shaft alignment optimization genetic algorithms

8/3/2019 Shaft Alignment Optimization Genetic Algorithms

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ABS TECHNICAL PAPERS 2003

Shaft Alignment Optimization with Genetic Algorithms 15

SHAFT ALIGNMENT OPTIMIZATIONWITH GENETIC ALGORITHMS

Davor verko, (AM),American Bureau of Shipping

ABSTRACT

A solution to the shaft alignment problem is a set of prescribed bearing offsets that ensure an acceptable load

distribution among the shaft-supporting bearings. Acceptable load distribution implies not only all positive bearingreactions under all operating conditions of the vessel, but also an acceptable relative-misalignment between the shaftand the bearing. In a marine environment the difficulty is not in finding a single suitable solution to the above criteria,

but rather in defining the optimal set of solutions capable of accommodating the extreme bearing disturbances - resultingmainly from hull deflections and thermal deviation. As the problem is stochastic, with an infinite number of satisfactorybearing offsets, it is appropriate to apply the Genetic Algorithm (GA) optimization procedure to search for the optimal

set of solutions, rather then rely on the plain trial and error approach or some of the step-by-step conventional searchalgorithms. With an ability to conduct parallel search throughout the solution space, the GA is particularly well suitedfor the problem at hand, as it has the capacity to simultaneously provide multiple sets of bearing offsets which satisfy

loading condition at bearings.

INTRODUCTION

Propulsion systems of modern vessels (Figure 1) are mostly diesel engine driven directly-coupled installations, the design of which results in increased disparity between structural flexibility

of the hull and the shafting. Namely, ships hulls have become more flexible with scantlingoptimization and with the increase in ships length, and as the demand for power has increased withthe ships size, the shafting diameters have become larger and the shafts stiffer (this is particularlytrue for VLCCs, ULCCs, large containerships). Consequently, the alignment of the propulsionsystem has become more sensitive to hull girder deflections, resulting in difficulties in analyzing thealignment and conducting the alignment procedure.

Figure 1 Directly coupled propulsion shafting - example

Accordingly, the frequency of shaft alignment related bearing damages has increasedsignificantly in recent years. The alignment related damages are mostly attributed to inadequate


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16 Shaft Alignment Optimization with Genetic Algorithms

analyses, changes in the design of the vessel, shipyards practices in conducting the alignment, and alack of well-defined analytical criteria. As the alignment analysis is the first step in the alignment

process, it is of important to define it with the largest possible error allowance, ensuring a relativelyrobust design with low sensitivity to disturbances affecting the propulsion shafting and the maindrive. Accounting for hull girder deflections is one of the most important issues in that process.

However, hull deflections are not of constant magnitude, but rather are a function of different vesselloading conditions as well as sea conditions the vessel operates in. Therefore, in order to definesatisfactory alignment for all expected operating conditions, the bearing offset has to be optimized soas to satisfy all expected alignment variations.

There were previous attempts in optimising the alignment applying different methods, e.g.Owen applied random search algorithm in search for optimal bearing offset. However, we believethat the genetic algorithms may be more suitable approach in shafting alignment optimisation for thereasons we elaborate below.

The shaft alignment problem is stochastic, with an infinite number of acceptable bearingoffsets. Finding a single good set of bearing offsets to satisfy bearing reaction requirements for onlyone or two different vessel conditions is not that difficult a task. However, the difficulty in finding a

solution significantly increases as the solution-space narrows when solutions have to satisfy anumber of additional criteria; such as hull deflections, hot and cold operation, bearing wear down,etc. The Genetic Algorithm (GA) optimization procedure is an appropriate tool to address exactlythese kinds of problems. The GA has the ability to conduct parallel searches throughout the solutionspace (which is its biggest advantage versus other search tools), and simultaneously yields a multipleset of bearing offsets that satisfy bearing loading requirements.

It is also to be noted that the primary goal of shaft alignment is to ensure an acceptable staticload distribution among the shaft-supporting bearings, which (what is generally accepted) will be a

prerequisite for satisfactory dynamic behavior of the propulsion system. Why do we not conduct thedynamic analysis only, if dynamic operation is what we eventually want to satisfy? The explanationis simple; the shaft alignment procedure is conducted and verified in static condition and the majorrole of the static alignment analysis is to provide information and data necessary for this procedure to

be conducted properly.

