Propeller excited vibratory forces in the shaft of a single screw tanker

Technische Hogeschool

REPORT No. 37M

Pelit

June 1960

by

Dr. Jr. J. D. VAN MANEN and Jr. R. WERELDSMA

STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE

AFDELING MACHINEBOUW - DROOGBAK 1A - AMSTERDAM

(NETHERLANDS' RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION)

ENGINEERING DEPARTMENT - DROOGBAK IA - AMSTERDAM

PROPELLER EXCITED VIBRATORY FORCES

IN THE SHAFT OF A SINGLE SCREW TANKER

REPORTS AND PUBLICATIONS OF THE NETHERLANDS RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION

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No. 1 S The determination of the natural frequencies of ship vibrations (Dutch).

By prof. ir H. E. Jaeger. May 1950.

No. 2 Confidential report, not published. July 1950.

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. V erbeek. November 1951.

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

By prof. ir J. W. Bonebak.ker, dr it W. J. Muller and it E. J. Diehl. February 1952.

No. 6 S Some tests on stayed and unstayed masts and a comparison of experimental results and calculated stresses

(Dutch).

By ir A. Verduin and ir B. Burghgraef. June 1952.

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

By it H. Visser. December 1952.

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

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

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

By prof. ir H. E. Jaeger, prof. it J. W. Bonebakkcr and J. Pereboom, in collaboration with A. Audige. October 1952.

No. 10 S On collecting ship service performance data and their analysis. By prof. it J. W. Bonebakker. January 1953.

No. 11 M The use of three-phase current for auxiliary purposes (Dutch). By ir J. C. G. van Wijk. May 1953.

No. 12 M Noise and noise abatement in marine engine rooms (Dutch). By "Technisch-Physische Dienst T.N.O. -T.H." April 1953.

No. 13 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 of heavy fuel oil for diesel engines (Dutch).

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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 J. B. Zabel. March 1956.

No. 17 M The application of new physical methods in the examination of lubricating oils.

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

No. 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 applied on board of these ships and their influence on the generating capacity (Dutch).

By it J. C. G. van Wijk.. February 1957. No. 19 M Crankcase explosions (Dutch).

By it J. H. Minkhorst. April 1957.

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

Suggestions about the riveting between steel and aluminium alloy ships' structures.

By prof. ir H. E. Jaeger. January 1955.

No. 21 S On stress calculations in helicoidal shells and propeller blades.

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

No. 22 S Some notes on the calculation of pitching and heaving in longitudinal waves.

By it J. Gerritsma. December 1955.

No. 23 S Second series of stability experiments on models of lifeboats. By it B. Burghgraef. September 1956.

No. 24 M Outside corrosion of and slagformation on tubes in oil-fired boilers (Dutch).

By dr W. J. Taat. April 1957.

-PROPELLER EXCITED VIBRATORY FORCES IN THE

SHAFT OF A SINGLE SCREW TANKER

Introduction

Initiated by the shipyard "Wilton-Fijenoord"

and to the order of the "Netherlands Research

Centre T.N.O. for Shipbuilding and Navigation"

an experimental research into the propeller excited dynamic forces in the propeller shaft was conducted

by the "Netherlands Ship Model Basin" at

Wage-ningen, Holland.

This research was under the auspices of the

Netherlands Research Centre T.N.O. and dealt

with by the Committee "Vibratory Forces in Shafts". This committee consists of representatives of Engine Builders and Ship Yards, Ship Owners, the Nether-lands Research Centre T.N.O. for Shipbuilding and Navigation and the Netherlands Ship Model Basin.

1. Description ofthe investigation

The dynamic forces, generated by the propeller

in the propeller shaft of a single screw ship, are

basically influenced by the following three factors:

The shape of the afterbody;

The clearances between propeller and screw

aperture;

The number of blades and the hydrodynamic

properties of the propeller blades.

Results of the investigations into the effect of the

shape of the afterbody on propulsion have been

published in ref. [1].

In ref. [2] the results are given of a research into

the influence of the clearances between propeller and screw aperture on the thrust and torque varia-tions of the propeller of a single screw cargo liner

(15,000 tons displacement, 8,600 B.H.P. metric, 116 r.p.m. and a ship speed of 16 knots).

The influence of the number of blades, the blade

type, the type of rudder and the draft of the ship on the dynamic performance of the propeller form the subject of the present work. The research was

*) Assistant Director Netherlands Ship Model Basin

Wagenin-gen Holland.

**) Head Instrumentation Department N.S.M.B. Wageningen

Holland.

by

Dr. Ir. J. D. VAN MANEN5) and Ir. R. WERELDSMA**)

Communication from the Netherlands Research Centre T.N.O. for Shipbuilding and Navigation Summary

In this experimental research the effect of type of screw blade, type of stern arrangement, power absorption,

draft and number of blades has been investigated for a single screw tanker with a block coefficient of 0.77. The

dynamic thrust, torque and bending moments, excited by the propeller in the shaft have been analysed.

carried out on a model, representing a 32,000 tons

deadweight single screw tanker to a

scale of 1

to 271A. Two different stern arrangements were examined. A conventional stern arrangement has

been tested in the load condition, a "Mariner"-stern arrangement both in the load and ballast conditions. The installed power is 12,660 shp at 105 r.p.m. and

the corresponding trial speed is 16.2 knots in the load condition and 17.0 knots in the ballast

con-dition. The details of the measuring technique,

ap-plied in this investigation, are reported in [3] and [4]. This technique is based on a special type of

correlation measurements according to the principle of "periodic sampling".

2. Data of the ship and the propellers investigated

The particulars of the ship in the two conditions

are given in table I.

TABLE I. Particulars of ship

The ballast condition is an extreme one with the

propeller blade tips in the top position partly out

of the water. This condition might simulate the

dynamic forces excited at a normal draft of the ship

in a seaway.

The body plan and stern arrangements are

pre-sented in Fig. 1.

Two stern arrangements were examined:

a conventional stern arrangement and

a "Mariner" stern arrangement.

