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Research Papers

Multimodal Tuning of an Axisymmetric Resonator

[+] Author and Article Information
Amir H. Behbahani

Department of Mechanical and
Aerospace Engineering,
University of California, Los Angeles,
Los Angeles, CA 90095

Robert T. M'Closkey

Professor
Department of Mechanical and
Aerospace Engineering,
University of California, Los Angeles,
Los Angeles, CA 90095
e-mail: rtm@seas.ucla.edu

1Present address: Biology and Biological Engineering, California Institute of Technology, Pasadena, CA.

2Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received June 19, 2018; final manuscript received March 19, 2019; published online May 2, 2019. Assoc. Editor: Douglas Bristow.

J. Dyn. Sys., Meas., Control 141(9), 091010 (May 02, 2019) (9 pages) Paper No: DS-18-1291; doi: 10.1115/1.4043331 History: Received June 19, 2018; Revised March 19, 2019

This paper reports an approach for the simultaneous elimination of the modal frequency differences within two pairs of modes in an axisymmetric resonator. Fabricated devices exhibit frequency detuning, which can be eliminated by strategically mass loading the resonator. Each pair of modes responds to the mass loading differently so models are developed to predict the postloading frequency differences. The models are incorporated into a search procedure to select deposition sites that simultaneously reduce the modal frequency difference within each pair. The proposed approach is successfully implemented on a resonator whose modal frequency differences are reduced below 200 mHz from an initial frequency difference of 23.5 Hz for a pair of modes at 13.8 kHz, and a 2.4 Hz difference for another pair of modes at 24.3 kHz.

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References

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Figures

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Fig. 1

Isometric view of the 1 cm diameter resonator. The large spoke positions are identified with the indices i ∈ {1, 2,…, 24} denoting the angular position, and k ∈ {1, 2, 3, 4} denoting the radial position. The electrodes labeled D1, D6, and D8 are used for exciting the resonator and the electrodes labeled S1S8 are used for measuring the radial velocities at points along the outermost ring.

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Fig. 2

A view of reflowed solder spheres in the area between the S6 and D1 electrodes. A 75 μm diameter solder sphere is on the upper right spoke and a reflowed 35 μm diameter solder sphere is on bottom left spoke.

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Fig. 3

Broadband frequency response using electrode D1 for excitation and electrode S7 for sensing. The n =1, 2, 3 pairs of modes are located near 6, 14, and 24 kHz, respectively.

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Fig. 4

Mode shapes for the n =2, 3 modes. Only one mode in each pair is shown.

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Fig. 5

(Top) The normalized magnitudes of the Fourier series coefficients for the spoke radial velocities at different layers for the n =2 modes. (Bottom) The normalized magnitudes of the Fourier series coefficients for the spoke tangential velocities at different layers for the n =2 modes.

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Fig. 6

(Top) The normalized magnitudes of the Fourier series coefficients for the spoke radial velocities at different layers for the n =3 modes. (Bottom) The normalized magnitude of the Fourier series coefficients for the spoke tangential velocities at different layers for n =3 modes.

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Fig. 7

Frequency response magnitude plot showing the n =2 modes prior to any mass deposition (the “as-fabricated” state of the resonator). The initial modal frequency difference is 23.49 Hz. The frequency response of an identified parametric model is also shown. The deviation between the model and empirical data is due to parasitic coupling which is present in the empirical frequency response but not the parametric model. The S2/D1 and S6/D8 measurement channels are shown (see Fig.1).

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Fig. 8

Frequency response magnitude plot showing the n =3 modes prior to mass deposition. The initial modal frequency difference is 2.43 Hz. The parasitic capacitive coupling creates the “floor” in the empirical frequency response. The S2/D1 and S5/D6 measurement channels are shown.

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Fig. 9

Elements of Cv for an n =2 mode with labels indicating the associated electrode (see Eq. (2)). A shift in phase aligns these elements on the real axis—these are the values that a graphed in Figs. 10 and 11.

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Fig. 10

Estimation of n =2 antinodes. The points are from the analysis of Cv and the dashed line is the model cos(2(θ−ψ)) where ψ is chosen to interpolate the points. Note that ψ2,2 − ψ2,1 = −44.9 deg.

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Fig. 11

Estimation of n =3 antinodes. Note that ψ3,2 − ψ3,1 = 30.4 deg.

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Fig. 12

The map of masses placed on the resonator in the deposition steps

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Fig. 13

Empirical frequency response of the n =2 pair of modes for the S2/D1 and S6/D8 measurement channels. The initial modal frequency difference of 23.49 Hz is reduced to 0.11 Hz after three rounds of mass deposition.

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Fig. 14

Empirical frequency response of the n =3 for the S2/D1 and S5/D6 measurement channels. The initial modal frequency difference of 2.43 Hz is reduced to 0.16 Hz after three rounds of deposition.

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