Precession of Vibrational Modes of a Rotating Hemispherical Shell

The precession of flexural vibrational modes of a rotating hemispherical thin shell is investigated. Niordson’s thin shell theory, which allows the stretch of the middle surface, is employed to derive the equations of bending vibration of a rotating shell. The shell is assumed to rotate at a low constant speed, so the centrifugal force is neglected and only the Coriolis inertial force is included in the equations of motion. The ratio of the rotating speed of the shell to the natural frequency of the first flexural mode, which is denoted by ε, is assumed small. The solutions of displacements are expanded in the power series of ε. The unperturbed (zero-order) equations, which represent the free vibration of the nonrotating shell, are proved to be self-adjoint. The perturbed frequency can be extracted from the equation of solvability condition directly without solving the perturbed system. The precession rate of the vibrational modes obtained theoretically in an analytical expression is verified by experiment. These results are helpful for the design of hemispherical resonant gyroscope.