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Chapter 11: Shafts and Associated Parts
Design issues to learn here:
Loading:
TorqueBending moment
Fatigue strength
Sizing
DeflectionLinear
Slope at bearings and gears
Keys
Whirling
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Loading: Needed first in calculations
Text Reference:
Figure 11.1, page425
(a) Gear forces
P1 and P2;
(b) free-body diagram with
torque and forces
(c) moment diagram in x-z and x-y planes
(d) torque diagram.
Min x-y plane Min x-z plane
Torsion
Combined bending
moment atx22xzxyx MMM +=
(Direction is somewhere
betweenx-y andx-z planes.)
(11.1)
a
p
N
hT
63025= (4.41)
horsepower
rpm
ForcerTP /=
)unitsSI(
PowerT=
radius
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Example 11.2 (Moment diagrams only)
Figure 11.3 Example 11.2. (a) Assembly drawing; (b) free-body diagram.Text Reference: Figure 11.3, page 430
Design for static strength is almost never relevant.
Problem: Given all forces, draw the bending moment
diagrams (2 planes) and torque diagram.
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Example 11.2 (cont.) Text Reference: Figure 11.3, page 430
(c)Min x-y plane (d)Min x-z plane
Figure 11.3
(e) torque diagram.
Note: Almost all shaft illustrations in
Hamrocks book are unrealistically long.
Real-world shafts are designed as short
as possible, often with no space
between gears and bearings.
Advantages of shorter shafts:
lower bending moments and
stresses
Smaller deflections
Lower cost.
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Stresses
332dM
=Bending stress ( =Mr/I) : (11.5)
3
16
d
T
=
Torsional stress ( = Tr/J) : (11.6)
Axial stress ( =P/A) : 24
d
P
=
Assume stress concentration Kfmultiplies the alternating part of
bending stress.
Assume stress concentration Kfsmultiplies the alternating part of
torsional stress.
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Fluctuating Stresses in shaft
Text Reference: Figure 11.4, page 432
(b) on oblique plane at angle .
Figure 11.4 Fluctuating stresses. (a)
on rectangular element
Given stresses on rectangular
element, determine stresses on
plane at an angle .
Force equilibrium
( ) 0sincossinsincoscos
=++
+++
AK
AKAKA
afm
afsmafsm
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Figure 11.5
Soderberg line for
shear stress.
Text Reference: Figure 11.5, page 433
SoderbergLine for Shear
Stress
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Fatigue
calculation
Figure 11.6 Eq. (11.29).
Text Reference: Figure 11.6, page 434
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Shaft fatigue formulae: MSST
Safety factor against fatigue: see next page.
From Mohr circle
3
32
d
Mx
=
3
16
d
Tyx
=
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Fatigue safety factor versus diameter
according to MSST
22
3
32
++
+
=
afse
y
mafe
y
m
ys
TKS
STMK
S
SM
Sdn
(11.34)
31
2232
++
+= afs
e
ymaf
e
ym
y
s TKS
STMK
S
SM
S
nd
(11.35)
Safety factor against fatigue (based on MSST):
To calculate diameter for a given safety factor, use
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Fatigue safety factor versus diameter
according to DET
22
3
++
+= afs
e
ymaf
e
ym
s
yK
S
SK
S
S
n
S
(11.36)
22
3
4
332
++
+
=
afse
ymaf
e
ym
ys
TKS
STMK
S
SM
Sdn (11.37)
31
22
4
332
++
+= afs
e
ymaf
e
ym
y
s TK
S
STMK
S
SM
S
nd
(11.38)
Safety factor against fatigue (based on DET):
To calculate diameter for a given safety factor, use
For DET, replace Eq. 11.31 with:
Derivation similar to MSST results in
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Diameter calculation: notes
31
2232
++
+= afs
e
ymaf
e
ym
y
s TK
S
STMK
S
SM
S
nd
(11.35)
Recall diameter for a given safety factor (MSST):
Stress concentration factorsKfandKfs depend on size
factorks, which depends on size d. Need iteration.
For a shaft that torques and rotates in only one direction:
Mean bending momentMm = 0.
Alternating torsion Ta = 0.
31
22
32
+
= maf
e
y
y
s TMK
S
S
S
nd
(11.35a)
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Example 11.4
Figure 11.7, page 438
31
2232
++
+= afs
e
ymaf
e
ym
y
s TKS
STMK
S
SM
S
nd
0 0
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Example 11.4
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Example 11.4
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Example 11.4
