A METHOD FOR TRACKING STRUCTURAL MODAL PARAMETERS IN REAL TIME

Abstract

Structural health monitoring relating to a real-time tracking method for structural modal parameters. The Natural Excitation Technique transforms structural random responses into free decaying responses used to calculate structural modal parameters by the Eigensystem Realization Algorithm combined with the stabilization diagram. Considering influence of environmental excitation level on the number of identified modes, the reference mode list is formed by union of modes obtained from response sets in a day. Then the modes can be tracked automatically according to rules of minimum frequency difference and maximum Modal Assurance Criterion (MAC). To avoid mode mismatch problem caused by absence of threshold, frequency differences and MACs between all modes from the latter response set and all reference modes are calculated and the mode will be tracked into the cluster corresponding to the specified reference mode only in the case that their frequency difference is smallest and the MAC is largest.

Claims

1. A method for tracking structural modal parameters in real time, wherein: step 1: extraction of modal parameters from different response sets (1) select a response set h as y(t)=[y.sub.1(t), y.sub.2(t), . . . , y.sub.z(t)].sup.T, t=1, 2, . . . , N, where N is number of sampling points, z is number of sensors to measure responses; transform the response set h into correlation function matrices r() with various time delays by Natural Excitation Technique: $\begin{array}{cc}r\left(\right)=\left[\begin{array}{cccc}{r}_{1,1}\left(\right)& r_{1,2}\left(\right)& \mathrm{.Math.}& r_{1,z}\left(\right)\\ r_{2,1}\left(\right)& r_{2,2}\left(\right)& \mathrm{.Math.}& r_{2,z}\left(\right)\\ \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}\\ r_{z,1}\left(\right)& r_{z,2}\left(\right)& \mathrm{.Math.}& r_{z,z}\left(\right)\end{array}\right]=E\left[y\left(t+\right).Math.{y\left(t\right)}^T\right]& \left(1\right)\end{array}$ where r.sub.ij() represents cross correlation function between response of measurement channel i and the response of measurement channel j; (2) construct block Hankel matrices H.sub.ms(k1) and H.sub.ms(k) with a correlation function matrix r() as $\begin{array}{cc}{H}_{m.Math..Math.s}\left(k-1\right)=\left(\begin{array}{cccc}r\left(k\right)& r\left(k+1\right)& \mathrm{.Math.}& r\left(k+s-1\right)\\ r\left(k+1\right)& r\left(k+2\right)& \mathrm{.Math.}& r\left(k+s\right)\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ r\left(k+m-1\right)& r\left(k+m\right)& \mathrm{.Math.}& r\left(m+s+k-2\right)\end{array}\right)& \left(2\right)\end{array}$ (3) set k=1, and then the Eigensystem Realization Algorithm is implemented on the matrices H.sub.ms (k1) and H.sub.ms (k) to calculate the modal parameters (frequencies, damping ratios and mode shapes) from the model orders ranging from to n.sub.u with the increment of , where is an even number; (4) preset the threshold of the frequency difference .sub.f,lim, the threshold of the damping difference .sub.,lim and the MAC threshold .sub.MAC,lim respectively; modes with their modal parameter dissimilarity satisfies the conditions df.sub.f,lim, d.sub.,lim and MAC.sub.MAC,lim, are considered as stable modes; then stable modes at successive model orders will be grouped into one cluster if their frequency difference is less than .sub.f,lim and the MAC exceeds .sub.MAC,lim; the clusters with their sizes of the number of stable modes in a cluster outnumber the limit n.sub.tol are selected as physical clusters; the averages of modal parameters in each physical cluster are defined as the representative values of physical modes, and then the representative values corresponding to physical clusters are considered as the identified modal parameters from the response set h, where the identified frequencies are f.sub.1,h, f.sub.2,h, . . . , f.sub.,h, the identified mode shapes are .sub.1,h, .sub.2,h, . . . , .sub.,h; step 2: tracking modal parameters identified from different response sets; (5) designate the union of the structural physical modes calculated from each response set in a day as the reference mode list, where the reference frequencies are marked as f.sub.1,ref, f.sub.2,ref, . . . , f.sub.,ref and the reference mode shapes are .sub.1,ref, .sub.2,ref, . . . , .sub.,ref; (6) track the structural physical mode j from the response set h into the cluster containing the reference mode if their dissimilarity of modal parameters satisfies the following four formulas: $\begin{array}{cc}\frac{.Math.f_{,\mathrm{ref}}-f_{j,h}.Math.}{\max \left(f_{,\mathrm{ref}},f_{j,h}\right)}\frac{.Math.f_{i,\mathrm{ref}}-f_{j,h}.Math.}{\max \left(f_{i,\mathrm{ref}},f_{j,h}\right)}.Math..Math..Math.i=1,2,\mathrm{.Math.}.Math.,& \left(3\right)\\ \frac{.Math.f_{,\mathrm{ref}}-f_{j,h}.Math.}{\max \left(f_{,\mathrm{ref}},f_{j,h}\right)}\frac{.Math.f_{,\mathrm{ref}}-f_{k,h}.Math.}{\max \left(f_{,\mathrm{ref}},f_{k,h}\right)}.Math..Math..Math.k=1,2,\mathrm{.Math.}.Math.,& \left(4\right)\\ \mathrm{MAC}\left({}_{,\mathrm{ref}},{}_{j,h}\right)\mathrm{MAC}\left({}_{i,\mathrm{ref}},{}_{j,h}\right).Math..Math..Math.i=1,2,\mathrm{.Math.}.Math.,& \left(5\right)\\ \mathrm{MAC}\left({}_{,\mathrm{ref}},{}_{j,h}\right)\mathrm{MAC}\left({}_{,\mathrm{ref}},{}_{k,h}\right).Math..Math..Math.k=1,2,\mathrm{.Math.}.Math.,.& \left(6\right)\end{array}$

Description

DESCRIPTION OF DRAWINGS

[0016] FIG. 1 presents the layout of fourteen vertical acceleration sensors of a bridge.

[0017] FIG. 2 shows the automatic tracking results according to this invention.

[0018] FIG. 3 shows the tracking results according to the threshold method.

DETAILED DESCRIPTION

[0019] The present invention is further described below in combination with the technical solution.

[0020] The bridge analyzed in the example is a single tower double cable plane asymmetric prestressed concrete cable-stayed bridge. As shown in FIG. 1, fourteen vertical acceleration sensors are arranged on the main girder to monitor the dynamic characteristics of the bridge. The vertical acceleration responses under ambient excitation are collected from Aug. 1, 2016 to Aug. 31, 2016, with the sampling frequency of 100 Hz. One hour responses from fourteen sensors are selected as a response set to estimate modal parameters.

[0021] The procedures are described as follows:

[0022] (1) The structural responses collected from 0:00-1:00 in Aug. 1, 2016 are selected as the response set h=1. The Natural Excitation Technique is used to transform the response set y(t)=[y.sub.1(t), y.sub.2(t), . . . , y.sub.14(t)].sup.T, t=1, 2, . . . , N into the correlation function matrices with various time delays , as shown in Eq (1).

[0023] (2) Set m=200, s=200. The correlation function matrices r() with =1399 and =2400 are used to build the block Hankel matrices H.sub.ms(0) and H.sub.ms (1), as shown in Eq. (2).

[0024] (3) Set the model orders range from =4 to n.sub.u=280, with the order increment of =4 and the order number of n.sub.u=70. Then the modal parameters (system eigenvalues .sub.i, natural frequencies f.sub.i, damping ratios .sub.i and mode shapes .sub.i) in each model order are calculated through the Eigensystem Realization Algorithm.

[0025] (4) The threshold of the frequency difference, the threshold of the damping difference and the MAC threshold are set as .sub.f,lim=5%, .sub.,lim=20% and .sub.MAC,lim=90% respectively. Modes with their modal parameter dissimilarity satisfies the conditions (df.sub.f,lim, d.sub.,lim and MAC.sub.MAC,lim) are stable. Then stable modes at successive model orders are grouped into one cluster if their frequency difference is less than .sub.f,lim and the MAC exceeds .sub.MAC,lim. The clusters with their sizes (the number of stable modes in a cluster) outnumber the limit n.sub.tol=0.5n.sub.u are selected as physical clusters. The averages of modal parameters in each physical cluster are defined as the representative values of physical modes, and then the representative values corresponding to =18 physical clusters are considered as the identified modal parameters from the response set h, where the identified frequencies are f.sub.1,1=0.378 Hz, f.sub.2,1=0.642 Hz, f.sub.3,1=0.750 Hz, f.sub.4,1=0.937 Hz, f.sub.5,1=0.998 Hz, f.sub.6,1=1.066 Hz, f.sub.7,1=1.266 Hz, f.sub.8,1=1.336 Hz, f.sub.9,1=1.519 Hz, f.sub.10,1=1.618 Hz, f.sub.11,1=1.692 Hz, f.sub.12,1=1.946 Hz, f.sub.13,1=2.018 Hz, f.sub.14,1=2.050 Hz, f.sub.15,1=2.245 Hz, f.sub.16,1=2.297 Hz, f.sub.17,1=2.586 Hz, f.sub.18,1=2.884 Hz.

[0026] (5) The union of the structural physical modes calculated from each response set in Aug. 1, 2016 is designated as the reference mode list, where the reference frequencies are f.sub.1,ref=0.378 Hz, f.sub.2,ref=0.642 Hz, f.sub.3,ref=0.750 Hz, f.sub.4,ref=0.937 Hz, f.sub.5,ref=0.998 Hz, f.sub.6,ref=1.066 Hz, f.sub.7,ref=1.266 Hz, f.sub.8,ref=1.336 Hz, f.sub.9,ref=1.519 Hz, f.sub.10,ref=1.618 Hz, f.sub.11,ref=1.692 Hz, f.sub.12,ref=1.946 Hz, f.sub.13,ref=2.018 Hz, f.sub.14,ref=2.050 Hz, f.sub.15,ref=2.245 Hz, f.sub.16,ref=2.297 Hz, f.sub.17,ref=2.586 Hz, f.sub.18,ref=2.884 Hz.

[0027] (6) The structural physical mode j from the response set h will be tracked into the cluster containing the reference mode if their dissimilarity of modal parameters satisfies Eqs (3-6). The tracking results are shown in FIG. 2.

[0028] To illustrate the superiority of the proposed method, the traditional threshold method is used to track the first modes changing with time, where the relative frequency difference and the MAC should satisfy |f.sub.,reff.sub.j,h|/max(f.sub.,ref, f.sub.j,h)5% and MAC (.sub.,ref,.sub.j,h)90% respectively. As shown in the crosses in FIG. 3, some modes cannot be tracked since the modal parameter differences between these modes and the reference modes do not meet |f.sub.,reff.sub.j,h|/max(f.sub.,ref, f.sub.j,h)5% and MAC (.sub.,ref,.sub.j,h)90%.