Cumulative effect of structural nonlinearities: chaotic dynamics of cantilever beam system with impacts Joseph Emans a, * , Marian Wiercigroch a , Anton M. Krivtsov b a Department of Engineering, Centre for Applied Dynamics Research, Fraser Noble Building, King’s College, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK b Department of Theoretical Mechanics, St. Petersburg State Polytechnical University, Politechnicheskaya Street 29, St. Petersburg 195251, Russia Accepted 22 June 2004 Abstract The nonlinear analysis of a common beam system was performed, and the method for such, outlined and presented. Nonlinear terms for the governing dynamic equations were extracted and the behaviour of the system was investigated. The analysis was carried out with and without physically realistic parameters, to show the characteristics of the system, and the physically realistic responses. Also, the response as part of a more complex system was considered, in order to investigate the cumulative effects of nonlinearities. Chaos, as well as periodic motion was found readily for the physically unrealistic parameters. In addition, nonlinear behaviour such as co-existence of attractors was found even at modest oscillation levels during investigations with real- istic parameters. When considered as part of a more complex system with further nonlinearities, comparisons with lin- ear beam theory show the classical approach to be lacking in accuracy of qualitative predictions, even at weak oscillations. Ó 2004 Elsevier Ltd. All rights reserved. 1. Introduction Beams are perhaps the most commonly used elements of engineering design, and there is a number of well estab- lished theories regarding them. These include Bernoulli, Euler, and Timoshenko [1], which provide analytical or approximate solutions. The dynamics of beams is very well understood for the vast majority of loading scenarios encountered. Yet even these well-researched elements may exhibit behaviour which is unpredictable, for example buck- ling, or during intermittent contact with another body, which can cause nonlinear behaviour manifesting in subhar- monic and superharmonic oscillations, coexistence of attractors and perhaps even chaos. If the beams are part of a larger system, significant errors can be incurred when considering the dynamic responses, both qualitatively and quantitatively, by neglecting even small nonlinearities. This is particularly true when the system contains other nonlinearities such as discontinuities or boundary conditions conducive of complex phenomenon. Many boundary conditions of beams can provide nonlinear behaviour to varying degrees, and various such arrangements 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.052 * Corresponding author. Chaos, Solitons and Fractals 23 (2005) 1661–1670 www.elsevier.com/locate/chaos