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Noah D. ManringMechanical and Aerospace Engineering,

University of Missouri–Columbia,Columbia, MO 65211

Designing the Shaft Diameterfor Acceptable Levels of StressWithin an Axial-PistonSwash-Plate Type HydrostaticPumpIn this research, the diameter of the shaft within an axial-piston swash-plate type hystatic pump is considered from a stress point of view. To analyze the loading of thethe components within the pump are studied using a force and torque diagram andshown that the loads are applied differently for three main sections of the shaft. Fromforce and torque diagram, the actual shaft loads are determined based upon the geoof the pump and the working pressure of the hydraulic system. Using well-accemachine design practices, governing equations for the shaft diameter are producethe various regions of loading along the shaft. These equations consider both bendintorsional stresses on the outer surface of the shaft. Results for the required shaft diaare then computed for a typical pump design and compared to the geometry of an ashaft. It is noted that stress concentrations can significantly alter these results and threquired shaft diameter can be reduced by applying the proper heat treatmentsincreasing the shaft strength. Finally, the designer is cautioned regarding the defledifficulties that can arise when the shaft diameter is reduced too much.@S1050-0472~00!01704-9#

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Introduction

Background. Axial piston machinery is used in fluid poweapplications for converting mechanical power to hydraulic powand visa versa. A most striking attribute of axial-piston hydstatic machinery has been its ability to convert large amountpower using relatively small components. This attribute is referto as power density and can best be illustrated by comparingsize of other machinery of similar power ratings to the size ofaxial-piston machine. For example, consider the size of a diengine which is used to convert chemical energy into mechanenergy, and the size of an electric motor which converts electrenergy into mechanical energy. Table 1 is generated to comthe power and size of these devices to an axial-piston puwhich converts mechanical energy into hydraulic energy. In ttable, the power rating is self explanatory, the volume refers tophysical space that the device takes up on a shop floor, andpower density is the ratio of output power to the volume. Anothway of considering the same information is to say, for machiof similar size, the piston pump transmits 27 times as much poas the diesel engine and 13 times as much power as an AC mThis comparison illustrates one of several motivations for ushydraulic power versus diesel or electrical power. While tpower density of hydrostatic equipment is noted as a posiattribute of these machines, it is often accompanied with msevere loading and higher stresses on internal parts than maexpected for other types of machinery. Since the input shaft ofaxial-piston pump is used to transmit the power between mechcal and fluid-power generating components, it is expected thashaft will undergo significant stresses during the normal operaof the machine.

From a design perspective, the shaft of the axial-piston mac

Contributed by the Stress Analysis & Failure Prevention Committee for publtion in the JOURNAL OF MECHANICAL DESIGN. Manuscript received Jul. 1999Associate Technical Editor: J. Gao.

Copyright © 2Journal of Mechanical Design

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is obviously central to the other working parts of the machinThat is to say that, the other machine parts are designed arothe shaft and that various machine dimensions are determbased upon the diameter of the shaft itself. For instance, onceshaft diameter is determined, the minimum piston pitch-circlealso roughly determined. Once the piston pitch-circle is detmined, the piston diameters themselves are also sized to tranthe specified amount of fluid flow. The piston diameters are thused to determine aspects of the slipper design and the locaand width of the ports within the head of the machine. All of this to say that, the shaft diameter must be properly designed awith the rest of the machine; because, if it is not, then any neechanges to the shaft diameter that manifest themselves aftemachine is designed and built will create necessary changes tof the other machine parts as well. In this event, the machdesign will have to be restarted from scratch. This scenario girise to the motivation for designing the shaft diameter propefrom an a-priori point of view.

