An exact generalised function approach to frequency response analysis of beams and plane frames with the inclusion of viscoelastic damping

Abstract

This paper concerns the frequency response analysis of beams and plane frames with an arbitrary number of Kelvin–Voigt viscoelastic dampers. Typical external and internal dampers are considered, as grounded translational, tuned mass, rotational and axial dampers, for bending and axial vibrations, respectively. Using the theory of generalised functions within a 1D formulation of equations of motion, exact closed-form expressions are derived for beam dynamic Green’s functions and frequency response functions under arbitrary polynomial load, for any number of dampers. For a plane frame, exact global frequency response matrix and load vector are built, with size depending only on the number of beam-to-column nodes, for any number of dampers and point/polynomial loads along the frame members. From the nodal displacement solution, the exact frequency response in all frame members is also obtained in closed analytical form. Numerical applications show many of the advantages of the proposed method.