Forced vibration analysis of flexible Euler-Bernoulli beams with geometrical discontinuities

This paper presents a novel framework for forced motion analysis of Euler-Bernoulli beam with multiple jumped discontinuities in the cross section. In this regard, the entire length of beam is partitioned into uniform segments between any two successive discontinuity points. Beam characteristics matrix can be derived based on the boundary conditions and the continuity conditions applied at the partitioned points. This matrix is particularly used to find beam natural frequencies and mode shapes. The governing ODE of motion and its state-space representation are then derived for the beam under a distributed dynamic loading condition. To clarify the implementation of the proposed method, a beam with two stepped discontinuities in the cross section is studied, and numerical simulations are provided to demonstrate the mode shapes and frequency response of beam for different stepped values. Results indicate that the added mass and stiffness significantly affects the mode shapes and natural frequencies.