Mechanical Analysis of Guide Rollers
Completed by: Ken Wilkinson
Completed on: October 19, 2015
Assumptions:
- The material properties determined by the group will be used for this analysis where possible.
Otherwise properties are taken for 1040 steel.
- Diameter is 320 mm for two rollers and assumed to be 160 mm for three rollers (Group Numbers).
- Density is 2020.5 kg/m^3 (Group Number).
- Length is 400mm (Customer Requirement).
- Elastic Modulus is 200 GPa for 1040 steel (metweb.com).
- Yield Stress is 415 MPa (matweb.com).
- Ultimate Stress is 620 MPa (matweb.com).
- The rollers will be treated as a solid shaft. In actuality there will be a hollowed roller slid over a shaft. In
terms of mechanical roller stress this will be the same, but for now critical features such as splines,
keyways, and pins are ignored. They will be done next phase.
- There is an additional 20 mm of non- work area added to each side. Presumably the wire will not
wrapped all the way to the edge of the roller, we will need a small length for mounting. This could be a
smaller diameter (as shown in sketch) or the full OD. 20 mm is taken from current roller.
- Wire tension is treated as a distributed load with wire tension assumed to be 25 N for the worst case
analysis. Pitch is assumed to be 1.515 mm for the smallest pitch setting.
- Tm = 10.99 N-m for two rollers and 0.78 N-m for three rollers (Group Numbers).
Set up of roller loading:
Figure 1: Guide Roller Loading Sketch
Determination of Max Loading:
Draw the Shear and Moment diagrams. Moment is the area under the Shear plot.
Figure 2: Shear and Bending Diagrams
Area under the curve:
𝐴𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 = 𝑏𝑎𝑠𝑒 ∗ ℎ𝑒𝑖𝑔ℎ𝑡
𝐿
𝐴𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 = 𝑋 ∗ 𝑤 (1)
2
Combining both yields
1
1 𝐿 𝐿
𝐴𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 = 𝑏𝑎𝑠𝑒 ∗ ℎ𝑒𝑖𝑔ℎ𝑡 = ∗ ∗ 𝑤
2
2 2 2
𝐿2
𝐴𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 = 𝑤 (2)
8
𝑀𝑚𝑎𝑥 = (4𝑋 + 𝐿)
At the center of the roller.
𝐿𝑤
(3)
8
There are also critical points at the edge of the work area, if there is a decrease in diameter, as this will
yield a stress concentration. This will be analyzed next phase as it is not a part of roller sizing.
In order to predict factor of safety, the Soderberg Failure Theorem will be used. This is the most
conservative failure theory, and will ensure the shaft survives.
When taken from Shigley’s Mechanical Design, Soderberg’s theorem states that:
1
1
1
16 1
1
2
2 2
2
2 2
=
(4(𝐾
𝑀
+
3(𝐾
𝑇
)
+
(4(𝐾𝑓 𝑀𝑚 ) + 3(𝐾𝑓𝑠 𝑇𝑚 ) ) ]
[
)
)
𝑓
𝑎
𝑓𝑠
𝑎
𝑛 𝜋𝑑3 𝑆𝑒
𝑆𝑦
Which gives the relation between diameter and design factor.
For the roller loading, torque is not alternating, and moment is fully reversing, so this simplifies to:
1
1
1
16 1
1
2 2
2 2
= 3 [ (4(𝐾𝑓 𝑀𝑎 ) ) + (3(𝐾𝑓𝑠 𝑇𝑚 ) ) ] (4)
𝑛 𝜋𝑑 𝑆𝑒
𝑆𝑦
For analysis of the work area critical point, the stress concentrations are 1 as there is no diameter
change or feature to experience a stress concertation.
Stress Analysis of two vs. three rollers:
In order to endurance limit:
𝑆𝑒 = 𝑘𝑎 𝑘𝑏 𝑘𝑐 𝑘𝑑 𝑘𝑒 𝑘𝑓 𝑆𝑒 ′
Assuming machined or cold drawn material
(𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝑘𝑎 = 4.51𝑆𝑢 −0.265
(𝑠𝑖𝑧𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝑘𝑏 = 1.51𝑑−0.157
(𝑙𝑜𝑎𝑑𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟) 𝑘𝑐 = 1 𝑓𝑜𝑟 𝑏𝑒𝑛𝑑𝑖𝑛𝑔
(𝑡𝑒𝑚𝑝𝑎𝑡𝑢𝑟𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) 𝑘𝑑 = 1 𝑖𝑓 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑎𝑡 𝑟𝑜𝑜𝑚 𝑡𝑒𝑚𝑝𝑢𝑟𝑎𝑡𝑢𝑟𝑒
(𝑟𝑒𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟) 𝑘𝑒 = 1 𝑢𝑛𝑙𝑒𝑠𝑠 𝑠𝑐𝑎𝑡𝑡𝑒𝑟 𝑑𝑎𝑡𝑎 𝑛𝑒𝑒𝑑𝑠 𝑡𝑜 𝑏𝑒 𝑡𝑎𝑘𝑒𝑛 𝑖𝑛𝑡𝑜 𝑎𝑐𝑐𝑜𝑢𝑛𝑡
(𝑚𝑖𝑠𝑐𝑒𝑙𝑙𝑎𝑛𝑒𝑜𝑢𝑠 𝑓𝑎𝑐𝑡𝑜𝑟) 𝑘𝑓 = 1
For ultimate stress less than 1400 MPa:
1
𝑆𝑒 = 𝑆𝑢
2
Solving for the two roller diameters:
𝑆𝑒,320 𝑚𝑚 𝑟𝑜𝑙𝑙𝑒𝑟 = 161.5 𝑀𝑃𝑎
𝑆𝑒,160 𝑚𝑚 𝑟𝑜𝑙𝑙𝑒𝑟 = 173.2 𝑀𝑃𝑎
The distributed load has magnitude
𝑤=
2 ∗ 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 ∗ cos 𝜃
(5)
𝑝𝑖𝑡𝑐ℎ
Where theta is the angle between the direction of tension and the y axis. For two rollers this value is
zero, and for three rollers this value is 15 degrees.
