WO2019169544A1 - 传感器数量不完备时结构模态识别的稀疏分量分析方法 - Google Patents

传感器数量不完备时结构模态识别的稀疏分量分析方法 Download PDF

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WO2019169544A1
WO2019169544A1 PCT/CN2018/078116 CN2018078116W WO2019169544A1 WO 2019169544 A1 WO2019169544 A1 WO 2019169544A1 CN 2018078116 W CN2018078116 W CN 2018078116W WO 2019169544 A1 WO2019169544 A1 WO 2019169544A1
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single source
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source point
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伊廷华
姚小俊
李宏男
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大连理工大学
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  • the invention belongs to the technical field of structural health monitoring data analysis, and relates to a method for structural modal identification when the number of sensors is incomplete, and specifically relates to a sparse component analysis method for structural modal recognition when the number of sensors is incomplete.
  • Modal recognition can acquire the dynamic characteristics of the structure and is one of the important techniques of structural dynamics.
  • the dynamic characteristics of the structure generally include the frequency, mode shape and damping ratio of the structure.
  • the process of identifying modal parameters from vibration data is consistent with the principle of blind source separation method. Therefore, the modal identification method based on blind source separation theory emerges. For field testing of large civil engineering structures, the number of installed sensors is sometimes less than the modal order to be identified, so the study of underdetermined blind source separation has very high practical value.
  • the second-order blind identification method for decomposing by constructing a Hankel matrix can reduce the number of sensors by using matrix expansion; the second-order blind identification method based on parallel factorization can keep the matrix decomposition unique in underdetermined conditions. Sex, thus solving the underdetermined problem.
  • these methods are mainly based on the cross-correlation matrix decomposition of the vibration signal, so the vibration signal needs to satisfy the stationarity assumption.
  • the vibration mode is firstly obtained by using the clustering characteristics of the vibration signal in the time-frequency domain, and then the modal response is reconstructed according to the sparse reconstruction method, and finally the frequency and damping ratio are obtained. Because it utilizes the sparse characteristics of the vibration signal in the time-frequency domain, it is not necessary to assume that the signal to be analyzed is a stationary signal, so it has greater advantages.
  • the sparse component analysis process includes single source point detection.
  • the purpose is to extract time-frequency points with only first-order modes participating in the contribution from all time-frequency points, thereby improving the accuracy of the mode estimation and reducing the amount of calculation.
  • the accuracy of single source point detection is low, which results in lower accuracy of mode shape estimation.
  • the modal response is sparsely reconstructed, the number of available sensors is small, which may result in the incomplete reconstruction of all modal responses, which may result in reduced or missing reproducibility. Therefore, it is necessary to improve the modal identification accuracy of the sparse component analysis method when the number of sensors is small.
  • a sparse component analysis method for structural modal identification when the number of sensors is incomplete the short-time Fourier transform of the structural acceleration response data is converted to the time-frequency domain, and only the first-order modality is detected by using the same direction of the real part and the imaginary part.
  • the time-frequency point of the contribution is the single source point, which is the initial result of the single source point; the initial result of the single source point detection is purified according to the single source point located near the peak of the power spectrum, and the single source point is clustered to obtain the mode shape.
  • Matrix constructing a generalized spectral matrix by using short-time Fourier transform coefficients, performing singular value decomposition on the generalized spectral matrix at a single source point, and treating the first singular value as the self-power spectrum of the single-order mode, by picking up the self-power
  • the peak of the spectrum acquires the frequencies of each order, and the inverse Fourier transform is used to convert the self-power spectrum to the time domain to extract the damping ratios of the respective orders;
  • the first step is to obtain the acceleration response of the structure at time t when the number of sensors is incomplete.
  • the single source point detection is based on the fact that the real and imaginary parts of the time-frequency coefficient have the same direction, using the following formula: Where Re ⁇ represents the real part of the extracted data, Im ⁇ represents the imaginary part of the extracted data, and ⁇ represents the threshold of single source point detection;
  • the detected single source point position is marked as (t k , ⁇ k ), and its value is:
  • Y(t k , ⁇ k ) [y 1 (t k , ⁇ k ), y 2 (t k , ⁇ k ),...,y l (t k , ⁇ k )] T ;
  • the third step is to average the logarithmic amplitudes of all sensor locations.
