An Application of the IMSC on a Non-linear Flexible Structure: Numerical Analysis and Experimental Validation

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1 , July 6-8, 2011, London, U.K. An Applaton of the IMSC on a Non-lnear Flexble Struture: Numeral Analyss and Expermental Valdaton G. Caulan, F. Resta, F. Rpamont Abstrat he ndependent modal ontrol to suppress the vbraton of nonlnear flexble strutures s appled n ths paper. ehnologal mprovements n the mehanal feld showed durng the reent years have led to hgh-performane systems wth low weght and, as a onsequene, hgh flexblty and low dampng. Here atve ontrol qukly bettered the tradtonal passve dampng systems. he struture nvestgated n ths paper s a mult-body flexble boom moved by hydraul atuators. he nonlnear system dynam was numerally modeled and a ontrol strategy, based on the use of the same atuators, was developed. Fnally a test rg was reated to expermentally valdate the proposed approah. Index erms Independent modal spae ontrol, nonlnear ontrol, vbraton suppresson I. INRODUCION In reent years vbraton ontrol has been nreasngly used not only n aerospae researh but also n many mehanal applaton felds. he need for weght reduton to mprove system performane led to strutures wth hgh flexblty and low dampng. hese strutures suffer fatgue and nstablty ssues rasng, as a onsequene, a number of safety questons. radtonal external passve ontrol methods are generally more nvasve, ntrodung mass nto the struture, and less effetve n a large range of frequenes. On the other hand, atve ontrol s an attratve soluton, espeally onsderng the extensve development of alulator hardware and the onsequent ost reduton. Among the dfferent ontrol tehnques proposed n reent deades, modal ontrol offers many advantages thanks to ts mmeday and onneton to the dynam desgn of the system. Moreover the mprovement of sensor and atuator tehnology and nnovatve algorthms allows the spllover lmts due to the trunated modes and unmodelled dynam to be partly overome. Manusrpt reeved Marh 05, 2011; revsed Aprl 12, 2011 hs work was supported n part by CIFA S.P.A. G. Caulan s wth the Mehanal Engneerng Department of Polterno d Mlano, Mlan, IALY (orrespondng author to provde phone: ; fax: ; e-mal: gabrele.aulan@me.polm.t). G. Resta s wth the Mehanal Engneerng Department of Polterno d Mlano, Mlan, IALY (e-mal: ferruo.resta@ polm.t). F. Rpamont s wth the Mehanal Engneerng Department of Polterno d Mlano, Mlan, IALY (e-mal: franeso.rpamont@ polm.t). In lterature modal ontrol was ntrodued by Balas [1] and Merovth [2] between the 1970s and 1980s. Balas studes deal wth vbraton suppresson n large spaeraft strutures, applyng the modal expanson theorem and modal ontrol. Some years later Merovth proposed an ndependent modal spae ontrol (IMSC) usng the modal flter tehnque for estmatng the modal oordnates [3]. he method was further mproved by Ba and Poh [4], who suggested a modfed ndependent modal spae ontrol (MIMSC) for optmal gan alulaton adoptng peoeletr atuators. Anyway Ln and Chu [5] demonstrated that modal ontrol, for a general dynam system wth omplex mode shapes, does not assure stablty even for the ontrolled modes. he spllover problems were partally ushoned by the use of dstrbuted sensor and atuators (Lee and Moon [6]) and some applatons of IMSC for the ontrol of flexble lnkage mehansm (Zhang [7] and Changjan [8]) an be found. In 2001, Inman too [9] dsussed the spllover ssue assoated wth modal ontrol, onludng that modern tehnology makes the problem manageable. Moreover, modal ontrol an be employed n onjunton wth FEM for model defnton, as shown by Skdmore [10] and Khulef [11]. hey studed an atve ontrol sheme for vbraton suppresson n a sngle beam/able struture, onsderng also ts large moton. hs paper deals wth ndependent modal ontrol on a mult-body (MB) non-lnear flexble boom. In partular suppresson of the vbraton aused by the large moton of the struture was nvestgated. Beause only a few modes atvely partpate n desrbng the system s dynam behavor, modal ontrol s partularly useful. he modal model requred for the ontrol law synthess an be defned both expermentally, through a modal parameter dentfaton ampagn, or numerally, by means of FEM. In the proposed work a FEM approah was followed. he model not only allowed ontrol/observer gan to be defned through pole plaement tehnques, but also permtted numeral smulatons. At the same tme a test rg was reated to valdate the defned ontrol law. Fnally a omparson between the behavour wth and wthout ontrol s proposed. II. HE SYSEM he boom s omposed of a number of segments, onneted to eah other wth revolvng jonts (Fg. 1). Eah segment s movement s generated by an hydraul atuator

