ME240 Winter 2000 Computer Assignment 3

Weebles Wobble - Due April 14

In this assignment, you will show how "Weebles wobble but they don't fall down"(TM). In addition, you will show just how fast Weebles wobble. The figure below shows a model of a Weeble which is basically a spherical base with the center of mass lying below the center of the sphere (if this looks familiar, this is your chance to redeem yourself on the exam). The Weeble wobbles by maintaining a rolling contact with the table. This particular Weeble has a radius of R=2cm, a mass of m=.05kg, a mass offset of d=1cm, and a moment of intertia about it's center of mass of Icm=.0005kg-m^2.

Weeble Model

PART I - Modeling

Here, you will analytically generate the equation of motion for the Weeble. Do not substitute in any numbers for parameters in part I.
  1. Draw a free body diagram of the Weeble and write the three equations of motion (sum forces in x and y and sum moments)
  2. Show that theta=0 is an equilibrium position for the Weeble.
  3. Using kinematics (assuming rolling without slipping), express the accelerations of the center of mass in terms of the angular velocity and angular acceleration of the weeble. Using these, find the equation of motion of the Weeble solely in terms of it's angular motion (theta and it's two time derivatives) and the physical parameters Icm, d, m, R, and g.
  4. Assuming the Weeble only wobbles a little bit, find the linearized equation of motion (for small angles).
  5. Find the natural frequency of vibration for small angles.

PART II - Nonlinear Vibration

Here, you will simulate the motion for the case where the Weeble is released from rest at some angle. Remember that angular velocities and accelerations must be in radians, not degrees.
  1. Write out the state-space representation of your nonlinear (large angle) equation of motion.Do not substitute in the numerical physical parameters.
  2. Using this representation, simulate the motion using Matlab for each of the following two sets of initial contitions:
    1. theta(0)=5 degrees, d/dt(theta)(0)=0.
    2. theta(0)=120 degrees, d/dt(theta)(0)=0.
    Simulate each case long enough to view several periods of oscillation.
  3. From these simulations, measure the frequency of oscillation. Compare the frequency of oscillation for the linearized case from Part I to the frequency obtained from the simulation cases. Explain any discrepancies. Does the frequency of oscillation vary with different initial conditions in the linearized case?

PART III - Damped Vibration

Now, we will examine what happens when we place the Weeble on a carpet rather than on a flat table. Because the carpet is "squishy" it applies a moment to the Weeble equal to -c*d\dt(theta) where c=.0006(N-m)/(rad/s) is the damping constant.
  1. Reclaculate the nonlinear equation of motion for the damped case.
  2. Linearize this equation and obtain expressions for the damped frequency of oscillation and the damping ratio (do not substitute in numbers yet).
  3. Write a state-space representation for the damped case and simulate the nonlinear damped equation of motion in Matlab for the same sets of initial conditions used in Part II.
  4. From the simulation results, measure the damped frequency of oscillation and estimate the damping ratio by looking at the decay rate of the amplitude of oscillation. Compute the damped frequency of oscillation and damping ratio from the linearized damped equation of motion and compare to your simulated cases. Explain any discrepancies.