A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems

Abstract

Citation: Liu, X. and Banerjee, J. R. (2017). A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems. Mechanical Systems and Signal Processing, 87, doi: 10.1016/j.ymssp.2016.10.017 This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent AbstractA highly efficient and accurate analytical spectral dynamic stiffness (SDS) method for modal analysis of plane elastodynamic problems based on both plane stress and plane strain assumptions is presented in this paper. First, the general solution satisfying the governing differential equation exactly is derived by applying two types of one-dimensional modified Fourier series. Then the SDS matrix for an element is formulated symbolically using the exact general solution. The SDS matrices are assembled directly in a similar way to that of the finite element method, demonstrating the method's capability to model complex structures. Any arbitrary boundary conditions are represented accurately in the form of the modified Fourier series. The Wittrick-Williams algorithm is then used as the solution technique where the mode count problem (J 0 ) of a fully-clamped element is resolved. The proposed method gives highly accurate solutions with remarkable computational efficiency, covering low, medium and high frequency ranges. The method is applied to both plane stress and plane strain problems with simple as well as complex geometries. All results from the theory in this paper are accurate up to the last figures quoted to serve as benchmarks.

An exact spectral dynamic stiffness theory for composite plate-like structures with arbitrary non-uniform elastic supports, mass attachments and coupling constraints

“…1(b)) is developed with the basic premise that the Kirchhoff plate theory is applied. (Of course, the same procedure is applicable to other SDS elements such as those for plane elasticity [31].) Based on the Kirchhoff plate theory (recalling Eq.…”

“…However, it should be noted that the capability of modelling non-uniform elastic supports and/or mass attachments and/or coupling constraints has been made possible only within the framework of the SDSM [27][28][29]31]. The classical DS elements in [32][33][34][35] can not be used to model plates with non-uniform BC and/or CC described in this paper.…”

“…It should be noted that the current SDS theory is complete general so that it can be applied to different SDS formulations based on different governing differential equations (GDE), e.g. [27][28][29]31]. Of course, the theory can be degenerated, as a special case, to be applied to classical dynamic stiffness method (DSM) based on different GDE, e.g., those in conjunction with refined theories like Carrera's Unified formulation [32][33][34][35][36], for more details see Section 2.2.3.…”

Dynamic stiffness formulation and free vibration analysis of specially orthotropic Mindlin plates with arbitrary boundary conditions

“…The reason for this can be attributed to the fact that the exact solution is readily available from the theory of elasticity for a simply-supported plate which makes it possible to develop the DSM in an exact sense to compute the natural frequencies and mode shapes of plates and plate assemblies in all frequency ranges. The absence of an exact solution for other types of plate boundary conditions substantially narrowed the range of applicability of the DSM in the past, but this restriction has recently been removed by some outstanding publications in recent years [11][12][13][14][15][16][17]. It should be recognized that seeking an exact solution for free vibration of plates with boundary conditions other than the simply-supported one is really very difficult task but, nevertheless, it is necessary because it is a fundamental step to make the DSM development sufficiently general when modelling complex structures.…”

“…On the question of the second point, the four edges of the plate can have any combination of boundary conditions, namely free, simplesupport or clamped. For instance, there can be simple support on one segment or edge of the plate whereas other segments of edges can be free or clamped, see [12][13][14]. However, on the final point, the authors concede that their method is applicable to rectangular plates only.…”