Abstract
Structural spectral elements are formulated using the analytical solution of the applicable elastodynamic equations and, therefore, mesh refinement is not needed to analyze high frequency behavior provided the elastodynamic equations used remain valid. However, for modeling complex structures, standard spectral elements require long and cumbersome analytical formulation. In this work, a method to build spectral finite elements from a finite element model of a slice of a structural waveguide (a structure with one dimension much larger than the other two) is proposed. First, the transfer matrix of the structural waveguide is obtained from the finite element model of a thin slice. Then, the wavenumbers and wave propagation modes are obtained from the transfer matrix and used to build the spectral element matrix. These spectral elements can be used to model homogeneous waveguides with constant cross section over long spans without the need of refining the finite element mesh along the waveguide. As an illustrating example, spectral elements are derived for straight uniform rods and beams and used to calculate the forced response in the longitudinal and transverse directions. Results obtained with the spectral element formulation are shown to agree well with results obtained with a finite element model of the whole beam. The proposed approach can be used to generate spectral elements of waveguides of arbitrary cross section and, potentially, of arbitrary order.
Highlights
Numerous hybrid waveguide finite element methods have been developed in recent years
The mass and stiffness matrices of the beam slice were extracted from the finite element analysis and postprocessed to assemble the transfer matrix according to Eq (11)
The applied forces are written as a function of the displacement field using the elastodynamic equations for the rod, and only longitudinal propagation modes were taken into account
Summary
Numerous hybrid waveguide finite element methods have been developed in recent years These methods were motivated by the need to minimize the computational cost required to analyze long waveguides with complex cross sections in mid frequencies. The wave model can be used to compute the spectral relations, the group and energy velocities, and the forced response [17,21] These wave approaches that use a finite element model of a slice of the waveguide are based on the periodic structure theory developed by Mead [19] in the early seventies. Because the spectral dynamic matrix is obtained, instead of the standard wave propagation solution, the proposed spectral elements can be combined with standard finite elements using a mobility approach
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