Axial shear vibrations of a cylinder of decreasing thickness

The loss of mass and stiffness due to ablation may significantly influence the vibrations of a solid propellant grain. This paper presents an analytical study of the axial shear vibrations of a long hollow cylinder that is subjected to time-dependent body forces in the axial direction. The outer surface of the cylinder is bonded to a rigid case, and the inner radius increases monotonically with time. An expression is determined for the shear stress at the bond interface. It is shown that the instantaneous frequency of the shear bond stress increases, and that its amplitude decreases toward burnout time. Various ablation rates are considered. Conventional methods of analysis, such as separation of variables and Fourier-Bessel analysis, are not directly applicable in this problem, since the boundary conditions are prescribed on a time-dependent surface. A modified Fourier-Bessel mode is defined that satisfies the boundary conditions. By substituting this mode into the equation of motion, a solution is obtained by asymptotic methods in the vicinity of the bond interface. The axial shear vibrations of an ablating viscoelastic cylinder are discussed briefly.