Mathematical Problems in Engineering
Volume 2007 (2007), Article ID 36261, 18 pages
doi:10.1155/2007/36261
Research Article
Modal Formulation of Segmented Euler-Bernoulli Beams
1Departamento de Matemática, Universidade Federal de Santa Maria, Avenida Roraima 1000, Santa Maria 97105-900, RS, Brazil
2Instituto de Matemática-Promec, Universidade Federal do Rio Grande do Sul, Avenida Bento Gonçalves 9500, Porto Alegre 91509-900, RS, Brazil
3Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Avenida Bento Gonçalves 9500, Porto Alegre 91509-900, RS, Brazil
Received 5 September 2006; Revised 8 December 2006; Accepted 11 February 2007
Academic Editor: José Manoel Balthazar
Copyright © 2007 Rosemaira Dalcin Copetti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.