Distribution of natural frequencies in certain structural elements

Abstract

The paper deals with the natural frequency distribution in beams, plates, and shallow spherical shells. Expressions are found for the distribution functions (number of modes not exceeding a given cutoff level), for these integral approximations, and for the frequency derivatives of the latter—the so−called smeared modal densities. Special attention is paid to the singularities in these densities (present in certain elements), which were noted by Bolotin and are referred to in literature as ’’condensation points.’’ It is shown that omission of the concept of lowest natural frequency leads to a physical paradox, and that the findings of most studies of modal densities should be revised. The method of integral approximations is applied to the problem of random vibrations of a shallow spherical viscoelastic panel. It is shown that the integral approximation of the averaged spectral density in the high−frequency range is equivalent to an infinite−system model, and, moreover that the physically inconsistent results reported in literature regarding infinite discontinuities in the spectral densities of damped shells are erroneous and due to application of asymptotic expressions, valid only at high frequencies, to the low−frequency range. Subject Classification: 40.22, 40.24, 40.26, 40.35.