Gibbs-phenomenon-free Fourier series for vibration and stability of complex beams

Abstract

The Gibbs phenomenon in Fourier series has long been recognized as a drawback in its applications, in particular when it is used to represent a function having discontinuities. One category, namely, intrinsic discontinuities, could be derived from the nature of the physical problems. Another, namely, inherent discontinuities, is created undesirably through the employment of Fourier series. To alleviate these drawbacks, two techniques are developed. The first aims at eliminating the inherent discontinuities. The size of the original domain [0, l] is first doubled with a virtual function φ0 defined over the neighboring domain [-l, 0]. The augmented domain [-l, l] is then further extended periodically. The virtual function φ0(y) is chosen so that continuities are achieved at y=0, ±l, ±2l, .... Consequently, no Gibbs phenomena will occur at the boundaries. The second aims at reproducing the intrinsic discontinuities in the representation function by incorporating piecewise cubic polynomials into the Fourier base function. The enlarged basis function, namely, the Gibbs-phenomenon-free-Fourier-series function, is able to represent accurately any specific boundary conditions and interior intrinsic discontinuous conditions. Examples of vibration and buckling analysis of complex beams show that the present method is versatile, accurate, and efficient.