PPLICION OF MC IN HE FREQUENCY DOMIN D. Fotsch and D. J. Ewins Dynamics Section, Mechanical Engineeing Depatment Impeial College of Science, echnology and Medicine London SW7 2B, United Kingdom BSRC he Modal ssuance Citeion (MC) has been used fo numeous yeas as a measue of coelation between test and analytical mode shapes. he fact that the MC consides only mode shapes usually means that a sepaate fequency compaison must be used in conjunction with the MC values to detemine the coelated mode pais. he mode shape coelation has geneally been displayed as the MC matix tabulated o plotted vesus expeimental mode numbe on the x axis and analytical mode numbe on the y axis. he natual fequency coespondence is usually displayed with a sepaate plot such as the expeimental natual fequency vesus the analytical natual fequency. ogethe, the two plots ae used to detemine the oveall coelation/coespondence of the modes. his pape pesents a new way of plotting the MC such that both the mode shape coelation as well as the natual fequency compaisons can be viewed simultaneously. Futhe, the new plot method is natually extended to the coelation of fequency esponse functions (FRFs) using the Fequency Domain ssuance Citeion (FDC) and a new technique called the Modal FRF ssuance Citeion (MFC). wo case studies of eal stuctues ae pesented to illustate the advantages of the new methods. NOMENCLURE ω ω analytical natual fequency fo mode expeimental natual fequency fo mode {φ a } analytical eigenvecto fo fequency ω H ( analytical FRF H ( expeimental FRF H ( analytical FRF at fequency ω H ( expeimental FRF at fequency ω INRODUCION he model coelation pocess usually employs a MC [1] plot, an auto-mc plot and a natual fequency compaison. he auto-mc, which compaes a set of modes with themselves, is used to detemine if the numbe and location of the chosen measuement DOFs ae sufficient to be able to distinguish the modes fom each othe. he existence of non-negligible off-diagonal coelation tems in the auto-mc is an indication of spatial aliasing. If the coesponding MC plot shows off-diagonal coelation, it cannot be detemined without the auto-mc whethe this is due to spatial aliasing o to genuine coespondence of the vaious modes involved. hus, the MC, the auto-mc and the natual fequency compaison plots must all be used to detemine the oveall coelation/coespondence of the modes. his pape pesents the FMC plot which is a new way of plotting the MC, auto-mc and fequency compaison simultaneously such that the mode shape coelation, the degee of spatial aliasing and the fequency compaison can be displayed in a single plot. he new plot method is natually extended to the coelation of fequency esponse functions (FRFs) using the Fequency Domain ssuance Citeion (FDC [2]) and a new technique intoduced hee as the Modal FRF ssuance Citeion (MFC). FMC: MC PLO WIH FREQUENCY SCLES n example of an oiginal MC plot fo a simple squae plate is shown in Figue 1. he use of the diffeent colou-filled o shaded ectangles to indicate the MC values is shown to be quite useful. Since the MC gives no infomation about the fequency coespondence and can sometimes show good coelation between modes that have significant
Fig. 1 Oiginal MC plot fequency sepaation, a natual fequency compaison plot is also equied, as shown in Fig. 2. Howeve, the fequency compaison equies the detemination of the coelated mode pais (CMPs), which is usually done Fig. 3 uto-mc of nalytical Modes It has been a goal of many eseaches in the aea of model coelation to find a way to display the fequency infomation along with the MC values to get a simultaneous indication of both the mode shape coelation and natual fequency compaison. his can be accomplished by dawing a cicle with a adius popotional to the value of the MC at the coodinates of each fequency pai. his is done fo both the MC and analytical auto-mc as shown in the FMC plot in Figue 4. Fig. 2 Fequency Compaison fo the CMPs by satisfying citeia fo the MC value and then subsequently fo the fequency sepaation. Futhe, the MC can show coelation between off-diagonal mode pais such as expeimental mode 5 and analytical mode 8 in Fig. 1. In ode to detemine if the off-diagonal coelation is genuine o due to spatial aliasing (if the numbe and location of the chosen measuement points ae sufficient to be able to distinguish the modes), the auto-mc, shown in Fig. 3, must be used. Fig. 4 FMC, MC plot with fequency scales With the auto-mc fo the analytical modes being epesented by the gey filled cicles and the MC by the black cicles, the fequency sepaation is shown by the hoizontal shift fom the gey to the black cicles. he addition of the fequency sepaation lines, in this case +10%, gives a visual scale fo the amount of fequency diffeence between the test and analytical modes. he change in cicle size fom the gey to the black cicles gives a visual measue of the change in the MC value fom the pefect value of MC=1. he FMC plot also shows the modal density o the elative
