Perturbed Boundary Condition Testing Concepts

Shumin Li, Stuart Shelley and David BrownUniversity of Cincinnati

Structural Dynamics Research LaboratoryCincinnati. OH 45221-0072

Perturbed Boundary Condition(PBC) TestingProcedures are being developed for measuringlarger databases for verification and/or updating ofFinite Element models. The PBC method can alsobe used to generate enhanced Unified MatrixPolynomial Approach (UMPA) models which canbe used to analytically describe the system. In thispaper the concepts of using the PBC method togenerate UA4PA models will be developed. Thesemodels will be used to describe the structuralcharacteristics of the system and to also describethe aeroelastic Jystem used to analyze aircraft flight

jlutter

Nomenclature

[Al :UMPA Model Response CoefficientPI :UMPA Model Input Coefficient[Cl :Damping Matrix[II Jndentity MatrixWI :Stiffness Matrix[M] :Mass Matrix[AM] :Mass Perturbation Matrix{F} :Force Vector{X) :Response VectorS :LaPlace Variable0 :Frequency

Introduction

The Perturbed Boundary Condition (PBC) testingmethod is a procedure which has been developed

for measuring a larger experimental database whichcan be used for verifying a component/systemmodel or for generating a model directly from theexperimental database. In this method a test articleor component is rested in a number of differentconfigurations, where each confibmration consists ofa modification of one or more of the systemsboundary conditions.

One of the historical problems associated with finiteelement model verification has been in measuringenough information to localize the errors in themodel. The PBC methods were originallydeveloped to address this problem. In a typicalexperimental modal test to gather data for a modelverification, the lowest 10 to 40 eigenvalues andeigenvectors of a system or component can bemeasured. The number depends upon thecharacteristics of the system (damping,nonlinearities, etc). In general, this is insufficient

Figure l--PBC Testing Procedure

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information for model verification. In order toimprove the problem the system can be brokendown into components and each component testedindividually and/or the system/components can betested using the PBC method. In the PBC methodthe system is tested in a number of drasticallydifferent configurations with the lowest IO-40modes determined for each configuration.(SeeFigure 1) The boundary conditions act to filter theoriginal modal space of the system into differentsubspaces. These subspaces can be used as adatabase for the model verification process.

The PBC method can also be used to generate aUnified Matrix Polynomial Approach (UMPA)model of the system. Generating a UMPA mode1 ofa system will be the main emphasis of this paper.Two different types of models will be discussed.The first case demonstrates how the PBC methodcan be used to improve an experimentally generatedmodal model or impedance model of asystem/component. The second case uses a specialadaptation of the PBC to generate a flight fluttermodel of an aircraft directly from flight data. Thisapproach yields a model of the aircraft structuralcharacteristics and a model of the aerodynamiccharacteristics which is a function of flightconditions (for example; air speed and density).This model can then be used to predict systemcharacteristics for untested flight conditions.

Background

In the early 1980s with the advent of Multiple-Input-Multiple-Output (MIMO) testing andparameter estimation procedures, improved sensorsand data acquisition systems,* were developed inorder to measure a more consistent database. Thiswas necessary in order to improve the performanceof the MIMO parameter estimation algorithmswhich assumed global modal parameters. Withthese new data acquisition systems all of theresponse transducers were pre-mounted on thestructure. All of the data was taken simultaneouslyor in a number of patches (switchable data sets)

over a relatively short time period. This improvedthe consistency of the data significantly byminimizing time variance effects. Even withMIMO testing procedures the number of reliablymeasured eigenvectors was limited to 10-40. Ingeneral, this was not enough measured informationto reliably locate the errors in a large finite elementmodel. However, with the development of MIMOdata acquisition systems it was possible to testmultiple configurations of a system in a timelyfashion. This led to the development of the PBCtesting procedure.

In the PBC testing procedure, selected boundaryconditions of the system are drastically changed andthe system is retested, Since these perturbationscause a drastic change in the systems response,each configuration reveals addition informationabout the system.

In the mid eighties, mass loaded boundary pointswere used as a means of measuring rotationaldegrees-of-freedomr3 at boundary points and as aninexpensive replacement method for a fixed basetestt4.51 One of the first PBC methods applied toFinite Element Modeling (FEM) verification wasapphed to large flexible structures by constrainingto ground interior points of a structure and verifyingthe constrained structure against a FEM 6-9 Theselarge flexible structures were difficult to test free-free because the frequencies of the deformationmodes were too close to the frequencies of thesupport system. Constraining the structureeliminated the support problem. It also isolatedsections of the structure, making it possible tolocate errors in the FEM.

In the early nineties procedures for utilizing a PBCdatabase for updating of finite elements modelswere being developedt~r. These techniques arecurrently at an early stage of development.

