Research Article Experimental Studies on Finite...

Research ArticleExperimental Studies on Finite Element ModelUpdating for a Heated Beam-Like Structure

Kaipeng Sun12 Yonghui Zhao1 and Haiyan Hu13

1State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and AstronauticsNanjing 210016 China2Shanghai Institute of Satellite Engineering Shanghai 200240 China3MOE Key Laboratory of Dynamics and Control of Flight Vehicle School of Aerospace Engineering Beijing Institute of TechnologyBeijing 100081 China

Correspondence should be addressed to Haiyan Hu hhyaenuaaeducn

Received 22 September 2014 Revised 6 December 2014 Accepted 11 December 2014

Academic Editor Tony Murmu

Copyright copy 2015 Kaipeng Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An experimental study was made for the identification procedure of time-varying modal parameters and the finite element modelupdating technique of a beam-like thermal structure in both steady and unsteady high temperature environments An improvedtime-varying autoregressive method was proposed first to extract the instantaneous natural frequencies of the structure in theunsteady high temperature environment Based on the identified modal parameters then a finite element model for the structurewas updated by using Kriging meta-model and optimization-based finite-element model updating method The temperature-dependent parameters to be updated were expressed as low-order polynomials of temperature increase and the finite elementmodel updating problem was solved by updating several coefficients of the polynomials The experimental results demonstratedthe effectiveness of the time-varyingmodal parameter identificationmethod and showed that the instantaneous natural frequenciesof the updated model well tracked the trends of the measured values with high accuracy

1 Introduction

Hypersonic vehicles are subject to very tough aerodynamicload and heating during their missions in the Earthrsquos atmo-sphere The aerodynamic heating is extremely importantbecause high temperature can affect the structural behaviorin several detrimental ways [1] The elevated temperature notonly degrades the ability of structure materials to withstandloads but also produces the thermal stresses in heatedstructures In addition the thermal stresses increase defor-mation change buckling loads and alter flutter behavior Forexample the studies [2ndash5] reported that even a minor changeof temperature would result in a significant alteration innatural frequencies of a beam because of the thermal stresseswhen the beam was constrained Thermal modal testingtechniques can provide quantitative analysis for the effect ofthe thermal load [6] Since the 1950s many experiments havebeen done in NASA Langley and Dryden research centers

for the metal and the composite panels [7 8] the X-15 wing[9] and the X-34 FASTRAC Composite Rocket Nozzle [10]The aforementioned studies have primarily focused on thedynamical properties of thermal structures in steady hightemperature environments (SHTEs)

As a matter of fact thermal structures in unsteady hightemperature environments (UHTEs) for example hyper-sonic vehicles subjected to unsteady aerodynamic heatinghave the characteristic of time-varying multiphysics fieldsThe identification of time-varying modal parameters is theforefront of inverse problems in structural dynamics Therehave been numerous theoretical and experimental studieson the identification of time-varying modal parameters forengineering structures The special issue [11] of MechanicalSystems and Signal Processing on the identification of timevarying structures and systems in 2014 offered a survey ofthe field in its current state reflected recent developmentsand also pointed out into the future in various aspects of

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 143254 15 pageshttpdxdoiorg1011552015143254

2 Shock and Vibration

the theories and applications Although recent years havewitnessed successful identifications of time-varying modalparameters of many engineering systems such as vehicle-bridge systems [12 13] machine condition monitoring sys-tems [14] flexible manipulators [15] and civil structures [1617] their applications to thermal structures in temperature-varying environment are still not available Yu et al [18]proposed an undetermined blind source separation methodto investigate the thermal effect on the modal parametersof a TC4 titanium-alloy column in a temperature-varyingenvironmentThey [19] also developed a time-varying modalparameter identification algorithm based on finite-data-window PAST and used it to investigate the effect of varyingtemperature and heating speed on the natural frequencies of atrapezoidal TA15 titanium-alloy plate To the best knowledgeof authors considerably less attention has been paid tothe finite element model updating (FEMU) for thermalstructures especially based on the identified time-varyingmodal parameters in UHTEs

Finite element (FE) modelling has received widespreadacceptance andwitnessed applications in various engineeringdisciplines Hence the FE model updating (FEMU) hasbecome a useful tool to improve the modelling assumptionsand parameters until a correlation between the analyti-cal predictions and experimental results satisfies practicalrequirements Mottershead and Friswell [20] comprehen-sively reviewed the model updating methods of structuralmodels In Marwalarsquos work [21] numerous computationalintelligence techniques were introduced and applied to FEmodel updating with in-depth comparisons As a mostwidely used method in FEMU the iterative method hasbeen formulated as an optimization problem and often basedon sensitivity analysis [22] and computational intelligencetechniques [23ndash25] Generally the optimization procedure isnonlinear and complex for a complicated FEMU problemIn recent years the particle swarm optimization (PSO)technique has been developed to implement the optimizationprocedure

The structural FE models with many geometric andphysical parameters to be updated may involve a largenumber of computations and need to be constructed byone of commercial finite element analysis packages suchas COMSOL ANSYS and NASTRAN Hence higher timeconsumption may be the disadvantage for the optimization-based algorithms due to their iterative strategy and repeatedanalysis in simulation models during the optimization pro-cess One way to overcome the difficulty of time consumptionand FE package-related problems during the optimization-based model updating is to replace the FE model by anapproximate surrogatereplacement meta-model that is fast-running and has fewer parameters involved Simpson [26]made a comparison of response surface and Kriging modelswhich are the two commonly used meta-models and drew aconclusion that both approximations predict reasonably wellwith the Kriging models having a slight overall advantagebecause of the lower root mean squared error values

The meta-model method for damage detection and relia-bility analysis has a long history However the Kriging meta-model (KMM) method for structural FEMU is somewhat

new especially for thermal structures in temperature-varyingenvironment As a continuation of authorsrsquo work [27 28]this paper presents KMM-PSO based FE model updatingand selection based on the experimentally identified time-invariant and time-varying modal parameters of a thermalstructure

The remainder is organized as follows In Section 2 theFE model updating and selection for a thermal structureis formulated and two objective functions are given forthe PSO An overview of PSO and a brief introductionto the time-varying autoregressive method for output onlyidentification are also presented The Kriging meta-modelis introduced to employ the PSO based FE model updatingand selection In Section 3 four groups of experiments arediscussed for the modal parameters of a beam-like structurein room temperature environment SHTE and UHTE InSection 4 the Kriging meta-model based FE model updatingand selection procedure is carried out to update the software-based FE model Finally some conclusions are drawn inSection 5

2 Formulation of FEMU forThermal Structures

21 Problem Description A linear thermal structure subjectto unsteady heating can be described in terms of the dis-tributed mass damping and stiffness matrices of the struc-ture in time domain via the following differential equation

Mx (119905) + C (119905) x (119905) + K (119905) x (119905) = f (119905) (1)

The real eigen-value problem for the 119895th mode reads

(K (119905) minus 120596

2

119895M)120593119895= 0 (2)

where the detailed meanings of the physical quantities in theabove two equations can be found in [28]

The traditional FE model updating process is achievedby identifying the correct mass and stiffness matrices whichare generally time-invariant This study however has to dealwith a time-variant FE model updating process because thesystem matrices change over time when the temperature-dependent material and temperature-dependent boundaryconditions are taken into consideration Afterwards usingthemeasured data the correctmass and stiffnessmatrices canbe obtained by identifying the correct material parameters ofthe structure and the appropriate boundary conditions underdifferent temperature conditions

For simplicity the thermal structure of concern is a kindof slender or thin-walled structures such as beams and platessubject to uniform heating with a uniformly distributedtemperature field119879(119909 119910 119911 119905) equiv 119879(119905) In other words the heattransfer process is neglected in the study Let 120579(119905) denote thetemperature increase in the thermal structure as follows

120579 (119905) = 119879 (119905) minus 119879ref (3)

where 119879ref is the reference temperature For the FEMU prob-lem of the thermal structure subject to unsteady heating the

Shock and Vibration 3

system parameters to be identified depend on the transienttemperature and can be expressed as

Γ119901= flowastℓ(120579) = P

0+ P1120579 + P2120579

2+ P3120579

3+ sdot sdot sdot (4)

where Γ119901is the parameter vector including the temperature-

dependent material and boundary parameters to be cor-rected and flowast

ℓis the general function of the temperature

increase 120579 In general flowastℓcan be expressed as a low-order poly-

nomial for example a linear quadratic or cubic functionin which ℓ represents the highest order of the polynomialThen the target parameters to be corrected change fromtemperature-dependent Γ

119901to a constant parameter vectorP

119896

119896 = 0 1 2 3

22 Objective Function for FEMU at Reference TemperatureIn a reference temperature environment (RTE) to correctlyidentify the moduli of elasticity that gives the updated FEmodel the following objective function which measures thedistance between the measured modal data and the modaldata predicted by FE model should be minimized

119869

0=

119873

sum

119895=1

120574

119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119891

meta119895

minus 119891

exp119895

119891

exp119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

(5)

where 1198690is the error function or objective function sdot is the

Euclidean norm 120574119895is the weighted factor for the 119895th mode

and 119873 is the number of measured modes respectively In(5) 119891meta119895

is the 119895th instantaneous natural frequency obtainedfrom the meta-model and 119891

exp119895

is the 119895th instantaneousnatural frequency identified from the measured responses ina thermal-structural experiment based on the identificationmethod for the time-invariant modal parameters

Thus the process of FEMUmay be viewed as an optimiza-tion problem as follows

find P0

min 119869

0

st P1198970le P0le P1199060

(6)

where 119897 and 119906 represent the lower and upper bounds of theparameter coefficient vector P

0 respectively

23 Objective Function for FEMU in a UHTE After anupdated model in the RTE is achieved the FEMU canbe performed in a UHTE which contextually means thatthe parameters to be corrected in this subsection becomethe constant parameter vector P

119896(119896 = 0 1 2 3) in (4)

It should be noted that P0here represents several special

parameters that cannot be identified in an RTEThe constantcoefficient of thermal expansion for example has no effecton the stiffness matrix of the FE model and thus cannot beidentified in an RTE but must be updated in a UHTE

