Hull vibrations of the cargo-passenger motor ship ORANJE NASSAU

REPORT No. 75 S August 1965 (S 3/39)

STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE

SHIPBUILDING DEPARtMENT EKELWEG 2, DELFT

HULL VIBRATIONS OF THE CARGO-PASSENGER

MOTOR SHIP "ORANJE NASSAU"

(SCHEEPSTRILLIÑGEN VAN HET VRACFIT- EN PASSAGIERSSCHIP M.S ,,ORANJE NASSAU")

by

Itt. W. VAN HOkSSÉN

Institute T.N.O. for Mechanical Constructions

Issued by the Council This report is not to be published uhless verbatim and unabridged

tions, undertaken on request of the Netherlands' Research Centre T.N.O. for Shipbuilding and

Navigation. It is to be regarded as a sequel to

report no. 58 S ,,Numerical calculatiOn of vertical hull vibrations of ships by discretizing the vibra tion system" byj. DE VEJEs. The method described therein has been applied and the results compared with measured values.

Measurements and calculations were carried but by the Institute T.N.O. for Mechanical Construc-tions.

The valuable assistance rendered by Messrs.

Koninldijke Nederlandsche Stoomboot

Maat-schappij N V, owners of the ship, by the builders Messrs. N.y. Scheepsbouwwerf Gebr. Pot and by the enginebuilders Messrs. Koninklijke

Machine-fabriek Gebr. Störk en Co. N.y. is gratefully

acknowledged.

CONTENTS

4 Bending stiffness ...

5 Shear stiffñess...

7 Numerical calculations ...

8 Comparisòn of shear correction factors 1.4

9 Conclusions 15

Acknowledgement 15

References... 15.

i Introduction

The main characteristics of the ship are given in

Table I. Figure 1 shows the ship after completion.

Figure 2, as an example, gives a cross-section,

indicating longitudinal scantlings. The general arrangement is shown in Figure 3.

The construction of the ship satisfies the require-ments of Lloyd's Register of Shipping, having lon-gitudinal framing in the double bottom and trans-verse framing elsewhere. Seams and frames of the side plating, as well the stringer angles of E-deck and F-deck are riveted, elsewhere electric welding is applied.

HULL VIBRATIONS OF THE CARGO-PASSENGER

MOTOR SHIP "ORANJE NASSAU"

by

IR. W. VAN HORSSEN

Summary

From known distributions of mass, bending stiffness and shear stiffness, five natural frequencies were calculated both of the vertical- and the horizontal vibrations, with the use of a discretizing method.

In these calculations the ship was considered as a Timoshenko beam.

Bending- and shear-stiffness have been calculated extensively in a number of cross-sections of the ship.

A correction was applied for the difference in virtual mass at the higher modes with respect to that at the 2-node. The influence of the rotary inertia has been taken into account in the form of an additional correction.

Part of the frequencies calculated was compared with measured ones. For the vertical vibration the effect of the shear stiffness was compared with the shear correction according to PRoI.sax [6].

Fig. 1. The cargo-passenger motor ship "Oranje Nassau"

;i UIIMIIUIINI a t, n nl iç t. n t,

t. tIttt

,w,

-I

p

t-

a.

- -,.

----. -

----C.uri.y N. V. Shr-p.bouaurrJ Po:

BIEZENO, KOCH and LEKKERKERKER calculated the natural frequencies and vibration profiles of

this ship [5]. They considered the ship as a beam

Table I. Main characteristics of the ship

Length o.a. 131.60 m

Length b.p. L = 120.00 m Breadth. moulded B = 17.20 m

Depth to main deck (F-deck) D = 9.90 m

Max. draught, moulded 6.90 m

Deadweight 5620 metric tons

Passengers 116

Max. continuous output of

main engine 4500 BHP (metric) at 140 r.p.m.

6

LONGITUD. 6x3.5 .312'

LONGITUD. 5x4'c.36' 1a5\

Fig. 2. Cross-section through engine room

with mass, bending stiffness and shear stiffness These stiffnesses were taken into account by elastic joints and shear joints. As it was still open to doubt whethér the beam model would be adequate espe-cially for the higher modes, it was decided to chèck the results i.e. both the natural frequencies and the vibration profiles by measuremçnts.

