Modal Parameter Identification Using Particle Swarm Optimization

Transcription

1 Modal Paamete Identification Using Paticle Swam Optimization M. El-Kafafy, A. Elsawaf, B. Peetes, T. Vampola, P. Guillaume Abstact In this pape, the applicability of Paticle Swam Optimization (PSO) to identify the modal paametes will be tested. PSO is a heuistic optimization method which does not equie the calculation of the eo deivatives with espect to the model paametes hence the Jacobian matix fomulation is not equied. The modal paametes will estimated by optimizing the modal model using PSO in ode to decease the eo between the modal model and the measued fequency esponse functions (FRFs) of the stuctue unde test. The applicability of PSO to optimize the modal model is evaluated by means of eal-life measuement example. Keywods Complex stuctues, Fequency esponse functions, Modal paamete identification, Paticle swam optimization. M I. INTRODUCTION ODAL analysis is cuently one of the key technologies used fo analysing the dynamic behaviou of complex stuctues such as cas, tucks, aicafts, bidges, offshoe platfoms and industial machiney A numbe of textbooks give a good oveview of the theoy and pactice in the domain of modal analysis [1-4]. Modal analysis is a pocess wheeby a stuctue is descibed in tems of its natual chaacteistics, which ae the esonance fequencies, damping atios and mode shapes - its dynamic popeties. These modal paametes ae the basic chaacteistics of any vibation (esonance) mode. A small foce exciting the stuctue at one of these esonance fequencies causes lage vibation esponses esulting in possible stuctual damage. The damping atios of the diffeent vibation modes contol the vibation level at the coesponding esonance fequency. The mode shape that is not global but local popety of the stuctue descibes how the stuctue will vibate when it is excited at a cetain esonance fequency. Fo the mode shape, local popety means that it depends on the numbe of measued degees of feedom (DOFs) and the locations of these DOFs with espect to the stuctue unde test. Indeed, we can also say it is global popety in the sense that it depends on the mass and stiffness M. El-kafafy is with the Helwan Univesity, Egypt and Vije Univesiteit Bussel (VUB), Bussels, Belgium. (melkafaf@vub.ac.be ). A. Elsawaf is with the Helwan Univesity, Egypt and Czech Technical Univesity in Pague, Czech Republic.(elsawafahmed@gmail.com) B. Peetes is with Siemens Industy Softwae, Leuven, Belgium (bat.peetes@siemens.com). T. Vampola is with Czech Technical Univesity in Pague, Czech Republic (Tomas.Vampola@fs.cvut.cz) P. Guillaume is with the Vije Univesiteit Bussel (VUB), Bussels, Belgium (patick.guillaume@vub.ac.be). distibution of the stuctue unde test. Indeed, getting accuate estimates fo these paametes helps to bette undestanding, modelling and contolling the dynamics of the vibatoy stuctues. In this contibution, based on successfully expeiences obtained with the application of Paticle Swam Optimization (PSO) in othe aeas [5-8], the authos decided to investigate the applicability of PSO technique in the field of modal paamete identification. PSO is a heuistic optimization method which is diffeent fom the optimization algoithms that ae being used in the modal paamete estimation community. PSO was oiginally contibuted by Kennedy and Ebehat [9] and was fist intoduced fo simulating social behaviou as a stylized epesentation of the movement of oganisms in a bid flock o fish school. PSO algoithm is a population based algoithm that makes few o no assumptions about the optimized poblem and can seach vey lage spaces of candidate solutions. The validation of applying PSO to modal paamete estimation in this pape will be done using a eal-life measuement fom aeospace application. The outline of the pape is as follows: in section II, a eview ove the modal paamete estimation techniques will be given. Then, the poblem statement will be descibed in section III. In section IV, a theoetical backgound about PSO will be given. Some validation esults will be shown in section V. Then, some concluding emaks ae given in section VI. II. MODAL PARAMETER ESTIMATION: A REVIEW Ove the last decades, a numbe of algoithms have been developed to estimate modal paametes fom measued fequency o impulse esponse function data. The algoithms have evolved fom vey simple single degee of feedom (SDOF) techniques to algoithms that analyse data fom multiple-input excitation and multiple-output esponses simultaneously in a multiple degee of feedom (MDOF) appoaches. In the time domain modal paamete identification techniques, the Complex Exponential (CE) algoithm is one of the ealiest modal analysis methods. The CE was impoved using least squae solution by Bown et al. [1], and it was called Least Squae Complex Exponential (LSCE) method. The LSCE method was developed fo MIMO systems as the Polyefeence Least Squae Complex Exponential (plsce) by Vold et al.[11]. Even though the plsce method uses FRFs as an input, it essentially opeates in the time domain. This is ISSN:

