Title.

Author(s)

Submitted to

Los AmmosNATIONAL LABORATORY

Experimental Program to Elucidate And Control Stimulated Brillouin4nd Raman Back Scattering in Long-Scale Plasmas

Juan C. FernandezJames A. CobbleDavid S. MontgomeryMark D. Wilke

International Atomic Energy AuthorityFusion Energy ConferenceYokoham% Japan10/19/98-10/23/98

_ –———.. .—...—r————=—.-— — .—_--.-_—.—-—.—. —

-—_— —...—_——--———.... ... --. .——- -=

=-:

Los Atamos Nat lord Laboratory, an aff irmative actioti equal opportunisty employer, is operated by the Universit y of California for the US Oepartment of E_mrgyundar cant ract W-74 05 -R4G36. W accetIt ante of this article, the publisher recognizes the t he US &wernrnant retsins a nonexclusive, royalty-free license to-..—..puplish or reporduce the published iorm of t hk cent ribution, or to “allowothers to do so, for U.S &vernment purpos=. The Los Alamos National Laboratory

requaata that the publisher identify this article as work performed under the auspices of the US Department of energy.Form No. 636 f%S72629 10/91

DISCLAIMER

This report was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.

DISCLAIMER

Portions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.

.

TEST-ANALYSIS CORRELATION AND FINITE ELEMENTMODEL UPDATING FOR NONLINEAR, TRANSIENT DYNAMICS

Fran~ois M. I-lemez’ and Scott W. Doebling2EngineeringSciences &Applications,ESA-EA, M/S P946Los Alamos National Laboratory, Los Alamos, NM 87545

ABSTRACT

This research aims at formulating criteria formeasuring the correlation between test data and finiteelement results for nonlinear, transient dynamics.After reviewing the linear case and illustrating thelimitations of modal-based updating when it is appliedto nonlinear experimental data, simple time-domain,test-analysis correlation metrics are proposed. Twoimplementations are compared: the conventionalleast-squares technique and the PrincipalComponent Decomposition that correlatessubspaces rather than individual time-domainresponses. Illustrations and discussions are providedusing the LANL 8-DOF system, an experimentaltestbed for validating nonlinear data correlation andmodel updating techniques.

NOMENCLATURE

The recommended ‘Standard Notation forModal Testing & Analysis” proposed in Reference [1]is used throughout this paper.

1. INTRODUCTION

The recondite nature of nonlinearity hasmade development of correct analytical models ofnonlinear systems a difficult task. Although anal~lcmethods and numerical tools are available formodeling specific types of nonlinearity, a systematicinvestigation of the formulation and resolution ofinverse problems for nonlinear dynamics has never,to the best of our knowledge, been addressed in theliterature. This work summarizes such an investigationand illustrates the issues of nonlinear finite element(FE) updating with a simple yet realistic testbed.

Handling nonlinear models is generallysomething analysts prefer to avoid because it toooften stretches to their limits our mathematical toolsand points to our lack of understanding of the physicsof structural dynamios. Nevertheless, improvingperformances requires to account for the true,nonlinear nature of structural systems and, therefore,it can be predicted that the field of conventionalmodal analysis will increasingly involve nonlinearity asstructural designs are increasingly being optimized.

When modal analysis is performed, structuralsystems are usually tested and analyzed under theassumption that the behavior remains linear in thefrequency range of interest. This fundamentalassumption makes it possible to interpret data in thefrequency domain because signals measured orsimulated through finite element analysis cangenerally be found periodic. On the other hand,analyzing nonlinear systems based on Fouriersuperposition or other modal transforms wouldtheoretically require the addition of higher-dimensional kernels, the definition of which dependson the type of nonlinearity encountered [2]. Anotherdrawback is that most transformations, such as Fouriertransforms, wavelets or Singular ValueDecomposition (SVD), are essentially linear tools:there is a theoretical limitation at analyzing nonlineardata with linear techniques. This motivates our choiceof attempting to instrument and correlate FE modelsin the time domain, even though this choice addsdifficulties that the frequency-domain approach allowsto bypass.

This research aims at formulating criteria formeasuring the correlation between test data and finiteelement results for nonlinear, transient dynamics. Weplace a particular emphasis on developing a

‘ TechnicalStaff Member,ESA-EngineeringAnalysisGroup(hemez@?lanLgov),MemberSEM.2SeniorTechnicalStaffMember,ESA-EngineeringAnalysisGroup(doebling@lanLgov),MemberSEM.

methodology that can handle any type and source ofnonlinearity, therefore, requiring both parametric andnon-parametric model updating. Typical examples ofnonlinearities we are interested in include materialnonlinearity, friction, impact and contact at theinterface between two components. These are typicalof nonlinearity sources dealt with in the automotiveand aerospace industries. The study of geometricalnonlinearity, on the other hand, is extensivelycovered in the literature: accurate models that handle,for example, large displacements and/or largedeformations can be obtained using the adequatevariational principles and FE discretizations [3].

[n this paper, the effectiveness of a ratherclassical time-domain, least-squares comparisonbetween measured and simulated responses isassessed using two implementations. The first onecorrelates directly the measured and simulatedsignals while the second one focuses morespecifically on correlating the subspaces to whichthese signals belong. Demonstration examples usingsimulated and real test data illustrate ideas proposedand offer a discussion of issues regarding theimplementation of these techniques.

2. EXPERIMENTAL TESTBED

The LANL 8-DOF (which stands for LosAlamos National Laboratory eight degrees offreedom) testbed consists of eight massesconnected by linear springs. The masses are free toslide along a center rod that provides support for thewhole system. Boundary conditions are unrestrained.Figure 1 shows the experimental testbed. It isinstrumented with eight accelerometers andexcitation is provided using either a hammer or ashaker. The first degree of freedom where theexcitation is applied provides a driving pointmeasurement for identification purposes. Theadvantage of the LANL 8-DOF testbed is that it canbe modeled fairiy accurately using a linear, mass-spring system. Nevertheless, the correlation of FEsimulations with test data illustrates the limitation of alinear modeling approach, as seen in Section 4.

Modal tests are performed on the nominalsystem and on a damaged version where the stiffnessof the fifth spring is reduced by 14Y0. A contactmechanism can also be added between masses 5and 6 to induce a source of contact/impact betweenthese two masses. The system is then tested undervarious excitation levels to assess the degree ofnonlinearity. Time-domain responses are measured at

each one of the 8 masses and modal parameters areidentified using a classical frequency-domain curvefitting algorithm.

Figure 1: LANL 8-DOF Testbed.

Table 1 compares the first five modesidentified with the nominal and damaged systems,both in the linear configuration (the contactlimpactmechanism is removed). Hammer excitations and dataaveraging are used for these series of tests. Largedamping ratios can be observed. It suggests that thesliding mechanism and the friction it generates play animportant role in the dynamics measured. These highdamping ratios are produced artificially when theidentification algorithm attempts to best-fit themeasured response using (implicitly) a proportionaldamping model. Also, the reduction of stiffnesstranslates, as expected, into a reduction of modalfrequencies. However, it can be stated fromfrequencies in Table 1 and a visual comparison ofmode shapes that this damage scenario has littleoverall effect on the response of the LANL 8-DOFsystem.

Table 1. Identified Modal Parameters Forthe Nominal and Damaged Systems.Nominal Modal Damaged Modal

Frequency Damping Frequency Damping

22.6 Hz 8.5’% 22.3 Hz 13.6’%044.5 Hz 4.3% 43.9 Hz 5.07065.9 Hz 3.3’3!0 64.8 Hz 3.5’7086.6 Hz 5.0’% 85.9 Hz 5.9?A099.4 Hz 2.6’% 99.7 Hz 3.6%

Next, test-analysis correlation (TAC) resultsare shown in Table 2 between the identified modalparametem of the damaged system and resultsobtained with the nominal (undamaged) FE model.The modal assurance criterion (MAC) illustrates theexcellent agreement between test and model vectorssince most values are above 98Y0, despite theunmodeled stiffness reduction and the effect offriction. Although it clearly shows a systematic error(the first three modes are very well correlated butexhibit a systematic 2.4?40frequency error in average),the linear FE model is believed to be a good startingpoint for the optimization.

Table 2. TAC Before FE Model Updating(Damaged System Vs. Nominal FE- Modei).

Identified FE Model Frequency MACFrequency Frequency Error

22.3 Hz 21.8 Hz -2.3Y0 99.77043.9 Hz 43.0 Hz -2.OYO 99.45%064.8 Hz 63.0 Hz -2.894. 99.47085.9 Hz 80.8 Hz -6.OYO 93.2’%.99.7 Hz 95.6 Hz -4.1 ‘%0 98.5%

3. TAC FOR LINEAR SYSTEMS

One method of obtaining a correctrepresentation is to create a finite element model of asystem and correlate this model with measurementdata taken from the system itself or some of itscomponents. Applied essentially to linear systems,this approach has been found quite effective whenmodal data are used in the correlation process. In thiscase, the equation of motion can be writlen as

[M(p)]{a(t)} + [K(p)]{u(t)} = {Fe(t)} (1)

which spells the equilibrium between inertia forces,internal (linear) forces and applied loading. Inequation (1), the mass and stiffness matrices dependon design variables {p} which express the parametricnature of FE representations. We W-II see in thefollowing that model updating is the procedure bywhich these variables {p} are optimized to minimizethe distance between test data and FE simulations.

