a frequency response function-based updating

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A frequency response function-based updatingtechnique for the finite element model of automotivestructuresD-C Lee1,2 and C-S Han1*

1Department of Mechanical Engineering, Hanyang University, Ansan, Republic of Korea2SPACE Solution Co. Ltd, Seoul, Republic of Korea

The manuscript was received on 31 March 2008 and was accepted after revision for publication on 18 June 2008.

DOI: 10.1243/09544070JAUTO866

Abstract: Todays automotive industry uses finite element analysis (FEA) in a huge variety ofapplications in order to optimize structures and processes before hardware is produced.Efficiencies can be enhanced and margins are reduced because the external loads andstructural properties are identified with higher confidence. The accuracy of FEA predictions hasbecome increasingly important and directly influences the competitiveness of a product on themarket. Because automotive structures are under dynamic environments, the correlation onthe basis of static deformations independent of the mass and damping parameters do notprovide a valuable reference from the view of the dynamic characteristics. In this paper, bysystematically comparing the results from analytical and experimental analysis techniques,finite element (FE) models can be validated by the deterministic and robust design on the basisof each tolerance of design parameters, and improved so that they can be used with moreconfidence in further analysis. Making use of different types of test datum, a recommendedprocedure is to use a sequence of analysis in which mass, stiffness, damping, and external

loading are validated and, if necessary, updated.

Keywords: virtual prototype, test-based technology, computer-aided engineering-basedtechnology, point mobility, frequency response, deterministic optimization design, robustoptimization design, finite element model

1 INTRODUCTION

In the design and optimization of vehicles, numerical

models for the prediction of the noise and vibration

behaviour play an important role. The validity and

reliability of these models can be drastically im-proved by application of model-updating techni-

ques. Model updating aims at the verification and

correction of numerical models of dynamic struc-

tures by means of comparison with experimentally

obtained data about the noise and vibration beha-

viour of the real structure. The use of quantitative

methods for assessing the differences between

dynamic tests and finite element analysis (FEA) has

been an active field for many years [16]. When FEA

was first introduced, it was treated cautiously by test-

oriented engineers; if a difference existed between

the different approaches, the finite element (FE)

model was suspected and major modelling revisions

were attempted to cure the discrepancy. With the

gains in reliability with FE methods, however, this

attitude has changed rapidly over the years; now the

goal is to find the actual physical difference, regard-

less of the source. Modal correlation has been used

to match eigenvalues and eigenvectors for the struc-

tures. These methods provide direct information on

the quality of the correlation, but have difficulty in

identifying specific physical properties. Also, these

methods had some difficulty when closely coupled

modes and finite damping occurred in the struc-tures. Another step forwards has been to adopt

*Corresponding author: Department of Mechanical Engineering,

Hanyang University, Sa-1 dong, Sangnok-gu, Ansan, Kyeonggi-

do, 425-791, Republic of Korea. email: [email protected];[email protected]

1781

JAUTO866 F IMechE 2008 Proc. IMechE Vol. 222 Part D: J. Automobile Engineering


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design optimization techniques to minimize the dif-

ferences between test and FEA results. The problemhere has been the difficulty in obtaining sensitivity

derivatives efficiently for the case of many eigenvec-

tors or response solutions. This has led to a mixture

of data comparisons, including combinations of

modal and forced-response samples. In the absence

of prototypes, analytical methods such as FEA are

very useful in resolving noise and vibration pro-

blems, by predicting the dynamic behaviour of the

automotive components and systems. FEA is a

simulation technique and involves making assump-

tions that affect analytical results. Acceptance and

use of these results are greatly enhanced through test

validation. The automotive development process

using computer-aided engineering (CAE) is shown

in Fig. 1. As the stages from the concept verification

plan to development verification plan are preceded,

the dependences on the CAE become larger and the

model correlations with test become more impor-

tant.

The aim of the model-updating technique pre-

sented by this paper is to reduce the errors in the

FE results by predicting changes to selected struc-

tural properties. This approach can bypass many ofthe previous difficulties by using the following meth-

ods.

