shafts and axles
TRANSCRIPT
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
Dr. A. Aziz Bazoune
King Fahd University of Petroleum & MineralsMechanical Engineering Department
CH-18 LEC 30 Slide 1
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
18-1 Introduction .92218-2 Geometric Constraints .92718-3 Strength Constraints .933
18-4 Strength Constraints Additional Methods .94018-5 Shaft Materials .94418-6 Hollow Shafts .94418-7 Critical Speeds (Omitted) .945
18-8 Shaft Design .950
CH-18 LEC 30 Slide 2
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
18-1 Introduction .92218-2 Geometric Constraints .92718-3 Strength Constraints .933
CH-18 LEC 30 Slide 3
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 4
Neglecting axial loads because they are comparatively very small at critical
locations where bending and torsion dominate. Remember the fluctuatingstresses due to bending and torsion are given by
Fatigue Analysis of shafts
a ma f m f
a ma fs m fs
M C M CK K
I I
T C T CK K
J J
Mm: Midrange bending moment, m: Midrange bending stressMa : alternating bending moment, a: alternating bending stress
Tm: Midrange torque, m: Midrange shear stressTa: alternating torque, m: Midrange shear stressKf: fatigue stress concentration factor for bendingKfs: fatigue stress concentration factor for torsion
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 5
For solid shaft with round cross section, appropriate geometry terms can be
introduced for C, IandJresulting in
Fatigue Analysis of shafts
32 32
16 16
a ma f m f
a ma fs m fs
M MK K
I I
T TK KJ J
Mm: Midrange bending moment, m: Midrange bending stressMa : alternating bending moment, a: alternating bending stress
Tm: Midrange torque, m: Midrange shear stressTa: alternating torque, m: Midrange shear stressKf: fatigue stress concentration factor for bendingKfs: fatigue stress concentration factor for torsion
CH-18 LEC 30 Slide 5
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I Fatigue Analysis of shafts
2 2 1/2 3 2 2 1/2 3
2 2 1/2 3 2 2 1/2 3
' 3 16 / 4( ) 3( ) 16 /
' 3 16 / 4( ) 3( ) 16 /
xya
xym
axa f a fs a
mxm f m fs m
d K M K AT d
d K M K T d B
Combining these stresses in accordance with the DE failure theory the von-
Mises stress for rotating round, solid shaft, neglecting axial loads are given by
2 2 2
3 3
' ' 16 161a m a m
e ut e ut e ut
S S n n nA nB
S S S S d S d S
The Gerber fatigue failure criterion
(18-12
where A and B are defined by the radicals in Eq. (8-12) as
2 2 1/2
2 2 1/2
4( ) 3( )
4( ) 3( )
f a fs a
f m fs m
A K M K T
B K M K T
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I Fatigue Analysis of shafts
1/31/2
2
281 1 e
e ut
BSnAd
S AS
(18-13
The critical shaft diameter is given by
or, solving for 1/n, the factor of safety is given by
1/22
3 21 8 1 1e
e ut
BSAn d S AS
(18-14
CH-18 LEC 30 Slide 7CH-18 LEC 30 Slide 7
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
Fatigue Analysis of shafts
(18-15)
where
CH-18 LEC 30 Slide 8
2 2
2 2
4( ) 3( )
4( ) 3( )
'
'
f a fs a
f m fs m
a
m
A K M K T
B K M K T
Ar
B
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
2
3
f a
fs m
A K M
B K T
Particular Case
Critical ShaftDiameter
CH-18 LEC 30 Slide 9
Fatigue Analysis of shafts
1/31/2
216
1 1 3f a fs m e
e f a ut
nK M K T Sd
S K M S
(18-16)
Safety Factor
1/22
3
1611 1
f a fs m e
e f a ut
K M K T S
n d S K M S
(18-17)
For a rotating shaft with constant bending and torsion, thebending stress is completely reversed and the torsion is
steady. Previous Equations can be simplified by settingMm= 0 andTa = 0, which simply drops out some of the terms.