Since the dynamics are linear, the equilibrium(1) can be transformed in the frequency domain usinga convolution operator (such as Laplace or Fouriertransform). The resulting equation relates the inputand output frequency response functions (FRF) ofthe system at any given sampling frequency Z as

([K(p)] - h [M(p)]) {u(L)} = {Fe(k)} (2)

Resonant frequencies L and mode shapes{0} are extracted from the homogeneous version ofequation (2). With orthogonality conditions added toequation (3) below, the mode shapes provide a basisfor the subspace to which the response {u(X)}belongs.

([K(p)] - X [M(p)]) {$}= O (3)

Experimentally, the system’s modalparameters are identified from measured FRFs ordirectly from time-domain data using identification

algorithms, a review of which can be found inReference [4]. Hence, equations (l-3) emphasize therelationships between FE matrices and quantitiesmeasured or identified during modal tests. They formthe basis of any TAC procedure: clearly, the system’scorrect representation is obtained when theseequations are verified as measured quantities replacethe FE outputs {a(t)}, {u(t)}, {u(s)} or (l.;{$}) inequations (1-3).

Note that this formalism extends withoutdifficulty to damped structural systems. In this case,modal parameters are complex quantities and thecomparison between their real and imaginary partsprovides information regarding the type of damping.The main difference between non-proportionally andproportionally damped systems is basically that thereal and imaginary parts of mode shape vectors arenot parallel in the former case. Note also thatexperimental procedures are available for dealing withdamped systems. While most techniques apply toproportional damping only, some recentdevelopments have proven efficient forreconstructing full-order, non-diagonal modaldamping matrices [5].

3.1 TAC & Linear FE Model Updating

Typical examples of TAC metrics that havebeen applied successfully to linear yet relativelysophisticated systems include minimum distancebetween identified and simulated frequencies, modeshapes [6] or frequency response functions [7], aswell as the minimization of modal residues [8-9], thedefinition of which is summarized below. Obviously,the equation of vibration (3) is violated when FEmodal parameters are replaced with test data as longas the parametric representation is erroneous. Thisinequality can be used for defining modal residuevectors {Rf(p,L)} that account for the out-of-balanceforces in the model as

[K(p)] {$} = % [M(p)] {@} + {Rf(p,A)} (4)

Vectors {Rf(p,%)} exhibit the largest entriesat degrees of freedom (DOF) where the equilibrium isviolated the most. This can be used as the basis foc1) Identifying the source of modeling erro~ and 2)Updating the model by minimizing a norm of vectors{Rf(p,%)}. This approach is referred to as force-based model updating since entries of residues{Rf(p,%)} in equation (4) are consistent with forces.The alternate approach of enforcing equation (3) is byallowing a mismatch of vectors that multiply the massand stiffness matrices

[K(P)] {w} = ~ [M(P)] {0} (5)

Then, a hybrid residue is defined as the differencebetween the mode shape {$} and the first inverseiterate {v} obtained by inverting the stiffness matrix inequation (5). Clearly, the FE model is in goodagreement with test data when the “inertia” modeshape {~) is equal to the “strain” mode shape {v},resulting in a minimum-norm residue {Rd(p,L)}

{Rd(p,L)} = {@}- {v} (6)

[n theory, these two modal residues areclosely related: multiplying equation (6) by thestiffness matrix and substituting equation (5) yields

[K(P)] {Rd(p,L)} = {Rf(p,L)} (7)

In practice, implementation constraints (in particular,the spatial incompatibility between measurement andFE discretizations) give rise to a variety of updatingand modal expansion techniques for which thecondition (7) is not necessarily satisfied. Therefore, amismatch between the updated models provided bythe minimization of residues {Rf(p,k)} and{Rd(p,k)} may be observed, as will be seen inSection 4, even though equation (7) proves that thesolution is unique.3 Note that similar residues may bedefined directly with the FRF input-output equation(2), offering a wide range of TAC and FE modelupdating techniques. Reviews and discussions ofstate-of-the-art updating methods can be found inReference [9] for hybrid residues and in References[10-1 1] for force-based residues.

3.2 Discussion of Modal Correlation

Obviously, formulating TAC in the frequencydomain offers more advantages than a time-domainapproach because: 1) Filtering and averaging areavailable to alleviate the repeatability issue; 2) Noisycomponents of the signal and rigid-body modes canbe filtered out and 3) Explicit relationships betweentest data and FE modeling are available in the form ofequations (2-3). For these reasons, model updatingprocedures for structural dynamics always attempt toenforce equations (2) or(3) and not the original, time-domain equation of motion (l). Another important

3 In a finite-dimensional space, all norms are equivalent.Therefore, equation (7) shows that minimizing residues{Rf(p,%)} or {Rd(p,k)} accounts for minimizing the sameerror using different norms. Thus, convergence patternsmay be different but the optimal solution reached uponconvergence must be the same in both cases.

justification is that correlating mode shapes isequivalent to matching the subspaces that themeasured and simulated responses belong to. Thisissue is discussed in Section 5.2 when it is attemptedto apply the same reasoning to nonlinear test data.

