Natural Frequency of Turbomachinery Blade by Perturbation Method

The present work focuses on deriving a perturbation formula for determining the natural frequencies of turbo-machinery blades. The blades are modelled as tapered cantilever beams. Cantilever beams are structural elements fixed at one end and free at the other, widely used in engineering applications, including turbine blades. Accurate prediction of natural frequencies is essential for the design and analysis of these structures, as it helps in understanding their dynamic behaviour and avoiding resonance-related failures. The formulation is done using the Lagrangian approach with use of Euler-Bernoulli beam theory. The complication due to taper is the varying cross-sectional area, which is captured in potential energy and kinetic energy. In the formulation, the Rayleigh-Ritz method is applied with admissible shape functions. The complication of taper is treated as a perturbation parameter over the uniform straight cantilever beam. Based on the perturbation theory and the Rayleigh-Ritz method, a closed-form formula is derived. The formula is derived as a correction due to taper over a uniform, non-rotating, straight cantilever beam. The accuracy of the formula is validated against literature and FEM simulations. Additionally, a web application is developed for practicing engineers.