首页 期刊 论文 学科刊群 资讯 加盟 关于
EN
image
Article(id=1227627589272007380, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1227620260010979924, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.05.006, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1657468800000, receivedDateStr=2022-07-11, revisedDate=1663257600000, revisedDateStr=2022-09-16, acceptedDate=null, acceptedDateStr=null, onlineDate=1770618825499, onlineDateStr=2026-02-09, pubDate=1716825600000, pubDateStr=2024-05-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770618825499, onlineIssueDateStr=2026-02-09, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770618825499, creator=13701087609, updateTime=1770618825499, updator=13701087609, issue=Issue{id=1227620260010979924, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='5', pageStart='729', pageEnd='902', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770617078068, creator=13701087609, updateTime=1770795280844, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228367696677499202, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1227620260010979924, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228367696677499203, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1227620260010979924, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=780, endPage=788, ext={EN=ArticleExt(id=1227627589511082715, articleId=1227627589272007380, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Analytical solution of one-dimensional periodic base band gap and its optimal design, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

The band gap characteristics of periodic structures provide a new idea for the field of seismic isolation in civil engineering，among which the one-dimensional periodic foundation structure has garnered significant attention due to its simple structure and economical applicability. In this paper，by studying the vibration characteristics of the one-dimensional periodic base structure，an approximate analytical solution for calculating the one-dimensional rubber-concrete periodic base band gap is derived，and on this basis，a one-dimensional rubber-concrete periodic foundation optimization design method based on the resonance zone of the superstructure is proposed. Numerical examples in the frequency domain and time domain show that the periodic foundation designed by this optimization method can ensure a good damping effect of its superstructure in a wide and continuous frequency range.

, correspAuthors=null, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Qiao-yun WU, Fu-jia XIANG, Ying-hong HUANG, Guo-qiang JING, Zhi-feng XU, Ying-xiong WU), CN=ArticleExt(id=1227627628618772522, articleId=1227627589272007380, tenantId=1146029695717560320, journalId=1225147924628267009, language=CN, title=一维周期基础的带隙解析解及其优化设计研究, columnId=0, journalTitle=振动工程学报, columnName=, runingTitle=null, highlight=null, articleAbstract=

周期结构的带隙特征为土木工程隔震领域提供了新思路，其中一维周期基础结构因其构造简单且经济适用而倍受关注。本文通过研究一维周期基础结构的振动特性，推导了用于计算一维橡胶⁃混凝土周期基础带隙的近似解析解，并在此基础上提出了基于上部结构共振区的一维橡胶⁃混凝土周期基础优化设计方法。频域和时域数值算例表明，由该优化方法所设计的周期基础可保证其上部结构在一个宽且连续的频段内均有较好的减震效果。

, correspAuthors=null, authorNote=null, correspAuthorsNote=
许峙峰（1990―），男，博士，特聘副教授。E-mail：。
, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=OCU0d2C5PM9WauOsJrPd3g==, magXml=JQgZghdf2rGRDopgJ7jkog==, pdfUrl=null, pdf=RbK+kXS72AckHbKQP7V7pA==, pdfFileSize=4545375, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=CXP2qwQZ4HReIChpEQysHg==, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=R20CpzPLk9UVUzHvftbshQ==, mapNumber=null, authorCompany=null, fund=null, authors=

吴巧云（1985—），女，博士，教授。E-mail：。

, authorsList=吴巧云, 项复佳, 黄映红, 荆国强, 许峙峰, 吴应雄)}, authors=[Author(id=1227675698601586701, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, orderNo=0, firstName=null, middleName=null, lastName=null, nameCn=null, orcid=null, stid=null, country=null, authorPic=null, dead=0, email=wuqiaoyun@wit.edu.cn, emailSecond=null, emailThird=null, correspondingAuthor=0, authorType=1, ext={EN=AuthorExt(id=1227675698698055698, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675698601586701, language=EN, stringName=Qiao-yun WU, firstName=Qiao-yun, middleName=null, lastName=WU, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=1, 2, address=1State Key Laboratory for Health and Safety of Bridge Structures, Wuhan 430034, China
2School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null), CN=AuthorExt(id=1227675698756775959, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675698601586701, language=CN, stringName=吴巧云, firstName=null, middleName=null, lastName=null, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=1, 2, address=1桥梁结构健康与安全国家重点实验室，湖北武汉　430034
2武汉工程大学土木工程与建筑学院，湖北武汉　430074, bio={"content":"

