gyroscopyГироскопия и навигацияGiroskopiya i Navigatsiya0869-70352075-0927AO «Концерн «ЦНИИ «Электроприбор»10.17285/0869-7035.0060gyroscopy-167Research ArticleСтатьиПроектирование механических резонаторов с учетом влияния рабочих параметров на их добротностьDesign to Operational Parameters Dependency on Quality Factor of Sensor Mechanical Resonatorshttps://orcid.org/0000-0001-8177-6895ШармаН. Г.SharmaG.N.

Шарма Н. Гириш. Научный сотрудник

Sharma, G.N. 

СундарараджанТ.SundararajanT.

Сундарараджан Т. Научный сотрудник

Sundararajan T. 

СингхГ. С.SinghG.S.

Сингх Гаутам Сачин. Старший преподаватель

Singh, G.S. 

Индийская организации космических исследований, подразделение инерциальных систем (г. Тируванантапурам, Керала)ИндияISRO Inertial Systems Unit, Thiruvananthapuram, KeralaIndiaКосмический центр им. Викрама Сарабхаи (г. Тируванантапурам, Керала)ИндияVikram Sarabhai Space Centre, Thiruvananthapuram, Kerala, IndiaIndiaИндийский технологический институт (г. Гувахати)ИндияIndian Institute of Technology, Guwahati, IndiaIndia202126092025292334Copyright © Шарма Н.Г., Сундарараджан Т., Сингх Г., 20252025Шарма Н.Г., Сундарараджан Т., Сингх Г.Sharma G., Sundararajan T., Singh G.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.gyroscopy.ru/jour/article/view/167

Важнейшим функциональным элементом любого высокоэффективного датчика резонансного типа является механический резонатор, мерой эффективности которого служит коэффициент добротности. В работе рассматриваются различные механизмы демпфирования в резонаторах гироскопов, в том числе термоупругое демпфирование (ТУД), рассеяние энергии колебаний в опоре резонатора, поверхностное рассеяние энергии, внутреннее трение в материале резонатора, демпфирование под воздействием среды и демпфирование, вызванное рассеянием энергии в электронике системы съема. Уменьшить демпфирование можно разными способами, в частности путем проектирования резонаторов различных конструкций, размеров и из различных материалов с учетом зависимости эксплуатационных характеристик от показателя добротности. Коэффициент добротности кольцевых микрорезонаторов может достигать сотен тысяч. Полусферические макрорезонаторы подходят для сверхвысоких значений добротности. Эксплуатация датчика при высоких температурах нежелательна из-за ТУД, а при отрицательных – ограничивается внутренним трением в материале. Увеличение рассеяния на несколько порядков наблюдается при нанесении тонкослойного металлического покрытия, что также обусловлено ТУД и внутренним трением в материале покрытия. Чтобы рассеяние энергии колебаний в опоре было минимальным, резонатор должен быть изготовлен с очень высокой точностью, поскольку данный тип демпфирования крайне чувствителен к малейшим дефектам производства. Чувствительность добротности к рабочему давлению различается в зависимости от конфигурации резонатора. Представленный в статье анализ поможет разработать полноценную стратегию проектирования, реализации и эксплуатации датчика, которая позволит добиться высоких эксплуатационных показателей. В статье определен также круг вопросов, требующих дальнейшего исследования с целью разработки серийных компактных кориолисовых вибрационных гироскопов (КВГ) со сверхвысокой добротностью.

The critical functional part of any high performance resonance based sensor is a mechanical resonator. The performance is measured by resonator quality factor (Q-factor). Damping mechanisms such as thermoelastic damping (TED), anchor loss, surface loss, material internal friction, fluid damping and electronics damping are covered in this review with more focus on gyroscope resonators. Dissipations can be reduced by different means. Hence, the effects of various design to operational parameters on the Q-factor for different configurations, sizes and materials are reviewed in detail. Micro scale ring resonators can achieve a Q-factor of the order of hundreds of thousands. Macro scale hemispherical resonators are suitable for ultrahigh Q-factors. High temperature sensor operation is not preferred because of TED, while sub-zero operation is limited by material internal friction. Few orders of dissipation increase are seen with thin film metallic coating due to TED and coating material internal friction. High precision fabrication is mandatory to achieve the designed minimum anchor loss as it is highly sensitive to fabrication imperfections. Q-factor sensitivity to operating pressure is different for different resonator configurations. This review study helps to build a comprehensive mechanical resonator design, realization and operation strategy to achieve high sensor performance. A roadmap on future research requirements for developing compact mass producible CVG type sensors with ultrahigh Q-factor is also highlighted.

