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Article(id=1228048671707689529, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.03.005, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1667145600000, receivedDateStr=2022-10-31, revisedDate=1670515200000, revisedDateStr=2022-12-09, acceptedDate=null, acceptedDateStr=null, onlineDate=1770719219376, onlineDateStr=2026-02-10, pubDate=1711555200000, pubDateStr=2024-03-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770719219376, onlineIssueDateStr=2026-02-10, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770719219376, creator=13701087609, updateTime=1770719219376, updator=13701087609, issue=Issue{id=1228048667874095618, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='3', pageStart='365', pageEnd='538', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770719218462, creator=13701087609, updateTime=1770795476854, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228368518803030940, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228368518803030941, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=402, endPage=413, ext={EN=ArticleExt(id=1228048671946764868, articleId=1228048671707689529, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Improved VMD and its application in instantaneous frequency identification of seismic responses of time-varying structures, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Under the action of strong seismic excitations，structures exhibit time-varying dynamic characteristics due to damage. Variational mode decomposition （VMD） can be used to analyze the instantaneous frequency variation of structural seismic responses and reveal the damage condition of structures during earthquake. When VMD is adopted for decomposing the non-stationary responses，there exists the problem of mode aliasing due to the artificially presetting the number of decomposed modes K and the quadratic penalty factor α. Aiming at solving this problem，an improved variational mode decomposition （IVMD） algorithm is proposed in this study，which，combined with Hilbert transform （HT），can more accurately identify the instantaneous frequencies of time-varying structures under non-stationary seismic excitations. The multiple signal classification （MUSIC） algorithm is used to determine the number of decomposition modes K. The comprehensive objective function is constructed based on the overall orthogonal coefficient and energy ratio coefficient，and the slap swarm algorithm （SSA） is used to optimize and determine the optimal quadratic penalty factor α. Based on the optimized parameters K and α，IVMD-HT is used to identify the instantaneous frequency of time-varying structures from the seismic responses. A simulated signal and the seismic responses of a 4-layer time-varying frame structure show that the accuracy of identified instantaneous frequencies by using the IVMD algorithm is higher than the identified ones by using the VMD algorithm. The feasibility of the proposed method is verified by using shaking table test data of a 12-story reinforced concrete frame structure model.

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强地震动作用下，结构动力特性因损伤将随时间变化，变分模态分解（Variational Mode Decomposition，VMD）可用于分析结构地震响应的瞬时频率变化规律，揭示地震过程中结构的损伤状态。针对人为预设分解模态数K和二次惩罚因子参数不准确导致VMD非平稳响应出现模态混叠的问题，提出了一种改进的VMD（Improved VMD，IVMD）算法，结合Hilbert变换（HT）可准确识别非平稳地震激励下时变结构的瞬时频率。采用多重信号分类（Multiple Signal Classification，MUSIC）算法确定分解模态数K，基于整体正交系数和能量比系数构建综合目标函数，并采用樽海鞘算法（Slap Swarm Algorithm，SSA）优化确定二次惩罚因子。在优化确定K和参数的基础上，结合IVMD和HT识别时变结构地震响应瞬时频率。典型模拟信号和地震激励下4层时变框架结构数值模拟响应表明，相比VMD算法，IVMD算法识别得到的瞬时频率精度更高；12层钢筋混凝土框架结构模型地震模拟振动台试验数据验证了所提方法的实用性。

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尚旭强（1996—），男，博士研究生。E-mail： 。
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王焱楠（1999—），女，硕士研究生。E-mail： 。

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王焱楠（1999—），女，硕士研究生。E-mail： 。

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王焱楠（1999—），女，硕士研究生。E-mail： 。

