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Using Fourier series to analyse mass imperfections in vibratory gyroscopes. (English) Zbl 1282.74034
Summary: When a vibrating structure is subjected to a rotation, the vibrating pattern rotates at a rate (called the precession rate) proportional to the inertial angular rate. This is known as Bryan’s effect and it is employed to calibrate the resonator gyroscopes that are used to navigate in outer space, the stratosphere and under the polar cap. We study Bryan’s effect for a non-ideal resonator gyroscope, using the computer algebra system (CAS) Mathematica to do the analysis involved, rendering this work accessible to undergraduate students with a working knowledge of college calculus and basic physics or mechanics (such as senior Engineering Mathematics students). In this paper the density of a slowly rotating vibrating annular disc is assumed to have small variations circumferentially, enabling a Fourier series representation of the density function. Using a CAS, the Lagrangian of the system of vibrating particles in the disc is calculated and, employing the CAS on the Euler-Lagrange equations, the equations of motion of the vibrating, rotating system are calculated in terms of “fast” variables, enabling us to demonstrate that the mass anisotropy induces a frequency splitting (beats). Unfortunately the fast variables are difficult to analyse (even with the aid of a CAS) and consequently a transformation from fast to slow variables is achieved. These slow variables are the principal and quadrature vibration amplitudes, precession rate and a phase angle. The transformation yields a system of four nonlinear ordinary differential equations (ODEs). This system of ODE demonstrates that the Fourier coefficients of the density function influence the precession rate and consequently a gyroscope manufactured from such a disc cannot use Bryan’s effect for calibration purposes. Indeed, the CAS visualises that a capture effect occurs with the precession angle that appears to vary periodically and not increase linearly (Bryan’s effect) as it would for a perfect structure. Keeping in mind that manufacturing imperfections will always be present in the real-world, the analysis shows how such density variations may be minimised. Using a symbolic manipulator such as Mathematica to do the “book-keeping” eliminates the plethora of technical detail that arises during calculations of a highly technical nature. This allows the aforementioned students to focus on the salient parts of the analysis, producing results that might have been beyond their capabilities without the aid of a CAS.
##### MSC:
 74H45 Vibrations in dynamical problems in solid mechanics 74S30 Other numerical methods in solid mechanics (MSC2010) 70E05 Motion of the gyroscope
##### Software:
MACSYMA; Mathematica
Full Text:
##### References:
 [1] Abouzahra, M. D.; Pavelle, R., Computer algebra applied to radiation from microstrip discontinuities, J. Symb. Comput., 10, 5, 525-528, (1990), accessed October 2013 · Zbl 0709.73067 [2] Benham, P. P.; Crawford, R. J., Mechanics of engineering materials, 381, (1991), Longman Scientific Harlow, Essex [3] Bryan, G. H., On the beats in the vibrations of a revolving cylinder or Bell, Proc. Camb. Philos. Soc. Math. Phys. Sci., 7, 101-114, (1890) [4] Caviness, B. F., Computer algebra: past and future, J. Symb. Comput., 2, 3, 217-236, (1986), accessed October 2013 · Zbl 0636.68033 [5] Eisenberger, M., Application of symbolic algebra to the analysis of plates on variable elastic foundation, J. Symb. Comput., 9, 2, 207-214, (1990), accessed October 2103 · Zbl 0713.73084 [6] Friedland, B.; Hutton, M. F., Theory and error analysis of vibrating member gyroscope, IEEE Trans. Autom. Control, AC-23, 4, 545-556, (1978) · Zbl 0381.93036 [7] Joubert, S. V.; Fay, T. H.; Voges, E. L., A storm in a wineglass, Am. J. Phys., 75, 7, 647-651, (2007) [8] Joubert, S. V.; Shatalov, M. Y.; Fay, T. H., Rotating structures and bryanʼs effect, Am. J. Phys., 77, 6, 520-525, (2009), accesed October 2103 [9] Joubert, S. V.; Shatalov, M. Y.; Fay, T. H., A CAS routine for obtaining eigenfunctions for bryanʼs effect, (Proceedings of the TIME2010 Conference, Malaga, Spain, (2010)), paper number D025 [10] Loper, E. J.; Lynch, D. D., Hemispherical resonator gyroscope, (1990), accessed September 2012 [11] Lynch, D. D., Vibratory gyro analysis by the method of averaging, (Proceedings of the 2nd Saint Petersburg International Conference on Integrated Navigation Systems, (1995)), 26-34 [12] Moses, J., Macsyma: a personal history, J. Symb. Comput., 47, 2, 123-130, (September 2012), accessed September 2013 [13] Rozelle, D. M., The hemispherical resonator gyro: from wineglass to the planets, (Proceedings of the 19th AAS/AIAA Space Flight Mechanics Meeting, Savannah, Georgia, (2009)), 1157-1178, accessed October 2013 [14] Shatalov, M. Y.; Joubert, S. V.; Coetzee, C. E., The influence of mass imperfections on the evolution of standing waves in slowly rotating spherical bodies, J. Sound Vib., 330, 127-135, (2011), accessed September 2012 [15] Spiegel, M. R., Theoretical mechanics, (1967), McGraw-Hill New York, pp. 148, 284
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