Shaft Alignment Optimization Genetic Algorithms

ABS TECHNICAL PAPERS 2003

SHAFT ALIGNMENT OPTIMIZATION WITH GENETIC ALGORITHMS Davor Šverko, (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 bearing reactions under all operating conditions of the vessel, but also an acceptable relative-misalignment between the shaft and 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 - resulting mainly from hull deflections and thermal deviation. As the problem is stochastic, with an infinite number of satisfactory bearing 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 search algorithms. With an ability to conduct parallel search throughout the solution space, the GA is particularly well suited for 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 directlycoupled 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 scantling optimization and with the increase in ship’s length, and as the demand for power has increased with the ships’ size, the shafting diameters have become larger and the shafts stiffer (this is particularly true for VLCCs, ULCCs, large containerships). Consequently, the alignment of the propulsion system has become more sensitive to hull girder deflections, resulting in difficulties in analyzing the alignment and conducting the alignment procedure.

Figure 1 Directly coupled propulsion shafting - example

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

Shaft Alignment Optimization with Genetic Algorithms

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analyses, changes in the design of the vessel, shipyards’ practices in conducting the alignment, and a lack 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 relatively robust design with low sensitivity to disturbances affecting the propulsion shafting and the main drive. 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 vessel loading conditions as well as sea conditions the vessel operates in. Therefore, in order to define satisfactory alignment for all expected operating conditions, the bearing offset has to be optimized so as 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 believe that the genetic algorithms may be more suitable approach in shafting alignment optimisation for the reasons we elaborate below. The shaft alignment problem is stochastic, with an infinite number of acceptable bearing offsets. Finding a single good set of bearing offsets to satisfy bearing reaction requirements for only one 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 a number 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 exactly these kinds of problems. The GA has the ability to conduct parallel searches throughout the solution space (which is its biggest advantage versus other search tools), and simultaneously yields a multiple set 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 static load 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 the dynamic analysis only, if dynamic operation is what we eventually want to satisfy? The explanation is simple; the shaft alignment procedure is conducted and verified in static condition and the major role 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 effective search mechanism. It is a randomised search algorithm based on evolutionary genetics (Appendix A). 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 system’s optimisation, e.g.: Okada and Neki 1992 optimised ship hull structure applying GAs, and Dai et al 1994 used GAs on marine propulsor design. Some GA applications to the ship propeller were attempted by Lee and Hajela 1996, who investigated GAs in helicopter rotor blade design. Others considered different approaches in optimisation, e.g.: Rahman and Caldwell 1995 applied a rulebased and rational design method on ship structures. But obviously none of the above approaches found wider application in the industry, possibly due to the following reasons: - optimisation normally requires highly sophisticated software (large number of criteria to optimise) 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 “optimal parameters” (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 the simple 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 genetic algorithm may not be better than the other methods can provide. The genetic algorithm with residual (GAR), as developed by Novkovic and Šverko, may regain the confidence in GA, as the algorithm requires no “parameter optimisation”, and it results in solutions 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 all satisfy imposed constraints, parameters and criteria. Multiple solutions are necessary as it is often imperative 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 for the genetic algorithm to perform. The GA program optimizes among several constraint functions (as defined by hull girder deflections - Figure 2). Constraints which bind the solution space are defined by hull deflection curvatures 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 to investigate 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 laden and ballast condition

