The blades of propellers, fans, compressor and turbines can be modeled as curved beams. In general, for industrial application, finite element method is employed to determine the modal characteristics of these structures. In the present work, a novel formula for determining the natural frequencies of a rotating circularly curved cantilever beam is derived. Rayleigh–Ritz approach is used along with perturbation method to obtain the analytical formula. In the first part of the work, a formula for natural frequencies of a non-rotating curved beam vibrating in its plane of curvature is presented. This formula is derived as a correction to the natural frequencies of its straight counterpart. The curvature is treated as a perturbation parameter. In the next part of the work, the effect of rotation on the curved beam is captured as an additional perturbation. Thus, the formula for a curved rotating beam is derived as a correction (involving two perturbation parameters) to the non-rotating straight beam. The results obtained using the derived formula are compared with the finite element method results. It is found that the frequency estimates from the formula are valid over a fairly large range of curvature and rotation speed. Thus, the derived formula can provide a faster alternative for design iterations in industrial applications.

Diatomic chain consisting of spring-mass system forms an attenuation bandgap due to the formation of the standing wave when the wavelength matches with the periodicity, known as Bragg-scattering. Edge-state phenomenon can be obtained by breaking the periodicity of the diatomic chain. Owing to this edge state phenomenon, the waves, and in turn the energy, is localized at the point of the asymmetry. In this work, an efficient vibrational energy harvesting technique is proposed exploiting this edge-state energy localization by inserting piezo-electric harvester at the junction of the asymmetry. To illustrate the performance of the proposed harvester under the broad-band noise, voltage-frequency curve for three different diatomic chain configurations, having same mass and stiffness, are analyzed. The performance metric, defined as the area under the voltage-frequency curve, shows 21.5 times higher performance can be obtained just by breaking the symmetry of a conventional diatomic chain. As it is also observed that the attenuation properties of the symmetric and asymmetric chain remain same; thus, the proposed edge-state energy harvester has a significant promise toward simultaneous energy harvesting and vibration control.

This work proposes a unique configuration of two-dimensional metamaterial lattice grid comprising of curved and tapered beams. The propagation of elastic waves in the structure is analyzed using the dynamic stiffness matrix (DSM) approach and the Floquet-Bloch theorem. The DSM for the unit cell is formulated under the extensional theory of curved beam considering the effects of shear and rotary inertia. The study considers two types of variable rectangular cross-sections, viz. single taper and double taper along the length of the beam. Further, the effect of curvature and taper on the wave propagation is analysed through the band diagram along the irreducible Brillouin zone. It is shown that a complete band gap, i.e. attenuation band in all the directions of wave propagation, in a homogeneous structure can be tailored with a suitable combination of curvature and taper. Generation of the complete bandgap is hinged upon the coupling of axial and transverse component of the lattice grid. This coupling emerges due to the presence of the curvature and further enhanced due to tapering. The double taper cross-section is shown to have wider attenuation characteristics than single taper cross-sections. Specifically, 83.36% and 63% normalized complete bandwidth is achieved for the double and single taper cross-section for a homogeneous metamaterial, respectively. Additional characteristics of the proposed metamaterial in time and frequency domain of the finite structure, vibration attenuation, wave localization in the equivalent finite structure are also studied.

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