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Article(id=1148106704539087356, tenantId=1146029695717560320, journalId=1146031787341344770, issueId=1148106698197295351, articleNumber=1003-3033(2025)02-0118-09, orderNo=null, doi=10.16265/j.cnki.issn1003-3033.2025.02.1216, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1726070400000, receivedDateStr=2024-09-12, revisedDate=1732118400000, revisedDateStr=2024-11-21, acceptedDate=null, acceptedDateStr=null, onlineDate=1751659569152, onlineDateStr=2025-07-05, pubDate=1740672000000, pubDateStr=2025-02-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1751659569152, onlineIssueDateStr=2025-07-05, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1751659569152, creator=13701087609, updateTime=1751659569152, updator=13701087609, issue=Issue{id=1148106698197295351, tenantId=1146029695717560320, journalId=1146031787341344770, year='2025', volume='35', issue='2', pageStart='1', pageEnd='252', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=0, createTime=1751659567641, creator=13701087609, updateTime=1757401525528, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1172190215188894212, tenantId=1146029695717560320, journalId=1146031787341344770, issueId=1148106698197295351, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1172190215188894213, tenantId=1146029695717560320, journalId=1146031787341344770, issueId=1148106698197295351, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=118, endPage=126, ext={EN=ArticleExt(id=1149767842909832019, articleId=1148106704539087356, tenantId=1146029695717560320, journalId=1146031787341344770, language=EN, title=Band gap characteristics and vibration isolation performance of combined periodic cross isolation trenches, columnId=1149733269173878863, journalTitle=China Safety Science Journal, columnName=Safety engineering technology, runingTitle=null, highlight=null, articleAbstract=

In order to effectively control low-frequency vibrations in engineering practice，the structure and the formation mechanism of the band gap were analyzed，and the finite element method was employed to investigate the influencing factor of the band gap in combined periodic cross-isolation trenches. Subsequently，this study analyzed vibration isolation performance using the model experiment. The results show that the band gap characteristics of the combined periodic cross-isolation trenches are mainly influenced by the periodic constant，the depth of the channel，the elastic modulus of soil and the filling medium. With the increase of the period constant，combined periodic cross-isolation trenches are more likely to obtain the low-frequency band gaps，but the bandwidth narrows. By increasing the depth，the low-frequency band gap with a larger bandwidth can be obtained，and the number of band gaps increases. With the increase of elastic modulus，the boundary frequency of the band gap migrates to high frequency synchronously，and a more low-frequency and wider band gap can be obtained by filling the medium. The experimental results show that the maximum attenuation amplitude of acceleration in the band gap range reaches 0.001 24 m/s2，the average amplitude attenuation ratio in the band gap is 0.206 2，and the maximum attenuation degree reaches 98.4%. The combined periodic cross-isolation trenches have remarkable vibration isolation performance.

, correspAuthors=Chunxin ZHAI, 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=Haiyang WANG, Chunxin ZHAI, Zhu WANG, Jinhao FENG, Xiaomao ZHANG), CN=ArticleExt(id=1148106717528847097, articleId=1148106704539087356, tenantId=1146029695717560320, journalId=1146031787341344770, language=CN, title=组合周期十字减振沟带隙特性及减振性能, columnId=1149733269727526997, journalTitle=中国安全科学学报, columnName=安全工程技术, runingTitle=null, highlight=null, articleAbstract=

为有效抑制工程实践中低频振动，在分析其能带结构及带隙形成机制的基础上，采用有限元方法研究组合周期十字减振沟带隙特性的影响因素，并结合模型试验分析其减振性能。结果表明：组合周期十字减振沟带隙特性主要受周期常数、减振沟深度、土体弹性模量和充填介质等因素影响；随着周期常数的增大，组合周期十字减振沟更易获得低频带隙，但是带宽变窄，而通过增大十字减振沟深度获得较大带宽的低频带隙，且带隙数量增多；随着弹性模量的增加，带隙的边界频率同步向高频迁移，通过填充介质获得更为低频且带隙较宽的带隙。试验结果显示，带隙范围内的加速度最大衰减幅度达到0.001 24 m/s2，带隙频段内的平均振幅衰减系数为0.206 2，且最大衰减程度达到98.45%，组合周期十字减振沟具有显著的减振性能。

