University of Notre Dame
Aerospace and Mechanical Engineering

AME 302: Differential Equations, Vibrations and Control
Homework 4

B. Goodwine
Spring, 2004		 Issued: February 20, 2004
Due: February 25, 2004

Unless otherwise indicated, all problems are from Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, seventh edition.

  1. Section 7.9, number 1 by diagonalization.
  2. Section 7.9, number 4 by undetermined coefficients.
  3. (Optional -- 20 points extra credit): Section 7.9, number 3 by undetermined coefficients.
  4. Section 7.9, number 5 by variation of parameters.
  5. Section 7.9, number 7 by any method.
  6. Write a computer program (you may use the code distributed in class, if you want, or you may write your own program in any computing language you desire) to simulate the vibration absorber problem. Use the following paramter values for all the simulations:
    1. mass one = 10;
    2. spring one constant = 1;
    3. mass two = 1;
    4. applied force magnitude = 5; and,
    5. applied force frequency = 2 rad/sec.
    Verify the following assertions and/or derivations from class:
    1. if mass two and spring two constant are tuned appropriately and the initial conditions are exactly correct, mass one will be stationary;
    2. even if mass two and the constant for spring two are tuned appropriately, the vibration absorber will be ineffective if the initial conditions are not what are required by the analysis;
    3. if mass two and constant for spring two are slightly mis-tuned, even if the initial conditions are correct, after a long enough period of time the absorber will be ineffective;
    4. if a damper is added between the two masses
      1. in steady state, smaller damper constants will result in a lower magnitude vibration of mass one; and,
      2. if the damping constant is smaller, it will take longer for the effect of initial conditions, i.e., the transient response, to decay.

2004-02-09
Last updated: February 20, 2004.