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Research Papers

Combination Parametric Resonance of Nonlinear Unbalanced Rotating Shafts

[+] Author and Article Information
M. S. Qaderi

Department of Mechanical Engineering,
Faculty of Engineering,
Kharazmi University,
Mofatteh Avenue,
Tehran 15719-14911, Iran
e-mail: m.saber.qaderi@gmail.com

S. A. A. Hosseini

Department of Mechanical Engineering,
Faculty of Engineering,
Kharazmi University,
Mofatteh Avenue,
Tehran 15719-14911, Iran
e-mail: ali.hosseini@khu.ac.ir

M. Zamanian

Department of Mechanical Engineering,
Faculty of Engineering,
Kharazmi University,
Mofatteh Avenue,
Tehran 15719-14911, Iran
e-mail: zamanian@khu.ac.ir

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 4, 2018; final manuscript received July 22, 2018; published online August 24, 2018. Assoc. Editor: Tsuyoshi Inoue.

J. Comput. Nonlinear Dynam 13(11), 111002 (Aug 24, 2018) (8 pages) Paper No: CND-17-1561; doi: 10.1115/1.4041029 History: Received May 04, 2018; Revised July 22, 2018

In this paper, dynamic response of a rotating shaft with geometrical nonlinearity under parametric and external excitations is investigated. Resonances, bifurcations, and stability of the response are analyzed. External excitation is due to shaft unbalance and parametric excitation is due to periodic axial force. For this purpose, combination resonances of parametric excitation and primary resonance of external force are assumed. Indeed, simultaneous effect of nonlinearity, parametric, and external excitations are investigated using analytical method. By applying the method of multiple scales, four ordinary nonlinear differential equations are obtained, which govern the slow evolution of amplitude and phase of forward and backward modes. Eigenvalues of Jacobian matrix are checked to find the stability of solutions. Both periodic and quasi-periodic motion were observed in the range of study. The influence of various parameters on the response of the system is studied. A main contribution is that the parametric excitation in the presence of nonlinearity can be used to suppress the forward synchronous vibration. Indeed, in the presence of combination parametric excitation, the energy is transferred from forward whirling mode to backward one. This property can be applied in control of rotor unbalance vibrations.

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References

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Figures

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Fig. 1

Schematic view of a rotting shaft under axial force

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Fig. 2

Frequency response curves of a1 and a2 for σ2=0, μe=0.01, and P=0.056

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Fig. 3

Frequency response curves of a1 and a2 for σ2=0.03, μe=0.01, and P=0.056

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Fig. 4

Frequency response curves of a1 and a2 for σ2=0.04, μe=0.01, P=0.056

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Fig. 5

Frequency response curves of a1 and a2 for σ2=0.06,μe=0.01, P=0.056

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Fig. 6

Time history and FFT for σ1=σ2=0, μe=0.01, and P=0.056

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Fig. 7

Time history and FFT for σ1=0.05,  σ2=0,μe=0.01, and P=0.056

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Fig. 8

Variation of amplitudes a1 and a2 with respect to P for σ2=0 and μe=0.01

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Fig. 9

Variation of amplitudes a1 and a2 with respect to eccentricity for σ2=0.04

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