Using Singular Value Decomposition to Compare Correlated Modal Vectors

Transcription

1 Using Singlar Vale Deomposition to Compare Correlated Modal Vetors James P. DeClerk Noise and Vibration Center General Motors North Amerian Operations Milford, Mihigan USA ABSTRACT A tool is needed to trak several mode shape estimates from iterative or repeated (variability) testing. Modal Assrane Criterion (MAC) is a well known modal vetor omparison tool, however it an ompare only two modal vetors at a time. A method to ompare mltiple samples of modal vetor is presented The approah begins with mode shape orrelation and orrespondene. Psedo orthogonality is reommended to establish orrespondene and failitate mapping of the orrelated mode shapes. The II mean vee t" or an d"" smg 1 ar vale deomposition" (SVD) approahes to vetor avera~ing are explored. The SVD approah yields a single nmber Vetor Spae Consisteny measre. A simple lmped-mass fll vehile model is sed to evalate the algorithm and stdy the vehile vibration sensitivity to variations in tire, sspension, engine mont and inherent body strtre stiffness. NOMENCLATURE [P] Transformation Matrix [M] Mass Matrix [<I>] Mode Shape Matrix [$] Mode Shape Vetor [\f/] Matrix of Mode Shapes Samples [A] Pole Matrix [ JH Hermitian Operator [ )+ Psedo Inverse Operator 1. INTRODUCTION There has been a signifiant amont of work related to sensitivity of natral freqeny to hanges in element mass and stiffness. Mh of this work was motivated by the pratie of modifying these parameters so that the model more arately predits the experimental data. Most implementations are based on mathing natral freqenies only; the mode shapes are assmed to trak along. Depending on the disrepany and the desired dynami performane, this may not be a good assmption. Mode shapes an be jst as, if not more sensitive than natral freqenies. Use of only one sample of experimental data is another ommon pratie assoiated with pdating finite element models. The need for more test samples was addressed by Ewins [1]. A more reent stdy [2] showed that onsistent testing tehniqe is reqired to minimize the test-test variation in order to be able to examine the sample variability isse. In order to inlde mode shape information in large sale variability stdies [2], a tehniqe is needed to ompare many mode shape samples. It is assmed that the respetive modal models span roghly the same vetor spae and similar modes are identified in eah model. Althogh diretly appliable, tilization of the Modal Assrane Criterion (MAC) to stdy mltiple samples of orrelated vetors is likely to be mbersome and onfsing. A maro or spatial qantifiation of mode shape variability in statistial terms wold be a more onvenient measre of test-to-test mode shape onsisteny. When onsidered along with pole information, this tool old also be sed to assess the sensitivities of modal models to parameter estimation tehniqes or physial modifiations of the system. This work presents two means of identifying overall system statistis and traking their sensitivity to 122

