Abstract

The advanced mechanical testing of microelectromechanical systems (MEMS) is necessary to provide feedback of measurements that can help the designer optimize MEMS structures and improve the reliability and stability of MEMS. We describe a digital image correlation (DIC) method for dynamic characterization of MEMS using an optical microscope with a high-speed complementary metaloxide semiconductor-based camera. The mechanical performance of a series of microgyroscopes is tested. The DIC method is employed to measure the microgyroscope in-plane displacement with subpixel accuracy. Use of the DIC method is less restrictive on the surface quality of the specimen and simplifies the measurement system. On the basis of a series of temporal digital images grabbed by a high-speed camera, the stability characteristic of the microgyroscopes is analyzed. In addition, the quality factors of the microgyroscopes are determined and agree well with other experimental methods.

© 2006 Optical Society of America

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  1. J. H. Zhao, Y. Du, M. Morgen, and P. S. Ho, "Simultaneous measurement of Young's modulus, Poisson ratio, and coefficient of thermal expansion of thin films on substrates," J. Appl. Phys. 87, 1575-1577 (2000).
    [CrossRef]
  2. N. A. Fawcett, "A novel method for the measurement of Young's modulus for thick-film resistor material by flexural testing of coated beams," Meas. Sci. Technol. 9, 2023-2026 (1998).
    [CrossRef]
  3. M. Qin and V. M. C. Poon, "Young's modulus measurement of nickel silicide on crystal silicon by a surface profiler," J. Mater. Sci. Lett. 19, 2243-2245 (2000).
    [CrossRef]
  4. B. T. Comella and M. R. Scanlon, "The determination of the elastic modulus of microcantilever beam using atomic force microscopy," J. Mater. Sci. 35, 567-572 (2000).
    [CrossRef]
  5. S. H. Wang, C. J. Tay, C. Quan, and H. M. Shang, "Determination of deflection and Young's modulus of a micro-beam by means of interferometry," Meas. Sci. Technol. 12, 1279-1285 (2001).
    [CrossRef]
  6. S. Nakano, R. Maeda, and K. Yamanaka, "Evaluation of the elastic properties of a cantilever using resonant frequencies," Jpn. J. Appl. Phys. 36, 3265-3266 (1997).
    [CrossRef]
  7. T. Yi and C. J. Kim, "Measurement of mechanical properties for MEMS materials," Meas. Sci. Technol. 10, 706-716 (1999).
    [CrossRef]
  8. C. Rembe and E. P. Tibken Bm Hofer, "Analysis of the dynamics in microactuators using high-speed cine photomicrography," J. Microelectromech. Syst. 10, 137-145 (2001).
    [CrossRef]
  9. C. Rembe and R. S. Muller, "Measurement system for full three-dimensional motion characterization of MEMS," J. Microelectromech. Syst. 11, 479-488 (2002).
    [CrossRef]
  10. M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, "Stroboscopic interferometer system for dynamic MEMS characterization," J. Microelectromech. Syst. 9, 409-418 (2000).
    [CrossRef]
  11. W. G. Knauss, I. Chasiotis, and Y. Huang, "Mechanical measurements at the micron and nanometer scales," Mech. Mater. 35, 217-231 (2003).
    [CrossRef]
  12. W. H. Peters, H. Zheng-Hui, M. A. Sutton, and W. F. Ranson, "Two-dimensional fluid velocity measurements by use of digital speckle correlation techniques," Exp. Mech. 24, 117-121 (1984).
    [CrossRef]
  13. M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill "Application of an optimized digital correlation method to planar deformation analysis," Image Vision Comput. 4, 143-150 (1986).
    [CrossRef]
  14. J. Moreland, "Micromechanical instruments for ferromagnetic measurements," J. Phys. D 36, R39-R51 (2003).
    [CrossRef]
  15. K. Yum, Z. Y. Wang, P. A. Suryavanshi, and M. F. Yu, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 96, 3933-3938 (2003).
    [CrossRef]
  16. J. W. Chon, P. Mulvaney, and J. E. Sader, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 87, 3978-3988 (2000).
    [CrossRef]
  17. H. Benaroya, Mechanical Vibration: Analysis, Uncertainties, and Control (Prentice-Hall, 1998).

