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Article(id=1228634335700513708, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228634329748796239, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.08.007, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1692374400000, receivedDateStr=2023-08-19, revisedDate=1703088000000, revisedDateStr=2023-12-21, acceptedDate=null, acceptedDateStr=null, onlineDate=1770858852553, onlineDateStr=2026-02-12, pubDate=1724774400000, pubDateStr=2024-08-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770858852553, onlineIssueDateStr=2026-02-12, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770858852553, creator=13701087609, updateTime=1770858852553, updator=13701087609, issue=Issue{id=1228634329748796239, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='8', pageStart='1269', pageEnd='1450', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770858851134, creator=13701087609, updateTime=1770859054135, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228635181259620818, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228634329748796239, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228635181263815123, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228634329748796239, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1330, endPage=1338, ext={EN=ArticleExt(id=1228634337264989109, articleId=1228634335700513708, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Analysis of transverse free vibration characteristics of modified Timoshenko beam on viscoelastic Pasternak foundction, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Based on the modified Timoshenko beam theory，the transverse vibration governing equation of the modified Timoshenko beam on the viscoelastic Pasternak foundation is established. The analytical solutions of the natural frequency and attenuation coefficient of the modified Timoshenko beam simply supported at both ends in the viscoelastic Pasternak foundation are derived with the reverberation-ray matrix method. The natural vibration characteristics of the modified Timoshenko beam on the viscoelastic Pasternak foundation under classical boundary conditions are calculated by dichotomy and golden section method. The effects of moment of inertia caused by shear deformation，beam length and different boundary conditions on the natural vibration characteristics of the structure are compared and analyzed. The results show that the natural frequency and attenuation coefficient of the modified Timoshenko beam on the viscoelastic Pasternak foundation are smaller than those of the classical Timoshenko beam； the shorter the beam，the more significant the influence of the moment of inertia caused by shear deformation on the natural frequency and attenuation coefficient of the structure，and the influence on the higher order is obviously greater than that on the low order； the stronger the boundary constraint condition，the more obvious the vibration energy attenuation.

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基于修正Timoshenko梁理论，建立黏弹性Pasternak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推导出黏弹性Pasternak地基上两端简支修正Timoshenko梁自振频率和衰减系数的解析解，结合二分法和黄金分割法计算了经典边界条件下黏弹性Pasternak地基上修正Timoshenko梁的自振特性，对比分析了考虑剪切变形引起的转动惯量、梁长及不同的边界条件对结构自振特性的影响。研究表明：黏弹性Pasternak地基上修正Timoshenko梁的各阶自振频率和衰减系数小于经典Timoshenko梁的各阶自振频率和衰减系数；梁越短，剪切变形引起的转动惯量对结构自振频率和衰减系数的影响越大，且对高阶的影响明显大于低阶；边界约束条件越强，振动能量衰减越明显。

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柳 伟（1990—），男，硕士，讲师。E-mail： 。

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柳 伟（1990—），男，硕士，讲师。E-mail： 。

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柳 伟（1990—），男，硕士，讲师。E-mail： 。

