Nk - Shaft Alignment

GUIDELINES ON SHAFTING ALIGNMENTGUIDELINES ON SHAFTING ALIGNMENT

Part A Part A GuidelinesGuidelines on Shafting Alignmenton Shafting AlignmentTaking into AccountTaking into AccountVariation in Bearing Offsets while Variation in Bearing Offsets while in Servicein Service

June 2006June 2006

Part BPart B Amendment of the Guidance to Amendment of the Guidance to the Rules and its Explanatory the Rules and its Explanatory NotesNotes

NIPPON KAIJI KYOKAINIPPON KAIJI KYOKAI

GUIDELINES ON SHAFTING ALIGNMENT

Copyright © 2006

All rights reserved

No part of this document may be reproduced in any from, or transmitted by any means, or otherwise, without prior written permission from the Society. For information contact Research Institute or Machinery Department, NIPPON KAIJI KYOKAI


Part A Guidelines on Shafting Alignment Taking into Account Variations in Bearing Offsets while in Service

Part A

June 2006

NIPPON KAIJI KYOKAI

PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE

Contents

1 Introduction············································································································· 1

1.1 Background ·················································································································· 1 1.2 Objectives····················································································································· 1

2 General Procedure for Shafting Alignment ·························································· 3

2.1 Alignment Calculations································································································· 3 2.2 Shafting Installation ················································· ···················································· 3 2.3 Verification of Shafting Alignment················································································· 4 2.4 Uncertainties Concerning Shafting Alignment ······························································ 4

3 Modelling of Shafting for Alignment Calculation ················································· 6

3.1 Number of Engine Bearings in Model··········································································· 6 3.2 Equivalent Circular Bar Representing Crankshaft ························································ 7 3.3 Loads···························································································································11

4 Determination of Initial Bearing Offsets ····························································· 12

4.1 Construction of Shafting Stiffness Matrix···································································· 12 4.2 Target Bearing Reactions ··························································································· 14 4.3 Calculation of Initial Bearing Offsets··········································································· 15

5 Optimization of Location of Intermediate Bearing············································· 19 6 Measurement of Hull Deflection ·········································································· 22

6.1 Items and Locations of Measurement ········································································ 22 6.2 Method of Measurement ···························································································· 23 6.3 Example of Measurement Results ············································································· 27

7 Prediction of Hull Deflection by FEM ·································································· 29

7.1 Objective ···················································································································· 29 7.2 FE Model ···················································································································· 29 7.3 Loads and Boundary Conditions ················································································ 30 7.4 Effects of Analysis Conditions ···················································································· 32 7.5 Effects of Added Stiffness of Main Engine on Predicted Deflection···························· 34

i


8 Prediction of Thermal Deformation of Engine Bedplate···································· 37

8.1 Temperature of Engine Structure in Running Condition ············································· 37 8.2 Thermal Deformation of Engine Bedplate ·································································· 37

9 Dynamic Components of Hull Deflection···························································· 40

9.1 Dynamic Hull Deflection Related to Ship Motions ······················································ 40 9.2 Deformation due to Thrust·························································································· 42

10 Determination of Final Bearing Offsets ······························································ 44

10.1 Prediction of Relative Displacement over Entire Length of Shafting Line ·················· 44 10.2 Discontinuity in Slope of Total Deflection Curve over Entire Length of Shafting Line· 46 10.3 Determination of Final Bearing Offsets in Shafting Installation ·································· 48

11 Confirmation of Bearing Reactions····································································· 49

11.1 Jack-up Method·········································································································· 49 11.2 Gauge Method············································································································ 55

12 Conclusions ·········································································································· 60

ii


1. Introduction

1.1 Background

The stiffness of recently designed marine propulsion shafting has been increasing remarkably, whereas hull

structures have become more likely to deform as a result of cutting-age optimized design of the scantlings and

the use of high tensile steel. Figure 1.1 shows that the output per rpm of propulsion shafting on tankers

registered with ClassNK has increased markedly since the latter half of the 1980s. This result is considered to

be attributable to increased main engine power combined with slower rotational speed.

The reduction in stiffness of hull and engine structures means that the offsets of the support bearings of

shafting are more likely to vary under different operating conditions. On the other hand, the increased stiffness

of shafting makes shafts less adaptable to any small departure of the bearing lines from the initial lines. This

combination is thought to be the main cause of recently reported cases of alignment related main bearing

damage.

1.2 Objectives

Studies on the deflection of the engine room double bottom and its effect on shafting alignment have been

conducted in Japan since the earlier 1970s. Due attention has been given to this problem at the design stage,

and the reaction of the bearings to such deflections were usually checked under the fully loaded condition as part

of normal alignment practice at shipyards in Japan. Consequently, few main engine bearing failures related to

shafting alignment have been experienced. However, the real causes of the reported bearing damages are not so

clear that they can be prevented from recurring. Moreover, discrepancies in shafting design and installation

0

100

200

300

400

500

600

1950 1960 1970 1980 1990 2000 2010

Year of build

Mai

n en

gine

out

put p

er re

volu

tion

(kW

/rpm

)

Fig. 1.1 Evolution of main engine output per revolution for tankers.

1


practice vary appreciably between shipyards. These are potential risks to causing shafting alignment related

problems.

From this perspective, ClassNK has carried out intensive research in this area during the past five years, with

the aim of providing the industry with consistent and comprehensive guidelines on shafting design and

installation.

These guidelines reflect our wealth of experience and the latest research achievement in this field and have

been developed to assist marine engineers ensure the integrity of shafting alignment from design to installation.

2


2. General Procedure for Shafting Alignment Shafting alignment is a process of calculation, installation, and confirmation, as well as readjusting, if necessary in order to ensure that the all bearings are appropriately loaded and no excessive bending present in any section. 2.1 Alignment Calculations When conducting an alignment calculation, the shafting is modeled using a continuous beam supported by bearings with due offsets, as shown in Fig. 2.1. The bearing reaction, bending, and shear stresses at each section are to be calculated in order to check if they are in compliance with predetermined acceptance criteria. 2.2 Shafting Installation During installation, the shafts including propeller shaft, intermediate shaft and crank shaft are decoupled from each other and laid down on the supports first, as shown in Fig. 2.2. Then, necessary adjustment of the height of each support, including possible temporary supports, are made to ensure that the calculated "GAP" and "SAG" between the mating flanges are realized. That is to say, although the appropriate bearing offsets can be determined by calculation, it is extremely difficult to check the offsets during installation; therefore, the gaps and sags are used as an indication of the bearing offsets actually realized. When shafts cannot be stably laid down alone, temporary supports or additional external forces provided by jacks may be added as long as they are taken into account in the calculations.

Fig. 2.1 Calculated shaft alignment.

Reference line

Bearing offset

SAG

SAG GAP

GAP

Jack force needed to stabilize the propeller shaft.

Temporary support

Fig. 2.2 Aligning shafts based on Gap and Sag method.

Specifically, the propeller shaft is laid down first, then its flange is taken as reference to adjust the height of

each support, including possible temporary supports, for the intermediate shaft to ensure that the calculated "GAP" and "SAG" between the mating flanges are realized. After the intermediate shaft has been laid down, its forward flange becomes a new reference for adjusting the position of the main engine by raising, lowering or 3


tilting the engine to ensure that the calculated "GAP" and "SAG" between the mating flanges are realized. The gaps and sags are measured using filler gauges. The designed gaps and sags should be, therefore, as large as possible in order to achieve a high accuracy of measurement.

2.3 Verification of Shafting Alignment When verifying the shafting alignment, the forward sterntube bearing and intermediate bearing are to be jacked up to see if they are producing satisfactory reactions. A jack up test on the main engine bearings, especially the two aftmost bearings, is also highly desirable. If circumstances do not allow such a test, alternatively the gauge method described in Chapter 11 is recommended, or at least crank web deflections should be measured in order to verify that they are in compliance with the criteria of the manufacturer.

2.4 Uncertainties Concerning Shafting Alignment 2.4.1 Various Errors The following errors will inevitably exist in the shafting alignment process, including calculation, installation, and verification:

- approximation of the shafting model for calculation (for example, the number of bearings taken into account, modeling of the sterntube bearing, and the diameter of the circular bar representing crankshaft in the model, etc.);

- the accuracy of the Gap, Sag method (namely possible discrepancies between calculation results and actual installation); and

- the accuracy of bearing reaction measurements.

Therefore, the results obtained should be judged after these errors have been properly estimated based on past experience. 2.4.2 Difference of Conditions between Shafting Installation and Typical Service Condition Shafting installation is usually performed in the so called launched condition with light draft and the main engine in a cold condition. However, when the vessel is in typical service condition, the draft, especially for tankers or bulk carriers, will change considerably, and the temperature in the main engine structure and nearby hull structure will rise. These changes will cause additional deflections of the hull as well as the main engine structure leading to the variations in bearing offsets. The bearing reactions will also change, accordingly.

It is natural to make the shafting alignment fitting for the typical service condition. Therefore, the initial bearing offsets should be compensated by taking account of the estimated variation as shown in Fig. 2.3 in case that unsatisfactory result is predicted if without such compensation. It can also be expressed by Eq. (2.1).

condition)Launched -condition loaded (Fully -condition Launched offset bearingInitially = (2.1)

Table 2.1 shows an example of calculated bearing offsets using Eq. (2.1). As can be seen from the table, the

initial bearing offsets should be compensated by an amount described by the term of "Fully loaded condition - Launched condition" in the right side of Eq. (2.1) represents the variation in bearing offsets between the shafting installation and the typical service condition in order to ensure that the shafting is satisfactory in its typical service condition.

4


-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

30000

Bearing location, distance from left end (mm)O

ffse

t (m

m)

0 5000 10000 15000 20000 25000

Launched conditionFully loaded conditionInitial bearing offsetShaft deflection line (Launched condition)Shaft deflection line (Fully loaded condition)Shaft deflection line (Initial bearing offset)

Fig. 2.3 Bearing offsets and shaft deflection lines under different conditions.

Table 2.1 Initial Bearing Offsets to Compensate for Bearing Height Variations in Service Condition

Bearing offset (mm) Bearing location (mm) Launched condition Fully loaded condition Initially bearing offset2830 0.400 3.000 -2.200 4630 0.000 2.500 -2.500 8465 0.050 2.000 -1.900 15295 0.900 1.500 0.300 22375 1.600 0.800 2.400 23375 1.600 0.700 2.500 24875 1.600 0.500 2.700 26375 1.600 0.700 2.500 27875 1.600 1.000 2.200

However, if the shafting alignment is predicted to be unsatisfactory under other conceivable operating conditions such as light ballast, then the compensation may need to be reduced to an extent by which all operating conditions can be accommodated.

5


3. Modelling of Shafting for Alignment Calculation

3.1 Number of Engine Bearings in Model Since the emphasis in traditional shafting alignment calculations has been placed on avoiding edge loading in sterntube bearings, little attention has been paid to main engine bearings. Figure 3.1 shows the number of engine bearings taken into account in the alignment calculations with respect to the number of engine cylinders in the actual practice of various shipyards. As can be seen from Fig. 3.1, in some cases only the aftmost three M/E bearings were incorporated into the calculation model, although this varies from yard to yard.

Fig. 3.1 The numbers of main bearings taken into account in current alignment calculations by builders.

Number of engine cylinders

Num

ber o

f bea

rings

incl

uded

However, the results of analysis suggests that the number of M/E bearings in the calculation model has appreciable effects on the calculated reactions of the aftmost three M/E bearings, while no noticeable effects can be seen on the sterntube and intermediate bearing reactions, as shown in Fig. 3.2, when the number of bearings is changed from three to nine for a seven cylinder engine. In particular, the reactions of the second and third aftmost M/E bearings will change remarkably when the number of M/E bearing changes from three to four. Supposing the reactions are true when nine (full number) M/E bearings are taken into account, the reaction of the second aftmost M/E bearing is overestimated, while the reaction of the third aftmost M/E bearing is underestimated when only three M/E bearings are incorporated into the calculation model. When the number of M/E bearings is increased to four, the result is reversed, namely, the reaction of the second aftmost M/E bearing is underestimated, while the reaction of the third aftmost M/E bearing is overestimated. The result is close to the true value when the number of M/E bearings is increased to five or more.

Although the above result was based on a seven-cylinder engine, the result is virtually independent of the number of cylinders, although it was used as an example here. That is to say that when the reaction of a M/E bearing needs to be accurately evaluated, two more M/E bearings beyond the bearing have to be incorporated in the calculation model. For example, if the reactions of five aftmost M/E bearings of a nine-cylinder engine have to be calculated precisely, then seven M/E bearings in total need to be included in the calculation model.

6


For a four-cylinder engine, if the reactions of the five aftmost M/E bearings have to be calculated precisely, then all six M/E bearings need to be taken into consideration. Since the aftmost three M/E bearings are most likely to be affected by the alignment, at least five M/E bearings must be used in the alignment calculation.

Fig. 3.2 Effect of numbers of main bearings taken into account in alignment calculations on bearing reactions.

Bearing number

Rea

ctio

n fo

rce

(kgf

)

ThreeFourFiveSixSevenEightNine

3.2 Equivalent Circular Bar Representing Crankshaft

3.2.1 Necessity of Determining Equivalent Circular Bar Representing the Crankshaft It is necessary to replace the crankshaft with a circular bar in the alignment calculations. However, a suitable method for determining the equivalent circular bar of a crankshaft has still not yet been clearly established, and in some cases, the crankshaft is simply replaced by a circular bar with a diameter equal to that of the crank journal. However, it is commonly considered that the bending stiffness of a crankshaft, especially in the case of long-stroke crankshafts, is generally lower than that of a circular bar with a diameter equal to the crankshaft due to the effect of webs, although it will vary slightly with crankshaft position. In order to prevent M/E bearings from alignment related damage, it is necessary to calculate the reactions of the M/E bearings accurately. Therefore, a circular bar equivalent in bending stiffness to the crankshaft has to be determined. In one example considered here, the diameter of the equivalent circular bar of a long-stroke crankshaft is determined as being about 60% of the journal diameter. Figure 3.3 shows a comparison of the calculated aftmost three bearing reactions between cases in which a circular bar represents the crankshaft with 100% and 60% of the journal diameter, respectively.

As can been seen from Fig. 3.3, when a circular bar with a diameter of 60% of the crank journal is adopted, the reaction of the aftmost M/E bearing decreases to almost half of that when the journal diameter is used, while the reaction of the second aftmost M/E bearing is expected to increase. That is to say, how the crankshaft is modelled will have a significant effect on the accuracy of determining the reactions of the aftmost M/E bearings.

7


- Equivalent Journal

-

Rea

ctio

n fo

rce

(kgf

)

- -

- Bearing number Fig. 3.3 Effect of crank equivalent diameter on bearing reactions.

3.2.2 Method for Determining Equivalent Diameter (1) Numerical Calculation by FEM The purpose of determining the equivalent diameter of the crankshaft is to ensure that the reaction of the bearings can be calculated correctly when bearing offsets change. Therefore, a detailed FE model of a crankshaft was first developed, as shown in Fig. 3.4(a). In the model, the left end was completely restrained, and an enforced vertical displacement was given at each bearing supporting point. The reactions of the supports were calculated using this model. At the same time, a FE model of a circular bar with identical boundary conditions in the detailed model was also developed, as shown in Fig. 3.4(b). The process of determining the equivalent diameter consists of gradually decreasing the diameter of the circular bar model until the calculated reactions become almost equal to those obtained from the detailed model.

0 1 2 3 4 5 6 0 1 2 3 4 5 6

(a)

All degrees of freedom restrained

(b)

Enforced vertical displacement

Fig. 3.4 (a) FE model of crankshaft, (b) FE model of circular bar.

The results show that a circular bar with a diameter of 60% of the journal diameter behaves in the same as the

crankshaft, as shown in Fig. 3.5.

8


Fig. 3.5 Comparison of bearing reaction between crank and circular bars of different diameters.

Bearing number

Rea

ctio

n fo

rce

(kgf

)

Crank Circular bar of diameter equal to journal

Round bar of diameter equal to 60% of journal

----

(2) Approximate Analytical Expression Assuming that the diameter of crankpin is equal to that of the crank journal, the equivalent diameter of a crankshaft can be calculated from the dimensions shown in Fig. 3.6 using Eq. (3.1).

t

l

L

dj

r

B

Fig. 3.6 Necessary dimensions of a crank throw for determining equivalent diameter.

9


jpww

eq dBBA

d41

11

11

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

+++

+= (3.1)

2

2

1

6501

1

23

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎠⎞

⎜⎝⎛+

=

Ld

.

Ll

Lrq

II

Ajw

jw

( )2

2

2

6501

113

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎠⎞

⎜⎝⎛++

=

Ld

.

Ll

Lrq

II

Bjwp

jw

ν

( )23

2

6501

113

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

Ld

.Llr

II

Bjjp

jp

ν

ν: the Poisson's Ratio of the crankshaft material

64

4j

jd

Iπ

=

32

4j

jpd

Iπ

=

12

3BtIw =

3BtIwp β=

1449623006058090008285700004057023

.tB.

tB.

tB. +⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=β

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

3

4

3

4

1

101

101

Btd

.dr

Btd

.q

j

j

j

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

3

4

3

4

2

4501

4501

Btd

.dr

Btd

.q

j

j

j

10


(3) Recommended Values by Engine Makers Where recommended equivalent diameters by the engine manufacturer are available, the recommended values are to be used. 3.3 Loads The selfweight of an equivalent circular bar should be compensated to the same extent as the crankshaft by increasing its specific gravity or by adding the selfweight differential to loads representing the piston weight.

Loads representing the propeller weight should be determined taking into account the buoyancy corresponding to the draft predicted in the condition in which the shafting alignment calculation is performed.

Loads representing the selfweight of the propeller shaft should also be modified by reducing the specific gravity to reflect the effect of the buoyancy of lubricating oil or sea water, depending on the type of lubricating system used for the sterntube bearings.

11


4. Determination of Initial Bearing Offsets

When aligning the shafting, the initial bearing offsets should be determined first. In other words, a decision

has to be made regarding whether a simple straight alignment is acceptable or appropriate offsets are necessary.

In practice, the initial bearing offsets are determined based on past experience at most shipyards.

Herein，a scientific approach to determining the bearing offsets based on a shafting stiffness matrix is

proposed. This approach is specifically designed to allow the offsets to be calculated directly from a straight

shafting line based on the target bearing reactions.

4.1 Construction of Shafting Stiffness Matrix

When a unit offset is given at a given bearing supporting point (positive downward, negative upward), reaction

forces will be generated at each bearing supporting point. These reaction forces are referred to by a bearing

reaction influential number (hereafter referred to as "reaction influential number"). The shafting stiffness

matrix is constructed by arraying these reaction influential numbers in a certain order. It is important to

understand that reactions used to construct the shafting stiffness matrix must be the effect only from the given

offset. When the result contains some components resulting from external loads or selfweight, these

components must be eliminated before the result can be used. Therefore, it is strongly recommended that the

reactions are calculated without any external loads, and that selfweight is neglected.

Figure 4.1 shows an example of a shafting alignment calculation model incorporating five M/E bearings and

representing the shafting of a VLCC. Cross-sectional data of the shafting in different sections are shown in

Table 4.1. Table 4.2 shows the bearing reactions at each bearing supporting point when a downward offset of

1.0 mm is applied to the No. 1 bearing.

