Chapter 9 Shaft Design Transmission shafts transmit torque from one location to another Spindles are short shafts Axles are non-rotating shafts Figure 9.1 is an example of a shaft with several features. It is a shaft for a Caterpillar tractor transmission 1. Figure 9.1: Example of a typical shaft design 1 From Frederick E. Giesecke, Technical Drawing, Chapter 13. 1
9.1 Shaft Loads Torsion due to transmitted torque Bending from transverse loads (gears, sprockets, pulleys/sheaves) o * a pulley and a sheave are essentially the same thing Steady or Fluctuating Steady transverse-bending load fully reversing bending stress (fatigue failure) 9.2 Attachments and Stress Concentrations Steps and shoulders are used to locate attachment (gears, sheaves, sprockets) Keys, snap rings, cross pins (shear pins), tapered pins Use generous radii to reduce stress concentrations Clamp collars Split collar Press fits and shrink fits Bearings may be located by the use of snap rings, but only one bearing is fixed Issues - axial location, disassembly, and element phasing (e.g., alignment of gear teeth for timing) MACHINE DESIGN - An Integrated Approach, 2ed by Robert L. Norton, Prentice-Hall 2000 bearing clamp collar step hub shaft hub key bearing snap ring taper pin press fit FIGURE 9-2 frame axial clearance sprocket step step gear press fit frame step sheave Various Methods to Attach Elements to Shafts Figure 9.2: Example of a shaft with various attachments and details 9.3 Shaft Materials Steel (low to medium-carbon steel) Cast iron Bronze or stainless steel Case hardened steel 2
9.3.1 Shaft Power Power is the time rate of change of energy (work). work = Force * distance or Torque * angle, so Power = Torque * angular velocity 9.4 Shaft Loading, Approaches to Analysis P wr = T orq ω (9.1) Most general form - A fluctuating torque and a fluctuating moment, in combination. If there are axial loads, they should be taken to ground as close to the load as possible. Given knowledge of the moments and the torques (i.e., mean and alternating components) Use the Design Steps for Fluctuating Stresses in Section 6.11 in combination with the multiaxial-stress issues addressed in Section 6.12. 9.5 Shaft Stresses Bending Stress Torsional Shear Stress M a c σ alt = k f I M m c σ mean = k fm I (9.2) (9.3) T a r τ alt = k fs J T m r τ mean = k fsm J (9.4) (9.5) 9.5.1 Shaft Failure in Combined Loading 9.6 Shaft Design 9.6.1 General Considerations 1. To minimize both deflections and stresses, the shaft length should be kept as short as possible and overhangs minimized. 2. A cantilever beam will have a larger deflection than a simply supported (straddle mounted) one for the same length, load, and cross section, so straddle mounting should be used unless a cantilever shaft is dictated by design constraints. (Figure 9-2 shows a situation in which an overhung section is required for serviceability.) 3. A hollow shaft has a better stiffness/mass ratio (specific stiffness) and higher natural frequencies than a comparably stiff or strong solid shaft, but will be more expensive and larger in diameter. 4. Try to locate stress-raisers away from regions of large bending moment if possible and minimize their effects with generous radii and relief. 5. General low carbon steel is just as good as higher strength steels (since deflection is typical the design limiting issue). 6. Deflections at gears carried on the shaft should not exceed about 0.005 inches and the relative slope between the gears axes should be less than about 0.03 degrees. 3
MACHINE DESIGN - An Integrated Approach, 2ed by Robert L. Norton, Prentice-Hall 2000 from ref. 2 from ref. 3 σ a S e from ref. 3 σ a S e 2 σa τm S + 2 e S = 1 ys 2 σa τa S + 2 e S = 1 es τ m τ a S ys (a) Combined stress fatigue-test data for reversed bending combined with static torsion (from ref. 4) FIGURE 9-3 S es (b) Combined stress fatigue-test data for reversed bending combined with reversed torsion (from ref. 5) Results of Fatigue Tests of Steel Specimens Subjected to Combined Bending and Torsion (From Design of Transmission Shafting, American Society of Mechanical Engineers, New York, ANSI/ASME Standard B106.1M-1985, with permission) Figure 9.3: Shaft failure in combined loading 4
