A hierarchical generalized formulation for the large-displacement dynamic analysis of rotating plates

A hierarchical formulation is presented for the large displacement modal and transient analysis of rotating plates having different thicknesses, sizes, and settings angle with respect to the spinning axis. After derivation of the governing equations of motion by the Principle of Virtual Displacements, the finite element discretization yields a set of nonlinear ordinary-differential equations for the generalized coordinates including gyroscopic terms, centrifugal/Euler accelerations and spin-softening effects. A Total Lagrangian approach is adopted. The modal analysis is performed by a two-step procedure. The static shape of the structure deformed by the centrifugal force is first defined. Subsequently, eigenfrequencies and mode shapes are computed by a classical eigenvalue problem past the static configuration previously computed. The transient analysis of the plates subject to a varying angular velocity is tackled by solving the nonlinear equations of motion by the Generalized-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} method coupled to the Newmark’s approximation for the velocity and acceleration fields. It has been numerically proven that accurate results can be obtained by the use of Murakami’s Zig-Zag Function which a-priori enforces the interlaminar discontinuity of the displacements’slopes in the Equivalent Single Layer axiomatic model, thus avoiding higher-order polynomial representations for the displacement fields, or the use of Layer-Wise theories.