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TECHNICAL PAPERS

# Axisymmetrical Snapping of a Spinning Nonflat Disk

[+] Author and Article Information
Jen-San Chen

Department of Mechanical Engineering,  National Taiwan University, Taipei, Taiwan 10617 R.O.C.jschen@ccms.ntu.edu.tw

Chi-Chung Lin

Department of Mechanical Engineering,  National Taiwan University, Taipei, Taiwan 10617 R.O.C.

J. Appl. Mech 72(6), 879-886 (Mar 05, 2005) (8 pages) doi:10.1115/1.2043188 History: Received June 03, 2004; Revised March 05, 2005

## Abstract

In this paper we study the steady-state deflection of a spinning nonflat disk, both theoretically and experimentally. Both the initial and the deformed shapes of the disk are assumed to be axisymmetrical. Von Karman’s plate model is adopted to formulate the equations of motion, and Galerkin’s method is employed to discretize the partial differential equations. In the case when the initial height of the nonflat disk is sufficiently large, multiple equilibrium positions can exist, among them the two stable one-mode solutions $P01$ and $P03$ are of particular interest. Theoretical investigation shows that if the disk is initially in the stressed position $P03$, it will be snapped to position $P01$ when the rotation speed reaches a critical value. Experiments on a series of copper disks with different initial heights are conducted to verify the theoretical predictions. Generally speaking, the experimental measurements agree well with theoretical predictions when the initial height is small. For the disks with large initial heights, on the other hand, the measured snapping speeds are significantly below the theoretical predictions. The circumferential waviness of the copper disks induced in the manufacturing process and the aerodynamic force at high rotation speed are two possible factors causing this discrepancy.

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Copyright © 2005 by American Society of Mechanical Engineers
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## Figures

Figure 1

(a) A spinning nonflat disk and (b) a disk element

Figure 2

Equilibrium positions as functions of rotation speed for H=3 and 5. The thick and thin lines represent the results from one- and two-term approximations, respectively. Solid and dashed lines represent stable and unstable positions, respectively

Figure 3

Equilibrium positions as functions of rotation speed for H=7 from a one-term approximation

Figure 4

Equilibrium positions as functions of rotation speed for H=10 from a two-term approximation

Figure 5

Strain energy contour for H=10 at Ω=0

Figure 6

Membrane stress resultant fields of position P03 at various speeds: (a) Nr and (b) Nθ

Figure 7

Maximum bending stress fields of position P03 at various speeds: (a) σrb and (b) σθb

Figure 8

Measurements of the radial variation of the nonflat disk prepared for experiment

Figure 9

Measurement of disk deflection as a function of rotation speed

Figure 10

Relation between snapping speed and initial height for eight disks

## Errata

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