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MECSOL 2022
8th International Symposium on Solid Mechanics
Exact modal analysis of time-dependent problems: a proof of concept in application to one-dimensional truss structures
Submission Author:
Ney Dumont , RJ
Co-Authors:
Ney Dumont
Presenter: Ney Dumont
doi://10.26678/ABCM.MECSOL2022.MSL22-0042
Abstract
We proposed about two decades ago to solve transient problems of potential and elasticity using an advanced mode superposition technique that applies to equilibrium-based finite elements. The developments combine and extend Pian’s hybrid finite element formulation and Przemieniecki’s suggestion of displacement-based, frequency-dependent elements, thus arriving at a hybrid finite element method (actually initially conceived in the frame of a variationally-based boundary element method) for the general analysis of transient problems. Starting from a frequency-domain formulation, we have shown that the traditional structural dynamics taught in the textbooks is just a first-order truncation of a power series for which there is an underlying complex-symmetric (if viscous damping is included), non-linear eigenvalue problem related to the lambda-matrices of a free-vibration analysis, with an {\it effective} stiffness matrix expressed as a frequency power series. In the present contribution, we show that whenever such {\it effective} stiffness matrix can be represented (preferably) as an analytical function of frequencies we can formulate the exact – not just an improved or generalized – modal analysis of a given structural problem. This exact modal analysis may also be carried out if the problem’s generalized stiffness and mass matrices can only be expressed numerically, albeit exactly within machine precision, for a given frequency number (which only in passing resembles a Laplace-transform analysis), although this may become computationally intensive. The analytical developments apply directly to some families of two- and three-dimensional finite elements. Still, we restrict our numerical applications to the simple truss problem including viscous damping as just a proof of concept. With this novel formulation an engineering structure – given its geometric and discretizing simplifications – can ultimately have its time response represented exactly.
Keywords
structural dynamics, Generalized modal analysis, Exact modal analysis, Generalized Eigenvalues and Eigenvectors Problems