The present report gives the results ofthese meas-urements and compares them with values obtained by a modified calculation (matrix method [8] in-stead of iteration method [5], [12]), based,

how-ever, on the above mentioned beam model. In

these calculations the mass-distribution during the measurements was inserted.

The fuel oil 'bunkers and the fresh water tanks were full, the cargo holds were empty.

The mean draught d of the ship was 4.24 rn, the displacement 5775 metric tons. During the

measurements of the natural frequencies the ship was laying at anchor with 24.7 m water under the

keel.

2 Measúrements

A mechanical vibration exciter was installed at the

tunneltop in hold 5, at the centreline roughly

16.7 mforward of the A.P.P. (approximately frame no. 25, see Figure 3). Although this loçation was far from ideal, it was the best one availáble at that time. Additional reinforcement was incorporated

under the bedplate of the exciter in the form of

extra cross beams, in order to transmit the exciting force into the ship and preventing local deforma-tions of the tunneltop.

The natural frequencies were' obtáined by meas-uring the vertical and the horizontal displaèements at a slowly increasing number of revolutions of the exciter.

The vibration profiles were obtained by meas-uring the displacements at 22 positions on E-deck and Fdeck (see Figure 3) at a constant number of exciter revolutions.

The naturàl frequencies are' given in Table II

(from [7]).

Table IL Measured natural frequencies

It proved to be impossible to excite the 4-rNV,

the higher horizontal modes and the torsional

mo4es. Probably this is closely connected with the location of the exciter.

The measured vibration profiles are given in

Figures 4 to 7..

The accuracy Of the measurements of the natural

fre4úencies was approximately 2%, that of the

measurements of the vibration profiles approximat-ely 10-15%. Vertical ' 2-nOdes 3-nodes 5-nodes 2.1,6 c/sec. 3.79 c/sec. 6.18 c/sec.

Frame n i -lo B deck

C deck)__±_

.-1 -

-=

't4

-50

3 Mass distribution

The mass consists of two components:

the mass of the ship, including outfit, machin-ery, oil, provisions, cargo, etc.

the so-called "virtual mass", or "added mass" of the water.

The ship was longitudinally divided into 20 seg-ments each with a length of 656 cm. The mass of

Fig. 4.. Measured vibiation profile f 2-node verticàl Fig. 3. General arrangement, indicating measuring pints

-each segment for .the measurement condition was

calculated by the yard. Moreover, the yrd pro-vided the calculations of the. virtual mass at the

2-node vertical vibration successively for the fully-loaded, half-loaded and the light ship condition,

with the aid of the data presented by PROHASKA [6].

The virtual mass for the measurement condition

8 i

Frame no.

Fig. 5. Measured vibration profile of 3-node vertical

Fig. 6. Measured vibration profile of 5-node vertical

Fig. 7. Measured vihratiòn profile of 2-node horizontal

E o D 4-, E o Bdeck A

deI

,'

between theabove-mentioned values as illustrated in Figure 8. The error caused by the interpolation

is very small. The resulting mass distribution is

given in Table III and Figure 9. The mass

calcula-tions were performed with the acceleration of

gravity g = 981 cm/sec2.

Table III. Mass distribution during measurement

The virtual mass at the higher vertical modes

can be derived by different methods:

-o 4-,

-i'

ship, etc. _kqfsec2

- cm

Fig. . Example of the interpolation of the virtual mäss

at the 2-node vertical vibration

LOCKWOOD TAYLOR[1] gives a reduction factor for. the 3-node vertical with respect to 'the 2-node Vertical. These reduction factors are given in Table IV.

Table IV. Reduction factor with respect to 2-NV for virtual

mass at vertical vibrations according to LÖCKWOOD TAYLOR

Thus the virtual mass at the 3-node vertical

follows from multiplying the virtual mass at the 2-node by factor 0.873 where LIB = 6.98 for. the ship concerned.

LEWIS [2] also calculated reduction factors for

the 3-node vertical. Recently the work OfLEWIS

has been extended by MUNAF (unpublished report). MUNAF'Sresults aré given by KRUPPA

[3]. These reduction factors are reproduced in

Table V.

Table V. Reduction fàctors'with respect to 2-NV for virtual mass at vertical vibrations according to LEwIs and MUNAF

1 2 3 4.5 67 8. 10_11 12_1314 151617 18j90

Aft segment no Forew&rd

Fig. 9. Mass distribution during measurement, vertical vibrations

L/B = length/breadth

6 7 8 9 10

3-Ny .0.837.