Table 4.1 Sectional Parameters of Sample Shafting

Arrangement

12

Did. (mm)Outer Inner Location (mm) I value (mm^4)

675.0 - 5.6000E+02 1.019025E+10874.6 - 1.4000E+03 2.872155E+10896.0 - 1.4235E+03 3.166574E+10918.4 - 2.2870E+03 3.492186E+10940.0 - 7.7850E+03 3.832492E+10942.0 - 9.5730E+03 3.865214E+10903.5 1.0003E+04 3.271015E+10865.0 - 1.0203E+04 2.748111E+10

1180.0 - 1.0483E+04 9.516953E+10725.0 - 1.4850E+04 1.356194E+10730.0 - 1.5740E+04 1.393995E+10725.0 - 2.1155E+04 1.356194E+10

1270.0 - 2.1294E+04 1.276982E+11980.0 - 2.1515E+04 1.276962E+11980.0 80 2.3375E+04 4.527463E+10536.0 - 2.7875E+04 4.051623E+09

Fig. 4.1 A shafting arrangement used to demonstrate how to create a shaft stiffness matrix.


Table 4.2 Reaction on Each Bearing Resulting from Lowering the No. 1 bearing by 1.0 mm

BearingNo. Location (mm) Offset (mm) Reaction (kgf)

No. 1 2.8300E+03

1.000 139295.220No. 2 4.6300E+03 0.000 -214398.670No. 3 8.4650E+03 0.000 81956.652No. 4 1.5295E+04 0.000 -9075.998No. 5 2.2375E+04 0.000 8643.455No. 6 2.3375E+04 0.000 -6589.810No. 7 2.4875E+04 0.000 213.664No. 8 2.6375E+04 0.000 -53.416No. 9 2.7875E+04 0.000 8.903

By repeating the above calculation for each bearing in turn, a matrix as shown in Table 4.3 can be obtained.

This result is precisely a shafting stiffness matrix that clearly takes the form of a square and symmetric matrix.

Table 4.3 Shaft Stiffness Matrix for Sample Shaft Arrangements

Offset (mm)

eaction (kgf)

δ1(1mm)

δ2(1mm)

δ3(1mm)

δ4(1mm)

δ5(1mm)

δ6(1mm)

δ7(1mm)

δ8(1mm)

δ9(1mm) R

13

139295.220 -214398.670 81956.652 -9075.998 8643.455 -6589.810 213.664 -53.416 8.903R1

-214398.670 342737.790 -147868.080 25863.061 -24630.482 18778.394 -608.860 152.215 -25.369R2 81956.652 -147868.080 86843.213 -29740.647 34253.719 -26115.195 846.745 -211.686 35.281R3

-9075.998 25863.061 -29740.647 25755.066 -64621.265 53184.964 -1724.440 431.110 -71.852R4

8643.455 -24630.482 34253.719 -64621.265 487261.830 -602809.510 204508.110 -51127.028 8521.171R5

-6589.810 18778.394 -26115.195 53184.964 -602809.510 895929.320 -459651.520 152728.030 -25454.671 R6

213.664 -608.860 846.745 -1724.440 204508.110 -459651.520 451271.610 -264078.490 69223.181R7 -53.416 152.215 -211.686 431.110 -51127.028 152728.030 -264078.490 255095.360 -92936.091R8

8.903 -25.369 35.281 -71.852 8521.171 -25454.671 69223.181 -92936.091 40699.447R9

That is to say that the bearing reactions (without the effect of external loads and selfweight) can be calculated

from the known set of offsets through the shafting stiffness matrix expressed in Eq. (4.1).

139295.220 -214398.670 81956.652 -9075.998 8643.455 -6589.810 213.664 -53.416 8.903

-214398.670 342737.790 -147868.080 25863.061 -24630.482 18778.394 -608.860 152.215 -25.36981956.652 -147868.080 86843.213 -29740.647 34253.719 -26115.195 846.745 -211.686 35.281-9075.998 25863.061 -29740.647 25755.066 -64621.265 53184.964 -1724.440 431.110 -71.8528643.455 -24630.482 34253.719 -64621.265 487261.830 -602809.510 204508.110 -51127.028 8521.171

-6589.810 18778.394 -26115.195 53184.964 -602809.510 895929.320 -459651.520 152728.030 -25454.671213.664 -608.860 846.745 -1724.440 204508.110 -459651.520 451271.610 -264078.490 69223.181-53.416 152.215 -211.686 431.110 -51127.028 152728.030 -264078.490 255095.360 -92936.091

8.903 -25.369 35.281 -71.852 8521.171 -25454.671 69223.181 -92936.091 40699.447

=

R1R2R3R4R5R6R7R8R9

δ 1δ 2δ 3δ 4δ 5δ 6δ 7δ 8δ 9

(4.1)


For simplicity, Eq.(4.1) may be rewritten in matrix and vector notation as shown in Eq.(4.2):

AδR = (4.2)

where

R={R1 R2…R9}T

139295.220 -214398.670 81956.652 -9075.998 8643.455 -6589.810 213.664 -53.416 8.903-214398.670 342737.790 -147868.080 25863.061 -24630.482 18778.394 -608.860 152.215 -25.369

81956.652 -147868.080 86843.213 -29740.647 34253.719 -26115.195 846.745 -211.686 35.281-9075.998 25863.061 -29740.647 25755.066 -64621.265 53184.964 -1724.440 431.110 -71.8528643.455 -24630.482 34253.719 -64621.265 487261.830 -602809.510 204508.110 -51127.028 8521.171

-6589.810 18778.394 -26115.195 53184.964 -602809.510 895929.320 -459651.520 152728.030 -25454.671213.664 -608.860 846.745 -1724.440 204508.110 -459651.520 451271.610 -264078.490 69223.181-53.416 152.215 -211.686 431.110 -51127.028 152728.030 -264078.490 255095.360 -92936.091

8.903 -25.369 35.281 -71.852 8521.171 -25454.671 69223.181 -92936.091 40699.447

A =

δ=｛δ1 δ2…δ9｝T

4.2 Target Bearing Reactions

The bearing reactions are to be calculated in the straight shafting line condition first. For small ships, even a

straight shafting alignment may in some cases meet the relevant requirement, but for large ships, the results from

the straight shafting line condition usually do not meet the required acceptance criteria. Figure 4.2 shows the

calculated deflection line, bending moment and shear force of the shafting model shown in Fig. 4.1.

Fig. 4.2 Deflection, bending moment and shear force for sample shaft arrangement under straight offset of bearings condition.

14


The calculated bearing reactions are shown in Fig. 4.3.

-150000

-100000

-50000

0

50000

100000No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9

Bearing Number

Reac

tion

forc

e (k

gf)

Bearing reaction. Negative values indicate upwardreactions while positive values indicate downwardreactions

Fig. 4.3 Bearing reaction forces for sample shaft arrangement under straight offset of bearings condition.

As can be seen in Fig. 4.3, the aft end bearing of the aftmost stern tube is overloaded, while the fore end

bearing of the aftmost stern tube is loaded in the wrong direction. Therefore, the alignment must be modified.

Assuming the target bearing reactions are known as shown in Table 4.4, an appropriate set of bearing offsets

should be determined so that the difference in the reaction forces between the target reactions and the reactions

under a straight condition, ΔR, can be generated.

Table 4.4 Target Bearing Reaction Forces for Sample Shaft Arrangement

Bearing No. Reactions of straight offset

(kgf)

Target reactions

(kgf)

ΔR = Target - Straight (kgf)

No. 1 -143537 -88333 55204 No. 2 62921 -17079 -80000 No. 3 -35917 -11917 24000 No. 4 -21989 -23989 -2000 No. 5 -48172 -28172 20000 No. 6 -8661 -26661 -18000 No. 7 -42226 -41726 500 No. 8 -51097 -51277 -180 No. 9 -15295 -15265 30

4.3 Calculation of Initial Bearing Offsets

The necessary initial offsets, δ, can be calculated backward from the required reactions, ΔR by Eq. (4.3) which is

a rewritten form of Eq. (4.2), solving for δ.

15


RAδ -1Δ= (4.3)

This can be expressed in following detailed form:

-1

139295.200 -214398.600 81956.600 -9075.900 8643.400 -6589.800 213.600 -53.400 8.900 -214398.600 342737.700 -147868.000 25863.000 -24630.400 18778.300 -608.800 152.210 -25.360

81956.600 -147868.000 86843.200 -29740.600 34253.700 -26115.100 846.700 -212.600 35.280 -9075.900 25863.000 -29740.600 25755.000 -64621.200 53184.900 -1724.400 431.110 -71.850 8643.000 -24630.400 34253.700 -64621.200 487261.800 -602809.500 204508.100 -51127.000 8521.100

-6589.800 18778.300 -26115.100 53185.900 -602809.500 895929.300 -459651.500 152728.000 -25454.600 213.660 -608.800 846.740 -1724.400 204508.100 -459651.500 451271.600 -264078.400 69223.100 -53.410 152.200 -211.680 431.110 -51127.000 152728.000 -264078.400 255095.300 -92936.000

8.900 -25.360 35.281 -72.850 8521.100 -25454.600 65223.100 -92936.000 40699.400

55204-8000024000-200020000

-18000500

-18030

However, because of the characteristic of the shafting stiffness matrix, the above equation will have infinitely

many solutions. Hence, a submatrix of this matrix by omitting both the first and last rows and first and last

columns is needed in order to solve this problem. This operation means that the constraints of the rigid

motions do not affect the bearings, including translation and rotation of the shafting line. The operation is

specifically to set both ends of the shafting line at an offset of zero. Accordingly, the solution for the No. 2 to

No. 8 offsets can be obtained as follows:

-1

-0.5225819432 -0.7255192463 -0.2621283778 0.1235714606 0.0892630911 0.0471291196 0.0955979752

δ2 δ3 δ4 δ5 δ6 δ7 δ8

=

=

342737.700 -147868.000 25863.000 -24630.400 18778.300 -608.800 152.210 -147868.000 86843.200 -29740.600 34253.700 -26115.100 846.700 -212.600

25863.000 -29740.600 25755.000 -64621.200 53184.900 -1724.400 431.110 -24630.400 34253.700 -64621.200 487261.800 -602809.500 204508.100 -51127.000 18778.300 -26115.100 53185.900 -602809.500 895929.300 -459651.500 152728.000

-608.800 846.740 -1724.400 204508.100 -459651.500 451271.600 -264078.400 152.200 -211.680 431.110 -51127.000 152728.000 -264078.400 255095.300

-8000024000-200020000

-18000500

-180

Therefore, the solution for all bearing offsets is as follows:

0

-0.52258194320 -0.72551924630 -0.26212837780 0.12357146060 0.08926309105 0.04712911959 0.01955979752

0

In practice, main engines are usually installed in a horizontal position. This can be realized by rotating the

whole shafting line by an angle equal to the slope of the engine section. The bearing offsets after rotation are 16


shown in Table 4.5 together with the respective calculated values. Fig. 4.4 is a graphical expression of the

calculated target bearing offsets.

Table 4.5 Calculated Target Bearing Offsets

Bearing No.

Bearing location

Bearing offsets as calculated

Bearing offsets with engine in nearly

horizontal position

No. 1 2830 0.0000 -0.5521 No. 2 4630 -0.5226 -1.0344 No. 3 8465 -0.7255 -1.1514 No. 4 15295 -0.2621 -0.5349 No. 5 22375 0.1236 0.0094 No. 6 23375 0.0893 -0.0025 No. 7 24875 0.0471 -0.0110 No. 8 26375 0.0196 -0.0050 No. 9 27875 0.0000 0.0091

Initail Bearing Offsets Obtained by Direct Calculation, (-) Upward Offset and (+)

Downward Offset

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.500 5000 10000 15000 20000 25000 30000

Bearing location, distance from left end (mm)

Off

set (

mm

)

Bearing offsets as calculated Bearing offsets with engine in nearlyhorizontal position

Fig. 4.4 Calculated target bearing offsets.

All translation or rotation of the shafting line as a whole will not affect the reactions of bearings as described

above, though small calculation errors could exist. Table 4.6 shows a comparison of the bearing reactions,

while Fig. 4.5 is their graphical expression. It can be seen from these results that the target bearing reactions

have been obtained.

17


Table 4.6 Comparison of Bearing Reactions between Offset Sets Shown in Table 4.5

Bearing reaction force (kgf)

Bearing No. Target Offsets as

calculated

Offsets with engine in nearly

horizontal position

No. 1 -88333 -88084 -88082 No. 2 -17079 -17088 -17089 No. 3 -11917 -11913 -11914 No. 4 -23989 -23989 -23988 No. 5 -28172 -28189 -28169 No. 6 -26661 -26624 -26672 No. 7 -41726 -41761 -41709 No. 8 -51277 -51254 -51290 No. 9 -15265 -15069 -15058

Bearing Reaction Force (kgf), (-) Upward and (+) Downward

-100000

-80000

-60000

-40000

-20000

0

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9

Bearing Number

Reac

tion

forc

e (k

gf)

Offsets as calculatedOffsets with engine in nearly horizontal position

Fig. 4.5 Comparison of bearing reactions between offset sets shown in Table 4.5.

18


5. Optimization of Location of Intermediate Bearing

The longitudinal placement of the intermediate bearing will have strong effect on the reaction of the aftmost

engine bearing when bearing offsets vary due to changes in ship draught. In fact, there was one case in which

damage to the aftmost engine bearing occurred shortly after the vessel was entered into service because the

intermediate bearing was installed too close to the aftmost engine bearing.

The location of the intermediate bearing has thus far been determined on the basis of experience, and there has

not been a rigorous, scientific approach to the placement of bearings.

It is generally effective to install the intermediate bearing as far as possible from the engine in order to reduce

its effect on the aftmost engine bearings when there is a change in the bearing offsets. On the other hand,

however, the influence number of the intermediate bearing itself will become greater because of the proximity of

the intermediate bearing to the sterntube bearing. A balanced judgment is needed, accordingly. The effect of

the location of the intermediate bearing on the influence number for each bearing was calculated in the example

shown in Fig. 5.1. The result of the calculation is shown in Fig. 5.2.

No.1 No.2 No.3 No.4 No.5 No.6

No.7 No.8 No.9

Fig. 5.1 An example of a shafting arrangement used to demonstrate how to optimize the location of the intermediate bearing.

19


Fig. 5.2 Influence number of intermediate bearing with respect to the other bearings.

-200000

-150000

-100000

-50000

0

50000

100000

150000

200000

5000 7000 9000 11000 13000 15000 17000 19000 21000

Intermediate Bearing Location (mm)

Reac

tion

Influ

ence

Num

ber

R41 R42R43 R44R45 R46R47 R48R49

Judging from the influence number of the intermediate bearing to itself, R44, as shown in Fig. 5.2., it may be

said that the level of sensitivity is the lowest for the current design position of the intermediate bearing, at 15,295

mm. However, if the influence on the first and second aftmost engine bearings is taken into consideration, the

intermediate bearing should be placed further aftward 2,795 mm to a position of 12,500 mm.

Thus, in order to consider the extent of influence on all bearings, a sensitivity index was proposed as

expressed in Eq. (5.1).

( )

number bearingteIntermedia :mionconsiderat into taken bearings totalofNumber :N

gith bearin to bearingteintermedia oft coefficien Influence :R

Rindexy Sensitivit

mi

N

1i

2mi∑

=

=

(5.1)

The result of applying Eq. (5.1) to the fore-mentioned example is shown in Fig. 5.3. The result suggests that

a unique optimal location for the intermediate bearing exists, in terms of minimum sensitivity. In this example,

the optimal point is 14,500 mm, moving aftward for 795 mm from the original design point.

20


21

LIM

Fig. 5.3 Optimal location of intermediate bearing obtained from sensitivity analysis.

0.00E+00

5.00E+09

1.00E+10

1.50E+10

2.00E+10

10000 11000 12000 13000 14000 15000 16000 17000 18000

Position of intermediate bearing (mm), LIM

Sesit

ivity

Inde

x

Initial design pointOptimized point

14,500


6. Measurement of Hull Deflection

6.1 Items and Locations of Measurement

6.1.1 Locations to Be Measured When a ship changes from light draught to full draught, the aft part of the hull and shafting are estimated to exhibit a hogging deflection as shown in Fig. 6.1.

Fig. 6.1 Illustration showing hull deflection from light to full draught: (a) light draught; (b) full draught.

(b)

(a)

In order to be able to investigate the effect of hull deflection on shafting alignment, it is ideal to measure the changes that take occur in each bearing offset beneath the shafting line from the foremost engine bearing to the propeller. However, the locations that can be practically measured are limited to points on the dotted red lines shown in Fig. 6.2 due to restrictions arising from the arrangement of the hull structure, engine, and shafting line. In other words, measurements are only possible on the tank top beneath the shafting line from the aft end of the engine to the fore sterntube bearing and alongside the engine. In this regard, some discrepancy may exist between the measured results and actual changes in the bearing offsets. Consequently, careful attention should be paid when applying the measured results in determining shafting alignment, which will be detailed hereinafter.

6.1.2 Measurement Items The principal items to be measured include the relative deflection of the tank top from one draft condition to another, represented in dark and red in Fig. 6.3, with respect to a reference line connecting two ends of the measured line. Another method, depicted in blue, is to measure the relative deflection at one end of the measurement line from one draft condition to another with respect to a reference point produced by a laser beam from the other end of the measurement line. As can be seen from the figure, when the line passing through the 22


two end points of the measurement line is being taken as a reference line, highly accurate measurement is necessary because the relative displacement could be very small. However, a better understanding of the entire profile of the displacement along the whole length of the measurement line can be gained by using this method and conducting measurements at multiple points. On the other hand, although comparatively large displacements can be measured using the reference point at one end produced by a laser beam emitted from the other end of the measurement line, the measurement of middle points will be difficult.

Possible measurement lines

Fig. 6.2 Possible locations for measuring hull deflections.

Reference lines

Displacements to be measured

Laser beam as reference line after deformation

Fig. 6.3 Illustration showing displacements to be measured and their respective reference lines.

6.2 Method of Measurement

6.2.1 Measurement Apparatus There are several measurement methods available that can be used to measure displacement depending on how the reference line is set and how the measurement is made. The so-called 'piano-wire method' has been commonly used so far when a line passing through two end points is established as the reference line, in which a tightened piano wire connecting the two end points is used as the reference line. When the piano wire method is applied, a nearly truly straight reference line can be established because piano wire is very thin and able to withstand comparatively large tension forces. Additionally, piano wire sagging arising from its selfweight can be calculated for correction, if necessary. However, the possibly greatest disadvantage of the piano-wire

23


method is that the measurement has to be made and read off manually. Therefore, it is difficult for the personnel conducting the measurement to repeat the measurements under varying conditions and to measure the displacement at multiple points simultaneously. In addition, problems arise with the accuracy of the measurements when the displacement is small.

Therefore, ClassNK Research Institute has developed a measuring apparatus using a steel strip with a very small cross sectional area as the reference line instead of piano wire in combination with highly precise laser type displacement sensors. This makes it possible to measure displacements at multiple points simultaneously with high precision. The system has already been successfully employed in actual measurements onboard a VLCC tanker for the first time in the world in August 2004.