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Railway Engineering Society, refType=null, unstructuredReference=BAI Haifeng. Research on the static response of tie on continuous elastic grade beam［J］. Journal of Railway Engineering Society， 2007， 24（5）： 22-27．, articleTitle=Research on the static response of tie on continuous elastic grade beam, refAbstract=null), Reference(id=1228634385159745902, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, 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=1228634386535477617, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, 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=1228634386657112436, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2009, volume=28, issue=3, pageStart=469, pageEnd=476, url=null, language=null, rfNumber=［3］, rfOrder=4, authorNames=Calim F F, journalName=European Journal of Mechanics， A： Solid, refType=null, unstructuredReference=Calim F F. Dynamic analysis of beams on viscoelastic foundation［J］. European Journal of Mechanics， A： Solid， 2009， 28（3）： 469-476．, articleTitle=Dynamic analysis of beams on viscoelastic foundation, refAbstract=null), Reference(id=1228634386749387126, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2010, volume=35, issue=2B, pageStart=125, pageEnd=132, url=null, language=null, rfNumber=［4］, rfOrder=5, authorNames=Attarnejad R, Shahba A, Semnani S J, journalName=Arabian Journal for Science and Engineering, refType=null, unstructuredReference=Attarnejad R， Shahba A， Semnani S J. Application of differential transform in free vibration analysis of Timoshenko beams resting on two-parameter elastic foundation［J］. Arabian Journal for Science and Engineering， 2010， 35（2B）： 125-132., articleTitle=Application of differential transform in free vibration analysis of Timoshenko beams resting on two-parameter elastic foundation, refAbstract=null), Reference(id=1228634386871021944, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2012, volume=331, issue=10, pageStart=2426, pageEnd=2442, url=null, language=null, rfNumber=［5］, rfOrder=6, authorNames=Ding H, Chen L Q, Yang S P, journalName=Journal of Sound and Vibration, refType=null, unstructuredReference=Ding H， Chen L Q， Yang S P. Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load［J］. Journal of Sound and Vibration， 2012， 331（10）： 2426-2442．, articleTitle=Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, refAbstract=null), Reference(id=1228634386963296635, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2003, volume=31, issue=3-4, pageStart=219, pageEnd=228, url=null, language=null, rfNumber=［6］, rfOrder=7, authorNames=Malekzadeh P, Karami G, Farid M, journalName=Computational Mechanics, refType=null, unstructuredReference=Malekzadeh P， Karami G， Farid M. DQEM for free vibration analysis of Timoshenko beams on elastic foundations［J］. Computational Mechanics， 2003， 31（3-4）： 219-228．, articleTitle=DQEM for free vibration analysis of Timoshenko beams on elastic foundations, refAbstract=null), Reference(id=1228634387068154238, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2014, volume=33, issue=1, pageStart=101, pageEnd=105, url=null, language=null, rfNumber=［7］, rfOrder=8, authorNames=彭丽, 丁虎, 陈立群, journalName=振动与冲击, refType=null, unstructuredReference=彭丽， 丁虎， 陈立群. 黏弹性三参数地基梁横向自由振动［J］. 振动与冲击， 2014， 33（1）： 101-105., articleTitle=黏弹性三参数地基梁横向自由振动, refAbstract=null), Reference(id=1228634387135263105, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2014, volume=33, issue=1, pageStart=101, pageEnd=105, url=null, language=null, rfNumber=［7］, rfOrder=9, authorNames=PENG Li, DING Hu, CHEN Liqun, journalName=Journal of Vibration and Shock, refType=null, unstructuredReference=PENG Li， DING Hu， CHEN Liqun. Transverse free vibration of a beam resting on a three-parameter viscoelastic foundation［J］. Journal of Vibration and Shock， 2014， 33（1）： 101-105., articleTitle=Transverse free vibration of a beam resting on a three-parameter viscoelastic foundation, refAbstract=null), Reference(id=1228634387219149188, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2013, volume=32, issue=2, pageStart=143, pageEnd=146, url=null, language=null, rfNumber=［8］, rfOrder=10, authorNames=彭丽, 丁虎, 陈立群, journalName=振动与冲击, refType=null, unstructuredReference=彭丽， 丁虎， 陈立群. 黏弹性Pasternak地基梁振动的复模态分析［J］. 振动与冲击， 2013， 32（2）： 143-146．, articleTitle=黏弹性Pasternak地基梁振动的复模态分析, refAbstract=null), Reference(id=1228634387282063752, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2013, volume=32, issue=2, pageStart=143, pageEnd=146, url=null, language=null, rfNumber=［8］, rfOrder=11, authorNames=PENG Li, DING Hu, CHEN Liqun, journalName=Journal of Vibration and Shock, refType=null, unstructuredReference=PENG Li， DING Hu， CHEN Liqun. Complex modal analysis for vibrations of a beam on a viscoelastic Pasternak foundation［J］. Journal of Vibration and Shock， 2013， 32（2）： 143-146．, articleTitle=Complex modal analysis for vibrations of a beam on a viscoelastic Pasternak foundation, refAbstract=null), Reference(id=1228634387349172619, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2013, volume=33, issue=5, pageStart=107, pageEnd=110, url=null, language=null, rfNumber=［9］, rfOrder=12, authorNames=彭丽, 丁虎, 陈立群, journalName=噪声与振动控制, refType=null, unstructuredReference=彭丽， 丁虎， 陈立群.黏弹性三参数地基上Timoshenko梁横向自由振动［J］. 噪声与振动控制， 2013， 33（5）： 107-110．, articleTitle=黏弹性三参数地基上Timoshenko梁横向自由振动, refAbstract=null), Reference(id=1228634387437253009, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2013, volume=33, issue=5, pageStart=107, pageEnd=110, url=null, language=null, rfNumber=［9］, rfOrder=13, authorNames=PENG Li, DING Hu, CHEN Liqun, journalName=Noise and Vibration Control, refType=null, unstructuredReference=PENG Li， DING Hu， CHEN Liqun. Transverse free vibration of a Timoshenko beam rested on transverse free vibration of a Timoshenko beam rested on three-parameter viscoelastic foundation［J］.Noise and Vibration Control， 2013， 33（5）：107-110．