{"title":"Analytical solutions for free vibrations of rectangular cuboid elastic lattices and their continuous approximations","authors":"H.P. Nguyen , Noël Challamel , C.M. Wang","doi":"10.1016/j.jsv.2024.118836","DOIUrl":null,"url":null,"abstract":"<div><div>This paper presents analytical solutions for the free vibration of an elastic cuboid lattice (rectangular parallelepiped) with sliding supports along its planar boundaries. The lattice model includes both central and angular interactions. The free vibration problem is solved by formulating a 3D difference eigenvalue problem, and exact solutions for the eigenfrequencies and eigenmodes are derived analytically using a trigonometric discrete displacement field. A cubic equation for the eigenfrequency squares is obtained, with solutions determined using Cardano's formula. The derived exact solutions for the finite cuboid lattice are validated by comparison with solutions obtained from a discrete algebraic method, calibrated for accurate stiffness and mass properties both within the lattice and at its boundaries. Furthermore, these exact solutions for the 3D lattice are benchmarked against analytical solutions for the corresponding 3D continuum, based on classical elasticity, lattice-based gradient elasticity, and lattice-based nonlocal elasticity theories, demonstrating their accuracy and reliability for free vibration analysis.</div></div>","PeriodicalId":17233,"journal":{"name":"Journal of Sound and Vibration","volume":"600 ","pages":"Article 118836"},"PeriodicalIF":4.9000,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Sound and Vibration","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022460X24005984","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/11/19 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"ACOUSTICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper presents analytical solutions for the free vibration of an elastic cuboid lattice (rectangular parallelepiped) with sliding supports along its planar boundaries. The lattice model includes both central and angular interactions. The free vibration problem is solved by formulating a 3D difference eigenvalue problem, and exact solutions for the eigenfrequencies and eigenmodes are derived analytically using a trigonometric discrete displacement field. A cubic equation for the eigenfrequency squares is obtained, with solutions determined using Cardano's formula. The derived exact solutions for the finite cuboid lattice are validated by comparison with solutions obtained from a discrete algebraic method, calibrated for accurate stiffness and mass properties both within the lattice and at its boundaries. Furthermore, these exact solutions for the 3D lattice are benchmarked against analytical solutions for the corresponding 3D continuum, based on classical elasticity, lattice-based gradient elasticity, and lattice-based nonlocal elasticity theories, demonstrating their accuracy and reliability for free vibration analysis.
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.
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