A Method for the Dbign of Ship' Propulsion Shaft Systems. William e Lehr, Jr. Edwin l Parker

7/28/2019 A Method for the Dbign of Ship' Propulsion Shaft Systems. William e Lehr, Jr. Edwin l Parker

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NPS ARCHIVE1961.06LEHR, W.

A METHOD FOR THE DBIGNOF SHIP' PROPULSION SHAFT SYSTEMS.

WILLIAM E LEHR, JR.EDWIN L PARKER


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LIBRARYU.S. NAVAL POSTGRADUATE SCHOOL

MONTEREY, CALIFORNIA


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A METHOD FOR THE DESIGN OF SHIP PROPULSIONSHAFT SYSTEMS

byWILLIAM E. LEHR, JR., LIEUTENANT, U.S. COAST GUARD

B.S., U.S. Coast Guard Academy(1955)and

EDWIN L PARKER, LIEUTENANT, U.S COAST GUARDB.S., U.S. Coast Guard Academy

(1954)

SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THEDEGREE OF NAVAL ENGINEER

AND THE DEGREE OFMASTER OF SCIENCE IN NAVAL ARCHITECTURE

AND MARINE ENGINEERINGat the

MASSACHUSETTS INSTITUTE OF TECHNOLOGYJune 1961

Signatures of Authors

Department of Naval Architectureand Marine Engineering, 20 May 1961Certified by

Accepted byThesis Advisor


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I . S. Naval rostgratluafe Scliod?Monterey, California

A METHOD FOR THE DESIGN OF SHIP PROPULSION SHAFT SYSTEMS,by WILLIAM E, LEHR, JR. and EDWIN L. PARKER. Submittedto the Department of Naval Architecture and Marine Engin-eering on 20 May 1961 in partial fulfillment of the re-quirements for the Master of Science degree in NavalArchitecture and Marine Engineering and the ProfessionalDegree, Naval Engineer,

ABSTRACT

An investigation was conducted to establish a minimumspan-length criteria for use in marine propulsion shaftingdesign.The investigation is conducted through computer studiesof families of synthesized shafting systems. Each system istreated as a continuous beam carrying concentrated and dis-tributed loads. In the studies span-length is systematicallyvaried. The sensitivity of the study systems to alignmenterrors is investigated using reaction influence numbers. Re-lative insensitivity to misalignment is judged on the basisof limiting values of allowable bearing pressures and allow-able difference in reactive loads at the reduction gearsupport bearingsThe results of this theoretical investigation indicatethe desirability of increased values for span-length fromthose frequently found in present practice. Shaft systemswith the following minimum span-lengths should be free frommost problems resulting from normal alignment errors and

the usual amount of bearing wear= (Span-lengths are ex-pressed as length to diameter ratio)For shafts with diameters 10 to 16 inches, L/D 14For shafts with diameters 16 to J50 inches, L/D 12In the conduct of the basic investigation several ad-ditional problems connected with shaft design were studied.A series of design nomograns for tallshai't sizing are de-rived from strength considerations . They are presented asa proposed aid for shaft design. The problem of fatigue

failure of tailshafts, at the propeller keyway, is con-sidered and a proposed method for corrective action Is given.Thesis Supervisor: S. Curtis Powell


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ACKNOWLEDGMENTS

The authors wish to acknowledge the technical assistanceand guidance given to them by H. Go Anderson, Manager, GearProduction Engineering^ Medium Steam Turbine, Generator andGear Department, General Electric Company; LCDR J. R. Baylis,USN, Associate Professor of Naval Engineering, MassachusettsInstitute of Technology, and Professor S, Co Powell,Associate Professor of Marine Engineering , Massachusetts In-stitute of Technology c The nomographic techniques used werebased on precepts presented by Professor D P Adams,Associate Professor of Engineering Graphics, MassachusettsInstitute of Technology.The authors also wish to express their appreciation to

Captain E So Arentzen, USN., Professor of Naval Construction,for his encouragement and inspiration without which thisthesis could not have been written

All computer work was carried out courtesy of theMassachusetts Institute of Technology Computation Center,Cambridge , Massachusetts


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TABLE OF CONTENTS

TitleAbstractAcknowledgmentsTable of ContentsList of FiguresList of Tables

Pageiiiiiiivvivii

1.0 INTRODUCTION1 . Background ................. 11 . Intent of Thesis Study 2

2 . NOMENCLATURE 35.0 THE EFFECT OF SHORT SHAFT SPANS

5.13.23.33.53.63.73.8

GeneralBearing Supports and Designed ReactionsComputed Bearing LoadsEffect on Stern BearingsCorrective Action for Stern Bearings .Effect on Reduction GearCorrective Action for Reduction Gear .Results of Analysis

4.0 CONSIDERATIONS FOR MINIMUM BEARING SPAN44

Need for Minimum Span .......Alignment and Load Conditions4.2.1 Bearing Loading .Gear Alignment . .Setting TolerancesOperational Wear .Foundation Flexure

444,4

2345

5.0 EVALUATION PROCEDURE FOR MINIMUM SPAN5.1 Development of Shaft Systems5.2 Computer Output .............5.3 Alignment for Comparison ....5.4 Comparison Procedure ........

567811

131516

17171818181919

23242526


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TABLE OF CONTENTS(continued

Page6.0 RESULTS OP MINIMUM SPAN EVALUATION

6 1 Three and Four Span Systems . 506 .2 Two Span Systems 306 .3 Five Span Systems ...... 326 . Summary 326 . Application of Results 327.0 STERN TUBE BEARINGS

7.1 Advantage of One Stern Tube Bearing 367.2 Location of Non-Weardown Bearing 378.0 LOW FREQUENCY CYCLIC STRESSES

8 . Background 398 . Stress Concentration 408.3 Steady Stress Reversal Theory 428.4 Application in Design 459.0 SIZING OF THE TAILSHAFT

9 1 Present Procedure 479.2 Development of Design Nomograms 4810 . CONCLUSIONS AND RECOMMENDATIONS 52

APPENDIX . o . 56aA. Recommended Design Procedure 57B. Shafting Systems Computer Program 6lC Selection of Shaft Systems for Study 65D. Strength and Vibration Requirements 71REFERENCES ............... 55


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LIST OF FIGURES

Figure Title *1 Typical Shaft System 6a2 Shaft Deflection Curve 6a2 After Stern Tube Bearing Weardown 8a4 Change in Reaction, Bearings 1, 2 and 2 15a5 Change in Reaction, Bearings 1, 2 and 2 15a6 Change in Reaction, Bearings 1, 2 and 2; 15b

Bearing 2 not in system.7 Schematic of Study Systems 22a8 Efiect of Low Cycle Steady Stress 42a

Reversal

A-l to A-6 Tailshaft Diameter Selection Nomograms 60a-60f


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LIST OF TABLES

Table Title Page

I Bearing Reactions and Influence Numbers 8II Bearing Reactions and Influence Numbers 9

III Study System Parameters 21,22IV Minimum Shaft and Span Length-Diameter 25Ratios

Tailshaft Safety Factor and Allowable 80Span Length Estimate


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1.0 INTRODUCTION1 . Background

The years since 1940 have seen a complete evolutionof the Navy's surface fleet, the advent of the superbulk carrier in the Merchant service, and the revolu-tionary change to nuclear propulsion in the submarine.These events, directly and indirectly, provided an im-petus for a great deal of development and research inthe field of marine turbines, reduction gears and pro-pellers. Each prototype of these components has hadthe benefit of developmental research incorporated inthe basic design procedure. In many eases research hasbeen carried to the extent of building shore based testunits to assist in the development. The result of thishas been to make available to the shipbuilding industryefficient and trouble-free units. On the other hand,propulsion shafting, the connecting link in the pro-pulsion system, has not been accorded the benefit ofsuch research. Fortunately, most shaft systems designedusing the existing criteria have provided excellentservice. However, the need for further considerationof shafting design practices has made itself conspicuousin numerous way, many of which have been covered com-


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prehensively in recent presentations. Shafting effectson reduction gear alignment (l), the relatively highcasualty rates of tallshafts (2), and the alignment pro-blems of shaft bearings (j5) are examples. In additionthere are numerous operational reports of shaft sealfailures and bearing failures.