, articleTitle=Transverse free vibration of a Timoshenko beam rested on transverse free vibration of a Timoshenko beam rested on three-parameter viscoelastic foundation, refAbstract=null), Reference(id=1228634387512750485, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2023, volume=42, issue=11, pageStart=1, pageEnd=10, url=null, language=null, rfNumber=［10］, rfOrder=14, 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=1228634387579859353, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2023, volume=42, issue=11, pageStart=1, pageEnd=10, url=null, language=null, rfNumber=［10］, rfOrder=15, 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=1228634387672134046, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2005, volume=33, issue=6, pageStart=711, pageEnd=715, url=null, language=null, rfNumber=［11］, rfOrder=16, authorNames=陈镕, 万春风, 薛松涛, journalName=同济大学学报, 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， 2005， 33（6）： 711-715., articleTitle=Timoshenko梁运动方程的修正及其影响, refAbstract=null), Reference(id=1228634387743437218, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2005, volume=33, issue=6, pageStart=711, pageEnd=715, url=null, language=null, rfNumber=［11］, rfOrder=17, authorNames=CHEN Rong, WAN Chunfeng, XUE Songtao, journalName=Journal of Tongji University, 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， 2005， 33（6）： 711-715., articleTitle=Modification of motion equation of Timoshenko beam and its effect, refAbstract=null), Reference(id=1228634387844100516, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2021, volume=48, issue=11, pageStart=142, pageEnd=149, url=null, language=null, rfNumber=［12］, rfOrder=18, authorNames=夏桂云, journalName=湖南大学学报, refType=null, unstructuredReference=夏桂云. Timoshenko梁的第二频谱分析［J］. 湖南大学学报， 2021， 48（11）： 142-149., articleTitle=Timoshenko梁的第二频谱分析, refAbstract=null), Reference(id=1228634387944763815, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2021, volume=48, issue=11, pageStart=142, pageEnd=149, url=null, language=null, rfNumber=［12］, rfOrder=19, authorNames=XIA Guiyun, journalName=Journal of Hunan University, refType=null, unstructuredReference=XIA Guiyun. Analysis on the second frequency spectrum of Timoshenko beam［J］. Journal of Hunan University， 2021， 48（11）： 142-149., articleTitle=Analysis on the second frequency spectrum of Timoshenko beam, refAbstract=null), Reference(id=1228634388037038506, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2011, volume=30, issue=1, pageStart=124, pageEnd=127, url=null, language=null, rfNumber=［13］, 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=1228634388112535984, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2011, volume=30, issue=1, pageStart=124, pageEnd=127, url=null, language=null, rfNumber=［13］, 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 metal aluminum alloy beam with Timoshenko beam theory［J］. Journal of Vibration and Shock， 2011， 30（1）： 124-127., articleTitle=Dynamic responses of a metal aluminum alloy beam with Timoshenko beam theory, refAbstract=null), Reference(id=1228634388192227761, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2020, volume=39, issue=3, pageStart=30, pageEnd=37, url=null, language=null, rfNumber=［14］, rfOrder=22, 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=1228634388255142323, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2020, volume=39, issue=3, pageStart=30, pageEnd=37, url=null, language=null, rfNumber=［14］, rfOrder=23, 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=1228634388322251190, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2023, volume=27, issue=4, pageStart=1209, pageEnd=1223, url=null, language=null, rfNumber=［15］, rfOrder=24, 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=1228634388389360057, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2015, volume=15, issue=15, pageStart=88, pageEnd=94, url=null, language=null, rfNumber=［16］, rfOrder=25, authorNames=徐梅玲, 叶茂, 付明科, journalName=科学技术与工程, refType=null, unstructuredReference=徐梅玲， 叶茂， 付明科， 等. 修正Timoshenko 梁自由振动及Euler 梁误差分析［J］. 科学技术与工程， 2015， 15（15）： 88-94., articleTitle=修正Timoshenko 梁自由振动及Euler 梁误差分析, refAbstract=null), Reference(id=1228634388452274620, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2015, volume=15, issue=15, pageStart=88, pageEnd=94, url=null, language=null, rfNumber=［16］, rfOrder=26, authorNames=XU Meiling, YE Mao, FU Mingke, journalName=Science Technology and Engineering, refType=null, unstructuredReference=XU Meiling， YE Mao， FU Mingke， et al. Systematic modal analysis of simply supported beam based on the modified motion equation of Timoshenko［J］. Science Technology and Engineering， 2015， 15（15）： 88-94., articleTitle=Systematic modal analysis of simply supported beam based on the modified motion equation of Timoshenko, refAbstract=null), Reference(id=1228634388531966401, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, doi=null, pmid=null, pmcid=null, year=2005, volume=83, issue=23-24, pageStart=1865, pageEnd=1877, url=null, language=null, rfNumber=［17］, rfOrder=27, 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］. 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postcode=null, companyName=null, departmentName=null, remark=1School of Civil Engineering, Lanzhou University of Information Technology, Lanzhou 730300,China), AuthorCompanyExt(id=1228634379409354937, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, companyId=1228634379392577719, 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=1228634379501629629, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, xref=2, ext=[AuthorCompanyExt(id=1228634379510018238, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, companyId=1228634379501629629, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2Geotechnical Engineering Research Institute, Xi’an University of Technology, Xi’an 710048,China), 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ArticleFig(id=1228634383637213499, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, language=EN, label=Fig.5, caption=The first six order modes of modified Timoshenko beam on viscoelastic Pasternak foundation under five boundary conditions, figureFileSmall=2LJ1uoR8qsLaNNatO+XdLg==, figureFileBig=HpjwDQIT6b1q+HjfDYEV7g==, tableContent=null), ArticleFig(id=1228634383721099582, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, language=CN, label=图5, caption=五种边界条件下黏弹性Pasternak地基上修正Timoshenko梁的前6阶模态, figureFileSmall=2LJ1uoR8qsLaNNatO+XdLg==, figureFileBig=HpjwDQIT6b1q+HjfDYEV7g==, tableContent=null), ArticleFig(id=1228634383800791361, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, language=EN, label=Tab.1, caption=