under

Figure 2 Hull girder deflections influence on shafting

Basically, GA optimisation works such that the software generates a desired number of solution strings, which are called population. This initially defined population is randomly defined and it may or may not contain any satisfactory solution. In the search for the solution the algorithm mimics 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 is applied (giving the fittest strings more chances to be selected into the mating pool for new generation), 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 in better 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 the mating pool through the selection process based on the individual string’s 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 binary coded, thus the allele is represented by a single bit) of the particular solution string is mutated (or not) 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 higher than would be required if convergence was desired. The convergence is not needed as we search for a 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 are considered 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 size o number of generations o mutation factors o number of crossover sites o Constraint functions (hull girder deflections) are defined o Prescribed thermal influence and diesel engine sag data is input The above information is specific to each particular shafting system, and can be amended as needed for the particular problem. - Selection: GA selects initial population: for given number of bearings the GA optimization routine 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 string’s 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 their chances of entering mating pool. - Selection: σ (sigma) selection (Tanese 1989) is applied to select pairs of strings to the mating pool 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 a desired 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 to the number of the bearings, o program is initiated to provide the desired number of solutions (10 is default number – this value 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 be conducted 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 move only as a rigid block. This avoids implausible solutions that may result from randomly selected extreme engine bedplate deflections. Also, by assuming rigid connection among engine bearings the search speed will be significantly faster, as the engine bearing offset will be evaluated at only two end bearings (the offset of the bearings in between is linearly interpolated). To account for actual engine flexibility the engine deflections can be accounted for in hull deflections. - Hull girder deflections can be estimated analytically or measured; hull deflection information is entered for two extreme conditions for which the alignment should be satisfactory (ballast and laden 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 the optimization process, and it will be considered in future research. The problem with bearing elasticity 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 correct the bearing offsets accordingly. The time required for GA to complete the search will depend on the complexity of the system. The more bearings in the system the more difficult it will be to find the combination of offsets to satisfy requirements. The same is true with constraints: the more stringent the constraint requirements 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 defining shortcuts 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 a typical single propulsion VLCC arrangement, with relatively short shafting and the low speed diesel engine 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 the shaft - main engine bearings: the aftmost three engine bearings are those particularly at risk to be damaged due to improper alignment. Figure 3 represents a discrete model of the propulsion shafting and diesel engine for the purpose 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, if hull 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 subject example, hull deflections are roughly estimated for evaluation purposes only. Low hull girder 20



deflections are intentionally chosen to show that even a small disturbance of the prescribed offset will potentially have some of the bearings unloaded in the system, if the initial bearing offset is not selected properly. Hull girder deflections can be estimated analytically or defined by measurements. ABS is currently involved in a long term project to measure hull girder deflections on several types of vessels of different sizes (i.e. VLCC’s, ULCC’s, bulk carriers and container vessels). The intention is to establish a data base of hull deflections which is to be used in alignment design to estimate hull deflections more accurately. The same measurements will be used by ABS to validate finite element models on vessels where hull girder deflection is analyzed numerically.

Bearing # 1 2 3 4 5 6 7 8 9 10 11

Hull deflection estimate [mm] Laden Ballast 0 0 0.5 -0.05 0.7 -0.07 1.2 -0.12 1 -0.1 0.8 -0.08 0.6 -0.06 0.4 -0.04 0.2 -0.02 -0.01 0.1 0 0

Table 1 Estimated hull girder deflections

If hull deflections corresponding to the ballast condition of the vessels are added on top of the prescribed bearing offsets the load distribution among bearings will not be satisfactory. The analysis shows 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 the prescribed bearing offsets the load distribution among bearings will not be satisfactory as well. The analysis 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 and the intermediate shaft bearing may easily unload as well. This implies that the original alignment design 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 the ballast to laden condition, but quite opposite, it worsens. Present practice in shaft alignment design normally does not include hull deflections; which is not surprising as the prediction of the alignment is quite difficult without proper analysis or measurements. The only means of controlling the alignment condition would be by bearing reaction measurements. However, bearing reaction measurements are seldom conducted with the vessel operating 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’ reaction measurements are not conducted as regular practice either. Now, if the alignment is not designed with sufficient tolerance to allow the propulsion shafting to accommodate eventual disturbances affecting 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 result in damage and eventual failure of bearings, but rather result in reduced life of the bearings and continuing 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 the alignment with another set of prescribed bearing offsets. However, without an optimization tool at hand 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 be provided in order to ensure the subject installation’s satisfactory alignment under both the ballast and loaded vessel conditions. Optimization may help investigate the solution simpler and faster than a trial 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 thus diversity in population (Novkovic, S. and Šverko, D. 1997). High mutation is desired as we are not interested in population convergence but in obtaining as many as possible highly diversified solutions. 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 engine raised above the zero offset line, and at the same time a very similar solution (bearing reaction 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 inclination gradient between the shaft and the stern tube bearing, however the stress in the shaft in that case will be higher. In cases without a forward stern tube bearing, the solution with M/E below the zero line will result in a very sensitive misalignment in the stern tube bearing and therefore may not be acceptable. 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 the maximum generations are reached if the number of solutions found is less then the preset number. o Deflection data is also provided and data can be amended as desired. Deflection data includes maximum hogging and maximum sagging deflections of the hull of the vessel, thermal rise at selected 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 GAR’s residue (Appendix A). By applying the same random seed in consecutive analyses with same input data, a repeatable analysis 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 on how 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

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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.



To show the diversity of GA’s performance, four solutions are extracted from the pool of satisfactory-prescribed displacements and shown in Appendix A. Two of four solutions are chosen with the main engine offset below the zero line, and the other two with the main engine above the zero line. The “best”, the most robust solution, is supposed to be selected by the designer, who should, in his/her decision-making, consider in particular the specifics of the shipbuilder’s production process and alignment procedures. This may need some further explanation: namely, the alignment is quite sensitive to relatively small disturbances in bearing offsets. If alignment procedure is fully conducted in the dry dock, for example, the GA optimization criteria shall be set to extract solution(s) which will correspond to particular condition in the dry dock, and need not result in satisfactory bearing reactions at the this stage of construction. However, when ship afloats the predicted hull deflections should improve alignment condition and result in acceptable load distribution among the bearings. In our test case we opted for the solution presented in Table 2. As we are not addressing any particular shipyard practices the selection is conducted as if the whole procedure is performed in dry dock, 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 and gap, bearing contact condition evaluation, etc. As mentioned above, in this particular case it is presumed that the alignment procedure is fully 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 be verified with relatively high accuracy. Table 3 shows a set of dry dock values for first of four selected 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 “optimal” solution so as to obtain a more preferable load distribution among bearings for the laden vessel condition.


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Generation: 9 String: 52 FITNESS: 1.100000 | SUPPORT REACTIONS | Total Total GA Max Hull Min Hull Thermal Engine | Ry[0] delRy Ry Ry Ry | Offset Offset defined Deflect. Deflect. Offset Sag. Sup. Node | (Max.Offs) (Min.Offs) (dy) | Max. Min. dy No No | [kN] [kN] [kN] [kN] [kN] | [mm] [mm] [mm] [mm] [mm] [mm] [mm] ------------------------------------------------------------------------------------------------------------------------------1 < 7> | 601.283 -56.872 518.533 544.996 544.411| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 2 < 14> | -41.678 87.605 106.331 46.072 45.927| 3.979 | 3.429 | 3.479 | 0.500 | -0.050 | 0.000 | 0.000 3 < 27> | 148.734 -20.861 34.172 124.780 127.873| 6.893 | 6.123 | 6.193 | 0.700 | -0.070 | 0.000 | 0.000 4 < 41> | 133.298 -108.513 275.984 32.933 24.785| 8.234 | 6.914 | 6.884 | 1.200 | -0.120 | 0.150 | 0.000 5 < 46> | 64.015 208.676 81.192 267.522 272.691| 8.149 | 7.049 | 7.003 | 1.000 | -0.100 | 0.150 | -0.004 6 < 48> | 286.255 -132.362 155.648 152.923 153.893| 7.954 | 7.074 | 7.012 | 0.800 | -0.080 | 0.150 | -0.008 7 < 50> | 272.916 26.788 285.205 301.263 299.704| 7.762 | 7.102 | 7.022 | 0.600 | -0.060 | 0.150 | -0.010 8 < 52> | 277.995 -5.009 345.582 265.960 272.986| 7.572 | 7.132 | 7.032 | 0.400 | -0.040 | 0.150 | -0.010 9 < 54> | 265.291 -1.102 143.036 274.995 264.188| 7.384 | 7.164 | 7.042 | 0.200 | -0.020 | 0.150 | -0.008 10 < 56> | 325.359 3.146 399.197 322.706 328.505| 7.298 | 7.188 | 7.052 | 0.100 | -0.010 | 0.150 | -0.004 11 < 58> | 96.318 -1.496 84.905 95.636 94.822| 7.208 | 7.208 | 7.058 | 0.000 | 0.000 | 0.150 | 0.000

Optimization with Genetic Algorithm

Optimization results: selected solutions from pool of 10 solutions (see Appendix A)

