Abstract <p>This investigation carries out a perturbation analysis of a viscously damped finite but imperfect cylindrical shell. This cylindrical shell is then trapped by a dynamically slowly varying loading history where the time variable is non – autonomously explicit in the formulation. Asymptotic expansions are utilized all through and the cylindrical shell is assumed to be simply – supported on both ends. The damping coefficient is of some order of imperfection magnitude while the loading history is assumed to be infinitely differentiable and has right hand derivatives of all orders at the initial time. Generally, the formulation becomes a one – small – parameter problem in the imperfection magnitude which is assumed small relative to unity. In the final analysis, a simple but implicit formula for determining the dynamic buckling load is obtained. It is observed that the dynamic buckling load depends, among other things, on the first derivative of the loading history evaluated at the initial time.</p>