An Analytical Method and Its Extension for Linear Modal Analysis of Beam-Type Systems Carrying Various Substructures

This study is concerned with an analytical solution and its extension on determining natural frequencies and mode shapes of beam-type systems carrying various substructures such as intermediate discontinuities and flexible foundations, etc. First, the method of separation of variables is utilized to devise the governing equation of the Timoshenko beams. Second, compatibility conditions attributed to various discontinuities as well as the complicated boundary conditions are expressed by coefficient matrices with the transfer matrix approach. Furthermore, flexible attachments and flexible foundations considered are described by frequency response functions (FRFs). Finally, the natural frequencies are determined by non-trivial solutions of the resulting matrix equation and the associated mode shapes are derived with Heaviside function. An important objective of this study is to demonstrate the applicability of the proposed methodology. This is achieved by selected numerical examples including a simple double-beam structure with elastic coupling. A new range of results is presented for beam-type structures which can be used as a benchmark to approximate solutions.