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Article(id=1228634330419884879, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228634329748796239, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.08.012, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1661788800000, receivedDateStr=2022-08-30, revisedDate=1671465600000, revisedDateStr=2022-12-20, acceptedDate=null, acceptedDateStr=null, onlineDate=1770858851293, onlineDateStr=2026-02-12, pubDate=1724774400000, pubDateStr=2024-08-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770858851293, onlineIssueDateStr=2026-02-12, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770858851293, creator=13701087609, updateTime=1770858851293, 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=1377, endPage=1385, ext={EN=ArticleExt(id=1228634330675737425, articleId=1228634330419884879, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=TMD control system based on kinetic energy transformation and the use of rotary inertia to virtualize translational mass, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

The traditional Tuned Mass Damper （TMD） has many issues，such as large additional mass requirement，limited installation space restrictions and large motion stroke required when the mass block vibrates. Based on the principle of translation-rotation mutual transformation and conservation of kinetic energy，the rotary inertia virtualizing translational mass based Tuned Mass Damper （RTMD） is proposed in this paper. The conceptual design of RTMD control system is carried out，and the motion equation of RTMD control system is established with the reference of a single degree of freedom system. The influence law of RTMD control system parameters on the vibration control effect of structure is analyzed. It is discovered that the control effect is closely related to the system’s physical mass ratio，inertial mass ratio，damping ratio，and such regulations can be applicable to general multiple degrees of freedom systems. The correctness of the design parameters of the RTMD control system and the realizability of the control system are verified by shaking table test of the design model. The results of time domain analysis and frequency domain analysis show that the dynamic response obtained from shaking table test is consistent with the dynamic response of numerical simulation，which verifies the correctness of the established theoretical equations of RTMD control system.

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针对调谐质量阻尼器（Tuned Mass Damper，TMD）通常需要较大的附加质量，安装空间受限以及质量块运动时需要较大的行程等问题，基于平动-转动运动形式相互转化和能量守恒原理，本文提出了利用转动惯量虚拟平动惯性质量的TMD控制系统（Rotary inertia virtualizing translational mass based Tuned Mass Damper，简称RTMD），进行RTMD控制系统的设计概念，以单自由度结构对象为例建立了附加RTMD控制系统的运动方程，分析了RTMD控制系统参数对结构振动控制效果的影响规律。结果表明控制效果与系统的质量比、惯质比、阻尼比等参数密切相关，相关规律也可以推广到一般多自由度结构体系一阶振动的调谐吸振减振控制。进行了模型振动台试验研究，时域分析和频域分析结果均表明，试验结果与基于理论模型的数值分析结果一致性良好，验证了RTMD控制系统理论模型的正确性、设计参数的合理性以及控制系统应用于实际问题的可行性。

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张春巍（1977—），男，博士，教授，博士生导师。E-mail： 。

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张春巍（1977—），男，博士，教授，博士生导师。E-mail： 。

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张春巍（1977—），男，博士，教授，博士生导师。E-mail： 。

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caption=试验结果与数值模拟结果对比, figureFileSmall=9cgPftPn+ey4NtP+fazc5Q==, figureFileBig=zcygXGRr2evcBM5Je+oSMA==, tableContent=null), ArticleFig(id=1228634369368191940, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634330419884879, language=EN, label=Tab.1, caption=

Analysis and comparison of three working cases

, figureFileSmall=null, figureFileBig=null, tableContent=
工况结构质量/kg实际质量比惯质比实际质量/kg虚拟质量/kg传递函数峰值
（a）1.160.020.30.02320.3485.125
（b）0.050.10.05800.1164.118
（c）0.05-0.0580-10.250
), ArticleFig(id=1228634369473049543, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634330419884879, language=CN, label=表1, caption=

三种工况分析对比

, figureFileSmall=null, figureFileBig=null, tableContent=
工况结构质量/kg实际质量比惯质比实际质量/kg虚拟质量/kg传递函数峰值
（a）1.160.020.30.02320.3485.125
（b）0.050.10.05800.1164.118
（c）0.05-0.0580-10.250
), ArticleFig(id=1228634369565324235, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634330419884879, language=EN, label=Tab.2, caption=