2 achieved by computing the impulse esponse functions (IRFs) fom the FRFs by invese Fouie tansfomation. In the fequency-domain modal paamete identification side, a vey popula implementation of the fequency-domain linea least squaes estimato optimized fo the modal paamete estimation is called Least Squaes Complex Fequency-domain (LSCF) estimato [12]. That method was fist intoduced to find initial values fo the iteative maximum likelihood method [13]. The LSCF estimato uses a discete-time common denominato tansfe function paameteization. In [14], the LSCF estimato is extended to a poly-efeence case (plscf). The plscf estimato uses a ight matix faction desciption (RMFD) model. Both of those estimatos have been developed fo handling modal data sets that ae typically chaacteized by a lage numbe of esponse DOFs, high modal density and a high dynamic ange. LSCF and plscf estimatos wee optimized both fo the memoy equiements and fo the computation speed. The main advantages of those estimatos ae thei speed and the vey clea stabilization chats they yield even in the case of highly noise-contaminated fequency esponse functions (FRFs). LSCF and plscf estimatos ae cuve fitting algoithms in which the estimation pocess is achieved without using infomation on the statistical distibution of the data. By taking knowledge about the noise on the measued data into account, the modal paametes can be deived using the so-called fequency-domain maximum likelihood estimato (MLE) with significant highe accuacy compaed to the ones developed in the deteministic famewok. MLE fo linea time invaiant systems was intoduced in [15] and it is extended to multivaiable systems in [16]. A multivaiable fequencydomain maximum likelihood estimato was poposed in [13] to identify the modal paametes togethe with thei confidence intevals whee it was used to impove the estimates that ae initially estimated by LSCF estimato. In [17], the polyefeence implementation fo MLE was intoduced to impove the stating values povided by plscf estimato. Both of the ML estimatos intoduced in [13, 17] ae based on a ational faction polynomial model, in which the coefficients ae identified. The modal paametes ae then estimated fom the coefficients in a second step. In these estimatos, the uncetainties on the modal paametes ae calculated fom the uncetainties on the estimated polynomial coefficients by using some lineaization fomulas. These lineaization fomulas ae staightfowad when the elation between the modal paamete and the estimated coefficients is explicitly known but can be quite involved fo the implicit case. Moeove, they may fail when the signal-to-noise atio is not sufficiently lage [18]. A combined deteministic stochastic modal paamete estimation appoach called Polymax Plus has been intoduced and successfully validated in [19-24]. This estimato combines the best featues of both the plscf estimato [14] of having a clea stabilization chat in a fast way and the MLE based estimation [13] of having consistent estimates of the modal paametes togethe with thei confidence bounds. A ecent maximum likelihood modal paamete identification method (ML-MM), which identify diectly the modal model instead of ational faction polynomial model, is intoduced and validated with simulated datasets and seveal eal industial applications in [25-31]. Basically, the design equiements to be met in the ML-MM estimato wee to have accuate estimate fo both of the modal paametes and thei confidence limits without using the lineaization fomulas which have to be used in case of identifying a ational faction polynomial models. And, meanwhile, to have a clea stabilization chat which enables the use to easily select the physical modes within the selected fequency band. Anothe advantage of the ML-MM estimato lies in its potential to ovecome the difficulties that the classical modal paamete estimation methods face when fitting an FRF matix that consists of many (i.e. 4 o moe) columns, i.e. in cases whee many input excitation locations have to be used in the modal testing. Fo instance, the high damping level in acoustic modal analysis equies many