EML 5060 Analysis in Mechanical Engineering 9/22/95
Closed book Van Dommelen 9:00-9:50 EST

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.

The velocity components in front of a body with a hole are:

$$u=2 y - {2y\over x^2+y^2} \qquad v = 2x + {2x\over x^2+y^2}.$$

The `streamlines' of a flow can be found from

$$v \, {\rm d}x - u \, {\rm d}y = 0.$$

Find the equation of the streamlines in this flow. Solution

2.

A damped spring mass system experiences an external force, resulting in an equation of motion

$$\ddot x + 4 \dot x +13 x = e^{-2t} + t +1.$$

Find the general motion of the mass x(t). Solution

3.

The equation governing a beam in a destabilizing force field is:

y'''' - y = 0.

Find the shape of the beam y(x) given the initial conditions

$$y(0)=y'(0)=y''(0)=0, \qquad y'''(0)=1.$$

Solution