REMODELLING OF VIBRATING SYSTEMS VIA FREQUENCY-DOMAIN-BASED VIRTUAL DISTORTION METHOD

Transcription

1 Ma³gorzata MECHANICS MRÓZ, Jan HOLNICKI-SZULC REMODELLING Vo. 24 No. 2 25OF VIBRATING SYSTEMS VIA FREQUENCY-DOMAIN-BASED VIRTUAL DISTORTION METHOD Ma³gorzata MRÓZ *, Jan HOLNICKI-SZULC * REMODELLING OF VIBRATING SYSTEMS VIA FREQUENCY-DOMAIN-BASED VIRTUAL DISTORTION METHOD SUMMARY The aer deas with Virtua Distortion Method (VDM) in frequency domain. VDM method is addressed to robems, where steady state resonse can be the base of the dynamica anaysis. The VDM methodoogy aows fast modification of the origina structures without need of modifications of their stiffness, daming and mass matrices. The resent work contains the methodoogy of the new aroach, some numerica resuts of comutations and otimization robem. Finay the anaysis of sensibiity is formuated. Keywords: structura remodeing, steady state dynamics, Virtua Distortion Method PRZEMODELOWYWANIE UK ADÓW DRGAJ CYCH ZA POMOC METODY DYSTORSJI WIRTUALNYCH W DZIEDZINIE CZÊSTOTLIWOŒCI Niniejszy artyku³ zajmuje siê Metod¹ Dystorsji Wirtuanych (MDW) w dziedzinie czêstotiwoœci. MDW odnosi siê do robemów dynamicznych, w których odowiedÿ jest ustaona w czasie. MDW ozwaa na szybk¹ modyfikacjê konstrukcji bez koniecznoœci modyfikacji ca³ej macierzy sztywnoœci, t³umienia czy masy. Praca zawiera sformu³owanie nowego odejœcia, wyniki obiczeñ numerycznych, a tak e sformu³owanie robemu otymaizacji i wra iwoœci konstrukcji da nowego odejœcia. 1. INTRODUCTION Numerica anaysis of dynamicay oaded mechanica systems is a cassica robem and enty of software ackages is avaiabe on the market. However, the design rocess of dynamicay resonding structures invoves a time consuming rocedure of system imroving, eading to desired fina resonse. Therefore, there is a need for numerica toos hefu in automatic redesign rocess of these structures. So-caed Virtua Distortion Method (VDM) aears to be romising aroach and has been aied in remodeing rocess of structures exosed to imact oads [3], where time-domain-based transient anaysis of dynamica resonse has been used. Anaogous aaratus has been successfuy used to sove the inverse dynamic robem of damage identification via anaysis of modification of eastic wave roagation trough a heathy and damaged structura eement [2]. The VDM methodoogy (restricted to inear resonses and using re-comuted so-caed infuence matrices) aows fast modification of the origina structures without need of modifications of their stiffness, daming and mass matrices. Aso, VDM aows numericay effective anaytica gradient comutation, what is crucia for efficient otimization rocess eading to soution of otima design or identification robem. The otimization rocess eading to soution of otima design or identification robem. The drawback of this mentioned time-domain-based VDM aroach is comutationa cost due to necessity of anaysis of the rocess evoution in time. There is a cass of robem where concet simiar to the mentioned above VDM aroach, but based on frequency-domain rather than time-domain resonse can be aied. This numericay economic method can be addressed to robems, where steady state resonse can be the base of the dynamica anaysis. For instance, the foowing tasks can be soved on the base of the VDM-F (Virtua Distortion Method in Frequency Domain) method: remodeing of vibrating system with harmonic excitation in order to reduce vibrations in a seected area, identification of materia/structura roerties on the base of monitored structura resonses for sames of harmonic excitations, detection and identification of damages (via inverse dynamic robem) on the base of monitored structura resonses for sames of harmonic excitations. The first mentioned above fied of aications corresonds aso to vibro-acoustic robems, with reativey high frequencies of excitations. Cassica FEM- based numerica toos are too exensive in this case and so-caed SEA (Statistica Energy Anaysis) [1] aroach with its drawbacks due to ack of accuracy has been roosed. There is a need for a combined (FEM-SEA) methodoogy abe to roose comromised techniques. The authors hoe that the discussed beow VDM-F aroach wi deveo to one of such roositions. 2. PROBLEM FORMULATION In order to resent basic formuas of the VDM-F method, et us focus on quick remodeing of truss structures under harmonic excitations. The structure is described with some arameters in which modifications coud be introduced. After modifications the structura resonse i.e. disacements and interna forces are recacuated and then infuence of the modifications is examined. * Institute of Fundamenta Technoogica Research, Warsaw, Poand, Smart Technoogy Centre, mzaremba@it.gov.; honicki@it.gov. 124