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 10
2'' 3
f aa
m fs m
K MrK T
(18-18)
Fatigue Analysis of shafts
CH-18 LEC 30 Slide 10
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 11
Shaft Diameter Equation for the DE-EllipticCriterion
2 2 1/2 3 2 2 1/2 3
2 2 1/2 3 2 2 1/2 3
' 3 16 / 4( ) 3( ) 16 /
' 3 16 / 4( ) 3( ) 16 /
xya
xym
axa f a fs a
mxm f m fs m
d K M K AT d
d K M K T d B
Remember
2 2 22 2 2
3 3
' ' 16 161a m a m
e y e y e y
S S n n nA nB
S S S S d S d S
The Elliptic fatigue-failure criterion is defined by
(18-12
where A and B are defined by
2 2 1/2
2 2 1/2
4( ) 3( )
4( ) 3( )
f a fs a
f m fs m
A K M K T
B K M K T
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 12
Shaft Diameter Equation for the DE-EllipticCriterion
Substituting for A and B gives expressions for d, 1/n and r:
(18-19
Critical Shaft Diameter
1/31
2 2 22/2
4 3 316
4f m fsf a fs a
e e
m
y y
K M K T
S S
K M
S
n K T
S
d
Safety Factor
2/2
2 2
12
34
643 3
1 1 f m fs m
y
a
e y
f fs a
e
K M K T
S S
K M K T
d S Sn
(18-20)
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 13
Shaft Diameter Equation for the DE-EllipticCriterion
2 2
2 2
4 3'
' 4 3
f a fs aa
mf m fs m
K M K T Ar
B K M K T
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
Shaft Diameter Equation for the DE-EllipticCriterion
CH-18 LEC 30 Slide 14
2
3
f a
fs m
A K M
B K T
Particular Case
Critical Shaft Diameter (18-21)
Safety Factor (18-22)
1/31/2
22
46
31 f a f
ye
s mK Tn
dK M
SS
1
2/2
3
2
164
13
fs m
y
f a
e
K M
Sn
T
d
K
S
For a rotating shaft with constant bending and torsion,the bending stress is completely reversed and thetorsion is steady. Previous Equations can be simplified
by settingMm = 0 andTa = 0, which simply drops outsome of the terms.
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
Shaft Diameter Equation for the DE-EllipticCriterion
CH-18 LEC 30 Slide 15
2
2
4 2'
' 33
f a f aa
m fs mfs m
K M K MAr
B K TK T
At a shoulderFigs. A-15-8 and A-15-9 provide information about Kt andKts.
For a hole in a solid shaft, Figs. A-15-10 and A-15-11 provide about Kt andKts .
For a hole in a solid shaft, use Table A-16 For grooves use Figs. A-15-14 and A-15-15
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
Shaft Diameter Equation for the DE-EllipticCriterion
CH-18 LEC 30 Slide 16
The value of slope at which the load line intersects the junction of
the failure curves is designated rcrit.
It tells whether the threat is from fatigue or first cycle yielding
Ifr> rcrit, the threat is from fatigue
Ifr< rcrit
, the threat is from first cycle yielding.
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
Shaft Diameter Equation for the DE-EllipticCriterion
CH-18 LEC 30 Slide 17
For the Gerber-Langer intersection the strength components Sa and
Sm are given in Table 7-10 as
22 2
1 1 12
1
yut e
me ut e
a y m
y m yacrit
m m m
SS SS
S S S
S S S
S S SSrS S S
(18-23)
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
Shaft Diameter Equation for the DE-EllipticCriterion
CH-18 LEC 30 Slide 18
For the DE-Elliptic-Langer intersection the strength components Sa
and Sm are given by
2
2 2
2y e
a
e y
m y a
a a
critm y a
S SS
S S
S S S
S S
r S S S
(18-24)
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 19
Note that in an analysis situation in which the diameter is known and the
factor of safety is desired, as an alternative to using the specialized equationsabove, it is always still valid to calculate the alternating and mid-rangestresses using the following Eqs.
and substitute them into the one of the equations for the failure criteria ,
Eqs. (7-48) to (7-51) and solve directly for n.
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
In a design situation, however, having the equations pre-solved for
diameter is quite helpful.
It is always necessary to consider the possibility ofstatic failure in
the first load cycle.
The Soderberg criteria inherently guards against yielding, as can
be seen by noting that its failure curve is conservatively within the
yield (Langer) line on Fig. 727, p. 348.
The ASME Elliptic also takes yielding into account, but is not
entirely conservative throughout its range. This is evident by
noting that it crosses the yield line in Fig. 727.
The Gerber and modified Goodman criteria do not guard against
yielding, requiring a separate check for yielding. A von Mises
maximum stress is calculated for this purpose.
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 23
To check for yielding, this von Mises maximum stress is compared to the
yield strength, as usual
For a quick, conservative check, an estimate for maxcan be obtained by
simplyadding a andm . (a + m ) will always be greater than or equal
to max, and will therefore be conservative.
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 25
Example
Solution
7-20
7-20
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 26
Figure 7-20
Figure 7-21
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
CH-18 LEC 30 Slide 28
18-22
18-14
18-24
Solderberg
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
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ME 307
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307MachineDesign I
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ME 307MachineDesign I
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ME 307MachineDesign I
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ME 307MachineDesign I
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ME 307MachineDesign I
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D A A i B Ch t 18 A l d Sh ft
ME 307MachineDesign I
C 8 C 30 Slid 36