Dynamic Stability of an Elastic Beam with Visco-Elasto-Damaged Translational and Rotational Supports

The phenomenon of dynamic instability is investigated in this paper for a beam constrained at its end sections by viscoelastic (Kelvin-Voigt) translational and rotational supports, additionally affected by a certain degree of damage, to understand the influence of viscoelasticity and damage in its response under a dynamic axial load. A calculation procedure is developed to investigate the regions of dynamic instability of such a constrained beam by using an exact approach for solving the unloaded case and by applying it to the well-known solution for the boundary frequency domains of the dynamic problem. Damage is supposed to affect the elastic response of the restraints and to follow the scalar isotropic model. With respect to viscoelasticity, the time variable is treated as a parameter to account for the different timescales, according to the two transient phenomena: viscoelasticity and instability caused by an external dynamic load. The regions of dynamic instability for several configurations of the constraints are shown as three-dimensional diagrams in which the aforementioned regions contained in the plane described by the dynamic component of the periodic load and the frequency of the same load are shown here to vary in time and with respect to the level of damage of the constraints. In particular, the first three natural frequencies of the beam for each studied configuration have been taken into account. As expected, for each natural frequency and each configuration, the instability domains settle into asymptotic values ascertaining that the presence of damage in time increases the instability of the beam, since it is proved to move the domains toward lower, more easily reachable values of the external load frequency.