Machine Description. The hydrostatic pump consists of several pistons within a common cylindrical block. The pistons anested in a circular array within the block at equal intervals abthe shaft centerline. As shown in Fig. 1, the cylinder block is hetightly against a port plate using the compressed force ofcylinder-block spring. A thin film of oil separates the port plafrom the cylinder block which, under normal operating conditionforms a hydrodynamic bearing between the two parts. The cy

ca-

Table 1 A comparison of power densities for three differentmachines

000 by ASME DECEMBER 2000, Vol. 122 Õ 553

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der block is connected to the shaft through a set of splines thatparallel to the shaft. A ball-and-socket joint connects the baseeach piston to a slipper. The slippers themselves are kept insonable contact with the swash plate by a retainer~not shown inFig. 1! where a hydrostatic and hydrodynamic bearing surfaseparates the slippers from the swash plate. The swash plate aa, may generally vary with time; however, for this research, tswash plate is considered to be held at a fixed angle which madiscretely adjusted for steady-state considerations. The entirechine is contained within a housing that is usually filled wihydraulic fluid.

While the port plate is held in a fixed position against the hethe shaft and the cylinder block rotate about the shaft centerlina constant angular speed,v. The reader should recall that the shaand cylinder block are connected through a set of splines thatparallel to the shaft. If the machine is operating as a pump,shaft drives the cylinder block. If the machine is operating amotor, the cylinder block drives the shaft. During this rotationmotion, each piston periodically passes over the dischargeintake ports on the port plate. Furthermore, because the slipare held against the inclined plane of the swash plate, the pisalso undergo an oscillatory displacement in and out of the cylinblock. As the pistons pass over the intake port, the piston wdraws from the cylinder block and fluid is drawn into the pistochamber. As the pistons pass over the discharge port, the padvances into the cylinder block and fluid is pushed out ofpiston chamber. This motion repeats itself for each revolutionthe cylinder block and the basic task of displacing fluid is accoplished. The intake and discharge flow of the machine are distuted using the head component which functions as a manifolthe rest of the hydraulic system.

Objectives. In this research, the diameter of the shaft withan axial-piston swash-plate type hydrostatic pump is considefrom a stress point of view. To analyze the loading of the shthe components within the pump are studied using a force

Fig. 1 General pump configuration

554 Õ Vol. 122, DECEMBER 2000

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torque diagram and it is shown that the loads are applied difently for three main sections of the shaft. From the force atorque diagram, the actual shaft loads are determined basedthe geometry of the pump and the working pressure of the hydrlic system. Using well-accepted machine design practices@1–3#,governing equations for the shaft diameter are produced forvarious regions of loading along the shaft. These equations csider both bending and torsional stresses on the outer surfacthe shaft. Results for the required shaft diameter are then cputed for a typical pump design and are compared to the geomof an actual shaft.

Shaft Loading

Force and Torque Diagram. Figure 2 shows a sectionedview of a single piston within the cylinder block as it operatwithin the pump. In this view, the reaction force between tswash-plate and thenth slipper is shown by the symbol,Rn . Fromthe left hand side of Fig. 2, it can be seen that the componenthis force in the downward direction is given by

Fn5Rn sin~a!, (1)

wherea is the swash plate angle of the pump. Equation~1! rep-resents the downward force exerted on the shaft by thenth piston.Summing these forces for all pistons within the pump yields ttotal force exerted on the shaft in the downward direction. Tresult is given by

F5(n51

N

Rn sin~a!, (2)

where N is the total number of pistons within the pump. Thtorque on the shaft is generated by the downward componenthe reaction force between thenth piston and the swash platemultiplied by the distance of thenth piston away from thez-axis.From Fig. 2, it can be seen that the piston is located a distaaway from thez-axis by the expression,r cos(un), wherer is thepiston pitch radius andun is the circular position of thenth piston.Multiplying this distance by the right hand side of Eq.~1! yieldsthe following result for the torque exerted on the shaft by thenthpiston:

Tn5Rn sin~a!r cos~un!. (3)

Summing these torques for all pistons within the pump yieldsfollowing result for the total torque exerted on the shaft

T5(n51

N

Rn sin~a!r cos~un!. (4)

Note: Figure 3 serves to graphically illustrate the total downwaforce given by Eq.~2! and the total torque given by Eq.~4!. Figure3 will be discussed later.

