The results are summarized in Tables 3 and 5 and
graphically presented in Fig. 7.
The average elasticity of a crank, 5.6. 1 0
mm/
ton, is calculated from the measured total stiffness
of 19.4 ton/mm.
The elasticities of the cranks appeared to be:
125%, 113%, 91%, 79%, 108%, 69%, 88%, 99%
and 96% of the mean value.
The average of these percentages is 96%, and
the difference of 4% is within the accuracy of the
measurement; this was demonstrated by the result
of two measurements for one crank showing a
dif-ference of 7%.
The measurement on crank 5 was carried
out
with a dial gauge as well as with
an inductive
dis-placement pick-up.
The difference between the results caused by
the axial elasticity of the journals viz. 0.0645. l0
mm/ton is too small to be measured, even if this
figure has to be increased by adding
a part of the
crankweh thickness to the length of the journal.
This corresponds with the fact that the measured
difference is 2% and therefore not significant.
The measurements were repeated for some cranks
in two crankshaft positions differing +120 and
.240 from the original position. Cranks 5 and 9
then showed differences in the order of 10%.
lo
It is reasonable to suppose that this difference is
to be attributed to the influence of the bearing
position. However, considering the accuracy of the
measuring method, it is also reasonable to suppose
that this difference is not significant either.
2.3
Crankshaft 3 (Fig. 3)
These measurements, carried out in the workshop,
were on an engine with dismantled reciprocating
parts. The bearing keepers were not fixed. The
axial load was applied by a hydraulic jack with a
built-in load pick-up.
The reaction of the load was supplied by the
astern pads of the thrustblock.
Four series of measurements were carried out,
the loads being 0, 7, 7, 0; 0, 14, 14, 0; 0, 21, 21, 0
and 0, 28, 28, 0 tons.
Between two equal successive loads, the load was
increased with 1 ton; thus the first measurement
was carried out with increasing load and the second
with decreasing load. The measurements were
repeated for two other shaft positions differing 1200
and 240° from the original position.
The compression of the crankshaft was
deter-mined by measuring the displacements with respect
to the floor at the forward and the rear end. This
measurement was carried out by means of dial
gauges located at the centre line with an accuracy
of 10-2 mm.
Further measurements were carried out on some
separately loaded cranks.
The compression of the cranks was measured
with dial gauges with an accuracy of 10
mm,
located at the centre line between the crankwebs.
All cranks were measured.
2.3.1 Results
The results are summarized in Tables 3 and 6, and
shown graphically in Fig. 8.
A simple statistical analysis of the results shows
an accuracy of about 5%. This accuracy was not
obtained for the measurements of the cranks.
The mean elasticity of a crank is 3.7 l0- mm/
ton; this value is calculated from the measured
total stiffness of 29 ton/mm.
The elasticities of the cranks appeared to be:
132%, 116%,93%,78%, 103%,84%,94%, 113%
and 111% of the mean value. The average, 102%,
does not differ significantly from the measured
mean value.
The measurements carried out at different shaft
positions do not show any difference in total
stiff-ness.
The differences shown by the cranks at different
shaft positions may be significant.
The measured elasticities of crank 6, which
showed the greatest difference, were 29, 3.1 and
3.4 l0
mm/ton. This means a mean deviation
from the mean value of 3.13. l0
mm/ton, which
is about 10%.
The results of additional measurements, carried
out with cranks 2 and 5 loaded individually and
set in T.D.C. in turn, showed a somewhat higher
stiffness than the results of the measurements with
the whole crankshaft.
Crank 5 showed no significant difference; crank
2, however, showed a difference of 13%, which may
be significant.
2.4
Crankshaft 4 (Fig. 4)
The axial load was applied by means of a
canti-lever and a jack.
Measurements were carried out in the workshop,
with axial loads of 1, 5, 10, 1, 15, 1 tons. The load
was determined with a dynamometer located
between the cantilever and the crankshaft. The
reaction of the load was supplied by the astern
pads of the thrust block. For each load the
measure-ments were repeated four times. The crankshaft
was turned 120° between any two measurements,
which means that the fourth reading is a check on
the first.
The accuracy of the dial gauges used was 1 0
mm.
The displacements with respect to the floor, as
well as the change of the web span of each crank,
were measured. Displacements of the shop's floor
due to the load could not he determined.
2.4.1 Results
The results are summarized in Tables 3 and 7, and
Fig. 9 shows them graphically.
The elasticity of the crankshaft appeared to be
47. 10
mm/ton. Because the accuracy of the
ap-paratus used is only 10
mm, the accuracy of the
results of the crank measurements is small. The
displacements of the cranks were not measured at
the centre line, but at the outer sides of the webs.
The measuring method is to be considered
insuf-ficient.
The differences of the readings obtained in the
different positions of the shaft are not significant.
The elasticities of the cranks are 90%, 90%,
80%, 80%, 80% and 80% of the mean elasticity
3
Discussion of the results
The results of the measurements have demonstrated
differences between the elasticities of the cranks of
a crankshaft. This is in agreement with the theory
which postulates an effect of crank angles on
elas-ticity. Therefore, in most published formulae, the
crank elasticity is related to the mutual angles of
the cranks.
In ref. 7 a method is developed in which the
mutual influence of the cranks is considered. Other
authors give approximate formulae in which the
average of the angles between the crank under
consideration and the adjacent cranks is used.
The total elasticity of the crankshaft should
equal the sum of crank elasticities, elasticities of
journals and elasticities of cylindrical parts. The
latter elasticities, which are easy to compute,
ap-pear to be very small compared with the crank
elasticities.
