Influences of material gradient and nonlinearity on the forced vibration of orthotropic shell structures

In this study, the influences of material gradient and nonlinearity on the forced vibration of orthotropic shell structures under external excitations are investigated for first time. The mathematical model of inhomogeneous orthotropic double?curved shallow shells is built using the Hamilton principle and von Karman?type nonlinearity. The basic equations are reduced to nonlinear ordinary differential equations using the Galerkin procedure. Using the multiscale method, the frequency?amplitude relations of double?curved shallow shells and the nonlinear frequency response of forced vibrations are obtained for first time. The reliability of the obtained expressions is checked by comparison with the literature data. In numerical analysis, the influence of inhomogeneity, orthotropy, nonlinearity, and the external excitation parameter on the frequency of forced vibrations is investigated in detail by performing unique numerical calculations taking into account various profiles of inhomogeneous orthotropic shallow spherical and hyperbolic paraboloidal (or hypar) shells.