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Article(id=1228295920496870060, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295917908980268, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2025.03.017, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1708012800000, receivedDateStr=2024-02-16, revisedDate=1712246400000, revisedDateStr=2024-04-05, acceptedDate=null, acceptedDateStr=null, onlineDate=1770778168080, onlineDateStr=2026-02-11, pubDate=1741536000000, pubDateStr=2025-03-10, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770778168080, onlineIssueDateStr=2026-02-11, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770778168080, creator=13701087609, updateTime=1770778168080, updator=13701087609, issue=Issue{id=1228295917908980268, tenantId=1146029695717560320, journalId=1225147924628267009, year='2025', volume='38', issue='3', pageStart='449', pageEnd='662', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770778167464, creator=13701087609, updateTime=1770949125482, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1229012967862235389, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295917908980268, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1229012967862235390, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295917908980268, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=604, endPage=611, ext={EN=ArticleExt(id=1228295920727556784, articleId=1228295920496870060, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Analysis of transverse free vibration characteristics of two-span continuously modified Timoshenko beams on viscoelastic four-parameter foundation, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In order to solve the transverse vibration problem of two-span continuously modified Timoshenko beam on viscoelastic four-parameter foundation，a new vibration governing equation is established by combining two-span continuously modified Timoshenko beam with viscoelastic four-parameter foundation. By using the echo matrix method，bisection and golden section method，the relation and difference between the natural vibration characteristics of two-span（equal-span，unequal-span）continuously modified Timoshenko beam and single-span modified Timoshenko beam on viscoelastic four-parameter foundation are analyzed. The results show that for the modified Timoshenko beam on the viscoelastic four-parameter foundation，the natural frequency of each order of the single-span beam is less than that of the two-span continuous beam，the even-order natural frequency and attenuation coefficient of the single-span beam are the same as the odd-order natural frequency and attenuation coefficient of the two equal-span continuous beam，and the odd-order natural frequency of the unequal-span two-span continuous beam is less than that of the two-span continuous beam. The even-order mode shapes of two equal-span continuous modified Timoshenko beams are symmetrical with respect to the supports in the middle of the span，and the odd-order modes are antisymmetric with respect to the mid-span.

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为解决黏弹性四参数地基上两跨连续修正Timoshenko梁的横向振动问题，将两跨连续修正Timoshenko梁与黏弹性四参数地基进行组合，建立新的振动控制方程，运用回传射线矩阵法，结合二分法和黄金分割法，分析了黏弹性四参数地基上两跨（等跨、不等跨）连续修正Timoshenko梁与单跨修正Timoshenko梁自振特性之间的联系与区别。结果表明：对于黏弹性四参数地基上的修正Timoshenko梁，单跨梁的各阶自振频率小于两跨连续梁的各阶自振频率，单跨梁的偶数阶自振频率和衰减系数与两等跨连续梁的奇数阶自振频率和衰减系数相同，不等跨两跨连续梁奇数阶自振频率小于两等跨连续梁奇数阶自振频率；两等跨连续修正Timoshenko梁的偶数阶振型关于跨中支座对称，奇数阶振型关于跨中支座反对称。

