APPLICATION OF LQR CONTROL TECHNIQUE TO OFFSHORE JACKET PLATFORM SUBJECTED TO EARTHQUAKE

Transcription

1 APPICAIO OF QR CORO ECHIQUE O OFFSHORE JACKE PAFORM SUBJECED O EARHQUAKE Abbas Ahmed Syed Khaja and Deepak Kumar Departmental of Ocean Engg. II-Madras, Chenna, Inda E-Mal: abbasgtl@yahoo.co.uk ABSRAC Offshore structures are dfferent than onshore structures n several sense. Very mportantly they dffer n atural frequency and Mass dstrbuton. atural frequency of onshore structures s much hgher than offshore structures. Generally, onshore structures have dstrbuted mass, whle offshore structure has the mass concentrated on the top of the platform. Such offshore structures are prone to extreme envronmental condtons. Under extreme envronmental condtons, offshore structures are vulnerable to damages because of hgher response. Structural workablty condtons get affected due to damage. Hence the vbratory behavour needs to be tackled wth avalable dampng technques. Among the avalable technques Sem-actve control nsures that structure should not exceed the response lmt for a wder frequency range. hs can be acheved by properly modellng the control parameters. Effcently chosen parameter guarantes the mnmzaton of responses for all types of envronmental forces. In ths study a 50 mts offshore structure s consdered. near control of the structure has been modelled for the earthquake force. he structure s subjected to El-Centro earthquake force. he response of the structure was well controlled by the near Quadratc Regulator (QR) Methodology. Keywords: tuned mass damper, offshore structure control, earthquake, sngle degree of freedom, sem-actve control, QR control. IRODUCIO Important structures have to be check for the all condtons of loadngs. he structural vbraton due to dynamc envronmental loads makes the structure unsafe. Extreme loads damage the structure and so lfe tme perod of the structure gets reduced. he protectve measure of structural control s studed by several researchers for onshore and offshore structures. For onshore structures plenty of work has been done (Dyke et al., [6]; & Spencer et al., [7]). For offshore structure manly the control s carred out on jacket platform. Floatng structure control s not studed because of ts hgh structural tme perod. Model s beng tested for the already recorded Earthquake force. But n realty the sensors mounted on the structure, sense the force and provde the data to the computer to evaluate the counter force to be provded for the actve part of control. In ths sngle degree of freedom (SDOF) structure has been modelled wth uned Mass Damper (MD) for an offshore jacket structure. he structure s subjected to the el-centro earthquake loadngs. he response of the structure s reduced by applyng the semactve control strategy. QR Methodology s utlzed to obtan the gan values. he obtaned gan value s used to mnmze the response of the structure. FORCE CACUAIOS Recorded acceleraton data were obtaned from the USGS for the elcentro earthquake for the tme perod of approxmately 53 seconds. he acceleratons are multpled by the mass to get the nertal force actng on the modal mass. Fgure-1 represents the nertal force actng on the structure. Wave condton s assumed to be calm sea state. Hence no force s added by the wave. Hgher acceleratons were only up to 30 seconds. he forces actng on the MD wll be lower snce the mass s just 2% of the man structure. Fgure-1. Force actng on the structure. EQUAIO OF MOIO he general equaton of moton s gven by mx cx kx F() t m - Modal Mass of the structure c - Dampng co-effcent k - Stffness co-effcent Ft () - Inertal force on the structure x, x, x - Acceleraton, Velocty & Dsplacement he nternal dampng forces of the structure are neglected as they are very less relatve to the sem-actve control forces. he mode shapes are obtaned from the SACS software. Modal masses are evaluated by usng normalzed (1) 17402

2 mode shape. Modal mass can be evaluated by followng formula (Rahul Rana., [2]) m M (2) m, - Modal mass and Actual mass - ormalsed Mode shape transpose and normalsed mode. he Fgure-2 shows the behavour of the structure due to nertal force generated by the earthquake acceleraton. he fgure represents sngle degree of freedom. Fgure-2. Structure behavour under nertal force. SAE-SPACE FORMUAIO he general form of representaton of the state space s gven by followng equatons. x( t) A( t) x( t) B( t) u( t) (3) y( t) C( t) x( t) D( t) u( t) (4) At ()- State matrx Bt ()- Input matrx () () Ct - Output matrx Dt - Drect transmsson matrx he equaton of moton of the sngle degree of freedom wthout dampng s gven n equaton 5 and MD s gven n equaton 6 my ky k( y y ) ExternalForce (5) m y k ( y y) c ( y y) (6) m - Modal Mass of tuned mass damper k - Stffness of SDOF y - Dsplacement of man mass y - Velocty of man mass y - Velocty of MD y - Acceleraton of man mass y - Acceleraton of MD he chosen state varables are x y; x y; x y ; x y (7) he state space matrces obtaned are gven below ( k k) / m 0 k / m 0 A k / m c / m k / m c / m 0 1/ m B * m* a 0 0 C D 0 (8a) (8b) (8c) (8d) MODE PARAMEERS he parameters are obtaned from SACS model. State values are evaluated accordngly. m= kg mt= kg k1= /mt ct= S/mt A e 7 B= * m* a