fequency spacing of the modes. In addition to aiding in the display of fequency sepaation between the test and analytical modes, the auto-mc shows the extent of the spatial aliasing. FMC: Case Study he fist case study is fo an assembly of two aeo engine casings fo which an FE model and extensive test data wee eadily available. Figue 5 shows the NSRN finite element model fo the aeo engine casing assembly. he shell model contained 10 ings of 60 nodes each (600 total nodes) and was used to calculate the fist 20 modes. est data wee taken fo 3 ings of 20 equally-spaced points in the adial diection but only fo the fist 44 modes. Fig. 6 Oiginal MC plot fo Casing ssembly Figue 7 shows the fequency compaison fo the coelated mode pais (which wee detemined fom Fig. 6) with +10% deviation lines. he Figue shows that with the exception of the one mode at about the expeimental fequency of 1100 Hz the fequency coespondence is within +10%. Fig. 5 NSRN Model fo ssembly of wo eo Engine Casings he oiginal MC plot is shown in Fig. 6. he Figue shows that the coelation is not vey good with only expeimental modes 3, 11 and 14 having MC values geate than 80%. Futhe, expeimental modes 2, 5 and 6 have MC values between 60% and 80% while the est of the modes have MC values below 60%. Note that seveal mode pais away fom the main diagonal show MC values between 20% and 40%, which could be eithe genuine coespondence of the modes o spatial aliasing. Fom the MC values shown in the figue it can be seen that thee ae (only) 12 potential coelated mode pais (MC values > 40%). Fig. 7 Fequency Compaison fo the CMPs he FMC plot is shown in Fig. 8 with the auto-mc epesented by gey filled cicles and the MC by black cicles. he fequency lines ae fo a spead +10%. Note the indication of some spatial aliasing by the off-diagonal gey cicles fom the auto-mc at about 1550 Hz and between 950 and 1000 Hz. he Figue shows that the geneal coelation is not vey good with only 4 of the mode pais (at analytical fequencies of about 100Hz, 250Hz, 800Hz and 1050Hz) having elatively high MC values compaed with the pefect auto- MC values of 1.0. Howeve, the fequency compaison fo the 4 coelated mode pais is within +10%. he Figue also shows that the off-diagonal coelation shown in the oiginal MC plot in Fig. 6 is due to spatial aliasing and not to genuine coelation. his can be
seen by the off-diagonal MC values at the analytical fequencies of about 1000 Hz, 1250 Hz and 1550 Hz shifted in fequency fom thei coesponding auto-mc off-diagonal values. egion of high modal density can be seen by the close spacing of the 7 modes of the auto-mc between about 950Hz and 1200Hz. and theefoe yields values between 0 and 1. esult of FDC=1 indicates pefect coelation, while 0 indicates no coelation. FRC he FRC is defined by [3]: FRC ( j, k ) = 2 { H ( ω )}{ H ( ω )} ({ H ( ω )}{ H ( ω )}){ H ( ω )}{ H ( ω )} ( ) (2) Fig. 8 FMC Plot fo eo Engine Casing ssembly whee H (ω ) is the analytical FRF fom a esponse at DOF j due to an excitation at DOF k and H (ω ) is the coesponding expeimental FRF. he FRC is analogous to the COMC [4] and theefoe yields values between 0 and 1 fo each DOF. esult of FRC=1 indicates pefect coelation, while 0 indicates no coelation, but fo a single DOF. Since FRFs contain the esponses fom many expeimental modes simultaneously, a significant discepancy in natual fequency between the expeimental and analytical values fo just a few of the modes may esult in poo FRC values. FREQUENCY DOMIN CORRELION he use of fequency esponse functions (FRFs) fo coelation instead of eigenvectos is attactive because expeimental FRFs ae moe easily obtainable than eigenvectos, which equie significant post measuement analysis of numeous FRFs. Futhe, thee ae