2003 (3)

W. G. Knauss, I. Chasiotis, and Y. Huang, "Mechanical measurements at the micron and nanometer scales," Mech. Mater. 35, 217-231 (2003).
[CrossRef]

J. Moreland, "Micromechanical instruments for ferromagnetic measurements," J. Phys. D 36, R39-R51 (2003).
[CrossRef]

K. Yum, Z. Y. Wang, P. A. Suryavanshi, and M. F. Yu, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 96, 3933-3938 (2003).
[CrossRef]

2002 (1)

C. Rembe and R. S. Muller, "Measurement system for full three-dimensional motion characterization of MEMS," J. Microelectromech. Syst. 11, 479-488 (2002).
[CrossRef]

2001 (2)

S. H. Wang, C. J. Tay, C. Quan, and H. M. Shang, "Determination of deflection and Young's modulus of a micro-beam by means of interferometry," Meas. Sci. Technol. 12, 1279-1285 (2001).
[CrossRef]

C. Rembe and E. P. Tibken Bm Hofer, "Analysis of the dynamics in microactuators using high-speed cine photomicrography," J. Microelectromech. Syst. 10, 137-145 (2001).
[CrossRef]

2000 (5)

J. H. Zhao, Y. Du, M. Morgen, and P. S. Ho, "Simultaneous measurement of Young's modulus, Poisson ratio, and coefficient of thermal expansion of thin films on substrates," J. Appl. Phys. 87, 1575-1577 (2000).
[CrossRef]

M. Qin and V. M. C. Poon, "Young's modulus measurement of nickel silicide on crystal silicon by a surface profiler," J. Mater. Sci. Lett. 19, 2243-2245 (2000).
[CrossRef]

B. T. Comella and M. R. Scanlon, "The determination of the elastic modulus of microcantilever beam using atomic force microscopy," J. Mater. Sci. 35, 567-572 (2000).
[CrossRef]

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, "Stroboscopic interferometer system for dynamic MEMS characterization," J. Microelectromech. Syst. 9, 409-418 (2000).
[CrossRef]

J. W. Chon, P. Mulvaney, and J. E. Sader, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 87, 3978-3988 (2000).
[CrossRef]

1999 (1)

T. Yi and C. J. Kim, "Measurement of mechanical properties for MEMS materials," Meas. Sci. Technol. 10, 706-716 (1999).
[CrossRef]

1998 (2)

N. A. Fawcett, "A novel method for the measurement of Young's modulus for thick-film resistor material by flexural testing of coated beams," Meas. Sci. Technol. 9, 2023-2026 (1998).
[CrossRef]

H. Benaroya, Mechanical Vibration: Analysis, Uncertainties, and Control (Prentice-Hall, 1998).

1997 (1)

S. Nakano, R. Maeda, and K. Yamanaka, "Evaluation of the elastic properties of a cantilever using resonant frequencies," Jpn. J. Appl. Phys. 36, 3265-3266 (1997).
[CrossRef]

1986 (1)

M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill "Application of an optimized digital correlation method to planar deformation analysis," Image Vision Comput. 4, 143-150 (1986).
[CrossRef]

1984 (1)

W. H. Peters, H. Zheng-Hui, M. A. Sutton, and W. F. Ranson, "Two-dimensional fluid velocity measurements by use of digital speckle correlation techniques," Exp. Mech. 24, 117-121 (1984).
[CrossRef]

Benaroya, H.

H. Benaroya, Mechanical Vibration: Analysis, Uncertainties, and Control (Prentice-Hall, 1998).

Chao, Y. J.

M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill "Application of an optimized digital correlation method to planar deformation analysis," Image Vision Comput. 4, 143-150 (1986).
[CrossRef]

Chasiotis, I.

W. G. Knauss, I. Chasiotis, and Y. Huang, "Mechanical measurements at the micron and nanometer scales," Mech. Mater. 35, 217-231 (2003).
[CrossRef]

Cheng, M.