  • the frequency coefficients y j (t, ⁇ i ) are sequentially connected to obtain a sequence Where N represents the number of frequency points used by the short-time Fourier transform;
  • Average the logarithmic magnitude of all sensor locations Calculate each sequence The logarithmic magnitude of each element in it, where Sequence The ⁇ th element in the middle, Amp j ( ⁇ ) represents the ⁇ th element in the logarithmic magnitude of the jth sensor position; Obtain an average logarithmic magnitude;
  • polynomial regression is used to calculate the trend term of the average log magnitude sequence Amp mean , and then the trend term is removed to obtain the sequence. Correct Perform statistical analysis to calculate the number of samples falling into each statistical interval; when the cumulative number of samples reaches 90% of the total number of samples, set the sample value of the corresponding statistical interval as the threshold, and the time-frequency point set represented by the sample below the threshold Marked as ⁇ ; the initial result of the single source point detection obtained in the second step Y(t k , ⁇ k ) falls into the point of the set ⁇ , and the purified single source point is obtained.
  • the fifth step is to use the hierarchical clustering method for the purified single source point. Classify and calculate the cluster center of each class, which is the mode matrix;
  • the generalized spectral matrix is constructed by using the time-frequency coefficient Y(t, ⁇ ) in the first step:
  • t i represents the ith time; the superscript * represents the conjugate of the complex number; E[ ⁇ ] represents the expectation of extracting the data;
  • the seventh step single source location
  • the included frequency indicator is in Singular value decomposition is performed on the generalized spectral matrix H yy to obtain the first singular value sequence s 1 at each frequency;
  • the values of the various single source points obtained in the fifth step on the first singular value sequence s 1 are regarded as the self-power spectra of the respective mode modes, and the frequency of each order is obtained by picking the peak frequency of s 1 , The damping ratio is extracted by shifting s 1 through the inverse Fourier transform to the time domain.
