A transient contact problem with moving boundaries for a thin elastic cylindrical shell and a rigid indenter bounded by a smooth convex surface is considered. The shell model is formulated as a 2D constrained continuum on the background of a general higher-order shell theory A closed mathematical formulation is given and a system of resolving equations is constructed. The solution to the problem of the transient function for the cylindrical shell is obtained. An algorithm for solving the spatial transient contact problem for an infinitely long cylindrical shell and absolutely rigid indenter in the case of a normal impact on the side surface of the shell is developed and implemented.

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