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 320mm for two rollers and assumed to be 160mm 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:
Combining both yields
At the center of the roller.
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:
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:
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:
Assuming machined or cold drawn material
For ultimate stress less than 1400 MPa:
Solving for the two roller diameters:
The distributed load has magnitude
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
Solving (4) for factor of safety yields:
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:
Using (5) and:
Yields two deflections where
Table 1: Deflection of Guide Roller Under Tension and Their Own Weight
Deflection of Roller Assemblyx / 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
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:
Solving using 8 equal sized lumps yields critical speeds of 470707.6 rpm for the two roller configuration and 117676.9rpm for the three roller configuration.
Summary/ Future Steps:
Feature / 2 Rollers / 3 RollersMechanical Stress Factor of Safety / 654 / 91
Maximum Deflection (nm) / 107 / 1653
Critical Speed (rpm) / 471000 / 118000
Rotational Speed Factor of Safety / 523 / 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.