AI 中文总结
本文提出用Dunkl微分算子替换标准导数来建模轻质弹性梁振动，形成反射耦合结构，推导精确模态解，并发现Dunkl参数可显著调制频率，为轻质结构优化提供解析基础。
AI 中文摘要
优化用于可再生能源应用的细长弹性结构需要非经典连续介质公式，这些公式能够考虑空间微观相互作用，同时不牺牲解析可处理性。在此，我们通过用Dunkl微分算子替换标准空间导数来扩展梁振动力学。这一修改引入了一种反射耦合的数学结构，该结构考虑了梁域上的空间宇称效应。我们将控制动力学方程表述为广义特征值问题，并在标准边界条件下推导出模态特性的精确解析表达式。经典极限证实了与经典欧拉-伯努利公式的精确收敛。参数分析表明，Dunkl参数充当反射诱导的调制参数，显著改变固有频率并改变高阶模态的模态特性。这些结果为轻质结构部件的动态优化提供了解析基线。
英文摘要
Optimizing slender elastic structures for renewable energy applications requires non-classical continuum formulations capable of accounting for spatial micro-interactions without sacrificing analytical tractability. Here, we extend beam vibration mechanics by replacing standard spatial derivatives with the Dunkl differential operator. This modification introduces a reflection-coupled mathematical structure that accounts for spatial parity effects across the beam domain. We formulate the governing dynamic equations into a generalized eigenvalue problem and derive exact analytical expressions for modal characteristics under standard boundary conditions. The classical limit confirms exact convergence to classical Euler-Bernoulli formulations. Parametric analyses reveal that the Dunkl parameter acts as a reflection-induced modulation parameter, significantly shifting natural frequencies and altering the modal characteristics of higher modes. These results provide an analytical baseline for dynamic optimization in lightweight structural components.
CommentsThe authors identified errors in the submitted version and wish to withdraw the current version to correct the manuscript before submitting a revised version, due to errors in certain sections of the paper
