Research Article Damage Detection of Bridges Using...

Research ArticleDamage Detection of Bridges Using Vibration Data by AdjointVariable Method

Akbar Mirzaee Mohsenali Shayanfar and Reza Abbasnia

Department of Civil Engineering Center of Excellence for Fundamental Studies in Structural EngineeringIran University of Science and Technology Narmak Tehran 16846-13114 Iran

Correspondence should be addressed to Mohsenali Shayanfar shayanfariustacir

Received 30 November 2013 Revised 28 April 2014 Accepted 29 April 2014 Published 4 June 2014

Academic Editor Mohammad Elahinia

Copyright copy 2014 Akbar Mirzaee et alThis 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

This research entails a theoretical and numerical study on a new damage detection method for bridges using response sensitivityin time domainThis method referred to as ldquoadjoint variable methodrdquo is a finite element model updating sensitivity based methodGoverning equation of the bridge-vehicle system is established based on finite element formulation In the inverse analysis the newapproach is presented to identify elemental flexural rigidity of the structure from acceleration responses of several measurementpoints The computational cost of sensitivity matrix is the main concern associated with damage detection by these methodsThe main advantage of the proposed method is the inclusion of an analytical method to augment the accuracy and speed of thesolutionThe reliable performance of themethod to precisely identify the location and intensity of all types of predetermined singlemultiple and random damages over the whole domain of moving vehicle speed is shown A comparison study is also carried out todemonstrate the relative effectiveness and upgraded performance of the proposed method in comparison to the similar ordinarysensitivity analysis methods Moreover various sources of errors including the effects of noise and primary errors on the numericalstability of the proposed method are discussed

1 Introduction

The main objective of developing the structural health mon-itoring (SHM) system for structures is to enhance struc-tural safety However in bridges SHM serves other eco-nomic benefits such as increasedmission reliability extendedlife of life-limited components reduced tests reduction inldquodown timerdquo increased equipment reliability customizationof maintenance actions and greater awareness of operatingpersonnel resulting in fewer accidents SHM also promisesto help reduce maintenance costs [1]

SHM algorithms are identified as static system identifica-tion (SI) and dynamic SI according to the types of structuralresponse used Dynamics-based SI techniques assess the stateof health of a structural component on the basis of the detec-tion and analysis of its dynamic response Such techniquescan be classified on the basis of the type of response beingconsidered for the investigations on the frequency or timedomain of interrogation and on the modality used to excitethe component [2]

The frequency domain SI and time-domain SI are morepractical than static SI as the static response for instancedisplacements of a structure are very difficult to measure inmost cases

The developments in the field of SI using vibration dataof civil engineering structures have been recently reviewedby several authors Some recent studies are briefly describedin the following

Doebling et al [3 4] have presented comprehensivereview of literature mainly focusing on frequency-domainmethods for damage detection in linear structures anddeclared that sufficient evidence exists to promote the useof measured vibration data for the detection of damage instructures using both forced-response testing and long-termmonitoring of ambient signals and there is a significant needin this field for research on the integration of theoreticalalgorithms with application-specific knowledge bases andpractical experimental constraints Another discussion onmethods of damage detection and location using naturalfrequency changes has been presented by Salawu [5] and

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 698658 17 pageshttpdxdoiorg1011552014698658

2 Shock and Vibration

his study showed that damage detection using vibrationfrequencies is not very reliable

Alampalli and Fu [6] and Alampalli et al [7] conductedlaboratory and field studies on bridge structures to investigatethe feasibility of measuring bridge vibration for inspec-tion and evaluation These studies focused on sensitivityof measured modal parameters to damage Cross diagno-sis using multiple signatures involving natural frequenciesmode shapesmodal assurance criteria and coordinatemodalassurance criteria was shown to be necessary to detect thedamages Casas and Aparicio studied concrete bridge struc-tures and investigated dynamic response as an inspection toolto assess bearing conditions and girder cracking [8] Theirstudy showed the need to investigate more than one naturalfrequency and also to determine mode shapes in order thatthe damage could be successfully detected and located

The frequency-domain SI algorithms have been morewidely developed and applied as the amount ofmeasured datais reduced dramatically after the transform thus they can behandled easily Unfortunately the effects of local damages onthe natural frequencies and mode shapes of higher modesare greater than lower ones but they are usually difficult tomeasure from experiments In addition structural dampingproperties cannot be identified in frequency domain SI

The time-domain SImay be an attractive one to overcomethe drawbacks of the frequency-domain SI For time-domainSI the forced vibration responses of the structure are neededin the identification However in some cases it is eitherimpractical or impossible to use artificial inputs to excitethe civil engineering structures so natural excitation mustbe measured along with the structural responses to assessthe dynamic characteristics [9 10] In recent years someresearchers have investigated both the problem of loadidentification (moving load and impact load) and modalparameters identification under operational conditions [1112] In addition identification of the structural parametersapplying a moving load has been considered in many papersLaw et al [13] presented a novel moving force and prestressidentification method based on the finite element and thewavelet-based methods for a bridge-vehicle system Jianget al [14] identified the parameter of a vehicle moving onmultispan continuous bridges Zhu and Law [15] presented amethod for damage detection of a simply supported concretebridge structure in time domain using the interaction forcesfrom the moving vehicles as excitation Majumder andManohar [16] proposed a time-domain approach for damagedetection in beam structures using vibration data induced bya vehicle moving on a bridge deck

Sensitivity based methods allow a wide choice of phys-ically meaningful parameters and this advantage has led totheir widespread use in damage detection The most impor-tant difficulty in sensitivity based SI methods is calculationof sensitivity matrix Calculation of this massive matrix isrepeated in each iteration and it is so time-consuming andhas a significant effect on the efficiency ofmethodDespite thehigh importance of calculation method of sensitivity matrixand the optimization of its performance in SI procedure thereis no literature on this regard In this paper computationalmethods for sensitivity matrix are discussed and a novel

Starting values

Numerical modelParameters beam stiffness

ExperimentTest specimen

Improved parameters

Minimization cost function

Identified model parameters

Figure 1 General flowchart of a FEM updating

sensitivity based damage detection method in time-domainreferred to as ldquoAdjoint variable methodrdquo is developed Com-putational algorithm of proposedmethod is presented and itsperformance is comparedwith the conventionalmethods andit is shown that the numerical cost is considerably reduced byusing the concept of adjoint variable