2 , July 6-8, 2011, London, U.K. through a knemat han assumed to be rgd. [ M ] δ + [ R ] δ + [ K ] δ = f g ( ) + [ Λ ] Fat (2) - δ s defned as δ = - [ ] K respetvely represent the nertal and elast matres, onsderng both the strutural and gravtatonal terms and the atuator ontrbuton; R s assumed to be proportonal to the elast and nertal ones. he proportonal oeffents are estmated from expermental data; f represents the onstant ontrbuton of the M and [ ] - the dampng term [ ] - g gravtatonal term. Owng to the low segment rotaton speed, entrfugal and Corols terms were assumed as neglgble. he followng paragraphs present and numerally/expermentally nvestgates a ontrol log for redung the strutural vbratons. Fg. 1. he test rg Owng to the length and the low flexonal stffness of the seton, the struture presents hgh flexblty produng, n normal operatng ondtons, low frequeny vbratons, generally assoated wth a low dampng rato. A numeral model, desrbed n [12], was developed as a tool both for defnng the ontrol log and for smulatng boom behavour. he boom knemats were solved usng the floatng frame of referene formulaton [13] to desrbe the large moton and the Fnte Element Method (FEM) to model the segment flexblty. Usng the Lagrange formulaton, the boom s non-lnear equaton of moton was obtaned ( ) = f (, ) + [ ] () t M Λat Fat (1) - represent the ndependent varable vetor, ontanng all the segment rotatons and nodal dsplaements M represents the nertal ontrbuton; - ( ) - f (, ) ontans all the non-lnear dampng, elast and gravtatonal terms F represents the large moton atuator fores and - at [ Λ at ] represents the knemat relatonshp between the atuator length and the ndependent oordnates vetor. Eq. (1) s a non-lnear equaton beause all the matres depend on the moton varables and n partular on the segment onfguraton. o study the vbraton problem and obtan system egenvalues, the non-lnear equaton an be evaluated and lneared n eah boom onfguraton. In a gener onfguraton of the system =, the (1) beomes III. ACIVE CONROL In operatng ondtons, the system desrbed earler s subjeted to a gener large moton whh auses, beause of the hgh flexblty of the segments, sgnfant vbraton levels. o suppress these vbratons ( δ ), ths paper proposes applyng a ontrol aton usng the same atuators used to move the ndvdual segments. In many pratal applatons, n fat, the atuator number and poston s assgned and annot be modfed. he vbraton ontrol log adopted s based on an ndependent modal approah [2,14,15]. Followng ths approah, the ontrol aton s alulated startng from the system vbratory state and a sutably defned gan matrx. In partular the vbratory state s desrbed by a set of modal oordnates representng the system dynams n the frequeny range of nterest. he modal oordnates, whh for a gener applaton annot be measured dretly (unless we use dstrbuted sensors [6]), have to be estmated by a modal observer. Below, the ndvdual steps are analyed n depth. he ontrol synthess an be alulated startng from (2), ntrodung the ontrol law vetor u and wthout onsderng the gravtatonal and external fores terms beause these ontrbutons don't modfy the value of the poles of the system [ ] δ + [ ] δ + [ ] δ = [ ] M R K Λ u (3) A. Independent modal ontrol he am of the ontrol fore u s to nrease the system s dampng rato n order to redue vbratons by mposng new values λ for the system s orgnal egenvalues λ. In partular the real part of the system s egenvalues was nreased, whle the magnary part remaned unhanged so as to avod havng to use sgnfant ontrol fores and mposng addtonal mehanal fatgue stress on the boom materal. As prevously desrbed, n the present work the moton