The measurements carried out on crankshafts 2
and 3 show good agreement between the overall
elasticity and the crank elasticities, and the same
applies to a lesser extent to crankshafts i and 4.
Therefore, the results of the latter measurements
should be considered less valuable.
The theory shows that an influence of the shaft
position can exist, since the bearing forces which
affect the crank elasticities depend on the shaft
position.
Summarizing it may be stated that the results
obtained have clearly shown a difference of the
crank stiffnesses; the influence of shaft position on
crank stiffness, however, appears to be less evident.
4Theoretical work
In this chapter, the elasticities of the cranks of the
shafts investigated will he calculated with the
for-mulae mentioned in refs. 1, 2, 3, 4 and 5.
The correspondence between measured and
cal-culated differences in crank stiffness will be
exam-ined.
4. 1 Calculation of crank elasticity
Application of an axial load in the centre line of
the crankshaft is only an approximation of the load
occurring with axial vibrations. In case of axial
vibrations exciting forces act on the crankpin, and
volume, inertia and damping forces occur
every-where in the crank.
The axial substitute, load Pa, in the centre line
of the journal produces a bending moment, PaR,
in the crankpin. The deformation of the crankpin
will penetrate in the crank-webs, which means that
the effective crankpin length is larger than p.
The crank-webs behave as free bars fixed at the
crankpins. The journals and the crankpins are
compressed, and the webs are loaded by shear
forces.
The journal will perform an angular rotation.
All the authors, mentioned in the references,
sup-pose the bearing clearance to he large enough to
permit free rotation.
The problem becomes more complicated, if one
considers two cranks connected at an angle a = 26.
An axially loaded free crank will show an angle of
rotation ç about an axis perpendicular to the plane
of the crank, at both ends. If two connected cranks
are supposed to be disconnected, the rotations in
the centre of the connecting journal can be resolved
in rotations p sin â about an axis in the bisecting
plane of the cranks, and in rotations ç cos â about
an axis perpendicular to it. The rotations
sin âhave no effect, since they have the same direction.
The rotations q cos â, however, have opposite
direc-tions. Reconnecting necessitates the cranks to be
turned back about the angle q cos â. This results
in a displacement of the end of l
cos 6, and thus
the journal is misaligned [7].
Consequently, stiff bearings will produce forces
which try to restore the alignment and elastic ones
will produce forces which try to improve
align-ment. These forces produce a bending moment in
the centre journal which is proportional to cos â,
as has been shown above. Therefore, this bending
moment is maximal if the cranks have the same
directions and zero for opposite directions.
Most authors take into account this bending
moment in their calculations of crank elasticity.
The problem is more complicated if more than two
cranks are considered. Bearing forces and internal
moments, which influence the cranks in further
locations of the shaft, may occur everywhere.
Most authors consider the effect of the adjacent
cranks only, neglecting the influence mentioned
above.
The authors of ref. 7, however, take the influence
into account in an exact manner.
4.2 Formulae used
Five of the six formulae mentioned in refs. 1, 2, 3,
4, 5 and 6 were used, since the formula of PooLE
[6] has been developed under the assumption that
the internal moment is known.
To enable comparison of results, the formulae
12
Since the constant factors are dimensionless, all
formulae give the axial elasticity in any
appropri-ate unit.
According par. 4.1, the formula of the axial
elasticity of a crank should at least contain the
angles between the crank under consideration and
the adjacent cranks. The authors of refs. 1, 2, 3, 4
and 5 use only one angle in their formulae, with
some or other assumption about the way in which
the effect of this crank, or of two half cranks, is
incorporated in the crankshaft. Therefore, the
for-mulae in the given form cannot be used for
calcula-tion of the elasticity of a crank making different
angles with adjacent cranks.
Even if the adjacent angles are equal, the result
can only be approximate because, firstly, it is
im-portant to know whether the angles have the same
or opposite directions and, secondly, because the
influence of other cranks cannot be neglected.
In the calculations, the effects of the angles were
taken into account in two ways, namely:
The elasticity was calculated on the basis of the
average angle and
The elasticity was calculated for each angle,
and the average of the elasticities was
deter-mined.
If the elasticities are used to calculate axial
vibra-tions, the first method is more convenient if the
lumped masses of the crankshaft are considered to
be concentrated in the journals.
If the lumped masses are considered to be
con-centrated in the crankpìns, as is usual for the
cal-culation of torsional vibrations it is worthwhile to
use the second method and to reckon with the
elasticity of each pair of adjacent half cranks
ac-cording to the enclosed angle.
It should be noted, however, that the
recipro-cating parts only very slightly participate in the
vibratory system and, that therefore, the
webs are the most important masses of the
crank-shaft.
Concentration of the masses in the webs, which
would be attractive from a theoretical point ofview,
is not recommended for reasons of computing
tech-nique.
The values of the bending elasticities of the
various crankparts used in the formulae are given
in Table 1.
B represents the angular rotation of the
crank-pin, fixed at one end, due to unit moment.
In ref. 2, an effective length equal to the actual
length plus one third of the web thickness is used.
The formula therefore contains a term Bh as an
additional elasticity.
Other authors [1], [3], [5] introduce a coefficient
for this purpose. In the calculation the actual length
must be used.
B
is the angular rotation of a bar with the
cross-section of the crank-web and a length equal
to the crank throw-radius due to unit moment.
Since the cross-section is not constant, an index
1, 2, 3 is added to B, if the cross-section under
consideration is at 0.5, 0.75 or i times the crank
throw-radius from the shaft centre line.