Segment n. Mass of the ship_(kgfsec2/cm> Virtual massat 2-NV

(kgf sec2/cm) Aft 43.9 0 2 135.6 10.8 3 163.7 67.0 4 300.0 193.0 5 403.0 358.0 6 288.0 514.0 7 303.0 632.0 8 468.0 690.0 9 629.1 . 716.0 l0 604.1 719.0 11 500.8 719.0 12 402.7 715.0 13 365.1 691.0 14 264.0 648.0 15 222.0 542.0 16 . 256.0 381.0 17. 227.5 207.0 18 155.9 72.5 19 125.6 11.2 20 28.3 0 Forward Total 5886.3 7886.5 L/B = length/breadth 5 7 8 9 Lewis 3-NV 0887 0898 0915 0927 0937 0945 4-NV 0.807 0.823 0.843 0.859 0.874 0.887 Munaf 5-NV 0.73,1 0.753 0.778 0.799 0.819 0.837 6-NV 0.661 0.691 0.720 0.747 0.773 0.798

t

lo

Thus the virtual mass of the vertical vibrations can also be derived from that at the 2-node by multiplying the 2-node virtual mass by factor

0.914 for the 3-node, 0.842 for the 4-node,

0.777 for the 5-node and 0.7 18 for the 6-node.

c. JoosEN and SPAREÑBERG [4] calculated the

vir-tual mass for a "ship" of infinite length with

rectangular cross-section as a function of the ratio of wave length to beam for some values of the ratio of draught to beam. If from previous calculations the wave lengths are known with a tolerable accuracy, reduction factors for higher modes with respect to the 2-node can be derived

from the results of JOO5EN and SPARENBERG.

With respect to the 2-node vertical these turned out to be about 0.89 for the 3-node, 0.78 for the

4-node, 0.69 for the 5-node and 0.64 for the

6-node.

As wave length was taken two times the length

between the extreme nodes of the calculated

profiles of Figure 12, divided by the number of crests in between these nodes.

In the numerical calculations the reduction factors

according to LEwIs and MUNAF were applied. These lead to slightly lower natural frequencies

than those according to LocKwooD TAYLOR or

JOOSEN and SPARENBERG.

The virtual mass for the horizontal vibrations was calculâted according to LOCKWOOD TAYLOR [1].

He gives for tbe".'irtua1 mass per unit length at

midship:

0.008443ad2 kgf sec2/cm2

and at the ends of the ship: 0.006485ayd2 kgfsec2/cm2

where d draught in m, y = specific weight in

kgf/dm3 and a is a reduction factor dependent on

the vibration mode. Reduction factor a can be

derived from Table VI.

Table VI. Reduction factors for virtual mass at horizontal

vibratiòns according to LOCKWOOD TAYLOR

By extrapolation is found

(d = 4,24 m and

Lid 28 for the ship during measurement)

a2 = 0.99 and a3 = 096

Accordingly, the virtual mass per unit length at the 2- and 3-node horizontal vibration has been calculated as given in Table VII.

Table VII. Virtual mass per unit length at horizontal'

vibrations

On the strength of Table VII, the virtual mass

belonging to each segment was taken as given in Table VIII. The virtual mass distribution used for horizontal vibrations is also given in Figure 10.

Table VIII. Horizontal vibrations, virtual mass distribu-tion during measuremeñt

For the horizontal vibrations the influence of the virtual mass is limited. Moreöver, the difference

between the virtual mass at the 2- and, 3-NH is

small. Therefore the virtual mass at the 4-, 5- and

6-NH was taken equal to that at the 3-NH. Midship (kgfsec'/cm') . Ends (kgf sec'/cm2) 2-NH 3-NH 0.154 0.149 0.118 0.115 Segment no. Virtual mass 2-NH (kgfsec2/cm) Virtual mass 3-NH (kgfsec'/crn) Aft 12.6 12.2 2 72.5 70.5 3 82.0 79.7 4 85.8 83.3 5 89.2 86.6 6 92.4 89.7 7 95.8 93.1 8 98.4 95.6 9 99.9 97.0 10 100.8 97.9 11 100.8 97.9 12 99.9 97.0 -13 98.4 95.6 14 95.8 93.1 15 924 89.7 16 89.2 86.6 17 85.8 83.3 18 82.0 79.7 19 72.5 70.5 20 12.6 12.2 Forward Tôtal 1658.8 1611.2 Lid = length/draught 12 14 16 18 20 2-NH 0.93 0.945 0.96 0.965 0.970 3-NH 0.89 0.905 0.92 0.93 0.94

virtual moss per segment fkgfse 100 90 80 70 60 2NH 3NFI 50 40 30 2 10 o 1 2 3 4 5 6 7 B 9 10 11 12 13 14 15 16 17 18 19 20

