A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

Abstract

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.