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Chapte 9 Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe Hao Chen, Zhi Sun and Limin Sun Additional infomation is available at the end of the chapte http://dx.doi.og/10.5772/45732 1. Intoduction Since the eection of the Stomaund Bidge of Sweden in 1956, cable-stayed bidge, as an efficient and economic bidge type to sumount a long-distance obstacle, has attacted moe and moe inteests both fom bidge engineeing community and fom the society and govenment. Nowadays, the cable stayed bidge is the most competitive type fo the bidge with the span of 300-1000 metes. Fo a cable suppoted bidge, which is geneally quite flexible and of low damping, its vibation unde ambient excitation (such as the wind and gound motion excitation) and opeational loading (such as the vehicle and tain loads) is quite citical fo its safety, seviceability and duability. Vibation contol countemeasues, such as the installation of the enegy dissipating devices, ae thus equied [1, 2]. Stuctual active contol, which applies a counte-foce induced by a contol device to mitigate stuctual vibation, has been widely poposed fo the vibation contol of cable stayed bidges and poven to be efficient by many eseaches [3-6]. Although the vibation esponse of a fully eected cable-stayed bidge should be contolled, a cable-stayed bidge unde constuction, which is of low damping and not as stable as the completed stuctue, is geneally moe vulneable to dynamic loadings. Duing the constuction stage, the cable pylons wee geneally eected fistly and the cable and main gides ae then hang on the pylons symmetically in a double-cantileve way. With the incease of the cantileve length, the bidge is moe and moe flexible. When the gide is on its longest double-cantileve state, the bidge is the most vulneable to the extenal distubance (such as the ambient wind fluctuation and gound motions). Moeove, if the cables, pylons and main gide of the bidge ae all steel components and thus the damping of the bidge is vey low, its vibation unde ambient excitations will be quite lage. The vibation eduction countemeasues ae thus in geat demand. Fedeic 2012 Sun et al., licensee InTech. This is an open access chapte distibuted unde the tems of the Ceative Commons Attibution License (http://ceativecommons.og/licenses/by/3.0), which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited.
196 Advances on Analysis and Contol of Vibations Theoy and Applications conducted seveal mock-up tests epesenting a cable-stayed bidge duing the constuction stage [7]. Since the contol objective was set to educe the gide vetical vibation esponse o cable paametic vibation esponse, the active tendon was installed as the contol device. While fo a cable stayed bidge unde uncontollable ambient excitations, stuctue will vibate not only in the vetical diection but also in the tansvese diection. Moeove, since the bidges ae geneally designed to cay the vetical loads, the unexpected tansvese loads, especially the tansvese dynamic loads, will induce stuctual safety and duability poblems. It is thus of cucial impotance to install some vibation eduction devices to contol bidge tansvese vibation. Fo this type of stuctue vibation contol poblem, the active mass dampe (AMD) o the active tuned mass dampe (ATMD) will be a competent candidate [8, 9]. In this chapte, the geneal pocedue and key issues on adopting an active contol device, the active mass dampe (AMD), fo vibation contol of cable stayed bidges unde constuction ae pesented. Taking a typical cable stayed bidge as the pototype stuctue; a lab-scale test stuctue was designed and fabicated fistly. A baseline FEM model was then setup and updated accoding to the modal fequencies measued fom stuctual vibation test. A numeical study to simulate the bidge-amd contol system was conducted and an efficient LQG-based contolle is designed. Based on that, an expeimental implementation of AMD contol of the tansvese vibation of the bidge model was pefomed. 2. Model stuctue desciption and vibation test The lab-scale bidge model studied in this chapte is designed accoding to a pototype cable-stayed bidge, the Thid Nanjing Yangtze Rive Bidge located in Jiangsu Povince of China. Since the pototype bidge is of all the chaacteistics of a moden cable-stayed bidge, the test model is assumed to be a good test bed to study the feasibility of active stuctual contol applied to cable-stayed bidges unde constuction. The test model was designed and fabicated to simulate the longest double cantileve state duing the constuction stage of the pototype bidge. Since stuctual dynamic esponse contol is the main focus of this chapte, the test model is peliminaily designed accoding to the dynamic scaling laws. Howeve, concening the estiction of test conditions, some modifications wee made duing the detailed design of the model bidge [10]. Fig. 1 shows the dimension of the designed model bidge. The bidge is composed of a 1.433 metes high cable pylon, a 3.08 metes long main gide, and six couples of stay cables. The coss section of the main gide is a ectangula of 16 mm wide and 10 mm high. At two ends of the main gide, the 3.6 kg and 3.8 kg weight AMD obits wee installed on the side span and main span espectively. The stay cables ae made of steel wie with the diamete of 1 mm. At the uppe end of each cable, an oiginal 30 cm long sping was installed and adjusted to simulate the cable foce. The cable foces wee adjusted to povide suppoting foce to the main gide and to foce it to match the designed layout of the test bidge. Table 1 shows the length, Young s Modulus and computed cable foce fo the cables. All of the components wee made of steel. Since the model consists of only one cable pylon and no othe pies, it is symmetic with espect to the cable pylon.
Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe 197 Figue 1. The elevation view (a) and side view (b) of the test model (unit: mm) Cable No. 1# 2# 3# Young s Modulus (Mpa) 451.7 327.0 216.0 Length with sping (m) 1.62 1.17 0.77 Sping Foce (N) 32.0 13.9 7.2 Table 1. Cable paametes Num. 1 2 3 4 5 6 7 8 Sensitivity (mv/g) 134.6 140.8 127.4 134.6 149.2 138.3 138.5 130.8 Location (m) 0.06 0.32 0.62 1.02 2.06 2.46 2.76 3.02 Table 2. Senso sensitivity and location distances fom the tip end of the side span
198 Advances on Analysis and Contol of Vibations Theoy and Applications Vibation tests unde foced excitations wee conducted to identify the dynamic chaacteistics of the bidge model. Eight acceleometes (as descibed in Table 2) wee distibuted along the main gide both on the side span and the main span to collect stuctual acceleation esponses at a sampling fequency of 50 Hz (as shown in Fig. 2a). A tansvese impulse foce was acted on the cantileve end of the side span to excite the stuctue. Concening that the bidge model is symmetical about the cable pylon and thus of epeated o close fequency modes, a modal identification algoithm of the capacity to identify the close modes, the wavelet based modal identification method developed by the eseach goup, is used [11]. Duing the analysis, the mothe wavelet function adopted is the complex Molet function with the cental fequency of 300 Hz and the scale incement is set to be 0.25 duing the analysis. Fig. 2b shows the wavelet scalgam of a set of esponse measuement on the tip end of the side span. As shown in the figue, stuctual tansvese vibation esponses wee dominated by two modes at the scale of 168.0 and 176.5, which coespond to the vibation modes of the natual fequencies of 1.701 Hz and 1.786 Hz, espectively. This figue also shows that the adopted modal identification method can sepaate these two close modes successfully. Stuctual modal paametes can then be estimated and the esults ae shown in Table 3. Figue 2. Collected (a) fee decay acceleation esponse and (b) its wavelet scalogam 3. Numeical modeling and model updating Fo stuctual active contol, a baseline numeical model is geneally equied fo contolle design. A FEM bidge model is thus setup in ANSYS accoding to the design diagam of the bidge model. The cable is modeled using a 3D uniaxial tension-only tuss element. Equivalent modulus fo the cables without sping ae computed using Enst fomula and then seies wound equivalent modulus fo the cable with sping can be established. Othe stuctual elements ae all modeled using 3D elastic beam element. Cables ae connected to the main gide using igid beam element. Additional masses wee modeled using isotopic mass element. The cable pylons and the main gide ae linked by coupling the hoizontal pojective intesection points of the lowest tansvese beam of the pylon with the main gide. The six DOFs at the feet of the pylon ae fixed.
Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe 199 Fequency Mode Shape The fist tansvese symmetical bending (TSB) mode The fist tansvese antisymmetical bending (TAB) mode Test 1.701 Hz 1.786 Hz Initial Model 1.848 Hz 1.856 Hz Updated Model 1.787 Hz 1.786 Hz Test [1.00 0.86 0.70 0.32 0.00 [1.00 0.84 0.72 0.28 0.00 0.30 0.70 0.90 0.99] -0.32-0.70-0.86-0.99] Initial Model [1.00 0.74 0.46 0.17 0.00 [1.00 0.74 0.47 0.17 0.00 0.13 0.38 0.60 0.81] -0.14-0.38-0.60-0.81] Updated Model [1.00 0.74 0.46 0.17 0.00 0.17 0.46 0.74 1.00 MAC* Initial Model 0.9620 0.9620 Updated Model 0.9722 0.9718 [1.00 0.74 0.46 0.17 0.00-0.17-0.46-0.74-1.00] * MAC (modal assuance citeia) is defined to be a coelation coefficient of two mode shape vectos. Table 3. The computed modal paametes befoe and afte model updating compaed with the tested mode paametes Taken eigenvalue analysis of the numeical model, stuctual natual fequencies and mode shape vectos can be computed. Table 4 shows the computed modal paametes of the tansvese bending modes. As shown in the table, since the sum of the effective mass of the fist tansvese anti-symmetic bending mode (TAB) and the fist tansvese symmetic bending mode (TSB) is 80.1% of the sum of the effective mass of all tansvese modes, these 2 tansvesal modes dominant the tansvese vibation of the bidge. Fig. 3 shows the mode shape of these two tansvese bending modes. No. Fequency (Hz) Mode desciption Effective mass (kg) 1 1.848 The fist TAB mode 6.00 2 1.856 The fist TSB mode 6.00 3 19.802 The second TAB mode 0.94 4 19.848 The second TSB mode 0.94 Table 4. The computed natual fequencies fo the fist 4 tansvese modes of the main gide Compaing the natual fequencies obtained fom the eigenvalue analysis on the numeical model and the vibation test on the test bidge, some diffeences can be obseved (as shown in Table 3). A model updating pocess is thus conducted to get an accuate baseline numeical model. The esults of the sensitivity analysis show that the vetical and tansvese vibation modes ae sensitive to the tip mass magnitude, while the Young s module of the cable is citical fo vetical bending modes. So these two paametes wee updated: the tip masses of two spans wee updated fom 3.8kg and 3.6kg to 4.16kg, espectively, and the sping modulus was updated fom 220N/m to 203N/m. Table 4 shows the natual fequencies and mode shapes of the updated model. As shown, the modal paametes of the fist TSB and TAB modes have a good match with the modal test esults.
200 Advances on Analysis and Contol of Vibations Theoy and Applications Figue 3. The baseline FEM model (a) setup in ANSYS and the computed mode shape (b) of the fist tansvese anti-symmetic and symmetic bending mode 4. Contol system simulation Based on the baseline numeical model updated accoding to dynamic measued stuctual modal paametes, a system simulation study is conducted. Fo a bidge-amd system, its govening equation of motion is T T MXt () CXt () KXt () F() t DT f() t (1) s s s e d mx () t f() t 0 (2) d d d f () t c [ x () t TDX ] b u() t (3) d d d d whee, M s, C s, and Ks ae the mass, damping and stiffness matices of the bidge stuctue; Xt () x 1 x i x n is a n-dimensional vecto, in which xi is the displacement esponse vecto of the ith DOF of the stuctue and n is the numbe of the DOF of the stuctue; Fe ( t)
Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe 201 is the excitation foce vecto acting on the stuctue; D 0 I 0 is the AMD location matix, in which 0 and I ae zeo and identity matix with appopiate sizes; fd( t ) is the actuation foce on the stuctue applied by the AMD;T is the tansfe matix fom the affiliated nodes of AMD to the pinciple nodes of the stuctue; md diag md1mdi mdl is the mass matix of the AMDs, in which m di is the mass of the ith AMD; xd( t ) ae absolute displacement of the AMD to the bidge stuctue, espectively; c d is the viscous coefficient of the AMD; b d is the foce-voltage coefficient of the AMD and ut () is the contol input voltage. Since the numeical model of a complex stuctue is geneally of a lage amount of DOFs, fo example in this study the FEM model obtained fom the last section is of 1188 DOFs, this will induce geat computation difficulty to design the contolle accoding to this so called full ode model. A educed ode model is