Aft segment no Foreword

Fig. 10. Virtual mass distribution during measurement, horizontal vibrations

For the calculations the mass of each segment was considered to be concentrated at the middle of the segment.

4 Bending stiffness

The bending stiffness of the hull was taken from calculations made by the yard. In calculating this stiffness only those parts were left out of

considera-tion which coùld reasonably be expected not to contribute to the bending stiffness. BIEZENO, KOCH and LEKKERKERKER [5] had discretized it in a

num-ber of elastic joints entirely in accordance with the discretizing methOd described in [8], [11], [12].

Between these elaStic joints the ship is corsidered

non-deformable. .

-Table IX gives the results, of the calculations of [5] in the form of the stiffness constants K of the elastic bending joints both for the vertical and the horizontal vibrations. In Figure 11 the numbering

and the situ4ting qf the. elastic joints and of the

masses are shown.

Young's modulus E has been taken 2.l2 108 kgf/cm2. Aft Foreword mj m2 frr ii' jmi8 rng m20 t2 t4 _2O o -6,8 K, _ky2 I/i g, q q, q,,y2 q,,

mass o bending joint rotary inertia shear joint

Fig. Il. Numbering of the elastic joints and concentrated

masses of the discretized system

Table IX. Stiffñess-constants K1 óf the bending:joints

5 Shear stiffness

The shear stiffness has also been taken from [5]. A paper, describing in detail the methods applied for the calculation of the shear Stiffness, is being prepared. In 'principle, shear stiffness s in a cross-section was calculated by the well-known formula:

where: r = shear stress D = shear force

A sectional area of longitudinal

con-structional members

G = shea,! molulus = 0.828. 106 kgf/cm2.

Shear stresses were calculated by the ele.nentary theory of thinral1ed multicellulâr beams..

Vertical (rad/kgf cm) Horizontal (rad/kgf cm) Aft K, 873.3 1434 .10-" 103.5 . 414.2 10-" K, K,'1, K, 199.4 2,73.9 97.85 188.5 148.8 180.9 74.38 116.7 K4 79.20 49.83 K4'1, K, K,'1, ' 9288 28.49 56.52 82.66 32.31 46.59 K6 26.96 21.00 K6'1, K, 50.76 25.04 51.95 37.41 18.73 37.51. K, K,'1, K, K,'1, K,, K,0'j, K 25.55 46.57 23.08 46.23 2297 46.58 25.16 18.33 35.83 18.02 36.27 18.44 37.48 18.80 59.20 37.72 K,, K12'1, K1, K,,', K1, K,4'1, K,, ' 32.02 70.81 32.20 61.56 39.30 93.71 5268 119.7 19.22 39.14 21.18 45.56 23.51 48.50 . 28.05 63.72 5633. 36.76 88.32 83.33 K,, K,,i1 '

o 00

12

To calculate the 'horizontal shear stiffness of "open" cross-sections BIEZENO, KOCH and

LEKKER-KERKER [5] developed a different method which accounted for bendingphenornena ofthedeckplates. Only for a restricted number of cross-sections the above-mentioned laborious method was applied.

An approximate method also derived in [5] was

applied for the remaining cross-section. Its use was

justified by comparing the shear stiffnesses

cal-culated by the first mentioned method and those calculated by the approximate one.

Finally the shear stiffness was discretized in

elastic joints entirely in accordance with the dis-cretizing method described by DE VRIE5 in [8]. Between these shear joints the ship is considered non-deformable. Table X gives the results in the form of stiffness constants qi of these shear joints, both for the vertical and the horizontal vibrations.

In Figure 11 the numbering and situating of the

shear joints is shown.