2 MECHANICS Vo. 24 No The genera form of equations of motion for a muti-degree of freedom system is as foows M ut &&() + C ut &() + K ut () = f() t (1) where M, C and K are mass, daming and stiffness matrices, resectivey and f (t) is the vector of externa forces. Each of the above mentioned matrices reresents a set of arameters which can be modified in a form ( M+ΔM) ut &&() + ( C+ΔC) ut &() + + ( K+ΔK) ut () = f() t where ΔM, ΔC, ΔK reresent changes to the mass, daming and stiffness matrices, resectivey. As a secific case we may choose to modify ony the stiffness and mass of structura comonents. A usefu too for redicting the resonse of a structure, given changes of some of its arameters (stiffness, Young moduus, mass, cross section) is the Virtua Distortion Method. In this aer the methodoogy and exame based on the new aroach-virtua Distortion Method in frequency domain is deveoed. The task is to demonstrate that the VDM-F based simuation of structura modifications eads to the same resuts as re-comuted dynamic resonse for the modified structure. We wi cacuate the infuence of changes in stiffness and mass on the resonse of a structure when the structure is excited with a harmonic force. Then, the robem is recacuated for few harmonic frequencies of excitation. A case without daming is considered in this work. It is aso assumed that a comonents are truss eements. First ony stiffness modification in eements was examined, then ony mass modification, and in the end mass and stiffness modifications were coued together. Finay an otimisation robem is formuated as an exame of ractica aication. 3. VIRTUAL DISTORTION METHOD IN FREQUENCY DOMAIN If the investigated structure is subjected to a harmonic excitation (2) f() t = f sin ( ω t) (3) then its resonse wi be comosed of two comonents: free vibrations resuting from the initiation of the externa excitation these vibrations wi be damed out and steady state vibrations due to excitation itsef. This work is focused on the case when the structure is in the steady state, negecting daming effect for simicity of the discussion. A eements vibrate with the same hase because there is no daming considered. Therefore the excitation (3) eads to the foowing dynamic resonse exressed by disacement ut () = usin ( ω t) (4) Changes in stiffness and mass distribution were modeed by virtua distortions denoting initia strains in structura eements and virtua forces in structura nods osciating with the same frequency as externa excitation: () t = sin ( ω t) () t = sin ( ω t) where the first quantity modes stiffness, whie the second one the mass distributions resectivey. Let us ca modified structure structure in which changes were made to the mass and stiffness matrix and modeed structure structure in which changes were made by virtua distortion, without changing mass and stiffness matrices. It is assumed in order to buid VDM equations that structure modeed by virtua distortions is identica with the modified structure. Equations of motion for modified and modeed structures can be obtained introducing in eqs. (1) and (2) comonents due to virtua distortions (5) (cf. [3]) (5) ˆ () T Mu&& t + G SGu ˆ () t = f () t (6) T () t () t () t Mu && + G S Gu å = f() t + () t (7) where: K = G T SG, S diagona matrix with eements Sii = EA i i / i comosed of the Young moduus E, eement cross section A and ength of eement, å = Gu, G denotes the disacement-strain transfer matrix. If the harmonic excitation is investigated the equations above take the foowing form (substituting eqs. (3) and (4) to eqs. (6) and (7)): 2 ˆ T ω Mu + G SGu ˆ = f (8) 2 T ω + Mu G S Gu å = f + (9) In the above equations we got rid of the time deendent members. The disacement deends ony on the frequency and the amitude of externa excitation and can be decomosed in the foowing form (cf. [3]) L, ω, ω i i in n ij j u = u + D + D (1) where: D ij Dik infuence matrix denoting amitude of disacement u i in node i generated by unit, harmonic force = 1, with frequency ω aied in node j, infuence matrix denoting amitude of disacement u i in node i generated by unit, harmonic strain distortion, with frequency ω aied in eement k. 125

