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.

References

References
1.
Shcheglov
,
K. V.
, and
Challoner
,
A. D.
,
2006
, “
Isolated Planar Gyroscope With Internal Radial Sensing and Actuation
,” U.S. Patent No. 7040163B2.
2.
Nitzan
,
S.
,
Ahn
,
C.
,
Su
,
T.-H.
,
Li
,
M.
,
Ng
,
E.
,
Wang
,
S.
,
Yang
,
Z.
,
O'Brien
,
G.
,
Boser
,
B.
,
Kenny
,
T.
, and
Horsley
,
D.
,
2013
, “
Epitaxially-Encapsulated Polysilicon Disk Resonator Gyroscope
,”
IEEE MEMS
, Taipei, Taiwan, Jan. 20–24, pp.
625
628
.
3.
Kim
,
D.
, and
M'Closkey
,
R.
,
2014
, “
A MEM Vibratory Gyro With Mode-Matching Achieved by Resonator Mass Loading
,”
IEEE/ION Position, Location and Navigation Symposium
, Monterey, CA, May 5–8, pp.
499
503
.
4.
Challoner
,
A.
,
Ge
,
H.
, and
Liu
,
J.
,
2014
, “
Boeing Disc Resonator Gyroscope
,”
IEEE/ION Position, Location and Navigation Symposium
, Monterey, CA, May 5–8, pp.
504
514
.
5.
Kim
,
D.
, and
M'Closkey
,
R. T.
,
2013
, “
Spectral Analysis of Vibratory Gyro Noise
,”
IEEE Sens. J.
,
13
(
11
), pp.
4361
4374
.
6.
Ge
,
H.
, and
M'Closkey
,
R.
,
2017
, “
Simultaneous Angular Rate Estimates Extracted From a Single Axisymmetric Resonator
,”
IEEE Sens. J.
,
17
(
22
), pp.
7460
7469
.
7.
Schwartz
,
D. M.
,
Kim
,
D.
,
Stupar
,
P.
,
DeNatale
,
J.
, and
M'Closkey
,
R. T.
,
2015
, “
Modal Parameter Tuning of an Axisymmetric Resonator Via Mass Perturbation
,”
J. Microelectromech. Syst.
,
24
(
3
), pp.
545
555
.
8.
Behbahani
,
A. H.
,
Kim
,
D.
,
Stupar
,
P.
,
DeNatale
,
J.
, and
M'Closkey
,
R. T.
,
2017
, “
Tailored Etch Profiles for Wafer-Level Frequency Tuning of Axisymmetric Resonators
,”
J. Microelectromech. Syst.
,
26
(
2
), pp.
333
343
.
9.
Bernstein
,
J.
,
Bancu
,
M.
,
Cook
,
E.
,
Henry
,
T.
,
Kwok
,
P.
,
Nyinjee
,
T.
,
Perlin
,
G.
,
Teynor
,
W.
, and
Weinberg
,
M.
,
2014
, “
Diamond Hemispherical Resonator Fabrication by Isotropic Glass Etch
,”
Solid-State Sensors, Actuators and Microsystems Workshop
, Hilton Head, SC, June 8–12, pp.
273
276
.
10.
Gallacher
,
B.
,
Hedley
,
J.
,
Burdess
,
J.
,
Harris
,
A.
, and
McNie
,
M.
,
2003
, “
Multimodal Tuning of a Vibrating Ring Using Laser Ablation
,”
Proc. Inst. Mech. Eng., Part C
,
217
(
5
), pp.
557
576
.
11.
Rourke
,
A.
,
McWilliam
,
S.
, and
Fox
,
C.
,
2002
, “
Multi-Mode Trimming of Imperfect Thin Rings Using Masses at Pre-Selected Locations
,”
J. Sound Vib.
,
256
(
2
), pp.
319
345
.
12.
Bisegna
,
P.
, and
Caruso
,
G.
,
2007
, “
Frequency Split and Vibration Localization in Imperfect Rings
,”
J. Sound Vib.
,
306
(
3–5
), pp.
691
711
.
13.
Choi
,
S.-Y.
, and
Kim
,
J.-H.
,
2011
, “
Natural Frequency Split Estimation for Inextensional Vibration of Imperfect Hemispherical Shell
,”
J. Sound Vib.
,
330
(
9
), pp.
2094
2106
.
14.
Fox
,
C.
,
1990
, “
A Simple Theory for the Analysis and Correction of Frequency Splitting in Slightly Imperfect Rings
,”
J. Sound Vib.
,
142
(
2
), pp.
227
243
.
15.
Rourke
,
A. K.
,
McWilliam
,
S.
, and
Fox
,
C. H. J.
,
2001
, “
Multi-Mode Trimming of Imperfect Rings
,”
J. Sound Vib.
,
248
(
4
), pp.
695
724
.
16.
Rao
,
S. S.
,
2007
,
Vibration of Continuous Systems
,
Wiley
,
Hoboken, NJ
.
17.
Lorentz
,
T.
,
Kim
,
D.
, and
M'Closkey
,
R. T.
,
2014
, “
A Novel Technique for Extracting Parametric Models From Mem Resonator Test Data
,”
International Symposium on Inertial Sensors and Systems (ISISS)
, Laguna Beach, CA, Feb. 25–26, pp.
1
4
.
18.
Bradley
,
S. P.
,
Hax
,
A. C.
, and
Magnanti
,
T. L.
,
1977
,
Applied Mathematical Programming
,
Addison-Wesley
, Boston, MA.
19.
The Mathworks, Inc.
,
2017
, “
MATLAB R2017b
,” The Mathworks, Natick, MA.
You do not currently have access to this content.