The PBC data can also be used to generate anexperimental model of the system. Two differentapproaches have been taken. One method is used to

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generate an enhanced modal model of the system,21while the second method is used to generated anenhanced Unified Matrix Polynomial Approach(UMPA) model of the system[3~51.

In this paper the UMPA approach will be reviewedin some detail and several examples will be given todemonstrate its characteristics. In the UMPA modelapproach the input-output data measured from anumber of PBC configurations are condensed intoa single UMPA model. This model completelydescribes the systems inputs and outputs at themeasurement points. The inputs consist of forcesapplied by an exciter system and the passive forcesdue the Perturbed Boundary Conditions. Theseforces are measured directly or indirectly. Figure 2is a schematic of a PBC test object showing theinputs and outputs acting on the system.

F

Figure 2--PBC-UMPA Model Schematic

Theory

In this section the basic theory for the developmentof an UMPA model from the PBC test data willgiven. Two different examples will be addressed.The first is the determination of an UMPA modelwhich can be used in place of a modal model or toextract a modal model of a system. The secondgenerates an UMPA model which can be used todescribe the flight flutter characteristics of anaircraft. In both cases the boundary conditions areperturbated in order to measure a more complete

model of the system.

Structural Dynamics Model

For the structural dynamic case the PBC method isused to apply passive forces at a number ofboundary points. These passive forces are used toprovide additional excitation to the system soadditional information is obtained about the system.The perturbed boundary condition can generatepassive forces due to stiffness changes, masschanges or changes due to connecting a subsystemat the boundary points. ln general, using simplechanges such as mass or stiffness changes makes iteasy to indirectly measure the passive forces bymonitoring the responses of the boundary point.For flexible structures stiffness changes areeffective while for stiff structures mass changes aremore effective. In this paper, mass modificationwill be discussed. Stiffhess changes can be handledin a similar fashion.

The matrix rational fraction power polynomialUMPA model of the PBC test data can be derivedsimilarly to that of a conventional test system(without PBC). In the Laplace domain, theequation of motion of an N degree of freedom,spatially complete system can be written as:

(S[Ml+S[Cl+[Xl)IX(s)l= P(s)1 (1)

After adding a mass perturbation, AM, this equationbecomes:

(~~~+~~l+~~~l+~~l)~~~~~l= {+I} (2)

The mass perturbation could be a single massaddition or multiple mass additions at multipleboundary points. Moving the inertial force causedby the additional mass to the right hand side of theequation gives:

[~[~l+~[~l+[~l](~~~)) ={W}-[s*AM]{x(s)} (3)= ct*

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where:

Equation (3) can be considered to be the equation ofmotion of the unmodified system excited by a totalforce which consists of the actual input force andthe passive or inertial force from dM. This meansthat adding a mass to a system is equivalent toadding a corresponding passive force to the system.For each additional test configuration, Equation (3)is appended with a set of equations of motion of thesame unmodified system excited by a different totalforce.

Let

Pzl -[4[AI] =[Clr-401 =[Kl[&I =[[I

(5)

Equation (3) becomes:

This is the Laplace domain UMPA (2,0) model forPBC test data. A more general UMPA (n,m) modelfor PBC test data is:

Since structures are often measured with moreresponse points than the number of modes existingin the frequency band, spatially complete systemsare generally the case. Therefore, the UMPA (2,0)model will be used in the following derivationwhich simplifies the discussion. Also, sincemeasurements are taken in the frequency domain,

all equations will be written in the frequencydomain instead of in the Laplace domain. Afrequency domain UMPA (2,0) model can bewritten as:

for k = 1. 2, _.., Ns

From the measured forces and responses, theunknown UMPA coefficient matrices, [AJs and[BJs, in Equation (8) can be estimated using apseudo-inverse procedure such as least-squares ortotal-least-squares. In the least squares method, theunknown coefficient matrices in Equation (8) canbe normalized with respect to [A,]:

Equation (8) becomes:

Re-writing the previous equation in matrix format:

The unknown coefficients are estimated as:

[;i, 4 &I=-[xb,) . Xb,)][

h)x(~,) .. (j~,)X(~,)+

jm,X(w,) .. j~dT4-F,(w,) .,. -~~kG) :

(12)

Notice that [A,] is normalized to an identity matrixhere instead of [A,] which is usually the case. Thisis because the resulting least squares problem datamatrix in Equation (12) is generally a numericallybetter conditioned matrix, assuming the system is

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positive definite. This means that the inverse ismore stable than the inverse of the data matrixformed from normalizing [A2]. This will directlyaffect the accuracy of the estimation of the UMPAcoefficients as discuss in [15]. However, in thesituation where the system is semi-definite, the rigidbody modes of the system are at the zero frequency.[A O ] is a singular matrix since the stiffness matrixof the system is a singular matrix. In this case it ispreferable to normalize [A>] to the identity matrix.

