3SHAFTThe term shaft usually refers to a component of circular cross-section that rotates and transmits power from a driving device, such as a motor or engine, through a machine.Shafts can carry gears, pulleys and sprockets to transmit rotary motion and power via mating gears, belts and chains.A shaft may simply connect to another via a coupling.A shaft can be stationary and support a rotating member such as the short shafts which support the non-driven wheels of automobiles, often referred to as spindles.Shigley’s Mechanical Engineering Design
4SHAFTA shaft is a rotating member usually of circular cross-section (solid or hollow), which transmits power and rotational motion.Machine elements such as gears, pulleys (sheaves), flywheels, clutches, and sprockets are mounted on the shaft and are used to transmit power from the driving device (motor or engine) through a machine.Press fit, keys, dowel, pins and splines are used to attach these machine elements on the shaft.The shaft rotates on rolling contact bearings or bush bearings.Various types of retaining rings, thrust bearings, grooves and steps in the shaft are used to take up axial loads and locate the rotating elements.Couplings are used to transmit power from drive shaft (e.g., motor) to the driven shaft (e.g. gearbox, wheels).Shigley’s Mechanical Engineering Design
5SHAFT Shaft design considerations include: 1. Size and spacing of components (as on a general assembly drawing), tolerances2. Material selection and material treatments3. Deflection and rigiditybending deflectiontorsional deflectionslope at bearingsshear deflection4. Stress and strengthstatic strengthfatiguereliability5. Frequency response6. Manufacturing constraintsShigley’s Mechanical Engineering Design
7Shaft- A rotating member used to transmit power. Shaft TypesShaft- A rotating member used to transmit power.Axle- A stationary member used as support for rotating elements such as wheels, idler gears, etc.Spindle- A short shaft or axle (e.g., head-stock spindle of a lathe).Stub shaft- 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 shaftsLine shaft- A shaft connected to a prime mover and used to transmit power to one or several machinesJackshaft- (Sometimes called countershaft). A short shaft that connects a prime mover with a line shaft or a machineFlexible shaft- A connector which permits transmission of motion between two members whose axes are at an angle with each otherShigley’s Mechanical Engineering Design
8Hollow Versus Solid Shafts Shaft DesignMost 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.Hollow Versus Solid ShaftsHollow 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 propeller shafts on rear wheel drive cars require lightweight shafts in order to handle speeds within the operating range of the vehicle.Shigley’s Mechanical Engineering Design
11Shaft DesignMaterial SelectionGeometric LayoutStress and strengthStatic strengthFatigue strengthDeflection and rigidityBending deflectionTorsional deflectionSlope at bearings and shaft-supported elementsShear deflection due to transverse loading of short shaftsVibration due to natural frequency
12Shaft MaterialsDeflection primarily controlled by geometry, not materialStress controlled by geometry, not materialStrength controlled by material property
13Shaft MaterialsShafts are commonly made from low carbon, CD or HR steel, such as ANSI 1020–1050 steels.Fatigue properties don’t usually benefit much from high alloy content and heat treatment.Surface hardening usually only used when the shaft is being used as a bearing surface.
16Shaft Materials Hot Rolled Hot rolling is a mill process which involves rolling the steel at a high temperature (typically at a temperature over 1700° F), which is above the steel’s recrystallization temperature. When steel is above the recrystallization temperature, it can be shaped and formed easily, and the steel can be made in much larger sizes. Hot rolled steel is typically cheaper than cold rolled steel due to the fact that it is often manufactured without any delays in the process, and therefore the reheating of the steel is not required (as it is with cold rolled). When the steel cools off it will shrink slightly thus giving less control on the size and shape of the finished product when compared to cold rolled.Uses: Hot rolled products like hot rolled steel bars are used in the welding and construction trades to make railroad tracks and I-beams, for example. Hot rolled steel is used in situations where precise shapes and tolerances are not required.
