Elastic waves in layered periodic curved beams

Abstract

A complete attenuation band in axial and flexural wave propagation can be noticed in a curved beam due to the coupling of the axial and transverse vibrations. However, the presence of a single peak in the attenuation frequency range was reported for the finite number of curved unit cells. This peak reduces the level of attenuation at the specific narrow frequency range within the attenuation band or disintegrates it into two bands. This significantly limits the use of curved beams for real-life structures. The origin of such a peak in the finite curved beam is not yet discussed in the present state of the art. In this paper, the underlying mechanics behind this phenomenon has been revealed from the receptance matrix of the unit cell. Furthermore, by introducing the layered periodic curved beam (LPCB) concept, the effect of this peak can be minimized. The unit cell of LPCB consists of two curved beams, which are formulated using the extensional theory of curved beams considering the effects of shear and rotary inertia. Additionally, a thorough analytical study has been conducted using the transfer matrix method in conjunction with the Floquet–Bloch theorem. The results evidenced that the bandwidth increases with the increase in bent angle for a homogeneous curved beam. The position of the first attenuation band of a homogeneous curved beam can be tuned to low frequency by decreasing its slenderness ratio. This could be utilized for low-frequency vibration control. Using a layered curved beam of different materials and geometry, it is possible to achieve complete normalized bandwidth of 183% through an appropriate selection of parameters. The response of the finite structure designed using periodic modulation in both geometry and material shows considerable attenuation in response in the bandgap frequencies.