3 , July 6-8, 2011, London, U.K. equaton was obtaned usng FEM dsretaton of the lnks. hs means that the varable vetor ontans, as mentoned n par. 2, all the nodal dsplaement of the struture, and ts dmenson s not ompatble wth a fullstate feedbak formulaton. Moreover, t should be onsdered that the struture dynam s ruled only by the frst modes, beause hgh frequeny modes usually have hgher dampng ratos and they an hardly be exted. For ths reason a redued modal system needs to be defned to desrbe boom moton usng a lmted set of oordnates, takng nto aount the well-known problem of spllover. Defnng the omplete modal oordnates vetor as q tot and the m redued modal oordnates vetor q, the followng oordnate hange an be performed [ ] δ = φ q φ q (4),tot tot φ,tot represents the n n egenvetor matrx of 1 [ M] [ K ], whle [ ] φ s an n m partton of φ,tot ontanng only the egenvetors of the modes onsdered n the ontrol formulaton. Applyng the (4) to the (3), the moton equaton beomes a set of ndependent modal equatons [ ] + [ ] + [ ] = [ ] [ ] m q r q k q φ Λ u (5) [ m ], [ r ] and [ ] obtaned by [ ] = [ ] [ ][ ] k are m m dagonal matres, m φ M φ (6) should be noted that (5) and (6) are only vald for systems wth real modes. Anyway the examned system dampng s very small and ts modes an be assumed real. o apply the state-spae ontrol approah to the redued modal system, the (5) an be wrtten n state-spae form 1 1 [ m] [ k] [ m] [ r] [] I [] 0 1 [ m] [ φ] [ Λ] [ 0] q = q q + u = A q+ B u q = q he ontrol fore s defned as [ ] q q u = G q (9) (7) (8), applyng the ndependent modal ontrol approah, the ontrol gan matrx s alulated as [12] ( ) G 1 1 q B 0 A U λ U (10) [ ] = q [ ] [ ][ ][ ] λ s a dagonal matrx ontanng the egenvetor - [ ] mposed for the ontrolled system - [ U ] s the rght egenvetor matrx of A q - q B represents the upper part of B q and must be nvertble; as shown n (7), the lower part s equal to ero when the system s desrbed by 2 nd order dfferental equatons In ths way, the (7) beomes [ ], q = A q B G q = A q (11) q q q he dsadvantage of the state-spae approah s that the gan matrx values lose physal meanng. For ths reason, the modal ontrol law proposed has been alulated mposng that the term [ φ] [ Λ] u of (5) doesn't ouple the system modes and leads to an nrease of the system dampng rato [ ] [ ] [ ] φ Λ u = g v q gp q (12) p g v have a prese physal meanng, beause they represent respetvely the nrease n stffness and dampng rato brought about by the ontrol. o ensure ndependent modal ontrol they must be dagonal, so that the ontrol law provdes an ndependent stffness and dampng ontrbuton on eah mode, keepng the same egenvetors as the unontrolled system. hs ondton an be obtaned only f the number of atuators s equal to the number of onsdered modal oordnates. In addton, the system must be ontrollable, so any row or olumn of [ φ] [ Λ ] must be non-ero. hs means that at least one ontrol fore must have a non-ero ontrbuton on every mode and eah ontrol fore must at on at least one mode. he ontrol fore an be alulated as g and [ ] 1 ([ ] [ ] ) [ ] ( ) u = φ Λ gv q + gp q q (13) = [ v] G G p q g and g p, the desred system egenvalues an be set for the ontrolled system. In partular g p an be set to ero to keep the same natural frequenes as the unontrolled system. Seletng [ ] v B. he modal observer Assumng that the exat modal oordnates are known,

4 , July 6-8, 2011, London, U.K. the method proposed guarantees a ompletely deoupled modal ontrol and avods spllover problems, even f the number of ontrolled modes s smaller than the number of struture modes. However, n most flexble struture ontrol applatons the modal oordnates are unknown and so have to be estmated usng a modal observer or flter [3]. In ths paper a modal observer has been onsdered (Fg. 2). C estmated modal oordnates measurements (ontrol fores) D measurements Boom (aeleratons) A q G o Fg. 2. he sheme of the modal observer B q measurements (ontrol fores) Aordng to the (7),the observer equaton an be wrtten as qˆ = A ˆ ˆ q q+ Bq u + Ko y y (14) [ ]( ) - ˆq s the estmator of the system modal oordnate vetor; K s the observer gan matrx, whh wll be - [ ] o defned later; - y and ŷ respetvely represent the measurements and estmated measurements vetors. y s an observer nput, whle ŷ s defned by [ ] [ ] yˆ = C q+ D u (15) he values of [ C ] and [ ] D, whh lnk the estmated measurements vetor to the observer states and ontrol fores, depend on sensor type (aelerometer, poston sensors, stran gauges, et.) and poston. Consderng the (15), and negletng the ontrbuton of [ D ], the observer equaton (14) beomes ( [ ][ ]) ˆ q o q [ o] [ A ] qˆ B u [ K ] y qˆ = A K C q+ B u + K y = + + o q o (16) Now t s possble to defne the last term of the observer, the gan matrx [ K o ], whh s alulated usng the pole plaement method. IV. NUMERICAL AND EXPERIMENAL ANALYSIS In the prevous paragraph the vbraton ontrol log was defned. o test the adopted soluton, a numeral and expermental ampagn was arred out. he test rg, ntrodued n