Influence analysis of mass ratio parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
质量比控制带宽/Hz共振振幅/dB
开始点结束点
无控--32.70
2.933.73-2.25
2.923.76-2.96
2.893.79-4.31
2.853.85-6.47
), ArticleFig(id=1228634369653404622, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634330419884879, language=CN, label=表2, caption=

质量比参数影响分析

, figureFileSmall=null, figureFileBig=null, tableContent=
质量比控制带宽/Hz共振振幅/dB
开始点结束点
无控--32.70
2.933.73-2.25
2.923.76-2.96
2.893.79-4.31
2.853.85-6.47
), ArticleFig(id=1228634369724707795, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634330419884879, language=EN, label=Tab.3, caption=

Influence analysis of inertial mass ratio parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
惯质比控制带宽/Hz共振振幅/dB
开始点结束点
无控--32.70
3.153.4814.10
3.113.5310.10
3.063.585.27
2.973.68-0.10
), ArticleFig(id=1228634369812788182, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634330419884879, language=CN, label=表3, caption=

惯质比参数影响分析

, figureFileSmall=null, figureFileBig=null, tableContent=
惯质比控制带宽/Hz共振振幅/dB
开始点结束点
无控--32.70
3.153.4814.10
3.113.5310.10
3.063.585.27
2.973.68-0.10
), ArticleFig(id=1228634369905062874, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634330419884879, language=EN, label=Tab.4, caption=

Influence analysis of damping ratio parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
阻尼比频率范围/Hz振幅峰值/dB频率范围/Hz共振振幅/dB频率范围/Hz振幅峰值/dB
开始点结束点开始点结束点开始点结束点
无控---2.723.0532.70---
2.723.0526.63.053.58-9.313.584.2826.6
2.723.0521.43.053.580.073.584.2820.4
2.723.0518.43.053.584.363.584.2817.1
2.723.0516.13.053.588.253.584.2814.5
), ArticleFig(id=1228634370039280607, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228634330419884879, language=CN, label=表4, caption=

阻尼比参数影响分析

, figureFileSmall=null, figureFileBig=null, tableContent=
阻尼比频率范围/Hz振幅峰值/dB频率范围/Hz共振振幅/dB频率范围/Hz振幅峰值/dB
开始点结束点开始点结束点开始点结束点
无控---2.723.0532.70---
2.723.0526.63.053.58-9.313.584.2826.6
2.723.0521.43.053.580.073.584.2820.4
2.723.0518.43.053.584.363.584.2817.1
2.723.0516.13.053.588.253.584.2814.5
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基于运动形式转化利用转动惯量虚拟平动惯性质量的TMD控制系统
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张春巍 1 , 石一峰 1 , 湛鑫 2 , 刘志虎 2 , 孙丽 2
振动工程学报 | 2024,37(8): 1377-1385
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振动工程学报 | 2024, 37(8): 1377-1385
基于运动形式转化利用转动惯量虚拟平动惯性质量的TMD控制系统
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张春巍1 , 石一峰1, 湛鑫2, 刘志虎2, 孙丽2
作者信息
  • 1沈阳工业大学建筑与土木工程学院，辽宁　沈阳　110870
  • 2沈阳建筑大学土木工程学院，辽宁　沈阳　110168

