Automatic Modal Analysis Myth or Reality?

Transcription

1 Automatic Moda Anaysis Myth or Reaity? Bart Peeters, Jenny Lau, Jeroen Lansots and Herman Van der Auweraer LMS Internationa, Leuven, Begium The increasing use of experimenta moda anaysis (EMA) as a standard too means that both experienced and inexperienced anaysts are faced with new chaenges: uncertainty about the accuracy of resuts, The route to automation sti requires discrimination methods to distinguish physica from mathematica poes, in particuar in the case of high-order or highy damped structures. This artice discusses an approach for automating the moda parameter estimation process and its industria vaidation. The vibration and acoustica behavior of a mechanica structure is determined by its dynamic characteristics. This dynamic behavior is typicay described with a inear system mode. The inputs to the system are forces (oads), and the outputs are the resuting dispacements or acceerations. System poes usuay occur in compex conjugate pairs, corresponding to structura vibration modes. The poe s imaginary part reates to the resonance frequency and the rea part to the damping. Structura damping is typicay very ow (a few percent of critica damping). The system s eigenvectors, expressed on the basis of the structura coordinates, correspond to characteristic vibration patterns or mode shapes. System identification from input-output measurements yieds the moda mode parameters. 1 This approach is now a standard part of the mechanica product engineering process. However, severa constraints make the system identification process for structura dynamics more compex than in eectrica engineering or process contro. A key issue is the difficuty of seecting the correct mode order and the corresponding vaidation of the obtained system poes. First of a, a continuous structure has an infinite number of modes. In practice, the anayst is interested ony in a imited number of these, up to a certain frequency or ony in a certain frequency band. Sti, mode orders of more than 100 are no exception. Furthermore, whie some of the modes are separated in resonance frequency, others may be very cose, eading to highy overapping responses. The standard approach of seecting a mode order and then deriving the corresponding poes is in genera not appicabe, and over-specification of the mode order is needed. Finay, the size of the probem often requires more than 1000 responses to be processed (e.g. a car body is discretized by more than 500 nodes and measured in three directions) and using arge data segments to reduce the measurement noise. The consequence of these constraints is that cassica system identification approaches, extracting the parameters of a discrete-time state-space mode or of an ARMA mode directy from the samped input-output data, are often neither practica nor feasibe. Specific procedures are then needed for moda anaysis. Moda anaysis users face the foowing chaenges: The ever increasing compexity of the tested structures: e.g. fuy assembed vehices instead of components, in-situ instead of aboratory measurements. The changing roe of testing in the product deveopment cyce, 2 impying a reduction of time avaiabe for testing and anaysis and a demand for increased accuracy adequate for use with hybrid or FE appications, Specific to the moda parameter estimation process itsef: inconsistency between estimates of different operators, the tedious task of seecting obvious poes in a stabiization diagram and the time-consuming iterations required to vaidate a moda mode. A key requirement for experimenta moda anaysis (EMA) is that a reiabe anaysis of compex datasets shoud be possibe Based on a paper presented at IMAC XXV, the 25th Internationa Moda Anaysis Conference, Orando, FL, February Figure 1. Stabiization diagram. Symbos have foowing meaning: o new poe; f stabe frequency; d stabe frequency and damping; v stabe frequency and eigenvector; s a criteria stabe. with minima, or even excuding, user interaction. This is the context of the methodoogy deveoped here. The foowing sections discuss: Using a stabiization diagram to sove the order determination probem. An automatic procedure is expanded to heaviy rey on the stabiization diagram concept. The methodoogy is vaidated using industria exampes. Stabiization Diagram The key difficuty in appying system identification for EMA of arge-scae structures is the seection of the mode order and of the corresponding system poes. In EMA, measured frequency response functions (FRFs) are curve-fit by a moda mode: 1 { } < > n n i i T { n } i < i H > =  + (1) i= 1 jw - i jw - i where: m Œ = FRF matrix containing the FRFs between a m inputs and a outputs n = number of modes H = compex conjugate transpose of a matrix { n i } Œ = mode shapes T m < i >Œ = moda participation factors i = poes The poes occur in compex conjugated pairs and are reated