The outline of the work is as follows inverse prob-lems along with model updating are briefly introduced inSection 2The basic theory of sensitivity analysis is addressedin Section 3 and the proposed algorithm will be presentedin Section 4 Numerical simulations along with comparisonstudies are presented in Section 5 with studies on the effectof different factors which may affect the accuracy of theproposed analysis in practice Conclusion will be drawn inthe last section

2 Finite Element Model Updating andInverse Problem

Since many algorithms of damage detection are based onthe difference between modified model before occurrenceof damage and after that problems such as parameteridentification and damage detection are closely related tomodel updating Discrepancy between twomodels is used fordetection and quantification of damage

A key step in model-based damage identification is theupdating of the finite element model of the structure in sucha way that the measured responses can be reproduced by theFE model A general flowchart of this operation is given inFigure 1The identification procedure presented in this paperis a sensitivity based model updating routine Sensitivitycoefficients are the derivatives of the system responses withrespect to the physical parameters or input excitation forceand are needed in the cost function of the flowchart ofFigure 1

Shock and Vibration 3

21 Finite ElementModeling of Bridge Vibration underMovingLoads For a general finite element model of a linear elastictime-invariant structure the equation of motion is given by

[119872] 119911119905119905

+ [119862] 119911119905

+ [119870] 119911 = [119861] 119865 (1)

where [119872] and [119870] are mass and stiffness matrices and[119862] is damping matrix 119885

119905119905

119885119905

and 119885 are the respectiveacceleration velocity and displacement vectors for the wholestructure and 119865 is a vector of applied forces with matrix [119861]

mapping these forces to the associated Dof rsquos of the structureA proportional damping is assumed to show the effect ofdamping ratio on the dynamic magnification factor Rayleighdamping in which the damping matrix is proportional tothe combination of the mass and stiffness matrices is usedConsider

[119862] = 1198860[119872] + 119886

1[119870] (2)

where 1198860

and 1198861

are constants to be determined from twomodal damping ratios If a more accurate estimation of theactual damping is required a more general form of Rayleighdamping the Caughey damping model can be adopted

The dynamic responses of the structures can be obtainedby direct numerical integration using Newmark method

22 Objective Functions The approach minimizes the dif-ference between response quantities (usually accelerationresponse) of the measured data and model predictions Thisproblem may be expressed as the minimization of 119869 where

119869 (120579) =1003817100381710038171003817119911119898 minus 119911(120572)

10038171003817100381710038172

= 120598119879

120598

120598 = 119911119898

minus 119911 (120572) (3)

Here 119911119898

and 119911(120572) are the measured and computed responsevectors 120572 is a vector of all unknown parameters and 120598 is theresponse residual vector

23 Penalty Function Methods When the parameters of amodel are unknown they must be estimated using measureddata The measured response is a nonlinear function of theparameters So minimizing the error between the measuredandpredicted responsewill produce a nonlinear optimizationproblem

Penalty functionmethod is generally used for modal sen-sitivity with a truncated Taylor series expansion in terms ofthe unknown parameters In this paper the truncated seriesof the dynamic responses in terms of the system parameter120572 are used to derive the sensitivity based formulation Theidentification problem can be expressed as follows to find thevector 120572 such that the calculated response best matches themeasured response that is

[119876] 119877 = (4)

where the selection matrix [119876] is a matrix with elementsof zeros or ones matching the Dof rsquos corresponding to themeasured response components Vector 119877 can be obtainedfrom (4) for a given set of 120572

120597d1(tnt)

1205971205721

120597d1(tnt)

1205971205722

120597d2(tnt)

1205971205721

120597d2(tnt)

1205971205722

120597di(tnt)

1205971205721

120597di(tnt)

1205971205722

120597dn(tnt)

1205971205721

120597dn(tnt)

1205971205722

120597d1(tnt)

120597120572i

120597d1(tnt )

120597120572n

120597d2(tnt)

120597120572i

120597d2(tnt)

120597120572n

120597di(tnt)

120597120572i

120597di(tnt)

120597120572n

120597dn(tnt)

120597120572i

120597dn(tnt)

120597120572n

120597d1(t1)

1205971205721

120597d1(t1)

1205971205722

120597d2(t1)

1205971205721

120597d2(t1)

1205971205722

120597di(t1)

1205971205721

120597di(t1)

1205971205722

120597dn(t1)

1205971205721

120597dn(t1)

1205971205722

120597d1(t1)

120597120572i

120597d1(t1)

120597120572n

120597d2(t1)

120597120572i

120597d2(t1)

120597120572n

120597di(t1)

120597120572i

120597di(t1)

120597120572n

120597dn(t1)

120597120572i

120597dn(t1)

120597120572n

middot middot middot
























Parameter

t

Dof

Figure 2 Three-dimensional sensitivity matrix

Let

120575119911 = minus [119876] 119877 = minus 119877cal (5)

where 120575119911 is the error vector in the measured output In thepenalty function method we have

120575119911 = [119878] 120575120572 (6)

where 120575120572 is the perturbation in the parameters and [119878]

is the two-dimensional sensitivity matrix which is one ofthe matrices at time 119905 in the three-dimensional sensitivitymatrix shown in Figure 2 [17] For a finite element modelwith119873 elements eachwith119872 systemparameters the numberof unknown parameters is 119873 times 119872 and 119873 times 119872 equationsare needed to solve the parameters Matrix [119878] is on theparameter-119905 plane in Figure 2 and we can select any rowof the three-dimensional sensitivity matrix say the 119894th rowcorresponding to the 119894thmeasurement for the purposeWhenwriting in full (5) can be written as

120575119911 =

(1199051

)

(1199052

)

(119905119897

)

minus

119877cal (1199051)119877cal (1199052)

119877cal (119905119897)

(7)


with 119897 ge 119873 times 119872 to make sure that the set of equation isoverdetermined Equation (6) can be solved by simple least-squares method as follows

120575120572 = [119878119879

119878]minus1

119878119879

120575119911 (8)

120572119895+1

= 120572119895

+ [119878119879

119895

119878119895

]minus1

119878119879

119895

( minus 119877cal) (9)

The subscript 119895 indicates the iteration number at which thesensitivity matrix is computed

One of the important difficulties in parameter estimationis ill-conditioning In the worst case this can mean that thereis no unique solution to the estimation problem and manysets of parameters are able to fit the data Many optimizationprocedures result in the solution of linear equations forthe unknown parameters The use of the singular valuedecomposition (SVD) [18] for these linear equations enablesill-conditioning to be identified and quantified The optionsare then to increase the available data which is often difficultand costly or to provide extra conditions on the param-eters These can take the form of smoothness conditions(eg the truncated SVD) minimum norm parameter values(Tikhonov regularization) or minimum changes from theinitial estimates of the parameters [19 20]

