Exact solutions of the Euler–Bernoulli equation for selected polynomially non-uniform beams used for acoustic black holes

IF 3.4 3区 工程技术 Q1 MECHANICS
Antonin Krpensky, Michal Bednarik
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引用次数: 0

Abstract

In this work, we present a method for obtaining exact analytical solutions to the Euler–Bernoulli equation for nonuniform beams with continuously varying rectangular cross-sections. The approach is based on factorizing the fourth-order equation into a system of second-order differential equations with variable coefficients. Focusing on polynomial expressions for the cross-sectional profile, we show that such factorization is possible only when the profile is described by a polynomial of at most third order. In the general cubic case, the resulting equation transforms into Heun’s differential equation; in degenerate cases, it reduces to the hypergeometric or Bessel equations, all of which admit closed-form solutions. To demonstrate the method’s applicability, we compute reflection coefficients for selected profiles relevant to Acoustic Black Holes and validate the analytical results using a Riccati-based numerical method, showing excellent agreement.
声学黑洞中所选多项式非均匀光束欧拉-伯努利方程的精确解
在这项工作中,我们提出了一种获得具有连续变化矩形截面的非均匀梁的欧拉-伯努利方程的精确解析解的方法。该方法基于将四阶方程分解为一组二阶变系数微分方程。聚焦于截面剖面的多项式表达式,我们证明了这种分解只有当剖面被至多三阶的多项式描述时才是可能的。在一般三次情况下,得到的方程转化为Heun微分方程;在简并的情况下,它简化为超几何方程或贝塞尔方程,它们都有闭解。为了证明该方法的适用性,我们计算了与声黑洞相关的选定剖面的反射系数,并使用基于riccati的数值方法验证了分析结果,结果显示出良好的一致性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
6.70
自引率
8.30%
发文量
405
审稿时长
70 days
期刊介绍: The International Journal of Solids and Structures has as its objective the publication and dissemination of original research in Mechanics of Solids and Structures as a field of Applied Science and Engineering. It fosters thus the exchange of ideas among workers in different parts of the world and also among workers who emphasize different aspects of the foundations and applications of the field. Standing as it does at the cross-roads of Materials Science, Life Sciences, Mathematics, Physics and Engineering Design, the Mechanics of Solids and Structures is experiencing considerable growth as a result of recent technological advances. The Journal, by providing an international medium of communication, is encouraging this growth and is encompassing all aspects of the field from the more classical problems of structural analysis to mechanics of solids continually interacting with other media and including fracture, flow, wave propagation, heat transfer, thermal effects in solids, optimum design methods, model analysis, structural topology and numerical techniques. Interest extends to both inorganic and organic solids and structures.
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