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instantaneous frequencies of a 12-story frame model identified by IVMD-HT algorithm, figureFileSmall=4Xjef9hDSmiwZQGC0ieDNQ==, figureFileBig=YTPSDzE4xAO68ExAG1sDBQ==, tableContent=null), ArticleFig(id=1228048706512024306, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048671707689529, language=CN, label=图15, caption=IVMD-HT算法识别的12层框架模型前3阶瞬时频率, figureFileSmall=4Xjef9hDSmiwZQGC0ieDNQ==, figureFileBig=YTPSDzE4xAO68ExAG1sDBQ==, tableContent=null), ArticleFig(id=1228048706583327477, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048671707689529, language=EN, label=Fig.16, caption=Moving mean of the first three order instantaneous frequencies under Q1 and Q2 cases, figureFileSmall=hcPS/75r2gbIdh+kc0+kwA==, figureFileBig=tE/H44ioH1jhDmFmwNdwEg==, tableContent=null), ArticleFig(id=1228048706650436344, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048671707689529, language=CN, label=图16, caption=Q1和Q2工况下前3阶瞬时频率移动均值, figureFileSmall=hcPS/75r2gbIdh+kc0+kwA==, figureFileBig=tE/H44ioH1jhDmFmwNdwEg==, tableContent=null), ArticleFig(id=1228048706721739515, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048671707689529, language=EN, label=Tab.1, caption=

RMSE of identified instantaneous frequencies with different value （K=3）

, figureFileSmall=null, figureFileBig=null, tableContent=
RMSE
总和2.411.79—
1阶0.350.300.33
2阶0.850.570.47
3阶1.210.92—
), ArticleFig(id=1228048706797236989, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048671707689529, language=CN, label=表1, caption=

不同取值下瞬时频率识别值的RMSE（K=3）

, figureFileSmall=null, figureFileBig=null, tableContent=
RMSE
总和2.411.79—
1阶0.350.300.33
2阶0.850.570.47
3阶1.210.92—
), ArticleFig(id=1228048706881123071, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048671707689529, language=EN, label=Tab.2, caption=

RMSE of identified instantaneous frequencies of a frame structure with different （K=4）

, figureFileSmall=null, figureFileBig=null, tableContent=
RMSE
总和1.090.75-
1阶0.110.100.11
2阶0.060.060.07
3阶0.220.220.50
4阶0.410.37-
), ArticleFig(id=1228048706952426241, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048671707689529, language=CN, label=表2, caption=

不同取值下框架结构瞬时频率识别值的RMSE（K=4）

, figureFileSmall=null, figureFileBig=null, tableContent=
RMSE
总和1.090.75-
1阶0.110.100.11
2阶0.060.060.07
3阶0.220.220.50
4阶0.410.37-
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改进VMD及其在时变结构地震响应瞬时频率识别中的应用
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王焱楠 , 尚旭强 , 黄天立
振动工程学报 | 2024,37(3): 402-413
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振动工程学报 | 2024, 37(3): 402-413
改进VMD及其在时变结构地震响应瞬时频率识别中的应用
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王焱楠 , 尚旭强 , 黄天立
作者信息
  • 中南大学土木工程学院，湖南　长沙　410075

    • 王焱楠（1999—），女，硕士研究生。E-mail： 。

通讯作者:

尚旭强（1996—），男，博士研究生。E-mail： 。
Improved VMD and its application in instantaneous frequency identification of seismic responses of time-varying structures
Yan-nan WANG , Xu-qiang SHANG , Tian-li HUANG
Affiliations
  • School of Civil Engineering，Central South University，Changsha 410075，China
出版时间: 2024-03-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.03.005
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摘要
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强地震动作用下，结构动力特性因损伤将随时间变化，变分模态分解（Variational Mode Decomposition，VMD）可用于分析结构地震响应的瞬时频率变化规律，揭示地震过程中结构的损伤状态。针对人为预设分解模态数K和二次惩罚因子参数不准确导致VMD非平稳响应出现模态混叠的问题，提出了一种改进的VMD（Improved VMD，IVMD）算法，结合Hilbert变换（HT）可准确识别非平稳地震激励下时变结构的瞬时频率。采用多重信号分类（Multiple Signal Classification，MUSIC）算法确定分解模态数K，基于整体正交系数和能量比系数构建综合目标函数，并采用樽海鞘算法（Slap Swarm Algorithm，SSA）优化确定二次惩罚因子。在优化确定K和参数的基础上，结合IVMD和HT识别时变结构地震响应瞬时频率。典型模拟信号和地震激励下4层时变框架结构数值模拟响应表明，相比VMD算法，IVMD算法识别得到的瞬时频率精度更高；12层钢筋混凝土框架结构模型地震模拟振动台试验数据验证了所提方法的实用性。

