ties to correlate the model with measured natu- ral frequency and modeshape .... reduced flexural stiffness of the rails due to the cut-outs in their walls. Figure 2.
CORRELATION OF A COARSE-MESH FINITE ELEMENT MODEL USING STRUCTURAL SYSTEM IDENTIFICATION AND A FREQUENCY RESPONSE ASSURANCE CRITERION D. J. Nefske and S. H. Sung Engineering Mechanics Department General Motors Research & Development Center Warren, Michigan, 48090-9055 USA
ABSTRACT A coarse-mesh finite element model of an engine cradle structure is experimentally correlated using the structural system identification technique. The coarsemesh model represents the cradle structure with beam and spring elements. The structural system identifcation code SSID is used to update the structural properties to correlate the model with measured natural frequency and modeshape data. A Frequency Response Assurance Criterion (FRAC) is developed to provide a measure of the correlation between the predicted and measured frequency response functions. The updated coarse-mesh model is shown to predict the lower frequency vibration response nearly as accurately as a fine-mesh plate model of the structure.
INTRODUCTION With contemporary CAE software for finite element modeling, a complex structure can be fairly easily meshed with numerous small-sized elements to generate a fine-mah model of the structure. For analyzing local phenomenon such as static stress or fatigue, a fine-mesh model of the structure may indeed be required. However, for vibration analysis, when global response is of primary interest, even a fine-mesh model may not accurately predict the dynamic response over a wide frequency range; as will be seen below for a detailed plate model of an engine cradle structure. In addition, computational times and storage associated with such a model are large and, because of its size, the model is difficult to modify. On the other hand, a coarse-mesh finite element model is attractive because it is much easier to work with and to modify than a fine-mesh model. For the engine cradle structure, such a model was developed from a coarse mesh of beam elements, sufficient to represent the basic geometry and expected global deformation. Spring elements were added to represent joint stiffnesses. Unfortunately, structural parameters such as joint stiffnesses are difficult to determine with certainty, thereby limiting the accuracy of the model. However, with current structural system identification and parameter estimation techniques [I], it may be possible to update these uncertain structural parameters based on either the experimentally measured response or that predicted by a detailed model of the structure. In this paper, the structural system identificaton technique described in [2]. is evaluated for updating the beam-type finite element model of the engine cradle structure described above. The SSID code [3,4], which is based on [2], is used to update the joint stiffnesses and beam
NOMENCLATURE [Cl [K] [M] {F) {Fo) {H,) {H,) (4”) {&} {V} {U,)
: Damping matrix of structure : Stiffness matrix of structure : Mass matrix of st;ucture : Time-dependent load vector : Frequency-dependent load vector : Predicted FRF vector : Measured FRF vector : Predicted modeshape vector : Measured modeshape vector : Time-dependent displacement vector : Frequency-dependent displacement vector w”, f, : Natural frequency of vibration w, f : Forced frequency of vibration [T] : Response sensitivity matrix (S,,] : Parameter error weighting matrix [S.,] : Response error weighting matrix r> To, rp : Parameter estimates u, uO, up : Response estimates
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The MSC/NASTRAN computer code [5,6] is used in this study to generate the finite element model and to solve Eqs. (2) and (3) for the natural frequencies, modeshapes, and FRF’s.
cross-sectional properties of the model to corm late the predicted response with measured natural frequency and modeshape data. A Frequency Response Assurance Criterion (FRAC) is defined BS an objective meawre of the correlation between the predicted and measured frequency response functions. The FRAC measure is applied to quantify the experimental correlation of the original beam model, the updated beam-spring model, and a fincmesh plate model of the engine cradle.
