Exact frequency response analysis of axially-loaded beams with viscoelastic dampers

A method is proposed for frequency response analysis of Euler–Bernoulli beams acted upon by a constant axial load, and carrying an arbitrary number of translational and rotational dampers with Kelvin–Voigt viscoelastic behavior. The method relies on the theory of generalized functions to handle the discontinuities of the response variables, within a standard 1D formulation of the equation of motion. Starting from the exact dynamic Green's functions of the bare beam, i.e. the beam without dampers, exact closed-form expressions are derived for the frequency response of the beam with dampers, subjected to harmonically varying, arbitrarily placed transverse point/polynomial loads, which hold for any number of dampers. Further, free vibration solutions are obtained from a characteristic equation built as determinant of a 4×4 matrix, regardless of the number of dampers. Numerical applications show the advantages of the proposed exact solutions.