吴巧云（1985—），女，博士，教授。E-mail：。

"}, bioImg=null, bioContent=

吴巧云（1985—），女，博士，教授。E-mail：。

, aboutCorrespAuthor=null)}, companyList=[AuthorCompany(id=1227675698182157298, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=1, ext=[AuthorCompanyExt(id=1227675698186351603, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698182157298, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1State Key Laboratory for Health and Safety of Bridge Structures, Wuhan 430034, China), AuthorCompanyExt(id=1227675698194740213, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698182157298, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1桥梁结构健康与安全国家重点实验室，湖北武汉　430034)]), AuthorCompany(id=1227675698287014907, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=2, ext=[AuthorCompanyExt(id=1227675698295403515, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698287014907, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China), AuthorCompanyExt(id=1227675698299597821, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698287014907, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2武汉工程大学土木工程与建筑学院，湖北武汉　430074)])]), Author(id=1227675698853244954, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, orderNo=1, firstName=null, middleName=null, lastName=null, nameCn=null, orcid=null, stid=null, country=null, authorPic=null, dead=0, email=null, emailSecond=null, emailThird=null, correspondingAuthor=0, authorType=1, ext={EN=AuthorExt(id=1227675698937131037, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675698853244954, language=EN, stringName=Fu-jia XIANG, firstName=Fu-jia, middleName=null, lastName=XIANG, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=2, address=2School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null), CN=AuthorExt(id=1227675699000045602, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675698853244954, language=CN, stringName=项复佳, firstName=null, middleName=null, lastName=null, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=2, address=2武汉工程大学土木工程与建筑学院，湖北武汉　430074, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null)}, companyList=[AuthorCompany(id=1227675698287014907, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=2, ext=[AuthorCompanyExt(id=1227675698295403515, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698287014907, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China), AuthorCompanyExt(id=1227675698299597821, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698287014907, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2武汉工程大学土木工程与建筑学院，湖北武汉　430074)])]), Author(id=1227675699075543078, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, orderNo=2, firstName=null, middleName=null, lastName=null, nameCn=null, orcid=null, stid=null, country=null, authorPic=null, dead=0, email=null, emailSecond=null, emailThird=null, correspondingAuthor=0, authorType=1, ext={EN=AuthorExt(id=1227675699155234856, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675699075543078, language=EN, stringName=Ying-hong HUANG, firstName=Ying-hong, middleName=null, lastName=HUANG, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=3, address=3China Power Construction Engineering Consulting Central Southern Co., Ltd., Wuhan 430071, China, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null), CN=AuthorExt(id=1227675699243315245, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675699075543078, language=CN, stringName=黄映红, firstName=null, middleName=null, lastName=null, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=3, address=3中国电力建设工程咨询中南有限公司，湖北武汉　430071, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null)}, companyList=[AuthorCompany(id=1227675698387678208, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=3, ext=[AuthorCompanyExt(id=1227675698396065792, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698387678208, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=3China Power Construction Engineering Consulting Central Southern Co., Ltd., Wuhan 430071, China), AuthorCompanyExt(id=1227675698404454401, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698387678208, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=3中国电力建设工程咨询中南有限公司，湖北武汉　430071)])]), Author(id=1227675699360755764, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, orderNo=3, firstName=null, middleName=null, lastName=null, nameCn=null, orcid=null, stid=null, country=null, authorPic=null, dead=0, email=null, emailSecond=null, emailThird=null, correspondingAuthor=0, authorType=1, ext={EN=AuthorExt(id=1227675699499167802, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675699360755764, language=EN, stringName=Guo-qiang JING, firstName=Guo-qiang, middleName=null, lastName=JING, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=1, address=1State Key Laboratory for Health and Safety of Bridge Structures, Wuhan 430034, China, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null), CN=AuthorExt(id=1227675699604025407, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675699360755764, language=CN, stringName=荆国强, firstName=null, middleName=null, lastName=null, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=1, address=1桥梁结构健康与安全国家重点实验室，湖北武汉　430034, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null)}, companyList=[AuthorCompany(id=1227675698182157298, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=1, ext=[AuthorCompanyExt(id=1227675698186351603, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698182157298, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1State Key Laboratory for Health and Safety of Bridge Structures, Wuhan 430034, China), AuthorCompanyExt(id=1227675698194740213, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698182157298, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1桥梁结构健康与安全国家重点实验室，湖北武汉　430034)])]), Author(id=1227675699696300101, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, orderNo=4, firstName=null, middleName=null, lastName=null, nameCn=null, orcid=null, stid=null, country=null, authorPic=null, dead=0, email=20120102@wit.edu.cn, emailSecond=null, emailThird=null, correspondingAuthor=0, authorType=1, ext={EN=AuthorExt(id=1227675699780186184, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675699696300101, language=EN, stringName=Zhi-feng XU, firstName=Zhi-feng, middleName=null, lastName=XU, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=2, address=2School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null), CN=AuthorExt(id=1227675699847295051, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675699696300101, language=CN, stringName=许峙峰, firstName=null, middleName=null, lastName=null, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=2, address=2武汉工程大学土木工程与建筑学院，湖北武汉　430074, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null)}, companyList=[AuthorCompany(id=1227675698287014907, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=2, ext=[AuthorCompanyExt(id=1227675698295403515, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698287014907, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China), AuthorCompanyExt(id=1227675698299597821, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698287014907, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2武汉工程大学土木工程与建筑学院，湖北武汉　430074)])]), Author(id=1227675699935375441, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, orderNo=5, firstName=null, middleName=null, lastName=null, nameCn=null, orcid=null, stid=null, country=null, authorPic=null, dead=0, email=null, emailSecond=null, emailThird=null, correspondingAuthor=0, authorType=1, ext={EN=AuthorExt(id=1227675700023455828, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675699935375441, language=EN, stringName=Ying-xiong WU, firstName=Ying-xiong, middleName=null, lastName=WU, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=4, address=4College of Civil Engineering of Fuzhou University, Fuzhou 350116, China, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null), CN=AuthorExt(id=1227675700094758998, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, authorId=1227675699935375441, language=CN, stringName=吴应雄, firstName=null, middleName=null, lastName=null, prefix=null, suffix=null, authorComment=null, nameInitials=null, affiliation=null, department=null, xref=4, address=4福州大学土木工程学院，福建福州　350116, bio=null, bioImg=null, bioContent=null, aboutCorrespAuthor=null)}, companyList=[AuthorCompany(id=1227675698488340488, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=4, ext=[AuthorCompanyExt(id=1227675698496729097, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698488340488, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=4College of Civil Engineering of Fuzhou University, Fuzhou 350116, China), AuthorCompanyExt(id=1227675698500923402, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698488340488, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=4福州大学土木工程学院，福建福州　350116)])])], keywords=[Keyword(id=1227675700224782427, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, orderNo=1, keyword=periodic foundation), Keyword(id=1227675700325445727, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, orderNo=2, keyword=band gap), Keyword(id=1227675700413526117, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, orderNo=3, keyword=approximate solution), Keyword(id=1227675700514189415, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, orderNo=4, keyword=vibration reduction), Keyword(id=1227675700598075498, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, orderNo=5, keyword=optimal design), Keyword(id=1227675700648407148, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, orderNo=1, keyword=周期基础), Keyword(id=1227675700715516013, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, orderNo=2, keyword=带隙), Keyword(id=1227675700778430575, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, orderNo=3, keyword=近似解), Keyword(id=1227675700841345137, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, orderNo=4, keyword=减震), Keyword(id=1227675700895871092, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, orderNo=5, keyword=优化设计)], refs=[Reference(id=1227675705392165063, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2013, volume=56, issue=null, pageStart=1271, pageEnd=1282, url=null, language=null, rfNumber=［1］, rfOrder=0, authorNames=Cheng Z B, Shi Z F, journalName=Engineering Structures, refType=null, unstructuredReference=Cheng Z B， Shi Z F. Novel composite periodic structures with attenuation zones［J］. Engineering Structures， 2013， 56： 1271-1282., articleTitle=Novel composite periodic structures with attenuation zones, refAbstract=null), Reference(id=1227675705497022665, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2018, volume=47, issue=4, pageStart=925, pageEnd=944, url=null, language=null, rfNumber=［2］, rfOrder=1, authorNames=Cheng Z B, Shi Z F, journalName=Earthquake Engineering & Structural Dynamics, refType=null, unstructuredReference=Cheng Z B， Shi Z F. Composite periodic foundation and its application for seismic isolation［J］. Earthquake Engineering & Structural Dynamics， 2018， 47（4）： 925-944., articleTitle=Composite periodic foundation and its application for seismic isolation, refAbstract=null), Reference(id=1227675705597685963, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2015, volume=24, issue=7, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=［3］, rfOrder=2, authorNames=Yan Y Q, Cheng Z B, Menq F Y, journalName=Smart Materials & Structures, refType=null, unstructuredReference=Yan Y Q， Cheng Z B， Menq F Y， et al. Three dimensional periodic foundations for base seismic isolation［J］. Smart Materials & Structures， 2015， 24（7）： 075006., articleTitle=Three dimensional periodic foundations for base seismic isolation, refAbstract=null), Reference(id=1227675705710932174, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2020, volume=211, issue=null, pageStart=110427, pageEnd=null, url=null, language=null, rfNumber=［4］, rfOrder=3, authorNames=Cheng Z B, Shi Z F, Palermo A, journalName=Engineering Structures, refType=null, unstructuredReference=Cheng Z B， Shi Z F， Palermo A， et al. Seismic vibrations attenuation via damped layered periodic foundations［J］. Engineering Structures， 2020， 211： 110427., articleTitle=Seismic vibrations attenuation via damped layered periodic foundations, refAbstract=null), Reference(id=1227675705794818257, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=1993, volume=86, issue=3, pageStart=141, pageEnd=143, url=null, language=null, rfNumber=［5］, rfOrder=4, authorNames=Sigalas M, Economou E N, journalName=Solid State Communications, refType=null, unstructuredReference=Sigalas M， Economou E N. Band structure of elastic waves in two dimensional systems［J］. Solid State Communications， 1993， 86（3）： 141-143., articleTitle=Band structure of elastic waves in two dimensional systems, refAbstract=null), Reference(id=1227675707149578451, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2013, volume=50, issue=null, pageStart=204, pageEnd=212, url=null, language=null, rfNumber=［6］, rfOrder=5, authorNames=Shi Z F, Huang J K, journalName=Soil Dynamics and Earthquake Engineering, refType=null, unstructuredReference=Shi Z F， Huang J K. Feasibility of reducing three-dimensional wave energy by introducing periodic foundations［J］. Soil Dynamics and Earthquake Engineering， 2013， 50： 204-212., articleTitle=Feasibility of reducing three-dimensional wave energy by introducing periodic foundations, refAbstract=null), Reference(id=1227675707250241750, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2012, volume=21, issue=11, pageStart=112003, pageEnd=null, url=null, language=null, rfNumber=［7］, rfOrder=6, authorNames=Xiang H J, Shi Z F, Wang S J, journalName=Smart Materials and Structures, refType=null, unstructuredReference=Xiang H J， Shi Z F， Wang S J， et al. Periodic materials-based vibration attenuation in layered foundations： experimental validation［J］. Smart Materials and Structures， 2012， 21（11）： 112003., articleTitle=Periodic materials-based vibration attenuation in layered foundations： experimental validation, refAbstract=null), Reference(id=1227675707329933528, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2010, volume=9, issue=1, pageStart=75, pageEnd=82, url=null, language=null, rfNumber=［8］, rfOrder=7, authorNames=Jia G F, Shi Z F, journalName=Earthquake Engineering and Engineering Vibration, refType=null, unstructuredReference=Jia G F， Shi Z F. A new seismic isolation system and its feasibility study［J］. Earthquake Engineering and Engineering Vibration， 2010， 9（1）： 75-82., articleTitle=A new seismic isolation system and its feasibility study, refAbstract=null), Reference(id=1227675707455762650, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2022, volume=35, issue=6, pageStart=1471, pageEnd=1480, url=null, language=null, rfNumber=［9］, rfOrder=8, authorNames=赵春风, 曾超, MO Y L, journalName=振动工程学报, refType=null, unstructuredReference=赵春风， 曾超， MO Y L. 