In addition to the above mentioned features of high precision, multi-point and automatic measurement, there is another merit to this approach in which possible dynamic components in displacement can be measured, as the measurement system allows continuous measurement to take place. The concept and mechanism of the measurement system is shown in Fig. 6.4. Figure 6.5. shows the apparatus that was installed onboard the above mentioned VLCC tanker.

Tightened steel strip

Laser displacement sensor Deflection line after change in condition (e.g. full loaded condition)

Initial deflection line (e.g. ballast condition)

Change in displacement between conditions

Fig. 6.4 Concept of a new hull deflection measurement system.

Since the tension force in the steel strip generated by weight can be regarded as being constant, the sagging of

the steel strip remains unchanged regardless of the draft or speed conditions of the vessel. Furthermore, since the purpose of the measurement system is to measure the deflection between two different conditions, once the first condition is set as the initial condition, only the change in deflection from the initial state needs to be recorded, therefore, it is unnecessary to know the actual extent of sagging of the steel strip or to make corrections for such sagging.

In order to eliminate the effect of possible lateral and axial vibration of the steel strip during measurement, a low pass filter with a cut-off frequency of 0.6 Hz was employed for each channel, after confirming that the resonant frequencies of the steel strip in these kinds of vibrations were higher than 5 Hz. The characteristic of the low pass filter is shown in Fig. 6.6.

The configuration of the measurement system was very simple as shown in Fig. 6.7. The laser displacement sensors were powered by a stable DC power source, and the extent of displacement was measured as an output voltage signal directly into a digital recorder.

6.2.2 Calibration of Measurement System 24


In addition to the output characteristics of individual laser displacement sensors that were provided by the sensor manufacturer, the output calibration of the entire measurement system was also calibrated with a dial gauge as shown in Fig. 6.8. The calibration was performed by adjusting the distance between the sensor and the target steel strip with the help of the dial gauge, both incrementally and decrementally at intervals of 0.05 mm (0.025 mm in one place), and recording the output of the sensor. The results of the calibration are shown in Fig. 6.9. As can be seen in Fig. 6.9(a), even a small change in displacement of 0.025 mm can be clearly detected from the corresponding variation in output voltage. In addition, it can be recognized that the linearity between displacement and the output voltage was significantly high.

Fig. 6.5 A new hull deflection measurement systeminstalled onboard. (a) For the aft part of hull under the intermediate shaft; (b) Alongside the engine; (c) Steel strip tightened by weight; (d) Close-up of laser displacement sensor; (e) Laser beam as reference line.

(a) (b)

(c)

(d)

Laser sensor

Strip

Laser beam Weight

Pulley

Laser sensor

Strip

Laser sensorStrip

(e)

Laser beam as reference line

25


Frequency (Hz)

Ana

logu

e ou

tput

vol

tage

(dB

)

Fig. 6.6 Frequency-response function of the laser displacement sensor at a response time of 500 ms.

Laser sensor Output：

1.0V/10.0mm DC Power

Digital data recorder

Input range：±5V

Tightened steel strip

Fig. 6.7 Diagram of the new hull deflection measurement system.

Laser beam

Fig. 6.8 Sensor calibration using dial gauge.

26


0.00.5

1.01.5

2.02.53.0

3.54.0

4.55.0

0.0 0.1 0.2 0.3 0.4 0.5

Displacement (mm)

Out

put v

olta

ge (V

)Fig. 6.9 Sensor calibration results. (a) Output voltage as the result of increasing and decreasing displacement; (b) Relationship between output voltage and displacement.

(a) (b)

6.3 Example of Measurement Results The measured results of deflections of the top plate of the double bottom in the subject VLCC are shown in Fig. 6.10 and Fig. 6.11. Figure 10 shows the data from shafting portion of the top plate while Fig. 11 shows the result along the engine side. It becomes evident from the results that hogging will be caused when the ship condition changes from light to deep draught. These results are in very good agreement with the results obtained by FEM analysis both quantitatively and qualitatively. In addition, these relative sagging deflections are further confirmed by the results obtained from using another measurement method shown in Fig. 6.5(e) where a laser beam is used as the reference line.

Measured deflection at shafting portion

00.10.20.30.40.50.60.70.80.9

1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Measuring location (mm)

Def

lect

ion

(mm

)

Full (19.5 m) - Light (9.0 m) Full (19.5 m) - Ballast (14.0 m) Full (19.5 m) - Ballast (16.0 m) Full (19.5 m) - Light (9.0 m)Full (19.5 m) - Ballast (14.0 m)Full (19.5 m) - Ballast (16.0 m)

Measured deflection at shafting portion of double bottom tank top plate

Fig. 6.10 Measured deflection of the double bottom top plate beneath the shafting.

27


Measured deflection at engine portionMeasured deflection at engine portion of double bottom tank top plate

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

Measuring location (mm)

Def

lect

ion

(mm

)

Full (19.5 m) - Ballast (14.0 m)

Full (19.5 m) - Ballast (16.5 m) Full (19.5 m) - Ballast (14.0 m)

Full (19.5 m) - Ballast (16.5 m)

Fig. 6.11 Measured deflection of the double bottom tank top plate on which the engine is seated.

28


7. Prediction of Hull Deflection by FEM

7.1 Objective In order to determine the effect of hull deflection on shafting alignment, it is important to predict the deflection of the engine room double bottom tank top due to changes in draught. Therefore, the onboard measurement of hull deflection as described in Chapter 6 is necessary. However, if the FE analysis can be proven to be a viable substitute for onboard measurement in predicting hull deflection, then such expensive and time-consuming onboard measurements can be avoided. In addition, it will enable the shafting alignment designs to take into account hull deflections due to changes in draught at the design stage. However, due consideration should also be given to the modelling range, loads and boundary conditions, and other factors involved, because there are few examples of FE analyses conducted of the aft-hull structure including the engine room using detailed models for recently designed ships.

The ClassNK Research Institute has conducted FE analysis on a VLCC oil carrier using a detailed aft-hull structure model including a detailed engine room model.

7.2 FE Model

7.2.1 The Subject Ship and Modelling Range The subject ship was a VLCC oil carrier (300,000 DWT) powered by a two-stroke, seven cylinder main diesel engine. The general arrangement and double bottom plan for the engine room are shown in Fig. 7.1. Two FE models, Model A and Model B, were constructed to investigate an optimum modelling range. Model A comprised the aft-hull structure from the engine room foreside bulkhead. Model B comprised the aft-hull structure from the foreside bulkhead (Frame (Fr.) 64) of the No. 5 cargo tank. The main diesel engine, accommodation space, rudder, and other equipment (auxiliary machinery and boilers, etc.) were not included.

E/R

Fr. 64 Fr. 53 Fr. 16 Fr. -7

No.5 C.O.T

FRAME S.P. 900 mm

4843

39

2934

2520

16

Fig. 7.1 General arrangement and double bottom plan of the engine room.

7.2.2 FE Model Model A and Model B were presented in Figure 7.3. The hull structure was modelled using shell elements, and the strength members (e.g. longitudinal frame, longitudinal beam, transverse frame, deck beam, etc.) in the aft-hull structure from Fr. 53 onwards were modelled using beam elements to reproduce their stiffness.

The element size of the aft-hull structure from Fr. 53 onwards was approximately equivalent to the 29


longitudinal frame spacing, while in the cargo hold part it was approximately equivalent to the transverse spacing. The analysis code used was Visual NASTRAN for Windows 2002.

Fr. 64

Fr. 16

Fr. 53

Fr. 16 x

y

z

Fig. 7.2 FE models.

(a) Model A (b) Model B

7.3 Loads and Boundary Conditions

7.3.1 Load Conditions The deflection of the engine room double bottom due to changes in draught can be obtained by subtracting the deflection observed in the light loading condition from that of the full loading condition. Therefore, FE analyses have to be carried out for two different load conditions having a relative difference, i.e. the draught condition and fully loaded tank condition. The draught and tank conditions during sea trials applied to these analyses are presented in Table 7.1. Since analyses were carried out assuming actual service conditions, the weight of the accommodation space, rudder, propeller, main diesel engine, and major equipment, shown in Table 7.2, were taken into account in the analyses. The hydrostatic pressure of seawater which acts on the outer shell of the ship and the liquid weight of the load which acts on each tank inner plate were applied to each element as pressure loads. The weights of equipment were applied to each node as concentrated loads.

Table 7.1 Draught and Tank Conditions

Light Load Full Load

Cargo

No. 5 C.O.T (C) 0% 72%

No. 5 C.O.T (P) 0% 100%

No. 5 C.O.T (S) 0% 100%

SLOP T. (P) 0% 100%

SLOP T. (S) 0% 100%

Ballast Water

A.P.T 96％ 0％

Draught (M) 6.02 18.62

30


Table 7.2 Load Conditions

Name Weight (T) Name Weight (T)

Accommodation Space 3040 Intermediate Shaft 38

Rudder (Light Load) 122 Main Diesel Engine 1032

Rudder (Full Load) 64 Boiler 78

Propeller (Light Load) 61 Generator 14

Propeller (Full Load) 58

Propeller Shaft 55 7.3.2 Boundary Conditions To establish an optimum boundary condition for predicting hull deflection, two boundary conditions were

applied as the following constraint ① and constraint ②. An optimum boundary condition was examined by comparing the analysis results resulting from each condition. Each boundary condition was applied to all nodes of the front bulkhead of each FE model, as shown in Fig. 7.3.

- Constraint ① This is an optimum boundary condition for cargo hold strength analysis. All constraints were applied to nodes in the front bulkhead of each model. Details of the boundary condition are as follow.

- Symmetry constraint was applied to all nodes in the front surface of the model. - A constraint in the X direction was applied to the node located at the intersection of the transverse

bulkhead with the upper-deck centre-line. - A constraint in the Y direction was applied to nodes located at the intersection of the longitudinal bulkhead

with the upper-deck, and at the intersection of the side shell with the upper-deck.

- Constraint ② - Absolute constraints were applied to all nodes in the foremost bulkhead of the model.

(a) Constraint ① (b) Constraint ②

：Absolute constraint

Front of model

：X direction constraint

：Y direction constraint

：Z direction symmetry constraint

Front of model

X

Y

Ｚ

Fig. 7.3 Boundary conditions. 31


7.4 Effects of Analysis Conditions

7.4.1 Effect of Modelling Range and Boundary Conditions To investigate the effect of the modelling range and boundary conditions, analyses using two models, model A

and B, and two boundary conditions, constraints ① and ②, were performed under the light load condition and the full load condition. In addition, a comparison of the relative displacements of deflection at the top of the engine room double bottom tank obtained from each analysis condition was carried out.

Deflections of each loading condition were provided based on the reference line which connected each displacement magnitude at the foremost point (Fr. 44) and aftmost point (Fr. 16) on the red line shown in Fig. 7.4. The relative displacements of the deflection due to changes in draught could be obtained by subtracting the deflection observed in the light load condition from the deflection observed in the full load condition. The relative displacement resulting from each model and each boundary condition are shown in Fig. 7.5. Since the red line measuring the magnitude of the displacement was not a straight line but was a discontinuous line divided into two portions consisting of the main engine installation portion and the intermediate shaft portion, the line in the chart had discontinuous points in the middle position.

4843

39

2934

2520

16

FRAME S.P. 900 mm

Fig. 7.4 Measurement lines.

0

1

2

3

4

5

-5000 0 5000 10000 15000 20000 25000 30000

Distance from aftmost bulkhead in engine room （mm）

Rela

tive

dis

plac

em

ent

（m

m）

Model A Constraint (1)

Model A Constraint (2)

Model B Constraint (1)

Model B Constraint (2)

Fig. 7.5 Relative displacements on double bottom tank top in engine room.

As Fig.7.5 indicates, there is a difference in the relative displacement of up to about 1.5 mm between

Constraint ① and Constraint ② in Model A. This difference can be expected to have been caused by the effect of deformation of Fr. 53 which applied one of the boundary conditions. The deformed shapes of Fr. 53 under each boundary constraint in Model A are presented in Fig. 7.6. 32


Fig. 7.6 Deformed shape of Frame 53.

X

Y

Ｚ (a) Model A Constraint ① (b) Model A Constraint ②

As can be seen from Fig. 7.6, the restrained bulkhead (Fr. 53) under Constraint ① was transformed in the X and Y directions. This showed that the deflection of the double bottom might be influenced by the deformation

of Model A caused by deformation of Fr. 53. Therefore, applying an absolute constraint like Constraint ② in order to disregard the deformation of Fr. 53 should not be assumed to be an optimum boundary condition for Model A.

Moreover, when thinking about the deformation of the entire ship, Fr. 53 would be constrained in the X and Y

directions by the foreside cargo hold structure. Therefore, it is unlikely that Constraint ①, which has freedom in the X and Y directions, would be an optimum boundary condition. Although Constraint ① has been used as an optimum boundary condition for the strength evaluation of the cargo hold structure, it would not necessarily be an optimum boundary condition for the aft-hull structure model having an asymmetric structure in the longitudinal direction.

On the other hand, the relative displacements of Constraints ① and ② of Model B corresponded well. This means that the influence of the boundary conditions has been counteracted as a result of extending the modelling range up to the cargo hold structure and taking the distance of the double bottom between the restrained bulkhead and engine room. Reproducing the deformed shape of Fr. 53 in Model B to Model A's boundary conditions would make it possible to provide a simplification of the analysis model. However, it will be difficult to reproduce the deformed shape of the bulkhead completely as a boundary condition. Therefore, when predicting the deflection of the engine room double bottom, it is necessary to construct not only the aft-hull structure but also the aftmost cargo hold structure. Further, a reliable result can be obtained by a simplified absolute constraint in the foremost bulkhead of model.

7.4.2 Effect of Load Conditions In the above-mentioned analyses, analyses were carried out under conditions that were as close to actual service conditions as possible. Therefore, the weight of the hull structure, liquid in the tanks, the accommodation space, rudder, propeller, main diesel engine, and major equipment had been taken into account in the analyses. However, since only the relative amount of deflection is necessary in shafting alignment calculation, it is sufficient just to determine the respective differences in deflection between the ballast and full load conditions as the boundary conditions. Therefore, unchanging loads between the light load condition and full load condition (i.e., the weight of the accommodation space, main diesel engine, and major equipment), and the weight of the rudder and the propeller which was thought to have hardly influence at all on the analysis results were omitted. Analyses were carried out using only the draught and tank condition shown in Fig. 7.1. The analysis results are presented in Fig. 7.7. Here, the load condition which was not omitted is referred to as Load A and the load 33


condition which was omitted is referred to as Load B.

0

1

2

3

4

5

-5000 0 5000 10000 15000 20000 25000 30000


Rel

ativ

e di

spla

cem

ent

（m

m）

Load A

Load B


As Fig. 7.7 indicates, the analysis result for Load B, which was the simplified load condition, was almost equal to the analysis result for Load A. Therefore, the load conditions shown in Table 7.2 can be omitted, and a reliable analysis result can be obtained by using only the relative differences in the respective load conditions such as the draught and tank condition.

7.5 Effects of Added Stiffness of Main Engine on Predicted Deflection

7.5.1 FE Model of Main Diesel Engine The main engine is expected to be the stiffest structure in the engine room and it is conceivable that its stiffness has some influence on deflection of the double bottom. Therefore, in order to estimate the influence of main engine stiffness on double bottom deflection, an FE model of a main engine was constructed and analyses of Model B including the main engine model (hereinafter called Model B+M/E) were carried out. The main engine was modelled using elements with the size of the holding down bolt spacing (about 200 mm) so as to take into consideration installation of the engine model on the exact position. The bed plates of the main engine bearings were modelled using solid elements, and other structures (e.g., the engine frame, cylinder cover, etc.) were modelled using shell elements. The FE model of the main engine is presented in Fig. 7.8.

Fig. 7.8 FE model for M/E structure.

34


7.5.2 Integration of the Main Engine and the Hull Structure When the main engine is actually integrated onto the engine seating surface on the double bottom, fluid resin is cast between the main engine and the seating surface. The resin layer mutually transfers deformations of the main engine and double bottom while absorbing the respective deformations of each. Therefore, in order to reproduce the integrated condition of the main engine accurately, solid elements with the material properties of the resin were applied to the resin layer. The appearance of the main engine integration is presented in Fig. 7.9.

Resin

M/E Fig. 7.9 Integration of M/E structure with hull in FE model using solid

element for resin layer on seating surface. 7.5.3 Effect on the Double Bottom The relative displacements of Model B and Model B+M/E on the double bottom tank top in the engine room are presented in Fig. 7.10. In addition, a detailed view of the relative displacement on the engine seating portion is presented in Fig. 7.11.

0

1

2

3

4

5

-5000 0 5000 10000 15000 20000 25000 30000


Rel

ativ

e dis

pla

cem

ent

（mm

） Model B　

Model B+M/E


35


0

0.5

1

1.5

10000 15000 20000 25000 30000


Rela

tive

dis

pla

cem

ent

（m

m）

Model B　

Model B+M/E

Fig. 7.11 A detailed view of relative displacements on double bottom tank top within the engine

portion of the shafting line.

As can be seen from Fig. 7.10, there is a slight difference between the relative displacements in Model B and Model B+M/E. This difference is attributable to the engine seating portion of the shafting line and it was hardly seen in other portions. In addition, as Fig. 7.11 also shows, the magnitude of the deflection in Model B+M/E clearly decreased at the engine seating portion of the shafting line. Namely, the difference in the relative displacements is caused by increases in the double bottom stiffness due to the integration of the main engine. Although the change of the relative displacement is only about 0.2 mm, it should never be disregarded because engine alignments of recent large diesel engines are very sensitive to even small changes in bearing offsets. Therefore, a reliable analysis can only be obtained by integrating a structural model of the main engine and the ship.

36


8. Prediction of Thermal Deformation of Engine Bedplate

8.1 Temperature of Engine Structure in Running Condition In the running condition, the upper structure of engine, which is located close to the combustion chambers has, in principal, a higher temperature than lower structures such as the bedplate, although the exact value depends on the size and output of the engine size. Figure 8.1 shows the approximate temperature ranges of the cylinder block, frame (column), and bedplate for a large-sized engine.

FRAME 40~50℃

BED PLATE 40~50℃

CYLINDER BLOCK 80~90℃

Fig. 8.1 An example of the distribution of the temperature in an engine structure in the running condition.

8.2 Thermal Deformation of Engine Bedplate Although thermal deformation of the engine structure will occur three dimensionally, only vertical displacement, however, will affect shafting alignment. Further, vertical displacement can be decomposed into parallel translation and hogging deformation, which will be explained later. The parallel rise in height of all bearing positions is caused mainly by the rise of the average temperature of the engine bedplate. On the other hand, hogging deformation is caused by a relatively larger longitudinal expansion of the upper parts of the engine structure, including the cylinder block, due to the higher temperature of these parts compared with the lower structural parts. The parallel rise of the bearing offsets which has less effect than hogging deformation on shafting alignment has already been taken into consideration during shafting alignment calculation at some shipyards. There have been few examples to permit sufficient consideration of the effect of hogging deformation on shafting alignment due to the difficulty in estimating the extent of such hogging deformation, despite such effect being evident from calculation results.

8.2.1 Parallel Rise of Bearing Offsets Caused by Thermal Expansion The parallel rise of bearing offsets is the result of a uniform rise in height in the vertical direction at each bearing position caused by an increase in the temperature of the engine bedplate. The parallel rise can be easily calculated after the distribution of the temperature in the bedplate is known. The amount of parallel rise in height, Δh, can be calculated by Eq. (8.1), where the temperature of the bedplate can be regarded as linearly distributed in the vertical direction.

