Eigenvalues of Trusses and Beams Using the Accurate Element Method

The accurate element method (AEM) is a method developed for the numerical integration of the ordinary differential equations. The differential equations are discretized by dividing the computational domain in elements and reducing the solution to nodal values, similar to the finite element method. The solution over the elements is approximated using high-degree interpolation functions. A high-degree interpolation function would require a large number of unknowns per element. A prominent attribute of the AEM is the methodology developed for eliminating unknowns inside the element. As a result, the salient feature of the AEM is the decoupling between the solution accuracy and the number of unknowns. Consequently, high accuracy solutions are obtained using a reduced computational cost compared to traditional methods, such as the finite element method. The AEM uses the same approach to solve initial value problems, boundary value problems or eigenvalue problems. This paper is focused on computing the eigenvalues and eigenvectors for the axial vibration of trusses and transverse vibration of beams. The results obtained using the AEM were compared against finite element results obtained using ANSYS. For both trusses and beams, the accuracy of the eigenvalues computed using the AEM was several orders of magnitude higher than that of the finite element analysis, while the computational time was approximately the same.

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