, correspAuthors=翟春鑫, authorNote=null, correspAuthorsNote=
**翟春鑫(2000—)，男，河南商丘人，硕士研究生，研究方向为工程安全与防灾减灾技术。E-mail：。
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王海洋 (1988—)，男，山东淄博人，博士，副教授，主要从事工程安全与防灾减灾技术等方面的研究。E-mail：。

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王海洋 (1988—)，男，山东淄博人，博士，副教授，主要从事工程安全与防灾减灾技术等方面的研究。E-mail：。

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王海洋 (1988—)，男，山东淄博人，博士，副教授，主要从事工程安全与防灾减灾技术等方面的研究。E-mail：。

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Rock and Soil Mechanics, 2022, 43(1): 110-118, 126., articleTitle=Study on band gap characteristics and vibration isolation of local resonance pile with periodic four component, refAbstract=null), Reference(id=1165682121604608885, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, doi=null, pmid=null, pmcid=null, year=2019, volume=53, issue=6, pageStart=169, pageEnd=175, url=null, language=null, rfNumber=[24], rfOrder=28, authorNames=李锁斌, 窦益华, 陈天宁, journalName=西安交通大学学报, refType=null, unstructuredReference=李锁斌, 窦益华, 陈天宁, 等. 局域共振型周期结构振动带隙形成机理[J]. 西安交通大学学报, 2019, 53(6): 169-175, 182., articleTitle=局域共振型周期结构振动带隙形成机理, refAbstract=null), Reference(id=1165682121894015862, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, doi=null, pmid=null, pmcid=null, year=2019, volume=53, issue=6, pageStart=169, pageEnd=175, url=null, language=null, rfNumber=[24], rfOrder=29, authorNames=LI Suobin, DOU Yihua, CHEN Tianning, journalName=Journal of Xi'an Jiaotong University, refType=null, unstructuredReference=LI Suobin, DOU Yihua, CHEN Tianning, et al. Formation mechanisms of vibration band gaps in locally resonant periodic structures[J]. Journal of Xi'an Jiaotong University, 2019, 53(6): 169-175, 182., articleTitle=Formation mechanisms of vibration band gaps in locally resonant periodic structures, refAbstract=null), Reference(id=1165682122095342455, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, doi=null, pmid=null, pmcid=null, year=2019, volume=36, issue=增1, pageStart=144, pageEnd=148, url=null, language=null, rfNumber=[25], rfOrder=30, authorNames=王维超, 刘泽, 于桂兰, journalName=工程力学, refType=null, unstructuredReference=王维超, 刘泽, 于桂兰. 周期十字空沟地震表面波屏障[J]. 工程力学, 2019, 36(增1): 144-148., articleTitle=周期十字空沟地震表面波屏障, refAbstract=null), Reference(id=1165682122200200056, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, doi=null, pmid=null, pmcid=null, year=2019, volume=36, issue=S1, pageStart=144, pageEnd=148, url=null, language=null, rfNumber=[25], rfOrder=31, authorNames=WANG Weichao, LIU Ze, YU Guilan, journalName=Engineering Mechanics, refType=null, unstructuredReference=WANG Weichao, LIU Ze, YU Guilan. Periodic barriers with cruciform trenches for seismic surface waves[J]. 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figureFileSmall=HtiOKDieW4Gi5NaojrCgog==, figureFileBig=T384GAWMa+5hM5bM76IuBA==, tableContent=null), ArticleFig(id=1165682112846902093, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, language=EN, label=Table 1, caption=

Structural parameters of the combined periodic cross-isolation trenchesm

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系数 a b c h H
结构
参数		 2 1.9 0.5 2.7 12a
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组合周期十字减振沟结构参数

, figureFileSmall=null, figureFileBig=null, tableContent=
系数 a b c h H
结构
参数		 2 1.9 0.5 2.7 12a
), ArticleFig(id=1165682112972731215, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, language=EN, label=Table 2, caption=

Material parameter

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弹性模量/MPa 密度/(kg·m-3) 泊松比
材料土体值 30 1 800 0.3
), ArticleFig(id=1165682113023062864, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, language=CN, label=表2, caption=

材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
弹性模量/MPa 密度/(kg·m-3) 泊松比
材料土体值 30 1 800 0.3
), ArticleFig(id=1165682113102754641, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, language=EN, label=Table 3, caption=

Material parameters of filling structure

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材料 弹性模量/
MPa		 密度/
(kg·m-3)		 泊松比
混凝土 30 000 2 500 0.2
钢 210 000 7 850 0.3
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填充结构材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
材料 弹性模量/
MPa		 密度/
(kg·m-3)		 泊松比
混凝土 30 000 2 500 0.2
钢 210 000 7 850 0.3
), ArticleFig(id=1165682113526379347, tenantId=1146029695717560320, journalId=1146031787341344770, articleId=1148106704539087356, language=EN, label=Table 4, caption=

Acceleration before and after isolation at different frequencies (part)

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频率/Hz 加速度(无减振沟)/
(m·s-2)		 加速度(有减振沟)/
(m·s-2)
109.88 0.000 73 0.000 33
114.77 0.000 39 0.000 10
117.21 0.001 41 0.000 25
126.98 0.001 31 0.000 07
186.81 0.000 54 0.000 01
192.91 0.000 74 0.000 08
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不同频率下减振前后的加速度(部分)

, figureFileSmall=null, figureFileBig=null, tableContent=
频率/Hz 加速度(无减振沟)/
(m·s-2)		 加速度(有减振沟)/
(m·s-2)
109.88 0.000 73 0.000 33
114.77 0.000 39 0.000 10
117.21 0.001 41 0.000 25
126.98 0.001 31 0.000 07
186.81 0.000 54 0.000 01
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组合周期十字减振沟带隙特性及减振性能
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王海洋 1, 2 , 翟春鑫 1, 2, ** , 王珠 1, 2 , 冯晋昊 1, 2 , 张小毛 3
中国安全科学学报 | 安全工程技术 2025,35(2): 118-126
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中国安全科学学报 | 安全工程技术 2025, 35(2): 118-126
组合周期十字减振沟带隙特性及减振性能
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王海洋1, 2 , 翟春鑫1, 2, ** , 王珠1, 2, 冯晋昊1, 2, 张小毛3
作者信息
  • 1 重庆交通大学 交通土建工程材料国家地方联合工程实验室，重庆 400074
  • 2 重庆交通大学 土木工程学院，重庆 400074
  • 3 中铁十八局集团有限公司，天津 300322

    • 王海洋 (1988—)，男，山东淄博人，博士，副教授，主要从事工程安全与防灾减灾技术等方面的研究。E-mail：。

通讯作者:

**翟春鑫(2000—)，男，河南商丘人，硕士研究生，研究方向为工程安全与防灾减灾技术。E-mail：。
Band gap characteristics and vibration isolation performance of combined periodic cross isolation trenches
Haiyang WANG1, 2 , Chunxin ZHAI1, 2, ** , Zhu WANG1, 2, Jinhao FENG1, 2, Xiaomao ZHANG3
Affiliations
  • 1 National & Local Joint Engineering Laboratory of Traffic Civil Engineering Materials，Chongqing Jiaotong University，Chongqing 400074，China
  • 2 School of Civil Engineering，Chongqing Jiaotong University，Chongqing 400074，China
  • 3 China Railway 18th Bureau Group Co.，Ltd.，Tianjin 300222，China
出版时间: 2025-02-28 doi: 10.16265/j.cnki.issn1003-3033.2025.02.1216
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摘要
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为有效抑制工程实践中低频振动，在分析其能带结构及带隙形成机制的基础上，采用有限元方法研究组合周期十字减振沟带隙特性的影响因素，并结合模型试验分析其减振性能。结果表明：组合周期十字减振沟带隙特性主要受周期常数、减振沟深度、土体弹性模量和充填介质等因素影响；随着周期常数的增大，组合周期十字减振沟更易获得低频带隙，但是带宽变窄，而通过增大十字减振沟深度获得较大带宽的低频带隙，且带隙数量增多；随着弹性模量的增加，带隙的边界频率同步向高频迁移，通过填充介质获得更为低频且带隙较宽的带隙。试验结果显示，带隙范围内的加速度最大衰减幅度达到0.001 24 m/s2，带隙频段内的平均振幅衰减系数为0.206 2，且最大衰减程度达到98.45%，组合周期十字减振沟具有显著的减振性能。

