A modal estimation based multitype sensor placement method

Transcription

1 A modal estimation based multitype senso placement method *Xue-Yang Pei 1), Ting-Hua Yi 2) and Hong-Nan Li 3) 1),)2),3) School of Civil Engineeing, Dalian Univesity of Technology, Dalian , China; 1) ABSTRACT A two-stage senso placement method, in which stain gauges ae placed togethe with tiaxial acceleometes to obtain moe accuate displacement mode infomation of a stuctue, is poposed. In stage one, the selection of acceleomete locations, the edundancy coefficient is used togethe with the modal assuance citeion to decease the edundancy infomation of the obtained displacement modes between the selected acceleomete locations. In stage two, the displacement modes of some estimated locations ae estimated based on the stain modes obtained fom the stain gauges. Detemination of diffeent candidate stain gauge locations affects the quality of the displacement mode estimation a lot. At last, the candidate stain gauge locations coesponding to the best estimation ae selected. The effectiveness of the poposed multitype senso placement method is demonstated by a numeical investigation using a benchmak model. Keywods: optimal senso placement; tiaxial acceleomete, stain gauge; candidate stain gauge locations; modal estimation 1. INTRODUCTION In SHM systems, stain gauges and acceleometes ae both widely used in pactice. Geneally, stain gauges ae used to measue the local defomations, and acceleometes ae used to obtain the global modal chaacteistics of the stuctue. It is meaningful to find an OSP method that compehensively utilizes the two types of sensos, in which stain gauges and acceleometes ae placed togethe instead of placed sepaately. (Zhang 2011) poposed an integated optimal placement method of displacement tansduces and stain gauges fo esponse econstuction, which uses the elationship between stains and displacements to pefom esponse estimation. 1) Phd Student 2) Pofesso 3) Pofesso

2 Futhemoe, a multitype senso placement method that uses the Kalman filte to obtain the best estimation of the esponses at key locations, in which stain gauges, displacement sensos and acceleation sensos ae placed togethe, was poposed by (Zhang 2016). The elationship between the stain mode and the displacement mode, which offes guidance egading the integated utilization of the two types of modal infomation obtained fom both types of sensos, was poposed by (Yam 1996). Howeve, few existing studies egading stain and acceleation senso placement methods focused on modal estimation, even though modal infomation is extemely impotant with espect to analyzing the dynamic chaacteistics of a stuctue. This pape pesents a multitype senso placement method in which stain gauges can be placed togethe with acceleometes to obtain the stuctual displacement mode infomation. The amount of displacement mode infomation contained in diffeent candidate stain gauge locations can be calculated by using the elationship between the displacement mode and the stain mode. The estimated locations ae detemined via compehensive consideation of the selected pefomance citeia and the pactical situation. In this method, the stain modes of measued locations ae utilized to estimate the displacement modes of the estimated locations without acceleometes, and the stain gauge locations ae selected accoding to the quality of the estimation. 2. THEORY FORMULATION OF THE MODAL ESTIMATION METHOD 2.1 selection of acceleomete locations The modal assuance citeion MAC method (Cane 1995) is used hee to select the initial acceleometes, and tiaxial acceleometes ae taken into account hee. The cosine of the angle between two mode shape vectos is used to evaluate the elationship between the two mode shape vectos. The MAC matix can be expessed as whee *,i and MAC T 2 *, i*, j *, i*, i *, j*, j j T T *, j ae the i th and j th columns of the taget mode shape matix, espectively. A cosine value nea zeo shows that the two coesponding mode shape vectos can be easily distinguished fom each othe. Afte fist seveal locations have been selected, evey time, only one acceleomete location that gets the smallest value of the maximum off-diagonal MAC tem is added to the existing placement. Consideing the continuity of the mode shapes, when two sensos ae placed too close to each othe, they usually contain simila modal infomation (Stephan 2012). Hee, the Fobenius nom is used to evaluate the similaity of mode shapes of diffeent locations, and the edundancy coefficient is defined as (1)