1.2 Intent of Thesis Study

It is the authors * contention that many of theabove problems will be alleviated, if the presentlyaccepted design procedures are complemented by considera-tion of minimum bearing spacing and low frequency cyclicstresses. Thus, it is the intent of this thesis to pre-sent the effect of these two considerations on the designof a shaft system and provide a series of convenientnomograms and tabular data which will permit the designerto quickly develop the preliminary characteristics of apropulsion shaft system.


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2,0 NOMENCLATURE

D - Outside diameter of shaft, (inches)d - Inside diameter of shaft, (inches)EoLo - Endurance limit of material, in-: air, (psi)F,S, - Factor of safetyF,S. .- Dynamic factor of safety This factor accountsfor the effects of dynamic loading and thrusteccentricity,I - Influence number of bearing (x) on bearing (y),~y or the change in reaction at bearing (y) for a1 mil deflection at bearing (x),J - Polar moment of inertia of shaft section , (inches )K, - Stress concentration factor in bending. In shaftsystems IC is applied to shaft flanges, oil holes,etc. For well designed axial keyways K. 1.K, - Stress concentration factor in torsion. In shaftsystems K. is applied to flanges, oil holes andkeywaysK, - The percentage of steady mean torque which makesup the alternating torque. Maximum alternatingtorque occurs at the torsional critical speed.

At speeds well removed from the torsional criticalswhich should be the ease for well designed systems,K-, will range from 0,05 to 0,25 depending upon thehull configuration and proximity of the propellerto the hull , struts , etc . The selection of avalue for K-, must be based on the designers ex-perience

L - Moment arm of the propeller assembly. It is theP distance from the center of gravity of the propellerto .the point of support in the propeller bearing,n - Ratio of inside to outside shaft diameters, d/DQ - Mean or Steady torque


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RPM - Revolutions per minuteR - Reaction in pounds at bearing (x)AR - Change in reaction, in pounds, at bearing (x)mmR - Reaction in pounds at bearing (x), all bearings

si on straight lineS, - Compressive stress due to bending, (psi)S - Steady compressive stress, (psi)SHP - Shaft horsepowerS - Resultant steady stress, (psi)S - Resultant alternating stress, (psi)S a - Steady shear stress, (psi)sS^ - Alternating shear stress, (psi)saS - Mean stress level of fluctuating steadystresses , (psiT - Propellor thrust , (lbs .W - Weight of propellor, (lbs.)YoPo - Yield Point of material, (psi)Y - Deflection in mils of bearing (x) from straightx line datum; + above datum, - below datum,


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3.0 THE EFFECT OF SHORT SHAFT SPANS3.1 General

At the present time classification rules in gen-eral make no mention of bearing spacing or of bearingloading, except to express the length of the bearingadjacent to the propeller as a function of shaft dia-meter. Most design procedures do limit indirectly themaximum bearing span by setting limits on allowablestresses, bearing load, and vibration considerations.However, as far as the authors have been able to de-termine, there are none that set a minimum on bearingspan, Thus a system such as shown in Figure 1 wouldsatisfy the classification rules and by most designprocedures would be considered a satisfactory shaftsystem.

That span is an important consideration in pro-ducing a satisfactory design is shown by a comparisonof intended loads with the computed bearing loads inthe system of Figure 1. The authors grant this considersonly one particular case; however , the system is repre-sentative of certain current practices and vividly points* itEquivalent Reduction Gear Diameter = 42.3Lineshaft Diameter * 21,9"Tailshaft Diameter - 23.8"


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up the problems encountered. To obtain the computedbearing loads, the shaft system composed of reductiongear, shaft, and propeller was treated as a continuousbeam carrying distributed and concentrated loads. Theinfluence line technique as developed by reference (4)and modified for use on the IBM-709 computer was ap-plied permitting an analytical solution of the continu-ous beam problem, (See Appendix B)

3.2 Bearing Supports and Designed Reactions

It is assumed in the solution that the bearingsact as zero clearance point supports at the mid-lengthof the bearings, with the exception of the after sterntube bearing. At the after stern tube bearing thesupport point is taken one shaft diameter forward ofthe aft end of the bearing. The effect of replacingthe bearing surface by a point support does not signi-ficantly affect the results obtained, except in the caseof the long stern tube bearings . For these bearings con-sideration must be given to the angle to which the sterntube has been bored and the state of weardown the bear-ing surfaces have attained. In the present case, it isassumed that the stern tube has been bored true tostraight line datum and the bearing surfaces initiallyhave sustained no weardown,


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Referring to the system shown in Figure 1, theintended bearing reactions with all bearings alignedon a straight line in the hot-operating condition wereapproximately as follows:

BearingNumber 1 2 5 4 5 6 7Load(lbs) 35,000 35,000 17,000 17,000 17,000 25,000 75,00

3.3 Computed Bearing Loads

The initial calculation using the influence linetechnique showed the forward stern tube bearing No. 6to be negatively loaded with all bearings on the straightline. In normal practice bearings have some diametricalclearance. Thus No. 6 journal would rise in its bearing.Effectively then, the bearing is not in the system, un-less the rise is greater than the diametrical clearance.A second calculation was made with the support at theforward stern tube bearing omitted from the system.Table I consists of values of bearing reactions and in-fluence numbers extracted from the computer output datafor the second calculation.


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5 6 776.1 D -7.2

-237.7 E 22.5452.4 L -42.9-734.2 E 119.0575.4 T -132.0

TABLE I. - BEARING REACTIONS AND INFLUENCE NUMBERSInfluence Numbers (lbs. change perOoOOl inch bearing rise)^ 12 3 4

1 2886.2 -5342.5 2761.1 -373.72 -5342.5 10369.5 -5978.8 1166.93 2761.1 -5978.8 4367.7 -1559.24 -373.7 1166.9 -1559.2 1381.65 76.1 -237.7 452.4 -734.26 DELETED PROM SYSTEM7 -7.2 22.5 -42.9 119.0 -132.0 D 40.0

Straight Line Bearing Reactions (lbs.)33902.0 35545.4 15076.9 19156.7 17755.3 0.0 99834.3

3.4 Effect on After Bearings

Figure 2 shows a plot of the shaft deflections basedupon data extracted from the computer output. It will benoted that at Bearing No, 6 the shaft is not resting onits support. The computer data indicated the journal isup 3.8 mils in the bearing. It is true that the journalwill probably settle on its bearing due to compression ofthe after stern tube bearing under load, but at best itwill be lightly loaded. Under these conditions of static


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INITIALCONDITION

PARTIALWEARDOWN

LOADDISTR.CONTACTSURFACE

AFTERWEARDOWN

FIG.-8a-


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loading any additional inertia loading caused by theship working in a seaway could result in the journalpounding in the forward stern tube bearing. This con-dition in proximity to the shaft seal would make itimpossible to maintain the integrity of the seal. Inaddition, as the after stern tube bearing begins towear in, the equivalent point support shifts forward.This is shown in Figure 3. When the bearing has wornto the configuration of the elastic curve, there isuniform distribution of load on the bearing. The pointsupport is then at the mid-length of the after sterntube bearing, as shown in Figure 3(c). A third cal-culation was made with all supports on a straight lineand the support point of the after bearing at its mid-length. For this calculation No. 6 bearing was in-cluded in the system. Table II lists the values ofbearing reactions and influence numbers for the afterthree bearing under these conditions.

