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Journal of Computational and Nonlinear Dynamics | Volume 7 | Issue 1 | Research Paper
Research Papers

Control of a Forced Impacting Hertzian Contact Oscillator Near Sub- and Superharmonic Resonances of Order 2

[+] Author and Article Information
Amine Bichri, Mohamed Belhaq

 University Hassan II-Casablanca, Laboratory of Mechanics, Casablanca, Morocco

J. Comput. Nonlinear Dynam 7(1), 011003 (Jul 22, 2011) (7 pages) doi:10.1115/1.4004309 History: Received March 13, 2011; Accepted May 23, 2011; Published July 22, 2011; Online July 22, 2011
Article
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The effect of a fast harmonic base displacement and of a fast periodically time varying stiffness on vibroimpact dynamics of a forced single-sided Hertzian contact oscillator is investigated analytically and numerically near sub- and superharmonic resonances of order 2. The study is carried out using averaging procedure over the fast dynamic and applying a perturbation analysis on the slow dynamic. The results show that a fast harmonic base displacement shifts the location of jumps, triggering the vibroimpact response, toward lower frequencies, while a fast periodically time-varying stiffness shifts the jumps toward higher frequencies. This result has been confirmed numerically for both sub- and superharmonic resonances of order 2. It is also demonstrated that the shift toward higher frequencies produced by a fast harmonic parametric stiffness is larger than that induced by a fast base displacement.
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Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

Grahic Jump Location
+Schematic model in the case of a rapid harmonic base motion
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Schematic model in the case of a rapid harmonic base motion
Figure 1

Schematic model in the case of a rapid harmonic base motion

Grahic Jump Location
+Amplitude-frequency response near the two-subharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.01, σ=0.5, and Ω−=8.
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Amplitude-frequency response near the two-subharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.01, σ=0.5, and Ω−=8.
Figure 2

Amplitude-frequency response near the two-subharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.01, σ=0.5, and Ω−=8.

Grahic Jump Location
+Amplitude-frequency response near the two-superharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.001, σ=0.1, and Ω−=8.
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Amplitude-frequency response near the two-superharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.001, σ=0.1, and Ω−=8.
Figure 3

Amplitude-frequency response near the two-superharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.001, σ=0.1, and Ω−=8.

Grahic Jump Location
+Schematic model in the case of a parametric stiffness
View Large  |  Save Figure  |  Download Slide (.ppt)
Schematic model in the case of a parametric stiffness
Figure 4

Schematic model in the case of a parametric stiffness

Grahic Jump Location
+Amplitude-frequency response near the two-subharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.01, σ=0.5, and Ω−=8.
View Large  |  Save Figure  |  Download Slide (.ppt)
Amplitude-frequency response near the two-subharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.01, σ=0.5, and Ω−=8.
Figure 5

Amplitude-frequency response near the two-subharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.01, σ=0.5, and Ω−=8.

Grahic Jump Location
+Amplitude-frequency response near the two-superharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.001, σ=0.1, and Ω−=8.
View Large  |  Save Figure  |  Download Slide (.ppt)
Amplitude-frequency response near the two-superharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.001, σ=0.1, and Ω−=8.
Figure 6

Amplitude-frequency response near the two-superharmonic resonance. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) for ξ=0.001, σ=0.1, and Ω−=8.

Grahic Jump Location
+Comparison of the frequency response near the two-subharmonic resonance for ξ=0.01, σ=0.5, Ω−=8 and for different values of a−
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of the frequency response near the two-subharmonic resonance for ξ=0.01, σ=0.5, Ω−=8 and for different values of a−
Figure 7

Comparison of the frequency response near the two-subharmonic resonance for ξ=0.01, σ=0.5, Ω−=8 and for different values of a−

Grahic Jump Location
+Comparison of the frequency response near the two-superharmonic resonance for ξ=0.001, σ=0.1, Ω−=8 and for different values of a−
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of the frequency response near the two-superharmonic resonance for ξ=0.001, σ=0.1, Ω−=8 and for different values of a−
Figure 8

Comparison of the frequency response near the two-superharmonic resonance for ξ=0.001, σ=0.1, Ω−=8 and for different values of a−

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