Transcript LEC 31 CH-18 Shafts and Axles
ME 307 Machine Design I Dr. A. Aziz Bazoune
Dr. A. Aziz Bazoune
King Fahd University of Petroleum & Minerals Mechanical Engineering Department Chapter 18: Axles and Shafts
ME 307 Machine Design I 18-1 Introduction ……….
922 18-2 Geometric Constraints ……….
927 18-3 Strength Constraints ……….
933 18-4 Strength Constraints – Additional Methods ……….
940 18-5 Shaft Materials ……….
944 18-6 Hollow Shafts ……….
944
18-7 Critical Speeds (Omitted) ……….
945 18-8 Shaft Design ……….
950 Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 2
ME 307 Machine Design I Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 3
ME 307 Machine Design I
Estimating Stress Concentration
The stress analysis process for concentrations .
fatigue is highly dependent on stress
Stress concentrations for size specifications shoulders and keyways are dependent on 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.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 4
ME 307 Machine Design I
Estimating Stress Concentration
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
.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 5
ME 307 Machine Design I
Estimating Stress Concentration
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 r/d
K t
0.02
2.7
0.05
2.1
0.1
1.7
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 .
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 6
ME 307 Machine Design I
Estimating Stress Concentration
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.
Fig. 7-9a.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 7
ME 307 Machine Design I
Estimating Stress Concentration
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.
Dr. A. Aziz Bazoune Fig. 7-9b.
Chapter 18: Axles and Shafts Fig. 7-9c.
CH-18 LEC 31 Slide 8
ME 307 Machine Design I Figure 7-9
Techniques 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 CH-18 LEC 31 Slide 9
ME 307 Machine Design I
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 fillet , for estimating K t values for the first iteration, an r/d ratio should be selected so K t values can be obtained. For the worst end of the spectrum, with r/d = 0.02 and D/d = 1.5
, K t 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 CH-18 LEC 31 Slide 10
ME 307 Machine Design I
A 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 K t = 2.2 for bending and K ts = 3.0 for torsion, assuming the key is in place.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 11
ME 307 Machine Design I
A 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 K t = 2.2 for bending and K ts = 3.0 for torsion, assuming the key is in place.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 12
ME 307 Machine Design I Table 7-1
First 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 CH-18 LEC 31 Slide 13
ME 307 Machine Design I
Fatigue Analysis of Shafts
The fatigue strength will be determined using: 1. Distortion-Energy-Gerber 2 Distortion-Energy-Elliptic Rotating Shaft under stationary bending and torsional moments
xa
xym
K f
32
M a
d
3
K fs
16
T m
d
3
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 14
ME 307 Machine Design I
Fatigue Analysis of Shafts
Gerber
Safety Factor
n
S e
'
a
n
S u
'
m
2 1
ASME-Elliptic
n
S e
'
a
2
n
Sy
'
m
2 1
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 15
ME 307 Machine Design I
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,
Problem 18-10
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 16
ME 307 Machine Design I
Problem 18-10
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 17
ME 307 Machine Design I
Problem 18-10
We have a design task of identifying bending moment and torsion diagrams which are preliminary to an industrial roller shaft design.
Gear Roller
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 18
ME 307 Machine Design I Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 19
ME 307 Machine Design I 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 CH-18 LEC 31 Slide 20
ME 307 Machine Design I
Problem 18-11
1. Using a 1035 hot rolled steel , estimate the necessary diameter at the locations of peak bending moment 2. Estimating the 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. 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 CH-18 LEC 31 Slide 21
ME 307 Machine Design I
Problem 18-11
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 22
ME 307 Machine Design I
Problem 18-11
From static Analysis
d
16
n
S y
4
M
2 3
T
2 1/2 1/3 0.6
in
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 23
ME 307 Machine Design I
Problem 18-11
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 24
ME 307 Machine Design I
Hollow Shafts
As an example equation shaft case: 18-21 is modified to take into account the hollow
d o
(1 16
n d d i o
4 ) 4
K M f a S e
2
K T fs m S y
2 1/2 1/3
where
d i
shaft.
and
d o
are respectively the inner and outer diameters of the
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.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 25
ME 307 Machine Design I
Shaft Design
One approach is (See Lab Handbook): 1.
Selecting a material (usually steel) 2.
3.
Drawing a free body diagram of the shaft Performing static equilibrium analysis and 4.
5.
6.
7.
Locating the critical area Performing static stress analysis to find a starting diameter size, d’ .
Using the value of d’ in calculating the endurance limit (a trial diameter can also be used) Estimating the critical value of the diameter, d , using DE-Gerber or DE ASME-elliptic methods 8.
9.
Repeat step 6 if d different from 10.
Performing deflection analysis d’ .
Building the rest of the shaft by considering the machine parts to be mounted on the shaft (bearings, gears, pulleys, …) 11.
Performing Dynamic analysis Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 26
ME 307 Machine Design I
Shaft Materials
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 .
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 27
ME 307 Machine Design I
N Shaft Design?
Y find safety Factor, n Y Shaft rotating?
N Find critical diameter, d N Shaft rotating?
Y Fatigue Analysis N Static Analysis Eq. 6-44 or 6-46 n Y Reversed bending & steady torque?
Eq. 18-17 or 18-22 Eq. 18-14 or 18-20
n Dr. A. Aziz Bazoune
Static Analysis Eq. 6-43 or 6-45 d Static Analysis Eq. 6-42 or 6-44 d' Fatigue Analysis Y Reversed bending & steady torque?
N
Chapter 18: Axles and Shafts
Eq. 18-16 or 18-21 N d d. NE. d’ Y d = Critical shaft diameter Deformation analysis Eq. 18-13 or 18-19 N
ME 307 Machine Design I Dr. A. Aziz Bazoune
Geometric Constraints
Unlike stress, which is a function of
local geometry and load geometry everywhere
determined.
.
Thus, The
,
deflection is a function of the task of deflection and rigidity analyses can be started only when the entire geometry of the shaft is 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.
Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 29
ME 307 Machine Design I
Stress Concentrations and Shaft 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 K t = 2.0 for profile key seats K t = 1.6 for sled runner keyseats
Sled runner keyseat Shoulders
Dr. A. Aziz Bazoune
Shoulder Profile keyseat Groove Shoulders
Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 30
ME 307 Machine Design I
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 .
Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 31
ME 307 Machine Design I
Geometric Constraints :
Shaft Deflection and Slopes
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.
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 32
ME 307 Machine Design I
Geometric Constraints :
Shaft Deflection and Slopes
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) .
Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 33
ME 307 Machine Design I Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts CH-18 LEC 31 Slide 34