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\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/12/08}\\
{\it Closed book \hfill Van Dommelen \hfill 7:30-9:30 am}
\end{center}

Solutions should be fully {\em derived} showing all intermediate
results, using class procedures.  Show all reasoning.  Bare answers
are absolutely not acceptable, because I will assume they come from
your calculator (or the math handbook, sometimes,) instead of from
you.  You must state what result answers what part of the question if
there is any ambiguity.  Answer exactly what is asked; you do not get
any credit for making up your own questions and answering those.  Use
the stated procedures.  Give exact, cleaned-up, answers where
possible.

One book of mathematical tables, such as Schaum{\rq}s Mathematical
Handbook, may be used, as well as a calculator, and a handwritten
letter-size formula sheet.  The Laplace tables of the book are
attached.  Do not use the convolution theorem unless unavoidable.

\begin{enumerate}
  
\item {\bf Background:} In the approximate Pohlhausen method for
  boundary layers, the following ODE arises
  \begin{displaymath}
    \frac{{\rm d}y}{{\rm d} x} = a - \frac{b y}{x}
  \end{displaymath}
  where $a$ and $b$ are given constants

  {\bf Question:} Solve the above ODE using the class procedures for
  this type of ODE.  Assume
  \begin{displaymath}
    x>0 \qquad y(1)=2
  \end{displaymath}

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Laplace transforms are a good way to solve
  dynamical systems, especially when their large-time behavior or
  stability is of interest.

  {\bf Question:} Solve
  \begin{displaymath}
    \frac{{\rm d}^2 y}{{\rm d} t^2} + y =
    \left\{
      \begin{array}{l}
        t \quad\mbox{for}\quad t<1 \\
        0 \quad\mbox{for}\quad t>1
      \end{array}
    \right. \qquad
    y(0) = \frac{{\rm d} y}{{\rm d} t}(0) = 0
  \end{displaymath}
  using the class Laplace transform procedures.  Make sure there is no
  funny mathematics in your final answer.  It must be phrased in
  simple terms that the instructor can understand.
\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} First order systems of ODE have the advantage
  that they can be used regardless of the size of your dynamic system
  and its nonlinearity.

  {\bf Question:} Consider a simple case; that of a mass spring-system
  with nonlinear damping:
  \begin{displaymath}
    \frac{{\rm d}^2 y}{{\rm d} t^2}
    + y^2 \frac{{\rm d} y}{{\rm d} t}
    + 4 y = 0
  \end{displaymath}
  \begin{enumerate}
  \item Explain how this second order equation can be reduced to the
    first order system
    \begin{displaymath}
      \frac{{\rm d} x_1}{{\rm d} t} = x_2
      \qquad
      \frac{{\rm d} x_2}{{\rm d} t} = -x_1^2 x_2 - 4x_1
    \end{displaymath}
  \item Explain why information about this nonlinear system can be
    obtained from solving the following, single, linear system:
    \begin{displaymath}
      \frac{{\rm d}}{{\rm d} t}
      \left(\begin{array}{c} y_1 \\ y_2 \end{array}\right)
      =
      \left(\begin{array}{rc} 0 & 1 \\ -4 & 0 \end{array}\right)
      \left(\begin{array}{c} y_1 \\ y_2 \end{array}\right)
    \end{displaymath}
    and identify $y_1$ and $y_2$.
  \item Solve the linear system using class procedures for constant
    coefficient first order systems.
  \item Classify the type of point and its stability.
  \item Verify that you got it right based on the determinant/trace
    plot.
  \item {\em Very neatly and quantitatively correct,} draw three
    representative solution curves.  Show the direction of time
    evolution using arrows.
  \item Is the linear system theoretically guaranteed to correctly
    predict the behavior of the nonlinear one?  {\em Sketch} in a
    separate phase plane what you expect for the nonlinear system.
  \end{enumerate}

\begin{rawhtml}
<A HREF="figures/3.jpg">Solution.</A>
\end{rawhtml}

\end{enumerate}

\end{document}
