Boundary layers and non-linear vibrations in an axially moving beam

The non-linear oscillations of a one-dimensional axially moving beam with vanishing flexural stiffness and weak non-linearities are analysed. The solution of the initial-boundary value problem for the partial differential equation that describes the motion of the beam when two parameters related to the flexural stiffness and the non-linear terms vanish is expanded into a perturbative double series. Two singular perturbation effects due to the small flexural stiffness and to the weak non-linear terms arise: (i) a boundary layer effect when the flexural stiffness vanishes, (ii) a secular effect. Some tests are performed to compare the first order perturbative solution with an approximate solution obtained by a finite difference scheme. The effect of the oscillation amplitude combined with the presence of small bending stiffness and axial transport velocity is investigated enlighting some interesting aspects of axially moving systems. The value of the perturbative series as a computational tool is shown.