Automatic method for tracking structural modal parameters

Abstract

Structural health monitoring relating to an automatic method for tracking structural modal parameters. First, Natural Excitation Technique is used to transform the random responses into correlation functions and Eigensystem Realization Algorithm combined with the stabilization diagram is used to estimate modal parameters from various response segments. Then, modes from the latter response segment are classified as traceable modes or untraceable modes according to correlations between their observability vectors and subspaces of the existing reference modes. Final, traceable modes will be grouped into specified clusters with the same structural characteristics on the basis of maximum modal observability vector correlation and minimum frequency difference. Meanwhile, union of the untraceable modes and existing reference modes are updated as the new reference modes which can be applied into the next tracking process. This can track the modal parameters automatically without artificial thresholds and the specified reference modes.

Claims

1. An automatic method for tracking structural modal parameters, wherein: step 1: extraction of modal parameters from different response segments select the random responses y(t)=[y.sub.1(t), y.sub.2(t), . . . , y.sub.z(t)].sup.T, t=1, 2, . . . , N recorded from acceleration sensors installed on an engineering structure as the response segment h, where z means a measuring point number and N is a number of samples; Natural Excitation Technique is used to obtain a correlation function matrix r(τ) with different time delays τ $\begin{array}{cc}r\left(\tau \right)=\left[\begin{array}{cccc}{r}_{1,1}\left(\tau \right)& r_{1,2}\left(\tau \right)& \mathrm{.Math.}& r_{1,z}\left(\tau \right)\\ r_{2,1}\left(\tau \right)& r_{2,2}\left(\tau \right)& \mathrm{.Math.}& r_{2,z}\left(\tau \right)\\ \mathrm{.Math.}& \mathrm{.Math.}& \ddots & \mathrm{.Math.}\\ r_{z,1}\left(\tau \right)& r_{z,2}\left(\tau \right)& \mathrm{.Math.}& r_{z,z}\left(\tau \right)\end{array}\right]& \left(1\right)\end{array}$ where r.sub.ij(τ) is a cross-correlation function between the measuring points i and j; construct Hankel matrix H.sub.ms(k−1) and H.sub.ms(k) by the correlation functions r(τ): $\begin{array}{cc}{H}_{ms}\left(k-1\right)=\left[\begin{array}{cccc}r\left(k\right)& r\left(k+1\right)& \mathrm{.Math.}& r\left(k+s-1\right)\\ r\left(k+1\right)& r\left(k+2\right)& \mathrm{.Math.}& r\left(k+s\right)\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ r\left(k+m-1\right)& r\left(k+m\right)& \mathrm{.Math.}& r\left(m+s+k-2\right)\end{array}\right]& \left(2\right)\end{array}$ set k=1, and then Eigensystem Realization Algorithm is performed on the matrix H.sub.ms(k−1) to calculate modal parameters with various modal orders, including the system eigenvalues, frequencies, damping ratios, mode shapes and the modal observability vectors, which range from the even number δ to n.sub.uδ with the order increment of δ; preset a tolerance limit of frequency difference e.sub.f,lim, a tolerance limit of damping difference e.sub.ξ,lim and a tolerance limit of Modal Assurance Criterion (MAC) e.sub.MAC,lim; the modes which are satisfied with the three tolerance limits are considered as stable modes; two stable modes in successive model orders will be grouped into one cluster if their frequency difference is less than e.sub.f,lim and the MAC is more than e.sub.MAC,lim, clusters with a number of stable modes exceeds the limit n.sub.tol are considered as physical clusters; the averages of the modal parameters in each physical cluster are considered as the identification results of a response segment h, including the system eigenvalue λ.sub.i,h, frequency f.sub.i,h, damping ratio ξ.sub.i,h mode shape vector φ.sub.i,h and modal observability vector w.sub.i,h=[φ.sub.i,h.sup.T λ.sub.i,h.sup.T . . . Δ.sub.i,h.sup.m-1φ.sub.i,h.sup.T].sup.T, where the superscript T indicates the transposition; step 2: track the modes from various response segments set h=1, the β modes estimated from the response segment h as initial reference modes, where frequencies f.sub.1=[f.sub.1,1, f.sub.2,1, . . . f.sub.