17Shaft Materials Cold Rolled Cold rolled steel is essentially hot rolled steel that has had further processing. The steel is processed further in cold reduction mills, where the material is cooled (at room temperature) followed by annealing and/or tempers rolling. This process will produce steel with closer dimensional tolerances and a wider range of surface finishes. The term Cold Rolled is mistakenly used on all products, when actually the product name refers to the rolling of flat rolled sheet and coil products.When referring to bar products, the term used is “cold finishing”, which usually consists of cold drawing and/or turning, grinding and polishing. This process results in higher yield points and has four main advantages:
18Shaft Materials Cold Rolled Cold drawing increases the yield and tensile strengths, often eliminating further costly thermal treatments.Turning gets rid of surface imperfections.Grinding narrows the original size tolerance range.Polishing improves surface finish.All cold products provide a superior surface finish, and are superior in tolerance, concentricity, and straightness when compared to hot rolled.Cold finished bars are typically harder to work with than hot rolled due to the increased carbon content. However, this cannot be said about cold rolled sheet and hot rolled sheet. With these two products, the cold rolled product has low carbon content and it is typically annealed, making it softer than hot rolled sheet.Uses: Any project where tolerances, surface condition, concentricity, and straightness are the major factors
19Shaft MaterialsCold drawn steel typical for d < 3 in.HR steel common for larger sizes. Should be machined all over.Low production quantitiesLathe machining is typicalMinimum material removal may be design goalHigh production quantitiesForming or casting is commonMinimum material may be design goal
20Issues to consider for shaft layout Axial layout of components Supporting axial loadsProviding for torque transmissionAssembly and DisassemblyFig. 7-1
21Axial Layout of Components Fig. 7-2Choose a shaft configuration to support and locate the two gears and two bearings.(b) Solution uses an integral pinion, three shaft shoulders, key and keyway, and sleeve. The housing locates the bearings on their outer rings and receives the thrust loads.(c) Choose fanshaft configuration.(d) Solution uses sleeve bearings, a straight-through shaft, locating collars, and setscrews for collars, fan pulley, and fan itself. The fan housing supports the sleeve bearings.
22Supporting Axial Loads Axial loads must be supported through a bearing to the frame.Generally best for only one bearing to carry axial load to shoulderAllows greater tolerances and prevents bindingFig. 7-3Fig. 7-4
23Providing for Torque Transmission Common means of transferring torque to shaftKeysSplinesSetscrewsPinsPress or shrink fitsTapered fitsKeys are one of the most effectiveSlip fit of component onto shaft for easy assemblyPositive angular orientation of componentCan design key to be weakest link to fail in case of overload
24Assembly and Disassembly Arrangement showing bearinginner rings press-fitted to shaftwhile outer rings float in thehousing. The axial clearanceshould be sufficient only toallow for machinery vibrations.Note the labyrinth seal on theright.Fig. 7-5Similar to the arrangement ofFig. 7–5 except that the outerbearing rings are preloaded.Fig. 7-6
25Assembly and Disassembly In this arrangement the inner ring of the left-hand bearing is locked to the shaft between a nut and a shaft shoulder. The locknut and washer are AFBMA standard. The snap ring in the outer race is used to positively locate the shaft assembly in the axial direction.Note the floating right-hand bearing and the grinding runout grooves in the shaft.Fig. 7-7This arrangement is similar toFig. 7–7 in that the left-hand bearing positions the entire shaft assembly.In this case the inner ring is secured tothe shaft using a snap ring.Note the use of a shield to prevent dirt generated from within the machine from entering the bearingFig. 7-8Shigley’s Mechanical Engineering Design
26SHAFTS: General Design Rules Keep shafts short, with bearings close to the applied loads. This will reduce deflections and bending moments, and increases critical speeds.Place necessary stress raisers away from highly stressed shaft regions if possible. If unavoidable, use generous radii and good surface finishes. Consider local surface-strengthening processes (shot-peening or cold-rolling).Use inexpensive steels for deflection-critical shafts because all steels have essentially the same modulus of elasticity.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 remaining factor is checked.Shigley’s Mechanical Engineering Design
27SHAFTS: Rules of Deflection Deflections should not cause mating gear teeth to separate more than about .005 in. They should also not cause the relative slope of the gear axes to change more than .03 deg.The shaft deflection across a plain bearing must be small compared to the oil film thickness.The shaft angular deflection at a ball or roller bearing should generally not exceed .04 deg. unless the bearing is self-aligning.Rule Of Thumb: Restrict the torsional deflection to 1° for every 20 diameters of length, sometimes less.Rule Of Thumb: In bending the deflection should be limited to .01 in. per foot of length between supports.Shigley’s Mechanical Engineering Design