Table X. Stiffness-constants q of the shear joints

6 Rotary inertia,

Because the influence of the rotary inertia is small, it was considered not worth-while to ascertain its exact numerical value in the different cross-sec-tions. Instead, the rotary inertia was supposed tO be equal to 3.3 e times moment of inertia I, known

from bending stiffness EI. Here e is the specific mass of steel. Factor 3.3 represents the material

which contributes to the rotary inettia, but not to the bending stiffness. This value was chosen on the strength of the supposition that it should be about

equal to the ratio of the total mass of the ship, without the virtual mass of the water, to that of those constructional parts that contribute to the

bending stiffness.

Table XI gives the rotary inertia thus calculated, cliscretized as described in [8].

Table XI. Discretized rotary inertia

7 'Numerical calculations

Except for the method of discretizing the mass, as mentioned before, the numerical calculations were performed entirely in accordance with [8].

First the calculations were performed using the 2-node virtual mass, ignoring the rotary inertia.

The results are collected in Table XII and in

Figures 12 and 13.

Table XII. Calculated natural frequencies with 2-node virtual mass and without rotary inertia

To the natural frequencies of Table XII an ap-proximative correction was applied, in accordance

with [8], to take intO account the difference in vir-tual mass at the higher mOdes. The results are

col-lected in Table XIII.

Table XIII. Calculated natural frequencies, corrected for virtual mass only

Vertical (kgf cm sec2) Horizontal (kgf cm sec2) Aft r, 16.79. 106 _29.46.10° T4 59.71 73.84 T6 132.84 172.17 l4522 194.95 153.86 193.82 'r,2 112.85 185.97 T» 94.73 152.07 69.98 98.94 63.90 36.75 10.35 2.47 Forward Vertical (cm/kgf) Horizontal (cm/kgf) Aft 1.200 . l0 0.6603. 10° q2'2 0.5226 0.2641 0.5046 0.1761 q4' 0.42 19 0.1598 0.2915 0.08998 0.2751 0.1278 0.2820 0.08900 0.2787 0.08780' q,i. 0.2787 0.1153 0.2734 0.1069 0.2664 0.08877 0.2784 0.09662 0.2930 0.07621 0.3329 0.1585 0.3469 0.1278 0.3290 0.1405 q1i 0.3127 0.2164 0.2888 0.2244 q19i, 0.5205 0.4 170 Forward Vibration mode Vertical (c/sec) Horizontal (e/sec) 2-node 2.26 3.21 3-node 4.18 '6.85 4-node 6.05 10.25 5-node 7.74 13.44 6-node 9.44 17.61 Vibration mode Vertical (c/see) Horizontal (e/sec) 2-node 2.26 3.21 3-node 4.27 6.88 4-node 6.29 10.29 5-node 8.15 13.49 6-node 10.19 17.66

J L ri O - 50 00 3-NV

Fig. 12. Calculated and measured vertical vibration profiles

An approximative correction for the rotary

inertia too was- calculated usihg Rayleigh's energy

method, as described in [8]. The results are col-lected in Table XIV.

Table XIV. Calculated natural frequencies, correcte4 for virtual mass and rotary inertia

4-NH

o

.--e--na

z

Fig.l3.Calculated and measured horizon tal vibration profiles

At higher modes the influence of the bending

stiffness on the natural frequencies decreases. This must be the reason why the perceiitual influence of the rotary inertia is more or less independent of the vibration mode.

The calculated natura! frequencies of Table XIV were compared with the measured ones of Table II

by calculating the difference as a percentage of

the meäsured ones. Table XV gives the results. Notwithstanding much pain has, been taken to make fair allowance for shear stiffness and rotary ihertia, the well-known phenomenon of increasing discrepancy between 'measu,rernents and calcula-tions with increasing vibration mode still exists.

Vibration mode Vertical (c/see) Horizontal (e/sec) 2-node 2.25 -3.16 3-node 4.23 6.76 4-node 6.24 10.11 5-node 8.09 13.30 6-node - 10.13 17.44

14

Table X-V. Difference between measured and calculated natural frequencies in % of the measured ones. (A plus sign means that the calculated natural frequency is the higher one)

8 Comparison of shear correction factors

Because the calculation of the shear stiffness dis-tribution as applied in this paper is rather cumber-some, much labour can be saved if the influence of

shear stiffness can be accounted for, within

rea-sonable accuracy, by correction factOrs.

Approximate shear correction factors are known from PRoliAsick [6] and LocKwooD TAYLOR [9], [10]. In this section the suitability of these

correc-tion factors is ascertained for the ship under

in-vestigation.

