A Multiple Scale Approach to Nonlinear Vibrations and Stability of Doubly Curved Shells/Panels Under External and Parametric Excitations

Abstract

This study focuses on nonlinear vibrations and stability of doubly curved shells/panels. The thin-walled shell structure is assumed to be acted upon by external loads as well as parametric excitation. Using the first-order shear deformation Fülgge theory assumptions, governing equations of motion are derived for a general doubly curved geometry. Nonlinear components which are compatible with the Von-Karman kinematic assumptions are retained in the formulation and the rest of nonlinear components are neglected due to the fact that strains are assumed to be small and rotations to be moderate. The formulation can be readily reduced to any geometry including cylindrical shells and spherical panels, as benchmark examples in this study. Employing the Galerkin discretization approach and applying a modification to the Volmir assumption, a reduced order form of the original system is determined, which is shown to be a great representative of the original system with much more computational advantages. Using the method of multiple scales, the reduced order nonlinear governing equations are solved to discover the primary resonance, forced vibration characteristics and stability margins of thin-walled shells/panels under the action of parametric excitation. Several interesting findings on nonlinear behavior of cylindrical shells and spherical panels (as benchmark geometries) are presented while giving a demonstration of the accuracy and computational advantages of the proposed model.