IMPROVEMENT OF A TAIL-PLANE STRUCTURAL MODEL USING VIBRATION TEST DATA. S. J. Guo

Transcription

1 IMPROVEMENT OF A TAIL-PLANE STRUCTURAL MODEL USING VIBRATION TEST DATA S. J. Guo Departmet of Aerospace, Civil ad Mechaical Egieerig, Uiversity of Hertfordshire, Hatfield, Hertfordshire, AL 9AB, U.K. ABSTRACT To achieve the best structural model improvemet usig vibratio test data, mai effort has bee made to locate poor modellig regios as a guidelie for subsequet model updatig. The method preseted ad used i this paper is the eergy error estimatio method. I the method the differece betwee aalytical ad test data based eergies at elemet scale is estimated to idicate ay poor structural mass ad stiffess modellig. As the result, poor modellig regios ca be distiguished from those correctly modelled ad the improvemet of origial structural model ca be carried out effectively ad accurately. To demostrate the applicatio of this method, a full-scale tail-plae structure has bee studied by usig simulated test modes as a simulated case ad usig measured modes as a practical case. I both cases poor modellig regios of the origial structural model have bee accurately located. Subsequetly, a sigificat improvemet of the structural model with the reductio of average frequecy error from origial 2.2% dow to.% for the simulated case ad from 4.6% to.8% for the practical case has bee achieved. Key Words: Tail-plae; Structural Model; Vibratio Test;. INTRODUCTION I the aalysis of structural dyamics, accurate modal predictio is essetial. For a practical structure of large size ad complex cofiguratio, however, modal errors due to simplificatio of structural modellig ad poor estimatio of structural parameters are almost ievitable. Poor aalytical modellig may also occur if udetected damage (fault) exists i a structure causig overestimatio of aalytical stiffess i the local regio. I order to improve the structural model efficietly, various methods based o available modal iformatio from vibratio test have bee developed i the last two decades. I the developmet of those methods, it has bee oted that the major difficulty towards achievig the correct ad uique solutio is caused by the isufficiecy (icompletio) of test modes. Oe type of the methods early developed to tackle this problem was proposed by Baruch [], Berma ad Nagy [2, 3], ad was based o a optimal method, e.g. the Lagrage multiplier method. These methods have the advatage of simple ad o-iterative model updatig procedure ad accurate matchig of the predicted modes with measured modes. However these methods make the origial system matrices recostructed without preservig their physical meaig of elemet coectio. Cosequetly the recostructed model would ot guaratee the modal correctio beyod the rage of measured modes. The secod type of developed methods icludes the two-respose method [4], the orthogoality ad eigedata costrait methods [5], ad the exteded corrected modal

