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Research Papers

# Forced Oscillations of a Continuous Asymmetrical Rotor With Geometric Nonlinearity (Major Critical Speed and Secondary Critical Speed)

[+] Author and Article Information
Imao Nagasaka

Department of Mechanical Engineering, Chubu University, Aichi 487-8501, Japannagasaka@isc.chubu.ac.jp

Yukio Ishida

Department of Mechanical Science and Engineering, School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japanishida@nuem.nagoya-u.ac.jp

Jun Liu

Department of Mechanical Science and Engineering, School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japanliu@nuem.nagoya-u.ac.jp

J. Vib. Acoust 130(3), 031012 (May 14, 2008) (7 pages) doi:10.1115/1.2890734 History: Received August 12, 2006; Revised October 26, 2006; Published May 14, 2008

## Abstract

Forced oscillations in the vicinities of both the major and the secondary critical speeds of a continuous asymmetrical rotor with the geometric nonlinearity are discussed. When the self-aligning double-row ball bearings support the slender flexible rotor at both ends, the geometric nonlinearity appears due to the stiffening effect in elongation of the shaft if the movements of the bearings in the longitudinal direction are restricted. The nonlinearity is symmetric when the rotor is supported vertically, and is asymmetric when it is supported horizontally. Because the rotor is slender, the natural frequency $pfn$ of a forward whirling mode and $pbn$ of a backward whirling mode have the relation of internal resonance $pfn:pbn=1:(−1)$. Due to the influence of the internal resonance, various phenomena occur, such as Hopf bifurcation, an almost periodic motion, the appearance of new branches, and the diminish of unstable region. These phenomena were clarified theoretically and experimentally. Moreover, this paper focuses on the influences of the nonlinearity, the unbalance, the damping, and the lateral force on the vibration characteristics.

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## Figures

Figure 2

Natural frequency diagram

Figure 7

Influence of dynamic unbalance of rotor

Figure 8

Resonance curve (secondary critical speed): (a) forward whirling motion and (b) backward whirling motion

Figure 9

Experimental setup: (a) photograph of experimental setup and (b) fundamental construction

Figure 1

Rotor model

Figure 3

Resonance curve of a linear system

Figure 4

Resonance curve of a vertical shaft

Figure 5

Resonance curve (major critical speed): (a) forward whirling motion and (b) backward whirling motion

Figure 6

Influence of a lateral force: (a) forward whirling motion and (b) backward whirling motion

Figure 10

Experimental result (major critial speed): (a) resonance curves of forward component, (b) resonance curves of backward component, (c) Point a⋅⋅(ω=772rpm), (d) Point b⋅⋅(ω=916rpm), (e) Point c⋅⋅(ω=842rpm)

Figure 11

Experimental result (secondary critial speed): (a) resonance curves of forward component, (b) resonance curves of backward component, point a⋅⋅(ω=362rpm), (d) Point b⋅⋅(ω=401rpm), and (e) Point c⋅⋅(ω=386rpm)

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