to the eigenfrequencies, w i [rad/s], and damping ratios x i [-] ( denotes a compex conjugate): i, i 2 = - xiwi ± j 1 - xi wi So in EMA, the probem of determining mode order bois down to deciding how many modes n to use for fitting the FRFs. Note that in cassica system identification iterature, many forma procedures exist to sove the probem of determining mode order. Modes of different order are identified and compared according to quaity criteria such as Akaike s fina prediction error or Rissanen s minimum description ength criterion. Most of these techniques were deveoped in the context of contro theory, where it is the aim to identify optima ow-order modes. But in structura dynamics, the order of the modes is typicay chosen much higher to reduce the bias on the estimates and to capture a reevant characteristics of the structure, even in the presence of arge amounts of measurement noise. As a consequence of order over-specification, the physicay meaningfu poes are competed with a set of mathematica poes, modeing mode, data and process noise but without having (2) SOUND AND VIBRATION/MARCH

2 Figure 2. Industria vaidation using a car body moda test and a sateite moda survey test. a reation to the structura probem. To address this probem, the concept of a stabiization diagram is introduced. The basic idea is that severa runs of the compete poe identification process are made by using modes of increasing order. Experience on a very arge range of probems shows that in such anaysis, the poe vaues of the physica eigenmodes aways appear at a neary identica frequency, whie mathematica poes tend to scatter around the frequency range. The poe vaues from a these anayses at different orders can be combined in one singe diagram, with the poe frequency as the horizonta axis and the soution order as the vertica axis. The poe is shown by a symbo in this diagram (Figure 1). Physica poes are readiy visibe in the diagram. To show that the frequency (damping and eigenvector, respectivey) of a poe fas within certain bounds of the vaues obtained at a ower system order, this is indicated by a symbo (for exampe by an f for frequency stabiization, d for damping and frequency or v for eigenvector and frequency). As typica stabiity criteria, the foowing vaues are used: 1% for frequency stabiity, 5% for damping stabiity, 2% for eigenvector stabiity. These defauts refect the accuracy of the estimates that can be expected in a wide range of industria moda anaysis appications. From such a diagram, it is not ony possibe to seect the optima system order, and for this order the vaid system poes, but it is even possibe to seect individua poes from different anayses. To this purpose, severa criteria can be used, such as the owest order at which a poe becomes stabe, the frequency or damping trend when potting a specific poe across the order etc. Of course, when seecting individua poes, a compete structura mode is not avaiabe. Recombining the poes into a new mode usuay soves this probem. In most moda parameter estimation methods, a stabiization diagram is constructed based on poe i and participation factor < i T > information. After interpreting the stabiization diagram, the mode shapes {n i } and the ower and upper residuas are obtained as the inear east-squares soution of Eq. 3: { } < > n n i i T { n } i < i H > [ LR] =  [ UR] (3) i jw - 2 = 1 i jw - i w With respect to Eq. 1, the ower and upper residuas m [ LR],[ UR] Œ, modeing the infuence of the out-of-band modes in the considered frequency band, have been added. Soving Eq. 3 aso soves the mode recombination probem that arises from seecting individua modes in the stabiization diagram. The process remains very interactive and requires good user experience, especiay for compex datasets. In some cases, it may even appear that the probem of determining mode order has been shifted to the probem of interpreting very uncear stabiization diagrams (see Industria Vaidation, beow). Automatic Moda Parameter Seection With the increasing use of moda anaysis as a standard too by many, incuding ess experienced users, the primary need is to automate the process. Researched soutions incude estimation methods that are much more robust with respect to the appearance of spurious poes. In this context, impressive resuts are obtained with PoyMAX, a discrete frequency-domain method that uses a east-squares approach to fit a rationa fraction poynomia mode to a mutipe input mutipe output (MIMO) FRF matrix. Exacty four years ago, an artice on PoyMAX appeared in Sound and Vibration. 3 The technica background on PoyMAX can aso be found esewhere. 4,5 The extension of PoyMAX to operationa moda anaysis (i.e. moda parameter estimation using operationa data obtained under immeasurabe excitation) was aso presented. 