组合周期  /  十字减振沟  /  带隙特性  /  减振性能  /  振幅衰减系数  /  低频带隙

In order to effectively control low-frequency vibrations in engineering practice，the structure and the formation mechanism of the band gap were analyzed，and the finite element method was employed to investigate the influencing factor of the band gap in combined periodic cross-isolation trenches. Subsequently，this study analyzed vibration isolation performance using the model experiment. The results show that the band gap characteristics of the combined periodic cross-isolation trenches are mainly influenced by the periodic constant，the depth of the channel，the elastic modulus of soil and the filling medium. With the increase of the period constant，combined periodic cross-isolation trenches are more likely to obtain the low-frequency band gaps，but the bandwidth narrows. By increasing the depth，the low-frequency band gap with a larger bandwidth can be obtained，and the number of band gaps increases. With the increase of elastic modulus，the boundary frequency of the band gap migrates to high frequency synchronously，and a more low-frequency and wider band gap can be obtained by filling the medium. The experimental results show that the maximum attenuation amplitude of acceleration in the band gap range reaches 0.001 24 m/s2，the average amplitude attenuation ratio in the band gap is 0.206 2，and the maximum attenuation degree reaches 98.4%. The combined periodic cross-isolation trenches have remarkable vibration isolation performance.

combination period  /  cross vibration isolation trenches  /  band gap characteristic  /  vibration isolation performance  /  amplitude attenuation ratio  /  low-frequency band gap
王海洋, 翟春鑫, 王珠, 冯晋昊, 张小毛. 组合周期十字减振沟带隙特性及减振性能. 中国安全科学学报, 2025 , 35 (2) : 118 -126 . DOI: 10.16265/j.cnki.issn1003-3033.2025.02.1216
Haiyang WANG, Chunxin ZHAI, Zhu WANG, Jinhao FENG, Xiaomao ZHANG. Band gap characteristics and vibration isolation performance of combined periodic cross isolation trenches[J]. China Safety Science Journal, 2025 , 35 (2) : 118 -126 . DOI: 10.16265/j.cnki.issn1003-3033.2025.02.1216
正文
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0 引言