From experiences gained in model updating with sim-ulated structures Li and Law [21] found that Tikhonovregularization can give the optimal solution when there is nonoise or very small noise in the measurement

24 Tikhonov Regularization Like many other inverse prob-lems (6) is an ill-conditioned problem In order to providebounds to the solution the damped least-squares method(DLS) is used and singular-value decomposition is used in thepseudoinverse calculation Equation (8) can be written in thefollowing form

120575120572 = (119878119879

119878 + 120582119877

119868)minus1

119878119879

120575119911 (10)

where 120582 is the nonnegative damping coefficient governingthe participation of least-squares error in the solution Thesolution of (10) is equivalent to minimizing the function

119869 (120575120572 120582119877

) = 119878120575120572 minus 1205751199112

+ 1205821198771205751205722 (11)

with the second term in (11) that provides bounds to thesolution When the parameter 120582

119877

approaches zero theestimated vector 120575120572 approaches the solution obtained fromthe simple least-squares method L-curve method is used inthis paper to obtain the optimal regularization parameter 120582

119877

25 Element Damage Index In the inverse problem of dam-age identification it is assumed that the stiffness matrix ofthe whole element decreases uniformly with damage and theflexural rigidity EI

119894

of the 119894th finite element of the beambecomes 120573

119894

EI119894

when there is damage [22] The fractionalchange in stiffness of an element can be expressed as

Δ119870119887119894

= (119870119887119894

minus 119887119894

) = (1 minus 120573119894

)119870119887119894

(12)

Sensitivity methods

Approximation approach

Forward finite difference

Central finite difference

Discrete approach

Analytical discrete approach

Semianalytical discrete approach

Continuum approach

Continuum-discrete method

Continuum-continuum

method

Figure 3 Different approaches to sensitivity analysis

where 119870119887119894

and 119887119894

are the 119894th element stiffness matrices ofthe undamaged and damaged beam respectively Δ119870

119887119894

is thestiffness reduction of the element A positive value of 120573

119894

isin

[0 1] will indicate a loss in the element stiffness The 119894thelement is undamaged when 120573

119894

= 1 and the stiffness of the119894th element is completely lost when 120573

119894

= 0The stiffness matrix of the damaged structure is the

assemblage of the entire element stiffness matrix 119887119894

119870119887

=

119873

sum119894=1

119860119879

119894

119887119894

119860119894

=

119873

sum119894=1

120573119894

119860119879

119894

119870119887119894

119860119894

(13)

where 119860119894

is the extended matrix of element nodal displace-ment that facilitates assembling of global stiffness matrixfrom the constituent element stiffness matrix

3 Sensitivity Analysis of TransientDynamic Response

The objective of sensitivity analysis is to quantify the effectsof parameter variations on calculated results Terms such asinfluence importance ranking by importance and domi-nance are all related to the sensitivity analysis

31 Methods of Structural Sensitivity Analysis When theparameter variations are small the traditional way to assesstheir effects on calculated responses is the employment ofperturbation theory either directly or indirectly via vari-ational principles The basic aim of perturbation theory isto predict the effects of small parameter variations withoutactually calculating the perturbed configuration but rather byusing solely unperturbed quantities

Various methods employed in sensitivity analysis arelisted in Figure 3 Three approaches are used to obtain thesensitivity matrix the approximation discrete and contin-uum approaches


32 Approximation Approach In the approximationapproach sensitivity matrix is obtained by either the forwardfinite difference or by the central finite difference method

If the design is perturbed to 119906 + Δ119906 where Δ119906 representsa small change in the design then the sensitivity of 120595(119906) canbe approximated as

119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906)

Δ119906 (14)

Equation (14) is called the forward difference method sincethe design is perturbed in the direction of +Δ119906 If minusΔ119906 issubstituted in (14) for Δ119906 then the equation is defined asthe backward differencemethod Additionally if the design isperturbed in both directions such that the design sensitivityis approximated by

119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906 minus Δ119906)

2Δ119906 (15)

then the equation is defined as the central difference method

33 Discrete Approach In the discrete method sensitivitymatrix is obtained by design derivatives of the discretegoverning equation For this process it is necessary to takethe derivative of the stiffness matrix If this derivative isobtained analytically using the explicit expression of thestiffness matrix with respect to the variable it is an analyticalmethod since the analytical expressions of stiffness matrixare used However if the derivative is obtained using a finitedifference method the method is called a semianalyticalmethod The design represents a structural parameter thatcan affect the results of the analysis

The design sensitivity information of a general perfor-mance measure can be computed either with the directdifferentiation method or with the adjoint variable method

331 Direct Differentiation Method The direct differentia-tion method (DDM) is a general accurate and efficientmethod to compute finite element response sensitivities to themodel parametersThis method directly solves for the designdependency of a state variable and then computes perfor-mance sensitivity using the chain rule of differentiation Thismethod clearly shows the implicit dependence on the designand a very simple sensitivity expression can be obtained

Consider a structure in which the generalized stiffnessand mass matrices have been reduced by accounting forboundary conditions Let the damping force be representedin the form of 119862(119887)119911

119905

where 119911119905

= 119889119911119889119905 denotes thevelocity vector Under these conditions Lagrangersquos equationof motion becomes the second-order differential equation as[23]

119872(119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 = 119865 (119905 119887) (16)

with the initial conditions

119911 (0) = 1199110

119911119905

(0) = 1199110

119905

(17)

If design parameters are just related to stiffness matrix wehave

[119872] 120597119911119905119905

120597119887119894 + [119862]

120597119911119905

120597119887119894 + [119870]

120597119911

120597119887119894

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(18)

in which 120597119911120597119887119894

120597119911119905

120597119887119894

and 120597119911119905119905

120597119887119894

are sensitiv-ity vectors of displacement velocity and acceleration withrespect to design parameter 119887119894 respectively Assume that

119884119905119905

=120597119911119905119905

120597119887119894 (19a)

119884119905

=120597119911119905

120597119887119894 (19b)

119884 =120597119911

120597119887119894 (19c)

So by replacing (19a) (19b) and (19c) to (18) we have

[119872] 119884119905119905

+ [119862] 119884119905

+ [119870] 119884

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(20)

The right side of (20) can be considered as an equivalentforce so (20) is similar to (16) and sensitivity vectors can beobtained by Newmark method

