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Research Papers

Actuation of Higher Harmonics in Large Arrays of Micromechanical Cantilevers for Expanded Resonant Peak Separation

[+] Author and Article Information
Nir Dick

School of Mechanical Engineering,
Faculty of Engineering,
Tel Aviv University Ramat,
Ramat Aviv,
Tel Aviv 69978, Israel
e-mail: dick.nir@gmail.com

Scott Grutzik

Component Science and Mechanics,
Sandia National Laboratories,
Albuquerque, NM 87185
e-mail: sjgrutz@sandia.gov

Christopher B. Wallin

Center for Nanoscale Science and Technology,
National Institute of Standards and Technology,
Gaithersburg, MD 20899;
Institute for Research in Electronics
and Applied Physics,
University of Maryland,
College Park, MD 20742
e-mail: christopher.wallin@nist.gov

B. Robert Ilic

Center for Nanoscale Science and Technology,
National Institute of Standards and Technology,
Gaithersburg, MD 20899
e-mail: robert.ilic@nist.gov

Slava Krylov

Mem. ASME
School of Mechanical Engineering,
Faculty of Engineering,
Tel Aviv University,
Ramat Aviv,
Tel Aviv 69978, Israel e-mail: vadis@eng.tau.ac.il

Alan T. Zehnder

Fellow ASME
Sibley School of Mechanical and Aerospace
Engineering,
Cornell University,
Ithaca, NY 14853
e-mail: ATZ2@cornell.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 15, 2017; final manuscript received February 13, 2018; published online April 26, 2018. Assoc. Editor: Miao Yu. This work is in part a work of the U.S. Government. ASME disclaims all interest in the U.S. Government's contributions.

J. Vib. Acoust 140(5), 051013 (Apr 26, 2018) (10 pages) Paper No: VIB-17-1498; doi: 10.1115/1.4039568 History: Received November 15, 2017; Revised February 13, 2018

A large array of elastically coupled micro cantilevers of variable length is studied experimentally and numerically. Full-scale finite element (FE) modal analysis is implemented to determine the spectral behavior of the array and to extract a global coupling matrix. A compact reduced-order (RO) model is used for numerical investigation of the array's dynamic response. Our model results show that at a given excitation frequency within a propagation band, only a finite number of beams respond. Spectral characteristics of individual cantilevers, inertially excited by an external piezoelectric actuator, were measured in vacuum using laser interferometry. The theoretical and experimental results collectively show that the resonant peaks corresponding to individual beams are clearly separated when operating in vacuum at the third harmonic. Distinct resonant peak separation, coupled with the spatially confined modal response, make higher harmonic operation of tailored, variable-length cantilever arrays well suited for a variety of resonant-based sensing applications.

