Recent developments in Code Aster to compute FRF and modes of VEM with frequency dependent properties

Transcription

1 Recent develoments in Code Aste to comute FRF and modes of VEM with fequency deendent oeties. Melette a, E. Pagnacco b and A. Ladie b a TAGET DELTA, 1 ue Adolhe Robet, 5800 Cosne su Loie, Fance b Laboatoie de Mécanique de Rouen, ISA de Rouen, Avenue de l Univesité, Saint Etienne du Rouvay, Fance nicolas.melette@tgdelta.com

2 This ae tackles the way how advanced caabilities ae intoduced in Code_Aste to take into account fequency deendent oeties of viscoelastic mateials. Comuting of fequency esonse functions and modes of vibation (eal o comlex) is addessed. Having fequency deendent modes is a ste fowad fo the modal ojection method and fo model udating with an exeimental modal basis as a efeence. An iteative method is oosed and imlemented in Code_Aste in ode to comute fequency deendent modes. The new caabilities ae then comaed with the standad aoaches in the case of an automotive windshield, whee the viscoelastic behavio of the olyvinylbutyal which comoses the laminate glass is consideed. 1 Intoduction ViscoElastic Mateials (VEM) ae intensively used to solve VH issues. Daming ads and anti flutte oducts educe the stuctue bone noise. Sealants and absobing mateials event the ai bone tansmission. Adhesives ae common solutions to assemble ats. Wheeas VEM ae well known, the finite element analysis of thei dynamic behavio is not staightfowad. Most of the time, the use of a finite element softwae equies to simlify VEM as elastic mateials. This simlification is not always the obvious thing to do and may lead to inaccuate esults. The fist at of this ae addesses the finite element analysis of stuctues comising VEM with fequency deendent dynamic oeties. Conventional aoaches ae discussed and a method is oosed to ovecome thei limitations. The method comutes fequency deendent modes using an iteative algoithm. The way how these modes can be used to imove the modal ojection method is descibed. Imlementation in Code_Aste is exlained. The second at descibes the alication of the method to the study of an automotive laminate windshield comising a olyvinylbutyal mateial with a stong fequency deendent viscoelastic behavio. Finite element analysis.1 State-of-the-at Fom a simulation oint of view, the fequency esonse of a stuctue comising at least one VEM is obtained by solving the dynamic equilibium equation: [ K ( )] [ M] u ( ) F ( ), (1) with [ K ( )], the comlex stiffness matix as: [ K ( )] [ K( )] i[ K ( )]. () The fequency deendence of the matix comes fom the use of fequency deendent comlex moduli to eesent viscoelastic behavios [1]. In many finite element codes, solving the system of Eq. (1) is not conventional because it needs to ealize the stiffness matix fo each fequency ste and to have comuting ocedues which ae able to deal with fequency deendent matices. Fo instance, ASTRAlike codes oose only a diect esonse aoach and it is not ossible to define moe than one fequency deendent behavio. This limitation is incomatible with today's stuctues comising seveal diffeent VEM. Concening Code_Aste, thee is neithe ossibility to define a fequency deendent behavio, no to solve Eq. (1) without any exta-develoment. The diect esonse aoach consists in solving Eq. (1) fo each fequency ste as: 1 ) [ K ( )] [ M ] F ( ) [0, ] u. (3) ( max This method has the advantage of comuting the exact esonse of the system. But, as it is necessay to comute and invese a comlex matix at each fequency ste, the comuting time can become ohibitive fo industial stuctues with seveal million degees of feedom. A few studies [-4] have shown the inteest to solve Eq. (1) with dedicated modal esonse methods in ode to imove the comutational efficiency while maintaining the accuacy of the esults. Modal esonse methods comute fequency esonses by ojecting the system of Eq. (3) on a modal basis [T ], with the assumtion: ( ) [ T] q ( ) u. (4) The ojection of the model on the consideed basis leads to a low ode model, that deceases significantly the numbe degees of feedom and consequently the comuting time of fequency esonses: T T 1 T q ( ) [ T] [ K ( )][ T] [ T] [ M][ T] [ T] F ( ). (5) Using oeties of elastic models (eal and fequency indeendent stiffness matix), the standad basis of the oula sectal decomosition method combines nomal modes solving of the classical eigenvalue oblem: [ K] [ M] 0 (6) and a static coection to ensue a coect eesentation of the low fequency contibution of tuncated high fequency nomal modes [5]. Classically, [T ] is comosed by the 1.5. nomal modes between 0. and a minimum of max Howeve, fequency deendence of the VEM dynamic oeties events fom using the sectal decomosition method, because the modal basis, ], is only valid [ 1, at the fequency ef, fo which it has been comuted. In actice, it has been demonstated [3] the validity domain of the basis can be extended in a ange aound the fequency ef. If the VEM of the model give inceasing moduli with esect to the fequency, the use of will lead to a lage validity domain than fo ef max any othe fequencies. Unfotunately, the validity domain of the basis may not extend ove all the fequency ange of inteest. Inaccuate esults may be obtained.