2 variable omponent properties. The theory and possible algorithm are presented followed by an example sing a simple lmped mass model that emlates the modal properties of a fll vehile modal model. 2. THEORY/ALGORITHM A two step algorithm for statistially omparing a several sample modal analysis reslts is presented. The steps are: 1) establish mode shape orrespondene and mapping and 2) ompte natral freqeny and vetor statistis Mode Shape Correspondene and Mapping The first task in establishing mode shape orrespondene is to selet a baseline vetor set to be sed as the basis for the "similar mode" gropings. Seletion of this set is arbitrary provided that eah sample mode set an be mapped to this basis. Two different approahes to establishing mode shape orrespondene are the Modal Assrane Criterion (MAC) and the Psedo Orthogonality Chek (POC). Thogh very easy to ompte, MAC often has diffilty niqely differentiating "spatially similar" modes. Alternatively POC will identify niqe orrespondene of similar modes, bt is more diffilt to ompte, bease it reqires a mass matrix. MAC an be sed aording to eqation 1 to establish a mapping matrix, [P], onsisting of ones and zeros. The "rond" operation is sed to enfore nsealed mapping. eqation 2. The mode sets mst be saled to nit modal mass prior to this allation. This approah will establish niqe orrespondene for similar mode sets. It will also map polarity disrepanies and map a nll vetor where there is no orrespondene to the base mode set. In analytial appliations the mass matrix is readily available, however, the lak of a mass matrix poses signifiant ompliation for psedo orthogonality orrelation of experimentally obtained modal vetors. An experimental mass matrix estimate an be ompted from nit mass saled experimental modal vetors. (<t>t(m)[<i>) =[I) [M]=[<~>rH[<~>r The mass matrix estimate [ M] old be established sing the base vetor set and then sed to ompte the mapping matrix for eah sample mode set. However, se of the baseline mass matrix estimate to proess a sample modal vetor set wold be invalid if there was a pertrbation to the sample mass distribtion. An alternative wold be to "orthogonalize" eah sample prior to ompting the mapping matrix. (3) If the mode shapes are not spatially niqe, this approah old yield an erroneos mapping matrix (e.g. some sample modes are not mapped to any base modes or mltiple sample modes are mapped to mltiple base modes). Sine all MAC vales are positive, this approah will not map the sample modes to have the same polarity as the sample modes. The "fix" operation (ronds toward ) old be sed instead of "rond", bt is more likely to establish no orrespondene at all. An alternative is to se psedo orthogonality to establish the sample mapping matrix, as shown in (1) One established, the mapping matrix is sed to order the sample poles (i.e. natral freqeny and damping) and mode shapes. [<I> ] = [<I> ][P. ] 1 ordered t I [A) =(A)(P) 1 ordered t 1 The mapped mode shapes an then be gathered aording to eqation 7. where ['l'j] is a olletion of jth mode from eah ordered sample mode set. At this point all of the (6) 123

3 sample vetors shold be saled the same way. Unit length saling will be reqired in ompting some vetor spae statistis Natral Freqeny and Mode Shape Vetor Statistis. Two approahes for determining vetor statistis are presented. The first is to apply onventional statistis to eah element of a sample set of vetors. The seond is a vetor spae approah sing singlar vale deomposition (SVD) of a sample set of vetors. The element average vetor (EA V) is obtained by averaging aross eah row of eva. { 4> i} EAv = averag~ 'l'i] between nity and maximm normalized singlar vale. Althogh not addressed in this paper, appliation of the two-dimensional Forier Transform (2FT) to a set of sample orrelated mode shape estimates old yield yet another approah to qantifying mode shape variability [4]. Similar to the SVD, the 2FT "lmps" or" averages" the deterministi portion of the sample mode shapes to the low order fntions. Spatial variation is represented in the higher order fntions. 3. EXAMPLE A twelve degree-of-freedom, lmped mass simlation of an atomobile (Figre 1) is sed to illstrate the onepts presented in the previos setion. We an desribe other statistis assoiated with this EA V by stdying the MAC vales between this vetor, the baseline vetor and eah sample vetor. The mean and variane of the set of MAC vales desribe the onsisteny to the set of sample vetors. For this appliation, one wold expet a flipped log-normal or Gamma distribtion. The spatial average vetor (SAV) is the first left eigenvetor of ['l'il It an be obtained from an SVD of ['l'j]. svoq 'l'j]) = [ j][ sj][ vj]h {4>j}SAV = {jl} By definition, [Uj] is a basis for hyperellipsoid ontaining the set of vetor samples and [~]is the weighting for eah orresponding basis vetor [3]. More statistial information abot ['l'j] an be obtained by examining [~]. If the sample vetors in ['l'j] are saled to nit length, the maximm element of [Sj] is bonded by the sqare of the nmber of samples, n 2 ; therefore, [Sj] old be normalized by dividing by n 2 The maximm normalized singlar vale is ths a measre of the onsisteny to the set of sample vetors. The remaining normalized singlar vales represent the spatial variane of the set of sample vetors. The sm of the remaining normalized singlar vales old be ompared to the variane or standard deviation of the vetors; however, it shold be noted that this vale is not an independent statistial measre bease it is eqal to the differene (9) Figre 1 Shemati of 12 DOF Lmped Mass Model The powertrain, body strtre, and tires are represented by two, six and for lmped masses respetively. The powertrain mont stiffness, inherent body stiffness, sspension stiffness, and tire stiffness are represented as lmped stiffness elements. Baseline vales for eah of these elements were tned sh that the system modes were similar to that of a fll vehile. The 1 samples for the eah variability simlation were obtained by applying random error/ variation to the stiffness element vales. For the low level variation stdy, the powertrain mont, body, tire and sspension stiffness vales has +/-%, +j-5%, +j- 8%, and + j-8% niform distribtion respetively relative to the baseline vales. These variation levels were hosen arbitrarily and shold not be assmed to represent typial variation for the physial omponents represented. Three levels of variation were applied to the stiffness element vales. The mid and high level variation was 3 and 5 times the low level variation, respetively. The mean and standard 124