β,1] and a modal observability matrix W.sub.1=[w.sub.1,1, w.sub.2,1, . . . , w.sub.β,1] are respectively marked as the reference frequency vector f.sub.ref=[f.sub.1,ref, f.sub.2,ref, . . . , f.sub.β,ref] and the reference observability matrix W.sub.ref=[w.sub.1,ref, w.sub.2,ref, . . . , w.sub.β,ref]; take singular value decomposition on the reference observability matrix W.sub.ref to obtain a reference modal subspace U.sub.1 and its orthogonal complement subspace U.sub.2: $\begin{array}{cc}{W}_{\mathrm{ref}}=U\Sigma V^H=\left[\begin{array}{cc}{U}_1& U_2\end{array}\right]\left[\begin{array}{cc}{\Sigma }_1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{V}_1^H\\ V_2^H\end{array}\right]& \left(3\right)\end{array}$ where the superscript H represents the complex conjugate transpose; since the modes estimated from the same response segment are uncorrelated, the rank of the matrix W.sub.ref equals to the order of the reference modes; for the response segment h=2, the α modes are calculated, where the frequencies f.sub.2=[f.sub.1,2, f.sub.2,2, . . . , f.sub.α,2] and the modal observability matrix W.sub.2=[w.sub.1,2, w.sub.2,2, . . . , w.sub.α,2]; for each mode j obtained from the response segment h=2, the correlations wMOC between the modal observability vector w.sub.j,2 and the reference modal subspace U.sub.1 as well as the orthogonal complement subspace U.sub.2 are respectively calculated as:
wMOC(U.sub.1,w.sub.j,2)=cos.sup.2(□[U.sub.1,w.sub.j,2])  (4)
wMOC(U.sub.2,w.sub.j,2)=cos.sup.2(□[U.sub.2,w.sub.j,2])  (5) where □ indicates the angle between the subspace and the vector; if wMOC(U.sub.1, w.sub.j,2)≥wMOC(U.sub.2, w.sub.j,2), mode j will be marked as traceable, otherwise mode j will be marked as untraceable; assuming that η(η≤α) modes can be selected as traceable modes from the α identified modes, the remaining α−η modes are untraceable; frequencies and a modal observability matrix of the traceable modes are respectively reformulated as {tilde over (f)}.sub.2=[{tilde over (f)}.sub.1,2, {tilde over (f)}.sub.2,2, . . . , {tilde over (f)}.sub.η,2] and {tilde over (W)}.sub.2=[{tilde over (w)}.sub.1,2, {tilde over (w)}.sub.2,2, . . . , {tilde over (w)}.sub.η,2]; the traceable mode l from the response segment h=2 and the reference mode χ will be grouped into one cluster if they satisfy: $\begin{array}{cc}\frac{.Math.f_{\chi ,\mathrm{ref}}-{\tilde{f}}_{\ell ,h}.Math.}{\max \left(f_{\chi ,\mathrm{ref}},{\tilde{f}}_{\ell ,h}\right)}\le \frac{.Math.f_{i,\mathrm{ref}}-{\tilde{f}}_{\ell ,h}.Math.}{\max \left(f_{i,\mathrm{ref}},{\tilde{f}}_{\ell ,h}\right)}i=1,2,\mathrm{.Math.},\beta & \left(6\right)\\ \frac{.Math.f_{\chi ,\mathrm{ref}}-{\tilde{f}}_{\ell ,h}.Math.}{\max \left(f_{\chi ,\mathrm{ref}},{\tilde{f}}_{\ell ,h}\right)}\le \frac{.Math.f_{\chi ,\mathrm{ref}}-{\tilde{f}}_{k,h}.Math.}{\max \left(f_{\chi ,\mathrm{ref}},{\tilde{f}}_{k,h}\right)}k=1,2,\mathrm{.Math.},\eta & \left(7\right)\\ \mathrm{MOC}\left(w_{\chi ,\mathrm{ref}},{\tilde{w}}_{\ell ,h}\right)\ge \mathrm{MOC}\left(w_{i,\mathrm{ref}},{\tilde{w}}_{\ell ,h}\right)i=1,2,\mathrm{.Math.},\beta & \left(8\right)\\ \mathrm{MOC}\left(w_{\chi ,\mathrm{ref}},{\tilde{w}}_{\ell ,h}\right)\ge \mathrm{MOC}\left(w_{\chi ,\mathrm{ref}},{\tilde{w}}_{k,h}\right)k=1,2,\mathrm{.Math.},\eta & \left(9\right)\end{array}$ where modal observability correlation (MOC) indicates the correlation of two modal observability vectors; the union of the α−η untraceable modes and the existing reference modes are updated as the new reference modes which can be used for the next tracking; the new reference frequency vector and the new reference modal observability matrix can be respectively expanded as f.sub.ref=[f.sub.1,ref, f.sub.2,ref, . . . , f.sub.β,ref, f.sub.β+1,ref, . . . , f.sub.β+α−η,ref] and W.sub.ref=[w.sub.1,ref, w.sub.2,ref, . . . , w.sub.β,ref, w.sub.β+1,ref, . . . w.sub.β+α−η,ref]; for the modal parameter from the response segments h=3, 4, . . . , the tracking procedures are the same as above.

Description

DESCRIPTION OF DRAWINGS

(1) FIG. 1 is the layout of vertical acceleration sensors for the main girder of a bridge.

(2) FIG. 2 is the results of automatically tracking the modal parameters of the main girder.

DETAILED DESCRIPTION

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

(4) As shown in FIG. 1, fourteen acceleration sensors are installed on the main girder of the bridge. The vertical acceleration responses under ambient excitation are recorded from Sep. 1, 2016 to Sep. 7, 2016 with the sampling frequency of 100 Hz. One hour responses are determined as a response segment for modal identification.

(5) The procedures are described as follows:

(6) (1) The structural random responses at 0:00-1:00 on Sep. 1, 2016 are determined as the response segment h=1, which can be represented as y(t)=[y.sub.1(t), y.sub.2(t), . . . , y.sub.14(t)].sup.T, t=1, 2, . . . , N. Then Natural Excitation Technique is used to obtain the correlation function matrix with different time delays, shown in Eq. (1).

(7) (2) Set m=250, s=250. The correlation functions r(τ) with τ=1˜499 and τ=2˜500 are respectively used to build Hankel matrices H.sub.ms(0) and H.sub.ms(1), shown in Eq. (2).

(8) (3) Set the minimum model order as δ=4, n.sub.u=70, and then the model order ranges to 280 with the order increment of δ=4. Eigensystem Realization Algorithm is performed on the Hankel matrices H.sub.ms(0) and H.sub.ms(1) to obtain the modal parameters from different model orders.

(9) (4) Set the tolerance limits of the frequency difference, the damping difference and MAC as e.sub.f,lim=5%, e.sub.ξ,lim=20% and e.sub.MAC,lim=90%, respectively. Modes which satisfy with the three tolerance limits are considered as stable modes. Two stable modes in the successive model orders will be grouped into one cluster if their frequency difference is less than e.sub.f,lim and the MAC is more than e.sub.MAC,lim. The physical clusters are those with the number of stable modes exceeds the limit n.sub.tol=0.5n.sub.u. The averages of the modal parameters in each physical cluster are considered as the identification results. Thus the β=15 modes with their frequencies less than 2.5 Hz are estimated as the initial reference modes, where the reference frequencies are f.sub.1,ref=0.387 Hz, f.sub.2,ref=0.648 Hz, f.sub.3,ref=0.754 Hz, f.sub.4,ref=0.932 Hz, f.sub.5,ref=0.985 Hz, f.sub.6,ref=1.060 Hz, f.sub.7,ref=1.278 Hz, f.sub.8,ref=1.321 Hz, f.sub.9,ref=1.513 Hz, f.sub.10,ref=1.605 Hz, f.sub.11,ref=1.685 Hz, f.sub.12,ref=1.954 Hz, f.sub.13,ref=2.000 Hz, f.sub.14,ref=2.038 Hz, f.sub.15,ref=2.212 Hz.

(10) (5) The α=16 modes are identified from the response segment h=2, where the frequencies are f.sub.1,2=0.386 Hz, f.sub.2,2=0.644 Hz, f.sub.3,2=0.755 Hz, f.sub.4,2=0.929 Hz, f.sub.5,2=0.983 Hz, f.sub.6,2=1.061 Hz, f.sub.7,2=1.257 Hz, f.sub.8,2=1.318 Hz, f.sub.9,2=1.503 Hz, f.sub.10,2=1.595 Hz, f.sub.11,2=1.676 Hz, f.sub.12,2=1.949 Hz, f.sub.13,2=1.998 Hz, f.sub.14,2=2.033 Hz, f.sub.15,2=2.220 Hz, f.sub.16,2=2.253 Hz.

(11) (6) Singular value decomposition is performed on the reference modal observability matrix W.sub.ref to obtain the reference modal subspace U.sub.1 and its orthogonal complement subspace U.sub.2. The correlations between modes identified from the response segment h=2 and the modal subspaces U.sub.1 and U.sub.2 are calculated, respectively. Modes j=1, . . . , 15 can be tracked by Eqs. (6-9) since wMOC(U.sub.1, w.sub.j,2)≥wMOC(U.sub.2, w.sub.j,2) is satisfied. Mode j=16 is untraceable as a result of wMOC(U.sub.1, w.sub.j,2)<wMOC(U.sub.2, w.sub.j,2) where wMOC(U.sub.1, w.sub.16,2)=0.238 and wMOC(U.sub.2,w.sub.16,2)=0.762. Then mode j=16 will be added into the existing reference mode list for the next tracking. The number of reference modes is 16 with f.sub.16,ref=2.253 Hz. For the response segments h=3, 4, . . . , repeat steps (1-4) to identify modal parameters and repeat step (6) to track modes. The tracking results are shown in FIG. 2.