7.Shafts and Shaft Components

8/10/2019 7.Shafts and Shaft Components

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MECopyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 7

Shafts andShafts

Components


2/90

ME Introduction

- A shaft is a rotating member, usually of circular cross section, used to transmit power or motion.

- It provides the axis of rotation, or oscillation, of elements such as gears, pulleys, flywheels, cranks, sprockets, and the like and controls the geometry oftheir motion.

- An axle is a non-rotating member that carries no torque and is used to support

rotating wheels, pulleys, and the like. The automotive axle is not a true axle; theterm is a carry-over from the horse-and-buggy era, when the wheels rotated onnon-rotating members.

- A non-rotating axle can readily be designed and analyzed as a static beam, andwill not warrant the special attention given in this chapter to the rotating shaftswhich are subject to fatigue loading.


3/90

ME Introduction

In this chapter, details of the shaft itself will be examined, including the following:

Material selection

Geometric layout

Stress and strength Static strength Fatigue strength

Deflection and rigidity Bending deflection Torsional deflection Slope at bearings and shaft -supported elements

Shear deflection due to transverse loading of short shafts

Vibration due to natural frequency


4/90

ME Introduction

- In deciding on an approach to shaft sizing, a stress analysis at a specific pointon a shaft can be made using only the shaft geometry in the vicinity of that

point. Thus the geometry of the entire shaft is not needed.

- In design it is usually possible to locate the critical areas, size these to meetthe strength requirements, and then size the rest of the shaft to meet therequirements of the shaft-supported elements.

- The deflection and slope analyses cannot be made until the geometry of theentire shaft has been defined. Thus deflection is a function of the geometryeverywhere , whereas the stress at a section of interest is a function of local

geometry .

- For this reason, shaft design allows a consideration of stress first. Then,after tentative values for the shaft dimensions have been established, thedetermination of the deflections and slopes can be made.


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ME Shaft Materials

- Deflection is not affected by strength, but rather by stiffness as represented bythe modulus of elasticity , which is essentially constant for all steels. For thatreason, rigidity cannot be controlled by material decisions, but only by

geometric decisions .

- Necessary strength to resist loading stresses affects the choice of materials andtheir treatments. Many shafts are made from low carbon, cold-drawn or hot-rolled steel, such as ANSI 1020-1050 steels.

- A good practice is to start with an inexpensive, low or medium carbon steelfor the first time through the design calculations.

- If strength considerations turn out to dominate over deflection, then a higherstrength material should be tried, allowing the shaft sizes to be reduced until

excess deflection becomes an issue.

- The cost of the material and its processing must be weighed against the needfor smaller shaft diameters.


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ME Shaft Materials

- Shafts usually dont need to be surface hardened unless they serve as theactual journal of a bearing surface.

- Typical material choices for surface hardening include carburizing grades ofANSI 1020, 4320, 4820, and 8620.

- Cold drawn steel is usually used for diameters under about 3 inches. Thenominal diameter of the bar can be left unmachined in areas that do not requirefitting of components.

- Hot rolled steel should be machined all over. For large shafts requiring muchmaterial removal, the residual stresses may tend to cause warping.

- If concentricity is important, it may be necessary to rough machine, then heattreat to remove residual stresses and increase the strength, then finish machineto the final dimensions by the process called cylindrical grinding.


7/90ME Shaft Materials

- For low production , turning is the usual primary shaping process. Aneconomic viewpoint may require removing the least material.

- High production may permit a volume-conservative shaping method (hot orcold forming, casting ), and minimum material in the shaft can become a designgoal.

- Cast iron may be specified if the production quantity is high, and the gears are

to be integrally cast with the shaft.

- Properties of the shaft locally depend on its history-cold work, cold forming,rolling of fillet features, heat treatment, including quenching medium, agitation,and tempering regimen.

- Stainless steel may be appropriate for some environments.


8/90ME Shaft Layout

- The general layout of a shaft to accommodate shaft elements, e.g. gears, bearings, and pulleys, must be specified early in the design process in order to perform a free body force analysis and to obtain shear-moment diagrams.

- The geometry of a shaft is generally that ofa stepped cylinder.

- The use of shaft shoulders is an excellentmeans of axially locating the shaft elementsand to carry any thrust loads.

- The figure shows an example of a steppedshaft supporting the gear of a worm-gearspeed reducer.

- Each shoulder in the shaft serves a specific purpose as shown.


9/90ME Shaft Layout

(a) Choose a shaf t configuration tosuppor t and l ocate the two gears andtwo bearings.

