On the transversal vibrations of an axially moving continuum with a time-varying velocity: Transient from string to beam behavior

In this paper an initial boundary value problem for a linear equation describing an axially moving stretched beam will be considered. The velocity of the beam is assumed to be time varying. Since the order of magnitude of the bending stiffness terms depends on the vibration modes and the frequencies involved a combination of two simplified models (that is, a string equation for the lower frequencies and a beam with string effect equation for the higher frequencies) will be used to describe the transversal vibrations of the system accurately. Based on the calculations of the natural frequencies the regions of applicability of these sub-models will be determined. A two time-scales perturbation method will be used to construct formal asymptotic approximations of the solutions. Non-resonant and some resonant cases will be studied for four different values of the relative errors. An important implication of the earlier results in the literature is that for these types of axially moving continua problems the use of only string-like models is not appropriate. To describe the dynamics of these types of axially moving continua problems correctly one has to include (small) bending stiffness in the model. In this paper it is explicitly shown how to work with a combined model that is a string model at the low frequencies and a tensioned beam model at the higher frequencies.