    • 张春巍（1977—），男，博士，教授，博士生导师。E-mail： 。

TMD control system based on kinetic energy transformation and the use of rotary inertia to virtualize translational mass
Chun-wei ZHANG1 , Yi-feng SHI1, Xin ZHAN2, Zhi-hu LIU2, Li SUN2
Affiliations
  • 1School of Architecture and Civil Engineering, Shenyang University of Technology, Shenyang 110870, China
  • 2School of Civil Engineering, Shenyang Jianzhu University, Shenyang 110168, China
出版时间: 2024-08-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.08.012
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摘要
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针对调谐质量阻尼器（Tuned Mass Damper，TMD）通常需要较大的附加质量，安装空间受限以及质量块运动时需要较大的行程等问题，基于平动-转动运动形式相互转化和能量守恒原理，本文提出了利用转动惯量虚拟平动惯性质量的TMD控制系统（Rotary inertia virtualizing translational mass based Tuned Mass Damper，简称RTMD），进行RTMD控制系统的设计概念，以单自由度结构对象为例建立了附加RTMD控制系统的运动方程，分析了RTMD控制系统参数对结构振动控制效果的影响规律。结果表明控制效果与系统的质量比、惯质比、阻尼比等参数密切相关，相关规律也可以推广到一般多自由度结构体系一阶振动的调谐吸振减振控制。进行了模型振动台试验研究，时域分析和频域分析结果均表明，试验结果与基于理论模型的数值分析结果一致性良好，验证了RTMD控制系统理论模型的正确性、设计参数的合理性以及控制系统应用于实际问题的可行性。

结构振动控制  /  TMD控制系统  /  平动-转动转化  /  转动惯量  /  振动台试验

The traditional Tuned Mass Damper （TMD） has many issues，such as large additional mass requirement，limited installation space restrictions and large motion stroke required when the mass block vibrates. Based on the principle of translation-rotation mutual transformation and conservation of kinetic energy，the rotary inertia virtualizing translational mass based Tuned Mass Damper （RTMD） is proposed in this paper. The conceptual design of RTMD control system is carried out，and the motion equation of RTMD control system is established with the reference of a single degree of freedom system. The influence law of RTMD control system parameters on the vibration control effect of structure is analyzed. It is discovered that the control effect is closely related to the system’s physical mass ratio，inertial mass ratio，damping ratio，and such regulations can be applicable to general multiple degrees of freedom systems. The correctness of the design parameters of the RTMD control system and the realizability of the control system are verified by shaking table test of the design model. The results of time domain analysis and frequency domain analysis show that the dynamic response obtained from shaking table test is consistent with the dynamic response of numerical simulation，which verifies the correctness of the established theoretical equations of RTMD control system.

