Stability of an Axially Moving Viscoelastic Beam

In this paper, we consider a viscoelastic flexible structure modeled as an Euler-Bernoulli beam. The beam is moving in the direction of its axis. This is one of the main features of this work. We will be dealing with variable intervals of integration and therefore the standard computation using differentiation under the integral sign is no longer valid. The boundary conditions are of 'cantilever' type: there is no displacement at one end and the other end is free. In fact, it is subject to a nonlinear force acting there. We prove that when the velocity of the beam is smaller than a critical value, the dissipation produced by the viscoelastic material is sufficient to suppress the transversal vibrations that occur during the axial motion of the beam. The rate of decay is shown to be exponential.