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Fourier Series and Orthogonal Functions
Published in George F. Simmons, Differential Equations with Applications and Historical Notes, 2016
Aside from the great practical value of trigonometric series for solving problems in physics and engineering, the purely theoretical part of this subject has had a profound influence on the general development of mathematical analysis over the past 250 years. Specifically, it provided the main driving force behind the evolution of the modern notion of function, which in all its ramifications is certainly the central concept of mathematics; it led Riemann and Lebesgue to create their successively more powerful theories of integration, and Cantor his theory of sets; it led Weierstrass to his critical study of the real number system and the properties of continuity and differentiability for functions; and it provided the context within which the geometric idea of orthogonality (perpendicularity) was able to develop into one of the major unifying concepts of modern analysis. We shall comment further on all of these matters throughout this chapter.
Chapter 13 Mathematical and statistical techniques
Published in B H Brown, R H Smallwood, D C Barber, P V Lawford, D R Hose, Medical Physics and Biomedical Engineering, 2017
Many physiological signals exhibit a marked periodicity. An obvious example is the electrocardiogram. Such signals can be described elegantly as a Fourier series. The Fourier series is a form of trigonometric series, containing sine and cosine terms. It was first developed by Jean-Baptiste Joseph Fourier, who published the basic principles as part of a work on heat conduction in Paris in 1822. The power and range of application of the methodology was immediately recognized, and it has become one of the most important mathematical tools in the armoury of the physicist and engineer.
Curve Fitting and Interpolation
Published in Ramin S. Esfandiari, Numerical Methods for Engineers and Scientists Using MATLAB®, 2017
So far in this chapter we have mainly discussed curve fitting and interpolation of data using polynomials. But in many engineering applications we encounter systems that oscillate, and consequently, the collected data exhibits oscillatory behavior. These types of systems are hence modeled via trigonometric functions 1, cos t, cos 2t, …, sin t, sin 2t, …. Fourier approximation/interpolation outlines the systematic use of trigonometric series for this purpose.
Damage detection for bridge structures under vehicle loads based on frequency decay induced by breathing cracks
Published in Structure and Infrastructure Engineering, 2023
The road irregularity model is simulated by trigonometric series superposition method: where is the amplitude, is the circle frequency, is the random phase angle, x is the local coordinate, N is the number of simulation points, and is the function of the spatial frequency of road surface irregularities: where is the coefficient of irregularity, and can be taken as a fixed value of 1.94. In the numerical simulation which represents a good road condition.
Volume of the hyperbolic cantor sets
Published in Dynamical Systems, 2020
Habibulla Akhadkulov, Yunping Jiang
It is obvious that every Dini continuous function is continuous and every α-Hölder continuous function is Dini continuous. Next we define a class of functions by using condition. Let be a continuous function. Denote by the second symmetric difference of ϕ; that is, where . We shall denote by the class of continuous functions satisfying the inequality where ζ satisfies condition and the constant depends only on ζ. Note that the class of real functions satisfying (1) with ζ replaced by 1, is called the Zygmund class [6]. The Zygmund class plays a key role in analysis of the trigonometric series. The class of real functions satisfying (1) with ζ replaced by where was investigated by Weiss and Zygmund [5]. They proved that such a class is a subclass of functions of bounded mean oscillation. Here we prove that the class is a subset of . More precisely we have the following
Flutter of high-dimension nonlinear system for a FGM truncated conical shell
Published in Mechanics of Advanced Materials and Structures, 2018
Y. X. Hao, S. W. Yang, W. Zhang, M. H. Yao, A. W. Wang
According to the studies of Noseir and Reddy [34], the rotation and in-plane inertia terms can be neglected because their effects are smaller than the radial inertia term in Eq. (14). Using Galerkin method and substituting the double trigonometric series into Eqs. (14a), (14b), (14d), and (14e), four algebraic equations with constant coefficients are deduced. Then, the midplane displacements u0,v0 and rotations of the transverse normal φθ and φx with respect to transverse displacements are obtained by solving these algebraic equations. Consequently, the high-dimensional second-order ordinary differential nonlinear equations with respect to transverse displacement w can be derived from Eq. (14c) as where wm denotes the amplitude of mth modal component, damping coefficient μ represents the effect of structural-damping μcm, and aerodynamic-damping μam, respectively. The coefficients Cκmi, Cami, and CTmi are the linear stiffness terms which are caused by structure, aerodynamic, and thermal loads, ζmijk is the nonlinear stiffness that includes cubic nonlinearities only. Because the radius R is not fixed, it needs integral operation manually or by Maple.