Механические резонаторыдобротностьтермоупругое демпфированиерассеяние энергии колебаний в опоре резонатораповерхностное рассеяниевнутреннее трениедемпфирование под воздействием средырассеяние энергии в электронике системы съема.Mechanical resonatorsquality factorthermoelastic dampinganchor losssurface lossinternal frictionfluid dampingelectronics damping.ReferencesGabrielson, T.B., Mechanical-thermal noise in micromachined acoustic and vibration sensors, IEEE Transactions on Electron Devices, 1993, vol. 40, no. 5, pp. 903–909.Gabrielson, T.B., Mechanical-thermal noise in micromachined acoustic and vibration sensors, IEEE Transactions on Electron Devices, 1993, vol. 40, no. 5, pp. 903–909.Tilmans, H.A.C., Elwebspoek, M., and Fluitman, J.H.J., Micro resonant force gauges, Sensors and Actuators A: Physical, 1992, vol. 30, no. 1–2, pp. 35–53.Tilmans, H.A.C., Elwebspoek, M., and Fluitman, J.H.J., Micro resonant force gauges, Sensors and Actuators A: Physical, 1992, vol. 30, no. 1–2, pp. 35–53.Zhou, X., Xiao, D., Hou, Z., Li, Q., Wu, Y., and Wu, X., Influences of the structure parameters on sensitivity and Brownian noise of the disk resonator gyroscope, Journal of Microelectromechanical Systems, 2017, vol. 26, no. 3, pp. 519–527.Zhou, X., Xiao, D., Hou, Z., Li, Q., Wu, Y., and Wu, X., Influences of the structure parameters on sensitivity and Brownian noise of the disk resonator gyroscope, Journal of Microelectromechanical Systems, 2017, vol. 26, no. 3, pp. 519–527.Ayazi, F., and Najafi, K., A HARPSS polysilicon vibrating ring gyroscope, Journal of Microelectromechanical Systems, 2001, vol. 10, no. 2, pp. 169–179.Ayazi, F., and Najafi, K., A HARPSS polysilicon vibrating ring gyroscope, Journal of Microelectromechanical Systems, 2001, vol. 10, no. 2, pp. 169–179.Choudhary, V., and Iniewski, K., MEMS: Fundamental Technology and Applications, CRC Press, 2013.Choudhary, V., and Iniewski, K., MEMS: Fundamental Technology and Applications, CRC Press, 2013.Wang, Z.H., Lee, J., and Feng, P.X.L., Spatial mapping of multimode brownian motions in high-frequency silicon carbide micro disk resonators, Nature Communications, 2014, vol. 5, no. 1, pp. 5158–5168.Wang, Z.H., Lee, J., and Feng, P.X.L., Spatial mapping of multimode brownian motions in high-frequency silicon carbide micro disk resonators, Nature Communications, 2014, vol. 5, no. 1, pp. 5158–5168.Hao, Z., Erbil, A., and Ayazi, F., An analytical model for support loss in micromachined beam resonators with in plane flexural vibrations, Sensors and Actuators A: Physical, 2003, vol. 109, no. 1, pp. 156–164.Hao, Z., Erbil, A., and Ayazi, F., An analytical model for support loss in micromachined beam resonators with in plane flexural vibrations, Sensors and Actuators A: Physical, 2003, vol. 109, no. 1, pp. 156–164.Izmailov, E.A., Kolesnik, M.M., Osipov, A.M., and Akimov, A.V., Hemispherical resonator gyro technology: Problems and possible ways of their solutions, Proc. St. Petersburg International Conference on Integrated Navigation Systems, Russia, 1999, pp. 61–69.Izmailov, E.A., Kolesnik, M.M., Osipov, A.M., and Akimov, A.V., Hemispherical resonator gyro technology: Problems and possible ways of their solutions, Proc. St. Petersburg International Conference on Integrated Navigation Systems, Russia, 1999, pp. 61–69.Joshi, S., Hung, S., and Vengallatore, S., Design strategies for controlling damping in micromechanical and nanomechanical resonators, EPJ Techniques and Instrumentation, 2014, vol. 5, no. 1, pp. 1–14.Joshi, S., Hung, S., and Vengallatore, S., Design strategies for controlling damping in micromechanical and nanomechanical resonators, EPJ Techniques and Instrumentation, 2014, vol. 5, no. 1, pp. 1–14.Imboden, M., and Mohanty, P., Dissipation in nanoelectromechanical systems, Physics Reports, 2014, vol. 534, no. 3, pp. 89–146.Imboden, M., and Mohanty, P., Dissipation in nanoelectromechanical systems, Physics Reports, 2014, vol. 534, no. 3, pp. 89–146.Thomson, W., On the dynamical theory of heat, Earth and Environmental Science: Transactions of the Royal Society of Edinburgh, 1853, vol. 20, no. 2, pp. 261–288.Thomson, W., On the dynamical theory of heat, Earth and Environmental Science: Transactions of the Royal Society of Edinburgh, 1853, vol. 20, no. 2, pp. 261–288.Bishop, J.E., and Kinra, V.K., Elastothermodynamic damping in laminated composites, International Journal of Solids and Structures, 1997, vol. 34, no. 9, pp. 1075–1092.Bishop, J.E., and Kinra, V.K., Elastothermodynamic damping in laminated composites, International Journal of Solids and Structures, 1997, vol. 34, no. 9, pp. 1075–1092.Zener, C., Internal friction in solids. I. Theory of internal friction in reeds, Physical Review, 1937, vol. 52, no. 3, pp. 230–235.Zener, C., Internal friction in solids. I. Theory of internal friction in reeds, Physical Review, 1937, vol. 52, no. 3, pp. 230–235.Zener, C., Wills, A.P., Mitchell, D.P., and Farwell, H.W., Internal friction in solids: II. General theory of thermoelastic internal friction, Physical Review, 1938, vol. 53, no. 1, pp. 90–99.Zener, C., Wills, A.P., Mitchell, D.P., and Farwell, H.W., Internal friction in solids: II. General theory of thermoelastic internal friction, Physical Review, 1938, vol. 53, no. 1, pp. 90–99.Berry, B.S., Precise investigation of the theory of damping by transverse thermal currents, Journal of Applied Physics, 1955, vol. 26, no. 10, pp. 1221–1224.Berry, B.S., Precise investigation of the theory of damping by transverse thermal currents, Journal of Applied Physics, 1955, vol. 26, no. 10, pp. 1221–1224.Lifshitz, R., and Roukes, M.L., Thermoelastic damping in micro and nano mechanical systems, Physical