Nonlinear Analysis: Real World Applications 7 (2006) 319 – 340 www.elsevier.com/locate/na
Time-averaged canonical perturbation of nonlinear vibrations of a spinning disk Natalie Baddour∗ , Jean W. Zu Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada, M5S 3G8 Received 17 January 2003; accepted 24 March 2005
Abstract This paper considers the dynamics of a spinning disk with the inclusion of in-plane inertia. The inclusion of the in-plane inertia couples the in-plane and transverse vibrations of the spinning disk. By considering one mode of each vibration, the problem can be reduced to a non-linear 2 DOF model. An analytical solution of this 2 DOF model is then developed using a time-averaged canonical perturbation approach and is compared with that of a numerical procedure. Through the use of these analytical and numerical tools, it becomes apparent that the inclusion of the in-plane inertia leads to the possibility of internal resonance between the oscillators, depending on the relationship between the natural frequencies of the two oscillators. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Nonlinear spinning disk coupled vibrations
1. Introduction Spinning disks can be found in many engineering applications. Common industrial applications include circular sawblades, turbine rotors, brake systems, fans, flywheels, gears, grinding wheels, precision gyroscopes and computer storage devices. Spinning disks may experience severe vibrations which could lead to fatigue failure of the system. Thus, the dynamics of spinning disks has attracted much research interest over the years. ∗ Corresponding author. Tel./fax: +1 416 978 1287.
E-mail address: baddour@mie.utoronto.ca (N. Baddour). 1468-1218/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2005.03.004