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A Comparative Evaluation of Some Solution Methods in Free Vibration Analysis of Elastically Supported Beams

This paper presents a comparison between two analytic methods for the determination of natural frequencies and mode shapes of Euler-Bernoulli beams. The subject under scrutiny is the problem of a beam supported by an arbitrary number of translational springs of varying stiffness, which is solved first by the method of Laplace transform and then by the Green’s function method. The two methods are compared on the level of algebraic equivalence of the resulting formulas and these are compared to the results obtained by FEM analysis for the case when the beam is supported with a single spring. It can be shown that both methods result in equivalent algebraic expressions, whose results are to be regarded as accurate for any given set of boundary conditions. This is also verified by means of FEM analysis, whose solutions converged to almost identical values. Hence, the two methods are found to be equally accurate for the calculation of natural frequencies and natural modes. This paper also brings a new formulation for the mode shape equation, obtained by the Laplace transform method, which to the best of the authors’ knowledge has not been reported in existing literature. Additional comments about the advantages and disadvantages of the two employed analytical methods are given from the standpoint of mathematical structure and practicality of implementation, which is also supplemented by a comment on the comparative advantages and disadvantages of FEM analysis.

Euler-Bernoulli beam ; Elastic support ; Free vibrations ; Green’s function ; Laplace transform

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