周期性基础地震波衰减性能及其试验验证［J］. 振动工程学报， 2022， 35（6）： 1471-1480., articleTitle=周期性基础地震波衰减性能及其试验验证, refAbstract=null), Reference(id=1227675707564814556, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2022, volume=35, issue=6, pageStart=1471, pageEnd=1480, url=null, language=null, rfNumber=［9］, rfOrder=9, authorNames=ZHAO Chunfeng, ZENG Chao, MO Y L, journalName=Journal of Vibration Engineering, refType=null, unstructuredReference=ZHAO Chunfeng， ZENG Chao， MO Y L. Attenuation performance of periodic foundation and its experimental verification for seismic waves［J］. Journal of Vibration Engineering， 2022， 35（6）： 1471-1480., articleTitle=Attenuation performance of periodic foundation and its experimental verification for seismic waves, refAbstract=null), Reference(id=1227675707661283551, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2014, volume=116, issue=4, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=［10］, rfOrder=10, authorNames=Yan Y Q, Laskar A, Cheng Z B, journalName=Journal of Applied Physics, refType=null, unstructuredReference=Yan Y Q， Laskar A， Cheng Z B， et al. Seismic isolation of two dimensional periodic foundations［J］. Journal of Applied Physics， 2014， 116（4）： 044908., articleTitle=Seismic isolation of two dimensional periodic foundations, refAbstract=null), Reference(id=1227675707803889890, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2019, volume=9, issue=4, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=［11］, rfOrder=11, authorNames=Witarto W, Wang S J, Yang C Y, journalName=AIP Advances, refType=null, unstructuredReference=Witarto W， Wang S J， Yang C Y， et al. Three-dimensional periodic materials as seismic base isolator for nuclear infrastructure［J］. AIP Advances， 2019， 9（4）： 045014., articleTitle=Three-dimensional periodic materials as seismic base isolator for nuclear infrastructure, refAbstract=null), Reference(id=1227675707879387364, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2019, volume=36, issue=增刊1, pageStart=324, pageEnd=328, url=null, language=null, rfNumber=［12］, rfOrder=12, authorNames=刘岩钊, 尹首浮, 于桂兰, journalName=工程力学, refType=null, unstructuredReference=刘岩钊， 尹首浮， 于桂兰. 周期格栅式表面波屏障的设计与性能研究［J］. 工程力学， 2019， 36（增刊1）： 324-328., articleTitle=周期格栅式表面波屏障的设计与性能研究, refAbstract=null), Reference(id=1227675707946496231, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2019, volume=36, issue=Sup1, pageStart=324, pageEnd=328, url=null, language=null, rfNumber=［12］, rfOrder=13, authorNames=LIU Yanzhao, YIN Shoufu, YU Guilan, journalName=Engineering Mechanics, refType=null, unstructuredReference=LIU Yanzhao， YIN Shoufu， YU Guilan. Design and investigation of periodic grid barriers for seismic surface waves［J］. Engineering Mechanics， 2019， 36（Sup1）： 324-328., articleTitle=Design and investigation of periodic grid barriers for seismic surface waves, refAbstract=null), Reference(id=1227675708034576618, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2012, volume=138, issue=7, pageStart=761, pageEnd=769, url=null, language=null, rfNumber=［13］, rfOrder=14, authorNames=Bao J, Shi Z F, Xiang H J, journalName=Journal of Engineering Mechanics, refType=null, unstructuredReference=Bao J， Shi Z F， Xiang H J. Dynamic responses of a structure with periodic foundations［J］. Journal of Engineering Mechanics， 2012， 138（7）： 761-769., articleTitle=Dynamic responses of a structure with periodic foundations, refAbstract=null), Reference(id=1227675708135239917, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2011, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=［14］, rfOrder=15, authorNames=包静, journalName=null, refType=null, unstructuredReference=包静. 周期性基础隔震结构的地震响应分析［D］. 北京： 北京交通大学， 2011., articleTitle=周期性基础隔震结构的地震响应分析, refAbstract=null), Reference(id=1227675708223320304, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2011, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=［14］, rfOrder=16, authorNames=Bao J, journalName=null, refType=null, unstructuredReference=Bao J. Seismic response of a structure with periodic isolated foundations［D］. Beijing： Beijing Jiaotong University， 2011., articleTitle=Seismic response of a structure with periodic isolated foundations, refAbstract=null), Reference(id=1227675708315594994, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=1989, volume=111, issue=1, pageStart=17, pageEnd=24, url=null, language=null, rfNumber=［15］, rfOrder=17, authorNames=Sackman J L, Kelly J N, Javid A E, journalName=Journal of Pressure Vessel Technology, refType=null, unstructuredReference=Sackman J L， Kelly J N， Javid A E. A layered notch filter for high-frequency dynamic isolation［J］. Journal of Pressure Vessel Technology， 1989， 111（1）： 17-24., articleTitle=A layered notch filter for high-frequency dynamic isolation, refAbstract=null), Reference(id=1227675708508532981, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2014, volume=57, issue=null, pageStart=143, pageEnd=151, url=null, language=null, rfNumber=［16］, rfOrder=18, authorNames=Shi Z F, Cheng Z B, Xiang H J, journalName=Soil Dynamics and Earthquake Engineering, refType=null, unstructuredReference=Shi Z F， Cheng Z B， Xiang H J. Seismic isolation foundations with effective attenuation zones［J］. Soil Dynamics and Earthquake Engineering， 2014， 57： 143-151., articleTitle=Seismic isolation foundations with effective attenuation zones, refAbstract=null), Reference(id=1227675708592419063, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2017, volume=60, issue=9, pageStart=3555, pageEnd=3573, url=null, language=null, rfNumber=［17］, rfOrder=19, authorNames=冯海新, 严君, 刘洪, journalName=地球物理学报, refType=null, unstructuredReference=冯海新， 严君， 刘洪， 等. 求解弹性波解耦方程的一种优化拟解析方法［J］. 地球物理学报， 2017， 60（9）： 3555-3573., articleTitle=求解弹性波解耦方程的一种优化拟解析方法, refAbstract=null), Reference(id=1227675708659527929, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2017, volume=60, issue=9, pageStart=3555, pageEnd=3573, url=null, language=null, rfNumber=［17］, rfOrder=20, authorNames=FENG Haixin, YAN Jun, LIU Hong, journalName=Chinese Journal of Geophysics, refType=null, unstructuredReference=FENG Haixin， YAN Jun， LIU Hong， et al. Optimized pseudo-analytical method for decoupled elastic wave equations［J］. Chinese Journal of Geophysics， 2017， 60（9）： 3555-3573., articleTitle=Optimized pseudo-analytical method for decoupled elastic wave equations, refAbstract=null), Reference(id=1227675708726636795, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, doi=null, pmid=null, pmcid=null, year=2012, volume=138, issue=7, pageStart=761, pageEnd=769, url=null, language=null, rfNumber=［18］, rfOrder=21, authorNames=Bao J, Shi Z F, Xiang H J, journalName=Journal of Engineering Mechanics, refType=null, unstructuredReference=Bao J， Shi Z F， Xiang H J. Dynamic responses of a structure with periodic foundations［J］. Journal of Engineering Mechanics， 2012， 138（7）： 761-769., articleTitle=Dynamic responses of a structure with periodic foundations, refAbstract=null)], funds=[Fund(id=1227675704960151738, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, awardId=BHSKL19-07-GF, language=CN, fundingSource=桥梁结构健康与安全国家重点实验室开放课题(BHSKL19-07-GF), fundOrder=null, country=null), Fund(id=1227675705056620733, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, awardId=52078395, language=CN, fundingSource=国家自然科学基金资助项目(52078395), fundOrder=null, country=null), Fund(id=1227675705132118207, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, awardId=武城建［2020］27号, language=CN, fundingSource=武汉市城建委科技计划项目(武城建［2020］27号), fundOrder=null, country=null), Fund(id=1227675705216004290, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, awardId=null, language=CN, fundingSource=湖北省首届青年拔尖人才计划项目, fundOrder=null, country=null)], companyList=[AuthorCompany(id=1227675698182157298, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=1, ext=[AuthorCompanyExt(id=1227675698186351603, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698182157298, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1State Key Laboratory for Health and Safety of Bridge Structures, Wuhan 430034, China), AuthorCompanyExt(id=1227675698194740213, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698182157298, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1桥梁结构健康与安全国家重点实验室，湖北武汉　430034)]), AuthorCompany(id=1227675698287014907, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=2, ext=[AuthorCompanyExt(id=1227675698295403515, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698287014907, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China), AuthorCompanyExt(id=1227675698299597821, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698287014907, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2武汉工程大学土木工程与建筑学院，湖北武汉　430074)]), AuthorCompany(id=1227675698387678208, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=3, ext=[AuthorCompanyExt(id=1227675698396065792, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698387678208, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=3China Power Construction Engineering Consulting Central Southern Co., Ltd., Wuhan 430071, China), AuthorCompanyExt(id=1227675698404454401, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698387678208, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=3中国电力建设工程咨询中南有限公司，湖北武汉　430071)]), AuthorCompany(id=1227675698488340488, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, xref=4, ext=[AuthorCompanyExt(id=1227675698496729097, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698488340488, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=4College of Civil Engineering of Fuzhou University, Fuzhou 350116, China), AuthorCompanyExt(id=1227675698500923402, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, companyId=1227675698488340488, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=4福州大学土木工程学院，福建福州　350116)])], figs=[ArticleFig(id=1227675701088809080, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.1, caption=One-dimensional periodic structure configuration, figureFileSmall=ppj+pJtll36Jg4MrsjAsIA==, figureFileBig=7pFwJprsLtvXccQosa/BzQ==, tableContent=null), ArticleFig(id=1227675701164306553, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图1, caption=一维周期结构构型, figureFileSmall=ppj+pJtll36Jg4MrsjAsIA==, figureFileBig=7pFwJprsLtvXccQosa/BzQ==, tableContent=null), ArticleFig(id=1227675701248192636, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.2, caption=The first several band gap approximations, figureFileSmall=FB5OYzVjGfx9cfIQkG7alg==, figureFileBig=0k1ye03Ze5R8Hp1RLKlkQQ==, tableContent=null), ArticleFig(id=1227675701327884415, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图2, caption=前数个带隙近似值, figureFileSmall=FB5OYzVjGfx9cfIQkG7alg==, figureFileBig=0k1ye03Ze5R8Hp1RLKlkQQ==, tableContent=null), ArticleFig(id=1227675702670061698, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.3, caption=Flow chart of optimization design method, figureFileSmall=5tGeF2yRz6jwqBUQCEwthw==, figureFileBig=dYu7qadgu5vGDpqogaOczw==, tableContent=null), ArticleFig(id=1227675702774919298, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图3, caption=优化设计方法的流程图, figureFileSmall=5tGeF2yRz6jwqBUQCEwthw==, figureFileBig=dYu7qadgu5vGDpqogaOczw==, tableContent=null), ArticleFig(id=1227675702867193988, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.4, caption=Configuration of the frame and the foundation, figureFileSmall=SRvnlGWErFAeJcoms8pXCA==, figureFileBig=hT0GkbxU3U26MiDG0GtowA==, tableContent=null), ArticleFig(id=1227675702959468679, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图4, caption=框架和基础的配置, figureFileSmall=SRvnlGWErFAeJcoms8pXCA==, figureFileBig=hT0GkbxU3U26MiDG0GtowA==, tableContent=null), ArticleFig(id=1227675703081103497, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.5, caption=Frequency responses and resonance regions of superstructure, figureFileSmall=T5Fp8iD//Ljp7Kfr0ArUFA==, figureFileBig=GqFXlKzcnHrTx/v+jwE6bw==, tableContent=null), ArticleFig(id=1227675703190155404, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图5, caption=上部结构的频率响应和共振区, figureFileSmall=T5Fp8iD//Ljp7Kfr0ArUFA==, figureFileBig=GqFXlKzcnHrTx/v+jwE6bw==, tableContent=null), ArticleFig(id=1227675703257264269, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.6, caption=Optimal designed band gap and resonance regions, figureFileSmall=wwoKH7fRXTuRERTirCwfSw==, figureFileBig=Tbz8r6iN7EgqZZiEhOYAWQ==, tableContent=null), ArticleFig(id=1227675703349538959, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图6, caption=优化设计的带隙和共振区, figureFileSmall=wwoKH7fRXTuRERTirCwfSw==, figureFileBig=Tbz8r6iN7EgqZZiEhOYAWQ==, tableContent=null), ArticleFig(id=1227675703433425041, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.7, caption=Sketch diagram of the three kinds of foundations configurations, figureFileSmall=ZlrAbvRjAuNT2OwkuHDjkA==, figureFileBig=/8dY16fQVWG+1deRXK6lxA==, tableContent=null), ArticleFig(id=1227675703508922516, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图7, caption=三种基础配置示意图, figureFileSmall=ZlrAbvRjAuNT2OwkuHDjkA==, figureFileBig=/8dY16fQVWG+1deRXK6lxA==, tableContent=null), ArticleFig(id=1227675703597002903, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.8, caption=Frequency responses of different foundations, figureFileSmall=URT47NHvYgJw8yQTKInuHQ==, figureFileBig=kKQTcza8cJaVrs98vDpYLA==, tableContent=null), ArticleFig(id=1227675703672500377, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图8, caption=不同基础的频率响应, figureFileSmall=URT47NHvYgJw8yQTKInuHQ==, figureFileBig=kKQTcza8cJaVrs98vDpYLA==, tableContent=null), ArticleFig(id=1227675703747997852, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Fig.9, caption=Acceleration time-histories of the superstructure-foundation coupled systems, figureFileSmall=Clne4cghRFvxwLAR205UEg==, figureFileBig=b4xmidrrf2b3jSAw50Eq9w==, tableContent=null), ArticleFig(id=1227675703848661151, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=图9, caption=上部结构-基础耦合系统的加速度时程, figureFileSmall=Clne4cghRFvxwLAR205UEg==, figureFileBig=b4xmidrrf2b3jSAw50Eq9w==, tableContent=null), ArticleFig(id=1227675703911575713, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Tab.1, caption=

Lower and upper bounds of the first five band gaps

, figureFileSmall=null, figureFileBig=null, tableContent=
频率S波P波
Eq.(18)Eq.(21)Eq.(22)Eq.(23)Eq.(24)Eq.(18)Eq.(21)Eq.(22)Eq.(23)Eq.(24)
f1-start/Hz7.078—7.078—7.72626.97—26.97—29.43
f1-end/Hz16.2816.28———62.0462.05———
f2-start/Hz19.29——19.2919.3673.50——73.5073.79
f2-end/Hz32.5732.57———124.09124.1———
f3-start/Hz34.29—34.29—34.31130.6—130.6—130.72
f3-end/Hz48.8648.86———186.1186.1———
f4-start/Hz50.05——50.0550.05190.6——190.6190.6
f4-end/Hz65.1565.1565.15——248.1248.2———
f5-start/Hz66.05—66.05—66.05251.6—251.6—251.6
f5-end/Hz81.4381.43———310.2310.2———
), ArticleFig(id=1227675704117096613, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=表1, caption=

前五个带隙的下界和上界

, figureFileSmall=null, figureFileBig=null, tableContent=
频率S波P波
Eq.(18)Eq.(21)Eq.(22)Eq.(23)Eq.(24)Eq.(18)Eq.(21)Eq.(22)Eq.(23)Eq.(24)
f1-start/Hz7.078—7.078—7.72626.97—26.97—29.43
f1-end/Hz16.2816.28———62.0462.05———
f2-start/Hz19.29——19.2919.3673.50——73.5073.79
f2-end/Hz32.5732.57———124.09124.1———
f3-start/Hz34.29—34.29—34.31130.6—130.6—130.72
f3-end/Hz48.8648.86———186.1186.1———
f4-start/Hz50.05——50.0550.05190.6——190.6190.6
f4-end/Hz65.1565.1565.15——248.1248.2———
f5-start/Hz66.05—66.05—66.05251.6—251.6—251.6
f5-end/Hz81.4381.43———310.2310.2———
), ArticleFig(id=1227675704217759913, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Tab.2, caption=

Specific parameters of the four-layer steel frame structure in the finite element simulation

, figureFileSmall=null, figureFileBig=null, tableContent=
单元类型单元材料杨氏模量/GPa密度/（kg∙m-3）泊松比
B33各向同性弹性200.07850.00.3
), ArticleFig(id=1227675704280674472, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=表2, caption=

四层钢框架结构有限元模拟具体参数

, figureFileSmall=null, figureFileBig=null, tableContent=
单元类型单元材料杨氏模量/GPa密度/（kg∙m-3）泊松比
B33各向同性弹性200.07850.00.3
), ArticleFig(id=1227675704347783338, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Tab.3, caption=

Parameters of the benchmark unit cell

, figureFileSmall=null, figureFileBig=null, tableContent=
层数材料杨氏模量/MPa密度/（kg∙m-3）泊松比层厚/m
1橡胶0.158612770.4630.2
2混凝土4000023000.20.2
), ArticleFig(id=1227675704435863725, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=表3, caption=

基准单元的参数

, figureFileSmall=null, figureFileBig=null, tableContent=
层数材料杨氏模量/MPa密度/（kg∙m-3）泊松比层厚/m
1橡胶0.158612770.4630.2
2混凝土4000023000.20.2
), ArticleFig(id=1227675704523944111, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Tab.4, caption=

Trial combinations of the layer thicknesses and number of unit cells

, figureFileSmall=null, figureFileBig=null, tableContent=
h1-trailh2-trailnH/mOF
1.16751.073924.4829-1.2597
0.77191.624324.7925-0.6949
0.86931.442324.6233-2.0685
1.02923.276514.3057-0.8755
), ArticleFig(id=1227675704620413105, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=表4, caption=

层厚和晶胞数量的试验组合

, figureFileSmall=null, figureFileBig=null, tableContent=
h1-trailh2-trailnH/mOF
1.16751.073924.4829-1.2597
0.77191.624324.7925-0.6949
0.86931.442324.6233-2.0685
1.02923.276514.3057-0.8755
), ArticleFig(id=1227675704695910580, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=EN, label=Tab.5, caption=

Parameters of the three kinds of foundations

, figureFileSmall=null, figureFileBig=null, tableContent=
基础h1/mh2/mnH/m单元类型
优化设计0.751.6524.80C3D8
基准0.200.20124.80C3D8
混凝土0.004.8014.80C3D8
), ArticleFig(id=1227675704792379574, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227627589272007380, language=CN, label=表5, caption=

三种基础的参数

, figureFileSmall=null, figureFileBig=null, tableContent=
基础h1/mh2/mnH/m单元类型
优化设计0.751.6524.80C3D8
基准0.200.20124.80C3D8
混凝土0.004.8014.80C3D8
)], attaches=null, journal=Journal(id=1225147830491308032, delFlag=0, nameCn=振动工程学报, nameEn=Journal of Vibration Engineering, nameHistory1=null, nameHistory2=null, issn=1004-4523, eissn=null, cn=32-1349/TB, coden=null, periodic=0, language=CN, oaType=null, ccby=null, superviseOffice=null, ownerOffice=null, pubOffice=null, editorOffice=null, officeType=null, aims=null, clcCode=null, officeProv=null, officeCity=null, officeAddr=null, officeZip=null, officeEmail=null, officePhone=null, editDirector=null, officeDirector=null, officeDirectorPhone=null, officeStaffNum=null, officeEmpNum=null, coverPicUrl=null, journalPrice=null, startedYear=null, abbrevIsoEn=Journal of Vibration Engineering, journalRemark=null, publicationField=null, createdTime=1770027604939, updatedTime=1770169610881, createdBy=18614031015, updatedBy=18614031015, firstLetterCn=J, firstLetterEn=J, subjectCode=Engineering, subjectName=null, subjectCodeEn=Engineering, subjectNameEn=null, picCn=null, picEn=null, jcr=null, cjcr=null, exts=[JournalExt(id=1225743346702925905, language=CN, name=振动工程学报, nameHistory1=null, nameHistory2=null, managedBy=中国科学技术协会, sponsoredBy=中国振动工程学会, publishedBy=, editorOffice=, officeProv=null, officeCity=null, officeAddr=, officeZip=, editDirector=, officeDirector=null, officePhone=null, coverPicUrl=null, journalRemark=, submitArticleUrl=null, websiteUrl=, createdTime=1770169587064, updatedTime=1770169587064, createdBy=13701087609, updatedBy=13701087609, submissionGuidelinesUrl=, submissionAuthorUrl=https://www.manuscripts.com.cn/zdgcxb, submissionEditorUrl=https://www.manuscripts.com.cn/zdgcxb, submissionReviewUrl=https://www.manuscripts.com.cn/zdgcxb, submissionCeEditorUrl=, submissionAeEditorUrl=, option={"copyright":""}), JournalExt(id=1225743346765840466, language=EN, name=Journal of Vibration Engineering, nameHistory1=null, nameHistory2=null, managedBy=, sponsoredBy=, publishedBy=, editorOffice=, officeProv=null, officeCity=null, officeAddr=, officeZip=, editDirector=, officeDirector=null, officePhone=null, coverPicUrl=null, journalRemark=, submitArticleUrl=null, websiteUrl=, createdTime=1770169587079, updatedTime=1770169587079, createdBy=13701087609, updatedBy=13701087609, submissionGuidelinesUrl=, submissionAuthorUrl=https://www.manuscripts.com.cn/zdgcxb, submissionEditorUrl=https://www.manuscripts.com.cn/zdgcxb, submissionReviewUrl=https://www.manuscripts.com.cn/zdgcxb, submissionCeEditorUrl=, submissionAeEditorUrl=, option={"copyright":""})], databaseList=null, tenantJournalId=1225147924628267009, websiteList=[Website(id=1225150618881404985, webName=null, webTitle=null, webDomain=null, webCopyrigh=null, webIpcNo=null, seoTitle=null, seoKeywords=null, seoDescription=null, tenantJournalId=null, journalId=1225147924628267009, journalNameCn=null, journalNameEn=null, grayFlag=null, tenantId=1146029695717560320, platformId=null, journalGroupId=null, journalGroupNameCn=null, journalGroupNameEn=null, type=1, domain=https://castjournals.cast.org.cn/joweb/zdgcxb/CN, language=CN, createTime=1770028269739, createBy=18614031015, updateTime=1770028293069, updateBy=18614031015, name=振动工程学报-中文, tplId=1146099689490845704, title=振动工程学报, delFlag=0, indexPage=/home, props=[WebsiteProps(id=1225151164178673750, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=articleTextType, value=kx, createTime=1770028399748, updateTime=1770028399748, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164157702227, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=banner, value=null, createTime=1770028399743, updateTime=1770028399743, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164203839577, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=grayFlag, value=0, createTime=1770028399754, updateTime=1770028399754, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164145119314, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=logo, value=https://castjournals.cast.org.cn/joweb/zdgcxb/EN/file/pic?fileId=L7mSU8YPwm66NWFMoTG4aQ==, createTime=1770028399740, updateTime=1770028399740, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164212228187, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=minRunFlag, value=0, createTime=1770028399756, updateTime=1770028399756, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164170285141, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=picServerUrl, value=https://castjournals.cast.org.cn/joweb/zdgcxb/CN/file/pic, createTime=1770028399746, updateTime=1770028399746, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164208033882, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=silenceFlag, value=0, createTime=1770028399755, updateTime=1770028399755, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164166090836, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=staticResourcePath, value=https://castjournals.cast.org.cn/joweb/cast_kjdb_cn_619/, createTime=1770028399745, updateTime=1770028399745, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164187062359, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=themeColor, value=null, createTime=1770028399750, updateTime=1770028399750, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151164195450968, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150618881404985, code=themeStyle, value=null, createTime=1770028399752, updateTime=1770028399752, creator=18614031015, updator=18614031015)]), Website(id=1225150619003039804, webName=null, webTitle=null, webDomain=null, webCopyrigh=null, webIpcNo=null, seoTitle=null, seoKeywords=null, seoDescription=null, tenantJournalId=null, journalId=1225147924628267009, journalNameCn=null, journalNameEn=null, grayFlag=null, tenantId=1146029695717560320, platformId=null, journalGroupId=null, journalGroupNameCn=null, journalGroupNameEn=null, type=1, domain=https://castjournals.cast.org.cn/joweb/zdgcxb/EN, language=EN, createTime=1770028269768, createBy=18614031015, updateTime=1770028309190, updateBy=18614031015, name=振动工程学报-英文, tplId=1146101810881728533, title=Journal of Vibration Engineering, delFlag=0, indexPage=/home, props=[WebsiteProps(id=1225151193366835296, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=articleTextType, value=kx, createTime=1770028406707, updateTime=1770028406707, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193350058077, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=banner, value=null, createTime=1770028406703, updateTime=1770028406703, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193387806819, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=grayFlag, value=0, createTime=1770028406712, updateTime=1770028406712, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193341669468, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=logo, value=https://castjournals.cast.org.cn/joweb/zdgcxb/EN/file/pic?fileId=L7mSU8YPwm66NWFMoTG4aQ==, createTime=1770028406701, updateTime=1770028406701, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193400389733, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=minRunFlag, value=0, createTime=1770028406715, updateTime=1770028406715, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193362640991, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=picServerUrl, value=https://castjournals.cast.org.cn/joweb/zdgcxb/EN/file/pic, createTime=1770028406706, updateTime=1770028406706, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193392001124, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=silenceFlag, value=0, createTime=1770028406713, updateTime=1770028406713, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193354252382, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=staticResourcePath, value=https://castjournals.cast.org.cn/joweb/cast_kjdb_en_623/, createTime=1770028406704, updateTime=1770028406704, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193371029601, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=themeColor, value=null, createTime=1770028406708, updateTime=1770028406708, creator=18614031015, updator=18614031015), WebsiteProps(id=1225151193379418210, tenantId=1146029695717560320, journalId=null, journalGroupId=null, siteId=1225150619003039804, code=themeStyle, value=null, createTime=1770028406710, updateTime=1770028406710, creator=18614031015, updator=18614031015)])], journalTitle=振动工程学报, weixinUrl=null, journalUrl=http://zdgcxb.csve.org.cn/, iacademicId=null, status=1, seqNo=null, journalTitleEn=Journal of Vibration Engineering, journalPhotoCn=null, journalPhotoEn=null, journalFirstLetter=J, journalRecommend=null, journalNew=null, journalCollection=null, jcrJf=null, cjcrJf=null, jcrJfStr=null, cjcrJfStr=null, submissionFirstDecision=null, sciSubjectClassification=null, casSubjectClassification=null, citeScore=null, totalCitationFrequency=null, icpCode=null, psCode=null, advertisingLicenseCode=null, copyrightInformation=null, country=null, option=, provinceCode=null, provinceName=null, collectFlag=false), detailUrlCn=https://castjournals.cast.org.cn/joweb/zdgcxb/CN/10.16385/j.cnki.issn.1004-4523.2024.05.006, detailUrlEn=https://castjournals.cast.org.cn/joweb/zdgcxb/EN/10.16385/j.cnki.issn.1004-4523.2024.05.006, pdfUrlCn=https://castjournals.cast.org.cn/joweb/zdgcxb/CN/PDF/10.16385/j.cnki.issn.1004-4523.2024.05.006, pdfUrlEn=https://castjournals.cast.org.cn/joweb/zdgcxb/EN/PDF/10.16385/j.cnki.issn.1004-4523.2024.05.006, aliStartDate=null, aliEndDate=null, collectionFlag=false, citedCount=null, citedUrl=null, reference=null)
收藏切换
一维周期基础的带隙解析解及其优化设计研究
收藏切换
PDF下载
吴巧云 1, 2 , 项复佳 2 , 黄映红 3 , 荆国强 1 , 许峙峰 2 , 吴应雄 4
振动工程学报 | 2024,37(5): 780-788
收起
收藏切换
振动工程学报 | 2024, 37(5): 780-788
一维周期基础的带隙解析解及其优化设计研究
全屏
吴巧云1, 2 , 项复佳2, 黄映红3, 荆国强1, 许峙峰2 , 吴应雄4
作者信息
  • 1桥梁结构健康与安全国家重点实验室，湖北武汉　430034
  • 2武汉工程大学土木工程与建筑学院，湖北武汉　430074
  • 3中国电力建设工程咨询中南有限公司，湖北武汉　430071
  • 4福州大学土木工程学院，福建福州　350116

    • 吴巧云（1985—），女，博士，教授。E-mail：。

通讯作者:

许峙峰（1990―），男，博士，特聘副教授。E-mail：。
Analytical solution of one-dimensional periodic base band gap and its optimal design
Qiao-yun WU1, 2 , Fu-jia XIANG2, Ying-hong HUANG3, Guo-qiang JING1, Zhi-feng XU2 , Ying-xiong WU4
Affiliations
  • 1State Key Laboratory for Health and Safety of Bridge Structures, Wuhan 430034, China
  • 2School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China
  • 3China Power Construction Engineering Consulting Central Southern Co., Ltd., Wuhan 430071, China
  • 4College of Civil Engineering of Fuzhou University, Fuzhou 350116, China
出版时间: 2024-05-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.05.006
文章导航
收藏切换
摘要
收起

周期结构的带隙特征为土木工程隔震领域提供了新思路，其中一维周期基础结构因其构造简单且经济适用而倍受关注。本文通过研究一维周期基础结构的振动特性，推导了用于计算一维橡胶⁃混凝土周期基础带隙的近似解析解，并在此基础上提出了基于上部结构共振区的一维橡胶⁃混凝土周期基础优化设计方法。频域和时域数值算例表明，由该优化方法所设计的周期基础可保证其上部结构在一个宽且连续的频段内均有较好的减震效果。

周期基础  /  带隙  /  近似解  /  减震  /  优化设计

The band gap characteristics of periodic structures provide a new idea for the field of seismic isolation in civil engineering，among which the one-dimensional periodic foundation structure has garnered significant attention due to its simple structure and economical applicability. In this paper，by studying the vibration characteristics of the one-dimensional periodic base structure，an approximate analytical solution for calculating the one-dimensional rubber-concrete periodic base band gap is derived，and on this basis，a one-dimensional rubber-concrete periodic foundation optimization design method based on the resonance zone of the superstructure is proposed. Numerical examples in the frequency domain and time domain show that the periodic foundation designed by this optimization method can ensure a good damping effect of its superstructure in a wide and continuous frequency range.

periodic foundation  /  band gap  /  approximate solution  /  vibration reduction  /  optimal design
吴巧云, 项复佳, 黄映红, 荆国强, 许峙峰, 吴应雄. 一维周期基础的带隙解析解及其优化设计研究. 振动工程学报, 2024 , 37 (5) : 780 -788 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.05.006
Qiao-yun WU, Fu-jia XIANG, Ying-hong HUANG, Guo-qiang JING, Zhi-feng XU, Ying-xiong WU. Analytical solution of one-dimensional periodic base band gap and its optimal design[J]. Journal of Vibration Engineering, 2024 , 37 (5) : 780 -788 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.05.006
正文
收起
引言
收起
与传统的被动地震控制系统相比，周期基础具有制造简单、能同时减轻水平和垂直地震反应等优点［1⁃5］。Shi等［6］提出了使用周期基础作为隔离器来抵抗地震动的想法。此后，不同形式的周期基础得到了迅速发展，从一维到二维和三维，从简单的单胞到多材料复杂单胞［7⁃12］。Xiang等［7］通过数值模拟和实验测试相结合的方式验证了带隙的减震效果。文献［8⁃9］通过有限元分析证明了一维和二维周期基础的地震衰减效应。文献［10⁃11］对二维和三维周期基础进行了实验研究，其结果表明在优化设计下周期基础可具有显著的减震性能。文献［13⁃14］提出了一种由混凝土层及橡胶层构成的层状周期基础，并用有限元方法分析了该基础对一个六层框架的减震作用。然而，一维周期基础隔震的理论研究在如下方面还有待完善：（1）周期基础的带隙计算方法有待完善。Sackman等［15］提出了一维周期基础第一个带隙的下界和上界的近似解，但该方法存在下界计算精度不准的问题。之后，Shi等［16］修改了Sackman近似解，但该修正只是一个曲线拟合结果，对于超出其数值研究范围的带隙预测结果的准确性有待提高。（2）周期基础带隙的覆盖范围不够全面。当前周期基础的隔震思路是通过扩大第一带隙的范围使第一带隙尽可能多地覆盖全部频率，这导致隔震效果不够理想。
本文通过研究一维周期基础结构的振动特性，推导了局部化因子、衰减系数、频率响应、平均频率响应等带隙有关重要参数的近似解析解，并在此基础上提出了基于上部结构频率响应的一维橡胶⁃混凝土周期基础优化设计方法。由该方法所设计的周期基础可保证上部结构在一个宽而连续的频段内均具有较好的减震效果。
1　一维橡胶⁃混凝土周期基础的带隙近似解析解
收起
一维橡胶⁃混凝土周期基础是一种由橡胶层和混凝土层组成的一维周期基础。相比其他类型的周期基础，一维橡胶⁃混凝土周期基础因其具有成本低、制造方便、隔震效果理想等多种优势而受到广泛关注。但工程上仍缺乏计算一维橡胶⁃混凝土周期基础带隙等关键参数的有效方法。为此，本文提出并验证了计算一维橡胶⁃混凝土周期基础的结构带隙、局部化因子、衰减系数和频率响应的近似解析式。基于该带隙解析解，提出了一种基于上部结构共振区的一维橡胶⁃混凝土周期基础优化设计方法，并通过数值算例证实了该优化设计方法的减震效果。
1.1　一维周期基础基于传递矩阵的带隙近似解析解
在图1所示的一维周期结构中，y方向上重复排列了n个单胞，若，则表示该结构为理想周期结构。每个单胞包含m个弹性层，其中，第k层的厚度为，单胞总厚度为，结构总高度为。第k层的弹性模量、泊松比和密度分别为，和。第j个单胞中第k层的局部坐标通过转换为全局坐标，其中，y为全局坐标，为第j个单胞中第k层的局部坐标。底部的特征为，顶部的特征为。可在底部x或y方向输入不同频率的入射谐波，其可为横波（S波）或纵波（P波）。
弹性波在第j个单胞中通过第k层传播的控制方程可由以下波动方程［17］描述：
式中　 为第j个单胞中第k层的位移；为波速，其中为波模量，对S波，；对P波，，其中，和为拉梅常数；为第k层材料的弹性模量。
谐波激励下的特解满足以下可分离变量的形式：
式中　 为波数；i为虚数单位；为频率；和为常数。
根据胡克定律，应力分量为：
其中：
针对谐波，通过使用传递矩阵法［18］，式（3）和（5）可重写为以下矩阵乘法形式：
式中　 为状态向量；为一个常数向量；可表示为：
将和代入式（6）中可得：
其中，第k层的传递矩阵为：
利用连续性条件，第j个单胞的顶部和底部之间的转换关系可通过如下递归表达式解得：
式中　 为该单胞的传递矩阵。由，可得行列式。
基于Floquet⁃Bloch理论，单胞的两个状态向量具有以下关系：
式中　 为该单胞的波数；和分别为单胞的输出和输入状态向量，均处于特征模式。
将式（11）代入式（10）中，得到以下特征形式：
式中　 I为单位矩阵。
由式（12）所解得的两个特征向量分别为和，对应的两个特征值分别为和，其满足：
式（13）的解为：
式中　 。
由式（13）可进一步推出以下条件有效：
上述状态向量的行为可分为以下两种情况讨论：
情况1为无衰减情况。此时，和形成一对共轭复数，即，。和两个实数满足。在这种情况下，状态向量将仅保持椭圆旋转，波可在没有任何能量损失的情况下通过。
情况2为衰减情况。此时 ，则是两个实数，是两个复数，，。根据特征值幂法可知，对于任意，最终会被变换至与平行的方向上，且n每增加1就会放大至原来的倍。此种情况下的所有入射波的频率组成了对应的带隙。若从底部到顶部跟踪波的传播，则入射波每通过一个单胞，其波幅就会衰减常数倍，该常数被称作衰减系数，即：
式中　 为局部化因子。
显然，在穿过整个周期结构后，入射波的总衰减幅度为。对理想一维周期结构，由于，输出波的振幅变成，这意味着禁止波传播。
1.2　一维橡胶⁃混凝土周期基础基于传递矩阵的带隙近似解析解
在本节推导中，定义第1层为橡胶层，第2层为混凝土层。对于一维橡胶⁃混凝土周期基础，方程（15）可改写为：
式中　 ；为橡胶层的速度常数；为橡胶层厚度；为混凝土层的速度常数；为混凝土层厚度。
此外，式（18）对于上标的对称性表明两层的顺序无关带隙的计算。
带隙边界的特征为。同时，若且足够小，可得，，则式（18）变成：
式（19）的解为带隙下界和上界的近似值，其可进一步表示为：
式中　 ，p为一个非负整数。
式标记了带隙上界，即：
值得注意的是，与Sackman的近似值一致［28］。
带隙下界的表达式为，即或。当和同时成立，通过使用和，可将以下两个方程作为带隙下界的隐式近似解：
为了进一步获得带隙下界的显式近似值，将和代入式（22）和（23）中，可得第p个带隙的起始频率为：
该近似解析解以第1层为橡胶层、第2层为混凝土层的一维橡胶⁃混凝土周期基础为例推导，式（24）得以成立的条件是，故本节提出的一维橡胶⁃混凝土周期基础基于传递矩阵的带隙近似解析解，适用于两个弹性层的厚度与其波速间比值（即波数）有较大差异的一维周期结构。本文提出的用于计算带隙上、下界的解析公式也可用于其他一维周期结构（例如橡胶⁃钢周期基础），但所计算的一维周期结构必须满足以下特征：（1）其单胞仅含两个弹性层；（2）两弹性层的厚度与其波速间比值（即波数）有较大差异。
2　带隙近似解析解算例验证
收起
为了验证本节所提出的带隙计算近似解，图2和表1分别给出并比较了由式（18）计算的基准单胞带隙的精确解和使用式（21）~（24）计算的相应近似解，其中频率值以工程频率f表示。在此处规定以下关于角频率和工程频率之间的对应关系为ω=2πf。
图2展示了前数个带隙的精确值与近似值的比较结果。表1表示的是前五个带隙的上、下界，第1列频率一共取了五个带隙，分别是每个带隙的起始和终点频率；第2~6列是S波下，用不同公式计算的带隙起始和终点位置；第7~11列是P波下，用不同公式计算的带隙起始和终点位置。
从图2和表1中可以看出，除了使用式（24）对第一带隙上界的估计有一定偏差外，其余近似解都具有足够的精度。由此证明上述关于一维橡胶⁃混凝土带隙的近似解析公式的准确性。
3　一维橡胶⁃混凝土周期基础的优化设计
收起
3.1　优化设计理念的提出
给定要设计的一维橡胶⁃混凝土周期基础和具有一系列共振区（RZ）的上层建筑，其中：所考虑的输入波（例如地震动）频段为 ，其中括号代表开区间；S波和P波的RZ集合分别用和表示，其中的第q个元素和的第r个元素分别表示为 和 ；S波和P波的BG集合分别用和表示，其中S波的第p个带隙和P波的第s个带隙分别表示为和 。
通过定义关于h1和h2的优化目标函数来衡量设计的优劣。经过优化设计的一维橡胶⁃混凝土周期基础，应确保上层结构的频率响应在地震动频段内的任何频率下都得到有效控制，这要求每个上部结构共振区都被相应的带隙覆盖。根据上述优化问题，本小节介绍了一种优化设计方法。该方法的核心思想为，在满足设计带隙包含全部上部共振区的情况下，找到的最优组成形式。优化设计方法的流程如图3所示。
3.2　算例设计
为了证明上文所提出的优化设计方法的实用性，将所提方法应用于表2中描述的四层钢框架结构的一维橡胶⁃混凝土周期基础设计中。四层钢框架结构长4.0 m，宽4.0 m，高4.0 m，梁柱截面为0.1 m×0.1 m，使用材料为钢材。基础设计为4.0 m×4.0 m方形基础，深度为4.0~5.0 m，使用材料为橡胶和混凝土，如表3所示。图4（a）和（b）分别显示了框架和基础的配置。
在设计中，输入波频段取为0.0~30.0 Hz，涵盖了大多数地震动的频段。设计目标是最小化上部结构在S波和P波组合下的频率响应。此时，每层的平均最大绝对频率响应定义为：
式中　 |FR2⁃b|，|FR3⁃b|，。FRi为第i层相对于基础底面的频率响应，即FRi⁃b=ui/ub，其中，ui为第i层最大位移（i=0时，u0为基础顶面最大位移），ub为基础底面最大位移。
最后，选定OF为：
式中下标“S”和“P”分别代表S波和P波的对应值；WP为P波的权重，考虑到上部结构更容易受到S波的影响，在以下分析中将WP设置为0.5。
首先，通过稳态动力学分析，计算表3描述的上部四层钢框架的频率响应函数（FRF）：其中，FRFi⁃t=20lg（FRi⁃t），FRFmax⁃t=max，，|FRF3⁃t|，，S波和P波的频率响应函数分别通过在x和y方向4个底部节点指定单位位移后，测量相应节点的输出位移获得。然后通过检查地震动频段中的频率响应是否大于相应的阈值来确定和，其中S波和P波的阈值分别设置为0.0和3.0（注：20lg（1.0）=0.0，20lg（1.414）≈3.0）。P波的阈值大于S波的原因是水平地震动通常更具破坏力。最终确定：=，（5.06，5.16），（6.44，7.37），和=，。图5（a）和（b）分别显示了S波和P波对应的四层钢框架上部结构的频率响应和共振区。
在MATLAB中编写了实现第3.1节中提出的优化设计方法的程序，并应用于基础的设计。表4为OFtrial的结果，得到（h1⁃trail，h2⁃trail，n）的最优组合为（0.7719，1.6243，2），最后经过实践考虑，确定（h1⁃opt，h2⁃opt，n）为（0.75，1.65，2），其中H和OF对应的值分别为4.8 m和-0.7657。图6（a）和（b）给出了相应的带隙和共振区，从图6中可以看出，所有共振区都分布在相应的带隙中。
3.3　算例频域分析和时域分析
为了更进一步验证所提出的设计方法在设计全频段的最优性，对周期基础的性能进行了有限元分析，包括频域分析和时域分析。构建并分析了三种上部结构⁃基础耦合系统（如图7所示），其中上部结构均为表3所述的四层钢框架，基础分别为由第3.2节中得到的优化设计基础、由12个基准单元组成的基准周期基础以及混凝土基础。基准单元参数见表1，三种基础的单元尺寸均为0.2 m×0.2 m×0.05 m，详细配置如表5所示，分析中的所有材料都是各向同性弹性的。