par. 2, was nstrumented wth - 3 load ells, to read total atuator fores; - 3 LVD sensors, to obtan boom onfguraton; - 1 aelerometer used as the observer nput and the boom vbraton ndator, loated at the end of the last segment. As sad n par. 2, the moton of the system s desrbed by a non lnear equaton. For ths reason all ontrol matres need to be updated at every step to make the ontrol log rgorous. hs updatng an be easly reprodued n the numeral smulatons, there are no lmts due to realtme ssues. However, the expermental applaton requres real-tme alulaton of ontrol matres and normal mroproessors do not have enough power or speed to handle the requred tme sale, so an approxmaton must be appled. In any ase, onsderng that the rotaton speed of the segment (and so the matrx varaton) s lower wth respet to the ontroller dynam, ths approxmaton doesn t ompromse ontrol effeny and two possble approahes an be adopted. he frst onssts n alulatng ontrol matres every tme steps, usng the ontrol board n a mult-taskng mode. A faster method alulates the ontrol fores every tme step whle the other method alulates them at longer ntervals. he seond approah s to pre-alulate the matres for a dsrete set of boom onfguratons. In ths way, at every tme step, the software smply needs to extrat the nearest onfguraton matres usng lnear nterpolaton. he atual boom onfguraton, as mentoned, an be determned from the atuator LVD output. he lmt of ths soluton s the memory avalable on the ontrol board. Consderng the small amount of memory requred by the present applaton, ths seond method was preferred and mplemented. he ontrolled system and observer pole are set as follows: - the magnary part of eah ontrolled and pole s set equal to the magnary part of the unontrolled system. hs s beause we are not nterested n modfyng the system frequenes sne ths s not the am of the present work and would lead to hgh ontrol fores. As mentoned, a ontrol fore nrease ould lead to hgh mehanal stress and fatgue problems. Instead, the magnary parts of the observer poles are set twe those of the unontrolled system, n order to make the observer dynams faster than the system dynams; - the real part of eah ontrolled and observed pole s plaed respetvely equal to 15% and 30% of the orrespondng magnary one, n order to set the ontrolled system dampng rato hgher than that of the unontrolled boom. For omparson, the possblty of usng a o-loated ontrol fore was analyed. hs soluton s the smplest one, beause t only requres the measurement of the atuator length and s the most natural soluton for addng dampng to the system. he resultng ontrol fore, onssts

5 , July 6-8, 2011, London, U.K. of a o-loated dervatve ontrol, that nreases the total dampng of the system by addng a dampng ontrbuton on the boom atuators. hs soluton s smpler than modal ontrol. In fat t requres low omputatonal effort and doesn t ompromse system stablty. However, ths soluton doesn t provde a large nrease n the dampng rato. In fat ths dervatve aton only works on atuator vbratons, wthout onsderng the vbratons of the segments due to ther flexblty. As a onsequene, even after optmaton, only a small dampng rato nrease s aheved. A. Numeral analyss hs paragraph presents some numeral results. All the smulatons refer to the numeral model of the test rg that was used for the experments. All the results (wth and wthout vbraton ontrol) refer to the same large movement of the boom (Fg. 3). ABLE I COMPARISON BEWEEN HE DAMPING RAIOS OF HE FIRSHREE MODES OF HE BOOM IN HORIZONAL CONFIGURAION WIH AND WIHOU CONROL Dampng ratos [%] No ontrol Modal ontrol Mode Mode Mode B. Expermental results hs paragraph presents the results obtaned on the test rg (Fg. 1). As n the numeral ase, the aeleraton of the end of the thrd segment s onsdered to estmate ontrol performane. Fg. 5 shows the expermental results, obtaned reprodung the same ondtons as n the numeral smulatons (movement, ontrol parameter, measurement nstruments). A good agreement wth the numeral data was obtaned. 