(b) Solu tion uses an integral pini on,three shaft shoul ders, key and keyway,and sleeve. The housinglocates the bearings on thei r outer r ingsand r eceives the thrust l oads.

(c) Choose fanshaf t conf iguration.

(d) Soluti on uses sleeve bear ings, astraight-thr ough shaf t, locating col lars,and setscrews for coll ars, fan pulley, and

fan i tsel f . The fan housing supports thesleeve bear ings.

The geometric configuration of a shaft to be designed is often simply a revisionof existing models in which a limited

number of changes must be made. If thereis no existing design to use as a starter,then the determination of the shaft layoutmay have many solutions.


10/90ME Shaft Layout

There are no absolute rules for specifying the general layout, but the followingguidelines may be helpful.

- Axial layout of components

- Supporting axial loads

- Providing for torque transmission

- Assembly and disassembly



Axial layout of components

- In general, it is best to support load-carrying components between bearings ,such as in Fig. 7 2a, rather than cantilevered outboard of the bearings, such asin Fig. 7 2c.

- Pulleys and sprockets often need to be mounted outboard for ease ofinstallation of the belt or chain. The length of the cantilever should be kept shortto minimize the deflection .

- Only two bearings should be used in most cases . For extremely long shaftscarrying several load-bearing components, it may be necessary to provide morethan two bearing supports with a special care for the alignment.

- Shafts should be kept short to minimize bending moments and deflections .Some axial space between components is desirable to allow for lubricant flowand to provide access space for disassembly of components with a puller.



Axial layout of components

- Load bearing components should be placed near the bearings , to minimizethe bending moment at the locations that will likely have stress concentrations,and to minimize the deflection at the load-carrying components.

- The primary means of locating the components is to position them against a shoulder of the shaft . A shoulder also provides a solid support to minimizedeflection and vibration of the component.

- When the magnitudes of the forces are reasonably low, shoulders can beconstructed with retaining rings in grooves, sleeves between components, orclamp-on collars .

- Where axial loads are very small, it may be feasible to do without the shoulders entirely , and rely on press fits, pins, or collars with setscrews tomaintain an axial location. See Fig. 7 2b and 7 2d



- In cases where axial loads are not negligible , e.g. for helical or bevel gears, ortapered roller bearings, it is necessary to provide a means to transfer the axialloads into the shaft, then through a bearing to the ground .

- Often, the same means of providing axial location, e.g., shoulders, retainingrings, and pins, will be used to also transmit the axial load into the shaft .

- It is generally best to have only one bearing carry the axial load , to allowgreater tolerances on shaft length dimensions, and to prevent binding if theshaft expands due to temperature changes. This is particularly important forlong shafts.

Supporting axial loads




- The figure shows example of shaftwith only one bearing carrying theaxial load against a shoulder , whilethe other bearing is simply press-fit onto the shaft with no shoulder.

- Tapered roller bearings used in amowing machine spindle . This designrepresents good practice for thesituation in which one or moretorque-transfer elements must bemounted outboard.




- The figure shows example of shaftwith only one bearing carrying the axialload against a shoulder , while the otherbearing is simply press-fit onto the shaftwith no shoulder.

- A bevel-gear drive in which both pinion and gear arestraddle-mounted.



Providing for Torque Transmission

- Most shafts serve to transmit torque from an input gear or pulley, through theshaft, to an output gear or pulley.

- The shaft must be sized to support the torsional stress and torsional deflection.The common torque-transfer elements , that used to provide a means oftransmitting the torque between the shaft and the gears, are:

Keys

Splines Setscrews Pins Press or shrink fits Tapered fits

- In addition to transmitting the torque, many of these devices are designed tofail i f the torque exceeds acceptable operating l imi ts, protecting moreexpensive components.


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ME Shaft Layout


- One of the most effective andeconomical means oftransmitting moderate to highlevels of torque is through a key that fits in a groove in the shaftand gear.

- Keyed components generallyhave a slip fit onto the shaft, soassembly and disassembly iseasy .


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ME Shaft Layout


- Splines are essentially stubbygear teeth formed on the outside of

the shaft and on the inside of thehub of the load-transmittingcomponent . Splines are generallymuch more expensive tomanufacture than keys, and are

typically used to transfer hightorques .

- One feature of a spline is that itcan be made with a reasonablyloose slip fit to allow for largeaxial motion between the shaft andcomponent while still transmittingtorque.


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ME Shaft Layout


- For cases of low torquetransmission, various means oftransmitting torque are available.These include pins, setscrews inhubs, tapered fits, and press fits.

