Modal identification method for non-proportionally damped structures based on extended sparse component analysis

Abstract

Data analysis for structural health monitoring relating to a method of modal identification for structures with non-proportional damping based on extended sparse component analysis. Hilbert transform constructs analytical signal of acceleration response. Analytical signal is transformed into time-frequency domain using short-time-Fourier transform. The criterion is taken as the correlation coefficient of adjacent frequency points is close to 1. Points contributed by only one mode are detected from the time-frequency plane. Phases calculated at single-source-points are used to remove local outliers through local outlier factor method. Amplitudes of complex-valued mode shapes are estimated by Hierarchical clustering of amplitudes for time-frequency coefficients at single-source-points. Averaged phases of grouped single-source-points are estimated phases of complex-valued mode shapes. Finally, complex-valued mode shapes are acquired. Modal responses are estimated by sparse reconstruction method. This method extends application range of sparse component analysis method, and can identify complex modes of non-proportionally damped structures.

Claims

1. A modal identification method for non-proportionally damped structures based on extended sparse component analysis, wherein the steps are as follows: step 1: constructing the analytical signals of the accelerations the acceleration of t instant is sampled as x(t)=[x.sub.1(t), x.sub.2(t), . . . , x.sub.m(t)].sup.T, where m is a number of sensors; [].sup.T represents the transpose of a matrix; the acceleration x(t) is transformed to {circumflex over (x)}(t) by Hilbert transform, which is represented by $\hat{x}\left(t\right)=\frac{1}{}{}_{-}^{+}\frac{x\left(\right)}{t-}d;$ an analytical signal of the acceleration is constructed by {tilde over (x)}(t)=x(t)+j{circumflex over (x)}(t), where j is the imaginary unit; step 2: obtaining the time-frequency representation of analytical signals the analytical signal {tilde over (x)}(t) of the acceleration is transformed into time-frequency domain using STFT; a time-frequency domain signal is represented as {tilde over (x)}(t,f)=[{tilde over (x)}.sub.1(t,f), {tilde over (x)}.sub.2(t,f), . . . , {tilde over (x)}.sub.m(t,f)].sup.T and f is frequency index; step 3: detecting SSPs using average correlation coefficients; a range of frequencies regarding a certain time period t.sub.k in the time-frequency plane are denoted as an analysis zone F; the correlation coefficient between the time-frequency coefficients of two sensor locations a and b is calculated using $R_{\mathrm{ab}}\left(t_k,F\right)=\frac{r_{\mathrm{ab}}\left(t_k,F\right)}{\sqrt{r_{\mathrm{aa}}\left(t_k,F\right)r_{\mathrm{aa}}\left(t_k,F\right)}},$ where $r_{\mathrm{ab}}\left(t_k,F\right)=\underset{fF}{.Math.}.Math.{\tilde{x}}_a\left(t_k,f\right).Math.{\tilde{x}}_b\left(t_k,f\right).Math.,$ a,b{1, . . . , m}; the correlation coefficients calculated from all sensor locations are averaged to get an average correlation coefficient $\stackrel{\_}{R}\left(t_k,F\right)=\frac{1}{M}\underset{a=1}{\overset{m}{.Math.}}\underset{b=1,ba}{\overset{m}{.Math.}}{R}_{\mathrm{ab}}\left(t_k,F\right)$ where M=C.sub.m.sup.2; C.sub.m.sup.2 represents the number of combination forms when picking two different unordered samples from m samples; the SSP detection criterion is denoted as R(t.sub.k,F)1, where is a threshold value; if the zone (t.sub.k,F) satisfies the detection criterion, the points in this zone are all marked as SSPs; the detected SSPs are marked as (t,f); step 4: removing the local outliers of the SSPs through local outlier factor method a first sensor location is chosen as the reference; the phase differences between each sensor location and the first sensor location are calculated by (t,f)=[.sub.1(t,f), . . . , .sub.m(t,f)], where .sub.k(t,f)=.sub.{tilde over (x)}.sub.1.sub.(t,f).sub.{tilde over (x)}.sub.k.sub.(t,f), k{1, . . . , m}; then the local outlier factors for the phase difference vector at (t,f) are calculated using local outlier factor method; the SSPs where the local outlier factors exceed 1 are removed; the updated SSPs are denoted as ({circumflex over (t)},{circumflex over (f)}); step 5: clustering the updated SSPs to obtain the amplitudes of the complex-valued mode shapes the amplitudes of the time-frequency coefficients at ({circumflex over (t)},{circumflex over (f)}) are grouped by the Hierarchical clustering method; the clustering centers are the amplitudes of each complex-valued mode shape which are denoted as ||=[|.sub.1|, |.sub.2|, . . . , |.sub.n|]; is the modal matrix and .sub.i, i=1, . . . , n is the mode shape vector; n is the number of modes; each group of SSPs after clustering are denoted as ({circumflex over (t)}.sub.1,{circumflex over (f)}.sub.1), . . . , ({circumflex over (t)}.sub.n,{circumflex over (f)}.sub.n); step 6: averaging the phases of each grouped SSPs the phase differences for each grouped SSPs are obtained as ({circumflex over (t)}.sub.1,{circumflex over (f)}.sub.1), . . . , ({circumflex over (t)}.sub.n,{circumflex over (f)}.sub.n); the mean values of phase differences for each grouped SSPs are calculated and denoted as ({circumflex over (t)}.sub.1,{circumflex over (f)}.sub.1), . . . , ({circumflex over (t)}.sub.n,{circumflex over (f)}.sub.n); step 7: assembling the complex-valued mode shapes using the estimated amplitudes and phases of the mode shapes each row of the modal matrix is normalized by dividing the first row to obtain the normalized matrix ||=[|.sub.1|, |.sub.2|, . . . , |.sub.n|]; finally, the complex-valued modal matrix is estimated as =[|.sub.1|e.sup.j({circumflex over (t)}.sup.1.sup.,{circumflex over (f)}.sup.1.sup.), |.sub.2|e.sup.j({circumflex over (t)}.sup.2.sup.,{circumflex over (f)}.sup.2.sup.), . . . , |.sub.n|e.sup.j({circumflex over (t)}.sup.n.sup.,{circumflex over (f)}.sup.n.sup.)].