SOLUTION ALGORITHM

Genetic algorithms (GAs, Holland 1975, Goldberg 1989) have proved to be an effectivesearch mechanism. It is a randomised search algorithm based on evolutionary genetics (AppendixA). GAs has been adapted for function optimization in a variety of ways.

There have been some attempts to apply GAs in ship structures and ship systems

optimisation, e.g.: Okada and Neki 1992 optimised ship hull structure applying GAs, and Dai et al1994 used GAs on marine propulsor design. Some GA applications to the ship propeller wereattempted by Lee and Hajela 1996, who investigated GAs in helicopter rotor blade design. Othersconsidered different approaches in optimisation, e.g.: Rahman and Caldwell 1995 applied a rule-

based and rational design method on ship structures. But obviously none of the above approachesfound wider application in the industry, possibly due to the following reasons:

- optimisation normally requires highly sophisticated software (large number of criteria tooptimise) customized for the particular problem,

- optimisation process requires very long computation time- optimisation methods seldom guarantee that an optimal (or close to optimal) solution would

be found.

So far, the GA optimisation applications have required an initial search for optimalparameters (e.g. Okada and Neki 1992), which may eventually mislead the GA to a suboptimal, or


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even far from optimal, solution. The quest for optimal parameters is the main disadvantage of thesimple GA, which eventually negates all the advantages it has against other optimisation methods.This may eventually discourage usage of the algorithm, as the results obtained by simple geneticalgorithm may not be better than the other methods can provide.

The genetic algorithm with residual (GAR), as developed by Novkovic and verko, may

regain the confidence in GA, as the algorithm requires no parameter optimisation, and it results insolutions which are often in close vicinity of the optimum.In this paper the shaft alignment optimisation problem is analysed applying GAR. The

following sections provide details on the shaft alignment optimisation algorithm.

SHAFT ALIGNMENT OPTIMIZATION

The goal of shaft alignment optimization is to provide a set of acceptable solutions which allsatisfy imposed constraints, parameters and criteria. Multiple solutions are necessary as it is oftenimperative to have the human evaluation as the final decisive factor in selecting the desired

alignment. Providing multiple solutions is an inherited characteristic, and a relatively simple task forthe genetic algorithm to perform.

The GA program optimizes among several constraint functions (as defined by hull girderdeflections - Figure 2). Constraints which bind the solution space are defined by hull deflectioncurvatures which normally represent the still water ballast and laden vessel condition. Sometimes,when maximum hogging and maximum sagging wave deflections are known, it may be advisable toinvestigate the extreme hull deflections influence on the alignment as well.

Typical hull girder deflections of a

VLCC vessel under ballast (above) and

laden(below)condition.

Behavior of the shafting under

laden and ballast condition

Figure 2 Hull girder deflections influence on shafting

Basically, GA optimisation works such that the software generates a desired number ofsolution strings, which are called population. This initially defined population is randomly definedand it may or may not contain any satisfactory solution. In the search for the solution the algorithmmimics the natural genetics process by applying so called selection, crossover and mutation

parameters in generating new solution space from the previous population. As the fitness function isapplied (giving the fittest strings more chances to be selected into the mating pool for newgeneration), every new generation is expected to have population of more fit individuals then the

previous. The more fit individual, in our case, will essentially contain bearing offsets which result inbetter load distribution among the bearings. Thus, all succeeding populations are essentially new,

and better offspring of the previous generations. Strings in current population are selected for themating pool through the selection process based on the individual strings fitness. Defining the


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fitness function is one of the basic criteria in solution finding. Moreover, mates (pairs of strings,selected for the mating pool in new generation creation), exchange portions of their strings(chromosomes) in the process called crossover, and each allele(in our case solution string is binarycoded, thus the allele is represented by a single bit) of the particular solution string is mutated (ornot) depending on the randomly applied probability of mutation. (A detailed description of the

algorithm is given in Appendix A.)In the shaft alignment problem the crossover and mutation rates are intentionally set higherthan would be required if convergence was desired. The convergence is not needed as we search fora set of acceptable solutions within the domain and not the single optimum.

The complexity and speed of optimization will depend on the number of variables that areconsidered in the optimization process. The parameters and alignment criteria which normally should

be considered are:- thermal expansion,- diesel engine bedplate prescribed sagging,- bearing wear down,- bearing elasticity is not considered due to its complexity (dependent on the contact area

/misalignment slope between the shaft and the bearing).Additional requirements may need to be satisfied too; e.g. the main engine flange allowable

moment and shear force is to be in accordance with engine designer recommendations.