1. The error measures are defined directly from the

solution vectors to avoid large, complicated, sym-

bolic equation entries and manually transcribed

data.

2. Frequency response solutions are used to avoid

the difficult task of calculating eigenvector deri-

vatives.

3. Constraint equations are built into the solution to

enforce test responses and produce faster con-vergence.

Test results show the feasibility of this approach,

and its practicality.

2 THEORETICAL CONSIDERATIONS FOR THEDYNAMIC CORRELATION

The automotive body structure made by the sheet-

metal-forming process influences the junction boun-

daries and gradual variation shapes on the panels

owing to the deformed shapes or strain distribu-

tions, which in turn influence the thinning effect of

panels or the geometric weak points. These material

and physical variations can change the structural

performances [711]. When taking into account these

parameters, the simulation model can give the rel-

iabilities of the prediction of structural responses.

In this paper, the parameter estimation method using

the frequency response is presented. The explana-

tion of the feasibility of parameter estimation on

the frequency domain can be explained as follows.

Starting with a global FE model and using the

direct frequency response, the resulting steady-state

matrix equations are

{v2MazjvBazKa

uf g~ Ff g 1

or, in terms of the dynamic stiffness matrix [Z], as

Zav ~ {v2MazjvBazKa

where [Ma], [Ba], and [Ka] are the mass, damping,

and stiffness matrices, respectively, for the analytical

model. Note that these matrices may be complex

and unsymmetric. The analytical solutions to the

loads, Fi5F(vi), are

Zav uai

~ Fif g 2

Fig. 1 Development process of the automotive structure using CAE

1782 D-C Lee and C-S Han

Proc. IMechE Vol. 222 Part D: J. Automobile Engineering JAUTO866 F IMechE 2008


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The objective of this procedure is to find the mod-

ifications to the FE properties that will produce

a new set of impedances [Za(v)], such that for each

selected frequency

Zxv uxi

& Fif g 3

where the arbitrary vector uxi

at each frequency

contains results that match the test data. The change

in impedance can be defined as

DZvi ~ Zxvi { Z

avi 4

The matrix changes in [Ma], [Ba], and [Ka] which

form the matrix [Za(v)], in turn, may be defined by

structural parameters, such as geometric dimen-

sions, material properties, or non-structural massdensities. Note that the potential number of para-

meters could grow to a large number if every element

in the analytical system could change independ-

ently. However, a unique solution is impossible.

The relationship to the LinEwins approach can

be shown by solving equation (3) by subtracting

equation (2) and substituting equation (4) to ob-

tain

Zaivi

uxi{uai

&{ DZivi u

xi

5

By inverting [Za

(v)] at each frequency, a basis foriteration is obtained as

uxi{uai

&{ Zaivi

{1

DZivi uxi

6

In other words, the changes in the responses are res-

tricted by the constant analytical structural proper-

ties in Zaivi

. The advantage of this method is that

the rows corresponding to the measured points

may be solved simultaneously with the unmeas-

ured points. The unmeasured points in the uxi

vectors are updated with each estimate of [DZi(vi)].The errors in the tested degrees of freedom become

the measure of convergence in the iterations. The

design parameters are the structural parameters to

be varied. Options include FE property data, scalar

element coefficients, and grid point locations. They

define the effects of the matrices [DK], [DC], and

[DM] indirectly through the FE sensitivity matrices.

The objective function may include a variety of

parameters, including the response errors, the limits

on the weight, and the modal properties. Current

tests have used a minimization of a simple sum over

key frequencies of the squared errors at importantpoints. Optimization constraints are the upper limits

of the magnitude of the dynamic characteristics for

selected points and selected frequencies. Indirect

constraints are available to limit the changes in the

design parameters for each iteration. An important

aspect of the optimization procedure is the linear-

ized iteration method using the sensitivity matrices.

