design of shafts and associated components

8/19/2019 Design of Shafts and Associated Components

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Design of Shafts and Associated Components

IntroductionThe term "shaft" applies to rotating machine members used for transmitting power or torque. The shaft is subject to torsion, bending, and occasionally axial loading.Stationary and rotating members, called axles, carry rotating elements, and are subjected

primarily to bending. Transmission or line shafts are relatively long shafts that transmittorque from motor to machine.

Countershafts are short shafts between the driver motor and the driven machine. Headshafts or stub shafts are shafts directly connected to the motor. otion or power can betransmitted through an angle without gear trains, chains, or belts by using flexibleshafting. Such shafting is fabricated by building up on a single central wire one or moresuperimposed layers of coiled wire.

!egardless of design requirements, care must be ta en to reduce the stress concentration

in notches, eyways, etc. #roper consideration of notch sensitivity can improve thestrength more significantly than material consideration. $qually important to the designis the proper consideration of factors nown to influence the fatigue strength of theshaft, such as surface condition, si%e, temperature, residual stress, and corrosiveenvironment.

High&speed shafts require not only higher shaft stiffness but also stiff bearing supports,machine housings, etc. High&speed shafts must be carefully chec ed for static anddynamic unbalance and for first&and second order critical speeds. The design of shafts insome cases, such as those for turbo&pumps, is dictated by shaft dynamics rather than byfatigue strength considerations.

The lengths of journals, clutches, pulleys, and hubs should be viewed critically becausethey very strongly influence the overall assembly length. #ulleys, gear couplings, etc.,should be placed as close as possible to the bearing supports in order to reduce the

bending stresses.

The dimensions of shafts designed for fatigue or static strength are selected relative tothe wor ing stress of the shaft material, the torque, the bending loads to be sustained,

and any stress concentrations or other factors influencing fatigue strength. Shaftsdesigned for rigidity have one or more dimensions exceeding those determined bystrength criteria in order to meet deflection requirements on axial twist, lateraldeflection, or some combination thereof. 'n increase in shaft diameter may also berequired to avoid unwanted critical speeds.

Stresses are only evaluated at critical locations( )ritical locations are usually*◦ +n the outer surface◦ here the bending moment is large


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◦ here the torque is present◦ here stress concentrations exist

Shear - / and bending -0/ stresses on the outer surface of a shaft, for a torque -T/ and bending moment - /*

12

345/156- d

T d d

T r

J T

π π τ ===

a/ For solid circular section:

12

155/426- d

M d d M

r I

M π π

σ ===

For hollow circular section:

o

io

i

o

i d d where

d T d

d d T d

d d T r

J T =

−=

−=

−== λ

λ π π π τ ,

/3-34

/-34

5/156/-- 21722

722

7

o

io

i

o

i d d

whered

M d

d d M d

d d M

r I

M =−

=−

=−

== λ λ π π π

σ ,/3-

15/-

155/426/-- 217

227

227

Principal Normal Stresses and Max Distortion Energ Failure criterion for non!rotating shaftsThe stress at a point on the shaft is normal stress -0/ in 8 direction and shear stress - /in 89 plane.

55

55

5

3 5555τ

σ σ τ

σ σ +

−=+

+= S and S

From Mohr Circle:


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5

5355

53

≤−+ fs

yp

N

S S S S S

ax :istortion $nergy theory*

5

55 1

≤+ fs

yp

N

S τ σ

#utting values of S 3 and S 5 and simplifying*

This is the design equation for non&rotating shaft

Design of rotating shafts and fatigue considerationThe most frequently encountered stress situation for a rotating shaft is to havecompletely reversed bending and steady torsional stress. ;n other situations, a shaft mayhave a reversed torsional stress along with reversed bending stress.

The most generali%ed situation the rotating shaft may have both steady and cycliccomponents of bending stress -0 av, 0 r / and torsional stress - av, r /.


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>sing these equivalent static stresses in our static design equation, the equation for rotating shaft is*

The term shaft usually refers to a relatively long member of round cross section thatrotates and transmits power. +ne or more members such as gears, sproc ets, pulleys,and cams are usually attached to the shaft by means of pins, eys, splines, snap rings,and other devices. These latter members as well as couplings and universal joints, whichserve to connect the shaft to its source of power or load, are among the ?associated

parts@. ' shaft can have a nonround cross section, and it need not rotate. ;t can bestationary and serve to support a rotating member, such as the short shafts -also called

spindles / that support the nondriving wheels of an automobile. The shafts supportingidler gears can be either rotating or stationary, depending on whether the gear is attachedto the shaft or supported by it through bearings.

Shafts supporting and driving vehicle wheels are also called axles. ;t is apparent thatshafts can be subjected to various combinations of axial, bending, and torsional loadsand that these loads may be static or fluctuating. Typically, a rotating shaft transmitting

power is subjected to a constant torque -producing a mean torsional stress/ together witha completely reversed bending load -producing an alternating bending stress/.