TABLE II - BEARING REACTION AND INFLUENCE NUMBERSInfluence Numbers (lbs. change per0.001 inch bearing rise)

Brg.No. 5 6 75 2140.9 -2115.6 781.46 -2115 06 2964.8 -1321.17 781.4 1321.1 642.4

Straight Line Bearing Reactions (lbs.)31547.0 -46530.0 135124.0


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A reasonable amount of weardown must take place be-fore a uniform load distribution will occur on the afterstern tube bearing. To determine the bearing loadingunder these conditions a value of weardown of 0.020 inchesat the mid -length of the bearing was chosen. Using thisvalue, Y? - -20.0 mils, and the influence numbers ofTable II the calculated reactions at bearings No. 5 andNo. 6 are:

(3-(l) R5 * R5sl+I7 .5Y7-+3l547+(78l.4)(-20.0)-l5,919.0 lbs.(3- (2) R6 - R6sl+I7-6y7-46550+(-1321.3)(-20.0)20,108 lbs.The negative reaction at the forward stem tube bearingNo. 6 indicates the journal will rise in that bearing andthe reaction would be zero. Therefore setting rI - 0.0the rise of the journal can be calculated.(3- (3) R6 - R6 + I6 _6Y6 0.0 - -20,108.0 + (2964. 8)Y6

Yg - 6.8 mils rise.An adjustment is now made to the reactions at bearingNo. 5 and No. 7 to reflect this rise of No. 6 journal.(3- (4) R* - r5 + i6 _5y6 15,919.0 + (-2115.6) (6.8) - 1,533.0 lbs(3- (5) R7 = R7gl + l6 _7

Y6

+ l7 _?

Y7

135,547.0 +(-1321.1) (6.8) + (642.4) (-20.0) - 113,293.0 lbs.

-10-


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Not only has the forward stern tube bearing become un-loaded but the adjacent lineshaft bearing No. 5 becomeslightly loaded. These conditions have negated any con-siderations the designer made, as far as strength,vibrations and bearing loads are concerned, based onall bearings carrying their designed load.

3.5 Corrective Action for Stem BearingsIt is possible to obtain the designed load reactions

for these three bearings through use of a "faired curvealignment" (3). However, the magnitude of the influencenumbers of Table II is unaffected by changes in verticalalignment. Since the magnitude is large in this area ofthe shaft, any appreciable error in initial setting orsubsequent weardown of the stern tube bearing can causeunloading or overloading of the other bearings in thegroup. For example, the influence of bearing No. 5 uponitself is 2140.9 lbs ./mil. Therefore a 5.0 mil error insetting would result in a 10,704 lbs. change in the re-action of the bearing.

Instead of using fair curve alignment, No, 6 bearingcan be positively loaded by increasing the span lengthbetween bearings No, 6 and 7 Additionally, since theinfluence numbers are inverse functions of span length,a significant reduction in their values for bearings


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No, 6 and 7 can be obtained. A softer tailshaft, re-latively less sensitive to alignment errors and afterstern tube bearing wear, is thus obtained.

Unfortunately, as the span between forward andafter stern tube bearings is increased, the distancebetween bearings No. 5 and 6 decreases. This resultsin a corresponding increase in sensitivity to misalign-ment between No, 5 and No, 6, The most obvious solutionto this dilemma is to remove the forward stern tube bear-ing entirely o In effect, this is physically what thesystem does. If the original design had considered sucha possibility the stern seal would have been located nearbearing No. 5. This of course assumes the stern tubecould be lengthened to accommodate such action.

A comparison of Tables I and II shows a full orderof magnitude decrease in the values of influence numberscan be obtained by such action. Thus the tailshaft isless sensitive to misalignment of bearing No. 5 and toweardown of the after stern tube bearing. Furthermoresome positive loading has been achieved at the bearingnext to the stern seal. As previously shown, it is small.An even greater span between the after stern tube bearingand the next forward bearing might be desirable from thestandpoint of loading this bearing. Such an increase inspan must be checked against strength, vibration and bear-


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ing load requirements. Based on these it might not bepossible to take such action without other adjustments.The point is that if the designer had considered a re-quired minimum span length in the initial design stage,both strength and vibration criteria, as well as loadand alignment criteria, could be met in this region.

3.6 Effect on Reduction Gear

Considering now the forward end of the system,additional alignment criteria must be met in the regionof the reduction gear. In general, most gear manu-facturers specify alignment and low speed gear bearingload conditions for satisfactory operation. These maytake the form of a maximum difference in static fore-and-aft bull gear bearing reactions (1) or maximum andminimum acceptable bearing loads based on unit pressure.In any case the shaft system must be compatible with therequirements of the installed gear, or if direct drive,to the propulsion plant.

In the system under consideration it was the intentof the designer to have the gear bearing reactions approximately equal for straight line alignment. Looking atTable I, it is seen this intent is satisfied. However,for the gear to be on the straight line in the hot-


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operating condition; it must be set some distance belowthe straight line in the cold-assembly condition. Thedistance which it is set below is made up of the riseof the bearings due to thermal expansion of the bearingsupports and the rise of the journals in their bearings.This rise takes place when going from the cold-assemblycondition to the hot -operating condition. Since assemblyand operating conditions vary and the support structureis complex, some tolerance must be allowed in predictingthis rise. Additionally, the erecting facility requiressome tolerance in setting the gear. Therefore, the de-sign must be such that the reduction gear and shaftcombination is able to absorb some misalignment.

There are various criteria that could be used toevaluate the effects of misalignment. The authors choseto use the changes in reactions due to parallel displace-ment of the low speed gear bearings . Using the influencenumbers of Table I, the reactions at bearings Nos. 1, 2and 3 were calculated for various offsets using thefollowing relationships

&- Rl- Bl.l +Il-lYl +I8-*T2 +13-iy3(5- (7) R2 - R2sl + I^gYj + I 2 .2Y2 + I>2Y5(?- (8) R3 = R33l + Ij.,^ + I 2 .5Y2 + I5 .5Y5

-14-


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The influence of bearing No, 3 is included, so itseffect may be calculated, when it unloads, and thejournal begins to rise in the bearing.

The calculated values for the reactions are plottedin Figure 4. It will be noted that bearing No. 3 willunload when the offset is 4.5 mils above straight linedatum. Similarly, at 7.0 mils below datum No. 2 bear-ing unloads. Now imposing the gear manufacturer'salignment criteria, which in this case was an allowabledifference in fore-and-aft gear bearing reactions of15,000 pounds, results in an allowable setting error of+2.0 mils, as shown in Figure 4. To attempt setting thegear with tolerances such as these would be completely un-realistic. If the gear could be properly positioned, itwould be impossible to maintain these tolerances underservice conditions.

Figure 5 is a plot illustrating the effect of mis-aligning, No. 5 bearing in the system and very vividlyshows the limited tolerances available for positioningof the spring bearings.

3.7 Corrective Action for Reduction Gear

To determine the effect of increased span betweenthe after gear bearing and the following lineshaft bear-


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T r006 004BELOW DATUM

1 r002 -^04 006ABOVE DATUM0.0(INCHES)

SIMULTANEOUS POSITION OFBRGS NO'S I &2 WITHRESPECT TO STRAIGHT LINE DATUMFIG. 4

60

50

40 -v>zo-30o-) Values of bending moments at selectedpoints

4) Values of shear stress at selectedpoints

5) Shaft deflections due to static load-ing.


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Thus information for 165 related shaft systems wasavailable for analysis by the authors.