The complete stiffness matrix [K] for frequency

response analysis consists of a superposition from

various sources according to

K ~ K1 zjg K1 zX

gS KS z K2 7

where [K1] is the structural stiffness matrix and [K2]

is the direct matrix input at nodes. gis the uniform

structural damping coefficient. gs is the structu-

ral damping coefficient of elements. Updating the

stiffness modelling with the displacement, the errorfunction

err~DuT Ce DuzDpT Cp Dp 8

should be minimized. In equation (8), Ce and Cp are

the diagonal weighting matrices for the selected

updating of the performance and the updating

parameters, respectively. The subscripts e and p

indicate the elastic and plastic performance indices,

respectively. Du is the difference between experi-

mental and analytical displacements: Du~

L uj

LpkDp~ ujexpn o

{ ujanan o

. DRkis the differ-

ence between updated and originally estimated

parameters: Dpk5pku2pka.

[K] are known as the functions of {Du} relative to

the structural rigidities due to the geometric dimen-

sions, and, therefore, {DF} can also be calculated in

terms of geometric dimensions. It can be assumed

that the tangent stiffness matrix, incremental defor-

mation, and load unbalance are functions of dimen-

sional parameters according to

K ~ K p Duf g~ Du p f g

DFf g~ DF p f g9

For design optimization of non-linear structural

performance, the sensitivity analysis is given by

equation (7).

LL[K]/LLpkis the derivative of the system stiffness

matrix with respect to the parameter and is given by

L K =Lpk~ K pkzD

pk { K pk pk

10

A frequency response updating technique for automotive structures 1783



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From the above equations, the undesired or des-

ired sensitivities d{Du}/drican influence the total

energy related to the elastic energy of the struc-

ture. Thus, the structural uncertainties must be con-

sidered in estimating the performances of struc-

ture. The structural uncertainties can be described

as the probabilistic distributed region of para-

meters. The probabilistic manifold set can be given

by some manifold sets from the nominal value and

distributions of uncertain parameters according to

V ai,bi,mi,Di ~ aiGpi{p

i

pi

mibi

& '

~ aiGDpipi

mibi

& '11

where V is the probabilistic manifold set of designparameters, piand p

i are the uncertainty para-

meters which have uncertainties related to geo-

metric dimensions or material properties, and the

nominal aiand biare called the scaling value of

uncertain parameters on the structural limits and

the arbitrary manifold, respectively. Gcan be

defined as the structural performance, (Dpi)/p* is

the sensitivity of uncertainty design parameters,

and miis the set of distributional random para-

meters.

The geometric parameters can be defined by set-

ting the basis vector of grid point changes to thedirections normal to the surfaces as

DGf g~ T Dpf g 12

where {DG} is the set of grid point changes, [T] is

the set of shape basis vectors, and {Dp} is the set

of scaled design parameter changes in the shape

dimension.

To quantify the correlation between predicted ana-

lytical results and test, use is made of the criterion

RAC xe,xa ~xef g

T xaf g 2

xef gT xef g

xaf g

T xaf g 13

where the response assurance criterion (RAC) is based

on a similar formulation used to correlate mode

shapes exist between 0 and 1. The response vectors

{x} may be the displacement, velocity, acceleration,

strain, or stress on the frequency domains. A sensitiv-

ity-based correlation algorithm requires the computa-

tion of design parameter variations for each config-

uration based on the corresponding analysis data. Thecorrelation equation can be given by

Q T Q Dpf g~ Q T DRf g

Q1

Q2 .

.

.

Qn

2666666437777775

TQ1

Q2.

.

.

Qn

2666666437777775DPf g~

Q1

Q2 .

.

.

Qn

2666666437777775

TDR1

DR2 .

.

.

DRnf g

8>>>>>>>>>>>:9>>>>>>=>>>>>>;

14

where [Q]and{DR} are the overall structural sensitivity

matrix and response error vectors between test and

analysis, respectively, {Dp} is the vector of design

parameter variations, and [Qn]and{DRn} represent the

corresponding overall structural sensitive matrix and

the corresponding error vector of the nth configura-

tion response, respectively.

