A METHOD OF ESTIMATING THE NUMBER OF MODES FOR THE SPARSE COMPONENT ANALYSIS BASED MODAL IDENTIFICATION

Abstract

Data analysis for structural health monitoring, relating to a method of estimating the number of modes for sparse component analysis based structural modal identification. First, structural responses are transformed into time-frequency domain using short-time Fourier transform method. Single-source-point detection method is applied to the time-frequency coefficients to pick out the single-source-points where only one mode makes contribution. The single-source-point vectors are normalized to the upper half unit circle. Three statistics are given to analyze the statistical property. The suggested number of subintervals is given. Through counting, the approximate probabilities in subintervals are calculated and then smoothed through the weighted average procedure. The local maximum values of the averaged probability curve are detected and the number of active modes is equal to the number of local maximum values.

Claims

1. A method of estimating the number of modes for the sparse component analysis based modal identification, wherein the steps are as follows: step 1: transforming sampled accelerations into time-frequency domain (1) accelerations of the structure are sampled and denoted as Acc(t)=[acc.sub.1(t), acc.sub.2(t), . . . , acc.sub.n(t)].sup.T, where n is number of sensors; then responses Acc(t) are transformed into time-frequency domain through short-time Fourier transform, which is noted as Acc(t, f); f is the frequency index; (2) detecting single-source-points; a single-source-point detection method is applied to select time-frequency points where only one mode is dominant; a principle of single-source-point detection is that directions formed by the real and the imaginary parts of time-frequency coefficients will not exceed a very small angle, which is called a threshold and noted as ; based on this property, the single-source-point detection can be accomplished through: $.Math.\frac{\mathrm{Re}.Math.{\left\{\mathrm{Acc}\left(t,f\right)\right\}}^T.Math.\mathrm{Im}.Math.\left\{\mathrm{Acc}\left(t,f\right)\right\}}{.Math.\mathrm{Re}.Math.\left\{\mathrm{Acc}\left(t,f\right)\right\}.Math..Math..Math.\mathrm{Im}.Math.\left\{\mathrm{Acc}\left(t,f\right)\right\}.Math.}.Math.>\cos \left(.Math..Math.\right)$ where Re{} and Im{} are the real and imaginary parts of a vector, respectively; detected single-source-points are marked as (t.sub.j, f.sub.j); therefore, the time-frequency coefficients of the single-source-points are denoted as ACC(t.sub.j, f.sub.j)=[Acc.sub.1(t.sub.j, f.sub.j), Acc.sub.2(t.sub.j, f.sub.j), . . . , Acc.sub.n(t.sub.j, f.sub.j)].sup.T; step 2: identifying number of active modes (3) two sensor locations k and l are chosen arbitrarily and corresponding single-source-points of these two locations are Acc.sub.k(t.sub.j,f.sub.j) and Acc.sub.l(t.sub.j, f.sub.j); (4) first, single-source-points of the locations k and l are arranged in column vectors, respectively; then, the single-source-point vectors are denoted as Acc1=[Acc.sub.k, Acc.sub.l].sup.T; Acc1 should be normalized to the upper half unit circle using: $A.Math..Math.