Homework3

Lecture Homework 3:

Due at beginning of Lecture on Thursday Feb. 1. Each pair should turn in a single report.

Educational Objective

- Compare experimental results to simulation results, and use differences to to improve the accuracy of the theoretical model.
- Use the improved theoretical model to perform a theoretical optimization of the design challange.
- In week 5, students will demonstrate optimized hardware. Hardware performance will be based upon both theoretical optimization, and fabrication skill.

Part A: Comparing Simulation to Experimental Results:

Use dimensions from your actual hardware setup.

1. Create a theoretical simulation model of the hardware in lab during week 3 (show above). This is the 2 shaft configuration with 4.2:1 gear ratio and the flywheel on the 2nd shaft.
  - Draw the necessary FBDs
  - Note: to save you time we are purposefully not asking you to write out all the equations. You may choose to use short cuts like effective friction and inertia.

- List assumptions made and justify why it is appropriate to make them.
- Make sure to factor in:
  - Masses and inertias of both shafts and pulleys
  - Friction in ball bearings
    - u = .0015 (coefficient of friction)
    - P_d = 12mm (ball bearing pitch diameter)
    - ~~Tension in belts.~~ For part A, you do not need to factor in belt tension.
  - Initial motor parameter estimates from HW 2 and prelab 3

Plot the experimental and theoretical velocity of the flywheel on the same plot.
Adjust simulation model to better match with experimental results. A key parameter to consider is your friction. Coulomb friction should be the dominant friction. For more accurate results you may want to incorporate viscous friction.
- Plot the experimental velocity, initial simulation velocity, and "adjusted" simulation velocity (on the same plot).
- Justify adjustments made and explain methods used.

1. Summarize your findings and theoretical model (half to full page). Topics to consider:
  - What is the source of the discrepancy between theory and experiment?
  - What is the effect of friction in the system?
  - What assumptions did you make, and whether or not they are justified.
  - Etc... These are just examples, you are expected write and discuss more.

Part B: Theoretical Optimization of Design Challenge

1. Draw the FBDs of the Challenge: 3 shafts, 2 belts, hanging weight, etc.
  - Make sure to factor in:
  - Masses and inertias of shafts and pulleys
  - Friction in ball bearings
    - u = .0015 (coefficient of friction)
    - P_d = 12mm (ball bearing pitch diameter)
  - Tension in the belts. See "dealing with normal forces" and diagram on the bottom of the page.

Using the FBDs above, derive the equations of motion and find the analytical (closed form) solution for:
- Angular velocity of the motor shaft with respect to time
- Height of the mass with respect to time
Simulate the configuration under a range of parameters: gear ratio and mass lifted. Calculate the cost function (see 'P' below) of each configuration. Plot the optimization results using the Matlab Mesh command.

- Lift the mass .75 meters. You do not need to lower the mass in simulation.
- Use a mass-lifting pulley with radius: p_1 = 3cm

Clearly state your theoretical optimal configuration and the maximum optimal performance value you expect (P)
- P = m/(t*E)
Turn in Matlab code with comments.

Dealing with Normal Forces: (correction from office hour)

In order to get an analytical solution, you will need to treat R1 and N1 as separate forces (as shown on the FBD). If you combine these two forces and take the vector magnitude (2 norm) you will get a square root that is very difficult to deal with. There are two reasons why treating R1 and N1 as separate forces is reasonable:

It lead's to a conservative estimate of the reaction forces
In reality, the balls in the bearing have play, so it is actually possible to have more than one normal forces on the shaft.

In addition, for the simulation, you may assume the angle of the tension (alpha) is small.