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References

Rhoads, J. , Shaw, S. , and Turner, K. , 2010, “ Nonlinear Dynamics and Its Applications in Micro- and Nanoresonators,” ASME J. Dyn. Syst., Meas. Control, 132(3), p. 034001. [CrossRef]
Lin, L. , Nguyen, C.-C. , Howe, R. T. , and Pisano, A. P. , 1992, “ Microelectromechanical Filters for Signal Processing,” IEEE Micro Electro Mechanical Systems (MEMS'92), An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots, Travemunde, Germany, Feb. 4–7, pp. 226–231.
Ho, G. K. , Abdolvand, R. , and Ayazi, F. , 2004, “ Through-Support-Coupled Micromechanical Filter Array,” 17th IEEE International Conference on Micro Electro Mechanical Systems (MEMS), Maastricht, The Netherlands, Jan. 25–29, pp. 769–772.
Elka, A. , and Bucher, I. , 2008, “ On the Synthesis of Micro-Electromechanical Filters Using Structural Dynamics,” J. Micromech. Microeng., 18(12), p. 125018. [CrossRef]
Chivukula, V. B. , and Rhoads, J. F. , 2009, “ MEMS Bandpass Filters Based on Cyclic Coupling Architectures,” ASME Paper No. DETC2009-87059.
Judge, J. , Houston, B. , Photiadis, D. , and Herdic, P. , 2006, “ Effects of Disorder in One- and Two-Dimensional Micromechanical Resonator Arrays for Filtering,” J. Sound Vib., 290(3–5), pp. 1119–1140. [CrossRef]
Small, J. , Arif, M. , Fruehling, A. , and Peroulis, D. , 2013, “ A Tunable Miniaturized RF MEMS Resonator With Simultaneous High Q (500-735) and Fast Response Speed (10-60),” J. Microelectromech. Syst., 22(2), pp. 395–405. [CrossRef]
Lin, Y. , Li, W.-C. , Kim, B. , Lin, Y.-W. , Ren, Z. , and Nguyen, C. T.-C. , 2009, “ Enhancement of Micromechanical Resonator Manufacturing Precision Via Mechanically-Coupled Arraying,” IEEE International Frequency Control Symposium, Joint With the 22nd European Frequency and Time Forum, Besancon, France, Apr. 20–24, pp. 58–63.
Kharrat, C. , Colinet, E. , Duraffourg, L. , Hentz, S. , Andreucci, P. , and Voda, A. , 2010, “ Modal Control of Mechanically Coupled Nems Arrays for Tunable RF Filters,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 57(6), pp. 1285–1295. [CrossRef]
Pachkawade, V. , Junghare, R. , Patrikar, R. , and Kraft, M. , 2016, “ Mechanically Coupled Ring-Resonator Filter and Array (Analytical and Finite Element Model),” IET Comput. Digital Tech., 10(5), pp. 261–267. [CrossRef]
Nguyen, C.-C. , 2007, “ MEMS Technology for Timing and Frequency Control,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 54(2), pp. 251–270. [CrossRef]
Acar, C. , and Shkel, A. , 2005, “ An Approach for Increasing Drive-Mode Bandwidth of MEMS Vibratory Gyroscopes,” J. Microelectromech. Syst., 14(3), pp. 520–528. [CrossRef]
Haronian, D. , and MacDonald, N. , 1996, “ A Microelectromechanics-Based Frequency-Signature Sensor,” Sens. Actuators, A: Phys., 53(1–3), pp. 288–298. [CrossRef]
Boisen, A. , and Thundat, T. , 2009, “ Design and Fabrication of Cantilever Array Biosensors,” Mater. Today, 12(9), pp. 32–38. [CrossRef]
Boisen, A. , Dohn, S. , Keller, S. , Schmid, S. , and Tenje, M. , 2011, “ Cantilever-like Micromechanical Sensors,” Rep. Prog. Phys., 74(3), p. 036101. [CrossRef]
Ziegler, C. , 2004, “ Cantilever-Based Biosensors,” Anal. Bioanal. Chem., 379(7–8), pp. 946–959. [PubMed]
Waggoner, P. , and Craighead, H. , 2007, “ Micro- and Nanomechanical Sensors for Environmental, Chemical, and Biological Detection,” Lab Chip, 7(10), pp. 1238–1255. [CrossRef] [PubMed]
Raiteri, R. , Grattarola, M. , Butt, H.-J. , and Skldal, P. , 2001, “ Micromechanical Cantilever-Based Biosensors,” Sens. Actuators, B: Chem., 79(2–3), pp. 115–126. [CrossRef]
Lavrik, N. V. , Sepaniak, M. J. , and Datskos, P. G. , 2004, “ Cantilever Transducers as a Platform for Chemical and Biological Sensors,” Rev. Sci. Instrum., 75(7), pp. 2229–2253. [CrossRef]
Rhoads, J. F. , DeMartini, B. E. , Shaw, S. W. , and Turner, K. L. , 2006, “ A SISO, Multi-Analyte Sensor Based on a Coupled Microresonator Array,” ASME Paper No. IMECE2006-13693.
DeMartini, B. , Rhoads, J. , Zielke, M. , Owen, K. , Shaw, S. , and Turner, K. , 2008, “ A Single Input-Single Output Coupled Microresonator Array for the Detection and Identification of Multiple Analytes,” Appl. Phys. Lett., 93(5), p. 054102. [CrossRef]
Yabuno, H. , Seo, Y. , and Kuroda, M. , 2013, “ Self-Excited Coupled Cantilevers for Mass Sensing in Viscous Measurement Environments,” Appl. Phys. Lett., 103(6), p. 063104. [CrossRef]