The Algorithm

The alignment optimization algorithm conducts the following tasks:- Shaft Alignment Analysis: define the influence coefficient matrix for the given propulsion

system. The ABS ShAl shaft alignment software is applied for that purpose.- Reading basic GA data:

o population sizeo number of generationso mutation factorso number of crossover siteso Constraint functions (hull girder deflections) are definedo Prescribed thermal influence and diesel engine sag data is input

The above information is specific to each particular shafting system, and can be amended asneeded for the particular problem.

- Selection: GA selects initial population: for given number of bearings the GA optimizationroutine randomly selects initial population of bearing offsets.

- Reaction Calculation: Influence coefficient matrix is multiplied by GA generated bearing offsets;bearing reactions are evaluated for every individual in the population- Fitness Evaluation: Fitness function is used to evaluate each strings fitness.

o fitness function is defined as a ratio between total reaction force and (for selected offset)calculated positive bearing reactions. The maximum fitness, when all bearing reactions are

positive will be 1 (one). Strings with all positive reactions gain 10% fitness to increase theirchances of entering mating pool.

- Selection: (sigma) selection (Tanese 1989) is applied to select pairs of strings to the matingpool where string will exchange their information and form new population of solutions

- Solution: If a satisfactory solution is found information is stored and the process continues until adesired number of solutions is obtained, or a maximum number of generations is reached:o maximum number of generations are defined initially for each particular system,


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o number of generations to run is specific to the particular problem number is proportional tothe number of the bearings,

o program is initiated to provide the desired number of solutions (10 is default number thisvalue may be changed as desired by the analyst)The program runs until one of the above requirements is fulfilled.

GA performance improvement:- One of the bearings in the system is fixed to avoid repeating the same solutions which would

result from rigid body translation. Moreover, with one bearing fixed the optimization can beconducted faster as a random search is now conducted with one bearing less.

- It is suggested to constrain the diesel engine bedplate deflection, by allowing it initially to moveonly as a rigid block. This avoids implausible solutions that may result from randomly selectedextreme engine bedplate deflections. Also, by assuming rigid connection among engine bearingsthe search speed will be significantly faster, as the engine bearing offset will be evaluated at onlytwo end bearings (the offset of the bearings in between is linearly interpolated). To account foractual engine flexibility the engine deflections can be accounted for in hull deflections.

- Hull girder deflections can be estimated analytically or measured; hull deflection information isentered for two extreme conditions for which the alignment should be satisfactory (ballast andladen condition, for example). The problem may be difficult to solve if the range of deflections,of two constraining deflection curvatures, is too wide, as the algorithm searches for a set of

prescribed displacements which satisfy both extremes.- Engine bedplate prescribed sag is also added as an input option in analysis.- Other disturbances, which may affect the alignment conditions, may be considered as well and

their effect on alignment investigated (e.g. bearing wear down).

Bearing elasticity is another complex issue that is not specifically addressed in theoptimization process, and it will be considered in future research. The problem with bearingelasticity is that it is not constant and it depends on the contact area between the shaft and the

bearing. The contact area varies as the bending curvature changes with bearing offset selection,which essentially requires an iterative procedure to evaluate the deformation of bearing and correctthe bearing offsets accordingly.

The time required for GA to complete the search will depend on the complexity of thesystem. The more bearings in the system the more difficult it will be to find the combination ofoffsets to satisfy requirements. The same is true with constraints: the more stringent the constraintrequirements are (e.g. wide hull deflection differences) the more demanding will be the search.Although the algorithm finds the solutions relatively fast, the further investigation in definingshortcuts in solution seeking is part of the future work.

The example below illustrates the optimization software application to a common VLCC.

Figure 3 Discrete model of the shafting


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APPLICATION OF THE SOFTWARE

The example used to evaluate the GAR performance on shaft alignment optimization is atypical single propulsion VLCC arrangement, with relatively short shafting and the low speed dieselengine as a main drive.

The particular problems these kinds of vessels may experience are:- after stern tube bearing damage due to the excessive misalignment between the bearing and theshaft

- main engine bearings: the aftmost three engine bearings are those particularly at risk to bedamaged due to improper alignment.

Figure 3 represents a discrete model of the propulsion shafting and diesel engine for thepurpose of shaft alignment analysis.

