ME 240
Introduction to Dynamics

Computer Assignment #4 : Brake Pedal Model Problem

System Parameters: Using the governing equations derived in HW#9, we will use Matlab to determine the time history of the motion of the brake pedal. Measurements on the pedal were used to determine that

m  =   5 kg
k0 =   1000 N-m
c0 =   0.7 N-m-s
k  =   500 N/m
a  =   0.10 m
l  =   0.30 m

1. Solve the nonlinear ODE for the rotation (theta) as a function of time by modifying the Matlab script provided tutorial program: for the following two force values
   • Small force --> 50 N
   • Large force --> 5000 N
   and plot the resulting solution (hand in the plots). The brake pedal is initially at rest. The length of calculation time is 0.5 second, starting from 0 to 0.5 second.
   We are going to use the default values for relative error (1e-3) and for absolute error (1e-6). You can adjust these values if you want more (or less) accurate results. Note that smaller these values are, more accurate the result is, but slower the calculation is. You can change the error values in 'options' command.
2. Using the linearized equations of motion and the code from Computer Assignment #2 Mass-Spring-Damper system : Complimentary solution , solve for the motion due to the same two forces:
   • Small force
   • Large force
   and plot the resulting solution (and hand in these plots).
   (HINT: For the linear solution, define a new variable psi = theta - theta_particular. The ODE for psi is homogeneous, with non-zero initial ``displacement''. The code for CA#2 then applies directly.)

   NOTE that Mass-Spring-Damper system : Complimentary solution is modified for this assignment, therefore, you will find it quite helpful.

3. For both force levels and methods of solution,
   • What is the final position theta?
   • By examining the spacing of the zero crossing of the theta - theta_final, estimate the natural frequency of the system.
4. Does the linear estimate provide for less than a 1% error in the final equilibrium position and the oscillation frequency?
5. The above brake pedal design was found to be to ``jittery'' by a test panel of drivers. The agreement was that the time taken to reach the final position was to long and that pedal vibrated at too high of a frequency. What design changes would you suggest? Describe the rationale for you decision.