EML 5060 Analysis in Mechanical Engineering 9/19/01
Closed book Van Dommelen 10:35-11:25am

Show all reasoning and intermediate results leading to your answer, or credit will be lost. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator. Follow the standard procedures; don't mess around at random.

1.

A weight W is hung from the ceiling by a net of ropes as shown. The equilibrium equations governing the tension forces in the ropes are:

where , and Using the systematic procedures used in this class, solve this system and find the general solution for the tension forces in the ropes in as simple and clear a form as possible. Keep the given order of the equations; do not use partial pivoting. Using the result, identify a rope that you could leave out and still be able to suspend any weight. Check that the tension forces in all the ropes are nonnegative: ropes cannot support compressive forces.

Solution.

2.

Air above the sea is moving with a wind velocity . The sea water is moving with a velocity . To study the motion of small particles that fall into the sea, special coordinates can be useful: Instead of describing the position of these particles using the conventional Cartesian coordinates x and y, it is easier to describe their position vector using the air and water velocities as basis vectors:

since changes in x' and y' are simply the amounts of time spend in air and water respectively. Find the formulas that give the coordinates x and y if we know x' and y'. Also find the formulas that give the coordinates x' and y' if we know x and y.

Solution.

3.

The two coupled pendulae shown satisfy the following equations of motion:

Assume Show that solutions of the form:

lead to an standard eigenvalue problem. Identify the matrix, eigenvector, and eigenvalue. Describe the motion(s) corresponding to the solution(s) of this eigenvalue problem.

Solution.