地震激励  /  时变框架结构  /  振动台试验  /  瞬时频率识别  /  变分模态分解

Under the action of strong seismic excitations，structures exhibit time-varying dynamic characteristics due to damage. Variational mode decomposition （VMD） can be used to analyze the instantaneous frequency variation of structural seismic responses and reveal the damage condition of structures during earthquake. When VMD is adopted for decomposing the non-stationary responses，there exists the problem of mode aliasing due to the artificially presetting the number of decomposed modes K and the quadratic penalty factor α. Aiming at solving this problem，an improved variational mode decomposition （IVMD） algorithm is proposed in this study，which，combined with Hilbert transform （HT），can more accurately identify the instantaneous frequencies of time-varying structures under non-stationary seismic excitations. The multiple signal classification （MUSIC） algorithm is used to determine the number of decomposition modes K. The comprehensive objective function is constructed based on the overall orthogonal coefficient and energy ratio coefficient，and the slap swarm algorithm （SSA） is used to optimize and determine the optimal quadratic penalty factor α. Based on the optimized parameters K and α，IVMD-HT is used to identify the instantaneous frequency of time-varying structures from the seismic responses. A simulated signal and the seismic responses of a 4-layer time-varying frame structure show that the accuracy of identified instantaneous frequencies by using the IVMD algorithm is higher than the identified ones by using the VMD algorithm. The feasibility of the proposed method is verified by using shaking table test data of a 12-story reinforced concrete frame structure model.