3 Ma³gorzata MRÓZ, Jan HOLNICKI-SZULC REMODELLING OF VIBRATING SYSTEMS VIA FREQUENCY-DOMAIN-BASED VIRTUAL DISTORTION METHOD In the formuas beow indices i, j, k run through a structura nodes whie indices, m, n through structura eements. It is ostuated that resonse of the structure modeed by virtua distortions has to be identica with the resonse of the modified structure. Therefore, for each eement which is modified the comiance of strains and forces in modeed system is required (cf. eqs. (8) and (9)) E ˆ nan EnAn n = ( n n) (11) n n and modification arameter μ, can be defined: μ n = = An where ˆn A n n n (12) and aows determination of the virtua quantities m and k modeing modifications of mass and stiffness matrices in the VDM-F rocedure. The matrices μ, ρ, A, L above are diagona. 4. RESULTS OF NUMERICAL COMPUTATIONS 4.1. Inut data for cacuations Dimensions: 1 m 1 m Aied force F = 15sin (ω*t) [N] Young moduus E = [Pa] Cross section S = 1 5 [m 2 ] Density ρ = 78 [kg/m 3 ] Eement number 4 modified eement Cross section of eement number 4 was decreased by 6%, μ =.4 (Fig. 1). n Gni ui = (13) Finay, it foows from (1), (12) and (13) ˆ L A n n GniDi GniDij + + j = L = A n n GniDi GniDij + + j n (14) It foows aso from eqs. (8) and (9) the requirement of second identity: 2 i = ΔMijω uj (15) Δ M ˆ ij = Mij Mij = =ρ An An n =ρan μn n ( ˆ ) ( 1) (16) and ρ denotes the materia density, whie k the ength of eement k. Finay it eads to L ( ) 2 i ij j jn n jk k = ΔM ω u + D + D = = ω ρ μ L An( n 1) n uj Djn n Djk k (17) From equations (14) and (17) can be written in the foowing form (negecting subscrits) å ( I ì) GD I ( I ì) GD 2 å 2 ω ña( I ì) LD ω ña( I ì) LD I å = L ( I ì) Gu = 2 L ω ña( I ì) Lu (18) 4.2. Eigenvaues Fig. 1. Truss structure testing exame Consistent mass matrix was used to obtain resuts shown in this aer, but aso eigenvaue robem with diagona mass matrix was cacuated for comarison. The own frequencies (Tab. 1) were extracted in order to choose the frequencies of excitations taken into considerations. Tabe 1. Own frequencies No. Lumed mass matrix Consistent mass matrix Resuts for mass and stiffness modification coued task The resuts of the VDM in frequency domain are shown together with resuts of steady state task without modification. Each tabe (Tabs. 2 5) contains maximum amitude of disacement for a free degrees of freedom for different vaues of frequencies. 126

4 MECHANICS Vo. 24 No Tabe 2. Amitude in 3 degree of freedom E E E E E E E E-3 Tabe 3. Amitude in 4 degree of freedom E E E E E E E E-2 Tabe 4. Amitude in 7 degree of freedom E E E E E E E E-3 Tabe 5. Amitude in 8 degree of freedom E E E E E E E E Comarison of resuts with FEM in time domain Comarison of resuts obtained trough the VDM-F simuation versus the direct, FEM based re-comuting done for the modified structure (transient anaysis) is resented (Tab. 6 and 7). Cacuations were made for the force vibrating with frequency ω = 2 Hz Comarison with steady state task Tabe 8 contains resuts obtained from steady-state FEM re-anaysis task and from VDM-F simuation in frequency domain. For the first case mass and stiffness matrices were modified, for second one changes were modeed by virtua distortions. 5. OPTIMIZATION PROBLEM AND ANALYSIS OF SENSIBILITY Otimization aims at finding the minimum of target function F deendent on chosen variabes caed decision variabes λ. Decision variabes coud be for instance mass, stiffness or cross-section area of structura eements. In order to demonstrative aicabiity of VDM-F et us search for such materia redistribution, determined by modifications of eements cross sections μ, that the strain amitude of the seected eement number 4 (Fig. 1) wi be minimized min 2 ( ) min( 4 ) f = (19) In this case decision variabe was modification arameter μ. The objective function (19) together with recomuted resonse u L, the infuence matrices D, D and reations (12), (16), (18) determine the otimization robem subjected to the contro arameters μ. Comutationa cost of gradient-based otimization technique deends mosty on the efficiency of sensitivity anaysis. In this case gradients can be deivered efficienty trough the foowing anaytica way df 4 2 m 4 i = 4 + dμ m μ μ i (2) Tabe 6. Comarison of cacuations using FEM and VDM/F D.O.F Structure without modifications FEM 3.19E E-3 4.2E E-3 Structure without modifications VDM 3.19E E-3 4.2E E-3 Modified structure VDM 4.52E E-3-2.7E E-3 Modified structure FEM 4.52E E-3 2.7E E-3 Tabe 7. Differences in ercentage D.O.F FEMno_mod/VDMno_mod.3%.%.3%.6% FEMno_mod/VDMmod % % 32.79% % VDMno_mod/VDMmod % % 32.77% % FEMmod/VDMmod.4% -.1% -.2%.8% no_mod cacuation for structure without modifications mod cacuation for structure with modifications 127