Modal data, however, is only relevant whendealing with linear systems and so confines anycorrelation done for nonlinear systems to the timedomain, forcing us to deal with the problems ofmultiple field measurement (combinations ofdisplacement, velocity and acceleration data mighthave to be measured), repeatability and noisymeasurements in addition to the already inadequaterepresentation of the system’s dynamics.

4. LINEAR, MODAL UPDATING OF THELANL 8-DOF TESTBED

In this Section, we discuss the main resultsobtained when the linear FE model of the LANL 8-DOF system is updated to match the identifieddynamics. As mentioned previously, our testbedexhibits a fair amount of friction which neither thefrequency-domain identification algorithm nor thelinear FE model account for. Hence, this experimentaims at demonstrating the limitations of inverse modalapproaches when the dynamic behavior involved isnonlinear.

Figures 2 and 3 illustrate the updatingobtained with the force-based and hybridformulations, respectively. In this example, the firstfive identified modes are used, all 8 elements areadjusted and the sensing configuration consists ofDOFS 1, 4 and 7 only, although all 8 outputs areavailable. Note that no measurement is connected tothe erroneous spring (attached to masses 5 and 6).This makes it more difficult to find the damagebecause test mode shapes must be expanded, whichundoubtfully introduces additional numerical errors. InFigures 2 and 3, each bar represents the percentageof adjustment brought to the corresponding springstiffness: an unambiguous damage identificationshould therefore be limited to a 14?4.reduction of thefifth stiffness. It can be observed that the two optimalsolutions are slightly different: this is due, asmentioned previously, to differences in handling the.reconstruction of mode shapes for the non-measuredcomponents 2, 3, 5, 6 and 8. The reader is referred toReferences [8] and [9] for discussions of mode shapeexpansion associated to force and hybrid residues,respectively. Also, References [12] and [13] present

.ry --- --

comparative studies of state-of-the-art techniques forsolving this critical problem.

10 -

35 -ggo

:-5 -

-lo -

-151, , , , , i

1 2 3 4 5 6 7FMe E!.menl Nunhr

Figure 2. Adjustments Brought to theLinear FE Model When Force-based Modal

Residues Are Minimized.

15 1

10-

85 -

$

go

s -5 -<

-lo-

-15, t , , ,

1 2 3 4 5 6 7Fti8WmlNunber

Figure 3. Adjustments Brought to theLinear FE Model When Hybrid Modal .

Residues Are Minimized.

In Figures 2 and 3, both updating techniquesidentify the correct amount of stiffness reduction atthe fifth spring. However, they also bring significantmodifications elsewhere in the model: we believe thatthis is a manifestation of the friction that optimizationalgorithms attempt to best-fit as stiffness adjustments.Table 3 lists the correlation obtained atler force-basedoptimization. A comparison with Table 2 shows a clearimprovement of the predictive quality of our linearmodel,

Table 3. TAC After FE Model Updating(Damaged System Vs. Adjusted FE Model).

Identified FE Model Frequency MACFrequency Frequency Error

22.3 Hz 22.4 Hz 0.3% 99.9’%043.9 Hz 44.2 Hz 0.6% 99.9%64.8 Hz 65.9 Hz 1.7$40 98.2%85.9 Hz 85.1 Hz -0.9% 97.29!099.7 Hz 99.8 Hz -0.1?!0 99.6’?40

Finally, we have also performed an initialcorrelation between the FE model and the responseof the nominal (undamaged) system. The result is thatthe stiffness of the first spring (connected to the

driving point measurement) is increased by 22% whileall other spring stiffnesses are increased by 5.27. inaverage. Hence, the initial modeling, too flexible evenwhen no damage is introduced, is improved but thisfirst step does not offset the ambiguous damageidentification results obtained when the model isoptimized to recover the damaged spring. Using alleight measurement points during the updatingprovides significantly better results.

We conclude that these results illustrate howmodal-based FE updating techniques can be usefultools for improving parametric models. Here, forexample, we learn that the driving point attachmentproduces a local stiffness that should be accountedfor in the modeling. At the same time, they show therapidity with which identification, TAC and updatingresults deteriorate when the dynamics of interestinvolve some source or level of nonlinearity.

5. TAC FOR NONLINEAR SYSTEMS

We now investigate the formulation of inverseproblems for nonlinear structural dynamics. Asbefore, we statt with a description of the equation ofmotion used for the (direct) FE simulations. Thecorrelation metrics are presented in Section 5.1 and itis shown that two different implementations can beproposed for solving basically the same inverseproblem.

[n this work, nonlinear structural dynamics aredescribed with the following equation of motion

[M(p)] {a(t)}+[K(p)]{u(t)}+{Fi(p,t)}={Fe(t)} (8)

Equation (8) states that the system is in equilibriumwhen the applied loading in the right-hand sidematches the combination of inertia and internal forcesin the Iet+hand side. The nonlinear internal forcevector {Fi(p,t)} accounts for any nonlinear, implicitfunction of the system’s state variables.