seismic excitation  /  time-varying frame structures  /  shaking table test  /  instantaneous frequency identification  /  VMD
王焱楠, 尚旭强, 黄天立. 改进VMD及其在时变结构地震响应瞬时频率识别中的应用. 振动工程学报, 2024 , 37 (3) : 402 -413 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.03.005
Yan-nan WANG, Xu-qiang SHANG, Tian-li HUANG. Improved VMD and its application in instantaneous frequency identification of seismic responses of time-varying structures[J]. Journal of Vibration Engineering, 2024 , 37 (3) : 402 -413 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.03.005
正文
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引言
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地震是一种难以预测且破坏性极强的自然灾害，常造成土木工程结构不同程度的破坏。结构一旦发生破坏，其自振频率、阻尼比等动力特性随之变化。利用先进的非平稳信号分析方法，分析非平稳地震激励下结构的地震动响应，准确提取结构的瞬时频率等时变动力特性，对于结构的损伤诊断和安全状态评估具有重要的价值［1］。
Loh等［2］采用子空间识别算法，分析了一栋7层钢筋混凝土框架结构、一栋9层层间隔震建筑和一座隔震桥梁的输入-输出地震响应数据，识别了结构的自振频率和阻尼比等动力特性，并将其应用于结构地震安全评估。Ni等［3］提出了一种基于结构地震输入-输出响应数据的快速贝叶斯频域模态参数识别方法，基于某12层框架结构模型振动台模拟地震试验数据和7层Van Nuys酒店建筑地震响应数据识别得到的结构模态参数，验证了方法的有效性和适用性。Ghahari等［4］指出，地震动输入数据常常难以获得，且考虑土-结构相互作用时，记录的自由场地震动响应或基础响应数据都不能作为输入用于基于输入-输出数据的模态参数识别方法，因此，盲源分离［5］、递归随机子空间识别［6］和频域分解［7］等仅基于输出响应的模态参数识别方法更适用于识别地震激励下时变结构的模态参数。
时频分析方法作为处理非平稳信号的有力工具，通过分析结构地震响应数据的时频谱图，提取结构瞬时频率等特征参数，可用于揭示结构在地震过程中的损伤状态［8］。樊海涛等［9］采用经验模态分解（EMD）从结构地震响应信号中提取结构模态响应，利用Hilbert变换识别了结构瞬时频率等时变特征，掌握了地震过程中结构损伤的发展规律。石春香等［10］采用EMD+HT方法，分析了深圳罗湖商务大厦模型模拟地震振动台响应数据，利用Hilbert阻尼谱研究了结构在地震过程中的破坏规律。陈隽等［11］采用EMD+HT方法识别了线性时变系统的瞬时频率和瞬时阻尼比特征，并将其应用于地震激励下12层短肢剪力墙结构振动台模拟地震试验数据，识别了模型结构在试验过程中动力性能的时变特性。
应该指出，由于EMD为经验算法，没有严格的数学理论基础，分解过程中常存在模态混叠等问题［12］。为更好地识别多分量非平稳信号的时变频率，提出了一种变分模态分解（VMD）方法［13］，该方法将信号同时分解为不同中心频率的窄带信号，具有较高的时频分辨率和噪声鲁棒性，已广泛应用于机械故障诊断［14］、模态参数识别［15-16］和风速预测［17］等非平稳信号分析领域，但VMD应用于非平稳地震激励下结构响应分析和损伤诊断方面的研究工作还较少。
VMD需人为预设分解模态数K和二次惩罚因子等参数，常采用多次试算方法确定参数取值，由此导致VMD在分解非平稳响应时常出现模态混叠问题。为此，许多学者提出了一些改进措施，以实现VMD参数优化选取。Huang等［18］和Ma等［19］应用尺度空间表示和尺度空间谱分割来估计分解模态数K。Lian等［20］采用能量损失、置换熵等指标反复搜索分解模态数K。唐贵基等［21］以包络熵为优化目标，采用粒子群算法同时搜寻K和的全局最优解。何勇等［22］以峭度和包络熵作为遗传算法的目标函数，对K和同步联合优化，并应用于轴承故障诊断。应该指出，同时对K和两个参数进行优化，计算效率低，且参数间相互影响导致分解模态数K出现偏差，即使参数取得最优值，也可能造成分解存在分量丢失或分量混叠等问题。因此，本文提出分别优化参数K和的思路，保证二者取值均为最优。此外，参数优化取值时常采用单一的局部目标函数［23］，这对于机械故障诊断是适用的，但不适用于非平稳地震激励下时变结构瞬时频率的识别问题。