Calculation parameters of modified Timoshenko beam

, figureFileSmall=null, figureFileBig=null, tableContent=
弹性模量剪切模量截面尺寸2梁长惯性矩密度截面剪切系数
4.322×10101.751×101060.4222700
), ArticleFig(id=1228634383884677443, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, language=CN, label=表1, caption=

修正Timoshenko梁的计算参数

, figureFileSmall=null, figureFileBig=null, tableContent=
弹性模量剪切模量截面尺寸2梁长惯性矩密度截面剪切系数
4.322×10101.751×101060.4222700
), ArticleFig(id=1228634383976952135, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, language=EN, label=Tab.2, caption=

Physical calculation parameters of viscoelastic foundation

, figureFileSmall=null, figureFileBig=null, tableContent=
弹簧系数阻尼系数剪切系数
1×1061×1041×107
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黏弹性地基的各项物理计算参数

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弹簧系数阻尼系数剪切系数
1×1061×1041×107
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The first eight order natural frequencies of simply supported viscoelastic foundation beam at both ends

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阶数l=30 ml=6 m
/%/%
112.80412.8040.0012.80412.8040.00
223.22823.1870.18434.390432.6990.39
376.32376.1450.231436.6181407.1952.05
4166.013165.5570.272649.4272531.5764.45
5286.854286.0000.303925.5013668.6676.54
6434.390432.6990.395215.1214791.2688.13
7604.326600.8230.586501.7935898.1389.28
8792.713785.9370.857780.5856992.43810.13
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两端简支黏弹性地基梁前8阶自振频率

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阶数l=30 ml=6 m
/%/%
112.80412.8040.0012.80412.8040.00
223.22823.1870.18434.390432.6990.39
376.32376.1450.231436.6181407.1952.05
4166.013165.5570.272649.4272531.5764.45
5286.854286.0000.303925.5013668.6676.54
6434.390432.6990.395215.1214791.2688.13
7604.326600.8230.586501.7935898.1389.28
8792.713785.9370.857780.5856992.43810.13
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The first eight order attenuation coefficients of simply supported viscoelastic foundation beam at both ends

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数l=30 ml=6 m
/%/%
10.82300.82300.000.82300.82300.00
20.82140.81800.410.79210.72608.34
30.81670.80371.590.75770.595821.38
40.80960.78193.420.74760.520530.38
50.80110.75525.730.75090.480935.96
60.79210.72608.340.75890.458839.55
70.78340.696511.090.76770.445541.97
80.77540.668013.850.77580.437143.66
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两端简支黏弹性地基梁前8阶衰减系数

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数l=30 ml=6 m
/%/%
10.82300.82300.000.82300.82300.00
20.82140.81800.410.79210.72608.34
30.81670.80371.590.75770.595821.38
40.80960.78193.420.74760.520530.38
50.80110.75525.730.75090.480935.96
60.79210.72608.340.75890.458839.55
70.78340.696511.090.76770.445541.97
80.77540.668013.850.77580.437143.66
), ArticleFig(id=1228634384593514842, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634335700513708, language=EN, label=Tab.5, caption=

The first eight order natural frequencies and attenuation coefficients of modified Timoshenko beam on viscoelastic Pasternak foundation under different boundary conditions

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阶数自由-自由固定-自由固定-固定固定-简支简支-简支
113.01020.8230812.95210.8235112.94320.8236812.93250.8234112.80360.82305
2895.95270.70329816.24890.78087794.08070.77087608.65400.77750432.69880.72604
31954.04490.571651840.23870.724571723.07470.749671574.48560.738631407.19500.59576
43083.87190.515722918.68670.683642768.24050.719162653.76630.705082531.57590.52051
54190.34260.467774015.21940.654823839.45950.693453756.77730.679923668.66740.48086
65282.77120.442805100.76180.637824921.83370.673844857.75920.662704791.26760.45877
76355.15870.436696180.03220.627396001.20150.660095950.85600.651145898.13760.44552
87420.68720.413737249.45630.621437077.95990.650417035.98820.643266992.43820.43706
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不同边界条件下黏弹性Pasternak地基上修正Timoshenko梁的前8阶自振频率和衰减系数

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阶数自由-自由固定-自由固定-固定固定-简支简支-简支
113.01020.8230812.95210.8235112.94320.8236812.93250.8234112.80360.82305
2895.95270.70329816.24890.78087794.08070.77087608.65400.77750432.69880.72604
31954.04490.571651840.23870.724571723.07470.749671574.48560.738631407.19500.59576
43083.87190.515722918.68670.683642768.24050.719162653.76630.705082531.57590.52051
54190.34260.467774015.21940.654823839.45950.693453756.77730.679923668.66740.48086
65282.77120.442805100.76180.637824921.83370.673844857.75920.662704791.26760.45877
76355.15870.436696180.03220.627396001.20150.660095950.85600.651145898.13760.44552
87420.68720.413737249.45630.621437077.95990.650417035.98820.643266992.43820.43706
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黏弹性Pasternak地基上修正Timoshenko梁横向自振特性分析
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柳伟 1 , 汪过兵 2 , 赵志鹏 1 , 赵晓军 1
振动工程学报 | 2024,37(8): 1330-1338
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振动工程学报 | 2024, 37(8): 1330-1338
黏弹性Pasternak地基上修正Timoshenko梁横向自振特性分析
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柳伟1 , 汪过兵2, 赵志鹏1, 赵晓军1
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