\mathrm{cc}.Math..Math.1.Math.\left(i\right)=\{\begin{array}{c}\frac{\mathrm{Acc}.Math..Math.1}{.Math.\mathrm{Acc}.Math..Math.1.Math.\left(i\right).Math.},A.Math..Math.\mathrm{cc}.Math..Math.1_k.Math.\left(i\right)0\\ -\frac{A.Math..Math.\mathrm{cc}.Math..Math.1}{.Math.A.Math..Math.\mathrm{cc}.Math..Math.1.Math.\left(i\right).Math.},\mathrm{Acc}.Math..Math.1_k.Math.\left(i\right)<0\end{array}$ where cc1(i) is normalized data of the i-th row vector in Acc1; (5) if the two elements in cc1 are treated as coordinates of a point in the Cartesian coordinates, coordinates of the arbitrary points are cc1(i)=[cc.sub.k(i), cc.sub.l(i)].sup.T, i=(1, 2, . . . , J), where J is the total number of points in cc1; three distance based statistics are constructed by Euclidean distance and Chebyshev distance between points in cc1 and the left end point [1, 0].sup.T, and the cosine distance between the points in cc1 and the center point [0,0].sup.T; the Euclidean distance is formulated as follows:
dist.sub.E(i)={square root over ((Acc.sub.k(i)+1).sup.2+Acc.sub.l(i).sup.2)} the Chebyshev distance is formulated as follows:
dist.sub.C(i)=max(|Acc.sub.k(i)+1|, |Acc.sub.l(i)|) the cosine distance is formulated as follows: ${\mathrm{dist}}_{}\left(i\right)=\arccos \left(\frac{{\mathrm{Acc}}_k\left(i\right)}{\sqrt{{{\mathrm{Acc}}_k\left(i\right)}^2+A.Math..Math.{{\mathrm{cc}}_i\left(i\right)}^2}}\right)$ the final statistic dist is determined from dist.sub.E, dist.sub.C and dist.sub.; (6) the statistic dist is sorted in descending order and then the sorted data is differentiated as (dist); the difference sequence is counted; when the accumulated sample size reaches 95% of the total sample size, a threshold is set and the samples beyond the threshold are removed; the remainder difference sequence is averaged to obtain the mean value .sub.mean; the maximum of the remainder difference sequence is .sub.max; the relation between the number and the length of the statistical intervals is: $P=\frac{\max \left(\mathrm{dist}\right)-\min \left(\mathrm{dist}\right)}{}$ where max() and min() are the maximum and minimum of a vector, respectively; when is equal to the mean value .sub.mean, the number of statistical subintervals is at a maximum and is denoted as P.sub.max; when is equal to the maximum value .sub.max, the number of statistical subintervals is at a minimum and is denoted as P.sub.min; therefore, the range for the suggested number of statistical subintervals is given as P[P.sub.min,P.sub.max]; (7) the statistical interval [max (dist)min(dist)] is divided into P subintervals with equal length; the number of samples in each subinterval is counted and denoted as p.sub.i, i=(1, 2, . . . , P); the approximate probability in each subinterval is calculated using Pr(i)=p.sub.i/P; the approximate probability curve is obtained through the weighted average procedure:
{circumflex over (P)}r(i)= 1/16(P(i2)+4P(i1)+6/P(i)+4P(i+1)+P(i+2)) where {circumflex over (P)}r is the approximate probability curve; (8) local maximum values of {circumflex over (P)}r are picked out and the number of active modes is equal to the number of local maximum values.

