Free vibration of rotating hollow spheres containing acoustic media

Abstract

When a vibrating structure is rotated with respect to inertial space, the vibrating pattern rotates at a rate proportional to the inertial rate of rotation. Bryan first observed this effect in 1890. The effect, called Bryan's effect in the sequel, has numerous navigational applications and could be useful in understanding the dynamics of pulsating stars and earthquake series in astrophysics and seismology. Bryan's factor (the coefficient of proportionality between the inertial and vibrating pattern rotation rates) depends on the geometry of the structure and the vibration mode number. The “gyroscopic effects” of a hollow isotropic solid sphere filled with an inviscid acoustic medium are considered here, but the theory is readily adapted to a hollow isotropic solid cylinder filled with an inviscid acoustic medium. A linear theory is developed assuming, among other mild conditions, that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. Thus centrifugal forces are considered to be negligible. Before calculating solutions for the displacement of a particle in the isotropic, spherical, distributed body, Bryan's factor is interpreted using a complex function. Here it is demonstrated that neither Bryan's effect nor Bryan's factor is influenced by including light, isotropic, viscous damping in the mathematical model. Hence damping is neglected in the sequel. Two scenarios are then identified. Firstly, we may assume that the acoustic medium is completely involved in the rotation (the spheroidal mode). Secondly, we may assume that the acoustic medium remains static with respect to the inertial reference frame (the torsional mode). We investigate the spheroidal mode using a numerical experiment that compares the rotational angular rate of a sphere (filled with an inviscid acoustic medium) with those of its vibrating patterns at both high and low vibration frequency.