Journal of Sound and < ibration (1999) 226(5), 923 } 940 Article No. jsvi.1999.2240, available online at http://www.idealibrary.com on DYNAMICAL BEHAVIOUR OF THE PLANAR NON-LINEAR MECHANICAL SYSTEM — PART I: THEORETICAL MODELLING N. JAKS [ IC D , M. BOLTEZ [ AR, I. SIMONOVSKI AND A. KUHELJ ;niversity of ¸jubljana, Faculty of Mechanical Engineering, As \ kerc \ eva 6, 1000 ¸jubljana, Slovenia (Received 21 August 1998, and in ,nal form 16 March 1999) A non-linear planar centrifugally excited oscillatory system was studied in its steady-state domain. The dynamic behaviour in phase space was analysed by a model based on the numerical integration of non-linear equations of motion. The integral of the correlation dimension and Lyapunov exponents were used as a quantitative measure to describe the motion of the model. The estimates of the correlation dimension for the values in real phase space and for those obtained by embedding numerical time histories show good agreement. In addition, the visualization procedure, as a qualitative measure, shows good agreement with the results of Lyapunov exponents and the correlation dimension of the model. Power spectral and bispectral analyses have further been used to analyze the behaviour of the model in the frequency domain. The dominance of the "rst mode was found, while other modes have signi"cantly lower power. In the calculated bicoherences a &&wall'' of values can be seen, which is attributed to the bicoherence's estimated sensitivity to the division by small number. This sensitivity resulted with an increase in the number of bicoherence peaks. A new approach to reduce the number of divisions by small number is proposed and its advantage over those found in the literature is given.  1999 Academic Press 1. INTRODUCTION The modelling of non-linear dynamics has been attracting increasing attention in recent years. Modern analytical tools*speci"cally the analyses in the phase space developed primarily in the basic scienti"c disciplines*are now entering into the applied technical sciences [1}11]. On the other hand, bispectral analysis is a part of the rapidly expanding "eld of Higher Order Statistics and can provide information on the quadratic-type non-linearities in the signal. The two parts of this paper endeavour to combine both approaches, using them "rstly on a theoretical model and secondly on real technical application. Dry friction frequently occurs in real technical systems. Its impact on the dynamics of the single-degree-of-freedom (d.o.f.) system is shown in references [1, 3, 4, 12, 13] and on systems with several d.o.f. in references [2, 5, 8}10, 14]. The 0022-460X/99/400923#18 $30.00/0  1999 Academic Press