Load draft Ballast draft

Length between

perpen-diculars 192.02 m 192.02 m

Breadth moulded 17.13 m 27.13 m

Draft on even keel . 10.285 m 6.40 m

Volume of displacement (mid) 41,200 rn:' 24,600 Blockcoefficient 0.77 a. c. . m3

0 APP 0 APP 8 9 -10 Fig; I,

Body pla, aud propeller arrangenient for stehi arrangement A and B

.MARINER' STERN ARRANGEMENT

CWL LOAD! CONDITION L BALLAST _CONDITION 14 1300 20 F PP

CONVENTIONAL STERN ARRANGEMENT

MARINER STERN

ARRANGEMENT

PITCHI DISTRIBUTION

'MI PER CENT

401.1

106 3

105.5_

,Fig.. 2a.. General gay of propeller 1, i, Ill and IV

SCREW 4 i., 6500mmi 1PaaR/0 = 0,760 5 0.507 p/A 0.456, 11 SCREW 11 Di t =F, 6500 mm P0716/04 0.802' A 946.=.0.531 AP/AO = 0.479 SCREW 1St 650.0 mm P0,7R/D,R1 0.782 Z 5 0/40, = 0.558 4p/40 0.4901 SCREW Ir b.= 6500 mm Poi a/D= 10.809 7 .= A D/A 0 = 015 6 4i A kipo = 9.06. ha fl _ t 05. 0 103.6

ilMilEMINIIM AMMER!-.Mi.

CIRSR 0.6' 0.66' 0.7 R 0.6 R tft

Nil

'al

in

Milk

AMOK

1 ,00.0

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'94.7_

MIMI

1 1

NEM

\ AS R; .91.5 \ \ 0.4 R

Rik

8 ' 0.3 R .875 ' 85.6

wIl

I I --T. -- 1 I.. 0.2R ... 1 i;Ugstifl

A=

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I UIJ.Y ....-... - , "L._ DSR

r ....

AIM

\ .. 0.6R \ \ 0.7R

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Tx

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R1%

-- --...

_{l___L__}

02R

IRS.

1111W

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, _

1.0 0955 0.9 07 R 0.4 0.3 0.2 1031 100.0 95.9 88.6 85.9 83.5 z AU/A0 = 5 5 = 105.1

4

In order to obtain an indication concerning the influences of the type of propeller on the dynamic performance behind a ship, 6 propeller models (I

to VI) were investigated.

Details of the propellers are given in Fig. 2a and

2b and in table 2.

Figs. 3a and 3b Show the distributions of the

axial wake component of the model with the

con-ventional and the "Mariner" stern arrangement

respectively.

Review of the test results

A review of the tests and different comparisons

are presented in table 3.

The measured quantities are given as a function

of the angular propeller position, fi, which is ex,

plained in Fig. 4.

Fig. 2b.. General plans. of propeller V and VI

TABLE 2. 'Characteristics of propellers.

TABLE 3 SCREW t 10 = 6 250m ni PO:71RA) 0V834 0.53,8 .A p/A 0 = 01.72 SCREW SEC = 165'00 mm PD7R/D. 0:790 T. 4 Aolaa = 0.4831 A p/A0o. 0:434 FIGURE NT 5. 5 1 b C 00 00 MFFuENCE OF THE BLADE TYPE

INFLUENCE OF POWER ABSORPTION _ I t

INFLUENCE OF SCREW APERTURE I

I

INFLUENCE OF NUMBER OF BLADES

--INFLUENCE OF DRAFT':., 095-0.9 R /88, WM.

In

107.0._ all1=1/10..

-TO 5,5 - 11.11i. ION 0.88 'Ft _,..3.. '0.7 4O _100.0 , I OAa -

NI

96.0 I I 1 t 1 '0.5 R

fp

82.0

'PIM

_,-.411.iialI 0.4 R IICAad 1111 8. 30S 88.1 '...---_,______i______' ----039 84.3

Will

0 24t

illIk

80'5, ....- ... I'L -- --

AI

Vie,Th

LU m SL95R wo. .100.1.1 .1 1054 --0.9R 4111116. , 0:8 R.

ill

105.1 1032'

ANEW

_, i . 0.7 12 ₁₉₅₀

-

r-- r 0.00 1 95.8

...-wl-\ .0 0.5 R

iiii

9 29

MN

1511

,

1 111 ,i 13° 30' 0.4R - _853.

Mil=

0.3 R

-I 11.

8

le

--.44 -.I,

lik

. .1 , 0 2 8, 85.8 oppi--.4-", 1

-""

'",'IL.m2g:SOF STERN 'CONDITIONCONDITION '.744..'

PER CENT I COMPARISONS

I CONVENTIONAL LOAD ma

U S.. CONVENTIONAL LOAD 100 I I

U N CONVENTIONAL LOAD

0 3 .MARMERT LORD 100

t--1

. _MARINER BALLAST 'DOI

SI I CONVENTIONAL 1040 TOO I

2E CONVENTIONAL LOAD 100

7 0 CONVENTIONAL LOAD MO

II CONVENTIONAL MAO TOO

tr V .MARINER. LOAD TOO

4 ,mARINER.' 114M4AST 1410

I 1

Propeller model no.. _II IV V VI

Diameter in mm 6500 6500 6500 6500 6250 6500

Number of blades.... . 5 5 5 5 6 4

Pitch at root in mm Hn 4219 4342 4268 5070 4245 4389

Pitch at blade tip in mm Ho

5187 5585 5585 5070 57301 5475

Pitch at 0.7 R in mm

H0.7 R

4937 5215 5260 5070' 5280 5135

Developed blade-area ratio Fa/F 0.507 0.531 0.564 0.558 0.538 0.483

PITCH DISTRIBUTION IN PER CENT 1.0 R 108.5 3. z I III D z 4

0.20 LOAD CONDITION 0.30 0 = 6500 mm 0.60 0.50 TO 0.70 080 0.60 0.80 0.70 osoi 0.20 "2 W.0.15 0.30 0 = 6500 mm

The definition and the plane of measurement of

the bending moment on the propeller shaft are also given in Fig. 4.

In Figs. 5 to 9 the thrust and torque variations are presented as percentages of the corresponding

average value. An analysis of the thrust and torque

records is given in the formulae, which represent:

the average value at the speed considered,

the absolute values of the amplitudes of the

harmonic components, and

the phase relation of the harmonic components with respect to the propeller position.

