Modal Formulation of Segmented Euler-Bernoulli Beams

We consider the obtention of modes and frequencies of segmented Euler-Bernoulli beams with internal damping and external viscous damping at the discontinuities of the sections. This is done by following a Newtonian approach in terms of a fundamental response of stationary beams subject to both types of damping. The use of a basis generated by the fundamental solution of a differential equation of fourth-order allows to formulate the eigenvalue problem and to write the modes shapes in a compact manner. For this, we consider a block matrix that carries the boundary conditions and intermediate conditions at the beams and values of the fundamental matrix at the ends and intermediate points of the beam. For each segment, the elements of the basis have the same shape since they are chosen as a convenient translation of the elements of the basis for the first segment. Our method avoids the use of the first-order state formulation also to rely on the Euler basis of a differential equation of fourth-order and it allows to envision how conditions will influence a chosen basis.