Large-amplitude vibration of functionally graded orthotropic double-curved shallow spherical and hyperbolic paraboloidal shells

Abstract

The purpose of this article is to study the large amplitude vibration behavior of functionally graded orthotropic double-curved shallow shells (FGODCSSs), such as the shallow spherical and hyperbolic paraboloidal shells. After mathematical modeling of the properties of the FG orthotropic material, von-Karman type non-linear basic relations are created, and at the next stage the non-linear equations of motion for double-curved shallow shells are derived. The non-linear basic partial differential equations of FGODCSSs are converted to non-linear ordinary differential equations using the principle of superposition and the Galerkin method. Then non-linear equations are solved by applying the method proposed by Grigolyuk [46] and get the expressions for the frequency-amplitude relationship and the ratio of the nonlinear frequency to the linear frequency for FGODCSSs. Using these expressions, the results are compared with the results in the literature, and after checking the reliability and accuracy of the proposed formulation, specific numerical calculations are performed. For specific analyzes, the homogenous and FG orthotropic shallow spherical and hyperbolic paraboloidal shells are used, and their large amplitude vibration behaviors are discussed in comparison with each other, and various examples reveal that the influence of heterogeneity is noticeable.