332 Adjoint Variable Method Sensitivity analysis can beperformed very efficiently by using deterministic methodsbased on adjoint functions The use of adjoint functions foranalyzing the effects of small perturbations in a linear systemwas introduced by Wigner [24]

Thismethod constructs an adjoint problem that solves theadjoint variable which contains all implicit dependent terms

For the dynamic response of structure the following formof a general performance measure will be considered

120595 = 119892 (119911 (119879) 119887) + int119879

0

119866 (119911 119887) 119889119905 (21)

where the final time 119879 is determined by a condition in theform

Ω(119911 (119879) 119911119905

(119879) 119887) = 0 (22)

It is presumed that (22) uniquely determines119879 at least locallyThis requires that the time derivative of Ω is nonzero at 119879 asfollows

Ω119905

=120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119911119905119905

(119879) = 0 (23)

When final time 119879 is prescribed before the response analysisthe relation in (22) needs not be considered

To obtain the design sensitivity of Ψ define a designvariation in the form

119887120591

= 119887 + 120591120575119887 (24)


Design 119887 is perturbed in the direction of 120575119887 with theparameter 120591 Substituting 119887

120591

into (21) the derivative of (21)can be evaluated with respect to 120591 at 120591 = 0 Leibnitzrsquos ruleof differentiation of an integral may be used to obtain thefollowing expression

1205951015840

=120597119892

120597119887120575119887 +

120597119892

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] + 119866 (119911 (119879) 119887) 1198791015840

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905

(25)

where

1199111015840

= 1199111015840

(119887 120575119887) equiv119889

119889120591119911 (119905 119887 + 120591120575119887)|

120591=0

=119889

119889119887[119911 (119905 119887)] 120575119887

1198791015840

= 1198791015840

(119887 120575119887) equiv119889

119889120591119879 (119887 + 120591120575119887)|

120591=0

=119889119879

119889119887120575119887

(26)

Note that since the expression in (21) that determines 119879depends on the design 119879 will also depend on the designThus terms arise in (25) that involve the derivative of 119879

with respect to the design In order to eliminate these termsdifferentiate (22) with respect to 120591 and evaluate it at 120591 = 0 inorder to obtain

120597Ω

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] +120597Ω

120597119911119905

[1199111015840

119905

(119879) + 119911119905119905

(119879) 1198791015840

]

+120597Ω

120597119887120575119887 = 0

(27)

This equation may also be written as

Ω119905

1198791015840

= [120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119905

119911119905119905

(119879)]1198791015840

= minus (120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887)

(28)

Since it is presumed by (23) that Ω119905

= 0 then

1198791015840

= minus1

Ω119905

(120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887) (29)

Substituting the result of (29) into (25) the following isobtained

1205951015840

= [120597119892

120597119911minus (

120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887))1

Ω119905

120597Ω

120597119911] 1199111015840

(119879)

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119911119905

1199111015840

119905

(119879)

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905 +

120597119892

120597119887120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

(30)

Note that1205951015840 depends on 1199111015840 and 119911

1015840

119905

at119879 as well as on 1199111015840 within

the integrationIn order to write Ψ

1015840 in (29) explicitly in terms of a designvariation the adjoint variable technique can be used In thecase of a dynamic system all terms in (16) can be multipliedby 120582119879

(119905) and integrated over the interval [0 119879] to obtain thefollowing identity in 120582

int119879

0

120582119879

[119872 (119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 minus 119865 (119905 119887)] 119889119905 = 0

(31)

Since this equation must hold for arbitrary 120582 which is nowtaken to be independent of the design substitute 119887

120591

into (31)and differentiate it with respect to 120591 in order to obtain thefollowing relationship

int119879

0

[120582119879

119872(119887) 1199111015840

119905119905

+ 120582119879

119862 (119887) 1199111015840

119905

+ 120582119879

119870 (119887) 1199111015840

minus120597119877

120597119887120575119887] 119889119905 = 0

(32)

where

119877 = 119879

119865 (119905 119887) minus 119879

119872(119887) 119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887) (33)

with the superposed tilde (sim) denoting variables that are heldconstant during the differentiation with respect to the designin (32)

Since (32) contains the time derivatives of 1199111015840 integrate

the first two integrands by parts in order to move the timederivatives to 120582 as follows

120582119879

119872(119887) 1199111015840

119905

(119879) minus 120582119879

119905

(119879)119872 (119887) 1199111015840

(119879) + 120582119879

119862 (119887) 1199111015840

(119879)

+ int119879

0

[120582119879

119905119905

119872(119887) minus 120582119879

119905

119862 (119887) + 120582119879

119870 (119887)] 1199111015840

minus120597119877

120597119887120575119887 119889119905

= 0

(34)

The adjoint variable method expresses the unknown termsin (30) in terms of the adjoint variable (120582) Since (34) musthold for arbitrary functions 120582(119905) 120582may be chosen so that thecoefficients of terms involving 119911

1015840

(119879) 1199111015840119905

(119879) and 1199111015840 in (30) and

(34) are equal If such a function 120582(119905) can be found then theunwanted terms in (30) involving 119911

1015840

(119879) 1199111015840119905

(119879) and 1199111015840 can be

replaced by terms that explicitly depend on 120575119887 in (34) and tobe more specific choose a 120582(119905) that satisfies the following

119872(119887) 120582 (119879) = minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911119905

(35)

119872(119887) 120582119905

(119879) = 119862119879

(119887) 120582 (119879) minus120597119892119879

120597119911

+ [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911

(36)

119872(119887) 120582119905119905

minus 119862119879

(119887) 120582119905

+ 119870 (119887) 120582 =120597119866119879

120597119911 0 le 119905 le 119879

(37)


Note that once the dynamic equations of (16) and (17) issolved and (22) is used to determine 119879 then 119911(119879) 119911

119905

(119879)120597Ω120597119911 120597Ω120597119911

119905

and Ω119905

may be evaluated Equation (23)can then be solved for 120582(119879) since the mass matrix 119872(119887) isnonsingular Having determined 120582(119879) all terms on the rightof (36) can be evaluated and the equation can be solvedfor 120582119905

(119879) Thus a set of terminal conditions on 120582 has beendetermined Since 119872(119887) is nonsingular (37) may then beintegrated from 119879 to 0 yielding the unique solution 120582(119905)Taken as a whole (35) through (37) may be thought of asa terminal value problem

Since the terms involving a variation in the state variablein (30) and (34) are identical substitute (34) into (30) toobtain