For this purpose the calculations of the vertical vibrations described before were repeated, using

the 2-node virtual mass and ignoring both shear

stiffness and rotary inertia. Shear correction factors were obtained by dividing the natural frequencies -mentioned in Table XII by the natural frequencies thus obtained. The results are given in Table XVI.

Table XVI. Shear correction factors, i.e. calculated natural frequencies with shear stiffijess divided by

those without shear stiffliess

According to PROHASKA [6] shear correction

faètor r2 for 2NV depends on LID, BID and ,

where: L = length between perpendiculars

D = depth to main deck B = breadth

= 0.15 to 0.17 fOr ships of ordinary shape 0.10 to 0.12 for ships with large super-structures.

For comparison was chosen

= 0.16, a rather

high value but a lower value would havegiven a

still higher shear correction factor. With L =

120m,B = 17.2mandD = 9.9m,itfollowsfrom

the graphs that for a ship with 2 to 4 decks and

double bottom, the shear correction factor

ac-cording to PROHASKA for NV will be r = 0.907.

LOCKWOOD TAYLOR [9], [10] introduced a fac-tor r, to be calculated from ship characteristics:

r = shear deflection / bending deflection

shear correction factor - l/v'TT

From the graph it follows that for 2N Vr = 0.2075 and therefore the shear correction factor according to LocKwooD TAYLOR for 2NV will be 0.910.

It must be concluded that PROHASKA and LOCK-WOOD TAYLOR give the same shear correction factor

for this ship and that both underestimated the

influence of shear on the natural frequency

con-siderably. Because it was rather disconcerting that both had misjudged the iafluence of shear so much,

it was decided to check the pertainingnumerical

calculation of the present work independently by

applying the iteration method [li] on the

tion ptofile going with the 2-node vertical vibra-tion. In applying the iteration method it is neces-sary to calculate the deformations produced by a given load. This was not done by using the influence numbers, but by calculating from equilibrium con-ditions the bending moments and the shear forces at the elastic joints. From these moments and forces the deformations of the joints, and therefore of the

whole system could be found easily. In conse-quence, starting from Tables III, IX and X the

2NV natural frequency was calculated in a man-ner entirely independent of the method applied in this paper.

These calculations confirmed beyond doubt the correctness of the numerical results obtained, based

only on data given in Tables III, IX and X.

In the calculations with the iteration method the

contribution to the 2NV vibratioñ profile of the

shear deformation as well as of the bending defor-mation were derived separately. Figure 14 shows these contributions separately as well as the total

vibration profile. From Figure 14 it can also be

concluded that the factor r defined by LocKwooD TAYLOR for this type of ship should have been

approximately r = 0.4 instead of r = 0.2075.

Y5 sheon deflection Yb bending dei rection

Yt totdi deflection

Fig. 14. Profile of 2-node vertical vibration and separate contributions by shear and bending deflection

2-NV + 3.7%

3-NV + 11.6%

5-NV ± 30.9%

2-NH

- 2.4%

Vibration mode Shear correction factor

2-NV 0.84

3-NV 0.67

4-NV 0.56

5-NV 0.47

9 Conclusions

Natural frequencies have been calculated with the elementary beam model. Rotary inertia and shear stiffness have been accounted for carefully.

The influence of rotary inertia proved to be

small, of the order of 2% or less. This applies as well to the horizontal as to the vertical vibrations, to the higher as well as to the lower modes.

The influence of shear stiffness proved to be of

importance at the 2-NV. At 3-NV this influence was about equal to that of the bending stiffness. At the higher vertical modes the shear stiffness

proved to be predominant.

The applied correction for the virtual mass at

the higher modes enlarged the difference between measured and calculated frequencies.

The calculated natural frequencies of the- 2-NV and horizontal vibrations respectively were 3.7% higher and 2.4% lower than the measured ones.

The calculated items of the 3-NV and 5-NV respectively surpassed the measured ones by 12

and 31%.

Comparing the measured and calculated vibra-tion profiles shows that,, within the accuracy of the measurements, a fair agreement exists for the lower frequencies.

It must be concluded that the elementary beam

model leads to an overestimation: of the higher

natural frequencies, from the 3-node on.

The deviations between calculations and

meas-urements lead to the conclusion that the present

calculation method is unsatisfactory for the higher modes. All efforts to obtain better 'results must be encouraged.

