Nullspace method based vibration analysis of functionally graded graphene origami-enabled auxetic metamaterial stepped cylindrical shells

In this work, a novel semi-analytical solution for the natural frequency of functionally graded (FG) stepped cylindrical shell made of graphene origami (GOri)-enabled auxetic metamaterials is presented. The vibration characteristic of the stepped shells is transformed into a nonstandard eigenvalue problem that is solved by using the nullspace method for the first time. A comprehensive study on the negative Poisson’s ratio affecting the natural frequencies of the FG shell is presented. Five types of GOri distribution patterns along the thickness direction are compared. The relationship between the maximum values of GOri volume fractions are calculated to ensure the equal mass of the shell. Based on the modified Chebyshev polynomials that satisfy the constraint conditions of each shell segment, the junction conditions between the contiguous shell segments are transformed into a matrix form related to the eigenvector restraints, and the characteristic equations are derived by using Lagrange equations. The influences of GOri parameters and boundary conditions on the natural frequencies of the FG stepped shells are highlighted. The results reveal that the process of Poisson’s ratio changing from positive to negative values correspond a reduction of effective stiffness followed by an increase of effective stiffness of the FG shells.