    • 柳 伟（1990—），男，硕士，讲师。E-mail： 。

Analysis of transverse free vibration characteristics of modified Timoshenko beam on viscoelastic Pasternak foundction
Wei LIU1 , Guo-bin WANG2, Zhi-peng ZHAO1, Xiao-jun ZHAO1
Affiliations
  • 1School of Civil Engineering, Lanzhou University of Information Technology, Lanzhou 730300,China
  • 2Geotechnical Engineering Research Institute, Xi’an University of Technology, Xi’an 710048,China
出版时间: 2024-08-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.08.007
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摘要
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基于修正Timoshenko梁理论，建立黏弹性Pasternak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推导出黏弹性Pasternak地基上两端简支修正Timoshenko梁自振频率和衰减系数的解析解，结合二分法和黄金分割法计算了经典边界条件下黏弹性Pasternak地基上修正Timoshenko梁的自振特性，对比分析了考虑剪切变形引起的转动惯量、梁长及不同的边界条件对结构自振特性的影响。研究表明：黏弹性Pasternak地基上修正Timoshenko梁的各阶自振频率和衰减系数小于经典Timoshenko梁的各阶自振频率和衰减系数；梁越短，剪切变形引起的转动惯量对结构自振频率和衰减系数的影响越大，且对高阶的影响明显大于低阶；边界约束条件越强，振动能量衰减越明显。

黏弹性Pasternak地基  /  修正Timoshenko梁  /  回传射线矩阵法  /  解析解  /  边界条件

Based on the modified Timoshenko beam theory，the transverse vibration governing equation of the modified Timoshenko beam on the viscoelastic Pasternak foundation is established. The analytical solutions of the natural frequency and attenuation coefficient of the modified Timoshenko beam simply supported at both ends in the viscoelastic Pasternak foundation are derived with the reverberation-ray matrix method. The natural vibration characteristics of the modified Timoshenko beam on the viscoelastic Pasternak foundation under classical boundary conditions are calculated by dichotomy and golden section method. The effects of moment of inertia caused by shear deformation，beam length and different boundary conditions on the natural vibration characteristics of the structure are compared and analyzed. The results show that the natural frequency and attenuation coefficient of the modified Timoshenko beam on the viscoelastic Pasternak foundation are smaller than those of the classical Timoshenko beam； the shorter the beam，the more significant the influence of the moment of inertia caused by shear deformation on the natural frequency and attenuation coefficient of the structure，and the influence on the higher order is obviously greater than that on the low order； the stronger the boundary constraint condition，the more obvious the vibration energy attenuation.