Implementing the FE representation (8) isnecessaty not only because systems we areinterested in are nonlinear but also becauseexcitation sources we consider present a lot of high-frequency dynamics. The modal superpositionapproach becomes ineffective because the loadingcan not be approximated from the low-frequencymodes and because time-domain responses fail to beperiodic. Hence, conventional modal analysis doesnot apply anymore. Numerical models undergo similardifficulties as computational and accuracy issues arise

-- . . --, . . .. .z.-.. ,. . .,,.., PT . . .. ..-C’.W. m-m-— - ,. , ----m.mx--- - -I-- Trr’. -

when extracting high-frequency modes, especiallywith large dimensional FE models.

5.1 TAC & Nonlinear FE Model Updating

In the following, we assume that time-domain,displacement measurements {utest(t)} are obtainedby instrumenting the system. This assumption ismade for simplicity. However, it can be verified easilythat all developments below apply to atiirarycombinations of displacement, velocity andacceleration measurements. (Note that higher-orderderivatives such as strains could also be employed.)

Since our objective is to generate a refinedand more accurate model, the natural TAC metrics toconsider are distances between test and simulationdata. Residue vectors are defined simply as

{R(p,t)} = {utest(t)} - {u(t)} (9)

The computational procedure consists of thefollowing steps: 1) For a parametric model defined bya design {p}, the FE response is simulated vianumerical integration of equations (8); 2) Residues (9)are calculated at prescribed DOFS and time samples;and 3) The cost function J(p) is minimized using anoptimization algorithm, where

J(p) = llR(p,t)ll + u lip - poll (lo)

It represents the 2-norm (Euclidean norm) of ourresidue vectors: note that the same definition appliesin the linear case with modal residues (4) or (6). h alsoincludes a minimum change term, or regularizationterm, that helps reducing the numerical ill-conditioning characteristic of inverse problems. Froman engineering point-of-view, it simply means that anoptimum design {p} is sought after that brings theleast possible change to the original design {PO}.The optimization procedure in Step 3 involvesmultiple FE simulations since time-domain responsesmust be calculated to evaluate the costs J(p) forvarious designs {p}. Our current implementationfeatures order-zero algorithms (the simplex method)and order-one algorithms (Gauss-Newton, BFGS andLevenberg-Marquardt, for which documentation canbe found in Reference [14]). Gradients are requiredwith order-one optimization methods and they arecurrently estimated with a centered finite differencescheme, which becomes computationally intensivewith large dimensional FE models.

5.2 Principal Component Decomposition

The correlation presented previously can beviewed as a rather conventional generalized least-squares (GLS) minimization. The GLS formulation hasbeen used for solving inverse problems in manyengineering applications for several decades. It is wellknown that its success is, to a great extent,conditioned by the ability of the math model to spanthe subspace to which the test data belongs. tt isinteresting to notice that this is exactly what modalcorrelation atlempts with linear systems since themeasured response belongs to a subspace spannedby the identified mode shapes.

Along these lines, the principal componentdecomposition (PCD) method developed andvalidated in Reference [15] attempts to generalize thenotion of mode shape for nonlinear systems. Ratherthan using the direct comparison (9), the SVD of time-domain data is first performed

( [U] ; ~] ; [V(t)] ) = svd( [u(t)] ) (11)

and the residue is basically defined as the distancebetween test and analysis right-singular vectors

{R(p,t)} = {Vtest(t)} - {V(t)} (12)

Since the right-singular vectors are orthogonal, theyprovide a basis of the multi-dimensional manifold towhich the nonlinear signals belong [16]. The PCDconsists of minimizing the distance between thesedecompositions rather than between the originalsignals. Our numerical results presented in Section 6feature the PCD implementation as it is derived inReference [15]. (For clarity, the presentation featuredin equations (11-12) is a simplified version of theactual method.)

In addition, the SVD offers a practical way offiltering out any measurement noise or rigid-bodymode contribution because these are typicallyassociated with singular values much smaller thanthose characteristic of the dynamics. However, weemphasize that the SVD can not be used fordetecting if test data are nonlinear. Althoughattractive, this idea is false: using ingenious initialconditions “for integrating the time-response of asimple 4-DOF mass-spring system (see Section 6.1),the PCD can be ‘fooled” and lead to believe that theresponse is nonlinear (because, often, the shapes ofright-singular vectors {V(t)} are characteristic ofwhether or not the system is nonlinear) when it is, infact, perfectly linear.