The bending moments in Figs. 5 to 9 are presented as percentages of the average propeller torque.

r. POSITIVE 142., POSITIVE p :ANGULAR 0.80 0.80 0.80 0.50_ 0.40

Fig. 3 b. Circumferential wake distributions

VERTICAL BENDING MOMENT

HORIZONTAL BENDING MOMENT

POSITION OF PRORELLER

Fig. 3 a. Wake distributions of the conventional stern arrangement in load condition and of the Mariner"-stern arrangement in load

and ballast conditions

Fig. 4. Definition of the symbols of vertical and horizonal bending

moments and the location of the plane of measurement

MARINER "

STERN ARRANGEMENT

0.8 BALL AST CONDITION

g...N Pt\ 14411/4.0.5 06 03

)

0.7 04 W 09 02 1 1.0 _ 0 0 _TV2

.

1 MARINER" STERN ARRANGEMENT

0 8_I.. LOAD CONDITION

03 0.6 0 4 W 1 0

7____Y

0.2 0.9 0 _./2 n 08 CONVENTIONAL STERN ARRANGEMENT 1.0AD CONDITION L 5k r 0 6 DV

I/

0 4 W .,.

/

1 07 0 2 0.9 n/2 MARINER" ., MARINER

CONVENTIONAL STERN ARRANGEMENT STERN ARRANGEMENT

STERN ARRANGEMENT

LOAD CONDITION BALLAST CONDITION

PLANE OF MEASUREMENT

OF IBEIIDING MOMENS

I

..

gji

rv/rlyvvrvvil

a 0- 0 a ..TC--25

Fig. 5. Effect of the blade type on the d) ;ramie propeller per f or mance

90°

leo°

PROPELLER POSIT ION

13

HORIZONTAL

BENDING MOMENT

180.

PROPELLER POSIT ION 0

2700 360° INDICATION PROP NUMBER OF BLADES STERN ARRANGEMENT CONDITION POWER

ABSORPTION PER CENT

I 5 CONVENTIONAL LOAD 100

----11 5 CONVENTIONAL LOAD 100 m 5 CONVENTIONAL LOAD LOAD 100 III 5 CONVENTIONAL 100 c4, a° _87i7 90 0 815 S1N(5 0 -83 0) 1.015 0194(50-1094 0) 0 1534510(50-1180) 1.242 SIN (50 -59.0 ) no° 2700 PROPELLER POSITION 13 1.352 SIN(100 .115 0) 0.30151N(15(3 - 36.5). 0.266 5IN( 20 0 117.0) MOONS 1289 SIN (101) 390 0 774 51N osp-149.0) 0168 SIN( NO 57.1, ) 6,T(3N5 0 957 SINN:115. 3115) 047? 504)100-102 0) 0 352 5151( 7013 40.5 ..T0NS 0.924 SIN (10 13 09.0) 0161 SIN (1513 -100) 0.076 SIN( 200-1400) MOONS 360° Ui 10 8940 8803 89.34 TORQUE VARIATIONS

VERTICAL BENDING MOMENT (PROPELLER WEIGHT

EXCLUDED a° ao. 180° 270° 360° PROPELLER POSITION p - 102 30 .1750 51N(5(1 -775) 3 6075iN(04070) 482 SIN(15(3 -44 5) 0 294 SIN( 20(3 1500) TONS 103 49 2 50551N(513-103 0) 2346510)1013 46 0) 522 SIN(15 0+1575 ) 0 392 510 (2013 -15) 00945 101139 +2415 518(5)3 -103 0) 3138 5IN (10 13 29 0) 0 652 SIN(1515 .150.0. 0 495 51N ( 200 - 53.0) TONS - 10181 .7 494 501(5)3 - 50 0) 1 782 S1N(10 .116 5) 0139 5.(15 - 60 5) SIN( 700 .170 0) TONS THRUST VARIATIONS 3610° 270 4 _25 910 f ( - -I I -I ) 0°

TABLE 4a. Effect of blade type on the dynamic propeller performance (torque and thrust)

Power absorption

per cent.

4. Discussion of the test results

a.

Effect of the blade type on the dynamic

per-formance of the propeller

In Fig. 5 a comparison of the results of the various

five bladed propellers (propeller I, II, III and IV;

see Fig. 2a and table 2) has been made.

In table 4a the amplitudes of the harmonic

com-ponents of the thrust and torque are given as

per-centages of the corresponding average value.

Table 4b shows the extreme values of the

hori-zontal and vertical bending moments as percentages of the average torque, that is given as the first term

of the corresponding formula in Fig. 5. The

con-stant vertical

bending moment excited by the

propeller weight can be estimated in this research at

30 per cent. of the average propeller torque at full power.

From these data it appears that the blade type

has only a minor influence on the dynamic propeller

Harmonic components as percentage of average value

Torque Thrust

TABLE 4b. Effect of blade type on the dynamic propeller performance (bending moment) Corres-ponding

figure

forces. The pattern of the vibrations generated is

the same for the various propeller blades. Only small phase shifts due to differences in skew back occur.

b. Effect of power absorption

Fig. 6 and table 5 show the results of the dynamic

measurements for the conventional stern arrange-ment with a five bladed propeller at one hundred

and at eighty per cent. power absorption.

It is found that in the two conditions tested a

decrease in power absorption results in:

a small increase in the relative and absolute values

of the harmonic components of the torque;

a decrease in the absolute values of the harmonic

components of the thrust;

no change in the relative values of the harmonic

components of the thrust.

In addition to these data it must be noted that the frequencies of the propeller excited vibratory forces,

being directly proportional to the r.p.m., are

con-TABLE 5. Effect of the power absorption on the dynamic propeller performance (torque and thrust)

orres-nding igurc 6 Prop. Num-ber of blades Stern arrangement Condi-tion Power absorption per cent.

Bendim; Moment as percentage of average torque

(propeller we'ght excluded)

Corres-ponding figure

Vertical Horizontal

Max. Min. Max. MM.