28Shaft design procedure for a shaft experiencing constant loading Determine the shaft rotational speed.Determine the power or torque to be transmitted by the shaft.Determine the dimensions of the power transmitting devices and other components mounted on the shaft and specify locations for each device.Specify the locations of the bearings to support the shaft.Propose a general form or scheme for the shaft geometry, considering how each component will be located axially and how power transmission will take place.Determine the magnitude of the torques throughout the shaft.Determine the forces exerted on the shaft.Produce shearing force and bending moment diagrams so that the distribution of bending moments in the shaft can be determined.Select a material for the shaft and specify any heat treatments etc.Determine an appropriate design stress, taking into account the type of loading (whether smooth, shock, repeated, reversed).Analyse all the critical points on the shaft and determine the minimum acceptable diameter at each point to ensure safe design.Specify the final dimensions of the shaft. This is best achieved using a detailed manufacturing drawing, and the drawing should include all the information required to ensure the desired quality. Typically this will include material specifications, dimensions and tolerances (bilateral, runout, datums etc.;), surface finishes, material treatments and inspection procedures.Shigley’s Mechanical Engineering Design
34General principles that should be observed in shaft design Keep shafts as short as possible with the bearings close to applied loads. This will reduce shaft deflection and bending moments and increase critical speeds.If possible, locate stress raisers away from highly stressed regions of the shaft. Use generous fillet radii and smooth surface finishes, and consider using local surface strengthening processes such cold rolling.If weight is critical use hollow shafts.Shigley’s Mechanical Engineering Design
35General principles that should be observed in shaft design Shigley’s Mechanical Engineering Design
36Shaft Design for Stress Stresses are only evaluated at critical locationsCritical locations are usuallyOn the outer surfaceWhere the bending moment is largeWhere the torque is presentWhere stress concentrations exist
37Shaft StressesStandard stress equations can be customized for shafts for convenienceAxial loads are generally small and constant, so will be ignored in this sectionStandard alternating and midrange stresses
39Shaft StressesCombine stresses into von Mises stresses
40Shaft StressesSubstitute von Mises stresses into failure criteria equation. For example, using modified Goodman line,Solving for d is convenient for design purposes
41Shaft StressesSimilar approach can be taken with any of the fatigue failure criteriaEquations are referred to by referencing both the Distortion Energy method of combining stresses and the fatigue failure locus name. For example, DE-Goodman, DE-Gerber, etc.In analysis situation, can either use these customized equations for factor of safety, or can use standard approach from Ch. 6.In design situation, customized equations for d are much more convenient.
44Shaft Stresses for Rotating Shaft For rotating shaft with steady bending and torsionBending stress is completely reversed, since a stress element on the surface cycles from equal tension to compression during each rotationTorsional stress is steadyPrevious equations simplify with Mm and Ta equal to 0
45Checking for Yielding in Shafts Always necessary to consider static failure, even in fatigue situationSoderberg criteria inherently guards against yieldingASME-Elliptic criteria takes yielding into account, but is not entirely conservativeGerber and modified Goodman criteria require specific check for yielding
46Checking for Yielding in Shafts Use von Mises maximum stress to check for yielding,Alternate simple check is to obtain conservative estimate of s'max by summing s'a and s'm
52Estimating Stress Concentrations Stress analysis for shafts is highly dependent on stress concentrations.Stress concentrations depend on size specifications, which are not known the first time through a design process.Standard shaft elements such as shoulders and keys have standard proportions, making it possible to estimate stress concentrations factors before determining actual sizes.
54Reducing Stress Concentration at Shoulder Fillet Bearings often require relatively sharp fillet radius at shoulderIf such a shoulder is the location of the critical stress, some manufacturing techniques are available to reduce the stress concentrationLarge radius undercut into shoulderLarge radius relief groove into back of shoulderLarge radius relief groove into small diameter of shaftFig. 7-9
69Deflection Considerations Deflection analysis at a single point of interest requires complete geometry information for the entire shaft.For this reason, a common approach is to size critical locations for stress, then fill in reasonable size estimates for other locations, then perform deflection analysis.Deflection of the shaft, both linear and angular, should be checked at gears and bearings.
70Deflection Considerations Allowable deflections at components will depend on the component manufacturer’s specifications.Typical ranges are given in Table 7–2
71Deflection Considerations Deflection analysis is straightforward, but lengthy and tedious to carry out manually.Each point of interest requires entirely new deflection analysis.Consequently, shaft deflection analysis is almost always done with the assistance of software.Options include specialized shaft software, general beam deflection software, and finite element analysis software.
77Adjusting Diameters for Allowable Deflections If any deflection is larger than allowed, since I is proportional to d4, a new diameter can be found fromSimilarly, for slopes,Determine the largest dnew/dold ratio, then multiply all diameters by this ratio.The tight constraint will be just right, and the others will be loose.
