{"title":"On group-theoretic eigenvalue vibration analysis of structural systems with C6v symmetry","authors":"Alphose Zingoni","doi":"10.1016/j.jsv.2024.118608","DOIUrl":null,"url":null,"abstract":"<div><p>In considerations of the linear vibration of symmetric systems, group theory allows the space of the eigenvalue problem to be decomposed into independent subspaces that are spanned by symmetry-adapted freedoms. These problems usually feature one or more <em>degenerate</em> subspaces (i.e. subspaces that contain repeating solutions). For such subspaces, the associated idempotents, as calculated from the character table of the symmetry group, are not capable of full decomposition of the subspace. In this paper, and based on group theory, simple algebraic operators that fully decompose the two degenerate subspaces of structural problems belonging to the C<sub>6v</sub> symmetry group are proposed. The operators are applied to the vibration of a spring-mass system, for which the results for natural frequencies are found to agree exactly with results from the literature. Their application to the vibration of a hexagonal plane grid reveals new insights on the character of the modes of degenerate subspaces. The overall conclusion is that, for problems belonging to the C<sub>6v</sub> symmetry group, the proposed operators allow the mixed modes of degenerate subspaces to be separated into two distinct symmetry categories, and are very effective in simplifying the actual computation of the repeating eigenvalues of these subspaces.</p></div>","PeriodicalId":17233,"journal":{"name":"Journal of Sound and Vibration","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022460X24003717/pdfft?md5=2a62f415b783244dae1b99e92cefd60e&pid=1-s2.0-S0022460X24003717-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Sound and Vibration","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022460X24003717","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ACOUSTICS","Score":null,"Total":0}
引用次数: 0
Abstract
In considerations of the linear vibration of symmetric systems, group theory allows the space of the eigenvalue problem to be decomposed into independent subspaces that are spanned by symmetry-adapted freedoms. These problems usually feature one or more degenerate subspaces (i.e. subspaces that contain repeating solutions). For such subspaces, the associated idempotents, as calculated from the character table of the symmetry group, are not capable of full decomposition of the subspace. In this paper, and based on group theory, simple algebraic operators that fully decompose the two degenerate subspaces of structural problems belonging to the C6v symmetry group are proposed. The operators are applied to the vibration of a spring-mass system, for which the results for natural frequencies are found to agree exactly with results from the literature. Their application to the vibration of a hexagonal plane grid reveals new insights on the character of the modes of degenerate subspaces. The overall conclusion is that, for problems belonging to the C6v symmetry group, the proposed operators allow the mixed modes of degenerate subspaces to be separated into two distinct symmetry categories, and are very effective in simplifying the actual computation of the repeating eigenvalues of these subspaces.
期刊介绍:
The Journal of Sound and Vibration (JSV) is an independent journal devoted to the prompt publication of original papers, both theoretical and experimental, that provide new information on any aspect of sound or vibration. There is an emphasis on fundamental work that has potential for practical application.
JSV was founded and operates on the premise that the subject of sound and vibration requires a journal that publishes papers of a high technical standard across the various subdisciplines, thus facilitating awareness of techniques and discoveries in one area that may be applicable in others.