- Press and shr ink f i ts forsecuring hubs to shafts are used

both for torque transfer and for preserving axial location. Theresulting stress-concentrationfactor is usually quite small.


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ME Shaft Layout


- Tapered f its between the shaftand the shaft-mounted device,such as a wheel, are often usedon the overhanging end of ashaft.

- At the early stages of the shaftlayout, the important thing is toselect an appropriate means oftransmitting torque. It isnecessary to know where the

shaft discontinuities, such as

keyways, holes, and splines, willbe in order to determine criticallocations for analysis .


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ME Shaft Layout

Assembly and Disassembly

- Consideration should be given to the method of assembling the componentsonto the shaft, and the shaft assembly into the frame.

- This generally requires the largest diameter in the center of the shaft, with progressively smaller diameters towards the ends to allow components to be slidon from the ends .

- If a shoulder is needed on both sides of a component, one of them must becreated by such means as a retaining ring or by a sleeve between twocomponents.

- The gearbox will need means to physically position the shaft into its bearings,and the bearings into the frame.

- This is typically accomplished by providing access through the housing to the bearing at one end of the shaft.


24/90

ME Shaft Layout


Ar rangement showing beari nginner r ings press-f i tted to shaf t

whi le outer r ings fl oat in thehousing. The axial clearanceshoul d be suff icient only toall ow for machi nery vibrations.

Ar rangement showing beari nginner r ings press-f i tted to shaf t andouter r ings are preloaded by thehousing of th e gearbox.


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ME Shaft Layout


I n th is arr angement th e inner r ing of the left-hand beari ng i s locked to the shaf t betweena nut and a shaf t shoulder. Th e snap r ing i n the outer race is used to posi tively locate theshaf t assembly in the axial di rection. Note: the fl oating r ight-hand beari ng and thegri nding runout grooves in the shaf t.


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ME Shaft Layout


This arr angement i s simi lar to theprevious one, in that the left-handbearing posi tions the entire shaf tassembly.

I n th is case the inner r ing i s secur ed tothe shaf t using a snap r ing.

Note the use of a shi eld to prevent dir tgenerated fr om within the machi nefrom enteri ng the beari ng.


27/90

ME Shaft Layout

- When components are to be press-fit to the shaft, the shaft should bedesigned so that it is not necessary to press the component down a long lengthof shaft. This may require an extra change in diameter, but it will reduce

manufacturing and assembly cost by only requiring the close tolerance for a short length .

- Consideration should also be given to the necessity of disassembling thecomponents from the shaft . This requires consideration of issues such as

accessibility of retaining rings, space for pullers to access bearings, openingsin the housing to allow pressing the shaft or bearings out, etc.


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ME Shaft Design for Stress

- It is not necessary to evaluate the stresses in a shaft at every point; a few potentially critical locations will suffice.

- Critical locations will usually be on the outer surface , at axial locations wherethe bending moment is large, where the torque is present, and where stressconcentrations exist .

- A free body diagram of the shaft will allow the torque at any section to bedetermined. The torque is often relatively constant at steady state operation. The

shear stress due to the torsion will be greatest on outer surfaces .

- The bending moments on a shaft can be determined by shear and bendingmoment diagrams . Since most shaft problems incorporate gears or pulleys thatintroduce forces in two planes, the shear and bending moment diagrams will

generally be needed in two planes . Resultant moments are obtained by summingmoments as vectors at points of interest along the shaft.

Critical Locations


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Critical Locations

- The normal stress due to bending moments will be greatest on the outersurfaces. In situations where a bearing is located at the end of the shaft, stressesnear the bearing are often not critical since the bending moment is small .

- Axial stresses on shafts due to the axial components transmitted through helicalgears or tapered roller bearings will almost always be negligibly small comparedto the bending moment stress. Consequently, it is usually acceptable to neglectthe axial stresses induced by the gears and bearings when bending is present in a

shaft .

- If an axial load is applied to the shaft in some other way, it is not safe to assumeit is negligible without checking magnitudes.


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Shaft stresses

- Bending, torsion, and axial stresses may be present in both midrange andalternating components.

- For analysis, it is simple enough to combine the different types of stresses intoalternating and midrange von Mises stresses .

- It is sometimes convenient to customize the equations specifically for shaftapplications.

- Axial loads are usually comparatively very small at critical locations where bending and torsion dominate, so they will be left out of the following equations.


