EML 5060 Analysis in Mechanical Engineering 9/20/02
Closed book Van Dommelen 11:50-12:40 pm

Show all reasoning and intermediate results leading to your answer, or credit will be lost. I must be able to see clearly how you derived everything, and you must state what the result you derived is in terms of what is asked. Answer exactly what is asked; you do not get credit for making up your own questions and answering those. You must use the systematic procedures followed in class, not mess around randomly until you get some answer. For example, you need to reduce matrices completely to echelon form where appropriate to the question, orthonormalize eigenvectors where appropriate, etcetera. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator.

1.

To solve the large systems of equations generated by numerical schemes, the general-purpose method is reduction to echelon form. Reduce the matrix

to echelon form. Solution.

2.

The degrees of freedom of a simple vibrating system are a linear displacement x and an angular one . To solve the vibration problem, we first make the following transformation to new coordinates q₁ and q₂:

Identify the basis vectors of the (q₁,q₂) coordinate system in terms of the coordinate system. Also give the transformation matrix from the (q₁,q₂) system to the system. Next, solving the (symmetric matrix) eigenvalue problem in the (q₁,q₂) coordinate systems, the solution can be written

where is some constant you can leave as . Find the transformation matrix from coordinates to coordinates.

Solution.

3.

The inertia matrix of a thin long solid body, , , etcetera, is found nuerically as

Find the basis vectors , , of a rotated Cartesian coordinate system in which the off-diagonal moments of inertia are zero. Also identify the remaining nonzero moments of inertia (called the principal moments of inertia.) Do you think my numerical integration may have been inaccurate? Solution.