The above propulsion shafting (Figure 3) is originally designed with following bearing offsets(Figure 4) and bearing reactions (Figure 5):

Figure 4 Bearing offset; Shaft deflection curve; Nodal slopes

Figure 5 Bearing reactions; Bending moment; Shear forces

Bearing reactions, bending moments and shear forces appear to be satisfactory. However, ifhull deflections are applied to the same system, the result of the analyses for two extreme cases of

hull girder deflections (Table 1) will not provide satisfactory bearing reactions. In the subjectexample, hull deflections are roughly estimated for evaluation purposes only. Low hull girder


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deflections are intentionally chosen to show that even a small disturbance of the prescribed offsetwill potentially have some of the bearings unloaded in the system, if the initial bearing offset is notselected properly.

Hull girder deflections can be estimated analytically or defined by measurements. ABS iscurrently involved in a long term project to measure hull girder deflections on several types of

vessels of different sizes (i.e. VLCCs, ULCCs, bulk carriers and container vessels). The intentionis to establish a data base of hull deflections which is to be used in alignment design to estimate hulldeflections more accurately. The same measurements will be used by ABS to validate finite elementmodels on vessels where hull girder deflection is analyzed numerically.

Hull deflection estimate[mm]

Bearing # Laden Ballast1 0 02 0.5 -0.053 0.7 -0.074 1.2 -0.125 1 -0.16 0.8 -0.087 0.6 -0.068 0.4 -0.049 0.2 -0.0210 0.1 -0.0111 0 0

Table 1 Estimated hull girder deflections

If hull deflections corresponding to the ballast condition of the vessels are added on top of theprescribed bearing offsets the load distribution among bearings will not be satisfactory. The analysisshows that the second aftmost bearing of the main engine may unload.

Ballast vessel hull girder deflections as estimated in Table 1

Total bearing offset Bearing reactions: M/E second aftmost bearing unloaded

Figure 6 Ballast vessel - Bearing offset disturbed by hull deflections; Bearing reactions - Unloaded M/E brg #2

In the case hull deflections of the laden vessel, when the same are added on top of theprescribed bearing offsets the load distribution among bearings will not be satisfactory as well. Theanalysis shows that the second aftmost bearing of the main engine may unload. Moreover, the

intermediate shaft bearing load is almost non-existent.


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Laden vessel hull girder deflections as estimated in Table 1 (should this read are estimated?

Total bearing offset Bearing reactions: M/E second aftmost bearing unloaded,intermediate shaft bearing very lightly loaded

Figure 7 Laden vessel - Bearing offset disturbed by hull deflections; Bearing reactions - Unloaded M/E brg #2

The above analyses show that the initially prescribed offset does not satisfy the alignment

requirements if hull deflections are considered, as the M/E second aftmost bearing is unloaded andthe intermediate shaft bearing may easily unload as well. This implies that the original alignmentdesign of the subject vessel is particularly sensitive to hull deflections. The significant problem in the

proposed bearing offset for the vessel in question is that the alignment does not improve from theballast to laden condition, but quite opposite, it worsens.

Present practice in shaft alignment design normally does not include hull deflections; whichis not surprising as the prediction of the alignment is quite difficult without proper analysis ormeasurements. The only means of controlling the alignment condition would be by bearing reactionmeasurements. However, bearing reaction measurements are seldom conducted with the vesseloperating in the ballast condition, and even less often for the laden condition. Moreover, unless there

is suspicion that something may be wrong in the system, the diesel engine bearings reactionmeasurements are not conducted as regular practice either. Now, if the alignment is not designedwith sufficient tolerance to allow the propulsion shafting to accommodate eventual disturbancesaffecting the system, one may expect to run into significant problems with alignment of the

propulsion shafting. In most cases the consequences are not extreme, and they not immediately resultin damage and eventual failure of bearings, but rather result in reduced life of the bearings andcontinuing problems in bearing performance.

In the subject shaft alignment design case, had the hull deflections be initially accounted for,the designer would have been able to predict eventual problems and eventually evaluate thealignment with another set of prescribed bearing offsets. However, without an optimization tool athand this process may be extremely time consuming and difficult, and thus it is seldom conducted in

the shipyards.

Optimization

The above analyses suggested that a different set of initially prescribed offsets should beprovided in order to ensure the subject installations satisfactory alignment under both the ballast andloaded vessel conditions. Optimization may help investigate the solution simpler and faster than atrial and error process conducted without support of the computer software.

