Files 2-Lectures LEC 31 CH-18 Shafts and Axles
Transcript of Files 2-Lectures LEC 31 CH-18 Shafts and Axles
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
Dr. A. Aziz BazouneKing Fahd University of Petroleum King Fahd University of Petroleum
& Minerals& MineralsMechanical Engineering Mechanical Engineering
DepartmentDepartment
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
18-1 18-1 Introduction ……….Introduction ……….92292218-218-2 Geometric Constraints ……….Geometric Constraints ……….92792718-318-3 Strength Constraints ……….Strength Constraints ……….93393318-418-4 Strength Constraints – Additional Methods Strength Constraints – Additional Methods ……….……….94094018-518-5 Shaft Materials ……….Shaft Materials ……….94494418-618-6 Hollow Shafts ……….Hollow Shafts ……….94494418-718-7 Critical Speeds (Omitted) ……….Critical Speeds (Omitted) ……….945945 18-818-8 Shaft Design ……….Shaft Design ……….950950
CH-18 LEC 31 Slide 2
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 3
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
The stress analysis process for fatigue is highly dependent on stress concentrations.
Stress concentrations for shoulders and keyways are dependent on size specifications that are not known the first time through the process.
Fortunately, since these elements are usually of standard proportions, it is possible to estimate the stress concentration factors for initial design of the shaft. These stress concentrations will be fine-tuned in successive iterations, once the details are known.
Estimating Stress Estimating Stress ConcentrationConcentration
CH-18 LEC 31 Slide 4
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
Shoulders for bearing and gear support should match
the catalog recommendation for the specific bearing or
gear.
A look through bearing catalogs shows that a typical
bearing calls for the ratio of D/d to be between 1.2 and
1.5.
For a first approximation, assume D/d =1.5 can be
assumed.
Fillet radius at the shoulder needs to be sized to avoid
interference with the fillet radius of the mating
component. There is a significant variation in typical
bearings in the ratio of fillet radius r/d versus bore
diameter, with typically ranging from around 0.02 to
0.06.
Estimating Stress Estimating Stress ConcentrationConcentration
CH-18 LEC 31 Slide 5
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
Figures A-15-8 and A-15-9 show that the stress
concentrations for bending and torsion increase
significantly in this range. For example, with D/d = 1.5
for bending
In most cases the shear and bending moment diagrams
show that bending moments are quite low near the
bearings, since the bending moments from the ground
reaction forces are small.
Estimating Stress Estimating Stress ConcentrationConcentration
CH-18 LEC 31 Slide 6
r/d 0.02 0.05 0.1
Kt 2.7 2.1 1.7
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
In cases where the shoulder at the bearing is found to be
critical, the designer should plan to select a bearing with
generous fillet radius, or consider providing for a larger fillet
radius on the shaft by relieving it into the base of the shoulder
as shown in Fig. 7-9a.
This effectively creates ahead
zone in the shoulder area that
does not carry the bending
stresses, as shown by the stress
flow lines.
Estimating Stress Estimating Stress ConcentrationConcentration
CH-18 LEC 31 Slide 7
Fig. 7-9a.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
A shoulder relief groove as shown in Fig. 7-9b can accomplish
a similar purpose. Another option is to cut a large-radius
relief groove into the small diameter of the shaft, as shown in
Fig. 7-9c.
Estimating Stress Estimating Stress ConcentrationConcentration
CH-18 LEC 31 Slide 8
Fig. 7-9b. Fig. 7-9c.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 9
Figure 7-9Techniques for reducing stress concentration at a shoulder supporting a bearing with a sharp radius. (a) Large radius undercut into the shoulder. (b) Large radius relief groove into the back of the shoulder. (c) Large radius relief groove into the small diameter.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 10
This has the disadvantage of reducing the cross-sectional
area, but is often used in cases where it is useful to provide a
relief groove before the shoulder to prevent the grinding or
turning operation from having to go all the way to the
shoulder.
For the standard shoulder filletFor the standard shoulder fillet, for estimating Kt values for
the first iteration, an r/d ratio should be selected so Kt values
can be obtained. For the worst end of the spectrum, with r/d
= 0.02 and D/d = 1.5, Kt values from the stress concentration
charts for shoulders indicate 2.7 for bending, 2.2 for torsion,
and 3.0 for axial.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 11
A keyway keyway will produce a stress concentration near a critical
point where the load transmitting component is located. The
stress concentration in an end-milled keyseat is a function of
the ratio of the radius r at the bottom of the groove and the
shaft diameter d. For early stages of the design process, it is
possible to estimate the stress concentration for keyways
regardless of the actual shaft dimensions by assuming a
typical ratio of r/d = 0.02. This gives Kt = 2.2 for bending and
Kts= 3.0 for torsion, assuming the key is in place.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 12
AA keyway keyway will produce a stress concentration near a critical
point where the load transmitting component is located. The
stress concentration in an end-milled keyseat is a function of
the ratio of the radius r at the bottom of the groove and the
shaft diameter d. For early stages of the design process, it is
possible to estimate the stress concentration for keyways
regardless of the actual shaft dimensions by assuming a
typical ratio of r/d = 0.02. This gives Kt = 2.2 for bending and
Kts= 3.0 for torsion, assuming the key is in place.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 13
Table 7-1First iteration estimates for stress concentration factors Kt
Warning: These factors are only estimates for use when actual dimensions are not yet determined. Do not use these once actual dimensions are available.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 14
Fatigue Analysis of ShaftsFatigue Analysis of Shafts
The fatigue strength will be
determined using:
1.Distortion-Energy-Gerber
2 Distortion-Energy-
Elliptic
3
32 a
xa f
MK
d
3
16 m
xym fs
TK
d
Rotating Shaft under stationary bending and torsional moments
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 15
Fatigue Analysis of ShaftsFatigue Analysis of Shafts
2' '
1a m
e u
n nS S
2 2' '
1a m
e
n nS Sy
Gerber
ASME-Elliptic
SafetySafety FactorFactor
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
Problem 18-10 Problem 18-10
CH-18 LEC 31 Slide 16
A geared industrial roll shown in the figure is driven at 300 rev/min by a force F acting on a 3-in-diameter pitch circle as shown. The roll exerts a normal force of 30 lbf/in of roll length on the material being pulled through. The material passes under the roll. The coefficient of friction is 0.40. Develop the moment and shear diagrams for the shaft modeling the roll force as a concentrated force at the center of the roll,
A geared industrial roll shown in the figure is driven at 300 rev/min by a force F acting on a 3-in-diameter pitch circle as shown. The roll exerts a normal force of 30 lbf/in of roll length on the material being pulled through. The material passes under the roll. The coefficient of friction is 0.40. Develop the moment and shear diagrams for the shaft modeling the roll force as a concentrated force at the center of the roll,
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
Problem 18-10 Problem 18-10
CH-18 LEC 31 Slide 17
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 18
We have a design task of identifying bending moment and torsion diagrams which are preliminary to an industrial roller shaft design.
Gear
Roller
Problem 18-10 Problem 18-10
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 19
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 20
This approach over-estimates the bending moment at C, torque at C but not at A.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 21
Problem 18-11 Problem 18-11
1. Using a 1035 hot rolled steel, estimate the necessary diameter at the locations of peak bending moment using a design factor of 2. These are likely to be fillets at both ends of the right hand bearing seat, where the bending moment is slightly less than the local extreme.
2. Estimating the fatigue stress-concentration factor as 2, and using a design factor of 2, what is the approximate necessary diameter of the bearing seat using the DE-elliptic fatigue failure criterion in Problem 18-10?
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 22
Problem 18-11 Problem 18-11
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 23
1/3
1/22 2164 3 0.6 in
y
nd M T
S
From static Analysis
Problem 18-11 Problem 18-11
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 24
Problem 18-Problem 18-11 11
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 25
As an example equation 18-21 is modified to take into account the hollow shaft case:
where di and do are respectively the inner and outer diameters of the shaft.
With this, one can consider that the stress-strength analysis is completed. You have obtained the minimum diameter at the critical section that can withstand the applied loads.