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Shaft stresses

The fluctuating stresses due to bending and torsion are given by:

I c M

K a

f a I c M

K m

f m

J cT

K a fsa J cT

K m fsm

where M m

and M a are the midrange and alternating bending moments, T

m and T

a

are the midrange and alternating torques, and K f and K fs are the fatigue stressconcentration factors for bending and torsion, respectively.

For round solid shaft:

3

32

d

M K

a f a

3

32

d

M K

m f m

3

16

d

T K a fsa 3

16

d

T K m fsm


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Shaft stresses

The von Mises stresses according to the distortion energy failure theory forrotating round, solid shafts, neglecting axial loads , are given by

21

2

3

2

3

2122 16332

3

/

m fsm f /

mmm d

T K d

M K

Note that the stress concentration factors are sometimes considered optional forthe midrange components with ductile materials, because of the capacity of theductile material to yield locally at the discontinuity.

21

2

3

2

3

2122 16

332

3

/

a fsa f /

aaa d

T K d

M K


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Shaft stresses

The equivalent alternating and midrange stresses can be evaluated using anappropriate failure curve on the modified Goodman diagram. The fatigue failure

criteria for the modified Goodman line is

ut

m

e

a

S

S

n

1

212221223 341341161 / m fsm f ut

/

a fsa f e

T K M K S

T K M K S d n

For design purposes, it is also desirable to solve the equation for the diameter.

hence

31

2122212234

134

116 / / m fsm f

ut

/

a fsa f e

T K M K S

T K M K S

nd


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Shaft stresses

The resulting equations for several of the commonly used failure curves aresummarized below. The names given to each set of equations identifies the

significant failure theory, followed by a fatigue failure locus name.

DE-Goodman


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Shaft stresses

DE-Gerber


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Shaft stresses

DE-ASM E Ell iptic


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Shaft stresses

DE-Soderberg


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Shaft stresses

- For a rotating shaft with constant bending and torsion, the bending stress iscompletely reversed and the torsion is steady. It can be simplified by setting M m

and T a equal to 0, which simply drops out some of the terms.

- It is always necessary toconsider the possibility of staticfailure in the first load cycle.

- The Soderberg criteriainherently guards against

yielding , as can be seen bynoting that its failure curve isconservatively within the yield(Langer) line.

- The ASME Elliptic also takes yielding into account, but is not entirely conservative.


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Shaft stresses

- The Gerber and modifiedGoodman criteria do not guard

against yielding, requiring aseparate check for yielding. A vonMises maximum stress iscalculated for this purpose.

2122max 3 / amam

21

2

3

2

3max

163

32 / am fsam f

d

T (T K d

M M K


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Shaft stresses

max

S n y y

- To check for yielding, this von Mises maximum stress is compared to the yieldstrength, as usual.

For a quick, conservative check, an estimate for max can be obtained bysimply adding a and m .

( a + m ) will always be greater than or equal to max, and will therefore beconservative.

Shaft stresses


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Shaft stresses

D /d = 42/28 = 1.50, r /d= 2.8/28 = 0.10

K t = 1.68

K ts = 1.42

Shaft stresses


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Shaft stresses

r = 2.8 mm, q = 0.85

r = 2.8 mm, q = 0.92

Shaft stresses


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Shaft stresses

Note: Torque is steady (constant) while the bending moment is completely reversed.

Shaft stresses


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Shaft stresses

Shaft stresses


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21

2

3

2

3

2122 16

332

3 /

m fsm f /

mmm d

T K d

M K

21

2

3

2

3

2122 16

332

3

/

a fsa f /

aaa d

T K d

M K

Shaft stresses


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Shaft stresses


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21

2

3

2

3max

163

32 /

am fsam f

d

T (T K d

M M K

Shaft stresses


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Estimating Stress Concentrations


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- The stress analysis process for fatigue is highly dependent on stressconcentrations.

- Shoulders for bearing and gear support should match the catalogrecommendation for the specific bearing or gear. A typical bearing calls forthe ratio of D/d to be between 1.2 and 1.5. For a first approximation, theworst case of 1.5 can be assumed.

- There is a significant variation in typical bearings in the ratio of filletradius versus bore diameter, with r /d typically ranging from around 0.02to 0.06. This is an area where some attention to detail could make a significantdifference.

- Fortunately, in most cases the shear and bending moment diagrams show that bending moments are quite low near the bearings, since the bending momentsfrom the ground reaction forces are small.