occasions when test data ae too noisy to extact the mode shapes accuately, leaving only the FRFs available fo coelation. he pimay tools used fo the coelation of FRFs ae the Fequency Domain ssuance Citeion (FDC) and the Fequency Response ssuance Citeion (FRC). Howeve, a thid tool intoduced hee as the Modal FRF ssuance Citeion (MFC) can be used as an intemediate step between the MC and FDC. FDC he FDC is given by [2]: FDC ( ω, = 2 ( ) ( ) { H( } { H( } ({ H( } { H( }){ H( } { H( } whee H ( is the analytical FRF at any analytical fequency, ω, and H ( is the expeimental FRF at any expeimental fequency, ω. he FDC is analogous to the MC (1) MFC he MFC is intoduced hee with the following definition: MFC ( ω, ω ) = 2 a { φ } { H ( ω )} a a { φ } { φ } ( { H ( ω )} { H ( ω )}) a } whee {φ is the analytical mode shape at any analytical fequency, ω, and H ( is the expeimental FRF at any expeimental fequency, ω. he MFC is analogous to the MC and theefoe yields values between 0 and 1. esult of MFC=1 indicates pefect coelation, while 0 indicates no coelation. he FDC and MFC have a distinct advantage ove the FRC in that they allow fo the spatial compaison of the analytical FRFs to the expeimental FRFs and the analytical mode shapes to the expeimental FRFs, espectively, at diffeent fequencies and theefoe fequency shift does not affect the coelation. he FDC and MFC plotted with fequency scales ae natual extensions of the FMC plot. (3)
he MFC has a distinct advantage ove both the FDC and the MC in that it allows fo the compaison of the data with the geatest fidelity---the analytical mode shapes and the expeimental FRFs. he analytical mode shapes, unlike the coesponding FRFs, equie no assumptions about damping. Similaly, the expeimental FRFs, unlike the coesponding mode shapes, equie no post pocessing of the measued data. Howeve, the MFC is only useful fo elatively lightly damped modes since the analytical mode shapes ae eal quantities fom an undamped eigen-solution and the expeimental FRFs ae complex quantities due to the inheent damping in the stuctue. heefoe, both the FDC and MFC wee chosen fo the case study that follows. FREQUENCY DOMIN CORRELION: Case Study he second case study is of an aeo engine casing fo which an FE model and both mode shape and FRF test data wee available. Figue 9 shows the NSRN finite element model fo the aeo engine casing. he shell model contained about 3700 nodes (18,500 DOFs) and was used to calculate the fist 19 modes. est data wee taken fo 2 ings of 12 equally-spaced points in the adial diection and 1 ing of 12 equally-spaced points in both the adial and axial diections. heefoe, a total of 48 FRFs wee taken fom which 14 mode shapes wee geneated. he coesponding analytical FRFs wee geneated using the FEM model. Fig. 10 Oiginal MC plot Figue 11 shows the conventional fequency compaison fo the coelated mode pais with +10% deviation lines. he Figue shows that all but the two coelated mode pais between analytical fequencies of 480 Hz and 540 Hz ae within +10% fequency sepaation. Fig. 11 Fequency Compaison fo the CMPs Fig. 9 NSRN Model of eo Engine Casing he oiginal MC plot is shown in Fig. 10. he Figue shows excellent coelation fo expeimental modes 1, 2 and 5 though 11 with MC values between 80% and 100%. Howeve, expeimental modes 3, 4, and 13 only have MC values between 60% and 80%. Futhe, expeimental mode 12 shows a high degee of coelation with analytical modes 12, 16 and 17. he significant off-diagonal coelation shown fo expeimental modes 9 though 14 could be fom aliasing o genuine loss of coelation. Fom the MC values shown in the figue it can be seen that thee ae 14 coelated mode pais (MC values > 40%). he FMC plot is shown in Fig. 12. he auto-mc fo the analytical modes is shown by the gey filled cicles, the MC is shown by the black cicles and the fequency lines ae fo a spead of +10%. he fequency shift (hoizontal fom the gey to the black cicles) is within +10% fo modes up to about 420 Hz. degee of spatial aliasing is shown by the auto-mc at analytical fequencies of about 430 Hz and between 500 Hz and 550 Hz. he coesponding MC values, with espective fequency shifts, show that it is indeed aliasing and not genuine coespondence. he off-diagonal coelation of modes 3 and 4 in the oiginal MC plot shown in Fig. 10 is shown hee by the concentic cicles at a fequency of about 90 Hz.
