Small Vibrations of Flexible Bars by Using the Finite Element Method with Equivalent Uniform Stiffness and Mass Methodology

Abstract

The object of this research is to calculate natural frequencies and mode shapes of a small oscillation vibration, superimposed on large static displacement, by using the finite element method in conjunction with the equivalent uniform stiffness and mass method. The equivalent uniform stiffness and mass matrices are determined from the differential equation by using Galerkin's method. The analysis herein is three-fold. Firstly, the large static equilibrium configuration of the deformed flexible beam is established. This is achieved by using the Euler-Bernoulli equation in conjunction with equivalent pseudo-linear and/or equivalent non-linear systems of constant stiffness. This concept and methodology were used quite extensively by the authors to study large deformation characteristics of various flexible beam problems of uniform and variable stiffness and of arbitrary loading conditions (see, for example, the work of Fertis and Afonta [1, 2], and Fertis and Lee [3, 4]). Secondly, the differential equation of motion for small amplitude vibrations from the large static equilibrium configuration is derived. Galerkin's technique is used to approximate the differential equation of motion. Finally, the natural frequencies of vibration and the associated mode shapes are determined by solving the resulting eigenvalue problem. A canned eigensolver from the University of Akron IMSL is used to extract the eigenvalues and eigenvectors. The concept of equivalent pseudo-linear systems of constant stiffness is used extensively in this research, in order to simplify the solution of these complex non-linear problems.