6 Very high system orders (more than 50) are ceary identified in a singe-step procedure, eading to extremey cear stabiization diagrams drasticay improving the quaity and the interpretabiity of the resut. It is a feature of the PoyMAX identification method to estimate the mathematica poes with a negative damping ratio. Therefore, these poes are readiy excuded before constructing the stabiization diagram. The fact that inherenty unstabe modes are identified is not a probem as a new mode is recomposed 3 after seecting the stabe poes from the stabiization diagram. Nevertheess, the route to automation sti requires discrimination methods to distinguish physica from mathematica poes. Probaby, the most natura way is trying to capture the decisions that an experienced moda anayst makes based on a stabiization diagram or by rues that can be impemented as an autonomous procedure. The seection of poes in a stabiization diagram has cassicay been done by an expert engineer visuay inspecting the symbos, which are based on simiarity in frequency, damping ratio or mode vector between poes beonging to subsequent mode orders (see above). Typica questions incude: How to seect poes in a stabiization diagram? How to speed up the iterative process of poe seection in stabiization diagram? How to ensure that consistent anayses are obtained from different peope from the same database? 18 SOUND AND VIBRATION/MARCH

3 An automatic seection procedure is the answer to these questions. The so-caed automatic moda parameter seection (AMPS) procedure is an inteigent rue-based approach in which the knowedge of experienced anaysts is captured. 7-9 The combination of PoyMAX and AMPS has the advantages that it is much faster than the time-consuming iterative process of manuay seecting poes. The combination provides ess-experienced anaysts access to expert knowedge bases and heps generate user-independent resuts. Industria Vaidation Proof of Concept. A benchmark anaysis was performed to evauate the resuts derived from the new AMPS too. The goa of the benchmark test was to provide a quaitative assessment of the too and to pace it within the spectrum ranging from novice to experienced moda anaysts. Eight peope were seected for the test, incuding four novices and four experts, a of whom had an engineering background. The novices received a short description of the task they had to carry out. In the test, a MIMO dataset from a fuy trimmed car body was used (Figure 2). It has two inputs and 264 measurement points distributed over the entire car body, eading to 528 FRFs. The parameter estimation was done with both a time-domain method (Poy-reference LSCE) and a frequency-domain method (PoyMAX) for the frequency band Hz. Both stabiization diagrams were created using a mode size of 64. An initia examination of the two stabiization diagrams (Figure 3) shows that the LSCE diagram is rather compicated and couded by spurious, mathematica poes, especiay at higher mode orders. But the PoyMAX diagram ceary shows the stabe poes throughout the entire frequency band. This resuted in arge differences in the number of poes seected by the various participants. The time taken by each participant to make the assessment was measured, and in genera, the LSCE task took about twice as ong to compete as the PoyMAX task. In addition, the experts spent about twice as much time in the assessment as the novices, who were so overwhemed by the compexity of the LSCE diagram that they quicky gave up. Figure 4 shows a frequency spectrum of the poe seection of a the test participants for the two stabiization diagrams. Users 1-4 are the novices; Users 5-8 are the experts. The vertica dotted ines show the seection made by the AMPS too. Reying on the LSCE diagram (Figure 3a), it was cear that the novices encountered difficuties in the Hz band; not ony did they miss poes, there was aso a wide variation in those that were seected. Even the experts did not find it easy in this frequency range, since the poes they seected did not ine up we, indicating differences in the frequencies (Figure 4a). Above 65 Hz, the experts agreed quite we, but the novices missed some of the poes competey. Reying on the PoyMAX diagram (Figure 3b), the majority of the participants seected a the poes. The resuts from a of the users (experts and novices) agreed much better, as indicated by the nicey aigned dots (Figure 4b). Figure 5 shows a comparison of the damping ratios of the seected poes. The crosses ( ) represent the AMPS seection. A custer of dots ( ) represents consensus over the seection by the different test participants,whie a scatter over the damping woud indicate, in case of novices, ack of experience and feeing for physica damping vaues. In genera, the participants demonstrated more consensus with respect to damping in the PoyMAX diagram (Figure 5b) than in the LSCE diagram (Figure 5a). AMPS argey agrees with this consensus. The sparse number of seections in the Hz band expains the differences. There