Furthermore a new problem arises when the tempera-ture-dependent parameter is approximated as a low-orderpolynomial Because it is impossible to knowapriori the exact

order of polynomial flowastℓin (4) a parameter vector expressed

by polynomials of different orders should be considered Ingeneral it is known that the higher the order is the smallerthe deviations between test and analysis are However thepurpose of FEMU is to predict the structural response topredict the effects of structural modifications or to serve asa substructure model to be assembled as part of a model ofthe overall structure From this viewpoint for simplicity onemay use a low order expansion for the material properties Ifthe order is fixed to the maximal value one would manuallyaccept or reject the high-order terms by identifying whetherthe coefficient is close enough to zero If any terms areneglected the deviations between test and analysis shouldbe reexamined Therefore the potential problems in theFEMU procedure are not only how to determine thoseparameters but also how to select the updated model Toavoid establishing many models for FEMU and manuallyselecting the updated model it is preferable to have anall-in-one procedure for both updating and selection ThePSO framework allows this simultaneous updating of allcompetingmodels and selection of the bestmodelHence thetwo aforementioned problems can be solved by minimisingan integrated objective (or fitness) function

A number of fitness or objective functions have beenavailable so far Most previous studies have sought a modelwith the fewest updating parameters needed to produce FEmodel results that are closest to measured results In thisstudy the Akaike information criterion (AIC) was used torepresent the integrated objective functionwith an additionalterm to treat ill-conditioned and noisy systems AIC can bedescribed by the following equation

119869

1=

1

119873sp

119873

sum

119895=1

119873sp

sum

119896=1

120574

119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119891

meta119895

(119905

119896) minus 119891

exp119895

(119905

119896)

119891

exp119895

(119905

119896)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

(7)

119869

2= 119873 log (119869

1+ 120582

2 10038171003817

1003817

1003817

L (P minus Plowast)1003817100381710038171003817

2

) + 120581119889 (8)

with

119889 = ℓ

1+ ℓ

2+ sdot sdot sdot + ℓ

119873119901 (9)

where119873sp is the number of samplings 119891meta119895

(119905

119896) and 119891exp

119895(119905

119896)

are frequencies at time 119905119896 and Plowast is the initial estimate of

parameter P respectively The second term in the bracketrepresents regularization where the parameter weightingmatrix L should be chosen to reflect the uncertainty in theparameter estimation and L = I represents the classicalTikhonov regularization Link [29] suggested that the factor120582

2 lies in the range of [0 03] High 120582 values are used if thereare many insensitive parameters and if the inverse problemis strongly ill-conditioned The second term of the aboveequation is known as the model complexity penalty termin which 120581 is a weighting factor and ℓ

119896(119896 = 1 2 119873

119901)

represents the highest order of the 119896th polynomial119873119901is the

dimension of the parameter vector


Start

Input training dataparameters P

Software-based modalanalyses

Output training resultsnature frequencies

Build Kriging meta-model

Output Kriging predictor

Yes

Initialize particle swarm

Stoppingcriteria

Yes

End

MSEacceptable

No

Kriging meta-model

PSO based FEMU

Nature frequencies fmeta

Set PSO parameters Pk(k = 0 1 2)

Calculate objective function in FEMUapproach for each particle swarm

Find the minimum of for all competing models

Evaluate value of

Output optimizedPk and its order

No

Update parameters(pbestk gbest velocity and position)

J2

J2

J2

Figure 1 Flow chart of Kriging meta-model (left dashed block) and particle swarm optimization based finite element model updating (rightdashed block)

Similarly to the FEMU in an RTE the optimizationprocess here may be expressed as follows

find P119896 119896 = 0 1 2 3

min 119869

2

st P119897119896le P119896le P119906119896

(10)

Figure 1 illustrates the detailed flow chart of the Krigingmeta-model and the particle swarm optimization based FEmodel updating and selection In the left dashed block aKriging meta-model is introduced to overcome the difficultyof time consumption because of the quantities of iterationsduring the updating process In the right dashed blockmodalparameters analyzed from Kriging meta-model and experi-mentally identified by time-varying autoregressive (TVAR)method are used to establish the objective function for thePSO based FE model updating and selection Therefore thenext 3 subsections briefly introduce the stochastic optimiza-tion technique the Kriging meta-modeling and the time-varying autoregressive method and their relevance to the FEmodel updating and selection for thermal structures

24 Stochastic Optimization Technique In this study PSOtechnique is employed to deal with the optimization prob-lems described in (6) and (10) PSO is a population-basedstochastic optimization technique developed by Eberhart andKennedy [30] in 1995 and inspired by the social behavior ofbird flocks and fish schools The PSO procedure begins witha group of random particles and then searches for optima byupdating generationsThe PSO shares many similarities withevolutionary computation techniques such as genetic algo-rithms (GAs) [24] However it has no evolution operatorssuch as crossover or mutation In the PSO particles updatethemselves with an internal velocity and also have memoryand one-way information sharing mechanism

Similarly to GA the algorithm begins with generating agroup of random particles called a swarm At each iterationthe particles evaluate their fitness (positions relative to thegoal) and share memories of their best positions with theswarm Subsequently each particle updates its velocity andposition according to its best previous position (denoted bypbest) and that of the global best particle (denoted by gbest)which has far been found in the swarm Let the position of aparticle be denoted by 119909

119896isin R119899 and let V

119896be its velocityThey

both are initially and randomly chosen and then iteratively


updated according to two formulae The following formulais used to update the particlersquos velocity and position asdetermined by Shi and Eberhart [31]

V119896= 119908 sdot V

119896+ 119888

1sdot 119903

1sdot (119901119887119890119904119905

119896minus 119909

119896)

+ 119888

2sdot 119903

2sdot (119892119887119890119904119905 minus 119909

119896)

119909

119896lArr997904 119909

119896+ V119896

(11)

where 119908 is an inertia coefficient that balances the global andlocal search 119903

1and 1199032are random numbers in the range [0 1]

updated at each generation to prevent convergence on localoptima and 119888

1and 119888

2are the learning factors that control

the influence of 119901119887119890119904119905119896and 119892119887119890119904119905 during the search process

Typically 1198881and 1198882are set to be 2 for the sake of convergence

[31]To avoid any physically unrealizable system matrix and

the thermal buckling which easily occurs at extremely hightemperatures artificial position boundaries should be set foreach particle There are four types of boundaries namelyabsorbing reflecting invisible and damping boundaries assummarized in [32] The damping boundary can providea much robust and consistent optimization performance ascompared with other boundary conditions and thus wasused in this study In addition to enforcing search-spaceboundaries after updating a particlersquos position it is alsocustomary to impose limitations on the distance where aparticle can move in a single step [33] which is done bylimiting the velocity to a maximum value with the purposeof controlling the global exploration ability of the particleswarm and preventing the velocity from moving towardsinfinity

In the implementation of FE model updating and selec-tion the parameters P

119896(119896 = 0 1 2 3) are usually set to be

positions of particles for stochastic optimization techniqueThe above updating process should be repeated until aspecified convergence value or total generation number isreachedThis way an optimal process for FE model updatingand selection can be achieved

25 Kriging Meta-Modeling For completeness a briefdescription of the Kriging meta-modeling is given in thissubsection Kriging was named after the pioneering workof D G Krige a South African Mining Engineer and wasformally developed by Matheron [34] Universal Krigingestimates the response 119884(x

lowast) at an untried site x

lowastas the

sum of a polynomial trend model f(xlowast)120573 and a systematic

departure term 119911(xlowast) representing low (large scale) and high

frequency (small scale) variations around the trend model[35]

119884 (xlowast) = f (x

lowast)120573 + 119911 (x

lowast) (12)

where f(xlowast) = [1198911

(xlowast) 119891

2(xlowast) sdot sdot sdot 119891

119872(xlowast)] and120573 = [120573

1120573

2

sdot sdot sdot 120573

119872]

T are the regression model and the regressioncoefficients respectively In the right hand of (12) the firstterm is the mean value which can be thought as a globallyvalid trend function And the second term 119911(x

lowast) is a Gaussian

distributed error term with zero mean and variance 1205902 Thecovariance matrix of 119911(x) is given by

V (119911 (x)) = 120590

2R119899119904times119899119904

(13)

In (13) each element119877119894119895ofR119899119904times119899119904

defined as119877(x119894 x119895) is the

spatial correlation function between any two of the 119899119904sample

points x119894and x119895

In the current simulation the term 119903 represents thedimension of vector x and the linear regression model ischosen for the mean part of the Kriging function [36]Accordingly 119891

119895(x) can be expressed as

119891

1(x) = 1 119891

2(x) = 119909

1

119891

3(x) = 119909

2 119891

1+119903(x) = 119909

119903

(14)

The Gaussian correlation function 119877 is taken as

119877 (x119894 x119895) =

119903

prod

119896=1

exp (minus119901119896

1003816

1003816

1003816

1003816

1003816

x(119896)119894

minus x(119896)119895

1003816

1003816

1003816

1003816

1003816

2

) (15)

where x(119896)119894

is the 119896th component of the 119894th sample point and119901

119896is the unknown correlation parameter which needs to be

fitted by optimizationTo construct the Kriging model the values of the

regression coefficients 120573 = [1205731120573

2sdot sdot sdot 120573

119872]

T must beapproximated by using the generalized least squares theoryat first Then the fitting correlation parameters 119901

119896can be

quantified by using the maximum likelihood estimationThegeneralized least-squares estimates of 120573 and 1205902 representedby 120573 and 2 respectively are given in detail as

120573 = (FTRminus1F)minus1

FTRminus1y

2=

(y minus F120573)TRminus1 (y minus F120573)119899

119904

(16)

where y and F are the 119899119904times 1 vector of output and the 119899

119904times 119899

119904

matrix at the sample inputs respectively which are expressedas

y =[

[

[

[

[

119910

1

119910

2

119910

119899119904

]

]

]

]

]

F =

[

[

[

[

[

119891

1(x1) 119891

2(x1) sdot sdot sdot 119891

119903+1(x1)

119891

1(x2) 119891


119903+1(x2)

d

119891

1(x119899119904) 119891

2(x119899119904) sdot sdot sdot 119891

119903+1(x119899119904)

]

]

]

]

]

(17)

With the vector r = [119877(xlowast x1) 119877(x

lowast x2) sdot sdot sdot 119877(x

lowast

x119899119904)]

