UNIVERSITY OF PITTSBURGH
MEMS 1029 Design Project #2 Shaft Design Jeremy Murphy 3/28/2013
P a g e | 1 I.
Introduction
A shaft is to be designed to meet the following specifications, arranged arranged as in Figure 1:
Figure 1: Shaft design layout.
Must transmit 50 W of power at 1200 RPM.
Bearings must have a lifetime of at least 20,000 hr. at a reliability of 0.90.
The coupling must be flexible, and attach to a drive shaft of 0.4997” +0.0000/0.0005”.
The sheave should have a diameter of around 150 mm.
Tension on the loose side of the belt will be 15% of that on the tight side.
This document will present the formal calculation process for determining a shaft of appropriate dimensions as well as all attached components with adequate safety parameters. parameters. It will then include a list of parts, numbers, and models of the shaft from manufacturer’s data including cost analysis.
II.
Shaft Strength Analysis
To begin, the power given in the problem statement must be converted into a torque applied to the shaft. This can be done with the equation for angular power:
Where: P = Power
T = Torque ω = Angular speed
This equation can be rearranged to solve for the torque, when given an angular speed N in rev/min, as:
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From the torque, the tensions on the sheave can be found via the equation for torque given combined forces F acting on a wheel of radius r:
Where F1 is the tight side tension and F 2 is the loose. The problem states that the loose side is 15% or 0.15 times the tight side tension, so Equation (2) can be rewritten as:
It will be convenient to express the total force as the addition of these two forces, which comes to F= 7.018 N. This force can be used in a beam bending analysis of the shaft. The free body diagram (FBD) is shown in Figure 2. RA
y
RB
F
x
A
300mm
Figure 2: Free body diagram for the shaft.
B
100mm
C
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This FBD assumes: (1) A 180-degree wrap angle over the sheave. (2) A distance of 100mm between the right-hand bearing and the sheave. (3) A negligible distance between the coupling and the left-hand bearing. (4) No reaction force at the coupling since it is flexible and close to A. (5) The sheave is close enough to t he end of the shaft to assume it is at the end for calculation’s sake. With these assumptions in mind, the static equilibrium equations can be established. Beginning with the moment about point A (with counter-clockwise taken as positive), the reaction force at point B can be solved for:
From there, the reaction force at A can be solved for using the y-direction force equilibrium.
th
Using Table A-9, #10, found in Shigley’s Mechanical Engineering Design, 9 Edition, which gives bending information for a simply-supported beam with an overhanding load, the maximum moment is found at point B and given by Equation (5) (Budyas 1017).
Where:
x = the x coordinate of point C. l = the distance to point C from the left side. a = The distance to the overhanging load from C. Variables x and l will be equal in this case, making the moment simply F*a, with a = 100mm. This gives a maximum moment of 0.718 N m.
P a g e | 4 As a starting point for determining shaft diameter, point C will be tested for fatigue and yielding since it is the point of maximum bending moment and contains a shoulder to hold the bearing in place, which will concentrate the local stress. For an initial test, the bending and shear stress concentration factors, Kt and Kts respectively, will be taken from Table 7-1 in Shigley, which gives first iteration estimates of stress concentration factors (Budyas 373). For a sharp shoulder fillet used to to hold bearings in place:
To perform a fatigue analysis, the bending moments and torques must be categorized into alternating and midline values, denoted by the subscripts a and m, respectively.
There will be no midline moment since it is continuously alternating as the shaft spins, and no alternating torque as it is constantly supplied from the motor. Next, a material must be chosen. A good simple starting choice is 1020 CD (cold-drawn) steel which has an ultimate tensile strength S ut= 470 MPa. The endurance limit can be found by using Equation 6-18 from Shigley , which gives the endurance limit with modification factors for individual designs (Budyas 287).
(Copy of Equation 6-18 from Shigley) Factors c-f can be assumed as one. The surface condition factor is given by Equation 6-19 in Shigley :
Where variables a and b are found in Table 6-2 on the following page. For a CD finish, strength in MPa, a=4.51, b=-0.265. Therefore, k a=0.883.
P a g e | 5 The size modification factor will be appropriately determined when the diameter of the shaft is chosen, so a good first estimate will be 0.9. Finally, the endurance limit can be estimated from Equation 6-8 in Shigley , which gives:
For steels with a tensile strength less than 1400 MPa (Budyas 282). This gives an endurance limit of 188 MPa. Using this along with the modification factors calculated calculated in Equation 6-18, 6 -18, the overall endurance limit is found to be S e=176.179 MPa. With this value established and the moments and torques calculated, the Distortion Energy Theory with Goodman Fatigue Failure Criterion can be applied. Goodman’s criterion was chosen because it is simple and conservative. This is given in Shigley as as Equation (7-8) to find the minimum diameter diameter of a shaft with a specified factor of safety, n (Budyas 368).
{ [() () ] [() () ] }
This equation, with a factor of safety of 2 and all other predetermined predetermined values plugged in, yields a minimum diameter of 5.29 mm. This diameter is rather small due to the low amount of power that the shaft transmits, so deflection limitations will likely be the governing factor in choosing an appropriate diameter. III.
