Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft T©The McGraw-Hil 35 Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition 352 Mechanical Engineering Design Figure 7-3 Tapered roller bearings used in a mowing machine spindle. This design represents good practice for the situation in which one or more torque transfer elements must be mounted outboard.(Source. Redrawn from material furnished by The Timken Company.) Figure 7-4 A bevelgear drive in which both pinion and gear are straddle-mounted.(Source: Redrawn from material furnished by Gleason Machine Division.) ·Pins ·Press or shrink fits ·Tapered fits In addition to transmitting the torque,many of these devices are designed to fail if the torque exceeds acceptable operating limits,protecting more expensive components. Details regarding hardware components such as keys,pins,and setscrews are addressed in detail in Sec.7-7.One of the most effective and economical means of transmitting moderate to high levels of torque is through a key that fits in a groove in the shaft and gear.Keyed components generally have a slip fit onto the shaft,so assembly and disassembly is easy.The key provides for positive angular orientation of the component,which is useful in cases where phase angle timing is important
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components © The McGraw−Hill 355 Companies, 2008 352 Mechanical Engineering Design • Pins • Press or shrink fits • Tapered fits In addition to transmitting the torque, many of these devices are designed to fail if the torque exceeds acceptable operating limits, protecting more expensive components. Details regarding hardware components such as keys, pins, and setscrews are addressed in detail in Sec. 7–7. One of the most effective and economical means of transmitting moderate to high levels of torque is through a key that fits in a groove in the shaft and gear. Keyed components generally have a slip fit onto the shaft, so assembly and disassembly is easy. The key provides for positive angular orientation of the component, which is useful in cases where phase angle timing is important. Figure 7–3 Tapered roller bearings used in a mowing machine spindle. This design represents good practice for the situation in which one or more torquetransfer elements must be mounted outboard. (Source: Redrawn from material furnished by The Timken Company.) Figure 7–4 A bevel-gear drive in which both pinion and gear are straddle-mounted. (Source: Redrawn from material furnished by Gleason Machine Division.)
356 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft ©The McGraw-Hill Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition Shafts and Shaft Components 353 Splines are essentially stubby gear teeth formed on the outside of the shaft and on the inside of the hub of the load-transmitting component.Splines are generally much more expensive to manufacture than keys,and are usually not necessary for simple torque transmission.They are typically used to transfer high torques.One feature of a spline is that it can be made with a reasonably loose slip fit to allow for large axial motion between the shaft and component while still transmitting torque.This is use- ful for connecting two shafts where relative motion between them is common,such as in connecting a power takeoff(PTO)shaft of a tractor to an implement.SAE and ANSI publish standards for splines.Stress concentration factors are greatest where the spline ends and blends into the shaft,but are generally quite moderate. For cases of low torque transmission,various means of transmitting torque are available.These include pins,setscrews in hubs,tapered fits,and press fits. Press and shrink fits for securing hubs to shafts are used both for torque trans- fer and for preserving axial location.The resulting stress-concentration factor is usu- ally quite small.See Sec.7-8 for guidelines regarding appropriate sizing and toler- ancing to transmit torque with press and shrink fits.A similar method is to use a split hub with screws to clamp the hub to the shaft.This method allows for disassembly and lateral adjustments.Another similar method uses a two-part hub consisting of a split inner member that fits into a tapered hole.The assembly is then tightened to the shaft with screws.which forces the inner part into the wheel and clamps the whole assembly against the shaft. Tapered fits between the shaft and the shaft-mounted device,such as a wheel,are often used on the overhanging end of a shaft.Screw threads at the shaft end then permit the use of a nut to lock the wheel tightly to the shaft.This approach is useful because it can be disassembled,but it does not provide good axial location of the wheel on the shaft. At the early stages of the shaft layout,the important thing is to select an appro- priate means of transmitting torque,and to determine how it affects the overall shaft layout.It is necessary to know where the shaft discontinuities,such as keyways,holes, and splines,will be in order to determine critical locations for analysis. Assembly and Disassembly Consideration should be given to the method of assembling the components onto the shaft,and the shaft assembly into the frame.This generally requires the largest diam- eter in the center of the shaft,with progressively smaller diameters towards the ends to allow components to be slid on from the ends.If a shoulder is needed on both sides of a component,one of them must be created by such means as a retaining ring or by a sleeve between two components.The gearbox itself will need means to physi- cally position the shaft into its bearings,and the bearings into the frame.This is typ- ically accomplished by providing access through the housing to the bearing at one end of the shaft.See Figs.7-5 through 7-8 for examples. Figure 7-5 Arrangement showing bearing inner rings press-fitted to shoft while outer rings float in the 中加00 <00000加9 housing.The axial clearance should be sufficient only to allow for machinery vibrations. Note the labyrinth seal on the right
