Short communication Stability of strongly nonlinear normal modes Geoffrey Recktenwald, Richard Rand * Department of Theoretical and Applied Mechanics, Cornell University, 207 Kimball Hall, Ithaca, NY 14850, USA Received 9 November 2005; accepted 11 November 2005 Available online 6 January 2006 Abstract It is shown that a transformation of time can allow the periodic solution of a strongly nonlinear oscillator to be written as a simple cosine function. This enables the stability of strongly nonlinear normal modes in multidegree of freedom sys- tems to be investigated by standard procedures such as harmonic balance. Ó 2005 Elsevier B.V. All rights reserved. PACS: 02.30.Hq; 05.45.Xt; 46.40.Ff Keywords: Stability of motion; Nonlinear normal mode 1. Introduction This work is concerned with the stability of periodic motions in multidegree of freedom conservative dynamical systems. As an example, consider the system with kinetic energy T and potential energy V, where T ¼ 1 2 _ x 2 þ 1 2 _ y 2 ; ð1Þ V ¼ 1 2 ðx 2 þ x 2 Þy 2 þ 1 4 x 4 þ 1 5 x 5 . ð2Þ This system is governed by the following equations of motion: € x þ x 3 þ x 4 þ xy 2 ¼ 0; € y þ x 2 y þ x 2 y ¼ 0. ð3Þ These equations possess an invariant manifold y = 0, on which lies a family of nonlinear normal modes (NNMs) which satisfy the following equations: € x þ x 3 þ x 4 ¼ 0; xð0Þ¼ A; _ xð0Þ¼ 0. ð4Þ 1007-5704/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.11.003 * Corresponding author. Fax: +1 607 255 2011. E-mail address: rhr2@cornell.edu (R. Rand). Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1128–1132 www.elsevier.com/locate/cnsns