structural vibration control  /  TMD control system  /  transformation of translational and rotational motions  /  rotary inertia  /  shaking table experiment
张春巍, 石一峰, 湛鑫, 刘志虎, 孙丽. 基于运动形式转化利用转动惯量虚拟平动惯性质量的TMD控制系统. 振动工程学报, 2024 , 37 (8) : 1377 -1385 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.08.012
Chun-wei ZHANG, Yi-feng SHI, Xin ZHAN, Zhi-hu LIU, Li SUN. TMD control system based on kinetic energy transformation and the use of rotary inertia to virtualize translational mass[J]. Journal of Vibration Engineering, 2024 , 37 (8) : 1377 -1385 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.08.012
正文
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引言
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调谐质量阻尼器（Tuned Mass Damper，TMD）由于安装简单、方便，维修更换容易，并且对结构功能的影响较小，无需外部能量输入便可以发挥作用而被广泛应用，可以较好地控制由风振或地震等不利条件引起的结构振动响应。但是TMD控制通常需要给结构附加一定的质量，有时所附加的质量很大才能对结构起到一定的控制效果。为了减小附加质量，同时保证原有的控制效果，各国学者开始对TMD装置进行更加深入的研究。Smith［1］通过力学与电学之间的力-电流类比，首次提出惯容器（Inerter）的概念，随后将惯容器应用在汽车的悬架系统中［2］，测试出含有惯容的悬架系统性能要优于普通的悬架系统。由于惯性元件可以放大速度和产生几倍甚至几百倍于自身的惯性质量等特征，此后被国内外学者逐渐关注、研究并应用到土木工程领域。Arakaki等［3］将滚珠丝杠放大速度的特性与黏滞阻尼器相结合，增大了阻尼器的有效黏滞阻尼，试验表明提高了阻尼器的耗能能力。Saito等［4］以单自由度结构为试验模型，研究了惯性黏滞阻尼器（Viscous Damper with Inertial Mass）在最优状态下的减振效果。基于质量放大效应的思路，Hwang等［5］提出了转动惯性质量的黏滞阻尼器 （Viscous Mass Damper，VMD），该装置除了具有放大阻尼的效应外，还会有额外的惯性质量生成，进一步提高了阻尼器的耗能能力。Lazar等［6］将传统TMD中的质量块用惯性元件替代，称之为调谐惯性阻尼器（TID），并进行试验，试验表明此装置具有较好的减振效果。刘良坤等［7］推导了TID在白噪声的作用下最优阻尼和最优刚度参数的解析式，并且得到了TID的最佳控制参数。Marian等［8］将惯容器和TMD相结合，设计出一种被动控制装置——调谐质量惯容阻尼器（Tuned Mass Damper Inerter，TMDI），这种被动控制装置利用了惯容器的质量放大效应，通过三自由度结构分析，发现TMDI附加在结构的减振效果比传统的TMD附加在结构的减振效果更优。莫凯程［9］对比了惯性阻尼器和黏滞阻尼器的减振效果，经过试验表明惯性阻尼器的减振效果要优于黏滞阻尼器。综上所述，利用惯性元件的质量放大效应特征，调整子结构的频率，与主结构的频率一致，达到抑制主结构振动响应的目的［10-11］，惯性元件与阻尼、弹簧等组合形成的惯性系统具有耗能增效的作用，与被动控制装置结合使用能够达到更好的结构振动抑制效果［12-14］。Zhang等［15-17］提出了牺牲转动速度以换取更大的转动惯性（即转动惯量）的方法，在悬吊结构摆振控制研究中进行了大量的实验尝试，相关方法与惯容出发点不同，但结果类似。
本文提出的RTMD控制系统是为了减小TMD附加给结构的实际质量需求，同时降低对附加结构安装空间的需求，降低控制代价的同时RTMD能够表现出更鲁棒的控制效果，还可以推广到结构振动主动控制的多个相关领域［18-19］。在提出转动惯量虚拟平动惯性质量的控制系统（RTMD）后，建立了单自由度结构附加RTMD控制系统模型，通过对参数的影响分析以及模型的试验验证，研究控制系统参数对结构振动控制效果的影响规律以及RTMD控制装置设计的合理性和有效性。
