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{\bf EML 5060\hfill Analysis in Mechanical Engineering Exam 1\hfill 11/12/92}\\
{\it Exam 2 \hfill Van Dommelen \hfill 2:45-4:10pm}
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\begin{enumerate}

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\item
(Vibrations, 19 points)
A spring mass system attached to an elastic foundation can be
written as a third order differential equation.
For a particular case, the equation
of motion is of the form
$$ x''' + 3 x'' + 3 x' + x = 0$$
where primes are derivatives with respect to time.
Find the solution if initially $x=0$, $x'=1$, and $x''=0$.

\item
(Solid mechanics, 19 points)
The elevation $u$ of a membrane near a corner
has a radial variation with distance from the corner
given by the differential equation
$$u'' + {1\over r} u' - {c\over r^2} u = 0$$
where $c>0$ is a constant depending on the angle of the corner.
Solve for $u(r)$.

\item
(Fluid mechanics, 21 points)
When we cut a slot in a wall and apply suction through the slot,
the boundary layer profile of the air approaching the slot
satisfies according to Schlichting the ODE:
$$f''' - f'^2 + 1 = 0$$
The solution satisfies $f'(\infty)=1$ and $f''(\infty)= 0$.
Reduce this equation to a separable first order one.  Because of time
constraints, you need not actually solve the first order equation.

\item
(Fluid mechanics, 21 points)
The velocity potential $\phi$ in steady
two-dimensional compressible flow satisfies
the equation
$$(a^2 - u^2)\phi_{xx} - 2 u v \phi_{xy} + (a^2 - v^2)\phi_{yy} = 0$$
in which $a$ is the speed of sound and
$u$ and $v$ are the velocity components in the $x$- and
$y$-directions, respectively.  The characteristics of this equation
are called `Mach-lines' and can be seen using Schlieren
optics.  Determine under which conditions Mach-lines can
occur, and give their direction.

\item
(Design, 20 points)
The electrostatic field $\vec E$ due to charged bodies
is conservative and follows from
a potential V. Each component of the field strength, as well as the
potential all satisfy the Laplace equation.
I want to suspend a positively charged particle freely in outer
space in stable
equilibrium
by surrounding it by a number of cleverly shaped charged bodies.
After 200 hours of Cray YMP processing, I still haven't found body shapes
which will give a stable suspension. The graduate students in my class
think it won't work.
But it seems obvious to me
that if I surround the particle at all sides by positively charged bodies
repulsing the particle, I am bound to trap it in the center.
Find out who is right and explain.

\end{enumerate}

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