5 Ma³gorzata MRÓZ, Jan HOLNICKI-SZULC REMODELLING OF VIBRATING SYSTEMS VIA FREQUENCY-DOMAIN-BASED VIRTUAL DISTORTION METHOD Tabe 8. Comarison of amitude for ω = 2 Hz D.O.F. Modeed structure Modified structure Change E E-3.5% E E-2.5% E E-3.8% E E-2.5% Five articuar comonents of the above formua can be determined via differentiation of the equations (1), (13) and (18). Substituting (1) to (13) we can get strain = + + L 4 G4i ui Dij j Dim m with derivatives and 4 = G 4iDim m 4 = G 4 j Dji i (21) (22) (23) Differentiating equations (18) we can get the foowing formuas å ( ) ( ) I ì GD I I ì GD 2 å 2 ω ña( I ì) LD ω ña( I ì) LD I å L å + + ì Gu GD å GD = 2 L å ( ) ω ñal u + D å + D ì (24) f Concuding, determination of gradient requires soution of the set of equations (24) with resect to m μ μ and i and then substitution of the obtained resuts (together μ with comonents (22), (23)) to the formua (2). The iterative technique based on the foowing steeest descent rue of modifications of contro arameters µ can be roosed ' f μ =μ α (25) μ ' where μ denotes the modified materia distribution in the next ste of iteration and α <,1>. It is imortant from comutationa oint of view that main matrices on the eft hand side in equations (18) and (24) describing VDM-F modeing and sensitivity are identica, what reduces significanty numerica cost. 6. SUMMARY AND CONCLUSIONS Virtua Distortion Method in frequency domain (VDM-F) is a usefu too to investigate dynamic robems. Static-ike infuence matrices was buid, ony once for each vaue of frequency. Based on VDM/F the otimization rocess in frequency domain is exected to be significanty faster comared to the one anayzed in time domain. Time domain tasks were much more time consuming because it was required to cacuate infuence matrices for a stes in the time eriod therefore VDM/F shoud mainy reduce comutationa time. In structure modeed by virtua distortions in frequency domain and modified structure in steady state task differences between resuts do not exceed.5%. Hence the VDM in frequency domain seems to be an effective method to cacuate vibrating structures oaded with harmonic excitations. It is ossibe to deveo agorithms to design and contro vibrating structures basing on VDM in frequency domain. Comarison of the maximum disacement for the case with eement 4 cross section changed by 6% and the maximum disacement for structures without modifications shows that the differences are about 42% for 3, 4 and 8 D.O.F. and about 33% for 7 D.O. F. Amost the same resuts were obtained from VDM/F, steady-state task and from FEM. Differences between do not exceed.1%. New aroach can be aicabe for remodeing structures subjected to harmonic excitations. The otimization robem can be considered, to find the otimum of the mass distribution in order to isoate or rotect a art of the structure from undesirabe vibrations. Another aication is to formuate inverse robem for identification of unknown structura characteristics. Acknowedgement The authors woud ike to gratefuy acknowedge the financia suort trough the FP5 Research Training Networks Project HPRN-CT (22 26) SMART SYSTEMS New Materias, Adative Systems and Their Noninearities: Modeing, Contro and Numerica Simuation 128

6 MECHANICS Vo. 24 No References [1] Lyon R.H., DeJong R.G.: Theory and Aication of Statistica Energy Anaysis. 2nd ed., 1995 [2] Koakowski P., Zieinski T.G., Honicki-Szuc J.: Damage Identification by the Dynamic Virtua Distortion Method. Journa of Inteigent Materia Systems and Structures, 15(6), 24, [3] Honicki-Szuc J., Paw³owski P., Wik³o M.: Design of Adative Structures dunder Random Imact Conditions. AMAS/ECOMAS/ STC Worksho on Smart Materias and Structures, Jadwisin, Setember 2 5, 23, ISBN , [4] Zieinski T.G.: Metoda Imusowych Dystorsji Wirtuanych z zastosowaniem do modeowania i identyfikacji defektów w konstrukcjach. Instytut Podstawowych Probemów Techniki Poskiej Akademii Nauk, 24, raca doktorska (in Poish) 129