3 . Comutation of fequency deendent modes Fo a fequency deendent eal stiffness matix,, the eigenvalue oblem of Eq. (6) is witten as: [ K ( )] [ ( )] ( ) [ M] ( ) 0 K, (7) whee the eigenfequency, (), and the eigenvecto, (), ae fequency deendent too. Fo a fixed, given fequency,, such eigensolutions can be obtained by solving the standad oblem of Eq. (6). Hence, the oosed method fo solving the fequency deendent oblem of Eq. (7) consists in using an iteative algoithm seaching fo: ( ) [ 1, ], (8) whee 1 and ae esectively the lowe and the ue cut-off fequencies of the eigenoblem and is the convegence citeion of the iteative algoithm. Fist, fo 1, the stiffness matix is ealized and the eigenoblem of Eq. (6) is solved. Then, is udated by th taking the value of the n eigenfequency fo which: n 1, min. (9) ext, the stiffness matix is ealized fo the new value of and the eigenoblem of Eq. (6) is solved again. The iteations will not sto while n. (10) When the ocedue conveges, n n ae extacted to fom the fequency deendent eigensolutions of Eq. (7) and the iteative algoithm will continue using as the guess value to comute the next n1 and eigensolution. Finally, the algoithm will sto when. It means all the fequency deendent eigensolutions have been comuted between 1 and. The comuting time of the oosed method is classically diven by the numbe of fequency deendent modes to be comuted, the numbe of degees of feedom of the model and the value of the convegence citeion,. It is also deendent on the numbe of nomal eigenvalues which ae comuted as solutions of Eq. (6) at each iteation. Having all the eigenvalues between 1 and fo each iteation is not necessay and could lead to ohibitive comuting times. In theoy, a minimum of two eigenvalues may be sufficient to un iteations: n to udate as descibed in Eq. (9) and n1 to continue when the ocedue conveges. In actice, this minimum numbe of eigenvalues will be detemined by the caabilities of the numeical solve fo Eq. (6). It will be discussed fo Code_Aste in section.4. The oosed method can be natually extended to the study of damed stuctues comising VEM. The fequency deendent eigenvalues and eigenvectos ae then comuted as solutions of the following oblem: [ K ( )] [ M] 0, (11) with 1, the comlex eigenvalues such that ~ 1. i with the modal stuctual daming facto, and 1,, the associated comlex modes. In this case, Eq. (8) is ewitten as: ~ ~ ( ) [ 1, ], (1) with Re( ), the coesonding eigenfequency in a stuctual daming model [6]. will be udated by th taking the value of the n eigenfequency fo which: n min 1, ~ ~. (13) Comuting fequency deendent eal o comlex modes should be a ste fowad fo comaison with exeimental modal bases. Indeed, when stuctues comising VEM ae tested, thei fequency deendent behavios ae hysically measued. The oosed method could hel to validate o to udate finite element models with exeimental efeences..3 Imovement of the modal ojection method The fequency deendent modes can be used to fom the basis of the modal ojection method fo esonse comutations of VEM. They will extend the validity domain of the basis ove all the fequency ange of inteest. In combination to a static coection, eithe eal modes will be used fo weakly damed stuctues, o comlex modes fo highly damed stuctues [3]. The static coection will be detemined with the stiffness matix ealized fo 0. Real modes will be efeed to educe comuting times, since numeical solves ae much moe efficient in this case. But fo highly damed stuctues, the ojection base comosed with such modes may be insufficient to obtain accuate fequency esonses. This can be imoved using the modified Modal Stain Enegy (MSE) method to educe the eos [7]. So, fequency deendent eal modes, ( ), become solutions of a modified fom of Eq. (7) in ode to take into account the daming matix (imaginay at of the stiffness matix) as: [ ( )] ( )[ K ( )] ( ) [ M] ( ) 0 K.(14) [ K ( )] and [ K ( )] ae defined by Eq. () and () is calculated by the following emiical fomula:

4 whee the tace fo an tace[ K ( )] ( ), (15) tace[ K( )] matix is defined as: A jj j1 tace[ A]. (16) In the iteative algoithm, the use of the modified MSE method leads to calculate ( ) fo each iteation and to seach fo the solutions of the esulting eigenvalue oblem: [ 0 K( )] ( )[ K ( )] [ M]. (17) Anothe method to imove the eal modes could be the use of esidual modes whose uose is to eesent the daming of VEM in the modal ojection basis [8]. The couling with fequency deendent modes has not yet been investigated..4 Imlementation in Code_Aste The aoach to fequency deendent modes descibed above has been imlemented in Code_Aste via a maco scit uses standad commands existing in Code_Aste fo the conventional modal method. Python codes have been added fo the new develoments: definition of a fequency deendent behavio, comutation of fequency deendent modes by the iteative algoithm, fequency esonse comutation with ealization of the stiffness matix at each fequency ste. In the iteations, modes ae comuted as solutions of Eq. (6) via the standa command. Ten eigensolutions ae seached aound fo each iteation. This numbe is a good comomise between secuing the convegence of the iteative algoithm and limiting the total comuting time. The use can define seveal VEM with diffeent behavios and choose the method to comute the esonses (diect, modal with constant eal modes o modal with fequency deendent modes). The fequency deendent modes can be obtained using the modified MSE o not. At this time of the imlementation, it is ossible to comute fequency deendent comlex modes, but they cannot be used as a modal basis fo esonse comutation because of existing ogamming locks in Code_Aste. This will be investigated in futhe woks. 3 Industial alication The fequency deendent modes aoach descibed above and imlemented in Code_Aste was alied to an automotive laminate windshield comising a olyvinylbutyal mateial between two glass layes. The objectives wee to validate the oosed aoach and to comae with the standad comutational methods fo both modes and esonses. 3.1 Descition of the model The thickness of the olyvinylbutyal was 0.76 mm. The thickness of each glass laye was.1 mm. The finite element mesh was ealized with 115 shell elements fo the glass and 178 solid elements fo the olyvinylbutyal (thee layes in the thickness). Shell and solid elements wee linked with coincident nodes (comatible meshes) and an offset is intoduced to define each shell mid laye. The aim was to comute velocity esonses of the windshield fo an unitay excitation u to 1000 Hz. The eimete of the windshield was clamed. Excitation and esonses wee defined on the ue glass laye, nomal to the windshield lane. omal velocities wee aveaged using the esonses fo 18 nodes as shown in Figue 1. Figue 1: FE model of the windshield (ed aow fo excitation foce, black aows fo esonses). The olyvinylbutyal was consideed as an isotoic VEM and the glass as an isotoic elastic mateial with a constant loss facto of 1%. The dynamic behavio of the VEM was eesented by the comlex shea modulus extacted fom the nomogam in Figue. Figue : Reduced fequency nomogam of a standad olyvinylbutyal [9]. 3. Comaison of modal bases With the objective to comute esonses u to 1000 Hz, modal bases wee detemined between 0 and 1500 Hz. Constant eal modes wee comuted using the standad Soensen method imlemented in Code_Aste [10], fo a stiffness matix ealized at the fequency of 1000 Hz. Using the oosed develoments, fequency deendent modes wee seached with an accuacy of 0.01 Hz, i.e

5 in Eq. (8). Comuting times ae given in Table 1 as atios to the time needed to have the constant eal modes. Table 1: Comuting times of modal bases with Code_Aste. Fequency deendent Real/Comlex Modified MSE Comuting time o Real o 1.0 Yes Real o 0.3 Yes Real Yes.5 Yes Comlex 8.7 Table 1 shows the exta cost of the modified MSE method is negligible comaed to the total cost of the fequency deendent eal modes. 119 constant eal modes wee comuted between 0 and 1500 Hz, wheeas 115 fequency deendent modes (eal o comlex) wee found. These fou additional modes aea because of the aoximation of the VEM as an elastic mateial. Twelve fequency deendent eal modes and the coesonding constant eal modes ae comaed in Table accoding to thei fequencies. mass-weighted omalised Coss Othogonality (CO) citeion [6]. eglecting the fequency deendence leads to eos of seveal Hetz in the ediction of the eigenfequencies of the windshield. The eo is educed aound the fequency of 1000 Hz, i.e. the one used to ealize the stiffness matix fo the comutation of the constant eal modes. The CO values ae highe than 0.98 fo all aied modes in the fequency ange of inteest. Theefoe, the stiffness of the olyvinylbutyal is not so influent on the modal shaes of the windshield. 3.3 Fequency esonse functions A seies of fequency esonse calculations using Code_Aste was efomed via diect and modal methods involving the above develoments. The efeence was defined as the diect method using fequency deendent oeties of the VEM. The fist study involved the comaison between the efeence and the conventional modal method using constant eal nomal modes and constant oeties. The modal method used constant values of the shea modulus and the loss facto of the olyvinylbutyal, calculated as aveages between 0 and 1000 Hz. The esulting esonses ae lotted in Figue 3 and indicate sevee eos of the modal aoach. Table : Comaison between twelve ais of modes. Eigenfequency (Hz) Mode ode Fequency deendent eal modes Constant eal modes Shift (Hz) Figue 3: Comaison with the efeence of the nomal mode ojection method using a constant stiffness matix. It is obvious the fequency deendence of the olyvinylbutyal must be taken into account in esonse comutations of the windshield. In the following, all hamonic esonse comutations have been efomed with ealization of the stiffness matix at each fequency ste fo comuting the esonse given by Eq. (5). Hence, the second study is efomed fom a fequency deendent stiffness and a modal basis comosed by constant eal modes and a static coection. Static coection was comuted fo the stiffness matix ealized at 0 Hz, wheeas modes wee comuted fo the matix ealized at 1000 Hz. The esulting esonse is comaed with the efeence in Figue 4, showing a geat imovement comaed to the fist study, which does not oject the tue fequency deendent stiffness matix on the modal basis but a constant stiffness matix. Modes wee chosen to be sead ove the fequency ange. Matching between modes was validated using a