Shaft Deflection Analysis
The bearings are a good first choice for deflection analysis, as they have a limit to the angular misalignment that they can tolerate as specified by their manufacturer. The equation for the beam deflection, found again from Table A-9 #10 in Shigley gives gives the deflection y of the beam with Modulus of Elasticity E and Second moment of cross-sectional area I from point A to B as (Budyas 1017):
Where a, l, and x are the same as stated previously. Bearing misalignment tolerances are often given in angular form, so the angle of the beam from the horizontal can be found by taking the derivative of Equation (9). When simplified, this derivative is:
Plugging in the numbers for the angular deflection at A and B, it is found that, in terms of I, the absolute values are:
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The deflection will be greater at B, so that point will be used to determine the minimum diameter. Shigley gives gives typical maximum allowable slopes in Table 7-2 for different types of bearings (Budyas 379).
To be extremely conservative as a first guess, and to allow for the selection of any type of ball bearing, the minimum tabulated value will be used, which is 0.001 rad. Plugging this value into equation 12 yields:
For a circular cross section:
So:
Hence, the deflection requirement does indeed lead to a higher minimum diameter. As a final test, the deflection of the pulley will be examined to see if it will further limit the minimum allowable diameter. The pulley is desired to be as straight in line as possible with the shaft to reduce alternating moments and off-center forces. Thus, a maximum deflection of 0.1 mm will be imposed. The deflection of the pulley at point C can be found, as before, using Table A-9 #10 from Shigley (Budyas (Budyas 1017). It is given as:
Plugging in the numbers as before gives:
Solving for I as in Equation (13):
The pulley deflection, therefore, will be the overall limiting factor in choosing a shaft diameter.
IV.
Bearing Life
The manufacturer’s catalog dynamic (c10) load rating based on desired hours of life under a given
load at 0.90 reliability is given by Equation (11-3) in Shigley (Budyas (Budyas 575). For a desired life in hours, L=20,000; speed in rpm, n=1 200; and a manufacturer’s rating life L 10 (usually 10^6 rev.):
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This application will then require a c10 rating of 79.25N.
V.
Actual Components
Mechanical goods supplier McMaster-Carr offers a wide variety of components well-suited for this application. Based on research into the components they offer to meet this need, a shaft diameter of 5/8” will be chosen, machined down to ½” for the mounting shoulders. These diameters translate to 15.88 mm and 12.7 mm respectively, which are close to the required minimum diameters and will allow for additional safety. The parts chosen are as follows: 1. Shaft: McMaster-Carr 5/8” diameter 24” long steel shaft, Model No. 1346K25.
This shaft will come to a total length of 609.6 mm. It will be trimmed down to 500 mm to allow for the 300 mm space between the bearings, 100 mm space to the sheave, and small additional sections outside of the pulley and bearings. This shaft is also made of AISI 1566 steel, which is stronger than the original assumed material and will further increase safety factors. 2. Bearings: 2x McMaster-Carr ½” bore Cast Iron Mounted Ste el Ball Bearings, Model No. 6244K51.
These bearings are rated for a dynamic load capacity of 2200 lb., which is 9786.08 N, far exceeding the design requirements. They can handle a maximum rotation speed of 6,299 rpm, also far exceeding requirements without being too costly. They come with pillow block housings to mount the shaft on the required surface. 3. Pulley: McMaster-Carr McMaster-Carr 5.95” Diameter Type A V-Belt Pulley, Model No. 6407A24 with ½” bore
Type JA Quick-Disconnect Bushing, Model No. 6086K111. This pulley comes to a diameter of 151.13 mm, close to the required 150 in the problem statement. It is paired with a quick-disconnect bushing that has a cap screw allowing it to tightly grip the shaft without the need for a shoulder to hold it in place axially. The bushing has a keyway on the inside that mates with a 1/8” (3.175mm) (3 .175mm) wide ANSI standard key to transmit power. The length required for this key can be calculated by considering the force at the surface of the shaft. This can be found by:
Which, for a shaft diameter of 12.7 mm at the pulley, comes to F=62.68 N.
P a g e | 8 The key thickness will be 3.175 mm, and a factor of safety n=3 will be imposed. McMaster-Carr key stock made of spring steel, AISI 1095 will be used. The yield stress will be assumed to be the minimum value listed in Shigley for for this type of steel to ensure a conservative estimate, which is S y = 500 MPa for this alloy quenched and tempered at 650 C. By the Distortion-Energy Theory:
For shear resistance, the length of the key can be found from Equation (18) (Budyas 395).
For crushing resistance, Equation (19) can be used (Budyas 395).