The shear correction according to PROHASKA [6] equaled that given by LocKwooD TAYLOR [9], [lo]. Both corrections proved to be too small by nearly a factor 2 for the ship under investigation.

Acknowledgement

The author wishes to express.his thanks to Mr. A. VERDUIN for the many useful discusions that have guided him during the preparation of the present report.

References

LOCKWOOD TAYLOR, J, Vibration of ships. Transactions of the Institution of Nava1 Architects, VoI.- 72, 1930, p. 162.

Lzwis, F. M., The -inertia of the -water surrounding a vibrating ship. Transactions of the Society of Naval

- Architects and Marine Engineers, Vol.37, 1929,-p. 1.

KRUPPA, C., Beitrag zum Problem der hydrodyna-mischen - Trägheitsgrössen bei elastischen Schiffs-schwingungen. Thesis Technische Universität Ber-lin, 1961. Públished in Schiffstechnik no. 9, 1962, no. 45 (January).

JOOSEN, W: P. A., and J. A. SPARENBERG, On the

lon-gitudinal reduction-factor for the added mass of vibrating ships with rectangular cróss-section.

Report no. 40 S, Netherlands' Research Centre T.N.O. for Shipbuilding and Navigation, 1961. Also in mt. Shipbuilding Progress, Vol. 8, no. 80, April

1961.

-BIEZENO, C. B., J. J. KOCH and J. G. LEKKERKERKER,

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schip. 1956 (unpublished report).

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(unpub-lished report).

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systçm. Report no. 58 S, Nétherlands' Research Centre T.N.O. for Shipbuilding añd Navigation, 1964. Also in mt. Shipbuilding Progress, Vol. 11, no. 118, June 1964.

Locscwoo TAYr.oR, J., Ship vibration periods.

Trans-- actions of, the North-East Coast Institution of' -En- -

-gineers andShipbuilders, VòI. 44, 1927-1928, p. 143.

TODD, F. H., Ship hull vibration Edward Arnold (Publishers) Ltd., London, 1961.

-BIEZENO, C. B.,and R. Ga.oiRL, Technische Dynamik. Berlin, 1953. - -

-KOCH, J. J., Eenige toepassingen van de leer der eigen-functies op vraagstukken uit de toègepaste mechá-nica. Thesis DeIft, 1929. --

-PUBLICATIONS OF THE NETHERLANDS' RESEARCH CENTRE T.N.O.

Reports _{FOR SHIPBUILDIÑG AND NAVIGATION}

No. i S The determination of the natural frequencies of shipvibrations (Dutch).

.

By prof. ir H. E. Jaeger. May i 950.

No. 3 S Practical possibilities of constructional applications of aluminium alloys to ship construction.

By prof. ir H. E. Jaeger. March 1951.

No. 4 S Corrugation of bottom shell plating in ships with all-welded or partially welded bottoms (Dutch).

By prof. ir H. E. Jaeger and ir H. A. Verbeek. November 1951.

No. 5 S Standard-recommendations for-measured mile and endurance trials of sea-going ships (Dutch).

By prof. ir J. W. Boneba/cker, dr ir W. J. Muller and ir E. J. Die/il. February 1952.

No. 6 5 Some tesos on stayed and unstayed masts and a comparison ofexperimental results and calculated stresses (Dutch).

By ir A. Verduin and ir B. Burghgraef. .June 1 952.

No. 7 M Cylinder wear in marine diesel engines (Dutch).

By ir H. Visser. December 1952.

No. 8 M Analysis aiid testing of lubricating oils (Dutch).

By ir R. N. M. A. Malotaux. and irj. G. Smit.July 1953.

No. 9 S Stability experiments on models ofDutch and French standardized lifeboats.

By prof. ir H. E. Jaeger, prof. ir J. W. Boneba/cker andj. Pereboom, in collaboration with A. Audigé. October 1952.

No. 10 S On collecting ship service performance data and their analysis.

v prof ir 7. W. Bonèbakker. January 1953.

No. 1 1 M The use of three-phase current for auxiliary purposes (Dutch).

Bv irj. C. G. van Wjk. May 1953.

No. 12 M Noise and noiseabatement in marine engine rooms (Dutch).

By " Tèchnisch-Physische Dienst T.N.O.- T.H." April 1953.

