Abstract

Abstract: In this talk, propagating waves in elastic bars will be considered in the spirit of asymptotic analysis. The case of bending-torsion coupling and that of shear correction will be discussed. The inclusion of shear deformation amounts to singular perturbation of the Euler-Bernoulli (EB) field equation. Timoshenko incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful ‘branch’ of the dispersion curve, which is mainly shear-wise polarized. Asymptotic formulae for dispersion, standing waves, and the density of modes will be given in terms of a small pertubation parameter. The second spectrum—in the light of the debate on its existence, or not—will be discussed. The so-called ‘Timoshenko beam equation’ appears to be dubious on its own and is not derivable by any known Lagrangian. Finally, the field equations of the bending-torsion coupled waveguide are modified by the inclusion of aerodynamic forces. The equations of motion are presented in terms of matrices explicitly dependent on the flow velocity. The temporal stability and the spatial stability are explored via a quartic and a quadratic equation, respectively. The problem can alternatively be posed as a •-matrix problem and a generalised eigenproblem, respectively. Illustrative examples will be given.