Coupled bending–torsional frequency response of beams with attachments: exact solutions including warping effects

This paper deals with the coupled bending–torsional vibrations of beams carrying an arbitrary number of viscoelastic dampers and attached masses. Exact closed analytical expressions are derived for the frequency response under harmonically varying, arbitrarily placed polynomial loads, making use of coupled bending–torsion theory including warping effects and taking advantage of generalized functions to model response discontinuities at the application points of dampers/masses. In this context, the exact dynamic Green’s functions of the beam are also obtained. The frequency response solutions are the basis to derive the exact dynamic stiffness matrix and load vector of a two-node coupled bending–torsional beam finite element with warping effects, which may include any number of dampers/masses. Remarkably, the size of the dynamic stiffness matrix and load vector is 8×8 and 8×1, respectively, regardless of the number of dampers/masses and loads along the beam finite element.