‘A AA I 1 . . . lIIA1. .,. lAAAl. m IAIAI ..,1 III 1 CXUIG -r, ltaullb UI u Ie ul_a tippludwl Wllrl

6. NONLINEAR UPDATING OF THE LANL8-DOF TESTBED

In this Section, we present an overview of theresults that have been obtained with the time-domainFE model correlation and updating procedure. InSection 6.1, a validation based on data simulatednumerically is discussed. In Section 6.2, the LANL 8-DOF testbed is analyzed in an attempt to identify thenonlinear contact/impact force.

6.1 Validation Using Simulation Results

The validation presented here employs the 4-DOF mass-spring system depicted in Figure 4 wherethe fourth spring exhibits a cubic stiffness. The “test”data are simulated numerically and feature a 15%stiffness reduction of the (linear) third springcombined to a 257. stiffness increase of the(nonlinear) fourth spring. Simulations run up to 0.2seconds and the response is sampled at 200 equally-spaced points in the [0;0.2] sec. interval and at DOFS1, 3 and 4. The second DOF is not measured whichleaves us the choice to either condense the FEmatrices down to the subset of measurement pointsavailable or to work with full-order matrices. Noexternal force is applied to the system; instead, thefirst mode shape of the associated linear model isused as initial condition for initializing the timeintegration. We emphasize that this system isnonlinear since it results from the combination of alinear model (where all four spring stiffnesses arelinear) and a nonlinear internal force applied to thefourth mass and proportional to the third power of thefourth displacement.

/kl B k3 k4

Ml M2 M3 M4

/ M

/

Figure 4. 4-DOF Nonlinear System.

During the correlation, the distance between‘measured” and simulated displacements at DOFS 1,3 and 4 is minimized. Figures 5 and 6 illustrate atypical comparison of displacement responses beforeand after model updating, respectively, Clearly, theFE responses (shown in dashed line) match moreclosely the “test” data (shown in solid line) after theoptimization. Table 4 compares the updating resultsobtained with four different implementations of ourtime-domain TAC technique. For each method, thefirst four rows list the percentages of stiffness

adjustment brought to the linear springs. The fifth rowlists the percentages of adjustment brought to thecubic spring stiffness. The correct answer is therefore-15% in row, and +25Y0 in row 5 and no adjustmentelsewhere in the model.

—n.1=-~-e~a4,

1‘0au25040f4n03 0.1 I

0.32 Q*4 0.14 =18 &2

?, I

05

Figure 5. Displacement Time History BeforeUpdating the 4-DOF Nonlinear Model.

03

Figure 6. Displacement Time History AfterUpdating the 4-DOF Nonlinear Model.

In Table 4, results of the GLS approach withFE model reduction (MR) are presented in columns 1and 2; in the second column, both displacement andvelocity data are used for updating the model. Whendisplacements only are inputted to the minimization,the solution expected is not reached. Nonetheless,the adjustment featured in column 1 reproducesalmost exactly the displacement “test” data, whichreminds us of the non-unicity of the solution. Toresolve this difficulty, velocity data are added and theoptimization then converges to the expected solution(see column 2). Results of the PCD approach usingfull-order matrices (FM) and model reduction arepresented in columns 3 and 4, respectively. Bothsimulations are based on displacement data only andboth provide excellent results. Comparing columns 3

and 4 shows that the solution is slightly deterioratedwhen MR is used. Condensing the FE matrices maybe necessary for initializing correctly the timeintegration procedure and should moreover lead tosignificant CPU time reductions with large FE models.

Table 4. Optimized 4-DOF FE ModeIsObtained With Various lmplementations~GLS, MR GLS, MR PCD, FM PCD, MR

D-data D, V-data D-data D-data6.4% 0.1% 0.2?40 1.6%4.370 0.570 1.470 5.9!40

-1.770 -14.9% -13.570 -12.170-3.070 -0.3?40 -1.070 -3.8Y05.970 24.6% 22.l% 20.8’%0

There is no noticeable advantage of oneapproach over the others in regard to computationalcost, which is not surprising considering the small sizeof this system. It is noticed, however, that order-oneoptimization algorithms require much fewer iterationsto reach convergence, the drawback being a highcomputational requirement (per iteration) forestimating the cost function’s gradients.

6.2 Application to the LANL 8-DOF Testbed

In this Section, model updating is applied tothe LANL 8-DOF testbed. The objective is tocorrelate transient, time-domain test data obtained byinstrumenting the system’s nonlinear configuration.Hence, the contact mechanism is now introduced.Figure 7 provides an illustration where it can be seenthat the clearance between the fifth and sixth massesis reduced to a small threshold, therefore, introducinga source of contactimpact during vibrations.

Figure 7. Contact Mechanism of the LANL8-DOF Testbed.

4 Symbols used in Table 4 are GLS, generalized least-squares; PCD, principal component decomposition; MR,model reduction; FM, full-order model (no reduction); D-data, displacement datq and V-data, velocity data.

Random, input excitations at the driving point(DOF 1) and the eight accelerations are measured at4,096 samples over a time period of 8 seconds. Dataare available at various force levels to identify thedegree of nonlinearity.