5.3 Alignment for Comparison

For each case the table of influence numbers wasapplied in conjunctionwith the maximum values of allowedmisalignment previously specified. In this manner thechanges in bearing reaction were calculated for eachsystem. These changes could have been applied to thestraight line reactions and the results compared withthe previously specified allowable values. For thestraight line alignment of the system the gear bearingreactions are not normally equal. It was decided a morerealistic base from which to evaluate the system was thealignment which causes equal gear bearing reactions.Equal reactions can be achieved by parallel movement ofthe gear bearings in the vertical direction. While thisis not the only way in which to achieve equal reactions,it was the one which the authors considered more ex-peditious. The amount of offset was solved for as follows

Rl " R2Y Yl x 2


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in the relationships

Rl - R131 + ^-l*! + VAR2 " R2S1 + h-2*l + Wfi

Combining and using the fact that Ion ^~\-o> by reciprocity,gives

1 * ul-l 2-2'Knowing the offset necessary to give equal gear bearing re-actions, the reactions for all bearings were calculated.

5.4 Comparison Procedure

With the shaft system aligned in this manner, the pro-cedure used for the comparison with allowable values was asfollows:

Condition I . +10.0 mil parallel deflection of lowspeed gear bearings . The changes in reaction at the gearsupports are

(10) AI^ - + 10.0(I1;L + I2-1 )and

(11) AR2 = + 10.0(l1 _2 + I22 )


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The difference in static fore-and-aft gear bearing reactions,remembering Ip.-i ~ *i p> is tnen

(12) (AR1 - AR2 ) + 10.0(1^ - I2 _2 )The change in reaction at bearing No. 3 is

(15) AR^ - + 10.0(1^ + I2 _3 )In a like manner the change in reactions at the other bearingsin the system were calculated using the applicable influencenumbers

Condition II . +10.0 mil deflection of intermediatelineshaft bearings. The difference in static fore-and-aftgear bearing reactions is

(14) (AR-l - AR2 ) * + 10.0(I3_1 - I>2 )The change in reaction at bearing No. 5 is

(15) AR^ - + 10.0 (I5-5 )

For systems with additional intermediate bearings thechanges can be calculated in the same manner using the appropriateinfluence numbers.

Condition III . 300.0 mils of weardown of the stern tubebearing. The difference in static fore-and-aft gear bearing


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reaction is

(16) (AR1 - AR2 ) = -500. 0(I4-1 - I4 _2 )

The change in reaction at bearing No. 2 is

(17) AR^ = -500.0(I4 _5 )

The change in reaction at the after stern tube bearing is

(18) AR4 = -300.0(I4 _4 ).

The above equations are for the two span system of Figure 7(a).For those systems with additional spans the appropriate in-fluence numbers are used and the changes at the other bearingsare calculated.

Before a system was considered as acceptable, it had tosatisfy the following criteria derived from the previously out-lined requirements?

1) (AR, - AR_) had to be less than the tabulatedlimit for difference in static fore-and-aftgear bearing reactions of Table III.

2) (R + AR), of the intermediate lineshaft bear-ings had to be such that the nominal bearingpressure is greater than 5.0 psi and lessthan 50.0 psi for a bearing with a length of


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1.5 diameters. For example, in the caseof bearing No. 3, this is

5.0(1.50D2 ) C (R + AR^) < 50.0(1.50D2 ).3) (R + AR), of the after stern tube bearing, had

to be such that the nominal bearing pressureis greater than 5.0 psi and less than 55.0psi for a bearing with a length of 4.0 dia-meters. For example, in the case of the twospan system this is

5.0(4.0D2 ) < (R4 + AR4 ) < 35.0(4,0D2 ).criteria had to be satisfied for Conditions I, II, and

III individually. The authors did not evaluate the systemsall three conditions imposed simultaneously, and perhaps

could be criticized for not covering the most stringent con-ditions. It is felt that the possibility of the maximum ofall three conditions occurring at the same time is quite re-


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6.0 RESULTS OF MINIMUM SPAN EVALUATION6.1 Three and Four Span Systems

The results of the evaluation showed that for threeand four span systems, Figure 7 (b) and (c), the follow-ing satisfied all conditions;

1) Shaft diameter of 10.00 to16.00 inches, a span lengthto diameter ratio equal toor greater than 14,

2) Shaft diameter of l6.00+ to30.00 inches, a span lengthto diameter ratio equal toor greater than 12.

6,2 Two Span Systems

For the two span system the limitation imposed on(AFL - AR_) was exceeded by the requirements of Con-dition III. This was true for all span length todiameter ratios investigated, with the exception ofthe shafts with diameters in the 26 to 30 inch range.For shafts with diameters of 26 to 30 inches, a spanlength to diameter ratio of 18 satisfied all conditions.


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This does not mean that shorter spans cannot be used sinceseveral courses of action are available. First, theallowable weardown of the stern tube bearing could bereduced; second, accept a higher value of (R - Rp) thanthe value used by the authors; third, the initial align-ment could be modified. The latter action would be therecommendation of the authors.

Basically, the procedure would be to initially alignthe system by offsetting the gear bearings to satisfythe following conditions?

1) The after gear bearings reaction R~greater than the forward gear bearing reactionR. by approximately 75$ of the tabulated limit.

2) The unit pressure on No. 3 bearing2equal to 15.0 psi; i.e., R_. 15(1.5D ).

Weardown of bearing No. 4 will increase the reactionat bearing No. 1 and decrease the reaction at bearingNo, 2. As weardown proceeds the reaction at No. 1 andNo. 2 will balance out; and after total weardown, thereaction at No. 1 will be greater than that at No. 2.For shaft systems 10.00 to 16.00 inches in diameterwith span length to diameter ratios of 18 and systems16.00 to 30.00 inches in diameter with span length todiameter ratios of 16, the final unbalance will be


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approximately the tabulated limit and all other require-ments will be satisfied.

6.2 Five Span Systems

The evaluation of several 5 span systems showed nofurther reduction of minimum span below that requiredin the 4 span systems was possible. The influence ofadjacent bearings on one another remains nearly constantwhen the number of spans is increased above three. Basedon this, it was deduced that the minimum span length for4 spans is applicable for all systems having a largernumber of spans.

6 . Summary

These results are summarized in Table IV for systemswith up to ten bearing spans,

6.5 Application of Results

The values in Table IV show only the minimum overalllength for constant diameter shafting with equal spanlengths . It is also applicable for use in systems withvarying diameters and unequally spaced bearings. Thediameter of the lineshaft will normally be smaller than


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that of the tailshaft since the stress due to bendingis lower in the lineshaft region. The shaft should beconsidered in two sections, and the applicable lengthto diameter ratio is obtained from Table IV by enteringwith the appropriate diameter. On the other hand, itmay be necessary to use unequal bearing spans becauseof obstructions, bulkheads or framing. The only re-quirement is that the shortest span should have a lengthto diameter ratio equal to or greater than the value ofTable IV, for the diameter used. Additionally thevalues of Table IV may be used in conjunction withhollow shafting by entering with an equivalent diameter,

tD , whenD (1-n ) D

Table IV shows that before the transition from a twospan system to one with three spans is possible j aminimum overall length of 36 and 42 shaft diameters isneeded for the upper and lower diameter ranges respectively.This means span lengths of 18 and 21 shaft diameters wouldhave to be used in the two span system. These may appearto be large when compared with present practices. How-ever, studies by the authors for these particular systemsindicated no difficulty from the standpoint of strength,vibration or bearing load, even with spans of 20 diametersfor the upper range, and 22 diameters for the lower range


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of shaft diameter. These are only guides for maximumallowable span lengths, and not hard fast rules. Eachsystem is different and must be analyzed in light ofthe designers strength and vibration requirements.

It should be kept in mind that incorporated in theresults are the specific limits of loading and geargeometry used by the authors. An attempt was made touse values which would approximate actual system para-meters, thereby making the results directly applicableto the majority of systems encountered. There is al-ways the exception when the foregoing results wouldhave to be modified. For example, all other parametersbeing equal, a shorter distance between gear bearingswould indicate a larger value for minimum span length.The authors feel that in most cases use of the valuesin Table IV will result in satisfactory operation inall res


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TABLE IVMINIMUM SHAFT AND SPAN LENGTH-DIAMETER RATIOS

D 10.00 to 16.00 InchesBrg.Spans 2545678 9 10Lms/D 18 14 14 14 14 14 14 14 141^ /D 36 42 56 70 84 98 112 126 140

D = 16,00+ to 30.00 InchesBrg.Spans 2 3 4 5 6 7 8 9 10WD 16 12 12 12 12 12 12 12 12Lmc/D 32 36 48 60 72 84 96 108 120

L - Minimum Span Length

L^ = Minimum Overall distance between the mid-length points of the after gear bearingand the aft stern tube bearing, for con-stant diameter shafts.


