PSI - Issue 71

Ishaan Tapas et al. / Procedia Structural Integrity 71 (2025) 477–483

478

natural frequencies that are within the range of expected external forces, engineers can minimize the risk of resonance and its potentially damaging effects. Resonant vibrations of turbomachinery blades are responsible for the catastrophic failure of turbomachines, and designers must avoid this condition to prevent failure. Determining the natural frequencies of turbomachinery blades enables the identification of critical speeds and safe operating speeds, which is essential for ensuring the reliability and performance of these machines. In this work, we propose to develop a closed-form formula in terms of a perturbation series to determine the natural frequencies of the turbomachinery blades. The blades are modelled as tapered cantilever beams under the Euler Bernoulli beam theory. Numerous studies have investigated the dynamics of turbomachine blades, focusing on their vibrational characteristics and the factors influencing their natural frequencies. A study about the out-of-plane free vibrations of a double-tapered Euler-Bernoulli beam mounted on a rotating hub was reported by Ozge Ozdemir Ozgumus and Metin O. Kaya, examining how parameters like hub radius, rotational speed, and taper ratios affect natural frequencies. The coupling of nonlinear dynamic equations for a rotating, double tapered cantilever Timoshenko beam was studied by Yinlei Huo and Zhongmin Wang. The authors derived the equations of motion using the Hamilton principle and providing insights into the influence of dimensionless parameters on natural frequencies. The effectiveness of the transfer matrix method to investigate the free flexural vibration behavior of tapered beams was reported by J.R. Banerjee and A. Ananthapuvirajah. Jung Woo Lee and Jung Youn Lee developed a transfer-matrix method for more precise solutions of tapered Bernoulli-Euler beams, focusing on the effects of taper ratios and rotation parameters. Additionally, J.R. Banerjee, H. Su, and D.R. Jackson examined the bending vibration dynamics of rotating tapered beams using the dynamic stiffness method, accounting for factors like centrifugal stiffening. Ajinkya Baxy and Abhijit Sarkar derived a novel formula for natural frequencies of rotating circularly curved cantilever beams, employing the Rayleigh-Ritz approach and perturbation methods. Lastly, J.A. Rodríguez conducted a fatigue analysis of steam turbine blades, revealing that resonance conditions significantly accelerate crack initiation and propagation. Nomenclature A Cross-sectional area A 0 Cross-sectional area at the root b Width of the beam d 1 Height of the beam at tip d 2 Height of the beam at the root d(x) Height of the beam at any point x along the beam E Young’s modulus I ZZ Area moment of inertia about z axis I 0 Area moment of inertia about the z-axis at the root L Length of the beam T Kinetic energy u(x) Axial displacement variable U Non-dimensional transverse displacement V Strain Energy w(x) Transverse displacement variable ρ Density ε xx Axial Strain ω Natural frequency of the beam ε Perturbation parameter η Non-dimensional axial coordinate β square of the non dimensional natural frequency φ Mode shape of uniform cantilever beam δ Non-dimensional natural frequency of uniform beam Lagrangian K 1 Zeroth term of perturbation formula