One-to-One Internal Resonance of Symmetric Crossply Laminated Shallow Shells

This paper presents the response of symmetric crossply laminated shallow shells with an internal resonance $ω2≈ω3,$ where $ω2$ and $ω3$ are the linear natural frequencies of the asymmetric vibration modes (2,1) and (1,2), respectively. Galerkin’s procedure is applied to the nonlinear governing equations for the shells based on the von Ka´rma´n-type geometric nonlinear theory and the first-order shear deformation theory, and the shooting method is used to obtain the steady-state response when a driving frequency Ω is near $ω2.$ In order to take into account the influence of quadratic nonlinearities, the displacement functions of the shells are approximated by the eigenfunctions for the linear vibration mode (1,1) in addition to the ones for the modes (2,1) and (1,2). This approximation overcomes the shortcomings in Galerkin’s procedure. In the numerical examples, the effect of the (1,1) mode on the primary resonance of the (2,1) mode is examined in detail, which allows us to conclude that the consideration of the (1,1) mode is indispensable for analyzing nonlinear vibrations of asymmetric vibration modes of shells.