I 5 conventional load 100

+58.4

-12.0

+35.0 +15.0

II 5 conventional load 100 +55.8

- 8.1

+34.3 +15.1

5

III 5 ccnventional load 100 +57.2

-10.0

+33.0 +18.0

IV 5 conventional load 100 +54.2

- 5.8

+35.0 +16.8

Num-Stern Condi- Power

Harmonic components as percentage of average value

c

Torque Thrust

Prop. her of

blades arran,cmcnt non absorption

per cent. 4th 1st Pc i 1st 2nd 3rd 2nd 3rd 4th I 5 conventional load 100 1.1 1.4 0.3 0.2 2.4 2.8 0.5 0.4 II 5 conventional load 80 1.6 1.6 0.4 0.3 2.5 2.8 0.7 0.2 1 st 2nd 3rd 4th 1st 2nd 3rd 4th II III

Iv

5 5 5 conventional conventional conventional conventional load load load load 100 100 100 100 0.9 1.1 1.7 1.4 1.6 1.4 1.1 1.0 0.3 0.3 0.5 0.2 0.3 0.2 0.4 0.1 1.7 2.4 2.4 2.4 3.6 2.8 3.1 1.7 0.5 0.5 0.6 0.1 0.3 0.4 0.5 0.1 5 Prop. Num-ber of blades Stern arrangement Condi-tion pattern, I I

-L

18

00

siderably affected by the degree of power

ab-sorption.

For the eighty per cent. power absorption

con-dition no bending moment measurements have been

made.

Effect of the type of stern arrangement

In Fig. 7a, b and table 6a, b, c, d comparisons have been made of the dynamic propeller forces, recorded for the conventional and the "Mariner"

stern arrangement.

From Fig. 7a and table Ga it appears that the

effect on thrust and torque is of secondary impor-tance if a five bladed propeller is used. If a

four-bladed propeller is used, it appears from Fig. 7b and

table 6b, that the amplitudes of the harmonic

thrust and torque components are smaller for the

Mariner stern arrangement than for the

conven-tional stern arrangement.

With respect to the bending moment on the shaft

it can be concluded from Fig. 7a, b and table 6c, d

that: the average values of the bending moment on

the shaft are not seriously affected by the type of

stern arrangement, the maximum and minimum

TORQUE 'VARIATIONS

1

9D° leo°

PROPELLER POSITION p

- 89 40 TO IS SIN.(5 -.105,0) ?'.289:5 IN (ISO +31.0) .!0,274,SINT15'31- /4910) ,0068 OIlS (2013 057,0) NIJONS

.7N73 I.266Sm0S -1120) I.205 SIN (100 ..51.0) ,0.307SINII9P - 2150"°(2C0 "). maoms'

THRUST VARIATIONS

It _ij I I 0 i

011.

90,0 leo°

PROPELLER POSITION p

-,0049 0.505515(52-40.310y 2. 84651N (10)1 55.0), 0522 SIN osp 650) 0159291N170117.5) TONS

4 5.9159' 2.104 SIN..(5 (3 - I, Dv 2.515 S, N(i O 014.0) 01646 SIN (15)1-03.0) km SINIZOP.'42.0)IONS

Fig, 6. Effect of the Power absorpti2n on ihe IIynanIic propeNer performance

values of the bending moment on the shaft are fof the Mariner stern arrangement smaller than for the

conventional stern arrangement if a five bladed

propeller is employed. An increase of these extreme

values has been recorded for the Mariner stern

arrangement compared with those of the conven-tional stern arrangement if a four bladed propeller is used.

From these results it can be concluded that from

the view point of propeller excited vibratory forces

the Mariner stern arrangement is

slightly more

favourable than a conventional one if a five bladed propeller is used.

If a four bladed propeller is fitted the choice of the stern arrangement depends on the importance attached by the designer to the amplitudes of the

different vibrations...

d. Effect of the number of blades

The number of blades of the propeller is the most

important parameter of the propeller .excited vi-bratory forces, not only for the amplitudes, gener-ated, but also for the frequencies generated as can

be seen from Figs. Sa, b, c and tables 7a, b,C.

2700 270° 340° 3 60 ° ,IINDICAVION PROP 1 NUMBER STERN' OF BLADES,' ARRANGEMENT II

CONDIrl'ON ABSORPTIONPOWER I

PER CENT 1 IC I I CONVENTIONAL LOAD , 100' rc 5 I .MARINER" LOAD 80 8 --s 0 -c.

34 4 0 0 TORQUE VARIATIONS _ 050 0 4 90° leo° PROPELLER 'POSITION ft - B9 <C 015516 (54-106) 289 SIN oo p 9 0 274 SIN(15 -149) 0 1695IN (2E4,7) M TO 95 5059 11311SIN ( 513- 717 ) 667SIN(10131073), 0075 (I5-95) 0191151N(20P155) NI TONS THRUST VARIATIONS

IL

/A

_

sitiTy

_o° 90° 180° PROPELLER PDS I T ION p 2;0° 360° _ .25 VERTICAL BENDING MOMENT (PROPELLER WEIGHT EXCLUDED) 103 19 SOSSIN(513-103) 846 SIN(10p46) 522S1N(1513167S) 392SIN(20p -175) TONS z 10( 11 3 342 SINOP -) 2 098SIN(1013,61) +0 259SIN(15P .121 ) 0 092 SIN(200 5°) TONS Fig. 7a.

Effect of the screw aperture on the dynamic performance of a five bladed propeller

INDICATION PROP NUMBER OF BLADES STERN 1 CONDIT iON 1 POWER

ABSORPTION PER CENT

31 5 CONVENTIONAL LOAD 100

-- I

It 5 MARINER . LOAD 100 I I I I 0. 90° 180° PROPELLER POSITION 13

HORIZONTAL BENDING MOMENT

00 90° 180° 2700 PROPELLER POSITION p 270° _ 50 25

.5

.5 /I / / 360° 0 I )

_.20 _., 1

rt(^

I 1 1 i I It t 1 I

It

1 1 t t ti I 1 t _. . I t t I 1 I I

II

I I I I I .5 1 1 I

IIt

I 1 I I Q 01° :00 913° TORQUE VARIAIHN leo° PROPELLER POSITION P 1 \ 1 \ \ 1 \ \ \ \ \ / I / \ / 8 /

\/

2170° 360° -18.22 5355i/4(0T-el 'T) ^1165 SIN (8 77 0) 0,512 SIN op ITS) 0 ITIISIN(TLp 114C)14.0000 632 SINC40 +1)2) 0 951 7.IN(11 335 0 ) 0 0 494 SIN Ott, 95.0 ) 0 114 SIN(1613113 5) 31 TONS THRUST VARIATIONS 360° 50 _25 / _50 0 VERTICAL BENDING MOMENT ( PROPELLER WEIGHT EXCLUDED) 9O° -101.30 17.12s114(0035 0) 150 SIN (8P 5) 0 03 SIN (12 p .108) 0.337s1N06p93 0) TONS .105.20 lo.ii e) 2 305 SIN (10 .7.0 + 1.166 SIN (12 0,03 7) SIN(161355 3) TONS Fig. 7b.

