University of Notre Dame
Aerospace and Mechanical Engineering

AME 302: Differential Equations, Vibrations and Control
Homework 7

B. Goodwine
Spring, 2004		 Issued: April 7, 2004
Due: April 14, 2004

  1. Sketch the Bode plot for each of the following transfer functions. In each case, sketch the individual terms separately (on the same plot is allowable, unless it becomes too cluttered) as well as the overall composite plot. Check your answer using the matlab bode() command.
    1. (10 points)

      \begin{displaymath} G(s) = \frac{10}{s+10} \end{displaymath}

    2. (10 points)

      \begin{displaymath} G(s) = \frac{(100(s+1)}{(s+10)(s+100)} \end{displaymath}

    3. (10 points)

      \begin{displaymath} G(s) = \frac{100}{s(s+10)} \end{displaymath}

    4. (10 points)

      \begin{displaymath} G(s) = \frac{s+100}{s(s+10)} \end{displaymath}

    5. (10 points)

      \begin{displaymath} G(s) = \frac{10}{s^2+s + 100} \end{displaymath}

    6. (10 points)

      \begin{displaymath} G(s) = \frac{10(s+1)}{s^2+s + 100} \end{displaymath}

    7. (10 points)

      \begin{displaymath} G(s) = \frac{10(s+1)}{(s+10)(s^2+ 5 s + 100)} \end{displaymath}

    8. (10 points)

      \begin{displaymath} G(s) = \frac{10(s+1)}{(s+10)(s^2+ 5 s + 1000)} \end{displaymath}

  2. For each of the following, determine the approximate gain and phase margins. Verify your answer in matlab using the margin() command. Also, using the matlab step() command, plot the step response of the closed loop transfer function, $\frac{Y(s)}{R(s)} = \frac{G(s)}{1+G(s)}$ to verify that the gain and phase margins give an accurate indication of stability.
    1. (10 points)

      \begin{displaymath} G(s) = \frac{K}{(s+5)(s^2+10s+100)} \qquad K = 100. \end{displaymath}

    2. (10 points)

      \begin{displaymath} G(s) = \frac{K}{(s+5)(s^2+10s+100)} \qquad K = 1000. \end{displaymath}

    3. (10 points)

      \begin{displaymath} G(s) = \frac{K}{(s+5)(s^2+10s+100)} \qquad K = 10000. \end{displaymath}


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