Shaft Equations:Shaft Equations:All of the following equations are general equations; you All of the following equations are general equations; you may need to use modifying factors such as: loading factors, may need to use modifying factors such as: loading factors, pulsating power source factors, safety factors, and stress pulsating power source factors, safety factors, and stress concentration factors.concentration factors.

Basic equations in torsion:Basic equations in torsion:Solid round shaft:Solid round shaft:

3

16

D

T

Hollow round shaft:Hollow round shaft:

)(

1644

0

0

iDD

DT

Basic equation in bending:Basic equation in bending:Solid shaft:Solid shaft:

3

32

D

M

Hollow shaft: Hollow shaft:

)(

3244io

o

DD

DM

Combined loading (solid shaft):Combined loading (solid shaft):

22

3max)8()8(

2TDFM

D

(18-4) max sheer(18-4) max sheer stress stress††

22

348)8(

4' TDFM

D

(18-5) Von Mises stress(18-5) Von Mises stress††

Torsional deflection:Torsional deflection:

4

32

DG

LT

RadiansRadiansFactors of Safety:Factors of Safety:

n

Sy

all 2

(18-6) Max sheer stress(18-6) Max sheer stress theory theory††

n

Sy'

(18-7)(18-7) Distortion energyDistortion energy††

T= torque (lb-in, N-m)T= torque (lb-in, N-m)F= axial load (lb, N)F= axial load (lb, N)Sy= yield strengthSy= yield strengthn = factor of safety n = factor of safety = sheer stress (psi, Pa)= sheer stress (psi, Pa)D= diameter of solid shaft (in, m)D= diameter of solid shaft (in, m)DDo o = outside diameter of solid shaft (in, m)= outside diameter of solid shaft (in, m)

DDii= inside diameter of solid shaft (in, m)= inside diameter of solid shaft (in, m)

M= bending moment (lb-in, N-m)M= bending moment (lb-in, N-m)L= length of shaft (in, m)L= length of shaft (in, m)G= sheer modulus (psi, Pa)G= sheer modulus (psi, Pa)

Use of ShaftsUse of ShaftsA machine is a device that converts some sort of energy into work. In many machines transfer of A machine is a device that converts some sort of energy into work. In many machines transfer of power (energy with respect to time) is needed in order to perform this task. Shafts are efficient power (energy with respect to time) is needed in order to perform this task. Shafts are efficient

devices for transferring power and can commonly be found in machines world wide.devices for transferring power and can commonly be found in machines world wide.Shaft DefinitionsShaft Definitions

Shaft- Shaft- A rotating member used to transmit power.A rotating member used to transmit power.Axle- Axle- A stationary member used as support for rotating elements such as wheels, idler gears, etc.A stationary member used as support for rotating elements such as wheels, idler gears, etc.

Spindle- Spindle- A short shaft or axle (e.g., head-stock spindle of a lathe).A short shaft or axle (e.g., head-stock spindle of a lathe).Stub shaft- Stub shaft- A shaft that is integral with a motor, engine or prime mover and is of a size, shape, and A shaft that is integral with a motor, engine or prime mover and is of a size, shape, and

projection as to permit easy connection to other shaftsprojection as to permit easy connection to other shaftsLine shaft- Line shaft- A shaft connected to a prime mover and used to transmit power to one or several A shaft connected to a prime mover and used to transmit power to one or several

machinesmachinesJackshaft- Jackshaft- (Sometimes called countershaft). A short shaft that connects a prime mover with a line (Sometimes called countershaft). A short shaft that connects a prime mover with a line

shaft or a machineshaft or a machineFlexible shaft- Flexible shaft- A connector which permits transmission of motion between two members whose axes A connector which permits transmission of motion between two members whose axes

are at an angle with each otherare at an angle with each otherShapesShapes

Most shafts are round but they can come in many different shapes including square and octagonal. Most shafts are round but they can come in many different shapes including square and octagonal. Keys and notches can also result in some unique shapes.Keys and notches can also result in some unique shapes.

Hollow Versus Solid ShaftsHollow Versus Solid ShaftsHollow shafts are lighter than solid shafts of comparable strength but are more expensive to Hollow shafts are lighter than solid shafts of comparable strength but are more expensive to

manufacture. Thusly hollow shafts are primarily only used when weight is critical. For example the manufacture. Thusly hollow shafts are primarily only used when weight is critical. For example the propeller shafts on rear wheel drive cars require lightweight shafts in order to handle speeds within propeller shafts on rear wheel drive cars require lightweight shafts in order to handle speeds within

the operating range of the vehicle.the operating range of the vehicle.