4 deviation of the stiffness element samples are given in Table 1. Graphial representations of the stiffness element distribtions are given in Figre 2. Eah set of stiffness samples were sed in the assembly of a sample stiffness matrix whih was sed in a system eigensoltion. The resltant modes were ordered sing psedo-orthogonality orrelation. Natral freqeny vales for the orrelated modes were gathered. The mean and standard deviation of the ompted natral freqenies are given in Table 2. The distribtions are also shown in Figre 3. The EAV and SAV were ompted for eah set of orrelated modes. These average vetors were ompared to eah sample vetor sing MAC. Statistis of these MAC vales were ompted and ompared. Preliminary reslts of this exerise showed no disernible differene between the EA V and SA V, therefore only omparisons sing the SA V are provided in this paper. Comparisons of the mean MAC(sample,SAV) and the normalized maximm singlar vale are given in Table 3. Similar omparisons of MAC vales, standard deviations and the sm of the remaining singlar vales is given in Table 4. Histograms of the MAC vales for eah ase are given in Figre 4. The normalized maximm singlar vale ompares very well to the mean MAC. In all, the normalized maximm singlar vale was a more onservative (less than) measre of onsisteny than the mean MAC. The variation (standard deviation) indiated by the sm of the remaining normalized singlar vales is somewhat similar to the standard deviations of the MAC samples. Signifiant orrelation is not expeted bease the sm of the remaining normalized singlar vales is not an independent statisti. 4. CONCLUSIONS Both the EA V and the SA V are potential methods to ompare many samples of modal vetors. The normalized maximm singlar vale is a good single qantity measre of sample vetor spae onsisteny. Stdying how these measres perform on dissimilar vetor spaes and atal experimental data is needed to validate their tility. Frther development of other spatial statistis wold also be desirable. 5. REFERENCES [1] Ewins, D. J. "One is not Enogh", Sond an Vibration, Agst [2) Cafeo, Dogett, Feldmaier, Lst, Nefske, and Sng, "A Design of Experiments Approah to Qantifying Test-to-Test Variability for a Modal Test", Proeedings of the th IMAC, Orlando, FL, [3) Conversation with Allyn Phillips (University of Cininnati) at IMAC 14, Febrary [4] Conversation and MATLAB stdies with Kevin Wittrp (GM NAO Noise and Vibration Center) September Low Error Stiffness Stiffness Differene between Sample Element Element Name Sample Mean and Standard Baseline Vale Deviation 1 Body1.16% 1.23E+5 2 Body2.9% 8.75E+4 3 Powertrain.24% 2.E+4 Mont 1 4 Powertrain -.6% 2.5E+4 Mont2 5 Tire -.75% 1.37E+3 6 Sspension -.57% 1.37E+3 Table 1 Mid Error High Error Differene between Sample Differene between Sample Sample Mean and Standard Sample Mean and Standard Baseline Vale Deviation Baseline Vale Deviation.5% 3.55E+5.9% 5.91E+5.4% 2.58E+5.7% 4.3E+5-2.5% 5.93E+4-4.2% 9.88E+4 2.4% 9.E+4 4.% 1.52E+5-1.5% 3.83E+3-2.5% 6.38E+3 2.3% 4.11E+3 3.8% 6.86E+3 Stiffness Element Statistis for 1 Samples with Uniform Random Variation for the Low, Mid, and High Error Test Cases 125