1　RTMD控制系统设计概念
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RTMD控制系统通过滚珠丝杠来实现平动-转动运动方式的相互转化，滚珠丝杠外部有套筒，套筒可以在滚珠丝杠上线性运动。套筒两端有凸起，套筒固定于质量块内部，保证质量块在线性运动时不与套筒滑移脱落。如图1所示，丝杠的长度应远大于质量块在受到外部激励之后平动的最大值（行程范围）。丝杠两端固定在被控结构上，滚珠丝杠与固定端用轴承连接，可以保证丝杠在固定端处的自由转动，丝杠穿过两固定端外接飞轮，飞轮与丝杠等速旋转。飞轮做成空心圆环状，可以减小控制系统的实际安装质量，加设飞轮是为了获得较大的转动惯量，产生大于自身实际质量几倍甚至几百倍的虚拟平动惯性质量效应。如图2所示，两侧固定端与质量块之间设置弹簧，提供恢复力，保证质量块在线性运动时的调谐机制，发挥TMD的正常功能。
2　系统运动方程
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以单自由度结构为例进行分析，相关结果可以推广到一般多自由度体系的一阶振动问题［20-21］。如图3所示，单自由度结构附加RTMD控制系统的运动方程为：
式中　主结构的质量、阻尼系数和刚度系数用，和表示；滚珠丝杠和飞轮产生的总虚拟平动惯性质量用表示；RTMD控制系统的质量、阻尼系数和刚度系数用，和表示。，分别为主结构和RTMD控制系统相对于地面的坐标，为地面参考坐标。
引入如下参数：主结构阻尼比，主结构频率，RTMD阻尼比，RTMD频率，物理质量比，转动惯量虚拟平动惯性质量与主结构质量比。为方便计算，假设系统为无阻尼系统，即。当地面受到加速度为的激励输入时，主结构和子结构的位移分别表示为，，和分别为主结构和子结构的位移传递函数，将，和分别代入式（1）中，可以得出和分别为：
根据传递函数（2）和（3），影响结构振幅大小的主要因素为，和等参数。一般情况下，当越大时（频率比和阻尼比满足最优情况时），RTMD系统控制结构振动的效果越好。但是在实际工程中，附加较大的质量会使结构的动力反应增大，安装空间也会受到限制，所以的取值一般有限。而引入虚拟质量（也就是增加）可以减轻附加结构的实际质量，解决实际工程中需要较大附加质量的问题，的大小直接影响结构响应振幅的大小。
假设结构质量，刚度，如图4（a）所示，以质量比为例，随着惯质比的增大，位移传递函数的幅值逐渐降低，表明虚拟质量对结构振动幅值产生影响。如图4（b）所示，当惯质比时，随着质量比的增大，位移传递函数幅值逐渐降低，减振控制效果也愈加明显。图4（a）中时RTMD控制系统的实际质量为0.0232 kg，图4（b）中时RTMD控制系统的实际质量为0.058 kg，由表1可以看出在不同的实际质量比情况下，工况（a）和（b）的传递函数幅值已经较为接近，但是工况（a）的实际质量明显小于工况（b）的实际质量。这证明了RTMD控制系统可以利用虚拟质量实现较大的平动惯性质量，从而解决实际工程所需TMD质量块较大的问题，且相比于工况（c），RTMD的控制效果明显优于TMD的控制效果。
图4（b）与（c）分别为RTMD和TMD控制下的结构位移传递函数，在实际质量比相同的情况下，图4（b）的传递函数数值均小于图4（c），以为例，图4（b）的，图4（c）的，由于RTMD系统中有虚拟质量的存在，相比于TMD控制系统的传递函数峰值下降了59.82%，进一步说明了RTMD的控制效果优于TMD控制。
3　RTMD系统参数影响分析
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为采用现代控制理论研究RTMD控制系统参数对结构振动控制效果的影响规律，将式（1）写成如下所示的矩阵表达形式：
式中
引入状态向量，将运动方程表示为状态方程：
式中
其中，为单位矩阵，为环境干扰的位置矩阵；表示外部荷载。
为了更加清晰地研究质量比、惯质比和阻尼比等参数对结构的振动控制影响情况，保持质量和刚度参数与第2节算例相同，根据系统的状态方程，绘出系统的频率特征曲线，分析RTMD控制系统参数对结构振动控制效果的影响规律。
3.1　质量比对控制效果的影响
质量比是控制系统的实际质量与被控结构的实际质量的比值，质量比的大小直接影响RTMD系统对结构振动的控制效果。当惯质比一定时，由图5看出，随着质量比的增加，结构振动的控制效果也越来越好。图中P点和Q点是RTMD系统控制结构振动的区间点，在此区间内RTMD控制有效，尤其是在结构的自振频率附近，减振效果最好。但是质量比较大（）时，结构的控制效果提升较小，即在控制代价很大的情况下，提升结构的控制效果将变得十分有限，所以本文引入虚拟平动惯性质量来进一步研究RTMD对结构振动的控制效果。