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7.0 STERN TUBE BEARINGS.7.1 Advantage of One Stem Tube Bearing

The reader perhaps has questioned that weardown of onlyone bearing was considered. It is granted that in multiplescrew arrangements with water-lubricated, intermediate,strut bearings, this would not be the case, The considera-tion of weardown at only one bearing does not invalidatethe general requirements for minimum spans for such asystem. On the other hand, it is proposed for all arrange-ments that the bearing adjacent to the stern seal be aninternal oil-lubricated bearing. There are a number ofvery strong arguments in favor of making the change fromthe water-lubricated bearing used in present practice.

1) The use of a non-weardown bearingaffords more positive control in positioningthe shaft relative to the stern seal. Sinceno weardown takes place, the relationshipwith the stern seal is maintained duringoperation. This would alleviate many pro-blems which the stern seal is noted for andwould also increase the life of the seal.

2) Locating this bearing within the ship,as opposed to within the stern tube, permits


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the use of a shorter stern tube. Thisis a consideration that arises in apply-ing the minimum span criteria.

3) The use of an oil -lubricatedbearing permits higher unit pressureand thus higher bearing loads withoutadverse effects.

4) For the lifetime of the shipmaintenance costs for an oil lubricatedbearing would be less, since bearingsurfaces would not have to be replaced.

This arrangement in a single screw ship would result inonly one stern tube bearing and was the reason weardownof only one bearing was considered. The evaluation studiesshowed this to be a feasible arrangement from the standpointof bearing loads and certainly should result in better sternseal operation.

7.2 Location of Non-Weardown Bearing

This bearing should be located in the fore-and-aftdirection as close to the stern seal as possible. Toaccomplish this it would be advantageous to have the tail-shaft to lineshaft flange forward of the bearing. During


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installation, it might be necessary to furnish temporarysupports for the lineshaft. Additional thought must begiven to removal of the tailshaft during routine dockings.This probably would be critical in the case of the two spansystem where clearing of the reduction gear might be dif-ficult unless sufficient axial distance is allowed forpulling the tailshaft.

It is the opinion of the authors that systems in-corporating the above arrangement will show a markedimprovement in stern seal operation for the life of the


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8.0 LCW FREQUENCY CYCLIC STRESSES8 o 1 Background

The problem of fatigue failure in way of the tailshaftkeyway has aroused considerable interest in recent years,and a definitive study of the problem was reported inreference (7). In that report the consideration giventhe use of stress concentration factors in shafting designwas of particular interest to the authors It is theirfeeling that a great majority of tailshaft' failures maybe traced to the non-application of these factors in theinitial design, notwithstanding the effects of corrosiveatmosphere, fretting, etc. Furthermore it is felt thatuse of the concentration factors with so-called steadystresses is justified from the standpoint of low frequencyfluctuations of the steady stresses.

The stress pattern in the tailshaft near the keywayis in general the result of steady shear, steady com-pressive, alternating shear, and alternating bending stresses Steady shear (S ) and steady compressive (S ) arecaused by mean shaft torque and thrust respectively. Al-ternating shear stress (S__) is the result of torquesavariations about the mean torque. Alternating bendingstress (S,) is caused by the lateral loading of the pro-


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peller and shaft plus any applied moment . The procedureat the present time is to combine the steady stress com-ponents using maximum shear theory to obtain the resultantsteady stress (S ) . Then in a similar fashion the re-sultant alternating stress (S ) is calculated. No con-rasideration is given to the phase of the alternating com-ponents, since it is assumed they will be in phaseperiodically. The resultant stresses are then correlatedto achieve a factor of utilization, or factor of safety,through use of the following relationship (8)s

S Suy; TT7 T E.L. P.S.or


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applied . It has been shown in laboratory tests thatthe presence of a stress raiser has only a minimaleffect on the level of continuously applied steadystress that will cause failure. On the other hand,when an alternating stress is applied, the level ofapplied stress for failure is significantly reducedin specimens with stress raisers.

The theoretical concept for this phenomenonrelates ability to carry an applied load to the dis-tribution of stress around the stress raiser . Forsteady stress the magnification of applied stressis believed to cause some localized yielding in theregion of the stress raiser. Because of the yieldinga redistribution of the stress pattern occurs, ef-fectively reducing the maximum level of stress in thepattern (9). The applied load can then be carried eventhough some yielding may occur. In the case of alter-nating stresses only highly localized yielding takesplace. An associated high stress level results. Underrepeated load application fatigw failure will occur.Equation 19 takes these effects into account by incor-porating the stress concentration factors with thealternating stresses. Its application has proven satis-factory in the design of many power transmission systemswhich operate at a continuous level of steady stress with


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a superimposed alternating stress. At the present timethis is the procedure used as strength criteria by theU, S. Navy to determine shaft diameter (10). Furthermore,many merchant designs incorporate a similar criteria as acheck on the adequancy of the required diameters specifiedby the classification societies.

The application of the foregoing procedure to shipshafting, however, does not always provide a fail-safesystem. In particular focus attention on the propellerkeyway. If it is assumed that the keyway is well designedand follows an easy taper in the axial direction, a stressconcentration factor in torsion only would be applied tothe alternating shear stress component. Theoretically,and within the accuracy of predicted values for the steadymean stress and alternating stress, a tailshaft with a de-finite factor of safety should be the result of applyingequation 20, Yet many tailshafts designed for strengthin this manner suffer fatigue crack failures in the regionof the keyway.

8.3 Steady Stress Reversal Theory

It is suggested that these failures are the resultof reversals of the large steady stresses. This is afactor which is not accounted for in the present procedure.Of course fluctuations of the steady stresses occurs in any


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power transmission system each time it is started andstopped. However, there are few systems operating ata comparable load level, that are started ahead,stopped, backed down and started ahead again as oftenas ships' shafting. It is theorized that each time aship is started, stopped, reversed, etc., constitutesreversal of the supposedly steady stresses. Thus thesteady stresses actually must be considered as alter-nating stresses of low cyclic frequency. It is hypothe-sized that stress concentration factors must be appliedto the steady stresses to eliminate fatigue failuresbecause of this.

The graphical representation of the above remarksmight illustrate the hypothesis more clearly. Figure 8(a)is an assumed fatigue curve for a metal similar to class2 steel. The endurance limit shown corresponds to thelevel of mean stress applied. Superimposed on this curveis a plot of the stress levels considered in the directapplication of Equation 20. It is seen the maximum appliedstress is equal to the sum of the continuously appliedsteady stress plus the alternating stress. It must be re-membered that the fatigue curve is not an analytical curve,but is the result of experimental data. The level of theendurance limit is a function of the magnitude of theapplied mean stress. Figure 8(b) illustrates the same

-4?-


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00

SQfUndWV SSBdlS 9NllVNa3J.1V sanindwv sssais ONiivNasnv

t 1 rSS381S XVW

_o

o

SS3UJ.S XVw-4^a-


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stress conditions as represented on a modified Goodmandiagram and is the graphical representation of Equation 19.