50/90



- In cases where the shoulder at the bearing is found to be critical, the designershould plan to select a bearing with generous fillet radius, or consider providing

for a larger fillet radius on the shaft by relieving it into the base of the shoulder.This effectively creates a dead zone in the shoulder area that does not carry the bending stresses, as shown by the stress flow lines.

Techniques for reduci ng str essconcentr ation at a shoul dersupporting a beari ng wi th a sharpradius.

(a) Large radius undercutinto the shoulder.



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- A shoulder relief groove can accomplish a similar purpose.

Techniques for reduci ng str essconcentr ation at a shoul dersupporting a beari ng wi th a sharpradius.

(b) Large radius rel ief gr oove intothe back of the shoulder.



52/90



- Another option is to cut a large-radius relief groove into the small diameter of theshaft. This has the disadvantage of reducing the cross-sectional area, but is often

used in cases where it is useful to provide a relief groove before the shoulder to prevent the grinding or turning operation from having to go all the way to theshoulder.

Techniques for r educi ng str essconcentr ation at a shoul dersupporti ng a beari ng with a sharpradius.

(c) Large radius rel ief groove intothe smal l diameter.



53/90



- For the standard shoulder fillet, for estimating K t values for the first iteration, anr /d ratio should be selected so K t values can be obtained.

F irst I teration Estimates for Stress Concentr ation F actors K t

Warni ng: Th ese factors are onl y estimates for use when actual dimensions arenot yet determi ned. Do not use these once actual dimensions are avail able.


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55/90



56/90



57/90



58/90


Start with Point I , where the bending


59/90


gmoment is high, there is a stressconcentration at the shoulder, and thetorque is present.

At I , M a = 468 Nm T m = 360 Nm

M m = T a = 0

Point I


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Note: use q =1.0 for conservative reason.

ut e S .S 50

well-rounded


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62/90



63/90


ut e S .S 50


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Note that this equation can be used directly

M m = T a = 0

Also check this diameter at the end of


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so c ec t s d a ete at t e e d othe keyway, just to the right of point I .

From moment diagram, estimate M at

end of keyway to be M = 443 Nm.

From torque diagram, T = 360 Nm

End of the keyway


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End-m ill keyseat

Table 7-1

Table 7-1

360

- The keyway turns out to be more critical than the shoulder. We can either increaseh d h h h l


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the diameter, or use a higher strength material.- Unless the deflection analysis shows a need for larger diameters, let us choose toincrease the strength.- Try 1050 CD, with S ut = 690 MPa.- Recalculate factors affected by S ut , i.e. k a S e; q K f a

Since the Goodman cr iterion i s conservative, we wil l accept this asclose enough to the requested 1.5.

Also check this diameter at the groove at


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g K .

From moment diagram, M at the groove is M = 283 Nm.

From torque diagram, T = 0.

Groove K


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The data for a specific retaining ring toobtain K f more accurately is needed.With a quick on-line search of aretaining ring specification using thewebsite www.globalspec.com .

This is low.

Appropriate groove specifications for a retaining ring for a shaft diameter of 42 mmare obtained as follows: width, a = 1.73 mm; depth, t = 1.22 mm; and corner radiusat bottom of groove, r = 0.25 mm.

Reta in ing r ing gro ove

with r /t = 0.25/1.22 = 0.205
http://www.globalspec.com/http://www.globalspec.com/


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and a /t = 1.73/1.22 = 1.42

n f > 1.5 OK

Quickly check if point M might be critical.


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Only bending is present, and the momentis small, but the diameter is small and the

stress concentration is high for a sharp

fi l let requir ed for a bearing .From the moment diagram,

M a = 113 Nm, M m = T m = T a = 0.

Point M

M

Sharp fillet (bearing)


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p ( g)

M m = T m = T a = 0


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- Deflection analysis at even a single point of interest requires complete geometryi f i f h i h f


74/90

ME Deflection Considerations

information for the entire shaft.

- Deflection of the shaft, both linear and angular, should be checked at gears and bearings.

- Allowable deflections will depend on many factors, and bearing and gear catalogsshould be used for guidance on allowable misalignment for specific bearings andgears.

- The allowable transversedeflections for spur gears aredependent on the size of theteeth, as represented by thediametral pitch P number ofteeth/pitch diameter.

As a rough guidel ine, typical rangesfor maxi mum slopes and tr ansversedef lections of the shaf t center l ineare given i n the table.

For shafts where the deflections may be sought at a number of different points


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- For shafts, where the deflections may be sought at a number of different points,integration using either singularity functions or numerical integration is

practical .

- In a stepped shaft , the cross sectional properties change along the shaft at eachstep, increasing the complexity of integration, since both M and I vary .