T the prediction at the unsampled location xlowastcan be

obtained as

119884 (xlowast) = f (x

lowast)

120573 + rTRminus1 (y minus F120573) (18)


For the FEMU problem of thermal structures parame-ters Γ

119901defined in (4) including the temperature-dependent

material and boundary parameters to be corrected are takenas input parameters of Kriging meta-model and the outputparameters are usually modal parameters such as naturalfrequencies or modal shapes

Before the Kriging predictor is used in structural FEmodel updating it should be verified to check whetherthe meta-model has enough accuracy Sacks et al statedthat the cross-validation and integrate mean square errorcan be utilized to assess the accuracy of a Kriging modelThe pointwise (local) estimate of actual error in Krigingapproximation was given by computing the mean squarederror (MSE) 120593(x) as follows [36]

120593 (x) =

2(1 + u (x)T (FTRminus1F) u (x) minus r (x)T Rminus1r (x))

u (x) = FTRminus1r (x) minus f (x) (19)

where 2 is the process variance defined in (16) and 1 is thevector of ones

26 Brief Description of the TVAR Method Identifying thetime-varying modal parameters is an important issue inthe FE model updating and selection for thermoelasticstructures Time-varying autoregressive method is one of themost popular time-frequency analysis methods for outputonly identification

This subsection deals with a TVAR process 119909(119905) (egdisplacement velocity or acceleration) of order 119901 in adiscrete-time as the following

119909 (119905) = minus

119901

sum

119894=1

119886

119897(119905) 119909 (119905 minus 119897) + 119890 (119905) (20)

where 119890(119905) is a stationary white noise process with zero meanand variance 1205902 and 119886

119897(119905) 119897 = 1 2 119901 are the TVAR

coefficientsUsing the basis function expansion and regression

approach the TVARprocess 119909(119905) of order119901 in a discrete-timecan be expressed in matrix form as

119909 (119905) = minusXT119905A + 119890 (119905)

(21)

where

AT= [119886

10 119886

1119898 119886

1199010 119886

119901119898]

XT119905= [119909 (119905 minus 1) 119892

0(119905) 119909 (119905 minus 1) 119892

119898(119905)

119909 (119905 minus 119901) 119892

0(119905) 119909 (119905 minus 119901) 119892

119898(119905)]

(22)

119890(119905) is a stationary white noise process with zero mean andvariance 1205902 119886

119897119896and 119898 are the weighted coefficients and the

dimension of the basis functions 119892119896(119905) 119896 = 0 1 119898

respectivelyThe recursive least square (RLS) estimation and exponen-

tial forgetting method with a constant forgetting factor are

used here such that the parameter estimation algorithm canbe written as

A119873+1

=

A119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

sdot (119909 (119873 + 1) + XT119873A119873)

P119873+1

=

1

120582

[P119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

XT119873P119873]

(23)

where the forgetting factor 120582 is chosen in the interval (0 1]and is typically close to one The initial value of A and P canbe selected as A

0= 0 P

0= 120583I where 120583 ≫ 1 and I is an

identity matrixOnce the TVAR coefficients are obtained the instanta-

neous natural frequencies can be derived from the conjugateroots 119904

119895(119905) 119904lowast119895(119905) of the time-varying transfer function corre-

sponding to the TVAR model as the following

119891

119895(119905) =

1

2120587Δ119905

radicln 119904119895(119905) sdot ln 119904lowast

119895(119905) (24)

where Δ119905 is the time-discretization step

3 Experimental Studies on a Cantilever Beam

In this study experimental modal analysis and operationalmodal analysis were carried out to obtain the time-invariantand time-varyingmodal parameters respectively in differenttemperature environments The experimental object here isa cantilever beam made of aluminum installed in a movablebox-type resistance furnace as shown as (a) and (b) inFigure 2 The beam was dynamically driven by a hammerimpact or a vibration shaker excitation in experiments Adouble-lug-type connector was used to connect the beamand the vibration shaker when the beam was heated by thefurnace where only the vibration shaker provided feasibleexcitations It should be emphasized that the mass of theconnector should not be neglected in modal analysis Forsimplicity hence the terms ldquobeam Ardquo and ldquobeam Brdquo areused hereinafter for the cantilever beam without the double-lug-type connector and with the double-lug-type connectorrespectively

Figure 2(c) shows the schematic framework of experi-mental setup where beam B is excited by a vibration shakerTable 1 lists four groups of experiments for different cases Forall groups the velocity responses of the beam were measuredby using a laser vibrometer as a noncontacted measurementtechnique

31 The First and Second Groups of Experiments In the firstand second groups of experiments the frequency responsesof beam A and beam B were measured via a hammer impactat room temperature respectively Without loss of generalitythe room temperature was assumed to be reference temper-ature Figure 3 illustrates the amplitude-frequency responses(AFRs) of themeasured frequency response functions (FRFs)of the two beams The first natural frequencies of beamA and beam B were 73125Hz and 7125Hz respectively


Resistance wire

Resistance wireBeam

Cantilever support

Temperature controlled tank

Thermocouple Double-lug-type connector

(a)

Resistance furnace

Temperaturecontrolled tank

Dynamic signal acquisitionand analysis system

Portable digitalvibrometer

Vibration exciter

(b)

Vibrationexciter


Beam section

Poweramplifier

CH1 CH0

Forcesensor

Velocity signals

PX I 6733


SmartOfficeK-type

thermocouplethermometer

Movable box-typeresistance furnace

Thermocouple

Laser

Double-lug-typeconnector

(c)

Figure 2 Experimental setup for thermal test of a beam (a) the aluminumbeam (b) the experimental setup and (c) the schematic framework

Beam ABeam B

Mag

nitu

de (d

B)

Frequency (Hz)

minus30

minus25

minus20

minus15

minus10

minus5

0

0 10 20 30 40 50

Figure 3 Amplitude-frequency responses of beam A and beam Bdriven by a hammer impact

while the second ones were 45875Hz and 32Hz respectivelyThe figure clearly shows that the added mass of the double-lug-type connector greatly reduced the second natural fre-quency of beam B but had a small influence on the firstnatural frequency of beam B due to the attachment position

0

0

10 20 30 40 50

0∘C200∘C300∘C

400∘C500∘C

minus30

minus25

minus20

minus15

minus10

minus5

Frequency (Hz)

Mag

nitu

de (d

B)

Figure 4 Amplitude-frequency responses of beam B driven by avibration shaker at different temperatures

32 The Third Group of Experiments In the third groupof experiments the beam B was subject to the heating offurnace Hence it was driven by a vibration exciter outsideof the furnace through a long steel rod connecting the beam


5005

1

2

3

4

5

6

10

15

20

25

30

100 150

Time (s)

Freq

uenc

y (H

z)

Frequency (Hz)

200 250 300

500 100 150

Time (s)200 250 300

times10minus4

Velo

city

(ms

)

minus3

0

3times10minus4

times10minus6

05 10 15 20 25 30

Pow

er sp

ectr

um (m

2s

2)

4

8

Figure 5 Complex Morlet transform scalogram of the velocity response

Table 1 Experiment descriptions

Number ofgroup Beam type Excitation type Temperature

environment1 Beam A Force-hammer RTE2 Beam B Force-hammer RTE3 Beam B Vibration shaker SHTE4 Beam B Vibration shaker UHTE

with a double-lug-type connector The heating of furnacewas controlled by the temperature controlled tank with twowindows displaying two temperatures that is the target tem-perature and the cavity temperature respectively Besides theactual temperature of the beam at different time instants wasmeasured by a K-type thermocouple thermometer as shownin Figure 2 The beam was heated in steady environmentsof high temperature at 200∘C 300∘C 400∘C and 500∘Crespectively For all the experiments in Sections 32 and 33the random excitation was provided by the shaker and thesampling frequency was set at 512Hz Figure 4 illustratesthe AFRs produced from velocity responses of beam B at

Table 2The first two natural frequencies (Hz) of beamB at differenttemperatures

Mode 0

∘C 200

∘C 300

∘C 400

∘C 500

∘C1 1025 9 8 75 7252 26 2275 225 2175 20

different temperatures and Table 2 lists the first two naturalfrequencies They demonstrate that the first two naturalfrequencies of the beam decreased with an increase of thetemperature

33The Fourth Group of Experiments In this group of exper-iments beam B was heated in an unsteady high temperatureenvironment The temperature was increased from the roomtemperature to about 500∘C At the same time of temperatureincrement the beam was subject to a random force fromthe vibration shaker and the velocity responses of the beamwere measured by using a laser vibrometer The measuredresponses were then used to extract the time-varying modal


0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)

Figure 6The first two instantaneous natural frequencies of beam Band transient temperature

parameters via the continuous wavelet transform (CWT)method and the TVAR method

Figure 5 shows the CWT scalogram of the velocityresponse using the Complex Morlet 33 as the wavelet basisThe top subfigure gives the signal waveform of the responsethe left bottom subfigure is the corresponding power spec-trum and the right bottom subfigure is the time-frequencyanalysis result with a color bar indicating themagnitude levelson the right Figure 6 illustrates the first two instantaneousnatural frequencies identified by using the TVAR algorithmand labeled in the left 119884-axis with respect to the measuredtemperature on the beam labeled in the right 119884-axis

With the comparison of Figures 5 and 6 both CWTmethod and TVAR method provided the good time-frequency representation of nonstationary dynamics butthe latter gave the result of much higher time-frequencyresolution In addition the TVAR method could provideparametric results which can be directly used in the nextFEMU procedure

4 Meta-Model Based FEMU

41 Numerical Simulation and Kriging-Based Meta-ModelingIn this study COMSOL a software of multiphysics was usedfor the FE based modal analysis of the beam under variousconditions of parameter combinations Figure 7 shows thedimension chart of the beam As illustrated in Figure 8 thegeometry model of beam B built in COMSOL contains twoparts that is the beam and the double-lug-type connector Inthe numerical simulations the connecting stiffness of the boltjoints was modeled by attaching an auxiliary surface betweenthe assembled parts defining the material properties of theauxiliary surface and connecting the assembled parts with themultipoint constraint (MPC) strategy

As well known the numerical simulation may give agood prediction for the natural frequencies of the beambut the results are not parameterized Furthermore the

135400

40

Φ65Unit (mm)

