Vibration of an Euler–Bernoulli beam on elastic end supports and with up to three step changes in cross-section

Compared to the bibliography on the transverse vibration of Euler–Bernoulli beams with one step change in cross-section, publications on beams with more than one step changes is not extensive. In this paper an analytical method is proposed to calculate the frequencies of beams with up to three step changes in cross-section. Combinations of the classical clamped, pinned, sliding, free, ‘general’ and ‘degenerate’ types of elastic end supports are considered. The frequency equations of stepped beams were expressed as fourth order determinant equated to zero. A scheme to calculate the elements of the determinant and a scheme to evaluate the roots of the determinant are presented. Special consideration is given to three types of stepped beams frequently encountered in engineering applications. The first three frequency parameters of beams with two and three step changes in cross-section are tabulated for selected sets of system parameters and 45 types of end supports. Computational difficulties were not encountered. The method proposed may be extended to tackle beams with any number of step changes in cross-section. The tabulated results may be used to judge the frequencies calculated by numerical methods.