5 Low Error Mid Error High Error Mode Baseline Differene between Sample Differene between Sample Differene between Sample Nmber Natral Sample Mean and Standard Sample Mean and Standard Sample Mean and Standard Freqeny (Hz) Baseline Deviation Baseline Deviation Baseline Deviation % % % % % % %.185.1%.54.% %.186.1%.57.% %.186.1%.58.% %.185.2%.51.2% % % % %.299.1% % %.366.3% % % % % %.479.5% % %.526.2% % Table 2 Natral Freqeny Statistis for the Low, Mid, and High Error Test Cases Low Error Mid Error High Error Mode Mean MAC Normalized Diff. Mean MAC Normalized Diff. Mean MAC Normalized Diff. Nmber (SAV,Sample) Max. Singlar (SA V,Sample) Max. Singlar (SA V,Sample) Max. Singlar Vale Vale Vale % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Table 3 Comparison of Mean MAC(SA V,sample) and the Normalized Maximm Singlar Vale for the Low, Mid, and High Error Test Cases Low Error Mid Error High Error Mode Standard Sm of Diff. Standard Sm of Diff. Standard Sm of Diff. Nmber Deviation Remaining Deviation Remaining Deviation Remaining MAC Singlar MAC Singlar MAC Singlar (SAV,Sample) Vales (SA V,Sample) Vales (SA V,Sample) Vales % % % % % % % % % % % % % % % % % % % % % % % % % % % 'ro % % % % % % % % Table 4 Comparison of Standard Deviation MAC(SAV,sample) and the Sm of the Remaining Singlar Vales for the Low, Mid, and High Error Test Cases 126

6 Sample Element Stiffness Vales ~----~----~----~----~ Distribtion of Stiffness Element Vales 1 f/1 tl) : s 5 (,) Sample Element Stiffness Vales 1~.-----~----~----~----~----~ 6 's Element Nmber Distribtion of Stiffness Element Vales 1 X 1 6 Stiffness Vale (Nim) Sample Element Stiffness Vales 1~.-----r-----~ ~----~ 6 's Element Nmber Distribtion of Stiffness Element Vales 1 X 1 6 Stiffness Vale (Nim) 6 ~4 ~ s Element Nmber X 1 6 Stiffness Vale (Nim) Figre 2 Sample History (left) and Histogram (right) of Stiffness Element Vales for the Low (top) Mid (middle) and High (bottom) Error Test Cases 127

7 Sample Natral Freqeny Vales ~----~----~------~----- Distribtion of Natral Freqeny Vales 1 VI II) f 5 ~ 6 1 ~~--~=-~-~~~~~--~~~~~ Sample Natral Freqeny Vales 1~.-----~----~----~----~----~ Mode Nmber Distribtion of Natral Freqeny Vales Natral Freqeny (I-t 6 [j'l4 ~ '----~--~----'----~-- j Sample Natral Freqeny Vales ~----~----,-----~----~ Mode Nmber Distribtion of Natral Freqeny Vales Natral Freqeny (I-t o 1 '------~----~ ' ~ Mode Nmber Natral Freqeny (1-t Figre 3 Sample History (left) and Histogram (right) of Natral Freqeny Estimates for the Low (top) Mid (middle) and High (botlom) Error Test Cases 128

8 Distribtion of Sample_SVD_MAC Vales 1 VI Ill ~ 5 8 Mode Nmber.7 MAC Vale Distribtion of Sample_SVD_MAC Vales 1 VI Ill ['!! ::I 5 1 Mode Nmber.7 MAC Vale Distribtion of Sample_SVD_MAC Vales 1 ~ ~ 5 Mode Nmber.7 MAC Vale Figre 4 Histogram of Modal Assrane Criterion Vales Between Eah Sample Vetor and the Spatial Average Vetor for the Low (top) Mid (middle) and High (bottom) Error Test Cases 129