M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill "Application of an optimized digital correlation method to planar deformation analysis," Image Vision Comput. 4, 143-150 (1986).
[CrossRef]

Chon, J. W.

J. W. Chon, P. Mulvaney, and J. E. Sader, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 87, 3978-3988 (2000).
[CrossRef]

Comella, B. T.

B. T. Comella and M. R. Scanlon, "The determination of the elastic modulus of microcantilever beam using atomic force microscopy," J. Mater. Sci. 35, 567-572 (2000).
[CrossRef]

Conant, R. A.

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, "Stroboscopic interferometer system for dynamic MEMS characterization," J. Microelectromech. Syst. 9, 409-418 (2000).
[CrossRef]

Du, Y.

J. H. Zhao, Y. Du, M. Morgen, and P. S. Ho, "Simultaneous measurement of Young's modulus, Poisson ratio, and coefficient of thermal expansion of thin films on substrates," J. Appl. Phys. 87, 1575-1577 (2000).
[CrossRef]

Fawcett, N. A.

N. A. Fawcett, "A novel method for the measurement of Young's modulus for thick-film resistor material by flexural testing of coated beams," Meas. Sci. Technol. 9, 2023-2026 (1998).
[CrossRef]

Hart, M. R.

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, "Stroboscopic interferometer system for dynamic MEMS characterization," J. Microelectromech. Syst. 9, 409-418 (2000).
[CrossRef]

Ho, P. S.

J. H. Zhao, Y. Du, M. Morgen, and P. S. Ho, "Simultaneous measurement of Young's modulus, Poisson ratio, and coefficient of thermal expansion of thin films on substrates," J. Appl. Phys. 87, 1575-1577 (2000).
[CrossRef]

Huang, Y.

W. G. Knauss, I. Chasiotis, and Y. Huang, "Mechanical measurements at the micron and nanometer scales," Mech. Mater. 35, 217-231 (2003).
[CrossRef]

Kim, C. J.

T. Yi and C. J. Kim, "Measurement of mechanical properties for MEMS materials," Meas. Sci. Technol. 10, 706-716 (1999).
[CrossRef]

Knauss, W. G.

W. G. Knauss, I. Chasiotis, and Y. Huang, "Mechanical measurements at the micron and nanometer scales," Mech. Mater. 35, 217-231 (2003).
[CrossRef]

Lau, K. Y.

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, "Stroboscopic interferometer system for dynamic MEMS characterization," J. Microelectromech. Syst. 9, 409-418 (2000).
[CrossRef]

Maeda, R.

S. Nakano, R. Maeda, and K. Yamanaka, "Evaluation of the elastic properties of a cantilever using resonant frequencies," Jpn. J. Appl. Phys. 36, 3265-3266 (1997).
[CrossRef]

McNeill, S. R.

M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill "Application of an optimized digital correlation method to planar deformation analysis," Image Vision Comput. 4, 143-150 (1986).
[CrossRef]

Moreland, J.

J. Moreland, "Micromechanical instruments for ferromagnetic measurements," J. Phys. D 36, R39-R51 (2003).
[CrossRef]

Morgen, M.

J. H. Zhao, Y. Du, M. Morgen, and P. S. Ho, "Simultaneous measurement of Young's modulus, Poisson ratio, and coefficient of thermal expansion of thin films on substrates," J. Appl. Phys. 87, 1575-1577 (2000).
[CrossRef]

Muller, R. S.

C. Rembe and R. S. Muller, "Measurement system for full three-dimensional motion characterization of MEMS," J. Microelectromech. Syst. 11, 479-488 (2002).
[CrossRef]

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, "Stroboscopic interferometer system for dynamic MEMS characterization," J. Microelectromech. Syst. 9, 409-418 (2000).
[CrossRef]

Mulvaney, P.

J. W. Chon, P. Mulvaney, and J. E. Sader, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 87, 3978-3988 (2000).
[CrossRef]

Nakano, S.