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components 356 © The McGraw−Hill Companies, 2008 Shafts and Shaft Components 353 Splines are essentially stubby gear teeth formed on the outside of the shaft and on the inside of the hub of the load-transmitting component. Splines are generally much more expensive to manufacture than keys, and are usually not necessary for simple torque transmission. They are typically used to transfer high torques. One feature of a spline is that it can be made with a reasonably loose slip fit to allow for large axial motion between the shaft and component while still transmitting torque. This is useful for connecting two shafts where relative motion between them is common, such as in connecting a power takeoff (PTO) shaft of a tractor to an implement. SAE and ANSI publish standards for splines. Stress concentration factors are greatest where the spline ends and blends into the shaft, but are generally quite moderate. For cases of low torque transmission, various means of transmitting torque are available. These include pins, setscrews in hubs, tapered fits, and press fits. Press and shrink fits for securing hubs to shafts are used both for torque transfer and for preserving axial location. The resulting stress-concentration factor is usually quite small. See Sec. 7–8 for guidelines regarding appropriate sizing and tolerancing to transmit torque with press and shrink fits. A similar method is to use a split hub with screws to clamp the hub to the shaft. This method allows for disassembly and lateral adjustments. Another similar method uses a two-part hub consisting of a split inner member that fits into a tapered hole. The assembly is then tightened to the shaft with screws, which forces the inner part into the wheel and clamps the whole assembly against the shaft. Tapered fits between the shaft and the shaft-mounted device, such as a wheel, are often used on the overhanging end of a shaft. Screw threads at the shaft end then permit the use of a nut to lock the wheel tightly to the shaft. This approach is useful because it can be disassembled, but it does not provide good axial location of the wheel on the shaft. At the early stages of the shaft layout, the important thing is to select an appropriate means of transmitting torque, and to determine how it affects the overall shaft layout. It is necessary to know where the shaft discontinuities, such as keyways, holes, and splines, will be in order to determine critical locations for analysis. Assembly and Disassembly Consideration should be given to the method of assembling the components onto 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 slid on from the ends. If a shoulder is needed on both sides of a component, one of them must be created by such means as a retaining ring or by a sleeve between two components. The gearbox itself 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. See Figs. 7–5 through 7–8 for examples. Figure 7–5 Arrangement showing bearing inner rings press-fitted to shaft while outer rings float in the housing. The axial clearance should be sufficient only to allow for machinery vibrations. Note the labyrinth seal on the right
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft ©The McGra-Hfl 357 Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition 354 Mechanical Engineering Design Figure 7-6 Similar to the arrangement of Fig.7-5 except that the outer bearing rings are preloaded. 咖I0 Figure 7-7 In this arrangement the inner ring of the lefthand 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 righthand bearing and the grinding runout grooves in the shaft. Figure 7-8 This arrangement is similar to Fig.7-7 in that the left-hand When components are to be press-fit to the shaft,the shaft should be designed bearing positions the entire so that it is not necessary to press the component down a long length of shaft.This shaft assembly.In this case the inner ring is secured to may require an extra change in diameter,but it will reduce manufacturing and assem- the shaft using a snap ring. bly cost by only requiring the close tolerance for a short length. Note the use of a shield to Consideration should also be given to the necessity of disassembling the compo- prevent dirt generated nents from the shaft.This requires consideration of issues such as accessibility of from within the machine from retaining rings,space for pullers to access bearings.openings in the housing to allow entering the bearing. pressing the shaft or bearings out,etc. 7-4 Shaft Design for Stress Critical Locations 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 where the bending moment is large,where the torque is present,and where stress concentrations exist.By direct comparison of various points along the shaft,a few critical locations can be identified upon which to base the design.An assessment of typical stress situations will help