2 costrait method [6]. Ulike the aforemetioed first type of methods, oly chose matrix elemets will be adjusted i these methods so that the origial cofiguratio of system matrices will be maitaied. I practice however, the success of those methods heavily deped upo the accuracy ad umber of available measured modes. That is a crucial limit of the methods whe applied to practical problems. Oe of the reasos for the above method limitatio may be that both the locatio ad magitudes of the modellig errors i a aalytical model eed to be idetified simultaeously durig model updatig. To avoid such difficulty, effort has bee made to tackle the problem i two steps, i.e. the modellig error localisatio followed by model correctio. The error magitudes of poorly modelled elemets are ot required at the first step of idetifyig their locatios. Hece the coditios required for accurate result ca be relaxed. By employig the sesitivity method for example, which may be classified as the third type of method, the sesitivity of eigedata errors [7] or orthogoality errors [8] to a small chage of idividual matrix elemet ca be calculated. A o-zero sesitivity value idicates that the elemets located i correspodig regio may eed improvemet for accurate modal predictio. I practice however, ot all modellig errors would be ecessarily idicated by o-zero sesitivity values. This is maily because the limited umber of available measured modes may be isesitive to some of the modellig errors. O the other had, ot all o-zero sesitivity values ecessarily represet the actual modellig errors. This is maily because some regios to which the respose is sesitive to modellig errors are actually correctly modelled. Normally a iterative performace is required for the model updatig to coverge. Because of the similarity of poor modellig localisatio to structural damage localisatio usig modal test data, some methods may be suitable for both applicatios. I the early 99s Hear ad Testa [9] used modal strai eergy for damage detectio i structures. Cotributio i developig the method has also bee preseted by Lim ad Kashagaki []. Later further effort has bee made to locate the structural damage based o the ratio of chage i the modal strai eergy caused by the damage []. Although this type of method is effective ad may be applied to locatig poor modellig, its accuracy of result, especially i the elemets adjacet to the damage regio, eeds further improvemet. I order to overcome this drawback, a eergy error estimatio method has bee proposed ad applied to both model improvemet [2] ad damage localisatio [3] by Guo ad Hemigway. This method is based o estimatig the aalytical ad modal test data based kietic ad strai eergies at fiite elemet scale. The differeces i magitude represet the existig eergy errors ad idicate poor mass ad stiffess modellig or iteral structural damage. I terms of modal strai eergy calculatio, this method is similar to the aforemetioed methods. However, this curret method also provides additioal measure to distiguish poor modellig or damage regios from those correctly modelled. This makes differece ad has the advatage over the previously developed similar methods. Further more from the estimated eergy error, a correctio factor for each elemet ca be created ad used to improve each idividual elemet ad hece the whole model. I this preset paper, effort has bee made to apply the eergy error estimatio method to improve the model of a full-scale tail-plae structure. Oe simulated example usig test data ad oe practical case usig measured modes have bee demostrated. I both cases poorly modelled regios of the origial structural model have bee located accurately. Subsequetly, sigificat improvemet of the structural model with reductio of average frequecy error from 2.2% dow to.% for the simulated test case ad from 4.6% to.8% for the practical case has bee achieved. 2

3 2. THE ENERGY ERROR ESTIMATION METHOD 2. MODELLING ERRORS INDICATED BY ENERGY ERRORS Usig the fiite elemet method, the stiffess ad mass models for a structural system are ormally represeted i matrices [K] a ad [M] a assembled from elemet models [K i ] a ad [M i ] a, i.e. [ K] a = [ Ki] a i = ad [ M] a = [ Mi] a i = () where represets the total umber of elemets i the system. I a desig process, uacceptable differece may be foud betwee the predicted ad measured modes of a structural system. I such case, it is usual to suspect that errors arise from the structural models uder the assumptio of reliable measured modes. The correct yet ukow stiffess ad mass models, [K] c ad [M] c, may be the represeted i terms of the origial stiffess ad mass matrices ad their error matrices, [ K i ] ad [ M i ], as follows [ K] c = ([ Ki] a + [ Ki ]) ad [ M] c = ([ Mi] a + [ Mi ]) (2) i = i = The modellig errors would cause differece betwee the aalytical ad test-data based eergies, which are defied here as the strai ad kietic eergy errors, S j ad T j, of the system as represeted below. T j T T S j = { ja [ Ki ] a{ ja { jt ([ Ki ] a + [ Ki ]){ jt 2 i= 2 i= (3a) T T = λ ja { ja [ M i ] a{ ja λ jt { jt ([ M i ] a + [ M i ]){ jt 2 i= 2 i= (3b) where λ ja ad λ jt are the jth aalytical ad measured eigevalues, { ja ad { jt the jth 2 λ ja 2 λ jt respectively. I the above equatios, sice each of the terms is scaled to uit aalytical ad test-data based mass-ormalised mode shape divided by ad value, both S j ad T j would be zero if [ K i ] ad [ M i ] were icluded. I the iitial aalysis stage however, [ K i ] ad [ M i ] are ukow, hece [K] C ad [M] C are ormally replaced by [K] a ad [M] a. I such case, the system eergy errors S j ad T j i a particular mode ca be approximated below ad become o-zero. S j T T T ({ ja [ Ki ] a{ ja { jt [ Ki ] a{ jt ) { jt [ Ki ] a{ j 2 i= i= i= (4a) 3

4 T j T T ( λ ja { ja [ M i ] a{ ja λ jt { jt [ M i ] a{ jt ) 2 i= i= T T λ jt { jt [ M i ] a{ j + λ j { jt [ M i ] a{ jt i= i= (4b) where λ j = λ ja - λ jt ad { j = { ja { jt represet the frequecy ad modal errors of the jth mode respectively. It is assumed here that a spatially complete set of measuremet is available. I practice, aalytical values will be employed to make the spatially icomplete measured modes expaded ad completed. Equatio (4) shows that the system eergy errors S j ad T j are the sum of elemet eergy errors S ij ad T ij i a particular mode, which are represeted below. T S ij { jt [ Ki ] a{ j (5a) T T T ij λ jt { jt [ M i ] a { j + λ j { jt [ M i ] a{ jt (5b) For those correctly modelled elemets, [ K i ] ad [ M i ] do t exist ad hece equatio (5) gives accurate represetatio of their eergy errors. Those eergy errors may ot be zero but small values due to modal error caused by poorly modelled elemets elsewhere. Oppositely, large eergy error values ormally arise from those poorly modelled elemets associated with [ K i ] ad [ M i ]. As more test modes are available ad used, these elemet eergy error values will be further icreased ad their differece from the correctly modelled elemets will be further elarged. The elemet eergy errors i m umber of test modes are represeted as follows: Ti = m T S i = { jt [ Ki ] a{ j (6a) j = m T T ( λ jt { jt [ M i ] a { j + λ j { jt [ M i ] a{ jt ) (6b) j = Eergy errors are thus used to idicate the possibility, or degree of, elemet modellig errors. 2.2 LOCALIZATION OF POORLY MODELLED ELEMENTS 2.2. Iitial Localizatio Accordig to equatios (5) ad (6), the total strai ad kietic eergy errors of a system modelled by usig umber of elemets ca be estimated by usig m umber of modes as follows: m m S = S ij ad T = T ij (7) i= j= i= j= A ratio of elemet agaist system eergy errors is the recommeded to evaluate stiffess ad mass modellig errors for each of the elemets i percetage 4

5 RKi = ( Si/ S ) (%) ad RMi = ( Ti/ T ) (%) (8) The RK i ad RM i, which are used as a idicator of locatig poor modellig, would have sigificatly higher magitude for a poorly modelled elemet tha correct oe. Their sig also idicates a uderestimatio (+) or overestimatio (-) of the stiffess or/ad mass i the aalytical model. Although the above RK i ad RM i, provide a useful idicator to localise poorly modelled elemets, the result is approximate ad a ucertaity remais as weather a small value of RK i ad RM i, idicates a small modellig error or just accuracy error? Such ucertaity is especially cocered i the regio adjacet to the poorly modelled elemets. This problem occurrig i the iitial stage of modellig error localizatio causes difficulty to achieve high accuracy ad reliability of locatig ad improvig poor modellig. I order to solve the problem, further ivestigatio has bee carried out ad a method to further distiguish the poorly modelled from correctly modelled elemets has bee proposed ad described below Fial Localisatio The method for further locatig poorly modelled elemets is based o the sig rather tha the magitude differece betwee the eergy errors of poorly ad correctly modelled elemets. From equatios (5-6), it is oted that the modal error { j plays major role i eergy error estimatio. This is because it cotais more explicit ad useful iformatio about the locatio ad magitude of modellig errors tha λ j. Sice { j i a poorly modelled regio is caused directly by a local modellig error, it should remai the same sig for ay mode. While i a correctly modelled regio, where { j is caused idirectly by modellig errors beyod the regio, the sig of { j ad hece S ij ad T ij may vary i differet mode. Such sig variatio ca be worked out from equatio (5) ad used as a idicator to distiguish the correctly modelled elemets from those poorly modelled oes. For example, if the sig of S ij ad/or T ij for the ith elemet chages whe usig a differet mode, this elemet stiffess ad/or mass may be idetified as correctly modelled. The elemet thus ca be excluded from the poor modellig regio regardless of its iitial RK i ad RM i values. If the sig remais the same as mode varies, the elemet would be idetified as poorly modelled ad thus should be icluded i the poor modellig regio for subsequet model improvemet. Although it has bee oted that some modes may play more sigificat role tha others i modellig error detectio, geerally speakig the more measured modes are available ad used, the more reliable result would be expected. I practice however, because of the ievitable measuremet errors ad icomplete measuremets i both spatial ad modal coordiates, this method may ot guaratee to filter off all correctly modelled elemets from the iitially located poorly modelled regios. Nevertheless this method provides a useful tool to ehace the accuracy ad reliability of the localizatio result. It is simple, efficiet ad accurate provided that the effect o { j by measuremet errors is less tha that by modellig errors. 2.3 IMPROVEMENT OF POOR ELEMENT MODELLING After localizig the poorly modelled elemets, model improvemet may be carried out with cofidece by evaluatig the stiffess ad/or mass error matrix [ K i ] ad [ M i ], 5

6 ad the modifyig the origial [K i ] a ad/or [M i ] a. Assumig [ K i ] ad [ M i ] are a fractio of [K i ] a ad [M i ] a of each elemet respectively, a approximatio of them may be obtaied by usig the estimated RK i ad RM i as follows: [ K i ] = FK i [K i ] a ad [ M i ] = FM i [M i ] a (9) where FK i = F k x RK i ad FM i = F m x RM i with F k ad F m represetig weightig factors for stiffess ad mass modellig improvemet respectively. A iitial model improvemet ca be achieved by substitutig the above estimated [ K i ] ad [ M i ] back ito equatio (2). Based o the improved model, further improvemet ca be carried out by repeatig the above procedure. As the model is further improved, the remaiig [ K i ] ad [ M i ] hece RK i ad RM i would become smaller. Such iterative implemet ca be carried out util the estimated RK i ad RM i coverge to a specified small value. Cosequetly the accuracy of predicted modes based o the improved model would be icreased to achieve a miimum differece from the measured modes. Such mode differece is assessed here by a average frequecy error f ad mode shape error Φ represeted as follows. m faj ftj m ( Φi ) j f = (%) ad Φ = ( ) (%) m j = ftj m j = i= ( Φ i ) tj () 3. APPLICATION TO A TAIL-PLANE STRUCTURE A full-scale tail-plae structure of 5.95 m i spa ad.83 m i maiframe legth as show i Figure was used for demostratio. The tail-plae is suspeded by a elastic bugee at the frot ed of its mai frame ad seated o two vertical beams at its back eds. I the curret demostratio of model improvemet, the tail-plae structural modellig was largely simplified by usig 3-D beam elemets as illustrated i Figure 2. Figure. A view of the tail-plae set-up for vibratio test 2 right side Y 9 Z 4 6 x

7 Figure 2. The aalytical beam model of the tail-plae structure The structure was represeted by a stiffess matrix of 92x92 order assembled from each of the beam elemets havig 2x2 order stiffess matrix. Similarly, a mass matrix of the same order assembled from odal mass matrix of 6x6 order was created to represet the mass properties of the structure. Cosiderig the symmetric cofiguratio of the structure, oly half of the structure was modelled. The modal aalysis for symmetric ad ati-symmetric modes was carried out separately. The first 9 modes with frequecy values listed i Table were obtaied as the aalytical results. Table. Predicted frequecies (Hz) of the tail-plae aalytical model Mode No Symmetric.77* 9.83* Ati-symmetric 4.77* * rigid-body mode 3. MODEL IMPROVEMENT USING SIMULATED TEST MODES (CASE ) This first example is aimed to demostrate the aalysis procedure of the method. To obtai a set of simulated modes as our test data, a test structural model was created. Takig the same cofiguratio ad beam modellig as the aalytical model show i Figure 2, the test structural model has % more mass i elemet, ad % less bedig ad torsio rigidity i elemets 9 &. To keep the cofiguratio symmetrical, the same differece as above was made o the opposite (right) side of the test model. I this example, although eight symmetrical modes of the test structure have bee predicted as show i Table, oly the six flexible modes, i.e. modes 3 8 were used as available test modes i the model improvemet procedure as described below. I this idealized example, the test modes are assumed to be spatially complete ad oise-free. Hece excellet modellig error locatig result is expected. 3.. Modellig error localizatio Usig the aalytical model ad the six test modes i equatios (6-8), a iitial eergy error distributio of the aalytical model was obtaied as show i Figure 3. The sigificatly large values of RK i ad RM i idicate that poor stiffess modellig is most likely to occur i elemets 9 ad, ad poor mass modellig i elemet. Due to symmetric cofiguratio, similar modellig errors are also expected o the opposite side of the aalytical model. It is oted that o-zero RK i ad RM i values also appear beyod those elemets. However they could be caused by approximatio of the method or very small modellig errors sice their values are sigificatly small. 7

8 I order to further distiguish the poorly modelled elemets from the correctly modelled oes, effort has bee made to clear the above diagram of iitial modellig error distributio. Usig Eq(5) i this secod stage, the sig of eergy error S ij ad T ij for each elemet ad mode was obtaied. Figure 4 shows that for all elemets apart from 9 &, the sig of S ij chages at least oce whe a differet mode is used. This reisures us that oly the stiffess of elemets 9 & is poorly modelled. Similarly, Figure 5 shows that oly elemets 6 & cotai mass modellig errors, while the other possible errors idetified iitially ca be cleared away from Figure 3. As the result, a much clearer view of modellig error distributio ca be obtaied as show i Figure 6. Comparig with the aalytical model, the egative RK 9 ad RK idicate that the test structure actually has smaller stiffess withi its elemets 9 ad regio. While the positive RM i idicates that more mass actually exists i the regio of elemet. Eergy Errors (%) mass stiffess elemet No. Figure 3. Eergy Errors Idicatig the Modellig Error Distributio Eergy error Sij mode 3 mode 5 mode 4 mode 6 mode 7 mode Elemet No Figure 4a. The sig variatio of S ij alog the tail-plae Eergy error Sij mode 3 mode 4 mode 5 mode 7 mode 6 mode 8 -

9 Figure 4b. The sig variatio of S ij alog the supportig beams Eergy error Τ ij.5 mode 3 mode 4 mode 5 mode 6 mode 7 mode Elemet No Figure 5a. The sig variatio of T ij alog the tail-plae Eergy error Tij.5 mode 3 mode 4 mode 5 mode 6 mode 7 mode Elemet No Figure 5b. The sig variatio of T ij alog the supportig beams Ee r gy Er r or s (%) mass stiffess elemet No.

10 Figure 6. A clear view of modellig error localizatio 3..2 Model Improvemet Havig localized the modellig errors as show i Figure 6, the subsequet model improvemet may be carried out by usig equatios (2), (8) & (9) i a iterative maer or miimisig f i a trial ad error approach. I this example, the above estimated RK i, RM i ad equatio (9) with weightig factors F k =.4 ad F m =.9 were used to evaluate [ K i ] ad [ M i ], which were the substituted ito equatio (2) to improve the origial [K] a ad [M] a. Sice the model could ot be completely corrected i oe performace as above, the same procedure were repeated based o the previously improved model util the remaiig errors became smaller tha a specified value. I this example, the iteratio was carried out util the evaluated F k x RK i ad F m x RM i became less tha 3% of their first iteratio values as show i Table 2. I the model improvemet, F m x RM 6 for elemet 6 was too small ad hece igored. The sum of F k x RK i ad F m x RM i from above iteratios gives a ratio of the fial evaluated [ K i ] agaist the origial [K] a ad [ M i ] agaist [M] a respectively. Accordig to equatios (2) & (9), the result shows that the origial stiffess for elemets 9 & should be reduced by % ad the origial mass i elemet should be icreased by 9.2% as fial improvemet. As a result, the modellig errors i the aalytical model have bee reduced from iitial % dow to less tha %. Table 2. Evaluated factors for model improvemet durig iteratio Iteratio No F k x RK 9 for elemet = -. F k x RK for elemet = -. F m x RM for elemet =.92 Comparig with the measured modes show i Table 3, mode errors of the improved aalytical model have bee largely reduced. For example, the average mode error for modes 3 8 has bee reduced from 2.23% to.9%. Table 3. Compariso betwee aalytical ad test frequecies (Hz) Mode No. * 2* origial f a error f (%) improved f a error f (%) test f t * rigid-body mode

11 3.2 MODEL IMPROVEMENT USING MEASURED MODES (CASE 2) I this secod example, improvemet of the origial aalytical model show i Figure 2 was carried out by usig oly four measured modes from a vibratio test of the tail-plae show i Figure. I the vibratio test, most of the 29 accelerometers were mouted alog the frot ad rear spars of the tail-plae as show i Figure 7 ad limited to measurig the vertical movemet. By applyig the quadratic iterpolatio method, the out of plae measuremets were used to obtai the itermediate umeasured displacemets, i.e. the out of plae vertical ad torsioal displacemets ad spa-wise bedig slopes. Due to the lack of i plae measuremets however, aalytical values were used as the displacemets at the itermediate umeasured DOFs, i.e. i plae traslatios ad yaw. However, it should be oted that these aalytical values do ot cotribute to ay eergy errors. Hece some compromise i the performace of the error localisatio ad accuracy of the model improvemet ca be expected x,y x,y 8 3 X 2 3 Y 2 5 x,y x,y accelerometer set i z-directio x,y -- i x, y-directio Figure 7. Top view of the measuremet poits o the tail-plae Table 4. Measured frequecies (Hz) from the tail-plae vibratio test Mode No Symmetric.76* 9.64* Ati-symmetric 5.9 * * rigid-body mode -- ot measured 3.2. Modellig error localizatio I this example, cosiderig the symmetric vibratio case ad half of the aalytical model, oly the four measured symmetric modes 3, 4, 7 & 8 were used i the aalysis. The rest of modes are listed oly for compariso. By applyig the EEE method, the total strai ad kietic eergy errors o both sides of the aalytical model were estimated from equatios (6) & (7) ad listed i Table 5. It shows that the strai eergy error RK i o the left side of the model is obviously larger tha that o the right side. Hece large stiffess modellig errors is likely to exist o the left side of the aalytical model. It also shows that the kietic eergy error RM i is much lower tha the RK i o both sides of the model. This

12 idicates that the overall mass modellig errors i the aalytical model is very small comparig with the stiffess errors. Table 5. Total eergy errors o both sides of the aalytical model Total eergy errors Left side Right side Strai S Kietic T.6.37 To have a detailed view of the modellig error distributio, RK i ad RM i i each elemet of the model were estimated by usig equatio (8) ad the results were show i Figure 8. It ca be see from Figure 8a that the stiffess errors are most likely to be located i the elemets 9 ad o the left side, ad also at the root elemets of the aalytical model. The egative sig of the errors idicate smaller stiffess i the actual structure tha that beig modelled. Figure 8b shows a eve distributio of RM i idicatig mass modellig errors of the model. Sice the maximum RM i value reaches oly about.5% of the sum of RM i values, which is very small comparig with the RK i show i Figure 8a, the mass modellig errors are igored i the subsequet model improvemet. Although the modellig error distributios obtaied so far are reasoable clear, it would be desirable to have a clearer view for more accurate localizatio of the modellig errors, especially i the regios with relatively low RK i values. At this stage, equatio (5) was used to get the sig variatio of elemet eergy error for each of the modes as show i Figure 9 (elemets beyod 3 are excluded due to their small errors as show i Figure 8). RKi (%) right side left side Elemet No Figure 8a. Stiffess modellig error distributio idicted by RK i 2 RM i (%) - -2 right side left side Elemet No. 2

13 Figure 8b. Mass modellig error distributio idicated by RM i.4 Eergy error Sij mode 3 mode 4 mode 7 mode Elemet No. Figure 9a. Elemetal strai eergy error for each mode Eergy error Tij mode 3 mode 4 mode 7 mode Elemet No. Figure 9b. Elemetal kietic eergy error for each mode Fig 9a shows that oly the sigs of strai eergy errors for elemets, 9 ~ 2 remai the same for all the modes. This idicates that oly those elemets were poorly modelled i terms of stiffess, ad the rest of elemets were correctly modelled. Figure 9b shows that oe of the kietic eergy errors remais the same sig for differet modes. This reisures that the origial mass modellig is good eough ad thus o improvemet is required i this case. Focusig o the stiffess modellig errors ad clearig out those fault errors iitially idetified as show i Figure 8a, a clearer view of locatios of stiffess modellig errors for the left side of the tail-plae was obtaied as show i Figure. The egative values of RK i show i Figure idicate that some local regios of the tail-plae actually have smaller stiffess tha that of the origial aalytical model. Havig a close look at the local structure where elemets 9 & were positioed as show i Figure 3

14 , the large stiffess modellig errors located i this regio ca be proved to be due to a local structural damage. The small modellig errors remaiig i the elemets & 2 may be due to the effect of the eighbourig structural damage ad approximatio of the method. The large modellig error of elemet was likely caused by the model simplificatio of the juctio betwee tail-plae ad its mai supportig beam as show i Figures. & 2. Those modellig errors caused sigificat differeces betwee the predicted modes from the origial model ad measured modes of the structure as show i Table Aalytical Model Improvemet First improvemet I order to improve the origial aalytical model, attetio was the focused o the poor modellig regios to evaluate the stiffess error matrix [ K i ] for elemets, 9 ad usig equatio (9). By substitutig the obtaied [ K i ] ito equatio (2), a improved [K] a of 2% stiffess (EI, GJ) reductio for elemet & 9, ad 22% reductio for elemet was obtaied. Cosequetly, the average mode errors of the improved aalytical model were reduced, from the origial f=4.6% to 3.8% ad from Φ=4.3% to 3.3% as show i Table 6. After the above first improvemet however, stiffess modellig errors still remaied i the aalytical model as show i Figure. elemet No RKi (%) origial st improvemet 2d improvemet Figure. Improved localizatio of stiffess modellig errors Table 6. Compariso betwee aalytical ad measured modes Mode Symmetric Modes Atisymmetric Modes Average No mode error origial f (Hz) f = 4.56 % Φ = 4.24 % st improved f by EEE method f = 3.8 % Φ = 3.25 % 2d improved f by EEE method f = 3.9 % Φ =.5 % Further improved f =.8 % measured f t

15 Figure. Structural damage o the left side of the tail-plae Secod improvemet If the above procedure was repeated based o the improved model, a further 2% stiffess reductio for elemets & 9 ad 4% reductio for elemet were estimated. Subsequetly a further reductio of the remaiig stiffess modellig errors was made after the secod improvemet as show i Figure. However, comparig with the measured modes agai as show i Table 6, it was oted that the average f was slightly icreased from 3.8% up to 3.9% although the average Φ was reduced further from 3.25% to.5%. It was also oted that such large reductio of stiffess would ot give a good represetatio of the actual structure. It was the suspected that some modellig errors remaied uidetified i the regios beyod those located so far. The failure to locate those errors may be maily because of the lack of measuremet alog the supportig frames of the tail-plae. Hece, a alterative approach for further model improvemet has to be cosidered Further improvemet I this example, it was assumed that the measured frequecies were more accurate ad reliable tha mode shapes. Thus, a approach based o the aalysis of frequecy sesitivity to the variatio of elemet stiffess parameters was used. I the aalysis, oly measured frequecies were used as the referece base to miimize the effect of iadequate measuremet of mode shapes o further model improvemet. Firstly we focused o the regio of juctio betwee the tail-plae ad its mai supportig frame. The aalytical model was updated by addig two more elemets betwee odes 5 ad 8 as show i Figure 2. This updated model should give a more realistic represetatio of the actual structure ad provide more parameters for sesitivity aalysis ad model improvemet i the regio. I the aalysis, although all measured frequecies were cosidered, attetio was paid maily to the first symmetric mode ad all the ati-symmetric modes, which had relatively large errors. It was foud that the first symmetric mode was sesitive to the bedig stiffess of the ew elemets - 5, 8 ad elemet 2-2 at the back. The first ati-symmetric mode was also sesitive to the ew elemets - 5, 8, but the third mode was more sesitive to elemets 9 2 & 2 2. From the sesitivity aalysis, bedig & torsio 5

