Keywords: distributed dynamic system, localized disturbances, eigenvalue expansion, modal space, complex frequencies, computational efficiency.

In this paper a finite Euler-Bernoulli beam on a viscoelastic foundation subjected to moving loads is considered. The beam and the viscoelastic foundation have uniform properties. It is assumed that this beam can contain concentrated masses, springs and dampers, described as localized disturbances.

When localized disturbances are considered, one has generally two ways to follow: either the disturbances will be inherent in parts of the vibration modes and frequencies or they will be included only in the modal space. In this paper the former approach will be identified as "the expansion over full vibration modes", and the latter one as "the expansion over simplified vibration modes". In the former approach the modal equations will be uncoupled. The difficulty is attributed to the vibration modes and frequency determination. The latter approach does not require complicated determination of the vibration modes, the difficulty arises from the fact that the governing equations in the modal space are coupled.

Further differences in these two methods are related to the disturbances type. When masses and, or springs are considered, then it is possible to maintain all the calculation within the real domain. Undamped full vibration modes can be determined exploiting the global dynamic stiffness matrix, where the localized disturbances are added directly into the diagonal terms [1]. The modal equations will be uncoupled and written in a standard form, incorporating now the viscous term. The disadvantage is that the free vibration calculation is quite difficult. In addition it must be repeated at any time when the structure suffers a small change.

On the other hand, when localized disturbances are not included in free vibration calculations, natural frequencies are known in advance and do not have to be calculated at any time the structure changes. Additional coupling terms in the modal space can be expressed using analytical vibration modes, therefore no discretisation is necessary. Nevertheless modal equations must be solved numerically.

When localized dampers are involved, the situation is much more complicated. Few works are available concerning modal expansion in such cases [2]. When full vibration modes are assumed, complex frequencies and modes shapes must be determined. Such results are important, because the finite element code usually fails to determine these results with sufficient accuracy.

The new contribution of this paper is a detailed comparison of the computation efficiency of the approaches named above.

1

Z. Dimitrovová, "A general procedure for the dynamic analysis of finite and infinite beams on piece-wise homogeneous foundation under moving loads", Journal of Sound and Vibration, 329, 2635-2653, 2010. doi:10.1016/j.jsv.2010.01.017

2

B. Yang, X. Wu, "Transient response of one-dimensional distributed systems: a closed form eigenfunction expansion realization", Journal of Sound and Vibration, 208(5), 763-776, 1997. doi:10.1006/jsvi.1997.1206

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £110 +P&P)