Effect of the screw aperture on the dynamic performance of a four bladed propeller

\

iso°

270°

PROPELLER POSITION

P

HORIZONTAL BENDING MOMENT

0 OI° 90 80° PROPELLER POSITION p 3 INDICATION PROP NUMBER OF BLADES STERN ARRANGEMENT CONDITION POWER

ABSORPTION PER CENT

41 4 CONVENTIONAL LOAD

---- SET

4 MARINER-LOAD 100 4 Lat' _..25 1 1 270° 380° 2170° leo° PROPELLER POSITION P ---360° 100 (

-TABLE Ga. Effect of the screw aperture on the dynamic propeller performance (torque and thrust)

TABLE 6b. Effect of the screw aperture on the dynamic propeller performance (torque and thrust)

TABLE 6c. Effect of the screw aperture on the dynamic propeller per (bending moment)

TABLE 6d. Effect of the screw aperture on the dynamic propeller performance (bending moment)

For the dynamic performance of the propeller a distinction must be made between the even- and the odd-bladed propellers due to the fact that for

the even-bladed propeller two blades are passing the

vertical plane simultaneously and for the

odd-bladed propellers the blades alternately pass the ver-tical above and below the propeller shaft.

In combination with thc peaks of the wake field coinciding with the vertical, the differences in the dynamic performance of the two propeller types

mentioned are clear.

With even-bladed propellers large torque and

thrust variations and small bending moment

varia- Cs-ing re res-ding ure

tions are experienced (see Fig. 8a, b and table 7a, d).

With odd-bladed propellers small torque and

thrust variations and large bending moment

varia-tions are recorded.

An increase in the even or odd number of blades

will lead to a decrease in the amplitudes of the

dynamic propeller forces due to the smaller

hydro-dynamic forces per screw blade (see in Fig. 8a the four and six bladed propeller). Fig. 8a shows that the horizontal bending moment of the six bladed

propeller is constant.

However, the above-mentioned characteristic properties of the dynamic propeller forces may not

Harmonic components as percentage of average value

Prop.

her of Stern Condi- _absorptionPower _ponding

Corrcs-Torque Thrust

blades arrangement tion

per cent. figure

1st 2nd 3rd 4th 1st 2nd 3rd 4th

II 5 conventional load NO 1.1 1.4 0.3 0.2 2.4 2.8 0.5 0.4

7a II 5 "Mariner" load 100 2.0 0.7 0.1 0.2 3.1 2.0 0.2 0.1

Harmonic components as percentage of average value 3rop. Num-her of blades Stern :arrangement .tion Power absorption per cent. Cor ppm figi Torque Thrust 1st 2nd 3rd 4th 1st 2nd 3rd 4th VI 4 conventional load 100 7.5 2.1 0.7 0.4 13.0 3.8 0.6 0.3 71 VI 4 "Mariner" load 100 5.3 1.1 0.6 0.2 9.7 2.7 1.1 0.3 Prop. Num-ber of blades Stern arrangement Cond i-tion Power absorption per cent.

Bending Moment as percentage of average torque

(propeller weight excluded )

Corres-ponding figure

Vertical Horizontal

Max. Min. Max. Min.

II II 5 5 conventional "Mariner" load load 100 100 +55.8 +43.8

-8.1

0 +34.3 +28.8 +15.1 +14.0 7a

Bending Moment as percentage of average torque

Prop. Num-her of blades Stern arrangement Con di-tion Power absorption per cent.

( propeller weight excluded) _Co

poll fil.

Vertical Horizontal

Max. Min. Max. Min.

VI VI 4 4 conventional "Mariner" load load 100 100

+29.4

+33.2 -1-18.8 +11.7 +25.0+22.2 +23.5+20.1 7 Conde-1 I

LU > 4LO LU > 0 LU 4 111 a Lii Lu 3 4 LII Lii 4 0 LU Li LU CL 0 --s _15 _010 r! A

I \

I \

i-\

I ,:\

/ \ " , \

,

T\ /

\

, . , \ , ,

ki

A \

/ 41

l k

i 'A

\ , \ \

N.

,!. \.., v

-wort

7 -.

\ 7,

,

/

`,

j/ \

\J

, \ , . \

,

..., I

,

\ ,

\

..._.; ..." ..._.... ..s...._,. - 880 11315SIN(5 -106.0) 219 SIN(10 0 39 0) 0 274 SINO5P-149 0160S1N(20P 57.0) MOONS 87 93 3 763 SINOP $BA) 0 710 01140 2 21 5) 0 285 SIN(1113 - 7,0) 03501104 p AI TONS 11822 585 SIN(I3 IN 0) 1 865 SIN(aM3 770) 0 5II2SIN(1213 111.S) 0 371 5,1(3 SP 840) M TONS _020 9100 1800 21700 3600 THRUST VARIATIONS PROPELLER POSITION 0309 2 SOS SIN(5P -103 0) 846 SiN(1013 460) 0 522 SIN(1SP +1675) 0 392 SIN(20P1 -17.5) TONS 121311 9 021SIN(60 SOO) 2.640 SINT1213 0200) 0 0 993 S1N(18P - 28.0) 0 0 143 S1N(24 ISOO)TONS 101.30 013 1205104p. es o) 3 850 5114(813 74.S) 0 618 51502P 5) 0 337 S1N(16P 90 0) TONS

Fig. 8a. Ef feel of the number of blades on lb

13IILIIIIC propeller performance (Load condition)