Fig. 12 FMC Plot fo eo Engine Casing he Modal FRF ssuance Citeion (MFC) plot is shown in Fig. 14. he MFC values wee detemined by compaing the analytical mode shapes at thei natual fequencies to the expeimental FRFs at thei esonant fequencies (ODSs). he auto-mfc fo the analytical mode shapes is shown by the gey filled cicles, the MFC between the analytical mode shapes and the expeimental ODSs is shown by the black cicles and the fequency lines ae fo a spead of +10%. Note the esemblance to both the FMC plot shown in Fig. 12 and the FDC plot shown in Fig. 13 fo fequencies up to about 420 Hz. he coelation is compaable to both (the MC and FDC) and fo fequencies at about 180 Hz, 330 HZ and 420 HZ it is not as good as the MC o FDC, which is pobably due to the damping pesent in the FRFs and not in the mode shapes. he offdiagonal coelation shown by the auto-mfc fo the analytical fequencies between 500 Hz and 550 Hz is again confimed to be aliasing and not genuine coelation by the coesponding MFC values. he fequency shift (hoizontal fom the gey to the black cicles) is again within +10% fo fequencies up to about 420 Hz. Fig. 13 shows the Fequency Domain ssuance Citeion (FDC), in the new plot fomat, fo the FRF coelation in this case. he FDC values wee detemined by compaing the analytical FRFs at thei esonant fequencies to the expeimental FRFs at thei esonant fequencies, a esponse vecto which is sometimes efeed to as the Opeating Deflection Shape (ODS). he auto- FDC fo the analytical FRFs is shown by the gey filled cicles, the FDC is shown by the black cicles and the fequency lines ae fo a spead of +10%. Note the esemblance to the FMC plot shown in Fig. 12 fo fequencies up to about 420 Hz. he coelation is compaable to the MC (FMC plot). he auto-fdc shows the spatial aliasing and fo the analytical fequencies between 500 Hz and 550 Hz the coesponding FDC values confim that it is aliasing and not genuine coelation. he fequency shift (hoizontal fom the gey to the black cicles) is within +10% fo FRFs up to about 420 Hz. Fig. 14 MFC, Mode Shape to FRF Coelation CONCLUSIONS In the case studies shown hee, the FMC plot has been shown to be vey useful in combining the infomation fom the fequency compaison with that of the MC and the auto-mc. In addition to combining the thee souces of infomation in one plot, the new plot fomat also shows modal density and helps to detemine whethe the high off-diagonal MC values ae fom genuine coelation o spatial aliasing. Fig. 13 New Plot Fomat fo FDC of FRFs he extension of the new plot fomat to the coelation of FRFs using both the FDC and the newly intoduced MFC was also shown to be quite pomising. In the one case study shown, both the FDC and MFC coelation yielded compaable infomation to the MC coelation. his esult is significant in that if eithe the FDC o MFC coelation can be used instead of the MC then the costly post measuement analysis of the expeimental FRFs to detemine
the mode shapes can be avoided. dditionally, fo lightly damped modes the MFC could be used instead of the FDC to avoid the analysis and assumptions equied fo detemining the analytical FRFs. CKNOWLEDGEMENS he authos would like to thank Rolls-Royce plc. fo poviding the financial and technical suppot fo this poject. REFERENCES [1] llemang, R J and Bown, D L Coelation Coefficient fo Modal Vecto nalysis, Poceedings of the 1 st Intenational Modal nalysis Confeence, pages 110-116, 1982 [2] Pascual, R., Golinval, J. C., Razeto, M., Fequency Domain Coelation echnique fo Model Coelation and Updating, Poceedings of the 15 th Intenational Modal nalysis Confeence, pages 587-592, 1997 [3] Heylen, W., vitabile, P., Coelation Consideations Pat 5 (Degee of Feedom Coelation echniques), Poceedings of the 16 th Intenational Modal nalysis Confeence, pages 207-214, 1998 [4] Lieven, N.., Ewins, D. J., Spatial Coelation of Mode Shapes, the Coodinate Modal ssuance Citeion (COMC), Poceedings of the 6 th Intenational Modal nalysis Confeence, pages 690-695, 1998