is a remarkabe improvement for both novices and experts in that band for the PoyMAX pot. It is cear that the association of the PoyMAX method and AMPS generates user-independent resuts and can be an educationa too for both novices and experts. More benchmark resuts and detais can be found Reference 9. Increased Productivity. Using AMPS, the iterative process of poe seection from a stabiization diagram is much ess time consuming. More poes are seected in just a few seconds, and the poe seection procedure can be made faster by using a arger band Figure 3. Car body stabiization diagrams; a) Poy-reference LSCE; b) PoyMAX. Figure 4. Car body poe seection resuts per user (Novices: 1-4, Experts 5-8); a) LSCE; b) PoyMAX. and a higher maximum mode order. This is iustrated using data from the body in white of a midsize car. The data consist of two inputs and as much as 2005 response degrees of freedom (DOFs). A compete moda anaysis was performed by an expert moda anayst over two weeks and resuted in 233 poes being found. The iterative procedure used in this type of moda anaysis cannot be appied to the whoe band of interest. Different frequency bands are anayzed and the poes, estimated from each frequency band, are then merged into one anaysis. With AMPS, the whoe frequency range (with 1437 spectra ines) of interest was treated at the same time with a mode size of 256. After 40 seconds, 112 poes were highighted and automaticay seected in the stabiization diagram. Figure 6 shows the resuts from a more detaied study of the range between 36 and 79 Hz. Not ony is the PoyMAX diagram much cearer, but AMPS is aso abe to find many more poes. Figure 7 shows the exceent quaity of the synthesized FRFs of the moda mode generated by the PoyMAX-AMPS seection. Most of the AMPS modes agree we with a manua expert SOUND AND VIBRATION/MARCH

4 Figure 5. Car body poe seection, frequency versus damping for a users ( ) and AMPS ( ); a) LSCE; b) PoyMAX. anaysis. The remaining modes need to be estimated iterativey. For a simpe structure, a modes are identified by AMPS, and the time gain achieves a significant 95%. On a more compex structure, where interactive moda anaysis is performed on severa smaer bands, an average 80% of the modes are identified with high confidence by AMPS. Taking the remaining interactive anaysis time into account, the overa productivity gain sti is an impressive 50%. Figure 6. Car body in white stabiization diagrams with AMPS resuts indicated by vertica ines; a) LSCE; b) PoyMAX. Outook and Concusions Simpified or even automated identification of the parameters of compex systems offer the key to a whoe series of mode-based engineering appications. In many probems, structura identification is cosey reated to detecting changes in the system dynamics. An exampe is the fight quaification of aircraft, requiring repeated in-fight moda anayses at different airspeeds. At each air speed, resonance frequencies and damping ratios of critica modes are checked to verify the absence of aero-eastic instabiity (futter). During the change from one fight condition to the next, the dynamics may change due to imminent futter, and the damping must be monitored continuousy.8,10 Another exampe is the use of changes in the moda system mode to detect structura damage or in assessing the integrity of a structure after forced oading during a quaification test. An exampe of such a structura heath monitoring (SHM) approach is found in Reference 11. As part of schedued major maintenance, each orbiter in the NASA Space Shutte feet undergoes moda testing on a reguar basis. It was demonstrated that automatic procedures coud reduce the data anaysis efforts from one month to one hour. In vibration-based SHM of civi engineering structures, ad-hoc automatic moda anaysis12 as we as statistica test methods are pursued.13 The same methodoogy can be appied to probems such as variant anaysis (product scatter) or end-of-ine product testing. Significant progress has been made in addressing the key probem of discriminating physica system poes from mathematica poes in the identification of the dynamics of compex structures. The stabiization diagram offers a heuristic approach. Nove mode estimation agorithms with cear stabiization behavior ease the 20 SOUND AND VIBRATION/MARCH 2008 Figure 7. Measured (red) and synthesized (green) FRFs. process dramaticay, opening the way to a fuy automated process. The combination of the PoyMAX moda parameter estimation method with the AMPS procedure means that the myth of automating moda anaysis is cose to becoming a reaity. We have demonstrated proof of concept, increased productivity and industria appicabiity. Acknowedgement This work was carried out within the EUREKA project E! 3341 FLITE2. The financia support of the Institute for the Promotion of Innovation by Science and Technoogy in Fanders (IWT) is gratefuy acknowedged.

