`Shock and VibrationVolume 15, Issue 6, Pages 655-664http://dx.doi.org/10.1155/2008/645014`

## On the Convergence of Nonlinear Modes of a Finite Element Model

Ramesh Balagangadhar1 and Joseph C. Slater2

1Goodrich Aerospace, Engineering Technologies, 101 Waco St, Troy, OH 45373, USA
2Department of Mechanical and Materials Engineering, Wright State University, Dayton, OH 45435, USA

Received 30 July 2005; Revised 6 August 2007

Copyright © 2008 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Convergence of finite element models is generally realized via observation of mesh independence. In linear systems invariance of linear modes to further mesh refinement is often used to assess mesh independence. These linear models are, however, often coupled with nonlinear elements such as CFD models, nonlinear control systems, or joint dynamics. The introduction of a single nonlinear element can significantly alter the degree of mesh refinement necessary for sufficient model accuracy. Application of nonlinear modal analysis [1,2] illustrates that using linear modal convergence as a measure of mesh quality in the presence of nonlinearities is inadequate. The convergence of the nonlinear normal modes of a simply supported beam modeled using finite elements is examined. A comparison is made to the solution of Boivin, Pierre, and Shaw [3]. Both methods suffer from the need for convergence in power series approximations. However, the finite element modeling method introduces the additional concern of mesh independence, even when the meshing the linear part of the model unless p-type elements are used [4]. The importance of moving to a finite element approach for nonlinear modal analysis is the ability to solve problems of a more complex geometry for which no closed form solution exists. This case study demonstrates that a finite element model solution converges nearly as well as a continuous solution, and presents rough guidelines for the number of expansion terms and elements needed for various levels of solution accuracy. It also demonstrates that modal convergence occurs significantly more slowly in the nonlinear model than in the corresponding linear model. This illustrates that convergence of linear modes may be an inadequate measure of mesh independence when even a small part of a model is nonlinear.