REMODELLING OF VIBRATING SYSTEMS VIA FREQUENCY-DOMAIN-BASED VIRTUAL DISTORTION METHOD Małgorzata MRÓZ and Jan HOLNICKI-SZULC Insttute of Fundamenta Technoogca Research, Swetokrzyska 21, -9 Warsaw, Poand Smart Technoogy Centre, Warsaw, Poand, e-ma: mzaremba@t.gov., honck@t.gov. Keywords: Structura remodeng, Steady state dynamcs, Vrtua Dstorton Method 1. Introducton Numerca anayss of dynamcay oaded mechanca systems s a cassca robem and enty of software ackages s avaabe on the market. However, the desgn rocess of dynamcay resondng structures nvoves a tme consumng rocedure of system mrovng, eadng to desred fna resonse. Therefore, there s a need for numerca toos hefu n automatc redesgn rocess of these structures. So-caed Vrtua Dstorton Method (VDM) aears to be romsng aroach and has been aed n remodeng rocess of structures exosed to mact oads [3], where tme-doman-based transent anayss of dynamca resonse has been used. Anaogous aaratus has been successfuy used to sove the nverse dynamc robem of damage dentfcaton va anayss of modfcaton of eastc wave roagaton trough a heathy and damaged structura eement [2]. The VDM methodoogy (restrcted to near resonses and usng re-comuted so-caed nfuence matrces) aows fast modfcaton of the orgna structures wthout need of modfcatons of ther stffness, damng and mass matrces. Aso, VDM aows numercay effectve anaytca gradent comutaton, what s cruca for effcent otmzaton rocess eadng to souton of otma desgn or dentfcaton robem. The otmzaton rocess eadng to souton of otma desgn or dentfcaton robem. The drawback of ths mentoned tme-doman-based VDM aroach s comutatona cost due to necessty of anayss of the rocess evouton n tme. There s a cass of robem where concet smar to the mentoned above VDM aroach, but based on frequency-doman rather than tme-doman resonse can be aed. Ths numercay economc method can be addressed to robems, where steady state resonse can be the base of the dynamca anayss. For nstance, the foowng tasks can be soved on the base of the VDM-F (Vrtua Dstorton Method n Frequency Doman) method: - remodeng of vbratng system wth harmonc exctaton n order to reduce vbratons n a seected area, - dentfcaton of matera/structura roertes on the base of montored structura resonses for sames of harmonc exctatons, - detecton and dentfcaton of damages (va nverse dynamc robem) on the base of montored structura resonses for sames of harmonc exctatons. The frst mentoned above fed of acatons corresonds aso to vbro-acoustc robems, wth reatvey hgh frequences of exctatons. Cassca FEM- based numerca toos are too exensve n ths case and so-caed SEA (Statstca Energy Anayss) [1] aroach wth ts drawbacks due to ack of accuracy has been roosed. There s a need for a combned (FEM-SEA) methodoogy abe to roose comromsed technques. The authors hoe that the dscussed beow VDM-F aroach w deveo to one of such roostons. 2. Probem formuaton In order to resent basc formuas of the VDM-F method, et us focus on quck remodeng of truss structures under harmonc exctatons. The structure s descrbed wth some arameters n whch modfcatons coud be ntroduced. After modfcatons the structura resonse.e. dsacements and nterna forces are recacuated and then nfuence of the modfcatons s examned. The genera form of equatons of moton for a mut-degree of freedom system s as foows: M u&& ( t) + C u& ( t) + K u( t) = f ( t) (1) where M, C and K are mass, damng and stffness matrces, resectvey and f(t) s the vector of externa forces. 1
