Stationary response of beams and frames with fractional dampers through exact frequency response functions

This paper proposes an original approach to the stochastic analysis of beams and plane frames with arbitrary number of fractional dampers, subjected to stationary excitations. External and internal, translational and rotational dampers are considered, with constitutive behavior modeled by the Riemann-Liouville fractional derivative. Starting from the Euler-Bernoulli formulation for bending vibration of a beam, and treating discontinuous response variables at the application points of dampers by the theory of generalized functions, it is shown that an appropriate use of dynamic Green's functions of the bare beam provides the exact frequency response to point or distributed polynomial load, in terms of four integration constants only, regardless of the number of dampers. Based on this result, exact closed-form expressions are built for the stationary response of a single beam and a plane frame, under stationary point/polynomial loads, for any number of dampers. The stationary response in every frame member is derived from a nodal displacement solution computed by an exact global frequency response matrix and a load vector, whose size depends only on the number of beam-to-column nodes, for any number of point/polynomial loads and dampers along the frame members. Solutions are built for the most general case of multiple dampers occurring simultaneously at the same point. Changes to consider single dampers at a given location are straightforward. Numerical applications show the advantages of the proposed method.