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components © The McGraw−Hill 357 Companies, 2008 354 Mechanical Engineering Design Figure 7–6 Similar to the arrangement of Fig. 7--5 except that the outer bearing rings are preloaded. Figure 7–7 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. When components are to be press-fit to the shaft, the shaft should be designed so that it is not necessary to press the component down a long length of 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 the components from the shaft. This requires consideration of issues such as accessibility of retaining rings, space for pullers to access bearings, openings in the housing to allow pressing the shaft or bearings out, etc. 7–4 Shaft Design for Stress Critical Locations 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 where the bending moment is large, where the torque is present, and where stress concentrations exist. By direct comparison of various points along the shaft, a few critical locations can be identified upon which to base the design. An assessment of typical stress situations will help. Figure 7–8 This arrangement is similar to Fig. 7--7 in that the left-hand bearing positions the entire shaft assembly. In this case the inner ring is secured to the shaft using a snap ring. Note the use of a shield to prevent dirt generated from within the machine from entering the bearing
358 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft T©The McGraw-Hil Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition Shafts and Shaft Components 355 Most shafts will transmit torque through a portion of the shaft.Typically the torque comes into the shaft at one gear and leaves the shaft at another gear.A free body diagram of the shaft will allow the torque at any section to be determined.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 bending moment diagrams.Since most shaft problems incorporate gears or pulleys that intro- duce forces in two planes,the shear and bending moment diagrams will generally be needed in two planes.Resultant moments are obtained by summing moments as vectors at points of interest along the shaft.The phase angle of the moments is not important since the shaft rotates.A steady bending moment will produce a com- pletely reversed moment on a rotating shaft,as a specific stress element will alter- nate from compression to tension in every revolution of the shaft.The normal stress due to bending moments will be greatest on the outer surfaces.In situations where a bearing is located at the end of the shaft,stresses near the bearing are often not critical since the bending moment is small. Axial stresses on shafts due to the axial components transmitted through heli- cal gears or tapered roller bearings will almost always be negligibly small compared to the bending moment stress.They are often also constant,so they contribute lit- tle to fatigue.Consequently,it is usually acceptable to neglect the 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 assume it is negli- gible without checking magnitudes. Shaft Stresses Bending,torsion,and axial stresses may be present in both midrange and alternating components.For analysis,it is simple enough to combine the different types of stresses into alternating and midrange von Mises stresses,as shown in Sec.6-14, p.309.It is sometimes convenient to customize the equations specifically for shaft applications.Axial loads are usually comparatively very small at critical locations where bending and torsion dominate,so they will be left out of the following equa- tions.The fluctuating stresses due to bending and torsion are given by Mac 0a=K I Mmc Om=Kj 1 7-1) Tac ta =Kfs Tmc tm=K: (7-2) where Mm and Ma are the midrange and alternating bending moments,T and Ta are the midrange and alternating torques,and Kf and Kfs are the fatigue stress concen- tration factors for bending and torsion,respectively. Assuming a solid shaft with round cross section,appropriate geometry terms can be introduced for c,I,and J resulting in 32Mo On=Kt πd3 Om =Kt 2Mmm nd3 (7-31 16T ta=Kfsπd 16Tm tm=K:Td形 (7-4
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components 358 © The McGraw−Hill Companies, 2008 Shafts and Shaft Components 355 Most shafts will transmit torque through a portion of the shaft. Typically the torque comes into the shaft at one gear and leaves the shaft at another gear. A free body diagram of the shaft will allow the torque at any section to be determined. 