Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model Joseph Páez Chávez a,b,⇑ , Marian Wiercigroch a a Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK b Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, P.O. Box 09-01-5863, Guayaquil, Ecuador article info Article history: Received 15 August 2012 Accepted 4 December 2012 Available online 2 January 2013 Keywords: Jeffcott rotor Non-smooth dynamical system TC-HAT Numerical continuation Grazing Hysteresis abstract We investigate complex dynamics occurring in a non-smooth model of a Jeffcott rotor with a bearing clearance. A bifurcation analysis of the rotor system is carried out by means of the software TC-HAT [25], a toolbox of AUTO 97 [6] allowing path-following and detection of bifurcations of periodic trajectories of non-smooth dynamical systems. The study reveals a rich variety of dynamics, which includes grazing-induced fold and period-doubling bifur- cations, as well as hysteresis loops produced by a cusp singularity. Furthermore, an analyt- ical expression predicting grazing incidences is derived. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Rotating machines are very popular in engineering applications, ranging from optical disk drives and washing machines to turbojet engines and power generators. The study of their dynamics has attracted much attention, and special emphasis has been put on the development and refinement of mathematical models for understanding different phenomena observed in rotor systems. In general, the performance of rotating machinery can be enhanced by increasing the speed of rotation and decreasing the radial clearance between the rotating and non-rotating parts. This, however, significantly increases the risk of intermittent contacts between the components of rotor systems due to forced vibrations, resulting not only in possible costly and catastrophic mechanical failures (e.g. in aircraft jet engines) but also in a threat to the health of workers (e.g. hand-arm vibration syndrome [12]). A common cause of vibration in rotating machinery is mass imbalance. This occurs when the principal axis of the moment of inertia of the rotating component does not coincide with the axis of rotation. Such eccentric rotors undergo periodic oscillation known as whirl. In practice, a rotor cannot be balanced perfectly, no matter what method is used, and the best achievable state of balance at the beginning of the operating life of a rotor tends to deteriorate with use. This fact has motivated many studies on rotor systems subjected to out-of-balance phenomena, see e.g. [10,11,15,17,21]. Most of the mathematical models used in these investigations are based on the Jeffcott rotor [13], which consists of a large unbalanced disk mounted midway between the bearing supports on a flexible shaft of negligible mass. Although the Jeffcott rotor model is an oversimplification of real rotors, it has proven to be very useful for understanding many important phenomena observed in real applications [7,9,26]. 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.12.007 ⇑ Corresponding author at: Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK. E-mail addresses: jpaez@espol.edu.ec (J. Páez Chávez), m.wiercigroch@abdn.ac.uk (M. Wiercigroch). Commun Nonlinear Sci Numer Simulat 18 (2013) 2571–2580 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns