China
Africa
Explore chapters and articles related to this topic
Basics of Numerical Calculation and Series
Published in Caio Lima Firme, Quantum Mechanics, 2022
A power series is an expansion of a function into an infinite sum of terms in which the variable x has the form of an infinitesimal polynomial. It is used to represent a determined function. For example, the power series of the function ex is: ex=1+x+x22!+x33!+x44!+x55!+...
Series Methods
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
Every power series (8.2) has an interval of convergence consisting of all values x for which the series converges. The interval of convergence is given in the form |x| < R, where R is called the radius of convergence of the power series (8.2). On its interval of convergence, a power series (8.2)converges absolutely, that is, the series ∑n=0∞|cn||xn|
Propagation characteristics of lamb waves in a functionally graded material plate with periodic gratings
Published in Mechanics of Advanced Materials and Structures, 2022
Chunlong Gu, Liansheng Ma, Zhiying Ou, Fangming Lei
The propagation characteristics of Lamb waves in the FGM plate with periodic gratings attached on both surfaces symmetrically are investigated in the article. The power series technique is employed to solve the governing differential equations with variable coefficients. By expanding the displacements of Lamb waves into the Fourier series along the propagation direction, the convergences of the power series and Fourier series are proved and the theoretical method presented in the article is verified by FEM. Subsequently, the effects of the structural parameters, such as the periodicity, mass, and length of the gratings, as well as gradient coefficient on the band structures are investigated. The numerical results show that the couplings between the different modes induced by FGM open the band gaps of Lamb waves. The frequency bands shift up as the periodicity of the gratings increases. However, this tendency is reverse with the increasing mass and length of the gratings. As the gradient coefficient increases, the frequency bands shift up and the width of the FBG decreases. The results obtained in the article provide the theoretical guidance for designing elastic wave filters with high-performance.