Active vibration suppression with using centralized actuators and fibres Ing. Tereza Kašpárková

Transcription

1 Actve vbraton suppresson wth usng centralzed actuators and fbres Ing. ereza Kašpárková Vedoucí práce: Doc. Ing. Zbynk Ška PhD Abstrakt Obvykle se k tluení vbrací kopoztových struktur pouívá velké noství aktuátor. ento lánek se na rozdíl od tohoto obvyklého zpsobu snaí tlut vbrace kopoztové struktury s poutí alého noství centralzovaných aktuátor.v lánku je popsán návrh typu senzor a aktuátor a jejch uíst"ní na základní struktu#e. K zp"tnovazebnéu #ízení je zde pouta etoda QR. Klíová slova Aktvní tluení vbrací, kopoztové struktury, QR. Introducton he coposte aterals are very popular because they have low weght hgh strength and possbltes of low teperature of expansblty. he planes, the bullet proof vests, soe constructon n space and the parts of cuttng or shapng achne are anufacture of coposte ateral. hs ateral have soe probles too. he bggest proble of ths ateral s lower dapng. he good soluton of ths proble s connecton of actve eleents to the coposte structure. he eleents are pezoelectrc patches and fbres. Mostly t s used a lot of eleents. hs concept have soe probles wth hgh electrc voltage, hgh prce and a lot of wre. hs probles leads to dea of new concept of soluton of the proble. Naely the use of only few lnear drves and fbres correctng the drves and structure.. he concept of dapng he coposte structures s four corned, thn - walled bea []. hs bea s ade fro carbon fbres. Hs length s 80 c and hs secton s 0x0 c. Each soluton are calculated for the freely hung bea. wo lnear drves are connected to each wall of the bea. he lnear drves are placed n the ddle of the wall of the bea. wo fbres connected each drves and sutable places of the bea (fgure ). he actve eleents (lnear drvers and fbres) are placed nsde the bea. he concept contan 6 sensors too. he sensors are acceleroeters. Fve acceleroeters easure the axs x, fve acceleroeters easure the axs y and sx acceleroeters easure the axs z. hey are placed on the outsde walls of the bea. Fg.. New concept red are actng eleents and blue are sensng eleents

2 3. he placeent of sensors and actuators he H Nor [] s used for optzaton of place for connectng sensors and fbres to the bea. It s calculated fro the state space for (equaton) [] x& A y C x + B x r d + B u, () where the atrx A s systes atrx, the atrx B s control nput atrx, the atrx B r s dsturbng atrx and C s output atrx. 3. he sensors he concept contan about two thousands and fve hundred places for connecton sensors. he H Nor of the th ode and kth sensor s n equaton k B C k, í K N, k KR () where N s the nuber of all ode and R s the nuber of all places for connecton sensors. he H placeent ndex f the th ode and kth sensor s n equaton 3 k k wk, í K N, k KR (3) where w k?0 s weght assgned and s transfer functon. he sensor placeent atrx s ade up fro sensor placeent ndexes n ths for n n k k k nk R R R nr (4) where the th row s set ndexes of sensors for th ode and the kth colun consst of ndexes of the kth sensors for every odes. he vector of sensor placeent ndexes s defned as [ ], where sk s rs su kth sensor ndexes over all odes S s s sr n (5) sk he ndex sk characterzes the portance of the kth sensor. he places wth the hghest ndexes are good for placng the sensors. k

3 he concept contan fve x-drecton sensors, fve y-drecton sensors and fve z-drecton sensors. he ndex sk s calculated for each three drecton (fgure, 3) and approprate places are chosen fro places wth bggest sk. Fg.. Index sk for drecton x and y Fg. 3. Index sk for drecton z 3. he actuators he bea s dvded to the two parts for fndng approprate places to connectng the fbres to the bea. he lnear drves ake two sets. Each sets are connected to sets of places between lnear drves and further end of the bea (fgure ). Both set of places for connectng fbres contan about one thousands place. he angles between fbre and the coordnate axes are very portant for actuator effcency and for every drves are dfferent. hs angel deterne the force rate n axes x, y, z. he atrx B s ade wth reflect angles between fbre and bea. he H Nors of the th ode wth the jth actuators s j B j C, í K N, j KS (6) where N s the nuber of all ode and S s the nuber of all places for connecton fbres. he H placeent ndex f the th ode and jth place for connectng fbre (actuators) s n equaton 7. j j wj, í K N, j KS (7)