3 DESIGN OPTIMIZATION PROCESS

The procedure can be divided into three parts: the

first part is an outer iteration improving design

parameters by sequential quadratic programming;

the second part is a deterministic analyser for linear

and non-linear structural performances; and the

third part is an inner iteration solving suboptimiza-

tion problems for reliability analysis. In the optimi-

zation procedure, the objective function to minimizeis the total elastic strain energy with a constraint

on the total available volume. Generally, a typical

optimization design problem of minimizing an ob-

jective function subject to a set of constraints can

be written

Minimize Yu, p

subject to Gku, p , k~1, . . . , m

mLjmjmUj, p

Ljpjp

Uj, j~1, . . . , n

15

In a realistic environment, the design parameters and

state parameters may fluctuate about their nominal

values. Thus, the distributions in the objective func-

tion and constraints due to the random parameter

must be considered in the feasible design stages. The

above optimization processes can be described as an

iterative search process that uses the following steps.

Step 1. Define an initial design pi50.

Step 2. Analyse the linear and non-linear char-

acteristics using a linear solver routine and a non-linear solver routine.




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Step 3. Compare the results of the analysis with

such requirements as allowable elastic or plastic

specifications.

Step 4. If the requirements are not met, perform

the optimization routine in order to set dp.

Step 5. Correct dpi+ 1 based on the fully stressed

design with b5 0.9 according to

dpiz1

new~ dpiz1

old

Upart

Uavg

b16

where

dpiz1

old~Gia

~ {+Yiz+Yi

2

+Yi{1 2 Si{1 !

dYi

+Yi

TGi !

The objective DRi(P) can be approximated for each

design P(i) using the series expansion

DYi~DYiPi

z

Xnj~1

L DY

LPjDpj 17

The gradient L(DY)i/LPjcan be obtained directly

from the results of FEA. If the gradient is known,

the search direction Dp can be obtained from the

solution of an approximate optimization problem.

Change the design parameters usingpi+ 15

pi+ dpi+ 1.

Step 6. If the requirements are satisfied, perform

the discrete design for the design parameters with

consideration of manufacturability. Otherwise, go

to step 2.

The deterministic structural optimization pro-

blem based on sequential quadratic programming

can be formulated as a non-linear constrained

optimization problem. Similarly, the general relia-bility-based structural optimization (RBSO) prob-

lem can also be formulated as a non-linear

constrained optimization problem in which the

constraint functions are reliability-measuring func-

tions. The reliability constraints ensure a more

feasible design guideline in the structural concept

design. Generally, the RBSO model includes two

types of design parameter: distributional random

parameters m and deterministic design parame-

ters p. In this paper, a new formulation that

considers the mean value and variation in the

performance function sequentially is proposed. The

objective-function-solving suboptimization step is

set to maximize or minimize a performance corres-

ponding to the substructure. The probability con-

straint related to the reliability of the performance

function is introduced. The Monte Carlo simulation

examines the variance of performance function and

constraint satisfaction probability at the optimal

points. The probability constraint has a suboptimi-

zation problem in terms of state parameters, design

parameters, and random parameters.

Let m5 {m1 m2 mn}T and p5 {p1 p2 pn}

T be the

distributional random and deterministic design

parameters, respectively. The RBSO problem can be

formulated as

Minimize W m, p

subject to ZiGkm, p ZUi, i~1, . . . , m

mLjmjmUj, p

Ljpjp

Uj, j~1, . . . , n

18

where W is the structural objective function andZi(Gk(m, p)) is the probability of the kth structuralperformance. ZUi is the required upper bound ofprobability. mLj and m

Uj are the lower and upper

bounds, respectively, of the jth distributional ran-dom design parameters; pLjand p

Uj are the lower

and upper bounds, respectively, of the jth determi-nistic design parameters. The reliability constraintsare assumed to be independent. To calculate theprobability ofZi(Gk(m, r)), the first-order reliabilitymethod (FORM) is used. At each design iteration,the FORM needs to determine a reasonable designassessment for each structural objective functionfor the next design iteration. To perform thegradient-based optimization iteration, the sensitiv-ity of objective function and constraints would beneeded. A suboptimization problem can be givenby

Minimize or maximize G u, m, p ~GzkX LG

Lp

Dp

subject to Q u, m, p 0

H u, m, p ~0

19

where kis the penalty factor with respect to each

constraint and determined by the user.