79Angular Deflection of Shafts For stepped shaft with individual cylinder length li and torque Ti, the angular deflection can be estimated fromFor constant torque throughout homogeneous materialExperimental evidence shows that these equations slightly underestimate the angular deflection.Torsional stiffness of a stepped shaft is
80Critical Speeds for Shafts A shaft with mass has a critical speed at which its deflections become unstable.Components attached to the shaft have an even lower critical speed than the shaft.Designers should ensure that the lowest critical speed is at least twice the operating speed.
81Critical Speeds for Shafts For a simply supported shaft of uniform diameter, the first critical speed isFor an ensemble of attachments, Rayleigh’s method for lumped masses gives
82Critical Speeds for Shafts Eq. (7–23) can be applied to the shaft itself by partitioning the shaft into segments.Fig. 7–12
83Critical Speeds for Shafts An influence coefficient is the transverse deflection at location I due to a unit load at location j.From Table A–9–6 for a simply supported beam with a single unit loadFig. 7–13
84Critical Speeds for Shafts Taking for example a simply supported shaft with three loads, the deflections corresponding to the location of each load isIf the forces are due only to centrifugal force due to the shaft mass,Rearranging,
85Critical Speeds for Shafts Non-trivial solutions to this set of simultaneous equations will exist when its determinant equals zero.Expanding the determinant,Eq. (7–27) can be written in terms of its three roots as
86Critical Speeds for Shafts Comparing Eqs. (7–27) and (7–28),Define wii as the critical speed if mi is acting alone. From Eq. (7–29),Thus, Eq. (7–29) can be rewritten as
87Critical Speeds for Shafts Note thatThe first critical speed can be approximated from Eq. (7–30) asExtending this idea to an n-body shaft, we obtain Dunkerley’s equation,
88Critical Speeds for Shafts Since Dunkerley’s equation has loads appearing in the equation, it follows that if each load could be placed at some convenient location transformed into an equivalent load, then the critical speed of an array of loads could be found by summing the equivalent loads, all placed at a single convenient location.For the load at station 1, placed at the center of the span, the equivalent load is found from
96Setscrews resist axial and rotational motion They apply a compressive force to create frictionThe tip of the set screw may also provide a slight penetrationVarious tips are availableFig. 7–15
97SetscrewsResistance to axial motion of collar or hub relative to shaft is called holding powerTypical values listed in Table 7–4 apply to axial and torsional resistanceTypical factors of safety are 1.5 to 2.0 for static, and 4 to 8 for dynamic loadsLength should be about half the shaft diameter
98Used to secure rotating elements and to transmit torque Keys and PinsUsed to secure rotating elements and to transmit torqueFig. 7–16
99Taper pins are sized by diameter at large end Small end diameter is Tapered PinsTaper pins are sized by diameter at large endSmall end diameter isTable 7–5 shows some standard sizes in inchesTable 7–5
100Keys come in standard square and rectangular sizes Shaft diameter determines key sizeTable 7–6
101KeysFailure of keys is by either direct shear or bearing stressKey length is designed to provide desired factor of safetyFactor of safety should not be excessive, so the inexpensive key is the weak linkKey length is limited to hub lengthKey length should not exceed 1.5 times shaft diameter to avoid problems from twistingMultiple keys may be used to carry greater torque, typically oriented 90º from one anotherStock key material is typically low carbon cold-rolled steel, with dimensions slightly under the nominal dimensions to easily fit end-milled keywayA setscrew is sometimes used with a key for axial positioning, and to minimize rotational backlash
102Head makes removal easy Projection of head may be hazardous Gib-head KeyGib-head key is tapered so that when firmly driven it prevents axial motionHead makes removal easyProjection of head may be hazardousFig. 7–17Shigley’s Mechanical Engineering Design
103Woodruff keys have deeper penetration Useful for smaller shafts to prevent key from rollingWhen used near a shoulder, the keyway stress concentration interferes less with shoulder than square keywayFig. 7–17
106Stress Concentration Factors for Keys For keyseats cut by standard end-mill cutters, with a ratio of r/d = 0.02, Peterson’s charts giveKt = 2.14 for bendingKt = 2.62 for torsion without the key in placeKt = 3.0 for torsion with the key in placeKeeping the end of the keyseat at least a distance of d/10 from the shoulder fillet will prevent the two stress concentrations from combining.