7. If plain (sleeve) bearings are to be used, the shaft deflection across the bearing length should be less than the oil-film thickness in the bearing. 8. If non-self-aligning rolling element bearings are used, the shaft s slope at the bearings should be kept to less than about 0.04 degrees. 9. If axial thrust loads are present, they should be taken to ground through a single thrust bearing per load direction. Do not split axial loads between thrust bearings as thermal expansion of the shaft can overload the bearings. 10. The first natural frequency of the shaft should be at least three times the highest forcing frequency expected in service, and preferably much more. (A factor of ten times or more is preferred, but this is often difficult to achieve). Designing for Fully Reversed Bending and Steady Torsion ASME Method (ANSI/ASME Standard for Design of Transmission Shafting B106.1M-1985. Uses the elliptical curve of Figure 9-3. Equations 9.5e and 9.6a,b. 9.6 can be applied only for constant torque fully reversed moment. No axial load More general loading cases require Equation 9.8. See Example 9.. 9.6.2 Shaft Deflection d = 32SafetyF actor M 3 a (k f ) π S 2 + 3 f 4 (T m ) S 2 (9.6) y Deflection is often the more demanding constraint. Many shafts are well within specification for stress but would exhibit too much deflection to be appropriate. 9.6.3 Keys and Keyways P gear d T a b l bearings are self-aligning so act as simple supports P9-03.pdf FIGURE P9-3 Shaft Design for Problems 9-6, 9-9. 9-11, and 9-12 Figure 9.4: Shaft with overhung gear Example -Homework Problem 9-2 5
9.6.4 Splines 9.6.5 Interference Fits Components can be attached to a shaft without a key or spline by using an interference fit. There are two methods used to assemble these components: press fit shrink (and/or expansion) fit The amount of interference is important The analysis of interference follows from the equations for pressure on thick-walled cylinders. A rule of thumb that is used is one to two thousands of diametral interference per unit of shaft diameter, e.g., a shaft of two inch diameter would have 0.004 inches of interference with an attached gear hub. Machinists use a simplified approach to this 1/1000 of interference for each inch of diameter. However, there is a formal approach Standards have been developed for these fits. Metric Preferred Metric Limits and Fits ANSI B4.2-1978. US Customary Preferred Limits and Fits for Cylindrical Parts ANSI B4.1-1967 9.7 Terms related to Fits and Tolerances ANSI B4.2-1978 9.7.1 Definitions D basic size of the hole d basic size of the shaft δ u upper deviation δ l lower deviation δ F Fundamental deviation D tolerance grade for the hole d tolerance grade for the shaft Tolerance the difference between the maximum and minimum size limits of the dimensions of a part Natural tolerance a tolerance equal to ± three standard deviations from the mean Clearance amount of space between an internal and external member Interference the amount of overlap between an internal and external member International Tolerance Grade Numbers (IT) designate groups of tolerances such that the tolerances for a particular IT number have the same relative level of accuracy, i.e., IT 9 Smaller numbers mean tighter tolerances, IT 6 through IT 11 are used for preferred fits. For a 32 mm hole we might use 32H7 The H establishes the fundamental deviation and the number 7 defines a tolerance grade of IT7. The grade number specifies a tolerance zone. For the mating shaft we might have 32g6 9.7.2 Table of Tolerance Grades 2 Lower and Upper Deviations For shaft letter codes c, d, f, g, and h 2 Shigley Table E-11, page 1188. 6
Table 9.1: International Tolerance Grades Basic Sizes All values in mm Tolerance Grades A < d B IT6 IT7 IT8 IT9 IT10 IT11 0-3 0.006 0.010 0.014 0.025 0.040 0.060 3-6 0.008 0.012 0.018 0.030 0.048 0.075 6-10 0.009 0.015 0.022 0.036 0.058 0.090 10-18 0.011 0.018 0.027 0.043 0.070 0.110 18-30 0.013 0.021 0.033 0.052 0.084 0.130 30-50 0.016 0.025 0.039 0.062 0.100 0.160 50-80 0.019 0.030 0.046 0.074 0.120 0.190 80-120 0.022 0.035 0.054 0.087 0.140 0.220 120-180 0.025 0.040 0.063 0.100 0.160 0.250 180-250 0.029 0.046 0.072 0.115 0.185 0.290 250-315 0.032 0.052 0.081 0.130 0.210 0.320 315-400 0.036 0.057 0.089 0.140 0.230 0.360 Upper deviation = fundamental deviation Lower deviation = upper deviation tolerance grade For shaft letter codes k, n, p,s, and u Lower deviation = fundamental deviation Upper deviation = lower deviation + tolerance grade Hole letter code is H Lower deviation = 0 Upper deviation = tolerance grade 9.7.3 Fundamental Deviations for Shafts Metric Series These are related to the tolerance grades. See the table below. Capital letters always refer to the hole (or bore) and lowercase letters are used for the shaft. 9.7.4 Fit Types Table 9.3 provides a linguistic description for commonly used references to fit types. 9.8 Flywheel Design One of the biggest issues with regard to flywheels is balancing. Because they are, by intention, devices with large inertias, balancing them to remove eccentric loading and thus lower the loading on bearings and other components is very important. Flywheels develop large stresses at their inter hub connection due to dynamic forces caused by the spinning. These stresses can lead to failure. Careful design is required to avoid catastrophic failure. 9.9 Critical Speeds There are three types of vibration that are encountered with shafts: 7