The authors of refs. 2 and 3 use correction terms
for the shearing of the webs and for the
compres-sion of crankpin and journal.
A then represents the area of the cross-section
of the crankpin, A, with an index, that of the web.
The authors of refs. 3 and 4 use a reduced length
of the web. This length reduction depends on the
crankpin diameter and the crank throw-radius.
The reduction is represented by ..
4.2.1
Formulae of DOREY [1]:
= R(pik1B+/c2B1)
where:
pi = 1.65 for fully-built crankshafts,
P2 = 1.625 for semi-built crankshafts,
k1 = 0.5 +
3Ç
and k2 = 0.1667 +
360
Allowance is made for the influence of the adjacent
cranks by linear interpolation between the extreme
positions.DOREY recommends to use an average angle for
the whole shaft. However, in this calculation the
formula is used for each crank separately.
4.2.2 Formula of DRAMINSKY and WARNING [2]:
f
k3\
k3\1= R
(Bp+Bh)
l-
+ B2(
-
+
p+a
2R
+
EA+
GA2
where:
k3 = i + cos a.
Bh is the elasticity of a shaft with a length of one
third of the width of the web, and having the same
cross-section as the crankpin.
For small values of a, the coefficient of B2 appears
to be negative. However, since the said authors use
a mean value for a, a negative contribution to the
elasticity of the webs will not occur in practical
calculations. In the calculation, the formula is used
4.2.3 Formula of ANDERSSON, OLSSON, GUSTAVS-so and BRÄMBERG [3]:
=
R2(P2k4B9 + (2A2k4 - /3AA3)
B1}
+
2(Rd)
EA9 GAW1where:
P2 = 1.1 for fully and semi built crankshafts,
k4 = 0.75-0.25 cos a,
= 1-0.28 dIR.
If the diameters of crankpin and journals are
un-equal, an average value can be used.
In this formula, all known effects, namely:
effec-tive crankpin length (pz), reduced web length (AA),
compression ofjournals and crankpin, and shearing
of the web, are incorporated.
4.2.4
Formula of GUGLIELMOTTI and MACLOTTA[4]:
= R2(ksBp+0.6Ag2ksBw3)
k5 is a factor shown graphically in Fig. 5; it almost
equals 2/3_1/3 COS a.1,0
r-0,5
Fig. 5 Relationship between k5 and a5
If the thicknesses of crankpin and journal are
not equal, an average value can be used.
4.2.5
Formula of JOHNSON and MCCLIMONT [5]:= R2(k6Ba+k7Bp+k8Bwi)
where:
k6 = 1/6 cos âk7 = l-1/6cosâ
k8 = 2/3_1/6 cos â
The formula is valid for two half cranks, the angle a
between the cranks being 2ò.
in this formula occurs the bending elasticity of
the journal; further it should be noted that in it
occurs cos â, in stead of cos a, which is used in the
other formulae.
4.3 Comparison of the formulae cited above
The coefficients of B9 and B50, for a = 0° and
a = 180°, in the several formulae are shown in
Table 2.
0° 30° 600 90° 120°
14
Table 2 Coefficients ofB and B0,
These figures show that the formula of
Gugliel-motti and Maciotta will yield the lowest elasticity.
It should be noted that the comparison is not
com-plete in that the formulae of Andersson and
Gu-glielmotti use a shorter length of the crankweb and
the formula of Draminsky represents the effective
length of the crankpin by a term.
The less significant correction for the change in
length of crankpin and journal and that for the
shearing of the web are not considered. All authors,
except R. POOLE [6], whose formula is not used in
this study, suppose the elasticity of the crankshaft
to be equal to the sum of the elasticities of the
various parts.
Consequently, the bending elasticity of the
journal does not occur in most formulae. Since,
however, bending moments do occur in thejournal,
journal elasticity plays a rôle too.
o E E o ¶ 12 10 8 6 4 2 o
Generally speaking it may be stated that no
more than the elasticity of a crankshaft is equal to
the sum of the crank elasticities, does the elasticity
of a crank equal the sum of elasticities of its parts.
Accordingly, from a theoretical point of view,
good agreement between calculated and measured
elasticities cannot he expected.
5
Comparison of calculated and measured
elasticities
The results of the measurements and the
calcula-tions of the crankshafts are shown in Table 3; those
of the cranks in Tables 4, 5, 6 and 7 and in Figs.
6, 7, 8 and 9.
This chapter compares the measured values with
those calculated, using the mean crank angle.
Furthermore are considered the differences
be-tween results of calculations using the mean crank
angle and the averages of results of calculations
using the angles of adjacent cranks.
5.1 Crankshafts
In the formulae of Dorey and Guglielmotti, the
elasticity ofjournals does not occur. In the formula
of Johnson, the bending elasticity occurs, but not
the axial elasticity. Therefore, with these formulae
the total elasticity of the crankshaft was computed
by adding the axial elasticity of the journals to the
results of the formulae. For all formulae, the
cal-Coefficient of B0 (00) B (180°) B (0°) B,,, (1800) Dorey 0.825 1.65 0.167 0.667 Draminsky and Warning 0.833 10.333
0.667 Andersson, Olsson, Gustavsson and Brämberg 0.55 1.1 0.167 0.667 Guglielmotti and Maciotta 0.33 1 0.200 0.600 Johnson and McClimont 0.833 1 0.500 0.667I.ZIT
2 3 4 5 6 7 8 9 10 - crsnkFig. 6 Crankshaft 1: Results of measurements and calculation
Draminsky Dorey Johnson Andersson Gu9lielrnotti Measurement O with journal without journal
12 o E E 10 C) 8 6 4 2 o 10 o E E a)
i:
culated axial elasticities were increased by those
of the cylindrical parts.
