Natural frequencies of a rotating curved cantilever beam: A perturbation method-based approach

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.