  • the present invention provides a modal parameter identification method based on sparse component analysis, which improves the single source point detection result and directly extracts the frequency and damping ratio from the time-frequency coefficient to increase the number of sensors. The accuracy of modal recognition in less cases.
  • the mass matrix, stiffness matrix and damping matrix are:
  • the excitation is excited by Gaussian white noise, and the sampling frequency is 100 Hz. Acceleration time-sampling is performed on two of the node positions.
  • ⁇ represents the circular frequency.
  • t i denotes the ith time; the superscript * denotes the conjugate of the complex number; E[ ⁇ ] denotes the expectation of extracting the data.

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Abstract

本发明属于结构健康监测技术领域，提供传感器数量不完备时结构模态识别的稀疏分量分析方法。对结构加速度响应数据进行短时傅里叶变换转换到时频域，利用实部和虚部方向相同检测出仅有一阶模态参与贡献的时频点即单源点，作为单源点的初始结果；依据单源点位于功率谱峰值附近对单源点检测的初始结果进行提纯，并对单源点进行聚类获得振型矩阵；利用短时傅里叶变换系数构造广义谱矩阵，对单源点处的广义谱矩阵进行奇异值分解，将第一个奇异值视为单阶模态的自功率谱，通过拾取自功率谱的峰值获取各阶频率，利用逆傅里叶变换将自功率谱转换到时域提取各阶阻尼比。本方法在传感器不完备的情况下获取结构的模态参数，提高稀疏分量分析方法的识别准确性。

Description

传感器数量不完备时结构模态识别的稀疏分量分析方法 技术领域
本发明属于结构健康监测数据分析技术领域，涉及传感器数量不完备时结构模态识别的方法，具体为传感器数量不完备时结构模态识别的稀疏分量分析方法。
背景技术
模态识别能够获取结构的动力特性，是结构动力学的重要技术之一。结构的动力特性一般包括结构的频率、振型和阻尼比。从振动数据中识别模态参数的过程与盲源分离方法的原理一致，因此基于盲源分离理论的模态识别方法应运而生。对于大型土木工程结构现场测试，安装的传感器个数有时会少于待识别的模态阶数，因此欠定盲源分离问题的研究具有非常高的实用价值。
针对欠定盲源分离问题，目前研究者们已提出了多种方法。例如，通过构造Hankel矩阵进行分解的二阶盲辨识方法，能够利用矩阵扩维降低对传感器个数的要求；基于平行因子分解法的二阶盲辨识方法，能够在欠定情况保持矩阵分解的唯一性，从而解决欠定问题。然而这些方法主要基于振动信号的互相关矩阵分解，因此振动信号需要满足平稳性假定。而基于稀疏分量分析的方法，首先利用振动信号在时频域内的聚类特性获得振型矩阵，再依据稀疏重构方法重构出各阶模态响应，最终得到频率和阻尼比。因其利用的是振动信号在时频域的稀疏特性，无需假定待分析信号为平稳信号，因此具有较大的优越性。
稀疏分量分析过程包含单源点检测，其目的是从所有时频点中抽取仅有一阶模态参与贡献的时频点，从而提高振型估计的精度，并减小计算量。然而，当传感器个数的较少时，单源点检测的精度较低，会导致振型估计的准确性较低。此外，模态响应在稀疏重构时，可用传感器个数较少会导致不能完全重构 出所有的模态响应，从而会导致重构的精度降低或者遗漏。因此，提高稀疏分量分析方法在传感器个数较少时的模态识别精度十分有必要。
发明内容
本发明的目的是提供一种改进的基于稀疏分量分析的模态识别方法，提高稀疏分量分析方法在传感器个数较少情况下的模态识别的精度。
本发明的技术方案：
一种传感器数量不完备时结构模态识别的稀疏分量分析方法，对结构加速度响应数据进行短时傅里叶变换转换到时频域，利用实部和虚部方向相同检测出仅有一阶模态参与贡献的时频点即单源点，作为单源点的初始结果；依据单源点位于功率谱峰值附近对单源点检测的初始结果进行提纯，并对单源点进行聚类获得振型矩阵；利用短时傅里叶变换系数构造广义谱矩阵，对单源点处的广义谱矩阵进行奇异值分解，将第一个奇异值视为单阶模态的自功率谱，通过拾取自功率谱的峰值获取各阶频率，利用逆傅里叶变换将自功率谱转换到时域提取各阶阻尼比；
分为估算振型矩阵、提取频率和阻尼比，具体步骤如下：
(一)估算振型矩阵
第一步，获取传感器数量不完备时结构在t时刻的加速度响应
Y(t)＝[y 1(t),y 2(t),…,y l(t)] T；采用短时傅里叶变换将时域加速度响应变换到时频域，其表达式变为Y(t,ω)＝[y 1(t,ω),y 2(t,ω),…,y l(t,ω)]，其中l为传感器的个数，ω表示圆频率；
第二步，获取单源点检测的初始结果并标记；
单源点检测的依据为时频系数的实部和虚部具有相同的方向，采用以下公 式：
Figure PCTCN2018078116-appb-000001
其中Re{·}表示所提取数据的实部，Im{·}表示所提取数据的虚部，Δβ表示单源点检测的阈值；
检测出的单源点位置标记为(t k,ω k)，其值为：