Table 8 gives the calculated figures of the
crank-shafts. These figures are expressed in percentages
of values measured. The mean deviation was added,
as far as this could be estimated from the few
measuring results available, with the formula:
S2(nl) =
(xj)2
where:
S = mean deviation
= calculated crankshaft elasticities in
percen-tage of values measured
n = number of items
Draminsky Dorey NJohnson A ride raso ri Measurement with journal without journalN
- . .yxj
x = arithmetic average =
.. -,--,--
NN__ /
/
AN
2 3 4 5 6 7 8 9 crankFig. 7 Crankshaft 2: Results of measurements and calculation
2 3 4 5 6 7 8 9
crank Fig. 8 Crankshaft 3: Results of measurements and calculation
6 4 2 o Draminsky / Dorey
/
/
/
Johnson Ande rcson Guglielmotti M casu re me n t ;:- with journal without journalFig. 9 Crankshaft 4: Results of measurements and calculation
Calculated total elasticity of the journals: 0.81 1 0- mm/ton
Calculated elasticity of the other cylindrical parts: 0.33. 10 mm/ton
In the results of the formulae of Dorey, Guglielmotti and Johnson, the elasticities of the journals have been incorporated. For all formulae, the results were increased by the elasticities of the other cylindrical parts.
Table 4 Measurcd and calculated elasticities of the cranks (without journals) of crankshaft I in l0- mm/ton
PT!2
Crankshaft Measured
Formulae
Dorcy Draminsky Andersson Guglielmotti Johnson
63.6 66.5 77.5 51.1 44.4 62.1 66.5 68.5 48.2 41.4 62.9 2 51.5 74.7 60.2 57.4 37.3 79.4 74.7 59.6 57.2 36.8 80.4 3 34.6 37.7 37.2 29.6 38.1 37.0 37.7 37.1 29.3 37.6 37.1 4 47.0 47.2 55.3 37.5 36.0 46.2 47.2 53.7 37.1 35.4 46.3 Crank
num eb r a Cat, Measured
Formulae
-l)orey Draminsky Andersson Guglielmotta Johnson
180° 180°
-
7.95 9.84 5.70 5.19 6.56 180° 7.95 9.84 5.70 5.19 6.56 2 126°-
6.44 7.76 5.03 4.46 6.07 72° 6.44 6.59 4.65 4.13 6.11 3 126°-
6.44 7.76 5.03 4.46 6.07 180° 6.44 6.59 4.65 4.13 6.11 4 108° 3.48 5.93 6.40 4.59 4.02 5.91 36° 5.93 4.34 4.25 3.61 6.04 5 108° 3.58 5.93 6.40 4.59 4.02 5.91 180° 5.93 5.34 4.25 3.61 6.04 6 108°-
5.93 6.40 4.59 4.02 5.94 36° 5.93 5.34 4.25 3.61 6.04 7 108° 3.74 5.93 6.40 4.59 4.02 5.91 180° 5.93 5.34 4.25 3.61 6.04 8 126°-
6.44 7.76 5.03 4.46 6.07 72° 6.44 6.59 4.65 4.13 6.12 9 126°-
6.44 7.76 5.03 4.46 6.07 180° 6.44 6.59 4.65 4.13 6.12 10 180° 5.18 7.95 9.84 5.70 5.19 6.56 180° 7.95 9.84 5.70 5.19 6.56 2 3 4 5 6 - crankTable 3 Measured and calculated values of the overall elasticities in l0- mm/ton
(The upper figures refer to the calculation with ear, the lower figures to the calculation with a)
16 12 o E E o >-C) X lo 8
Calculated total elasticity of the journals: 0.58- lO-° mm/ton
Calculated elasticity of the other cylindrical parts: 0.85- lO mm/ton
Table 6 Measured and calculated elasticities of the cranks (without journals) of crankshaft 3 in lO° mm/ton
Calculated total elasticity of the journals: 0.71- lO-e mm/ton
Calculated elasticity of the other cylindrical parts: 0.36- l0° mm/ton
fable 7 Measured and calculated elasticitics of the cranks (without journals) of crankshaft 4 in l0- mm/ton
Calculated total elasticity of the journals: 0.49- 10-° mm/ton Calculated elasticity of the other cylindrical parts: 0.35. lO-° mm/ton
number a aav Measured Dorey Draminsky Andersson Gugliclinotti Johnson
1800 1500 6.92 9.87 10.15 7.63 4.93 9.38 120° 9.87 9.43 7.35 4.75 9.39 2 120° 6.27 8.60 7.88 6.75 4.31 8.89 120° 8.60 7.88 6.75 4.31 8.89 3 80° 5.05 6.93 3.70 5.13 3.27 8.36 40° 6.93 3.91 5.21 3.24 8.45 4 80° 4.40 6.93 3.70 5.13 3.27 8.36 ¡20° 6.93 3.91 5.21 3.24 8.45 5 120° 6.00 8.60 7.88 6.75 4.31 8.89 120° 8.60 7.88 6.75 4.31 8.89 6 80° 3.85 6.93 3.70 5.13 3.27 8.36 40° 6.93 3.91 5.21 3.24 8.45 7 80° 4.90 6.93 3.70 5.13 3.27 8.36 120° 6.93 3.91 5.21 3.24 8.45 8 120° 5.50 8.60 7.88 6.75 4.31 8.89 120° 8.60 7.88 6.75 4.31 8.89 9 150° 5.33 9.87 10.15 7.63 4.93 9.38 1800 9.87 9.43 7.35 4.75 9.39 cranK
number a aav Measured
Formulae
Dorey Draminskv Andersson Guglielmotti Johnson
180° 1 150° 4.93 4.84 5.43 3.80 5.08 4.19 120° 4.84 5.14 3.65 4.90 4.19 2 120° 4.32 4.28 4.54 3.38 4.44 4.04 120° 4.28 4.54 3.38 4.44 4.04 3 80° 3.48 3.53 2.92 2.70 3.38 3.87 40° 3.53 3.02 2.76 3.35 3.90 4 80° 2.92 3.53 2.92 2.70 3.38 3.87 120° 3.53 3.02 2.76 3.35 3.90 5 120° 3.83 4.28 4.54 3.38 4.44 4.04 120° 4.28 4.54 3.38 4.44 4.04 6 80° 3.14 3.53 2.92 2.70 3.38 3.87 40° 3.53 3.02 2.76 3.35 3.90 7 80° 3.51 3.53 2.92 2.70 3.38 3.87 120° 3.53 3.02 2.76 3.35 3.90 8 120° 4.23 4.28 4.54 3.38 4.44 4.04 120° 4.28 4.54 3.38 4.44 4.04 9 150° 4.14 4.84 5.43 3.80 5.08 4.19 180° 4.84 5.14 3.65 4.90 4.19 ranK