37


⎟⎠⎞

⎜⎝⎛ Δ+Δ

=Δ2

21 TThh α (8.1)

where α is the thermal expansion coefficient of the material of the bedplate, h is the height from the engine seating plane to the bearing center, and ΔT1 and ΔT2 represent the increases in temperature at the engine seating plane and bearing center, respectively.

In cases where measured results on sister engines are available or the manufacturer has recommend values for the parallel rise in height, the values obtained can be applied.

Cold Warm

Fig. 8.2 Vertical uniform displacement at the main bearing center arising from thermal expansion of the bedplate.

Uniform displacement at main bearing center

h

Δh

h+Δ

h

8.2.2 Hogging Deformation of Engine Bedplate Caused by Thermal Expansion The hogging deformation of an engine bedplate forms a convex upward profile deformation along the bearing center line and is caused by a non-uniform distribution of the temperature over the whole structure of the engine, as shown in Fig. 8.3. Warm Cold

Fig. 8.3 Hogging displacement at main bearing center arising as a result of thermal expansion of the entire engine structure.

38


Fig. 8.4 shows the calculated results of an example using a FE model of an engine structure, giving a temperature increase of 15oC from bottom to top and allowing the structure to deform freely. In this example, a maximum hogging deformation of about 0.1 mm was calculated. For specific engines, however, it is necessary to use more accurate temperature distribution data and constraints from the hull structure.

Fig. 8.4 FE model of an engine structure showing hogging displacement at the main bearing center arising from thermal expansion.

Thus, in this sense, it is better to consider the effect of the constraints of the hull even in cases where data on

separate engine hogging is available from the manufacturer. The most accurate method to determine the hogging deflection of an engine bedplate caused by thermal

expansion is onboard measurement. This consists of measuring the relative deflections that occurs from cold to warm (hot) condition under given draught (loading) conditions.

39


9. Dynamic Components of Hull Deflection

9.1 Dynamic Hull Deflection Related to Ship Motions

9.1.1 Hull Deflection due to Ship motion

A dynamic component in the hull deflection was detected during the onboard measurement conducted of the

subject VLCC by the Society. This is the first time that such a phenomenon could be confirmed by virtue of a

measurement system capable of measuring hull deflections automatically and continuously. Figure 9.1 and 9.2

show the time history and power spectrum of the measured hull deflection beside the main engine in both the

stop and full speed condition, respectively. The waves in the sea area where the measurements were carried out

were comparatively high due to the effects of a typhoon during the time when the measurements were taken.

As can be seen from these results, the dominant periodic variation in hull deflection has a frequency of about of

0.06 Hz, while the frequency in full speed condition was a little higher than in the stop condition. The

amplitude of the variations in the full speed condition is much greater than in the stop condition. Such periodic

fluctuation in hull deflection is considered to be related to the ship motions of pitching and heaving in waves

because of its lowness of frequency. Because the resonance frequencies of the undesired lateral and axial

vibration of the tightened steel strip are much higher than the frequencies of ship motions, it was unlikely that

the strip could have been affected by such ship motions. The subsequent two peaks in the power spectrum

diagrams at frequencies of around 0.45 and 0.9 Hz are considered to represent two and three-noded vertical hull

vibration respectively. It is, however, negligible in comparison to the variation in deflection due to rigid ship

motions. In addition, it is confirmed that the variation of deflection due to rigid ship motions is hardly affected

by loading condition and would not appear when the sea is calm, for example, in a protected area in close to

shore.

13 s

Dis

plac

emen

t

Time

0.058 Hz

0.45 Hz 0.91 Hz

Pow

er sp

ectru

m

Frequency (Hz)

(a) Time history

(b) Power spectrum

Fig. 9.1 Time history of displacement at the longitudinal middle of the engine and its power spectrum when the ship was in stop condition.

40


12 s

Time

Dis

plac

emen

t

0.068 Hz 0.46 Hz

0.96 Hz

Pow

er sp

ectru

m

Frequency (Hz)

(a) Time history

(b) Power spectrum

Fig. 9.2 Time history of displacement at the longitudinal middle of the engine and its power spectrum when the ship was at a full speed of about 18 knots.

9.1.2 Magnitude of Ship Motion Induced Hull Deflection

The magnitude of the above mentioned ship motion induced hull deflection will vary, depending on sea

conditions, size and speed of the ship, and other factors. In the example of the VLCC on which the

measurements discussed here were carried out, the amplitude of the maximum displacement is 0.3 mm in

shafting portion of the measurement line and 0.2 mm in engine side portion of the measurement line, respectively.

This maximum displacement is almost equivalent to half of the displacement caused by a difference of about 10

m in draught, which is about 0.6 mm. Figures 9.3 and 9.4 show the fluctuations of the measured displacements in

the engine side portion and the shafting portion of the measurement line. This is the first time for such a

dynamic component in deflection to be detected because of the measurement system's ability to measure

continuously and to such a high degree of precision. The dynamic component could be a main contributory

factor to the shafting alignment related engine bearing failures reported so far.

Fig. 9.3 Dynamic relative displacement over the length of the engine.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

30 40 50 60 70 80 90Time (S)

Dis

plac

emen

t (m

m)

5Ch6Ch8Ch9Ch10Ch

-0.4-0.3-0.2

-0.10

0.10.2

0.30.4

-2000 0 2000 4000 6000 8000 10000 12000

Location (mm)

Disp

lace

men

t (m

m)

Hog1Sag1Hog2Sag2

41


Fig. 9.4 Dynamic relative displacement over the span between the aft end of the engine and the fore stern tube bearing.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

30 40 50 60 70 80 90Time (S)

Di

-0.4-0.3-0.2-0.1

00.10.20.30.4

-2000 0 2000 4000 6000 8000 10000 12000

Location (mm)

Dis

plac

emen

t (m

m)

Hog12Ch 3Ch 4Ch Sag1

Hog2

9.2 Deformation due to Thrust

Another factor likely to cause hull deflection while in service is thrust. To verify this, a FE model subjected to

thrust corresponding to normal navigation condition was developed, as shown in Fig. 9.5. The calculated result

from the FE model is shown in Fig. 9.6 and Fig. 9.7. Figure 9.6 shows the absolute hull deflections with and

without thrust, while Fig. 9.7 shows the differences between the two conditions, i.e., the relative deflection from

the stop condition to normal navigation condition. As can be seen from Fig. 9.7, thrust will cause a hogging

deflection as there is an increase in draught. This effect, however, can be ignored in design, because it is very

minimal, being only 0.2 mm over the entire span from the propeller to the fore side of the engine in the VLCC

case.

spla

cem

m)

Sag2

ent (

m

Thrust

Fig. 9.5 FE model subjected to thrust.

42


0.0

2.03.0

4.05.0

6.07.0

05000100001500020000250003000035000

Distance from foremost main bearing (mm)

spla

cem

ent (

mm

)

With thrust

Without thrust

1.0D

i

Fig. 9.6 Absolute displacements of cases with and without thrust.

With - Without

0

0.1

0.2

0.3

0.4

05000100001500020000250003000035000


Dis

plac

emen

t (m

m)

Fig. 9.7 Relative displacement of the case with thrust to the case without thrust.

43


10. Determination of Final Bearing Offsets

10.1 Prediction of Relative Displacement over Entire Length of Shafting Line

10.1.1 Prediction of Relative Displacement over Entire Shafting Length by Measurements The change in bearing reaction forces due to variations in the bearing offsets can be calculated, provided that the relative displacement to a reference straight line passing through both end bearing supporting points can be measured as shown in Fig. 10.1. However, only the displacements of the top of the bottom plating at shafting portion of a shafting line within the engine room and along side the main engine can be actually measured, as shown in Fig. 10.1(b).

In order to recreate the entire relative displacement over the shafting span from these separately measured displacements, it is necessary to assume that all bearing supporting points are on a smooth curve. The 'smooth curve' mentioned here is defined as a curve whose differential function varies continuously from point to point along the curve. If this assumption is proved acceptable, the whole relative displacement can be recreated by joining these separately measured displace lines with the same slope at their joining points, as show in Fig. 10.1(c). In addition, as shown in the figure, the displacement within the stern tube is approximated as a straight line.

It is noteworthy that the curves mentioned here are curves that join the bearing supporting points, and are not the shafting lines themselves.

Actually measurable displacements Displacements necessary to calculate bearing reactions

Fig. 10.1 Total displacement can be recreated by connecting separately measured displacements at shafting and engine portion of the shafting line, having the different displacement curves have the same slopes at each connecting points. (a) Original bearing support line. (b) Bearing support line after deformation. (c) Entire bearing support line recreated from separate sets of measured displacements.

(a)

(b)

(c)

44


It can be is easily understood that all bearing supporting points will lie on a smooth curve, assuming that all bearing supporting points were initially on a straight line in an elastic body, as shown in Fig. 10.2. A straight line a-b in an elastic body will become a curve a'-b' after the elastic body is deformed under an external force or enforced restraints, as shown in Fig. 10.2(b). Since the derivative of displacement in the Y direction with respect to x denotes shear strain, the derivative must have a unique value at any point. Therefore, the displacement curve in the Y direction must be continuous and smooth.

x

y

z

a b

(a)

a' b'

(b)

v a'-b'

x (c) Fig. 10.2 (a) A straight line a-b in an elastic body. (b) The line a-b becomes a curve

a'-b' after the body is deformed. (c) The curve a'-b' must be smooth and continuous, since the differential dv/dx represents shear strain.

10.1.2 Prediction of Relative Displacement over Entire Shafting Length by Calculations Another way other besides measurement to estimate the relative displacement over the entire shafting length is to use a FE model that integrates the engine structure with the hull structure, as shown in Fig. 10.3. Since the FEM can be used to estimate the entire relative displacement directly, there is no need to recreate the entire relative displacement from separately obtained displacements. Furthermore, once the FE model has been validated, it can be used to evaluate the potential effect of any structural alteration. Therefore, it is desirable to perform FE analysis as the circumstances permit.

However, the bearing offsets obtained do not necessarily lie on a smooth line due to the mesh size. In such a case, a polynomial of the 3rd to 6th degree fitted to the data could be used in the shafting alignment, considering the deflection curve of a cantilever beam with non-uniform loads.

45


Fig. 10.3 FE model integrated main engine structure with hull used to calculate the displacement at each bearing support point.

10.2 Discontinuity in Slope of Total Deflection Curve over Entire Length of Shafting Line Figure 10.4 is a close-up view of the deflection of the aft engine part of the structure obtained from the model shown in Fig. 10.3. As can be seen from the result, while the deflection curve of the top of the double bottom beneath the shafting line is quite smooth, there is clearly a discontinuity at the aft engine end where the deflection curve consists of the point on the top of the double bottom in the shaft portion and the supporting points of the engine bearing in the engine portion of the double bottom.

3.00

3.50

4.00

4.50

5.00

600080001000012000140001600018000


Dis

plac

emen

t (m

m) Center Line

Bearing Line

Aftermost end of the engine structure

Fig. 10.4 FEM results showing discontinuity between slopes of deflection curves of double bottom and main bearing centers.

A discontinuity in the slope of the total deflection curve over the entire shafting line could cause adverse effects on bearing reactions. Fig. 10.5 shows two sets of bearing heights that differ from each other only in the presence of a slope continuity at the longitudinal boundary between the hull and engine structures. While the deflection in the engine portion of the curve in red smoothly joins the deflection in the shaft portion of the curve, the slope of the curve representing the engine portion of the deflection in blue is discontinuous at the longitudinal boundary between the hull and engine structures. The bearing reaction forces are calculated for the two sets of bearing offsets, and are shown in Fig. 10.6. The shafting model used in this calculation is the same as that used in previous chapters. 46


47

As can be seen from the results shown in Fig. 10.6, the bearing reactions will change significantly if the

discontinuity in the slope of the deflection curve exists despite the magnitude of the total hog that remains unchanged. In this example, the first aftmost engine bearing becomes overloaded and the second aftmost engine bearing becomes loaded in the wrong direction.

Therefore, it is desirable to ascertain whether there is any discontinuity in the slope of the deflection curve, using an integarated engine-hull FE model.

Then let's consider why such a discontinuity in the slope of deflection cuve is generated. As mentioned before in section 10.1.1, a smooth curve in an elastic body will remain smooth after the body is deformed. This can be explained by the fact that the deflection curve at the center line on the double bottom top plate is relatively smooth, as shown in Fig. 10.6. It can be easily understood that when assuming there is a straight line directly beneath the double bottom top, then the sraight line will become a smooth curve after hull deformation.

Bearing reaction force (kgf), (-) Upward and (+) Downward-200000

-150000

-100000

-50000

0

50000No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9

Bearing number

Rea

ctio

n fo

rce

(kgf

)

OriginalSmooth deflectionDeflection with breaking point

Fig. 10.6 The effect of slope discontinuity to bearing reactions

continuity at the longitudinal boundary between the hull and engine structures. Fig. 10.5 Two sets of bearing heights that differ from each other only in slope

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.00 5000 10000 15000 20000 25000 30000

Bearing location, distance from left end (mm)

Off

set (

mm

)

Original offset (mm) Final smooth offsetSmooth hull deflection (mm) Straight engine portionFinal breaking offset Quadratic polynomial (Smooth hull deflection)


Furthermore, the deflection line of the center line on the top plate of the double bottom will also become a smooth line, neglecting small local deformations between the double bottom top plate and the assumed straight line.

On the other hand, the deflection at the supporting points of the engine bearings in the engine portion of the line can not be regarded as coming from the same straight line as the center line on the top plate of the double bottom, as shown in Fig. 10.7. While the change in offsets of the intermediate bearing and stern tube bearings comes from the center line on the top plate of the double bottom, the change in offsets of the engine bearings comes from the engine seating lines on the top plate of the double bottom. It is noted from the FE analysis results that the displacemet of the top plate of the double bottom varies in the transverse direction, although the magnitude of the fluctuation depends on the location of the cross section. Therefore, the discontinuity in the slope of the deflection curve is considered to arise from these variations in displacement in the transverse direction.

Fig. 10.7 Illustration showing how the slope discontinuity is developed

Schematic sectional deflection curve of the top plate of the double bottom showing variation over the width

Center line on the top plate of the double bottom

Engine seating lines on the top plate of the double bottom Shafting line

10.3 Determination of Final Bearing Offsets in Shafting Installation Based on the background described above, the final bearing offsets in a shafting installation should be determined so that the shafting alignment can meet all relavent requirements, even after adding both static and dynamic changes in bearing offsets to it.

Therefore, specifically, the first step is to determine the bearing offsets as in usual practice. Then the next step is to re-calculate the shafting alignment taking into account the predicted changes in bearing offset to check whether the alignment still meets the relevant requirements. If the re-calculated result is satisfactory, then the initially obtained bearing offsets can be set as the final parameters for the shafting installation. If the re-calculated result is not satisfactory, then the initially obtained bearing offsets should be readjusted until the relevant requirements are met. 48


11. Confirmation of Bearing Reactions

11.1 Jack-up Method

The Jack-up method is a method in which the reaction of a bearing in question is measured from the jack load

that is set as close to the bearing as possible through a separately determined modification factor (see Fig. 11.1).

The jack load is estimated from the relationship between the jack load and the jack displacement which is

recorded during the jack up and jack down process. In the jack up test, the dial gauge used to measure the jack

displacement should be properly secured so that it is affected by neither the rise of the shaft nor the deformation

of the floor plate.

Hydraulic jack

Dial gauge

Fig. 11.1 Jack-up measurement of the bearing load.

11.1.1 Theoretical Jack up Process

The jack up process can be theoretically simulated through calculation. It is demonstrated using a shafting

model shown in Fig. 11.2. The simulation is performed by calculating the relationship between the lift up and

the reaction at the jack point. The main steps in the simulation are as follows.

500 mm

Jack

Fig. 11.2 A shafting model used to demonstrate the theoretical jack-up process.

- An additional supporting point is generated by setting the jack directly under the shaft at a point 500 mm

49


away from the intermediate bearing of which the reaction to be measured. The reaction of this newly added

supporting point is, however, zero at this point.

- The jack is gradually lifted, in increments of 0.002 mm in this case, until the bearing load becomes zero.

The total lift is 0.01587 mm in this case.

- Because the bearing has been completely unloaded, it is no longer a supporting point. Therefore, the

calculation described below is performed using the beam model without the supporting point as shown in Fig.

11.3. Since the number of supporting points is reduced by one, the relationship between the jack lift and the

jack load (the slope) is also changed. The result of the simulation is shown in Fig. 11.4.

500 mm

Jack

Fig. 11.3 A state where the load of the bearing in question is just becoming zero.

0.000.010.020.030.040.050.060.070.080.090.100.110.12

0 5000 10000 15000 20000 25000 30000

Jack load (kgf)

Shaf

t lift

at j

ack

poin

t (m

m)

Intermediate bearing loadedIntermediate bearing unloadedLinear extrapolation (Intermediate bearing unloaded)

P0

R'JRJ

Fig. 11.4 Theoretically calculated jack-up result of the shafting model.

Point P0 in the above figure represents a state in which the reaction of the intermediate bearing has just

become zero. The jack load at this point is represented by R'J. By fitting the relationship between the jack lift

and the jack load after this point to a linear curve and then extrapolating the line to where the jack lift is zero, the

jack load when the jack lift is zero, Rj, can be obtained. This load from which the bearing load is usually

calculated is, in other words, the jack load when the jack is being set and the bearing in question is removed. 50


In this example,

kgf.'R J 524488=

kgf RJ 24450=

11.1.2 Determination of Bearing Load from Jack up Test

The bearing load RB can be calculated from the obtained jack load RB J using the following equation (11.1).

JB RCR ×= (11.1)

where, C is the correction factor calculated as follows:

BJ

BB

IIC −=

where,

IBB: The reaction influence number of the bearing when the jack is regarded as a supporting point.

IBJ: The reaction influence number of the bearing to the jack supporting point.

In order to better understand the jack up method，Eq. (11.1) is derived below. Figure 11.5 shows three

important states during the jack up process using the calculated shafting lines.

0.01

6 m

mδh

Fig. 11.5 Deflection curves of the shaft line at three important states during the jack-up process.

51


State 1： The initial state, at this point the reaction of the bearing is RB. The jack is set underneath the shaft

but the jack load remains zero. This state is represented by the solid black line in Fig. 11.5.

B

State 2： After gradually lifting the jack up by a lift of δ, the bearing load becomes zero while the bearing

remains in contact with the shaft. This state is represented by the solid red line in Fig. 11.5. At this

point the jack load is R'J.

State 3： Supposing that the bearing has been removed at State 2, the jack is lowered by an amount δ back to the

original setting position. This state is represented by the dotted red line in Fig. 11.5. During this

process the bearing supporting point is supposed to have lowered by an amount h. At this point the

jack load is RJ.

State 4： Supposing that the bearing has been set again at the supporting point, the bearing is lifted by an

amount h, until the jack load becomes zero. During this process the jack, of course, remains in

contact with the shaft. The bearing load at this point is exactly the load RB that is to be measured.

In other words, state 4 is identical to state 1.