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 bending moment diagrams. Since most shaft problems incorporate gears or pulleys that introduce forces in two planes, the shear and bending moment diagrams will generally be needed in two planes. Resultant moments are obtained by summing moments as vectors at points of interest along the shaft. The phase angle of the moments is not important since the shaft rotates. A steady bending moment will produce a completely reversed moment on a rotating shaft, as a specific stress element will alternate from compression to tension in every revolution of the shaft. The normal stress due to bending moments will be greatest on the outer surfaces. In situations where a bearing is located at the end of the shaft, stresses near the bearing are often not critical since the bending moment is small. Axial 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 bending moment stress. They are often also constant, so they contribute little to fatigue. Consequently, it is usually acceptable to neglect the 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 assume it is negligible without checking magnitudes. Shaft Stresses Bending, torsion, and axial stresses may be present in both midrange and alternating components. For analysis, it is simple enough to combine the different types of stresses into alternating and midrange von Mises stresses, as shown in Sec. 6–14, p. 309. It is sometimes convenient to customize the equations specifically for shaft applications. 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. The fluctuating stresses due to bending and torsion are given by σa = Kf Mac I σm = Kf Mmc I (7–1) τa = Kf s Tac J τm = Kf s Tmc J (7–2) where Mm and Ma are the midrange and alternating bending moments, Tm and Ta are the midrange and alternating torques, and Kf and Kf s are the fatigue stress concentration factors for bending and torsion, respectively. Assuming a solid shaft with round cross section, appropriate geometry terms can be introduced for c, I, and J resulting in σa = Kf 32Ma πd3 σm = Kf 32Mm πd3 (7–3) τa = Kf s 16Ta πd3 τm = Kf s 16Tm πd3 (7–4)
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft T©The McGraw-Hill 359 Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition 356 Mechanical Engineering Design Combining these stresses in accordance with the distortion energy failure theory, the von Mises stresses for rotating round,solid shafts,neglecting axial loads,are given by c。=(o2+3r/ 32KfMa +3 16KfsTa πd3 πd3 7-51 m=(o品+3r)/2 (7-61 Note that the stress concentration factors are sometimes considered optional for the midrange components with ductile materials,because of the capacity of the ductile material to yield locally at the discontinuity. These equivalent alternating and midrange stresses can be evaluated using an appropriate failure curve on the modified Goodman diagram(See Sec.6-12,p.295, and Fig.6-27).For example,the fatigue failure criteria for the modified Goodman line as expressed previously in Eq.(6-46)is TSut Substitution of o and o from Eqs.(7-5)and (7-6)results in 日-{传K+3K,五+KMP+3K,四 For design purposes,it is also desirable to solve the equation for the diameter. This results in d=({K,MP+3K. KF+3K,r 1/3 Similar expressions can be obtained for any of the common failure criteria by sub- stituting the von Mises stresses from Eqs.(7-5)and(7-6)into any of the failure criteria expressed by Eqs.(6-45)through(6-48),p.298.The resulting equations for several of the commonly used failure curves are summarized below.The names given to each set of equations identifies the significant failure theory,followed by a fatigue failure locus name.For example,DE-Gerber indicates the stresses are combined using the distortion energy (DE)theory,and the Gerber criteria is used for the fatigue failure. DE-Goodman -9{长[KMP+3K,P+[sKMP+3K月 7-7 d-({長4K,MP+3Kmz +2MF+e) (7-8)
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components © The McGraw−Hill 359 Companies, 2008 356 Mechanical Engineering Design Combining these stresses in accordance with the distortion energy failure theory, the von Mises stresses for rotating round, solid shafts, neglecting axial loads, are given by σ a = (σ2 a + 3τ 2 a ) 1/2 = 32Kf Ma πd3 2 + 3 16Kf sTa πd3 2 1/2 (7–5) σ m = (σ2 m + 3τ 2 m) 1/2 = 32Kf Mm πd3 2 + 3 16Kf sTm πd3 2 1/2 (7–6) Note that the stress concentration factors are sometimes considered optional for the midrange components with ductile materials, because of the capacity of the ductile material to yield locally at the discontinuity. These equivalent alternating and midrange stresses can be evaluated using an appropriate failure curve on the modified Goodman diagram (See Sec. 6–12, p. 295, and Fig. 6–27). For example, the fatigue failure criteria for the modified Goodman line as expressed previously in Eq. (6–46) is 1 n = σ a Se + σ m Sut Substitution of σ a and σ m from Eqs. (7–5) and (7–6) results in 1 n = 16 πd3 1 Se 4(Kf Ma) 2 + 3(Kf sTa) 21/2 + 1 Sut 4(Kf Mm) 2 + 3(Kf sTm) 21/2 For design purposes, it is also desirable to solve the equation for the diameter. This results in d = 16n π 1 Se 4(Kf Ma) 2 + 3(Kf sTa) 21/2 + 1 Sut 4(Kf Mm) 2 + 3(Kf sTm) 21/2 1/3 Similar expressions can be obtained for any of the common failure criteria by substituting the von Mises stresses from Eqs. (7–5) and (7–6) into any of the failure criteria expressed by Eqs. (6–45) through (6–48), p. 298. The resulting equations for several of the commonly used failure curves are summarized below. The names given to each set of equations identifies the significant failure theory, followed by a fatigue failure locus name. For example, DE-Gerber indicates the stresses are combined using the distortion energy (DE) theory, and the Gerber criteria is used for the fatigue failure. DE-Goodman 1 n = 16 πd3 1 Se 4(Kf Ma) 2 + 3(Kf sTa) 21/2 + 1 Sut 4(Kf Mm) 2 + 3(Kf sTm) 21/2 (7–7) d = 16n π 1 Se 4(Kf Ma) 2 + 3(Kf sTa) 21/2 + 1 Sut 4(Kf Mm) 2 + 3(Kf sTm) 21/2 1/3 (7–8)