1205951015840

=120597119892

120597119887120575119887 + int

119879

0

[120597119866

120597119887+

120597119877

120597119887] 119889119905120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

equiv120597120595

120597119887120575119887

(38)

Every term in this equation can now be calculated Theterms 120597119892120597119887 120597119866120597119887 and 120597Ω120597119887 represent explicit partialderivatives with respect to the design The term 120597119877120597119887however must be evaluated from (33) thus requiring 120582(119905)Note also that since design variation 120575119887 does not depend ontime it is taken outside the integral in (38)

Since (38) must hold for all 120575119887 the design derivativevector of 120595 is

119889120595

119889119887

=120597119892

120597119887(119911 (119879) 119887)

+ int119879

0

[120597119866

120597119887(119911 119887) +

120597119877

120597119887(120582 (119905) 119911 (119905) 119911

119905

(119905) 119911119905119905

(119905) 119887)] 119889119905

minus1

Ω119905

[120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]120597Ω

120597119887

(39)

34 Continuum Approach In the continuum approach thedesign derivative of the variational equation is taken beforeit is discretized If the structural problem and sensitivityequations are solved as a continuum problem then it iscalled the continuum-continuum method The continuumsensitivity equation is solved by discretization in the sameway that structural problems are solved Since differentiationis taken at the continuum domain and is then followed bydiscretization this method is called the continuum-discretemethod

35 Sensitivity Method Selection The advantage of the finitedifference method is obvious If structural analysis can beperformed and the performance measure can be obtained

as a result of structural analysis then the expressions in(14) and (15) are virtually independent of the problem typesconsidered

Major disadvantage of the finite difference method is theaccuracy of its sensitivity results Depending on perturbationsize sensitivity results are quite different For a mildlynonlinear performancemeasure relatively large perturbationprovides a reasonable estimation of sensitivity results How-ever for highly nonlinear performances a large perturbationyields completely inaccurate results Thus the determinationof perturbation size greatly affects the sensitivity result Andeven though it may be necessary to choose a very small per-turbation numerical noise becomes dominant for a too-smallperturbation size That is with a too-small perturbation noreliable difference can be found in the analysis results

The continuum-continuum approach is so limited andis not applicable in complex engineering structures becausevery simple classical problems can be solved analytically

The discrete and continuum-discrete methods are equiv-alent under the conditions given below using a beam asthe structural component It has also been argued thatthe discrete and continuum-discrete methods are equivalentunder the conditions given below [23]

First the same discretization (shape function) used in theFEA method must be used for continuum design sensitivityanalysis Second an exact integration (instead of a numericalintegration) must be used in the generation of the stiffnessmatrix and in the evaluation of continuum-based designsensitivity expressions Third the exact solution (and not anumerical solution) of the finite elementmatrix equation andthe adjoint equation should be used to compare these twomethods Fourth the movement of discrete grid points mustbe consistent with the design parameterization method usedin the continuum method

In this paper two different analytical discrete methodsincluding direct differential method (DDM) and adjoint vari-able method (ADM) are presented and efficiency of proposedmethod is investigated when compared with DDMmethod

4 Proposed Method

While structural vibration responses are used for damagedetection assuming 119866 = 0 (37) is a free vibration of beamwith terminal conditions Solving (37) for a single degree offreedom system is as follows

119898120582119905119905

minus 119888120582119905

+ 119896120582 = 0

with terminal conditions 120582 (119879) (119879)

120582119879

(119905) = 119890120585120596(119905minus119879)

(1198601

sin (120596119863

119905) + 1198611

cos (120596119863

119905))

1198601

= (120582119905

(119879)

120596119863

minus120585

radic1 minus 1205852120582 (119879)) cos (120596

119863

119879)

+ 120582 (119879) sin (120596119863

119879)

1198611

=120582 (119879)

cos (120596119863

119905)minus 1198601

tan (120596119863

119879)

(40)


in which

120585 =119888

2119898120596=

119888

119888crlt 1 120596

119863

= 120596radic1 minus 1205852 (41)

When time 119879 is known the coefficients of the characteristicequation of 1198791015840 and thereupon Ω will be zero so the terminalconditions are as follows

120582 (119879) = 0 (42)

120582119905

(119879) = 119872minus1

(119887) times (minus120597119892119879

120597119911) (43)

Substitute (42) into (43) to obtain

1198601

=120582119905

(119879)

120596119863

cos (120596119863

119879)

1198611

= minus120582119905

(119879)

120596119863

sin (120596119863

119879)

(44)

Note that 120597119892120597119911 like 1198601

and 1198611

is dependent on time 119879 soterminal values for different amounts of119879 are not similar andadjoint equation should be calculated for all amounts of 119879

separately So

120582119879

(119905) = 119890120585120596(119905minus119879)

(120582119905

(119879)

120596119863

cos (120596119863

119879) sin (120596119863

119905)

minus120582119905

(119879)

120596119863

sin (120596119863

119879) cos (120596119863

119905))

= 119875119879

119891 (119905) + 119876119879

119892 (119905)

(45)

in which

119875119879

= 119890minus120585120596119879

120582119905

(119879)

120596119863

cos (120596119863

119879)

119891 (119905) = 119890120585120596119905 sin (120596

119863

119905)

119876119879

= minus119890minus120585120596119879

120582119905

(119879)

120596119863

sin (120596119863

119879)

119892 (119905) = 119890120585120596119905 cos (120596

119863

119905)

(46)

41 Sensitivity Matrix for Physical Parameter Using (39) andassuming 119879 is known and 119866 = 0 because of using structuralvibration data (47) can be obtained

119889120595

119889119887= int119879

0

120597119877

120597119887119889119905 (47)

In this equation

119877 = 119879

119865 (119905) minus 119879

119872119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887)

119862 = 1198860

119870 (119887) + 1198861

119872(48)

is Rayleigh damping matrix so

120597119877

120597119887= minus120582119879119886

0

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887 (49)

And finally component of sensitivity matrix in time 119879 is

119889120595

119889119887(119879) = int

119879

0

(minus1205821198791198860

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887) 119889119905 (50)

In a multidegree of freedom problem solving the aboveequations directly is not possible and for this purposechange the variables as follows

120582 = [120601] 119884 (51)

In this equation matrix [120601] forms vibration modes (modalmatrix) and terminal conditions of above equations are

119884 (119879) = 119872minus1

[120601]119879

[119898] 120582 (119879) (52)

119884119905

(119879) = 119872minus1

[120601]119879

[119898] 120582119905

(119879) (53)

By inserting (51) in (37) and multiplying [120601]119879 in both sides

the new equation in modal space is

[119872] 119884119905119905

minus [119862] 119884119905

+ [119870] 119884 = 0 (54)

Each of [119872] [119862] and [119870] matrices are diagonal so

119872119894

119884119905119905119894

minus 119862119894

119884119905119894 + 119870119894

119884119894

= 0 (55)

119889120595

119889119887(119879) = minus int

119879

0

⟨119884⟩ times [120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

+ ⟨119884⟩ times [120601]119879

times [120597119896

120597119887] times 119911 119889119905

(56)

Consider

[120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

= 119911119911119905

[120601]119879

times [120597119896

120597119887] times 119911 = 119911119911

(57)

Equation (56) can be reduced to the following equation

119889120595

119889119887(119879) = minusint

119879

0

⟨119884⟩ times 119911119911119905

+ ⟨119884⟩ times 119911119911 119889119905 (58)

From (45) variable 119884 in modal space can be written as

119884 = 119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905) (59)

Replacing (59) in (58) a new expression is derived to calculatethe sensitivity as follows

119889120595

119889119887(119879)

= minusint119879

0

(119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911119905

+ (119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911 119889119905

(60)


Equation (60) can be rewritten as follows

119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)

Consider following parameters

119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)

So (61) is presented as

119889120595

119889119887(119879) = minus ⟨119875 (119879)⟩ times (119860 + 119862) minus ⟨119876 (119879)⟩ times (119861 + 119862)

(63)

The solution of (63) is directly too time-consuming becausein each time step all terms in (63) should be recalculatedTherefore an incremental solution is developed as follows

119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)

Similar to (65) for other parameters we have

120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)

And finally the sensitivity expression in time 119879 + Δ119879 is asfollows

119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)

42 Computational Algorithm The computational algorithmthat leads to the determination of sensitivity matrix is asfollows

Step 1 Calculate 120582119905

(119879) from (43)

Step 2 Calculate 120596 120596119863

and 120601 from and consider 119894 = 1

Step 3 For the 119894th element calculate 120597119870120597119887 119911119911119905

and 119911119911 andconsider 119895 = 1

Step 4 For the 119895th sensor and the corresponding Dofcalculate 120582

119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)

Step 7 Calculate 120575119860 120575119861 120575119862 and 120575119863 from ((65)sim(66))

Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899

lt 119879final consider 1198790

= 119879119899

and 119879119899

= 119879119899

+ Δ119905 andgo to Step 5 otherwise go to next step

Step 10 If 119895 lt number of sensors consider 119895 = 119895 + 1 and goto Step 4 otherwise go to next step

Step 11 If 119894 lt number of elements consider 119894 = 119894 + 1 and goto Step 3 otherwise finish


Sensors

Element numberMoving vehicle

Direction of measured response for identification

10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Figure 4 Multispan bridge model used in detection procedure

43 Procedure of Iteration for Damage Detection The initialanalytical model of a structure deviates from the true modeland measurement from the initial intact structure is used toupdate the analytical model The improved model is thentreated as a reference model and measurement from thedamaged structure will be used to update the referencemodel

When response measurement from the intact state ofthe structure is obtained the sensitivities are computedfrom the proposed algorithm or direct differentiate method(20) based on the analytical model of the structure andthe well knowing input force and velocity The vector ofparameter increments is then obtained from (8) or (10) usingthe computed and experimentally obtained responses Theanalytical model is then updated and the correspondingresponse and its sensitivity are again computed for the nextiteration When measurement from the damaged state isobtained the updated analyticalmodel is used in the iterationin the same way as that using measurement from the intactstate Convergence is considered to be achieved when thefollowing criteria are met as follows

1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1

1003817100381710038171003817Response119894+1 minus Response119894

10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)

The final vector of identified parameter increments corre-sponds to the changes occurring in between the two statesof the structure The tolerance is set equal to 1 times 10

minus6 in thisstudy except otherwise specified

Equation (6) has been popularly used in the form ofthe first-order approximation of the increment on the left-hand side of the equation The higher-order term of theTaylor expansion has been omitted in the computation Theiterative computation described above on the updating of thesensitivity and the system aims at reducing error due to suchan omission particularly with large local damages

5 Numerical Results

To illustrate the formulations presented in the previoussections we consider the system shown in Figures 4 and 8and capabilities of the proposed method are investigated

The relative percentage error (RPE) in the identifiedresults is calculated from (69) where sdot is the normofmatrixand 119864Identified and 119864True are the identified and the true elasticmodulus respectively Consider

RPE =

1003817100381710038171003817119864Identified minus 119864True1003817100381710038171003817

1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)

Since the true value of elastic modulus is unknown RPE canjust be used for investigating the efficiency of method

51 Multispan Model A three-span bridge as shown inFigure 4 is studied to illustrate the proposed method Itconsists of 30 Euler-Bernoulli beam elements with 31 nodeseach one with two Dof rsquos The mass density of material is78 times 10

3 kgm3 and the elastic modulus of material is 21 times

107Ncm2 The total length of bridge is 30m and height and

width of the frame section are respectively 200 and 200mmThe first five undamped natural frequencies of the intactbridge are 3773 5517 6697 1342 and 196485Hz Rayleighdampingmodel is adoptedwith the damping ratios of the firsttwo modes taken to be equal to 005The equivalent Rayleighcoefficients 119886

0

and 1198861

are respectively 01 and 4804 times 10minus5

The transverse point load 119875 has a constant velocity 119881 =

119871119879 where 119879 is the traveling time across the bridge and 119871 isthe total length of the bridge

For the forced vibration analysis an implicit time inte-gration method called ldquothe Newmark integration methodrdquois used with the integration parameters 120573 = 14 and120574 = 12 which leads to the constant-average accelerationapproximation

Speed parameter is defined as

120572V =119881

119881cr (70)

in which119881cr is critical speed (119881cr = (120587119897)radicEI120588)119881 is movingload speed and 120588 is mass per unit length of beam

511 Damage Scenarios Five damage scenarios of singlemultiple and random damages in the bridge without mea-surement noise are studied and they are shown in Table 1

Local damage is simulated with a reduction in the elasticmodulus of material of an element The sampling rate is10000Hz and 450 data of the acceleration response (degree of


Table 1 Damage scenarios for multispan bridge

Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multi 8 13 and 29 11 4 and 7 NilM1-3 Multi 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil

Table 2 RPE of DDMmethod for model 1

Damage scenario Speed parameter01 02 03 04 05 06 07 08 09

M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007

indeterminacy is 15) collected along the z-direction at nodes5 15 and 25 are used in the identification

Scenario 1 studies the single damage scenario The itera-tive solution converges in all speed parameter ranges with amaximumRPE of 0088 inDDMmethod and 00354 in ADMmethod

Scenarios 2 and 3 are on multiple damages with differentamount of measured responses for the identification andScenarios 4-5 are on random damages with different averagefor the identification These scenarios also converge in allspeed parameter ranges Onemore scenario withmodel erroris also included as in Scenario 6 This scenario consists of nosimulated damage in the structure but with the initial elasticmodulus ofmaterial of all the elements underestimated by 5in the inverse identification

Using both described methods including DDM andproposed method the damage locations and amount areidentified correctly in all scenarios (Figure 5) and the RPEparameter is shown in Tables 2 and 3

Further studies on Scenario 6 shows that both methodsare sensitive to the initial model error and for the maximum20 initial error can be converged and a relatively good finiteelement model is therefore needed for the damage detectionprocedure

512 Effect of Noise Noise is the random fluctuation in thevalue of measured or input that causes random fluctuation inthe output value Noise at the sensor output is due to eitherinternal noise sources such as resistors at finite temperaturesor externally generatedmechanical and electromagnetic fluc-tuations [6]

To evaluate the sensitivity of results to suchmeasurementnoise noise-polluted measurements are simulated by addingto the noise-free acceleration vector a corresponding noisevector whose root-mean-square (rms) value is equal toa certain percentage of the rms value of the noise-free

0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number

Original modelDetected model

minus004

minus002

0

002

004

times105

Figure 5 Detection of damage location and amount in elements 37 19 25 and 28 and distribution of error in different elements withADM scheme

data vector The components of all the noise vectors are ofGaussian distribution uncorrelated and with a zero meanand unit standard deviation Then on the basis of the noise-free acceleration 119885

119905119905119899119891 the noise-polluted acceleration 119885

119905119905119899119901

of the bridge at location 119909 can be simulated by

119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891

) times 119873level times 119873unit (71)


Table 3 RPE of ADMmethod for model 1


M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)

Figure 6 RPE contours with respect to noise level and loops

where RMS(119885119905119905119899119891

) is the rms value of the noise-free accel-eration vector 119885

119905119905119899119891times 119873level is the noise level and 119873unit is

a randomly generated noise vector with zero mean and unitstandard deviation [14]

In order to study the effect of noise on stability ofsensitivity methods Scenario 2 (speed ratio of moving loadis considered to be constant and equal to 05) is consideredand different levels of noise pollution are investigated andRPE changes with increasing number of loops for the iterativeprocedure have been studied

Results are illustrated in Figure 6 for DDM and ADMmethods respectively

These contours show that both ADM and DDMmethodsare sensitive to the noise and if the noise level becomes greaterthan 13 these methods lose their effectiveness and are notable to detect damage So in cases with noise level greaterthan 13 a denoising tool alongside sensitivity methodsshould be used

513 Efficiency of ProposedMethod In order to compare andquantify the performance of different methods and evaluate

the proposed method relative efficiency parameter (REP) isdefined as follows

REP =STDDMSTADM

(72)

in which ST is the solution time of SI method In fact thisparameter represents the computation cost of method

Figure 7 shows REP changes with respect to the speedparameter in different scenarios

Table 4 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 21599and 123739 and its average is 46580 therefore the adjointvariable method is extremely successful and computationalcost for this method is about 215 of other sensitivity basedfinite element model updating methods

52 PlaneGridModel Aplane gridmodel of bridge is studiedas another numerical example to illustrate the effectivenessof the proposed method The finite element model of thestructure is shown in Figure 8The structure ismodeled by 46frame elements and 32 nodes with three Dof at each node for


Table 4 REP ranges in different scenarios

Damage scenario Max REP Min REP AverageM1-1 123739 49093 76744M1-2 35953 22271 27166M1-3 54912 45801 49990M1-4 60214 2287 46553M1-5 38383 21599 31221M1-6 76027 32449 47804Total 123739 21599 46580

Table 5 Damage scenarios for grid model

Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM2-1 Single 41 7 NilM2-2 Multi 3 26 35 and 40 9 14 3 and 8 NilM2-3 Multi 5 7 12 15 24 and 37 4 11 6 2 10 and 16 NilM2-4 Random All elements Random damage in all elements with an average of 5 NilM2-5 Random All elements Random damage in all elements with an average of 15 NilM2-6 Estimation of undamaged state All elements 5 reduction in all elements Nil

02468101214

0103

0507

09

12ndash1410ndash128ndash106ndash8

4ndash62ndash40ndash2

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05

Figure 7 REP changes in different scenarios with respect to speedparameter

the translation and rotational deformationsThemass densityof material is 78 times 10

3 kgm3 and the elastic modulus ofmaterial is 21 times 10

7Ncm2 The first five undamped naturalfrequencies of the intact bridge are 4559 9277 18174 25973and 39907Hz Rayleigh damping model is adopted with thedamping ratios of the first two modes taken to be equalto 005 The equivalent Rayleigh coefficients 119886

0

and 1198861

arerespectively 01 and 2364 times 10

minus5

521 Damage Scenarios Five damage scenarios of singlemultiple and random damages in the bridge without

measurement of the noise are studied and they are shown inTable 5

The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 10) collected alongthe 119911-direction at nodes 4 11 21 and 27 are used

Similar to the previous model Scenario 1 studies thesingle damage scenarioThe iterative solution converges in allspeed parameter ranges with a maximum RPE of 00006 inDDMmethod and 00011 in ADMmethod

Scenarios 2 and 3 are on multiple damages with differentamount of measured responses for the identification andScenarios 4-5 are on random damages with different averagefor the identification These scenarios also converge in allspeed parameter ranges Onemore scenario withmodel erroris also included as Scenario 6 This scenario consists of nosimulated damage in the structure but with the initial elasticmodulus of material of all the elements under-estimated by5 in the inverse identification

Using both described methods including DDM andproposed method the damage locations and amount areidentified correctly in all the scenarios (Figure 9) and the RPEparameter is shown in Tables 6 and 7

522 Effect of Noise In order to study effect of noise onstability of sensitivity methods scenario 3 (speed ratio ofmoving load is considered to be constant and equal to05) is considered and different levels of noise pollution areinvestigated and RPE changes with increasing number ofloops for the iterative procedure has been studied

