Applied Mathematical Modelling 31 (2007) 54–77 www.elsevier.com/locate/apm
Nonlinearly coupled in-plane and transverse 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, Ont., Canada M5S 3G8 Received 1 January 2003; received in revised form 1 March 2005; accepted 2 August 2005 Available online 21 September 2005
Abstract Previous nonlinear spinning disk models neglected the in-plane inertia of the disk since this permits the use of a stress function. This paper aims to consider the effect of including the in-plane inertia of the disk on the resulting nonlinear dynamics and to construct approximate solutions that capture the new dynamics. The inclusion of the in-plane inertia results in a nonlinear coupling between the in-plane and transverse vibrations of the spinning disk. The full nonlinear partial differential equations are simplified to a simpler nonlinear two degrees of freedom model via the method of Galerkin. A canonical perturbation approach is used to derive an approximate solution to this simpler nonlinear problem. Numerical simulations are used to evaluate the effectiveness of the approximate solution. Through the use of these analytical and numerical tools, it becomes apparent that the inclusion of in-plane inertia gives rise to new phenomena such as internal resonance and the possibility of instability in the system that are not predicted if the in-plane inertia is ignored. It is also demonstrated that the canonical perturbation approach can be used to produce an effective approximate solution. Ó 2005 Elsevier Inc. All rights reserved. Keyword: 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. Many authors have investigated the vibrations of spinning disks using linear theory. The original papers are by Lamb and Southwell [1] and by Southwell [2] where the disk was modelled as a spinning membrane with added bending stiffness. Another popular approach in the literature is to model the spinning disks as a pure membrane with no bending stiffness [3–6]. The incorporation of both the bending stiffness of the disk and the effect of rotation leads to a fourth order PDE that is difficult to solve. As a result, various researchers have *
Corresponding author. Tel./fax: +1 709 753 3628. E-mail address: baddour@mie.utoronto.ca (N. Baddour).
0307-904X/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2005.08.004