-.50 -025 I I I 1 1 -25 0° leo° PROPELLER POSITION 13 _025 I 0° 9100 180° PROPELLER POSITION fls 270° 360° INDICATION PROP NUMBER OF BLADES STERN ARRANGEMENT CONDITION POWER

ABSORPTION PER CENT

It 5 CONVENTIONAL LOAD 100

--y 6 CONVENTIONAL LOAD 100 yr 4 CONVENTIONAL LOAD 100 TORQUE VARIATIONS VERTICAL

BENDING MOMENT ( PROPELLER

WEIGHT EXCLUDED) 2;0° 360° 1 I I I o° so. 1800 270° 360° PROPELLER POSITION 13 _50 HORIZONTAL BENDING MOMENT 0 1 \\ 30.0 \ I \ I 1 .Ii I I _I I -p

LL, 2;0°

A'

I

A

I

,

A

'ii I

r

r

1 1 1

A

1 1 i I 1 1 l i I /

I,

\

/

I / \

/'

I \ / ... \_,/ I / ,... I\ ... 270° PROPELLER POSITION p so° TORQUE VARIATIONS 180. THRUST VARIATIONS t-1 I 1 CI I I 1 I I t I I I is \ i I I I I I I I \ t I 1 I I

',OSP 01.342 SINSp -555: 2,88S,N(13P.96.1)13.259 siN

p t21, 0.092SIN(Z: ) TONS P.a .805516(8 p .3T) 1.166 515(12131035)..3590INOSPS5.3) TONS Fig. 8b.

Effect of the number

of

blades on the dynamic propeller performance (Load condition)

360° VERTICAL BENDING MOMENT (PROPELLER WEIGHT EXCLUDED) N N I ot, so° leo° 2;0° PROPELLER POSITION p 160° 2170° PROPELLER POSITION p INDICATION PROP NUMBER OF BLADES STERN ARRANGEMENT CONDITION POWER

ABSORPTION PER CENT

LI 5 _MARINER " LOAD

---- ITC

4 MARINER " LOAD 100 i 1 1 1 I 1 i k I 1 1 , _5 a _25 0 PROPELLER POSITION p -9, 3 Sir.

0lD-0.0.087sIN(isp-9s)..o.i9es15(2o piss) m TONS

HORIZONTAL BENDING MOMENT p.,) .0.911 SIN (op .$) 5.(l2 p ..)0J34 515(16 r3t193) U TONS ..25 \ I 1 1

.\ A A

1 I

I

180° (15 0

10J w --, I u, I 1 1 1 1 o 1 5 TORQUE 'VARIATIONS

/

_

VERTICAL BENDING 'MOMENT

( PkoRELLER WEIGHT t))CLUDEO)! obg log. PROPELLER POSITIOTN 7270°

HORIZONTAL 'BENDING MOMENT

leftr

no°

360°

PROPELLER POSITION

(3

Fig. 8c, Effect of the number of blades On the dynande popeller performance (Ballast condition),

a La .i 360° INDICATION PROP NUMBER OF BLADES STERN ARRANGEMENT CONDITION POWER

ABSORPTION PER CENT

II 5 MARINER" BALLAST 100

---- 3LE

4 .. MARINER "

__

BALLAST 100 5.49151N(513 -.85 6) 0 1.938 5.(l 0 0 .172-5) , 0.945 5IN050 -1310) 0 0.0 40 SIN(200 - 41.0) 10115 .9.204 501(4 0 99.0) 1.941 5184(8p 9m) 1.351 sio 02/3 46.0 .., '285 510416 0 0 85.0) IONS -_ I 0° 90° I 80° 2700 3600 PROPELLER POSITION (3 -69.34 , 2 725 SINN .900) 1.092S1N(100 i7C.5) 0.231SIN (150 -MO 5) 0,2275101(700 20 2) /A TONS BOOS 4193 SIN (03.1ao a) t 1.123SIN (60 0117.5) 0. 0.426 SIN (l2 0 .116.0) 0.1955IN (I60 615) 16.70045 THRUST VARIATIONS 15

'I'1

I,v PI I / I I k .10 1 / I

II

/ 1

II 1

/ I I 11 1' I

I'I'

/ ' I Ll' I. I I / 1 I l'' / / 1,

,

0 I I III

A

\ I 1 5 ,Il 1 il

.

I a I ___A__ to a 1 1 \\ .\--;\ / / 1 1 1 / 1 / T., \ \ \ , / \ \ ,..---= I._ _1.. I __ 90° 180° 270° 360° PROPELLER POSITION pi -- 105.80 - 104.,71 / ( -0°

TABLE 7a. Effect of the numb(' of blades on the dynamic propeller performance (torque and thrust) Prop. II VI Num-ber of blades Stern arrangem conventi convent; conventi

TABLE 7b. Effect of the number of blades on the dynamic propeller performance (torque and thrust)

TABLE 7c. Effect of the number of blades on the dynamic propeller performance (torque and thrust)

TABLE 7d. Effect of the number of blades on the dynamic propeller Performance (bending moment)

TABLE 7e. _{Effect of the number of blades on the dynamic propeller perfornzance (bending moment)}

s-ng ,Ilt . Condi-tion Power absorption per cent.

Harmonic components as percentage of average value

Corres-ponding figure Torque Thrust 1st 2nd 3rd 4th 1st 2nd 3rd 4th °nal load 100 1.1 1.4 0.3 0.2 2.4 2.8 0.5 0.4 onal load 100 4.3 0.8 0.3 0.0 9.0 2.6 1.0 0.1 8a onal load 100 7.5 2.1 0.7 0.4 13.0 3.8 0.6 0.3

Harmonic components as percentage of average value

Prop. Num-ber of Stern . Cond 1- Power absorption Corres-ponding Thrust

blades arrangement non per cent. Torque figure

list 2nd 3rd 4th lit 2nd 3rd 4th

II 5 "Mariner" load 100 2.0 0.7 0.1 0.2 3.1 2.0 0.2 0.1

8b VI 4 "Mariner" load 100 5.3 1.1 0.6 0.2 9.7 2.7 1.1 0.3

Harmonic components as percentage of average value

P Num-her of Stern . Cond- Power absorption Corres pondin Thrust

blades arrangement tion per cent. Torque figurt

1st 2nd 3rd 4th 1st 2nd 3rd 4th

II 5 "Mariner" ballast _{100 3.1 1.2 0.3 0.3 5.2 1.8 0.8 0.0}

8c VI 4 "Mariner" ballast 100 4.8 1.3 0.5 0.2 8.8 1.9 1.3 1.3

Bending Moment as percentage of average torque rop. Num-ben of blades Stern arrangement Condi-tion Power absorption per cent.