6 lage fo models with a geate numbe of degees of feedom. Figue 4: Comaison with the efeence of the modal aoach using tue fequency deendent stiffness matix, constant eal modes and static coection. Comaed to the efeence, lage diffeences in amlitude ae obseved at all the esonant eaks u to 700 Hz with a maximum diffeence of 4 db at the fist esonant eak. This illustates the constant eal modes ae not sufficient in modeling the olyvinylbutyal. A thid study was efomed to assess the caability of the fequency deendent modes in eesenting the viscoelastic behavio of the olyvinylbutyal. Resonses wee comuted by the oosed aoach using two diffeent modal bases. One was comosed by fequency deendent eal modes. The othe one was comosed by fequency deendent beta-modes which wee obtained by the modified MSE. In both cases, modes wee combined with a static coection which was comuted again fo the stiffness matix ealized at 0 Hz. The esulting esonses ae comaed with the efeence in Figue 5. Figue 5: Comaison with the efeence of the oosed aoach using fequency deendent modes. Both modal bases educe stongly the diffeences with the efeence. The diffeences ae lowe than 1.5 db with the eal modes, and lowe than 0.5 db with the beta-modes in all the fequency ange. This confims the imotance of the fequency deendent modes in modeling viscoelastic mateials with stong fequency deendent behavios. In tems of comutational efficiency, the diect solution took 6 minutes wheeas the modal aoach equied 10 minutes with the eal modes and 1 minutes with the betamodes. This imovement in comutation time may be 4 Conclusion An iteative method fo comuting fequency deendent modes of stuctues with VEM has been develoed and imlemented in Code_Aste. These modes can be used as a modal basis fo fequency esonse calculations. They will extend the validity of the basis ove all the fequency ange by taking into account the fequency deendence of the stiffness matix. Caabilities of Code_Aste have been imoved by imlementing a maco command to define VEM with fequency deendent behavios and comute esulting fequency esonses via a diect o a modal aoach. A efomance study was caied out to demonstate the efficiency of the fequency deendent modes in modeling an automotive laminate windshield comising a olyvinylbutyal mateial with a stong fequency deendent viscoelastic behavio. Refeences [1] A. D. ashif, D. I. G. Jones, J. P. Hendeson, Vibation Daming, John Wiley and Sons (1985) [] A- Sandwich Shells, Comutational Methods and Exeimental IMAC (000) [3]. Melette, S. Gemès, F. van Hee, L. Jézéquel, D. edictions of stuctues containing high daming ISMA, Leuven (004) [4] Resonse Comutation of Stuctues with Viscoelastic Daming Mateials using a Modal Aoach - Descition of the Method and Alication on a Ca Doo Model Teated with High Daming Foam 5 th Symosium on Automobile Comfot, Le Mans (008) [5] M. Géadin, D. Rixen, Théoie des Vibations, Alications à la Dynamique des Stuctues, Editions Masson (1996) [6] D.J. Ewins, Modal Testing: Theoy and Pactice, Reseach Studies Pess LTD. (1986) [7] B- Modified MSE Method fo Viscoelastic Systems: A Weighted Stiffness Ma ASME Jounal of Vibation and Acoustics 117, 6-31 (1995) [8] A. Bobillot, E. Balmès, Iteative Techniques fo Eigenvalue Solutions of Damed Stuctues Couled with Fluids, 43d AIAA/ASME/ASCI/AHS/ASC Stuctues, Stuctual Dynamics and Mateials Confeence, Denve (00) [9] oeties on the vibational daming of a ISMA, Leuven (00) [10] Solveus modaux et ésolution du oblème généalisé (GEP)Refeence documentation R of Code_Aste (011)