S. Nakano, R. Maeda, and K. Yamanaka, "Evaluation of the elastic properties of a cantilever using resonant frequencies," Jpn. J. Appl. Phys. 36, 3265-3266 (1997).
[CrossRef]

Peters, W. H.

M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill "Application of an optimized digital correlation method to planar deformation analysis," Image Vision Comput. 4, 143-150 (1986).
[CrossRef]

W. H. Peters, H. Zheng-Hui, M. A. Sutton, and W. F. Ranson, "Two-dimensional fluid velocity measurements by use of digital speckle correlation techniques," Exp. Mech. 24, 117-121 (1984).
[CrossRef]

Poon, V. M. C.

M. Qin and V. M. C. Poon, "Young's modulus measurement of nickel silicide on crystal silicon by a surface profiler," J. Mater. Sci. Lett. 19, 2243-2245 (2000).
[CrossRef]

Qin, M.

M. Qin and V. M. C. Poon, "Young's modulus measurement of nickel silicide on crystal silicon by a surface profiler," J. Mater. Sci. Lett. 19, 2243-2245 (2000).
[CrossRef]

Quan, C.

S. H. Wang, C. J. Tay, C. Quan, and H. M. Shang, "Determination of deflection and Young's modulus of a micro-beam by means of interferometry," Meas. Sci. Technol. 12, 1279-1285 (2001).
[CrossRef]

Ranson, W. F.

W. H. Peters, H. Zheng-Hui, M. A. Sutton, and W. F. Ranson, "Two-dimensional fluid velocity measurements by use of digital speckle correlation techniques," Exp. Mech. 24, 117-121 (1984).
[CrossRef]

Rembe, C.

C. Rembe and R. S. Muller, "Measurement system for full three-dimensional motion characterization of MEMS," J. Microelectromech. Syst. 11, 479-488 (2002).
[CrossRef]

C. Rembe and E. P. Tibken Bm Hofer, "Analysis of the dynamics in microactuators using high-speed cine photomicrography," J. Microelectromech. Syst. 10, 137-145 (2001).
[CrossRef]

Sader, J. E.

J. W. Chon, P. Mulvaney, and J. E. Sader, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 87, 3978-3988 (2000).
[CrossRef]

Scanlon, M. R.

B. T. Comella and M. R. Scanlon, "The determination of the elastic modulus of microcantilever beam using atomic force microscopy," J. Mater. Sci. 35, 567-572 (2000).
[CrossRef]

Shang, H. M.

S. H. Wang, C. J. Tay, C. Quan, and H. M. Shang, "Determination of deflection and Young's modulus of a micro-beam by means of interferometry," Meas. Sci. Technol. 12, 1279-1285 (2001).
[CrossRef]

Suryavanshi, P. A.

K. Yum, Z. Y. Wang, P. A. Suryavanshi, and M. F. Yu, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 96, 3933-3938 (2003).
[CrossRef]

Sutton, M. A.

M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill "Application of an optimized digital correlation method to planar deformation analysis," Image Vision Comput. 4, 143-150 (1986).
[CrossRef]

W. H. Peters, H. Zheng-Hui, M. A. Sutton, and W. F. Ranson, "Two-dimensional fluid velocity measurements by use of digital speckle correlation techniques," Exp. Mech. 24, 117-121 (1984).
[CrossRef]

Tay, C. J.

S. H. Wang, C. J. Tay, C. Quan, and H. M. Shang, "Determination of deflection and Young's modulus of a micro-beam by means of interferometry," Meas. Sci. Technol. 12, 1279-1285 (2001).
[CrossRef]

Tibken Bm Hofer, E. P.

C. Rembe and E. P. Tibken Bm Hofer, "Analysis of the dynamics in microactuators using high-speed cine photomicrography," J. Microelectromech. Syst. 10, 137-145 (2001).
[CrossRef]

Wang, S. H.

S. H. Wang, C. J. Tay, C. Quan, and H. M. Shang, "Determination of deflection and Young's modulus of a micro-beam by means of interferometry," Meas. Sci. Technol. 12, 1279-1285 (2001).
[CrossRef]

Wang, Z. Y.

