Vibrations of multi-span non-uniform beams with arbitrary discontinuities and complicated boundary conditions

Multi-span flexural beams with multiple discontinuities are widely used in engineering, and to analyze the free vibrations has important significance to the efficient design of such systems. In this paper, a general solution of vibrations of multi-span beams with arbitrary discontinuities and complicated boundary conditions is presented. The entire beam is first divided into several segments according to the positions of discontinuities. The compatibility conditions are considered to account for the discontinuity configurations induced by various attachments. The characteristic equation of the beam is derived using separation of variables in conjunction with the transfer matrix method. The analytic model accounts for the geometrical characteristics of the beam with unequal span, variable crosssection as well as the arbitrariness of discontinuities. Complicated boundary conditions, i.e., elastically supported and lumped mass boundary, are also included. Finally, numerical examples are carried out to analyze the effects of support stiffness and number of constraints on the modal characteristics of beams. From the comparison of results with those obtained with the finite element method, the validity and practicability of the present procedure are validated.