Structural System Identification The structural system identification procedure in [2] is adopted to update the structural parameters in the finite element model. Basically, this procedure involves minimizing the error function, J = {u - U,]=[s”.]-I{” - u,)
THEORETICAL BACKGROUND Finite Element Andy& The governing finite-element equations of mc+ tion of a structure can be expressed in the matrix form
w1tii1+ Kx~) + Lilly) = IWI
+ir - ~.l=[s,,l-‘I~ - p.1
(4) where {t) is the updated parameter estimate which is obtained from the minimization. Here, {r.) is the initial parameter estimate and {u.) is the target response which is being used to correlate the model (modal frequency, modeshape, or FRF data). {II) is the predicted response for the updated parameter estimate {r) and is approximated by the Taylor series expansion,
(1)
where [Ml, [Cl, and [K] are mass, damping, and stiffness matrices, respectively. These matrices are functions of the structural parameters of the finite element model, such as plate thicknesses, beam crosssectional properties, spring stiffnesses, elas tic moduli, etc. These structural parameters can be updated using the structural system identification technique. {F(t)) represents the forcing input which is prescribed. Two types of analyses are typically used in updating a finite element model: (a) normal mode analysis and (b) frequency response analysis. In normal mode analysis, {F(t)} = 0 and [Cl is assumed to be 0, so that Eq. (1) reduces to, WI - u3Ml){4,)
= 0
{“I= I”p)+rltr- r&d
where the subscript p refers to the prior estimate, and [TJ = [&I~/&~] is a sensitivity matrix of partial derivatives of the response with respect to the structural parameters. In Eq. (4), [S..] and [S,,] are matrices which weight the relative importance of the response and parameter errors. For the case where one has data on the uncertainty in the target response or the initial parameter estimates, these matrices can be constructed to represent that uncertainty. The SSID code [3,4] i s b a s e d o n t h e s e e q u a t i o n s a n d i s used in conjunction with a MSC/NASTRAN finite element model to obtain an updated finite element model, as illustrated in Fig. 1. In this case, the design variables are the structural parameters which are updated.
n=1,2,... ( 2 )
where f. = wJ2n are the natural frequencies and (6”) are the corresponding modeshape vectors of the structure. In this case, the structural parameters can be updated to correlate fn and {&) with the measured natural frequency and modeshape data. In frequency response analysis, {F(t)} = {F.) exp(iwt) and {V(t)) = {L’.) exp(iwt), so that Eq. (1) reduces to,
@-1+4q - ~2[~1wol = jF.1
(5)
Correlation Measures The accuracy of the finite element model can be evaluated by comparisons with measured natural frequency, modeshape, or frequency response function data. Modeshapes are typically compared using the well-known Modal Assurance Criterion (MAC) defined as [7],
(3)
where {UO) is the displacement response vector and f = w/22 is the forcing frequency. {F.) represents the applied load and, for a single forcing input of magnitude IF*\, the frequency response function (FRF) is {H,) = {U,)/lF.I. In this case, the structural parameters can be updated to correlate {HI] with the measured FRF data at any forcing frequency f.
where (6”) is the predicted modeshape vector at the natural frequency fn and {a] is the corresponding measured modeshape vector. The MAC
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takes on values from 0 to 1 (a MAC value of 1 means perfect modeshape correlation) and can be evaluated for either the entire set or a partial set of modeshape vector data.
closely represents the actual geometry of the cradle, including the non-uniform crow-sectional variation of the frame, the cut-outs in the walls of the longitudinal rails, and the unsymmetrical brackets at the junctions. On the other hand, the beam model has only a sufficient number of beam elements to represent the overall geometry and the expected global deformation of the structure. The structural properties of the beam elements are prescribed to account for the cross-sectional variation of the frame and, to some extent, for the unsymmetrical brackets. However, they do not account for the joint flexibility at the four junctions between the cross-members and the rails or for the reduced flexural stiffness of the rails due to the cut-outs in their walls.
Figure 1. Structural System IDentification (SSID) procedure A Frequency Response Assurance Criterion (FRAC) is proposed here to quantify the accuracy of the finite element model FRF prediction. In analogy to the MAC definition, the FRAC is defined as,
where (HJ) is the predicted FRF vector and {H,) is the measured FRF vector. The FRAC is evaluated as a function of the forcing frequency f. While the MAC corr&tes the individual modes, the FRAC correlates the frequency response, which will depend on the participation of all of the individual modes. As with the MAC, the FRAC takes on values from 0 to 1 (a FRAC value of 1 means perfect FRF correlation) and can be evaluated for either the entire set or a partial set of FRF data.