viscoelastic Pasternak foundation  /  modified Timoshenko beam  /  reverberation-ray matrix method  /  analytical solution  /  boundary condition
柳伟, 汪过兵, 赵志鹏, 赵晓军. 黏弹性Pasternak地基上修正Timoshenko梁横向自振特性分析. 振动工程学报, 2024 , 37 (8) : 1330 -1338 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.08.007
Wei LIU, Guo-bin WANG, Zhi-peng ZHAO, Xiao-jun ZHAO. Analysis of transverse free vibration characteristics of modified Timoshenko beam on viscoelastic Pasternak foundction[J]. Journal of Vibration Engineering, 2024 , 37 (8) : 1330 -1338 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.08.007
正文
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引言
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黏弹性地基梁在土木工程领域应用广泛［1-2］，如机场、铁路、公路、高层建筑基础中均有涉及。而结构的自振特性是反映结构动力特性的重要物理量，通过对其进行准确高效的研究不仅对结构设计和施工计算提供理论基础，而且对避免发生共振、疲劳问题和对结构的减振有重要的帮助，因此，地基梁的动力学特性在工程领域及学术界备受关注［3-5］。
目前各种地基上经典的Timoshenko梁研究较为丰富，Malekzadeh等［6］采用微积分法研究了弹性地基上Timoshenko梁的自振特性的数值解；彭丽等［7-9］运用复模态方法研究了Pasternak地基上Timoshenko梁的自振特性及任一初始激励条件下外激励的响应；余云燕等［10］ 求解了黏弹性Pasternak地基上Timoshenko梁在不同的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。这些理论虽然有较大的研究空间，但梁在高跨比比较大、局部悬空、局部高度承载情况下，由于未考虑剪切变形引起转动惯量的影响，导致经典Timoshenko梁存在挠度关于时间的四阶导数项、第二频谱、物理意义不明确等问题，使计算结果有较大偏差。因此，陈镕等［11］对传统的Timoshenko梁的运动方程进行了修正，发现修正Timoshenko梁对高频段有较大影响，指出考虑梁剪切变形引起的转动惯量后，时间的四阶导数项自然会消失；夏桂云［12］利用固有频率和临界频率的关系论证了Timoshenko梁产生第二频谱的原因，通过实例验证Timoshenko梁第二频谱的存在，因此，准确、合理地对其进行计算具有重要的意义。吴晓等［13］应用Timoshenko梁修正理论推导了泡沫铝合金梁的自振频率表达式，并求解了在简谐荷载作用下强迫振动的解析解；王家乐等［14］基于修正Timoshenko梁理论，采用复模态分析法推导多种边界条件下弹性地基梁振动超越方程及模态函数；Li等［15］推导出分数阶标准固体黏弹性地基上修正Timoshenko梁的运动控制方程，得到了自振频率的解析解；徐梅玲等［16］采用分离变量法，给出Euler梁模型相对于修正Timoshenko梁模型的误差计算公式。但上述研究未充分考虑地基土体颗粒之间相互剪切的连续性，也未分析剪切变形引起的转动惯量、梁长及边界条件对黏弹性地基上修正Timoshenko梁自振频率、衰减系数和模态的影响。
本文将修正Timoshenko梁理论与黏弹性Pasternak地基进行组合，建立新的黏弹性地基梁振动控制方程，运用回传射线矩阵法解耦，得到黏弹性Pasternak地基中两端简支修正Timoshenko梁自振频率和衰减系数的解析解，对比分析了考虑剪切变形引起的转动惯量、梁长和不同的边界条件对结构自振特性的影响。从而为黏弹性地基梁振动分析和计算提供理论基础。
1　振动控制方程及方程的解
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黏弹性地基上修正Timoshenko梁的力学模型如图1所示。地基与修正Timoshenko梁的相互作用采用考虑土体连续性的巴氏（Pasternak）模型，即在Winkler地基模型的基础上通过一层只能产生横向剪切变形而不可压缩的剪切层来实现。建立整体坐标系，引入2个对偶局部坐标系和。
对图1取微段隔离体，其受力情况如图2所示。
根据达朗贝尔原理，对微段隔离体建立竖向力及力矩平衡略去高阶项，得：
式中　 ，分别为密度、横截面惯性矩；和分别为截面剪力和弯矩， ，，其中和分别为梁的总挠度和截面转角，，，和分别为弹性模量、剪切模量、横截面面积和截面剪切系数；为黏弹性地基梁的地基竖向反力，，其中，和分别为土体弹簧系数、土体阻尼系数和地基剪切系数。
整理式（1），得到黏弹性Pasternak地基上修正Timoshenko梁的振动控制方程为：
对式（2）进行Fourier变换有：
式中　顶标“”和“-”表示频域中的变量；； ，，，，为圆频率，和分别为自振频率和衰减系数；，，，为截面的回转半径，为纵波波速，为横波波速。
求解式（3）得：
式中　 ，为待定的入射波波幅；，为待定的出射波波幅；，为波数，满足：
式中　 ；；。
对应于波数和，与的比值为：
弯矩和剪力在频域中的表达式为：
式中　 ，，。
2　经典边界条件下的自振频率及模态求解
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2.1　两端简支条件下黏弹性地基梁的自振频率方程
以两端简支的黏弹性修正Timoshenko梁为例，运用回传射线矩阵法推导自振频率方程，其自由振动的边界条件为：
将式（4），（7）代入式（8），并整理成矩阵形式有：
式（9）可写为：
式中　 ，分别为出射波和入射波波幅向量；为散射矩阵。三者表达式如下：
入射波波幅向量和出射波波幅向量之间的相位关系为：
式中　 ；为传播矩阵，。
与中的元素完全相同，只是各元素的排列位置有所调整，引入置换矩阵：
将式（12）代入式（11）得：
将式（14）代入式（10）得：
式中　 为回传射线矩阵；为单位矩阵。
要使式（15）中的有非零解，则其系数行列式为零，则：
利用指数函数与三角函数的关系，有：
忽略其奇异解，式（17）可解：
式中　 l为梁长。
将式（18）代入式（5）求解复系数一元二次方程，解得：
进一步得到两端简支边界条件下黏弹性Pasternak地基上修正Timoshenko梁的自振频率为：
衰减系数为：
由线性代数，有：
式中　 为矩阵的伴随矩阵；为矩阵的模。
当中的实部取自振频率，虚部取衰减系数时，式（22）可表示为：
式中为第k阶频率。
设是出射波波幅向量的维数，为的任意正整数，矩阵的第列记为，则：
由式（24）可知，的每一个非零列均可作为黏弹性地基梁在自由振动时非零出射波的波幅向量，求得以后，代入式（14）可求得，将和代入式（4）中，将各节点处的位移归一化处理后，得到黏弹性地基梁的模态曲线。