由表2可知，随着质量比的增加，控制带宽也随之增加，表明较大的附加质量可以在更宽的频率范围内对结构有更为理想的控制效果。
3.2　惯质比对控制效果的影响
惯质比是RTMD控制系统的虚拟质量与被控结构的实际质量的比值。当控制系统的质量比一定时，惯质比的大小直接影响结构振动的控制效果。以下分析中设定惯质比为0.01，0.02，0.04，0.08四种情况，得到结构的动力响应情况如图6所示。
由图6可见，结构在无控状态下，当频率处在结构的自振频率时，位移幅值达到最大，此时结构处在最不利的振动状态。如表3所示，相比于无控制状态，当结构加入RTMD控制系统之后，振幅在共振频率范围内RTMD控制状态分别下降了18.6，22.6，27.43，32.8 dB。
上述结果说明RTMD控制系统的减振效果优良。当惯质比较小（，）时，RTMD控制系统产生的惯性力也相对较小，对结构的控制效果相对有限，仅在有限的频率带宽范围内降低结构的振幅。当惯质比较大（，）时，此时控制系统产生的惯性力相对较大，从图6中可看出相比于小惯质比，大惯质比的控制效果更佳，同时控制频率带宽也相应增加，说明在更宽的频率范围内RTMD控制系统可以更加有效地控制主结构的振动。外部激励频率在接近结构自振频率时，控制效果达到最佳，避免结构在最不利的激励作用时产生较大振幅。此外，比较图6与5所示结果，发现相同的频率范围内惯质比参数比质量比参数更能拉宽控制系统的有效带宽，间接说明了RTMD控制系统的有效性和合理性。
3.3　阻尼比对控制效果的影响
阻尼比的大小直接影响结构的耗能能力，研究RTMD控制系统的阻尼比对于评价结构的控制效果有重要的意义。假设RTMD控制系统的阻尼比为，，，四种情况，得到阻尼比对结构振幅的影响如图7所示。
被控结构在输入频率为 Hz时振幅会达到最大，此时为“共振”状态，当频率逐渐远离自振频率时结构响应的振幅逐渐降低。加入RTMD控制系统后，被控结构在“共振”频率范围内时的控制效果最好，四种阻尼比对应的振幅分别下降了42.01，32.63，28.34，24.45 dB。在表4中当频率在2.72~3.05 Hz和3.58~4.28 Hz范围时，控制系统为低阻尼比时结构产生的振幅要大于高阻尼比时结构的振幅，并且随着阻尼比越来越大，振幅波动逐渐减小并且峰值会逐渐降低。在外部激励频率逐渐接近结构自振频率的过程中，当RTMD控制系统的阻尼较低时，不能有效消耗主结构的能量，因此在上述频率范围时RTMD在低阻尼比情况下主结构的振幅要大于高阻尼比情况下主结构的振幅。
当外部激励频率在3.00~3.39 Hz范围内时，结构进入“共振”区间，由图7相位角图可以看出，在此频率范围内时相位角迅速接近90°，当达到结构自振频率时，RTMD控制系统与结构相位完全相反达到90°，此时是控制效果最佳状态。随着控制系统的阻尼比增加，减振效果越来越差，所以设计控制装置时，选择阻尼比较小的材料。但是选择小阻尼比时，对频率比的敏感性会增加，因此在进行RTMD控制系统的设计时，需要确定最优的频率和阻尼比。
根据Rana等［22］给出的正弦激励作用下，以结构响应位移为优化目标的TMD控制参数优化公式：
求解出RTMD控制系统的最佳频率和阻尼比分别为19.702 rad/s和0.135，得出在最优参数下，控制系统结构的伯德图如图8所示。
根据图8可知，用最优参数确定RTMD控制系统的刚度和阻尼后，被控结构的振幅在3.11~3.96 Hz频率范围内下降约43.86%，减振效果明显且振幅波动较为平缓。在上述频率范围之外，RTMD几乎没有对结构产生控制作用，并且在非共振区也没有对结构产生明显的负面影响。在图5和7中，可以明显地看出当RTMD控制系统所在的频率位于结构有控制效果的频率范围之外时，此时由于控制系统的刚度和阻尼没有取得最优值，RTMD控制系统会给结构带来一定的负面效果，使被控结构的振幅大于无控状态时结构的振幅。综上所述，为达到理想的减振效果，在设计RTMD控制系统时，要合理选择最佳的阻尼比和频率比，使得RTMD系统的减振效果和控制频带处于一个最优的组合范围内，这也提示我们今后对RTMD控制系统开展可变阻尼设计研究的意义。
4　试验研究
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为了验证RTMD控制系统设计的合理性和系统的有效性，下面将对RTMD控制系统进行振动台试验验证。首先建立系统动力学模型，然后根据振动台试验数据与模拟数据进行对比分析。依据D’Alembert原理建立RTMD控制系统的动力学方程为：。
4.1　模型配置