excitation locations to get sufficient excitation of the modes. All the peviously mentioned modal paamete identification methods ae based on fitting a mathematical model (e.g. polynomial-based models o modal model) to the measued data (i.e. FRFs). This fitting can be done eithe in a linea-least squaes sense (e.g. plscf) o in a non-linea least squaes sense (e.g. MLE o ML-MM). In case of non-linea least squaes-based estimatos [13, 28, 32, 33], a non-linea optimization algoithm is equied since the cost function to be minimized is nonlinea in the paametes of the model. In system identification community Levenbeg- Maquadt method [33, 34], which combines Gauss-Newton and Gadient descent methods, is commonly used to minimize the cost function of these estimatos. III. PROBLEM STATEMENT The modal model is consideed as one of the impotant objectives of any modal estimation pocess, by which we ae chaacteizing the system dynamics in tems of its modal paametes (i.e. poles, mode shapes and paticipation factos). This model poposes that the fequency esponse function matix of the system can be fomulated in its modal fom as follows [3]: Nm T * H L L H Ωk N * on i 1 Ωk Ωk (1) LR UR Ω H Ω k NoNi with with 2 k the fequency esponse function matix N o the numbe of the measued outputs and numbe of the measued inputs, N i the the polynomial Ωk jk basis function in case of using continuous-time fomulation (s- ISSN:

3 domain), k the cicula fequency in ad/sec., T 1 N L i, No 1 ψ, λ the mode shape, the paticipation facto and the pole coesponding to the th NoNi mode. LR and NoNi UR ae the lowe and uppe esidual tems. Since the modal model fomulation uses a limited numbe of modes to model the FRF matix within the analysis fequency band, the lowe and uppe esidual tems ae used to compensate fo the esidual effects that come fom the out-of-band modes. Equation 1 consides a displacement FRF, while often the acceleation FRF is measued in modal analysis tests o sometime velocity FRFs like in case of using the Lase Dopple Vibomete. In such cases, equation 1 should be coected as follows: 2 H Ω Ω H Ω Vel k k Dis k H Ω Ω H Ω Accel k k Dis k with Dis, Vel and HAccel (2) H H the displacement, velocity and acceleation FRFs. In case of opeational modal analysis (OMA), the uppe and lowe esidual tems (opeational esiduals) ae diffeent fom the esiduals in case of expeimental modal analysis (EMA) used in equation 1. They wee detemined by veifying the asymptotic behaviou of the output specta of a single-degee of feedom system excited by a white noise in [35]. Once the modal model is deived fo a cetain stuctue, a numbe of applications of modal analysis can be instigated using this modal model. In the following, some of the applications in which the modal model can be used Coelation of FEM Damage detection Stuctual modification Sensitivity analysis Foced esponse pediction Substuctue coupling Active and semi-active contol It is obvious that all the above-mentioned applications heavily depend on the extacted modal paametes. In othe wod, a successful applicability of all those applications depends on the quality of the modal model. The quality of the estimated modal model mainly depends on the quality of the measued data and on the asymptotic popeties of the modal paamete estimato used to extact the modal model paametes fom the measued data. As mentioned in section II, in the liteatue thee ae seveal modal paamete estimatos ae successfully intoduced and validated to obtain accuate modal model. In this contibution, the objective now is to ty PSO in optimizing the modal model pesented by equation 1 with the aim to have accuate modal paametes (i.e., L T,, LR, UR) fo the stuctue unde test within the analysis fequency band. Theefoe, to optimize the modal model in equation 1 the following cost function (equation 3) will be minimized using PSO: NN N f o i, E, 2 (3) PSO k l k l1 k1 with N the numbe of the fequency lines and f E, Hˆ Ω, H is the eo between the l k l k l k modal model Hˆ lω, k measued FRFs l epesented by equation 1 and the H at fequency line k. So, the cost k function is simply the sum of the squaes of the absolute value of the eo ove all the inputs, outputs and fequency lines. So, the modal paametes of the modal model in equation 1 will be tuned by the PSO algoithm in the way which minimizes the cost function descibed above by equation 3. To educe the computational time taken by PSO, the poles and the paticipation factos L ae taken as the paametes to be optimized by PSO, and then the mode shapes and the esidual tems ( LR & UR ) ae calculated in a linea least