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Avoidance of Numerical Singularities in Free Vibration Analysis of Euler-Bernoulli Beams using Green Functions

This paper investigates the reliability of an algorithm that implements the Green function method in free vibration analysis of Euler-Bernoulli beams. The investigation is concerned with the robustness of the algorithm with respect to the occurrence of numerical singularities in the calculation procedure of mode shapes. The problem is studied for beams supported with an arbitrary number of intermediate translational springs, which can be understood as a generalization of the cases when the beam is without elastic supports and when the beam rests on intermediate rigid supports. The problem of numerical singularities arises from the fact that the elements of the modal vector have to be expressed in terms of an "arbitrarily" chosen referential element of that vector, whose value can vanish if it coincides closely enough with a node of the sought mode shape. The problem is generally tackled here with the introduction of a fictitious spring of a vanishingly small stiffness, and the robustness of the algorithm depends crucially on the appropriate placement of that spring. This paper presents several useful guidelines for the implementation of computer code based on these principles and its reliability is demonstrated through examples.

numerical singularity ; Green functions ; free vibr ations ; Euler-Bernoulli beam ; spring support

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