RTMD控制系统试验模型主要由质量块、滚珠丝杠套筒、圆盘（飞轮）和螺旋弹簧四部分组成，如图9所示。滚珠丝杠的导程为2 cm，半径为0.8 cm，圆盘的半径为3 cm，厚度为0.2 cm，质量块的尺寸为7 cm×7 cm×7.64 cm，质量块质量为2.55 kg，弹簧刚度为740 N/m，根据设计飞轮与丝杠旋转产生的虚拟质量为3.3 kg。
4.2　自由振动试验
由于阻尼的存在会引起结构不同的动力特性及复杂的动力响应，通过理论模拟分析和振动台试验验证来综合确定RTMD控制系统的阻尼参数。给质量块一个初始状态（初始位移），测出质量块的位移时程曲线如图10所示。由图10可以看出，通过数值模拟与试验得到的质量块位移响应一致，根据对数衰减率计算出阻尼比为14%。由于试验中控制装置的两端固定支座、滚珠丝杠与套筒之间、质量块与振动台台面之间存在不可避免的机械摩擦阻力，所以在1.2 s之后试验曲线不再像模拟曲线那样产生多个周期的往复振动。图11为频域曲线，在自由衰减运动状态下试验值和模拟值的曲线趋势一致，在共振处达到峰值。
4.3　强迫振动试验
通过自由振动试验，计算出RTMD试验模型系统的阻尼系数之后，仍然借助于Quancer shake table Ⅲ振动台进行强迫振动试验研究。输入简谐激励，其中为幅值，为频率。本文采用幅值，频率为1.80，1.85和1.90 Hz三种工况进行试验研究。当外部激励作用在RTMD的质量块上时，质量块的运动带动丝杠和飞轮旋转，如图12所示。
通过输入不同频率的正弦激励作用，由图13可知，质量块的行程在接近自振频率时达到最大值72.1 mm，当偏离此频率时质量块的行程逐渐降低，这与传统的TMD系统的工作机理一致。
5　结论
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本文提出利用转动惯量虚拟平动惯性质量的TMD，即RTMD控制系统的概念，采用RTMD控制以单自由度结构为代表的受控对象，进行参数影响分析，获得控制系统参数影响规律的基础上，进行了控制系统模型的振动台试验验证，得到以下主要结论：
（1）参数分析结果表明，当质量比和惯质比较大、阻尼比较小时，RTMD的控制效果较好，当激励频率接近结构的自振频率时，RTMD的控制效果最佳。
（2）振动台试验验证了RTMD控制系统理论模型的正确性和合理性。通过时域分析和频域分析得到试验值和模拟值高度一致的结果，表明此装置可以产生预期的虚拟平动惯性质量效应，并且产生的虚拟质量效应已经超过了质量块的原有实际物理质量。
（3）通过RTMD与TMD两种系统传递函数的对比，可知两者在达到类似控制效果的前提下，RTMD控制系统的实际质量要远远低于TMD控制系统的实际质量，可以达到减轻控制系统实际物理质量的目的，并且RTMD系统的鲁棒性更优。
本文研究结果表明基于转动惯量虚拟平动惯性质量的方法是可行的，RTMD控制系统的原理完全不同于现有的惯容装置，现有文献报导的惯容装置（如TMDI）均是一个额外附加的元件，需要与TMD质量块采用不同的连接方式（如并联，串联等）结合起来使用，共同工作。本文提出的RTMD系统是一个独立、完整、高度集成的控制系统，不需要与TMD并联、串联等，是直接安装在被控结构上，即可进行工作的TMD系统。在本文方法和概念模型的基础上，未来还有很多技术问题值得深入研究，例如通过设计不同的飞轮尺寸和导程，获得不同量级的虚拟平动惯性质量等。
基金
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  • 国家自然科学基金资助项目(52361135807)
  • 国家重点研发计划资助项目(2019YFE0112400)
文献
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2024年第37卷第8期
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文章信息
doi: 10.16385/j.cnki.issn.1004-4523.2024.08.012
  • 接收时间：2022-08-30
  • 首发时间：2026-02-12
  • 出版时间：2024-08-28
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  • 收稿日期：2022-08-30
  • 修回日期：2022-12-20
基金
国家自然科学基金资助项目(52361135807)
国家重点研发计划资助项目(2019YFE0112400)
作者信息
    1沈阳工业大学建筑与土木工程学院，辽宁　沈阳　110870
    2沈阳建筑大学土木工程学院，辽宁　沈阳　110168
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