Normal modes of oscillations of a rotating membrane with a rigid central insert (an application of Heun functions)

The eigenvalues problem eigenvalues for the equation of the transverse oscillations of a homogeneous annular membrane with a rigid insert, rotating with a constant angular velocity about its central axis, is considered. Exact analytical expressions for the eigenfunctions in terms of special functions (local Heun functions), as well as normalization integrals, are found. An explicit expression for the time-invariant shape of the membrane during regular precession of its rotation axis is obtained.