Summary
Free vibration of a periodically supported (multispan) beamin via a simplified Bresse-Timoshenko theory is studied by the Krein's method suggested in 1933 for the Bernoulli-Euler beams. Approximate differential equations are utilized with both shear deformations and rotary inertia included, but with the term representing the joint action of these effect omitted. Detailed analytical and numerical analysis are performed for the natural frequencies of beams with different boundary conditions at their ends. Following Krein, the continuity requirements at the intermediate supports are treated as equations in finite differences and solved exactly.
As in the classical Bernoulli-Euler beam, the natural frequencies fall into periodically spaced bands, with each band containing a number of frequencies equal to that of spans. The shear deformations and rotary inertia shift the classical frequency bands to the left, this effect being more pronounced for higher bands.
Extensive numerical results are reported for three-, five- and ten-span beams. Comparison with previously reported results (obtained by straightforwarded analysis) for a three-span beam shows excellent agreement.
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Abramovich, H., Elishakoff, I. Application of the Krein's method for determination of natural frequencies of periodically supported beam based on simplified Bresse-Timoshenko equations. Acta Mechanica 66, 39–59 (1987). https://doi.org/10.1007/BF01184284
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DOI: https://doi.org/10.1007/BF01184284