In the radial motion of an elastic cylindrical shell following impulsive pressure all round, imperfections in the uniformity can cause development of flexure through interaction of (fluctuating) circumferential force with flexural curvature. Flexural modes of rather high order (several nodes in the circumference) can be strongly excited, especially those having a frequency one half that of the breathing mode (nonlinear autoparametric excitation). Almost all the original energy can be transferred, over a number of cycles, from the breathing mode to one or two high flexural modes.

In the motion of an elastic cylindrical shell following impulsive pressure, marked deviations from linear theory may occur through interaction of circumferential membrane force with flexural curvature. Energy is transferred to flexural modes of rather high order. The resulting flexural stress is significantly higher than that given by linear theory.

When a closed spherical shell is suddenly enveloped by a uniform impulse, the response is a purely radial motion of each shell element. If the impulse is slightly nonuniform, this membrane “breathing mode” may be dynamically unstable due to the interaction of the membrane stress with flexural curvature. The linear solution for axisymmetric motion divides into two sets of modes. One set, the membrane modes, is essentially extensional; the second set, the composite modes, is neither purely extensional nor purely inextensional. The linear response to an arbitrary radial velocity distribution indicates that breathing mode stability may be considered with respect to perturbations exciting only composite modes. Using this result the nonlinear study shows that a cyclic energy exchange may occur between the breathing mode and rather high-order composite modes. The ensuing deflections and stresses may be considerably in excess of those determined by consideration of the breathing mode only.

The effect of the spatial distribution of impulsive loads on dynamic snap-through of a shallow circular arch is considered in detail. The Budiansky-Roth criterion is used to establish critical magnitudes of the load for many distributions by numerical integration of an approximate set of equations obtained by Galerkin’s method. It is necessary to distinguish between snap-through on the initial oscillation (immediate) and snap-through occurring during a finite time of the response. Critical magnitudes for both cases are compared to a distribution independent lower bound obtained from an analysis of the critical points. For all distributions considered the lower bound is a less conservative estimate of the critical magnitude for finite time snap-through than for immediate snap-through. The effect of small damping on this conclusion, however, remains an open question.