2.2　其他边界条件下黏弹性地基梁的自振频率方程
固定-简支边界条件下黏弹性Pasternak地基上修正Timoshenko梁，其自由振动的边界条件为：
散射矩阵为：
为：
式（27）是关于自振频率及衰减系数的二维复数超越方程。经计算，其他边界条件下的自振频率方程也是隐式超越方程，在数学上此类问题只有数值解而没有解析解。考虑到以上因素，根据回传射线矩阵法的列式特点提出将二分法和黄金分割法（简称求根法）结合起来进行迭代求解复杂超越方程，即分别对的实部和虚部进行循环，当的模小于预先给定的误差时，取出对应的实部和虚部，得到其他边界条件下黏弹性地基上修正Timoshenko梁的自振频率和衰减系数。
3　算例分析
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黏弹性Pasternak地基上修正Timoshenko梁计算模型如图1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［17］中的数值，如表2所示。
3.1　不同梁理论对黏弹性Pasternak地基梁自振特性的影响
以两端简支的黏弹性Pasternak地基上修正Timoshenko梁为例，利用表1和2中的计算参数，改变梁长，得出不同梁理论下黏弹性Pasternak地基梁前8阶自振频率和衰减系数的精确解如表3和4所示。其中黏弹性Pasternak地基上经典Timoshenko梁（P-T）和修正Timoshenko梁（P-MT）理论的区别在于振动控制方程中是否考虑了剪切变形引起的转动惯量影响。表中误差为相对误差，其自振频率相对误差计算公式为，其衰减系数相对误差计算公式为。
由表3可知，黏弹性Pasternak地基上经典Timoshenko梁和修正Timoshenko梁理论第1阶自振频率相同，第2阶之后，随着阶数的增大，两者频率值的差距逐渐增大，并且修正Timoshenko梁的各阶自振频率小于经典Timoshenko梁的各阶自振频率。当梁长l=30 m时，高跨比为0.05，其7，8阶自振频率相对误差为0.58%和0.85%，当梁长l=6 m时，高跨比为0.25，其7，8阶自振频率相对误差分别为9.28%和10.13%。从相对误差的角度来看，在黏弹性Pasternak地基上考虑剪切变形引起转动惯量的修正Timoshenko梁与不考虑其影响的经典Timoshenko梁自振频率计算结果在高跨比比较大的高频段振动有较大偏差。所以在涉及黏弹性地基梁具体工程计算中，在梁对高跨比比较大的高频段计算结果有重大影响的分析中（诸如冲击等问题），应采用修正Timoshenko梁理论。
由表4可知，黏弹性Pasternak地基上经典Timoshenko梁和修正Timoshenko梁理论第1阶衰减系数相同，第2阶之后，随着阶数的增大，两者衰减系数值的差距逐渐增大，并且修正Timoshenko梁的各阶衰减系数小于经典Timoshenko梁的各阶衰减系数。梁长为6 m时，第2阶衰减系数相对误差值与梁长为30 m时第6阶衰减系数相对误差值相等，且梁长为6 m时，其7，8阶衰减系数相对误差分别高达41.97%和43.66%。所以在工程实践中涉及振动能量损耗控制时，应考虑剪切变形引起转动惯量的影响，否则会带来较大偏差。
图3和4为不同梁长情况下黏弹性Pasternak地基上经典Timoshenko梁与修正Timoshenko梁的前6阶模态。从图3和4可知，l=30 m时，黏弹性Pasternak地基上经典Timoshenko梁与修正Timoshenko梁的第1阶振型曲线之间的差距最大，振型峰值相同，从第2阶开始，随着阶数的增大，其振型曲线之间的差距逐渐减小，但振型峰值逐渐增大。l=6 m相对于l=30 m的短梁，黏弹性Pasternak地基上经典Timoshenko梁与修正Timoshenko梁振型峰值增大得越明显，表明梁越短，考虑剪切变形引起转动惯量的修正Timoshenko梁对振型峰值的影响越大。所以在工程结构抗震计算中，按照修正Timoshenko梁设计是偏于安全的。
3.2　边界条件对黏弹性Pasternak地基上修正Timoshenko梁自振特性的影响
黏弹性Pasternak地基上修正Timoshenko梁的各项物理参数采用表1和2中的数值，通过回传射线矩阵法，结合二分法和黄金分割法，计算了黏弹性Pasternak地基上修正Timoshenko梁在不同边界条件下前8阶自振频率和衰减系数的数值解如表5所示。
由表5可知，五种边界条件下黏弹性Pasternak地基上修正Timoshenko梁的前8阶自振频率由大到小依次排序为：自由-自由>固定-自由>固定-固定>固定-简支>简支-简支；衰减系数由大到小依次排序为：固定-固定>固定-简支>固定-自由>简支-简支>自由-自由，表明约束越强，振动能量衰减越明显。
图5为五种边界条件下黏弹性Pasternak地基上修正Timoshenko梁的前6阶模态。由图5可见，将各节点处的位移归一化处理后，其振型峰值变化并没有统一的规律，但不同边界条件下的振型曲线差异十分明显。
4　结论
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将回传射线矩阵法推广至黏弹性地基梁的振动分析中，求解了黏弹性Pasternak地基中两端简支修正Timoshenko梁自振频率和衰减系数的解析解，对比分析了考虑剪切变形引起的转动惯量、梁长和不同的边界条件对结构自振特性的影响。得出以下结论：
（1）黏弹性Pasternak地基上修正Timoshenko梁的各阶自振频率和衰减系数小于经典Timoshenko梁的各阶自振频率和衰减系数；梁越短，对黏弹性Pasternak地基上修正Timoshenko梁自振频率和衰减系数的影响就越大，所以在黏弹性地基梁具体工程计算中涉及振动能量损耗控制时，或在梁对高跨比比较大的高频段计算结果有重大影响的分析中（诸如冲击等问题），应采用考虑剪切变形引起转动惯量的修正Timoshenko梁理论，否则会带来较大偏差。
（2）梁越短，考虑剪切变形引起转动惯量的修正Timoshenko梁对振型峰值的影响越大。所以在工程结构抗震计算中，按照修正Timoshenko梁设计是偏于安全的。
（3）不同边界条件下黏弹性Pasternak地基上修正Timoshenko梁的前8阶自振频率大小关系为：自由-自由>固定-自由>固定-固定>固定-简支>简支-简支；其衰减系数大小关系为：固定-固定>固定-简支>固定-自由>简支-简支>自由-自由，表明约束越强，振动能量衰减越明显。
（4）运用回传射线矩阵法可以求解埋置结构的高阶自振频率、模态等动力参数，列式统一，求解稳定，易于编程，且具有较高的计算效率和精度，从而为工程实践及工程设计提供理论基础。
基金
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  • 甘肃省科技计划资助项目(23JRRA1374)
  • 2022年甘肃省高等教育教学成果培育项目(197)
文献
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参考文献 引证文献
排序方式：
［1］
白海峰. 基于连续弹性地基梁的轨枕静力响应研究［J］. 铁道工程学报， 2007，24（5）： 22-27.
BAI Haifeng. Research on the static response of tie on continuous elastic grade beam［J］. Journal of Railway Engineering Society， 2007， 24（5）： 22-27．