针对上述VMD算法存在的不足，本文提出一种改进VMD（IVMD）算法，并应用于非平稳地震激励下时变结构响应瞬时频率识别。首先，考虑到分解模态数K对VMD结果的决定性作用，基于MUSIC功率谱确定；然后，考虑到仅对二次惩罚因子进行优化可提高计算效率，采用整体正交系数和能量比系数构造综合目标函数，并基于樽海鞘算法（SSA）优化确定；最后，基于优化确定的参数K和，采用IVMD算法分解地震响应得到固有模式函数（IMF），并结合Hilbert变换识别结构瞬时频率。采用模拟信号、地震激励下4层时变框架结构的数值模拟响应和12层框架结构模型振动台模拟地震试验数据验证了所提方法的有效性和实用性。
1　VMD算法
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VMD算法采用一种非递归的变分模型，预设分解模态数K，将多分量信号同时分解为个中心频率为的单分量调幅-调频信号，即固有模式函数（IMF）分量。基于每个IMF分量围绕其中心频率有最紧密带宽的基本假定，构造的带宽方程：
式中　 为虚数单位；表示卷积计算；为分解得到的第k个IMF分量；为的中心频率；为对时间t求偏导数；为脉冲函数。将的解析信号与相乘并将其频谱移至基带，计算解调信号梯度范数的平方，构成约束变分模型。引入拉格朗日乘子对约束进行控制，二次惩罚因子提高变分问题的收敛性，从而转化为无约束变分问题：
式中　 L表示增广拉格朗日量；<>表示求内积。
通过乘法器交替方向法（ADMM）交替求解，和，寻找增广Lagrange函数的鞍点，得到最优解：
式中　 ，和分别为，和的傅里叶变换；n为迭代次数。
方程（4）计算模态分量的过程，相当于对残余信号分量进行维纳滤波。VMD在频域对每个IMF分量进行更新，对进行傅里叶逆变换得到IMF分量的时域表示。对于所有，拉格朗日乘子由下式更新：
式中　 为递归步长，当无需精确重构原始信号时，。
不断按式（3），（4）和（5）迭代更新，直到满足以下收敛准则：
式中　 为设定的判别精度。
2　基于改进VMD算法识别时变结构瞬时频率
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VMD算法分解信号时，采用的分解模态数K和二次惩罚因子对分解结果影响较大，其中K直接决定分解结果的准确性，K取值过小或过大将分别导致模态混叠或冗余噪声模态；二次惩罚因子控制每个IMF的频带宽度和噪声含量。目前，VMD算法分解信号时采用的参数K和具有随机性，且缺乏选择标准，从而影响了IMF分量用于识别瞬时频率的准确性。因此，确定优化的参数K和对基于VMD识别时变结构地震响应的瞬时频率至关重要。
2.1　确定及验证分解模态数K
傅里叶频谱常用来确定VMD的分解模态数K［15］，然而，傅里叶变换处理非平稳信号时，其频谱中存在大量虚假峰值，此外噪声干扰引起频谱中出现较多波动［24］，因此，傅里叶频谱常无法准确确定分解模态数K。相比傅里叶频谱，MUSIC功率谱更平滑，分辨率更高［25］，能很好地处理非平稳信号并减小噪声干扰，基于MUSIC功率谱能更准确地确定分解信号的分解模态数K。
MUSIC功率谱的基本思想是对信号的阶相关矩阵作特征分解，并把特征向量张成信号空间和噪声空间，然后根据信号和噪声空间的正交性进行功率谱估计：
式中　 ；为的共轭转置向量；为信号个数。在处呈现一个尖峰，因此，可根据MUSIC功率谱的峰值数确定分解模态数K。
为验证MUSIC功率谱确定分解模态数K的准确性，构造一个具有时变频率的非平稳信号，该信号由3个频率按正弦变化的单分量信号组成：
式中　 为白噪声，其功率为信号功率的10%，信号的信噪比为14 dB，信号时长为10 s，采样频率为 Hz。
图1所示为信号的MUSIC功率谱、傅里叶频谱和短时傅里叶谱。由图1（a）可见，MUSIC功率谱有3个明显的峰值，不存在虚假峰值且几乎无波动，由此确定分解模态数K=3，真实反映了信号的分量组成。由图1（b）可见，傅里叶频谱中有7个明显的峰值（其中峰值①，③，④和⑥为虚假峰值）且波动较大，由此确定分解模态数K=7，这与信号的分量组成不符。由图1（c）可见，短时傅里叶谱模糊不清，无法得出分解模态数K。由此可知，MUSIC功率谱有抑制噪声、分解非平稳信号的能力，可用于准确确定非平稳信号分解模态数K。
2.2　优化确定二次惩罚因子