Review B, 2000, vol. 61, no. 8, pp. 5600–5609.Lifshitz, R., and Roukes, M.L., Thermoelastic damping in micro and nano mechanical systems, Physical Review B, 2000, vol. 61, no. 8, pp. 5600–5609.Wong, S.J., Fox, C.H.J., McWilliam, S., Fell, C.P., and Eley, R., A preliminary investigation of thermoelastic damping in silicon rings, Journal of Micromechanics and Microengineering, 2004, vol. 14, no. 9, pp. S108–S113.Wong, S.J., Fox, C.H.J., McWilliam, S., Fell, C.P., and Eley, R., A preliminary investigation of thermoelastic damping in silicon rings, Journal of Micromechanics and Microengineering, 2004, vol. 14, no. 9, pp. S108–S113.De, S.K., and Aluru, N.R., Theory of thermoelastic damping in electrostatically actuated microstructures, Physical Review B, 2006, vol. 74, no. 14, p. 144305_1–13.De, S.K., and Aluru, N.R., Theory of thermoelastic damping in electrostatically actuated microstructures, Physical Review B, 2006, vol. 74, no. 14, p. 144305_1–13.Yi, Y., and Matin, M.A., Eigenvalue solution of thermoelastic damping in beam resonators using a finite element analysis, Journal of Vibration and Acoustics: Transactions of the ASME, 2007, vol. 129, no. 4, pp. 478–483.Yi, Y., and Matin, M.A., Eigenvalue solution of thermoelastic damping in beam resonators using a finite element analysis, Journal of Vibration and Acoustics: Transactions of the ASME, 2007, vol. 129, no. 4, pp. 478–483.Rajagopalan, J., and Saif, M.T.A., Single degree of freedom model for thermoelastic damping, Journal of Applied Mechanics: Transactions of the ASME, 2007, vol. 74, no. 3, pp. 461–468.Rajagopalan, J., and Saif, M.T.A., Single degree of freedom model for thermoelastic damping, Journal of Applied Mechanics: Transactions of the ASME, 2007, vol. 74, no. 3, pp. 461–468.Prabhakar, S., and Vengallatore, S., Theory of thermoelastic damping in micromechanical resonators with two-dimensional heat conduction, Journal of Microelectromechanical Systems, 2008, vol. 17, no. 2, pp. 494–502.Prabhakar, S., and Vengallatore, S., Theory of thermoelastic damping in micromechanical resonators with two-dimensional heat conduction, Journal of Microelectromechanical Systems, 2008, vol. 17, no. 2, pp. 494–502.Prabhakar, S., Paidoussis, M.P., and Vengallatore, S., Analysis of frequency shifts due to thermoelastic coupling in flexural-mode micromechanical and nanomechanical resonators, Journal of Sound and Vibration, 2009, vol. 323, no. 1, pp. 385–396.Prabhakar, S., Paidoussis, M.P., and Vengallatore, S., Analysis of frequency shifts due to thermoelastic coupling in flexural-mode micromechanical and nanomechanical resonators, Journal of Sound and Vibration, 2009, vol. 323, no. 1, pp. 385–396.Kinra, V.K., and Milligan, K.B., A second-law analysis of thermoelastic damping, Journal of Appled Mechanics: Transactions of the ASME, 1994, vol. 61, no. 1, pp. 71–76.Kinra, V.K., and Milligan, K.B., A second-law analysis of thermoelastic damping, Journal of Appled Mechanics: Transactions of the ASME, 1994, vol. 61, no. 1, pp. 71–76.Chandorkar, S.A., Candler, R.N., Duwel, A., Melamud, R., Agarwal, M., Goodson, K.E., and Kenny, T.W., Multimode thermoelastic dissipation, Journal of Applied Physics, 2009, vol. 105, no. 4, pp. 1–13.Chandorkar, S.A., Candler, R.N., Duwel, A., Melamud, R., Agarwal, M., Goodson, K.E., and Kenny, T.W., Multimode thermoelastic dissipation, Journal of Applied Physics, 2009, vol. 105, no. 4, pp. 1–13.Ghaffari, S., and Kenny, T.W., Thermoelastic dissipation in composite silicon MEMS resonators with thin film silicon dioxide coating, MRS Online Proceedings Library, 2012, vol. 1426, pp. 193–198.Ghaffari, S., and Kenny, T.W., Thermoelastic dissipation in composite silicon MEMS resonators with thin film silicon dioxide coating, MRS Online Proceedings Library, 2012, vol. 1426, pp. 193–198.Duwel, A., Candler, R.N., Kenny, T.W., and Varghese, M., Engineering MEMS resonators with low thermoelastic damping, Journal of Microelectromechanical Systems, 2006, vol. 15, no. 6, pp. 1437–1445.Duwel, A., Candler, R.N., Kenny, T.W., and Varghese, M., Engineering MEMS resonators with low thermoelastic damping, Journal of Microelectromechanical Systems, 2006, vol. 15, no. 6, pp. 1437–1445.COMSOL Multiphysics, Ver 5.3, COMSOL Inc., Massachusetts, U.S.A.COMSOL Multiphysics, Ver 5.3, COMSOL Inc., Massachusetts, U.S.A.Roszhart, T.V., The effect of thermoelastic internal friction on the Q of micromachined silicon resonators, IEEE 4th Technical Digest on Solid-State Sensor and Actuator Workshop, Hilton Head, SC, USA, 1990, pp. 13–16.Roszhart, T.V., The effect of thermoelastic internal friction on the Q of micromachined silicon resonators, IEEE 4th Technical Digest on Solid-State Sensor and Actuator Workshop, Hilton Head, SC, USA, 1990, pp. 13–16.Yasumura, K.Y., Stowe, T.D, Chow, E.M., Pfafman, T., Kenny, T.W., Stipe, B.C., and Rugar, D., Quality factor in micron and submicron thick cantilevers, Journal of Microelectromechanical Systems, 2000, vol. 9, no. 1, pp. 117–125.Yasumura, K.Y., Stowe, T.D, Chow, E.M., Pfafman, T., Kenny, T.W., Stipe, B.C., and Rugar, D., Quality factor in micron and submicron thick cantilevers, Journal of Microelectromechanical Systems, 2000, vol. 9, no. 1, pp. 117–125.Candler, R.N., Li, H., Lutz, M., Park, W., Partridge, A., Yama, G., and Kenny, T.W., Investigation of energy loss mechanisms in micromechanical resonators, Proc. 12th International Conference on Solid state Sensors, Actuators and Microsystems, USA, 2003, pp. 332–335.Candler, R.N., Li, H., Lutz, M., Park, W., Partridge, A., Yama, G., and Kenny, T.W., Investigation of energy loss mechanisms in micromechanical resonators, Proc. 12th International Conference on Solid state Sensors, Actuators and Microsystems, USA, 2003, pp. 332–335.Guo, F.L., and Rogerson, G.A., Thermoelastic coupling effect on a micro-machined beam machined beam resonator, Mechanics Research Communications, 2003, vol. 30, no. 6, pp. 513–518.Guo, F.L., and Rogerson, G.A., Thermoelastic coupling effect on a micro-machined beam machined beam resonator, Mechanics Research Communications, 2003, vol. 30, no. 6, pp. 513–518.Houston, B.H., Photiadis, D.M., Vignola, J.F., Marcus, M.H., Lui, X., Czaplewski, D., Sekaric, L., Butler, J., Pehrsson, P., and Bucaro, J.A., Loss due to transverse thermoelastic currents in microscale resonators, Material Science and Engineering A, 2004, vol. 370, no. 1, pp. 407–411.Houston, B.H., Photiadis, D.M., Vignola, J.F., Marcus, M.H., Lui, X., Czaplewski, D., Sekaric, L., Butler, J., Pehrsson, P., and Bucaro, J.A., Loss due to transverse thermoelastic currents in microscale resonators, Material Science and Engineering A, 2004, vol. 370, no. 1, pp. 407–411.Sun, Y., Fang, D., and Soh, A.K., Thermoelastic damping in micro-beam resonators, International Journal of Solids and Structures, 2006, vol. 43, no. 10, pp. 3213–3229.Sun, Y., Fang, D., and Soh, A.K., Thermoelastic damping in micro-beam resonators, International Journal of Solids and Structures, 2006, vol. 43, no. 10, pp. 3213–3229.Candler, R.N., Duwel, A., Varghese, M., Chandorkar, S.A., Hopcroft, M.A., Park, W.T., Kim, B., Yama, G., Partridge, A., Lutz, M., Kenny, T.W., Impact of geometry on thermoelastic dissipation in micromechanical resonant beams, Journal of Microelectromechanical Systems, 2006, vol. 15, no. 4, pp. 927–934.Candler, R.N., Duwel, A., Varghese, M., Chandorkar, S.A., Hopcroft, M.A., Park, W.T., Kim, B., Yama, G., Partridge, A., Lutz, M., Kenny, T.W., Impact of geometry on thermoelastic dissipation in micromechanical resonant beams, Journal of Microelectromechanical Systems, 2006, vol. 15, no. 4, pp. 927–934.Darvishian, A., Nagourney, T., Cho, J.Y., Shiari, B., and Najafi, K., Thermoelastic dissipation in micromachined birdbath shell resonators, Journal of Microelectromechanical Systems, 2017, vol. 26, no. 4, pp. 758–772.Darvishian, A., Nagourney, T., Cho, J.Y., Shiari, B., and Najafi, K., Thermoelastic dissipation in micromachined birdbath shell resonators, Journal of Microelectromechanical Systems, 2017, vol. 26, no. 4, pp. 758–772.Wong, S.J., Fox, C.H.J, and McWilliam, S., Thermoelastic damping of the in-plane vibration of thin silicon rings, Journal of Sound and Vibration, 2006, vol. 293, no. 1–2, pp. 266–285.Wong, S.J., Fox, C.H.J, and McWilliam, S., Thermoelastic damping of the in-plane vibration of thin silicon rings, Journal of Sound and Vibration, 2006, vol. 293, no. 1–2, pp. 266–285.Duwel, A., Gorman, J., Weinstein, M., Borenstein, J., and Ward, P., Experimental study of thermoelastic damping in MEMS gyros, Sensors and Actuators A: Physical, 2003, vol. 103, no. 1–2, pp. 70–75.Duwel, A., Gorman, J., Weinstein, M., Borenstein, J., and Ward, P., Experimental study of thermoelastic damping in MEMS gyros, Sensors and Actuators A: Physical, 2003, vol. 103, no. 1–2, pp. 70–75.Hwang, C., Some experiment on the vibration of a hemispherical shell, Journal of Applied Mechanics, 1966, vol. 33, no. 4, pp. 817–824.Hwang, C., Some experiment on the vibration of a hemispherical shell, Journal of Applied Mechanics, 1966, vol. 33, no. 4, pp. 817–824.Fan, S., Liu, G., and Wang, Z., On vibration of hemispherical shell using finite element method, Applied Mathematics and Mechanics, 1991, vol. 12, no. 12, pp. 1169–1175.Fan, S., Liu, G., and Wang, Z., On vibration of hemispherical shell using finite element method, Applied Mathematics and Mechanics, 1991, vol. 12, no. 12, pp. 1169–1175.Choi, S.Y., Na, Y.H., and Kim, J.H., Thermoelastic damping of inextensional hemispherical shell, Proc. of World Academy of Science, Engineering and Technology, 2009, vol. 56, pp. 198–203.Choi, S.Y., Na, Y.H., and Kim, J.H., Thermoelastic damping of inextensional hemispherical shell, Proc. of World Academy of Science, Engineering and Technology, 2009, vol. 56, pp. 198–203.Heidari, A., Chan, M.L., Yang, H.A., Jaramilo, G., Taheri-Tehrani, P., Fonda, P., Najar, H., Yamazaki, K., Lin, L., and Horsley, D.A., Micromachined polycrystalline diamond hemispherical shell resonators, Proc. 2013 Transducers and Eurosensors: The 17th International Conference on Solid-State Sensors, Actuators and Microsystems, China, 2013, pp. 2415–2418.Heidari, A., Chan, M.L., Yang, H.A., Jaramilo, G., Taheri-Tehrani, P., Fonda, P., Najar, H., Yamazaki, K., Lin, L., and Horsley, D.A., Micromachined polycrystalline diamond hemispherical shell resonators, Proc. 2013 Transducers and Eurosensors: The 17th International Conference on Solid-State Sensors, Actuators and Microsystems, China, 2013, pp. 2415–2418.Darvishian, A., Shiari, B., Cho, J.Y., Nagourney, T., and Najafi, K., Investigation of thermoelastic loss mechanism in shell resonators, Proc. ASME International Mechanical Engineering Congress and Exposition, Canada, 2014, pp. 39331–39336.Darvishian, A., Shiari, B., Cho, J.Y., Nagourney, T., and Najafi, K., Investigation of thermoelastic loss mechanism in shell resonators, Proc. ASME International Mechanical Engineering Congress and Exposition, Canada, 2014, pp. 39331–39336.Sorenson, L., Shao, P., and Ayazi, F., Bulk and surface thermoelastic dissipation in micro hemispherical shell resonators, Journal of Microelectromechanical Systems, 2015, vol. 24, no. 2, pp. 486–502.Sorenson, L., Shao, P., and Ayazi, F., Bulk and surface thermoelastic dissipation in micro hemispherical shell resonators, Journal of Microelectromechanical Systems, 2015, vol. 24, no. 2, pp. 