Research Papers

Nonlinear Models for Magnet Placement in Individually Actuated Magnetic Cilia Devices

[+] Author and Article Information
Nathan Banka

Ultra Precision Control Laboratory,
Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195

Santosh Devasia

Ultra Precision Control Laboratory,
Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195
e-mail: devasia@uw.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 1, 2017; final manuscript received September 20, 2017; published online December 22, 2017. Assoc. Editor: Zongxuan Sun.

J. Dyn. Sys., Meas., Control 140(6), 061011 (Dec 22, 2017) (12 pages) Paper No: DS-17-1233; doi: 10.1115/1.4038534 History: Received May 01, 2017; Revised September 20, 2017

This paper presents a model for predicting the optimal magnet placement in magnetic cilia devices that achieve individual control via localization of the driving magnetic field. In this configuration, each cilium is controlled by a magnetic field source which is limited in spatial extent, and the cilia are spaced sufficiently far apart that the control remains uncoupled. An implementation is presented using an electromagnetic field source to attain large-deformation actuation (transverse deflections of 47% of the length). The large deformations are achieved by exploiting the nonlinear response of a flexible cantilever in a nonuniform magnetic field. However, the same nonlinearities also pose a modeling challenge: the overall performance is sensitive to the location of the electromagnet and the location that produces the largest deflections is nonlinearly dependent on the strength of the magnetic field. The nonlinear displacement of the cilium is predicted using a finite element model of the coupled magnetic–structural equations for static inputs at varying field strengths and magnet positions. The deflection at the model-predicted optimal placement is within 5% of the experiment-predicted optimal placement. Moreover, actuator placement using a model that does not include the nonlinearities is estimated to result in performance loss of about 50% peak deflection. This result emphasizes the importance of capturing nonlinearities in the system design.

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Grahic Jump Location
Fig. 1

Conceptual schematic showing the localization of the magnetic force leading to individual controllability. Because the magnetic field falls off as the inverse square of distance, it has much more influence on the near cilium than on the far cilium.

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Fig. 2

Schematic of the experimental system showing top and side views. The artificial cilium is a flexible magnetic cantilever that is deflected due to a field produced by the electromagnet located at Xm = (xm, ym, zm).

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Fig. 3

Control of the current i through the electromagnet coil is achieved using an analog feedback loop (push-pull amplifier) implemented using op-amps. Measuring the voltage Vfb = iRg across the gain resistor provides an estimate of the coil current. The voltage signal V is prefiltered with an active low-pass filter with 31 Hz cutoff frequency.

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Fig. 4

Image processing sequence for a single frame of video: (a) graylevel image showing the manually identified magnet and base positions Xm and Xb, (b) the gray level image is processed to enhance contrast, (c) thresholding, (d) large connected regions are selected and smoothed, (e) the region touching the manually labeled base position is retained, shown here overlaid on the original image. Due to low contrast, the algorithm misidentifies shadows as part of the cilium. (f) Shape identification via adaptive Hough transform [20,21].

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Fig. 5

Top view of the model representation (right) of the experimental system (left). A field source at (xm, ym) produces a field Hc which induces a magnetization M in a rectangular cantilever (i.e., artificial cilium). The interaction of the magnetization M with the applied field Hc generates a magnetic body force density fb on the cantilever, resulting in a deflection field U and corresponding tip deflection ytip.

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Fig. 6

Numerical solution in the presence of instability. At point A, the stiffness matrix is singular. The load control method (top) is similar to the experimental procedure, but fails to traverse the large jump in response from point A to point B, leading to divergence or slow convergence. In the arclength method (bottom), a spherical constraint is used so that the solutions follow the unstable path from point A to point C to point B without discontinuity: (a) load control and (b) arclength method.

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Fig. 7

Detail of the arclength incrementation procedure. Starting from a previous solution at (k−1V*, k−1U*), the center of the constraint surface at (kV+, kU+) is found by projecting along the tangent to the equilibrium path. Then the first trial solution, (kV(0), kU(0)), is the intersection of the tangent and the circle. The iterative procedure given by Eqs. (28) and (33) follows the circle from (kV(0), kU(0)) to the equilibrium at (kV*, kU*).

