Generalized Fourier Series

2014 ◽  
pp. 163-201 ◽  
Author(s):  
Christian Constanda
Keyword(s):  
Fourier Series ◽  
Generalized Fourier Series

Modeling the Quasigeoid Heights on Local Areas of the Earth Surface by the Results of Expansion into a Generalized Fourier Series

2020 ◽  
Vol 28 (4) ◽  
pp. 82-94
Author(s):  
V.F. Kanushin ◽  
◽  
I.G. Ganagina ◽  
D.N. Goldobin ◽  
◽  
...  
Keyword(s):  
Fourier Series ◽  
Orthonormal System ◽  
Local Area ◽  
Spherical Functions ◽  
Trigonometric Functions ◽  
Computing Technology ◽  
Generalized Fourier Series ◽  
The Earth ◽  
Local Areas ◽  
Discrete Values

The article presents two methods of modeling discrete heights of a quasigeoid on a local area of the earth’s surface using a gen-eralized Fourier series. The first method is based on modeling the characteristics of the earth’s gravitational field on a plane and involves the use of a two-dimensional Fourier transform by an orthonormal system of trigonometric functions. The second method consists in the expansion of the quasigeoid heights in a Fourier series by an orthonormal system of spherical functions on a local area of the earth’s surface. The errors of approxima-tion of the obtained discrete values of the quasigeoid heights on the local territory are analyzed. It is shown that with the modern computing technology, the most accurate and technologically simple way to model the quasigeoid heights on local areas is to expand them into a Fourier series by an orthonormal system of spherical functions.

Stability of the neutral equilibrium of columns with a rotational end restraint by the generalized Fourier series

2019 ◽  
Vol 25 (4) ◽  
pp. 961-967
Author(s):  
Yan-Ping Zhao ◽  
Lin Li ◽  
Ming Jin
Keyword(s):  
Fourier Series ◽  
Potential Energy ◽  
Energy Functional ◽  
Buckling Mode ◽  
Generalized Fourier Series ◽  
Order Variation ◽  
Neutral Equilibrium ◽  
Post Buckling ◽  
The Stability ◽  
End Restraint

In this paper, stability of the neutral equilibrium and initial post-buckling of a column with a rotational end restraint is analyzed based on Koiter initial post-buckling theory. The potential energy functional is written in terms of the angle. By the generalized Fourier series of the disturbance angle, it is proved that the second-order variation of the potential energy is semi-positive definite at the neutral equilibrium. The stability of the neutral equilibrium is determined by the sign of the fourth-order variation for the buckling mode. For all values of the stiffness of the rotational end restraint, the neutral equilibrium is stable and the bifurcation equilibrium is upward in the initial post-buckling.

The Generalized Fourier Series Method

2020 ◽  
Author(s):  
Christian Constanda ◽  
Dale Doty
Keyword(s):  
Fourier Series ◽  
Series Method ◽  
Generalized Fourier Series ◽  
Fourier Series Method

scholarly journals In-Plane Vibration Analysis of Annular Plates with Arbitrary Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xianjie Shi ◽  
Dongyan Shi ◽  
Zhengrong Qin ◽  
Qingshan Wang
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Wide Spectrum ◽  
Arbitrary Boundary ◽  
Different Boundary Conditions ◽  
Plane Vibration ◽  
Arbitrary Boundary Conditions ◽  
Annular Plates ◽  
Generalized Fourier Series

In comparison with the out-of-plane vibrations of annular plates, far less attention has been paid to the in-plane vibrations which may also play a vital important role in affecting the sound radiation from and power flows in a built-up structure. In this investigation, a generalized Fourier series method is proposed for the in-plane vibration analysis of annular plates with arbitrary boundary conditions along each of its edges. Regardless of the boundary conditions, the in-plane displacement fields are invariantly expressed as a new form of trigonometric series expansions with a drastically improved convergence as compared with the conventional Fourier series. All the unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz technique. Unlike most of the existing studies, the presented method can be readily and universally applied to a wide spectrum of in-plane vibration problems involving different boundary conditions, varying material, and geometric properties with no need of modifying the basic functions or adapting solution procedures. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current solution for predicting the in-plane vibration characteristics of annular plates subjected to different boundary conditions.

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