A study of the coupling effect of inertia terms and vibration modes on the dynamic stability of simply supported cylinders of finite or infinite length subjected to uniform radial impulsive pressure (as in space reentry or underwater explosion). In the prebuckling stage, the shells exhibit symmetrical motion; during buckling they oscillate, but if the oscillation is bounded rather than uncontrolled, the system is called dynamically stable. The well-known analogy between thermal stress and equivalent loading is used to derive an equivalent thermal stress problem. The coupling of radial and tangential inertia terms does not seem to be pronounced for radial pressure. The consideration of nonlinear terms in the stability equation is essential. Appendixes give the derivation of the problem, the two inertia terms, and the nonlinear effect. (Presented at the 5th U.S. National Congress of Applied Mechanics, Minneapolis, June 1966.) 18 pp. Refs.