Vibration analysis of non-uniform Timoshenko beams coupled with flexible attachments and multiple discontinuities

In this paper, a general analytical solution is used to study free vibration of non-uniform Timoshenko beams coupled with flexible attachments and multiple discontinuities in engineering applications. Based on separation of variables in conjunction with the transfer matrix approach, the compatibility conditions across discontinuity points and arbitrary boundary conditions of the beams are devised in a systematic manner. Consequently, regardless of the number and types of discontinuities, the frequency equation can be conveniently established with a four-order determinant which results in significant saving of the computational effort. Furthermore, treating flexible attachments as complicated boundaries and describing all their relevant mechanical characteristics with appropriate transfer functions, the flexibility of such attachments can also be correctly integrated into the characteristic equation. Results of this study are validated with those from the Finite Element Method (FEM) and available results in the literature which reveal the accuracy and application of the proposed method. Parametric studies are further presented to analyze the effect of parameter properties of attached discontinuities on the natural frequencies of the beams.