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Research Papers

Forced Vibrations of a Very Slender Continuous Rotor With Geometrical Nonlinearity (Harmonic and Subharmonic Resonances)

[+] Author and Article Information
Imao Nagasaka

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

Yukio Ishida

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

J. Vib. Acoust 132(2), 021004 (Mar 16, 2010) (9 pages) doi:10.1115/1.4000841 History: Received October 31, 2006; Revised July 05, 2007; Published March 16, 2010; Online March 16, 2010

Abstract

When both ends of an elastic continuous rotor are supported simply by double-row self-aligning ball bearings, the geometrical nonlinearity appears due to the stiffening effect in the elongation of the rotor if the movement of the bearings in the longitudinal direction is restricted. As the rotor becomes more slender, the geometrical nonlinearity becomes stronger. In this paper, we study on unique nonlinear phenomena caused by both of the nonlinear spring characteristics and an initial axial force in the vicinity of the major critical speed $ωc$ and twice $ωc$ in a very slender continuous rotor. When the rotor is supported horizontally, the difference in support stiffness and the asymmetrical nonlinearity appear as a result of shifting from the equilibrium position. By the influences of the internal resonance and the initial axial force, the nonlinear resonance phenomena become very complicated. For example, the peak resonance splits into two peaks, and these two peaks leave each other and then one becomes a hard spring type while the other becomes a soft spring type, respectively. Moreover, almost periodic motions and chaotic vibrations appear. In this paper, we prove the above phenomena theoretically and experimentally.

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Figures

Figure 1

Theoretical model

Figure 2

Natural frequency diagrams

Figure 3

Forward and backward harmonic components of resonance (case without an initial axial force)

Figure 4

Effects of the initial axial forces (forward component)

Figure 5

Forward and backward components of a subharmonic resonance of order 1/2 (case with an initial axial force)

Figure 6

Experimental setup

Figure 7

Experimental results (case with a tension)

Figure 8

Theoretical results (case with a tension)

Figure 9

Experimental results (case with a compression)

Figure 10

Theoretical results (case with a compression)

Figure 11

Potential energy

Figure 12

Poincaré map (T0=−5, case with a compression) (ac)

Figure 13

Largest Lyapunov exponent (case with a compression)

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