Although all DOFS are measured duringvibration tests, we assume that data are available atDOFS 1, 5 and 6 only. Therefore, the correlationconsists of matching these three acceleration time-histories with results from the numerical simulation.Since the correlation involves three DOFS only,model reduction is implemented to condense the FEmatrices and force vectors. The particular techniquechosen preserves exactly the lowest frequencies andmode shapes of the linear model. Figure 8 illustratesthe correlation before FE model updating.

Acdemucm$ 2-9fc+aFE Mcdel Ur.iab.a

4.0sI I0 O.a? 0.04 O.w O.w 0.1 0.12 0.14 0.18Om

I

4.C6 I I0 O.a? 0.04 0.C6 0.08 0.1 0.t2 0.14 0.16

O.w

.

.

4.C6 I I0 0.02 0.04 0.00 O.ca 0,1 0.12 0.14 0.16

l-mwhsx=n& (s&dnlw.25sc llashd M.3. FEMl

Figure 8. Acceleration Time History BeforeUpdating the LANL 8-DOF Testbed.

As in Section 4, our modeling of this systemis perfectly linear, except for the addition of an internalforce vector. First, we attempt to represent thenonlinearity as an internal force triggered whencontact or penetration are detected during timeintegration. The amounts of force applied to masses5 and 6 are opposite in direction (such that the twomasses are pushed away from each other) andproportional to the depth of penetration. Therefore,this simple contact modeling is parametrized by theamount of penetration allowed and the stiffness ofthe reaction force. However, attempts to updatethese parameters have proven unsuccessful so far.

Therefore, a second approach is pursuedusing a non-parametric force vector. Arbitrary internalforces are applied to each one of the eight masses ofthe system and TAC is used for estimating theseforce levels at prescribed time samples.

The overall procedure goes as follows.Unknowns of the optimization are the eight forcecomponents. Correlation is based on the first 90acceleration measurements that span the timewindow [0;0.1 68] sec. For the numerical simulation,FE matrices and force vectors are reduced to the sizeof the TAC model (DOFS 1, 5 and 6 only) and theresponse of the condensed model is integrated intime using 10 sampling points between any twomeasurements. As the FE response is integrated intime, the internal force vector is optimized.

-1OOO(NmLwc.Lh?dalheDoF8c401eMde41

o 0.02 O.M 0.05 003 0.1 0.12 0.14 0.16Tmeh Saada

4ml ,-, 1,,

Ii

3-4m1-=$t4umbemcenc4eoleooFK401et.kdd)

a. Iom 0.04 0.26 0.00 0.1 0.12 0.14 0.16

llmeh Sexads

Figure 9. Time-Histories of Internal ForcesObtained Via FE Model Correlation.

Figure 9 shows the reconstruction of internalforce as optimization are performed for each timeinterval containing three consecutive measurements.In other words, 30 optimization are performed, oneevery 0.0056 sec. This result is obtained when theGLS cost function is optimized (no SVD is involvedhere). Obviously, no clear interpretation of this forcingfunction can be made. However, it can be observedthat components at DOFS 1 and 3 remain practicallyzero. In addition, components of the internal forcevector seem separated in two subsets: one is formedby DOFS 1-4 and the other one consists of DOFS 5-8.Notice how components 2 and 4 record largeamplitudes around 0.08 sec. while components 5-8reach their minima at this time sample. This may be aside effect of our contact mechanism locatedbetween masses 5 and 6.

It can also be observed that, to the exceptionof DOF 1 where the external force is applied, all DOFSfeature significant levels of internal force even thoughthey are not directly involved with the nonlinearmechanism. This might be a manifestation of thesystem’s friction. However, these results are

preliminary and further investigation is requiredbefore any plausible assessment of these forces canbe made. No matter what the source of the internalforce turns out to be, Figure 10 shows a clearimprovement of the correlation with test data whenthis force vector is included in the FE model.

AaAmlkm Ane$FEMc&4Ll@dng0.C6

:08

-0.26 I 1

‘“”~4.C6 I I

0 0.02 Om 0.C6 0.08- 0.1 0.12 0.14 0.16

-A J0 0.02 0.04 0.26 0.0s 0.1 0.12 0.14 0.16

Tbmhse.mds (did Dlw. tE=stdaskd!lr03 =FEMl

Figure 10. Acceleration Time History AfterUpdating the LANL 8-DOF Testbed.

We emphasize that results presented hereare a first step: non-parametric model updating doesnot necessarily yield a sound understanding of thesystem’s physics. For all practical purposes, a secondidentification would be required to generate a useful,parametrized nonlinear model.