确定K值后，VMD算法分解结果的准确性主要取决于二次惩罚因子。取值偏小，则各阶IMF受噪声干扰大；取值偏大，则信号过度分解。为得到仅包含单一频率成分且受噪声干扰小的IMF，本节基于VMD算法分解的后验信息构建目标函数，采用优化算法实现的确定。
2.2.1　构造目标函数
包络熵、峭度、相关性等目标函数常用于机械故障诊断领域，然而，这些目标函数仅针对单一IMF。基于分解得到的全部IMF，构造综合目标函数评估所有IMF的性能，可提高识别时变结构各阶瞬时频率的准确性。
（1） 整体正交系数P
整体正交系数P表示IMF之间的正交性，其表达式如下：
式中　 ，和分别为第i，j个IMF和信号的频域序列；m表示离散变量；为信号长度。
信号分解完全时，各阶IMF分量正交，整体正交系数P值最小。应该指出，由于实际信号中不可避免地存在噪声，极易出现整体正交系数P最小时，相应的取值过小，由此导致各阶频率的能量聚集性差。
（2） 能量比系数E
为解决单一整体正交系数易受噪声干扰的问题，引入能量比系数E，其定义为各阶IMF能量与原始信号能量之比，表达式如下：
根据帕塞瓦等式，式（10）有频域和时域两种表达式。当信号分解不完全时，能量比系数E值偏小。反之，信号过度分解时，能量比系数E值偏大。
（3） 综合目标函数F
当整体正交系数与能量比系数的取值合适时，既可保证模态正交性，也可避免能量损失。因此构造综合目标函数F，表达式如下：
式中　 和分别为归一化整体正交系数和能量比系数的倒数。F表示各IMF之间正交系数和分解过程中能量损失之和，当F取最小值时，减轻了噪声影响且各阶IMF分量正交，从而避免了模态混叠和过度分解。通过寻找综合目标函数最小值Fmin，即可确定，Fmin对应的即为最优值，表达式如下：
2.2.2　验证目标函数
为验证构造的综合目标函数F的有效性，图2给出了信号的整体正交系数P、能量比系数倒数R和综合目标函数F与取值的关系曲线，取值范围为［500，2.5×105］。由图2可知，整体正交系数P在B点取最小值，但此时取值偏小，VMD算法分解得到的模态分量受噪声干扰大，单一P值无法确定的最优值。能量比系数倒数R在C点取最小值，但此时取值偏大，模态混叠严重，单一R也无法确定的最优值。本文构造的综合目标函数F，在介于B和C两点之间的A点取最小值，相应的为最优值，此时，VMD算法分解结果噪声干扰小且模态混叠程度轻。
由图2可知，当综合目标函数F在突变区间取最大值时，对应的值用于VMD时会造成过度分解，由此表明的取值偏大。尽管图2中D点处综合目标函数F取值较小且变化平稳，但对应的值过大导致VMD严重过度分解，从此处选取的值不合适。因此，本文进一步将的搜寻取值范围限定在范围内，以保证取值为局部最优而非全局最优，并提高了搜索效率。
2.2.3　SSA优化搜索二次惩罚因子
樽海鞘算法（SSA）［26］是一种群体智能优化算法，具有搜索速度快、有效避免前期搜索不充分而陷入局部最优的特点。该算法将种群个体分为领导者和追随者，前面的个体领导后面的个体相继排列成“链”状，迭代更新个体位置以搜寻最优解。本文采用SSA算法优化搜索二次惩罚因子的最优值。
针对信号，2.1节中已确定分解模态数K=3，本节采用综合目标函数F和SSA算法优化搜索二次惩罚因子的最优值，的搜寻范围为。图3所示为综合目标函数F的迭代曲线。由图3可知，SSA算法迭代18次后收敛，此时Fmin为0.9952，对应的最优值为34000，与网格优化结果一致。SSA算法迭代收敛用时112.6 s，网格优化（搜寻步长为1）算法迭代收敛用时3424.9 s，SSA算法节约计算时间约96%，搜寻效率高。
综上所述，采用MUSIC算法确定分解模态数K，构造综合目标函数F，采用SSA算法优化搜索二次惩罚因子，实现了VMD参数的优化确定，本文称其为改进变分模态分解（IVMD）算法。
2.3　Hilbert变换识别时变结构瞬时频率
基于IVMD算法分解结构响应得到IMF，基于Hilbert变换即可计算得到结构的各阶瞬时频率，计算公式如下：
式中　 为第个模态分量的瞬时频率；为的Hilbert变换。
针对信号，基于2.1节和2.2节中确定的VMD算法分解最优参数，即K=3，，利用IVMD对其分解，并将分解后的IMF进行HT识别得到各阶瞬时频率，如图4所示。由图4可知，识别的信号前2阶瞬时频率与理论值完全吻合；由于噪声干扰，识别的第3阶瞬时频率存在一定的波动，但其均值与理论值基本一致。此外，由于HT的端点效应，信号端部识别得到的各阶瞬时频率误差较大。
2.4　IVMD-HT算法识别时变结构瞬时频率流程
图5给出了基于IVMD-HT算法识别非平稳信号瞬时频率的流程图，其基本步骤包括：（a） 根据式（7）的MUSIC功率谱估计，得到光滑、分辨率高的MUSIC功率谱，根据谱峰确定信号的分解模态数K；（b） 设置二次惩罚因子的搜寻范围，基于综合目标函数F并采用SSA算法搜寻的最优值，Fmin对应的值即为最优；（c） 利用最优K和，对信号进行IVMD得到一系列IMF，并采用HT识别信号各阶IMF的瞬时频率。其中，步骤（a）和（b）两部分即为IVMD算法。
