Gyroscopic effects in vibrating fluid-filled spheres subjected to inertial rotation

Abstract

In symmetric distributed structures subjected to vibration and an inertial rotation, the vibrating patterns turn in the direction of revolution at a rate proportional to the inertial angular rate. This effect has numerous applications in navigational instruments, such as hemispherical rotational sensor. It is also important for astrophysics and seismology to understand the dynamics of pulsating stars and earthquake series. The coefficient of proportionality between the inertial and vibrating pattern rates depends on the geometry of structure, mode number, etc, and plays a crucial role in this study. In this paper the authors consider gyroscopic effects in hollow solid spheres filled with an inviscid fluid. The dynamics of the sphere are considered in terms of linear elasticity. Two limiting cases of the fluid motion are considered: in the first case, we suppose that the fluid is fully involved in the rotation; in the second, the fluid does not rotate relative to the inertial reference frame. It is also assumed that the angular rate is constant and much smaller than the lowest eigenvalue of the system. Hence centrifugal effects, proportional to the square of the angular rate, are considered to be negligible. The effects of structure prestress due to gravitational forces are also neglected. Two types of nonaxisymmetric modes of the system are considered, namely spheroidal and torsional. A numerical experimental observation is made that, for lower eigenvalues and lower circumferential wave numbers, the difference between the modulus of the rotational angular rates of the fluid-filled sphere and those of its vibrating patterns is small. However, this difference is large for higher modes and eigenvalues of the system