thus equied. In this study, the citical modal eduction method is adopted fo this pupose because this method can geatly educe stuctual DOFs and the educed ode model is of clea physical meaning [12, 13]. Concening a stuctue whose vibation is dominated by the fist modes, its dynamic esponse can be appoximately expessed as Xt () Y () t and the full ode model can i i1 thus be educed to an -ode educed ode model. The govening equation of motion fo this educed ode model is T T T T T T T T T M Yt () ( C c DTTD) Yt () KYt () DT cx () t F() t DT but () (4) s s d s d d e d i mx () t cx () t ctdy but () (5) d d d d d d whee, Yt () is the modal esponse vecto supeposed by the modal esponses of the th pinciple modes; is the genealized mode shape matix composed of the mode shape T T T vectos of the fist pinciple modes; Ms Ms, Cs Cs and Ks Ks ae T stuctual genealized mass, damping and stiffness matices, espectively; Fe Fe is the genealized excitation vectos. In this study, since the vibation esponses of the fist two tansvesal vibation modes ae the most accountable fo stuctual tansvese vibation, stuctual modal mass, stiffness and damping matices ae computed using the modal paametes of these two modes. Coespondingly, the govening equation of motion fo the educed ode bidge-amd system can be deived. Fo this educed ode bidge-amd system, its state space equation is x A x B ue w (6) z z z z C x D u E w (7) y y y y CxDu Ew (8)
202 Advances on Analysis and Contol of Vibations Theoy and Applications whee, x is an a dimensional state-space vecto, a 2 l, and l ae the numbe of DOF fo the educed ode stuctue and the numbe of the installed AMD, espectively; A is an a a system matix; u is the contol input vecto fo the l AMDs; B is an a l AMD location matix; w is the genealized modal excitation vecto; E is an a excitation matix; z is an l dimensional contol output vecto; y is a q dimensional obseve output vecto. The system matices can be expessed as: Y x Y ; x d 0 I 0 A M K M C M D T c 1 1 * 1 T T T s s s s d 1 1 0 md cdh md cd ; 0 0 1 1 B Ms Hb d ; E M s ; 1 md b d 0 C z I ; 0 IA D z I ; 0 IB E z 0 ; 0 IE C y I ; D 0 IA y I ; E 0 IB y 0 ; 0 IE * T whee C C S Hc d H T ; H TD I ; 0 and I ae zeo and identity matix with appopiate sizes; and ae appopiately selected weighting matix to adjust the optimize objective of the contolle, espectively. The contol output vecto z and the obseve output vecto y ae the displacement, velocity and acceleation esponses of the bidge stuctue o the AMDs. z z y The contol output and obseve output matices C, D, C, and D y ae detemined accoding to the senso and actuato placement. The senso placement should basically A I satisfy the following obsevability citeia ank a y, whee is an abitay C complex numbe [14]. Fo a system with n modes of epeated o close fequency, n sensos ae equied fo full state esponse measuement. In this study, to povide edundant channels to collect stuctual esponse, eight acceleometes, the same as afoementioned in the modal test, ae installed along the main gide duing the contol pocess. The actuatos ae placed on the gide by checking the following contollability citeia a ank A I, B. In this study, diffeent schemes fo AMD placement will be discussed in detail in the following sections. 5. Contolle design The design of a contolle is vey impotant fo the success of stuctual active vibation contol. In this study, the LQG contol algoithm is adopted since this contol algoithm can offe excellent contol pefomance and is of good obustness as shown in some peliminay
Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe 203 study [10]. Duing the contolle design pocess, the excitation is assumed to be a stationay white noise, and the following cost function is set as the contol objective: J lim t ( T T E z ) 0 Qz u Ru dt t (9) whee z, system output vaiables, ae set to be the tansvese displacement o acceleation esponse at the tip ends; Q, a squae matix of the same ode as z, equals to the multiplication of an identity matix with a paamete q; u is a contol foce vaiable; R, the active foce weight matix, is set to be an identity matix of the same ode as the numbe of AMD applied. The design of the LQG contolle is to adjust the weight paamete q via optimizing the pefomance of the system with compensato unde the limitation