Piston Reaction. The reaction between thenth piston and theswash plate may be determined by summing the forces whichacting on the piston in thex-direction and setting them equal t

Fig. 2 Force and torque diagram for loads exerted on the shaft

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the piston’s time rate-of-change of linear momentum. Writing tequation, and rearranging terms, yields the following result forreaction force between thenth piston and the swash plate:

Rn5M pxn1PnAp

cos~a!, (5)

whereM p is the mass of a single piston,xn is the piston’s accel-eration in thex-direction,Pn is the fluid pressure within thenthpiston chamber, andAp is the pressurized area of a single pisto

Piston Kinematics. From Fig. 2, it can be seen that the postion of the nth piston-slipper ball joint in thex-direction is de-scribed by the equation

xn5r tan~a!sin~un!. (6)

Differentiating this equation twice with respect to time yields tnth piston’s acceleration in thex-direction. This result is given by

xn52r tan~a!v2 sin~un!, (7)

wherev is the angular speed of the pump shaft. Note: in Eq.~7!the swash-plate angle,a, and the angular speed of the pump,v,are assumed to be constants.

Piston Pressure. A remaining quantity of interest for thisanalysis is the fluid pressure within thenth piston chamber whichis denoted by the symbol,Pn . To begin this sub-topic, it shouldbe noted that previous research has addressed the complexlem of evaluating the fluid pressure within a single piston cham@4#. In this research, a model for describing the fluid presswithin the nth piston chamber was presented; and, due tonon-analytical nature of the model, a numerical solution was gerated. In these studies, it was observed that the piston preremained essentially constant while the piston transversed direover one of the ports; but, when the piston crossed over fromport to another it would occasionally experience pressure spikethe transition zone. It was determined that these spikes rewhen the volumetric compression or expansion of the chamexceeds the flow capacity of the fluid, which either enters or ethe piston chamber depending upon the direction of flow potenIt was also determined that the pressure spikes could be enated by altering the geometry of the port-plate slotting oroperating at adequately high enough pressures. Since indtypically designs port plates to avoid creating pressure spikethe piston chamber, it is reasonable to assume that the presspike conditions rarely occur and that smooth transitions betwports are most commonly observed. If this is the case, thenpressure within thenth piston chamber may be graphically d

Fig. 3 Piston pressure profile

Journal of Mechanical Design

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scribed by the piston pressure profile which is shown in Fig. 3this figure, the pressure is shown by the outside profile aroundport plate. The downward reaction forces between the pistonsthe swash plate are graphically shown in Fig. 3 as well forinstantaneous position of the cylinder block. In Fig. 3 it canseen that the piston pressure remains constant over the intakedischarge ports of the pump~Pd or Pi! and that the piston pressure changes smoothly from one pressure to the other as the pcrosses top or bottom dead center. The average angular dispment over which the pressure transition occurs is shown byangular dimension,g. This dimension is commonly referred to athe pressure carry-over angle because it describes how farpressure is ‘‘carried over’’ at the top and bottom dead centeUsing Fig. 3, a discontinuous expression for the approximate psure within thenth piston chamber may be written as

Pn55Pd 2p/21g,un,p/2

Pd2m~un2p/2! p/2,un,p/21g

Pi p/21g,un,3p/2

Pi1m~un23p/2! 3p/2,un,3p/21g

, (8)

where m5(Pd2Pi)/g. The motive for presenting this result ithat it can be used analytically in this work. For a more accurprediction of this quantity, a numerical solution for the fluid presure within thenth piston chamber must be sought; and, treader is referred to previous research@4# for a suggested methodof solving the numerical problem.

