University of Notre Dame
Aerospace and Mechanical Engineering

AME 437: Control Systems Engineering
Homework 7

B. Goodwine
Spring, 2003		 Issued: March 17, 2003
Due: March 21, 2003

Unless otherwise indicated, all problems are from the course text.

  1. The following exercises are to develop an intuition regarding the response of a ``normalized second order system.'' Although I'm usually pretty flexible regarding such things, sometimes formality is necessary (like if you are making a report for a board of directors). As such, in this homework, all of your plots should be completely labeled (electronically, not by hand), and if there are multiple elements to each plot, each curve should be appropriately identified. Also, if the standard notation utilizes Greek letters, then they should be used in the plot, as is illustrated in the following figure.
    1. Write a computer program (C, C++ or FORTRAN) to numerically compute the step response of the normalized second order transfer function
      \begin{displaymath} T_{N2} = \frac{\omega_n^2}{s^2 + 2 \zeta \omega_n s + \omega_n^2}, \end{displaymath}

      using the fourth order Runge-Kutta method to compute the solution to the differential equation. Submit your program code.

    2. Let $\omega_n = 1$. On the same graph, plot the response from $t = 0$ to $t = 10$ for $\zeta = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 0.9$.

    3. For any three of these values of $\zeta$, plot the response for that value of $\zeta$ and verify that the percentage overshoot is actually what is predicted by Figure 6.4 in the course text.

    4. For the same three values of $\zeta$ and using the same three plots as the previous problem, verify that the settling time, $t_s$ is as predicted in section 6.4.4 in the course text.

    5. For the same three values of $\zeta$ and using the same three plots as the previous problem, verify that the peak time, $t_p$ is as predicted in section 6.4.2 in the course text.

    6. Let $\zeta = 0.2$. Plot the response for $t = 0$ to $t = 10$ for $\omega_n = 1, 2, 3, 5, 10$.

    7. Again, pick three of the responses and, as before, verify the percentage overshoot, the settling time and the peak time.

  2. 6.8.4:2

A lawyer and an engineer were fishing in the Caribbean. The lawyer said, "I'm here because my house burned down, and everything I owned was destroyed by the fire. The insurance company paid for everything."

"That's quite a coincidence," said the engineer. "I'm here because my house and all my belongings were destroyed in an earthquake, and my insurance company also paid for everything."

The lawyer looked somewhat confused. "How do you start an earthquake?" he asked.

Last updated: March 17, 2003.
B. Goodwine (goodwine@controls.ame.nd.edu)