Thickness = 3

Figure 7 The dimension chart of the beam

simulation processmay be time-consumingThemain idea ofmeta-modeling is to construct a parameterizedmathematicalmodel between the input parameters and the output results bya number of numerical simulations and then use themodel topredict other output results In this study the Krigingmethodwas employed to construct the meta-model for buildingaccurate global approximation in a given design space

In this study several parameters such as the density120588

119887of the beam the density 120588

119888of the connector the added

mass 119898119886of the long steel rod the density 120588as and elastic

modulus 119864as of the auxiliary surface were assumed to betemperature-independent while the elastic modulus 119864

119887of

the beam material and the added stiffness 119870119886of the long

rod were taken as temperature-dependent parameters Asmentioned earlier the temperature-dependent parameterscan be expressed as low-order polynomials that is 119864

119887and119870

119903

yield the following polynomials of temperature increase 120579

119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870

1198860are the elastic modulus and the stiffness

of the long rod at the reference temperature 119879ref respectivelyand 119864

119887119896and 119870

119886119896(119896 = 1 2) are the coefficients independent

of temperature In the study 120588119887was taken as 225275 kgm3

according to measured mass and volume of the beam and120588

119888of the connector was taken as 79496 kgm3 in the same

way The value of 120588as was taken as a constant of 2000 kgm3since the thin auxiliary surface did not have any significantinfluence on the modal parameter The parameters to beupdated in the next 3 subsections are (1) 119864

1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)

42 First Step of FEMU For the first step of FEMU of beamB under a hammer impact at room temperature 119864

1198870and 119864as

were used as updating parameters of the KMM based FEMUTo simulate the initial model of beam B to be updated theinitial values of 119864

1198870and 119864as were taken as 65GPa and 95 times

104 Pa The modal analysis via FEM method was performedon the initial model to obtain the initial natural frequenciesThe initial values of the first two natural frequencies and thecorresponding differences are shown in Table 3

The updating parameters share the same region for thetraining data in the Kriging meta-modeling To constructthe Kriging meta-model valid over a range of parametersthe moduli of elasticity 119864

1198870and 119864as were restricted to vary

from 50 to 70GPa and 8 times 104 to 1 times 105 Pa respectively Thedesign of experiment (DOE) is a key problem in decidinghow to select the inputs at which the deterministic computer


Top view

Beam

Connector

Auxiliary surface

Three views of beam B

0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z

Figure 8 The geometry model of beam B

Table 3 Natural frequency differences of beam B under a hammer impact before and after model updating

Mode Measured (Hz) Initial (Hz) Error () Updated (Hz) Error ()1 7125 7480 498 7125 0002 32000 33556 486 32000 000

555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)

Figure 9 Predicted values of the first natural frequency

codes are run in order to most efficiently control or reducethe statistical uncertainty of the computed predictions Inthis study the rectangular grid method [36] was used to dealwith the DOE problemThis was easily done by using the fullmultiparameter sweep [37] and specifying all combinationstype in COMSOL A total of 25 experiments were carried outThe sampled parameter values and corresponding naturalfrequencies computed from FE models were used as thetraining data of the Kriging meta-model A 50 times 50 uniform

555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)

Figure 10 Predicted values of the second natural frequency

mesh grid in the region covered by the design sites wasgenerated to evaluate the predictor Figures 9 and 10 illustratethe mesh plots of the predicted values of the first andsecond natural frequencies at the grid points respectivelyThe horizontal axes are parameters selected while the verticalaxis gives the predicted response (natural frequency) at anypoint or location

To check the accuracy of the Kriging meta-model theMSEs were computed for each mode as shown in Figures 11


Table 4 Natural frequency differences of beam B under vibration shaker excitation before and after model updating


555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)

Figure 11 Mean squared errors of the first natural frequency

and 12 They demonstrate that all MSE values were close tozero so that the created meta-model had a high regressionaccuracy

Then the FEMU was performed with the FE modelreplaced by the Kriging meta-model The residuals betweenthe accurate (predicted by Kriging meta-model) and themeasured (by experiment) natural frequencies were used inthe optimized objective function expressed in (5) A single-objective optimization algorithm with equal weight for eachnatural frequency was implemented to achieve the best min-imization of natural frequency residuals The optimizationalgorithm used to minimize the objective function is animproved PSO method in MATLAB In implementing thePSO for the FE model updating the population was takenas 50 119888

1and 119888

2were set to be 2 and 119908 was set to be 1 The

tuning minimization process was over when the toleranceswere achieved or a predefined number of iterations werereached Table 3 shows the updated natural frequencies andtheir differences of beamB and illustrates good results of PSObased FE model updating The final updated results for theparameter were 119864

1198870= 5878GPa and 119864as = 9524 times 104 Pa

As shown in Table 3 the errors between the first two naturalfrequencies measured and those obtained from the initialFE model were about 5 on average When the KMM-PSObased FEmodel updating algorithmwas used the errors werereduced to very small values

43 Second Step of FEMU For the second step of FEMUof beam B under vibration shaker excitation at room tem-perature 119898

119886and 119870

1198860were used as updating parameters of

the KMM-PSO based FEMUThe numerical modal analysisthe correspondingmeta-modeling and the sequential FEMU

555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)

Figure 12 Mean squared errors of the second natural frequency

procedure are almost the same as the first step of FEMU inSection 42 and in consequence the detailed processes are notrepeated in this subsection The initial values of 119898

119886and 119870

1198860

were set as 02 kg and 2 times 103Nm and the correspondingupdated values were 014548 kg and 209812Nm respec-tively Table 4 shows the initial and updated values of the firsttwo natural frequencies and the corresponding differencesAs listed in the table the errors between the first two naturalfrequencies measured and those obtained from the initial FEmodel were about 58 on average When the KMM-PSObased FEmodel updating algorithmwas used the errors werereduced to tiny values

44 Third Step of FEMU For the third step of FEMU ofbeam B under vibration shaker excitation in a UHTE theparameters to be updated are temperature-dependent 119864

119887119896

and 119870

119886119896(119896 = 1 2) were used as updating parameters

of the KMM-PSO based FEMU The time-varying modalparameters identified in Section 33 were used to establish theobjective function in (8) In addition the linear functions oftemperature increase for the parameters were used as initialguesses and are shown in Figures 13 and 14 respectivelyFigure 6 also illustrates the first two instantaneous naturalfrequencies computed via the FE method with these initialparameters

Asmentioned earlier the temperature-dependent param-eters to be identified can be expressed as lower-order poly-nomials of the temperature increase but the exact orderis unknown beforehand Hence the best order and thecoefficients of these polynomials should be simultaneouslyidentified Without loss of generality a simple case wasconsidered to verify the proposed method For this purpose


025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)

Figure 13 Temperature-dependent elastic modulus of the beammaterial

05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Figure 14 Temperature-dependent stiffness of the long rod

119864

119887and 119870

119886were assumed to be linear or quadratic functions

of the temperature increase The process of ensuring thepositivity of Youngrsquos modulus for instance can be recalled asfollows By appropriately selecting the maximal and minimalvalues of119864Δ119864

119896≜ 120574(119864maxminus119864min)119879

119896

max 119896 = 1 2 was definedwhere 119896 represents the order and 120574 is a weighting coefficientIn practice 120574 asymp 1 was taken for a low order and 120574 asymp 01 wasset for a high order to prevent strong nonlinearity of materialproperty Afterwards the bounds of 119864 with an increase oftemperature were checked by using theMonte Carlo method

Different from the typical FEMU procedure the methodproposed here includes FE model selection For instancethe beams were modelled by four competing models 119872

119896

119896 = 1 4 as listed in Table 5 Here each particle had4 dimensions such that all the competing models shouldbe searched in a 4-dimensional space and then all of the

Table 5 Model parameterization

Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Figure 15 Global fitness and global best model number duringupdating

particles were built to compute the fitness function 1198692in (8)

so as to find the smallest value of fitness function Althoughall the models should be searched in the same space eachmodel was actually constrained to a particular subspace of thespace Afterwards the most important issue for the FEMU intime domain is the implementation of the PSO algorithm Itwas assumed that the values of the polynomial coefficients forthe thermal-structural properties were restricted to differentintervals and varied in their corresponding intervals Table 6presents the initial values of the parameters and the lower andupper bounds of their intervals The factor 1205822 and weightingmatrix L in the objective function 119869

2were set as 1205822 = 5times10

minus4120581 = 8 times 10

minus4 and L = diag([1 3]) respectively After substi-tuting the objective function 119869

1with equal weight for each

natural frequency into 119869

2 the multiobjective optimization

algorithmwas implemented to achieve the best minimizationof 1198692Figure 13 shows the identified values of Youngrsquos modulus

119864

119887of the beam compared with the initial values Figure 14

shows the identified values of the added stiffness 119870119886of the

long rod compared with the initial valuesFigure 15 shows the convergence of the objective function

119869

2over the 50 iterations of the algorithm and illustrates

that the PSO algorithm rapidly converged to the ultimateminimum error within the first 22 iterations Figure 15 alsoillustrates the convergence behavior of the global best modelover 50 iterations It is quite clear from Figure 15 that theobjective function played a significant role in updating themodel parameters The global best model in the simulation


Table 6 Particle swarm optimization parameters

PSO parameter Initial Lower bound Upper bound Identified119864

1198871(GPasdot∘Cminus1) minus0054 minus02 0 minus42463119890 minus 2

119864

1198872(GPasdot∘Cminus2) 0 minus9119890 minus 5 9119890 minus 5 minus20803119890 minus 5

119870

1198861(Nsdotmminus1 sdot ∘Cminus1) minus0002 minus5119890 minus 2 0 minus21041119890 minus 3

119870

1198862(Nsdotmminus1 sdot ∘Cminus2) 0 minus9119890 minus 5 9119890 minus 5 minus89016119890 minus 5

Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)

Figure 16 Mean absolute percentage errors of the first two naturalfrequencies

began with1198721 then changed to119872

3and119872

1 afterwards went

to1198723 and subsequently remained unchanged for the rest of

the simulation The result indicates that 1198723was the global

best model That is 119864119887and 119870

119886were a quadratic polynomial

and a linear polynomial of the temperature increase respec-tively to balance the model errors and complexity

To have a quantitative discussion the mean absolutepercentage error (MAPE) is defined as

MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)

where 119873sp is the total number of samplings and 119910

119896and

119910

119896denote the true values computed by the direct modal

analysis and the identified value at the 119896th time instantrespectively Figure 16 illustrates the MAPEs of the first twonatural frequencies The average error between the initialvalues of the first two natural frequencies and the true valueswas 18 When the KMM-PSO based FE model updatingand selection method were used the error was reduced to01 on average

Overall the KMM-PSO based FEMU approach proposedin this study updates the model of a thermal structure inboth RTE and UHTE well On the other hand the proposedmethod can greatly reduce the computation time comparedto other FEMU algorithms based on direct FE modeling andmodal analysis by using commercial FE analysis packagesTable 7 gives the comparison of computational time of the

Table 7 Comparison of computational time

Method Number ofMA

Time ofMA Total time

The proposed method 10646400 3785 s 3925 sFEMU based on direct MAin COMSOL 10646400 616 d 616 d

MA modal analysis s seconds d days

proposed method with the FEMU algorithm based on directmodal analyses in COMSOL It should be pointed out thatthe FEMU algorithm based on direct modal analysis inCOMSOLwas not actually carried out and the correspondingcomputation time was predicted according to the callingnumbers of Kriging predictor function and the actual time(about 5 s) of one run of FE modal analysis in COMSOL

5 Conclusions

The paper presents the experimental study for the identifica-tion of time-varying modal parameters and the applicationof the Kriging meta-model for the finite element modelupdating and selection of a beam-like thermal structure inboth steady and unsteady high temperature environmentsThe time-invariant natural frequencies were identified fromthe vibration test in room temperature and were used toestablish the objective function for the FEMU in an RTEAs the thermal structure in unsteady high temperatureenvironments exhibits the characteristics of time-varyingmultiphysics fields a KMM-PSO based FE model updatingand selection was proposed based on the experimentallyidentified time-varying modal parameters of the thermalstructure in a UHTE The presented TVAR method wellextracted the instantaneous natural frequencies of the ther-mal structure in temperature-varying environment from theoutput responses of the structure onlyThe KMM-PSO basedFEMU approach proposed in this study well updated themodel of the thermal structure in both steady and unsteadyhigh temperature environments The integrated method wastime-saving and feasible to industry The study presentsa preliminary investigation into the use of Kriging as astatistical-based approximation technique for modeling thecomplicated thermal structure with local joint and featuresof time-varying multiphysics fields

Abbreviations

AFR Amplitude-frequency responseAIC Akaike information criterionCWT Continuous wavelet transform


FE Finite elementFEMU Finite element model updatingFRF Frequency response functionsGA Genetic algorithmsKMM Kriging meta-modelMAPE Mean absolute percentage errorMPC Multipoint constraintMSE Mean squared errorPSO Particle swarm optimizationRLS Recursive least squareRTE Reference temperature environmentSHTE Steady high temperature environmentTVAR Time-varying autoregressiveUHTE Unsteady high temperature environment

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 11472128 11302098 and11290151 the Funding of Jiangsu Innovation Program forGraduate Education under Grant CXLX13 130 and theResearch Fund of State Key Laboratory of Mechanics andControl of Mechanical Structures (NUAA) under Grants0114G01 and 0113Y01

References

[1] E A Thornton Thermal Structures for Aerospace ApplicationsAIAA Reston Va USA 1996

[2] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007

[3] J D Hios and S D Fassois ldquoIdentification of a global modeldescribing the temperature effects on the dynamics of a smartcomposite beamrdquo in Proceedings of the International Conferenceon Noise and Vibration Engineering (ISMA rsquo06) pp 3279ndash3293Leuven Belgium September 2006

[4] J D Hios and S D Fassois ldquoStochastic identification oftemperature effects on the dynamics of a smart composite beamassessment of multi-model and global model approachesrdquoSmart Materials and Structures vol 18 no 3 Article ID 0350112009

[5] S KazemiradMH Ghayesh andMAmabili ldquoThermal effectson nonlinear vibrations of an axially moving beam with anintermediate spring-mass supportrdquo Shock andVibration vol 20no 3 pp 385ndash399 2013

[6] H Cheng H B Li R H Jin Z Q Wu and W Zhang ldquoThereview of the high temperature modal test for the hypersonicvehiclerdquo Structure amp Environment Engineering vol 39 no 3 pp52ndash59 2012

[7] L F Vosteen and K E Fuller ldquoBehavior of a cantilever plateunder rapid heating conditionsrdquo NACA RM L55E20 1955

[8] M W Kehoe and H T Snyder ldquoThermoelastic vibration testtechniquesrdquo NASA Technical Memorandum 101742 1991

[9] R R McWithey and L F Vosteen ldquoEffects of transient heatingon the vibration frequencies of a prototype of the X-15 wingrdquoNASA Technical Note D-362 1960

[10] A M Brown ldquoTemperature-dependent modal testanalysiscorrelation of X-34 FASTRACcomposite rocket nozzlerdquo Journalof Propulsion and Power vol 18 no 2 pp 284ndash288 2002

[11] L Garibaldi and S Fassois ldquoMSSP special issue on the iden-tification of time varying structures and systemsrdquo MechanicalSystems and Signal Processing vol 47 no 1-2 pp 1ndash2 2014

[12] S-D Zhou W Heylen P Sas and L Liu ldquoParametric modalidentification of time-varying structures and the validationapproach of modal parametersrdquoMechanical Systems and SignalProcessing vol 47 no 1-2 pp 94ndash119 2014

[13] A Bellino S Marchesiello and L Garibaldi ldquoExperimentaldynamic analysis of nonlinear beams under moving loadsrdquoShock and Vibration vol 19 no 5 pp 969ndash978 2012

[14] R Yan and R X Gao ldquoHilbert-huang transform-based vibra-tion signal analysis for machine health monitoringrdquo IEEETransactions on Instrumentation and Measurement vol 55 no6 pp 2320ndash2329 2006

[15] K Liu and X Sun ldquoSystem identification and model reductionfor a single-link flexible manipulatorrdquo Journal of Sound andVibration vol 242 no 5 pp 867ndash891 2001

[16] C S Huang S L Hung W C Su and C L Wu ldquoIdentifi-cation of time-variant modal parameters using time-varyingautoregressive with exogenous input and low-order polynomialfunctionrdquoComputer-Aided Civil and Infrastructure Engineeringvol 24 no 7 pp 470ndash491 2009

[17] W C Su C Y Liu and C S Huang ldquoIdentification of instanta-neous modal parameter of time-varying systems via a wavelet-based approach and its applicationrdquo Computer-Aided Civil andInfrastructure Engineering vol 29 no 4 pp 279ndash298 2014

[18] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014

[19] K Yu K Yang and Y Bai ldquoExperimental investigation on thetime-varying modal parameters of a trapezoidal plate in tem-perature-varying environments by subspace tracking-basedmethodrdquo Journal of Vibration and Control 2014

[20] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[21] T Marwala Finite-Element-Model Updating Using Computa-tional Intelligence Techniques Springer London UK 2010

[22] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[23] T Marwala ldquoFinite element model updating using wavelet dataand genetic algorithmrdquo Journal of Aircraft vol 39 no 4 pp709ndash711 2002

[24] R Perera S-E Fang and A Ruiz ldquoApplication of particleswarm optimization and genetic algorithms to multiobjectivedamage identification inverse problems with modelling errorsrdquoMeccanica vol 45 no 5 pp 723ndash734 2010

[25] M Luczak S Manzato B Peeters K Branner P Berring andM Kahsin ldquoUpdating finite element model of a wind turbineblade section using experimental modal analysis resultsrdquo Shockand Vibration vol 2014 Article ID 684786 12 pages 2014


[26] T W Simpson ldquoComparison of response surface and krigingmodels in the multidisciplinary design of an aerospike nozzlerdquoTech Rep NASACR-1998-206935 1998

[27] K P Sun H Y Hu and Y H Zhao ldquoIdentification of time-varying modal parameters for thermo-elastic structure subjectto unsteady heatingrdquoTransactions of Nanjing University of Aero-nautics and Astronautics vol 31 no 1 pp 39ndash48 2014

[28] K Sun Y Zhao and H Hu ldquoIdentification of temperature-dependent thermalmdashstructural properties via finite elementmodel updating and selectionrdquo Mechanical Systems and SignalProcessing vol 52-53 pp 147ndash161 2015

[29] M Link ldquoUpdating of analytical modelsmdashbasic procedures andextensionsrdquo inModal Analysis and Testing J M Silva and NMMaia Eds pp 281ndash304 Springer Dordrecht The Netherlands1999

[30] R Eberhart and J Kennedy ldquoA new optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[31] Y Shi and R C Eberhart ldquoA modified particle swarm opti-mizerrdquo in Proceedings of the IEEE World Congress on Compu-tational Intelligence and the 1998 IEEE International Conferenceon Evolutionary Computation Proceedings pp 69ndash73 IEEEAnchorage Alaska USA May 1998

[32] T Huang and A S Mohan ldquoA hybrid boundary conditionfor robust particle swarm optimizationrdquo IEEE Antennas andWireless Propagation Letters vol 4 no 1 pp 112ndash117 2005

[33] R C Eberhart and Y Shi ldquoComparing inertia weights and con-striction factors in particle swarm optimizationrdquo in Proceedingsof the 2000 Congress on Evolutionary Computation IEEE LaJolla Calif USA 2000

[34] G Matheron ldquoPrinciples of geostatisticsrdquo Economic Geologyvol 58 no 8 pp 1246ndash1266 1963

[35] T Goel R T Hafkta andW Shyy ldquoComparing error estimationmeasures for polynomial and kriging approximation of noise-free functionsrdquo Structural and Multidisciplinary Optimizationvol 38 no 5 pp 429ndash442 2009

[36] S N Lophaven H B Nielsen and J Soslashndergaard ldquoDACEA Matlab Kriging toolbox version 20rdquo Tech Rep IMM-TR-2002-12 Technical University of Denmark Kongens LyngbyDenmark 2002

[37] COMSOL Multiphysics Corporation ldquoStructural Mechanics-Verification Modelsrdquo 2013

