Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 21 (2007) 973–993 Quantifying chaotic responses of mechanical systems with backlash component Tegoeh Tjahjowidodo , Farid Al-Bender, Hendrik Van Brussel Mechanical Engineering Department, Division PMA, Katholieke Universiteit Leuven, Celestijnenlaan 300B, B3001 Heverlee, Belgium Received 9 June 2005; received in revised form 4 October 2005; accepted 7 November 2005 Available online 15 December 2005 Abstract This paper presents a detailed numerical and experimental study of a mechanical system comprising a backlash element that shows chaotic behaviour in a certain excitation range. It aims to quantify the chaotic behaviour of the responses and correlate the quantification parameters to the parameters of the (non-linear) system, in particular the backlash size. The motivation for this investigation is to be able ultimately to identify the parameters of non-linear systems without necessarily being able to ensure periodic behaviour. Application of surrogate data tests is utilised to prove the presence of the chaotic behaviour in the response. The Simple Non-linear Noise Reduction is applied to the resulting data to have a better interpretation of the chaotic in the response. r 2005 Elsevier Ltd. All rights reserved. Keywords: Backlash; Chaotic; Surrogate data test; Noise reduction 1. Introduction When excited by a periodic input, a non-linear system may exhibit two types of steady-state response behaviour: periodic or non-periodic, i.e. chaotic. In case of a periodic response, various techniques are available for the characterisation of non-linear systems. Volterra and Wiener series [1] are very suitable for systems exhibiting superharmonic response, while other techniques taken as an extension of linear ARMA theory, namely NARMA and NARMAX have also been developed and proposed for the purpose of identification of various systems [1,2]. The application of the Hilbert Transform (HT) can be effective for practical analysis of the system parameter identification [3–6] for the case of the system with geometric non- linearities. Wavelet analysis was in particular shown to be another identification approach, which offers significant improvement in comparison with HT technique, though it suffers from memory hunger and processing time [5,7]. ‘‘Well-behavedness’’ of a non-linear system is however not always guaranteed, so that under certain excitation conditions, which are not always under our control, chaotic behaviour might occur, rendering the aforementioned techniques inapplicable. In order to be able to deal with such cases, new methods are needed ARTICLE IN PRESS www.elsevier.com/locate/jnlabr/ymssp 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.11.003 Corresponding author. Tel.: +32 16 322515; fax: +32 16 322987. E-mail address: tegoeh.tjahjowidodo@mech.kuleuven.be (T. Tjahjowidodo).