GAR software is applied taking into consideration the following data (Figure 8):


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Figure 8 GA input data

o Population size is 500 strings.o Program is set to run 50 generations as it searches for all positive bearing reactions satisfying the

deflection data tabulated on the left of the front-end window (Figure 8).

o Mutation gradient factor is set to 15, which will eventually ensure very high mutation, and thusdiversity in population (Novkovic, S. and verko, D. 1997). High mutation is desired as we arenot interested in population convergence but in obtaining as many as possible highly diversifiedsolutions. The same purpose of increased diversity is assigned to the mutation division factor. Diversified solutions are desired because very different bearing offsets may still satisfy the

bearing reactions. Namely, satisfactory bearing reactions may be obtained with the engineraised above the zero offset line, and at the same time a very similar solution (bearingreaction wise) may be obtained with the main engine (M/E) lowered below zero offset line.

Solution with M/E lowered below zero line will eventually result in smaller inclinationgradient between the shaft and the stern tube bearing, however the stress in the shaft in thatcase will be higher.

In cases without a forward stern tube bearing, the solution with M/E below the zero line willresult in a very sensitive misalignment in the stern tube bearing and therefore may not beacceptable.

Particular details about each solution will be discussed below.o Maximum number of solutions is set to 10. Number can be changed as desired. Program will stop

the search if 10 satisfactory bearing offsets are found. The program will search until themaximum generations are reached if the number of solutions found is less then the presetnumber.

o Deflection data is also provided and data can be amended as desired. Deflection data includesmaximum hogging and maximum sagging deflections of the hull of the vessel, thermal rise atselected bearings, and prescribed bedplate sag.


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o Random seed which will initialize initial population is set to 12345. This number is arbitrary any number is satisfactory. Random seed defines initial probabilities of mutation and crossover,which in generations afterwards is taken from GARs residue (Appendix A).By applying the same random seed in consecutive analyses with same input data, a repeatableanalysis will be ensured. When random seed is set to zero the program uses a system clock to

initiate first population and analysis cannot be replicated identically.

Solutions obtained by GAR are tabulated in a format that provides detailed information onhow a particular change in the bearing offset condition affects the alignment. Namely:o Bearing reactions calculated for:

zero offset reactions reaction difference which when applied to zero-offset solution provides the desired bearing

load (i.e. all positive reactions) maximum hogging bearing reactions maximum sagging bearing reactions even keel bearing reactions

o Bearing offset includes thermal condition and bedplate prescribed sagging: maximum hogging bearing offset maximum sagging bearing offset GAR generated bearing offset Deflection data (max. hogging, max, sagging, thermal and prescribed bedplate sag)

Figure 9 Selected prescribed displacement solution which satisfies hull deflections between two boundaries; i.e.minimum expected hull deflections, maximum hull deflections and hull deflections between the two.


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To show the diversity of GAs performance, four solutions are extracted from the pool ofsatisfactory-prescribed displacements and shown in Appendix A. Two of four solutions are chosenwith the main engine offset below the zero line, and the other two with the main engine above thezero line. The best, the most robust solution, is supposed to be selected by the designer, whoshould, in his/her decision-making, consider in particular the specifics of the shipbuilders

production process and alignment procedures. This may need some further explanation: namely, thealignment is quite sensitive to relatively small disturbances in bearing offsets. If alignment procedureis fully conducted in the dry dock, for example, the GA optimization criteria shall be set to extractsolution(s) which will correspond to particular condition in the dry dock, and need not result insatisfactory bearing reactions at the this stage of construction. However, when ship afloats the

predicted hull deflections should improve alignment condition and result in acceptable loaddistribution among the bearings.

In our test case we opted for the solution presented in Table 2. As we are not addressing anyparticular shipyard practices the selection is conducted as if the whole procedure is performed in drydock, and thus we want to allow as much provision for eventual error in estimated hull deflections.GAR-defined prescribed displacement and respective bearing reactions (Figure 9), shall be further

analyzed to define all details necessary to fully support the alignment procedure; such as sag andgap, bearing contact condition evaluation, etc.

As mentioned above, in this particular case it is presumed that the alignment procedure isfully conducted in dry dock. Therefore, GA defined bearing offsets are actually values which are to

be applied to the bearings while the vessel is in dry dock. Obtained reactions may therefore beverified with relatively high accuracy. Table 3 shows a set of dry dock values for first of fourselected solutions (see also Appendix A).

Table 4 shows bearing reactions when estimated minimum hull deflections (ballast condition)are added to the initially prescribed displacements (Figure 9).

Table 5 shows bearing reactions when estimated maximum hull deflections (laden condition)are added to the initially prescribed displacements (Figure 9). In this case we selected optimalsolution so as to obtain a more preferable load distribution among bearings for the laden vesselcondition.


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Dry dock condition offset and bearing reactions

Reactions Offset

Ry

(dy)

[kN]

GA

Define

dDy

[mm]

1 544.411 0

2 45.927 3.479

3 127.873 6.193

4 24.785 6.884

5 272.691 7.003

6 153.893 7.012

7 299.704 7.022

8 272.986 7.032

9 264.188 7.042

10 328.505 7.052

11 94.822 7.058

12

34

56

78

910

11

BearingReactions[k

N]

BearingOffset*100

[mm]

0

100

200

300

400

500

600

700

800

Bearing Reactions [kN]

Bearing Offset * 100 [mm]

Table 3 Dry dock - Bearing reactions for prescribed offset

Ballast vessel offset and bearing reactions

Reactions Offset

Ry

(dy)

[kN]

GA

Defined

Dy

[mm]

1 544.996 0

2 46.072 3.429

3 124.78 6.123

4 32.933 6.914

5 267.522 7.049

6 152.923 7.074

7 301.263 7.102

8 265.96 7.132

9 274.995 7.164

10 322.706 7.188

11 95.636 7.208

12

34

56

78

9

10

11

BearingReactions[kN]

BearingOffset*100[mm]

0

100

200

300

400

500

600

700

800

Bearing Reactions [kN]

Bearing Offset * 100 [mm]

Table 4 Ballast vessel hull deflections - Bearing reactions and total bearing offset


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Laden vessel offset and bearing reactions

Reactions Offset

Ry

(dy)

[kN]

GA

Defined

Dy[mm]

1 518.533 0

2 106.331 3.979

3 34.172 6.893

4 275.984 8.234

5 81.192 8.149

6 155.648 7.954

7 285.205 7.762

8 345.582 7.572

9 143.036 7.384

10 399.197 7.298

11 84.905 7.208

12

34

56

78

910

11

0

100

200

300

400

500

600

700

800

900

Beari ng Reactions [kN]

Beari ng Of f set * 100 [mm]

Table 5Laden vessel hull deflections - Bearing reactions and total bearing offset

For the estimated hull deflections the bearing reactions in all three cases: even keel (drydock), ballast, and laden, are satisfactory. The solution is relatively robust for predicted hulldeflections, and no bearing unloading is expected.

Another important issue to be investigated is the misalignment slope between the shaft and

the tail shaft bearing. The misalignment shall be reduced by slope boring if the shaft exerts excessive pressure on the bearing shell. ABS shaft alignment software is used in the bearing contactinvestigation.

Dry dock condition no slope boringContact pressure 497 [MPa]

Dry dock condition with slope boringContact pressure reduced to 139 [MPa]


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Slope boringrequirements for thedry dock conditionwould satisfy ballastcondition as well.

Slope boringrequirements for thedry dock conditionwould satisfy loadedcondition also.

Misalignment slope is0.15 [mrad] which isbelow industrysaccepted requirementsfor slope reduction byslope boring orinclination.


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CONCLUSION

The genetic algorithm has proved its ability to find a desired number of acceptable solutionswithin given constraints, and the solution is found in a relatively short time. All the benefits ofconducting the shaft alignment optimization are immediately obvious from the presented example. It

is noticed that the original alignment, as defined by taking the conventional approach in conductingalignment, will not result in a satisfactory static loading condition for the estimated hull deflectionsapplied. Using a conventional approach the second aftmost main engine bearing, and possibly theintermediate shaft bearing may be unloaded. Unloading of the main engine bearing confirms the very

problems currently experienced a considerable number of propulsion installations. These all giveeven more credibility to the proposed method, which can provide satisfactory solutions to a

potentially dangerous problem.The biggest obstacle we are facing is the prediction of the hull girder deflections. The

solution to the problem very much depends on our ability to evaluate hull deflections accuratelyenough, and with sufficient confidence. One possible way of doing this is to establish a generic data

base of hull girder deflections for certain categories of vessels and using it when vessels of similar

designs are evaluated. Data can be obtained either analytically or by measurements. The ABS hasalready taken steps in that direction.

Relatively accurate hull deflection prediction and optimized alignment would allow shipdesigners to confidently prescribe alignment for a vessels dry dock condition. The alignment

procedure could then be conducted fully in the dry dock where we can control alignment conditionmuch more accurately. This would significantly increase the precision of the whole process, asalignment measurement in the dry dock would be possible with very little disturbance affecting thesystem (no hull deflections), and thus, with an firm referent line the alignment may be conductedmuch closer to the on analytically predicted.

Further research will also be necessary in investigating bearing elasticity influence on theoffset change. Genetic algorithm may be particularly suitable for this task due to complexity of the

problem, as the bearing contact area changes with deflection curvature of the shaft, and thus theconsequence is constantly changing bearing elasticity. Also investigation into the robustness towardsdeviation of the offset at any individual bearing will be important to optimize. This issue is moredifficult to optimize than the hull deflections, as the change in offset at only one bearing generatesmuch larger disturbance to the alignment then the smooth transition caused by hull deflections (goodexamples of offset change at only one bearing would be temperature change below the bearing, orware-down of the bearing liner).

It should be stated that analytical models seldom represent actual conditions of a vessel, andthis creates lots of difficulties in alignment rectification if required. This however, gives even morecredibility to an alignment optimization which may ensure a more robust initial alignment, one that is

much less susceptible to unpredicted disturbances in the system.


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REFERENCES

Dai, C., Hambric, S., Mulvihill, L., Tong, S.S., and Powell, D. (1994), A prototype MarinePropulsor Design Tool Using Artificial Inteligence and Numerical Optimization Techniques,SNAME Transactions, 102: 57-69

Goldberg, D.E. (1987): Simple Genetic Algorithm and the minimal deceptive problem in Davis,L., editor, Genetic Algorithm and Simulated Annealing, Pitman, London

Holland, J.H. (1975): Adaptation in Natural and Artificial Systems, Ann Arbor, University ofMichigan Press

Lee, J. and Hajela, P. (1996): Parallel Genetic Algorithm Implementation in Multidisciplinary RotorBlade Design,Journal of Aircraft, 33, 5: 962-969

Novkovic, S. and verko, D. (1997): Genetic Waste and the Role of Diversity in Genetic

Algorithm Simulations, Proceedings of the Second Workshop on Economics with Heterogeneous Interacting Agents, Ancona, May 30-31

Novkovic, S. and verko, D. (1998): The Minimal Deceptive Problem Revisited: The Role ofGenetic waste, Computers and Operations Research, 25,11:895-911

Novkovic, S. and verko, D. (2003 - in review): A Genetic Algorithm With Self-GeneratedRandom Parameters,Journal of Computing and Information Technology

Okada, T. and Neki, I. (1992): Utilization of Genetic Algorithm for Optimizing the Design of ShipHull Structures,J.S.N.A. Japan, 171: 43-55

Owen, S.E. ( ): Optimization and Stochastic Modeling Applied to Propulsion Shafting Alignment,SNAME Transactions

Rahman, M.K. and Caldwell, J.B (1995): Ship Structures: Improvement by Rational DesignOptimization,Int. Shipbuilding Progr., 42, 429: 61-102

Tanese, R. (1989):Distributed Genetic Algorithms for Function Optimization, Doctoral Dissertation,University of Michigan, Ann Arbor, MI.


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APPENDIX A

Genetic Algorithm

In genetic algorithm (GA) optimisation of the propulsion shafting alignment, the shaft

alignment bearing offsets are represented by a population of n strings of finite length, l, whoseelements (alleles) are coded, usually in binary alphabet {0,1}. Algorithm functioning is guided bythree standard operators, namely selection, crossover and mutation.Selection operator is a random process of selection of a mating pool of individual strings, which will

provide genetic information to the future generation, where better than average performing stringshave a greater probability of selection into the mating pool. Performance is measured by a fitnessfunction (f). The process is an artificial version of the survival of the fittest. Here, less than average

performing strings, measured by their fitness value, have a greater chance of receding.Crossoveroperator will exchange genetic material between a pair of strings with probabilitypc, and a

point of crossoverk, somewhere in the interval [1, l-1]. Alleles are then exchanged between positionsk+1 and l. For example, if we have two strings of length l= 7, crossing at k= 4. Two new strings

emerge as follows:

0110|101

0000|1110000|000

1111|111

MUTATION

CROSSOVER

poolmaitngforselectedarestringstwo

Mutation changes the allele with probabilitypm, turning 0 into 1, and vice versa. This operator is a

source of population diversification. In the example above the mutation is applied after two stringsare selected for the mating pool and their chromosomes exchanged information during the crossover.Alleles two, three and seven are mutated in this example.