The effect of low frequency variations of the steadystresses is similarly shown in Figures 8(c) and (d). Aship operates more often in the ahead condition than inthe astern so there will be some mean stress level aboutwhich the steady stress fluctuates. The value of meanstress has been arbitrarily selected in the figures. Ifstress concentration factors are applied to the portionof steady stress in excess of the mean value, thefluctuating curve shown in Figure 8(c) is obtained. Theoriginal alternating stress is still present and must beincluded causing a further increase in the magnitude ofthe maximum applied stress. The cumulative effect of thefluctuating steady stress and the alternating stress mayresult in a maximum applied stress in excess of the en-durance limit . Rather than having a system with an in-definite life, the designer may find his system has adefinite predictable life span. A fatigue failure willoccur as soon as sufficient starts and stops, or cyclesof steady stress, have taken place.

Depending upon the magnitude of the maximum appliedstress, fatigue failure conceivably could occur after afew hundred or more cycles. It is not difficult tovisualize a ship's shafting system accumulating this manysteady stress reversals in a few years operation. Hence,

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an unpredicted early failure due to fatigue mayoccur.

8,4 Application in Design

The most difficult problem in the attempt to in-corporate the foregoing in the present design procedureis the determination of a mean steady stress level aboutwhich the steady stress will vary. A statistical studyof a number of ship's bell books coupled with a shaftstress analysis could provide the answer. Once the meanstress level has been determined, the appropriate stressconcentration factors can be applied, and the shaft maybe designed so as to avoid fatigue failure during thelife of the ship.

Summarizing, it is hypothesized that the factor ofsafety, a better term would be factor of utilization,predicted by the presently used working stress equationmay be in error, since it is not based on a considerationof the low frequency variations of steady stress. Thisshould not be construed as questioning the validity ofthe equation but rather the validity of the term steadystress.

It is theorised that this error can be corrected byapplying stress concentration factors to the fluctuatingsteady stresses. This results in the working stressequation taking the form


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1/

\| (i-p)(s^) /\)[(kb )(sb+psc )]% ^kt )(ssatps3 )] 2(21) fTST " Yl\ + ETTT^ ~~

(Maximum Steady Stress) - (Mean Value of Steady Stress)p " (Maximum Steady Stress)

This equation will require some increase in diameterover that specified by the presently used equation 20 toachieve the same factor of safety. Through the use of thismodified form of the equation a designer should be able toprovide, barring other effects, shafting with a moreaccurately predicted service life.

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9.0 SIZING OF THE TAILSHAFT9.1 Present Procedure

The maximum stress level in a shaft system can beexpected In the region of the tailshaft. This is a re-sult of the large cantilevered propeller weight intro-ducing bending stresses. Thus in propulsion shaft designthe first step normally involves the selection of anadequate tailshaft diameter. At the present time the de-signer must estimate an initial diameter or compute aminimum required diameter by use of a classification rule.A strength calculation based on that estimate is made toensure that the cross section can carry the expected loadwith some factor of safety. If the Initial estimate wasgood and the desired factor of safety is obtained no .further calculation Is required. This is not always theease and several trials may be required.

It is possible to carry out an analytical solutionfor shaft diameter using Equation 20 and the relationshipsfor the steady and alternating stresses as a function ofdiameter and the applied load. Unfortunately, such adirect calculation requires the solution of a sixth orderpolynomial In diameter (D) which is little improvementover the trial and error procedure. A simple solution of

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the polynomial is available through use of nomographictechniques . In this manner a direct solution is avail-able.

9.2 Development of Design Nomograms

In the development of the nomograms it was assumedthat the designer would have the following two sets ofparameters available. The first are characteristicsspecified by the particular hull and ship's speed.

1. Shaft Horsepower2. Shaft RPM3. Propeller Weight4. Propeller Thrust5. Propeller Overhang

The second set of parameters are design variables and aredependent upon material used and the designer's criteriaand experience.

1. Endurance limit of material2. Yield point of material3. Ratio of inside to outsidediameter4. Stress concentration factorfor bending5. Stress concentration factorfor torsion6. Percentage of steady mean torquewhich is equivalent to alter-nating torque

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7. Desired factor of safety8. Dynamic factor of safety toaccount for inertia loadingand eccentricity of thrust.

With these known parameters, the stresses in thetailshaft can be expressed as functions of the requireddiameter

Steady Shear stress,

(22 ) Sfl . 521,000 (SHP)S (1-iT) D^(RPM)

Steady Compressive stress.

(25) s - lf$$1-n )DAlternating Shear stress,

(24) S ots = K. (S


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(26) D6 - N(D5 ) - L(D2 ) + M - 0.0where

N 2(F.S."TFT(1-n )(E.L.) (F.S, d )W L )2 + (642,000 K^SHP/RPM) 1

-i21.273 T(F.S.)(1-n )(Y.P.)

M N< 642,000(SHP)(F.S.)(l-n*)(RPM)(Y P o )

Figure A-6 is the graphical solution of Equation 26. Toenter the figure values of L, M, and N must be known. Theycan be calculated directly $ however ae & time saver, theperipheral diagrams Figures A-l to A-5 have been developedto facilitate a simple graphical solution.

In the development of the diagrams no limitations asfar as possible combinations of parameters were specifiedwith the exception of Figure A-6. For Figure A-6 an upperlimit was placed on the values of M and N. This was donesince it is inconceivable that a material with the lowestvalue of yield point would be used with a combination ofthe highest SHP, propeller weight, etc.

It should be noted that the diagrams as developeddo not include a consideration of the effects of steady

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stress fluctuations . It would have been most desirableto have included this effect. The authors decided inthe absence of an accurate estimate of the mean steadystress level not to include it in the preliminary de-sign stage. It was the opinion of the authors that in-clusion of this effect should be in the form of a checkon the adequacy of the design. In which case, thestress components can be calculated using the diameterselected. Then using Equation 21 and the designer'sestimate of the mean steady stress level a factor ofsafety can be calculated, A comparison of this factorof safety with the intended factor used in the selectionof the diameter would indicate the adequacy of the de-sign. In this manner consideration is given to bothhigh and low frequency cyclic stresses.


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10.0 CONCLUSIONS AND RECOMMENDATIONS

The power transmission system for a ship is an im-portant, integral part of the propulsion plant. As suchit requires a comprehensive design procedure to ensurean optimum trouble-free system. To obtain a good designthe shafting, bearings, reduction gear, and propellermust be considered as a single integrated unit. Specifi-cally, a recommended procedure would include:

lo The integrated system should be treated as acontinuous beam carrying both distributed and concen-trated loads and carried by point supports at the bear-ing locations. Using this arrangement a solution ofthe continuous beam problem should be obtained withparticular attention to support reactions, deflectionsand bending moments.

2. Minimum span lengths , or maximum number ofsupport bearings, should be selected on the basis ofinsensitivity to initial misalignment errors and wear-down of the water-lubricated bearings. Any method usedto .judge the degree of insensitivity should consider theeffects of misalignment on allowable bearing pressuresand change in reactions at the reduction gear bearings.

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Table IV can be used as a guide in span length selection.It is recommended that for a given shaft diameter andover-all length, the values listed in Table IV be consideredas minimum span lengths.

5. In general each design should include a com-prehensive study of strength and vibration characteristics.Both required shaft diameter and maximum span length areset by strength and vibration requirements . Maximum spanlengths of 20 to 22 diameters are possible. However , eachdesign must be checked t ensure that critical whirlingfrequency criteria are met if spans of this length areused. Required tailshaft diameter can be obtained throughan application of equation (20) or through use of the pro-cedure of Appendix A.

4. In connection with required shaft diameter, con-sideration should be given to the adverse effect of lowfrequency fluctuations of the supposedly steady stresses.To aid such a consideration, it is recommended that astatistical study be undertaken to facilitate predictionof the mean level of steady stress. The study could con-sist of a survey of the bell books of several types ofships now in operation. A quantitative summary of aheadand astern engine orders would then be available from whichpredicted levels of mean stress could be derived. With thif


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information it would be possible to correlate endurancelimit, low cycle stress fluctuations and maximum appliedstress.