K. Yum, Z. Y. Wang, P. A. Suryavanshi, and M. F. Yu, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 96, 3933-3938 (2003).
[CrossRef]

Yamanaka, K.

S. Nakano, R. Maeda, and K. Yamanaka, "Evaluation of the elastic properties of a cantilever using resonant frequencies," Jpn. J. Appl. Phys. 36, 3265-3266 (1997).
[CrossRef]

Yi, T.

T. Yi and C. J. Kim, "Measurement of mechanical properties for MEMS materials," Meas. Sci. Technol. 10, 706-716 (1999).
[CrossRef]

Yu, M. F.

K. Yum, Z. Y. Wang, P. A. Suryavanshi, and M. F. Yu, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 96, 3933-3938 (2003).
[CrossRef]

Yum, K.

K. Yum, Z. Y. Wang, P. A. Suryavanshi, and M. F. Yu, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 96, 3933-3938 (2003).
[CrossRef]

Zhao, J. H.

J. H. Zhao, Y. Du, M. Morgen, and P. S. Ho, "Simultaneous measurement of Young's modulus, Poisson ratio, and coefficient of thermal expansion of thin films on substrates," J. Appl. Phys. 87, 1575-1577 (2000).
[CrossRef]

Zheng-Hui, H.

W. H. Peters, H. Zheng-Hui, M. A. Sutton, and W. F. Ranson, "Two-dimensional fluid velocity measurements by use of digital speckle correlation techniques," Exp. Mech. 24, 117-121 (1984).
[CrossRef]

Exp. Mech. (1)

W. H. Peters, H. Zheng-Hui, M. A. Sutton, and W. F. Ranson, "Two-dimensional fluid velocity measurements by use of digital speckle correlation techniques," Exp. Mech. 24, 117-121 (1984).
[CrossRef]

Image Vision Comput. (1)

M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill "Application of an optimized digital correlation method to planar deformation analysis," Image Vision Comput. 4, 143-150 (1986).
[CrossRef]

J. Appl. Phys. (3)

K. Yum, Z. Y. Wang, P. A. Suryavanshi, and M. F. Yu, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 96, 3933-3938 (2003).
[CrossRef]

J. W. Chon, P. Mulvaney, and J. E. Sader, "Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids," J. Appl. Phys. 87, 3978-3988 (2000).
[CrossRef]

J. H. Zhao, Y. Du, M. Morgen, and P. S. Ho, "Simultaneous measurement of Young's modulus, Poisson ratio, and coefficient of thermal expansion of thin films on substrates," J. Appl. Phys. 87, 1575-1577 (2000).
[CrossRef]

J. Mater. Sci. (1)

B. T. Comella and M. R. Scanlon, "The determination of the elastic modulus of microcantilever beam using atomic force microscopy," J. Mater. Sci. 35, 567-572 (2000).
[CrossRef]

J. Mater. Sci. Lett. (1)

M. Qin and V. M. C. Poon, "Young's modulus measurement of nickel silicide on crystal silicon by a surface profiler," J. Mater. Sci. Lett. 19, 2243-2245 (2000).
[CrossRef]

J. Microelectromech. Syst. (3)

C. Rembe and E. P. Tibken Bm Hofer, "Analysis of the dynamics in microactuators using high-speed cine photomicrography," J. Microelectromech. Syst. 10, 137-145 (2001).
[CrossRef]

C. Rembe and R. S. Muller, "Measurement system for full three-dimensional motion characterization of MEMS," J. Microelectromech. Syst. 11, 479-488 (2002).
[CrossRef]

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, "Stroboscopic interferometer system for dynamic MEMS characterization," J. Microelectromech. Syst. 9, 409-418 (2000).
[CrossRef]

J. Phys. D (1)

J. Moreland, "Micromechanical instruments for ferromagnetic measurements," J. Phys. D 36, R39-R51 (2003).
[CrossRef]

Jpn. J. Appl. Phys. (1)

