In this work we consider segmented Euler-Bernoulli beams that can have an internal damping of the type Kelvin-Voight and external viscous damping at the discontinuities of the sections. In the literature, the study of this kind of beams has been sufficiently studied with proportional damping only, however the effects of non-proportional damping has been little studied in terms of modal analysis. The obtaining of the modes of segmented beams can be accomplished with a the state space methodology or with the classical Euler construction of responses. Here, we follow a newtonian approach with the use of the impulse response of beams subject both types of damping. The use of the dynamical basis, generated by the fundamental solution of a differential equation of fourth order, allows to formulate the eigenvalue problem and the shapes of the modes in a compact manner. For this, we formulate in a block manner the boundary conditions and intermediate conditions at the beam and values of the fundamental matrix at the ends of the beam and in the points intermediate. We have chosen a basis generated by a fundamental response and it derivatives. The elements of this basis has the same shape with a convenient translation for each segment. This choice reduce computations with the number of constants to be determined to find only the ones that correspond to the first segment. The eigenanalysis will allow to study forced responses of multi-span Euler-Bernoulli beams under classical and non-classical boundary conditions as well as multi-walled carbon nanotubes (MWNT) that are modelled as an assemblage of Euler-Bernoulli beams connected throughout their length by springs subject to van der Waals interaction between any two adjacent nanotubes.
Axisymmetric compressive buckling of multi-walled carbon nanotubes under different boundary conditions