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柳伟（1990—），男，硕士，讲师。E-mail：
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rfNumber=[1], rfOrder=0, authorNames=雷晓燕, 邢聪聪, 吴神花, journalName=振动工程学报, refType=null, unstructuredReference=雷晓燕，邢聪聪，吴神花. 轨道结构中高频振动特性分析[J]. 振动工程学报，2020,33(6):1245-1252., articleTitle=轨道结构中高频振动特性分析, refAbstract=null), Reference(id=1228312970787225906, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2020, volume=33, issue=6, pageStart=1245, pageEnd=1252, url=null, language=null, rfNumber=[1], rfOrder=1, authorNames=LEI Xiaoyan, XING Congcong, WU Shenhua, journalName=Journal of Vibration Engineering, refType=null, unstructuredReference=LEI Xiaoyan，XING Congcong，WU Shenhua. Mid-and high-frequency vibration characteristics of track structures[J]. Journal of Vibration Engineering，2020,33(6):1245-1252., articleTitle=Mid-and high-frequency vibration characteristics of track structures, refAbstract=null), Reference(id=1228312970887889206, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2019, volume=32, issue=5, pageStart=38, pageEnd=46, url=null, language=null, rfNumber=[2], rfOrder=2, authorNames=刘飞禹, 吴杰杰, 陈江, journalName=中国公路学报, refType=null, unstructuredReference=刘飞禹，吴杰杰，陈江，等.考虑水平摩阻的Pasternak路基基层及路面变形分析[J]. 中国公路学报，2019,32(5):38-46., articleTitle=考虑水平摩阻的Pasternak路基基层及路面变形分析, refAbstract=null), Reference(id=1228312971017912630, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2019, volume=32, issue=5, pageStart=38, pageEnd=46, url=null, language=null, rfNumber=[2], rfOrder=3, authorNames=LIU Feiyu, WU Jiejie, CHEN Jiang, journalName=China Journal of Highway and Transport, refType=null, unstructuredReference=LIU Feiyu，WU Jiejie，CHEN Jiang，et al.Analysis of pavement and subbase on Pasternak foundation considering horizontal friction[J]. China Journal of Highway and Transport，2019,32(5):38-46., articleTitle=Analysis of pavement and subbase on Pasternak foundation considering horizontal friction, refAbstract=null), Reference(id=1228312971114381625, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2023, volume=54, issue=2, pageStart=244, pageEnd=253, url=null, language=null, rfNumber=[3], rfOrder=4, authorNames=肖旻, 王正中, 吴浪, journalName=水利学报, refType=null, unstructuredReference=肖旻，王正中，吴浪，等.寒区预制混凝土衬砌梯形渠道弹性冻土地基梁模型[J]. 水利学报，2023,54(2):244-253., articleTitle=寒区预制混凝土衬砌梯形渠道弹性冻土地基梁模型, refAbstract=null), Reference(id=1228312971256987966, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2023, volume=54, issue=2, pageStart=244, pageEnd=253, url=null, language=null, rfNumber=[3], rfOrder=5, authorNames=XIAO Min, WANG Zhengzhong, WU Lang, journalName=Journal of Hydraulic Engineering, refType=null, unstructuredReference=XIAO Min，WANG Zhengzhong，WU Lang，et al.Elastic frozen soil foundation beam model for trapezoidal canal lined with concrete precast slabs in cold regions[J]. Journal of Hydraulic Engineering，2023,54(2):244-253., articleTitle=Elastic frozen soil foundation beam model for trapezoidal canal lined with concrete precast slabs in cold regions, refAbstract=null), Reference(id=1228312971357651267, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2023, volume=45, issue=5, pageStart=912, pageEnd=920, url=null, language=null, rfNumber=[4], rfOrder=6, authorNames=禹海涛, 卫一博, journalName=岩土工程学报, refType=null, unstructuredReference=禹海涛，卫一博. 穿断层分段柔性接头隧道纵向地震响应解析解[J]. 岩土工程学报，2023,45(5):912-920., articleTitle=穿断层分段柔性接头隧道纵向地震响应解析解, refAbstract=null), Reference(id=1228312971475091783, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2023, volume=45, issue=5, pageStart=912, pageEnd=920, url=null, language=null, rfNumber=[4], rfOrder=7, authorNames=YU Haitao, WEI Yibo, journalName=Chinese Journal of Geotechnical Engineering, refType=null, unstructuredReference=YU Haitao，WEI Yibo. Analytical solution for longitudinal seismic response of tunnels with segmental flexible joints crossing faults[J]. Chinese Journal of Geotechnical Engineering，2023,45(5):912-920., articleTitle=Analytical solution for longitudinal seismic response of tunnels with segmental flexible joints crossing faults, refAbstract=null), Reference(id=1228312971567366476, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2015, volume=47, issue=12, pageStart=25, pageEnd=27, url=null, language=null, rfNumber=[5], rfOrder=8, authorNames=刘希成, journalName=煤炭工程, refType=null, unstructuredReference=刘希成. 弹性地基梁理论在软弱地基柱下条形基础中的应用[J]. 煤炭工程，2015,47(12):25-27., articleTitle=弹性地基梁理论在软弱地基柱下条形基础中的应用, refAbstract=null), Reference(id=1228312971680612689, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2015, volume=47, issue=12, pageStart=25, pageEnd=27, url=null, language=null, rfNumber=[5], rfOrder=9, authorNames=LIU Xicheng, journalName=Coal Engineering, refType=null, unstructuredReference=LIU Xicheng. Application of elastic foundation beam theory in strip foundation under column of soft ground[J]. Coal Engineering，2015,47(12):25-27., articleTitle=Application of elastic foundation beam theory in strip foundation under column of soft ground, refAbstract=null), Reference(id=1228312971798053203, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2015, volume=37, issue=3, pageStart=302, pageEnd=316, url=null, language=null, rfNumber=[6], rfOrder=10, authorNames=夏桂云, 曾庆元, journalName=力学与实践, refType=null, unstructuredReference=夏桂云，曾庆元. 深梁理论的研究现状与工程应用[J]. 力学与实践，2015,37(3):302-316., articleTitle=深梁理论的研究现状与工程应用, refAbstract=null), Reference(id=1228312971915493719, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2015, volume=37, issue=3, pageStart=302, pageEnd=316, url=null, language=null, rfNumber=[6], rfOrder=11, authorNames=XIA Guiyun, ZENG Qingyuan, journalName=Mechanics in Engineering, refType=null, unstructuredReference=XIA Guiyun，ZENG Qingyuan. Timoshenko beam theory and its applications[J]. Mechanics in Engineering，2015,37(3):302-316., articleTitle=Timoshenko beam theory and its applications, refAbstract=null), Reference(id=1228312972024545626, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2005, volume=33, issue=6, pageStart=711, pageEnd=715, url=null, language=null, rfNumber=[7], rfOrder=12, authorNames=陈镕, 万春风, 薛松涛, journalName=同济大学学报（自然科学版）, refType=null, unstructuredReference=陈镕，万春风，薛松涛，等.Timoshenko梁运动方程的修正及其影响[J]. 同济大学学报（自然科学版），2005,33(6):711-715., articleTitle=Timoshenko梁运动方程的修正及其影响, refAbstract=null), Reference(id=1228312972095848797, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2005, volume=33, issue=6, pageStart=711, pageEnd=715, url=null, language=null, rfNumber=[7], rfOrder=13, authorNames=CHEN Rong, WAN Chunfeng, XUE Songtao, journalName=Journal of Tongji University（Natural Science）, refType=null, unstructuredReference=CHEN Rong，WAN Chunfeng，XUE Songtao，et al.Modification of motion equation of Timoshenko beam and its effect[J]. Journal of Tongji University（Natural Science），2005,33(6):711-715., articleTitle=Modification of motion equation of Timoshenko beam and its effect, refAbstract=null), Reference(id=1228312972196512096, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2021, volume=48, issue=11, pageStart=142, pageEnd=149, url=null, language=null, rfNumber=[8], rfOrder=14, authorNames=夏桂云, journalName=湖南大学学报（自然科学版）, refType=null, unstructuredReference=夏桂云. Timoshenko梁的第二频谱分析[J]. 