MaterialMaterial ProcessingProcessing TipsTips

o To resist wear, case-hardening methods such as, nitriding, cyaniding, flame and induction To resist wear, case-hardening methods such as, nitriding, cyaniding, flame and induction hardening can be used.hardening can be used.

o Cold-drawn steel bars have better physical properties than hot-rolled bars of similar steels. Cold-drawn steel bars have better physical properties than hot-rolled bars of similar steels. Cold-drawing causes residual surface stresses that offset higher endurance strength due to Cold-drawing causes residual surface stresses that offset higher endurance strength due to hot-rolling.hot-rolling.

o Cutting keyways and slots in the shaft may cause warping due to the relief of surface Cutting keyways and slots in the shaft may cause warping due to the relief of surface stresses.stresses.

o Peening and other processes that produce surface compressive stresses counteract the Peening and other processes that produce surface compressive stresses counteract the effect of fatigue stress.effect of fatigue stress.

Rules of DeflectionRules of Deflection

o Deflections should not cause mating gear teeth to separate more Deflections should not cause mating gear teeth to separate more than about .005 in. They should also not cause the relative slope than about .005 in. They should also not cause the relative slope of the gear axes to change more than .03 deg.of the gear axes to change more than .03 deg.

o The shaft deflection across a plain bearing must be small The shaft deflection across a plain bearing must be small compared to the oil film thickness.compared to the oil film thickness.

o The shaft angular deflection at a ball or roller bearing should The shaft angular deflection at a ball or roller bearing should generally not exceed .04 deg. unless the bearing is self-aligning.generally not exceed .04 deg. unless the bearing is self-aligning.

o Rule Of Thumb: Restrict the torsional deflection to 1° for every 20 Rule Of Thumb: Restrict the torsional deflection to 1° for every 20 diameters of length, sometimes less.diameters of length, sometimes less.

o Rule Of Thumb: In bending the deflection should be limited to .01 Rule Of Thumb: In bending the deflection should be limited to .01 in. per foot of length between supports.in. per foot of length between supports.

General PrinciplesGeneral Principles

o Keep shafts short, with bearings close to the applied loads. This will reduce deflections and Keep shafts short, with bearings close to the applied loads. This will reduce deflections and bending moments, and increases critical speeds.bending moments, and increases critical speeds.

o Place necessary stress raisers away from highly stressed shaft regions if possible. If Place necessary stress raisers away from highly stressed shaft regions if possible. If unavoidable, use generous radii and good surface finishes. Consider local surface-unavoidable, use generous radii and good surface finishes. Consider local surface-strengthening processes (shot-peening or cold-rolling).strengthening processes (shot-peening or cold-rolling).

o Use inexpensive steels for deflection-critical shafts because all steels have essentially the Use inexpensive steels for deflection-critical shafts because all steels have essentially the same modulus of elasticity.same modulus of elasticity.

o Early in the design of any given shaft, an estimate is usually made of whether strength or Early in the design of any given shaft, an estimate is usually made of whether strength or deflection will be the critical factor. A preliminary design is based on that criterion; then, the deflection will be the critical factor. A preliminary design is based on that criterion; then, the remaining factor is checked.remaining factor is checked.

Common Means of Securing ShaftsCommon Means of Securing Shafts††

o Press and shrink fitsPress and shrink fits

o Cotter and washerCotter and washer

o Nut and washerNut and washer

o SleeveSleeve

o Shaft shoulderShaft shoulder

o Ring and grooveRing and groove

o SetscrewSetscrew

o Split hub or tapered two-Split hub or tapered two-piece hubpiece hub

o Collar and screwCollar and screw

o PinsPins

Common Elements Used To Transfer TorqueCommon Elements Used To Transfer Torque††

o KeysKeys

o SplinesSplines

o SetscrewsSetscrews

o PinsPins

o Press or shrink fitsPress or shrink fits

o Tapered fitsTapered fits

Keyed ShaftsKeyed Shafts

www.robotcombat.comwww.robotcombat.com

www.theadamscompany.com/ shafts.htm www.theadamscompany.com/ shafts.htm

REFERENCE:REFERENCE:

† † Shigley, Joseph Edward, and Charles R. Mischke. Shigley, Joseph Edward, and Charles R. Mischke. Mechanical Engineering Design. Fifth Edition. Mechanical Engineering Design. Fifth Edition. Boston: McGraw Hill, 2002.Boston: McGraw Hill, 2002.

POSTER BY:POSTER BY:

AUSTIN HOWARDAUSTIN HOWARD

BRADY CALVERTBRADY CALVERT

ERIK VAN PATTENERIK VAN PATTEN