Dunkl-Based Modeling of Vibrational Modes in Lightweight Elastic Beams
Abstract
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.
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BibTeX
@misc{bouguerne2026dunklbased,
title = {{Dunkl-Based Modeling of Vibrational Modes in Lightweight Elastic Beams}},
author = {Hacene Bouguerne},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09369},
eprint = {2609.09369},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09369},
}
RIS
TY - UNPB
TI - Dunkl-Based Modeling of Vibrational Modes in Lightweight Elastic Beams
AU - Hacene Bouguerne
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.09369
UR - https://arxiv.org/abs/2609.09369
AB - 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.
ER -