3 he MD s placed where the maxmum dsplacement occurs. hs helps n effectve reducton of the responses. he Fgure-3 represents the general setup wth MD placed at top of the structure. he rccat equaton provdes the soluton to obtan the value of P 1 A P PA PBR B P Q 0 (11) near gan s mentoned as ( k ) s gven as 1 k R B S k and the on-lnear gan (12) he rccat equaton for the non-lnear compensaton s gven as S A Bk A Bk S SB R B S 1 ( ) ( ) ( ) Q 0 he fnal compensatory control force s gven as (13) Fgure-3. Structure behavour under nertal force. SEMI-ACIVE CORO MEHODOOGY he control algorthms adopted for actve control part of the structure s QR technque. he control has to be desgned for lnear and non-lnear values of the structure. he multloop feedback gans for lnear and nonlnear wll reduce the vbratonal responses. QR methods general form for a fnte tme () s gven as 1 0, x Sx + x x u x k0 J Q R (9) he state feedback law s gven by the followng equatons for the lnear part u And kx (10a) 1 ( k R B PB) B PA (10b) Where S and Q are symmetrc and non-negatve defnte matrx and R s a symmetrc and postve defnte u R B P R B S X (14) 1 1 ( ) In ths case snce the model s consdered to be lnear. So only lnear values have been obtaned for control. he followng gan values have been obtaned for Q 1000*( c'* c) & R1e 5. K P 1.0e e SIMUIK MODE he smulaton has been carred out wth the Smulnk model shown below. he feedback s shows the evaluated control gans to the system whch reduces the responses of the structure. he gan can also be ncluded n state-space functon block n whch parameter can be taken as A Bk ntegratng (Mohamed zrb.,[3]).. he equaton3 can also be solved by Fgure-4. Smulnk model

4 RESUS AD DISCUSSIO Fgure-5. Passve control of the structure wth MD. Fgure-6. Sem-Actve control of the structure wth MD. Fgure-7. Wthout control, passve and sem-actve control

5 Percentage control n passve control depends on mass of control devce. In ths case mass rato of the MD s taken as 2%. For the same mass rato, sem-actve control (QR) s used to obtan the control response. Uncontrolled and controlled response usng sem-actve control s shown n Fgure-7. he percentage control obtaned s 53% n comparson wth passve control. Clearly there s ncrease n percentage control n semactve control over passve control. he obtaned smulaton results n the above Fgures-5 & 6 show the passve and sem-actve control of the structure. he passve control shows a maxmum response of 2.82mts at 29.5secs. Sem-Actve control gves a maxmum response of 1.325mts at 23.4secs. COCUSIOS he effect of earthquake response s studed on jacket platform of 50mts. State-space Sngle Degree of Freedom (SDOF) Model has been developed wth MD. Smulaton has been carred out for passve and semactve control of the system subjected to elcentro earthquake force. Only lnear part of the control was studed wth near Quadratc Regulator (QR). Dsplacement response of jacket deck n surge drecton s obtaned usng passve control, sem-actve control and wthout any control. he control was effectvely appled and responses were reduced as desred. Percentage reducton n the response between passve dampng and sem-actve dampng was 53.05%. [5] Shehata E. Abdel Raheem Study of nonlnear response of steel fxed offshore platform under envronmental loads. Arab J Sc Eng- 39. pp [6] Dyke S.J., Spencer Jr., B.F., San, M.K. and Carlson, J.D Expermental verfcaton of sem-actve structural control strateges usng acceleraton feedback. Proc. 3rd Int. Conf. on Moton and Vb. Control, Chba, Japan, III, pp [7] Spencer Jr., B.F., Carlson, J.D., San, M.K. and Yang, G On the current status of magnetorheologcal dampers: Sesmc protecton of full-scale structures. Proc. Amercan Control Conf. pp , Albuquerque, M. REFERECES [1] Kenj Kawano, K. Venkataramana and Kohe Furukawa Earthquake engneerng, enth world conference. Balkerna, Rotterdam. pp [2] Rahul Rana Response control of structures by tuned mass dampers and ther generalzatons. Paper o Eleventh world conference on earthquake engneerng. [3] Mohamed Zrb, af Almutar, Mohamed Abdel- Rohman and Mohamed erro on-lnear and robust control schemes for offshore steel jacket platform. onlnear Dynamcs, Vol. 35. pp [4] Davorn Hrovat, Pnhas Barak and Mchael Rabns Sem-actve versus passve or actve tuned mass dampers for structural control.j. Eng. Mech.109. pp