Precession of elastic waves in vibrating isotropic spheres and transversely isotropic cylinders subjected to inertial rotation

Abstract

It was found by G. Bryan in 1890 that vibrating pattern of a rotating ring follows to a direction of the inertial rotation of this ring with an angular rate of the vibrating pattern smaller than the inertial rate. In 1979 E. Loper and D. Lynch proposed a hemispherical vibrating bell gyroscope utilising the Bryan’s effect, which can measure an inertial angular rate and angle of rotation about the symmetry axis of the hemispherical shell. All these works exploited the precession properties of thin vibrating shells subjected to an inertial rotation around their axes of symmetry. In 1985 V. Zhuravlev generalized the abovementioned results and shown that the Bryan’s effect has a three dimensional nature, i.e. that a vibrating pattern of an isotropic spherically symmetric body, arbitrary rotating in 3-D space, follows the inertial rotation of the solid body with a proportionality factor depending on the vibrating mode. This result had a qualitative nature without classification of vibrating modes and calculation of the corresponding proportionality factors. In the present paper radial and torsional vibrational modes are considered on the basis of an exact solution of 3-D equations of motion of an isotropic body in spherical coordinates. The solutions are obtained by means of a three potentials method in the spherical Bessel and associated Legendre functions. The proportionality factors of corresponding vibrating modes are calculated. The effects of gyroscopic forces on wave propagation in a transversely isotropic cylinder due to the inertial rotation are considered. The solutions are expressed in Bessel functions for different modes and corresponding Bryan’s proportionality factors are calculated