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近年来，随着我国城市化进程的快速推进，工业活动引起的振动灾害日益严峻，其生成的低频高强振动波成为危害职业健康的因素之一[1]，会对人们身心健康及生活环境造成严重威胁[2-3]。低频振动波具有较长的波长和周期，并主要以表面波的形式进行传播，而基于周期结构的带隙特征，利用振动波的反射、折射等特性，在波的传播过程中设置周期减振沟可有效衰减和隔离表面波。聚焦于组合周期十字减振沟的低频带隙特性，通过研究其带隙特性影响因素和减振性能，为工程实践中低频振动的有效抑制提供理论和试验支持。
周期结构理论起源于FLOQUET[4]对一维Mathieu's方程的研究，随后RAYLEIGH[5]又研究了一维周期结构中弹性波的传播问题，提出有限周期结构衰减域的概念，BLOCH[6]将Floquet理论进行了推广并应用至三维周期结构中，最终形成Bloch-Floquet定理[7-8]。此后大量学者将此理论推广应用至波在周期结构中的传播特征研究中，如振动波[9-10]、声波、电磁波[11]等，工业上统一将其称为声子晶体、光子晶体和周期结构[12]。周期结构具有典型的带隙特征，其表示为波动在周期结构内部传播的圆频率与波矢之间的关系，结合Bloch-Floquet定理求解典型单元的特征值即可得到周期结构的带隙特征[13]。研究周期结构带隙特征大都集中在声子晶体领域，MINIACI等[14]研究了十字形开孔声子晶体板的带隙特征并通过超声波试验进行了验证；ZHANG Yafang等[15]设计了不同材料组合的声子晶体管并计算了其带隙特征，表明该种结构能够打开低频带隙;姜博龙[16]采用平面波展开法计算了混凝土异形截面桩的平面内和平面外带隙，并分析了影响因素。
基于周期结构的带隙特性，许多学者设计优化了波屏障等减振措施。首先，WOODS[17]通过隔振试验，得出周期结构桩想要达到良好的减振效果，需满足桩径大于屏蔽波波长的1/6的要求；MA Meng等[18]借助室内试验，验证了周期结构屏障的减振效果，并明确减振性能和桩间距的关系。除试验研究，诸多学者通过数值模拟的方式研究周期性结构的带隙特性。HUANG Jiankun等[19]提出一种分层周期结构波屏障，该屏障由2种不同的材料交叉拓展而成，该结构在45~75 Hz产生较宽的带隙，研究了不同材料组分与几何参数对于铁路荷载下减振效果的影响；BRÛLÉ等[20]开展了全尺寸现场试验，并结合数值模拟，证明了布拉格周期结构可有效降低50 Hz的振动波;MINIACI等[21]通过在土体中开十字孔、圆孔和加入圆柱谐振体的方法设计了3种不同的周期结构，研究其对表面波以及导波的衰减作用，十字孔结构可对地震波的衰减有显著作用;高盟等[22]将传统波阻板进行正方形排列和三角形排列形成周期结构波阻板，通过数值模拟的方法，研究几何参数以及形状对带隙的影响，并通过正交分析得到最优组合形式；陈晓斌等[23]分析了四组元局域共振周期桩的带隙特征，并且相较于三组元结构，其减振效果提升近30%；李锁斌等[24]明确二维局域共振周期结构的振动带隙形成机制，为周期结构带隙的主动设计奠定了基础。虽然目前国内外关于周期结构减振屏障取得了大量的研究成果，但还缺乏对组合周期十字减振低频带隙特性影响因素的系统研究，尚不明确其带隙形成机制和减振性能。
鉴于此，笔者在分析组合周期十字减振沟能带结构及带隙形成机制的基础上，利用数值模拟系统研究周期十字减振沟带隙特性的影响因素，并利用工程振动控制试验台验证分析其减振性能，以期为组合周期十字减振沟优化设计及应用提供参考。
1 结构设计与带隙特性及形成机制
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1.1 结构设计
组合周期十字减振沟具备典型的带隙特征，通过在土体中周期性挖掘十字形空腔即可形成，其单胞结构模型和单胞x-y平面如图1所示，其中，H为土体厚度，h为空沟深度，a为单胞周期常数，b为十字空腔长度，c为十字形空腔宽度，单位均为m。
对于均匀的各向同性线弹性波介质，在不考虑阻尼且初始应力为0的假设下，在笛卡尔坐标系下建立弹性波动方程：
ρ u ¨ = ▽ [ ( λ + 2 μ ) ( ▽ · u ) ] - ▽ × ( μ ▽ × u )
式中：ρ为质量密度，kg/m3； u ¨为位移对时间的二阶导数▽为倒三角算子；u为位移矢量； λ和 μ为拉梅常数。
根据Bloch-Floquet定理，简化式(1)可得下式周期性边界条件：
u ( r + a ) = u ( r ) e x p [ i ( k · a ) ]
式中：r为位置矢量；k为波矢量。
由于模型单胞结构在x-y平面内具有周期性，将无限自由度的组合周期十字减振沟周期结构在其边界添加周期性边界条件。利用有限元软件COMSOL Multiphysics进行带隙计算，将k沿图2中第一不可约Brillouin区的Γ-X-M-Γ参数化扫描，即可得到该结构在不同波矢条件下所对应的特征频率，再绘制波矢特征频率曲线图，得到其频散曲线。
