Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements

Abstract

This paper presents a formulation of an approximate spectral element for uniform and tapered rotating Euler-Bernoulli beams. The formulation takes into account the varying centrifugal force, mass and bending stiffness. The dynamic stiffness matrix is constructed using the weak form of the governing differential equation in the frequency domain, where two different interpolating functions for the transverse displacement are used for the element formulation. Both free vibration and wave propagation analysis is performed using the formulated elements. The studies show that the formulated element predicts results, that compare well with the solution available in the literature, at a fraction of the computational effort. In addition, for wave propagation analysis, the element shows superior convergence.

Vibration analysis of a cracked rotating tapered beam using the p-version finite element method

“…Attarnejad and Shahba [18] used the basic displacement functions obtained by solving the governing static differential equation of flapwise motion of rotating tapered Euler-Bernoulli beams to establish the finite element formulations. Vinod et al [19] presented an approximate spectral finite element with two different interpolating functions for free vibration and wave propagation analysis of uniform and tapered rotating beams. Banerjee et al [20] utilized the dynamic stiffness method to study the free vibration of rotating tapered Euler-Bernoulli beams, whose height and/or breadth vary linearly along the beam length.…”

Basic Displacement Functions in Analysis of Centrifugally Stiffened Tapered Beams

“…Since obtaining an exact closed-form solution either seems impossible or involves cumbersome mathematical operations, many researchers have used approximate methods such as the Rayleigh-Ritz method [8], Frobenius series solution [9][10][11][12][13], finite element method [14][15][16][17][18][19][20][21], Galerkin method [22,23] and differential transform method [24][25][26][27]. Hodges [28] proposed an approximate formula for calculating the fundamental natural frequency of a uniform rotating beam clamped at the root.…”

High-order non-reflecting boundary conditions for dispersive waves in polar coordinates using spectral elements

“…The formulation used by Lindquist et al [13] remedied this limitation by using a high-order treatment of space (SE) and time (Runge-Kutta) to show the benefits of using the high-order boundary (G-N) scheme. Spectral elements methods used recently by Mitra and Gopalakrishnan [14] and Vinod et al [15] to solve wave propagation problem and by Dorao and 0096-3003/$ -see front matter Published by Elsevier Inc. doi:10.1016/j.amc.2011. 12.023 Jakobsen [16] for incompressible flow.…”