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.

References

References
1.
Satir
,
P.
, and
Christensen
,
S.
,
2007
, “
Overview of Structure and Function of Mammalian Cilia
,”
Annu. Rev. Physiol.
,
69
, pp.
377
400
.
2.
Brennen
,
C.
, and
Winet
,
H.
,
1977
, “
Fluid Mechanics of Propulsion by Cilia and Flagella
,”
Annu. Rev. Fluid Mech.
,
9
(
1
), pp.
339
398
.
3.
Ding
,
Y.
, and
Kanso
,
E.
,
2015
, “
Selective Particle Capture by Asynchronously Beating Cilia
,”
Phys. Fluids
,
27
(
12
), p.
121902
.
4.
Gilpin
,
W.
,
Prakash
,
V. N.
, and
Prakash
,
M.
,
2017
, “
Vortex Arrays and Ciliary Tangles Underlie the Feeding-Swimming Trade-Off in Starfish Larvae
,”
Nat. Phys.
,
13
, pp.
380
386
.
5.
den Toonder
,
J.
,
Bos
,
F.
,
Broer
,
D.
,
Filippini
,
L.
,
Gillies
,
M.
,
de Goede
,
J.
,
Mol
,
T.
,
Reijme
,
M.
,
Talen
,
W.
,
Wilderbeek
,
H.
,
Khatavkar
,
V.
, and
Patrick Anderson
,
P.
,
2008
, “
Artificial Cilia for Active Micro-Fluidic Mixing
,”
Lab Chip
,
8
(
4
), pp.
533
541
.
6.
Oh
,
K.
,
Chung
,
J.
,
Devasia
,
S.
, and
Riley
,
J.
,
2009
, “
Bio-Mimetic Silicone Cilia for Microfluidic Manipulation
,”
Lab Chip
,
9
(
11
), pp.
1561
1566
.
7.
Keißner
,
A.
, and
Brucker
,
C.
,
2012
, “
Directional Fluid Transport Along Artificial Ciliary Surfaces With Base-Layer Actuation of Counter-Rotating Orbital Beating Patterns
,”
Soft Matter
,
8
(19), pp.
5342
5349
.
8.
Khaderi
,
S.
,
Craus
,
C.
,
Hussong
,
J.
,
Schorr
,
N.
,
Belardi
,
J.
,
Westerweel
,
J.
,
Prucker
,
O.
,
Rühe
,
J.
,
Den Toonder
,
J.
, and
Onck
,
P.
,
2011
, “
Magnetically-Actuated Artificial Cilia for Microfluidic Propulsion
,”
Lab Chip
,
11
(
12
), pp.
2002
2010
.
9.
Vilfan
,
M.
,
Potočnik
,
A.
,
Kavčič
,
B.
,
Osterman
,
N.
,
Poberaj
,
I.
,
Vilfan
,
A.
, and
Babič
,
D.
,
2010
, “
Self-Assembled Artificial Cilia
,”
Proc. Natl. Acad. Sci. U.S.A.
,
107
(
5
), pp.
1844
1847
.
10.
Shields
,
A.
,
Fiser
,
B.
,
Evans
,
B.
,
Falvo
,
M.
,
Washburn
,
S.
, and
Superfine
,
R.
,
2010
, “
Biomimetic Cilia Arrays Generate Simultaneous Pumping and Mixing Regimes
,”
Proc. Natl. Acad. Sci. U.S.A
,
107
(
36
), pp.
15670
15675
.
11.
Downton
,
M.
, and
Stark
,
H.
,
2009
, “
Beating Kinematics of Magnetically Actuated Cilia
,”
EPL (Europhys. Lett.)
,
85
(
4
), p.
44002
.
12.
Khaderi
,
S.
,
Baltussen
,
M.
,
Anderson
,
P.
,
den Toonder
,
J.
, and
Onck
,
P.
,
2010
, “
Breaking of Symmetry in Microfluidic Propulsion Driven by Artificial Cilia
,”
Phys. Rev. E
,
82
(
2
), p.
027302
.
13.
Khaderi
,
S.
,
den Toonder
,
J.
, and
Onck
,
P.
,
2012
, “
Fluid Flow Due to Collective Non-Reciprocal Motion of Symmetrically-Beating Artificial Cilia
,”
Biomicrofluidics
,
6
(
1
), p.
014106
.
14.
Gauger
,
E. M.
,
Downton
,
M. T.
, and
Stark
,
H.
,
2009
, “
Fluid Transport at Low Reynolds Number With Magnetically Actuated Artificial Cilia
,”
Eur. Phys. J. E
,
28
(
2
), pp.
