The nonlinear, damped dynamic response of a simply supported, nearly circular cylindrical shell due to an exponentially decaying, uniform radial pressure is computed for peak pressures and total impulses between the static and dynamic buckling limits. The analysis employs Fourier series expansions of the dependent variables in the circumferential coordinate, finite difference approximations of the axial derivatives, and Newmark's beta-method for the time integration. All nonlinear coupling between the retained modes is included. The inclusion of damping reduces the maximum amplitudes of the dominant modes by approximately 30%, but stresses larger than the yield stress can occur at peak pressures and total impulses well below the dynamic buckling limit. The amplitudes of the static buckling modes are found to be linearly proportional to the magnitude of the initial imperfections, but the parametrically excited modes are unaffected by the initial imperfections.