5. Consideration should be given to the installationof a non-weardown bearing adjacent to the stern seal. Thisbearing should replace the presently used water-lubricatedbearing. In this manner more definite support in the sealarea will be provided.

6, The theoretical investigation for minimum spanlengths revealed that extra difficulties may be expectedwith two span systems. These close coupled systems shouldbe avoided whenever possible. It would be desirable touse overall system lengths suitable for three spans in theinterest of design simplicity. Whenever two span systemsmust be used, it will be necessary to specify an optimumsystem alignment as well as minimum span length to getthe most desirable shafting system.

If the recommended design procedure is carried outit should be possible to design shaft systems which willprovide optimum operating characteristics. They will beimmune to excessive gear tooth wear, stern seal difficulties,and support bearing problems. In addition some improvementshould be obtained as far as tailshaft liner and fatiguefailures are concerned.


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REFERENCES

(1) H. C= Anderson and JJ, Zrodowski, "Co-ordinatedAlignment of Line Shaft, Propulsion Gear andTurbines", Transactions SNAME, I960.(2) H No Pemberton and G. P. Smedley, "An Analysis ofRecent Screwshaft Casualties", NEC Institutionof Engineers and Shipbuilders, Vol, 76,Part 6, April i960.(3) R. E. Koshiba, J. J, Francis and R.A. Woolacott,"The Alignment of Main Propulsion Shaft Bear-ings", SNAME, New England Section, January1956.(4) E. T. Antkowiak, "Calculation of Ship PropulsionShafting Bearing Reactions for IBM-65p Computer"Boston Naval Shipyard, Development ReportNo. R-ll, October 1957(5) J. M. Labbert-on and L. S, Marks, "Marine Engineers

Handbook", McQraw Hill, New York, 19^5(6) G. Mann and W Do Markle, "Propulsion ShaftingArrangement", NavE. Thesis, Massachusetts In-stitute of Technology, 196Q.(7) No Ho Jasper and L. A Rupp, "An Experimental andTheoretical Investigation of Propeller Shaft-Failures", Transactions SNAME, 1952.(8) Co R. Soderberg, "Factor of Safety and Working

Stress", APM-S2-2, 1928(9) Ho J. Grover, S, A Gordon and L. R. Jackson,"Fatigue of Metals and Structures", NAVER 00-25-534, U.S. Government Printing Office, 195*1.(10) U.S. Navy Bureau of Ships, "Propulsion Shafting",DDS 4301, 1 May 1957.(11) E Panagopulos, "Design-Stage Calculations ofTorsional, Axial, and Lateral Vibrations of

Marine Shafting", Transactions SNAME, 1950

;r-


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(12) H. L. Seward, "Marine Engineering, Vol. I",Society of Naval Architects and Marine En-gineers, New York, 1942.(13) "Rules for the Classification and Constructionof Steel Vessels", American Bureau of Shipping,New York, 1958.(14) H. C. Anderson, D. E.Bethune and E. H. Sibley,"Shafting System Programs MGE-402, Programmedfor an IBM-704, DF59MSD-202, General ElectricCompany, April 8, i960.

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APPENDIX

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APPENDIX ARECOMMENDED PRELIMINARY DESIGN PROCEDURE

Set up the following tabular form:

Hull Parameters1. Shaft horsepower2 RPM ftftftOOft3. Propeller Weight (W )4 Thrust (T ) . ftooftoooo5. Propeller Overhang (L ) .

XT

6. Overall length (L ) ft

00009000 o

o ft a o o

ooooo9 6 O ft

000

Design Parametersj. Diameter ra^io \yi ) oo............oo2. Percent Steady Torque (K.,) .5. Stress Concentration bending (ic)4. Stress Concentration torsion (Kt )5. Yield Point of Material (Y.P.)6. Endurance Limit of Material in Air(E.L.)7. Dynamic Factor of Safety desired8. Factor of Safety desired .......

ft o

ft O ft O

O ft o o o

ft O ft

O O O ft

hp.ft ft RPM

lbs.ft ft ft lbs.ft o o ins.00 , ins

jpsipsi

Enter Figure A-l with (Y.P.) and (n) connect and markthe V-Soale, then connect (T) and (F.S.) and markU-Scale. Connect V and U and read L-Scale. L


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Enter Figure A-2 with (K.) and SHP connectand mark intersection on U-Scale. ConnectU with (Kn) entry and mark intersection onSHP scale. Connect this point with the RPMentry and read A-Scale. A Enter Figure A-5 with (K^) and F.S. d )connect and mark the U-Scale, Connectthis point with (L ) and mark intersectionon V-Scale. Connect this point with (W )and read B-Scale. B Enter Figure A-4 with A and B, from inter-section of A and B follow circle to left-hand vertical scale. Connect this pointwith (F.S.) and mark intersection onS-Seale. Connect S with (E.L.) and readC-Scale. C Enter Figure A-5 with (Y.P.) and RPMconnect and mark S -Scale. Connect(F,S.) and SHP and mark T-Scale. Con-

*nect T and S Scale and read D Scale. D

* This is a dummy variable and shouldnot be confused with Diameter (D).

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Perfor

1.

m the following

4nd-n4 ) -N - C/(l-n4 )N2/4 =

D

calculations:

2. a

3. 9

4. K

5. 10(l-n4 )E2

M m N2/4 - E2

6.

7.

N

M

Enter Figure A-6 with L and N and con-nect with a straight line. Locate theintersection of this line with thevalue of M. Follow the vertical linefrom this point and read the valuefor Tailshaft diameter. DiameterDivide L by the Diameter* L/D Enter Table IV of the paper in theappropriate diameter range and de-termine the number of bearing spansthat can be used. This is only afirst approximation, sinoe the de-signer will want to adjust the line-

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shaft diameter to meet his own criteria ofstrength and vibration. However, the re-quirements for minimum span must still bemet for the diameter used.

The above procedure did not consider the low fre-quency cyclic stresses. If the designer wishes toincorporate this consideration he may do so by useof Equation 21.

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14

13

124ii 410

9

8

7 -E

6

544

3

2-=

I

THRUSTFACTOR OF SAFETTYIELD POINT10 TO 00 RATIO

[I 27 3(FS.)f

[[YP)(I-N ZJ

v.X'-*

3040bO-60

^70= t-- zf80E" Q|-90 d- >-

-100

iE

-60a-


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642,000 SHP(KTK,) x 10'*


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25 ^>a*wm

ft o

000

n

AoOo

R,

R.n

If we now take the inverse of the coefficient matrixand multiply both sides of the above matrix equationgives:

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Letting a., represent the elements of the inversematrix

A,

0,

R,

\

alla12a13al4a]a2la22a25a24a25a51a52a55a34a35L4la42a45a44a45 *

a o o ooo9

oooWata2ate3ata4ato5

lmO

OOOa2ma3ma4m

O o

o o

O O mm

4o


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This means that assuming no external forces orweight

ri - 4*5i + 4^2 + 5*55 . . . + 4%Oo - ^21 + 4a22 + cT5a23 + 4a2n

The remainder of the program write-up of the two re-ferences is correct.

Statistical data, as obtained from the computer calculation, for the cases studied is on file with theDepartment of Naval Architecture.

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APPENDIX CSELECTION OF SHAFT SYSTEMS FOR STUDY

1. It was necessary to synthesize groups of typicalshaft system components from statistical data. If line-shaft and tailshaft diameter, shaft horsepower, and RPMare known it is possible to use the information inreference (6) for this purpose. The four referenceparameters were selected in the following manner,

Lineshaft and Tailshaft Diameters - A series ofshaft diameters, ranging from 10 to 30 inches inclusive,was arbitrarily specified. In order to reduce the numberof cases to be studied and to simplify the study process,it was assumed that both the tailshaft and lineshaftwould be of the same diameter. This assumption leads toan inefficient design in any real design problem sincethe lineshaft will generally be subjected to smallerstress levels than the tailshaft. It is possible touse smaller lineshaft diameters than tailshaft dia-meters because of this. However by choosing a singleoverall diameter which is strong enough to carry theapplied loads on the tailshaft, a more conservativeoverall design is specified. Furthermore, since anyalignment criteria is dependent on shaft stiffness and

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bearing loading, a larger diameter will tend to makepredicted values of minimum span lengths somewhatlarger than those actually required. One of the ob-jectives of the thesis study was the establishmentof a minimum allowable span criteria. In this respectthe use of larger diameters will result in a slightlyconservative criteria. Solid shafts only were studiedin the interest of simplicity.

