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

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
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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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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