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Technical Brief

Bryan's Factor for Truncated Spherical and Conical Shells

[+] Author and Article Information
K. Y. Narasimhan

Lockheed Martin Space Systems,
Sunnyvale, CA 94088-3504
e-mail: k.y.narasimhan@gmail.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 8, 2013; final manuscript received January 29, 2014; published online March 18, 2014. Assoc. Editor: Jeffrey F. Rhoads.

J. Vib. Acoust 136(3), 034502 (Mar 18, 2014) (5 pages) Paper No: VIB-13-1312; doi: 10.1115/1.4026674 History: Received September 08, 2013; Revised January 29, 2014

Closed-form expressions are derived for the Bryan's factor of truncated spherical and conical shells through a Galerkin procedure. Results lead to the value obtained by G. H. Bryan for the case of the ring, thereby demonstrating accuracy of the method. It is shown that the Bryan's factor depends only on the shape of the structure and the modes of vibration. The material properties are required to determine the resonating frequency and the Q-factor.

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References

Bryan, G. H., 1890, “On the Beats in the Vibrations of a Revolving Cylinder or Bell,” Proceedings of the Cambridge Philosophical Society, Vol. VII, Cambridge University Press, Cambridge, UK, pp. 101–111.
Joubert, S. V., Shatalov, M. Y., and Fay, T. H., 2009, “Rotating Structures and Bryan's Effect,” Am. J. Phys., 77(6), pp. 520–525. [CrossRef]
Joubert, S. V., Shatalov, M. Y., and Fay, T. H., 2010, “A CAS Routine for Obtaining Eigenfunctions for Bryan's Effect,” Technology and Its Integration Into Mathematics Education Conference (TIME2010), Malaga, Spain, July 6–10.
Joubert, S. V., Shatalov, M. Y., and Manzhirov, A.V., 2013, “Bryan's Effect and Isotropic Nonlinear Damping,” J. Sound Vib., 332(23), pp. 6169–6176. [CrossRef]
Narasimhan, K. Y., and Rozelle, D. M., 2013, “Hemispherical Resonator Gyros Used in SBIRS GEO-3 to GEO-6 Satellites (From Wine Glass to Planets),” Lockheed Martin Report.
Heidari, A., Chan, M. L., Yang, H. A., Jaramillo, G., Taheri-Tehrani, P., Fonda, P., Najar, H., Yamazaki, K., Lin, L., and Horsley, D. A., 2013, “Hemispherical Wineglass Resonators Fabricated From Microcrystalline Diamond,” J. Micromech. Microeng., 23(12), p. 125016. [CrossRef]
Zotov, S. A., Trusov, A. A., and Shkel, A. M., 2012, “Three-Dimensional Spherical Shell Resonator Gyroscope Fabricated Using Wafer-Scale Glassblowing,” JMEMS, 21(3), pp. 509–510. [CrossRef]
Sorenson, L. D., Gao, X., and Ayazi, F., 2012, “3-D Micromachined Hemispherical Shell Resonators With Integrated Capacitive Transducers,” IEEE 25th International Conference on Micro Electro Mechanical Systems (MEMS), Paris, January 29–February 2, pp. 168–171. [CrossRef]
Flugge, W., 1973, Stresses in Shells, Springer, New York.
Galerkin, B. G., 1915, “Rods and Plates,” Vestnik Inzhenerov, 19(1), pp. 897–908.

Figures

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Fig. 1

The spherical coordinate system

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Fig. 2

Microscale HRG made of polycrystalline diamond

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Fig. 3

Truncated spherical shell parameters

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Fig. 4

Bryan's factor for truncated spherical shell

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Fig. 5

Coordinate system for truncated conical shell

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Fig. 6

Truncated conical shell parameters

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Fig. 7

BF for truncated conical shell for varying α between 0 degree and 180 degrees

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