squaes sense as implicit functions of the optimized poles and paticipation factos using equation 1. IV. PARTICLE SWARM OPTIMIZATION Recently, paticle swam optimization (PSO) has attacted a lot of attention because it s easy to implement, obust, fast convegence, and fo its ability to solve many optimization poblems. PSO algoithm optimizes a poblem using a population (swam) of candidate solutions (paticles). Paticles have thei own positions, and fly aound in the poblem solution space looking fo best fitness value. Those paticles ae initially scatteed in the solution space with initial ( ) ( ) positions. The position and velocity v of a paticle at the geneation, ae iteatively enhanced in the solution space towads the optimum solution. Each movement of a ( ) paticle is influenced by its local best position b and the global best position ( ) g obtained fom all candidates in the solution space. When the pocess epeated fo sufficient numbe, the best solutions eventually will be found. Equation (4), shows the mathematical fomula used fo updating the positions and the velocities of the paticles [36]; v ( 1) ( ) ( 1) ( 1) v ( ) ( ) ( ) v acc 1 and1 b ( ) ( ) acc 2 and2 g (4) is the constiction coefficient, acc 1 and acc 2 ae acceleation coefficients, and 1 and and 2 ae andom numbes between and 1. The values used in this study fo, acc 1 and acc 2 ae.729, 2.5 and 2.5 espectively [36]. Figue 1 shows a flow chat fo the optimization pocedues. ISSN:

4 -3. g/n db -11. Initialize paticles with andom positions. Each one has and velocity v Defined each paticle best position b, and the global best position g fo all paticles Update the position and the velocity of each paticle Eq. (4) No Stat Fo each paticle, calculate PSO k, Maximum iteation numbe eached End Yes Figue 1: Optimization pocedues flow chat F F 1. Amplitude FRF back:phd:+x/f2:fed:+z FRF wing:vvd:+z/f2:fed:+z. Figue 2: Typical FRFs fo the tested business jet (the fequency axis is hidden fo confidentiality) V. VALIDATION RESULTS A. Inflight Dataset Example In this section, the poposed PSO fo modal paamete estimation will be validated using expeimentally measued FRFs that wee measued duing a business jet in-flight testing. These types of FRFs ae known to be highly contaminated by noise. Duing this test, both the wing tips of the aicaft ae excited duing the flight with a sine sweep excitation though the fequency ange of inteest by using otating fans. The foces ae measued by stain gauges. Next to these measuable foces, tubulences ae also exciting the plane esulting in athe noisy FRFs. Figue 2 shows some measued fequency esponse functions (FRFs), which clealy show the noisy chaacte of the data. Duing the flight, the acceleations wee measued at nine locations while both the wing tips wee excited (two inputs). The PSO needs an initial guess about the numbe of the modes within the analysis band. The plscf estimato is applied to the measued FRFs to have a clue about how many modes ae expected in the desied band. It was found that thee ae 13 physical vibation modes within the analysis band. The modal model (1) is then optimized by minimizing the cost function (3) using the PSO technique. To stat the PSO, lowe and uppe bounds have to be defined fo all the paametes that have to be optimized (i.e. the poles and paticipation factos). The pole consists of eal and imaginay pats. Fo each mode, the undamped esonance fequency is /2 and the damping atio is Re / n. Assuming lowe and uppe bounds fo the poles can be made easie if the pole fo each mode is witten as a function of the esonance fequency and damping atio. The pole can be witten as a function of and as 2 n n j 1 n. So, instead of optimizing the poles, the fequencies and n damping atios will be optimized. The esonance fequency fo all the modes is bounded by the minimum and maximum fequency of the analysis band. The damping atio is allowed to vay between and 1% since the stable systems have to have a positive damping and 1 % is a logical value fo the damping atio fo the mechanical stuctues. The eal and imaginay pats of paticipation factos ae allowed to vay between -1 and 1. It should be said that modal paametes estimation using PSO in such complex cases will depend on the use s expeience and, in most of the cases, the pocedue must be epeated a few times modifying the lowe and uppe bounds fo the optimized paametes until the esults convege to optimum values. Cost Function Iteations Figue 3: Deceasing of the eo at diffeent iteation ISSN:

