A method of estimating the number of modes for the sparse component analysis based modal identification

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)=[acc1(t), acc2(t), . . . , accn(t)]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 Δα;
• based on this property, the single-source-point detection can be accomplished through:
• 
• Re
• 
• {
• Acc
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• t
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• T
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• 
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• 
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• 
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• >
• cos
• 
• (
• Δ
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• α
• )
• where Re{⋅} and Im{⋅} are the real and imaginary parts of a vector, respectively;
• detected single-source-points are marked as (tj, fj);
• therefore, the time-frequency coefficients of the single-source-points are denoted as ACC(tj, fj)=[Acc1(tj, fj), Acc2(tj, fj), . . . , Accn(tj, fj)]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 Acck(tj,fj) and Accl(tj, fj);
• (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=[Acck, Accl]T;
• Acc1 should be normalized to the upper half unit circle using:
• A
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• where Âcc1(i) is normalized data of the i-th row vector in Acc1;
• (5) if the two elements in Âcc1 are treated as coordinates of a point in the Cartesian coordinates, coordinates of the arbitrary points are Âcc1(i)=[Âcck(i), Âccl(i)]T, i=(1, 2, . . . , J), where J is the total number of points in Âcc1;
• three distance based statistics are constructed by Euclidean distance and Chebyshev distance between points in Âcc1 and the left end point [−1, 0]T, and the cosine distance between the points in Âcc1 and the center point [0,0]T;
• the Euclidean distance is formulated as follows:
• distE(i)=√{square root over ((Acck(i)+1)2+Accl(i)2)}
• the Chebyshev distance is formulated as follows:
• distC(i)=max(|Acck(i)+1|, |Accl(i)|)
• the cosine distance is formulated as follows:
• dist
• θ
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• (
• i
• )
• =
• arccos
• (
• Acc
• k
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• (
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• (
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• 2
• +
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• the final statistic dist is determined from distE, distC and distθ;
• (6) the statistic dist is sorted in descending order and then the sorted data is differentiated as Δ(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 Δmean;
• the maximum of the remainder difference sequence is Δmax;
• the relation between the number and the length of the statistical intervals is:
• P
• =
• max
• 
• (
• dist
• )
• -
• min
• 
• (
• dist
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• δ
• where max(⋅) and min(⋅) are the maximum and minimum of a vector, respectively;
• when δ is equal to the mean value Δmean, the number of statistical subintervals is at a maximum and is denoted as Pmax;
• when δ is equal to the maximum value Δmax, the number of statistical subintervals is at a minimum and is denoted as Pmin;
• therefore, the range for the suggested number of statistical subintervals is given as P∈[Pmin,Pmax];
• (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 pi, i=(1, 2, . . . , P);
• the approximate probability in each subinterval is calculated using Pr(i)=pi/P;
• the approximate probability curve is obtained through the weighted average procedure:
• {circumflex over (P)}r(i)= 1/16(P(i−2)+4P(i−1)+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.