Closed-Form Solutions for Vibrations of Nearly Vertical Strings and Beams by Means of Asymptotic Methods

Top tensioned strings and beams are often used in civil and marine applications. Typically these members have constant cross sections, and a pronounced, usually linear, tension variation, due to the effects of gravity. In this paper simple, approximate formulas for the natural frequency of such strings are derived, based on asymptotic techniques, while for the tensioned beam case approximate closed-form results are developed by the Wentzel-Kramers-Brillouin method. Both derivations are shown in reasonable detail. While similar work is known for a beam with varying axial tension this is believed to be the first time that a single analytic expression is developed for the full length of the beam. A simple example in which the bottom tension is only 9% of the top tension is analyzed for cases with and without bending stiffness, and the solutions have been compared to the exact solution for the string case and to the results from three finite-element programs for the beam case. The accuracy was found to be very good, even in this situation, in which the tension variation is large.