Coleman M.

In the vibration control of flexible structures, boundary feedback schemes are employed in order to damp vibrations and achieve stabilization. A knowledge of the system’s vibration spectrum is crucial to this process. We have developed a perturbation approach that, when coupled with various asymptotic methods, yields highly accurate estimates for the vibration spectra of Euler-Bernoulli beam and Kirchhoff thin plate problems. Here, that method is extended to a stand-alone method, applicable to the Timoshenko beam equations. Of course, many such problems are solvable using commercially available software packages these days. The disadvantage here is that, as they generally employ FEM or similar numerical methods, these packages do not offer any of the analytical or physical insights that are provided by asymptotic methods. Thus, for example, our method allows us to see the similarities and differences between the Euler-Bernoulli, Rayleigh and Timoshenko beams. However, by their nature, asymptotic methods are least reliable at the low end of the spectrum where the most “important” frequencies – i.e., those corresponding to the greatest vibration energies – occur. This is especially true of Timoshenko beam problems. Here, then, the perturbations allow us vastly to improve these low-end estimates, to the point that excellent agreement with numerical results is obtained in every case we have tried, including that of a built-up box beam with very large values for the flexural rigidity and shear stiffness (this last being an example where the basic asymptotic results vary greatly from the numerical results, well into the mid-range of the spectrum).