Figure 2.
Finite-element models of cradle
Figure 3 shows the measured modeshapes and natural frequencies of the first four modes of the cradle (81. The predicted natural frequencies from the plate model and from the original beam model are compared with the measured natural frequencies in Table I. The table also gives the MAC value of the predicted modes. The plate model predicts the natural frequency of the first two modes very accurately, but the third and fourth mode are reversed in frequency-order and have lower MAC values. On the other hand, the beam model significantly over-estimates the natural frequency of the first three modes, although the natural frequency
APPLICATION TO ENGINE CRADLE Finite Element Analysis The fine-mesh plate model and the coarse-mesh beam model which represent the engine cradle structure are shown in Fig. 2. The plate model has 16,044 degrees-of-freedom, and the beam model has 456 degrees-of-freedom. The plate model very 599
of the fourth mode is fairly accurately predicted. The MAC values of the modes predicted by the beam model are high, except for the fourth mode.
converged in reverse frequency-order, with a lower MAC value for the fourth mode. By prescribing both the natural frequencies and modeshapes in Update B, all four modes converged in the carrect frequency-order, with reasonably high MAC wdua. Table I. Predicted vs. measured modes of original models
Figure 3. Measured modes and updated structural parameters in FEM Structural
System
Identification
To account for the flexibility of the weld connections at the four junctions between the crow members and the rails, torsional spring elements were added to represent the joint stiffnesses in the three rotational directions. A FEM sensitivity analysis was then conducted to identify the joint coefficients and the beam inertia properties of the rails that most affected the first four modes. In the analysis, the joint coefficients at the four junctions were linked together. Similarly, the inertia properties of both longitudinal rails were linked tw gether. The parameters with the highest senitivity values are shown in Fig. 3 and were identified as the joint coefficients & and’& and the vertical bending moment-of-inertia I,, of the rails. The four modes were insensitive to K,, and the vertical bending mode wa5 insensitive to all of the joint coefficients. Based on the sensitivity analysis, Ks, KG, Ivv were chosen as the uncertain structural pammeters to update in order to correlate the model with the four measured natural frequencies and modeshapes in Fig. 3. Table II contains the results for two updates, In Update A, only the measured natural frequencies of the modes were used to update the model, while in Update B both the measured natural frequencies and modeshapes were used to update the model. The frequency results for Up date A show that the first two modes converged in the correct order, while the third and fourth modes
Table II. Predicted vs. measured modes of updated beam model
The updated joint coefficient and inertia property values are given in Table III. To obtain COTrelation of the vertical bending mode in Fig. 4b, the rlul moment-of-m&la IV,, had to be approximately halved to account for the added flexibility in bending due to the cut-outs in the rail walls. Unfortunately, the remaining inertia properties of the rails could not be updated, because the first four modes were insensitive to them. To update these inertia properties would require additional modal data. In addition, the joint co&cients and inertia properties were linked and proportionally updated in this study. In a more detailed study, these properties could be updated on an individ-
models, including the plate model, are more accurate in predicting the vertical and fore-aft response (Figs. 4 and 6, respectively) than in predicting the lateral response (Fig. 5). This effect is attributed to the lower accuracy of the lateral modes predicted by all of the models, as can be seen from Table Il.
ual basis to obtain a more accurate model. Finally, it should be noted that the accuracies of the updated property values in Table III were not directly validated, although they are physically reasonable and produce models which accurately prediet the modes. Table III. Updated structural parameters
Updated
FEM
Correlation
Figures 46 compare the predicted FRF response of the four models in Tables I and II versus the measured FRF response. The response is compared at the right-rear rail for vertical excitation at the left-front rail (Locations ‘0’ and ‘I’, respectively, in Fig. 2). Figures 4b-6b compare the prediction of the original beam model with the measured response and show that the predicted response deviates from the measured response above 50 Hz. On the other hand, the predictions of either updated beam model in Figs. 4c-6c and 4d6d axe nearly as accurate as the predictions of the fine-mesh plate model in Figs. 4a-6a. All of the
Figure 5. Predicted vs. measured FRF in lateral direction (- predicted, ---- measured)
Figure 6.
