Spectral realization of the method of matched sections for thin-plate vibration

Abstract

The paper further develops the method of matched sections as a new universal numerical technique. Like the finite element method, FEM, it supposes that: a) the domain is represented as a mesh of simple elements; b) algebraic relations between the main parameters are established from the governing differential equations; c) relationships from all elements are assembled into one global matrix. The relations between the main parameters are established similarly to those for the corresponding 1D task, so any 2D problem is considered a combination of two 1D problems—one is x-dependent, and the other is y-dependent. In all technical aspects, this paper resembles our previous one, which was devoted to the static analysis of plate bending. It operates by the same governing parameters, the same structure of the dependencies between them as well as the organization of the calculation scheme. The only difference consists in connection equations (analogy of element interpolation functions), which establish the relationship between inlet parameters (about the left and lower sides of the element) and outlet parameters (the left and upper sides). They are derived as the frequency-dependent ones (frequency is explicitly used in them) and in the liming case of very long (narrow) plates coincide with the known beam frequency-dependent solution. Several practical examples demonstrate the efficiency of the proposed method.