Exact frequency response of two-node coupled bending-torsional beam element with attachments

Abstract

• Coupled bending-torsional dynamics of beams with dampers/masses is addressed. • Exact analytical frequency response is built for any harmonic point/polynomial loads. • Exact two-node 6 × 6 dynamic stiffness matrix and 6 × 1 load vector are built in closed form. • Dynamic stiffness matrix and load vector hold same size for any number of dampers/masses and loads. • Modal frequency responses and modal impulse responses are derived by complex modal analysis. This paper addresses the frequency response of coupled bending-torsional beams carrying an arbitrary number of in-span viscoelastic dampers and attached masses. Using the elementary coupled bending-torsion theory, along with appropriate generalized functions to treat the discontinuities of the response variables at the application points of dampers/masses, exact analytical expressions are derived for the frequency response of the beam under harmonically-varying, arbitrarily-placed point/polynomial loads. On this basis, the exact 6 × 6 dynamic stiffness matrix and 6 × 1 load vector of a two-node coupled bending-torsional beam finite element, with any number of in-span dampers/masses and harmonic loads, are obtained in a closed analytical form. Finally, the modal frequency response functions of the beam are built by a complex modal analysis approach, upon deriving pertinent orthogonality conditions for the modes. In this context, the modal impulse response functions are also obtained for time-domain analysis under arbitrary loads.