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

Abstract

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.

Analytical solution and free vibration analysis for beams with step changes in stiffness

“…Numerical methods predominantly deal with the dynamic issues of beams with geometrical discontinuity, such as the finite element method (FEM) (Jaworski et al, 2008), composite element method (CEM) (Lu et al, 2009;Zeng et al, 1998), Adomian decomposition method (ADM) (Mao et al, 2011(Mao et al, , 2015Keshmiri et al, 2018;Wattanasakulpong et al, 2016), differential quadrature element method (DQEM) (Wang and Wang, 2013;Bambill et al, 2015;Wang et al, 1997), discrete singular convolution (DSC) (Duan and Wang, 2013;Wei, 2001;Duan et al, 2014) and Chebyshev-Ritz method (Lee, 2015;Akbarzadeh et al, 2018). Nevertheless, analytical solutions are invaluable for understanding the interplay between the dynamic responses of the entire beam and the stiffness variation of the partial beam and identifying the most important parameters for the problem.…”

“…, 1997), discrete singular convolution (DSC) (Duan and Wang, 2013; Wei, 2001; Duan et al. , 2014) and Chebyshev–Ritz method (Lee, 2015; Akbarzadeh et al. , 2018).…”