Ryan, T. , Judge, J. , Vignola, J. , and Glean, A. , 2012, “ Noise Sensitivity of a Mass Detection Method Using Vibration Modes of Coupled Microcantilever Arrays,” Appl. Phys. Lett., 101(4), p. 043104. [CrossRef]
Glean, A. A. , Vignola, J. F. , Judge, J. A. , and Ryan, T. J. , 2013, “ Impact of Mass Ratio and Bandwidth on Apparent Damping of a Harmonic Oscillator With Subordinate Oscillator Array,” ASA International Congress on Acoustics (ICA), Montreal, QC, Canada.
Torres, F. , Uranga, A. , and Barniol, N. , 2014, “ Multi-Cantilever Oscillator,” Procedia Eng., 87, pp. 32–35. [CrossRef]
Spletzer, M. , Raman, A. , Sumali, H. , and Sullivan, J. , 2008, “ Highly Sensitive Mass Detection and Identification Using Vibration Localization in Coupled Microcantilever Arrays,” Appl. Phys. Lett., 92(11), p. 114102. [CrossRef]
Thiruvenkatanathan, P. , and Seshia, A. A. , 2012, “ Mode-Localized Displacement Sensing,” J. Microeletromech. Syst., 21(5), pp. 1016–1018. [CrossRef]
Oguchi, T. , Hayase, M. , and Hatsuzawa, T. , 2005, “ Micromachined Display Device Using Sheet Waveguide and Multicantilevers Driven by Electrostatic Force,” IEEE Trans. Ind. Electron., 52(4), pp. 984–991. [CrossRef]
Guillon, S. , Saya, D. , Mazenq, L. , Perisanu, S. , Vincent, P. , Lazarus, A. , Thomas, O. , and Nicu, L. , 2011, “ Effect of Non-Ideal Clamping Shape on the Resonance Frequencies of Silicon Nanocantilevers,” Nanotechnol., 22(24), p. 245501. [CrossRef]
Judge, J. A. , Woods, T. J. , and Vignola, J. F. , 2009, “ Considerations for Use of Square-Paddle Resonators for Arrays of Micro- and Nanoscale Devices,” ASME Paper No. DETC2009-87441.
Choubey, B. , Boyd, E. , Armstrong, I. , and Uttamchandani, D. , 2012, “ Determination of the Anisotropy of Young's Modulus Using a Coupled Microcantilever Array,” J. Microeletromech. Syst., 21(5), pp. 1252–1260. [CrossRef]
Sabater, A. , and Rhoads, J. , 2015, “ Dynamics of Globally and Dissipatively Coupled Resonators,” ASME J. Vib. Acoust., 137(2), p. 021016. [CrossRef]
Sato, M. , Hubbard, B. , and Sievers, A. , 2006, “ Colloquium: Nonlinear Energy Localization and Its Manipulation in Micromechanical Oscillator Arrays,” Rev. Mod. Phys., 78(1), p. 137. [CrossRef]
Buks, E. , and Roukes, M. , 2002, “ Electrically Tunable Collective Response in a Coupled Micromechanical Array,” J. Microeletromech. Syst., 11(6), pp. 802–807. [CrossRef]
Krylov, S. , Lulinsky, S. , Ilic, B. R. , and Schneider, I. , 2014, “ Collective Dynamics of Arrays of Micro Cantilevers Interacting Through Fringing Electrostatic Fields,” ASME Paper No. DETC2014-34904.
Krylov, S. , Lulinsky, S. , Ilic, B. , and Schneider, I. , 2014, “ Collective Dynamics and Pattern Switching in an Array of Parametrically Excited Micro Cantilevers Interacting Through Fringing Electrostatic Fields,” Appl. Phys. Lett., 105(7), p. 071909. [CrossRef]
Ono, T. , Tanno, K. , and Kawai, Y. , 2014, “ Synchronized Micromechanical Resonators With a Nonlinear Coupling Element,” J. Micromech. Microeng., 24(2), p. 025012. [CrossRef]
Baskin, J. , Park, H. , and Zewail, A. , 2011, “ Nanomusical Systems Visualized and Controlled in 4D Electron Microscopy,” Nano Lett., 11(5), pp. 2183–2191. [CrossRef] [PubMed]
Linzon, Y. , Ilic, B. , Lulinsky, S. , and Krylov, S. , 2013, “ Efficient Parametric Excitation of Silicon-on-Insulator Microcantilever Beams by Fringing Electrostatic Fields,” J. Appl. Phys., 113(16), p. 163508. [CrossRef]
Meirovitch, L. , 2001, Fundamentals of Vibrations, McGraw-Hill, Boston, MA. [PubMed] [PubMed]
Gutschmidt, S. , and Gottlieb, O. , 2012, “ Nonlinear Dynamic Behavior of a Microbeam Array Subject to Parametric Actuation at Low, Medium and Large Dc-Voltages,” Nonlinear Dyn., 67(1), pp. 1–36. [CrossRef]
Kambali, P. , Swain, G. , and Pandey, A. , 2016, “ Frequency Analysis of Linearly Coupled Modes of MEMS Arrays,” ASME J. Vib. Acoust., 138(2), p. 021017. [CrossRef]
Lifshitz, R. , and Cross, M. , 2003, “ Response of Parametrically Driven Nonlinear Coupled Oscillators With Application to Micromechanical and Nanomechanical Resonator Arrays,” Phys. Rev. B, 67(13), p. 134302. [CrossRef]
Dick, A. , Balachandran, B. , and Mote , C., Jr. , 2008, “ Intrinsic Localized Modes in Microresonator Arrays and Their Relationship to Nonlinear Vibration Modes,” Nonlinear Dyn., 54(1–2), pp. 13–29. [CrossRef]
Blocher, D. B. , 2012, “ Optically Driven Limit Cycle Oscillations in MEMS,” Ph.D. thesis, Cornell University, Ithaca, NY.