Table 2 Optimal solution




Dry dock condition offset and bearing reactions Reactions Offset 800

544.411 45.927 127.873 24.785 272.691 153.893 299.704 272.986 264.188 328.505 94.822

600 500 400 300 200 100 0 1

2

3 4 5 Bearing Reactions [kN] 6 Bearing Offset * 100 [mm]

7

8

9 10 Bearing Reactions [kN]

1 2 3 4 5 6 7 8 9 10 11

700

11

Bearing Offset * 100 [mm]

Ry (dy) [kN]

GA Define d Dy [mm] 0 3.479 6.193 6.884 7.003 7.012 7.022 7.032 7.042 7.052 7.058

Table 3 Dry dock - Bearing reactions for prescribed offset

Ballast vessel offset and bearing reactions Reactions Offset 800

544.996 46.072 124.78 32.933 267.522 152.923 301.263 265.96 274.995 322.706 95.636

700 600 500 400 300 200 100 0 1

2

3 4 Bearing Reactions [kN] 5

Bearing Offset * 100 [mm] 6

7 8 9 10 11

Bearing Offset * 100 [mm]

1 2 3 4 5 6 7 8 9 10 11

GA Defined Dy [mm] 0 3.429 6.123 6.914 7.049 7.074 7.102 7.132 7.164 7.188 7.208

Bearing Reactions [kN]

Ry (dy) [kN]

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


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Laden vessel offset and bearing reactions Reactions Offset 900

Ry (dy) [kN] 1 2 3 4 5 6 7 8 9 10 11

518.533 106.331 34.172 275.984 81.192 155.648 285.205 345.582 143.036 399.197 84.905

GA Defined Dy [mm] 0 3.979 6.893 8.234 8.149 7.954 7.762 7.572 7.384 7.298 7.208

800 700 600 500 400 300 200 100 0 1

2

3 4 5 Bear i ng Reacti ons [kN] Bear i ng Of f set * 100 [mm]

6

7

8

9 10

11

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 (dry dock), ballast, and laden, are satisfactory. The solution is relatively robust for predicted hull deflections, 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 contact investigation. Dry dock condition no slope boring Contact pressure 497 [MPa]

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



Slope boring requirements for the dry dock condition would satisfy ballast condition as well.

Slope boring requirements for the dry dock condition would satisfy loaded condition also. Misalignment slope is 0.15 [mrad] which is below industry’s accepted requirements for slope reduction by slope boring or inclination.


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CONCLUSION The genetic algorithm has proved its ability to find a desired number of acceptable solutions within given constraints, and the solution is found in a relatively short time. All the benefits of conducting 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 conducting alignment, will not result in a satisfactory static loading condition for the estimated hull deflections applied. Using a conventional approach the second aftmost main engine bearing, and possibly the intermediate 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 give even 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 accurately enough, 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 has already taken steps in that direction. Relatively accurate hull deflection prediction and optimized alignment would allow ship designers to confidently prescribe alignment for a vessel’s dry dock condition. The alignment procedure could then be conducted fully in the dry dock where we can control alignment condition much more accurately. This would significantly increase the precision of the whole process, as alignment measurement in the dry dock would be possible with very little disturbance affecting the system (no hull deflections), and thus, with an firm referent line the alignment may be conducted much closer to the on analytically predicted. Further research will also be necessary in investigating bearing elasticity influence on the offset 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 the consequence is constantly changing bearing elasticity. Also investigation into the robustness towards deviation of the offset at any individual bearing will be important to optimize. This issue is more difficult to optimize than the hull deflections, as the change in offset at only one bearing generates much larger disturbance to the alignment then the smooth transition caused by hull deflections (good examples of offset change at only one bearing would be temperature change below the bearing, or ware-down of the bearing liner). It should be stated that analytical models seldom represent actual conditions of a vessel, and this creates lots of difficulties in alignment rectification if required. This however, gives even more credibility 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 Marine Propulsor 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 of Michigan Press Lee, J. and Hajela, P. (1996): “Parallel Genetic Algorithm Implementation in Multidisciplinary Rotor Blade Design”, Journal of Aircraft, 33, 5: 962-969 Novkovic, S. and Šverko, 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 Šverko, D. (1998): “The Minimal Deceptive Problem Revisited: The Role of ‘Genetic waste’”, Computers and Operations Research, 25,11:895-911 Novkovic, S. and Šverko, D. (2003 - in review): “A Genetic Algorithm With Self-Generated Random Parameters”, Journal of Computing and Information Technology Okada, T. and Neki, I. (1992): “Utilization of Genetic Algorithm for Optimizing the Design of Ship Hull 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 Design Optimization”, 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, whose elements (alleles) are coded, usually in binary alphabet {0,1}. Algorithm functioning is guided by three 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 strings have a greater probability of selection into the mating pool. Performance is measured by a fitness function (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. Crossover operator will exchange genetic material between a pair of strings with probability pc, and a point of crossover k, somewhere in the interval [1, l-1]. Alleles are then exchanged between positions k+1 and l. For example, if we have two strings of length l = 7, crossing at k = 4. Two new strings emerge as follows: two strings areselected for maitng pool

1 1 1 | 1 1 1 1 ⎫ CROSSOVER ⎯→1 1 1 | 0 0 0 0 ⎬ ⎯⎯⎯⎯ 0 0 0 | 0 0 0 0⎭ ⇓

MUTATION

1 0 1 | 0 1 1 0

Mutation changes the allele with probability pm, 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 strings are 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 creates individual strings in each new generation by exchanging chromosomes between pairs of randomly selected mates from the old generation. The algorithm makes copies of strings with probability of selection 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 more likely to contribute genetic material to the offspring of the next generation, while poorly performing strings 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 to eliminate the need for a trial and error approach to the search of complex spaces, yet most of its applications 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 most applications the parameters are fixed throughout the run, which may create a problem (insufficient diversification, or convergence to suboptimal solution) if parameters are not “optimally” selected. It has been acknowledged that variable parameter setting is more effective. It is found (Novkovic and Šverko 1997, 1998, 2003) that random provision of parameters, created throughout the run of the GA itself, results in satisfactory resolution of the initial parameter selection. The Novkovic - Šverko 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, and doing away with fine-tuning of probabilities of crossover and mutation. Genetic algorithm simulations (Holland 1975) have been applied in a number of fields, either to capture the dynamics of adjustment, which reflects adaptive behavior and learning, or as an effective search 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 output printed in Table 6 below. A sample of 15 strings representing shaft alignment system is extracted for the purpose of explaining GAR coding and decoding. Initial population

Initial population is randomly selected Initial parameters are defined for first generation and system continues to run by selected parameters randomly from genetic residue One 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 it ensures faster convergence towards acceptable solutions.

String In this particular case each string contains five segments: e.g. four bearings and genetic residue Genetic residue

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’, ‘C’ and ‘P’ at the end of the string

Bearings Maximum bearing vertical movement is given as an input. This allows GA to dynamically allocate 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 of 1/1000 [mm]. Additional 9 alleles are fixed for every system and represent: 1 allele - Sign of the offset +/1 allele – Keep or remove bearing 1 allele – Allow position change 5 allele – size of the position change 1 allele – direction of position change Each of the four bearing is decoded (starting from right to left) and obtained number is normalized to respective values as explained above. If any segment of the system is considered as a rigid body (diesel engine for example) the vertical 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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Bearing reactions

Bearing reactions are calculated for obtained bearing offsets:

{Re actions}1x11 = [Inf .Mat ]11x11 ⋅ {Offset}1x11 reaction forces are product between the influence coefficient matrix and selected bearing offsets. Fitness Fitness is then evaluated and assigned to each string. Fitness function is defied as ratio between total bearing load and total positive load:

Fitness =

Total _ Load Total _ positive _ load

Accordingly, maximum fitness is equal to 1 (one). Stings with all positive reactions gain 10% fitness to increase their chances of entering mating pool Selection 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 is found.

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Sonja Novkovic and Davor Sverko


Population size (popsize) = 500 Maximum # of generation (maxgen) = 200 -----------------------------------------------------------------------------------------------------------------------------------Population Report: Generation 0 # residue string brg 4 brg 3 brg 2 brg 1 -----------------------------------------------------------------------------------------------------------------------------------1) 001101010001001111000010010000 0010110000000000000000 0011101000000000000000 0110001000000000000000 1011000000000000000000| F= 2) 010100101101101101001111111111 0111000010000000000000 1001110100000000000000 1001011000000000000000 1111001010000000000000| F= 3) 010111011111010101100110010101 1101010001001110100001 0011100101001010010001 0101011101110111001100 1000101100100010010110| F= 4) 101111100110010100010110001001 1111101011001011001001 0101011010100111100011 1110001111000010101100 1100011000010100101111| F= 5) 100001011110011011101010011100 0011110111111100111100 1111011001001000001100 1011111111000000000011 1010011111100101110101| F= 6) 001110001011100100101100011001 1011111100010111001111 1101111100111100111111 1001000011100110110000 1011001111101001010110| F= 7) 110010111011010001101111111100 0101011011110000011100 0010011110111001000111 1111110011100101000110 0010001110001110110001| F= 8) 001111110110110011010001001000 0110011110110011111101 0100010010001010111101 0101111010011110000100 0111100101111100000101| F= 9) 000110010111001110000011001110 1101011110101111111011 1001100010001101011001 0000010101111000001100 1001100111110010100100| F= 10) 001110111111111010010001010110 0001001100010010111000 1101011101101011111001 0111000101010111101011 0010010100101011001100| F= 11) 101100100100101100111111010110 0000001001110011101111 0101110011011001111011 0000011011010001010101 0001011101100000011110| F= 12) 110010110101011000000111001101 0110000011000010100011 0101010111010110101101 1011010010000101001000 0110110101101010100011| F= 13) 010111111000010011100111001111 0001111001100110111101 1100000001010011000010 1010101100001010001001 0111001001000011101110| F= 14) 101000100101000111011100011101 0111001011010101101110 0100000001010101101010 0011001110010110000011 0000101011101111011011| F= 15) 001101001011110111011001110100 1111001001010010001001 0111101100001001011001 0010111000001110100010 0100110010010101011000| F= ……….

Analysis performed with Pseudo RNG with seed

No Rotation on the bit level (No shift_bits)

Number of parallel populations = 1 Length of chromosome = 118 Length of active string = 88 Length of waste = 30 Number of noncoded segments = 0 Length of noncoded segments = 0 Number of tail segments = 0 Number of crossing sites = 7

GAR PROGRAM V2000 - All Rights Reserved

GAR OPTIMIZATION

Bearing offset coding example (initial population)

Table 6 GA Coding

0.981530 0.981530 0.535900 0.180852 0.000000 0.169809 0.127426 0.149787 0.242587 0.000000 0.040429 0.280660 0.334464 0.000000 0.103452

R=0.141 R=1.000 R=0.396 R=0.384 R=0.653 R=0.775 R=0.997 R=0.070 R=0.201 R=0.084 R=0.960 R=0.451 R=0.453 R=0.779 R=0.614

fitness C=0.309 C=0.426 C=0.835 C=0.580 C=0.608 C=0.893 C=0.819 C=0.701 C=0.805 C=0.978 C=0.175 C=0.344 C=0.076 C=0.279 C=0.967

P=0.207 P=0.324 P=0.367 P=0.744 P=0.523 P=0.221 P=0.796 P=0.247 P=0.099 P=0.234 P=0.697 P=0.795 P=0.373 P=0.634 P=0.205

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Optimization with Genetic Algorithm

Optimization results: four solutions are extracted for evaluation

Table 7 Four selected solutions


Shaft Alignment Optimization with Genetic Alogrithims

Generation: 59 String: 226 FITNESS: 1.100000 | SUPPORT REACTIONS | Total Total GA Max Hull Min Hull Thermal Engine | Ry[0] delRy Ry Ry Ry | Offset Offset defined Deflect. Deflect. Offset Sag. Sup. Node | (Max.Offs) (Min.Offs) (dy) | Max. Min. dy No No | [kN] [kN] [kN] [kN] [kN] | [mm] [mm] [mm] [mm] [mm] [mm] [mm] ------------------------------------------------------------------------------------------------------------------------------1 < 7> | 601.283 -36.848 538.557 565.020 564.435| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 2 < 14> | -41.678 74.064 92.790 32.531 32.386| 0.458 | -0.092 | -0.042 | 0.500 | -0.050 | 0.000 | 0.000 3 < 27> | 148.734 -44.142 10.890 101.498 104.592| -0.504 | -1.274 | -1.204 | 0.700 | -0.070 | 0.000 | 0.000 4 < 41> | 133.298 -118.447 266.050 22.999 14.851| 0.099 | -1.221 | -1.251 | 1.200 | -0.120 | 0.150 | 0.000 5 < 46> | 64.015 248.593 121.110 307.440 312.609| 0.027 | -1.073 | -1.119 | 1.000 | -0.100 | 0.150 | -0.004 6 < 48> | 286.255 -148.223 139.786 137.061 138.032| -0.166 | -1.046 | -1.108 | 0.800 | -0.080 | 0.150 | -0.008 7 < 50> | 272.916 29.998 288.416 304.474 302.914| -0.357 | -1.017 | -1.097 | 0.600 | -0.060 | 0.150 | -0.010 8 < 52> | 277.995 -5.611 344.980 265.358 272.384| -0.546 | -0.986 | -1.086 | 0.400 | -0.040 | 0.150 | -0.010 9 < 54> | 265.291 -1.226 142.912 274.872 264.065| -0.733 | -0.953 | -1.075 | 0.200 | -0.020 | 0.150 | -0.008 10 < 56> | 325.359 3.510 399.561 323.070 328.869| -0.818 | -0.928 | -1.064 | 0.100 | -0.010 | 0.150 | -0.004 11 < 58> | 96.318 -1.669 84.731 95.462 94.649| -0.906 | -0.906 | -1.056 | 0.000 | 0.000 | 0.150 | 0.000

Generation: 77 String: 359 FITNESS: 1.100000 | SUPPORT REACTIONS | Total Total GA Max Hull Min Hull Thermal Engine | Ry[0] delRy Ry Ry Ry | Offset Offset defined Deflect. Deflect. Offset Sag. Sup. Node | (Max.Offs) (Min.Offs) (dy) | Max. Min. dy No No | [kN] [kN] [kN] [kN] [kN] | [mm] [mm] [mm] [mm] [mm] [mm] [mm] ------------------------------------------------------------------------------------------------------------------------------1 < 7> | 601.283 -25.224 550.181 576.644 576.059| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 2 < 14> | -41.678 51.274 70.000 9.741 9.596| 0.398 | -0.152 | -0.102 | 0.500 | -0.050 | 0.000 | 0.000 3 < 27> | 148.734 -28.789 26.243 116.851 119.945| -0.290 | -1.060 | -0.990 | 0.700 | -0.070 | 0.000 | 0.000 4 < 41> | 133.298 -129.084 255.414 12.362 4.214| 0.391 | -0.929 | -0.959 | 1.200 | -0.120 | 0.150 | 0.000 5 < 46> | 64.015 258.333 130.849 317.179 322.348| 0.323 | -0.777 | -0.823 | 1.000 | -0.100 | 0.150 | -0.004 6 < 48> | 286.255 -152.181 135.829 133.104 134.074| 0.130 | -0.750 | -0.812 | 0.800 | -0.080 | 0.150 | -0.008 7 < 50> | 272.916 30.799 289.217 305.275 303.715| -0.061 | -0.721 | -0.801 | 0.600 | -0.060 | 0.150 | -0.010 8 < 52> | 277.995 -5.761 344.830 265.208 272.234| -0.249 | -0.689 | -0.789 | 0.400 | -0.040 | 0.150 | -0.010 9 < 54> | 265.291 -1.257 142.881 274.841 264.034| -0.436 | -0.656 | -0.778 | 0.200 | -0.020 | 0.150 | -0.008 10 < 56> | 325.359 3.601 399.652 323.161 328.960| -0.521 | -0.631 | -0.767 | 0.100 | -0.010 | 0.150 | -0.004 11 < 58> | 96.318 -1.712 84.688 95.419 94.605| -0.609 | -0.609 | -0.759 | 0.000 | 0.000 | 0.150 | 0.000


Shaft Alignment Optimization with Genetic Alogrithims

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Shaft Alignment Optimization Genetic Algorithms

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