Thickness Shear Vibrations of an Ablating Rocket

The lost of nonentun and stiffness due to ablation nay 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 tine 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 tine. An expresaion is determined for the shear stress at the bond-interface. It is shown that thf frequency of the shear-bond stress increases, and that ita amplitude decreases towards bur.nout time. The shear stress is studied for various ablation ratea. Conventional methods of analysis, such aa 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 aatisfies the boundary conditions. By substituting this mode into the equation of motion, a solution la obtained by asymptotic methods in the vicinity of the bond-Interface. The analysis la extended to include the axial shear vibrations of an ablating viscoelastic cylinder. Vlscoelastlcity la Introduced by means of the relaxation function In shear. List of Major Symbols •(t).b m,n q.(C/p)Vf/b r,e,« t tr Vn e.c n n DCO Fl(xJ,t) G CW-CgCf) H(t) o o «CO M>'«<) n n VÜC (T),l] n n ■ inner radius at t 0 ■ Inner and outer radlua ■ mode numbers diownsionless burning time ■ polar coordinates time ■ burning time displacement component ■ constant Fourier-Bessel coefficients ■ constanta ■ dimensionless creep function ■ body force per unit mass ■ shear modulus ■ shear relaxation function ■ Meavlalde unit function • Besael functions of order tero ■ dlawnslonless body force • time dependent Fourier-Bessel coefficients mode of free vibration ■ ablation mode of free vibration W(R,) x r > n