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Bauman Moscow State Technical University.   El № FS 77 - 48211.   ISSN 1994-0408

Using a Combination of FEM and Perturbation Method in Frequency Split Calculation of a Nearly Axisymmetric Shell with Middle Surface Shape Defect

# 05, May 2016
DOI: 10.7463/0516.0839190
Article file: SE-BMSTU...o174.pdf (1706.68Kb)
authors: D.S. Vakhlyarskiy1,*, A.M. Guskov1, M.A. Basarab1, V.A. Matveev1

1 Bauman Moscow State Technical University, Moscow, Russia

This paper proposes a method to calculate the splitting of natural frequency of the shell of hemispherical resonator gyro. (HRG). The paper considers splitting that arises from the small defect of the middle surface, which makes the resonator different from the rotary shell. The presented method is a combination of the perturbation method and the finite element method. The method allows us to find the frequency splitting caused by defects in shape, arbitrary distributed in the circumferential direction. This is achieved by calculating the perturbations of multiple natural frequencies of the second and higher orders. The proposed method allows us to calculate the splitting of multiple frequencies for the shell with the meridian of arbitrary shape.
A developed finite element is an annular element of the shell and has two nodes. Projections of movements are used on the axis of the global cylindrical system of coordinates, as the unknown. To approximate the movements are used polynomials of the second degree. Within the finite element the geometric characteristics are arranged in a series according to the small parameter of perturbations of the middle surface geometry.
Movements on the final element are arranged in series according to the small parameter, and in a series according to circumferential angle. With computer used to implement the method, three-dimensional arrays are used to store the perturbed quantities. This allows the use of regular expressions for the mass and stiffness matrices, when building the finite element, instead of analytic dependencies for each perturbation of these matrices of the required order with desirable mathematical operations redefined in accordance with the perturbation method.
As a test task, is calculated frequency splitting of non-circular cylindrical resonator with Navier boundary conditions. The discrepancy between the results and semi-analytic solution to this problem is less than 1%. For a cylindrical shell is made a comparison of results with solution in ANSYS commercial complex - a difference is less than 1%. For a hemispherical shell was found the frequency splitting. The comparison has shown that a discrepancy between the results and ANSYS solution is less than 1%. The solution of this problem allows us to estimate further a mutual influence of defects of different nature (shape, thickness, density, modulus of elasticity, etc.) on splitting the frequency of the HRG. This is an urgent problem in terms of balancing the HRG resonators.