;n addition to satisfying strength requirements, shafts must be designed so thatdeflections are within acceptable limits. $xcessive lateral shaft deflection can hamper gear performance and cause objectionable noise. The associated angular deflection can

be very destructive to nonAself&aligning bearings -either plain or rolling/. Torsionaldeflection can affect the accuracy of a cam& or gear&driven mechanism.


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three or more bearings must be used to provide adequate support and rigidity, precisealignment of the bearings in the supporting structure must be maintained.

Shaft axial positioning and provision for carrying thrust loads normally require that oneand only one bearing ta e thrust in each direction. Sometimes a thrust load is sharedamong two or more plain thrust bearings. ;n this case there must be sufficient axialclearance to ensure against ?binding@ under any operating conditions. #roductiontolerances may be such that only one bearing will carry the thrust until after initial?wearing&in.@ ;t is important that the members supporting the shaft bearings besufficiently strong and rigid.

Mounting Parts onto $otating ShaftsSometimes members li e gears and cams are made integral with the shaft, but moreoften such members -which also include pulleys, sproc ets, etc./ are made separatelyand then mounted onto the shaft. The portion of the mounted member in contact withthe shaft is the hub. The hub is attached to the shaft in a variety of ways. ;n


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Shaft Design for StressCritical Locations;t is not necessary to evaluate the stresses in a shaft at every point( a few potentiallycritical locations will suf ce. )ritical locations will usually be on the outer surface at axial locations where the bending moment is large where the tor!ue is present and where stress concentrations exist . Iy direct comparison of various points along the

shaft, a few critical locations can be identi ed upon which to base the design.

ost shafts will transmit torque through a portion of the shaft. Typically the torquecomes into the shaft at one gear and leaves the shaft at another gear. The torque is oftenrelatively constant at steady state operation. The shear stress due to the torsion will begreatest on outer surfaces.

The bending moments on a shaft can be determined by shear and bending momentdiagrams. Since most shaft problems incorporate gears or pulleys that introduce forcesin two planes, the shear and bending moment diagrams will generally be needed in two

planes. !esultant moments are obtained by summing moments as vectors at points of interest along the shaft. ' steady bending moment will produce a completely reversedmoment on a rotating shaft, as a speci c stress element will alternate from compressionto tension in every revolution of the shaft. The normal stress due to bending momentswill be greatest on the outer surfaces. ;n situations where a bearing is located at the endof the shaft, stresses near the bearing are often not critical since the bending moment issmall.


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'xial stresses on shafts due to the axial components transmitted through helical gears or tapered roller bearings will almost always be negligibly small compared to the bendingmoment stress. They are often also constant, so they contribute little to fatigue.)onsequently, it is usually acceptable to neglect the axial stresses induced by the gearsand bearings when bending is present in a shaft. ;f an axial load is applied to the shaft insome other way, it is not safe to assume it is negligible without chec ing magnitudes.

Shaft StressesIending, torsion, and axial stresses may be present in both midrange and alternatingcomponents.


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These equivalent alternating and midrange stresses can be evaluated using anappropriate failure curve on the modi ed Moodman diagram.


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KK.. -x/where

#$%'SM$ $lliptic

K -xi/

K. -xii/

#$%Soderberg

KK-xiii/

KK..-xiv/


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'S $ $lliptic also ta es yielding into account, but is not entirely conservativethroughout its entire range. This is evident by noting that it crosses the yield line -see

page 5FC/. The Merber and modi ed Moodman criteria do not guard against yielding,requiring a separate chec for yielding. ' von ises maximum stress is calculated for this purpose.

K.. -xv/To chec for yielding, this von ises maximum stress is compared to the yield strength,as usual.

KKKKKKKK -xvi/


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case of 3.P can be assumed. Similarly, the llet radius at the shoulder needs to be si%edto avoid interference with the llet radius of the mating component.

There is a signi cant variation in typical bearings in the ratio of llet radius versus borediameter, with r 6d typically ranging from around 7.75 to 7.74. ' quic loo at the stressconcentration charts -


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Table CA3 summari%es some typical stress concentration factors for the rst iteration inthe design of a shaft. Similar estimates can be made for other features. The point is tonotice that stress concentrations are essentially normali%ed so that they are dependent onratios of geometry features, not on the speci c dimensions. )onsequently, by estimatingthe appropriate ratios, the rst iteration values for stress concentrations can be obtained.These values can be used for initial design, then actual values inserted once diametershave been determined.

(ig. )%*+ Techni!ues for reducing stress concentration at a shoulder supporting abearing with a sharp radius. , a - arge radius undercut into the shoulder. , b - argeradius relief groo/e into the back of the shoulder. , c - arge radius relief groo/e into the

small diameter

design of shafts and associated components

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