Higher order beam equations
Paper in proceedings, 2012
This paper considers the dynamic equations of circular cylindrical beams. The method is based on the three dimensional theory, adopting the generalized Hamilton's principle. By adopting a power series expansion method in the radial coordinate, together with a Fourier series expansion in the circumferential direction, this procedure results in sets of equations of motion together with consistent sets of end boundary conditions. These are derived in a systematic fashion up to arbitrary order, and are believed to be asymptotically correct. As such, the equations of motion are hyperbolic. Among the derived equation set are recursion relations, from which it is possible to express higher order displacement and stress fields in terms of lower order displacement fields.
Results are obtained for all Fourier modes, among which axisymmetric, torsional and flexural modes are special cases. Special attention is paid towards the flexural mode. Using different truncation orders of the present theory, comparisons may be performed with classical theories such as the Euler-Bernoulli and the Timoshenko theories, besides the exact theory.
Numerical examples are presented for dispersion curves of an infinite beam for the three lowest modes. Here various truncation orders are presented, as well as the exact theory. It is clear that higher accuracy is obtained as more terms are used. The lowest mode curve is accurately captured in the lower frequency range for all theories. Higher order truncations are indistinguishable from the exact curves in the presented range.
Concerning eigenfrequencies, the three lowest frequencies for two different beams are presented for different truncation orders, classical theories as well as the exact theory for simply supported ends. As for the dispersion curves, the series expansion results converge to the exact results as the power series orders are increased. It is clear that more accurate results are obtained for lower frequencies and slender beams as expected. The Timoshenko theory is astonishingly accurate while the Euler-Bernoulli theory confirms the well known fact that this theory renders reasonably accurate results for slender beams in the low frequency spectra.
Various plots on mode shapes and stress distributions are compared for the fundamental frequency for a simply supported beam. Here the curves using the lowest series expansion theory are very close to the exact results. There are more pronounced differences between theories for the mode shapes and the stress distributions, compared to the eigenfrequency calculations and the dispersion curves. The differences are most prominent for the stress distributions.