  1. Heidari A., Chan M-L., Yang H-A., Jaramillo G., Taheri-Tehrani P., Fonda P., Najar N., Yamazaki K., Lin L., Horsley D. A. Hemispherical wineglass resonators fabricated from the microcrystalline diamond. Journal of Micromechanics and Microengineering, 2013, vol. 23, no. 12, 8 p. DOI: 10.1088/0960-1317/23/12/125016
  2. Pai P., Chowdhury F.K., Mastrangelo C.H., Tabib-Azar M., MEMS-Based hemispherical resonator gyroscopes. Conference: Sensors, 2012 IEEE. DOI: 10.1109/ICSENS.2012.6411346
  3. Lunin B.S., Matveyev V.A., Basarab M.A. Volnovoy tverdotel'nyy giroskop. Teoriya i tekhnologiya [Hemispherical resonator gyroscope. Theory and technology]. Moscow, Radiotekhnika Publ., 2014. 176 p. (in Russian).
  4. Hwang R.S., Fox C.H.J., and McWilliam S. The in-plane vibration of thin rings with in-plane profile variations. Part I: General background and theoretical formulation. Journal of Sound and Vibration, 1999, vol. 220, no. 3, pp. 497-516. DOI: 10.1006/jsvi.1998.1963
  5. Fox C.H.J., Hwang R.S., and McWilliam S. The in-plane vibration of thin rings with in-plane profile variations. Part II: Application to nominally circular rings. Journal of Sound and Vibration, 1999, vol. 220, no. 3, pp. 517-539. DOI: 10.1006/jsvi.1998.1962
  6. Yilmaz E. and Bindel D. Effects of imperfections on solid-wave gyroscope dynamics. Proc. IEEE Sensors, Baltimore, MD, USA, Nov. 2013, pp. 1331–1334.
  7. Sato K. Free flexural vibrations of an elliptical ring in its plane. J. Acoust. Soc. Am., 1975, vol. 57, no. 1, pp.113-115. DOI: 10.1121/1.380420
  8. Brigham G.A. In-plane free vibrations of tapered oval rings. The Journal of the Acoustical Society of America. 1973. vol. 54. no. 2. pp. 451-460.
  9. Novozhilov V.V. Teoriya tonkikh obolochek [Thin shells theory]. St Petersburg, SPnU Publ., 2010. 380 p. (in Russian).
  10. Karachun V.V., Mel'nik V.N. Dynamic equations for rotational shell with zero Gaussian curvature and spontaneous shape of meridian line. Vestnik dvigatelestroyeniya, 2009, no. 3, pp.29-36. (in Russian).
  11. Basarab M.A., Kravchenko V.F., Matveyev V.A., Pustovoyt V.I. Atomic functions in the problem of determining Rayleigh function and precession coefficient for hemispherical resonator gyro. Doklady Akademii Nauk, 2001, vol. 376, no. 4. pp. 474-479. (in Russian).
  12. Collatz L. Eigenwertaufgaben mit technischen anwendungen. Akademische verlagsgesellschaft Geest & Portig K.-G., 1963. 500 s.
  13. Merkur'yev I.V., Podalkov V.V. Dinamika mikromekhanicheskogo i volnovogo tverdo-tel'nogo giroskopov [The dynamics of MEMS and hemispherical resonator gyroscopes]. Moscow, Fizmatlit Publ., 2009. 228 p. (in Russian).
  14. Astakhov S.V. Nelineynyye effekty v dinamike volnovogo tverdotel'nogo i mikromekhanicheskogo giroskopov v usloviyakh medlenno menyayushchikhsya parametrov. Diss. kand. tekhn. nauk. [Nonlinear effects in the dynamics of hemispherical resonator and MEMS gyroscopes in case of slowly varying parameters: cand. tech. sci. diss.]. Moscow, 2012. 157 p. (in Russian).
  15. Donnik A.S. Vliyaniye geometricheskoy neodnorodnosti i uprugoy anizotropii mate-riala na tochnostnyye kharakteristiki volnovogo tverdotel'nogo giroskopa: diss. kand. tekhn. nauk. [Influence of geometrical inhomogeneity and material elastic anisotropy on accuracy of a hemispherical resonator gyroscope: cand. tech. sci. diss.]. Moscow, 2006. 131 p. (in Russian).
  16. Lunin B.S. Fiziko-khimicheskiye osnovy razrabotki polusfericheskikh rezonatorov volnovykh tverdotel'nykh giroskopov [Physical-and-chemical bases of designing hemispherical resonators of solid-state wave gyroscopes]. Moscow, MAI Publ., 2005. 224 p. (in Russian).
  17. Biderman V.L. Mekhanika tonkostennykh konstruktsiy. Statika [Mechanics of thin-walled structures. Statics]. Moscow, Mashinostroyeniye Publ., 1977. 488 p. (in Russian).
  18. Heidari A., Chan M., Yang H., Jaramillo G., Taheri-Tehrani P., Fonda P., Najar H., Yamazaki K., Lin L., Horsley D. Hemispherical wineglass resonators fabricated from the microcrystalline diamond. Journal of Micromechanics and Microengineering, 2013, vol. 23, no. 12, pp. 125016-23(8). DOI: 10.1088/0960-1317/23/12/125016
  19. Kozubnyak S.A. Splitting of natural frequencies of cylindrical resonator gyro due to non-ideal shape. Vestnik MGTU im. N.E. Baumana. Ser. Priborostroyeniye. = Ser. Instrument Engineering, 2015. no. 3, pp. 39–49. (in Russian). DOI: 10.18698/0236-3933-2015-3-39-49
  20. Lankaster P. Teoriya matrits [Theory of matrices]. Moscow, Nauka Publ., 1973. 280 p. (in Russian).
  21. Madelung E. Matematicheskiy apparat fiziki [Mathematical apparatus of physics]. Moscow, Fizmatlit Publ., 1949. 618 p. (in Russian).
  22. Naraykin O.S., Sorokin F.D., Kozubnyak S.A Natural frequencies splitting of a ring resonator of a solid-state gyroscope, caused by shape imperfection. Vestnik MGTU im. N.E. Baumana. Ser. Mashinostroyeniye = Ser. Mechanical Engineering, 2012, no. 6, pp. 176-185. (in Russian).
  23. Golovanov A.I., Tyuleneva O.N., Shigabutdinov A.F. Metod konechnykh elementov v statike i dinamike tonkostennykh konstruktsiy [Finite element method in statics and dynamics of thin-walled structures]. Moscow, Fizmatlit Publ., 2006. 392 p. (in Russian).
  24. Matveyev V.A., Basarab M.A., Lunin B.S. Approximation of density distribution of solid-state wave gyro resonator with respect to measured disbalance parameters. Pribory i sistemy. Upravleniye, kontrol', diagnostika = Instruments and Systems: Monitoring, Control, and Diagnostics, 2015, no. 10, pp. 9-16. (in Russian).
  25. Basarab M.A., Lunin B.S., Matveyev V.A., Chumankin Ye.A. HRG resonator balancing by chemical etching. Giroskopiya i navigatsiya, 2015, vol. 88, no. 1, pp.61-70. (in Russian).
  26. Basarab M.A., Lunin B.S., Matveyev V.A., Chumankin Ye.A. Static balancing of cylindrical resonators of wave solid-state gyroscopes. Giroskopiya i navigatsiya, 2014, vol. 85, no. 2, pp. 43-51. (in Russian).
  27. Chang Chia-Ou, Chang Guo-En, Chou Chan-Shin, et al. In-plane free vibration of a single-crystal silicon ring. Int. Journal of Solids and Structures, 2008, vol. 45, pp. 6114-6132. DOI: 10.1016/j.ijsolstr.2008.07.033
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