Summary of Technical Calculations

Team 16601: Glass Cutting Machine Guide Rollers

Each section of this document will focus on a certain area of the machine. They will begin by capturing the initial calculations the team carried out in order to develop understanding of the system. These are things such as feasibility calculations. The document will then go into detail about sizing, and expected results, where applicable, and give information as to the calculations that were made to inform design decisions. This document aims to show the high level results of the various analysis, rather than the specifics of each given calculation. A list of the documents containing further detail is given for each section.

  1. Stress Analysis
  2. Loading Calculations
  3. Roller Stress
  4. Frame Base Stress
  5. Dynamic Analysis
  6. Inertia Calculations
  7. Excitation Frequency
  8. Roller Vibration
  9. Base Frame Vibration
  10. Bearing Life Calculations
  11. Power Analysis
  1. DS264 Maximum Cuts
  2. Motor Sizing
  3. Efficiency

1.a: Loading Calculations

Contributing Documents: Mechanical Analysis of Guide Rollers, Torque Derivations, Torque Calculations Revisited

The distributed load on the rollers due to the wire tension is:

Max bending moment is:

Where L is cutting length, X is the excess roller length, and w is the distributed load.

The equation for torque was derived to be:

The equation has 3 coefficients of friction that are unknown or unmeasured, but can be used for qualitative assessments with good estimates for them.

1.b: Roller Stress:

Contributing Documents: Mechanical Analysis of Guide Rollers, Mechanics Analysis

Design Factor for Roller due to static loading. The following equation is good for a rough idea of how the roller will respond to the given stress. The Mechanics Analysis sheet can perform these calculations. It is based on the Soderberg Theorem.:

1.c: Frame Base Stress:

Contributing Documents: Base Plate Analysis, Frame Base (Solidworks Part)

Once the fixture had been designed, FEA simulation was carried out using Solidworks Simulation, for several different loading conditions, with results displayed below.

Fixture / Support / Max Deflection (mm) / Max Stress (MPa)
4 Corners / None / 12.061 / 4607
4 Corners / 4" Support / 0.568 / 1082
4 Corners / Double Thickness / 2.182 / 3916
Both Sides / None / 1.356 / 612
Both Sides / 4" Support / 0.281 / 767
Both Sides / Double Thickness / 0.183 / 392
Bottom / n/a / 0.00001667 / 1.04

There exists a singularity in the FEA that causes the max stress to be artificially high. The results demonstrate that the Frame Base will survive loading.

2.a: Inertia Calculations:

Contributing Documents: Roller Inertia Reduction Feasibility

An equation for inertia was derived. This was used to help the team understand what effect changing inertia would cause. Decreasing inertia will decrease the torque required to spin the rollers, and therefore use less power. This also has the potential to decrease vibrational stability.

2.b: Excitation Frequency:

Contributing Documents: Vibration Analysis, Base Plate Analysis

For our system:

During design, this is the natural frequency that should be avoided, as any part with this natural frequency will be excited at resonance, causing force and displacement magnification.

2.c: Roller Vibration Stability:

Contributing Documents:Vibration Stability

The natural frequency for the roller can be estimated by the following equation. This is based on estimating the equivalent stiffness for a rod with circular cross section.

Ideally this natural frequency will be much higher than the excitation frequency, in order to prevent any mass imbalance from being an issue. The estimated natural frequency was found to be 319 Hz, which is far above the predicted operating frequency.

2.d: Base Frame Vibration:

Contributing Documents: Vibration Analysis, Base Plate Analysis

Solidworks Simulation was used to analyze the natural frequencies and mode shapes of the base plate after cutting the aperture in the middle for slurry.

The primary mode of vibration is displayed below, which occurs at the first natural frequency.:

Fixture / Support / wn_1 / wn_2
4 Corners / None / 21.484 / 48.312
4 Corners / 4" Support / 77.284 / 104.05
4 Corners / Double Thickness / 39.473 / 90.257
Both Sides / None / 65.379 / 76.68
Both Sides / 4" Support / 152.05 / 176.77
Both Sides / Double Thickness / 129.74 / 150.68

2.e: Bearing Life Analysis:

Contributing Documents: Bearing Analysis.

This equation can be used to predict the life of the bearing before 10% of the balls will fail. This is considered the standard number to use as life for a bearing. The Bearing Analysis spreadsheet is set-up to calculate the stress based on a variety of input conditions. For the predicted bearing load of any given bearing. After determining shaft diameters, several bearings were compared. Ultimately the bearing selected was based on the bearing with the lowest cost per life cycle, along with a relatively high life. The predicted life for the selected bearing (the NUP 310) was 455,587 hours, which should last the length of life of the machine.