湖南大学学报（自然科学版），2021,48(11):142-149., articleTitle=Timoshenko梁的第二频谱分析, refAbstract=null), Reference(id=1228312973568049508, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2021, volume=48, issue=11, pageStart=142, pageEnd=149, url=null, language=null, rfNumber=[8], rfOrder=15, authorNames=XIA Guiyun, journalName=Journal of Hunan University（Natural Sciences）, refType=null, unstructuredReference=XIA Guiyun. Analysis on the second frequency spectrum of Timoshenko beam[J]. Journal of Hunan University（Natural Sciences），2021,48(11):142-149., articleTitle=Analysis on the second frequency spectrum of Timoshenko beam, refAbstract=null), Reference(id=1228312973639352681, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2020, volume=39, issue=3, pageStart=30, pageEnd=37, url=null, language=null, rfNumber=[9], rfOrder=16, authorNames=王家乐, 夏桂云, journalName=振动与冲击, refType=null, unstructuredReference=王家乐，夏桂云. Winkler地基上修正Timoshenko梁振动分析[J]. 振动与冲击，2020,39(3):30-37., articleTitle=Winkler地基上修正Timoshenko梁振动分析, refAbstract=null), Reference(id=1228312973723238764, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2020, volume=39, issue=3, pageStart=30, pageEnd=37, url=null, language=null, rfNumber=[9], rfOrder=17, authorNames=WANG Jiale, XIA Guiyun, journalName=Journal of Vibration and Shock, refType=null, unstructuredReference=WANG Jiale，XIA Guiyun. Vibration analysis for a modified Timoshenko beam on Winkler elastic foundation[J]. Journal of Vibration and Shock，2020,39(3):30-37., articleTitle=Vibration analysis for a modified Timoshenko beam on Winkler elastic foundation, refAbstract=null), Reference(id=1228312973832290672, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2022, volume=44, issue=4, pageStart=751, pageEnd=758, url=null, language=null, rfNumber=[10], rfOrder=18, authorNames=余云燕, 孔嘉乐, 陈进浩, journalName=地震工程学报, refType=null, unstructuredReference=余云燕，孔嘉乐，陈进浩，等.变截面修正Timoshenko梁自振频率的回传射线矩阵法分析[J]. 地震工程学报，2022,44(4):751-758., articleTitle=变截面修正Timoshenko梁自振频率的回传射线矩阵法分析, refAbstract=null), Reference(id=1228312973941342581, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2022, volume=44, issue=4, pageStart=751, pageEnd=758, url=null, language=null, rfNumber=[10], rfOrder=19, authorNames=YU Yunyan, KONG Jiale, CHEN Jinhao, journalName=China Earthquake Engineering Journal, refType=null, unstructuredReference=YU Yunyan，KONG Jiale，CHEN Jinhao，et al.Natural frequency of a modified Timoshenko beam with variable cross-section with the method of reverberation ray matrix[J]. China Earthquake Engineering Journal，2022,44(4):751-758., articleTitle=Natural frequency of a modified Timoshenko beam with variable cross-section with the method of reverberation ray matrix, refAbstract=null), Reference(id=1228312974025228664, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2011, volume=30, issue=1, pageStart=124, pageEnd=127, url=null, language=null, rfNumber=[11], rfOrder=20, authorNames=吴晓, 孙晋, 黄翀, journalName=振动与冲击, refType=null, unstructuredReference=吴晓，孙晋，黄翀，等.用Timoshenko梁修正理论研究泡沫金属铝合金梁的动力响应[J]. 振动与冲击，2011,30(1):124-127., articleTitle=用Timoshenko梁修正理论研究泡沫金属铝合金梁的动力响应, refAbstract=null), Reference(id=1228312974130086266, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2011, volume=30, issue=1, pageStart=124, pageEnd=127, url=null, language=null, rfNumber=[11], rfOrder=21, authorNames=WU Xiao, SUN Jin, HUANG Chong, journalName=Journal of Vibration and Shock, refType=null, unstructuredReference=WU Xiao，SUN Jin，HUANG Chong，et al.Dynamic responses of a foam metal aluminum alloy beam with Timoshenko beam theory[J]. Journal of Vibration and Shock，2011,30(1):124-127., articleTitle=Dynamic responses of a foam metal aluminum alloy beam with Timoshenko beam theory, refAbstract=null), Reference(id=1228312974230749567, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2023, volume=27, issue=4, pageStart=1209, pageEnd=1223, url=null, language=null, rfNumber=[12], rfOrder=22, authorNames=LI M L, WEI P J, ZHOU X L, journalName=Mechanics of Time-Dependent Materials, refType=null, unstructuredReference=LI M L，WEI P J，ZHOU X L. Wave propagation and free vibration of a Timoshenko beam mounted on the viscoelastic Pasternak foundation modeled by fraction-order derivatives[J]. Mechanics of Time-Dependent Materials，2023,27(4):1209-1223., articleTitle=Wave propagation and free vibration of a Timoshenko beam mounted on the viscoelastic Pasternak foundation modeled by fraction-order derivatives, refAbstract=null), Reference(id=1228312974318829955, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2014, volume=34, issue=4, pageStart=148, pageEnd=152, url=null, language=null, rfNumber=[13], rfOrder=23, authorNames=郑仰坤, 袁向荣, journalName=噪声与振动控制, refType=null, unstructuredReference=郑仰坤，袁向荣. 不同跨径比的两跨连续梁的模态分析试验[J]. 噪声与振动控制，2014,34(4):148-152., articleTitle=不同跨径比的两跨连续梁的模态分析试验, refAbstract=null), Reference(id=1228312974411104647, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2014, volume=34, issue=4, pageStart=148, pageEnd=152, url=null, language=null, rfNumber=[13], rfOrder=24, authorNames=ZHENG Yangkun, YUAN Xiangrong, journalName=Noise and Vibration Control, refType=null, unstructuredReference=ZHENG Yangkun，YUAN Xiangrong. Modal analysis test of a two-unequal-span continuous beam[J]. Noise and Vibration Control，2014,34(4):148-152., articleTitle=Modal analysis test of a two-unequal-span continuous beam, refAbstract=null), Reference(id=1228312974515962249, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2016, volume=33, issue=12, pageStart=48, pageEnd=52, url=null, language=null, rfNumber=[14], rfOrder=25, authorNames=张盼, 袁向荣, 刘辉, journalName=实验技术与管理, refType=null, unstructuredReference=张盼，袁向荣，刘辉，等.基于视频图像法的两跨连续梁振动研究[J]. 实验技术与管理，2016,33(12):48-52., articleTitle=基于视频图像法的两跨连续梁振动研究, refAbstract=null), Reference(id=1228312974612431242, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2016, volume=33, issue=12, pageStart=48, pageEnd=52, url=null, language=null, rfNumber=[14], rfOrder=26, authorNames=ZHANG Pan, YUAN Xiangrong, LIU Hui, journalName=Experimental Technology and Management, refType=null, unstructuredReference=ZHANG Pan，YUAN Xiangrong，LIU Hui，et al.Research on vibration of two-span continuous beam based on video image method[J]. Experimental Technology and Management，2016,33(12):48-52., articleTitle=Research on vibration of two-span continuous beam based on video image method, refAbstract=null), Reference(id=1228312974725677454, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2012, volume=13, issue=1, pageStart=1, pageEnd=2, url=null, language=null, rfNumber=[15], rfOrder=27, authorNames=吴晶, 袁向荣, journalName=长春工程学院学报（自然科学版）, refType=null, unstructuredReference=吴晶，袁向荣. 两等跨连续梁的模态分析试验[J]. 长春工程学院学报（自然科学版），2012,13(1):1-2., articleTitle=两等跨连续梁的模态分析试验, refAbstract=null), Reference(id=1228312974838923666, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2012, volume=13, issue=1, pageStart=1, pageEnd=2, url=null, language=null, rfNumber=[15], rfOrder=28, authorNames=WU Jing, YUAN Xiangrong, journalName=Journal of Changchun Institute of Technology（Natural Sciences Edition）, refType=null, unstructuredReference=WU Jing，YUAN Xiangrong. Experiment of modal analysis to two-span continuous beams[J]. Journal of Changchun Institute of Technology（Natural Sciences Edition），2012,13(1):1-2., articleTitle=Experiment of modal analysis to two-span continuous beams, refAbstract=null), Reference(id=1228312974952169876, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2017, volume=30, issue=1, pageStart=149, pageEnd=154, url=null, language=null, rfNumber=[16], rfOrder=29, authorNames=周盛林, 李凤明, journalName=振动工程学报, refType=null, unstructuredReference=周盛林，李凤明. 