Let u* and m* denote the optimal state parameter

and random parameter, respectively, of the subopti-mization problem, then the KuhnTucker necessary




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conditions can be given by

LG

LuzlT

LH

LuzgT

LQ

Lu~0

LG

LmzlT

LH

LmzgT

LQ

Lm~0

H u1, m1, p1 ~0

gTQ u1, m1, p1 ~0 at gw020

The optimal random parameter m* is the most

probable violation point of the current design.

LLH/LLm and LLG/LLm can be calculated by differ-

entiating the state equation with respect to random

parameters. The change in the suboptimization

solution G* with respect to the design parameters

can be given by

dG1~LG

LpdrzlT

LH

LpdrzgT

LQ

Lpdr 21

4 SIMULATION

Frequency in the dynamic characteristics of a vehicle

is one of the most important factors influencing the

noise, vibration, and harshness quality of passenger

comfortability. In particular, the driver is sensitive to

noise and vibration in the frequency range of such

braking noises as squeal, moan, and groan. To control

these kinds of vibration and to understand the current

phenomena in more detail, a well-correlated FE

model may be necessary [1215]. While the decision

as to what the parameters are is relatively simple, the

way in which they are applied to a realistic model is a

Fig. 2 Design procedure of automotive structurethrough the model correlation (CAD, compu-ter-assisted design)

Fig. 3 Schematic sensor location for the frequency response test




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different problem in terms of feasibility and reliability.

To update the simulation model, Fig. 2 shows the

design procedure of automotive structure.

The FE simulation of the coupled torsion beam

axle (CTBA) structure requires making valid assump-

tions. These assumptions need to be made in such a

fashion that they do not compromise the accuracy of

the results obtained. In order to ensure that these

assumptions are adequate, analytical models are cor-

related with actual test data. The process of correl-

ating analytical models is important if the model

is going to be used to suggest design modifica-

tions to reduce noise, vibration, and harshness con-

cerns.

The CTBA evaluation consists of three excitation

configurations that are conducted under the mount-

ing boundary conditions, which defines the trans-

lational directions at the brake mounting point.

Figure 3 shows the schematic sensor location for

the frequency response test. The accelerometers are

moved to measure the frequency response corre-

sponding to modal model geometry points. The

modal model of the CTBA is generated from its

geometric dimensions, which consist of 30 geometry

points. The geometry points are located symmetri-

cally about the centre-line. The driving points or

excitation points are selected on the basis of the

mode shapes that need to be excited such as the first

and second bending and torsion modes. The drivingpoints are selected to avoid nodes of the mode

shapes of interest. Figure 4 shows the design flow of

the model correlation.

Multi-criteria optimization considering thicknesses

of four domains based on the element distributional

domains shown in Fig. 5, the thickness of trailing arm,

and the sector shape dimensions was performed for

the model update. The structural damping ratio was

set at 3 per cent for the frequency range of interest.

Table 1 shows the material and physical properties of

the CTBA shown in Fig. 6, which is the FEA model of

the CTBA assembly. The optimization problem is tofind the thicknesses and sector shape dimensions that

minimize the relative errors between the test results

(ft) and the simulation results (fa) under a given

frequency range, which includes the dynamic beha-

viour under the frequencies of interest according to

Minimize Yp ~X

fti{wifa

i p for X4

i~1

wi~1

subject to 0:95 ti 0ti1:05 ti 022

The geometrical constraints of the open sector shape

dimension of torsion beam are

GLjconfiguration vector Gf gGuj, j~1, . . . , n

where {G} is the move vector of the bead shape.

The CTBA is modelled using shell and solid elements

with appropriate thickness values along the beam

length. The body mounting points are modelled usingrigid body elements and spring elements. The results of

Fig. 4 Design flow for the model correlation (DOE,design of experiments; DSA, design sensitivityanalysis; CG, centre of gravity; FRF, frequencyresponse function)




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point mobility are shown in Fig. 7(a), which explains

the mobility trends on the selections of design

parameters in the validation phase (ALT 3, with

consideration of three thinning effect zones; ALT 2,

with consideration of two thinning effects; ALT 1, with

consideration of one thinning effect). Figure 7(b)

shows the results between the selection of thinning

domain and thinness of torsion beam through the

model-updating scheme. The dynamic test set-up and

durability simulations are shown in Fig. 8. Table 2

show the deterministic optimum and robust optimumon the penalty factor, k51.1.

