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\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 10/18/96 }\\
{\it Closed book \hfill Van Dommelen \hfill 9:00-9:50am }
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used.

\begin{enumerate}

\item
In the numerical solution of the simplest one-dimensional
heat conduction problem, you need to solve a linear system
of equations with a `tridiagonal' matrix of the form:
\begin{displaymath}
A =
    \left(\begin{array}{rrrrr}
               2 & -1 &  0 &  0 &  0 \\
              -1 &  2 & -1 &  0 &  0 \\
               0 & -1 &  2 & -1 &  0 \\
               0 &  0 & -1 &  2 & -1 \\
               0 &  0 &  0 & -1 &  2
          \end{array}\right)
\end{displaymath}
The work in solving this five by five system
is not as much as you would expect due to all those zeros.
They remain zeros.
Show that this is true by solving $A {\bf x} = {\bf b}$ if ${\bf b}$ is the
unit vector
\begin{displaymath}
    {\bf b}^T = (1,0,0,0,0).
\end{displaymath}
Use Gaussian elimination without partial pivoting.  The inverse matrix
$A^{-1}$ is bad news since it does not have all these zeros.  (The
solution vector ${\bf x}$ is the first column of $A^{-1}$).
\begin{rawhtml}
  <a href="figures/1.gif">Solution</a>
\end{rawhtml}

\item
The two spring-mass systems below have identical masses $m=1$ and
spring constants $k_1$.  Unfortunately, they resonate since there is
an external disturbance of frequency $\omega=\sqrt{k_1}$ present. In the
hope of eliminating the resonance, you provide some mutual support
to the masses by attaching a second spring with stiffness $k_2$ between
them.  The new equations of motion are:
\begin{displaymath}
\left(\begin{array}{c} \ddot x_1 \\ \ddot x_2 \end{array}\right) +
\left(\begin{array}{cc} k_1+k_2 & -k_2 \\ -k_2 & k_1 + k_2 \end{array}\right)
\left(\begin{array}{c} x_1 \\ x_2 \end{array}\right) +
 =
\left(\begin{array}{c} 0 \\ 0 \end{array}\right).
\end{displaymath}
Substitute in the assumptions
\begin{displaymath}
\begin{array}{c}
    x_1 = A \cos(\omega t + \phi), \\
    x_2 = B \cos(\omega t + \phi),
\end{array}
\end{displaymath}
where $A$, $B$, and $\phi$ are constants, to get an eigenvalue problem for
the frequency $\omega$.  Answer the question whether the
additional spring eliminates the resonating frequency.  If not, give
the physical reason why not.

(lost picture showing from left to right: wall, spring k1, mass m,
spring k2, mass m, spring k1, wall)

\begin{rawhtml}
  <a href="figures/2.gif">Solution</a>
\end{rawhtml}

\item
In a slowly rotating coordinate system,
the relative velocity vector ${\bf v}^T = (u,v,w)$ of a mass
satisfies the equations of motion
\begin{displaymath}
\frac{d{\bf v}}{dt} =
\left(\begin{array}{ccc}
          0        & -\omega_3 & \omega_2  \\
          \omega_3 & 0         & -\omega_1 \\
         -\omega_2 & \omega_1  & 0
      \end{array}\right){\bf v},
\end{displaymath}
assuming that no forces act on the mass.
Diagonalize the matrix and then solve the above differential equations
for the diagonal matrix, in other words, solve
\begin{displaymath}
\frac{d{\bf v}'}{dt} =
\left(\begin{array}{ccc}
          d_1 &  0  &  0  \\
           0  & d_2 &  0  \\
           0  &  0  & d_3
      \end{array}\right){\bf v}',
\end{displaymath}
Show that the solution  is periodic in time.
Explain why you are not (or should not be) surprised for physical reasons.
\begin{rawhtml}
  <a href="figures/3.gif">Solution</a>
\end{rawhtml}

\end{enumerate}

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