486–502.Kim, B., Hopcroft, M.A., Candler, R.N., Jha, C.M., Agarwal, M., Melamud, R., Chandorkar, S.A., Yama, G., and Kenny, T.W., Temperature dependence of quality factor in MEMS resonators, Journal of Microelectromechanical Systems, 2008, vol. 17, no. 3, pp. 755–766.Kim, B., Hopcroft, M.A., Candler, R.N., Jha, C.M., Agarwal, M., Melamud, R., Chandorkar, S.A., Yama, G., and Kenny, T.W., Temperature dependence of quality factor in MEMS resonators, Journal of Microelectromechanical Systems, 2008, vol. 17, no. 3, pp. 755–766.Nayfeh, A.H., and Younis, M.I., Modeling and simulations of thermoelastic damping in microplates, Journal of Micromechanics and Microengineering, 2004, vol. 14, no. 12, pp. 1711–1717.Nayfeh, A.H., and Younis, M.I., Modeling and simulations of thermoelastic damping in microplates, Journal of Micromechanics and Microengineering, 2004, vol. 14, no. 12, pp. 1711–1717.Ghaffari, S., Ng, J.E., Ahn, C.H., Yang, Y., Wang, S., Hong, V.A., and Kenny, T.W., Accurate modelling of quality factor behaviour of complex silicon MEMS resonator, Journal of Microelectromechanical Systems, 2015, vol. 24, no. 2, pp. 276–288.Ghaffari, S., Ng, J.E., Ahn, C.H., Yang, Y., Wang, S., Hong, V.A., and Kenny, T.W., Accurate modelling of quality factor behaviour of complex silicon MEMS resonator, Journal of Microelectromechanical Systems, 2015, vol. 24, no. 2, pp. 276–288.Prabhakar, S., and Vengallatore, S., Thermoelastic damping in bilayered micromechanical beam resonators, Journal of Micromechanics and Microengineering, 2007, vol. 17, no. 3, pp. 532–538.Prabhakar, S., and Vengallatore, S., Thermoelastic damping in bilayered micromechanical beam resonators, Journal of Micromechanics and Microengineering, 2007, vol. 17, no. 3, pp. 532–538.Sun, Y., Yang, J., and Jiang, Y., A theoretical analysis of thermoelastic damping model in laminated trilayered circular plate resonators, World Journal of Mechanics, 2014, no. 4, pp. 102–111.Sun, Y., Yang, J., and Jiang, Y., A theoretical analysis of thermoelastic damping model in laminated trilayered circular plate resonators, World Journal of Mechanics, 2014, no. 4, pp. 102–111.Jimbo, Y., and Itao, K., Energy loss of a cantilever vibrator, Journal of Horological Institute of Japan, 1968, vol. 47, pp. 1–15.Jimbo, Y., and Itao, K., Energy loss of a cantilever vibrator, Journal of Horological Institute of Japan, 1968, vol. 47, pp. 1–15.Sharma, G.N., Sundararajan, T., and Gautam, S.S., Effect of geometric imperfections on anchor loss and characterisation of a Coriolis vibratory gyroscope resonator with high quality factor, Gyroscopy and Navigation, 2020, vol. 11, no. 3, pp. 206–213.Sharma, G.N., Sundararajan, T., and Gautam, S.S., Effect of geometric imperfections on anchor loss and characterisation of a Coriolis vibratory gyroscope resonator with high quality factor, Gyroscopy and Navigation, 2020, vol. 11, no. 3, pp. 206–213.Yang, J., Ono, T., and Esashi, M., Mechanical behavior of ultrathin microcantilever, Sensors and Actuators A: Physical, 2000, vol. 82, no. 1, pp. 102–107.Yang, J., Ono, T., and Esashi, M., Mechanical behavior of ultrathin microcantilever, Sensors and Actuators A: Physical, 2000, vol. 82, no. 1, pp. 102–107.Photiadis, D.M., and Judge, J.A., Attachment losses of high Q oscillators, Applied Physics Letters, 2004, vol. 85, no. 3, pp. 482–484.Photiadis, D.M., and Judge, J.A., Attachment losses of high Q oscillators, Applied Physics Letters, 2004, vol. 85, no. 3, pp. 482–484.Hao, Z., and Ayazi, F., Support loss in the radial bulk-mode vibrations of center-supported micromechanical disk resonators, Sensors and Actuators A: Physical, 2007, vol. 134, no. 2, pp. 582–593.Hao, Z., and Ayazi, F., Support loss in the radial bulk-mode vibrations of center-supported micromechanical disk resonators, Sensors and Actuators A: Physical, 2007, vol. 134, no. 2, pp. 582–593.Rabinovich, D., Givoli, D., and Becache, E., Comparison of high order absorbing boundary conditions and perfectly matched layers in the frequency domain, International Journal for Numerical Methods in Biomedical Engineering, 2010, vol. 26, no. 10, pp. 1351–1369.Rabinovich, D., Givoli, D., and Becache, E., Comparison of high order absorbing boundary conditions and perfectly matched layers in the frequency domain, International Journal for Numerical Methods in Biomedical Engineering, 2010, vol. 26, no. 10, pp. 1351–1369.Bindel, D.S., Quevy, E., Koyama, T., Govindjee, S., Demmel, J.W., and Howe, R.T., Anchor loss simulation in resonators, Proc. 18th IEEE International Conference on Microelectromechanical Systems, USA, 2005, pp. 133–136.Bindel, D.S., Quevy, E., Koyama, T., Govindjee, S., Demmel, J.W., and Howe, R.T., Anchor loss simulation in resonators, Proc. 18th IEEE International Conference on Microelectromechanical Systems, USA, 2005, pp. 133–136.Basu, U., and Chopra, A.K., Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation, Computer Methods in Applied Mechanics and Engineering, 2003, vol. 192, no. 11–12, pp. 1337–1375.Basu, U., and Chopra, A.K., Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation, Computer Methods in Applied Mechanics and Engineering, 2003, vol. 192, no. 11–12, pp. 1337–1375.Frangi, A., Bugada, A., Martello, M., and Savadkoohi, P.T., Validation of PML based models for the evaluation of anchor dissipation in MEMS resonators, European Journal of Mechanics A / Solids, 2013, vol. 37, pp. 256–265.Frangi, A., Bugada, A., Martello, M., and Savadkoohi, P.T., Validation of PML based models for the evaluation of anchor dissipation in MEMS resonators, European Journal of Mechanics A / Solids, 2013, vol. 37, pp. 256–265.Thakar, V., and Rais-Zadeh, M., Optimization of tether