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



the theories and applications Although recent years havewitnessed successful identifications of time-varying modalparameters of many engineering systems such as vehicle-bridge systems [12 13] machine condition monitoring sys-tems [14] flexible manipulators [15] and civil structures [1617] their applications to thermal structures in temperature-varying environment are still not available Yu et al [18]proposed an undetermined blind source separation methodto investigate the thermal effect on the modal parametersof a TC4 titanium-alloy column in a temperature-varyingenvironmentThey [19] also developed a time-varying modalparameter identification algorithm based on finite-data-window PAST and used it to investigate the effect of varyingtemperature and heating speed on the natural frequencies of atrapezoidal TA15 titanium-alloy plate To the best knowledgeof authors considerably less attention has been paid tothe finite element model updating (FEMU) for thermalstructures especially based on the identified time-varyingmodal parameters in UHTEs

Finite element (FE) modelling has received widespreadacceptance andwitnessed applications in various engineeringdisciplines Hence the FE model updating (FEMU) hasbecome a useful tool to improve the modelling assumptionsand parameters until a correlation between the analyti-cal predictions and experimental results satisfies practicalrequirements Mottershead and Friswell [20] comprehen-sively reviewed the model updating methods of structuralmodels In Marwalarsquos work [21] numerous computationalintelligence techniques were introduced and applied to FEmodel updating with in-depth comparisons As a mostwidely used method in FEMU the iterative method hasbeen formulated as an optimization problem and often basedon sensitivity analysis [22] and computational intelligencetechniques [23ndash25] Generally the optimization procedure isnonlinear and complex for a complicated FEMU problemIn recent years the particle swarm optimization (PSO)technique has been developed to implement the optimizationprocedure

The structural FE models with many geometric andphysical parameters to be updated may involve a largenumber of computations and need to be constructed byone of commercial finite element analysis packages suchas COMSOL ANSYS and NASTRAN Hence higher timeconsumption may be the disadvantage for the optimization-based algorithms due to their iterative strategy and repeatedanalysis in simulation models during the optimization pro-cess One way to overcome the difficulty of time consumptionand FE package-related problems during the optimization-based model updating is to replace the FE model by anapproximate surrogatereplacement meta-model that is fast-running and has fewer parameters involved Simpson [26]made a comparison of response surface and Kriging modelswhich are the two commonly used meta-models and drew aconclusion that both approximations predict reasonably wellwith the Kriging models having a slight overall advantagebecause of the lower root mean squared error values

The meta-model method for damage detection and relia-bility analysis has a long history However the Kriging meta-model (KMM) method for structural FEMU is somewhat

new especially for thermal structures in temperature-varyingenvironment As a continuation of authorsrsquo work [27 28]this paper presents KMM-PSO based FE model updatingand selection based on the experimentally identified time-invariant and time-varying modal parameters of a thermalstructure

The remainder is organized as follows In Section 2 theFE model updating and selection for a thermal structureis formulated and two objective functions are given forthe PSO An overview of PSO and a brief introductionto the time-varying autoregressive method for output onlyidentification are also presented The Kriging meta-modelis introduced to employ the PSO based FE model updatingand selection In Section 3 four groups of experiments arediscussed for the modal parameters of a beam-like structurein room temperature environment SHTE and UHTE InSection 4 the Kriging meta-model based FE model updatingand selection procedure is carried out to update the software-based FE model Finally some conclusions are drawn inSection 5

2 Formulation of FEMU forThermal Structures

21 Problem Description A linear thermal structure subjectto unsteady heating can be described in terms of the dis-tributed mass damping and stiffness matrices of the struc-ture in time domain via the following differential equation

Mx (119905) + C (119905) x (119905) + K (119905) x (119905) = f (119905) (1)

The real eigen-value problem for the 119895th mode reads

(K (119905) minus 120596

2

119895M)120593119895= 0 (2)

where the detailed meanings of the physical quantities in theabove two equations can be found in [28]

The traditional FE model updating process is achievedby identifying the correct mass and stiffness matrices whichare generally time-invariant This study however has to dealwith a time-variant FE model updating process because thesystem matrices change over time when the temperature-dependent material and temperature-dependent boundaryconditions are taken into consideration Afterwards usingthemeasured data the correctmass and stiffnessmatrices canbe obtained by identifying the correct material parameters ofthe structure and the appropriate boundary conditions underdifferent temperature conditions

For simplicity the thermal structure of concern is a kindof slender or thin-walled structures such as beams and platessubject to uniform heating with a uniformly distributedtemperature field119879(119909 119910 119911 119905) equiv 119879(119905) In other words the heattransfer process is neglected in the study Let 120579(119905) denote thetemperature increase in the thermal structure as follows

120579 (119905) = 119879 (119905) minus 119879ref (3)

where 119879ref is the reference temperature For the FEMU prob-lem of the thermal structure subject to unsteady heating the



Γ119901= flowastℓ(120579) = P

0+ P1120579 + P2120579

2+ P3120579








119896

119896 = 0 1 2 3


119869

0=

119873

sum

119895=1

120574

119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119891

meta119895

minus 119891

exp119895

119891

exp119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

(5)





exp119895



find P0

min 119869

0

st P1198970le P0le P1199060

(6)


0 respectively


119896(119896 = 0 1 2 3) in (4)







119869

1=

1

119873sp

119873

sum

119895=1

119873sp

sum

119896=1

120574

119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119891

meta119895

(119905

119896) minus 119891

exp119895

(119905

119896)

119891

exp119895

(119905

119896)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

(7)

119869

2= 119873 log (119869

1+ 120582

2 10038171003817

1003817

1003817

L (P minus Plowast)1003817100381710038171003817

2

) + 120581119889 (8)

with

119889 = ℓ

1+ ℓ


119873119901 (9)


(119905

119896) and 119891exp

119895(119905

119896)




119896(119896 = 1 2 119873

119901)




Start






Yes


Stoppingcriteria

Yes

End

MSEacceptable

No

Kriging meta-model

PSO based FEMU





Evaluate value of


No


J2

J2

J2



find P119896 119896 = 0 1 2 3

min 119869

2

st P119897119896le P119896le P119906119896

(10)









V119896= 119908 sdot V

119896+ 119888

1sdot 119903

1sdot (119901119887119890119904119905

119896minus 119909

119896)

+ 119888

2sdot 119903

2sdot (119892119887119890119904119905 minus 119909

119896)

119909

119896lArr997904 119909

119896+ V119896

(11)




1and 119888











lowastas the




119884 (xlowast) = f (x

lowast)120573 + 119911 (x

lowast) (12)


(xlowast) 119891


119872(xlowast)] and120573 = [120573

1120573

2


119872]




V (119911 (x)) = 120590

2R119899119904times119899119904

(13)







119891

1(x) = 1 119891

2(x) = 119909

1

119891

3(x) = 119909

2 119891

1+119903(x) = 119909

119903

(14)


119877 (x119894 x119895) =

119903

prod

119896=1

exp (minus119901119896

1003816

1003816

1003816

1003816

1003816

x(119896)119894

minus x(119896)119895

1003816

1003816

1003816

1003816

1003816

2

) (15)

where x(119896)119894






119872]


119896can be



FTRminus1y

2=


119904

(16)


119904times 119899

119904


y =[

[

[

[

[

119910

1

119910

2

119910

119899119904

]

]

]

]

]

F =

[

[

[

[

[

119891

1(x1) 119891


119903+1(x1)

119891

1(x2) 119891


119903+1(x2)

d

119891

1(x119899119904) 119891

2(x119899119904) sdot sdot sdot 119891

119903+1(x119899119904)

]

]

]

]

]

(17)



lowast

x119899119904)]


obtained as

119884 (xlowast) = f (x

lowast)







120593 (x) =






119909 (119905) = minus

119901

sum

119894=1

119886

119897(119905) 119909 (119905 minus 119897) + 119890 (119905) (20)


119897(119905) 119897 = 1 2 119901 are the TVAR



119909 (119905) = minusXT119905A + 119890 (119905)

(21)

where

AT= [119886

10 119886

1119898 119886

1199010 119886

119901119898]

XT119905= [119909 (119905 minus 1) 119892

0(119905) 119909 (119905 minus 1) 119892

119898(119905)

119909 (119905 minus 119901) 119892

0(119905) 119909 (119905 minus 119901) 119892

119898(119905)]

(22)







A119873+1

=

A119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

sdot (119909 (119873 + 1) + XT119873A119873)

P119873+1

=

1

120582

[P119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

XT119873P119873]

(23)


0= 0 P

0= 120583I where 120583 ≫ 1 and I is an





119891

119895(119905) =

1

2120587Δ119905


119895(119905) (24)







Resistance wire

Resistance wireBeam

Cantilever support



(a)

Resistance furnace




Vibration exciter

(b)

Vibrationexciter


Beam section

Poweramplifier

CH1 CH0

Forcesensor

Velocity signals

PX I 6733


SmartOfficeK-type



Thermocouple

Laser


(c)


Beam ABeam B

Mag

nitu

de (d

B)

Frequency (Hz)

minus30

minus25

minus20

minus15

minus10

minus5

0

0 10 20 30 40 50



0

0

10 20 30 40 50

0∘C200∘C300∘C

400∘C500∘C

minus30

minus25

minus20

minus15

minus10

minus5

Frequency (Hz)

Mag

nitu

de (d

B)




5005

1

2

3

4

5

6

10

15

20

25

30

100 150

Time (s)

Freq

uenc

y (H

z)

Frequency (Hz)

200 250 300

500 100 150

Time (s)200 250 300

times10minus4

Velo

city

(ms

)

minus3

0

3times10minus4

times10minus6

05 10 15 20 25 30

Pow

er sp

ectr

um (m

2s

2)

4

8







Mode 0

∘C 200

∘C 300

∘C 400

∘C 500

∘C1 1025 9 8 75 7252 26 2275 225 2175 20




0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)








135400

40

Φ65Unit (mm)

Thickness = 3








119887of



119887and119870

119903


119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870



119887119896and 119870






1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)


1198870and 119864as








Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Γ119901= flowastℓ(120579) = P

0+ P1120579 + P2120579

2+ P3120579








119896

119896 = 0 1 2 3


119869

0=

119873

sum

119895=1

120574

119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119891

meta119895

minus 119891

exp119895

119891

exp119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

(5)





exp119895



find P0

min 119869

0

st P1198970le P0le P1199060

(6)


0 respectively


119896(119896 = 0 1 2 3) in (4)







119869

1=

1

119873sp

119873

sum

119895=1

119873sp

sum

119896=1

120574

119895

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119891

meta119895

(119905

119896) minus 119891

exp119895

(119905

119896)

119891

exp119895

(119905

119896)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

(7)

119869

2= 119873 log (119869

1+ 120582

2 10038171003817

1003817

1003817

L (P minus Plowast)1003817100381710038171003817

2

) + 120581119889 (8)

with

119889 = ℓ

1+ ℓ


119873119901 (9)


(119905

119896) and 119891exp

119895(119905

119896)




119896(119896 = 1 2 119873

119901)




Start






Yes


Stoppingcriteria

Yes

End

MSEacceptable

No

Kriging meta-model

PSO based FEMU





Evaluate value of


No


J2

J2

J2



find P119896 119896 = 0 1 2 3

min 119869

2

st P119897119896le P119896le P119906119896

(10)









V119896= 119908 sdot V

119896+ 119888

1sdot 119903

1sdot (119901119887119890119904119905

119896minus 119909

119896)

+ 119888

2sdot 119903

2sdot (119892119887119890119904119905 minus 119909

119896)

119909

119896lArr997904 119909

119896+ V119896

(11)




1and 119888











lowastas the




119884 (xlowast) = f (x

lowast)120573 + 119911 (x

lowast) (12)


(xlowast) 119891


119872(xlowast)] and120573 = [120573

1120573

2


119872]




V (119911 (x)) = 120590

2R119899119904times119899119904

(13)







119891

1(x) = 1 119891

2(x) = 119909

1

119891

3(x) = 119909

2 119891

1+119903(x) = 119909

119903

(14)


119877 (x119894 x119895) =

119903

prod

119896=1

exp (minus119901119896

1003816

1003816

1003816

1003816

1003816

x(119896)119894

minus x(119896)119895

1003816

1003816

1003816

1003816

1003816

2

) (15)

where x(119896)119894






119872]


119896can be



FTRminus1y

2=


119904

(16)


119904times 119899

119904


y =[

[

[

[

[

119910

1

119910

2

119910

119899119904

]

]

]

]

]

F =

[

[

[

[

[

119891

1(x1) 119891


119903+1(x1)

119891

1(x2) 119891


119903+1(x2)

d

119891

1(x119899119904) 119891

2(x119899119904) sdot sdot sdot 119891

119903+1(x119899119904)

]

]

]

]

]

(17)



lowast

x119899119904)]


obtained as

119884 (xlowast) = f (x

lowast)







120593 (x) =






119909 (119905) = minus

119901

sum

119894=1

119886

119897(119905) 119909 (119905 minus 119897) + 119890 (119905) (20)


119897(119905) 119897 = 1 2 119901 are the TVAR



119909 (119905) = minusXT119905A + 119890 (119905)

(21)

where

AT= [119886

10 119886

1119898 119886

1199010 119886

119901119898]

XT119905= [119909 (119905 minus 1) 119892

0(119905) 119909 (119905 minus 1) 119892

119898(119905)

119909 (119905 minus 119901) 119892

0(119905) 119909 (119905 minus 119901) 119892

119898(119905)]

(22)







A119873+1

=

A119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

sdot (119909 (119873 + 1) + XT119873A119873)

P119873+1

=

1

120582

[P119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

XT119873P119873]

(23)


0= 0 P

0= 120583I where 120583 ≫ 1 and I is an





119891

119895(119905) =

1

2120587Δ119905


119895(119905) (24)







Resistance wire

Resistance wireBeam

Cantilever support



(a)

Resistance furnace




Vibration exciter

(b)

Vibrationexciter


Beam section

Poweramplifier

CH1 CH0

Forcesensor

Velocity signals

PX I 6733


SmartOfficeK-type



Thermocouple

Laser


(c)


Beam ABeam B

Mag

nitu

de (d

B)

Frequency (Hz)

minus30

minus25

minus20

minus15

minus10

minus5

0

0 10 20 30 40 50



0

0

10 20 30 40 50

0∘C200∘C300∘C

400∘C500∘C

minus30

minus25

minus20

minus15

minus10

minus5

Frequency (Hz)

Mag

nitu

de (d

B)




5005

1

2

3

4

5

6

10

15

20

25

30

100 150

Time (s)

Freq

uenc

y (H

z)

Frequency (Hz)

200 250 300

500 100 150

Time (s)200 250 300

times10minus4

Velo

city

(ms

)

minus3

0

3times10minus4

times10minus6

05 10 15 20 25 30

Pow

er sp

ectr

um (m

2s

2)

4

8







Mode 0

∘C 200

∘C 300

∘C 400

∘C 500

∘C1 1025 9 8 75 7252 26 2275 225 2175 20




0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)








135400

40

Φ65Unit (mm)

Thickness = 3








119887of



119887and119870

119903


119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870



119887119896and 119870






1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)


1198870and 119864as








Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start






Yes


Stoppingcriteria

Yes

End

MSEacceptable

No

Kriging meta-model

PSO based FEMU





Evaluate value of


No


J2

J2

J2



find P119896 119896 = 0 1 2 3

min 119869

2

st P119897119896le P119896le P119906119896

(10)









V119896= 119908 sdot V

119896+ 119888

1sdot 119903

1sdot (119901119887119890119904119905

119896minus 119909

119896)

+ 119888

2sdot 119903

2sdot (119892119887119890119904119905 minus 119909

119896)

119909

119896lArr997904 119909

119896+ V119896

(11)




1and 119888











lowastas the




119884 (xlowast) = f (x

lowast)120573 + 119911 (x

lowast) (12)


(xlowast) 119891


119872(xlowast)] and120573 = [120573

1120573

2


119872]




V (119911 (x)) = 120590

2R119899119904times119899119904

(13)







119891

1(x) = 1 119891

2(x) = 119909

1

119891

3(x) = 119909

2 119891

1+119903(x) = 119909

119903

(14)


119877 (x119894 x119895) =

119903

prod

119896=1

exp (minus119901119896

1003816

1003816

1003816

1003816

1003816

x(119896)119894

minus x(119896)119895

1003816

1003816

1003816

1003816

1003816

2

) (15)

where x(119896)119894






119872]


119896can be



FTRminus1y

2=


119904

(16)


119904times 119899

119904


y =[

[

[

[

[

119910

1

119910

2

119910

119899119904

]

]

]

]

]

F =

[

[

[

[

[

119891

1(x1) 119891


119903+1(x1)

119891

1(x2) 119891


119903+1(x2)

d

119891

1(x119899119904) 119891

2(x119899119904) sdot sdot sdot 119891

119903+1(x119899119904)

]

]

]

]

]

(17)



lowast

x119899119904)]


obtained as

119884 (xlowast) = f (x

lowast)







120593 (x) =






119909 (119905) = minus

119901

sum

119894=1

119886

119897(119905) 119909 (119905 minus 119897) + 119890 (119905) (20)


119897(119905) 119897 = 1 2 119901 are the TVAR



119909 (119905) = minusXT119905A + 119890 (119905)

(21)

where

AT= [119886

10 119886

1119898 119886

1199010 119886

119901119898]

XT119905= [119909 (119905 minus 1) 119892

0(119905) 119909 (119905 minus 1) 119892

119898(119905)

119909 (119905 minus 119901) 119892

0(119905) 119909 (119905 minus 119901) 119892

119898(119905)]

(22)







A119873+1

=

A119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

sdot (119909 (119873 + 1) + XT119873A119873)

P119873+1

=

1

120582

[P119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

XT119873P119873]

(23)


0= 0 P

0= 120583I where 120583 ≫ 1 and I is an





119891

119895(119905) =

1

2120587Δ119905


119895(119905) (24)







Resistance wire

Resistance wireBeam

Cantilever support



(a)

Resistance furnace




Vibration exciter

(b)

Vibrationexciter


Beam section

Poweramplifier

CH1 CH0

Forcesensor

Velocity signals

PX I 6733


SmartOfficeK-type



Thermocouple

Laser


(c)


Beam ABeam B

Mag

nitu

de (d

B)

Frequency (Hz)

minus30

minus25

minus20

minus15

minus10

minus5

0

0 10 20 30 40 50



0

0

10 20 30 40 50

0∘C200∘C300∘C

400∘C500∘C

minus30

minus25

minus20

minus15

minus10

minus5

Frequency (Hz)

Mag

nitu

de (d

B)




5005

1

2

3

4

5

6

10

15

20

25

30

100 150

Time (s)

Freq

uenc

y (H

z)

Frequency (Hz)

200 250 300

500 100 150

Time (s)200 250 300

times10minus4

Velo

city

(ms

)

minus3

0

3times10minus4

times10minus6

05 10 15 20 25 30

Pow

er sp

ectr

um (m

2s

2)

4

8







Mode 0

∘C 200

∘C 300

∘C 400

∘C 500

∘C1 1025 9 8 75 7252 26 2275 225 2175 20




0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)








135400

40

Φ65Unit (mm)

Thickness = 3








119887of



119887and119870

119903


119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870



119887119896and 119870






1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)


1198870and 119864as








Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











V119896= 119908 sdot V

119896+ 119888

1sdot 119903

1sdot (119901119887119890119904119905

119896minus 119909

119896)

+ 119888

2sdot 119903

2sdot (119892119887119890119904119905 minus 119909

119896)

119909

119896lArr997904 119909

119896+ V119896

(11)




1and 119888











lowastas the




119884 (xlowast) = f (x

lowast)120573 + 119911 (x

lowast) (12)


(xlowast) 119891


119872(xlowast)] and120573 = [120573

1120573

2


119872]




V (119911 (x)) = 120590

2R119899119904times119899119904

(13)







119891

1(x) = 1 119891

2(x) = 119909

1

119891

3(x) = 119909

2 119891

1+119903(x) = 119909

119903

(14)


119877 (x119894 x119895) =

119903

prod

119896=1

exp (minus119901119896

1003816

1003816

1003816

1003816

1003816

x(119896)119894

minus x(119896)119895

1003816

1003816

1003816

1003816

1003816

2

) (15)

where x(119896)119894






119872]


119896can be



FTRminus1y

2=


119904

(16)


119904times 119899

119904


y =[

[

[

[

[

119910

1

119910

2

119910

119899119904

]

]

]

]

]

F =

[

[

[

[

[

119891

1(x1) 119891


119903+1(x1)

119891

1(x2) 119891


119903+1(x2)

d

119891

1(x119899119904) 119891

2(x119899119904) sdot sdot sdot 119891

119903+1(x119899119904)

]

]

]

]

]

(17)



lowast

x119899119904)]


obtained as

119884 (xlowast) = f (x

lowast)







120593 (x) =






119909 (119905) = minus

119901

sum

119894=1

119886

119897(119905) 119909 (119905 minus 119897) + 119890 (119905) (20)


119897(119905) 119897 = 1 2 119901 are the TVAR



119909 (119905) = minusXT119905A + 119890 (119905)

(21)

where

AT= [119886

10 119886

1119898 119886

1199010 119886

119901119898]

XT119905= [119909 (119905 minus 1) 119892

0(119905) 119909 (119905 minus 1) 119892

119898(119905)

119909 (119905 minus 119901) 119892

0(119905) 119909 (119905 minus 119901) 119892

119898(119905)]

(22)







A119873+1

=

A119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

sdot (119909 (119873 + 1) + XT119873A119873)

P119873+1

=

1

120582

[P119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

XT119873P119873]

(23)


0= 0 P

0= 120583I where 120583 ≫ 1 and I is an





119891

119895(119905) =

1

2120587Δ119905


119895(119905) (24)







Resistance wire

Resistance wireBeam

Cantilever support



(a)

Resistance furnace




Vibration exciter

(b)

Vibrationexciter


Beam section

Poweramplifier

CH1 CH0

Forcesensor

Velocity signals

PX I 6733


SmartOfficeK-type



Thermocouple

Laser


(c)


Beam ABeam B

Mag

nitu

de (d

B)

Frequency (Hz)

minus30

minus25

minus20

minus15

minus10

minus5

0

0 10 20 30 40 50



0

0

10 20 30 40 50

0∘C200∘C300∘C

400∘C500∘C

minus30

minus25

minus20

minus15

minus10

minus5

Frequency (Hz)

Mag

nitu

de (d

B)




5005

1

2

3

4

5

6

10

15

20

25

30

100 150

Time (s)

Freq

uenc

y (H

z)

Frequency (Hz)

200 250 300

500 100 150

Time (s)200 250 300

times10minus4

Velo

city

(ms

)

minus3

0

3times10minus4

times10minus6

05 10 15 20 25 30

Pow

er sp

ectr

um (m

2s

2)

4

8







Mode 0

∘C 200

∘C 300

∘C 400

∘C 500

∘C1 1025 9 8 75 7252 26 2275 225 2175 20




0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)








135400

40

Φ65Unit (mm)

Thickness = 3








119887of



119887and119870

119903


119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870



119887119896and 119870






1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)


1198870and 119864as








Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














120593 (x) =






119909 (119905) = minus

119901

sum

119894=1

119886

119897(119905) 119909 (119905 minus 119897) + 119890 (119905) (20)


119897(119905) 119897 = 1 2 119901 are the TVAR



119909 (119905) = minusXT119905A + 119890 (119905)

(21)

where

AT= [119886

10 119886

1119898 119886

1199010 119886

119901119898]

XT119905= [119909 (119905 minus 1) 119892

0(119905) 119909 (119905 minus 1) 119892

119898(119905)

119909 (119905 minus 119901) 119892

0(119905) 119909 (119905 minus 119901) 119892

119898(119905)]

(22)







A119873+1

=

A119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

sdot (119909 (119873 + 1) + XT119873A119873)

P119873+1

=

1

120582

[P119873minus P119873X119873(120582 + XT

119873P119873X119873)

minus1

XT119873P119873]

(23)


0= 0 P

0= 120583I where 120583 ≫ 1 and I is an





119891

119895(119905) =

1

2120587Δ119905


119895(119905) (24)







Resistance wire

Resistance wireBeam

Cantilever support



(a)

Resistance furnace




Vibration exciter

(b)

Vibrationexciter


Beam section

Poweramplifier

CH1 CH0

Forcesensor

Velocity signals

PX I 6733


SmartOfficeK-type



Thermocouple

Laser


(c)


Beam ABeam B

Mag

nitu

de (d

B)

Frequency (Hz)

minus30

minus25

minus20

minus15

minus10

minus5

0

0 10 20 30 40 50



0

0

10 20 30 40 50

0∘C200∘C300∘C

400∘C500∘C

minus30

minus25

minus20

minus15

minus10

minus5

Frequency (Hz)

Mag

nitu

de (d

B)




5005

1

2

3

4

5

6

10

15

20

25

30

100 150

Time (s)

Freq

uenc

y (H

z)

Frequency (Hz)

200 250 300

500 100 150

Time (s)200 250 300

times10minus4

Velo

city

(ms

)

minus3

0

3times10minus4

times10minus6

05 10 15 20 25 30

Pow

er sp

ectr

um (m

2s

2)

4

8







Mode 0

∘C 200

∘C 300

∘C 400

∘C 500

∘C1 1025 9 8 75 7252 26 2275 225 2175 20




0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)








135400

40

Φ65Unit (mm)

Thickness = 3








119887of



119887and119870

119903


119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870



119887119896and 119870






1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)


1198870and 119864as








Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Resistance wire

Resistance wireBeam

Cantilever support



(a)

Resistance furnace




Vibration exciter

(b)

Vibrationexciter


Beam section

Poweramplifier

CH1 CH0

Forcesensor

Velocity signals

PX I 6733


SmartOfficeK-type



Thermocouple

Laser


(c)


Beam ABeam B

Mag

nitu

de (d

B)

Frequency (Hz)

minus30

minus25

minus20

minus15

minus10

minus5

0

0 10 20 30 40 50



0

0

10 20 30 40 50

0∘C200∘C300∘C

400∘C500∘C

minus30

minus25

minus20

minus15

minus10

minus5

Frequency (Hz)

Mag

nitu

de (d

B)




5005

1

2

3

4

5

6

10

15

20

25

30

100 150

Time (s)

Freq

uenc

y (H

z)

Frequency (Hz)

200 250 300

500 100 150

Time (s)200 250 300

times10minus4

Velo

city

(ms

)

minus3

0

3times10minus4

times10minus6

05 10 15 20 25 30

Pow

er sp

ectr

um (m

2s

2)

4

8







Mode 0

∘C 200

∘C 300

∘C 400

∘C 500

∘C1 1025 9 8 75 7252 26 2275 225 2175 20




0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)








135400

40

Φ65Unit (mm)

Thickness = 3








119887of



119887and119870

119903


119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870



119887119896and 119870






1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)


1198870and 119864as








Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5005

1

2

3

4

5

6

10

15

20

25

30

100 150

Time (s)

Freq

uenc

y (H

z)

Frequency (Hz)

200 250 300

500 100 150

Time (s)200 250 300

times10minus4

Velo

city

(ms

)

minus3

0

3times10minus4

times10minus6

05 10 15 20 25 30

Pow

er sp

ectr

um (m

2s

2)

4

8







Mode 0

∘C 200

∘C 300

∘C 400

∘C 500

∘C1 1025 9 8 75 7252 26 2275 225 2175 20




0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)








135400

40

Φ65Unit (mm)

Thickness = 3








119887of



119887and119870

119903


119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870



119887119896and 119870






1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)


1198870and 119864as








Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 50 100 150 200 250 3005

10

15

20

25

30

TrueUpdated

Initial

Freq

uenc

y (H

z)

Time (s)

250

300

350

400

450

500

550

Tem

pera

ture

(∘C)








135400

40

Φ65Unit (mm)

Thickness = 3








119887of



119887and119870

119903


119864

119887(120579) = 119864

1198870+ 119864

1198871120579 + 119864

1198872120579

2

119870

119886(120579) = 119870

1198860+ 119870

1198861120579 + 119870

1198862120579

2

(25)

where 1198641198870

and 119870



119887119896and 119870






1198870and 119864as (2) 119898119886

and1198701198860 and (3) 119864

119887119896and119870

119886119896(119896 = 1 2)


1198870and 119864as








Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Top view

Beam

Connector

Auxiliary surface


0

010

0

01

02

03

04

minus10

minus10

minus20minus30

10minus3

10minus3

xy

z

x

y

z




555

665

7

885

95

1065

7

75

8

9

f1

(Hz)

times1010

times104

Eas (Pa)Eb0

(Pa)



555

665

7

885

95

10

30

9

f2

(Hz)

29

31

32

33

34

35

times1010

times104

Eas (Pa)Eb0

(Pa)







555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












555

665

7

885

95

100

3

4

6

5

9

1

2

MSE

times10minus11

times1010

times104

Eas (Pa)Eb0

(Pa)




1and 119888






119886and 119870



555

665

7

8

85

95

100

3

4

9

1

2

MSE

times10minus12

times1010

times104

Eas (Pa)Eb0

(Pa)



119886and 119870

1198860



119887119896

and 119870





025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










025

30

35

40

45

50

55

60

65

50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)

Youn

grsquos m

odul

us (G

Pa)


05

10

15

20

25

Stiff

ness

(kN

m)

0 50 100 150 200 250 300 350 400 450 500 550

UpdatedInitial

Temperature (∘C)


119864

119887and 119870



119896≜ 120574(119864maxminus119864min)119879

119896



119896



Modelidentity

Number ofparameters

Max order(119864119887 119870

119886)

Parametersymbols

119872

12 (1 1) 119864

1198871 119870

1198861

119872

23 (1 2) 119864

1198871 119870

1198861 119870

1198862

119872

33 (2 1) 119864

1198871 119864

1198872 119870

1198861

119872

44 (2 2) 119864

1198871 119864

1198872 119870

1198861 119870

1198862

Glo

bal fi

tnes

s

Number of iteration

1

2

3

4

Glo

bal b

est m

odel

num

ber

0 10 20 30 40 50minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1











119864




119869







119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119864


119870


119870


Order of frequency

InitialUpdated

001 2

02

04

06

08

10

12

14

16

18

20

MA

PE (

)



3and119872

1 afterwards went







MAPE =

1

119873sp

119873sp

sum

119896=1

1003816

1003816

1003816

1003816

119910

119896minus 119910

119896

1003816

1003816

1003816

1003816

119910

119896

times 100 (26)


119896and

119910





Method Number ofMA





5 Conclusions


Abbreviations






Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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