失谐连续双跨梁结构振动特性的理论和实验研究[J]. 振动工程学报，2017,30(1):149-154., articleTitle=失谐连续双跨梁结构振动特性的理论和实验研究, refAbstract=null), Reference(id=1228312975036055958, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2017, volume=30, issue=1, pageStart=149, pageEnd=154, url=null, language=null, rfNumber=[16], rfOrder=30, authorNames=ZHOU Shenglin, LI Fengming, journalName=Journal of Vibration Engineering, refType=null, unstructuredReference=ZHOU Shenglin，LI Fengming. Theoretical and experimental studies on the vibration characteristics of disordered two-span beams[J]. Journal of Vibration Engineering，2017,30(1):149-154., articleTitle=Theoretical and experimental studies on the vibration characteristics of disordered two-span beams, refAbstract=null), Reference(id=1228312975119942041, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2015, volume=36, issue=3, pageStart=279, pageEnd=292, url=null, language=null, rfNumber=[17], rfOrder=31, authorNames=LI F M, SONG Z G, journalName=Applied Mathematics and Mechanics, refType=null, unstructuredReference=LI F M，SONG Z G. Vibration analysis and active control of nearly periodic two-span beams with piezoelectric actuator/sensor pairs[J]. Applied Mathematics and Mechanics，2015,36(3):279-292., articleTitle=Vibration analysis and active control of nearly periodic two-span beams with piezoelectric actuator/sensor pairs, refAbstract=null), Reference(id=1228312975237382557, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2023, volume=42, issue=11, pageStart=1, pageEnd=10, url=null, language=null, rfNumber=[18], rfOrder=32, authorNames=余云燕, 付艳艳, 张伟, journalName=振动与冲击, refType=null, unstructuredReference=余云燕，付艳艳，张伟. 黏弹性Pasternak地基上两跨连续Timoshenko梁横向自振特性分析[J]. 振动与冲击，2023,42(11):1-10., articleTitle=黏弹性Pasternak地基上两跨连续Timoshenko梁横向自振特性分析, refAbstract=null), Reference(id=1228312975317074336, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2023, volume=42, issue=11, pageStart=1, pageEnd=10, url=null, language=null, rfNumber=[18], rfOrder=33, authorNames=YU Yunyan, FU Yanyan, ZHANG Wei, journalName=Journal of Vibration and Shock, refType=null, unstructuredReference=YU Yunyan，FU Yanyan，ZHANG Wei. Analysis of transverse natural frequency of two-span continuous Timoshenko beam on viscoelastic Pasternak foundation[J]. Journal of Vibration and Shock，2023,42(11):1-10., articleTitle=Analysis of transverse natural frequency of two-span continuous Timoshenko beam on viscoelastic Pasternak foundation, refAbstract=null), Reference(id=1228312975405154723, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, doi=null, pmid=null, pmcid=null, year=2005, volume=83, issue=23-24, pageStart=1865, pageEnd=1877, url=null, language=null, rfNumber=[19], rfOrder=34, authorNames=KARGARNOVIN M H, YOUNESIAN D, THOMPSON D J, journalName=Computers and Structures, refType=null, unstructuredReference=KARGARNOVIN M H，YOUNESIAN D，THOMPSON D J，et al.Response of beams on nonlinear viscoelastic foundations to harmonic moving loads[J]. Computers and Structures，2005,83(23-24):1865-1877., articleTitle=Response of beams on nonlinear viscoelastic foundations to harmonic moving loads, refAbstract=null)], funds=[Fund(id=1228312970564927783, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, awardId=23JRRA1374, language=CN, fundingSource=甘肃省科技计划项目(23JRRA1374), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1228312961819803800, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, xref=1., ext=[AuthorCompanyExt(id=1228312961823998105, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, companyId=1228312961819803800, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1.School of Civil Engineering，Lanzhou University of Information Science and Technology，Lanzhou 730300，China), AuthorCompanyExt(id=1228312961832386713, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, companyId=1228312961819803800, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1.兰州信息科技学院土木工程学院，甘肃　兰州　730300)]), AuthorCompany(id=1228312961941438622, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, xref=2., ext=[AuthorCompanyExt(id=1228312961949827230, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, companyId=1228312961941438622, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2.Geotechnical Engineering Research Institute，Xi’an University of Technology，Xi’an 710048，China), AuthorCompanyExt(id=1228312961962410147, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, companyId=1228312961941438622, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2.西安理工大学岩土工程研究所，陕西　西安　710048)])], figs=[ArticleFig(id=1228312967284981990, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=EN, label=Fig.1, caption=Mechanical model and local coordinate system of two-span continuously modified Timoshenko beam on viscoelastic four-parameter foundation, figureFileSmall=EClffjqL8Th4+zAPXGmOvw==, figureFileBig=g/8cO9Ung4XjRSUB2Nlo+Q==, tableContent=null), ArticleFig(id=1228312967398228201, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=CN, label=图1, caption=黏弹性四参数地基上两跨连续修正Timoshenko梁的力学模型及局部坐标系, figureFileSmall=EClffjqL8Th4+zAPXGmOvw==, figureFileBig=g/8cO9Ung4XjRSUB2Nlo+Q==, tableContent=null), ArticleFig(id=1228312967536640240, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=EN, label=Fig.2, caption=Force analysis diagram of micro-segment isolator, figureFileSmall=o7sQoBC2NeKfQsh1DxZ5lg==, figureFileBig=NufSQ2nqM295wuC9Ta9OHA==, tableContent=null), ArticleFig(id=1228312967620526325, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=CN, label=图2, caption=微段隔离体受力分析图, figureFileSmall=o7sQoBC2NeKfQsh1DxZ5lg==, figureFileBig=NufSQ2nqM295wuC9Ta9OHA==, tableContent=null), ArticleFig(id=1228312967737966839, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=EN, label=Fig.3, caption=Schematic diagram of olar coordinate search, figureFileSmall=4W06ro/yENhi3AV199Yk5Q==, figureFileBig=DonmUeaLO6Z3eWJAU14D2g==, tableContent=null), ArticleFig(id=1228312969176613117, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=CN, label=图3, caption=极坐标搜索示意图, figureFileSmall=4W06ro/yENhi3AV199Yk5Q==, figureFileBig=DonmUeaLO6Z3eWJAU14D2g==, tableContent=null), ArticleFig(id=1228312969327608064, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=EN, label=Fig.4, caption=The first eight order modes of the structure under various working conditions, figureFileSmall=hV4eWMYb/cV2HPtUBdm4oQ==, figureFileBig=Wlayj3gLnLJcPjWsy7+C3g==, tableContent=null), ArticleFig(id=1228312969453437190, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=CN, label=图4, caption=各工况下结构的前8阶模态, figureFileSmall=hV4eWMYb/cV2HPtUBdm4oQ==, figureFileBig=Wlayj3gLnLJcPjWsy7+C3g==, tableContent=null), ArticleFig(id=1228312969575072012, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=EN, label=Tab.1, caption=

