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Research Papers

Discontinuity-Induced Bifurcations in Systems With Hysteretic Force Interactions

[+] Author and Article Information
Harry Dankowicz1

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801danko@illinois.edu

Mark R. Paul

Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061mrp@vt.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 4(4), 041009 (Aug 25, 2009) (6 pages) doi:10.1115/1.3192131 History: Received July 23, 2008; Revised March 31, 2009; Published August 25, 2009

This paper presents the application of the discontinuity-mapping technique to the analysis of discontinuity-induced bifurcations of periodic trajectories in an example hybrid dynamical system in which changes in the vector field associated with the crossing of a discontinuity-surface depend on the direction of crossing. The analysis is motivated by a hysteretic model of the capillary force interactions between an atomic-force-microscope cantilever probe tip and a nanoscale sample surface in the presence of a thin liquid film on the tip and the surface and operating in intermittent-contact mode. The analysis predicts the sudden termination of branches of periodic system responses at parameter values corresponding to grazing contact with the onset of the hysteretic force interactions. It further establishes the increase beyond all bounds of the magnitude of one of the eigenvalues of the linearization of a suitably defined Poincaré mapping, indicating the destabilizing influence of near-grazing contact.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

A schematic of an example oscillator. Two distinct regimes of interaction between the mass and its environment are represented by the additional spring with stiffness K and zero load when q=d0. Here, don denotes the displacement of the mass at which the spring is initially engaged and doff denotes the displacement of the mass at which the spring is disengaged.

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Figure 2

A sample periodic trajectory of the example oscillator with Fd≈6.93. Here, dots refer to zero-crossings of the event functions corresponding to the onset and cessation of interactions with the additional linear spring and the Poincaré sampling trigger function. The dashed line refers to the instantaneous change in x3 from 2π to 0 that results from its definition.

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Figure 3

Periodic solutions that achieve grazing contact with hon=0 in state space. Here, the dashed trajectory is governed entirely by the foff vector field, whereas the solid trajectory includes a segment governed by fon.

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Figure 4

The solid trajectory in Fig. 3 is a solution to the original hybrid dynamical system with hysteresis shown in the upper panel as well as the alternative piecewise smooth dynamical system without hysteresis shown in the lower panel

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Figure 5

The discontinuity-mapping D:x0↦x2 accounts for the correction to the local flow. Here, solid curves correspond to the Φoff flow and dashed curves correspond to the Φon flow.

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Figure 6

Branch of periodic trajectories emanating from the solid trajectory in Fig. 3 characterized by the corresponding values of q−don and θ at the intersection with hPoincaré=0 (when ignoring the mode transition associated with a crossing with hon=0) as well as the largest-in-magnitude eigenvalue. Here, the dashed curves are the predictions from the discontinuity-mapping analysis, whereas the solid curves are obtained from the numerical continuation.

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