Description

DETAILED DESCRIPTION

(1) The present invention is further described below in combination with the technical solution.

(2) The numerical example of a 3 degree-of-freedom mass-spring model is employed. The mass, stiffness, and damping matrix are given as follows:

(3) $M=\left[\begin{array}{ccc}3& & \\ & 2& \\ & & 1\end{array}\right],K=\left[\begin{array}{ccc}4& -2& \\ -2& 4& -2\\ & -2& 10\end{array}\right],C=\left[\begin{array}{ccc}0.1856& 0.2290& -0.9702\\ 0.2290& 0.0308& -0.0297\\ -0.9702& -0.0297& 0.1241\end{array}\right]$

(4) The initial condition is given as x(0)=[1, 0, 0].sup.T. Then the free vibration responses at the three nodes of the system are acquired. The acceleration responses of the three nodes are sampled with the sampling rate 100 Hz.

(5) Step 1: Constructing the analytical signals of the accelerations

(6) The acceleration of t instant is sampled as x(t)=[x.sub.1(t),x.sub.2(t),x.sub.3(t)].sup.T, where [].sup.T represents the transpose of a matrix. The acceleration is transformed to {circumflex over (x)}(t) by Hilbert transform, which is represented by

(7) $\hat{x}\left(t\right)=\frac{1}{}{}_{-}^{+}\frac{x\left(\right)}{t-}d.$
The analytical signal of the acceleration is constructed by {tilde over (x)}(t)=x(t)+j{circumflex over (x)}(t), where j is the imaginary unit.

(8) Step 2: Obtaining the time-frequency representation of analytical signals

(9) The analytical signal {tilde over (x)}(t) of the acceleration is transformed into time-frequency domain using STFT. The time-frequency domain signal is represented as {tilde over (x)}(t,f)=[{tilde over (x)}.sub.1(t,f), {tilde over (x)}.sub.2(t,f), {tilde over (x)}.sub.3(t,f)].sup.T and f is frequency index.

(10) Step 3: Detecting SSPs using average correlation coefficients.

(11) A range of frequencies regarding a certain time period in the time-frequency plane are denoted as an analysis zone F. The correlation between the time-frequency coefficients of two sensor locations a and b is calculated using

(12) $r_{\mathrm{ab}}\left(t_k,F\right)=\underset{fF}{.Math.}.Math.{\tilde{x}}_a\left(t_k,f\right).Math.{\tilde{x}}_b\left(t_k,f\right).Math.,$
a,b{1, 2, 3}. The correlation coefficient of r.sub.ab(t.sub.k,F) is calculated by

(13) 0$R_{\mathrm{ab}}\left(t_k,F\right)=\frac{r_{\mathrm{ab}}\left(t_k,F\right)}{\sqrt{r_{\mathrm{aa}}\left(t_k,F\right)r_{\mathrm{aa}}\left(t_k,F\right)}}.$
The sensor locations are combined in pairs using any two locations and the correlation coefficients are calculated for the corresponding locations. The correlation coefficients for all the combination forms are averaged to get the average correlation coefficient R(t.sub.k,F). The SSP detection criterion is denoted as R(t.sub.k,F)1, where =10.sup.4. If the zone (t.sub.k,F) satisfies the detection criterion, the points in this zone are all marked as SSPs. The detected SSPs are marked as (t,f).