2.5　分析与讨论
为验证本文算法获取优化的参数K和后识别瞬时频率的准确性，人为选取了不同的参数K和，图6给出了VMD算法识别的信号的瞬时频率。由图6可知，当K随机取值，=34000时，尽管取最优值，但K取值偏小或偏大分别出现缺少高阶频率或存在冗余噪声频率的问题，K值不准确直接导致结果偏差。当K=3，随机取值，偏小时，各阶瞬时频率的识别结果受噪声干扰大；偏大时，尽管识别了第2阶频率，但由于模态混叠，第3阶和第2阶频率完全重叠，第3阶频率无法被识别。由此表明，人为随机对K和取值对瞬时频率的识别结果影响很大，会导致不能获取清晰准确的频率信息；采用优化取值的参数K和，可更准确地识别信号的瞬时频率。
为定量评价信号瞬时频率的识别精度，定义均方根误差RMSE：
式中　 为瞬时频率识别值；为瞬时频率理论值；N为响应信号的长度。
当K取值准确（即K=3）时，表1对比了不同取值下VMD识别得到的信号各阶瞬时频率的RMSE值。由表1可知，当采用优化确定的参数（即=34000）时，识别得到的各阶瞬时频率的RMSE值最小，结果最准确。由此表明，采用优化取值参数的IVMD算法，可解决VMD算法参数随机取值导致瞬时频率识别不准、精度不高的问题。
3　数值模拟
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为验证IVMD-HT算法识别地震激励下时变结构瞬时频率的有效性和准确性，构建了一个具有时变刚度的4层剪切型框架结构，如图7所示。结构参数取值为：各层质量；各层阻尼系数；第3，4层刚度系数为常值，，，第1，2层刚度系数随时间变化，。
地震激励Q采用图8所示的El Centro地震波，激励时长为100 s，采样频率为250 Hz。采用四阶龙格-库塔（Runge-Kutta）进行结构地震响应分析，并添加10%的白噪声。
图9所示为El Centro地震激励下时变框架结构顶层的加速度响应及其傅里叶频谱和MUSIC功率谱。响应时长为100 s，其中5~20 s信号能量最大。图9（b）所示的傅里叶频谱中存在虚假峰值和波动干扰，峰值多而杂乱，难以准确确定分解层数K；图9（c）所示的MUSIC功率谱中可清楚地观察到4个峰值，由此确定分解模态数K=4。
确定分解模态数K后，设置SSA算法的种群个数为10，最大迭代次数为100，搜索范围为。基于综合目标函数F搜寻最优值，其迭代收敛曲线如图10所示。由图10可知，迭代23次时收敛，最优值，对应的值为10142.9。
采用确定的优化参数K=4，=10142.9，基于IVMD-HT算法识别得到地震激励下时变框架结构的瞬时频率，如图11所示。此外，图11中还给出了根据“冻结法”确定的瞬时频率理论值，即假设结构物理参数在每个时间间隔内保持不变，得到每个时间间隔内的瞬时频率。由图11可见，识别得到的框架结构瞬时频率与理论瞬时频率基本吻合，无模态混叠。其中，第2阶瞬时频率识别值与理论值吻合最好；第3，4阶瞬时频率在部分时刻存在突变，这是由于HT计算瞬时频率的公式误差造成的［11］。采用移动平均（Moving Average）法，即选定窗口长度为80，将窗口内的瞬时频率数据进行算数平均处理后，第3，4阶瞬时频率的移动均值与理论值基本一致。
当K=4时，表2给出了不同取值，VMD算法识别得到的地震激励下时变框架结构瞬时频率的RMSE值。由表2可知，当采用优化确定的参数（即=10142.9）时，识别得到的结构瞬时频率的RMSE值最小，结果最准确；当随机取值偏小（即=2000）时，识别的瞬时频率精度较低；当随机取值偏大（即=25000）时，模态混叠导致第4阶瞬时频率无法被识别。由此表明，采用优化取值参数的IVMD-HT算法，可用于识别地震激励下时变框架结构的瞬时频率，且结果最准确，精度最高。
4　试验验证
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为试验验证IVMD算法识别地震激励下时变结构瞬时频率的有效性和准确性，采用同济大学提供的12层钢筋混凝土框架模型模拟地震振动台试验数据［27］。模型缩尺比例为1/10，总高度3.6 m，每层高0.3 m，平面结构尺寸为0.6 m×0.6 m，如图12所示。模型试验采用了4种地震波，测试了61个工况，分别在模型底层、2层、4层、6层、8层、10层和12层的X，Y，Z方向布置了23个加速度传感器。模型的详细几何尺寸、物理参数和模拟地震振动台试验情况参见文献［27］。本研究仅分析Q1和Q2两种工况下模型在X向El Centro地震激励下结构顶层的加速度响应，Q1和Q2工况地震激励加速度峰值（PGA）分别为0.646g和0.775g，信号采样频率为255.102 Hz。