B

In the process of transition from state 3 to state 4, the change in the bearing and jack loads can be expressed as

follows:

Bearing: 0 + IBBh = RBB

Jack: RJ＋IBJh = 0

Therefore，

JBJ

BBB R

IIR −=

Since the reaction influence numbers for the shafting model shown in Fig. 11.5 are:

IBJ = -1538261 (kgf/mm)

IBB = 1535661 (kgf/mm)

Therefore, the jack correction factor is:

99830.IIC

BJ

BB =−=

In addition, RJ is 24450 kgf as mentioned in 11.1.1, thus the bearing load RB is obtained as follows: B

kgf 2440824450.RB =×= 99830

This result completely agrees with that directly calculated using the beam model described earlier.

The bearing load can also be expressed in Eq. (11.2), if we consider the process of transition from state 1 to

state 2 in the same way as above.

JJJ

JBB 'R

II

R −= (11.2)

52


where,

IJJ: The reaction influence number of the jack when the jack is regarded as a supporting point.

IJB: The reaction influence number of the jack to the baring.

Equation (11.2) theoretically leads to the same result as that obtained from Eq. (11.1). In practice, however, R'J is difficult to determine accurately from the measured relationship between the jack lift and jack load, because the real jack load-lift curve will form a loop referred to as an hysteresis curve due to the unavoidable friction in the hydraulic jack. Equation (11.2) is thus less used in practice.

11.1.3 Analysis Method of Jack up Recording Curves

The hydraulic pressure is higher in the lifting process than in the lowering process for a given external load,

because friction between the cylinder and the rod is unavoidable. Due to this reason, a loop called an hysteresis

curve is generated during the jack up and jack down process as schematically shown in Fig. 11.6. To eliminate

the effect of the friction, the middle line of the loop should be used to determine the jack load.

2DI

JRR

R+

= (11.3)

Jack load

Increasing

Decreasing

Average line

RD RJ RI

Ver

tical

lift

of sh

aft a

t jac

k po

int

Fig. 11.6 Hysteresis curve of actual jack-up result due to unavoidable friction in hydraulic jack.

It is important to confirm the sudden change, known as 'break point', that appears in the slope of the jack

load-lift curve. The break point comes about because the number of the supporting points in the shafting decreases by one when the bearing load becomes zero. If such a break point does not appear and the jack load-lift curve looks like that shown in Fig. 11.7, there is a high possibility that the bearing load is nearly zero at the original position.

53


Jack load

Increasing

DecreasingV

ertic

al li

ft of

shaf

t at j

ack

poin

t

Fig. 11.7 Jack-up result showing no break point is a sign that the bearing being unloaded.

As noted above, the break point comes about because the number of the supporting points in the shafting decreases by one when the bearing load becomes zero. The load of the bearing immediately next to the jack first becomes zero and the first break point appears. When the jack continues to rise, then another bearing close to the jack will also become unloaded and the slope of the jack load-lift curve will become steeper, as shown in Fig. 11.8. If the jack continues to rise further, at some point, the upper clearance of the nearest bearing to the jack will finally disappear, causing a downward bearing reaction, and slope of the jack lift-load curve will suddenly decrease. Therefore, in order to avoid excessive jack up, it is important to finish the jack-up test as soon as sufficient data for determining RJ has been gathered, after the first break point has appeared.

Jack load

No bearing unloaded

Ver

tical

lift

of sh

aft a

t jac

k po

int

First bearing unloaded

Second bearing unloaded

Shaft touched first bearing upper limit

Fig. 11.8 Each slope in the jack-up result corresponds to a different shafting model.

54


11.2 Gauge Method

11.2.1 Mechanism of Gauge Method

The bending moment along the shafting will change when the reactions of the bearings change due to the change

in the bearing offsets. Since the change in bending moment at any cross section is linearly proportional to the

change in bearing offsets or bearing reactions, the change in bearing offsets or bearing reactions can be reversely

calculated from the change in bending moment at several cross sections which are measured. Supposing that the No. 4，No. 5 and No. 6 bearings in Fig. 11.9 rise by 1.0 mm, respectively, the change in the

bending moment at cross sections 10,000 mm, 17,000 and 18,000 mm from the left end, respectively, is calculated and shown in Table 11.1. These values are called moment influence numbers, and the matrix composed of these moment influence numbers as its elements is referred to as the moment influence number matrix.

No.1 No.2 No.3 No.4 No.5 No.6

M1 M2 M3

Fig. 11.9 A shafting model for demonstrating gauge method

Table 11.1 Bending Moment Influence Number Matrix Offset (mm)

Moment (kgf-mm) δ4 (1mm) δ5 (1mm) δ6 (1mm)

M1 (at 10000 mm) -28154234 17284736 -12732321

M2 (at 17000 mm) 18606641 -187185 -6315427 M3 (at 18000 mm) 5815648 44923370 -43329799

Therefore, the relationship between the change in bending moment and the change in bearing offsets can be given in Eq. (11.4). -28154234 17284736 -12732321

18606641 -187185 -63154275815648 44923370 -43329799

δ4δ5δ6

M1M2M3

= (11.4)

55


The change in the bearing reaction for a given change in bearing offsets of bearings No. 4，No. 5, and No. 6 in Fig. 11.9 is calculated and shown in Table 11.2, while the corresponding shafting deflection line is shown in Fig. 11.10. In practice, it is not possible to determine the change in the bearing reactions because the change in the bearing offsets is usually unknown. Table 11.2 Calculated Bearing Reaction Changes due to Changes in Bearing Height

Original bearing condition Changed bearing conditionBearing

location (mm) Bearing height(mm)

Bearingreaction (kgf)

Bearing height(mm)

Bearingreaction (kgf)

2830 0.400 -88341 0.400

-93142

4630 0.000 -17067 0.000 -33878465 0.050 -11496 0.050 -27408

15295 0.900 -24346 1.400 -420822375 1.600 -34528 2.600 -11779823375 1.600 1631 3.000 71796

56

Fig. 11.10 Shaft deflection curves before and after change in bearing height.

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

The change in the bearing reactions is therefore determined using the above mentioned method, assuming that

changes in the bending moment at three cross sections M1, M2, and M3 have been measured. The changes in the bearing offsets can be obtained from Eq. (11.5) by substituting the relevant values into Eq. (11.4).

(11.5)

-28154234 17284736 -1273232118606641 -187185 -6315427

5815648 44923370 -43329799

δ4δ5δ6

M1M2M3

=

-1

i

k

−5.237308941× 10−8 −3.17599646×10−8 2.001875716×10−8

−1.381009923× 10−7 −2.322276437× 10−7 7.442829116×10−8

−1.502094535× 10−7 −2.450312114× 10−7 5.677366995×10−8

y

{.i

k

−14617630274537

−12830525

y

{

i

k

0.49999999331.0000000781.400000089

y

{

=

=

3.0

4.00 5000 10000 15000 20000 25000

Bearing location, distance from shaft left end (mm)

Bea

ring

high

t (m

m)

Original bearing conditionChanged bearing conditionOriginal shafting lineChanged shafting line


The changes in the bearing reactions can be determined from Eq. (11.6) using the reaction influence number matrix and the relative changes in the bearing offsets with reference to the stern tube center line, assuming that the stern tube bearing offsets do not change.

57

(11.6)

The reactions after the changes in the bearing offsets can therefore be obtained by adding the changes to the

initial bearing reactions, as shown in Eq. (11.7).

i

k

139295 −214398 81956 −9074 8435 −6213−214398 342736 −147866 25858 −24035 17705

81956 −147866 86840 −29734 33426 −24622−9074 25858 −29734 25741 −62936 501458435 −24035 33426 −62936 287393 −242282−6213 17705 −24622 50145 −242282 205268

y

{

.

i

k

000

0.51.01.4

y

{

i

k

−4800.213681

−15911.820137.5−83269.870165.7

y

{

Δ R 1Δ R 2Δ R 3Δ R 4Δ R 5Δ R 6

=

=

=

=

-8 83 41-1 70 67-114 96-2 43 46-3 45 28

163 1

i

k

−4800.213681

−15911.820137.5−83269.870165.7

y

{

+

+

R1R2R3R4R5R6

Ri1Ri2Ri3Ri4Ri5Ri6

ΔR1ΔR2ΔR3ΔR4ΔR5ΔR6

=

-93141-3386

-27408-4208

-11779771797

(11.7)

This result is completely in agreement with that directly calculated, as shown in Table 11.2. This method has the major advantage of being able to determine the reactions of the aft stern tube bearing or

bearings inside the engine, on which the jack up test is difficult. However, as can be seen in Fig. 11.11, the bending moment caused by the change in the bearing offsets only varies linearly between any two adjacent bearings. Therefore, the bending moment at any third cross section, M3, can be calculated from the previously known bending moment at any other two different cross sections M1 and M2. Therefore, only one, in case two bending moments in the adjacent span have already been used, or at most two independent bending moments between any two adjacent bearings can be used in the calculation. On the other hand, because the relatively easy measurement of the bending moments is only possible before and after the intermediate bearing, it is necessary to simplify the shafting calculation model by reducing the number of engine bearings taken into account and by using the stern tube bearing as the reference line. Furthermore, because the measurement of the


bending moment requires highly specialized knowledge and technique in addition to expensive equipment, the method is only employed when the reactions of the aftmost engine bearings need to be determined with high accuracy.

M1M2

M3

Fig. 11.11 Bending moments in shaft caused by misalignment only.

11.2.2 Measurement Method of Bending Moment

The mechanism of bending moment measurement consisting of using strain gauges to measure the axial strain

from bending. It is common to use four strain gauges to form a Wheatstone bridge in order to gain a larger

output and to cancel out the effect of the thrust induced axial strain, as shown in Fig. 11.12. It is important to

glue the two sets of two strain gauges directly opposite to each other by appropriately marking the positions.

G1

G2

G3

G4

13

1423

24M

G1

G2

G3

G4

13

14 23

24

Vin

Vout

Fig. 11.12 Bending moment measurement using the strain gauge technique.

Because the shaft is turning, a wireless system called a telemetry system is usually needed to measure bending moment, as shown in Fig. 11.13. In the telemetry system, the firmly glued strain gauges are connected to transmitters with their own power source. Therefore, the gauges and the transmitters rotate together with the shaft. Signals from the transmitters are sent to a data recorder via a receiver after being received by a receiving

58


antenna installed around the shaft.

Strain gauge

Receiver Data recorder

Receiving antenna

Transmitter Shaft

Fig. 11.13 A telemetry system used to measure bending moment.

Photo 11.1 shows the glued strain gauges and installed components of the telemetry system.

Strain gauges glued onto shaft

Transmitter

Receiving antenna

Batteries for transmitter

(a) (b)

Photo 11.1 (a) Strain gauges glued onto shaft. (b) Mounted essential components comprising telemetry system.

59


12. Conclusions

These guidelines provide methods and procedures for the design, installation and confirmation of shafting

alignment in dealing with the effects of the changes that occur in bearing offsets between different load and

operating conditions, after detailing the phenomena and showing the extent of the changes through

measurements and calculations performed on an actual VLCC. The ultimate goal of the guidelines is to help

prevent improper shafting alignment related damages of bearings and shafts. Although this goal can be

achieved during the initial installation - bearing load measurement - readjustment cycle, the cost will be huge in

the event of a major alteration of the shafting alignment at the sea trial stage. Therefore, appropriate procedures

are necessary that take into account actual operating conditions for the shafting alignment. These may be

summarized as follows:

Using an equivalent circular bar to model the crankshaft, incorporating all engine bearings in the shafting

alignment calculation to improve the accuracy of the calculations.

•

•

•

•

•

Optimizing the longitudinal location of the intermediate bearing to reduce the sensitivity of the shafting to

changes in bearing offset.

Predicting the changes in bearing offsets including the dynamic components between the installation and the

representative operating conditions and compensating for these changes in the initial bearing offsets, if

necessary.

Directly confirming the reactions of the forward stern tube bearing, intermediate bearing, and aftmost engine

bearing in the engine warm and design draught condition.

If there are any recommendations by the engine manufacturer, reducing the load of the aftmost engine bearing

to nearly zero in the installation condition as a rough measure of the necessary initial compensation, and

complying with the recommendation, accordingly.

60


Part B Amendment of the Guidance to the Rules and its Explanatory Notes

Part B June 2006

NIPPON KAIJI KYOKAI

Foreword

In the propulsion system of ships, large loads due to the weight of the propeller act on the aft edge of the shaft, which is a major characteristic of the shafting system. Bending moment due to such loads results in edge loading of the aftmost bearing. Consequently, Classification Societies have developed rules on shafting alignment taking into consideration the strength of the aftmost bearing. ClassNK’s Rules have also been developed with a focus on the aftmost bearing. Hence, shafting alignment calculations for propulsion shafting are required for shafting where the length of the aftmost bearing is shorter than the length normally set by the Rules or where the propeller shaft is intended to be classed as a Kind 1C type shaft. On the other hand, there has been a growing number of incidents of engine bearing damage reported in recent large 2-stroke cycle main engines. The cause of such damage is generally thought to be attributable to an increase in bearing loads, but may also be connected with a tendency for the span between engine bearings to become shorter. In view of the contact that exists between the shaft and bearings, this tendency makes the shafting more sensitive to changes in bearing offsets. Among the various cases of bearing damage reported, there have been cases in which an engine bearing becomes unloaded due to the effects of changes in temperature and hull girder deflection. Therefore, in the design of shafting alignment, it is important to give the shaft as much flexibility as possible with due consideration given to the effects of temperature changes and hull girder deflections as well as to reduce the edge loading of the aftmost bearing. ClassNK has recently amended its Rules and Guidance on Shafting Alignment. These revised requirements will apply to ships that submit an application for a Classification Survey during Construction to the Society on or after 1 July 2006. According to the amended Rules, shafting alignment calculations will be required for all shafting having an oil-lubricated propeller shaft with a diameter of 400 mm or more. Furthermore, the Annex D6.2.13 to the Guidance provides clear descriptions of models of shafting, conditions to be calculated, the evaluation of calculation results and other related matters. This document gives a description of the main changes made in this revised version of the Guidance with respect to the calculation of shafting alignment and provides an explanation of the details of such calculations and the reason for the revisions made. It is hoped that this information will prove useful for designers at shipbuilders and engine manufacturers as well as surveyors and other parties with an interest in effective shafting alignment.

NIPPON KAIJI KYOKAI

June 2006

CONTENTS GUIDANCE FOR CALCULATION OF SHAFT ALIGNMENT 1.1 General 1

1.1.1 Application 1 1.1.2 Calculation Sheet of Shaft Alignment 1

1.2 Model of Shafting 2 1.2.1 Loads 2 1.2.2 Bearings 2 1.2.3 Equivalent Diameter of Crankshaft 2 1.2.4 Shafting with Reduction Gear 2

1.3 Load Condition and Evaluation of Calculation Results 3 1.3.1 Light Draught Condition (Cold Condition) 3 1.3.2 Light Draught Condition (Hot Condition) 4 1.3.3 Full Draught Condition (Hot Condition) 4

1.4 Matters relating to Shaft Alignment Procedures 6 1.4.1 Sags and Gaps between Shaft Coupling Flanges 6 1.4.2 Procedures for Measuring Bearing Loads 6

EXPLANATORY NOTES 1.1 General 7

1.1.1 Application 7 1.1.2 Calculation Sheet of Shaft Alignment 8

1.2 Model of Shafting 9 1.2.1 Loads 9 1.2.2 Bearings 10 1.2.3 Equivalent Diameter of Crankshaft 11 1.2.4 Shafting with Reduction Gear 11

1.3 Load Condition and Evaluation of Calculation Results 12 1.3.1 Light Draught Condition (Cold Condition) 12 1.3.2 Light Draught Condition (Hot Condition) 13 1.3.3 Full Draught Condition (Hot Condition) 14

1.4 Matters relating to Shaft Alignment Procedures 17 1.4.1 Sags and Gaps between Shaft Coupling Flanges 17 1.4.2 Procedure for Measuring Bearing Loads 17

References 18 APPENDIX A (DERIVATION OF δ B2 AND δ B3) 1 Approximate Calculation of Relative Displacement of the Hull 19 2 Reaction Influence Numbers determined by Relative Displacement Model 21 3 Hull Deflection that results in Engine Bearings Becoming Unloaded 23

3.1 In the case of elastic support 23 3.2 In the case of rigid support 23

4 Calculation Example 27

PART B AMENDMENT OF THE GUIDANCE TO THE RULES AND ITS EXPLANATORY NOTES

1

GUIDANCE FOR CALCULATION OF SHAFT ALIGNMENT 1.1 General 1.1.1 Application

-1 This Guidance applies to shaft alignment calculations required in Sections D6.2.10, D6.2.11 and D6.2.13 of the Rules. With regard to the paragraphs in 1.3 of this Guidance, the application is to be in accordance with Table 1.1.1-1.

Table 1.1.1-1 Application of Section 1.3 of the Guidance

Paragraphs 1) 2) Type of main propulsion machinery 1.3.1 1.3.2 1.3.3 3)

Two-stroke cycle diesel engine ● ● ● Four-stroke cycle diesel engine ● ● - Steam turbine ● ● -

Notes 1) ●: Applicable -: Not applicable 2) 1.3.1: Light draught condition (cold condition) 1.3.2: Light draught condition (hot condition) 1.3.3: Full draught condition (hot condition) 3) Only applicable to oil tankers, ships carrying dangerous chemicals in bulk, bulk

carriers, and general dry cargo ships, where: ● an oil tanker is a ship defined in 1.3.1(11), Part B of the Rules;

● a ship carrying dangerous chemicals in bulk is a ship defined in 2.1.43, Part A of the Rules;

● a bulk carrier is a ship defined in 1.3.1(13), Part B of the Rules; and ● a general dry cargo ship is a ship defined in 1.3.1(15), Part B of the Rules.

-2 Notwithstanding the provisions of sub-paragraph 1.1.1-1 above, paragraphs 1.1.2, 1.2.1 and 1.3.1 (excluding 1.3.1-4) below are to apply to shaft alignment calculations required in D6.2.10 and D6.2.11, where the main propulsion shafting comprises a oil-lubricated propeller shaft with a diameter less than 400 mm.

-3 An alternative method of calculation different from that described in this Guidance may be employed subject to prior acceptance by the Society.

1.1.2 Calculation Sheet of Shaft Alignment

Calculation sheets for shaft alignment that include the following data are to be submitted for approval: (a) Diameter (outer and inner) and length of shafts (b) Length of bearings (c) Concentrated loads and loading points (d) Support points (e) Bearing offsets from reference line (f) Reaction influence numbers (g) Bending moments and bending stresses (h) Bearing loads and nominal bearing pressure (i) Relative inclination of the propeller shaft and aftmost stern tube bearing or maximum

bearing pressure in the aftmost stern tube bearing (j) Deflection curves for the shafting (k) Sags and gaps between shaft coupling flanges (l) Procedures for measuring bearing loads (in cases where such measurement is required)


2

1.2 Model of Shafting 1.2.1 Loads

-1 Static loads are to be used in the shaft alignment calculations.

-2 The buoyancy force working on the shafting is to be considered as a load. The tensile force due to the cam shaft drive chain specified by the engine manufacturer is also to be considered as a load for the engine.

1.2.2 Bearings

-1 When only one support point is assumed in the aftmost stern tube bearing, its location is to be at L/4 or D/3 from the aft end of the bearing. When two support points are assumed, their locations are to be at the each end of the aftmost stern tube bearing. When three or more support points are assumed, their locations may be decided by the designer. The location of the support point in each bearing other than the aftmost stern tube bearing is to be the center of the bearing.

