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Research Papers

Drifting Impact Oscillator With a New Model of the Progression Phase

[+] Author and Article Information
Olusegun K. Ajibose, Ekaterina Pavlovskaia, Alfred R. Akisanya

Center for Applied Dynamic Research,  School of Engineering, University of Aberdeen, Aberdeen, AB24 UE, Scotland, UK

Marian Wiercigroch1

Center for Applied Dynamic Research,  School of Engineering, University of Aberdeen, Aberdeen, AB24 UE, Scotland, UKm.wiercigroch@abdn.ac.uk

Györygy Károlyi

 Department of Structural Mechanics, Budapest University of Technology and Economics, Müegyetem rkp. 3, 1111, Budapest, Hungary

1

Corresponding author.

J. Appl. Mech 79(6), 061007 (Sep 17, 2012) (9 pages) doi:10.1115/1.4006379 History: Received January 25, 2011; Revised March 09, 2012; Posted March 15, 2012; Published September 17, 2012; Online September 17, 2012

In this paper, a new model of the progression phase of a drifting oscillator is proposed. This is to account more accurately for the penetration of an impactor through elasto-plastic solids under a combination of a static and a harmonic excitation. First, the dynamic response of the semi-infinite elasto-plastic medium subjected to repeated impacts by a rigid impactor with conical or spherical contacting surfaces is considered in order to formulate the relevant force-penetration relationship during the loading and unloading phases of the contact. These relationships are then used to develop a physical and mathematical model of a new drifting oscillator, where the time histories of the progression through the medium include both the loading and unloading phases. A nonlinear dynamic analysis of the system was performed and it confirms that the maximum progressive motion of the oscillator occurs when the system exhibits period one motion. The dynamic response for both contact geometries (conical or spherical) show a topological similarity for a range of the static loads.

Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Models of a drifting oscillators: (a) ideal with no elastic properties [2], (b) linear with elastic-viscous properties [1]

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Figure 2

Penetration of a medium by (a) conical and (b) spherical impactors (pile-up exaggerated). Adapted from Ref. [16].

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Figure 3

(a) Typical force-penetration relationships during both phases of the impact. (b) Graphical representation of Pr (X1 , Xp , Xf ).

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Figure 4

Physical model of an impactor colliding with an elasto-plastic medium

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Figure 5

Time histories of the conical impactor for various values b of the static load. Displacement x of the mass is shown by solid lines, displacement z of the slider by dashed lines. (b) is a blow-up of a small part of (a).

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Figure 6

Bifurcation diagram of the relative velocity yz′ as a function of the static load b in case of the conical impactor

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Figure 7

Progression z during 150 cycles of forcing as a function of the static force b for the conical impactor

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Figure 8

(a) Trajectory of the conical impactor in the (x − z, yz′) phase plane, and the Poincaré map. (b) Velocity y of the impactor as a function of time τ for b = 0.1.

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Figure 14

(a) Trajectory of the spherical impactor in the (x − z, yz′) phase plane, and the Poincaré map. (b) Velocity y of the impactor as function of time τ for b = 0.1.

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Figure 15

(a) Trajectory of the spherical impactor in the (x − z, yz′) phase plane, and the Poincaré map. (b) Velocity y of the impactor as function of time τ for b = 0.2.

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Figure 16

(a) Chaotic trajectory of the spherical impactor for b = 0.05. (b) Trajectory of the spherical impactor in the (x − z, yz′) phase plane for b = 0.35.

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Figure 9

(a) Trajectory of the conical impactor in the (x − z, yz′) phase plane, and the Poincaré map. (b) Velocity y of the impactor as a function of time τ for b = 0.2.

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Figure 10

(a) Chaotic attractor of the conical impactor for b = 0.05. (b) Trajectory of the conical impactor in the (x − z, yz′) phase plane for b = 0.35.

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Figure 11

Time histories of the spherical impactor for various values b of the static load. Displacement x of the mass is shown by solid lines, displacement z of the slider by dashed lines. (b) is a blow-up of a small part of (a).

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Figure 12

Bifurcation diagram of the relative velocity yz′ as a function of the static load b in case of the spherical impactor

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Figure 13

Progression z during 150 cycles of forcing as a function of the static force b for the spherical impactor

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