- Fortunately, only the gross geometric dimensions need to be included, as thelocal factors such as fillets, grooves, and keyways do not have much impact ondeflection .

- Many shafts will include forces in multiple planes, requiring either a threedimensional analysis, or the use of superposition to obtain deflections in two

planes which can then be summed as vectors.

- A deflection is lengthy and tedious to carry out manually. Consequently,

practically all shaft deflection analysis will be evaluated with the assistance ofsoftware. This is practical if the designer is already familiar with using the software and with how to properly model the shaft .

- Once deflections at various points have been determined, if any value is larger


76/90


p , y gthan the allowable deflection at that point, a new diameter can be found from

where yall is the allowable deflection at that station and nd is the design factor.Similarly, if any slope is larger than the allowable slope all , a new diameter can

be found from

where (slope) all is the allowable slope. As a result of these calculations, determinethe largest d new /d old ratio, then multiply all diameters by this ratio.

41 /

all

old d old new y

ynd d

41 /

all

old d old new slope

dy/dxnd d


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Another potential problem however is called critical speeds : at certain


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ME Critical Speeds for Shafts

- Another potential problem, however, is called critical speeds : at certainspeeds the shaft is unstable, with deflections increasing without upper bound.

- Designers seek first critical speeds at least twice the operating speed.

- The shaft, because of its own mass, has a critical speed. The ensemble ofattachments to a shaft likewise has a critical speed that is much lower than theshafts intrinsic critical speed. When geometry is simple, as in a shaft ofuniform diameter, simply supported , it can be expressed as

where m is the mass per unit length, A the cross-sectional area, and thespecific weight.

A gEI

l

m EI

l

22

1

- Two different methods of calculating critical speed will be introduced, i.e.,


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Rayleighs method and Dunkerleys equation .

- For an ensemble of attachments, Rayleighs method for lumped masses gives

where wi is the weight of the ith location and yi is the deflection at theith body location.

- It is possible to partition the shaft into

segments and placing its weight force atthe segment centroid as seen in thefigure.

Rayleighs method

21

ii

ii

yw

yw g

Rayleighs method


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- We can use influence coefficients . An influence coefficient is the transversedeflection at location i due to a unit load at location j on the shaft ( ij ).

From the tables in the appendix

ii22ii

222

a x26

a x 6

i jii j

i ji j

j

iij

xalx EIl

xl a

xbl EIl

xb

F

y

Rayleighs method


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For three loads the influence coefficients may be displayed as

Maxwells reciprocity theorem states that there is a symmetry about the maindiagonal, composed of 11, 22, and 33, of the form ij = ji. This relation reducesthe work of finding the influence coefficients.

From the influence coefficients above, the deflections y1, y2, and y3 become:

1331221111 F F F y

3333223113 F F F y

2332222112 F F F y

Rayleighs method


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333

2

3322

2

2311

2

13 ym ym ym y

The forces F i can arise from weight attached wi or centrifugal forces mi 2 yi ofeach component, e.g., the gears attached on the shaft.

The deflections can be written with inertial forces as

133

2

3122

2

2111

2

11 ym ym ym y

233

2

3222

2

2211

2

12 ym ym ym y

Dunkerleys equation


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If we had only a single mass m1 alone, the critical speed would be given by

111

1112

11

1

g w

m

2

33

2

22

2

11

2

3

2

2

2

1

111111

2

33

2

22

2

11

2

1

1111

Likewise for m2 or m3 acting alone, we will obtain:

If we order the critical speeds such that 1 < 2 < 3. So the first, or fundamental,critical speed 1 can be approximated by

This idea can be extended to an n-body shaft:n

i ii 1

22

1

11


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There are two loads from the weight off


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gears, from

xbl EIl

xb

F

y

i ji j

j

iij

222

6

Eq. 7-24

Rayleighs method


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Rayleigh s method

21

ii

ii

yw

yw g

Dunkerleys equation


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111

1112

11

1

g

wm

Dunkerleys equation


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n

i ii 1

22

1

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The shaft, because of its own mass, has a critical speed. The ensemble ofattachments to a shaft likewise has a critical speed that is much lower than the


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attachments to a shaft likewise has a critical speed that is much lower than theshafts intrinsic critical speed. When geometry is simple, as in a shaft of uniformdiameter, simply supported, it can be expressed as

This value should be added in the critical speed approximated byDunkerleys equation

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7.Shafts and Shaft Components

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Transcript of 7.Shafts and Shaft Components