Time domain green functions for the homogeneous Timoshenko beam

Abstract

In this paper, wave splitting technique is applied to a homogeneous Timoshenko beam. The purpose is to obtain a diagonal equation in terms of the split fields. These fields are calculated in the time domain from an appropriate set of boundary conditions. The fields along the beam are represented as a time convolution of Green functions with the excitation. The Green functions do not depend on the wave fields but only on the parameters of the beam. Green functions for a Timoshenko beam are derived, and the exponential behaviour of these functions as well as the split modes are discussed. A transformation that extracts the exponential part is performed. Some numerical examples for various loads are presented and compared with results appearing in the literature.

A time domain algorithm for the reflection of waves on a viscoelastically supported Timoshenko beam

“…The influence of the fast wave, arriving at time s = 0, on the displacement and the shear force is very small and for that reason the main effects of multiple scattering are present from time s = 4.25; this is the arrival time of the part of the incoming slow wave that has been once reflected into a slow wave and arrives again at the reflection boundary as a slow wave, and ki --+ 0 the support will vanish and the unrestrained solutions will be approached. Finally, the change of shape of the shear wave, from the incident one, in the solutions corresponding to the unrestrained beam, is due only to dispersion [16]; similar dispersive effects are apparent in the other cases. The bending angle and the bending moment, Figure 8, are influenced by both the fast and the slow wave; therefore the effects of multiple reflection appear from time s = 2.0.…”

“…The incoming fields are propagated through the unrestrained region x E [O, 1] to the reflection boundary. This propagation is carried out by applying the Green's function techniques described in [16]. Thereafter, the reflected fields at x = 1 are calculated by using the relation (3.1), together with the numerical solutions of the reflection equation -these are presented in Figures 9 and 10.…”

“…The corresponding rotation angle and the bending moment are given in Figure 8 4 • In the unrestrained case, i.e. a free beam, this scattering problem was treated in [16], and the case where the suspension is semi-infinite was treated in [6]. These results are reproduced here to enable a comparison with the finite suspension solution presented here.…”

“…The splitting operator used throughout this paper is the one introduced in [16], which in turn is modified from the one initially introduced in [25]. The operators are collected in Appendix A of this paper; see in particular equations (A.5) and (A.6).…”

“…In order to suppress any numerical instabilities that could otherwise occur, this exponential factor should be extracted in the numerical treatment of the reflection kernel. So it is desirable that the reflection kernel is transformed by extracting this exponential divergence, inherited from the wave splitting operator [16], by writing…”