Aerodynamic Model

Application of the PBC concept to flutter estimationwas conducted as part of a NASA - Ames/Universityof Cincinnati Joint Research Interchange project[ 161. For the aerodynamic case the passive forcesacting on the boundary points are due to theaerodynamic forces. These forces are theaerodynamic forces which are a function of thesystem response, such as wing bending and torsionvibration. The model has the same form as thestructural dynamics case or

(13)In this case the structural model consists of thegeneral UMPA model with N, coefficients A ofsize N, by N,, and Nb coefficients B of size N, byN;. The aerodynamic model consists of N,coefficients F, and a scaler multiplier or weighingcoefficrent, w,~, associated with each F coefficientwhich is a function of the operating condition. Thiswould generally be a function of the dynamicpressure, for flight flutter applications, which isdetermined by flight speed and air density. The sizeof the matrix coefficients is determined by thenumber of channels of experimentally measuredresponse data, N,, and the number of channels ofmeasured input force, N,. The number of systempoles estimated is equal to N, times the larger ofN, and N,.

For this case the As, Bs and F,s are the unknowns

and the response of the aircraft X(jo), appliedforces, F, and the flight conditions, w,, (air speed,density, dynamic pressure) are the measuredinformation. By assuming A, is equal to an identitymatrix the remaining As, Bs and F,s can bedetermined in a least square sense. In order togenerate enough equations to solve for theunknowns, the aircraft is excited at a number ofdifferent frequencies with externally generatedforcing functions and must be flown at severaldifferent flight conditions in order to perturb theboundary conditions.

Generally, the larger the number of responsemeasurement channels that are available the moreaccurate the estimation results will be. Processing alarge number of channels, however, increases thecomputational burden and can generate a largenumber of computational poles which requiresignificant user judgement and effort to sort out.This can make the estimation process unwieldy forthe flight test application which requires that flutterpredictions be made quickly since the decision toproceed to the next test point relies on theprediction, It has been demonstrated that spatialfilters can condense the data from a large number ofresponse channels to a number of channels equal tothe number of modes contributing to the response,without compromising estimation accuracy, Thisallows the accuracy of a large response channelcount estimation solution to be achieved with theeffort of a small estimation solution. The spatialfiltering may be performed in either the time orfrequency domain, operating on the measured timeresponses or the frequency response function data.Details of this technique may be found in [I 71.

The experimental procedure for determining theUMPA model is to fly the plane at a number offlight conditions which are approaching the flutterconditions and measure the responses and appliedforces acting on the system for each of theseconditions. This data is used with the UMPA/PBCmethod to generate a closed loop model of theaircraft in flight which is a function of the flight

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conditions. The flight conditions associated withthe onset of flutter can then be predicted from theclosed loop model by analytically varying the flightcondition until the system goes unstable. The flightconditions change the w,, terms in Equation 13which describes the close loop response of theaircraft.

Computational Examples

In this section several numerically generatedcomputational examples will be given. The firstexample is a structural dynamics example which isgenerated to illustrate the effect of mass loadingboundary points to obtain better estimates ofresiduals at the boundary points, The secondexample is an aerodynamic example where acomputer simulation is used to construct a data setwhere the aerodynamic forces acting on a &ulatedmodel of the aircraft are used as the passive forcesfor the perturbed boundary conditions,

Structural Dynamics Example

A fifteen degree-of-freedom, lumped mass systemis used as the structural dynamics example. In thisexample the lumped mass system simulates fivelocal modes at five boundary points.(points 1 1 - 15 inFigure 3) This simulates the influences of residualterms at connection points. The residual termsconsist of local modes with eigen-frequencieshigher than the testing bandwidth.

The external forces applied to the structure in orderto determine the UMPA model were applied atinterior points not at the connection points. Passiveforces were generated at the connection points byadding masses at these points. The masses wereadded in severa stages: 1) an UMPA model wasdetermined by using data taken from the originalunmodified system; 2) mass was added to oneconnection point and the model recomputed; 3)

mass was added to all the points simultaneously andthe model estimated. The model was estimatedfrom data taken in a frequency band below thefrequencies of the local modes. This is shown inFigure 4 where the response of the original systemis shown in addition to the response of the systemwith mass added to all boundary points (points I I15). The effect ofadding the masses was to pull thelocal modes into the measurement bandwidth.

An UMPA model was determined by exciting onPoints l-5 and measuring the responses at all fifteenpoints. Both baseline and proportional noise wasadded to the data in order to simulate measurementnoise. An example of the noise is shown in Figure5. The noise is greater than would normally existin a typical test.

Figure 3 -- 15 DOF Lumped Mass System

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Figure 4--Composite Frequency ResponseFunction of Original System and ModifiedSystem.