Description

DETAILED DESCRIPTION

[0026] The present invention is further described below in combination with the technical solution.

[0027] The numerical example of a 6 degree-of-freedom in-plane lumped-mass model is employed. The mass for the first floor is 3 kg, and the masses for the other floors are 1 kg. The stiffness for the first floor is 2 kN/m, and the stiffnesses for the rest floors are 1 kN/m. The Rayleigh damping is adopted as C=M+K, where the factors are =0.05 and =0.004. The model is excited in the sixth floor by an impulse, and the free decayed response is sampled with a sampling rate of 100 Hz. The procedures are described as follows:

[0028] (1) The accelerations of the structure are sampled and denoted as Acc(t)=[acc.sub.1(t), acc.sub.2(t), . . . , acc.sub.6(t)].sup.T. Then, the responses Acc(t) are transformed into time-frequency domain through short-time Fourier transform, which are noted as Acc(t, f)=[acc.sub.1(t, f), acc.sub.2(t, f), . . . , acc.sub.6(t, f)].sup.T. f is the frequency index;

[0029] (2) The single-source-points are detected using:

[00005]$.Math.\frac{\mathrm{Re}.Math.{\left\{\mathrm{Acc}\left(t,f\right)\right\}}^T.Math.\mathrm{Im}.Math.\left\{\mathrm{Acc}\left(t,f\right)\right\}}{.Math.\mathrm{Re}.Math.\left\{\mathrm{Acc}\left(t,f\right)\right\}.Math..Math..Math.\mathrm{Im}.Math.\left\{\mathrm{Acc}\left(t,f\right)\right\}.Math.}.Math.>\cos \left(.Math..Math.\right)$

where Re{} and Im{} are the real and imaginary parts of a vector, respectively; is 2. The detected single-source-points are marked as (t.sub.j, f.sub.j). Therefore, the time-frequency coefficients of the single-source-points are denoted as Acc(t.sub.j, f.sub.j)=[Acc.sub.1(t.sub.j, f.sub.j), Acc.sub.2(t.sub.j, f.sub.j), . . . , Acc.sub.6 (t.sub.j,f.sub.j)].sup.T;

[0030] (3) Two sensor locations 5 and 6 are chosen and the corresponding single-source-points of these two locations are Acc.sub.5(t.sub.j, f.sub.j) and Acc.sub.6(t.sub.j, f.sub.j).

[0031] (4) First, the single-source-points of the 5.sup.th and 6.sup.th locations are arranged in column vectors, respectively. Then, the single-source-point vectors are denoted as Acc1=[Acc.sub.5, Acc.sub.6].sup.T. Acc1 should be normalized to the upper half unit circle using:

[00006]$A.Math..Math.\mathrm{cc}.Math..Math.1.Math.\left(i\right)=\{\begin{array}{c}\frac{\mathrm{Acc}.Math..Math.1}{.Math.\mathrm{Acc}.Math..Math.1.Math.\left(i\right).Math.},A.Math..Math.\mathrm{cc}.Math..Math.1_k.Math.\left(i\right)0\\ -\frac{A.Math..Math.\mathrm{cc}.Math..Math.1}{.Math.A.Math..Math.\mathrm{cc}.Math..Math.1.Math.\left(i\right).Math.},\mathrm{Acc}.Math..Math.1_k.Math.\left(i\right)<0\end{array}$

where cc1(i) is the normalized data of the i-th row vector in Acc1.

[0032] (5) If the two elements in cc1 are treated as the coordinates of a point in the Cartesian Coordinates, the coordinates of the arbitrary points are cc1(i)=[cc.sub.5(i), cc.sub.6 (i)].sup.T, i=(1, 2, . . . , J), where J is the total number of points in cc1. Three distance based statistics are given formed by the Euclidean distance and the Chebyshev distance between the points in cc1 and the left end point [1,0].sup.T, and the cosine distance between the points in cc1 and the center point [0,0].sup.T.

The distance dist.sub.E is selected as the statistic dist.

[0033] (6) The statistic dist is sorted in descending order and then the sorted data is differentiated as (dist). The difference sequence is counted. When the accumulated sample size reaches 95% of the total sample size, a threshold is set and the samples beyond the threshold are removed. The remainder difference sequence is averaged to obtain the mean value .sub.mean. The maximum of the remainder difference sequence is .sub.max. The relation between the number and the length of the statistical intervals is:

[00007]$P=\frac{\max \left(\mathrm{dist}\right)-\min \left(\mathrm{dist}\right)}{}$

where max() and min() are the maximum and minimum of a vector, respectively. When is equal to the mean value .sub.mean, the number of statistical subintervals is at the maximum and is denoted as P.sub.max. When is equal to the maximum value .sub.max, the number of statistical subintervals is at the minimum and is denoted as P.sub.min. Therefore, the range for the suggested number of statistical subintervals is given as P[P.sub.min,P.sub.max].

[0034] (7) The number of statistical subintervals is chosen as P=(P.sub.min+P.sub.max)/2. The statistical interval [max (dist)min(dist)] is divided into P subintervals with equal length. The number of samples in each subinterval is counted and denoted as p.sub.i, i=(1, 2, . . . , P). The approximate probability in each subinterval is calculated using Pr(i)=p.sub.i/P. The approximate probability curve is obtained through the weighted average procedure:

{circumflex over (P)}r(i)= 1/16(P(i2)+4P(i1)+6P(i)+4P(i+1)+P(i+2))

where {circumflex over (P)}r is the approximate probability curve.

[0035] (8) Six local maximum values of {circumflex over (P)}r are picked out and the number of active modes is six.