Uncovering Physical Structural Model Behind Finite Difference Model for Vibration Analysis of Plates

Abstract

This paper uncovers a physical structural model for the finite difference model (FDM) for vibration analysis of plates. The physical model comprises a grid (or net) of rigid bar segments joined by rotational springs as well as a torsional spring system within the grid panels. We have named the physical model as the Hencky bar-net model (HBM) after Prof. Heinrich Hencky of Darmstadt TH who first proposed a similar discrete bar model in 1920. Based on the principle of minimum potential energy, it will be shown herein that the HBM has the same set of algebraic governing equations of motion as that of the FDM. Owing to the equivalency of the HBM and the FDM, the stiffnesses of elastic springs for the HBM at the edges may be calibrated by comparing its edge conditions with the FDM’s. Two plate vibration problems were solved as illustrative example problems to demonstrate the accuracy and efficiency of HBM. Also discussed are the interesting features of HBM which one could exploit for structural analyses.