(14) Step 4: Removing the local outliers of the SSPs through local outlier factor (LOF) method

(15) The first sensor location is chosen as the reference. The phase differences between each sensor location and the first sensor location are calculated by (t,f)=[.sub.1(t,f), . . . , .sub.3(t,f)], where .sub.k(t,f)=.sub.{tilde over (x)}.sub.1.sub.(t,f).sub.{tilde over (x)}.sub.k.sub.(t,f), k{1, . . . , 3}. Then the local outlier factors for the phase difference vector at (t,f) are calculated using LOF method. The SSPs where the local outlier factors exceed 1 are removed. The updated SSPs are denoted as ({circumflex over (t)},{circumflex over (f)}).

(16) Step 5: Clustering the updated SSPs to obtain the amplitudes of the complex-valued mode shapes

(17) The amplitudes of the time-frequency coefficients at ({circumflex over (t)},{circumflex over (f)}) are grouped by the Hierarchical clustering method. The clustering centers are the amplitudes of each complex-valued mode shape which are denoted as ||=[|.sub.1|, |.sub.2|, |.sub.3|]. is the modal matrix and .sub.ii=1, . . . , 3 is the mode shape vector. Each group of SSPs after clustering are denoted as ({circumflex over (t)}.sub.1,{circumflex over (f)}.sub.1), . . . , ({circumflex over (t)}.sub.n,{circumflex over (f)}.sub.n).

(18) Step 6: Averaging the phases of each grouped SSPs

(19) The phase differences for each grouped SSPs are obtained as ({circumflex over (t)}.sub.1,{circumflex over (f)}.sub.1), . . . , ({circumflex over (t)}.sub.n,{circumflex over (f)}.sub.n). The mean values of phase differences for each grouped SSPs are calculated and denoted as ({circumflex over (t)}.sub.1,{circumflex over (f)}.sub.1), . . . , ({circumflex over (t)}.sub.n,{circumflex over (f)}.sub.n).

(20) Step 7: Assembling the complex-valued mode shapes using the estimated amplitudes and phases of the mode shapes

(21) Each row of the modal matrix is normalized by dividing the first row to obtain the normalized matrix

(22) $.Math.\stackrel{\_}{}.Math.=\left[.Math.{\stackrel{\_}{}}_1.Math.,.Math.{\stackrel{\_}{}}_2.Math.,.Math.{\stackrel{\_}{}}_3.Math.\right]=\left[\begin{array}{ccc}1& 1& 1\\ 0.9904& 1.4734& 0.9442\\ 8.0556& 0.4437& 0.2228\end{array}\right].$
Finally, the complex-valued modal matrix is estimated as

(23) $\begin{array}{c}=\left[.Math.{\stackrel{\_}{}}_1.Math.e^{j\stackrel{\_}{}\left({\hat{t}}_1,{\hat{f}}_1\right)},.Math.{\stackrel{\_}{}}_2.Math.e^{j\stackrel{\_}{}\left({\hat{t}}_2,{\hat{f}}_2\right)},.Math.{\stackrel{\_}{}}_n.Math.e^{j\stackrel{\_}{}\left({\hat{t}}_3,{\hat{f}}_3\right)}\right]\\ =\left[\begin{array}{ccc}1.0000& 1.0000& 1.0000\\ -0.1915-0.9717j& -1.4713+0.0789j& 0.9440+0.0197j\\ 0.6035+8.0329j& -0.3872+0.2165j& 0.2014+0.0954j\end{array}\right].\end{array}$

(24) Step 8: Identifying the frequency and damping ratio

(25) The inverse of complex modal matrix is calculated. The modal responses are calculated by q(t)=.sup.1{tilde over (x)}(t) where .sup.1 is the inverse matrix of . If the accelerations of node 1 and node 2 are adopted, the number of sensors is less than the number of modes. The modal responses are estimated using the cost function

(26) $H_{}\left(q\right)=L-\underset{i=1}{\overset{L}{.Math.}}\exp \left(\frac{-{.Math.q_i.Math.}^2}{2{}^2}\right)$
under the constraint condition {tilde over (x)}=q. The cost function is minimized by the graduated nonconvex optimization method to estimate the modal responses. q.sub.i is the i-th element in q. L is the total number of the elements in q. is the decrement factor, which is given as =[0.5, 0.4, 0.3, 0.2, 0.1]. The final identified modal frequencies are 0.1360 Hz, 0.2472 Hz, and 0.5001 Hz. Damping ratios of each mode are 3.1898%, 1.6403%, and 1.7333%.