Figure 1.2.2-1 Location of single support point in aftmost stern tube bearing. -2 Either rigid support or elastic support may be acceptable for the type of support used.

-3 When the thrust shaft is integrated with the crankshaft, not less than five main bearings of the engine are to be considered in the shaft alignment calculation.

Figure 1.2.2-3 Number of main engine bearings to be considered. 1.2.3 Equivalent Diameter of Crankshaft

When evaluating the shafting of a two-stroke cycle diesel engine, the equivalent diameter of the crankshaft specified by the engine manufacturer is to be used in the shaft alignment calculation, in order to give due consideration to the lesser bending stiffness that exists in the actual crankshaft compared with simply using the diameter of the crank journal in the model.

1.2.4 Shafting with Reduction Gear

For shafting with a reduction gear such as that found in main steam turbine or geared diesel engines, the shafting from the propeller to the wheel gear is to be considered in the shaft alignment calculation.

#1 #2 #3 #4 #5

Aftmost engine bearing

X

L

X = L/4 or D/3X: Location of single support point from

bearing aft end L: Length of aftmost stern tube bearing D: Diameter of propeller shaft

Aftmost stern tube bearing

Propeller shaft

D


3

1.3 Load Condition and Evaluation of Calculation Results 1.3.1 Light Draught Condition (Cold Condition)

-1 Shaft alignment calculations are to be performed assuming that the ship is in the light draught condition and the main propulsion machinery is in the cold condition. In cases where the shafts are coupled before launching, the shaft alignment calculation is to be performed for the coupled condition instead of for the light draught condition without taking the buoyancy force on the propeller into account.

-2 When the aftmost stern tube bearing consists of oil-lubricated white metal, an evaluation is to be made of the nominal bearing pressure together with either the relative inclination between the propeller shaft and aftmost stern tube bearing or the maximum bearing pressure in the aftmost stern tube bearing, either of which is determined in order to prevent edge loading on the bearing. Each calculated value is to be within the allowable limit shown in Table 1.3.1-2.

Table 1.3.1-2 Allowable Limits for Aftmost Stern Tube Bearing (Oil-lubricated White Metal)

Allowable limit Notes Nominal bearing pressure 0.8 MPa Relative inclination between the propeller shaft and the aftmost stern tube bearing

3 × 10-4 rad Applicable where number of support points is one or two. For two support points, relative inclination is to be calculated at each end of the bearing (see Figure 1.3.1-2(a)).

Maximum bearing pressure 40 MPa Applicable where the maximum bearing pressure is calculated (see Figure 1.3.1-2(b)).

Figure 1.3.1-2(a) Relative inclination. Figure 1.3.1-2(b) Maximum bearing pressure. -3 The bending moment (absolute value) calculated at any bearing is not to be more than the value

determined for the aftmost stern tube bearing.

-4 In principle, the bearing load calculated at each bearing is to be a positive value. However, for the aftmost bearing of a two-stroke cycle diesel engine, a bearing load of zero may be accepted (negative value is not acceptable) subject to the agreement of the engine manufacturer. The direction of the bearing load is shown in Figure 1.3.1-4.

Figure 1.3.1-4 Direction of bearing load.

Propeller shaft

Aftmost stern tube bearing

Relative inclination

Pressure distribution

Maximum bearing pressure

Bearing load (positive)

Shaft

Bearing load (negative)

Shaft


4

1.3.2 Light Draught Condition (Hot Condition)

-1 Shaft alignment calculations are to be performed assuming that the ship is in the light draught condition and the main propulsion machinery is in the hot condition. In this case, the increases in offset specified by the manufacturer for the engine bearings and the bearings in the reduction gear are to be considered for the hot condition.

-2 The full immersion condition of the propeller may also be taken into account in the calculation. -3 When the shafts are coupled before launching, the calculation is to be performed under the

assumption that there is no change in the bearing offsets from the reference line between the conditions before and after launching.

-4 The bearing load calculated at each bearing is to be a positive value. -5 For shafting with a reduction gear, the difference in the bearing loads between the fore and aft

bearings of the wheel gear in the hot condition is to be within the allowable limit specified by the manufacturer.

-6 The bending moment due to eccentric thrust of the propeller may be taken into account in the calculation.

1.3.3 Full Draught Condition (Hot Condition)

-1 The shaft alignment for oil tankers, ships carrying dangerous chemicals in bulk, bulk carriers, and general dry cargo ships is to be designed so as to satisfy the following criteria in order that all engine bearings are fairly evenly loaded, even under the hull deflection that occurs when the ship is in the full draught condition. The extent of relative displacement due to the difference between hull deflection that occurs in the light draught condition and hull deflection that occurs in the full draught condition which results in the second or third aftmost engine bearing becoming unloaded, as measured at the aftmost bulkhead of the engine room (calculated as δ B2 and δ B3, respectively), is to be greater than the allowable lower limit δ BM shown in Fig. 1.3.3-1 (a).

Figure 1.3.3-1(a) Allowable lower limit, δ BM, for δ B2 and δ B3.

The relative displacements, δ B2 and δ B3 , above are to be calculated using the equations shown in sub-paragraphs (1) or (2) below, depending on the type of bearing support adopted (elastic or rigid support) that is used to calculate the reaction influence numbers in the alignment calculations.

(1) In the case of elastic support, δ B2 and δ B3 can be obtained with i =2 and 3, respectively:

L (mm)Distance from the support point of the aftmost engine bearing to the aftmost bulkhead of the engine room (see Figure 1.3.3-1(b)).

5 7 9 11 13 15 x 103

10

8

6

4

2

0

δ BM

(mm

)


5

δ Bi = - R i / S i

where, i: Engine bearing number as counted from the aft of the engine R i: Reaction force at engine bearing No. i determined by the calculation in 1.3.2 (kN) S i: Influence number for engine bearing No. i when the hull deflection at the aftmost bulkhead

of the engine room becomes -1 mm; obtained from the following equation (kN/mm):

∑∑−

=−+

−

=−+ +−=

1

5.1 ,1

1

1 ,1 )5.05.1(

b

annnib

a

nnnibi xCxCS

LXx nn /=

n: Support point number of the shafting (counted from the aft of the shafting) a: Number of the nearest support point forward of the aftmost bulkhead of the engine room

(counted from the aft of the shafting) b: Support point number of the aftmost engine bearing (counted from the aft of the shafting) X n : Distance from the support point b to the support point n (mm) L: Distance from the support point b to the aftmost bulkhead of the engine room (mm) C m, n : Influence number at the support point m when the relative displacement at the support

point n becomes -1 mm (kN/mm) (see Figure 1.3.3-1(b)).

Figure 1.3.3-1(b) Engine bearing numbers and support point numbers.

(2) In the case of rigid support, δ B2 and δ B3 can be obtained by solving the following simultaneous

equations (1) and (2), respectively:

S1 δ B2 + (C1, 1 -K) δ 1 + C1, 3 δ 3 + C1, 4 δ 4 + C1, 5 δ 5 = C1, 2 R 2 /K

S2 δ B2 + C2, 1δ 1 + C2, 3 δ 3 + C2, 4 δ 4 + C2, 5 δ 5 = (C2, 2 -K) R 2 /K S3 δ B2 + C3, 1 δ 1 + (C3, 3 -K) δ 3 + C3, 4 δ 4 + C3, 5 δ 5 = C3, 2 R 2 /K

S4 δ B2 + C4, 1 δ 1 + C4, 3 δ 3 + (C4, 4 -K) δ 4 + C4, 5 δ 5 = C4, 2 R 2 /K

S5 δ B2 + C5, 1 δ 1 + C5, 3 δ 3 + C5, 4 δ 4 + (C5, 5 -K) δ 5 = C5, 2 R 2 /K

S1 δ B3 + (C1, 1 -K) δ 1 + C1, 2 δ 2 + C1, 4 δ 4 + C1, 5 δ 5 = C1, 3 R 3 /K

S2 δ B3 + C2, 1 δ 1 + (C2, 2 - K) δ 2 + C2, 4 δ 4 + C2, 5 δ 5 = C2, 3 R 3 /K

S3 δ B3 + C3, 1 δ 1 + C3, 2 δ 2 + C3, 4 δ 4 + C3, 5 δ 5 = (C3, 3 - K) R 3 /K

S4 δ B3 + C4, 1 δ 1 + C4, 2 δ 2 + (C4, 4 -K) δ 4 + C4, 5 δ 5 = C4, 3 R 3 /K

S5 δ B3 + C5, 1 δ 1 + C5, 2 δ 2 + C5, 4 δ 4 + (C5, 5 -K) δ 5 = C5, 3 R 3 /K

Aftmost bulkhead of E/R

Intermediate shaftPropeller shaft

Crankshaft i

LX 1

X 4 Aftmost engine bearing m, n = 1 2 3 4 (=a) 5 (=b) 6 7 8 9

(1)

(2)


6

where, K: Stiffness of bearing support, given as K = 5,000 (kN/mm) Si : Influence number for engine bearing i (See (1) above) C i, j : Influence number for engine bearing i when the relative displacement at engine bearing j

becomes -1 mm (kN/mm). (The numbers i and j are counted from the aft of the engine.) δ i ( i =1, 2, 3, 4, 5): Elastic relative displacement at each engine bearing resulting from the relative

displacement δB2 and δB3 . (δ i is unknown.) -2 Notwithstanding the provisions of sub-paragraph 1.3.3-1, the Society may examine and accept

alternative criteria, provided that a document is submitted that makes it possible to evaluate the condition of the engine bearings when the ship is in the full draught condition.

-3 Other documents, such as those showing the results of structural analysis to evaluate the extent of hull deflection, may be required by the Society in cases where the stern hull construction is considered to be unconventional.

1.4 Matters relating to Shaft Alignment Procedures 1.4.1 Sags and Gaps between Shaft Coupling Flanges

Sags and gaps between shaft coupling flanges in the uncoupled condition are to be calculated under the condition that the bearing offsets from the reference line are those used in the calculation described in paragraph 1.3.1 above.

Figure 1.4.1 Sag and gap between shaft coupling flanges.

1.4.2 Procedure for Measuring Bearing Loads

In cases where the bearing load is measured using the jack-up technique, a document describing the measurement procedures followed, including jack-up positions, load correction factors, and expected jack-up loads, is to be prepared. The immersion of the propeller at the time of the measurement is also to be considered in the bearing loads measured.

SAG

GAP 1 GAP 2

GAP = GAP 1 + GAP 2


7

EXPLANATORY NOTES 1.1 General

1.1.1 Application

-1 In the design of shafting alignment, it is important to give the shaft as much flexibility as possible against changes in bearing offsets caused by the effects of temperature and hull girder deflections as well as to reduce the loads that concentrate at the aft end of the aftmost bearing due to the weight of the propeller. Therefore, the following measures are generally taken to accommodate these offsets: installation of the main engine below a predetermined reference line, installation of the intermediate shaft bearing at a location far away from the main engine to some extent, and other steps deemed suitable for the design adopted.

In the Society’s previous version of the Guidance to the Rules, shafting alignment calculations were required only for oil-lubricated propeller shafts, where the length of the stern tube bearing is shorter than that determined by the Rules or where the shaft is intended to be classed as a Kind 1C type shaft. However, because shafting alignment is originally to be examined without any relation to these conditions, it was determined in the new version of the Guidance to the Rules that the calculations are to be applied to all ships with oil-lubricated propeller shafts excluding small ships, as described below.

Figure 1.1 shows the relationship between the diameter of propeller shafts and main engine output. As can be seen from this figure, most of the oil-lubricated systems include a propeller shaft of 400 mm or more. In the range of 300 - 400 mm, the region of the oil-lubricated systems overlaps that of seawater-lubricated systems. However, comparatively few cases of damage are found in engine bearings as well as stern tube bearings in the oil-lubricated systems in this range. For this reason, small ships having a propeller shaft less than 400 mm in diameter are exempt from the application of the calculation requirement, even if the shaft is of an oil-lubricated type (see 6.2.13, Part D of the Rules).

Figure 1.1 Relationship between propeller shaft diameter and main engine output.

Shafting alignment is affected by changes in temperature and hull deflection. In the calculation sheets submitted until now, the examination of temperature changes has generally depended on the shipbuilders’ design; that is, some shipbuilders had examined only cold condition while other shipbuilders examined both cold and hot conditions. However, no shipbuilders had examined the

0

200

400

600

800

1000

1200

100 1,000 10,000 100,000

Main engine output (kW)

Dia

met

er o

f pro

pelle

r sha

ft (m

m) Oil-lubricated shaft (Kind 1C)

Seawater-lubricated shaft


8

effects of hull deflections due to changes in ship draught for the reason that highly accurate estimates were difficult and few measurement examples existed.

In the amended Guidance, the calculation conditions to be applied were determined according to the type of main engine installed considering the effects of changes in temperature and draught, as shown in Table 1.1.1-1 in the Guidance.

As can be seen from the table, the Guidance requires alignment calculations for both cold and hot conditions under the ship’s light draught condition, regardless of the type of main engine employed. In addition, when a two-stroke cycle diesel engine is installed, calculations are also required aimed at preventing unloading of the engine bearings due to hull deflection under full load condition (hot condition). The calculation equations used in this case are applied to ships with large differences in draught, such as tankers, bulk carriers, and the like, but not to container ships, pure car carriers, or similar types of ships. Also, paragraph 1.3.3 in the revised Guidance does not apply to two-stroke cycle geared diesel engines, because provisions for two-stroke cycle engines were prepared assuming a direct coupled engine.

-2 Where the length of the stern tube bearing is made shorter than that determined by the Rules or

where the propeller shaft is classed as a Kind 1C type shaft, provisions quite similar to those set in the previous version of the Guidance are applied to shafting systems having an oil-lubricated propeller shaft of less than 400 mm in actual diameter.

-3 The shafting alignment design mostly depends upon the knowledge and know-how of shipbuilders

and engine manufactures, and thus may not necessarily be based on the same or common criteria. Therefore, even if the criteria proposed differ from those stated in the Guidance, the Society will approve a design where the criteria are found to be acceptable.

1.1.2 Calculation Sheet of Shaft Alignment

The diameter and length of shafts, bearing length, concentrated loads, loading points, bearing support points, and bearing offsets from reference line are input data for shafting alignment calculations. It is also necessary to check the calculations themselves. Therefore, these data should be specified in the drawing.

The reaction influence number (also known as “reaction influence coefficient”) is an important parameter that relates to the flexibility of the shaft. As shown in Figure 1.2.2-3 in the Guidance, the alignment calculations are to be carried out considering five or more bearings from the aft of the engine, and all reaction influence numbers are to be specified in the drawing. However, a simplified description that includes only five engine bearings is acceptable in cases where the actual calculation is made considering five or more engine bearings (for example, all engine bearings).

At a minimum, the bending moment acting on the shaft and the displacement of the shaft are to be illustrated in the calculation results, and bearing loads (bearing reactions) are to be shown in an accompanying table. The relative slope between the propeller shaft and the aftmost stern tube bearing is to be shown with the position of the bearing support in the figure. In cases where the bearing load (or pressure) acting on the aftmost stern tube bearing is calculated as a distributed load (or pressure), the maximum bearing pressure is to be shown with a figure illustrating the distribution.

The sag and gap between coupling flanges and the jack-up load in the bearing load measurement (where required by the Guidance) are important information to be confirmed by the field surveyor. Therefore, a document describing the measurement procedure employed including the calculated values is also to be submitted to the Society. Moreover, the calculated values (target values) are to be indicated with the applicable tolerances.


9

1.2 Model of Shafting 1.2.1 Loads

-1 In the amended version of the Guidance, a static load, i.e., the load acting on the shafting system when the shaft is in the stand-still condition, is to be used in the shaft alignment calculations. These provisions are the same as those in the previous version of the Guidance. However, there are some cases where the bearing strength and certain other factors cannot be assessed accurately using only a static load. Such examples are described below.

Bending moment due to hydrodynamic propeller forces The center of effort of the propeller mean thrust force differs from the actual center of the propeller due to the non-uniform distribution of the ship’s wake. Figure 2.1 shows a schematic of this effect. When a propeller blade rotates in the wake, it generates a larger thrust force in the upper side of the propeller center than in the lower side This leads to the generation of an eccentric thrust that act on the propeller resulting in the generation of bending moment, MP, in the illustrated direction.

Figure 2.1 Bending moment due to hydrodynamic propeller forces.

In general, the bending moment, MP, can be said to be a kind of “safety enhancing” moment, because it acts so that edge loading on the aft portion of the aftmost bearing may be reduced. MP also causes a comparatively small change in load on the intermediate shaft bearings and engine bearings. However, in a shafting system that includes reduction gears, small changes in the inclination of the gear shaft may possibly affect the strength of the gear teeth. Therefore, in order to examine the strength of the gear teeth in detail, it is necessary to confirm the changes in load that take place on the gear shaft bearings using alignment calculations for the hot condition considering the effect of MP.

Bending moment that acts on thrust shaft In a large two-stroke cycle diesel engine fitted with a crankshaft that is integrated with the thrust shaft, the thrust pads are, in most cases, not arranged over the entire circumferential plane of the thrust collar. In this case, a bending moment, MT, will act on the thrust shaft while the ship is underway, as shown in Figure 2.2.

Figure 2.2 Bending moment acting on the thrust shaft.

This MT need not be input in the alignment calculation because the calculation required by the Guidance is for the static condition. MT may also act as a kind of “safety enhancing” moment in view of the effect that it has on preventing the unloading of the second aftmost engine bearing

Large thrust force

Small thrust force

MP

MT

Thrust collar

Thrust

Thrust pads

Arrangement of thrust pads



10

caused by hull deflection. However, unloading of the aftmost engine bearing might occur in a ship in the light design draught condition. In order to prevent such a situation, it is important to confirm the load of the aftmost bearing by the alignment calculation considering MT.

-2 Consideration needs to be given to buoyancy when evaluating the load that acts on a propeller. Since the extent to which the propeller becomes immersed varies depending on the draught condition of the ship, the load is to be obtained by subtracting the effect of buoyancy due to immersion of the propeller from the weight of the propeller for that condition. In addition, it is also recommended that buoyancy due to the lubricant in the stern tube also be considered, even though such buoyancy will only have a small effect on the calculation results.

If the camshaft of the engine is driven by the crankshaft using a chain, the tension of the chain has a comparatively large effect on the load of the aftmost and second aftmost engine bearings in cases where the tension acts on the thrust collar. Consequently, this tension also needs to be included in the calculation.

1.2.2 Bearings

-1 When performing a static alignment calculation, a model with one or two support points is used in most cases for the aftmost stern tube bearing. The position of the support points is usually assumed to be L/4 or D/3 (in the case of one-point support) or both ends of the bearing (in the case of two-point support), which is considered to be normal practice in alignment design.

However, it is difficult to determine uniformly fixed positions for the support points in the Guidance because the position will depend on the design of the alignment. Hence, a support point position that differs from the above fixed positions will also be acceptable, unless it is significantly distant from the normally standard position.

-2 With respect to the support condition of the bearings, a rigid (simple) support approach is used in

many alignment calculations, whereas there are few examples in which elastic (spring) support is used. An oil-film support is needed for dynamic calculations. However, the effect of the oil-film need not be considered in the Guidance, as the Guidance focuses on static calculations.