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Fig. 8

Handling of initial conditions when the experimental cilium is initially deflected: (a) a finite element mesh overlaid on a photograph of the initial deflection, for comparing the model to a single experiment and (b) the rotated coordinates (x′, y′) which are used to compare multiple different experiments; each placement is measured in a frame rotated by the cilium initial angle θ for that experiment as shown

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Fig. 9

Measured radial magnetic field profile (i.e., the radial component of Hc(x)) and best-fit model profile with a fit parameter kc = 0.722 at an estimated height from the electromagnet top of zmeas = 4.09 mm

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Fig. 10

Experimental frequency response (•) and best fit via nonlinear least squares (—). The output amplitude is the peak-to-peak tip response, ypp

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Fig. 11

Model () and experimental (▲) results from the model fitting. Error bars indicate the range of estimated tip deflections for that voltage level.

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Fig. 12

Typical measured input signal, i.e., the coil current i in Eq. (1), for placement experiments

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Fig. 13

Model () and experimental (▲) results for the placement (xm, ym) = (5.9, 5.74) shown in the inset (). This placement is just outside the critical placement boundary, resulting in low response. Error bars indicate the range of estimated tip deflections for that voltage level. The shaded band indicates the model-predicted deflection with 5% variation in the transverse magnet position ym. For load reversal (see Fig. 6(b)), the model data points () follow the reversed path while the model line () shows the predicted response when the input voltage V is increased.

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Fig. 14

Model () and experimental (▲) results for the placement (xm, ym) = (14.3, 3.72) shown in the inset (). This experiment exhibited saturation due to contact and is near the optimal placement at low voltage. Error bars indicate the range of estimated tip deflections for that voltage level. The shaded band indicates the model-predicted deflection with 5% variation in the transverse magnet position ym.

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Fig. 15

Model () and experimental (▲) results for the placement (xm, ym) = (10, 5.34) shown in the inset (). This experiment exhibited saturation due to contact and jump discontinuity due to negative stiffness. Error bars indicate the range of estimated tip deflections for that voltage level. The shaded band indicates the model-predicted deflection with 5% variation in the transverse magnet position ym.

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Fig. 16

Model () and experimental (▲) results for the placement (xm, ym) = (13.4, 8.18) shown in the inset (). This placement is near the optimum at high voltage and the response includes jump discontinuity, saturation due to contact, and bistability. Error bars indicate the range of estimated tip deflections for that voltage level. The shaded band indicates the model-predicted deflection with 5% variation in the transverse magnet position ym.

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Fig. 17

Cantilever deflections predicted by the model (contours) and measured experimentally (•): (a) V = 0.2 V, (b) V = 0.58 V, and (c) V = 1.19 V

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Fig. 18

Outlines of the placements with deflections in the top 10% at the three input voltage levels shown in Fig. 17 for the experimental () and model (– –) results. For each outlined region, the weighted average (xm□, ym□) computed from Eq. (42) (, ) and the placement (xm△, ym△) with the highest observed deflection (, △) are shown.

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Fig. 19

Trends in optimal placement (xm□, ym□) and maximal tip deflection ytip□ with increasing input amplitude, as predicted bythe model (—) and estimated from the experimental data (•). For the model-predicted tip deflection (bottom), the line indicates the experimental tip deflection linearly interpolated (via Delaunay triangulation [31]) at the optimal placement predicted by the model data.

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Fig. 20

Comparison of static response in air (▲) and in water () with the magnet placement at Xm = (11.36, 7.22). This placement is near the optimal placement for V = 1.19 V shown in Fig.18.

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Fig. 21

Comparison of frequency response in air (—) and in water () with measured peak-to-peak input voltage 1.13 V. The peak-to-peak tip amplitude ypp was estimated as in Sec. 3.7.2. The magnet placement was at Xm = (11.36, 7.22), which is near the optimal placement for V = 1.19 V shown in Fig. 18. At low frequencies (less than 2 Hz), the response is relatively constant and is similar in air and in water.

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Fig. 22

Experimental response at the low-input optimal placement XL△=(13.4, 3.26) () and high-input optimal placement XNL△=(7.8, 6.68) (). Neglecting the nonlinear effects leads to 50% reduction in tip displacement at high voltage.



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