Average Force and Torque. The general expressions for thdownward force and torque exerted on the shaft are given in E~2! and~4!. These equations describe the instantaneous loadsare exerted on the shaft which tend to oscillate at certain dominfrequencies depending upon the number of pistons withinpump and the rotational speed of the shaft. If the pump issigned with a sufficiently large number of pistons, the amplituof the oscillations can be reduced and the frequency of the olations can be increased. In general, it should be noted that, f9-piston pump, the amplitude of the oscillations which occurthe instantaneous downward-force are less than610% of the av-erage force while the amplitude of the oscillations which occurthe instantaneous torque are less than61.5% of the averagetorque. Again, the size of these amplitudes will vary dependupon the number of pistons which are used in the pump desigit is assumed that the oscillatory effects of the instantaneous fand torque are small compared to their average effects, thenoscillatory effects may be neglected and the average forcetorque may be used to consider the loading of the shaft. UsEqs.~2! and~4!, the average quantities of force and torque maycomputed using the integral-averaging technique. This technyields the following general forms:

F5N

2p E0

2p

Rn sin~a!dun ,

(9)

T5N

2p E0

2p

Rn sin~a!r cos~un!dun .

Substituting the results of Eqs.~5!, ~7!, and~8! into Eq. ~9!, andperforming the discontinuous integration of the pressure teryields the following results for the average force and torqueerted on the shaft:

F5NAp~Pd1Pi !tan~a!

2, T5

NAp~Pd2Pi !r tan~a!

p.

(10)

Note: these results have been linearized for small values ofg. It isinteresting to observe that these results are completely indedent of the piston mass,M p , and the angular speed of the shav, which says that the average effects of piston inertia on

DECEMBER 2000, Vol. 122 Õ 555

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quantities of shaft force and torque are zero. In fact, this is trueany arrangement of pistons; however, if the pistons are spaevenly within a circular array about the centerline of the shaftcan be shown that the instantaneous and average effects opiston inertia are identically zero. This may be proven by usthe instantaneous forms of Eqs.~2! and ~4! with the Lagrangetrigonometric identities@5#. Note: unlike a centrifugal pump, thtorque exerted on the shaft of a hydrostatic pump is indepenof the angular shaft speed. This is because the discharge prewithin a hydrostatic pump is generated by compressing the flas opposed to a centrifugal pump that uses the Bernoulli tradebetween fluid velocity and pressure. If it is assumed that thecharge pressure of the pump,Pd , is much, much greater than thintake pressure,Pi , then the intake pressure may be ignored athe average force and torque results may be written as

F51

2NApPmaxtan~a!, T5

r

pNApPmaxtan~a!, (11)

where Pd has been replaced by a maximum discharge pressPmax, since this is the quantity of interest in this research.

Internal Bending-Moment and Torque Diagrams. Figure 4shows a free-body-diagram of the shaft with the average forcetorque loads applied as shown in Eq.~11! and with the supportbearing reactions. In this figure, it is shown that there are thregions along the shaft which experience different types of loing. In Region 1, the shaft experiences an internal bendmoment due to the downward force,F; but, there is no torqueexerted in this section of the shaft since the torque loads areplied only between the cylinder-block spline and the input splof the shaft. In Region 2, there is an internal bending-momacting on the shaft plus a torque load resulting from the intorque to the pump,T. In Region 3, there is no internal bendingmoment; however, the torque load continues to exist in this adue to the input torque on the shaft. Using Fig. 4, it can be shothat the internal bending-moment and the torque within each mregion of the shaft are given by the following expressions:

Fig. 4 Internal bending moment and torque diagrams for theshaft


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x8

~L11L2! D , M350,

T150, T25T, T35T, (12)

where M1 , M2 , and M3 are the internal bending-momentsRegions 1, 2, and 3 respectively, andT1 , T2 , and T3 are thetorques exerted on the shaft in these same regions. Note: in tequationsx8 is the distance along the shaft beginning from the lbearing reaction.

