Coupled tension–torsion vibration of repetitive beam-like structures

Equivalent continuum stiffness properties, derived from the eigenanalysis of a single cell of a planar beam-like repetitive structure, have previously been employed within well-known dynamic theories, such as Euler–Bernoulli and Timoshenko for flexural vibration, suitably modified, to predict natural frequencies of vibration. Here the approach is applied to two structure types that exhibit tension–torsion coupling. The first is modelled on a NASA deployable structure with a cross-section of equilateral triangular form, but with asymmetric triangulation on the faces. The second is a related, more symmetric, structure but with pre-twist. The simplest tension–torsion dynamic theory due to Di Prima is employed, and this is extended to more general end conditions. This combined periodic structure/substitute continuum approach provides excellent agreement with predictions from the finite element method, especially for the lower modes of vibration; typically, agreement is within ±1% for the lowest 8–10 natural frequencies for the longer, 30-cell structures considered here, the majority of these being torsional modes, and within ±1% for the lowest 4–5 modes for the shorter, ten-cell, structures. This level of accuracy is attainable so long as a single wavelength spans 2–3 cells of the repetitive structure.