Triply coupled vibrational band gap in a periodic and nonsymmetrical axially loaded thin-walled Bernoulli–Euler beam including the warping effect

The propagation of triply coupled vibrations in a periodic, nonsymmetrical and axially loaded thin-walled Bernoulli–Euler beam composed of two kinds of materials is investigated with the transfer matrix method. The cross-section of the beam lacks symmetrical axes, and bending vibrations in the two perpendicular directions are coupled with torsional vibrations. Furthermore, the effect of warping stiffness is included. The band structures of the periodic beam, both including and excluding the warping effect, are obtained. The frequency response function of the finite periodic beam is simulated with the finite element method. These simulations show large vibration-based attenuation in the frequency range of the gap, as expected. By comparing the band structure of the beam with plane wave expansion method calculations that are available in the literature, one finds that including the warping effect leads to a more accurate simulation. The effects of warping stiffness and axial force on the band structure are also discussed.