Calculation parameters of modified Timoshenko beam

, figureFileSmall=null, figureFileBig=null, tableContent=
参数取值
弹性模量E/Pa4.321 × 1010
剪切模量G/Pa1.751 × 1010
横截面面积A/m21.5 × 1.5
梁长l/m12
横截面惯性矩Iz/m40.422
密度ρ/(kg·m-3)2700
截面剪切系数k'π2/12
), ArticleFig(id=1228312969692512525, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=CN, label=表1, caption=

修正Timoshenko梁的计算参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数取值
弹性模量E/Pa4.321 × 1010
剪切模量G/Pa1.751 × 1010
横截面面积A/m21.5 × 1.5
梁长l/m12
横截面惯性矩Iz/m40.422
密度ρ/(kg·m-3)2700
截面剪切系数k'π2/12
), ArticleFig(id=1228312969801564433, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=EN, label=Tab.2, caption=

Physical calculation parameters of viscoelastic four-parameter foundation

, figureFileSmall=null, figureFileBig=null, tableContent=
参数取值
土体弹簧系数kv/(N·m-2)106
土体阻尼系数βv/(N·s·m-2)105
地基剪切系数Gv/N107
地基水平摩阻系数H/N108
), ArticleFig(id=1228312969902227731, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=CN, label=表2, caption=

黏弹性四参数地基的各项物理计算参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数取值
土体弹簧系数kv/(N·m-2)106
土体阻尼系数βv/(N·s·m-2)105
地基剪切系数Gv/N107
地基水平摩阻系数H/N108
), ArticleFig(id=1228312970015473947, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=EN, label=Tab.3, caption=