1.2 带隙特性及形成机制分析
采用COMSOL Multiphysics固体力学模块计算其带隙特性。根据已有研究[20]，十字减振沟的长度b>0.85倍的周期常数时会产生结构表面波带隙，并且土体厚度大于10倍周期常数便可保证表面波带隙计算的准确性[25]，而十字形空腔宽度c对表面波带隙影响较小，结构参数和材料属性见表1和表2。对模型进行网格划分与参数化扫描后，计算得到十字减振沟的能带结构如图3所示，可见十字空腔的结构设计在一定尺寸下将打开2条带隙，图中斜线区域内侧边缘线为声线，带隙用灰色区域表示。从图3可以看出，模型的2条完全带隙均处于30 Hz以内，第1条完全带隙在第6能带与第7模态之间，范围为14.18~17.07 Hz；第2条完全带隙在第7与第8能带之间，范围为18.21~19.89 Hz，结构的带隙总宽度为4.89 Hz。因此，组合周期十字减振沟的低频带隙范围相对于低频振动具有显著优势。
为进一步研究组合周期十字减振沟带隙形成机制，分别在COMSOL Multiphysics软件中提取图3中A、B、C、D特殊点处的振型模态，D点为声线上的一个点，B点为第2带隙的下边界，C点为第1带隙的下边界，4种不同模态的特征频率分别为24.81、18.21、14.08、17.96 Hz，分别为2、1、2.1、0.9，其模态振型如图4所示。对比研究单胞振型模态与能带结构，相应的运动主要来源于方形块的平面外弯曲，类似于梁的弯曲振动。晶胞中相邻的一对块体反向振动，而对角的一对则同向振动，十字减振沟间发生大量位移振动，结构间产生明显的局域共振现象，振动波能量被耗散，从而产生特定频率下的带隙。此外，因为十字空腔由比较窄的土壤连接，十字空腔的刚性降低，在能带上方的A点模型的弯曲运动占据主导地位。
2 带隙影响因素分析
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通过上述理论分析，组合周期十字减振沟带隙特征除了受结构参数的影响，还将随着材料参数的变化而变化。此外，为达到较宽的带隙，要求十字空腔具有足够大的填充率。因此，随着填充率的增大，以及十字空腔结构深度的增大，结构的稳定性会有所下降，为保证结构稳定性，会填充不同的介质。为此，采用数值模拟研究周期常数、十字空腔高度、土体弹性模量以及充填介质4个因素对带隙的起始频率、截止频率和带隙宽度的影响。
2.1 周期常数
按表1和表2所示在组合周期十字空腔结构参数和材料属性不变的条件下，随着周期常数的增大，组合周期十字减振沟的带隙特性变化如图5所示。从图5可以看出，随着单胞常数的不断增加，带隙的起始频率和截止频率均向低频移动，且下降趋势逐渐变缓，但截止频率的下降幅度相对更大，结构完全带隙的带隙宽度从7 Hz逐渐减至1 Hz。在周期常数从0.5 m增加到3 m的过程中，起始频率和截止频率分别从64.9和72.9 Hz降低到10.8和11.8 Hz，降低幅度分别达到83.36%和83.82%，声锥高度从108.1 Hz减低至17.9 Hz，降低83.44 %。这是由于随着周期单胞常数的增加，十字空腔内部结构逐渐增大，结构柔度增大，刚度降低。同时，声锥高度也与周期常数和剪切波速有关，周期常数增加会导致声锥高度的减小，两者同时变化会导致带隙向低频方向移动。因此，在需要抑制低频振动的应用中，在保证结构刚度与稳定性的情况下可选择较高的周期常数。
2.2 十字空腔高度
十字空腔高度对于完全带隙起着基本作用，控制其他参数不变，通过改变h/a分别为1.15、1.25、1.35、1.5，得到空腔高度对十字减振沟带隙的影响，如图6所示。由图6可知：随着十字空腔高度逐渐增加，当h/a≤1.2时，带隙的截止频率不受影响，基本保持在20 Hz，而起始频率逐渐向低频移动，从19 Hz降低到15.5 Hz，因此，带隙宽度逐渐从1 Hz增加到4 Hz。当h/a从1.25变化至1.5的过程中，能带结构出现2条带隙，且随着十字空腔高度的增大，第1带隙的起始和截止频率与第2带隙的起始频率均呈现先下降后上升的趋势，而第2带隙的截止频率不受影响，同样保持20 Hz不变。这是因为表面波模态内部十字空腔结构向平面外弯曲，由于空腔被较窄的土层连接，随着深度的越大，减振沟的刚度越小，结构弯曲振动占主导，导致表面波模态变丰富，特征频率变低。此外，2条带隙宽度均从h/a=1.35时开始缩减，第1条带隙宽度从2.89 Hz降低到2.7 Hz，第2条从2.2 Hz直至带隙消失。当h/a=1.5时，能带结构变回1条完全带隙。因此，h/a=1.35表面波带隙宽度最大，在实际应用中可以控制特定空腔高度以实现较宽且低频的带隙。
2.3 土体弹性模量
