Euler-Bernoulli beams frorn a symmetry standpoint-characterization of equivalent equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:345

Article

Soh, Celestin Wafo

关键词: Euler-Bernoulli equation;    equivalence problem;    lie symmetry;   

DOI  :  10.1016/j.jmaa.2008.04.023

来源: Elsevier

【 摘 要 】

We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2A(1), A(1) circle plus A(2), 3A(1), or A(3,3) circle plus A(1). Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A(3,3) circle plus A(1) corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250]. Published by Elsevier Inc.

【 授权许可】

Free