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Research Papers

# The Effect of Axisymmetric Nonflatness on the Oscillation Frequencies of a Rotating Disk

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

Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, V6T 1Z4, Canadaraminkh@mech.ubc.ca

Stanley G. Hutton

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

J. Vib. Acoust 132(5), 051012 (Sep 01, 2010) (8 pages) doi:10.1115/1.4001516 History: Received November 07, 2009; Revised March 23, 2010; Published September 01, 2010; Online September 01, 2010

## Abstract

This study examines the frequency characteristics of thin rotating disks subjected to axisymmetric nonflatness. The equations of motion used are based on Von Karman’s plate theory. First, the eigenfunctions of the stationary disk problem corresponding to the stress function and transverse displacement are found. These eigenfunctions produce an equation that can be used in Galerkin’s method. The initial nonflatness is assumed to be a linear combination of the eigenfunctions of the transverse displacement of the stationary disk problem. Since the initial nonflatness is assumed to be axisymmetric, only eigenfunctions with no nodal diameters are considered to approximate the initial runout. It is supposed that the disk bending deflection is small compared with disk thickness, so we can ignore the second-order terms in the governing equations corresponding to the transverse displacement and the stress function. After simplifying and discretizing the governing equations of motion, we can obtain a set of coupled equations of motion, which takes the effect of the initial axisymmetric runout into account. These equations are then used to study the effect of the initial runout on the frequency behavior of the stationary disk. It is found that the initial runout increases the frequencies of the oscillations of a stationary disk. In the next step, we study the effect of the initial nonflatness on the critical speed behavior of a spinning disk.

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## Figures

Figure 9

Change in the critical speed of the (0,4) mode, assuming different shapes of runout (the legends show the shape of initial nonflatness)

Figure 4

Nondimensionalized oscillation frequencies of the stationary disk when the nonflatness is assumed to be in the shape of the (1,0) mode

Figure 5

Nondimensionalized oscillation frequencies of the stationary disk when the nonflatness is assumed to be in the shape of the (2,0) mode

Figure 6

Nondimensionalized oscillation frequencies of the stationary disk when the nonflatness is assumed to be in the shape of the (3,0) mode

Figure 7

Change in the critical speed of the (0,2) mode, assuming different shapes of runout (the legends show the shape of initial nonflatness)

Figure 8

Change in the critical speed of the (0,3) mode, assuming different shapes of runout (the legends show the shape of initial nonflatness)

Figure 1

Oscillation frequencies versus rotation speed when the runout is zero (solid lines) and when W00R=100 (broken lines)

Figure 2

Oscillation frequencies versus rotation speed when the runout is zero (solid lines) and when W10R=100 (broken lines)

Figure 3

Nondimensionalized oscillation frequencies of the stationary disk when the nonflatness is assumed to be in the shape of the (0,0) mode

## Errata

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