Impact-Induced Internal Resonance Phenomena in Nonlinear Doubly Curved Shallow Shells with Rectangular Base

The problem of the low-velocity impact of an elastic sphere upon a nonlinear doubly curved shallow shell with a rectangular platform is investigated. The approach utilized in the present paper is based on the fact that during impact only the modes strongly coupled by some internal resonance condition are initiated. Such an approach differs from the Galerkin method, wherein resonance phenomena are not involved. Since is it assumed that shell’s displacements are finite, then the local bearing of the shell and impactor’s materials is neglected with respect to the shell deflection in the contact region. In other words, the Hertz’s theory, which is traditionally in hand for solving impact problems, is not used in the present study; instead, the method of multiple time scales is adopted, which is used with much success for investigating vibrations of nonlinear systems subjected to the conditions of the internal resonance. The influence of impactor’s mass on the phenomenon of the impact-induced internal resonance is revealed.