S. Nakano, R. Maeda, and K. Yamanaka, "Evaluation of the elastic properties of a cantilever using resonant frequencies," Jpn. J. Appl. Phys. 36, 3265-3266 (1997).
[CrossRef]

Meas. Sci. Technol. (3)

T. Yi and C. J. Kim, "Measurement of mechanical properties for MEMS materials," Meas. Sci. Technol. 10, 706-716 (1999).
[CrossRef]

S. H. Wang, C. J. Tay, C. Quan, and H. M. Shang, "Determination of deflection and Young's modulus of a micro-beam by means of interferometry," Meas. Sci. Technol. 12, 1279-1285 (2001).
[CrossRef]

N. A. Fawcett, "A novel method for the measurement of Young's modulus for thick-film resistor material by flexural testing of coated beams," Meas. Sci. Technol. 9, 2023-2026 (1998).
[CrossRef]

Mech. Mater. (1)

W. G. Knauss, I. Chasiotis, and Y. Huang, "Mechanical measurements at the micron and nanometer scales," Mech. Mater. 35, 217-231 (2003).
[CrossRef]

Other (1)

H. Benaroya, Mechanical Vibration: Analysis, Uncertainties, and Control (Prentice-Hall, 1998).

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Figures (9)

Fig. 1
Fig. 1

(a) Part of the structure of the first microgyroscope sample. (b) Working principle of the vibratory microgyroscope.

Fig. 2
Fig. 2

Schematic of bilinear interpolation.

Fig. 3
Fig. 3

Attenuation of the free vibration.

Fig. 4
Fig. 4

Measurement result of one point on the first microgyroscope sample vibrating at its resonant frequency of 2065 Hz . (a) Displacement–time curves obtained by the proposed method. (b) Closeup of the subarea S in (a). (c) Velocity–time curve derived from (b).

Fig. 5
Fig. 5

Spectrum of Fig. 4(a) by the FFT method.

Fig. 6
Fig. 6

Damping motion curve of the first microgyroscope.

Fig. 7
Fig. 7

(a) Motion of one point on the second microgyroscope (resonant frequency 3906 Hz ). (b) Motion of one point on the third microgyroscope (resonant frequency 6966 Hz ).

Fig. 8
Fig. 8

Motion of one spring (folder beams) on the second microgyroscope during half of its vibration cycle. (a) Structure of the spring. (b)–(e) The reconstruction motions of the spring corresponding to time t = 0 , 1∕32, 3∕32, 5 / 32 ms , respectively.

Fig. 9
Fig. 9

(a) Motion in the y direction of one point on the first microgyroscope vibrating at excited frequency 2065 Hz . (b) Spectrum distribution of Fig. 9(a).

Tables (1)

Tables Icon

Table 1 Summary of Measurement Results of Mechanical Characteristics of Three Microgyroscope Samples

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

C ( x , y , t i ) = x = w w y = - w w [ f ( x , y , t i ) f ( x , y , t i ) m ] 2 [ f ( x , y , t i + 1 ) f ( x , y , t i + 1 ) m ] 2 x = - w w y = - w w [ f ( x , y , t i ) f ( x , y , t i ) m ] 2 x = - w w y = - w w [ f ( x , y , t i + 1 ) f ( x , y , t i + 1 ) m ] 2 ,
U ( u , v , t i ) = i = 0 N 1 U ( u , v , i ) .
f ( x , y , t i + 1 ) = ( 1 a ) ( 1 b ) f ( x 1 , y 1 , t i + 1 ) + b ( 1 a ) × f ( x 1 , y 2 , t i + 1 ) + a ( 1 b ) f ( x 2 , y 1 , t i + 1 ) + a b f ( x 2 , y 2 , t i + 1 ) ,
a = x x 1 x 2 x 1 , b = y y 1 y 2 y 1 .
x ( t ) = A exp ( ζ ω n t ) sin ( ω d t + α ) ,
ω d = ( 1 ζ 2 ) 1 / 2 ω n .
ζ = 1 2 π m ln A n A n + m ,

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