Predicted vs. measured FRF in fore-aft direction (- predicted, --- measured)
The FRAC rnea~ure was evaluated as a function of the forcing frequency using Eq. (7) and is shown in Fig. 7 for the four models. Figure 7b confirms that the original beam model is only valid at frequencies below approximately 50 Hz, where the FRAC value is close to 1. On the other hand, Figs. 7c and 7d show that either updated beam model is accurate in the frequency
Figure 4. Predicted vs. measured FRF in vertical direction (- predicted, - measured) 601
correlation of the original beam model, the two updated beam models, and a fine-mesh plate model of the engine cradle structure. The FRAC was shown to be in agreement with subjective enlution of the correlations. It demonstrated that the original beam model was only accurate up to approximately 50 Bz, while both updated beam models were nearly as accurate as the fine-mesh plate model up to approximately 150 Hz. By up dating the individual structural properties of the beam model and by correlating with additional modal data, it may be possible to extend the range of accuracy of the beam model to even higher frequencies.
range below approximately 150 Hz, which is the same frequency limit as for the fine-mesh plate model in Fig. ‘la. While the plate model exhibits the highest average FRAC value (0.89) over the O-200 Ez frequency range, either updated beam model exhibits an average FRAC value which is only slightly lower (0.64 and 0.66, respectively). Also the average FRAC value shows that the finite element model from Update B is slightly more accurate than the model from Update A.
ACKNOWLEDGEMENTS This study was conducted under a Cooperative Research and Development Agreement (CRADA) between General Motors Corporation and Sandia National Laboratories. REFERENCES F&well, M. I and Mottersbead, J.E., “Finite Element Model Updating in Structural Dynamics,” Kluwer Academic Publishers, The Netherlands, 1995. Collins, J. D, Hart, G. C., Hasselman, T. K. and Kennedy, B., ‘Statistical Identification of Structures,” AIAA Journal, Vol. 12, No. 2, pp. 165190, Feb. 1974. Engineering Mechanics Associates, “SSID - A Computer Code for Structural System Identification,” Technical Report No. TR-911134-1, Sandia National Labs, Albuquerque NM, March 1991. Engineering Mechanics Associates, “SSID User’s Manual (Version l.l),” Technical R.eport No. TR-91-1134-2. Sandia National Labs, Albuquerque, NM, March 1991. 5. MaeNed, R. II. ed., “The NASTRAN Theoretical Manual,” Level 15.5, The MacNealSchwendler Corporation, 1972. 6. “MSC/NASTRAN User’s Manual,” Version 67, The MacNeal-Schwendler Corporation, Los Angeles California, 1991. 7. Ewins, D. J., “Modal testing: Theory and Practice,” Research Studies Press Ltd., Letchworth England, 1966. 8. Cafeo, J . A . , Feldmaier, D . A . a n d Doggett, S. J., General Motors Research & Development Center, personal communication of measured modal data.
Figure 7. Frequency Response Assurance Criterion (FRAC) for cradle models CONCLUSION The structural system identification technique was applied to experimentally correlate a coarsemesh finite element model of an engine cradle structure. The coarse-mesh model WBS developed from beam elements, sufficient to represent the overall geometry and the expected global deformation of the structure. Torsional spring elements were added to represent the joint stiffnesses at the junctions. The SSID code was applied to uniformly update the joint coefficients and the rail inertia properties to correlate the model with measured natural frequency and modeshape data. Two updated models with slightly different property values were obtained. A slightly more accurate model was obtained when using both natural frequency and modeshape data to update the model than when using natural frequency data alone. A Frequency Response Assurance Criterion (FRAC) wa5 proposed as an objective measure of the correlation between the predicted and measured frequency response functions. The FRAC measure was applied to quantify the experimental
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