110Retaining RingsRetaining rings are often used instead of a shoulder to provide axial positioningFig. 7–18
111This requires sharp radius in bottom of groove. Retaining RingsRetaining ring must seat well in bottom of groove to support axial loads against the sides of the groove.This requires sharp radius in bottom of groove.Stress concentrations for flat-bottomed grooves are available in Table A–15–16 and A–15–17.Typical stress concentration factors are high, around 5 for bending and axial, and 3 for torsionShigley’s Mechanical Engineering Design
112Limits and FitsShaft diameters need to be sized to “fit” the shaft components (e.g. gears, bearings, etc.)Need ease of assemblyNeed minimum slopMay need to transmit torque through press fitShaft design requires only nominal shaft diametersPrecise dimensions, including tolerances, are necessary to finalize design
113TolerancesClose tolerances generally increase costRequire additional processing stepsRequire additional inspectionRequire machines with lower production ratesFig. 1–2
114Standards for Limits and Fits Two standards for limits and fits in use in United StatesU.S. Customary (1967)Metric (1978)Metric version will be presented, with a set of inch conversions
115Nomenclature for Cylindrical Fit Upper case letters refer to holeLower case letters refer to shaftBasic size is the nominal diameter and is same for both parts, D=dTolerance is the difference between maximum and minimum sizeDeviation is the difference between a size and the basic sizeFig. 7–20
116Tolerance Grade Number Tolerance is the difference between maximum and minimum sizeInternational tolerance grade numbers designate groups of tolerances such that the tolerances for a particular IT number have the same relative level of accuracy but vary depending on the basic sizeIT grades range from IT0 to IT16, but only IT6 to IT11 are generally neededSpecifications for IT grades are listed in Table A–11 for metric series and A–13 for inch series
119DeviationsDeviation is the algebraic difference between a size and the basic sizeUpper deviation is the algebraic difference between the maximum limit and the basic sizeLower deviation is the algebraic difference between the minimum limit and the basic sizeFundamental deviation is either the upper or lower deviation that is closer to the basic sizeLetter codes are used to designate a similar level of clearance or interference for different basic sizes
120Fundamental Deviation Letter Codes Shafts with clearance fitsLetter codes c, d, f, g, and hUpper deviation = fundamental deviationLower deviation = upper deviation – tolerance gradeShafts with transition or interference fitsLetter codes k, n, p, s, and uLower deviation = fundamental deviationUpper deviation = lower deviation + tolerance gradeHoleThe standard is a hole based standard, so letter code H is always used for the holeLower deviation = 0 (Therefore Dmin = 0)Upper deviation = tolerance gradeFundamental deviations for letter codes are shown in Table A–12 for metric series and A–14 for inch series
121Fundamental Deviations – Metric series Table A–12
122Fundamental Deviations – Inch series Table A–14
123Specification of FitA particular fit is specified by giving the basic size followed by letter code and IT grades for hole and shaft.For example, a sliding fit of a nominally 32 mm diameter shaft and hub would be specified as 32H7/g6This indicates32 mm basic sizehole with IT grade of 7 (look up tolerance DD in Table A–11)shaft with fundamental deviation specified by letter code g (look up fundamental deviation dF in Table A–12)shaft with IT grade of 6 (look up tolerance Dd in Table A–11)Appropriate letter codes and IT grades for common fits are given in Table 7–9
124Description of Preferred Fits (Clearance) Table 7–9
125Description of Preferred Fits (Transition & Interference) Table 7–9
126Procedure to Size for Specified Fit Select description of desired fit from Table 7–9.Obtain letter codes and IT grades from symbol for desired fit from Table 7–9Use Table A–11 (metric) or A–13 (inch) with IT grade numbers to obtain DD for hole and Dd for shaftUse Table A–12 (metric) or A–14 (inch) with shaft letter code to obtain dF for shaftFor holeFor shafts with clearance fits c, d, f, g, and hFor shafts with interference fits k, n, p, s, and u
129Stress in Interference Fits Interference fit generates pressure at interfaceNeed to ensure stresses are acceptableTreat shaft as cylinder with uniform external pressureTreat hub as hollow cylinder with uniform internal pressureThese situations were developed in Ch. 3 and will be adapted here
130Stress in Interference Fits The pressure at the interface, from Eq. (3–56) converted into terms of diameters,If both members are of the same material,d is diametral interferenceTaking into account the tolerances,
131Stress in Interference Fits From Eqs. (3–58) and (3–59), with radii converted to diameters, the tangential stresses at the interface areThe radial stresses at the interface are simplyThe tangential and radial stresses are orthogonal and can be combined using a failure theory
132Torque Transmission from Interference Fit Estimate the torque that can be transmitted through interference fit by friction analysis at interfaceUse the minimum interference to determine the minimum pressure to find the maximum torque that the joint should be expected to transmit.