Length calculation parameters of each span beam under seven working conditions

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工况l12/ml23/ml/m跨度比
工况1012120
工况2111120.09
工况3210120.20
工况439120.33
工况548120.50
工况657120.71
工况766121.00
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7种工况下各跨梁的长度计算参数

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工况l12/ml23/ml/m跨度比
工况1012120
工况2111120.09
工况3210120.20
工况439120.33
工况548120.50
工况657120.71
工况766121.00
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The first eight order natural frequencies and attenuation coefficients of the structure under various working conditions

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工况参数自振频率/Hz
1阶2阶3阶4阶5阶6阶7阶8阶
工况1121.01850436.86940887.097701410.066701965.973202533.680203103.238303670.43040
δn8.130157.880427.580997.303447.074856.896116.758886.65362
工况2189.49210620.968701076.023201651.747002256.645102871.058703486.103104098.18490
δn8.079977.826957.532767.260257.035726.860566.726536.62394
工况3223.36320665.064401245.305201888.834902533.680202736.414403253.810903922.07110
δn8.037387.744617.425237.140066.896117.145836.810956.66738
工况4263.57420780.916601410.066701621.933702207.638802947.815303670.430403784.93150
δn8.000337.663237.303447.410027.078776.848146.653626.84264
工况5316.02160887.097701087.380601746.859302533.680202668.880303423.587404233.75080
δn7.960397.580997.579747.228826.896117.067816.763226.57230
工况6383.74350720.098001189.090301917.362402133.255403055.829803286.635704026.78420
δn7.914427.711057.473337.141267.161626.810716.858286.66318
工况7436.86940610.884501410.066701576.185302533.680202654.998303670.430403757.69660
δn7.880427.776797.303447.387376.896117.051806.653626.80016
), ArticleFig(id=1228312970334241060, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295920496870060, language=CN, label=表4, caption=

各工况下结构的前8阶自振频率和衰减系数

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工况参数自振频率/Hz
1阶2阶3阶4阶5阶6阶7阶8阶
工况1121.01850436.86940887.097701410.066701965.973202533.680203103.238303670.43040
δn8.130157.880427.580997.303447.074856.896116.758886.65362
工况2189.49210620.968701076.023201651.747002256.645102871.058703486.103104098.18490
δn8.079977.826957.532767.260257.035726.860566.726536.62394
工况3223.36320665.064401245.305201888.834902533.680202736.414403253.810903922.07110
δn8.037387.744617.425237.140066.896117.145836.810956.66738
工况4263.57420780.916601410.066701621.933702207.638802947.815303670.430403784.93150
δn8.000337.663237.303447.410027.078776.848146.653626.84264
工况5316.02160887.097701087.380601746.859302533.680202668.880303423.587404233.75080
δn7.960397.580997.579747.228826.896117.067816.763226.57230
工况6383.74350720.098001189.090301917.362402133.255403055.829803286.635704026.78420
δn7.914427.711057.473337.141267.161626.810716.858286.66318
工况7436.86940610.884501410.066701576.185302533.680202654.998303670.430403757.69660
δn7.880427.776797.303447.387376.896117.051806.653626.80016
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黏弹性四参数地基上两跨连续修正Timoshenko梁的横向自振特性分析
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柳伟 1 , 汪过兵 2
振动工程学报 | 2025,38(3): 604-611
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振动工程学报 | 2025, 38(3): 604-611
黏弹性四参数地基上两跨连续修正Timoshenko梁的横向自振特性分析
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柳伟1 , 汪过兵2
作者信息
  • 1.兰州信息科技学院土木工程学院，甘肃　兰州　730300
  • 2.西安理工大学岩土工程研究所，陕西　西安　710048

通讯作者:

柳伟（1990—），男，硕士，讲师。E-mail：
Analysis of transverse free vibration characteristics of two-span continuously modified Timoshenko beams on viscoelastic four-parameter foundation
Wei LIU1 , Guobing WANG2
Affiliations
  • 1.School of Civil Engineering，Lanzhou University of Information Science and Technology，Lanzhou 730300，China
  • 2.Geotechnical Engineering Research Institute，Xi’an University of Technology，Xi’an 710048，China
出版时间: 2025-03-10 doi: 10.16385/j.cnki.issn.1004-4523.2025.03.017
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摘要
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为解决黏弹性四参数地基上两跨连续修正Timoshenko梁的横向振动问题，将两跨连续修正Timoshenko梁与黏弹性四参数地基进行组合，建立新的振动控制方程，运用回传射线矩阵法，结合二分法和黄金分割法，分析了黏弹性四参数地基上两跨（等跨、不等跨）连续修正Timoshenko梁与单跨修正Timoshenko梁自振特性之间的联系与区别。结果表明：对于黏弹性四参数地基上的修正Timoshenko梁，单跨梁的各阶自振频率小于两跨连续梁的各阶自振频率，单跨梁的偶数阶自振频率和衰减系数与两等跨连续梁的奇数阶自振频率和衰减系数相同，不等跨两跨连续梁奇数阶自振频率小于两等跨连续梁奇数阶自振频率；两等跨连续修正Timoshenko梁的偶数阶振型关于跨中支座对称，奇数阶振型关于跨中支座反对称。

黏弹性四参数地基  /  两跨连续修正Timoshenko梁  /  回传射线矩阵法  /  横向振动  /  模态

In order to solve the transverse vibration problem of two-span continuously modified Timoshenko beam on viscoelastic four-parameter foundation，a new vibration governing equation is established by combining two-span continuously modified Timoshenko beam with viscoelastic four-parameter foundation. By using the echo matrix method，bisection and golden section method，the relation and difference between the natural vibration characteristics of two-span（equal-span，unequal-span）continuously modified Timoshenko beam and single-span modified Timoshenko beam on viscoelastic four-parameter foundation are analyzed. The results show that for the modified Timoshenko beam on the viscoelastic four-parameter foundation，the natural frequency of each order of the single-span beam is less than that of the two-span continuous beam，the even-order natural frequency and attenuation coefficient of the single-span beam are the same as the odd-order natural frequency and attenuation coefficient of the two equal-span continuous beam，and the odd-order natural frequency of the unequal-span two-span continuous beam is less than that of the two-span continuous beam. The even-order mode shapes of two equal-span continuous modified Timoshenko beams are symmetrical with respect to the supports in the middle of the span，and the odd-order modes are antisymmetric with respect to the mid-span.