3.a: DS264 Maximum Cuts

Contributing Documents: EE Hand-off Document

The DS264 workpiece allowed length on the rollers is given as 820 mm. The wire width is given as 100-160 µm. The machine can cut wafers as thin as 100 µm. Therefore, for a given wafer number one needs the same amount of wires (technically subtract one wire but who cares…). Since the maximum length is given, the amount of cuts that machine can make can be determined by adding the width of the wire and wafer and dividing that amount into the maximum workpiece length. Assuming 100 µm wire and a 100 µm wafer, that leaves 200 µm required per cut. Diving 820 mm by 200 µm provides 4100 cuts maximum. This number is crucial for scaling the machine and sizing the motors correctly. At 4100 cuts there are 5 wires per millimeter so not a trivial figure.

3.b: Motor Sizing

Contributing Document: EE Hand-off Document

To size the motors, two methods were used to get similar numbers. First off, the 4100 cuts maximum is a crucial assumption. The second number is the DS264 motor continuous torque rating of 500 Nm, which is a given. Usually, a PMDC (Permanent Magnet Direct Current) servo motor is sized with a 20% margin for continuous load torque, therefore one could guess that the DS264 designers estimated a torque of about 400 Nm (rounded to 25%). However, subtracting the 20% margin is not necessary since one would have to add it back again when sizing the motor again. Method 1 differs by including the inertia in the torque estimate where method 2 ignores it. Either way, due to the inertia being a negligible part of the total motor torque, the numbers are very similar.

Assumptions:
-Inertia of DS-264 Guide Rollers[1] : 1.664 kgm2
-Inertia of Current Design: 0.8 kgm2
-Acceleration of Wire (measured value): 2 m/s2
-Radius of Guide Rollers: 0.16 m
-Scaling of cuts on machine is linear

Method 1: Guide rollers sized with Inertia in mind
Assumptions:
-All of remaining torque goes into cutting
-DS-264 Motors sized for 4100 cuts

Method 2: Guide roller motors sized for continuous torque.
Assumptions:
-Motors were sized for continuous torque just for 4100 cuts
-Inertia ignored because it only really makes a difference on acceleration. On deceleration the inertia actually helps regenerate energy.


Either way, the torque estimate is pretty close to the same number. The inertia does not play a large role. These numbers include whatever safety margin the DS264 had unless the scaling is not linear. The scaling would not be linear if there are some sort of variables that come into play at a small or large number of cuts. Therefore, the motors would be sized now on their continuous torque rating being similar to these numbers. Also, RPM and motor type come into play.

The motor type chosen was a direct drive motor, which is advantageous because of its ability to work under high inertia mismatch. Inertia mismatch occurs when a regular motor has a small inertia in its rotor and tries to control a large inertial load. Generally, for most applications it is recommended to be at a less than a 10:1 ratio or even 5:1. By using a direct drive, a gearbox is not needed for the guide roller application to handle the large inertial mismatch.

3.c: Efficiency

Contributing Document:Motor Efficiency Analysis, EE-Handoff Document

The guider roller profile is shown below.

To estimate average power for the whole cycle, the following equation can be used. The variable I is inertia of the rollers, ɑris the acceleration rate, ɑfis the deceleration rate, tris the ramp up time, ts is the steady time, tfis the ramp down time, Tf is the torque due to friction estimated in section 3a above, and Vf is the final or steady velocity at the top of the profile. Finally, T is the total time of one forward cycle T=tr+ts+tf. It is important to keep all units in radians, seconds, and Nm or make sure to add unit conversions.

This equation is an estimate of average power needed for the application. Assuming tr= tf=7.5 sec, ts=20 sec, ɑr=ɑf=12.5rad/sec2 (2 m/s2), Tf =36 Nm, T=35 sec, Vf=94.25 rad/sec (900 RPM), and I=0.8 kgm2 for the current design, and average power of 2.67 kW. Even though this is small, it is important to still size the motor for the continuous torque or Tf.

Even though this average power can be calculated, there really is no way to calculate the motor efficiency without simulation software such as Motion Analyzer from Rockwell, or doing very tough calculations using the motor specs to try and guess an efficiency. Luckily, PMDC motors are inherently one of the most efficient type of motors, so it is not absolutely crucial to determine exact efficiency.

[1] Inertia of guide rollers calculated by standard equation 0.5Mr2. Mass was 130 kg and radius 0.16 m.