1/31/222
4
164 3
(1 )
f a fs mo
e yi o
K M K Tnd
S Sd d
Hollow ShaftsHollow Shafts
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 26
One approach is (See Lab Handbook):
1. Selecting a material (usually steel)
2. Drawing a free body diagram of the shaft
3. Performing static equilibrium analysis and
4. Locating the critical area
5. Performing static stress analysis to find a starting diameter size, d’.
6. Using the value of d’ in calculating the endurance limit (a trial diameter can also be used)
7. Estimating the critical value of the diameter, d, using DE-Gerber or DE-ASME-elliptic methods
8. Repeat step 6 if d different from d’.
9. Building the rest of the shaft by considering the machine parts to be mounted on the shaft (bearings, gears, pulleys, …)
10.Performing deflection analysis
11.Performing Dynamic analysis
Shaft DesignShaft Design
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 27
Shafts are usually made of ductile materials.
Small shafts with diameters less than 3.5 in (90 mm)
are usually made of Cold Drawn carbon steel (AISI
1018-1050).
Larger diameter shafts are machined from Hot
Rolled steel.
Heat treated steels are also used when higher
strengths are necessary.
Shaft MaterialsShaft Materials
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 28
Shaft
Design?
Find critical diameter, d
Shaft rotating?N
Static AnalysisEq. 6-42 or 6-44
Static AnalysisEq. 6-43 or 6-45
d = Critical shaftdiameter
d. NE. d’
Y
N
find safety Factor, n
Y N
Static AnalysisEq. 6-44 or 6-46
n
Fatigue Analysis
Y
N Y
Shaft rotating?
n
Reversedbending & steady
torque?
N
Eq. 18-17 or 18-22 Eq. 18-14 or 18-20
Y Reversedbending & steady
torque?
N
Eq. 18-16 or 18-21 Eq. 18-13 or 18-19
d d'
Fatigue Analysis
d
N
Complete shaftgeometry & perform
Deformation analysis
Y
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 29
Geometric ConstraintsGeometric Constraints
Unlike stress, which is a function of local geometry and load, deflection is a function of the geometry everywhere. Thus, The task of deflection and rigidity analyses can be started only when the entire geometry of the shaft is determined.
However the approach described in section 18-2, which is based on bearing slope constraints as limiting, may be used first assuming a uniform diameter shaft and using equations 18-1 and 18-2 to find the diameters at the bearings.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 30
Shoulders
ShouldersSled runner keyseat
ShoulderGroove
Profile keyseat
Stress Concentrations and Shaft Stress Concentrations and Shaft Geometry Geometry
Shaft shoulders are used to position and provide necessary thrust supports for elements such as bearings, gears, pulleys,… Provisions must be made for torque-transfer elements such as keys, splines, pins The theoretical stress concentration factors for shoulders, grooves and transverse holes can be obtained from appendix [A15+]. Others are
Kt = 2.0 for profile key seatsKt = 1.6 for sled runner keyseats
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 31
Shaft Geometry Shaft Geometry
To determine the entire geometry of the shaft one has to rely on existing models. Some of these models are given in figures 18-1 through 18-8 of the Textbook. More shaft configurations can be found in the FAG handbooks of the Design of Rolling Bearing Mountings.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 32
The transverse deflection of the elastic curve of the shaft
can be determined by any one of the methods studied in
Chapter 5.
The superposition method, which utilizes Appendix A-9, is
recommended. For complex shaft geometry the numerical
integration or computer program may be used.
Geometric ConstraintsGeometric Constraints: : Shaft Deflection Shaft Deflection and Slopesand Slopes
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 33
1. The slope at ball bearings should be limited at 0.25 deg the
slope at roller bearings and long journal bearings should be a
lot less. For details on acceptable slopes refer to FAG and
SKF catalogs.
2. For machinery shafting, the deflection should be no greater
than 0.001 in/ft (0.075 mm/m) of shaft length between
bearing supports.
3. For shafts mounting good quality spur gears, the deflection
at the gear mesh should not exceed 0.005 in. (0.125 mm) or
F/200 (F is the gear face width in inches) and the slope
should be limited 0.0286 deg.
4. For shafts mounting good quality bevel gears, the deflection
at the gear mesh should not exceed 0.003 in. (0.076 mm).
Geometric ConstraintsGeometric Constraints: : Shaft Deflection Shaft Deflection and Slopesand Slopes
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts
ME 307 Machine Design I
CH-18 LEC 31 Slide 34