5 Figue 3 shows the deceasing of the cost function as a function of the numbe of iteations. In this case study, thee ae 9 outputs and 2 inputs and 13 modes. Fo this data set, the numbe of paametes to be optimized by PSO is 78 paametes: the fequency and the damping atio fo each mode ( 2Nm 2 13) plus the eal and the complex pats of the paticipation factos fo each mode ( 2N N ). The Maximum numbe of iteations taken was 1 iteations, and the PSO takes about 5 minutes to achieve those iteations. To check the accuacy of the estimated (optimized) modal model, a simple but vey popula way to validate the model is to compae the obtained model to the measuements. Figue 4 shows the quality of the fit between the measued and synthesized FRFs calculated based on the obtained modal model. It can be seen fom this figue that the PSO is able to convege to a modal model that closely fits the measued data, which indicates that the model epesents well the dynamic of the system unde test in the analysis band. In Figue 5, the auto Modal Assuance Citeion (auto MAC) of the estimated mode shapes is shown. It shows that the identified modes ae not coelated except fo some highe fequency modes. This coelation of the highe fequency modes is due to the spatial aliasing since the numbe of the measued outputs is only 9. Figue 6 shows some of the identified mode shapes. VI. CONCLUSION The estimation of modal paametes using Paticle Swam Optimization (PSO) was tied and its application to eal measued data showed that it can be used with a dependency on the use s expeience and the quality of the defined lowe and uppe bounds of the optimized paametes. In some cases, the pocedue has to be epeated few times modifying the defined lowe and uppe bounds of the paametes to each an optimum solution. The PSO does not equie the calculation of eo deivatives and Jacobian matix which might be taken as an advantage fo the method. On the othe hand, the quality of the solution and the calculation time of the PSO in such application have been found to be highly dependent on the quality of the defined bounds fo the paametes. Fo the futue wok, the PSO will be investigated fo the modal paamete estimation using moe industial applications. ACKNOWLEDGMENT This publication was suppoted by the Euopean social fund within the fame wok of ealizing the poject "Suppot of inte-sectoal mobility and quality enhancement of eseach teams at Czech Technical Univesity in Pague", CZ.1.7/2.3./3.34. Peiod of the poject s ealization Also, the financial suppot of the IWT (Flemish Agency fo Innovation by science and Technology) and Siemens Industy Softwae, though the Innovation mandate IWT poject 14539, is gatefully acknowledged. m i Phase, deg. Phase, deg. Phase, deg. Phase,deg Measued FRF (9,2) Synthesized FRF (9,2) Fequency, Hz Measued FRF (1,1) Synthesized FRF (1,1) Fequency, Hz Measued FRF (4,1) Synthesized FRF (4,1) Fequency, Hz Measued FRF (5,1) Synthesized FRF (5,1) Fequency, Hz Figue 4: Some typical synthesized FRF compaed with the measued ones ISSN:

6 Figue 5: Auto modal assuance citeion (MAC) of the identified mode shapes Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 1 Figue 6: Gaphical epesentation of some typical estimated mode shapes REFERENCES [1] Ewins, D., Modal Testing: Theoy and Pactice. 1986, Hetfodshie: Reseach Studies Pess. [2] He, J. and Z. Fu, Modal Analysis. 21, Oxfod: A division of Reed Educational and Pofessional Publishing Ltd. [3] Heylen, W., S. Lammens, and P. Sas, Modal Analysis Theoy and Testing. 1997, Hevelee: Katholieke Univesiteit Leuven, Depatment Wektuigkunde. [4] Maia, N. and J. Silva, Theoetical and expeimental Modal analysis. 1997, Hetfodshie: Reseach Studies Pess LTD. [5] Elsawaf, A., F. Ashida, and S.-i. Sakata, Optimum Stuctue Design of a Multilaye Piezo-Composite Disk fo Contol of Themal Stess. Jounal of Themal Stesses, (9): p [6] Meteed, H., et al., Vibation Contol of MR-Damped Vehicle Suspension System Using PID Contolle Tuned by Paticle Swam Optimization. SAE Intenational Jounal of Passenge Cas - Mechanical Systems, (2). [7] Elsawaf, A., F. Ashida, and S.