of enegy supply. The design of the contolle elies on the full state feedback vecto X. Since limited sensos ae mounted on the stuctue, this full state vecto cannot be diectly measued but be estimated fom the senso measuements. When the excitation foces w and the measuement noise v ae uncoelated white noise pocess, the Kalman-Bucy filte is employed to get an estimation of the state vecto X [15]. To obtain a good contolle fo expeimental implementation, a seies of numeical analysis with diffeent value of weight paamete q ae conducted. Duing the numeical analysis, stuctual modal damping coefficients ae set to be the same as the eal measued modal damping atios. A scaled El Cento eathquake time histoy, whose dominant fequency band coves the fist 2 tansvese modal fequencies of the bidge, is adopted to excite the bidge in the numeical study. Two AMD placement stategies ae employed fo the compaison of optimal actuato location. These two stategies ae the stategy of one AMD placed at the tip end of the main span (named S1) and the stategy of two AMDs placed at the tip ends of both spans (named S2). The AMDs ae simulated to be the two electic sevotype AMD cats povided by Quanse Inc. with the following expession of the actuation foce f 1 ( t) 8.246 x ( t) 1.42 u( t) (10) d d f 2 ( t) 12.576 x ( t) 1.73 u( t) (11) d To simulate a moe pactical contol condition duing expeimental implementation, the following constaint condition is adopted: 1) The discete digital computation is employed fo the contolle computation with the sampling fequency of 500Hz; 2) The pecision of the A/D convete is set to be 12-bits and the ange of the input voltage is set to be 5 V; 3) The measuement noise of a oot mean squae (RMS) value of 0.015 V is added into each channel of the acceleation esponses, which coesponds to the 0.3% of voltage ange of the A/D convete; 4) The maximum actuation voltage is set to be 5 V with the coesponding RMS voltage of 1.67 V and the maximum actuation displacement is set to be 0.08 m with the coesponding RMS displacement of 0.027 m. d
204 Advances on Analysis and Contol of Vibations Theoy and Applications Figue 4. The atio of the contolled acceleation RMS value to uncontolled RMS value at the tip ends of the side span (a) and main span (b) with espect to the weighting paamete q fo S1 contol Fo S1 contol, a seies of numeical studies simulating the contol system with one AMD cat installed on the tip end of the main span of the bidge, whose actuation foce is expessed as Eq. (10), ae conducted when the weighting paamete q vaies fom 0.01 to 10. Fig. 4 shows the elative RMS atio of the contolled acceleation esponse to the uncontolled acceleation esponse at the tip end of both main span and side span with the vaying of q. As shown in the figue, q = 0.398 ae set fo S1 to achieve an optimal contol pefomances. This figue also tells that fo S1 contol, the tip acceleation esponse of the main span can be well contolled; howeve, a good contol pefomance fo the tip acceleation at the side span cannot be achieved by adjusting the weight paamete. Fig. 5 shows the contolled and uncontolled tip acceleation esponse of the side span and the main span when q is set to be 0.398. This figue also tells that the well designed contolle can geatly educe the acceleation esponse of the tip end of the main span but cannot mitigate the vibation of the side span. Fo S2 contol, the contol system with two AMD cats, whose actuation foce expessions ae shown as Eq. (10) and (11), installed on the tip ends of both spans of the bidge, is simulated. Numeical studies ae conducted when the weighting paamete q vaies fom 0.01 to 100. Fig. 6 shows the elative RMS atio of the contolled acceleation esponse to the uncontolled acceleation esponse at the tip end of both spans with the vaying of q. As shown in the figue, when q = 9.1, the contol system povides the most optimal contol pefomances. This figue also tells that fo S2 contol, the tip acceleation esponse of both the main span and the side span can be well educed. Fig. 7 shows the contolled and uncontolled tip acceleation time esponse of the side span and the main span when q is set to be 9.1. This figue also tells that the well designed contolle can geatly educe the acceleation esponse at the tip end of both spans.
Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe 205 Figue 5. Excitation (a), diving voltage (b), main span tip acceleation (c), and side span tip acceleation (d) time histoies of the bidge unde El Cento seismic excitation fo S1 contol
206 Advances on Analysis and Contol of Vibations Theoy and Applications Figue 6. The atio of the contolled acceleation RMS value to uncontolled RMS value at the tip ends of the side span (a) and main span (b) with espect to the weighting paamete q fo S2 contol The contol pefomance compaison of these two AMD placing stategies tell that fo the cable-stayed bidge studied, which is of two dominant tansvese vibation modes with close fequencies, the single AMD contol stategy (S1) can only educe stuctual vibation esponse of the AMD-instumented span, and the double AMD contol stategy (S2) can achieve a good contol pefomance fo educing stuctual esponse of both spans. These obsevations ae veified by checking the contollability citeia. If the contol system is of two close eigenvalues, at least two actuatos ae equied to ensue the system is contollable. Moeove, since stuctual dominant tansvese modes ae anti-symmetic and symmetic bending modes, the shift of the AMD position along one span of the bidge will only popotionally vay the coefficients of B. If two AMDs ae placed at one span of the bidge, the coesponding two columns of B ae linea dependant. Theefoe, the two AMDs must be placed at both the side span and the main span espectively to ensue the contollability of the bidge. 6. Expeimental implementation To veify the feasibility of the AMD contol fo tansvese vibation eduction of cablestayed bidge unde constuction, an expeimental study on the fabicated test model is conducted in the Bidge Testing Laboatoy of Tongji Univesity. Duing the expeiment, the S2 contol stategy was adopted accoding to the conclusion obtained fom the above numeical simulation study. Fig. 8 shows the layout of the expeimental setup. As shown in the figue, the contol system includes a data acquisition system, a cental contol compute, and two AMD cats. The data acquisition system consists of eight acceleometes, whose sensitivity is checked using dynamic calibation method; Dspace signal amplifie and filte; a geneal pupose data acquisition and contol boad MultiQ-3, which has 8 single ended analog inputs, 8 analog outputs, 16 bits of digital input, 16 bits of digital output, 3 pogammable times and up to 8 encode inputs decoded in quadatue (option 2E to 8E). The cental contol compute is of 512 Mb memoy and 1.0 GHz Intel Celeon pocesso. The
Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe 207 Figue 7. Diving voltage of the main span (a) and side span (b), and tip acceleation time histoies of main span (c) and side span (d) of the bidge unde El Cento seismic excitation fo S2 contol
208 Advances on Analysis and Contol of Vibations Theoy and Applications AMD cats (as shown in Fig. 8) ae electic sevo type [16]. They ae 0.645 kg weight. Thei tack length is 32 cm and the max safe input voltage is 5V. The system is designed unde the constaint of avoiding the AMD to knock the baffle. If the AMD position exceeds the safe ange of the obit, the system will be shut down. Figue 8. The expeimental setup (a) and the AMD devices (b) Duing the expeiment, the tip ends of the main gide wee pulled tansvesely using steel wies fistly to geneate an initial displacement in this diection. The wies wee then cut suddenly and the bidge stated to vibate due to this initial potential enegy impoted. Seveal seconds late, the powe of the AMD contol system was tuned on and stuctual vibation esponse was ecoded. Thee case studies wee conducted to check the
Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe 209 pefomance of the AMD contol system unde diffeent excitation schemes. Fo the fist case, case C1, the tip ends of both the main span and the side span wee pulled in the same diection. Afte the steel wies wee cut, the tansvese symmetic bending mode of the bidge was excited. Fo case C2, the tip ends of the two spans wee pulled in the opposite diection to excite the tansvese anti-symmetic bending mode of the bidge. Case C3 simulated bidge fee vibation unde an initial displacement of the main span. Both antisymmetic and symmetic tansvese bending modes of the bidge would thus be excited. Fo test case C1, Fig. 9 shows the AMD diving voltage, ecoded tip acceleation esponses with o without AMDs, and thei Fouie spectums. As shown in these figues, when the powe of the AMD cats was still tuned off, stuctual acceleation esponses wee aleady educed. When the powe is tuned on, the fee decay atios of stuctual esponses wee futhe inceased. That veifies the pefomance of the active contol system on stuctual esponse eduction. The compaison of the uncontolled and AMD contolled Fouie spectum magnitude of stuctual esponses also veifies this statement. Moeove, as shown, when the AMD cats wee installed, the peaks of Fouie spectum wee shifted