geometry to achieve low anchor loss in Lame mode resonators, Proc. Joint Europian Frequency and Time Forum and International Frequency Control Symposium, Czech Republic, 2013, pp. 129–132.Thakar, V., and Rais-Zadeh, M., Optimization of tether geometry to achieve low anchor loss in Lame mode resonators, Proc. Joint Europian Frequency and Time Forum and International Frequency Control Symposium, Czech Republic, 2013, pp. 129–132.Lee, J.E.Y., Yan, J., and Seshia, A.A., Quality factor enhancement of bulk acoustic resonators through anchor geometry design, Proc. 22nd International Conference on Solid State Sensors, Actuators and Microsystems, Germany, 2008, pp. 536–539.Lee, J.E.Y., Yan, J., and Seshia, A.A., Quality factor enhancement of bulk acoustic resonators through anchor geometry design, Proc. 22nd International Conference on Solid State Sensors, Actuators and Microsystems, Germany, 2008, pp. 536–539.Lee, J.E.Y., Yan, J., and Seshia, A.A., Study of lateral mode SOI-MEMS resonators for reduced anchor loss, Journal of Micromechanics and Microengineering, 2011, vol. 21, no. 4, p. 045010_1–10.Lee, J.E.Y., Yan, J., and Seshia, A.A., Study of lateral mode SOI-MEMS resonators for reduced anchor loss, Journal of Micromechanics and Microengineering, 2011, vol. 21, no. 4, p. 045010_1–10.Wang, K., Wong, A.C., and Nguyen, C.T.C., VHF free-free beam high Q micromechanical resonators, Journal of Microelectromechanical Systems, 2000, vol. 9, no. 3, pp. 347–360.Wang, K., Wong, A.C., and Nguyen, C.T.C., VHF free-free beam high Q micromechanical resonators, Journal of Microelectromechanical Systems, 2000, vol. 9, no. 3, pp. 347–360.Darvishian, A., Shiari, B., Guohong, H., and Najafi, K., Effect of substrate thickness on quality factor of mechanical resonators, Proc. IEEE International Symposium on Inertial Sensors and Systems, USA, 2015, pp. 1–4.Darvishian, A., Shiari, B., Guohong, H., and Najafi, K., Effect of substrate thickness on quality factor of mechanical resonators, Proc. IEEE International Symposium on Inertial Sensors and Systems, USA, 2015, pp. 1–4.Cho, J.Y., and Najafi, K., A high Q all fused silica solid stem wineglass hemispherical resonator formed using micro blow torching and welding, Proc. 28th IEEE International Conference on MEMS, Portugal, 2015, pp. 821–824.Cho, J.Y., and Najafi, K., A high Q all fused silica solid stem wineglass hemispherical resonator formed using micro blow torching and welding, Proc. 28th IEEE International Conference on MEMS, Portugal, 2015, pp. 821–824.Darvishian, A., Shiari, B., Cho, J.Y., Nagourney, T., and Najafi, K., Anchor loss in hemispherical shell resonators, Journal of Microelectromechanical Systems, 2017, vol. 26, no. 1, pp. 51–66.Darvishian, A., Shiari, B., Cho, J.Y., Nagourney, T., and Najafi, K., Anchor loss in hemispherical shell resonators, Journal of Microelectromechanical Systems, 2017, vol. 26, no. 1, pp. 51–66.Jeanroy, A., Bouvet, A., and Remillieux, G., HRG and marine applications, Gyroscopy and Navigation, 2014, vol. 5, no. 2, pp. 67–74.Jeanroy, A., Bouvet, A., and Remillieux, G., HRG and marine applications, Gyroscopy and Navigation, 2014, vol. 5, no. 2, pp. 67–74.Sorenson, L., and Ayazi, F., Effect of structural anisotropy on anchor loss mismatch and predicted case drift in future micro-hemispherical resonator gyros, Proc. IEEE/ION Position, Location and Navigation Symposium, USA, 2014, pp. 493–498.Sorenson, L., and Ayazi, F., Effect of structural anisotropy on anchor loss mismatch and predicted case drift in future micro-hemispherical resonator gyros, Proc. IEEE/ION Position, Location and Navigation Symposium, USA, 2014, pp. 493–498.Zotov, S.A., Simon, B.R., Prakhobko, I.P., Trusov, A.A., and Shkel, A.M., Quality factor maximization through dynamic balancing of tuning fork resonators, IEEE Sensors Journal, 2014, vol. 14, no. 8, pp. 2706–2714.Zotov, S.A., Simon, B.R., Prakhobko, I.P., Trusov, A.A., and Shkel, A.M., Quality factor maximization through dynamic balancing of tuning fork resonators, IEEE Sensors Journal, 2014, vol. 14, no. 8, pp. 2706–2714.Bindel, D.S., and Govindjee, S., Elastic PMLs for resonator anchor loss simulation, International Journal for Numerical Methods in Engineering, 2005, vol. 64, no. 6, pp. 789–818.Bindel, D.S., and Govindjee, S., Elastic PMLs for resonator anchor loss simulation, International Journal for Numerical Methods in Engineering, 2005, vol. 64, no. 6, pp. 789–818.Heidari, A., Chan, M.L., Yang, H.A., Jaramillo, G., Tehrani, P.T., Fonda, P., Najar, H., Yamazaki, K., Lin, L., and Horsley, D.A., Micromachined polycrystalline diamond hemispherical shell resonators, Proc. 17th International Conference on Solid State Sensors, Actuators and Microsystems, Spain, 2013, pp. 2415–2418.Heidari, A., Chan, M.L., Yang, H.A., Jaramillo, G., Tehrani, P.T., Fonda, P., Najar, H., Yamazaki, K., Lin, L., and Horsley, D.A., Micromachined polycrystalline diamond hemispherical shell resonators, Proc. 17th International Conference on Solid State Sensors, Actuators and Microsystems, Spain, 2013, pp. 2415–2418.Hwang, R.S., Fox, C.H.J., and McWilliam, S., Free vibrations of elliptical rings with circumferentially variable thickness, Journal of Sound and Vibration, 1999, vol. 228, no. 3, pp. 683–699.Hwang, R.S., Fox, C.H.J., and McWilliam, S., Free vibrations of elliptical rings with circumferentially variable thickness, Journal of Sound and Vibration, 1999, vol. 228, no. 3, pp. 683–699.Choi, S.Y., and Kim, J.H., Natural frequency split estimation for inextensional vibration of imperfect hemispherical shell, Journal of Sound and Vibration, 2011, vol. 330, no. 9, pp. 2094–2106.Choi, S.Y., and Kim, J.H., Natural frequency split