4 where w j?0 s weght assgned and s transfer functon. he actuator placeent atrx s ade up fro actuator placeent ndexes n ths for n n j j j nj S S S ns (8) where th row s set ndexes of actuators for th ode and the jth colun consst of ndexes of the jth actuators for every odes. he vector of actuator placeent ndexes s defned, where s rs su jth actuator ndexes over all odes as [ ] a a a as n (9) he ndex characterzes the portance of the jth place for connectng fbre to bea. he places wth the hghest ndexes are good for connectng fbres. he concept contan 6 fbres and eght lnear drves. he ndex s calculated for each eght drves and approprate places for connectng two fbres to bea are chosen fro places wth bggest.soe fgure wth for the frs set of places are n the fgure 4,5. j Fg. 4. Index for drves and Fg. 5. Index for drves 3 and 4

5 4. he control he actuators are under control of the state space feedback control [3]. he state space defnton of syste (equaton ) s necessary for the control synthess. he control nput s defne by the state space feedback as u K x (0) where K s the gan of feedback control. Fro equatons and 0 follow x& ( A B K) u + B d () he QR [4] s ethod of state space feedback control. hs ethod s used to fndng the gan K. he gan K s deterned as nzaton of quadratc cost functon J n equaton r J ( x Q x + u u) 0 dt () where Q and represent weghts on the dfferent states and control channels. he atrx Q have to be syetrc se-postve defnte and have to be syetrc postve defnte. he atrx Q [5] s where H s the vector of weght of the state. he gan atrx K s Q H H (3) K B P (4) where P s calculated fro Rccat equaton PA + A P + Q PB B P 0 (5) he control s used to dapng 6 odes so the vector H have sze x6. he weghts of the control channels s he results he fnal results are coposed fro the results of placeent and the results of control. he results of placeent are the optu places for sxteen sensors and sxteen fbres. he results of the control s the gan atrx K. he gan atrx s odfed wth respect to the coordnate systes on the bea wall. he fnal results are n the fgure 6. hey are transfer functon of controlled and uncontrolled syste

6 Fg. 6. ransfer functon to the tree another sensors 6. he concluson he paper presents the new concept of actve vbraton suppresson of coposte bea. Eght lnear drves wth sxteen fbres connectng the drves and the structure are used as actuaton syste. 6 one-axs acceleroeters are connected to the bea as a sensors. he ethod uses only a few actuators. hs s the an advantage wth respect to the concept wth hgh nuber of dstrbuted pezoelectrc actuators. he approprate placng of sensors and fbre connecton s very portant for the effcency. he control law synthess s realzed by the state space feedback control usng QR. 7. he Acknowledgeents he authors apprecate the knd support of the grant "Research of new prncple of echancal and boechancal systes wth a ntellgent behavour" (D0/08/H068) he lst of the sybols x syste state (odal state) [] u control nput [kg..s - ] d dsturbng nput [kg..s -, kg. s - ] y easure output [.s - ] A syste atrx [-]

7 B nput control atrx [-] B r dsturbng atrx [-] C output atrx [-] syste H Nor [-] B nor of nput control atrx [-] C nor of output atrx [-] w weght atrx [-] odal dapng [-] M natural frequence [/s] N H nor placeent ndex [-] N vector of placeent ndexes [-] O placeent atrx [-] K gan atrx [-] J quadratc cost functon [-] Q weght atrx [-] weght atrx [-] H vector of weght of the state [-] coeffcent [-] k coeffcent [-] j coeffcent [-] N nuber of ode [-] S nuber of places for actuator [-] R nuber of places for sensor [-] References [] Kašpárková,., Ška Z, Actve vbraton soppresson of coposte bea, th Workshop on Appled Mechancs, Proceedng, 00,. [] awronsk, W. K., Advanced Structural Dynacs and Actve Control of Structures, Mechancal Engneerng Seres, Sprnger, 004 [3] Kejval, J., luení vbrací stroj poloaktvní dynacký hlte. DsertaQní práce, RVU, Praha, 00 [4]. E. Stavroulaks,. Foutstz, E. Hadjgeorgou, D. Marnova, C. C. Banotopoulos, Desgn and robust optal control of sart beas wth applcaton on vbratons suppresson, Advances n Engneerng Software, 36, 005, [5] Ška, Z., Aktvní a poloaktvní snování echanckého ktání stroj. HabltaQní práce, RVU, Praha, 004