No. i 3 M Investigation of cylinder wear in diesel engines by means of laboratory machines (Dutch).

By ir H. Visser. December 1954.

No. 14 M The purification ofheavy fuel dil for diesel engines (Dutch).

By A. Bremer. August 1953.

No. 15 S Investigation of the stress distribution in corrugated bulkheads with vertical troughs.

By prof ir H. E. Jaeger, ir B. Burghgraef and I. van der Ham. September 1954.

No. 16 M Analysis and testing of lubricating oils II (Dutch).

By ir R. N. M. A. Malotaux and drs .7. B. Zabel. March 1956.

No. I 7 M The application ofnew physical methods in the examination oflubricating oils.

By ir R. N. M. A. Malotaux and dr F. van Zeggeren. March 1957.

No. I 8 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 applied on board of these ships änd their in-fluence on the generating capacity (Dutch).

By ir J. C. G. van Wgk. Februar I 957. No. 19 M Crankcase explosions (Dutch).

By ir J. H. Minlchorst. April 1957.

No. 20 S An analysis of the application of aluminium alloys in ships' structures.

Suggestions about the riveting betweeîì steel and aluminium alloy ships' structures. By prof. ir H. E. Jaeger. January 1955.

No. 2 1 S On stress calculations in helicoidal shells and propeller blades.

By dr ir J. W. Cohen. July 1955.

No. 22 S Somenotes on the calculation ofpitching and heaving in longitudinal waves. By ir j. Gerriisma. December 1955.

No. 23 S Second series of stability experiments on models of lifeboats. By ir B. Burghgraef. September 1 956.

No. 24 M Outside corrosión of and slagformation on tubes in oil-fired boilers (Dutch). Bydr W.j. That. April 1957.

No. 25 S Experisnental determination of damping, added mass and added mass moment of inertia of a shipmodel.

By ir J. Gerritsma. October 1957.

No. 26 M Noise measurements and noise reduction in ships.

By ir G. J. van Os and B. van Strenbrugge. May 1957.

No. 27 S Initial metacentric height of small seagoing ships and the inaccuracy and unreliability of calculated curves of righting levers.

By frof. ir J. W. Bonebakker. December 1957.

No. 28 M Influence of piston temperature on piston fouling and piston-ring wear in diesel engines using residual fuels.

By ir H. Visser. June 1959.

No. 29 M The influence of hysteresis on the value of the modulus of rigidity of steel.

By ir A. Hoppe and ir A. M. Hens. December 1959.

No. 30 S An experimental analysis of shipmotions in longitudinal regular waves.

By ir J. Gerritsma. December 1958.

No. 31 M Model tests concerning damping coefficients and the increase in the moments of inertia due to entrained water

of ship's propellers.

By N.J. Visser. October 1959.

No. 32 S The effect of a keel on the rolling characteristics of a ship. By irJ. Gerritsma. July 1959.

No. 33 M The application of new physical methods in the examination of lubricating oils. (Continuation of report No. 17 M.)

By ir R. N. M. A. Malotaux and dr F. van Zeggeren. November 1959.

No.34 S Acoustical principles in ship design. By ir J. H. Janssen. October 1959. No. 35 S Shipmotions in longitudinal waves.

By ir J. Gerritsma February 1960.

No. 36 S Experimental determination of bending moments for three models of different fullness in regular waves.

By ir J.Ch. De Does. April 1960.

No. 37 M Propeller excited vibratory forces in the shaft of a single screw tanker

By dr ir J. D. van Manen and ir R. Wereidsina. June 1960. No. 38 5 Beasnknees and other bracketed connections.

By prof. ir H. E. Jaeger and ir 3.3. W. Nibbering. january 1961.

No. 39 M Crankshaft coupled free torsional-axial vibratiOns of a ship's propulsion system. By ir D. van Dort and N. J. Vi.cser. September 1963.

No 40 S On the longitudinal reduction factor for the added mass of vibrating ships with rectangular cross-section By ir W. P. A. Joosen and dr J. A. Sparenberg. April 1961.

No. 41 S Stresses in fiat propeller blade models determined by the moiré-method. By ir F. K. Ligtenberg. June 1962.

No. 42 S Application of modern digital computers in naval-architecture.

By ir H. J. Zunderdorp. June 1962.

No. 43 C Raft triáIs arid ships' trials with some underwater paint -systems.