viscoelastic four-parameter foundation  /  two-span continuously modified Timoshenko beam  /  return ray matrix method  /  transverse vibration  /  mode
柳伟, 汪过兵. 黏弹性四参数地基上两跨连续修正Timoshenko梁的横向自振特性分析. 振动工程学报, 2025 , 38 (3) : 604 -611 . DOI: 10.16385/j.cnki.issn.1004-4523.2025.03.017
Wei LIU, Guobing WANG. Analysis of transverse free vibration characteristics of two-span continuously modified Timoshenko beams on viscoelastic four-parameter foundation[J]. Journal of Vibration Engineering, 2025 , 38 (3) : 604 -611 . DOI: 10.16385/j.cnki.issn.1004-4523.2025.03.017
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地基梁作为众多工程构件的基本模型，被广泛应用于铁路轨道[1]、公路路面[2]、输水渠道[3]、隧道管棚[4]、建筑工程中的地下条形基础[5]等，其动力学特性分析在工程领域及学术界备受关注。目前梁的弯曲振动已有多种理论，其中初等梁理论因求解方便、应用最广而成为经典，但其对梁的高阶振动、高度局部承载、高跨比较大等情况，存在静力问题计算挠度偏小和动力问题高估振动频率等不足[6]。Timoshenko梁理论的振动控制方程解耦后存在挠度关于时间的4阶导数项物理意义不明确、第二频谱、截面剪切修正系数多解等问题[6-8]。为解决这些问题，陈镕等[7]对经典的Timoshenko梁运动方程进行修正，指出考虑梁剪切变形引起的转动惯量后，时间的4阶导数项自然会消失，解决了经典Timoshenko梁理论一个振型对应两个振动频率的问题。夏桂云[8]利用固有频率和临界频率的关系论证了Timoshenko梁产生第二频谱的理论原因，通过实例验证Timoshenko梁第二频谱的存在，并从理论上预测存在第二频谱现象的其他结构。王家乐等[9]指出当地基梁为深梁或者计算结构高阶自振频率时（诸如冲击等问题），应采用修正Timoshenko梁理论。修正Timoshenko梁理论考虑了梁的剪切变形及其所引起的转动惯量的影响，在深梁和高频振动特性的分析方面优于经典Timoshenko梁理论。基于修正Timoshenko梁理论，余云燕等[10]运用回传射线矩阵法求解了三种经典边界条件下变截面修正Timoshenko梁的自振频率。吴晓等[11]应用Timoshenko梁修正理论求解了泡沫铝合金梁的自振频率表达式及其在简谐荷载作用下强迫振动的解析解。LI等[12]推导出分数阶标准固体黏弹性地基上修正Timoshenko梁的振动控制方程，分析土体参数对修正Timoshenko梁波速、自振频率的影响。但以上研究以单跨梁为主。
对两跨连续梁自振特性的研究，多以弹性地基模型和简单梁理论为主。郑仰坤等[13]利用MIDAS有限元分析软件和DASP设备分别对连续梁进行模态分析，得出不同跨径比下连续梁的前3阶振型及其频率。张盼等[14]采集连续梁振动视频并将其转化为数字图像，经MATLAB程序读取整个振动过程梁边缘的数据，通过DASP系统对观测数据进行模态分析，得到两跨连续梁的前2阶频率和振型。吴晶等[15]用DASP软件对两跨连续梁进行试验模态分析，得出该连续梁的前2阶频率及其阻尼比。周盛林等[16]基于Bernoulli-Euler梁理论，采用模态分析法获得双跨梁的频率方程，由此求解出双跨梁的自振频率。LI等[17]采用模态展开法研究了两跨梁的自由振动特性，对中间支座错位的两跨梁振动特性进行分析。由以上研究可以看出，对两跨连续梁的研究采用有限元分析和试验分析的居多。余云燕等[18]求解了黏弹性Pasternak地基上Timoshenko梁在不同约束条件下单跨及两跨连续地基梁的自振频率、衰减系数和模态，但未考虑地基水平摩阻的影响，也未分析跨度比对黏弹性四参数地基上两跨连续修正Timoshenko梁的影响。
回传射线矩阵法（MRRM）不仅能精确计算复杂杆系结构的初期瞬态响应，而且能准确计算结构的自振频率和振型，尤其在高阶自振频率和振型的计算上更有优势。故本文基于修正Timoshenko理论，建立黏弹性四参数地基上修正Timoshenko梁的横向振动控制方程，运用回传射线矩阵法解耦后，结合二分法和黄金分割法，分析黏弹性四参数地基上两跨（等跨、不等跨）连续修正Timoshenko梁与单跨修正Timoshenko梁自振特性之间的联系与区别。
1　振动控制方程及方程的解
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黏弹性四参数地基上两跨连续修正Timoshenko梁的力学模型如图1（a）所示，土体与修正Timoshenko梁的相互作用采用黏弹性Pasternak地基模型，并考虑地基水平摩阻的影响，建立整体坐标系(x，v)，将地基梁划分为2个单元3个节点，节点的编号如图1（a）所示，对地基梁的每个单元JK引入2个对偶局部坐标系(x，y)JK和(x，y)KJ，原点分别在节点J和节点K，如图1（b）所示。
对图1（a）取微段隔离体，其受力情况如图2所示。图中，ys表示梁中心轴至梁顶端的距离。黏弹性地基梁在弯曲变形时，梁一般绕其中性轴转动，造成梁底相对于地基有一定的水平变形，如果地基与梁间的接触面比较粗糙，将对梁有水平约束作用。设梁底宽为w，地基与梁底的水平剪切系数为τ，梁中性轴距底边为yx，梁的截面转角为ϕ，梁底相对滑动为yxϕ。
地基水平摩阻对梁中性轴产生的分布弯矩为：
式中，P表示地基梁在单位长度上受到的水平摩阻力；H为地基水平摩阻系数。
在对偶局部坐标系下，对微段隔离体建立竖向力及力矩平衡方程，并略去高阶项，得：
式中，qv (x，t)为黏弹性地基梁的竖向反力，，其中，v为梁的总挠度，kv、βv、Gv分别为土体弹簧系数、土体阻尼系数、地基剪切系数；V (x，t)和Mz (x，t)分别为梁的截面剪力和弯矩，，其中，A、ρ、E、G、Iz、k'分别为梁的横截面面积、密度、弹性模量、剪切模量、横截面惯性矩、截面剪切系数。
整理式（2），得到黏弹性四参数地基上两跨连续修正Timoshenko梁的振动控制方程为：
设：
式中，“^”表示频域中的变量；；ω为圆频率，，其中和δn分别为自振频率和衰减系数。
将式（4）、（5）代入式（3）中，直接进行Fourier正变换得：
求解式（6）得梁的总挠度和截面转角在频域中的表达式分别为：
式中，a1 (ω)、a2 (ω)为待定的入射波波幅；d1 (ω)、d2 (ω)为待定的出射波波幅；k1，k2为波数，满足：
式中，，其中，，为截面的回转半径，波波速，为横波波速。
对应于波数kj，与的比值为：
弯矩和剪力在频域中的稳态解表达式分别为：
式中，βj =-iEIzkj gj，γj = k'AG (ikj - gj)，j = 1，2。
在频域中，对所有节点建立局部坐标系下的力平衡和位移协调条件如下：
将式（7）、（10）代入式（11），得到不同约束条件下各节点的散射关系为：
式中，aJ和dJ分别表示节点J的局部入射波波幅向量和局部出射波波幅向量，SJ表示节点J的局部散射矩阵，它们的表达式分别为：
将所有节点散射矩阵组集到一起，从而得到总体散射关系为：
式中，a和d分别为总体入射波波幅向量和总体出射波波幅向量，，；S为总体散射矩阵，表达式为：
注意到当一个出射波由杆件JK的J点产生，沿着该杆件向K点传播，从K点这一端来看，该波为入射波。因此，在同一杆件中，对一端而言的入射波波幅向量aJK和对另一端而言的出射波波幅向量dJK之间存在一个相位关系，引入传播矩阵PJK，有aJK =PJKdJK，其中，lJK为杆件JK的长度，diag(…)表示对角矩阵。
将向量aJK组集到整体入射波波幅向量a中，dJK也同样组集到整体出射波波幅向量d中，有
式中，；P为传播矩阵，。
与d中的元素完全相同，只是各元素的排列位置有所调整，引入置换矩阵U：
其中：
将式（17）代入式（16）得：
将式（20）代入式（14）得：
式中，R = SPU为回传射线矩阵；I为单位矩阵。
要得到非零的出射波波幅向量d，则[I - R]的行列式必须等于0，从而得到关于圆频率ω 的特征方程为：
式（22）为关于自振频率及衰减系数δn的非线性复数隐式超越方程，由线性代数理论可知：
式中，det[I - R]为矩阵[I - R]的模；adj[I - R]为矩阵[I - R]的伴随矩阵。
当det[I - R]中ω的实部取自振频率，虚部取衰减系数δn时，式（23）为：
在如图3所示的极坐标ρ-o-θ下，令，，由dρ和dθ组成的每个局域运用二分法和黄金搜索法进行迭代求解，即用二分法搜索出行列式实部和虚部变号的点，再用黄金分割法搜索行列式模的极小值点，adj| I -R(ω)|的每一个非零列可看作黏弹性地基梁自由振动时非零出射波幅向量d (ω)。当行列式的模小于预先给定的误差时，停止迭代，则结构的自振频率，衰减系数δn = ρsinθ，可通过式（20）求得a，则结构中任意点处的横向位移可由式（7）求出，将各点位移归一化后即可得到结构的振型曲线。实际计算中，利用MATLAB语言编制相关程序。
2　算例分析
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黏弹性四参数地基上两跨连续修正Timoshenko梁的力学计算模型如图1（a）所示，修正Timoshenko梁的计算参数如表1所示，土体的计算参数参考文献[19]中的数值，如表2所示。
黏弹性四参数地基上两跨连续修正Timoshenko梁的各项物理参数采用表1和2中的数值，7种工况下各跨梁的长度计算参数如表3所示。其中，工况1为单跨梁，工况2~6为不等跨两跨连续地基梁，工况7为两等跨连续地基梁。根据回传射线矩阵法求解各工况下结构的前8阶自振频率和衰减系数的数值解如表4所示。
由表4可知，工况1的各阶自振频率明显小于其他工况下的各阶自振频率，表明黏弹性四参数地基上单跨修正Timoshenko梁的各阶自振频率小于两跨连续修正Timoshenko梁的各阶自振频率，但其随阶数增长的速度明显大于两跨连续修正Timoshenko梁。工况1的偶数阶自振频率和衰减系数与工况7的奇数阶自振频率和衰减系数相同，表明黏弹性四参数地基上单跨修正Timoshenko梁的偶数阶自振频率和衰减系数与两等跨连续修正Timoshenko梁的奇数阶自振频率和衰减系数相同，这一结果与文献[18]的结论一致。工况2~6的奇数阶自振频率小于工况7的奇数阶自振频率，其偶数阶自振频率大于工况7的偶数阶自振频率，表明黏弹性四参数地基上不等跨两跨连续修正Timoshenko梁奇数阶自振频率小于两等跨连续修正Timoshenko梁奇数阶自振频率，但其偶数阶自振频率大于两等跨连续修正Timoshenko梁偶数阶自振频率。针对工况2~6，第1阶自振频率随着跨度比的增大而增大，衰减系数随着跨度比的增大而减小，从第2阶开始，随着跨度比的增大，自振频率和衰减系数的变化并没有统一的规律，但在不同跨度比时结构的自振频率和衰减系数差异十分明显，表明跨度比对黏弹性四参数地基上两跨连续修正Timoshenko梁自振特性的影响较大。
图4为各工况下结构的前8阶模态。由图4可知，工况3、工况5和工况7的第5阶自振频率和衰减系数完全相同，工况4和工况7的第3阶及第7阶自振频率和衰减系数也完全相同，但其对应的振型曲线未完全重合，表明即使跨度比变化时结构具有相同的自振频率和衰减系数，但振型曲线受跨度比的影响不完全相同。针对工况7的结构振型曲线，其偶数阶关于跨中支座对称，其奇数阶关于跨中支座反对称，工况1的奇数阶振型关于跨中支座对称，偶数阶关于跨中支座反对称，而工况2~6的结构振型曲线变化较为复杂，并没有统一的规律，表明跨度比对黏弹性四参数地基上两跨连续修正Timoshenko梁的模态影响较大。同时，黏弹性四参数地基上两等跨连续梁的振型曲线与单跨梁的振型曲线有本质的区别。
3　结论
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将回传射线矩阵法推广至黏弹性四参数地基上两跨连续修正Timoshenko梁的振动分析中，结合二分法和黄金分割法，分析了黏弹性四参数地基上两跨（等跨、不等跨）连续修正Timoshenko梁与单跨修正Timoshenko梁自振特性之间的联系与区别。得出以下结论：
（1）对于黏弹性四参数地基上修正Timoshenko梁，单跨梁的各阶自振频率小于两跨连续梁的各阶自振频率，但其随阶数增长的速度明显大于两跨连续梁；单跨梁的偶数阶自振频率和衰减系数与两等跨连续梁的奇数阶自振频率和衰减系数相同。
（2）黏弹性四参数地基上不等跨两跨连续修正Timoshenko梁奇数阶自振频率小于两等跨连续修正Timoshenko梁奇数阶自振频率，但其偶数阶自振频率大于两等跨连续修正Timoshenko梁偶数阶自振频率。