Average Load Point. The instantaneous load point for thdownward force,F, along thex-axis relative to the origin ofthe coordinate system shown in Fig. 2 is given by the centroexpression

X51

F (n51

N

Fnxn , (13)

whereFn , F, and xn are given in Eqs.~1!, ~2!, and ~6! respec-tively. Rearranging terms and taking the integral average of bsides of the equation yields the following expression:

E0

2p

FXdun5E0

2p

Fnxndun5Ap tan2~a!r ~Pd2Pi !g. (14)

Sincea andg are much less than one, it can be seen that the rhand side of Eq.~14! is small and perhaps negligible. SinceF isclearly not small, the only way for Eq.~14! to be satisfied is forthe centroidal dimension,X, to be small or nearly zero. If weassume thatX is zero, then we can say that the average load pofor the downward force,F, is located at the origin of thex, y, zcoordinate system as shown in Fig. 2. Another way of describthis point is to notice that there is a plane which always contathe centers of the piston-slipper ball joints. The average load pis then located at the intersection of this plane and the centeof the shaft. By placing the load point in this position, the dimesionsL1 andL2 shown in Fig. 4 can be determined based uponload carrying points of the right and left bearings.

Governing Shaft-Equation

Goodman Line. Figure 5 shows a schematic of the Goodmline. This diagram is based upon experiments which have incated that a fluctuating applied stress, comprised of a mean analternating component, tends to reduce the life of the shaft cpared to a steady applied stress which is only comprised of a mcomponent. If a combination of mean and alternating stressesapplied to the shaft, experiments have shown that they shoexist below the Goodman line if the shaft is to exhibit an infinlife. The Goodman line of Fig. 5 is shown for a generalized siation where the desired safety factor of one’s calculation,nf , canbe chosen. The equation of the Goodman line may be expreby

sa85Sf

nf2

Sf

Sutsm8 , (15)

wheresa8 is the von Mises alternating stress component,sm8 is thevon Mises mean stress component,Sf is the fatigue strength of thematerial, andSut is the ultimate tensile strength of the materiaThe von Mises stress is defined as a uniaxial tensile stresscreates the same distortion energy as the actual combinatioapplied stresses. For a two dimensional stress state which is cbined of only one normal stress and a shear stress, the von Mequivalent stress is shown in Fig. 6 and the mean and alternacomponents of this stress may be separately expressed as

sa85Asa213ta

2 and sm8 5Asm2 13tm

2 , (16)


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wheresa andta are the alternating normal and shear componeof the actual stress state andsm andtm are the mean normal andshear components of the actual stress state.

Shaft Stresses. The maximum applied stresses on the surfaof the shaft are different for the three main sections of the shafshown in Fig. 4. In Region 1, the only stresses on the shaft surare due to bending and, because the shaft is rotating, these strare alternating about a mean value of zero. In Region 2,stresses on the shaft surface result from the alternating benstress and the mean torsional stress due to the shaft torquRegion 3, the only stresses in the shaft are the mean torsistresses. For Region 1, the alternating and mean componenstress may be written as

s1a5kf

32M1

pd13 , s1m

50,(17)

t1a50, t1m

50,

wherekf is a stress fatigue concentration-factor,M1 is the internalmoment on the shaft which is described in Eq.~12!, andd1 is theshaft diameter at the location of interest within Region 1. UsiEq. ~16!, it can be seen that the von Mises equivalent stresseRegion 1 may be written as

s1a8 5kf

32M1

pd13 and s1m

8 50. (18)

Fig. 5 Goodman line based upon experiments

Fig. 6 Equivalent von Mises stress in two dimensions


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Similarly, the alternating and mean components of stress onsurface of the shaft in Region 2 may be written as

s2a5kf

32M2

pd23 , s2m

50,(19)

t2a50, t2m

5kf sm

16T

pd23 ,

whereM2 is the internal moment on the shaft which is describin Eq. ~12!, d2 is the shaft diameter at the location of interewithin Region 2, kf sm

is the mean shear-stress fatigu

concentration-factor, andT is the shaft torque shown in Eq.~11!.Using Eq. ~16!, it can be seen that the von Mises equivalestresses in Region 2 may be written as

s2a8 5kf

32M2

pd23 and s2m

8 5kf sm

16T

pd23). (20)