number a ea5 Measured
Formulae
Dorey Draminsky Andersson Guglielmotti Johnson
180° 150° 8.82 8.79 11.88 6.98 6.80 7.89 120° 8.79 11.08 6.74 6.55 7.91 2 120° 7.35 7.73 9.38 6.22 5.94 7.55 120° 7.73 9.38 6.22 5.94 7.55 3 90° 6.08 6.68 5.97 5.19 4.85 7.26 60° 6.68 5.97 5.19 4.85 7.29 4 900 5.88 6.68 5.97 5.19 4.85 7.26 120° 6.68 5.97 5.19 4.85 7.29 5 120° 6.90 7.73 9.38 6.22 5.94 7.55 120° 7.73 9.38 6.22 5.94 7.55 6 5.71 8.79 11.88 6.98 6.80 7.89 180° 8.79 11.08 6.74 6.55 7.91
18
Table 8 Calculated crankshaft elasticities in percentages of values measured
These results make a chaotic impression. Crankshaft
3 gives the best results; this could be expected from
the measuring method applied to this crankshaft.
The results of crankshaft 2 are bad, in spite of the
accuracy of the method. This might be due the fact
that crankshaft 2 is fully-built, while the formulae
are possibly more appropriate to semi-built
shafts.Only Dorey's formula distinguishes both types, but
the difference is no more than about 1%.
The results indicate that, whichever formula is
used, the discrepancy between calculated and
mea-sured values is in the order of 20%. The results of
semi-built crankshafts 1, 3 and 4 are considerably
better.
The formulae of Dorey and of Johnson and
McClimont give reasonable approximations for
these shafts.
The practically developed formulae seem to be
superior to those developed theoretically.
5.2 Gran/cs
In the calculation of the elasticities with the
for-mulae of Draminsky and Andersson the term
con-cerning the journal elasticity was neglected, because
nearly all the experimental data refer to
measure-ments of stiffnesses of cranks not including journals.
Tables lO, 11, 12 and 13 show the proportional
Table 9 Mean values of the relationship per calculation method
relationships between calculations and
measure-ment results.
Table 9 shows, for each crankshaft, the mean
values of the relationship per calculation method.
Except for crankshaft 1, of which not all cranks
were calculated, the results show a good agreement
with the results of the whole crankshaft. This could
be expected, since both for the calculation and for
the measurement the elasticity of the crankshaft is
closely related to the sum of the crank elasticities.
The figures of the crankshafts mentioned in
Tables 10, 11, 12 and 13 show much more
varia-tion of the relavaria-tionships between the calculated and
measured values.
5.2.1
Crankshaft i
Only four cranks were measured, viz, nos. 4, 5, 7
and 10.
The agreement between cranks 4, 5 and 7 is due
to the same mean value of the crankangles of these
cranks and, therefore, the same calculated values.
Accordingly, the differences are due only to the
differences of the measurement results.
Since all the figures mentioned exceed 100%,
and also the elasticity of the crankshaft was more
than the sum of the crank elasticities, the accuracy
of the measurements seems to he doubtful.
5.2.2
Crankshaft 2
The results indicate that the considerable
differ-ences between the measured and
calculated values
are due to the formulae used. Regarding
crank-shaft type 2 neither the mean value, nor the effect
of the position of the crank, has been evaluated
correctly by the authors concerned.
Author Crankshaft 1 Crankshaft 2 Crankshaft 3 Crankshaft 4 RMS
Dorey 105 145 109 lOO 115 22 Draminsky 122 117 108 118 116 6 Andersson 80 111 86 80 89 15 Guglielmotti 71 72 110 77 82 19 Johnson 98 156 105 97 114 28 95 121 104 94 103 13 RM 20 33 10 16 16 Author Crank-shaft i Crank-shaft 2 Crank-shaft 3 Crank-shaft 4 Dorey 162 153 107 116 Draminsky 181 118 104 134 Andersson 123 117 84 92 Guglielmotti 109 75 107 88 Johnson 154 167 106 114
Author Crank 4 Crank 5 Crank 7 Crank 10 Average RMS
Dorey 170 165 159 153 162 7
Draminsky 184 179 171 190 181 8
Andersson 132 128 123 110 123 9
Guglielmotti 116 112 107 100 109 7
Johnson 170 167 158 127 154 18
Table 11 Relationship between calculated and measured values ol crankshaft 2 in percentages
Table 12 Relationship between calculated and measured values of crankshaft 3 in percentages
Table 13 Relationship between calculated and measured values of crankshaft 4 in percentages
5.2.3
Crankshaft 3
The figures of table 12 show that the formulae
seem to be much more suitable for crankshafts of
type 3. Nevertheless, the differences found must be
due to incorrectness of the formulae, since it
isunlikely that measuring faults will exceed 10%.
