EML 5060 Analysis in Mechanical Engineering 11/3/96
Closed book Van Dommelen 9:00-9:50am

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used.

1.

In a model for turbulent flows, you need to solve the following system of two ordinary differential equations:

$${{\rm d} x\over {\rm d}t} = x y \qquad {{\rm d} y\over {\rm d}t} = 2 y^2 - x^2$$

Eliminate the time t between these two equations, then find y as a function of x. Solution

2.

The potential V across the condensor of a serial CL circuit satisfies the differential equation:

$$\ddot V + 4 V = e(t).$$

Find V(t) if the applied potential takes the form $e(t)=1/\cos(2t)$.Assume $V(0) = \dot V(0) = 0$. Solution

3.

In a two degree-of-freedom dynamic system, one of the degrees of freedom can be eliminated to produce the following fourth order equation for the remaining unknown x(t):

$$\stackrel{\textstyle{....}}x + 13 \stackrel{\textstyle{..}}x + 36 x = 0 .$$

Solve this equation subject to the initial conditions:

$$x(0) = 0 \quad \stackrel{\textstyle{.}}x(0) = 2 \quad \stack... ...style{..}}x(0) = 0 \quad \stackrel{\textstyle{...}}x(0) = -13.$$

Solution