Axially Moving Beams with Varying Length: Wave Reflection and Propagation Approach

The present paper analyses the dynamic of a clamped-free beam with axially transport of mass and varying length. The transverse vibration of the beam is described in term of harmonic propagating and decaying waves in dispersive medium. Reflection and propagation matrices for non-homogeneous and time-varying boundary conditions are given and a simple model that capture the dynamic behaviour of the system is defined. The analysis has proved that this method could be a valuable choice to gain important and concise results such as continuous change in the frequencies with time, increase or decrease of the transverse deformed shapes, energy evaluation within the domain with respect to the time. Eventually, an approximate formula to compute the continuous change in the natural frequencies together with investigations about the nature of the time rate of change of the vibrational energy are provided.