5 References 1. Heyen, W., Lammens, S., Sas, P., Moda Anaysis Theory and Testing. Dept. of Mech. Eng., K.U. Leuven, Begium, Van der Auweraer, H., Requirements and Opportunities for Structura Testing in View of Hybrid and Virtua Modeing, Proc. of ISMA 2002 Int. Conf. on Noise and Vibration Eng., Leuven, Begium, September Peeters, B., Lowet, G., Van der Auweraer, H., Leuridan, J., A New Procedure for Moda Parameter Estimation, Sound and Vibration, January Guiaume, P., Verboven, P., Vananduit, S., Van der Auweraer, H., Peeters, B., A Poy-Reference Impementation of the Least-Squares Compex Frequency-Domain Estimator, Proc. of IMAC 21, the Int. Moda Anaysis Conf., Kissimmee, FL, February Peeters, B., Van der Auweraer, H., Guiaume, P., Leuridan, J., The Poy- MAX Frequency-Domain Method: A New Standard for Moda Parameter Estimation? Shock and Vibration, Vo. 11, pp , Peeters, B., Van der Auweraer, H., Vanhoebeke, F., Guiaume, P., Operationa Moda Anaysis for Estimating the Dynamic Properties of a Stadium Structure During a Footba Game, Shock and Vibration, Vo. 14, No. 4, pp , LMS Internationa, LMS Test.Lab Moda Anaysis, Leuven, Begium, Scionti, M., Lansots, J., Goethas, I., Vecchio, A., Van der Auweraer, H., Peeters, B., De Moor, B., Toos to Improve Detection of Structura Changes from In-Fight Futter Data, Proc. 8th Int. Conf. on Recent Advances in Structura Dynamics, Southampton, UK, Juy Lansots, J., Rodiers, B., Peeters, B., Automated Poe-Seection: Proofof-Concept and Vaidation, Proc. of ISMA 2004 Int. Conf. on Noise and Vibration Eng., Leuven, Begium, September Verboven, P., Cauberghe, B., Guiaume, P., A Structura Heath Monitoring Approach for Futter Testing, Proc. of ISMA 2002 Int. Conf on Noise and Vibration Eng., Leuven, Begium, September Pappa, R. S., James III, G. H., Zimmerman, D. C., Autonomous Mode Identification of the Space Shutte Tai Rudder, Journa of Spacecraft and Rockets, Vo. 35, No. 2, pp , Peeters, B., De Roeck, G., One-Year Monitoring of the Z24-Bridge: Environmenta Effects Versus Damage Events, Earthquake Engineering and Structura Dynamics, Vo. 30, No. 2, pp , Bassevie, M., Abdeghani, M., Benveniste, A., Subspace-Based Faut Detection Agorithms for Vibration Monitoring, Automatica, Vok. 36, No. 1, pp , The authors can be reached at: bart.peeters@msint.com. SOUND AND VIBRATION/MARCH