Again, the alternating and mean components of stress on theface of the shaft in Region 3 may be written as

s3a50, s3m

50,(21)

t3a50, t3m

5kf sm

16T

pd33 ,

where,d3 is the shaft diameter at the location of interest withRegion 3. Using Eq.~16!, it can be seen that the von Miseequivalent stresses in Region 3 may be written as

s3a8 50 and s3m

8 5kf sm

16T

pd33). (22)

Shaft Equations. By substituting Eqs.~18!, ~20!, and ~22!into Eq. ~15!, and by using the results of Eqs.~11! and ~12!, theminimum shaft diameter for the three main sections of the shmay be written as

d1>H Fkf

L2

Sf

x8

~L11L2!J1/3

,

d2>H FS kf

L1

SfF12

x8

~L11L2!G1kf sm

r

Sut

)

p D J 1/3

, (23)

d3>H Fkf sm

r

Sut

)

p J 1/3

,

where

F5nf16NApPmaxtan~a!

p. (24)

Stress Concentration FactorsEquation~23! shows results that depend upon the stress c

centration factors,kf andkf sm. These factors are used to accou

for the stress concentrations which occur along the shaft at crsections where the smooth contour of the shaft is disrupted.amples of such disruptions include grooves for o-ring seals, sfor bearings, notches for key-ways, or splines for the cylindblock interface. At these locations of abrupt geometry change,actual stresses may be two to three times what would normallcharacterized by a smooth shaft with no disruptions. Strconcentration-factors for typical geometries have been studiedcollected in standard textbooks. See Norton@6#, Spotts and Shoup@7#, or Shigley and Mitchell@8# for tables of stress concentrationfactors. These references show that reasonable values for sconcentration factors range between 1 and 3. Note: these faare dimensionless. In general, the value of a stress concentrfactor depends upon the notch radius and its depth; and, e


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more importantly, it depends upon the actual diameter of the csection being examined. Since the dependency ofkf andkf sm

onthe shaft diametersd1 , d2 , andd3 is typically nonlinear, there isnot a closed-form solution for Eq.~23! in the vicinity of a stressconcentration. From Eq.~23! it can be seen that the presence ostress concentration requires that the shaft diameter be deslarger than it normally would be if the stress concentration wabsent. By ignoring stress concentrations~i.e., settingkf andkf sm

equal to unity! a minimum requirement for the shaft diametefrom a stress point of view, may be established along the enlength of the shaft. The user of these results must keep in mhowever, that in the vicinity of an abrupt geometry change,actual shaft diameter may need to be 10 to 40 percent ladepending upon the physical characteristics of the stress contration itself.

ResultsSolutions to Eq.~23! are graphically represented in Fig. 7 fo

three different fatigue safety-factors:nf51, 2, and 3. In thesesolutions, the fatigue strength of the shaft material,Sf , has beenset equal to 350 MPa, the ultimate tensile strength of the smaterial,Sut , has been set equal to 700 MPa, the concentrafactorskf and kf sm

have been set equal to unity, the maximuoperating pressure,Pmax, has been set equal to 42 MPa, and othgeometrical features have been used from a standard pump dwhich is typically shown in Fig. 1. The pump parameters thwere used in the calculations of Eq.~23! are given in the Appen-dix. The contour lines in Fig. 7 illustrate the minimum shaft dameter that is needed along the shaft to satisfy the specified sfactor against fatigue failure. Note: these contour lines are a mmum diameter. As shown in Fig. 7, larger diameters exist in ctain locations along the actual shaft and this means that the shstronger than it needs to be in these areas. Though these ldiameters are not needed for strength, they may be needeother mechanical functions like, fitting into a standard-size being or, guiding a cylinder block spring which wraps around tshaft. As one would expect, a fatigue safety-factor of 1 yieldssmallest required shaft diameter while a fatigue safety-factor oyields the largest required shaft diameter. The magnified porof Fig. 7 shows a critical region along the shaft where a fatigfailure could possibly occur. From this view, it can be seen tthe current shaft is designed at best with a safety factor oagainst a fatigue failure. The actual stress concentration withis region may cause the design to be nearly unacceptable—would have to be checked using stress concentration studiethe particular geometry shown. Another critical region is shoto be the ‘‘customer end’’ of the pump where the right-hand-sof the shaft is actually coupled to a motor. Here we see thatshaft must be designed to withstand the mean torsional stressated by the input torque,T. Again, this region appears to bmarginal for a safety factor of 2 and is clearly unacceptable fosafety factor of 3.