estimation for inextensional vibration of imperfect hemispherical shell, Journal of Sound and Vibration, 2011, vol. 330, no. 9, pp. 2094–2106.Pan, Y., Qu, T., Wang, D., Wu, S., Liu, J., Tan, Z., Yang, K., and Luo, H., Observation and analysis of the quality factor variation behaviour in a monolithic fused silica cylindrical resonator, Sensors and Actuators A: Physical, 2017, vol. 260, pp. 81–89.Pan, Y., Qu, T., Wang, D., Wu, S., Liu, J., Tan, Z., Yang, K., and Luo, H., Observation and analysis of the quality factor variation behaviour in a monolithic fused silica cylindrical resonator, Sensors and Actuators A: Physical, 2017, vol. 260, pp. 81–89.Gretarsson, A.M., and Harry, G.M., Dissipation of mechanical energy in fused silica fibers, Review of Scientific Instruments, 1999, vol. 70, no. 10, pp. 4081–4087.Gretarsson, A.M., and Harry, G.M., Dissipation of mechanical energy in fused silica fibers, Review of Scientific Instruments, 1999, vol. 70, no. 10, pp. 4081–4087.Picard, A., Barat, P., Borys, M., Firlus, M., and Mizushima, S., State of the art mass determination of 28Si spheres for the Avogadro project, Metrologia, 2011, vol. 48, no. 2, pp. S112–S119.Picard, A., Barat, P., Borys, M., Firlus, M., and Mizushima, S., State of the art mass determination of 28Si spheres for the Avogadro project, Metrologia, 2011, vol. 48, no. 2, pp. S112–S119.Mihailovich, R.E., and MacDonald, N.C., Dissipation measurements in vacuum operated single crystal silicon micro resonators, Sensors and Actuators A: Physical, 1995, vol. 50, no. 3, pp. 199–207.Mihailovich, R.E., and MacDonald, N.C., Dissipation measurements in vacuum operated single crystal silicon micro resonators, Sensors and Actuators A: Physical, 1995, vol. 50, no. 3, pp. 199–207.Ahamed, M.J., Senkal, D., and Shkel, A.M., Effect of annealing on mechanical quality factor of fused quartz hemispherical resonator, Proc. IEEE International Symposium on Inertial Sensors and Systems, USA, 2014, pp. 1–4.Ahamed, M.J., Senkal, D., and Shkel, A.M., Effect of annealing on mechanical quality factor of fused quartz hemispherical resonator, Proc. IEEE International Symposium on Inertial Sensors and Systems, USA, 2014, pp. 1–4.Nagourney, T., Cho, J.Y., Darvishian, A., Shiari, B., and Najafi, K., Effect of metal annealing on the Q-factor of metal coated fused silica micro shell resonators, Proc. IEEE International Symposium on Inertial Sensors and Systems, Hapuna, HI, U.S.A., 2015, pp. 1–5.Nagourney, T., Cho, J.Y., Darvishian, A., Shiari, B., and Najafi, K., Effect of metal annealing on the Q-factor of metal coated fused silica micro shell resonators, Proc. IEEE International Symposium on Inertial Sensors and Systems, Hapuna, HI, U.S.A., 2015, pp. 1–5.Saito, D., Yang, C., Heidari, A., Najar, H., Lin, L., Horsley, D.A., Microcrystalline diamond cylindrical resonators with quality factor up to 0.5 million, Applied Physics Letters, 2016, vol. 108, no. 5, p. 051904_1-5.Saito, D., Yang, C., Heidari, A., Najar, H., Lin, L., Horsley, D.A., Microcrystalline diamond cylindrical resonators with quality factor up to 0.5 million, Applied Physics Letters, 2016, vol. 108, no. 5, p. 051904_1-5.Bernstein, J.J., Bancu, M.G., Bauer, J.M., Cook, E.H., Kumar, P., Newton, E., Nyinjee, T., Perlin, G.E., Ricker, J.A., Teynor, W.A., and Weinberg, M.S., High Q diamond hemispherical resonators: fabrication and energy loss mechanisms, Journal of Micromechanics and Microengineering, 2015, vol. 25, no. 8, p. 085006 _1-12.Bernstein, J.J., Bancu, M.G., Bauer, J.M., Cook, E.H., Kumar, P., Newton, E., Nyinjee, T., Perlin, G.E., Ricker, J.A., Teynor, W.A., and Weinberg, M.S., High Q diamond hemispherical resonators: fabrication and energy loss mechanisms, Journal of Micromechanics and Microengineering, 2015, vol. 25, no. 8, p. 085006 _1-12.Pan, Y., Wang, D.Y., Wang, Y.Y., Liu, J.P., Wu, S.Y., Qu, T.L., Yang, K.Y., and Luo, H., Monolithic cylindrical fused silica resonators with high Q-factors, MDPI Sensors, 2016, vol. 16, no. 8, p. 1185_1-14.Pan, Y., Wang, D.Y., Wang, Y.Y., Liu, J.P., Wu, S.Y., Qu, T.L., Yang, K.Y., and Luo, H., Monolithic cylindrical fused silica resonators with high Q-factors, MDPI Sensors, 2016, vol. 16, no. 8, p. 1185_1-14.Guzzo, P.L., Shinohara, A.H., and Raslan, A.A., A comparative study on ultrasonic machining of hard and brittle materials, Journal of Brazilian Society of Mechanical Science and Engineering, 2004, vol. 26, no. 1, pp. 56–61.Guzzo, P.L., Shinohara, A.H., and Raslan, A.A., A comparative study on ultrasonic machining of hard and brittle materials, Journal of Brazilian Society of Mechanical Science and Engineering, 2004, vol. 26, no. 1, pp. 56–61.Palasantzas, G., Surface roughness influence on the quality factor of high frequency nano resonators, Journal of Applied Physics, 2008, vol. 103, no. 4, p. 046106_1-3.Palasantzas, G., Surface roughness influence on the quality factor of high frequency nano resonators, Journal of Applied Physics, 2008, vol. 103, no. 4, p. 046106_1-3.Matveev, V.A., Lunin, B.S., and Basarab, M.A., Solid-State Wave Gyro Navigation System, Harbin: Harbin Institute of Technology Press, 2013.Matveev, V.A., Lunin, B.S., and Basarab, M.A., Solid-State Wave Gyro Navigation System, Harbin: Harbin Institute of Technology Press, 2013.Shiari, B., and Najafi, K., Surface effect influence on the quality factor of microresonators, Proc. 17th International Conference on Solid state Sensors, Actuators and Microsystems, Spain, 2013, pp. 1715–1718.Shiari, B., and Najafi, K., Surface effect influence on the quality factor of microresonators, Proc. 17th International Conference on Solid state Sensors, Actuators and Microsystems, Spain, 2013, pp. 