Control of Impact Microactuators for Precise Positioning

[+] Author and Article Information
Xiaopeng Zhao

Department of Engineering Science and Mechanics, MC 0219  Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061xzhao@vt.edu

Harry Dankowicz

Department of Engineering Science and Mechanics, MC 0219  Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061

Note that an impacting trajectory is said to be periodic if it is periodic in all the state variables except q1, which changes by a discrete amount during every sliding episode. Indeed, since the vector field is independent of q1, its time history has no affect on the system behavior.

J. Comput. Nonlinear Dynam 1(1), 65-70 (May 09, 2005) (6 pages) doi:10.1115/1.1951781 History: Received February 10, 2005; Revised May 09, 2005

Electrically driven impact microactuators generate nanoscale displacements without large driving distances and high voltages. These systems exhibit complex dynamics because of inherent nonlinearities due to impacts, friction, and electric forces. As a result, dramatic changes in system behavior, associated with so-called grazing bifurcations, may occur during the transition between impacting and nonimpacting dynamics, including the presence of robust chaos. For successful open-loop operating conditions, the system design is limited to certain parameter regions, where desired system responses reside. The objective of this paper is to overcome this limitation to allow for a more precise displacement manipulation using impact microactuators. This is achieved through a closed-loop feedback scheme that successfully controls the system dynamics in the near-grazing region.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

The schematic of the impact microactuator. Figure reproduced from 5 with permission from the publisher.

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Figure 2

Schematic diagram illustrating the intersections of trajectory segments with the Poincaré surface P in the presence (upper curve) and in the absence (lower curve) of impact, respectively. Here, the dashed line represents a virtual trajectory corresponding to the vector field fstick in the absence of impacts. At the intersections, q2 possesses a local maximum value q2max. If we define the penetration as q2max−δ, then an impacting trajectory has a positive penetration while the nonimpacting trajectory has a negative penetration.

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Figure 3

Schematic bifurcation scenarios associated with the switching between impacting motions and nonimpacting motions (G=grazing contact, SN=saddle-node bifurcation, PD=period-doubling bifurcation). Here, solid curves correspond to stable periodic motions and dashed curves to unstable periodic motions. The black regions correspond to impacting chaotic attractors.

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Figure 4

Periodic impacting motion for parameter values near grazing in the absence of control [Panel (a)] and subset of values of c3 and c5, such that ξn>0 for n⩽90 when c7=0.89 [Panel (b)]. Enlargement of the boxed area in (a) shows a short sliding phase immediately after impact [cf. impacting periodic orbits on upper branch in Fig. 3(a)]. The vertex of the shaded wedge in (b) lies at c3=−0.0523838 and c5=0.0208373.

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Figure 5

Steady-state response in the presence of control for V=V*+4×10−4 and ω=0.5: (a) Poincaré map sampled at the intersections and (b) movements of the front stopper walls relative to the zero-voltage position. We note that the Poincaré map of the periodic orbit shown in Fig. 4 is a single point with positive penetration.

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Figure 6

Bifurcation diagrams of near-grazing dynamics in the presence of control [cf. Fig. 3(a) for the uncontrolled case]: (a) prediction from the composite Poincaré mapping and (b) direct numerical simulations (over a larger interval in V−V*)




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