为更精确地识别地震激励下模型的瞬时频率，研究仅考虑模型的前3阶模态，采用截止频率为15 Hz的6阶切比雪夫Ⅰ型低通滤波器对模型地震响应信号进行重采样，重采样频率为100 Hz。图13所示为Q1和Q2两种工况下的模型顶层加速度响应及其傅里叶频谱和MUSIC功率谱。图13（b），（e）所示的傅里叶频谱中存在虚假峰值和波动干扰，峰值多而杂乱，难以准确确定分解模态数K；图13（c），（f）所示的MUSIC功率谱中可清楚地观察到3个峰值，由此确定分解模态数K=3。
确定分解模态数K后，设置SSA算法的种群个数为10，最大迭代次数为100，搜索范围为。基于综合目标函数F搜寻最优值，其迭代收敛曲线如图14所示。由图14可知，Q1和Q2工况下分别迭代57次和58次后收敛，最优值分别为4246.7和6125.7。
Q1和Q2工况下，分别采用确定的优化参数组合［K=3，=4246.7］和［K=3，=6125.7］，基于IVMD-HT算法识别得到12层框架模型的前3阶瞬时频率，如图15所示。由图15可见，Q1和Q2工况下框架模型前3阶瞬时频率成分被准确地识别。
由文献［27］可知，模型结构在模拟地震振动台试验的过程中出现了损伤，其频率具有时变特性。采用移动平均法（窗口长度为100）处理瞬时频率曲线，消除HT计算瞬时频率时出现的负频率问题，得到的瞬时频率移动均值（见图16）真实地反映了地震激励下模型结构频率的时变规律。由图16可见，Q2工况下识别得到的模型瞬时频率值低于Q1工况，且波动幅度较大。原因在于，相比Q1工况，Q2工况下模型损伤更严重，结构刚度下降更多，这与试验现象基本一致。由此表明，采用优化参数的IVMD算法，可识别地震激励下框架结构损伤引起的频率时变特征，对比不同损伤工况下结构瞬时频率的变化趋势即可得到结构的损伤程度。
5　结论
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提出了一种基于IVMD-HT算法识别时变结构地震响应瞬时频率的方法。采用模拟信号、地震激励下4层时变框架结构数值模拟响应和12层框架结构模型地震模拟振动台的试验数据验证了所提方法的有效性和实用性。主要结论如下：
（1） 相比傅里叶频谱和短时傅里叶谱，基于MUSIC功率谱确定的VMD模态数K更准确；基于整体正交系数和能量比系数构建的综合目标函数，结合SSA算法可更有效地优化确定二次惩罚因子。
（2） 基于优化确定K和参数的IVMD算法，解决了人为预设参数时VMD存在的模态混叠和噪声干扰等问题，提高了计算效率；提高了结合HT识别时变结构地震响应瞬时频率的精度。
（3） 应该指出，二次惩罚因子的优化搜索范围仍需通过试算人为确定。
基金
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  • 国家自然科学基金资助项目(52078486)
  • 国家自然科学基金资助项目(U1734208)
  • 中南大学中央高校基本科研业务费专项资金资助项目(2023ZZTS0410)
文献
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doi: 10.16385/j.cnki.issn.1004-4523.2024.03.005
  • 接收时间：2022-10-31
  • 首发时间：2026-02-10
  • 出版时间：2024-03-28
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  • 收稿日期：2022-10-31
  • 修回日期：2022-12-09
基金
国家自然科学基金资助项目(52078486)
国家自然科学基金资助项目(U1734208)
中南大学中央高校基本科研业务费专项资金资助项目(2023ZZTS0410)
作者信息
    中南大学土木工程学院，湖南　长沙　410075

通讯作者:

尚旭强（1996—），男，博士研究生。E-mail： 。
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https://castjournals.cast.org.cn/joweb/zdgcxb/CN/10.16385/j.cnki.issn.1004-4523.2024.03.005
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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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