Plugging (5) into (3) and solving for maximum moments yields values of
𝑀𝑎,2 𝑟𝑜𝑙𝑙𝑒𝑟𝑠 = 792 𝑁𝑚
𝑀𝑎,3 𝑟𝑜𝑙𝑙𝑒𝑟𝑠 = 765 𝑁𝑚
Solving (4) for factor of safety yields:
𝑛2 𝑟𝑜𝑙𝑙𝑒𝑟𝑠 = 654
𝑛3 𝑟𝑜𝑙𝑙𝑒𝑟𝑠 = 91
The factor of safety is much higher for the 2 roller configuration, however the factor of safety for both
roller configurations is so high that either one should be more than adequate for the mechanical stress
that the system is experiencing. Even if something was overlooked, either roller should be adequate. It is
more likely the system will mechanically fail where it is coupled to the motor.
Deflection Analysis:
Taking the elastic curve for a distributed load from Beer and Johnston:
𝛿=
−𝑤
(𝑥 4 − 2𝐿𝑥 3 + 𝐿3 𝑥) (6)
24𝐸𝐼
Using (5) and:
𝑤𝑤𝑒𝑖𝑔ℎ𝑡 =
𝑟𝑜𝑙𝑙𝑒𝑟 𝑚𝑎𝑠𝑠
𝑟𝑜𝑙𝑙𝑒𝑟 𝑙𝑒𝑛𝑔𝑡ℎ
Yields two deflections where
𝛿𝑛𝑒𝑡 = √𝛿𝑡𝑒𝑛𝑠𝑖𝑜𝑛 2 + 𝛿𝑤𝑒𝑖𝑔ℎ𝑡 2
Table 1: Deflection of Guide Roller Under Tension and Their Own Weight
Deflection of Roller Assembly
x
2 rollers
3 rollers
d tension
d weight
d total
d tension
d weight
d total
0
0.00
0.00
0.00
0.00
0.00
0.00
0.025
-21.21
-1.02
21.23
-327.78
-16.39
328.19
0.05
-41.49
-2.00
41.54
-641.21
-32.07
642.02
0.075
-60.03
-2.90
60.10
-927.71
-46.40
928.87
0.1
-76.13
-3.68
76.22
-1176.63
-58.84
1178.10
0.125
-89.25
-4.31
89.35
-1379.27
-68.98
1381.00
0.15
-98.92
-4.78
99.04
-1528.85
-76.46
1530.76
0.175
-104.86
-5.07
104.98
-1620.53
-81.04
1622.56
0.2
-106.85
-5.16
106.98
-1651.42
-82.59
1653.48
0.225
-104.86
-5.07
104.98
-1620.53
-81.04
1622.56
0.25
-98.92
-4.78
99.04
-1528.85
-76.46
1530.76
0.275
-89.25
-4.31
89.35
-1379.27
-68.98
1381.00
0.3
-76.13
-3.68
76.22
-1176.63
-58.84
1178.10
0.325
-60.03
-2.90
60.10
-927.71
-46.40
928.87
0.35
-41.49
-2.00
41.54
-641.21
-32.07
642.02
0.375
-21.21
-1.02
21.23
-327.78
-16.39
328.19
0.4
0.00
0.00
0.00
0.00
0.00
0.00
deflection magnitude (nanometers)
Deflection Magnitude of Guide Roller
1800
1600
1400
1200
1000
800
600
400
200
0
"2 Rollers"
"3 Rollers"
0
0.1
0.2
0.3
0.4
0.5
x (meters)
Figure 3: Deflection Magnitude of Guide Rollers
Critical Speed Estimate
Using Rayleigh’s Method, the critical speed is determined by breaking the shaft up into several lumps
and plugging into:
𝑔 ∑ 𝑤𝑖 𝑦𝑖
𝜔𝑐𝑟𝑖𝑡 = √
∑ 𝑤𝑖 𝑦𝑖 2
Solving using 8 equal sized lumps yields critical speeds of 470707.6 rpm for the two roller configuration
and 117676.9 rpm for the three roller configuration.
Summary/ Future Steps:
Feature
Mechanical Stress Factor of Safety
Maximum Deflection (nm)
Critical Speed (rpm)
Rotational Speed Factor of Safety
2 Rollers
654
107
471000
523
3 Rollers
91
1653
118000
65
Overall, the two roller system is better mechanically. It has much higher factors of safety for both stress
due to loading and the critical speed of the shaft, as well as a lower max deflection. It should be noted,
however, that both are sufficiently mechanically strong by a huge margin. Neither configuration will fail
along the maximum moment point, or reach sufficient speed or deflection to cause a problem.
The most likely place for the roller to fail mechanically will be the motor interface or where the roller is
attached to the inner shaft. The purpose of the analysis in this document was to compare the rollers in
terms of sizing. Once the team has a configuration selected, more analysis will need to be done on these
other critical points to ensure mechanical life.