Moreover, the consequence of a measuring fault
would result in a deviation of the figures of a crank
in the same direction. This, however, does not
occur.
5.2.4
Crankshaft 4
The results are affected very unfavourably by the
calculation of crank 6. In the calculation of the
first as well as that of the end crank, it is assumed
that no internal moment occurs in the end journal.
Considerations of symmetry show that no
inter-nal moment occurs between two adjacent cranks
making an angle of 180°. The mean crank angle
of the end crank is determined by the assumption
that this crank is adjacent to a crank, at an angle
of 180°.
The results of this calculation indicate that this
assumption is not always justified. Experimental
data show that an internal moment mostly occurs
at the thrustshaft side of the crankshaft.
To a lesser extent, the results of the calculations
of crankshafts 2 and 3 are also effected
unfavour-ably by the calculation of the end crank.
5.3 Comparison of calculated elasticities
As mentioned before, for all calculations of the
elasticities two different methods were followed,
viz.:
Using the mean crank angle, thus:
y =f{h/2(ai+a+1)}
Calculating the elasticity for each angle and
using the average of these elasticities, thus:
y = '/z{f(a) +f(ai+i)}
In these formulae, f is the symbol for any
appro-priate function. It is evident that, if f is
a linear
function of a, the results of both methods will be
equal. This occurs with the formula of Dorey. In
the other formulae, a occurs as cos a; except in
the formula of Johnson and McClimont, which
contains cos '/2a.
Therefore use of these formulae will yield
dif-ferences between their results.
Table 14 shows the proportional relationship
between the results of the methods mentioned, for
all the different cases of the crankshafts under
con-sideration.
The results of the formula of Dorey are not
men-tioned, because they show no difference.
Generally, the differences are small compared with
those between results of calculation and
measure-ment.
Author Crank 1 Crank 2 Crank 3 Crank 4 Crank 5 Crank 6 Crank 7 Crank 8 Crank 9 Aver-ageR1
Iiorcy 143 137 137 158 143 180 141 156 185 153 18 Draminsky 147 126 73 84 131 96 76 143 190 118 39 Andersson 110 108 102 117 113 133 105 123 143 117 14 Guglielmotti 71 69 65 74 72 85 67 78 93 75 9 Johnson 136 142 166 190 148 217 171 162 176 167 25 Author Crank 1 Crank Crank 2 3 Crank 4 Crank 5 Crank 6 Crank Crank 7 8 Crank 9 Aver-age RMS -Dorey 98 99 101 121 112 112 101 101 117 107 9 Draminsky 110 105 84 100 119 93 83 107 131 104 15 Andersson 77 78 78 92 88 86 77 80 92 84 7 Guglielmotti 103 103 97 116 116 108 96 105 123 107 9 Johnson 85 93 111 133 105 123 110 96 101 106 14Author Crank i Crank 2 Crank 3 Crank 4 Crank 5 Crank 6 Average RMS
Dorey 100 105 110 114 112 154 116
Draminsky 135 128 98 102 136 208 134 36
Andersson 79 85 85 88 90 122 92 14
Guglielmotti 77 81 80 83 86 119 88 16
20
Table 14 Relationship between the results of both methods in percentages
Both methods show no difference of the results,
whether the angle between the adjacent cranks is
equal either to a+ai+1 or to aai+i. None of the
formulae under consideration have a possibility to
incorporate the difference between these cases. It
is nevertheless evident that there should he a
dif-ference in the elasticities. This is clearly shown if a
crankshaft is considered of which all cranks have
equal angles in the same directions with the
pre-ceding crank. For reasons of symmetry, internal
moments in the journals of this crankshaft cannot
occur.
When the angles are +a and a alternately,
internal moments, with vectors in the bisecting
plane of the cranks, will occur. In neither case does
a bearing force exist. Bearing forces only
exist if theabsolute values of the angles aj and aj+i are
dif-ferent. This is another point that has not been
expressed in the formulae used.
6
Results of calculations and
measure-ments of cranks
These results are shown in Figs. 6, 7, 8 and 9.
From comparison of these figures it appears that
the results of crankshaft 3 show the best similarity.
It is difficult to decide which formula gives the
best results. All formulae show considerable
differ-ences with the mean values.
In the formulae of Draminsky and Johnson, the
effect of the crankangle is not taken into account
properly.
Obviously, in Draminsky's formula too much
and in Johnson's formula too little value is
at-tached to the influence of the crankangle.
7
Conclusions
1. The formulae given in the literature yield dif.
9 Acknowledgement
ferences between results of calculations that are
too big to consider any of these formulae
reliable. No formula, moreover, fits all the data
of measurement.
2. The calculation results of fully-built crankshaft
2 are considerably worse than those of
semi-built crankshafts 1, 3 and 4.
For accurate calculations, internal moments
and bearing forces must be taken into account.