RPM Selection - Values of RPM were arbitrarilychosen within the range of values found in presentday ships.

Shaft Horsepower - For each combination of RPMand diameter a corresponding value for SHP was calcu-lated. The calculation was made through an applicationof the following empirical formula from reference (13).

3D o 95 64 SHPD m u.Sfc RpM

or

In this manner some measure of correlation was achievedbetween the entering parameters for use with the statisti-cal data. Thus it was possible to obtain dimensions fora propeller and reduction gear which were compatiblewith each other and the assumed shaft diameter.


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2. Once shaft diameter, RPM and SHP were known thefollowing specific information was obtained for eachshaft system.

Reduction Gear Dimensionsa. Gear shaft diameter - D gsb. Gear weight - Wgc. Equivalent gear diameter - D.

This is the diameter of a solid steel shaft whichhas a stiffness, in bending, equal to the stiffnessof the reduction gear unit

d. Length between gear support bearings - L,e. Length of bull gear face - Lff Concentrated gear weight - W

When an equivalent gear diameter is used to replacethe stiffness of the reduction gear, the equivalentshaft will have a smaller weight than the total weightof the gear unit . This difference in weight can becalculated and added to the equivalent shaft as a con-centrated weight.

AWg - wg - (L)D2 Lfp = Density of Steel,

lbs./in.


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Propeller Dimensionsa. Propeller weight - Wb. Propeller overhang - L

This is the distance from the center of gravity ofthe propeller to the shaft support point in the afterstern tube bearing. The support point was assumed oneshaft diameter forward of the outer extremity of thestern bearing,

c Mass moment of inertia factor for the2propeller about its axis - W rPAs noted in reference (12), the mass moment of in-

ertia of a propeller can be approximated by the formulaW r2

I = ; r = radius of gyration ofs the propeller

3. With the above data a series of representative shaftsystems was available for study. The study system para-meters are shown in Table III . It should be noted thateach of the final synthesized systems was checked inaccordance with the strength criteria of Appendix D.This was done to ensure the adequacy of the arbitraryshaft diameters in light of the corresponding propellerand reduction gear dimensions . The only unknown informationstill to be determined for each system is allowable spanlengths


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NOMENCLATURE PECULIAR TO APPENDIX D

2A Shaft cross-section area ~w~> lnE - Modulus of Elasticity - 29 x 10 , psi.f = Frequency of Lateral Vibration, epm.g - Acceleration due to gravity = 586 - *l6m - secI Moment of inertia of shaft in bendingI = Mass moment of inertia of propeller about

a diameter (increased 60$ for entrainedwater) ^ r21,0 p2g

7TD 4J Polar moment of inertia of shaft ^o"-* in L = Length between tailshaft support points, inches.L = Length of lineshaft spans, inches.Mt - Torsional Moment - 22ig|2 ^12 x SHP , m.-lbs.Mma:K= Bending Moment, in-lbsm Propeller mass (increased J>0%> for entrainedWwater) 1.3 _m mi,w , f 33,000 x SHP x p.c.T Thrust - ^6l.34 V^s (1-t)p.c. = Propulsive coefficient.


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t = Thrust deduction.

VKTS" sPeed in Knots.SHP Shaft horsepower.

ttDw Weight of shaft per inch pjr-2ST= Shaft mass per Inch - J

p Density of steel 0,282 lbs ./in. ,

-70-


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APPENDIX DSTRENGTH AND VIBRATION REQUIREMENTS

1. The usual design process is concerned with theprovision of adequate shaft diameters for requiredstrength, and limits on the maximum length betweensupports to preclude the existence of vibrationcriticals in the range of operating RPM. An appli-cation of a strength and vibration criteria such asthat outlined in reference (10) will satisfy these re-quirements. In the development of a minimum spancriteria consideration of strength and vibration re-quirements do not enter directly. However a check hadto be made on the compatibility of maximum and minimumspan criteria; i.e., the minimum span must not be great-er than the maximum allowable span required by strengthand vibration considerations.

It was also necessary to make a direct shaft strengthcalculation for each of the synthesized study systems.This was done t6 ensure a large enough shaft crosssection to carry the loads of the various components.

2. The maximum bending stress occurs at the supportpoint in the after stern tube bearing. It is caused by


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the large overhung propeller weight and the effectof thrust eccentricities . All of the other basicstresses are common to the entire shaft length. Thusa shaft cross-section of sufficient size to carry thestresses at the after support point should be adequatefor the remainder of the system. For each of thesynthesized shaft systems, equation (20) was appliedat that support point to check the adequacy of theshaft diameter. To apply equation (20) it was necessaryto compute values for the various steady and alternatingcomponents

Steady Shear Stress

Sor

s 2J

dd) s s . **$&)Steady Compressive Stress

S - ^c AThe following parameters were assumed for all

cases i

Propulsive coefficient, p.c. 0.65Speed in knots, V, ._ 20Thrust deduction, t - o 2


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or(2d) S - 16.85 (2)(|i)V

Alternating Shear Stress

(3d) ssa - 0.05 ss - 1 ' 61^ 10^gg)In all cases it was assumed that the alternating

component of shear stress would be equivalent to 5$of the steady component

Alternating Bending Stressq HiOj.lb Dbb " 21

y

For the after support point it was assumed thatM was made up of the following parts

W L = Moment caused by propeller overhangC XT2 2

p u P Moment caused by shaft overhang

M = 2W L 9 additional moment from thrustoc p p eceenticityor 2 2

sb - ^t^plp + pV1-)For all eases the following data, with reference to

stress concentration factors and type of shaft material,


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was specified:

Class Bs steel:Yield Point, YP. - 40,000 psiFatigue Limit, F.L, 54,000 psi

Stress concentration factors

Bending, k, (at keyway) =1.0Torsion, k. (at keyway) 1.9

IJpon inserting, in equation (20), the values computedfrom the above equations and assumptions, a safetyfactor for each of the basic study systems was computed.A shaft diameter giving a safety factor of approximately2 was considered satisfactory. The results of thesecalculations are listed in Table V

3 Allowable maximum tailShaft lengths were estimatedthrough application of the following equation (11) forcalculating frequency of lateral vibrations.

30\ / 11f * tH/ T o L T i/f L I? 7T 4\ Ix (Lp + ) + rnl^ + ) + *(- +^ +^or upon rearranging

r 3-1900EI L

2 5r* Hlff 8


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The above equation was solved for length, L, foreach study system via a trial and error method. Theseresults are tabulated In Table V,

4, In the lineshaft region it was possible to calcu-late a maximum allowable span based on strength require-ments for the given shaft diameter. Equation (20) canagain be applied Values for the steady shear, steadycompressive., and alternating stresses as calculated inthe shaft sizing procedure can be used directly. How-ever It Is necessary to recalculate a value for thealternating bending stress,, The bending stress in thelineshaft was calculated by assuming that each line-shaft span acts like a built-in beam carrying a uni-formly distributed load. The accuracy of this assumptionwas verified in several instances. It was found thatan actual shaft span has a bending moment, at the shaftsupports, within +8$ of that predicted through appli-cation of the built-in

Documents

A Method for the Dbign of Ship' Propulsion Shaft Systems. William e Lehr, Jr. Edwin l Parker