在频域分析中，将不同频率的单位谐波位移统一分配到底面，其中沿x轴或y轴的谐波位移分别代表S波或P波。采用“稳态直接分析步”计算框架的频响函数，其中S波和P波的频率范围为0.02~30.0 Hz，频率增量为0.02 Hz。三种基础的频响函数模拟结果如图8（a）~（f）所示。可以看出，最优地基对S波和P波的减震性能非常出色，以至于地震动频段中几乎所有的频响都小于0，并且大多数S波甚至低于-40（20lg（1/100）=-40），大多数P波低于-20（20lg（1/10）=-20），而基准基础对频率在共振区内而在带隙外的入射波没有起到衰减作用。可见，无论在横波输入还是纵波输入的情况下，最优地基优于混凝土地基和基准地基，证明确保上部结构的共振区完全被相应的带隙覆盖是一种良好的一维橡胶⁃混凝土周期基础设计策略，能够保证良好的减震效果。
在时域分析中，将El Centro地震记录的加速度分配到地基底部，其中N⁃S方向、垂直方向和E⁃W方向的相应加速度分别被分配到地基底部x方向、y方向和z方向。本次分析选择“隐式动态分析步”，总时长为35 s，采用0.0005 s的固定时间增量。得到El Centro波下上部结构⁃基础耦合系统的加速度时程，如图9所示。其中，优化设计基础的顶部加速度峰值为2.7 m/s2，基准基础结构的顶部加速度峰值为4.18 m/s2，混凝土基础顶部加速度峰值为12.91 m/s2，基准基础结构的顶部加速度峰值为混凝土基础顶部加速度峰值的32.4%，优化设计基础的顶部加速度峰值为基准基础顶部加速度峰值的64.6%。一方面，通过比较图9（a），（d），（g），（j），（m）和图9（b），（e），（h），（k），（n）可以看出，与混凝土基础相比，优化设计基础可以在有效降低上部结构水平加速度响应的同时滤除高频响应，而基准基础也可以滤除高频水平响应，但水平加速度衰减效果不明显。另一方面，通过比较图9（c），（f），（i），（l），（o）可以观察到，优化设计基础可以在不显著增加相应幅度的情况下过滤高频竖向响应，而基准基础将增大竖向响应。可以看出周期基础在隔离地震加速度方面的有效性，同时证实了本文所提周期基础优化设计方法的实用性。
4　结论
收起
本文推导了用于计算一维橡胶⁃混凝土周期基础带隙的近似解析解，并在此基础上提出了其局部化因子、衰减系数、频率响应、平均频率响应等参数的近似解析解，通过算例证实了该近似解析解计算带隙的精确性；提出了一种基于上部结构共振区的一维橡胶⁃混凝土周期基础优化设计方法，时域和频域数值算例表明，由该方法所设计的周期基础可保证其上部结构在一个宽且连续的频段内均有较好的减震效果。
基金
收起
  • 桥梁结构健康与安全国家重点实验室开放课题(BHSKL19-07-GF)
  • 国家自然科学基金资助项目(52078395)
  • 武汉市城建委科技计划项目(武城建［2020］27号)
  • 湖北省首届青年拔尖人才计划项目
文献
收起
参考文献 引证文献
排序方式：
［1］
Cheng Z B， Shi Z F. Novel composite periodic structures with attenuation zones［J］. Engineering Structures， 2013， 56： 1271-1282.
［2］
Cheng Z B， Shi Z F. Composite periodic foundation and its application for seismic isolation［J］. Earthquake Engineering & Structural Dynamics， 2018， 47（4）： 925-944.
［3］
Yan Y Q， Cheng Z B， Menq F Y， et al. Three dimensional periodic foundations for base seismic isolation［J］. Smart Materials & Structures， 2015， 24（7）： 075006.
［4］
Cheng Z B， Shi Z F， Palermo A， et al. Seismic vibrations attenuation via damped layered periodic foundations［J］. Engineering Structures， 2020， 211： 110427.
［5］
Sigalas M， Economou E N. Band structure of elastic waves in two dimensional systems［J］. Solid State Communications， 1993， 86（3）： 141-143.
［6］
Shi Z F， Huang J K. Feasibility of reducing three-dimensional wave energy by introducing periodic foundations［J］. Soil Dynamics and Earthquake Engineering， 2013， 50： 204-212.
［7］
Xiang H J， Shi Z F， Wang S J， et al. Periodic materials-based vibration attenuation in layered foundations： experimental validation［J］. Smart Materials and Structures， 2012， 21（11）： 112003.
［8］
Jia G F， Shi Z F. A new seismic isolation system and its feasibility study［J］. Earthquake Engineering and Engineering Vibration， 2010， 9（1）： 75-82.
［9］
赵春风， 曾超， MO Y L. 周期性基础地震波衰减性能及其试验验证［J］. 振动工程学报， 2022， 35（6）： 1471-1480.
ZHAO Chunfeng， ZENG Chao， MO Y L. Attenuation performance of periodic foundation and its experimental verification for seismic waves［J］. Journal of Vibration Engineering， 2022， 35（6）： 1471-1480.
［10］
Yan Y Q， Laskar A， Cheng Z B， et al. Seismic isolation of two dimensional periodic foundations［J］. Journal of Applied Physics， 2014， 116（4）： 044908.
［11］
Witarto W， Wang S J， Yang C Y， et al. Three-dimensional periodic materials as seismic base isolator for nuclear infrastructure［J］. AIP Advances， 2019， 9（4）： 045014.
［12］
刘岩钊， 尹首浮， 于桂兰. 周期格栅式表面波屏障的设计与性能研究［J］. 工程力学， 2019， 36（增刊1）： 324-328.
LIU Yanzhao， YIN Shoufu， YU Guilan. Design and investigation of periodic grid barriers for seismic surface waves［J］. Engineering Mechanics， 2019， 36（Sup1）： 324-328.
［13］
Bao J， Shi Z F， Xiang H J. Dynamic responses of a structure with periodic foundations［J］. Journal of Engineering Mechanics， 2012， 138（7）： 761-769.
［14］
包静. 周期性基础隔震结构的地震响应分析［D］. 北京： 北京交通大学， 2011.
Bao J. Seismic response of a structure with periodic isolated foundations［D］. Beijing： Beijing Jiaotong University， 2011.
［15］
Sackman J L， Kelly J N， Javid A E. A layered notch filter for high-frequency dynamic isolation［J］. Journal of Pressure Vessel Technology， 1989， 111（1）： 17-24.
［16］
Shi Z F， Cheng Z B， Xiang H J. Seismic isolation foundations with effective attenuation zones［J］. Soil Dynamics and Earthquake Engineering， 2014， 57： 143-151.
［17］
冯海新， 严君， 刘洪， 等. 求解弹性波解耦方程的一种优化拟解析方法［J］. 地球物理学报， 2017， 60（9）： 3555-3573.
FENG Haixin， YAN Jun， LIU Hong， et al. Optimized pseudo-analytical method for decoupled elastic wave equations［J］. Chinese Journal of Geophysics， 2017， 60（9）： 3555-3573.
［18］
Bao J， Shi Z F， Xiang H J. Dynamic responses of a structure with periodic foundations［J］. Journal of Engineering Mechanics， 2012， 138（7）： 761-769.
2024年第37卷第5期
PDF下载
82
41
引用本文
BibTeX
文章信息
doi: 10.16385/j.cnki.issn.1004-4523.2024.05.006
  • 接收时间：2022-07-11
  • 首发时间：2026-02-09
  • 出版时间：2024-05-28
补充材料
相关文章
文章信息
作者
出版历史
  • 收稿日期：2022-07-11
  • 修回日期：2022-09-16
基金
桥梁结构健康与安全国家重点实验室开放课题(BHSKL19-07-GF)
国家自然科学基金资助项目(52078395)
武汉市城建委科技计划项目(武城建［2020］27号)
湖北省首届青年拔尖人才计划项目
作者信息
    1桥梁结构健康与安全国家重点实验室，湖北武汉　430034
    2武汉工程大学土木工程与建筑学院，湖北武汉　430074
    3中国电力建设工程咨询中南有限公司，湖北武汉　430071
    4福州大学土木工程学院，福建福州　350116

通讯作者:

许峙峰（1990―），男，博士，特聘副教授。E-mail：。
参考文献
分享链接
https://castjournals.cast.org.cn/joweb/zdgcxb/CN/10.16385/j.cnki.issn.1004-4523.2024.05.006
分享至
全文二维码

扫描看全文

引用本文
BibTeX
本文的引用情况
2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
关闭全屏
相关链接
论文
最新文章
热读文章
热点专题
内参
蓝皮书
产业发展报告
传播力报告
期刊分级目录
资讯
学术新闻
行业新闻
学术会议
行业会议
关于
关于我们
联系我们
北京市海淀区学院南路86号
100081
010-62199257
qkjq@cast.org.cn

中国科学技术协会版权所有 京ICP 备10216604号-4 海淀分局备案 1101084647
科技导报社主办 中国科协信息中心技术支持