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Research Papers

Vibration Characteristics of Rotating Thin Disks—Part II: Analytical Predictions

[+] Author and Article Information
Ramin M. H. Khorasany

Stanley G. Hutton

Department of Mechanical Engineering,  The University of British Columbia, Vancouver, BC, V6T 1Z4, Canada

J. Appl. Mech 79(4), 041007 (May 16, 2012) (10 pages) doi:10.1115/1.4005810 History: Received April 22, 2010; Revised July 23, 2011; Posted February 01, 2012; Published May 16, 2012

This paper is concerned with the geometric nonlinear analysis of the lateral displacement of thin rotating disks when subjected to a space fixed stationary force. Of particular interest is the development of the stationary wave and the effect of this wave on the frequency response of the disk as a function of its rotational speed. The predictions of this analysis are compared with experimental data obtained in a companion paper (Khorasany and Hutton, “Vibration Characteristics of Rotating Thin Disks—Part I: Experimental Results,” ASME J. Appl. Mech., 79(4), p. 041006). The governing equations are based on Von Kármán plate theory. A Galerkin solution of the governing non linear equations is developed. The eigenfunctions derived from the linear analysis of a stationary disk are used as approximations to the spatial response of the disk, and the eigenfunctions of the biharmonic equation as approximations for the stress function. Using the developed solution, the equilibrium configuration of the disk under the application of a space fixed force is found. In order to facilitate the prediction of the frequency response, as a function of disk rotational speed, the governing nonlinear equations are linearized around the equilibrium solution. The linearized equations are then used to find the eigenvalues of the spinning disk under the application of a space fixed force. The effect of different levels of nonlinearity on the disk frequencies is studied and compared with experimental results. The analysis is shown to produce an accurate representation of the measured response. Of particular interest is the disk response at speeds close to and above the linear critical speed. In this region, both the analysis and the experimental results display frequency “lock-in” behavior in which the frequency of backward travelling waves becomes constant for supercritical speeds. No speed exists for which backward travelling waves have zero frequency. Thus, critical speeds do not exist in the presence of geometric nonlinearities.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 2

Nonlinear frequencies of the backward (solid lines) and forward (broken lines) traveling waves of (a) (0,2), (b) (0,3), and (c) (0,4) modes of the stationary disk

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Figure 3

(a) The particular part of the stress function, (b) the nonlinear radial stress, and (c) the nonlinear hoop stress for three different angular directions when w0/w0hh=4

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Figure 4

Nonlinear frequencies of the spinning 1.27 mm-thick disk when w0/w0hh=0.1 (a) numerical results (broken and solid lines show the linear and nonlinear results, respectively) and (b) experimental results (run-up case)

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Figure 5

A comparison between the numerical and experimental results (run-up case) for the dc amplitude of oscillations when w0/w0hh=0.1 at the location of the applied external force

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Figure 1

Linear natural frequencies of the disk versus rotation speed

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Figure 7

A comparison between the numerical and experimental results (run-up case) for the dc amplitude of oscillations when w0/w0hh=0.6

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Figure 8

Analysis of the effect of magnitude of nonlinearity on development of the stationary wave, w0/w0hh=0.4,  0.6, and  2.0

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Figure 9

Calculated linear (broken lines) and nonlinear frequencies (solid lines) of the spinning disk when w0/w0hh=2.0

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Figure 6

Nonlinear frequencies of the spinning disk when w0/w0hh=0.6 (a) numerical results (b) experimental results (run-up case)

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