16 stiffess of the ew elemets -5, -8 was determied to be 6.53 N.m2 ad the bedig stiffess of elemet 9-2 was reduced from 6.62 N.m2 to N.m2. Followig the above stiffess updatig, the mode errors were further reduced. A compariso betwee the frequecies of the origial model, improved model usig EEE method ad further improved model usig sesitivity method is show i Table Z Y x elemet ode Figure 2. A updated aalytical model for further improvemet 4. CONCLUDING REMARKS The eergy error estimatio method provides a feasible ad efficiet tool for model improvemet based o a set of icomplete measured modes. The method is applicable to locatig the regios of aalytical modellig errors or structural damage. From the ivestigatio of this curret paper, the followig coclusios ca be extracted. By usig this method, both the stiffess ad mass modellig errors of a aalytical model ca be idetified ad localised. I the iitial localisatio of modellig errors, the result was approximate due to the effect of the limit of the method ad available measured modes. However a additioal measure proposed i this method ca icrease the localisatio accuracy. This is the major advatage of the method i terms of accuracy. Although the method depeds upo the quality of measured modes, the regios with sigificat modellig errors such as the damage regio ca be distiguished from good modellig regios. I the regios such as the supportig frames of the tail-plae where few measuremets were made, the estimated eergy errors wo t be accurate eough to idicate modellig errors. I such case, sesitivity aalysis based o measured frequecies provides a effective alterative. The model improvemet is restricted to oly those regios where modellig errors have bee idetified. This esures the reliability ad accuracy of the improved model. The improved aalytical model retais its origial matrix size ad cofiguratio ad thus retais the physical descriptio of the system. Fially it has bee oted i this method that some modes may play more sigificat role tha others to locate modellig errors. This may raise a cocer of mode sesitivity of the method. Geerally speakig, the effect of mode o the accuracy of result would deped 6

17 upo the structural cofiguratio ad modellig error locatios. Although i practice we ormally do t have much choice of the very limited available measured modes, a ivestigatio ito this modal sesitivity problem is recommeded, but beyod the scope of this curret paper. REFERENCES. M. BARUCH 978 AIAA Joural, Vol. 6, Optimal procedure to correct of stiffess ad flexibility matrices usig vibratio tests. 2. A. BERMAN 979 AIAA Joural, Vol. 7, Mass matrix correctio usig a icomplete set of measured modes. 3. A. BERMAN ad E.J. NAGY 983 AIAA Joural, Vol. 2, Improvemet of a large aalytical model usig test data. 4. S.R. IBRAHIM et al 989 Proceedigs of the 7 th IMAC, Las Vegas, A direct two-respose approach for updatig aalytical dyamic models of structures with emphasis o uiqueess. 5. W.M. TO, R.M. LIN ad D.J. EWINS 99 Proceedigs of the 8 th IMAC, A criterio for the localizatio of structural modificatio sites usig modal data. 6. S.J. GUO 993 PhD thesis, Uiversity of Hertfordshire. Aalytical dyamic model improvemet usig vibratio test data 7. J.C. CHEN, L.F. PERETTI ad J.A. GARBA 984 AIAA Paper, 84-5, Spacecraft structural system idetificatio by modal test. 8. S.J. GUO ad N.G. HEMINGWAY 99 Proceedigs of the 9 th IMAC, Improvemet of a FE model with dyamic reductio usig test data. 9. G. HEARN ad R.B. TESTA 99 Joural of Structural Egieerig ASCE 7, Modal aalysis for damage detectio i structures.. T.W. LIM ad T.A.L. KASHANGAKI 994 AIAA Joural 32, Structural damage detectio of space truss structures usig best achievable eige-vectors.. Z.Y.SHI, S.S.LAW ad L.M. ZHANG 998 Joural of Soud & Vibratio 28(5), Structural damage localizatio from modal strai eergy chage. 2. S.J. GUO ad N.G. HEMINGWAY 995 Joural of Mechaical Egieerig Sciece (Part C) Vol. 29, A eergy error estimatio method for improvig aalytical models usig vibratio modal test data. 3. S.J. GUO ad N.G. HEMINGWAY 2 Proceedigs of It. Coferece o Advaced Problems i Vibratio, Structural damage localisatio usig eergy error estimatio method. 7