-3 In order to construct an exact model for alignment calculation, it is desirable to include all engine

bearings in the model. However, there have been numerous examples of calculation done until now that have used a reduced number of engine bearings for the reason that there was little difference in the load acting on the intermediate shaft bearing. Figure 2.3 shows the effect that the number of bearings included in the calculation has on the bearing loads.

Figure 2.3 Effect of the number of engine bearings included in the calculation of bearing loads.

No. 1 - No. 3 No. 1 - No. 4No. 1 - No. 5No. 1 - No. 6No. 1 - No. 7No. 1 - No. 8No. 1 - No. 9

Engine bearings calculated (7-cylinder engine)

No. 1 (aftmost engine bearing) No. 2 No. 3 Identification number of engine bearings

Bear

ing

load

s


11

According to this figure, the loads on the aftmost and the second aftmost engine bearings change significantly with an increase in the number of engine bearings included in the calculation. It can be seen that five or more bearings are necessary in order to obtain a reasonably accurate calculation of the bearing loads. It was thus determined from this fact that at least five engine bearings from the aft of the engine are to be included in the calculation model.

1.2.3 Equivalent Diameter of Crankshaft

When conducting an alignment calculation, the actual complex shape of the crankshaft needs to be replaced with a simple beam (round bar). In the previous version of the Guidance, the use of an equivalent shaft to represent the crankshaft was not prescribed. Consequently, there have been cases when a simple beam with a diameter equivalent to the crank journal had been used when carrying out alignment calculations.

However, the diameter of the equivalent crankshaft model in recent two-stroke diesel engines used for propulsion has tended to become considerably smaller as the piston stroke has become longer. In large, long stroke engines, the diameter has become approximately 60% of the journal diameter. Because the bending stiffness of the equivalent shaft can differ markedly from that of the crank journal, a brief description of the equivalent crankshaft model has been added in the revised version of the Guidance in order that the most suitable equivalent diameter may used in the alignment calculations.

In principle, calculation of the equivalent diameter is to be conducted in accordance with the instructions of the engine manufacturer. An approximation equation developed by the Society [1] is also available that may be used in addition to the guidelines of the manufacturer.

1.2.4 Shafting with Reduction Gear

The calculation model for a shafting system with a reduction gear, such as a main steam turbine system or a geared diesel engine, generally ranges from the propeller to the fore bearing of the wheel gear shaft. This approach has been newly described in the Guidance.


12

1.3 Load Condition and Evaluation of Calculation Results

1.3.1 Light Draught Condition (Cold Condition)

-1 Shaft alignment calculation for the cold condition is the basic calculation method used when determining the positioning and installation of the shafting system. The calculation procedure consists of first determining the positions of the bearings so that the proper values for the bearing loads may be attained when the shafts are coupled in the cold condition. Next, the amount of sag and gap between the coupling flanges is calculated in order that the bearings may be installed in the calculated positions.

Coupling of the shafts is generally done in the light ballast condition (the draught condition in which the propeller blades are exposed above the surface of the sea) after launching; however, in some cases it is done while the ship is in the drydock condition before launching. The approach adopted depends on the shipbuilder. It is therefore specified that application of the Guidance would also include such situations.

-2 The strength of the aftmost stern tube bearing can be examined even by the alignment calculation intended for the cold condition, because the effect of temperature and hull deflection on the bearing load is small. If an exact assessment is required, the strength is to be determined based on the dynamic loads in the hot condition. However, bearing damage can be avoided by evaluating edge loading under the cold condition. Therefore, the allowable limit of the relative slope between the propeller shaft and the aftmost stern tube bearing (or maximum bearing pressure) has been prescribed for the cold condition, based on past practical experience and data summarized by the Japan Institute of Marine Engineering [2].

In principle, the relative slope is to be calculated at the position of the support point shown in Figure 1.2.2-1 in the Guidance. However, it should be noted that the bending moment due to hydraulic propeller force, shown in Figure 2.1 of these Explanatory Notes, is not to be considered in the calculation, because the allowable limit of the relative slope is prescribed for the static condition.

The allowable limit of the nominal bearing pressure is the same as in the previous version of the Guidance.

-3 The provisions of 1.3.1-3 in the revised Guidance are the same as in the previous version of the Guidance. In a marine propulsion system, it is generally given that the bending moment of the shaft becomes greatest due to the weight of the propeller at the position of the aftmost stern tube bearing. This type of typical shafting system is the object considered in the Guidance.

-4 In ships with large differences in draught, the load of the aftmost engine bearing tends to increase due to the influence of hull deflections, whereas that of the second aftmost engine bearing tends to decrease (refer to paragraph 1.3.3 below). To prevent engine bearing damage that may occur in such a situation, some shipbuilders install the shafting so that the aftmost engine bearing becomes unloaded in the light ballast condition (cold condition) if the engine is of a recent two-stroke cycle type.

Unloading of the aftmost bearing in the cold condition is acceptable because the strength of the bearing is examined by the alignment calculation for the hot condition. However, careful attention should be paid to this calculation for the cold condition. It should be noted that in alignment calculations commonly used, the top clearance of the bearing is not considered. Therefore, if the calculation results for the aftmost bearing load are negative, it is necessary to carry out the calculations again excluding the bearing, in order to confirm the displacement of the shaft at the position of the bearing. If the calculated displacement falls within the range for the bearing top clearance, the result means that the bearing is unloaded (see Figure 3.1).

Moreover, calculation results that include a negative bearing load even though the bearing is unloaded means that other bearings are taking up the load for the minus portion, and that the


13

load of other bearings differ from the actual respective values for each bearing. In order to prevent such a situation from occurring, the Guidance specifies that calculation results with a negative bearing load are not acceptable.

Figure 3.1 Negative bearing load.

1.3.2 Light Draught Condition (Hot Condition)

-1 This calculation condition is used to examine the strength of the intermediate shaft bearings, engine bearings, and other bearings in the ship’s ballast condition when the propulsion machinery is in the hot condition. It is assumed here that the hull girder deflection is the same as that in the condition described in paragraph 1.3.1 above and that the differences in the change in bearing offsets are only attributable to differences in temperature.

Temperature changes cause changes in the offsets of all bearings such as the stern tube bearings, intermediate shaft bearings, gear shaft bearings, engine bearings, and the like. However, since large offset changes are seen especially on gear shaft bearings and engine bearings, temperature changes on at least these bearings are to be considered in the calculation. The amount of the offset change is to be in accordance with the manufacturer’s instructions, as it varies depending on the manufacturer and their respective models. For main engines, an increase in temperature of from 20 to 55°C is generally considered and, as shown in Figure 3.2, the resulting change in the bearing offset becomes larger as the scale of the engine (bore of the cylinder) becomes greater.

Figure 3.2 Relationship between change in bearing offset and cylinder bore.

-2 Buoyancy acting on the propeller changes depending on the extent of immersion of the propeller. However, the effect of buoyancy is such a degree that the load on the aftmost stern tube bearing changes only slightly, and changes in the loads on the intermediate shaft bearings and engine bearings are negligible. Hence, the degree of propeller immersion to be considered in the Guidance is, for example, to be either full immersion for the ballast condition or half immersion for the light ballast condition, whichever is acceptable.

-3 Assuming that the bearing offset changes to some extent as a result of the influence of the change in hull girder deflection that occurs before and after launching, in cases where shafts are coupled before launching, the alignment calculation for the hot condition is to be carried out considering the influence of the deflection as well as the change in temperature. However, according to the results of measurements obtained up to now, the change in engine room double bottom deflection

(+) (+)

(-)

(+) (+)

Unloading R1

R2 R3 R2’

R3’

0.00

0.10

0.20

0.30

0.40

300 400 500 600 700 800 900

Cha

nge

in b

earin

g of

fset

(mm

)

Cylinder bore (mm)


14

before and after launching is comparatively small. Therefore, calculations carried out assuming the hull deflection condition prior to launching may be acceptable, if consideration is given to the effects of propeller buoyancy and the extent of thermal increase.

-4 Unloading of a bearing in the static condition, which means the detachment of the shaft from the bearing, indicates the possibility that, when dynamic loads act on the shaft, the bearing may be hit repeatedly by the shaft, resulting in eventual failure of the bearing. Therefore, all bearing loads are to be positive in the running condition of the engine.

-5 In cases where a shafting system with reduction gear is used (see Figure 3.3), the gear teeth rather than the bearings are susceptible to misalignment of the shafting. The contact condition between the wheel and the pinion depends on the difference in the bearing loads on the fore and aft side of the gear. The manufacturer of the gear box will usually determine the allowable limit for the hot condition. The allowable limit depends on the manufacturer. Therefore, the Guidance specifies that the system is to be so designed as not to exceed the allowable limit determined by the manufacturer.

Figure 3.3 Geared propulsion system.

-6 As shown in Figure 2.1 of the Explanatory Notes, bending moment due to hydrodynamic propeller force (upward moment caused by eccentric thrust force) is generated while the ship is underway, resulting in a change in the load on the aftmost stern tube bearing. This bending moment has a comparatively small effect on the loads of the intermediate shaft bearings or engine bearings, which are to be examined in the calculation for light draught condition (hot condition). Although the Society’s Guidance primarily assumes a static condition, it is acceptable to take this upward moment by eccentric thrust force into account when performing calculations for the hot condition.

1.3.3 Full Draught Condition (Hot Condition)

-1 Damage to engine bearings due to hull deflection have been seen in ships with large differences in draught such as oil tankers, bulk carriers, and the like. A significant feature of the damage was unloading of the second aftmost engine bearing, which was generally found to be attributable to hull deflection behind the main engine. However, investigations by the Society also found that unloading could take place at the third aftmost engine bearing as well as the second aftmost engine bearing, depending on the design of the engine bearing offsets.

The Guidance specifies that the shafting alignment is to be designed so that the parameters, δ B2

and δ B3 , are not less than the allowable limit shown in Figure 1.3.3-1(a) in the Guidance. The two parameters represent the difference in the extent of relative displacement of the hull due to hull deflection that occurs between the “zero” or baseline displacement of the light draught, hot condition, and the point at which hull deflection results in the second and third aftmost engine bearings, respectively, becoming unloaded in the full draught, hot condition. This difference in extent is measured at the aftmost bulkhead of the engine room. Said another way, it is an inverse measure of the extent of displacement that must take place until unloading occurs. Hence, the greater the value (i.e., the greater the margin before displacement results in unloading), the less likelihood that unloading of the respective bearing due to deflection will occur. The value of these parameters can be obtained using approximation equations provided by the Society. The equations were developed based on the results of past structural analyses and measurements in which relative displacement, δ , behind the main engine is expressed using the following equation:

Propeller shaft Intermediate shaft Thrust shaft

Gear shaft


15

(1) where, X : distance from support point of aftmost engine bearing (mm), L : distance from support point of aftmost engine bearing to aftmost bulkhead of engine room (mm), δ B : relative displacement of hull at position of aftmost bulkhead of engine room (mm).

Figure 3.4 Calculation model for determining relative displacement of hull. If Equation (1) is assumed, the change in offsets of the intermediate shaft bearings and stern tube bearings can be obtained by δ B. When relative displacement δ changes, the change in the reaction of the second aftmost engine bearing, for example, can be calculated from changes in each bearing offset behind the main engine and the respective reaction influence number with respect to the second aftmost engine bearing. In cases where the second aftmost engine bearing becomes unloaded, δ can be calculated inversely from the change in load (reaction) that occurs in the second aftmost engine bearing and the reaction influence number for that bearing. δB2 can be obtained from δ at that time using Equation (1). The approximation equations for δB2 and δB3 in the Guidance were derived using such a calculation process. The values of the reaction influence numbers considered in the calculations will vary somewhat depending on the type of support used for the bearings, which may either consist of rigid support (simple support) or elastic support. Therefore, the Guidance provides equations to approximate δB2 and δB3 for both cases (see Appendix A).

Bearing damage is not necessarily generated under a static draught when the ship is fully loaded, but rather possibly generated by the influence of waves at that time. Figure 3.5 shows the values of δB2 or δB3 , whichever are less, calculated in a shafting system where bearing damage due to hull deflection has never occurred. The lower limit of δB2 or δB3 was determined as shown in Equation (2) using L (mm) so that the line of the lower limit would be below these calculated data.

(2)

When δB2 (or δB3 ) exceeds the lower limit, the difference between δB2 (or δB3 ) and the lower limit means that there is a sufficient a safety margin with respect to hull deflection up to the point of unloading of the respective engine bearing. It can be said that δB2 (or δB3 ) is a parameter that indicates the flexibility of the shaft to hull deflection.

δ B (X /L )1.5 (X ≤ L) δ B {1.5 (X /L ) - 0.5} (X ≥ L)δ = {

Relative displacement

δ B

δ = δ B {1.5(X/L)-0.5}

δ = δ B (X/L)1.5 L

Aftmost bulkhead of E/R Aftmost engine bearing

Y

X

1 (mm) (L ≤ 9000)

- 8 (mm) (9,000 ≤ L) L

1000Lower limit of δ B2 (or δ B3 ) =


16

Figure 3.5 Values of δB2 or δB3 , whichever is less.

-2 In addition to the method described in sub-paragraph 1.3.3-1 above, there may be an appropriate

method for evaluating the strength of the bearings based on the results of structural analysis or full-scale measurements. Therefore, if a document is submitted that includes an evaluation of the engine bearings in the full load condition, the Society will examine the document and approve it, where acceptable.

-3 The criteria described in sub-paragraph 1.3.3-1 above apply for ships with a standard stern structure. If the ship has an unconventional stern structure, the approximation equation on relative displacement of the hull can not be applied for the ship. Accordingly, in cases where the stern hull structure is considered to be unconventional, a document including the results of structural analysis and other relevant data on the hull deflection may be required by the Society. The structural analysis is to be done to evaluate the relative displacement that occurs when the ship’s draught changes from the light ballast condition (or ballast condition) to the full load condition.

Bulk Carrier Tanker

δ B2

or δ

B3 ,

whi

chev

er is

less

(mm

)

15 6 7 8 9 10 11 12 13 14 x1000L (mm)

0

5

10

15

20

22

Lower limit


17

1.4 Matters relating to Shaft Alignment Procedures 1.4.1 Sags and Gaps between Shaft Coupling Flanges

Any sags and gaps between coupling flanges are measured from the aftmost coupling to the foremost coupling in turn to determine the positions of each bearing. The calculation sheet is to include expected values of the sags and gaps with the relevant tolerances, and is to be submitted to the site surveyor when the measurements are performed.

1.4.2 Procedure for Measuring Bearing Loads

In general, bearing loads are measured using the jack-up method. The ‘jack-up method’ is a technique used to obtain the bearing load from a jack load using a hydraulic jack that is placed adjacently to the bearing to be measured. An outline of the measurement method is shown in Figure 4.1.

As the oil pressure of the jack rises, the plotted point, that is the relationship between the jack load and displacement, shifts from the initial state of A to the next state, B, in which the bearing load becomes unloaded. When the oil pressure is raised further, the plotted point reaches the state C with a change in the gradient of the jack-up curve. As oil pressure decreases from this state, the plotted point returns to the state of A via states of D and E. The resulting curve of the change in jack load plotted during the process of lifting and lowering the shaft by the hydraulic jack forms a hysteresis curve, as shown in Figure 4.1.

Figure 4.1 Procedure for applying jack-up method. The bearing load, RB , can be obtained by the following equation using jack load, RJ.

(3)

C is a load correction factor which is given by (4)

where, IBB : reaction influence number of bearing support itself IJB : reaction influence number of jack support with respect to bearing support.

Dial gauge

Hydraulic jackJack load

Displ

acem

ent

A

B

C

Increasing

Decreasing

D

E

C = IBB IJB

RB = C RJ .

B (Bearing: unloaded)

A (Jack: no load)

C

Bearing support

Jack


18

Attention should be paid to how to read RJ. In the event that Equation (4) is used to determine a load correction factor, RJ should be the average of jack loads RI and RD . This average can be obtained by extending two straight lines BC and DE to the position at which the displacement becomes zero (see Figure 4.2).

Although the bearing load can be obtained from the jack load R′J which is read without extending the lines BC and DE, it should be noted that, in this case, the definition of load correction factor is different from Equation (4). That is,

(5) (6)

Where, IBJ : reaction influence number of bearing support with respect to jack support = IJB IJJ : reaction influence number of jack support itself.

However, a number of measurements are made using Equations (3) and (4) based on past experience.

Figure 4.2 Bearing load measurement using the jack-up method.

References [1] Guidelines on Shafting Alignment, Part A, “Guidelines on Shafting Alignment Taking into Account

Variation in Bearing Offsets while in Service ”, ClassNK, 2006. [2] H. Yoshii, “Shaft Alignment ”, Journal of the JIME, Vol. 39, No. 9, p. 597, 2004.

RD RJ RI R′JJack load

Displ

acem

ent

IJJ

I’JJ

RJ = RI + RD 2

C′ = IBJ IJJ

RB = C′ R′J .

I′JJ : Reaction influence number of jack support itself (in case where the bearing to be measured is excluded).


19

APPENDIX A DERIVATION OF δ B2 AND δ B3

1 Approximate Calculation of Relative Displacement of the Hull

A slope alignment approach is generally adopted when designing the installation of a marine propulsion system in which the main engine is installed below a predetermined reference line. When hull deflection takes place due to an increase in draught in such a shafting system, the relative positions of each bearing to each other will change, as shown in Figure 1.1(a). Accurately determining these changes in position will have a direct effect on determining the most suitable alignment of the bearings and shafting. Hence, it is desirable to determine the relative difference in position or displacement that occurs from the initial state of the ship in the light ballast condition and after deflection in the fully loaded condition. The resulting relative displacement, as illustrated in Figure 1.1(b), is based on the state of the shafting prior to deflection of the supporting hull structures, assuming that the relative displacement of the engine portion of the shafting (the crankshaft) is negligible. It is useful to set the initial condition as a baseline against which to measure the relative displacement that results in the fully laden state. Hereinafter, the approximate calculation of relative displacement of the hull, δ , behind the main engine is examined using the XY-coordinate system shown in Figure 1.1(b), in which the aftmost engine bearing is defined as the origin of the coordinate system. It is assumed that the changes in each bearing offset behind the engine correspond to the relative displacement of the relevant part of the supporting hull structure.

Figure 1.1 Change in bearing offsets due to hull deflection.

Although relative displacement, δ , along the length of the shafting aft from the aftmost engine bearing to the aft bulkhead of the engine room can be determined by finite element analysis, it is also found to be roughly proportional to the n power of X. Use of such a relation can help simplify the process of determining the relative displacement for a given alignment.

Figure 1.2 Comparison between FE analysis and several Xⁿ curves.