Table 9.2: Fundamental Deviations for Shafts Metric Series basic dimension Clearance Transition Interference A < d B Upper Deviation Letter Lower-Deviation Letter c d f g h k n p s u 0-3 -0.060-0.020-0.006-0.002 0 0 +0.004 +0.006 +0.014 +0.018 3-6 -0.070-0.030-0.010-0.004 0 +0.001 +0.008 +0.012 +0.019 +0.023 6-10 -0.080-0.040-0.013-0.005 0 +0.001 +0.010 +0.015 +0.023 +0.028 10-14 -0.095-0.050-0.016-0.006 0 +0.001 +0.012 +0.018 +0.028 +0.033 14-18 -0.095-0.050-0.016-0.006 0 +0.001 +0.012 +0.018 +0.028 +0.033 18-24 -0.110-0.065-0.020-0.007 0 +0.002 +0.015 +0.022 +0.035 +0.041 24-30 -0.110-0.065-0.020-0.007 0 +0.002 +0.015 +0.022 +0.035 +0.048 30-40 -0.120-0.080-0.025-0.009 0 +0.002 +0.017 +0.026 +0.043 +0.060 40-50 -0.130-0.080-0.025-0.009 0 +0.002 +0.017 +0.026 +0.043 +0.070 50-65 -0.140-0.100-0.030-0.010 0 +0.002 +0.020 +0.032 +0.053 +0.087 65-80 -0.150-0.100-0.030-0.010 0 +0.002 +0.020 +0.032 +0.059 +0.102 80-100 -0.170-0.120-0.030-0.012 0 +0.003 +0.023 +0.037 +0.071 +0.124 100-120 -0.180-0.120-0.036-0.012 0 +0.003 +0.023 +0.037 +0.079 +0.144 120-140 -0.200-0.145-0.043-0.014 0 +0.003 +0.027 +0.043 +0.092 +0.170 140-160 -0.210-0.145-0.043-0.014 0 +0.003 +0.027 +0.043 +0.100 +0.190 160-180 -0.230-0.145-0.043-0.014 0 +0.003 +0.027 +0.043 +0.108 +0.210 180-200 -0.240-0.170-0.050-0.015 0 +0.004 +0.031 +0.050 +0.122 +0.236 200-225 -0.260-0.170-0.050-0.015 0 +0.004 +0.031 +0.050 +0.130 +0.258 225-250 -0.280-0.170-0.050-0.015 0 +0.004 +0.031 +0.050 +0.140 +0.284 250-280 -0.300-0.190-0.056-0.017 0 +0.004 +0.034 +0.056 +0.158 +0.315 280-315 -0.330-0.190-0.056-0.017 0 +0.004 +0.034 +0.056 +0.170 +0.350 315-355 -0.360-0.210-0.062-0.018 0 +0.004 +0.037 +0.062 +0.190 +0.390 355-400 -0.400-0.210-0.062-0.018 0 +0.004 +0.037 +0.062 +0.208 +0.435 8
Table 9.3: Fit Types and their description Type of fit Reference Description Symbol Clearance Loose running fit For wide commercial tolerances or allowances H11/c11 on external members Free running fit Not for use where accuracy is essential, but good for large temperature variations, high running speeds, or heavy journal pressures H9/d9 Close running fit For running on accurate machines H8/f8 and for accurate location at moderate speeds and journal pressures Sliding fit Where parts are not intended to run H7/g6 freely, but must move and turn freely and locate accurately Locational clearance fit Provides snug fit for location of stationary H7/h6 (snug fit) parts, but can be freely assembled and disassembled Transition Locational transitional fit For accurate location, a compromise between clearance and interference H7/k6 Locational transitional fit For more accurate location where H7/n6 (wringing fit) Interference Locational transitional fit (tight fit) Medium Drive Fit Force Fit greater interference is permissible For parts requiring rigidity and alignment with prime accuracy of location but without special bore pressure requirements For ordinary steel parts or shrink fits on light sections, the tightest fit usable with cast iron Suitable for parts which can be highly stressed or for shrink fits where the heavy pressing forces required are impractical H7/p6 H7/s6 H7/u6 Lateral vibration Shaft whirl Torsional vibration 9.10 Couplings Many applications require us to connect one shaft to another axially. This is done with the use of couplings. Note that the possibility of getting the two shafts perfectly aligned (linearly and angularly) is essentially zero, so couplings are typically designed to accomodate some misalignment. Couplings come in many shapes, sizes, and degrees of misalignment. One type of coupling you might be familiar with is the universal joint, see Figure 9.5. A recent inovation used with front wheel drive is the CV (constant velocity) joint. Another type used widely for connections to electric motors is a flexible coupling, see Figure 9.6. 9.11 Summary While shafting can be purchased as a stock item, most applications require some customization of the layout and dimensioning to accommodate the attachment of components and bearings. Almost all shafts are 9
Figure 9.5: Typical automotive universal joint Figure 9.6: Small flexible couplings designed for high cycle fatigue (HCF), and are made of steel, since it has an fatigue limit. One is cautioned to applied the shaft diameter design equations presented in Norton (Equation 9.6 & Equation 9.8) properly since specific requirements must be met to apply these equations. Many other factors come into play during the shaft design process. These may include: keyways and keys splines couplings shaft vibrations and balancing flywheels 10