Analytic trapezoidal Fourier pelement for vibrating plane problems
Abstract
A trapezoidal Fourier pelement for the inplane vibration analysis of twodimensional elastic solids is presented. Trigonometric functions are used as enriching functions instead of polynomials to avoid illconditioning problems. The element matrices are analytically integrated in closed form. With the additional Fourier degrees of freedom (d.o.f.s), the accuracy of the computed natural frequencies is greatly increased. One element can predict many modes accurately. Since a triangle can be divided into three trapezoidal elements, the range of application is much wider than the previously derived rectangular Fourier pelement. Numerical examples show that convergence is very fast with respect to the number of trigonometric terms. Comparison of natural modes calculated by the trapezoidal Fourier pelement and the conventional finite elements is carried out. The results show that the trapezoidal Fourier pelement produces much higher accurate modes than the conventional finite elements with the same number of d.o.f.s. For a benchmark problem, the condition number of the mass matrix using Legendre pelement increases rapidly and it becomes nonpositive with 22 terms. The condition numbers of the Fourier pelement matrices are consistently much lower than those of the Legendre pelement.
 Publication:

Journal of Sound Vibration
 Pub Date:
 March 2004
 DOI:
 10.1016/S0022460X(03)002633
 Bibcode:
 2004JSV...271...67L