Keywords: free vibration analysis, Ritz method, multi-step beam, hierarchical functions, h-method, p-method, Bardell’s polynomials, trigonometric functions.

The free vibration analysis of homogeneous uniform beams, plates and multi-step beams by the Ritz method, using local Bardell’s polynomials and trigonometric functions, is studied in this paper. The first part of the paper presents a comparative study of the convergence of hierarchical sets under both p- and h-refinements. To this end, a beam and plate are modelled using a single element with a varied number of local functions and multiple elements with a fixed number of local functions. In the second part the above sets of local hierarchical functions are applied for the free vibration analysis of a multi-step beam. With the aim to improve the accuracy of the fundamental mode when using local trigonometric functions, a set of modified local trigonometric functions is proposed to facilitate the satisfaction of the global natural boundary conditions. The use of modified local trigonometric functions with the satisfaction of global natural boundary conditions is shown to significantly improve the accuracy and convergence of the fundamental mode while also converging for higher modes. Moreover, it is also shown to converge under h-refinements in contrast to the divergence observed when using standard trigonometric functions.

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