Combining the survival of the fittest principle with randomized search, the GA createsindividual strings in each new generation by exchanging chromosomes between pairs of randomlyselected mates from the old generation. The algorithm makes copies of strings with probability ofselection proportional to their performance (measured by a fitness function - unique to the

problematic one is solving), so that the more successful strings (those of higher fitness) are morelikely to contribute genetic material to the offspring of the next generation, while poorly performingstrings are more likely to recede.

Potential problems in most of the GA algorithms performances mainly result from the parameter selection. Generally speaking the GA as an effective search technique is intended toeliminate the need for a trial and error approach to the search of complex spaces, yet most of itsapplications include trial and error to determine the three essential parameters - mutation rate,crossover rate and seed for the random number generator (Okada and Neki 1992). In mostapplications the parameters are fixed throughout the run, which may create a problem (insufficientdiversification, or convergence to suboptimal solution) if parameters are not optimally selected. Ithas been acknowledged that variable parameter setting is more effective. It is found (Novkovic andverko 1997, 1998, 2003) that random provision of parameters, created throughout the run of the GAitself, results in satisfactory resolution of the initial parameter selection.

The Novkovic - verko version of the GA algorithm uses the chromosome portions, which do

not translate into fitness (genetic residual), and assigns to genetic residual a diversifying function,


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essentially a control over the GA parameters, providing random parameter setting along the way, anddoing away with fine-tuning of probabilities of crossover and mutation.Genetic algorithm simulations (Holland 1975) have been applied in a number of fields, either tocapture the dynamics of adjustment, which reflects adaptive behavior and learning, or as an effectivesearch algorithm in optimization problems. However, not to many applications are seen in

optimizing ship hull structures.The tabulated instructions below explain GAR coding, and fitness evaluation as per outputprinted in Table 6 below. A sample of 15 strings representing shaft alignment system is extracted forthe purpose of explaining GAR coding and decoding.

Initial population is randomly selectedInitial

population

Initial parameters are defined for first generation and system continues to run by selectedparameters randomly from genetic residueOne pair of strings is initiated to zero bearings offset. The zero-offset string has relatively

high fitness (particularly when compared with totally random selected solutions), thus itensures faster convergence towards acceptable solutions.String

In this particular case each string contains five segments: e.g. four bearings and geneticresidue

Geneticresidue Genetic residue contains three parameters in binary format: random number, probability of

mutation and probability of crossover. Decoded values of parameters are printed as R, Cand P at the end of the string

BearingsMaximum bearing vertical movement is given as an input. This allows GA to dynamicallyallocate size of the active string (thus saves on the memory and increases the search speed).

Allowable maximum offset shall be normally selected considering hull deflection margin.In the subject example 13 alleles are necessary for decoding bearing maximum offset of 5[mm]. Decoded value is normalized so to get the range from +/- 5 [mm] with increment of1/1000 [mm].Additional 9 alleles are fixed for every system and represent:

1 allele - Sign of the offset +/-1 allele Keep or remove bearing1 allele Allow position change5 allele size of the position change1 allele direction of position change

Each of the four bearing is decoded (starting from right to left) and obtained number isnormalized to respective values as explained above.

If any segment of the system is considered as a rigid body (diesel engine for example) thevertical position of all bearings between the two edge supports is linearly interpolated.

Actual system consists of 11 bearings:Bearing number one (aft stern tube bearing) is fixed and initiated to zero offset.Bearings 4 to 11 are considered connecting the rigid body (diesel engine). Thus the

bearings 5 to 10 are linearly interpolated within the offsets evaluated for bearings 4 and 11.Offset of the bearings 2, 3, 4 and 11 are generated by GA and decoded accordingly.


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Bearingreactions Bearing reactions are calculated for obtained bearing offsets:

{ } [ ] { } 1111111111 .Re xxx OffsetMatInfactions =

reaction forces are product between the influence coefficient matrix and selected bearingoffsets.

FitnessFitness is then evaluated and assigned to each string. Fitness function is defied as ratio

between total bearing load and total positive load:

loadpositiveTotal

LoadTotalFitness

__

_=

Accordingly, maximum fitness is equal to 1 (one).

Stings with all positive reactions gain 10% fitness to increase their chances of enteringmating poolSelection

String of the higher fitness have greater chance to get selected.Selected pair of string is crossed and mutated in accordance with the respective

probabilities.New generation of offspring is generated and process is repeated until desired solution isfound.


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shaft alignment optimization genetic algorithms

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