［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］
Calim F F. Dynamic analysis of beams on viscoelastic foundation［J］. European Journal of Mechanics， A： Solid， 2009， 28（3）： 469-476．
［4］
Attarnejad R， Shahba A， Semnani S J. Application of differential transform in free vibration analysis of Timoshenko beams resting on two-parameter elastic foundation［J］. Arabian Journal for Science and Engineering， 2010， 35（2B）： 125-132.
［5］
Ding H， Chen L Q， Yang S P. Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load［J］. Journal of Sound and Vibration， 2012， 331（10）： 2426-2442．
［6］
Malekzadeh P， Karami G， Farid M. DQEM for free vibration analysis of Timoshenko beams on elastic foundations［J］. Computational Mechanics， 2003， 31（3-4）： 219-228．
［7］
彭丽， 丁虎， 陈立群. 黏弹性三参数地基梁横向自由振动［J］. 振动与冲击， 2014， 33（1）： 101-105.
PENG Li， DING Hu， CHEN Liqun. Transverse free vibration of a beam resting on a three-parameter viscoelastic foundation［J］. Journal of Vibration and Shock， 2014， 33（1）： 101-105.
［8］
彭丽， 丁虎， 陈立群. 黏弹性Pasternak地基梁振动的复模态分析［J］. 振动与冲击， 2013， 32（2）： 143-146．
PENG Li， DING Hu， CHEN Liqun. Complex modal analysis for vibrations of a beam on a viscoelastic Pasternak foundation［J］. Journal of Vibration and Shock， 2013， 32（2）： 143-146．
［9］
彭丽， 丁虎， 陈立群.黏弹性三参数地基上Timoshenko梁横向自由振动［J］. 噪声与振动控制， 2013， 33（5）： 107-110．
PENG Li， DING Hu， CHEN Liqun. Transverse free vibration of a Timoshenko beam rested on transverse free vibration of a Timoshenko beam rested on three-parameter viscoelastic foundation［J］.Noise and Vibration Control， 2013， 33（5）：107-110．
［10］
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.
［11］
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， 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， 2005， 33（6）： 711-715.
［12］
夏桂云. Timoshenko梁的第二频谱分析［J］. 湖南大学学报， 2021， 48（11）： 142-149.
XIA Guiyun. Analysis on the second frequency spectrum of Timoshenko beam［J］. Journal of Hunan University， 2021， 48（11）： 142-149.
［13］
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 metal aluminum alloy beam with Timoshenko beam theory［J］. Journal of Vibration and Shock， 2011， 30（1）： 124-127.
［14］
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.
［15］
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.
［16］
徐梅玲， 叶茂， 付明科， 等. 修正Timoshenko 梁自由振动及Euler 梁误差分析［J］. 科学技术与工程， 2015， 15（15）： 88-94.
XU Meiling， YE Mao， FU Mingke， et al. Systematic modal analysis of simply supported beam based on the modified motion equation of Timoshenko［J］. Science Technology and Engineering， 2015， 15（15）： 88-94.
［17］
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.
2024年第37卷第8期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.08.007
  • 接收时间：2023-08-19
  • 首发时间：2026-02-12
  • 出版时间：2024-08-28
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  • 收稿日期：2023-08-19
  • 修回日期：2023-12-21
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甘肃省科技计划资助项目(23JRRA1374)
2022年甘肃省高等教育教学成果培育项目(197)
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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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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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