1715–1718.Numata, K., Yamamoto, K., Ishimoto, H., Otsuka, S., Kawabe, K., Ando, M., and Tsubono, K., Systematic measurement of the intrinsic losses in various kinds of bulk fused silica, Physical Letters: A, 2004, vol. 327, no. 4, pp. 263–271.Numata, K., Yamamoto, K., Ishimoto, H., Otsuka, S., Kawabe, K., Ando, M., and Tsubono, K., Systematic measurement of the intrinsic losses in various kinds of bulk fused silica, Physical Letters: A, 2004, vol. 327, no. 4, pp. 263–271.Penn, S.D., Harry, G.M., Gretarsson, A.M., Kittelberger, S.E., Saulson, P.R., Schiller, J.J., Smith, J.R., and Swords, S.O., High quality factor measured in fused silica, Review of Scientific Instruments, 2001, vol. 72, no. 9, pp. 3670–3673.Penn, S.D., Harry, G.M., Gretarsson, A.M., Kittelberger, S.E., Saulson, P.R., Schiller, J.J., Smith, J.R., and Swords, S.O., High quality factor measured in fused silica, Review of Scientific Instruments, 2001, vol. 72, no. 9, pp. 3670–3673.Ageev, A., Palmer, B.C., Felice, A.D., Penn, S.D., and Saulson, P.R., Very high quality factor measured in annealed fused silica, Classical and Quantum Gravity, 2004, vol. 21, no. 16, pp. 3887–3892.Ageev, A., Palmer, B.C., Felice, A.D., Penn, S.D., and Saulson, P.R., Very high quality factor measured in annealed fused silica, Classical and Quantum Gravity, 2004, vol. 21, no. 16, pp. 3887–3892.Walther, H., Internal friction in solids, The Scientific Monthly, 1935, vol. 41, no. 3, pp. 275–277.Walther, H., Internal friction in solids, The Scientific Monthly, 1935, vol. 41, no. 3, pp. 275–277.White, B.E., and Pohl, R.O., Internal friction of subnanometer α-SiO films, Physical Review Letters, 1995, vol. 75, no. 24, pp. 4437–4439.White, B.E., and Pohl, R.O., Internal friction of subnanometer α-SiO films, Physical Review Letters, 1995, vol. 75, no. 24, pp. 4437–4439.Li, T.J., and Bellon, L., Dissipation of micro-cantilevers as a function of air pressure and metallic coating, Europhysics Letters, 2011, vol. 98, no. 1, p. 14004_1-4.Li, T.J., and Bellon, L., Dissipation of micro-cantilevers as a function of air pressure and metallic coating, Europhysics Letters, 2011, vol. 98, no. 1, p. 14004_1-4.Sandberg, R., Molhave, K., Boisen, A., and Svendsen, W., Effect of gold coating on the Q-factor of a resonant cantilever, Journal of Micromechanics and Microengineering, 2005, vol. 15, no. 12, pp. 2249–2253.Sandberg, R., Molhave, K., Boisen, A., and Svendsen, W., Effect of gold coating on the Q-factor of a resonant cantilever, Journal of Micromechanics and Microengineering, 2005, vol. 15, no. 12, pp. 2249–2253.Sosale, G., Prabhakar, S., Frechette, L., and Vengallatore, S., A microcantilever platform for measuring internal friction in thin films using thermoelastic damping for calibration, Journal of Microelectromechanical Systems, 2011, vol. 20, no. 3, pp. 764–773.Sosale, G., Prabhakar, S., Frechette, L., and Vengallatore, S., A microcantilever platform for measuring internal friction in thin films using thermoelastic damping for calibration, Journal of Microelectromechanical Systems, 2011, vol. 20, no. 3, pp. 764–773.Sosale, G., Das, K., Frechette, L., and Vengallatore, S., Controlling damping and quality factors of silicon microcantilevers by selective metallization, Journal of Micromechanics and Microengineering, 2011, vol. 21, no. 10, p. 105010_1-7.Sosale, G., Das, K., Frechette, L., and Vengallatore, S., Controlling damping and quality factors of silicon microcantilevers by selective metallization, Journal of Micromechanics and Microengineering, 2011, vol. 21, no. 10, p. 105010_1-7.Newell, W.E., Miniaturization of tuning forks, Science: New Series, 1968, vol. 161, pp. 1320–1326.Newell, W.E., Miniaturization of tuning forks, Science: New Series, 1968, vol. 161, pp. 1320–1326.Aoust, G., Levy, R., Bourgeteau, B., and Traon, O.L., Viscous damping on flexural mechanical resonators, Sensors and Actuators A, 2015, vol. 230, pp. 126–135.Aoust, G., Levy, R., Bourgeteau, B., and Traon, O.L., Viscous damping on flexural mechanical resonators, Sensors and Actuators A, 2015, vol. 230, pp. 126–135.Rodriguez, J., Chandorkar, S.A., Glaze, G.M., Gerrard, D.D., Chen, Y., Heinz, D.B., Flader, I.B., and Kenny, T.W., Direct detection of anchor damping in MEMS tuning fork resonators, Journal of Microelectromechanical Systems, 2018, vol. 27, no. 5, pp. 800–809.Rodriguez, J., Chandorkar, S.A., Glaze, G.M., Gerrard, D.D., Chen, Y., Heinz, D.B., Flader, I.B., and Kenny, T.W., Direct detection of anchor damping in MEMS tuning fork resonators, Journal of Microelectromechanical Systems, 2018, vol. 27, no. 5, pp. 800–809.Wan, Q., Gu, H., Fan, B., Zhao, H., Xu, D., and Guo, S., A high symmetry polysilicon micro hemispherical resonating gyroscope with spherical electrodes, Proc. IEEE Sensors, Glasgow, UK, 2017, p. 10_1-3.Wan, Q., Gu, H., Fan, B., Zhao, H., Xu, D., and Guo, S., A high symmetry polysilicon micro hemispherical resonating gyroscope with spherical electrodes, Proc. IEEE Sensors, Glasgow, UK, 2017, p. 10_1-3.Zotov, S.A., Simon, B.R., Sharma, G., Han, J., Prikhodko, I.P., Trusov, A.A., and Shkel, A.M., Investigation of energy dissipation in low frequency vibratory MEMS demonstrating a resonator with 25 minutes time constant, Proc. 2014 Solid-State Sensors, Actuators and Microsystems Conference, USA, 2014, pp. 76–79.Zotov, S.A., Simon, B.R., Sharma, G., Han, J., Prikhodko, I.P., Trusov, A.A., and Shkel, A.M., Investigation of energy dissipation in low frequency vibratory MEMS demonstrating a resonator with 25 minutes time constant, Proc. 2014 Solid-State Sensors, Actuators and Microsystems Conference, USA, 2014, pp. 76–79.

The authors declare that there are no conflicts of interest present.