An analytical approach to study the dynamic characteristic of beams carrying any type of attachments with arbitrary distributions under elastic constraint boundary supports

An analytical method to study the dynamic characteristic of free vibration of beams carrying any type of attachments with arbitrary distributions on elastic constraint boundary supports is developed in the paper. To obtain an exact solution of the governing function, the displacement function is expressed as a modified Fourier series based on the Euler-Bernoulli beam differential equation, which consists of a standard Fourier cosine series plus several supplementary series used to improve uniform convergence of the series representation. Compared with other techniques, the current method offers a unified solution to entire situations of beams carrying various types of attachments, regarding different distributions and arbitrary boundary conditions. The results of different numerical examples are compared with the results of the references to illustrate the excellent accuracy of the current solution and validate the methodology. Furthermore, the proposed analytical method can be directly extended to caculate the natural frequencies of beam on Pasternak soil and with distribution attachment varying with its length which is never studied before