Each of the above mentoned matrces reresents a set of arameters whch can be modfed n a form: ( M + M ) u&& ( t) + ( C + C) u& ( t) + ( K + K ) u( t) = f ( t) (2) where M, C, K reresent changes to the mass, damng and stffness matrces, resectvey. As a secfc case we may choose to modfy ony the stffness and mass of structura comonents. A usefu too for redctng the resonse of a structure, gven changes of some of ts arameters (stffness, Young moduus, mass, cross secton) s the Vrtua Dstorton Method. In ths aer the methodoogy and exame based on the new aroach-vrtua Dstorton Method n frequency doman s deveoed. The task s to demonstrate that the VDM-F based smuaton of structura modfcatons eads to the same resuts as recomuted dynamc resonse for the modfed structure. We w cacuate the nfuence of changes n stffness and mass on the resonse of a structure when the structure s excted wth a harmonc force. Then, the robem s recacuated for few harmonc frequences of exctaton. A case wthout damng s consdered n ths work. It s aso assumed that a comonents are truss eements. Frst ony stffness modfcaton n eements was examned, then ony mass modfcaton, and n the end mass and stffness modfcatons were coued together. Fnay an otmsaton robem s formuated as an exame of ractca acaton. 3. Vrtua Dstorton Method n frequency doman If the nvestgated structure s subjected to a harmonc exctaton: f ( t) = f snω t (3) then ts resonse w be comosed of two comonents: free vbratons resutng from the ntaton of the externa exctaton these vbratons w be damed out and steady state vbratons due to exctaton tsef. Ths work s focused on the case when the structure s n the steady state, negectng damng effect for smcty of the dscusson. A eements vbrate wth the same hase because there s no damng consdered. Therefore the exctaton (3) eads to the foowng dynamc resonse exressed by dsacement: u( t) = u snω t () Changes n stffness and mass dstrbuton were modeed by vrtua dstortons denotng nta strans n structura eements and vrtua forces n structura nods oscatng wth the same frequency as externa exctaton: ( t) = snω ( ) snω t = t t (5) where the frst quantty modes stffness, whe the second one the mass dstrbutons resectvey. Let us ca modfed structure - structure n whch changes were made to the mass and stffness matrx and modeed structure structure n whch changes were made by vrtua dstorton, wthout changng mass and stffness matrces. It s assumed n order to bud VDM equatons that structure modeed by vrtua dstortons s dentca wth the modfed structure. Equatons of moton for modfed and modeed structures can be obtaned ntroducng n eqs. (1) and (2) comonents due to vrtua dstortons (5) (cf. [3]): ˆ T EA Mu&& ( t) + B B u( t) = f ( t) (6) L T Mu&& ( t) + B S Gu( t) ( t) = f ( t) + ( t) (7) where: K = B T SG, S dagona matrx wth eements S = E A comosed of the Young moduus E, eement cross secton A, B transfer matrx = Gu G -denotes the dsacement-stran transfer matrx. If the harmonc exctaton s nvestgated the equatons above take the foowng form (substtutng eqs. (3) and () to eqs. (6) and (7)): ω Mu ˆ + B SGu = f (8) 2 T ˆ 2
2 T ω Mu + B S Gu = f + (9) In the above equatons we got rd of the tme deendent members. The dsacement deends ony on the frequency and the amtude of externa exctaton and can be decomosed n the foowng form (cf. [3]): L, ω, ω j j u = u + D + D (1) where: Dj - nfuence matrx denotng amtude of dsacement u n node generated by unt, harmonc force wth frequency ω aed n node j, nfuence matrx denotng amtude of dsacement D 1 =, u n node generated by unt, harmonc stran dstorton, wth frequency ω aed n eement. In the formuas beow ndces, j, k run through a structura nodes whe ndces, β, γ through structura eements. It s ostuated that resonse of the structure modeed by vrtua dstortons has to be dentca wth the resonse of the modfed structure. Therefore, for each eement whch s modfed the comance of strans and forces n modeed system s requred (cf. eqs. (8) and (9)): and modfcaton arameter, can be defned: ˆ ( E A ) E A = (11) Â = = (12) A where: = G u (13) Fnay, t foows from (1), (12) and (13): ( ) ( ) j j j j ˆ L L A G D β β G D A G D β β G D + + = + + (1) It foows aso from eqs. (8) and (9) the requrement of second dentty: = M ω u 2 j j (15) ( ˆ ) ( ) M = Mˆ M = ρ A A = ρ A 1 j j j and ρ denotes the matera densty, whe the ength of eement. Fnay t eads to: ( ) ( 1) ( ) L L = M ω u + D + D = ω ρ A u + D + D (17) 2 2 j j j jk k j j jk k From equatons (1) and (17) can be wrtten n the foowng form (negectng subscrts): (16) ( ) ( ) ( ) ω ( ) ( ) ( ) L I GD I I GD I Gu = 2 2 ω ω 2 L ρa I LD ρa I LD I ρa I Lu (18) and aows determnaton of the vrtua quanttes m and modeng modfcatons of mass and stffness matrces n the VDM-F rocedure. The matrces ρ,, A, L above are dagona.. Resuts of numerca comutatons.1. Inut data for cacuatons Dmensons: 1m x 1m Aed force F=15sn (omega*t) [N] Young moduus E=2,1e11 [Pa] Cross secton S=1e-5 [m 2 ] Densty ro=78 [kg/m 3 ] Eement number -modfed eement 3
Cross secton of eement number was decreased by 6%, =. Fg. 1 Truss structure testng exame.2. Egenvaues Dagona mass matrx was used to obtan resuts shown n ths aer, but aso egenvaue robem wth dagona mass matrx was cacuated for comarson. The own frequences were extracted n order to choose the frequences of exctatons taken nto consderatons. Tabe 1. Own frequences umed mass matrx [Hz] consstent mass matrx [Hz] 1 31, 359,1 2 7,16 813,1 3 765,26 13 992,53 133.3. Resuts for mass and stffness modfcaton -coued task The resuts of the VDM n frequency doman are shown together wth resuts of steady state task wthout modfcaton. Each tabe contans maxmum amtude of dsacement for a free degrees of freedom for dfferent vaues of frequences. Tabe 2. Amtude n 3 degree of freedom Omega [Hz] Modfcaton [m] wthout modfcaton [m] 1 5,2252E- 3,628898E- 2 9,667579E- 6,86583E- 8,58817E- 8,16559E- 6 6,29329E- 5,76819E- Tabe 3. Amtude n degree of freedom Omega [Hz] Modfcaton [m] wthout modfcaton [m] 1 1,96335E-3 1,3636E-3 2 3,27371E-3 2,157829E-3 2,17231E-3 2,21123E-3 6 7,71528E- 7,3718E- Tabe. Amtude n 7 degree of freedom Omega [Hz] Modfcaton [m] wthout modfcaton [m] 1 3,533E-,79627E- 2 5,292E- 7,2283E- 3,29E- 6,711153E- 6,138775E-5 1,166E- Tabe 5. Amtude n 8 degree of freedom Omega [Hz] Modfcaton [m] wthout modfcaton [m] 1 2,23726E-3 1,683276E-3 2 3,9515E-3 2,898E-3 1,7858E-3 1,8963E-3 6 1,358597E- 1,29893E-.. Comarson of resuts wth FEM n tme doman Comarson of resuts obtaned trough the VDM-F smuaton versus the drect, FEM based re-comutng done for the modfed structure (transent anayss) s resented. Cacuatons were made for the force vbratng wth frequency ω=2hz. Tabe 6. Comarson of cacuatons usng FEM and VDM/F D.O.F. 3 7 8 structure wthout modfcatons FEM 5,96E- 2,11E-3 6,99E- 2,E-3 structure wthout modfcatons VDM 6,9E- 2,16E-3 7,2E- 2,9E-3 modfed structure VDM 9,67E- 3,3E-3 5,2E- 3,95E-3 modfed structure FEM 9,59E- 3,33E-3 5,19E- 3,87E-3
Tabe 7. Dfferences n ercentage D.O.F. 3 7 8 FEMno_mod/VDMno_mod 2% 2% % 2% FEMno_mod/VDMmod 38% 38% 33% 38% VDMno_mod/VDMmod 37% 37% 3% 37% FEMmod/VDMmod 1% 3% 1% 2% no_mod - cacuaton for structure wthout modfcatons mod - cacuaton for structure wth modfcatons.5. Comarson wth steady state task Tabe 8 contans resuts obtaned from steady-state FEM re-anayss task and from VDM-F smuaton n frequency doman. For the frst case mass and stffness matrces were modfed, for second one changes were modeed by vrtua dstortons. Tabe 8. Comarson of amtude for omega=2hz D.O.F. Modeed structure [m] Modfed structure [m] Change 3 9,6676E- 9,6676E-,% 3,27E-3 3,27E-3,% 7-5,29E- -5,29E-,% 8 3,952E-3 3,952E-3,% 5. Otmzaton robem and anayss of sensbty Otmzaton ams at fndng the mnmum of target functon F deendent on chosen varabes caed decson varabes. Decson varabes coud be for nstance mass, stffness or cross-secton area of structura eements. In order to demonstratve acabty of VDM-F et us search for such matera redstrbuton, determned by modfcatons of eements cross sectons, that the stran amtude of the seected eement number (Fg. 1.) w be mnmzed: mn 2 ( f ) mn ( ) = (19) In ths case