Free response of a continuous vibrational system with attachments and/or discontinuities using segmented operational Tau method

It has been widely known that for complicated beam-like structures with various types of attachments and/or discontinuities analytical techniques are not always applicable. In this paper, a very efficient numerical method based on the Tau method is proposed to tackle the mentioned problem. A general form of the linear vibrational eigen-equation, based on the Euler–Bernoulli bending theory, together with its boundary conditions and continuity equations is considered. The problem is then formulated using a segmented form of the operational Tau method which is called the segmented operational Tau method. To investigate the reliability and accuracy of the proposed method some vibrational problems are solved and compared with the analytical solutions providing the exact frequencies and mode shapes. For a complicated case of a non-uniform beam with various types of attachments, since there was no analytical solution, results are validated with the finite element method. This paper has provided a platform for solving free vibrational problems of many complicated constrained systems through a very simple, highly accurate technique.