(propeller weight excluded)

Corr,

pondi figu 1

Vertical Horizontal

Max. Min. Max. Min.

II 5 conventional _load 100 +55.8

- 8.1

_{+34.3 +15.1}

V 6 conventional load 100 +27.1 +18.1 _+25.8 +25.8 8a

VI 4 conventional load 100

_+29.4

_{+18.8 +25.0 +23.5}

Prop.

Num-ber of Stern Cond i- absorptionPower

Bending Moment as percentage of average torque d

(propeller weight exclude)

Corms-ponding Vertical

blades arrangement non per cent. Horizontal _figure

Max. Min. Max. Min.

II 5 "Mariner" load ₁₀₀ +43.8 _{0 +28.8 +14.0}

8b

VI 4 "Mariner" load 100 _+33.2

_+11.7

_{+22.2 +20.1}

16

TABLE 7f. Effect of the number of blades on the dynamic propeller performance (bending moment)

appear in extreme circumstances as can be seen from Fig. 8c and table 7c, f.

For the "Mariner" rudder arrangement in the

ballast condition the bending moment variations of

the four and the five bladed propeller are nearly

equal and the extreme differences in the amplitudes of the thrust and torque variations such as recorded

in the load condition do not exist.

As mentioned in section 2 of the paper this bal-last condition is an extreme one. However, these

extreme circumstances might occur at normal

drafts with the ship in a seaway, if the propeller tips periodically project above the water.

e. Effect of draft

In Fig. 9 and table 8 comparisons of the propeller

excited vibratory forces have been made for the

load and ballast condition of the ship with the

mariner" stern arrangement.

TABLE 8a. Effect of draft on the dynamic propeller performance (torque

and thrust)

Po abso per

From the comparison made in Fig. 9a and table 8a it appears that for the five bladed propeller the amplitudes of the harmonic components of thrust and torque are larger in the ballast condition than

in the load condition. The variations of the bending

moment on the shaft are only slightly affected by the draft. However, the effect of the draft on the

mean values of the vertical and horizontal bending moment is considerable. The latter phenomena can

lead to

a decrease in the dynamic load on

the

propeller shaft.

From Fig. 9b and table 8b it appears that the

effect of the draft on the thrust and torque

varia-tions of the four bladed propeller

is practically

negligible. Both the variations and the mean values

of the bending moments excited by a four bladed propeller are considerably affected if the propeller

tips project periodically above the water.

TABLE 8b. Effect of draft on the dynamic propeller performance (torque and thrust)

Power absorption

per cent.

Harmonic components as percentage of average value Torque 1st 2nd I 3rd 4th Thrust 1st 2nd 3rd 4th Corres-ponding figure

Bending Moment as percentage of average torque

Prop. Num-bet of blades Stern arrangement Condi-tion Power absorption per cent.

(propeller weight excluded)

Corres-ponding figure Vertical Horizontal Max. MM. Max. MM. II 5 "Mariner" ballast 100

+ 7.7

-28.2

+20.8

+ 8.8

_Sc VI 4 "Mariner" ballast 100 +13.1

-34.1

+26.0

+11.5 wer ption cent.

Harmonic components as percentage of average value

Co por fi; Torque Thrust 1st 2nd 3rd 4th 1st 2nd 3rd 4th 00 00 2.0 3.1 0.7 1.2 0.1 0.3 0.2 0.3 3.1 5.2 2.0 1.8 0.2 0.8 0.1 0.0 . . II 5 "Mariner" load II 5 "Mariner" ballast VI 4 "Mariner" load 100 5.3 1.1 0.6 0.2 9.7 2.7 1.1 0.3 9b VI 4 "Mariner" ballast 100 4.8 1.3 0.5 0.2 8.8 1.9 1.3 1.3

Num- _Stern

Condi-Prop. ber of

blades arrangement tion

Num- _Stern

Condi-Prop. ber of

blades arrangement tion

rres-ding tire a I I I i

UI U. 0 sQ.69 1.8115/N(5p-71.S) 0.667SIN(1013107.5) 0.087019 (150,9S.5) 119/1SIN(2013,S5.5) 'MONS 0. z 15134 2.725S1N(S p -90) 1.092S1N (10 13 17o.0) 0.231519 (is p-11o.s )

9.0.,sm(zo o.a) u.Toms

4 50 Lii 4 corj,, [25 0. la 5

a1 %

--\ A \ 1 \ 're / I I 1

.,s_

.., 'I ...-I 5.,. ..\ 1 -1 I . \

I

\

\ \ / , , \ .' \ / \ / \ I /

\I

h

../

\ i \ i

\1

,-.. ,.. 90° Fig. 94:

Effect of the drat/ on the dynamic performance of a like bladed ,propeller

HORIZONTAL GENOINO MOMENT INDICATION

----

PROP NUMBER

OS

STERN' ARRANGEMENT CONDITION -POWER

ABSORPTION PER CENT

II -5 _ MARINER LOAD 100 =

--- U

MARINER" BALLAST 100 TORQUE VARIATIONS VERTICAL tENDING MOMENT (PROPELLE-R WEIGHT EntUDEC) o° so° io-o° 360° PROPELLER POSITION p 2700 69 - goo 180° 1PROPELLER POSITION p I -I 0-6 so° leo° [PROPELLER POSITION p -i01,31 1342SIN(S11-,66.5) 2.0911SIN(/013 II 0) i0,259 SIN(1513 121.0 0.092S114(2013.10) TONS z 10580 S49/ SIN (5p=.o) t93I0(io_o41720)

0.e45 siN (isp i3Lo)

0.0400I9(20p-m.o) TONS

1.

