  Timoshenko's theory for bending vibrations of a beam has been extensively
studied since its development nearly one hundred years ago. Unfortunately there
are not many analytical results. The results above the critical frequency
inclusive haeve been tested only recently. Here an analytical expression for
the solutions of the Timoshenko equation for free-free boundary conditions,
below the critical frequency, is obtained. The analytical results are compared
with recent experimental results reported of aluminum and brass beams The
agreement is excellent, with an error of less than 3%, for the aluminum beam
and of 5.5% for the brass beam. Some exact results are also given for
frequencies above the critical frequency.
