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Research Papers

Frequency Response Characteristics of Parametrically Excited System

[+] Author and Article Information
Qinkai Han1

School of Jet Propulsion, Bei Hang University, Beijing 100191, Chinahanqinkai@hotmail.com

Jianjun Wang

School of Jet Propulsion, Bei Hang University, Beijing 100191, Chinawangjianjun@buaa.edu.cn

Qihan Li

School of Jet Propulsion, Bei Hang University, Beijing 100191, China

1

Corresponding author.

J. Vib. Acoust 132(4), 041004 (May 20, 2010) (11 pages) doi:10.1115/1.4000772 History: Received December 05, 2008; Revised July 31, 2009; Published May 20, 2010; Online May 20, 2010

The frequency response characteristic of a general time-invariant system has been extensively analyzed in literature. However, it has not gained sufficient attentions in the parametrically excited system. In fact, due to the parametric excitation, the frequency response of time-periodic system differs distinctly from that of the time-invariant system. Utilizing Sylvester’s theorem and Fourier series expansion method, commonly used in the spectral decomposition for matrix, the frequency response functions (FRFs) of a single-degree-of-freedom (SDOF) parametrically excited system are derived briefly in the paper. The external resonant condition for the system is obtained by analyzing the specific expressions of FRFs. Then, a spur-gear-pair with periodically time-varying mesh stiffness is selected as an example to simulate the frequency response characteristics of parametric system. The effects of parametric stability, periodic mesh stiffness parameters (mesh frequency and contact ratio), and damping are considered in the simulation. It is shown from both theoretical and simulation results that the frequency response of parametric system has the following properties: there are multiple FRFs even for a SDOF periodic system as the forced response contains many frequency components and each FRF is corresponding to a certain response spectrum; the system has multiple external resonances. Besides the resonance caused by the external driving frequency equals to the natural frequency, the system will also be external resonant if external frequency meets the combination of natural frequency and parametric frequency. When the system is in external resonant state, the dominant frequency component in the response is the natural frequency; damping makes the peak values of FRFs drop evidently while it has almost no impact on the FRFs in nonresonant regions.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Dynamic model of a spur-gear-pair

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Figure 2

The stable and unstable regions for the system under three damping states (dark areas are the unstable regions and the others are the stable regions): (a) ζ=0, (b) ζ=0.02, and (c) ζ=0.05

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Figure 3

The convergence for the FRFs of parametric system: (a) |H2d| and (b) |H3d|

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Figure 4

The time-domain forced responses of geared system with external frequency, respectively, equals to the peak frequencies of the FRF |H2d|: (a) ω=0.1864; (b) ω=0.6136; (c) ω=0.9864; (d) ω=1.4136; (e) ω=1.7864; (f) ω=2.2136; (g) ω=2.5864; (h) ω=3.0136; (i) ω=3.3864; (j) ω=3.8136; (k) ω=4.1864; and (l) ω=4.6136

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Figure 5

Seven types of FRFs in frequency band 0–5

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Figure 6

The zoom-in plots of six types of FRFs near four resonant frequencies: (a) ω=0.1864, (b) ω=0.6136, (c) ω=0.9864, and (d) ω=1.4136

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Figure 7

Waterfall plot of the FRF |H−3d| with contact ratio ε=1.5: m=−7, ▽; m=−6, *; m=−5, ◇; m=−4, *; m=−3, +; m=−2, ×, m=−1, ◻; m=0, ○; m=1, △; m=2, ◁; m=3, ▷

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Figure 8

Waterfall plot of FRF |H1d| with mesh frequency ω¯=1: m=−6, *; m=−5, ◇; m=−4, *; m=−3, +; m=−2, ×; m=−1, ◻; m=0, ○; m=1, ▽; m=2, △; m=3, ◁; m=4, ▷

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Figure 9

The effects of damping on the FRF |H1d| in two frequency bands: (a) external excitation frequency in 0.5–0.75 and (b) external excitation frequency in 0.9–1.1

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