Homework #8
Assigned: Mar. 24, 2009
Due: Mar. 31, 2009
Points: 40

1. 20 points
   By following the derivations we did in the lectures for the minimum characterization of the free vibration problem of a taut string and a beam and their Rayleigh quotients, repeat this for the axial vibration of a bar. Start with the kinetic and potential energies for an uloaded bar, use Hamilton's principle to write the dynamic equation of motion, use separation of variables technique, and obtain the eigenvalue problem. Then, proceed with the minimum characterization theorem and prove it. The last step is to obtain the Rayleigh quotient.
   For extra credit of 20 points write down the problem of designing strongest column for specified volume of material and solve it analytically as much as you can. Numerical solution will fetch you additional extra credit!
2. 20 points
   Submit a precise mathematical statement of your project problem in the form of a calculus of variations problem. You should write it in the standard form by clearly identifying the objective function, design variable(s), governing equation for state variable(s), resource constraints, and other constraints. Do not forget to write the data needed for the problem.