A Revisit of Spinning Disk Models, Part I: Derivation of Equations of Motion

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Applied Mathematical Modelling 25 (2001) 541±559

www.elsevier.nl/locate/apm

A revisit of spinning disk models. Part I: derivation of equations of motion N. Baddour, J.W. Zu

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Department of Mechanical Engineering and Industrial Engineering, University of Toronto, 5 King's College Road, Toronto, Ontario, Canada M5S 3G8 Received 1 November 1999; received in revised form 30 August 2000; accepted 5 October 2000

Abstract Previous models of spinning disks have focused on modelling the disk as a spinning membrane. The e ect of bending sti ness was then incorporated by adding the appropriate term to the previously derived spinning membrane equation. A pure spinning plate model does not exist in the literature. Furthermore, in both existing linear and nonlinear models of spinning disks, the in-plane inertia and rotary inertia of the disk have been ignored. This paper revisits the derivation of the equations of motion of a spinning plate. The derivation focuses on the use of Hamilton's principle with linear Kirchho and nonlinear von Karman strain expressions. In-plane and rotary inertias of the plate are automatically taken into account. The use of Hamilton's principle guarantees the correct derivation of the corresponding boundary conditions. The resulting equations and boundary conditions are discussed. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Spinning disk; Hamiltonian; Modelling; Derivation of equations

1. Introduction Spinning disks can be found in many engineering applications. Common industrial applications include circular sawblades, turbine rotors, brake systems, fans, ¯ywheels, 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. The ®rst step in investigating the vibrations of spinning disks is to setup a suitable mathematical model that captures the essential physics of the problem while remaining tractable. Assumptions about the system under investigation must be made at the stage of developing a model for the system. It is obvious that di erent assumptions will lead to di erent equations of motion and thus di erent solutions and techniques are required to obtain these solutions. In this paper, existing models used to analyze spinning disks will be thoroughly reviewed. Subsequently, new linear and nonlinear equations of motion with corresponding boundary conditions will be derived via Hamilton's principle. *

Corresponding author. Tel.: +1-416-978-0961; fax: +1-416-978-7753. E-mail addresses: baddour@mie.utoronto.ca (N. Baddour), zu@mie.utoronto.ca (J.W. Zu).

0307-904X/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 0 ) 0 0 0 6 5 - 2


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