, Noise Floor I

Figure 5 - - Typical FRF Measurement

The eigenvalues of the UMPA model are shown inFigure 6 for the original system. The solid linecorresponds to the true eigen-frequencies and theestimated values are shown as the plottedpointsThe local modes could not be determinedfrom this data set.

0 2 4 6 6 10 12 14 16MC&

Figure 6 -- Estimates of Natural FrequenciesFrom Conventional Test Data

In Figure 7, mass was added to one of theconnection points and the eigenvalues werecomputed from the UMPA model of this mass

Figure 7 -- Estimates of Natural FrequenciesWith One Mass Addition at Point 12

loaded system. The local mode at the location ofthe mass addition is predicted. In Figure 8, mass isadded to all five locations and the UMPA modelreestimated. For this case all of the local modeswere predicted.

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0 2 4 6 6 10 12 14 16Mode

Figure 8 -- Natural Frequencies for MultipleModifications at the Five Connection Points

Aerodynamic Example

Two analytical cases are used to demonstrate thisprocedure. A two degree of freedom system wassimulated in a Simulink model of the flightconditions, The data from this simulation was usedto evaluate the method. The data used to generatethe model was taken at flight speeds significantlylower than the flutter speed. This data was used tosimulate the type of data measured during a flighttest of an aircraft. An UMPA model of thestructural and aerodynamic characteristics of thesystem was determined horn this simulated data set.

The UMPA model was used to predict thefrequency and damping of the flutter modes overa range of flight speeds and these predictionscompared to the Simulink simulation results for thesame flight speeds. A comparison of the UMPA

Figure 9: Variation of Flutter Model Dampingand Frequency with Velocity

model with the simulation is shown in Figure 9.Inthis figure the eigenvalues of the two-degree offreedom system predicted from the UMPA/PBCmodel are plotted as a function of the flight speed.The actual eigenvalues are also plotted, but the twocurves overlay and cannot be distinguished. Themodel accurately predicts the flutter condition. Atthe flutter condition the damping of one of themodes approaches zero.

A ten degree-of-freedom example with added inputand output noise was run to simulate a morerealistic case. Again, Simulink was used to generatethe simulation data. An UMPA model was

q 4 1.6 LB z 2.2 2~4 23 23 3Frequency HtiZ

Figure 10: Experimental and Synthesized FRF- Velocity = 1000

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determined from the multiple perturbed boundarysimulated conditions. The data corresponds to anumber of flight conditions. In Figure 10, acomparison between an FRF synthesized from theestimated UMPA model and an FRF estimated fromthe simulated flight data for the close loopfrequency response function is shown at an airspeedof 1000 KM/Hour. This figure indicates that theUMPA model does a excellent job of approximatingthe flight data. Recall that the data from all flightconditions is utilized simultaneously in theestimation of the UMPA model which can then beused to synthesize FRFs for any flight condition.

L8,L.~ I%o IOW ,500

VelOCily

Figure 11: Pole Trajectories Predicted byUMPA Method

The variation of frequency and damping of allsystem poles as a function of flight speed predictedby the estimated UMPA/PCB model is shown inFigure 11. Also shown is the actual frequency anddamping of the two flutter modes. Again this dataindicates excellent agreement. This example wasused to demonstrate a possible method ofgenerating a flight flutter model directly from flightdata. The flight data would be collected by flyingthe aircraft at a number of flight conditions as closeto the flutter condition as possible. The aircraftwould be excited with in-flight exciters and theflight conditions, externally applied forces, and

response would be recorded and processed togenerate an UMPA-PBC model of the aircraft.This model would be used to predict the flutterconditions.

Conclusions

In this paper the concepts of using a PerturbedBoundary Condition(PBC) Testing method fordeveloping a system model was presented. In thismethod a system model is developed by measuringthe input and response data of the system andfitting this data to a Unified Matrix PolynomialApproach (UMPA) model of the system. In orderto supplement the input data, the boundary pointsare perturbed and the passive forces due to theperturbation are measured. These forces aremeasured at points of interest and help to betterdefine the influence of these points on the systemcharacteristics. Two analytical examples werepresented in the paper. The first example was astructural dynamics example where the PBC testmethod was used to help measure the influence oflocal modes at potential connection points of asystem. These residual modes were estimated bymaking a simple mass perturbation at theconnection points. In the second example anaeroelastic model of an aircraft was obtained wherethe aerodynamic forces were considered as thepassive inputs to the structure. The aeroelasticmodel can be used to predict the flutter conditionsfor the aircraft. Excellent results were obtained inboth of these analytical cases.

The parameter estimation aspects of the PBCprocedure have been studied in detail and theinfluence of various types of measurement noisecharacterized. The next step is to apply thismethod to several experimental cases.

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