Fig. 7 Diameter profiles required for three different factors ofsafety


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Discussion and ConclusionsThis paper has presented equations that may be used to d

the shaft diameter for an axial-piston swash-plate type hydrospump for the purposes of preventing a fatigue failure. Fromphysical loads that are generated on the shaft, it is clear that tregions which are characterized by differently applied stressesbe identified. For this reason, three governing results are presein Eq. ~23! for designing the shaft diameter within each majregion of the shaft. Within each of these results, a common lfunction, F, appears which describes the severity of the loadand thereby tends to drive the diameter of the shaft from a fatifailure point of view.

In the results of this work, the effect of stress concentratioalong the shaft have been neglected by setting the stconcentration-factors,kf and kf sm

, equal to unity. In general, ithas been noted that these factors are determined based upofinal geometry of the shaft and that they can cause the requdiameter of the shaft to increase in the notched areas by as mas 10 to 40% depending upon the severity of the geometry chaStress concentration factors can be reduced in magnitude by ua large radius to make the geometry changes more gradual. Nsince this is a fatigue analysis, all stress concentrations shoulrigorously considered before the design is finalized. See the sdard text books in the References section of this paper for taof stress concentration factors.

Within axial-piston pumps, it is often desirable to make tshaft as thin as possible for the purposes of reducing the ovpump size. Equation~23! shows that from a stress point of viewthe diameter of the shaft may be reduced by increasing the mrial strength properties,Sf andSut . This may be accomplished bheat treating the shaft appropriately and by using a steel wsufficient carbon content for achieving the desired strengthcommon steel used for shaft design is SAE 4140. When thisterial is quenched and tempered at 200°C, the ultimate tenstrength of the material may get as high as 1,800 MPa andassociated fatigue strength will be roughly half of the ultimatensile strength. Though heat treating the shaft may increasstrength and cause it to be less susceptible to a fatigue failuredesigner should remember that heat treating a shaft does notthe shaft’s modulus of elasticity,E, and therefore, for the samshaft geometry, the deflection characteristics stay the same.

The reader will recall that the deflection of the shaft in tz-direction is given by the following governing equation:

d2z

dx82 5M ~x8!

EI~x8!, (25)

where M (x8) is the internal bending moment of the shaft aI (x8) is the area moment of inertia given by

I ~x8!5pd~x8!2

64, (26)

where d(x8) is the shaft diameter at the locationx8 along theshaft. Note: both the internal bending moment and the areament of inertia are shown in Eq.~25! to vary explicitly with theaxial location along the shaft,x8. The internal bending moment othe shaft,M (x8), may be evaluated without knowing the diametof the shaft itself and, in fact, this information is shown in Fig.On the other hand, the shaft diameter must be known a prbefore the area moment of inertia,I (x8), may be determined. Byreducing the shaft diameter, the area moment of inertia will adecrease and the shaft deflection will increase. If the shaft detion increases too much, the deflection itself may becomecritical issue in designing the shaft diameter rather than theplied stress. Excessive shaft deflections can cause the subearings and the cylinder-block spline interface to wear adversIf the shaft deflections are unreasonable, they may causecylinder-block to tip away from the port plate in which case t


tt

n 1,

or

1,

, or

, 2,

r 3

haft-

g

sh-

ng,’’

machine cannot function properly. The reader should note thawork of this paper only addresses the shaft diameter from a spoint of view. It is recommended that a shaft-deflection studydone prior to completing the shaft design and that these studieviewed in light of the published limits from bearing manufactuer’s regarding misalignment and the functional requirementsthe machine.

Appendix

Ap5789 mm2 N59L15156 mm r 567 mmL25107 mmL35134 mm a50.297 rad

Nomenclature

Ap 5 pressurized area of a single pistond1,2,3 5 shaft diameter at a location of interest within Regio

1, 2, or 3E 5 modulus of elasticity for steelF 5 instantaneous load exerted on the shaft by all pisto

within the pumpF 5 average load exerted on the shaft by all pistons

within the pumpFA 5 left bearing reactionFB 5 right bearing reactionFn 5 downward force acting on the shaft from thenth pis-

tonI 5 area moment of inertia

kf 5 stress fatigue concentration-factorkf sm 5 mean shear-stress fatigue concentration-factor

L1,2,3 5 length of the shaft in Region 1, 2, or 3M p 5 mass of a single piston

M1,2,3 5 internal bending-moment in Region 1, 2, or 3m 5 pressure transition slope on the valve plateN 5 total number of pistons within the pumpn 5 piston counter; i.e., thenth piston

nf 5 fatigue safety factorPd 5 discharge pressure of the pumpPi 5 intake pressure of the pump

Pmax 5 maximum discharge pressure of the pumpPn 5 fluid pressure within thenth piston chamberRn 5 reaction force between thenth piston and the swash

plater 5 piston pitch-radius

Sf 5 fatigue strength of the shaft materialSut 5 ultimate tensile strength of the shaft material

T 5 instantaneous torque exerted on the shaft by all pistons within the pump


theressbys ber-of

ns

-

T1,2,3 5 torque on the shaft in Region 1, 2, or 3T 5 average torque exerted on the shaft by all pistons

within the pumpTn 5 torque exerted on the shaft by a single pistonX 5 centroidal position of the downward force acting on

the shaftx8 5 distance along the shaft relative to the left bearing

reaction-pointxn 5 position of thenth piston-slipper ball joint in the

x-directiona 5 swash-plate angleg 5 pressure carry-over angle on the valve plate

un 5 circular position of thenth pistonsa 5 alternating component of normal stresssa8 5 von Mises alternating stress componentsm 5 mean component of normal stresssm8 5 von Mises mean stress component

s1,2,3a 5 alternating component of normal stress in Regions2, or 3

s1,2,3m 5 mean component of normal stress in Regions 1, 2,3

s1,2,3a8 5 von Mises alternating stress component in Regions

2, or 3s1,2,3m

8 5 von Mises mean stress component in Regions 1, 23

ta 5 alternating component of shear stresstm 5 mean component of shear stress

t1,2,3a 5 alternating component of shear stress in Regions 1or 3

t1,2,3m 5 mean component of shear stress in Regions 1, 2, oF 5 load function for the shaftv 5 shaft speed of the pump

References@1# ANSI/ASME Standard, 1985, Design of Transmission Shafting, B106.1M@2# Loewenthal, S. H., 1978, Proposed Design Procedure for Transmission S

ing Under Fatigue Loading, Technical Note TM-78927. NASA.@3# Kececioglu, D. B., and Lalli, V. R., 1975, Reliability Approach to Rotatin

Component Design, Technical Note TN D-7846. NASA.@4# Manring, N. D., 1998, The torque on the input shaft of an axial-piston swa

plate type hydrostatic pump. ASME J. Dyn. Syst., Meas., Control,120, pp.57–62.

@5# Damtew, F. A., 1998, ‘‘The Design of Piston-bore Springs for OverwhelmiInertial Effects Within Axial-piston Swash-plate Type Hydrostatic MachinesM.S. Thesis. University of Missouri—Columbia, Columbia, MO.

@6# Norton, R. L., 1998,Machine Design—An Integrated Approach, Prentice-Hall,Upper Saddle River, NJ 07458.

@7# Spotts, M. F., and Shoup, T. E., 1998,Design of Machine Elements, 7th ed.Prentice-Hall, Upper Saddle River, NJ 07458.

@8# Shigley, J. E., and Mitchell, L. D., 1983,Mechanical Engineering Design, 4thed., McGraw-Hill Book Company, New York, NY.


shaft

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