4. Key: McMaster-Carr 1/8” x 1/8” Spring Steel key stock, 12” long, Model No. 98535A120, cut to 27mm (1 1/16”).
5. Coupling: McMaster-Carr McMaster-Carr Ultra-Flex Double-Loop Shaft Coupling, ½” bore, Model No. 6183K61.
This coupling is not only o nly cheap, but allows huge angular misalignment between the drive shaft and designed shaft. 6. Retaining Rings: 2x McMaster-Carr ½” Diameter Black-Finish Steel Self-Locking Retaining Rings,
Model No. 98430A138. These retaining rings will be used to hold the bearings in place against the shoulders. They will be appropriate given the light loads and remove the need for additional diameter change to accommodate regular retaining rings.
VI.
Final Shaft Analysis
With the components of the shaft selected, the critical locations must be checked with the actual parameters to ensure that no failure will occur. o ccur. Beginning with the shoulder support for the bearing at point C as before, the ratio of the fillet radius to diameter should be:
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The true stress concentration factors, K f and Kfs, for bending and shear, respectively, are given by Equations (21) and (22).
Where q and qs are the notch sensitivities for bending and shear, respectively. respectively. These can be found from Figures 6-20 and 6-21 in Shigley given given the notch radius and tensile strength of the steel, which is reported to be 1100 MPa (Budyas 295-296). Referring to the charts,
When plugged into Equations (21) and (22) with K t and Kts the same as in the first iteration, these give values of:
The surface condition factor, k a will be the same. The shape factor, k b, can now be calculated as:
Per Shigley Equation Equation 6-20 (Budyas 288). This gives k b = 0.945, close to the initial guess. The yield strength is estimated again at half the tensile strength, or 550 MPa. Therefore Equation 6-18 from puts the overall endurance limit at 458.44 MPa. The Goodman Failure Criterion will be used Shigley puts again, this time by calculation the alternating and midline Von-Mises Stresses via Equations (24) and (25), respectively, which are Equations 7-5 and 7-6 from Shigley with with terms that are nonzero and simplified (Budyas 368).
√
These equations yield alternating and midline Von-Mises Stresses of 8.574 MPa and 3.400 MPa, respectively. The Goodman Criterion, with a factor of safety n, expressed in terms of these stresses is:
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Giving a factor of safety of 45.88, which is more than satisfactory for safety concerns. Since the bearing at A will have less of a load, and the pulley is held on without diameter change, this result is appropriate for the whole shaft.
It may be important to consider the effects of thermal expansion, especially if the bearings are tightly secured axially. An average thermal expansion coefficient, α, for steel is 10.8*10 -6/°C. This can simply be multiplied by the temperature change expected to find the strain related to the elongation to the member. Assuming a very conservative 50 °C temperature increase, the strain is 5.4*10 -4 mm/mm, which will have a negligible effect on shaft length at any point. The final consideration is the shaft critical speed that will cause uncontrolled increasing deflections. The equation to find this speed is given as Equation 7-22 in Shigley (Budyas (Budyas 383).
Where: ω1 = critical speed
l = length of shaft g= acceleration due to gravity E= Modulus of Elasticity I= second moment of cross sectional area A= cross sectional area γ= specific weight
With g= 9.8 m/s 2, I as previously defined, A being π/4*d2, and γ being 76.5 kN/m3 for carbon steels,
So the design is well below the critical shaft speed. Since all of the safety concerns are far exceeded, this shaft meets all of the design requirements.
P a g e | 11 VII.
Final Design
Solid models of the final design follow. Unfortunately, McMaster-Carr McMaster-Carr did not offer a model of their sheave nor a detailed drawing, so an attempt to recreate it is shown. The coupling also did not come with a model, and would be very complicated to draw in SolidWorks, so a picture is included to illustrate what it would look like. It would attach to the left side opposite the sheave. A bill of materials follows those models.
Ultra-Flex Coupling (From McMaster-Carr McMaster-Carr website)
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5
2
3
6
Table 1: Bill of Materials. All Parts from McMaster-Carr
Part No. 1
2
3
4 5 6 7
Description 5/8” Diameter, 24” Long Steel Shaft 2x ½” bore Cast Iron Mounted Steel Ball Bearings Quick-Disconnect Bushing Style JA, ½” Bore, 1/8” x 1/16” Keyway V-Belt Pulley 5.95” Outer Diameter Standard Key Stock, 1/8” x 1/8”, 12” long 2x ½” Bore Self -Locking -Locking External Retaining Rings Ultra-Flex Double-Loop Shaft Coupling, ½” to ½” Bore
Model No. 98535A120
Cost $18.00
6244K51
$65.68
6086K111
$11.78
6407A24
$42.70
98535A120
$0.97
98430A138
$0.10 ($9.52 per 100)
6183K61
$19.83
Total
$159.06
P a g e | 13 VIII.
Conclusion
The shaft designed far exceeds the specifications of the problem statement. statement. The main reason for this is the excessively low power transmitted. The other main reason is the availability of parts. It was incredibly difficult to find a pulley that was 150 mm in diameter and would fit on anything smaller than a ½” shaft. This shaft however, is not excee dingly expensive and will conceivably last indefinitely indefinitely under the defined operating conditions.
IX.
References
1.) Budynas, R and J. Nisbett. Shigley’s Mechanical Engineering Design, Ninth Edition. New York: McGraw Hill Companies, 2011. Digital.