decson varabe was modfcaton arameter m. The objectve functon (19) together wth recomuted resonse u L, the nfuence matrces D, D and reatons (12), (16), (18) determne the otmzaton robem subjected to the contro arameters. Comutatona cost of gradent-based otmzaton technque deends mosty on the effcency of senstvty anayss. In ths case gradents can be devered effcenty trough the foowng anaytca way: df m = 2 + d m (2) Fve artcuar comonents of the above formua can be determned va dfferentaton of the equatons (1), (13) and (18). Substtutng (1) to (13) we can get stran: wth dervatves: L ( ) = G u + D + D (21) j j m m = G D m m and = G jd j (22) (23) Dfferentatng equatons (18) we can get the foowng formuas: 5
( ) ( ) L I GD I I GD Gu + GD + GD = ω ( ) ( ) ( ) 2 ω 2 ω 2 L + + ρa I LD ρa I LD I ρal u D D (2) f m Concudng, determnaton of gradent requres souton of the set of equatons (2) wth resect to and and then substtuton of the obtaned resuts (together wth comonents (22), (23)) to the formua (2). The teratve technque based on the foowng steeest descent rue of modfcatons of contro arameters can be roosed. where ' f = (25) ' denotes the modfed matera dstrbuton n the next ste of teraton and <,1>. It s mortant from comutatona ont of vew that man matrces on the eft hand sde n equatons (18) and (2) descrbng VDM-F modeng and senstvty are dentca, what reduces sgnfcanty numerca cost. 6. Summary and concusons Vrtua Dstorton Method n frequency doman (VDM-F) s a usefu too to nvestgate dynamc robems. Statc-ke nfuence matrces was bud, ony once for each vaue of frequency. Based on VDM/F the otmzaton rocess n frequency doman s exected to be sgnfcanty faster comared to the one anayzed n tme doman. Tme doman tasks were much more tme consumng because t was requred to cacuate nfuence matrces for a stes n the tme erod therefore VDM/F shoud many reduce comutatona tme. In structure modeed by vrtua dstortons n frequency doman and modfed structure n steady state task dfferences between resuts do not exceed 35%. Hence the VDM n frequency doman seems to be an effectve method to cacuate vbratng structures oaded wth harmonc exctatons. It s ossbe to deveo agorthms to desgn and contro vbratng structures basng on VDM n frequency doman. Comarson of the maxmum dsacement for the case wth eement cross secton changed by 6% and the maxmum dsacement for structures wthout modfcatons shows that the dfferences are about 38% for 3, and 8 D.O.F. and about 3% for 7 D.O. F. Amost the same resuts were obtaned from VDM/F, steadystate task and from FEM. There are no dfferences between ths two souton rocedures. New aroach can be acabe for remodeng structures subjected to harmonc exctatons. The otmzaton robem can be consdered, to fnd the otmum of the mass dstrbuton n order to soate or rotect a art of the structure from undesrabe vbratons. Another acaton s to formuate nverse robem for dentfcaton of unknown structura characterstcs. 7. Acknowedgement The authors woud ke to gratefuy acknowedge the fnanca suort trough the FP5 Research Tranng Networks Project HPRN-CT-22-28 (22-26) SMART SYSTEMS New Materas, Adatve Systems and Ther Nonneartes: Modeng, Contro and Numerca Smuaton References [1] Lyon, R. H., DeJong, R. G., Theory and Acaton of Statstca Energy Anayss, second edton 1995. [2] Koakowsk, P., Zensk, T.G., Honck-Szuc, J., Damage Identfcaton by the Dynamc Vrtua Dstorton Method, Journa of Integent Matera Systems and Structures, 15(6),. 79-93, 2. [3] Honck-Szuc J., Pawłowsk P., Wkło M.:Desgn of Adatve Structures under Random Imact Condtons. AMAS/ECOMAS/STC Worksho on Smart Materas and Structures, Jadwsn, Setember 2-5, 23, ISBN 3-5-22331-2,.5-67 [] Zensk, T.G, Metoda Imusowych Dystorsj Wrtuanych z zastosowanem do modeowana dentyfkacj defektów w konstrukcjach, raca doktorska, Instytut Podstawowych Probemów Technk Poskej Akadem Nauk, 2 (n osh) 6