设定结构参数不变，仅改变土体的弹性模量，对于带隙的影响特征如图7所示。从图7中可以看出，当土体弹性模量较低时，带隙集中于低频范围内，这是由于较小的结构刚度对低波阻隔作用更强，随着土体弹性模量的增加，完全带隙的起始和截止频率均向高频移动，土体弹性模量从20 MPa增加到120 MPa的过程中，带隙的起始和截止频率分别增加149.63%和143.29%，但截止频率增加程度相对更为迅速，因此，带隙宽度从2.9 Hz增加到6.22 Hz。这是因为随着土体弹性模量的增加，土体相对变硬，结构刚度变大，导致完全带隙向高频移动。因此，土体属性的变化对带隙具有显著影响，对于弹性模量较高的土体，周期十字减振沟对小于20 Hz的低频振动衰减能力较弱;当土体弹性模量小于40 MPa时，组合周期十字减振沟对低频表面波振动具有较好的衰减作用，在实际应用过程中，需要考虑不同应用场地的土体材质，进而确定被保护对象的频率范围。
2.4 十字减振沟充填介质
从现实应用的角度出发，使用Q235A型钢及C30混凝土进行介质填充，其构形分别如图8b和图8c所示，其长度b为1.9 m，宽度c为0.5 m，空心十字截面的厚度为0.1 m，并且填充高度都为0.8 m，其中，土体材料参数见表2，混凝土和钢材料参数见表3。
分别计算空心十字钢及实心十字混凝土能带结构如图9所示，从图9中可以看出，空心十字钢截面产生2条带隙，带隙范围分别为12.39~16.39和19.1~20 Hz，带隙宽度分别为4和0.9 Hz。十字实心混凝土截面产生1条带隙，带隙范围为12.44~17.59 Hz，带隙宽度为5.51 Hz，实心十字混凝土填充拥有更宽的带隙。相比于未填充十字空腔结构，2种充填结构的起始频率均比十字空腔的起始频率低，并且十字实心混凝土截面的带隙宽度更大。这是因为填充介质后，整体结构的刚度及质量均增大，根据 f = K / J / 2 π，其中，K为等效刚度，J为等效质量。当等效刚度的增加趋势小于等效质量时，组合周期十字减振沟的频率逐渐降低，而声锥曲线不变，进而使得频带降低。因此，在应用时，可以考虑填充不同的介质以应对更低频的振动灾害。
3 组合周期十字减振沟减振性能分析
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3.1 组合周期十字减振沟试验概况
为进一步研究组合周期十字减振沟带隙范围内减振性能，利用工程振动试验台进行试验。该试验台整体安装尺寸的长×宽×高为2 m×1.5 m×0.9 m，并采用5 cm厚的聚苯乙烯泡沫板紧密贴合在其底部及内壁四周，以降低周围的反射波对试验结果的影响。试验土体取自重庆轨道交通15号线2期土建工程施工现场，风干后采用分层压实的方法填充于试验箱中。激振系统包括激振器、功率放大器。拾振系统包括压电式加速度传感器、信号接收器及带有数据分析软件的电脑等组成，试验设置激振器施加脉冲荷载，电压幅值输入为4 000 mV，周期为0.2 s，占空比为10%。
试验布设如图10所示，受振动试验平台尺寸以及人力物力的影响，根据试验平台尺寸采用1∶10小尺寸开展试验研究。此外，为确保小尺寸十字空腔的稳定性，十字空腔边界由3 mm厚较硬的有机玻璃材料薄层制成[18]，这一额外的层会使完全带隙移到稍高的频率，但可确保足够的土壤压实，避免因弯曲扭转变形导致十字空腔被破坏，如图11a所示。设置其单胞周期常数为0.2 m，且由于十字空腔宽度过小时会导致土体不稳定，且压实难度增大，因此，在试验时选择十字空腔宽度b为0.9倍的周期常数0.18 m，c取0.25倍周期常数0.05 m，h取0.27 m，H取12倍周期常数。压实布设完成后的十字空沟试验模型如图11b所示，设置无十字减振沟工况和有十字减振沟工况进行对照试验。
为便于分析试验模型的带隙特征，根据实测数据，得到土壤密度为1 610 kg/m3，弹性模量为29.47 MPa，泊松比为0.36，计算得到周期十字减振沟的完全带隙和方向带隙计算结果如图12和图13a所示，图中每一条曲线代表一阶频带，完全频率范围为183.00~200.16 Hz，存在3条k沿Γ-X的方向带隙，频率范围分别是108.85~112.15、133.22~ 156.64和172.58~200.16 Hz。
3.2 试验结果及减振性能分析
通过分析地表加速度实测数据，得到不同频率下应用十字减振沟前后的加速度，见表4，并绘制振动加速度频谱曲线，如图13b所示，并对比其与方向带隙计算结果，在布置周期十字减振沟后，地表加速度幅值在带隙频率范围内明显降低，加速度频域曲线的衰减区间要比方向带隙的能带结构更宽。首先加速度幅值在108.66 ~118.43 Hz出现较大程度衰减，平均衰减幅度达到75.96%，并且在109.88和114.77 Hz达到谷值，衰减幅度分别达到76.81%和75.75%，117.21 Hz达到最大衰减幅度，达0.001 16 m/s2。振动波在124.53~157.50 Hz则有更大程度的衰减，整体平均衰减幅度达到82.41%，并且最大衰减幅度在124.53 Hz达到了97.69%，当频率为126.98 Hz时，加速度达到最大衰减值，为0.001 24 m/s2。此外，振动波在172.15~200.00 Hz范围内同样有较强的抑制，平均衰减幅度达到72.58%，最大加速度降低幅值在192.91 Hz达到0.000 66 m/s2，并且地表加速度在186.81 Hz达到最低值，为0.000 53 m/s2，此时降低幅度达到98.45%。并且对于100 Hz以内以及非带隙区间内的振动波，由于结构的局域共振作用，对应于加速度频谱图中该范围内同样会有衰减，但带隙区间内振动衰减幅度大于带隙外的衰减程度。因此，验证了带隙减振效应的有效性，组合周期十字减振沟对振动波具有显著的减振性能。