The errors can be generated by the residual forces

of the constraint conditions. These values of three

measured points over all three frequencies were the

functions chosen to be minimized by the optimizer.

Some of many test variations were as follows.

1. The stiffness property difference in the test run

was reduced from 20 per cent to 5 per cent.

2. A constant damping factor was used for all

models, instead of varying the damper elements.

3. Different constraints combinations of using realor imaginary components of the response errors

as well as requested magnitudes and phase angles

were tried.

4. Objective functions to be minimized included

combinations and summations.

If the solution was linear with respect to the design

parameters, unique solutions could be expected. The

number of design parameters is the number of

unknown parameters. The potential number of

known coefficients is equal to the number of test

degrees of freedom multiplied by the number of

sample frequencies multiplied by 2. It is recom-

mended that an over-determined system is used to

allow for possible redundant data. Of course, a large

number of frequencies in a small range with only a

few active modes will not provide much indepen-dent information. However, the use of many sample

test points on the structure and the use of multiple

Fig. 5 Thinning domains of forming process

Table 1 Material properties and thicknesses

Part Material name Thickness (mm)

Torsion beam SAPH440 2.8Trailing arm SAPH440 2.7Spring bracker SAPH440 3.0Damper bracker SAPH440 3.0

Spindle plate SAPH440 10.0 Fig. 6 CTBA model for simulation




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load cases will help. If the design changes are large,

the non-linearities may produce several different

reasonable solutions.

5 CONCLUSION

This procedure is an efficient and accurate tool forreliable prediction of structural performance including

static and dynamic specifications. The advantage of

the presented procedure increases such accuracies of

durability designs as the damage locations and fatigue

life. The correlation method presented in this paper

offers an efficient approach to robust design by

defining the robustness of the objective and the

constraint functions. The optimum evaluated from

robust design is the optimum intensive to the

Fig. 7 Results of point mobility at point 4 through the model-updating scheme




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variations in design variables satisfying the imposed

constraints. The design process is practical for struc-

tural designs. The robust optimum is evaluated

through the sequential stage optimization which

follows the deterministic optimization. Robust design

has to be performed considering the tolerances of the

design parameters and the robustness of objective

function is enhanced by decreasing the weighting

factor, while the robustness of the constraint function

is enhanced by increasing the penalty factor. The

factors should be selected depending on the char-

acteristics of the problem.

Fig. 8 (a) CTBA test set-up and (b) results of durability simulation




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ACKNOWLEDGEMENTS

This work is financially supported by the Ministry ofEducation and Human Resources Development, theMinistry of Commerce, Industry and Energy, and theMinistry of Labor of the Republic of Korea through

the fostering project of the Laboratory of Excellency.

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APPENDIX

Notation

[C] damping matrix

[K] stiffness matrix

[M] mass matrix

p design parameter

[Q] structural sensitivity matrix

[T] shape basis vector[Z] dynamic stiffness matrix

{DP} design parameter variation

{DR} response error vectors

V probabilistic manifold set

Table 2 Deterministic optimum and robust optimum (k5 1.1)

Design parameter (units) Initial value Lower bound Upper boundDeterministicoptimum Robust optimum

Torsion beam (mm)

Z1 2.8 2.66 2.94 2.8 2.75Z2 2.8 2.66 2.94 2.8 2.74

Z3 2.8 2.66 2.94 2.75 2.70Z4 2.8 2.66 2.94 2.63 2.65

Trailing arm (mm) 2.7 2.56 2.83 2.8 2.75Shape dimension (mm),

shaped dimensionin 0 5 3.0 3.17out 0 5 3.0 3.17

Mass (kg) 25 22.76 23.01



a frequency response function-based updating

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