Since these moments and forces are statically
indeterminate, any accurate calculation will be
very complicated. The bearing
elasticities
coming into the picture cannot be determined
easily. However, they are possibly not
impor-tant.
It is not certain that the effect of the
crank-angle on the elasticity can be accounted for
with acceptable approximation by one figure
(e.g. the mean value of angles between cranks).
In a simple suitable formula, possibly at least
two parameters (e.g. both signed angles) should
occur.
The accuracy of the measurements cannot be
determined with the formulae used.
8
Future work
Development of an accurate formula of the
crank-shaft elasticity requires better knowledge of the
effect of shape, as well as of mutual angles of
cranks. This can be done by:
measurements at full scale crankshafts;
measurements at crankshaft models;
measurements at cranks, or half cranks;
calculation of crankshafts consisting of cranks
whose influence numbers are known;
theoretical investigation of forces and deflexions
occurring in a crank;
theoretical and experimental investigation of
the effective bearing stiffness of a crankshaft
under service conditions.
The Netherlands Ship Research Centre TNO
ac-knowledges the cooperation of the Koninklijke
Maatschappij "De Schelde" N.y., the Dok en
Werf Maatschappij Wilton Fijenoord N.y., the
Koninklijke Rotterdamse Lloyd N.y. and the
Machinefabriek P. Smit N.y., which all enabled
the investigations here reported to be carried out.
Crankshaft CaD Draminsky Andcrsson Guglielmotti Johnson
1800 36° 108° 120 108 112 98 2 180° 120° 150° 105 104 104 100 2 120° 40° 80° 95 98 101 99 3 180° 120° 150° 105 104 104 100 3 120° 40° 80° 97 98 101 99 4 180° 120° 150° 107 108 104 100 4 120° 60° 9Q0 100 100 100 100
References
DOREY, 1939. Transactions of the North East Coast In-stitution of Marine Engineers, vol. LV, pp. 303-294.
DRAMINSKY and WARNING, 1940. Bulletin Technique du
Bureau Veras, numéro spécial. "Vibrations axiales des arbres manivelles".
1942. Motortechnische Zeitschrift, Heft 2, pp. 49-52. "Axialschwíngungen von Kurbeiwellen".
ANDERSSON, OLSSON, GUSTAVSS0N and BRÄMBERG, 1963.
International Shipbuilding Progress, vol. 10, No. 107, pp. 235-253. "Axial vibrations and measurements of
stresses of crankshafts".
GUGLIELMOTTI and MAclorrA, 1962. CIMAC,
Copen-hagen, pp. 633-692. "Recherches expérimentales pour l'étude de vibrations axiales de vilebrequins". JoHNsoN and MCCLIMONT, 1963. Transactions of the Institute of Marine Engineers, Vol. 75, No. 4, pp. 121-167. "Machinery induced vibrations".
POOLE, 1941. Proceedings of the Institute of Mechanical
Engineers, pp. 167-2 18. "The axial vibration of diesel engine crarikshafts".
VAN DORT and VIsSER, 1963. Netherlands Research Centre TNO for Shipbuilding and Navigation, Re-port No. 39 M. "Crankshaft coupled free torsional axial vibrations of a ship's propulsion system".
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I S The determination of the natural frequencies of ship vibra-tiOns (Dutch). H. E. Jaeger, 1950.
3 S Practical possibilities of constructional applications of alu-minium alloys to ship construction. H. E. Jaeger, 1951. 4 S Corrugation of bottom shell plating in ships with all-welded
or partially welded bottoms (Dutch). H. E. Jaeger and H. A. Verbeek, 1951.
5 S Standard-recommendations for measured mile and endur-ance trials of sea-going ships (Dutch). J. W. Bonebakker, w. j. Muller and E. J. Diehl, 1952.
6 S Some tests on stayed and unstayed masts and a comparison of experimental results and calculated stresses (Dutch). A. Verduin and B. Burghgraef, 1952.
7 M Cylinder wear in marine diesel engines (Dutch). H. Visser,
1952.
8 M Analysis and testing oflubricating oils (Dutch). R. N. M. A. Malotaux andJ. G. Smit, 1953.
9 S Stability experiments on models of Dutch and French stan-dardized lifeboats. H. E. Jaeger, J. W. Bonebakker and J. Pereboom, in collaboration with A. Audigé, i 952.
1 0 S On collecting ship service performance data and their analysis.
J. W. Bonebakker, 1953.
1 1 M The use ofthree-phase current for auxiliary purposes (Dutch).
J. C. G. van Wijk, 1953.
12 M Noise and noise abatement in marine engine rooms (Dutch). Technisch-Physische Dienst TNO-TH, 1953.
13 M Investigation of cylinder wear in diesel engines by means of laboratory machines (Dutch). H. Visser, 1954.
14M The purification of heavy fuel oil for diesel engines (Dutch).
A. Bremer, 1953.
15 S Investigations of the stress distribution in corrugated bulk-heads with vertical troughs. H. E. Jaeger, B. Burghgraef and
I. van der Ham, 1954.
16 M Analysis and testing of lubricating oils II (Dutch). R. N. M. A. Malotaux and J. B. Zabel, 1956.
17 M The application of new physical methods in the examination
of lubricating oils. R. N. M. A. Malotaux and
F. vanZeggeren, 1957.
18 M Considerations on the application of three phase current on board ships for auxiliary purposes especially with regard to fault protection, with a survey of winch drives recently ap-plied on board of these ships and their influence on the gene-rating capacity (Dutch). J. C. G. van Wilk, 1957.