Y(t k,ω k)＝[y 1(t k,ω k),y 2(t k,ω k),…,y l(t k,ω k)] T；
第三步，对所有传感器位置的对数幅值进行平均
对所有传感器位置的时频系数做相同的如下处理：第j个传感器位置的时频系数为y j(t,ω)，将各频率截面ω i,i＝1,2,,N对应的时频系数y j(t,ω i)依次连接得到序列
Figure PCTCN2018078116-appb-000002
其中N表示短时傅里叶变换采用的频点数；
对所有传感器位置的对数幅值进行平均：采用
Figure PCTCN2018078116-appb-000003
计算各序列
Figure PCTCN2018078116-appb-000004
中每个元素的对数幅值，其中
Figure PCTCN2018078116-appb-000005
为序列
Figure PCTCN2018078116-appb-000006
中的第τ个元素，Amp j(τ)表示第j个传感器位置的对数幅值中的第τ个元素；
Figure PCTCN2018078116-appb-000007
得到平均对数幅值；
第四步，采用多项式回归计算平均对数幅值序列Amp mean的趋势项，然后将趋势项去除，得到序列
Figure PCTCN2018078116-appb-000008
对
Figure PCTCN2018078116-appb-000009
进行统计分析，计算落入各统计区间的样本个数；当累积样本个数达到总样本个数90％时，将相应统计区间的样本值设为阈值，阈值以下的样本代表的时频点集合标记为Ω；剔除第二步得到的单源点检测的初始结果Y(t k,ω k)中落入集合Ω的点，得到提纯的单源点
Figure PCTCN2018078116-appb-000010
第五步，使用层次聚类方法对提纯后的单源点
Figure PCTCN2018078116-appb-000011
进行分类，并计算各个类的聚类中心，即为振型矩阵；
(二)提取频率和阻尼比
第五步，利用第一步中的时频系数Y(t,ω)构造广义谱矩阵：
Figure PCTCN2018078116-appb-000012
式中：
Figure PCTCN2018078116-appb-000013
t i表示第i个时刻；上标*表示求复数的共轭；E[·]表示提取数据的期望；
第七步，单源点位置
Figure PCTCN2018078116-appb-000014
包含的频率指标为
Figure PCTCN2018078116-appb-000015
在
Figure PCTCN2018078116-appb-000016
处对广义谱矩阵H yy进行奇异值分解，得到各频率处的第一个奇异值序列s 1；
第八步，将第五步得到的各类单源点在第一个奇异值序列s 1上的值视为各阶模态的自功率谱，通过拾取s 1的峰值频率得到各阶频率，通过将s 1经过逆傅里叶变换转到时域提取阻尼比。
本发明的有益效果：本发明提供了一种基于稀疏分量分析的模态参数识别方法，通过对单源点检测结果进行提纯，并直接从时频系数中提取频率和阻尼比，提高传感器个数较少情况下模态识别的准确性。
具体实施方式
以下结合技术方案，进一步阐明本发明的实施方式。
取一个三自由度弹簧质量块***，质量矩阵、刚度矩阵和阻尼矩阵分别为：
Figure PCTCN2018078116-appb-000017
激励采用高斯白噪声激励，采样频率为100Hz，对其中两个节点位置进行加速度时程采样。
一、振型矩阵估计
(1)采样得到结构在t时刻的加速度响应Y(t)＝[y 1(t),y 2(t)] T。采用短时傅里叶变换将时域的加速度响应Y变换到时频域，其表达式变为 Y(t,ω)＝[y 1(t,ω),y 2(t,ω)]，这里ω表示圆频率。
(2)依据
Figure PCTCN2018078116-appb-000018
获取单源点检测初始结果，其中Re{·}表示提取数据的实部，Im{·}表示提取数据的虚部。检测出的单源点位置记为(t k,ω k)，则单源点处的值为：Y(t k,ω k)＝[y 1(t k,ω k),y 2(t k,ω k)] T。
(3)第1个传感器位置的时频系数为y 1(t,ω)，将各频率截面ω i,(i＝1,2,…,N)对应的时频系数y 1(t,ω i)依次连接得到序列
Figure PCTCN2018078116-appb-000019
对y 2(t,ω)做相同的处理。采用
Figure PCTCN2018078116-appb-000020
计算各序列
Figure PCTCN2018078116-appb-000021
中每个元素的对数幅值，其中
Figure PCTCN2018078116-appb-000022
为序列
Figure PCTCN2018078116-appb-000023
中的第τ个元素，Amp j(τ)表示第j个传感器位置的对数幅值中的第τ个元素。将两个传感器位置的对数幅值进行平均：
Figure PCTCN2018078116-appb-000024
得到平均对数幅值。
(4)采用多项式回归计算平均对数幅值序列Amp mean的趋势项，并将趋势项去除，得到序列
Figure PCTCN2018078116-appb-000025
对
Figure PCTCN2018078116-appb-000026
进行统计分析，计算落入各统计区间的样本个数。当累积样本个数达到总样本数的90％时，将相应统计区间的样本值设为阈值，阈值以下的样本代表的时频点集合标记为Ω。剔除步骤(2)得到的单源点检测的初始结果Y(t k,ω k)中落入集合Ω的点，得到提纯的单源点
Figure PCTCN2018078116-appb-000027
(5)使用层次聚类方法将提纯后的单源点
Figure PCTCN2018078116-appb-000028
分成3类，并计算各个类的聚类中心，得到归一化的振型矩阵：
Figure PCTCN2018078116-appb-000029
二、提取频率和阻尼比
(6)利用步骤(1)中的时频系数Y(t,ω)构造广义谱矩阵：
Figure PCTCN2018078116-appb-000030
式中：
Figure PCTCN2018078116-appb-000031
t i表示第i个时刻；上标*表示求复数的共轭；E[·]表示提取数据的期望。
(7)在各频率指标
Figure PCTCN2018078116-appb-000032
处对广义谱矩阵H yy进行奇异值分解，得到第一个奇异值序列s 1。
(8)步骤(5)得到的各类单源点在s 1上的值视为各阶模态的自功率谱，通过拾取s 1的峰值频率得到各阶频率，通过将s 1经过逆傅里叶变换转到时域提取阻尼比。频率的识别结果为：f n1＝3.2959Hz，f n2＝10.8099Hz，f n3＝11.7813Hz。阻尼比的识别结果为：ξ 1＝0.0474，ξ 2＝0.0290，ξ 3＝0.0112。

Claims (1)

  1. 一种传感器数量不完备时结构模态识别的稀疏分量分析方法，分为估算振型矩阵、提取频率和阻尼比，其特征在于，步骤如下：
    (一)估算振型矩阵
    第一步，获取传感器数量不完备时结构在t时刻的加速度响应Y(t)＝[y l(t),y 2(t),…,Y l(t)I T；采用短时傅里叶变换将时域加速度响应变换到时频域，其表达式变为Y(t，ω)＝[y 1(t，ω)，y 2(t，ω)，…，y l(t，ω)]，其中l为传感器的个数，ω表示圆频率；
    第二步，获取单源点检测的初始结果并标记；
    单源点检测的依据为时频系数的实部和虚部具有相同的方向，采用以下公式：
    Figure PCTCN2018078116-appb-100001
    其中Re{·}表示所提取数据的实部，Im{·}表示所提取数据的虚部，Δβ表示单源点检测的阈值；
    检测出的单源点位置标记为(t k,ω k)，其值为：
    Y(t k，ω k)＝y 1(t k，ω k)，y 2(t k，ω k)，…y l(t k，ω k)] T；
    第三步，对所有传感器位置的对数幅值进行平均
    对所有传感器位置的时频系数做相同的如下处理：第j个传感器位置的时频系数为y j(t,ω)，将各频率截面ω i,i＝1,2,…,N对应的时频系数y j(t,ω i)依次连接得到序列
    Figure PCTCN2018078116-appb-100002
    其中N表示短时傅里叶变换采用的频点数；
    对所有传感器位置的对数幅值进行平均：采用
    Figure PCTCN2018078116-appb-100003
    计算各序列
    Figure PCTCN2018078116-appb-100004
    j＝1，2，…，l中每个元素的对数幅值，其中
    Figure PCTCN2018078116-appb-100005
    为序列
    Figure PCTCN2018078116-appb-100006
    中的第τ个元素，Amp j(τ)表示第j个传感器位置的对数幅值中的第τ个元素；
    Figure PCTCN2018078116-appb-100007
    得到平均对数幅值；
    第四步，采用多项式回归计算平均对数幅值序列Amp mean的趋势项，然后将 趋势项去除，得到序列
    Figure PCTCN2018078116-appb-100008
    对
    Figure PCTCN2018078116-appb-100009
    进行统计分析，计算落入各统计区间的样本个数；当累积样本个数达到总样本个数90％时，将相应统计区间的样本值设为阈值，阈值以下的样本代表的时频点集合标记为Ω；剔除第二步得到的单源点检测的初始结果Y(t k,ω k)中落入集合Ω的点，得到提纯的单源点
    Figure PCTCN2018078116-appb-100010
    第五步，使用层次聚类方法对提纯后的单源点
    Figure PCTCN2018078116-appb-100011
    进行分类，并计算各个类的聚类中心，即为振型矩阵；
    (二)提取频率和阻尼比
    第五步，利用第一步中的时频系数Y(t,ω)构造广义谱矩阵：
    Figure PCTCN2018078116-appb-100012
    式中：
    Figure PCTCN2018078116-appb-100013
    t i表示第i个时刻；上标*表示求复数的共轭；E[·]表示提取数据的期望；
    第七步，单源点位置
    Figure PCTCN2018078116-appb-100014
    包含的频率指标为
    Figure PCTCN2018078116-appb-100015
    在
    Figure PCTCN2018078116-appb-100016
    处对广义谱矩阵H yy进行奇异值分解，得到各频率处的第一个奇异值序列s 1；
    第八步，将第五步得到的各类单源点在第一个奇异值序列s 1上的值视为各阶模态的自功率谱，通过拾取s 1的峰值频率得到各阶频率，通过将s 1经过逆傅里叶变换转到时域提取阻尼比。
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