270° 360° too° 270° PROPELLER POSITION p

A A A A

6 _.5 THRUST VARIATIONS

/1A

A

r r r r

I S.,

_ .10 45 -10 I o t ° 9O° i+o° 270° 36C' PROPELLER POSITION p 88.87 0052 SIN (a. 74 5) c SIN (a p ai 0) .94 SIN (lip 98 C) 3 130 055 (lapI/9 IcNs

0850 0 G Ign ,tri (4p loo,) a 1123 siN

) 51)82 *5170) El, (16p r/ TONS THRUST VARIATIONS 10071 a 2,4SIN (oa 99 0) 1 341 SIN (8P 35,,10 1213 .4,0 sO°o° leo° PROPELLER POSITION p 0555 l3215550(LIIOI0) 2 9,15,1 M3+8,7) 11065. (,,P. 100 0) 0 n3590I060.55 ,) 1.0 1 395 Sial(1005 0) 10 NS _50 25

/--,

/.,

/

A / \ / A / A 0

/

A / A

/ AS

/ A / / A

/

A

/

\

/ A

/1

/

1 / / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 8 / 1 / 1 / i -25

/

I 1 /

If

I \ / I ,/

If

IL,/ s.... L. _.25 .

Fig. 9b. Effect of the draft on the dynamic performance of a four bladed propeller

90° iso° PROPELLER POSITION 13 270° 31:1° INDICATION PROP NUMBER OF BLADES STERN ARRANGEMENT CONDITION POWER

ABSORPTION PER CENT

4 . MARINER" LOAD BALLAST 100 100 4 _MARINER" TORQUE VARIATIONS VERTICAL BENDING MOMENT (PRrPELLER WEIGHT EXCLUDED) 900 160° 270. 360° PROPELLER POSITION 0

HORIZONTAL BENDING MOMENT

;700 360° -I -N N / I -0

-_L

-TABLE 8c. Effect of draft on the dynamic propeller performance (bending moment)

TABLE 8d. Effect of draft on the dynamic propeller performance (bending moment)

5. Conclusions and RC C0117 men d a t ians

From the results of the tests the following

con-clusions can be drawn:

The effect of the number of blades on the

dynamic propeller forces is of primary impor-tance compared with the effects of blade type,

power absorption, type of stern arrangement

and draft.

In general large thrust and torque variations

are coupled with small variations in the bending

moments with a four bladed propeller. With a five bladed propeller small thrust and torque variations are coupled with large variations in the bending moments in the shaft.

If the frequency, equal to four times the r.p.m.,

is not expected to be critical from the view point

of axial or torsional shaft vibrations the

appli-cation of a four bladed propeller

is

recom-mended, due to the smaller variations in the

bending moments.

Six bladed propellers are not recommended

from the viewpoint of efficiency.

Extreme draft conditions of the ship, which

may occur in a seaway, disturb considerably the general picture of the dynamic propeller forces as concluded in a and b.

The results of this research underline once more

the conclusion made in ref. [1] that an appreciable reduction of the propeller excited vibratory forces in the shafts of single screw ships can only be

ob-tained by an extreme change in the lines of the

afterbody such as for instance the so called Hogner cigar shaped afterbody.

For further experimental research the N.S.M.B. has designed and constructed a new dynamometer

with which the dynamical thrust and torque as well as the bending moments on the shaft in two planes

can be recorded. In this way the two components,

the propeller shear force and the thrust excentricity, which are building up the bending moments can be

analysed.

For a theoretical analysis of these experimental

data wake measurements with a five hole pitot tube as described by Pien [5] will be carried out. A quasi stationary analysis of the screw propellers examined

in these wake patterns will be carried out on an

electronic digital computer according to Lerbs'

in-duction factor method. Finally dynamic effects

will be accounted for according to two dimensional

flutter data as done by Ritger and Breslin [6].

Acknowledgement

The authors wish to express their thanks to Ir. J.

E. Woltjer, Director of Drydock and Shipyard

"Wilton-Fijenoord" Ltd., for his stimulating

initia-tion of, and his interest in, this research. Refeeences

Marten, J. D. van and Kam/s, The effect of shape of

after-body on propulsion." Paper presented at Annual Meeting of the S.N.A.M.E., New York, N.Y., November 12-13,

1959.

"Report on self propulsion test and instantaneous torque- and

thrust measurements, carried out with ship model 1735 fitted

either with a four or five bladed propeller, for the single

screw motor cargo liner "Cuxhaven." Netherlands Research

Centre for Shipbuilding and Navigation (Machine

con-struction research ). N.S.M.B. Tank Test Report no. 77.

Wageningen, 1959.

Maurn, J. D. ran and Wrreldsma, R.: "Dynamic measurements

on propeller models." Joint Communication from the

N.S.M.B., Wageningen, and the Netherlands Research

Centre T.N.O. for Shipbuilding and Navigation. Intern.

Shipb. Progress, 1959.

Wrreldsme, R.: "Measurements of very small periodic signals

with unfavourable signal-noise ratio (to be published)." Pien, P. C.: "Five-hole spherical pitot tube.". D.T.M.B.-Report

no. 1229, 1958.

Ritgrr, P. D. and Breslin, J. P.: "A theroy for the quaisi-steady and unsteady thrust and torque of a propeller in a ship wake." Stevens Inst. of Techn., E.T.T.-Rapport no. 686,

1958. Prop. N um-. of Stern Condi-tion Power absorption

Bending Moment as percentage of average torque (propeller weight excluded)

Corres-ponding

Vertical Horizontal

blades arrangement per cent. figure

Max. Min. Max. MM.

II 5 "Mariner" load 100 +43.8 0 +28.8 +14.0 9a II 5 "Mariner" ballast 100

+ 7.7

-28.2

+20.8

+ 8.8

Prop. Num-her of blades Stern arrangement . Condi-tion Power absorption per cent.

Bending Moment as percentage of average torque

(propeller weight excluded)

Corres-ponding figure Vertical Horizontal Max. MM. Max. MM. VI 4 "Mariner" load 100 +33.2 +11.7 +22.2 +20.1 9b VI 4 "Mariner" ballast 100 +13.1

-34.1

+26.0 +11.5 a., J.: ber I