EML 5060Analysis in Mechanical Engineering Exam 111/12/92
Exam 2 Van Dommelen 2:45-4:10pm

1.

(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.

2.

(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).

3.

(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.

4.

(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.

5.

(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.