Abstract
Dynamic Finite Element formulation is a powerful technique that combines the accuracy of the exact analysis with wide applicability of the finite element method. The infinite dimensionality of the exact solution space of plate equation has been a major challenge for development of such elements for the dynamic analysis of flexible two-dimensional structures. In this research, a framework for such extension based on subset solutions is proposed. An example element is then developed and implemented in MAT LAB software for numerical testing, verification, and validation purposes. Although the presented formulation is not exact, the element exhibits good convergence characteristics and can be further enriched using the proposed framework.
Highlights
Dynamic stiffness modeling is a well-established technique in vibrational analysis of structural elements
These methods seek to propose formulations that have the accuracy of the exact solutions and wider applicability of the finite element methods by incorporating some form of the closed form solutions of governing equations instead of polynomials used by classic finite element method (FEM)
Two of the most famous dynamic stiffness formulations, mainly applied to various beam-structures, are dynamic stiffness matrix (DSM) and Dynamic Finite Element (DFE) methods, which produce accurate results with much coarser mesh compared to FEM formulations
Summary
Dynamic stiffness modeling is a well-established technique in vibrational analysis of structural elements. The other problem in the case of DFE formulation comes from evaluating the resulting integral equations, written in terms of the dynamic shape functions, to develop element matrices Since these functions are transcendentally dependent on frequency, the integrations are difficult to handle given the current computational power of computers especially for infinite dimensional domains. Casimir et al [16] developed the DSM for rectangular thin plates and demonstrated that accurate results can be achieved with limited number of elements Their method involved solving infinite dimensional matrices, limited to simple cases, and provided problem specific formulation that must be reformulated for each new configuration. An example 4-node 12-DOF element is developed and validated against FEM, where its accuracy and efficiency are demonstrated through the free flexural vibration analysis
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