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
snap ring
clamp
collar
taper
pin
key
hub
bearing
hub
shaft
step
bearing
press
fit
step
step
step
press
fit
axial
clearance
frame
frame
sprocket
gear
FIGURE 9-2
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
sheave
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
P wr = T orq ∗ ω
9.4
(9.1)
Shaft Loading, Approaches to Analysis
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
σalt
σmean
Ma c
I
Mm c
= kf m
I
= kf
(9.2)
(9.3)
Torsional Shear Stress
τalt
τmean
9.5.1
9.6
9.6.1
Ta r
J
Tm r
= kf sm
J
= kf s
(9.4)
(9.5)
Shaft Failure in Combined Loading
Shaft Design
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
σa
Se
σa
Se
from ref. 3
2
2
 σa 
 τa 
 S  +S  =1
 e
 es 
2
2
 τm 
 σa 
 S  +S  =1
 e
 ys 
τa
Ses
τm
Sys
(a) Combined stress fatigue-test data for reversed
bending combined with static torsion (from ref. 4)
(b) Combined stress fatigue-test data for reversed
bending combined with reversed torsion (from ref. 5)
FIGURE 9-3
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
v
s
u
u
32Saf
etyF
actor
Ma 2 3 T m 2
3
(kf
) + (
)
d=t
π
Sf
4 Sy
(9.6)
More general loading cases require Equation 9.8.
See Example 9..
9.6.2
Shaft Deflection
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
FIGURE P9-3
P9-03.pdf
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
9.7.1
Terms related to Fits and Tolerances ANSI B4.2-1978
Definitions
D – basic size of the hole
d – basic size of the shaft
δu – upper deviation
δl – lower deviation
δF – 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
A<d≤B
0-3
3-6
6-10
10-18
18-30
30-50
50-80
80-120
120-180
180-250
250-315
315-400
IT6
0.006
0.008
0.009
0.011
0.013
0.016
0.019
0.022
0.025
0.029
0.032
0.036
IT7
0.010
0.012
0.015
0.018
0.021
0.025
0.030
0.035
0.040
0.046
0.052
0.057
Tolerance Grades
IT8
IT9
0.014 0.025
0.018 0.030
0.022 0.036
0.027 0.043
0.033 0.052
0.039 0.062
0.046 0.074
0.054 0.087
0.063 0.100
0.072 0.115
0.081 0.130
0.089 0.140
IT10
0.040
0.048
0.058
0.070
0.084
0.100
0.120
0.140
0.160
0.185
0.210
0.230
IT11
0.060
0.075
0.090
0.110
0.130
0.160
0.190
0.220
0.250
0.290
0.320
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
A<d≤B
0-3
3-6
6-10
10-14
14-18
18-24
24-30
30-40
40-50
50-65
65-80
80-100
100-120
120-140
140-160
160-180
180-200
200-225
225-250
250-280
280-315
315-355
355-400
Clearance
Upper Deviation Letter
c
d
f
g
-0.060 -0.020 -0.006 -0.002
-0.070 -0.030 -0.010 -0.004
-0.080 -0.040 -0.013 -0.005
-0.095 -0.050 -0.016 -0.006
-0.095 -0.050 -0.016 -0.006
-0.110 -0.065 -0.020 -0.007
-0.110 -0.065 -0.020 -0.007
-0.120 -0.080 -0.025 -0.009
-0.130 -0.080 -0.025 -0.009
-0.140 -0.100 -0.030 -0.010
-0.150 -0.100 -0.030 -0.010
-0.170 -0.120 -0.030 -0.012
-0.180 -0.120 -0.036 -0.012
-0.200 -0.145 -0.043 -0.014
-0.210 -0.145 -0.043 -0.014
-0.230 -0.145 -0.043 -0.014
-0.240 -0.170 -0.050 -0.015
-0.260 -0.170 -0.050 -0.015
-0.280 -0.170 -0.050 -0.015
-0.300 -0.190 -0.056 -0.017
-0.330 -0.190 -0.056 -0.017
-0.360 -0.210 -0.062 -0.018
-0.400 -0.210 -0.062 -0.018
h
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
Transition
Interference
Lower-Deviation Letter
k
n
p
s
u
0 +0.004 +0.006 +0.014 +0.018
+0.001 +0.008 +0.012 +0.019 +0.023
+0.001 +0.010 +0.015 +0.023 +0.028
+0.001 +0.012 +0.018 +0.028 +0.033
+0.001 +0.012 +0.018 +0.028 +0.033
+0.002 +0.015 +0.022 +0.035 +0.041
+0.002 +0.015 +0.022 +0.035 +0.048
+0.002 +0.017 +0.026 +0.043 +0.060
+0.002 +0.017 +0.026 +0.043 +0.070
+0.002 +0.020 +0.032 +0.053 +0.087
+0.002 +0.020 +0.032 +0.059 +0.102
+0.003 +0.023 +0.037 +0.071 +0.124
+0.003 +0.023 +0.037 +0.079 +0.144
+0.003 +0.027 +0.043 +0.092 +0.170
+0.003 +0.027 +0.043 +0.100 +0.190
+0.003 +0.027 +0.043 +0.108 +0.210
+0.004 +0.031 +0.050 +0.122 +0.236
+0.004 +0.031 +0.050 +0.130 +0.258
+0.004 +0.031 +0.050 +0.140 +0.284
+0.004 +0.034 +0.056 +0.158 +0.315
+0.004 +0.034 +0.056 +0.170 +0.350
+0.004 +0.037 +0.062 +0.190 +0.390
+0.004 +0.037 +0.062 +0.208 +0.435
Type of fit
Clearance
Transition
Interference
Table 9.3: Fit Types and their description
Reference
Description
Loose running fit
For wide commercial tolerances or allowances 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
Close running fit
For running on accurate machines
and for accurate location at moderate
speeds and journal pressures
Sliding fit
Where parts are not intended to run
freely, but must move and turn freely
and locate accurately
Locational clearance fit Provides snug fit for location of station(snug fit)
ary parts, but can be freely assembled
and disassembled
Locational transitional fit
For accurate location, a compromise between clearance and interference
Locational transitional fit For more accurate location where
(wringing fit)
greater interference is permissible
Locational transitional fit For parts requiring rigidity and align(tight fit)
ment with prime accuracy of location
but without special bore pressure requirements
Medium Drive Fit
For ordinary steel parts or shrink fits
on light sections, the tightest fit usable
with cast iron
Force Fit
Suitable for parts which can be highly
stressed or for shrink fits where the
heavy pressing forces required are impractical
Symbol
H11/c11
H9/d9
H8/f8
H7/g6
H7/h6
H7/k6
H7/n6
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