Dynamic Stability of Elastic Rectangular Plates with Viscoelasto-Damaged Constraints

The problem of dynamic instability of rectangular plates with complex constraints at the edges is here solved starting from a theory in the literature; the suggested approach allows for the definition of generic, non perfect constraint configurations for the regions of instability due to in-plane compression loads as diagrams that are expressed in function of the applied dynamic force and its frequency. The novelty of the work stands in (1) the study of viscoelasticity, combined with damage, at the constraints, which allows for the discussion of the performance of a rectangular plate with regards to dynamic instability at varying degrees of damping and damage, and (2) the analytical approach developed to solve the associated eigenvalue problem for rectangular plates with generic viscoelasto-damaged constraints at two opposite edges and simply supported at the other edges. Regarding the second point, the differential equation of motion of such a system is derived and, via the variable separation method, a solution for the transverse displacements is sought among those satisfying the generic boundary conditions at the four edges. Namely, the constraints are assigned in a homogeneous way, which has particularly convenient results for computational purposes. By imposing that the determinant of the resulting system vanishes, a closed-form solution is found for the free vibration problem. The associated boundaries of the unstable regions for specific constraint configurations are discussed.