Abstract

This research presents the numerical analysis of the triply coupled flap-wise, cord-wise and torsional vibrations of flexible rotating blades. Euler-Bernoulli bending and St. Venant torsion beam theories are considered to derive the governing differential equations of motion. Based on Finite Element Methodology (FEM), the cubic "Hermite" shape functions are implemented where the solution of the equations results in a linear engine problem. Then, the Dynamic (frequency dependent) Trigonometric Shape Functions (DISF's) for beam's uncoupled displacements are derived. The application of the Dynamic Finite Element (DFE) approach to the solution of the governing equations is then presented. The DFE formulation, based on the weighted residual method and the DTSF's results in a nonlinear engine problem representing eigenvalues and engine modes of the system. The applicability of the DFE method is then demonstrated by illustrative examples, where a Wittrick-Williams root counting technique is used to find the system's natural frequencies. The DFE approach, an intermediate method between FEM and "Exact" formulation, is characterized by higher convergence rates, and can be advantageously used when multiple natural frequencies and/or higher modes of beam-like structures are to be evaluated.

Highlights

  • The study of the dynamic behaviour of flexible structures is an intrinsic part of design of such systems

  • Finite Element Methodology (FEM) is one of the most powerful numerical methods widely used to solve the problems of complicated geometry and/or mechanical properties, where the analytical solutions are not always available

  • The FEM results can be compared with the exact analytical solution, and the convergence of the FEM method can be evaluated

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Summary

Introduction

The study of the dynamic behaviour of flexible structures is an intrinsic part of design of such systems. Many engineering problems, such as buckling and vibration analyses of flexible structures, lead to one of the following equations; = [A]{x} = X{x} [^(A)]{x} = 0. In vibration analysis of flexible structures, the eigenvalues and eigenvectors are equivalent to natural frequencies and mode shapes, respectively, where X= co^. In vibration analysis of a beam there are many different approaches using the FEM (see for example [6,17,28,29,30]). Each of these methods transforms the differential equations of motion into an algebraic eigenvalue problem. In the DFE, the solutions of the differential equations governing the uncoupled vibrations of the beam are chosen as the basis functions of approximation space.

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