to the left-hand-side, which meant that the natual fequencies of the system wee deceased. Table 5 shows the peak and RMS acceleation esponse of the stuctue with and without AMD. As shown in the table, afte the AMD was mounted, the peak and RMS acceleations ecoded at the tip point of the side span wee educed 44.1 % and 82.1 %, espectively. Fo the tip point of the main span, the ecoded peak and RMS acceleations wee educed 31.1 % and 81.4 %, espectively. Fo test cases C2 and C3, the peak and RMS acceleation esponses of the stuctue with and without AMD wee ecoded. As shown in Table 5, the tansvese vibation esponses fo these sensomounted points wee also efficiently educed: Fo case C2, stuctual peak and RMS acceleation esponses wee espectively educed 28.3% and 65.4% fo the tip point of the side span and 22.0% and 68.4% fo the tip point of the main span; Fo case C3, stuctual peak and RMS acceleation esponses wee educed 65.8% and 85.6% espectively fo the tip point of the side span and 40.5% and 76.7% fo the tip point of the main span. Moeove, concening that the contolle is designed via numeical studies on a educed ode model, the good contol pefomances obtained on the bidge model tells that the contol algoithm adopted in this study is of good obustness. Case No AMD Peak acc. of senso 1 With AMD No AMD RMS acc. of senso 1 With AMD No AMD Peak acc. of senso 8 With AMD No AMD RMS acc. of senso 8 With AMD C1 4.431 2.479 1.826 0.326 4.312 2.970 1.810 0.336 C2 4.198 3.012 1.341 0.464 3.563 2.779 1.327 0.419 C3 5.186 1.773 1.800 0.260 5.833 3.473 1.820 0.424 Table 5. Peak and RMS acceleation at the tip ends of the bidge with and without AMDs fo the expeimental cases
210 Advances on Analysis and Contol of Vibations Theoy and Applications Figue 9. The diving voltage (a) and tip acceleation (b) time histoies and Fouie spectum (c) of esponses fo contol case C1 7. Concluding emaks In this chapte, the active mass dampes ae implemented fo vibation contol of a lab-scale cable stayed bidge in double cantileve constuction state. The esults of both numeical
Tansvese Vibation Contol fo Cable Stayed Bidge Unde Constuction Using Active Mass Dampe 211 simulation and expeimental study veified that the poposed AMD contol technique is applicable and efficient fo the contol of tansvese vibation of cable-stayed bidge unde constuction. Fo the cable-stayed bidge studied in this chapte, stuctual vibation test showed that the bidge was of two dominant tansvese bending modes with close fequencies. The numeical study veified that fo the contol of such a stuctue with epeated fequencies; at least two AMDs should be installed fo a good contol pefomance. Moeove, the placement of those two AMDs should be caefully studied. Fo the contol of a linea, time-invaiant system, an accuate and complete system models ae geneally equied. Howeve, fo the bidge-amd system studied in this chapte, it is vey difficult, if not impossible, to set up such a numeical model due to the complex layout of the bidge stuctue. Since stuctual vibation esponses ae geneally govened by some dominant vibation modes and the objective of stuctual vibation contol is to educe but not to completely estain stuctual vibation esponses, this study veified that a educed ode model constucted fom the citical modes is good enough fo the contolle design to achieve an excellent vibation eduction pefomance. Consideing the diffeences between the numeical model and the eal stuctue, the contol algoithm adopted in this study must be obust to the vibation popety change of the contolled stuctue. The esults of expeimental studies show that the vibation of the test model can be well contolled using the contolle designed fom the numeical studies. That means the LQG contol algoithm is of good obustness fo eal application. This study is an initial wok on AMD contol of tansvese vibation of a cable stayed bidge unde constuction befoe it can be used fo eal applications. Although the expeimental study veified the efficiency of the adopted AMD contol fo stuctual esponse eduction unde given excitation, some pimay issues fo eal application of the AMD contol technique, such as how to deal with time delay, how to educe the equiement on powe supply of the contol system, and et al., ae not addessed in this study. Futhe laboatoy studies o field applications on some eal bidges will be conducted in the coming futue to discuss these issues and make the technique moe efficient and pactical fo eal applications. Autho details Hao Chen Institute of Engineeing Mechanics, China Eathquake Administation, Sanhe, Hebei, China Zhi Sun and Limin Sun State Key Laboatoy fo Disaste Reduction in Civil Engineeing, Tongji Univesity, Shanghai, China Acknowledgement This eseach is patially suppoted by the National High-tech R&D Pogam of China (863 Pogam) (Gant No. 2006AA11Z109), and Shanghai Rising Sta Tacking Pogam (Gant No. 09QH1402300). These suppots ae geatly appeciated.
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