3 R j 1 3i 3j F 3i F 3j F (2) whee R j is the edundancy coefficient between the mode shapes of the i th and the j th locations. When the value of R j is close to one, the two coesponding locations shae almost the same modal infomation, which is not acceptable in the senso placement. A nea one edundancy coefficient value needs to be avoided in evey senso selection step. 2.1 Selection of stain gauge locations fo modal estimation Sometimes, afte the initial acceleomete locations have been detemined accoding to the pefomance citeia, the numbe of acceleometes can be deceased due to vaious pactical factos. The displacement modes of these deleted locations can be estimated based on the stain modes obtained fom the stain gauges. Then, the linea dynamical equation of the FE model can be witten as M C K f (3) whee M, C and K ae the mass, damping and stiffness matixes of the system, espectively; f is the input vecto; is the nodal displacement vecto of all nodes in the global coodinate system, and a dot ove the vecto epesents the deivative with espect to time. The elationship between stain mode and displacement mode can be obtained B Bq q (4) B (5) whee is the stain vecto of stains of candidate locations in the whole FE model; denotes the stain mode matix of the candidate stain locations; B is the tansfomation matix between the stains and the nodal displacement of all nodes, which is diectly affected by the candidate stain locations; denotes the displacement mode matix of all nodes; q denotes the modal coodinates. Pactically, the stain modes obtained fom stain measuements may diffe fom the computed esult fom the FE model. The stain modes obtained fom the measuement data can be expessed as B v (6) whee v is a noise matix in which the i th column vi can be seen as a stationay Gaussian noise with zeo mean and a covaiance Cov( v ) ii. When pefoming the i

4 displacement mode estimation, in the displacement mode shape matix, only the ows coesponding to the estimated locations need to be estimated. The ight side of Eq. (5) can be tansfomed into whee B B B Nd Nd (7) N d is the numbe of ows of the displacement mode matix ; ows of coesponding to the estimated locations; Nd ows of ; denotes the N d denotes the emaining B denotes the columns of B coesponding to the estimated locations; N d B denotes the emaining d N columns of B. The selection of stain gauges involves selecting some ows in the stain mode shape matix of all the candidate stain gauge locations; Eq. (6) can be e-witten as Nd Nd P B PB Pv (8) whee P is a selection matix that consists of only zeos and ones and the numbe of ows of P is equal to the numbe of selected stain gauges. In Eq. (8), when the numbe of selected stain gauges is geate than o equal to the numbe of ows of the displacement mode matix, the least squaes estimation of the displacement modes of estimated locations can be obtained 1 N N B P PB B P P B ˆ T T T T d d (9) whee ˆ is the estimation of ; The estimation vecto ˆ i and the covaiance matix of the estimation vecto ˆ i ae expessed as 1 ˆ T T T T Nd Nd i i i B P PB B P P B (10) Cov( ˆ ) ( B P PB ) (11) 2 T T 1 i i Whee the subscipt i denotes i th column of the matix. Consideing the situation in which the diagonal tems of Cov( ˆ ) epesent the eo of the estimation in each i location of the i th modes, the tace of the covaiance matix Cov( ˆ ) i is used to evaluate the estimation uncetainty eo ˆ t T T 1 i i B P PB (12) The uncetainty of the estimation matix ˆ can be expessed as

5 eo ˆ t T T B P PB 1 (13) Consideing the existence of the pediction eo, the ows of B containing nothing but zeos can be deleted befoe calculating to educe the dimension of B, which can impove the quality of the displacement mode estimation. Finally, the best selection of the stain gauge locations is obtained fom the selection matix coesponding to the smallest value of estimation uncetainty. 3. NUMERICAL STUDIES To demonstate the effectiveness of the poposed multitype senso placement method, a bidge benchmak model (Caicedo 2006) is used fo numeical investigation in this section. The fist 10 mode shapes obtained fom the FE model ae adopted hee fo senso placement investigation. The initial candidate stain gauge locations on each beam element ae at 1/3 and 2/3 the longitudinal length of the beam, and the stain gauges ae placed on the flange of the beam element. Afte the initial 8 acceleomete locations haven selected accoding to the MAC and the edundancy infomation, 3 acceleometes fom the initial acceleomete locations ae andomly deleted. To estimate the tanslation modes of the 3 estimated locations, 9 stain gauges ae needed hee to ensue that when estimating the tanslation modes, the numbe of selected stain gauges is geate than o equal to the numbe of ows of the tanslation modes. Similaly, the MAC is used to evaluate the displacement mode infomation fom the senso placement, and the change of the value of the maximum off-diagonal MAC tem is used as a citeion to evaluate the pefomance of the poposed senso placement. Table 1 lists the values of the maximum off-diagonal MAC tems of diffeent selected senso locations. The displacement modes of the final senso locations consist of the displacement modes of the 5 acceleomete locations and the displacement modes estimated based on the stain modes of the 9 stain gauges. When calculating the values of the off-diagonal MAC tems of diffeent selected senso locations, the diagonal tems ae also defined as zeo. Compaing the situation of 5 emaining locations with the situation of initial 8locations, the value of the maximum off-diagonal MAC tem obviously inceases because the 3 sensos ae andomly deleted fom the initial acceleometes and the emaining senso locations cannot satisfy the MAC well. Compaing the situation of final locations with the situation of 5 emaining locations, the value of the maximum off-diagonal MAC tem obviously deceases, which indicates that the stain gauges help much in impoving the pefomance of the senso placement. Compaing the situation of final locations with the situation initial 8locations, maximum off-diagonal MAC values ae simila, which demonstates that the obtained multitype senso placement pefoms well unde the MAC when compaed to the initial acceleometes.

6 As a esult, the poposed senso placement method is effective in helping to obtain moe accuate displacement mode infomation of the stuctue unde MAC. Table 1 Values of maximum off-diagonal MAC tems 8 initial 5 emaining Diffeent senso locations locations locations Maximum off-diagonal MAC tem value Final locations CONCLUSIONS In this pape, the poblem of the compehensive placement of stain gauges and tiaxial acceleometes is solved using a modal estimation theoy. The displacement modes of the estimated locations without acceleometes can be estimated by the stain modes obtained fom some selected stain gauge measuements. A bidge benchmak model is used hee fo a numeical investigation. The obtained senso placement satisfies the MAC well, which demonstates the effectiveness of the poposed senso placement method. In this pape, the stain modes ae utilized to estimate the displacement modes of the estimated locations; and the stain gauge locations need to contain as much displacement mode infomation as possible without being at the midpoints. REFERENCES Zhang XH, Zhu S, Xu YL, Hong XJ. (2011), Integated optimal placement of displacement tansduces and stain gauges fo bette estimation of stuctual esponse. Int. J. Stuct. Stab. Dyn., 11(3), Zhang CD, Xu YL. (2016), Optimal multi-type senso placement fo esponse and excitation econstuction. J. Sound Vib., 360, Yam LY, Leung TP, Li DB, Xue KZ. (1996), Theoetical and expeimental study of modal stain analysis. J. Sound Vib., 191(2), Cane TG, Dohmann CR. (1995), A modal test design stategy fo model coelation, 13th Intenational Modal Analysis Confeence, Nashville. Stephan C. (2012), Senso placement fo modal identification. Mech. Syst. Signal Poc., 27, Caicedo JM, Catbas N, Gul M, Zauin R, Dyke SJ. (2006), Benchmak poblem fo bidge health monitoing: Definition pape, The 4th Wold Confeence on Stuctual Contol and Monitoing, San Diego, Califonia, USA.