19 M Crankcase explosions (Dutch). J. H. Minkhorst, 1957. 20 S An analysis of the application of aluminium alloys in ships'
structures. Suggestions about the riveting between steel and aluminium alloy ships' structures. H. E. Jaeger, 1955. 2 1 S On stress calculations in heliocoidal shells and propeller
blades. J. W. Cohen, 1955.
22 S Some flotes on the calculation of pitching and heaving in longitudinal waves. J. Gerritsma, 1955.
23 5 Second series of stability experiments on models of lifeboats.
B. Burghgraef, 1956.
24 M Outside corrosion of and slagformation on tubes m oil-fired boilers (Dutch). W. J. Taat, 1957.
25 S Experimental determination of damping, added mass and added mass moment of inertia of a shipmodel. J. Gerritsma,
1957.
26 M Noise measurements and noise reduction in ships. G. J. van Os and B. van Steenbrugge, 1957.
27 5 Initial metacentric height of small seagoing ships and the
inaccuracy and unreliability of calculated curves of righting levers. J. W. Bonebakker, 1957.
28 M Influence of piston temperature on piston fouling and piston-ring wear in diesel engines using residual fuels. H. Visser,
1959.
29 M The influence of hysteresis on the value of the modulus of rigidity of steel. A. Hoppe and A. M. Hens, 1959.
30 S An experimental analysis of shipmotions in longitudinal re-gular waves. J. Gerritsma, 1958.
31 M Model tests concerning damping coefficient and the increase in the moment of inertia due to entrained water of ship's propellers. N. J. Visser, 1960.
32 S The effect of a keel on the rolling characteristics of a ship. J. Gerritsma, 1959.
33 M The application of new physical methods in the examination of lubricating oils (Contin. of report 17 M). R. N. M. A. Malotaux and F. van Zeggeren, 1960.
34 5 Acoustical principles in ship design. J. H. Janssen, 1959. 35 S Shipmotions in longitudinal waves. J. Gerritsma, 1960. 36 S Experimental determination of bending moments for three
models of different fullness in regular waves. J. Ch. de Does,
1960.
37 M Propeller excited vibratory forces in the shaft of a single screw tanker. J. D. van Manen and R. Wereldsma, 1960. 38 S Beamknees and other bracketed connections. H. E. Jaeger
andJ.J. W. Nibbering, 1961.
39 M Crankshaft coupled free torsional-axial vibrations of a ship's propulsion system. D. van Dort and N. J. Visser. 1963. 40 S On the longitudinal reduction factor for the added mass of
vibrating ships with rectangular cross-section. W. P. A. Joosen andJ. A. Sparenberg, 1961.
41 S Stresses in flat propeller blade models determined by the moiré-method. F. K. Ligtenberg, 1962.
42 S Application of modern digital computers in naval-architec-ture. H. J. Zunderdorp, 1962.
43 C Raft trials and ships' trials with some underwater paint
systems. P. de Wolf and A. M. van Londen, 1962.44 S Some acoustical properties of ships with respect to noise control. Part I. J. H. Janssen, 1962.
45 S Some acoustical properties of ships with respect to noise control. Part II. j. H. Janssen, 1962.
46 C An investigation into the influence of the method of applica-tion on the behaviour of anti-corrosive paint systems in sea-water. A. M. van Londen, 1962.
47 C Results of an inquiry into the condition of ships' hulls in relation to fouling and corrosion. H. C. Ekama, A. M. van
Londen and 1'. de Wolf, 1962.
48 C Investigations into the use of the wheel-abrator for removing rust and miliscale from shipbuilding steel (Dutch) . Interim
report. J. Remmelts and L. D. B. van den Burg, 1962. 49 5 Distribution of damping and added mass along the length of
a shipmodel. J. Gerritsma and W. Beukelman, 1963. 50 5 The influence of a bulbous bow on the motions and the
pro-pulsion in longitudinal waves. J. Gerritsma and W. Beukel-man. 1963.
51 M Stress measurements on a propeller blade of a 42,000 ton tanker on full scale. R. Wereldsma, 1964.
52 C Comparative investigations on the surface preparation of shipbuilding steel by using wheel-abrators and the application of shop-coats. H. C. Ekama, A. M. van Londen and J.
Rem-melts, 1963.
53 S The braking of large vessels. H. E. Jaeger, 1963.
54 C A study of ship bottom paints in particular pertaining to the behaviour and action of anti-fouling paints. A. M. van
Lon-den, 1963.
55 S Fatigue of ship structures. J. J. W. Nibbering, 1963. 56 C The possibilities of exposure of anti-fouling paints in Curaçao,
Dutch Lesser Antilles. P. de Wolf and M. Meuter-Schriel,
1963.
57 M Determination of the dynamic properties and propeller ex-cited vibrations of a special ship stern arrangement. R. We-reldsma, 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, directly-coupled engines. C. Kapsenberg, 1964.
60 S Natural frequencies of free vertical ship vibrations. C. B. Vreugdenhil, 1964.
61 S The distribution of the hydrodynamic forces on a heaving and and pitching shipmodel in still water. J. Gerritsma and W.
Beukelman, 1964.
62 C The mode of action of anti-fouling paints: Interaction be-tween 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 Mitchell and V. A.
Ogale, 1965.
64 C Barnacle fouling on aged anti-fouling paints; a survey of pertinent literature and some recent observations. P. de Wolf,
1964.
65 S The lateral damping and added mass of a horizontally oscil-lating shipmodel. G. van Leeuwen, 1964.
66 S Investigations into the strength of ships' derricks. Part I. F. X. P. Soejadi, 1965.