7. CONCLUSION

This work presents an investigation of thecorrelation of test data to finite element models fornonlinear, transient dynamics. Modal techniques inthe frequency domain are discussed and it is shownthat they fail to correlate a linear model when theexperimental data involves significant friction.Therefore, the correlation is formulated in the timedomain and the merits of two implementations arecompared using test data from the LANL 8-DOFnonlinear testbed.

The preliminary results obtained are veryencouraging and additional full-scale testing isplanned with these and other methods. In particular,the optimal control-based formulation proposed inReference [17] seems very promising because itenables the non-parametric identification ofunmodeled nonlinear dynamics. It is currently beinginterfaced with our modeling and correlation software.

...—— — .-. . .. - .-.,.-C-A !7, .-. Z,S, S -- --! - ,.. -,7., .. ..- .5-.T - , . . -.-.’ 7=----- -------

ACKNOWLEDGMENTS

The authors would like to recognize thecontribution of Paula Beardsley, Graduate student atthe University of Washington, Seattle, Washington,who produced some of the results presented in thispaper during a Summer 1998 internship at LANL.Also, the expertise and guidance of Dr. Thomas D.Burton, Chairman of the Department of MechanicalEngineering at Texas Tech University, Lubbock,Texas, is acknowledged and gratefully appreciated.

REFERENCES

[1] Lieven, N.A.J., and Ewins, D.J., “AProposal For Standard Notation and Terminology inModal Analysis,” 10th IA44C, Feb. 2-5, 1992, SanDiego, California, pp. 1414-1419.

[2] Masri, S.F., Miller, R.K., Saud, A.F., andCaughey, T.K., “Identification of Nonlinear VibratingStructures: Part I - Formulation,” ASME Jowna/ ofApplied Mechanics, Vol. 54, Dec. 1987, pp. 918-922.

[3] Zienkiewicz, O.C., and Taylor, R.L., TheFinite Element Method, 4th Edition, McGraw-Hill,London, U.K., 1989.

[4] Maia, N.M.M., and SiIva, J. M. M.,Theoretical and Experimental ModalAnalysis, Ed., Research Studies Press, Ltd., Wiley& Sons, New York, 1997.

[5] Alvin, K.F., Peterson, L.D., and Park,K.C,, “Extraction of Normal Modes and Full ModalDamping From Complex Modal Parameters: AMAJowna/, Vol. 35, No. 7, July 1997, pp. 1187-1184.

[6] Berman, A., and Nagy, E.J., “Improvementof a Large Analytical Model Using Test Data; AMAJourna/, Vol. 21, No. 8, Aug. 1983, pp. 1168-1173.

[7] Imregun, M., and Visser, W.J., “A Reviewof Model Updating Techniques,” Shock and VibrationDigest, Vol. 23, No. 1, 1991, pp. 19-20.

[8] Farhat, C., and Hemez, F.M., “UpdatingFinite Element Dynamic Models Using an Element by

Element Sensitivity Methodology,” AMA Journa/, Vol.31, No. 9, Sept. 1993, pp. 1702-1711.

[9] Chouaki, A.T., Ladevbze, P., and Proslier,L., “Updating Structural Dynamics Models WithEmphasis on the Damping Properties,” AMA Journa/,Vol. 36, No. 6, June 1998, pp. 1094-1099.

[10] Hemez, F.M., “Can Model Updating Tellthe Truth?,” 16th /MAC, Santa Barbara, California,Feb. 2-5, 1998, pp. 1-7.

[11] Mottershead, J.E., and Friswell, M.I.,“Model Updating in Structural Dynamics: A Survey,”Journal of Sound and Vibration, Vol. 162, No. 2,1993, pp. 347-375.

[12] Hemez, F.M., “Comparing Mode ShapeExpansion Methods For Test-Analysis Correlation,”12th /MAC, Jan. 31-Feb. 3, 1994, Honolulu, Hawaii,pp. 1560-1567.

[13] Levine-West, M., Kissil, A., and Milman,M., “Evaluation of Mode Shape ExpansionTechniques on the Micro-Precision InterferometerTruss,” 12th /MAC, Jan. 31-Feb. 3, 1994, Honolulu,Hawaii, pp. 212-218.

[14] Jacobs, D.A.H., The State of the Artin Numerical Analysis, Ed., Academic Press,London, U.K., 1977.

[15] Hasselman, T.K., Anderson, M.C., andWenshui, G., “Principal Components Analysis ForNonlinear Model Correlation, Updating andUncertainty Evaluation,” 16th IM4C, Santa Barbara,California, Feb. 2-5, 1998, pp. 664-651.

[16] Mees, A. I., Rapp, P.E., and Jennings,L.S., “Singular Value Decomposition and EmbeddingDimension,” Physics/ Review A, Vol. 36, No. 1, July1987, pp. 340-346.

[17j Dippery, K.D., and Smith, S.W., “AnOptimal Control Approach to Nonlinear SystemIdentification,” 16th /MAC, Santa Barbam, California,Feb. 2-5, 1998, pp. 637-643.