GENERALIZED FINITE ELEMENT METHOD FOR VIBRATION ANALYSIS OF BARS

Abstract

The vibration analysis is an important stage in the design of mechanical systems and structures subject to dynamic loads like wind and earthquake. The Finite Element Method (FEM) is commonly used in vibration analysis and its approximated solution can be improved using two refinement techniques: h and p-versions. The h-version of FEM gives good results for the lowest frequencies but demands great computational cost to work up the accuracy for the higher frequencies. The accuracy of the FEM can be improved applying the polynomial p refinement. Some enriched methods based on the FEM have been developed in last 20 years seeking to increase the accuracy of the solutions for the higher frequencies with lower computational cost. The purpose of this paper is to present a formulation of the Generalized Finite Element Method (GFEM) to free and transient vibration analysis of bars. The Generalized Finite Element Method is developed by enriching the standard Finite Element Method space, whose basis performs a partition of unity, with knowledge about the differential equation being solved. The proposed method combines the best features of GFEM and enriched methods: (a) efficiency, (b) hierarchical refinements and (c) the introduction of boundary conditions following the standard finite element procedure. In addition the enrichment functions are easily obtained. The main features of the GFEM are discussed and the partition of unity functions and the local approximation spaces are presented. The efficiency and convergence of the proposed method for vibration analysis of bars are checked. The results obtained by the GFEM are compared with those obtained by the analytical solution, some enriched methods and the h and p versions of the Finite Element Method.