231
242
.
15.
Khaderi
,
S.
,
Hussong
,
J.
,
Westerweel
,
J.
,
den Toonder
,
J.
, and
Onck
,
P.
,
2013
, “
Fluid Propulsion Using Magnetically-Actuated Artificial Cilia–Experiments and Simulations
,”
RSC Adv.
,
3
(
31
), pp.
12735
12742
.
16.
Rahbar
,
M.
,
Shannon
,
L.
, and
Gray
,
B. L.
,
2014
, “
Microfluidic Active Mixers Employing Ultra-High Aspect-Ratio Rare-Earth Magnetic Nano-Composite Polymer Artificial Cilia
,”
J. Micromech. Microeng.
,
24
(
2
), p.
025003
.
17.
Chen
,
C.-Y.
,
Chen
,
C.-Y.
,
Lin
,
C.-Y.
, and
Hu
,
Y.-T.
,
2013
, “
Magnetically Actuated Artificial Cilia for Optimum Mixing Performance in Microfluidics
,”
Lab Chip
,
13
(
14
), pp.
2834
2839
.
18.
Coq
,
N.
,
Ngo
,
S.
,
Du Roure
,
O.
,
Fermigier
,
M.
, and
Bartolo
,
D.
,
2010
, “
Three-Dimensional Beating of Magnetic Microrods
,”
Phys. Rev. E
,
82
(
4
), p.
041503
.
19.
Banka
,
N.
,
Ng
,
Y. L.
, and
Devasia
,
S.
,
2017
, “
Individually-Controllable Magnetic Cilia: Mixing Application
,”
ASME J. Med. Devices
,
11
(
3
), p.
031003
.
20.
Illingworth
,
J.
, and
Kittler
,
J.
,
1987
, “
The Adaptive Hough Transform
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
9
(
5
), pp.
690
698
.
21.
Li
,
H.
,
Lavin
,
M. A.
, and
Le Master
,
R. J.
,
1986
, “
Fast Hough Transform: A Hierarchical Approach
,”
Comput. Vision, Graph., Image Process.
,
36
(
2–3
), pp.
139
161
.
22.
Simkin
,
J.
, and
Trowbridge
,
C.
,
1979
, “
On the Use of the Total Scalar Potential on the Numerical Solution of Fields Problems in Electromagnetics
,”
Int. J. Numer. Methods Eng.
,
14
(
3
), pp.
423
440
.
23.
Carpentier
,
A.
,
Galopin
,
N.
,
Chadebec
,
O.
,
Meunier
,
G.
, and
Guérin
,
C.
,
2014
, “
Application of the Virtual Work Principle to Compute Magnetic Forces With a Volume Integral Method
,”
Int. J. Numer. Modell.: Electron. Networks, Devices Fields
,
27
(
3
), pp.
418
432
.
24.
Mikhlin
,
S.
,
1965
,
Multidimensional Singular Integrals and Integral Equations
,
Pergamon Press
, New York.
25.
Banka
,
N.
,
2017
, “
Individually-Controllable Magnetic Artificial Cilia for Microfluidic Manipulation Tasks
,”
Ph.D. thesis
, University of Washington, Seattle, WA.https://digital.lib.washington.edu/researchworks/handle/1773/40244
26.
Engel
,
A.
, and
Friedrichs
,
R.
,
2002
, “
On the Electromagnetic Force on a Polarizable Body
,”
Am. J. Phys.
,
70
(
4
), pp.
428
432
.
27.
Bathe
,
K.-J.
,
1996
,
Finite Element Procedures
,
Prentice Hall, Upper Saddle River, NJ
.
28.
Ritto-Corrêa
,
M.
, and
Camotim
,
D.
,
2008
, “
On the Arc-Length and Other Quadratic Control Methods: Established, Less Known and New Implementation Procedures
,”
Comput. Struct.
,
86
(
11
), pp.
1353
1368
.
29.
Meirovitch
,
L.
,
2001
,
Fundamentals of Vibrations
,
McGraw-Hill
, New York.
30.
Bombard
,
A. J.
,
Knobel
,
M.
,
Alcantara
,
M. R.
, and
Joekes
,
I.
,
2002
, “
Evaluation of Magnetorheological Suspensions Based on Carbonyl Iron Powders
,”
J. Intell. Mater. Syst. Struct.
,
13
(
7–8
), pp.
471
478
.
31.
Amidror
,
I.
,
2002
, “
Scattered Data Interpolation Methods for Electronic Imaging Systems: A Survey
,”
J. Electron. Imaging
,
11
(
2
), pp.
157
176
.
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