此外，为进一步探究组合周期十字减振沟在带隙频率范围内实际减振性能，计算不同频率下的振幅衰减系数T，如图14所示。首先整个带隙频率范围内的平均T为0.206 2，随着向带隙中心频段靠近，振动波在108.85~112.15 Hz的带隙范围内受到大量衰减，T从0.453 7逐步减小到0.221 6，减振性能逐渐增强。在133.22~156.64 Hz的带隙范围内，T在155.06 Hz达到谷值，振动波衰减程度达到92.1%，整段频率的平均T为0.198 4。此外，在172.58~200.16 Hz的方向和完全带隙重叠频段内，振动同样得到有效抑制，此频段的平均T达到0.198 8，最大衰减程度同样在186.81 Hz处取得，衰减程度为98.45%。研究结果表明：组合周期十字减振沟能够较大程度地抑制振动波的传播，再一次证明了其具有良好的减振性能。
4 结论
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1) 周期十字减振沟带隙特性受周期常数和十字空腔高度影响较大，高周期常数可获得低频带隙，但带宽较窄，而带隙数量随着十字空沟高度的增加而增多，且带隙均呈现先增宽后缩窄的变化趋势，因此，在减弱低频振动的场景中可考虑高周期常数和十字空腔高度。
2) 选取高弹性模量填充介质可获得高频、宽带隙特征，选取小于40 MPa的弹性模量填充介质可以阻隔30 Hz以下的表面波，且实际应用时需要考虑土体材质，进而确定被保护对象的频率范围。相比于未填充十字空腔结构，充填介质结构的带隙宽度及起始频率均低于十字空腔，可在应对更为低频的振动灾害中选取填充合适的介质。
3) 通过开展组合周期十字减振沟的减振性能试验，加速度频谱图衰减区间与带隙范围基本吻合，十字减振沟应用后整体加速度幅值明显低于无十字减振沟的情况，振动波加速度最大衰减幅度达到0.001 24 m/s2，带隙频段内的平均T为0.206 2，且最大衰减程度达到98.45%，表明组合周期十字减振沟具有显著的减振性能。
基金
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  • 重庆市自然科学基金资助(CSTB2023NSCQ-MSX0180)
  • 重庆交通大学交通土建工程材料国家地方联合工程实验室开放基金资助(LHSYS-2021-006)
  • 安徽省建筑声环境重点实验室开放基金资助(AAE2022KFO2)
  • 重庆交通大学研究生科研创新项目(2023S0026)
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doi: 10.16265/j.cnki.issn1003-3033.2025.02.1216
  • 接收时间：2024-09-12
  • 首发时间：2025-07-05
  • 出版时间：2025-02-28
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  • 收稿日期：2024-09-12
  • 修回日期：2024-11-21
基金
重庆市自然科学基金资助(CSTB2023NSCQ-MSX0180)
重庆交通大学交通土建工程材料国家地方联合工程实验室开放基金资助(LHSYS-2021-006)
安徽省建筑声环境重点实验室开放基金资助(AAE2022KFO2)
重庆交通大学研究生科研创新项目(2023S0026)
作者信息
    1 重庆交通大学 交通土建工程材料国家地方联合工程实验室，重庆 400074
    2 重庆交通大学 土木工程学院，重庆 400074
    3 中铁十八局集团有限公司，天津 300322

通讯作者:

**翟春鑫(2000—)，男，河南商丘人，硕士研究生，研究方向为工程安全与防灾减灾技术。E-mail：。
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https://castjournals.cast.org.cn/joweb/zgaqkxxb/CN/10.16265/j.cnki.issn1003-3033.2025.02.1216
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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
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