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Research Papers

Ground-Based Vibration Response of a Spinning, Cyclic, Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects

[+] Author and Article Information
Hyunchul Kim, I. Y. Shen

Department of Mechanical Engineering, University of Washington, Seattle, WA 98195-2600

J. Vib. Acoust 131(2), 021007 (Feb 18, 2009) (13 pages) doi:10.1115/1.3025847 History: Received October 22, 2007; Revised June 29, 2008; Published February 18, 2009

This paper is to study ground-based vibration response of a spinning, cyclic, symmetric rotor through a theoretical analysis and an experimental study. The theoretical analysis consists of three steps. The first step is to analyze the vibration characteristics of a stationary, cyclic, symmetric rotor with N identical substructures. For each vibration mode, we identify a phase index n and derive a Fourier expansion of the mode shape in terms of the phase index n. The second step is to predict the rotor-based vibration response of the spinning, cyclic, symmetric rotor based on the Fourier expansion of the mode shapes and the phase indices. The rotor-based formulation includes gyroscopic and centrifugal softening terms. Moreover, rotor-based response of repeated modes and distinct modes is obtained analytically. The third step is to transform the rotor-based response to ground-based response using the Fourier expansion of the stationary mode shapes. The theoretical analysis leads to the following conclusions. First, gyroscopic effects have no significant effects on distinct modes. Second, the presence of gyroscopic and centrifugal softening effects causes the repeated modes to split into two modes with distinct frequencies ω1 and ω2 in the rotor-based coordinates. Third, the transformation to ground-based observers leads to primary and secondary frequency components. In general, the ground-based response presents frequency branches in the Campbell diagram at ω1±kω3 and ω2±kω3, where k is phase index n plus an integer multiple of cyclic symmetry N. When the gyroscopic effect is significantly greater than the centrifugal softening effect, two of the four frequency branches vanish. The remaining frequency branches take the form of either ω1+kω3 and ω2kω3 or ω1kω3 and ω2+kω3. To verify these predictions, we also conduct a modal testing on a spinning disk carrying four pairs of brackets evenly spaced in the circumferential direction with ground-based excitations and responses. The disk-bracket system is mounted on a high-speed, air-bearing spindle. An automatic hammer excites the spinning disk-bracket system and a laser Doppler vibrometer measures its vibration response. A spectrum analyzer processes the hammer excitation force and the vibrometer measurements to obtain waterfall plots at various spin speeds. The measured primary and secondary frequency branches from the waterfall plots agree well with those predicted analytically.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

First eight mode shapes of the eight-bracket disk

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Figure 2

Waterfall plot simulating primary and secondary frequencies of (0,4)L mode from 0 rpm to 4800 rpm

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Figure 3

Waterfall plot simulating primary and secondary frequencies of (0,4)H mode from 0 rpm to 4800 rpm

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Figure 4

Waterfall plot simulating primary and secondary frequencies of 10th and 11th modes from 0 rpm to 4800 rpm, λ12=1.14×10−5

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Figure 5

Waterfall plot simulating primary and secondary frequencies of 14th and 15th modes from 0 rpm to 4800 rpm

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Figure 6

Waterfall plot simulating primary and secondary frequencies of 10th and 11th modes from 0 rpm to 4800 rpm, λ12=57

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Figure 7

Experimental setup of eight-bracket disk on air-bearing spindle

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Figure 8

Measured waterfall plot simulating primary and secondary frequencies of 10th and 11th modes from 0 rpm to 4800 rpm

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Figure 9

Measured Campbell diagram for all vibration modes

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