Figures

Grahic Jump Location
Fig. 1

(a) Schematic illustration of the cantilever array device. The array contains N silicon cantilevers with linearly varying length, width b, thickness h, pitch B, and an overhang Lo. n = 1, 2 … N represents the beam number within the array. The first cantilever n = 1 is the longest and of the length L1 = Lmax ≈ 500 μm. The last beam n = N is the shortest and of length LN = Lmin ≈ 350 μm. The beams deflect in the out-of-plane z direction. The length of the beams varies linearly between L1 ≈ 500 μm and L100 ≈ 350 μm. The difference in length between any two adjacent cantilevers is 1.515 μm. (b) Top view of the array. Clamped edges are schematically illustrated with diagonally hatched areas.

Grahic Jump Location
Fig. 2

Finite element results showing three-dimensional snapshots of several natural modes that are associated with the corresponding natural frequencies fi. (a) f5 = 26.515 kHz, (b) f20 = 28.865 kHz, and (c) f80 = 43.313 kHz illustrate cantilevers vibrating at the first harmonic. (d) f105 = 163.210 kHz, (e) f120 = 178.536 kHz, and (f) f180 = 266.109 kHz depict the cantilevers vibrating at the second harmonic.

Grahic Jump Location
Fig. 3

Finite element results showing several normalized modal amplitudes of the array. Rows 1, 2, and 3 correspond to the first, second, and third harmonics, respectively.

Grahic Jump Location
Fig. 4

Natural frequencies of the array obtained by the FE analysis. Three propagation bands corresponding to the first, second, and third harmonics of the cantilevers, respectively, are shown. Inset depicts the frequency curve corresponding to the first propagation band of the array. The isolated dot in the inset corresponds to the upper cutoff frequency fU(1)=f100 of the first propagation band.

Grahic Jump Location
Fig. 5

RO model results. Spectral response of the L25 = 463.64 μm cantilever, calculated for γ = 0.01, Q = 1000 and using the stiffness matrix given by Eq. (13). Spectral response at the (a) first and (b) second harmonics. Insets show enlarged region corresponding to smaller frequency sweep interval.

Grahic Jump Location
Fig. 6

RO model results showing modal patterns of the array calculated for γ = 0.01, Q = 1000 and using the stiffness matrix given by Eq. (13). Modal patterns at the (a) first and (b) second harmonics. Inset shows smaller frequency interval. Gray levels represent normalized vibrational amplitudes at each drive frequency with the values varying between 0 and 1.

Grahic Jump Location
Fig. 7

Schematic of the experimental setup

Grahic Jump Location
Fig. 8

Optical micrograph of a laser beam focused onto a micromechanical beam. The scale bar is ≈ 50 μm.

Grahic Jump Location
Fig. 9

Measured spectral response of five different cantilevers within the array at the (a)–(e) first and (f)–(j) second harmonic. Insets in (a) and (f) show zoomed-in regions corresponding to smaller frequency sweep intervals. The nominal lengths of the beams are L1 = 500 μm, L15 = 478.79 μm, L25 = 463.64 μm, L35 = 448.49 μm, and L45 = 443.34 μm.

Grahic Jump Location
Fig. 10

(a) Measured spectral response of cantilever L25 vibrating at the third harmonic with a drive frequency sweep time of ≈ 20 s. Measured frequency spectra as a function of the drive voltage for peaks highlighted by dashed boxes at (b) 436.681 kHz ± 2.2 Hz and (c) 491.694 kHz ± 2.6 Hz (mean ± error from Lorentzian fit). Inset shows a linear dependence of the photodiode output on the drive voltage. The error bars, calculated from the Lorentzian fit, are smaller than the marker size. The solid line represents a linear fit.

Grahic Jump Location
Fig. 11

Experimental (triangular markers) and theoretical (FE model, black dots) values of natural frequencies as a function of mode number. The three propagation bands correspond to the first, the second, and the third harmonics of the cantilevers, respectively. Inset shows the lower part of the frequency curve corresponding to the first harmonic of the cantilevers. Frequency uncertainties, calculated from the Lorentzian functional fit, were 22 Hz, 11 Hz, and 181 Hz for the for the first, second, and third bands, respectively.

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