-i. Sakata, Hybid constained optimization fo design of a piezoelectic composite disk contolling themal stess. Jounal of Theoetical and Applied Mechanics Japan, : p [8] Elsawaf, A., et al., Paamete identification of Magnetoheological dampe using paticle swam optimization, in Poc. of the Second Intl. Conf. on Advances In Civil, Stuctual and Mechanical Engineeing- CSM : Bimingham, UK. p [9] Kennedy, J. and R. Ebehat, Paticle Swam Optimization, in Poceedings of IEEE Intenational Confeence on Neual Netwoks p [1] Bown, D.L., et al., Paamete estimation techniques fo modal anlysis. SAE Tansactions, pape No , 1979: p [11] Vold, H., et al., A multi-input modal estimation algoithm fo minicomputes. SAE Tansactions, (1): p [12] Van de Auweae, H., et al., Application of a fast-stabilization fequency domain paamete estimation method. Jounal of Dynamic System, Measument, and Contol : p [13] Guillaume, P., P. Veboven, and S. Vanlanduit. Fequency-domain maximum likelihood identification of modal paametes with confidence intevals. in the 23d Intenational Semina on Modal Analysis Leuven, Belgium. [14] Guillaume, P., et al. A poly-efeence implementation of the leastsquaes complex fequency domain-estimato. in the 21th Intenational Modal Analysis Confeence (IMAC). 23. Kissimmee (Floida). [15] Schoukens, J. and R. Pintelon, Identification of linea systems: A pactical guide to accuate modeling. 1991, Oxfod: Pegamon Pess. [16] Guillaume, P., Identification of multi-input multi-output systems using fequency-domain models, in dept. ELEC. 1992, Vije Univesiteit Bussel (VUB): Bussels. [17] Caubeghe, B., P. Guillaume, and P. Veboven. A fequency domain poly-efeence maximum likelihood implementation fo modal analysis. in 22th Intenational Modal Analysis Confeence. 24. Deabon (Detoit). [18] Pintelon, R., P. Guillaume, and J. Schoukens, Uncetainty calculation in (opeational) modal analysis. Mechanical Systems and Signal Pocessing, (6): p [19] El-kafafy, M., P. Guillaume, and B. Peetes, Modal paamete estimation by combining stochastic and deteministic fequency-domain appoaches. Mechanical System and signal Pocessing, (1-2): p [2] El-kafafy, M., et al. Advanced fequency domain modal analysis fo dealing with measuement noise and paamete uncetainty. in the IMAC-XXX Jacksonville, FL, USA. [21] El-kafafy M., et al. Polymax Plus estimato: bette estimation of the modal paametes and thei confidence bounds. in the Intenational Confeence on Noise and Vibation Engineeing (ISMA) 214. Leuven, Belgium. [22] Peetes, B., M. El-Kafafy, and P. Guillaume. The new PolyMAX Plus method: confident modal paamete estimation even in vey noisy cases. in the Intenational Confeence on Noise and Vibation Engineeing (ISMA) 212. Leuven, Belgium. [23] Peetes, B., M. El-Kafafy, and P. Guillaume. Dealing with uncetainty in advanced fequency-domain opeational modal analysis in the eleventh Intenational Confeence on Computational StuctuTechnology,CST212, Dubovnik, Coatia. [24] Peetes, B., et al. Uncetainty popagation in Expeimental Modal Analysis. in the poceedings of Intenational Modal Analysis Confeence - IMAC XXXII 214. Olando - Floida - USA. ISSN:

7 [25] Accado G., et al. Expeimental Acoustic Modal Analysis of Automotive Cabin in the Intenational Modal Analysis Confeence IMAC XXXIII 215. Olando, Floida, USA.: Spinge. [26] El-Kafafy, M., Design and validation of impoved modal paamete estimatos, in Mechanical engineeing dept.. 213, Vije Univesiteit Bussel (VUB): Bussels. [27] El-Kafafy, M., et al. A fast maximum likelihood-based estimation of a modal model. in the Intenational Modal Analysis Confeence (IMAC XXXIII) Olando, Floida, USA Spinge. [28] El-Kafafy, M., T. De Toye, and P. Guillaume, Fast maximumlikelihood identification of modal paametes with uncetainty intevals: A modal model fomulation with enhanced esidual tem. Mechanical Systems and Signal Pocessing, (1-2): p [29] El-Kafafy, M., et al., Fast Maximum-Likelihood Identification of Modal Paametes with Uncetainty Intevals: a Modal Model-Based Fomulation. Mechanical System and Signal Pocessing, : p [3] El-Kafafy, M., et al. A fequency-domain maximum likelihood implementation using the modal model fomulation. in 16th IFAC Symposium on System Identification, SYSID 212. Bussels. [31] Peetes, B., et al. Automotive cabin chaacteization by acoustic modal analysis. in the poceedings of the JSAE Annual congess Japan. [32] Hemans, L., H. Van de Auweae, and P. Guillaume. A fequencydomain maximum likelihood appoach fo the extaction of modal paametes fom output-only data. in ISMA23, the intenational Confeence on Noise and Vibation Engineeing Leuven, Belgium. [33] Pintelon, R. and J. Schoukens, System Identification: A Fequency Domain Appoach. 21, Piscataway: IEEE Pess. [34] Eykhoff, P., System identification: paamete and state estimation. 1979, Bistol: John Wiley & Sons Ltd [35] Peetes, B., et al., Opeational modal analysis fo estimating the dynamic popeties of a stadium stuctue duing a football game. Shock and Vibation, (4): p [36] Clec, M. and J. Kennedy, The paticle swam: Explosion, stability, and convegence in a multidimensional complex space. IEEE Tansactions on Evolutionay Computation - TEC, 22. 6(1): p ISSN: