Coupled bending and torsional free vibrations of beams with in-span supports and attached masses

This paper proposes an exact approach to coupled bending and torsional free vibration analysis of beams with monosymmetric cross section, featuring an arbitrary number of in-span elastic supports and attached masses. The proposed method relies on the elementary coupled bending-torsion theory and makes use of the theory of generalized functions to handle the discontinuities of the response variables. Based on a simple procedure, exact natural frequencies and closed-form eigenfunctions are calculated from a characteristic equation built as determinant of a 6×6 matrix, for any number of supports/masses. Likewise, the exact dynamic stiffness matrix of the beam is obtained in a closed form, with size 6×6 regardless of the number of supports/masses, to be assembled for frame analysis. In addition, the orthogonality condition among the eigenfunctions is derived. Two numerical examples show the advantages of the proposed exact method.