(b) X

YRelative displacement (δ )


(a)

Base line

M/E

Before deflection (Light ballast condition)

After deflection (Fully loaded condition)

Rel

ativ

e di

spla

cem

ent δ

(m

m)

0

1

2

3

4

5

6

7

8

0 5 10 15

X: Distance from the aftmost engine bearing (m)

FEM n = 1.4 n = 1.5 n = 1.6 n = 2.0 n = 3.0


20

Figure 1.2 shows a comparison between the result of finite element analysis and several Xⁿ curves having various values of n, with respect to the relative displacement of the shafting from the aftmost engine bearing to the aftmost bulkhead of the engine room in a 300,000 DWT oil tanker. The figure indicates that an n of 1.5 gives the best approximation of the relative displacement. Similar results were also seen in other ships. Hence, it is thought that the relative displacement of the hull supporting the shafting from the aftmost engine bearing to the aftmost bulkhead of the engine room can be expressed as a rough estimate using an n of 1.5 (that is, X 1.5 ). The hull structure on the aft side of the engine room aftmost bulkhead includes a highly rigid stern frame, which makes it possible to approximate the liner change in displacement of this part of the hull. Therefore, the change in offsets of the stern tube bearings behind the aftmost bulkhead of the engine room is given by the tangent to a curve with an exponent of X1.5 at the position of the aftmost bulkhead. Based on the approximation method mentioned above, relative displacement of the hull, δ , due to the increase in draught can be expressed by Equation (1). Figure 1.3 shows the calculation model used to determine the changes in each bearing offset behind the engine due to this relative displacement of the hull.

where,

X : distance from support point of aftmost engine bearing (mm), L : distance from support point of aftmost engine bearing to engine room aftmost bulkhead (mm), δ B : relative displacement at the position of engine room aftmost bulkhead (mm).

Figure 1.3 Concept of relative displacement model used to calculate the changes

in each bearing offset behind the aftmost engine bearing.

In Equation (1) the relative displacement, δ , behind the engine is expressed with the distance, L , from the aftmost engine bearing to the aftmost bulkhead of the engine room, while δ B represents the extent of relative displacement at the position of the bulkhead. Hence, δ B can be regarded as a parameter to determine the relative displacement curve of the hull.

δ B (X / L )1.5, ( X ≤ L ) δ B { 1.5 (X / L ) – 0.5 }, (L ≤ X )δ = {

Relative displacement

δ B

δ = δ B {1.5(X/L)-0.5}

δ = δ B (X/L)1.5

L

Aftmost bulkhead of E/R Aftmost engine bearing

Y

X

(1)


21

2 Reaction Influence Numbers determined by Relative Displacement Model

A coefficient known as a "reaction influence number" is commonly used in alignment calculations for marine shafting. A reaction influence number is defined as a coefficient that indicates the relative extent of change that occurs in a bearing reaction when the offset of a particular bearing changes for a given unit value. For example, when the offsets of support points Nos. 1 to 4 change in the shafting system shown in Figure 2.1(a), the amount of change in the reaction of support point No. 6 (the second aftmost engine bearing), ∆R6 , is expressed using reaction influence number, C m, n , as follows:

where,

δ n : amount of change in the offset of support point n , C m, n : amount of change in the reaction of support point m when support point n is displaced

downward by 1 mm (reaction influence number).

Figure 2.1 Calculation of bearing reaction using reaction influence numbers.

Equation (2) can be rewritten using the relative displacement model of the hull described in section 1. This is because the changes in the offsets of support points Nos. 1 to 4 are obtained by Equation (1). Consequently, Equation (2) can be rewritten as follows using δ B and Xn, where Xn is the distance from support point No. 5 (the aftmost engine bearing) to support point n in which:

In Equation (3), ∆R 6 is expressed as the product of δ B and S 6 , whereas in Equation (2) ∆R 6 is expressed as the summation of the product of δ n and C 6, n (n = 1 to 4). Since both δ B and δ n

indicate relative displacements, S 6 also represents the same meaning as the reaction influence

∆R6 = C6, 1 δ 1 + C6, 2 δ 2 + C6, 3 δ 3 + C6, 4 δ 4

= C6, n δ n . Σ n =1

4 (2)

Aftmost bulkhead of E/R

Intermediate shaft Propeller shaft

Change in bearing reaction = ∆R 6


1 2 3 4 5 6 7 8 9

1 2

34 C 6, 4

C 6, 3

C 6, 2

C 6, 1

δ 1 δ 2 δ 3

δ 4

(a)

Intermediate shaft Propeller shaft

S 6 δ B

(b)

(1 2 3 4 5)(No. of engine bearing)

1 2 3 4 5 6 7 8 9

∆R 6 = δ B S 6 , S 6 = C 6, n {1.5 (X n / L ) - 0.5} + C 6, n (X n / L ) 1.5. Σ

n = 1

3

Σ n = 4

4

(3)

(4)


22

numbers, C 6, n (n = 1 to 4). If we assume that the whole of the relative displacements of the support points, δ n, can be replaced with the relative displacement of the hull, δ B , at the position of the engine room aftmost bulkhead, as shown in Figure 2.1(b), S 6 can be regarded as an equivalent reaction influence number that reflects the total combined effect of each reaction influence number, C 6, n (n = 1 to 4), on that point. In the following equation, this equivalent influence number is generalized in order to obtain the influence numbers for each engine bearing. Now, assuming that S i is the increase of reaction at engine bearing No. i (from aft) when the hull is displaced downward by 1 mm at the position of the engine room aftmost bulkhead, then this S i is expressed by following equation:

where, n : number of support point (counted from the aft of the shafting), a : number of the nearest support point forward of the aftmost bulkhead of engine room (counted

from the aft of the shafting), b : number of support point of the aftmost engine bearing (counted from the aft of the shafting), X n : distance from support point b to n, L : distance from support point b to aftmost bulkhead of engine room, C m, n : amount of increase in reaction (reaction influence number) at support point m , when support

point n is displaced downward by 1 mm.

In Equation (5), it should be noted that the subscript i in S i indicates the number of the bearing as counted from the aft of the engine. S 6 in Equation (4) corresponds to S 2 (i =2) in Equation (5), where a and b are 4 and 5, respectively.

Σ n =1

a-1

Σ n =a

b-1 S i = C b + i-1, n (1.5 x n - 0.5) + C b + i-1, n x n 1.5, (5)

x n = X n /L


23

3 Hull Deflection that results in Engine Bearings Becoming Unloaded

Using the reaction influence number S i , connecting with hull deformation, we can determine the amount of relative displacement of the hull that would cause the ith engine bearing to become unloaded. There are two cases in which either elastic support (spring support) or rigid support (simple support) are used as the support condition of the bearings to calculate the reaction influence numbers, C m, n . Each case is examined below.

3.1 In the case of elastic support

In this case, δ B , which is the relative displacement at which the engine bearings become unloaded, can be obtained using a simplified calculation method. Let R i be the reaction of engine bearing No. i (from aft) before hull deformation and let δ B i be the relative displacement of the hull (at the position of the aftmost bulkhead of the engine room) at which engine bearing No. i becomes unloaded after hull deflection, then

Hence,

Since unloading of engine bearings due to hull deflection is often limited to the second or third aftmost bearing in most cases, both δ B2 and δ B3 are to be calculated and whichever is less is to be compared with the lower limit.

3.2 In the case of rigid support

Figure 3.1 shows how to deal with the stiffness of an engine bearing support in the state before hull deflection. In cases where rigid support is considered in the calculation, bearing reaction Ri

corresponding to the given offset is easily obtained for each respective bearing, and the displacement of the support point due to the load is zero, as shown in Figure 3.1(a). However, if the effect of the stiffness (spring) is considered here, the support points are in fact displaced downward slightly by the load, as shown in Figure 3.1(b), which actually involves a change in the bearing reaction Ri for each engine bearing.

Figure 3.1 Initial condition of engine bearings.

Assuming that reaction R i does not change even if the effect of the stiffness (spring) is considered, the displacement of each support point after loading is expressed with the stiffness K (= constant) of each support point using the following equation:

In the discussion below, we examine the change in bearing reaction when the hull deflects from the initial condition shown in Figure 3.1(b) due to an increase of draught. Figure 3.2 shows the change in the support point (contact point with the shaft) that occurs before and after hull deflection. As can be seen from this figure, it is assumed that the change in the vertical position of the support point shown here is not caused by the deflection of hull structure below the engine, but by the elastic deformation of the engine bed due to the increase in the bearing load.

∆R i = 0 – R i = δ Bi S i .

δ Bi = – R i / S i . (6)

R 1 R 2 R 3 R 4 R 5

(a) Rigid support (b) Elastic support

K

h 1 h 2 h 3 h 4 h 5

h i = R i / K

h i = R i / K . (7)


24

Figure 3.2 Displacement of support points due to elastic deformation of the engine bed.

In paragraph 3.1 above, a simple calculation could be used to give the bearing reactions of each bearing after hull deflection. However, in order to obtain the bearing reactions after hull deflection using reaction influence numbers based on rigid support, the equations that include the reactions and displacements of the support points as variables need to be solved for specific conditions. Setting the number of engine bearings at five and denoting the relative displacement of the hull at the position of the engine room aftmost bulkhead by δ B , changes in the reactions of each engine bearing can be expressed as follows:

where,

i : number of support point (engine bearings) counted from the aft of the engine, R i : reaction of support point i (counted from the aft of the engine) before hull deflection, R′i : reaction of support point i (counted from the aft of the engine) after hull deflection, δ i : elastic displacement of support point i (counted from the aft of the engine), C i, j : amount of increase of reaction (reaction influence number) at support point i when support

point j is displaced downward by 1 mm. (i, j : number of support point counted from the aft of the engine).

(Note that the meaning of the subscripts in Equation (8) differ from those in Equation (2).)

On the other hand, from the relationship:

Equation (8) can be rewritten as:

R′1 – R 1 = S 1 δ B + C 1, 1 δ 1 + C 1, 2 δ 2 + C 1, 3 δ 3 + C 1, 4 δ 4 + C 1, 5 δ 5 , R′2 – R 2 = S 2 δ B + C 2, 1 δ 1 + C 2, 2 δ 2 + C 2, 3 δ 3 + C 2, 4 δ 4 + C 2, 5 δ 5 , R′3 – R 3 = S 3 δ B + C 3, 1 δ 1 + C 3, 2 δ 2 + C 3, 3 δ 3 + C 3, 4 δ 4 + C 3, 5 δ 5 , R′4 – R 4 = S 4 δ B + C 4, 1 δ 1 + C 4, 2 δ 2 + C 4, 3 δ 3 + C 4, 4 δ 4 + C 4, 5 δ 5 , R′5 – R 5 = S 5 δ B + C 5, 1 δ 1 + C 5, 2 δ 2 + C 5, 3 δ 3 + C 5, 4 δ 4 + C 5, 5 δ 5 ,

(8)

(9)R′i – R i = K δ i ,

S 1 δ B + (C 1, 1 – K ) δ 1 + C 1, 2 δ 2 + C 1, 3 δ 3 + C 1, 4 δ 4 + C 1, 5 δ 5 = 0, S 2 δ B + C 2, 1 δ 1 + (C 2, 2 – K ) δ 2 + C 2, 3 δ 3 + C 2, 4 δ 4 + C 2, 5 δ 5 = 0, S 3 δ B + C 3, 1 δ 1 + C 3, 2 δ 2 + (C 3, 3 – K ) δ 3 + C 3, 4 δ 4 + C 3, 5 δ 5 = 0, S 4 δ B + C 4, 1 δ 1 + C 4, 2 δ 2 + C 4, 3 δ 3 + (C 4, 4 – K ) δ 4 + C 4, 5 δ 5 = 0, S 5 δ B + C 5, 1 δ 1 + C 5, 2 δ 2 + C 5, 3 δ 3 + C 5, 4 δ 4 + (C 5, 5 – K ) δ 5 = 0.

(10)

h i

Bearing offset input

δ iR i

R′i

Reference line

(After hull deflection)

(Before hull deflection)

Tank top of engine room double bottom


25

In Equation (10), we consider the situation that the second aftmost engine bearing becomes unloaded after hull deflection, which is expressed by the following equation:

This indicates that the support point of the second aftmost engine bearing moves upward so that the displacement, h 2 , which is due to the bearing load R 2 before hull deflection return to the original state, in other words, the restoring force of the spring may be zero.

Let δ B2 be the relative displacement, δ B , which causes the second aftmost engine bearing to become unloaded. Then, the following simultaneous equations with five variables, i.e., δ B2, δ 1, δ 3 , δ 4 , and δ 5 , need to be solved in order to find δ B2 , since δ 2 is not a variable:

where, K = 5,000 kN/mm (constant value).

Figure 3.3 illustrates the state of the shafting and relative displacement of the engine bearings before and after hull deflection.

Figure 3.3 Reactions and displacements of support points when stiffness of engine bearings is considered.

Simultaneous Equations (12) can be expressed in matrix form as follows:

where,

δ 2 = – h 2 = – R 2 / K . (11)

S 1 δ B2 + (C 1, 1 – K ) δ 1 + C 1, 3 δ 3 + C 1, 4 δ 4 + C 1, 5 δ 5 = C 1, 2 R 2 /K , S 2 δ B2 + C 2, 1 δ 1 + C 2, 3 δ 3 + C 2, 4 δ 4 + C 2, 5 δ 5 = (C 2, 2 – K ) R 2 /K , S 3 δ B2 + C 3, 1 δ 1 + (C 3, 3 – K ) δ 3 + C 3, 4 δ 4 + C 3, 5 δ 5 = C 3, 2 R 2 /K , S 4 δ B2 + C 4, 1 δ 1 + C 4, 3 δ 3 + (C 4, 4 – K ) δ 4 + C 4, 5 δ 5 = C 4, 2 R 2 /K , S 5 δ B2 + C 5, 1 δ 1 + C 5, 3 δ 3 + C 5, 4 δ 4 + (C 5, 5 – K ) δ 5 = C 5, 2 R 2 /K ,

(12)

(a) Before hull deflection

R 1 R2 R3 R4 R 5

(b) After hull deflection

δ B2 R’2 = 0 : δ 2 = – h 2

R’1R’2

R’3 R’4 R’5

[D2 ] [δ B2 ] = h 2 [C 2 ] , (13)


26

Hence,

where, [D2] INV denotes the inverse matrix of [D2].

Thus, we can obtain δ B2 as follows:

where, ([D2] INV) L1 denotes the elements (vector) of the first row of [D2]

INV. Similarly, the relative displacement δ B3 which results in the third aftmost engine bearing to become unloaded can be obtained by solving the following simultaneous equations with five variables (the matrix representation is omitted):

In a shafting alignment with constant offsets of the engine bearings, the calculation of δ B2 is sufficient to check the strength of all engine bearings, because the load on the second aftmost engine bearing decreases as draught increases. However, in some cases where the engine bearing offsets are unequal, the load on the third aftmost engine bearing could decrease. Consequently, both δ B2

and δ B3, in general, should be calculated and whichever is less is to be compared with the lower limit.

[D2 ] = ,

S 1 C 1, 1K C 1, 3 C 1, 4 C 1, 5S 2 C 2, 1 C 2, 3 C 2, 4 C 2, 5S 3 C 3, 1 C 3, 3K C 3, 4 C 3, 5S 4 C 4, 1 C 4, 3 C 4, 4K C 4, 5S 5 C 5, 1 C 5,3 C 5,4 C 5,5K

δ B2δ 1 δ 3 δ 4 δ 5

[δ B2 ] = ,

C 1, 2 C 2, 2KC 3, 2 C 4, 2 C 5, 2

[C 2 ] = ,

h 2 = R 2 / K ,

C i, i K = C i. i – K ( i = 1, 2, 3, 4, 5 ).

[δ B2 ] = h 2 [D2] INV [C 2] , (14)

δ B2 = h 2 ([D2] INV) L1

[C 2] , (15)

S 1 δ B3 + (C 1, 1 – K ) δ 1 + C 1, 2 δ 2 + C 1, 4 δ 4 + C 1, 5 δ 5 = C 1, 3 R 3 /K , S 2 δ B3 + C 2, 1 δ 1 + (C 2, 2 – K ) δ 2 + C 2, 4 δ 4 + C 2, 5 δ 5 = C 2, 3 R 3 /K , S 3 δ B3 + C 3, 1 δ 1 + C 3, 2 δ 2 + C 3, 4 δ 4 + C 3, 5 δ 5 = (C 3, 3 – K ) R 3 /K , S 4 δ B3 + C 4, 1 δ 1 + C 4, 2 δ 2 + (C 4, 4 – K ) δ 4 + C 4, 5 δ 5 = C 4, 3 R 3 /K , S 5 δ B3 + C 5, 1 δ 1 + C 5, 2 δ 2 + C 5, 4 δ 4 + (C 5, 5 – K ) δ 5 = C 5, 3 R 3 /K .

(16)


27

4 Calculation Example

Figure 4.1 and Table 4.1 show the dimensions of the shafting prepared to test and verify the calculations and the reaction influence numbers, respectively. The values of the reaction influence numbers were obtained based on the assumption of rigid support for the bearings, therefore paragraph 1.3.3-1.(2) of the Annex to the Guidance was applied in the calculation of δ B2 and δ B3 . The reactions and offsets of the bearings under the hot condition are shown in Table 4.2.

Figure 4.1 Dimensions of shafting.

Table 4.1 Reaction Influence Numbers (kN /mm)

Table 4.2 Bearing Reactions and Offsets (Hot Condition)

Results of Calculations

Table 4.3 shows the results of the calculations of Si (i =1 to 5). It can be understood from this table that the engine bearings most notably affected by hull deflection are the aftmost and second aftmost engine bearings, and further, that the load on the second aftmost engine bearing decreases whereas the load on the aftmost engine bearing increases.

The values of δ B2 and δ B3 are shown in Table 4.4. This table shows that the second aftmost engine bearing will first become unloaded when the ship’s draught increases and δ B reaches 7.26 mm.

Figure 4.2 shows a comparison between δ B2 and the allowable limit in the case of L = 10,400 mm.

Figure 4.2 Comparison between δ B2 and

the allowable limit.

1370 2555 5665 5540 870 1290 1290 1290

SB1 SB2 SB3 IB1 1 2 3 4 5

E/R aft bulkhead 10400

SB1 SB2 SB3 IB1 1 2 3 4 5 SB1 -1094.4 1760.31 -710.86 60.88 -58.32 44.00 -2.00 0.50 -0.08SB2 1760.31 -2956.05 1338.87 -193.96 185.79 -140.17 6.38 -1.59 0.27SB3 -710.86 1338.87 -778.71 217.35 -244.91 184.77 -8.41 2.10 -0.35IB1 60.88 -193.96 217.35 -175.85 436.06 -357.35 16.26 -4.06 0.681 -58.32 185.79 -244.91 436.06 -3498.3 4574.60 -1761.8 440.43 -73.402 44.00 -140.17 184.77 -357.35 4574.60 -7194.4 4005.52 -1340.2 223.353 -2.00 6.38 -8.41 16.26 -1761.8 4005.52 -3991.1 2353.52 -618.224 0.50 -1.59 2.10 -4.06 440.43 -1340.2 2353.2 -2283.1 832.485 -0.08 0.27 -0.35 0.68 -73.40 223.35 -618.22 832.48 -364.72

SB1 SB2 SB3 IB1 1 2 3 4 5 Reaction (kN) 446.69 15.16 69.89 95.33 226.39 131.83 256.28 314.09 90.80Offset (mm) 0.35 0.00 -0.15 0.20 0.40 0.40 0.40 0.40 0.40

S 1 S 2 S 3 S 4 S 5 73.91 -66.79 3.04 -0.75 0.14

Table 4.3 S i (kN/mm)

δ B2 δ B3 7.26 19.14

Table 4.4 δ B2 and δ B3 (mm)

5 7 9 11 13 15x103

10

8

6

4

2

0

L (mm)

δ B2 (

mm

)

Allowable limit

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