（3）黏弹性四参数地基上，随着跨度比的增大，不等跨两跨连续修正Timoshenko梁的高阶自振频率和衰减系数变化并没有统一的规律，但在不同跨度比时结构的自振频率和衰减系数差异十分明显。
（4）虽然跨度比变化时结构具有相同的自振频率和衰减系数，但是振型曲线受跨度比的影响并非完全重合。
（5）黏弹性四参数地基上，两等跨连续修正Timoshenko梁的偶数阶振型关于跨中支座对称，其奇数阶振型关于跨中支座反对称，单跨修正Timoshenko梁的奇数阶振型关于跨中支座对称，偶数阶振型关于跨中支座反对称，随着跨度比的变化，不等跨两跨连续修正Timoshenko梁的振型曲线变化较为复杂，并没有统一的规律。
（6）本文计算方法可求解任意边界条件下及复杂参数结构的横向自振特性，从而为工程实践提供理论基础。
基金
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  • 甘肃省科技计划项目(23JRRA1374)
文献
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参考文献 引证文献
排序方式：
[1]
雷晓燕，邢聪聪，吴神花. 轨道结构中高频振动特性分析[J]. 振动工程学报，2020,33(6):1245-1252.
LEI Xiaoyan，XING Congcong，WU Shenhua. Mid-and high-frequency vibration characteristics of track structures[J]. Journal of Vibration Engineering，2020,33(6):1245-1252.
[2]
刘飞禹，吴杰杰，陈江，等.考虑水平摩阻的Pasternak路基基层及路面变形分析[J]. 中国公路学报，2019,32(5):38-46.
LIU Feiyu，WU Jiejie，CHEN Jiang，et al.Analysis of pavement and subbase on Pasternak foundation considering horizontal friction[J]. China Journal of Highway and Transport，2019,32(5):38-46.
[3]
肖旻，王正中，吴浪，等.寒区预制混凝土衬砌梯形渠道弹性冻土地基梁模型[J]. 水利学报，2023,54(2):244-253.
XIAO Min，WANG Zhengzhong，WU Lang，et al.Elastic frozen soil foundation beam model for trapezoidal canal lined with concrete precast slabs in cold regions[J]. Journal of Hydraulic Engineering，2023,54(2):244-253.
[4]
禹海涛，卫一博. 穿断层分段柔性接头隧道纵向地震响应解析解[J]. 岩土工程学报，2023,45(5):912-920.
YU Haitao，WEI Yibo. Analytical solution for longitudinal seismic response of tunnels with segmental flexible joints crossing faults[J]. Chinese Journal of Geotechnical Engineering，2023,45(5):912-920.
[5]
刘希成. 弹性地基梁理论在软弱地基柱下条形基础中的应用[J]. 煤炭工程，2015,47(12):25-27.
LIU Xicheng. Application of elastic foundation beam theory in strip foundation under column of soft ground[J]. Coal Engineering，2015,47(12):25-27.
[6]
夏桂云，曾庆元. 深梁理论的研究现状与工程应用[J]. 力学与实践，2015,37(3):302-316.
XIA Guiyun，ZENG Qingyuan. Timoshenko beam theory and its applications[J]. Mechanics in Engineering，2015,37(3):302-316.
[7]
陈镕，万春风，薛松涛，等.Timoshenko梁运动方程的修正及其影响[J]. 同济大学学报（自然科学版），2005,33(6):711-715.
CHEN Rong，WAN Chunfeng，XUE Songtao，et al.Modification of motion equation of Timoshenko beam and its effect[J]. Journal of Tongji University（Natural Science），2005,33(6):711-715.
[8]
夏桂云. Timoshenko梁的第二频谱分析[J]. 湖南大学学报（自然科学版），2021,48(11):142-149.
XIA Guiyun. Analysis on the second frequency spectrum of Timoshenko beam[J]. Journal of Hunan University（Natural Sciences），2021,48(11):142-149.
[9]
王家乐，夏桂云. Winkler地基上修正Timoshenko梁振动分析[J]. 振动与冲击，2020,39(3):30-37.
WANG Jiale，XIA Guiyun. Vibration analysis for a modified Timoshenko beam on Winkler elastic foundation[J]. Journal of Vibration and Shock，2020,39(3):30-37.
[10]
余云燕，孔嘉乐，陈进浩，等.变截面修正Timoshenko梁自振频率的回传射线矩阵法分析[J]. 地震工程学报，2022,44(4):751-758.
YU Yunyan，KONG Jiale，CHEN Jinhao，et al.Natural frequency of a modified Timoshenko beam with variable cross-section with the method of reverberation ray matrix[J]. China Earthquake Engineering Journal，2022,44(4):751-758.
[11]
吴晓，孙晋，黄翀，等.用Timoshenko梁修正理论研究泡沫金属铝合金梁的动力响应[J]. 振动与冲击，2011,30(1):124-127.
WU Xiao，SUN Jin，HUANG Chong，et al.Dynamic responses of a foam metal aluminum alloy beam with Timoshenko beam theory[J]. Journal of Vibration and Shock，2011,30(1):124-127.
[12]
LI M L，WEI P J，ZHOU X L. Wave propagation and free vibration of a Timoshenko beam mounted on the viscoelastic Pasternak foundation modeled by fraction-order derivatives[J]. Mechanics of Time-Dependent Materials，2023,27(4):1209-1223.
[13]
郑仰坤，袁向荣. 不同跨径比的两跨连续梁的模态分析试验[J]. 噪声与振动控制，2014,34(4):148-152.
ZHENG Yangkun，YUAN Xiangrong. Modal analysis test of a two-unequal-span continuous beam[J]. Noise and Vibration Control，2014,34(4):148-152.
[14]
张盼，袁向荣，刘辉，等.基于视频图像法的两跨连续梁振动研究[J]. 实验技术与管理，2016,33(12):48-52.
ZHANG Pan，YUAN Xiangrong，LIU Hui，et al.Research on vibration of two-span continuous beam based on video image method[J]. Experimental Technology and Management，2016,33(12):48-52.
[15]
吴晶，袁向荣. 两等跨连续梁的模态分析试验[J]. 长春工程学院学报（自然科学版），2012,13(1):1-2.
WU Jing，YUAN Xiangrong. Experiment of modal analysis to two-span continuous beams[J]. Journal of Changchun Institute of Technology（Natural Sciences Edition），2012,13(1):1-2.
[16]
周盛林，李凤明. 失谐连续双跨梁结构振动特性的理论和实验研究[J]. 振动工程学报，2017,30(1):149-154.
ZHOU Shenglin，LI Fengming. Theoretical and experimental studies on the vibration characteristics of disordered two-span beams[J]. Journal of Vibration Engineering，2017,30(1):149-154.
[17]
LI F M，SONG Z G. Vibration analysis and active control of nearly periodic two-span beams with piezoelectric actuator/sensor pairs[J]. Applied Mathematics and Mechanics，2015,36(3):279-292.
[18]
余云燕，付艳艳，张伟. 黏弹性Pasternak地基上两跨连续Timoshenko梁横向自振特性分析[J]. 振动与冲击，2023,42(11):1-10.
YU Yunyan，FU Yanyan，ZHANG Wei. Analysis of transverse natural frequency of two-span continuous Timoshenko beam on viscoelastic Pasternak foundation[J]. Journal of Vibration and Shock，2023,42(11):1-10.
[19]
KARGARNOVIN M H，YOUNESIAN D，THOMPSON D J，et al.Response of beams on nonlinear viscoelastic foundations to harmonic moving loads[J]. Computers and Structures，2005,83(23-24):1865-1877.
2025年第38卷第3期
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doi: 10.16385/j.cnki.issn.1004-4523.2025.03.017
  • 接收时间：2024-02-16
  • 首发时间：2026-02-11
  • 出版时间：2025-03-10
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  • 收稿日期：2024-02-16
  • 修回日期：2024-04-05
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甘肃省科技计划项目(23JRRA1374)
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    1.兰州信息科技学院土木工程学院，甘肃　兰州　730300
    2.西安理工大学岩土工程研究所，陕西　西安　710048

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柳伟（1990—），男，硕士，讲师。E-mail：
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2种不同金属材料的力学参数
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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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