One-to-one internal resonance of symmetric crossply laminated shallow shells

This paper presents the response of symmetric crossply laminated shallow sh ells with an internal resonance omega (2) approximate to omega (3), where o mega (2) and omega (3) are the linear natural frequencies of the asymmetric vibration modes (2, 1) and (1,2), respectively. Galerkin's procedure is ap plied to the nonlinear governing equations for the shells based on the von Karman-type geometric nonlinear theory and the first-order shear deformatio n theory, and the shooting method is used to obtain the steady-state respon se when a driving frequency Omega is near omega (2). In order to take into account the influence of quadratic nonlinearities, the displacement functio ns of the shells are approximated by the eigenfunctions for the linear vibr ation mode (1, 1) in addition to the ones for the modes (2, 1) and (1, 2). This approximation overcomes the shortcomings in Galerkin 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 its to conclude tha t the consideration of the (1,1) mode is indispensable for analyzing nonlin ear vibrations of asymmetric vibration modes of shells.