14 No ontrol No ontrol Modal ontrol Modal ontrol aeleraton [m/s2] 8 6 Aeleraton modulus [m/s 2 ] 4 Fg. 3. Large moton referene for the boom segments: startng and fnal onfguraton (a) and rotaton referene (b) 2 0 (a) tme [s] (b) Frequeny [H] Fg. 4 ompares the system vbratons wth and wthout modal ontrol, hghlghtng the nrease n dampng. he aeleraton of the end of the thrd segment s onsdered as ndatng ontrol performane. On the left, the fgure shows the aeleraton tme hstory durng the large moton desrbed n Fg. 3. he tme hstory also nludes stat aeleraton due to gravty. On the rght t shows, on a logarthm sale, the aeleraton spetrum wth the applaton of an external fore on the frst segment end. hs smulaton was arred out n the fnal onfguraton of the large moton shown n Fg. 3. he external fore onssts of a frequeny sweep up to 7 H, that nvolves the frst three natural frequenes of the struture. he fgure shows that modal ontrol auses a large reduton of vbratons near the ontrolled natural frequenes. However, system response nreases between these. aeleraton [m/s2] No ontrol Modal ontrol (a) tme [s] Aeleraton modulus [m/s 2 ] No ontrol Modal ontrol (b) Frequeny [H] Fg. 4. Numeral aeleraton of the end of the thrd segment: tme hstory (a) and spetrum (b) In any ase, as able 1 also shows, modal ontrol an ensure a large nrement n the boom dampng rato. Fg. 5. Expermental aeleraton of the end of the thrd segment: tme hstory (a) and spetrum (b) In order to omplete the numeral-expermental analyss, n Fg. 6 a omparson s shown. Aeleraton modulus [m/s 2 ] Numeral model Expermental test rg (a) Frequeny [H] Aeleraton modulus [m/s 2 ] Numeral model Expermental test rg (b) Frequeny [H] Fg. 6. Numeral-expermental spetra omparson wthout (a) and wth (b) modal ontrol V. CONCLUSIONS In ths paper a ontrol strategy was mplemented to suppress the vbratons ndued n a non-lnear hghlyflexble boom by ts large moton. For ths knd of struture the modal soluton remans an attratve one for modellng the system dynams wth a redued number of degrees of freedom n a foused range of frequenes. As a onsequene the non-lnear atve ontrol log adopted s based on the ndependent modal ontrol theory, llustrated usng the 2 nd order equaton typal of mehanal systems. In the same way, a modal observer was mplemented to estmate the non-measurable modal oordnates. he ontrol fore s appled to the system through the same hydraul atuators responsble for the boom s large moton. Frst ths methodology was tested usng numeral

6 , July 6-8, 2011, London, U.K. smulaton, showng that an nrease of the dampng rato of up to about 15% ould be aheved wthout affetng the large moton and redung materal stress durng operaton. Fnally a test rg was reated and the methodology was expermentally valdated. he expermental results show a very good agreement wth the numeral ones and the modal ontrol aton provdes an hgh reduton of test rg vbratons. REFERENCES [1] M.J. Balas (1978). Atve ontrol of flexble systems. Journal of Optmaton heory and Applatons, Volume 25(3), pp [2] L. Merovth (1983). Comparson of ontrol tehnques for large flexble systems. Journal of Gudane, Control and Dynams, Volume 6(4), pp [3] L. Merovth and H. Baruh (1985). Implementaton of modal flters for ontrol of strutures. Journal of Gudane Control and Dynams, Volume 8(6), pp [4] A. Ba and S. Poh (1990). Expermental mplementaton of the modfed ndependent modal spae ontrol method. Journal of Sound and Vbraton, Volume 139(1), pp [5] Y.H. Ln and C.L. Chu (1995). A new desgn for ndependent modal spae ontrol of general dynam systems. Journal of Sound and Vbraton, Volume 180, pp [6] C.K. Lee and F.C. Moon (1990). Modal sensors/atuators. Journal of Appled Mehans, ransatons ASME, Volume 57(2), pp [7] X. Zhang, H. Lu and W. Cao (1996). Atve vbraton ontrol of flexble mehansms. Chnese Journal of Mehanal Engneerng, Volume 32(1), pp [8] S. Changjan, Z. Xanmn and S. Yunwen (2000). Complex mode atve vbraton ontrol of hgh-speed flexble lnkage mehansms. Journal of Sound and Vbraton, Volume 234(3), pp [9] D.J. Inman (2001). Atve modal ontrol for smart strutures. Phlosophal ransatons of the Royal Soety Seres A, Volume 359, pp [10] G.R. Skdmore and W.L. Hallauer Jr (1985). Modal-spae atve dampng of a beam-able struture: theory and experment. Journal of Sound and Vbraton, Volume 101(2), pp [11] Y.A. Khulef (2001). Vbraton suppresson n rotatng beams usng atve modal ontrol. Journal of Sound and Vbraton, Volume 242(4), pp [12] F: Resta, F. Rpamont, G. Caulan and M. Ferrar (2010). Independent modal ontrol for nonlnear flexble strutures. Journal of Sound and Vbraton, Volume 329(8), pp [13] A.A. Shabana (1997). Flexble Multbody Dynams: Revew of Past and Reent Developments. Multbody System Dynams, Volume 1(2), pp [14] L. Merovth (1990). Dynams and Control of Strutures, J. Wley & Sons, New York, ISBN [15] B. Fredland (1986). Control System Desgn, MGraw Hll, New York, ISBN