Figure 10 shows that both ADM and DDM methods aresensitive to the noise and if the noise level becomes greaterthan 2 and 17 for ADM method and DDM methodrespectively these methods lose their effectiveness and arenot able to detect damage So in cases with noise level greaterthan mentioned values a denoising tool such as wavelettransform alongside sensitivity methods should be used The


Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40

Figure 8 Plane grid bridge model used in detection procedure



M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011

wavelet transform is mainly attractive because of its ability tocompress and encode information to reduce noise or to detectany local singular behavior of a signal [25]

523 Efficiency of Proposed Method Figure 11 shows REPchanges with respect to the speed parameter in differentscenarios Table 8 shows that in different scenarios andfor different speed parameters the efficiency parameter isbetween 14998 and 31370 and its average is 21173 therefore

the adjoint variable method is extremely successful andcomputational cost for this method is about 472 of othersensitivity based finite element model updating methods

6 Conclusion

A new damage detection method based on finite elementmodel updating and sensitivity technique using accelerationtime history data of a bridge deck affected by amoving vehicle


Table 8 REP ranges in different scenarios for model 2


0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105

Figure 9 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme

with specified load named ldquoADMrdquo method is presentedThe updating procedure can be regarded as a parameteridentification technique which aims to fit the unknownparameters of an analytical model such that the modelbehaviour corresponds as closely as possible to the measuredbehaviour

Newmark method is used to calculate the structuraldynamic response and its dynamic response sensitivitymatrix is calculated by adjoint variable method In orderto solve ill-posed inverse problem Tikhonov regularizationmethod is used and L-curve method is implemented to findoptimum value of the regularization parameter

In proposed method an incremental solution for adjointvariable equation developed that calculates each element ofsensitivity matrix separatelyThemain advantage is inclusion

of an analytical method to augment the accuracy and speedof the solution

Numerical simulations demonstrate the efficiency andaccuracy of the method to identify location and intensityof single multiple and random damages in different bridgemodels

Comparison studies confirmed that computational costfor this method is much lower than other traditional sensitiv-ity methods For modern practical engineering applicationsthe cost of damage detection analysis is expensive So thismethod is feasible for large-scale problems

Similar to other sensitivity methods the drawback ofproposed method is its low stability against input measure-ment noise which can be easily improved by using low-passdenoising tools such as wavelets

Nomenclature

119872 119862 and 119870 The structural massdamping and stiffnessmatrices of the bridge

119911 119911119905

119911119905119905

Nodal displacementvelocity and accelerationvectors respectively

119865 = 1198651

(119905) 1198652

(119905) 119865119873119865

(119905)119879 Vector of applied forces

119870119887119894

and 119887119894

The 119894th element stiffnessmatrices of the undamagedand damaged beam

Δ119870119887119894

The stiffness reduction ofthe element

[119861] Mapping force matrix tothe associated Dof of thestructure

1198860

and 1198861

Rayleigh dampingcoefficients

119911119898

and 119911(120572) The measured andcomputed response vectors

120598 Response residual vector[119876] Matrix with elements of

zeros or ones matching theDof corresponding to themeasured responsecomponents

[119878] Sensitivity matrix120572 Vector of all unknown

parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6



Figure 11 REP changes in different scenarios with respect to speedparameter for model 2

120582119877

Regularization parameter120573119894

isin [0 1] Loss in the element stiffness120595 General performance measure119879 Final time119887 Design parameter119887120591

Perturbation of designparameter in the direction of 120575119887with the parameter 120591

120582 Adjoint variable119889120595119889119887 Sensitivity of performance with

respect to design parameter

120585 Damping ratio[120593] Modal matrixRPE Relative percentage of errorREP Relative efficiency parameter119864Identified and 119864True Identified and the true elastic

modulus119871 Total length of the bridge119881 Velocity of traveling load120572V Speed parameter119881cr Critical speed120588 Mass per unit lengthrms Root-mean-square119885119905119905119899119891

and 119885119905119905119899119901

Noise-free acceleration andnoise-polluted acceleration

ST Solution time of systemidentification method

Conflict of Interests

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

References

[1] M Prashant and P R Ganguli Structural Health MonitoringUsing Genetic Fuzzy Systems Springer London UK 2011

[2] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013

[3] S W Doebling C R Farrar M B Prime and D W ShevitzDamage Identification and Health Monitoring of Structural AndMechanical Systems from Changes in Their Vibration Character-istics A Literature Review Los AlamosNational Laboratory LosAlamos NM USA 1996


[4] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[5] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[6] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation 1994

[7] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995

[8] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural Engineering vol120 no 8 pp 2437ndash2449 1994

[9] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003

[10] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009

[11] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007

[12] WX Ren andZH Zong ldquoOutput-onlymodal parameter iden-tification of civil engineering structuresrdquo Structural Engineeringamp Mechanics vol 17 pp 1ndash16 2004

[13] S S Law S Q Wu and Z Y Shi ldquoMoving load and prestressidentification using wavelet-based methodrdquo Journal of AppliedMechanics vol 75 pp 021014-1ndash0021014-7 2008

[14] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004

[15] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007

[16] L Majumder and C S Manohar ldquoA time-domain approach fordamage detection in beam structures using vibration data witha moving oscillator as an excitation sourcerdquo Journal of Soundand Vibration vol 268 no 4 pp 699ndash716 2003

[17] Z R Lu and S S Law ldquoFeatures of dynamic response sensitivityand its application in damage detectionrdquo Journal of Sound andVibration vol 303 no 1-2 pp 305ndash329 2007

[18] G H Golub and C F van Loan Matrix Computations JohnsHopkins Baltimore Md USA 3rd edition 1996

[19] P C Hansen ldquoAnalysis of discrete ill-posed problems by meansof the L-curverdquo SIAM Review vol 34 pp 561ndash580 1992

[20] P C Hansen ldquoRegularization tools a MATLAB package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 pp 1ndash35 1994

[21] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010

[22] X Q Zhu and H Hao ldquoDamage detection of bridge beamstructures under moving loadsrdquo Research Program ReportSchool of Civil and Resource Engineering The University ofWestern Australia 2007

[23] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005

[24] E P Wigner ldquoEffect of small perturbations on pile periodrdquoManhattan Project Report CP-G-3048 1945

[25] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo MechanicalSystems and Signal Processing vol 40 pp 645ndash666 2013

International Journal of

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Control Scienceand Engineering

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



his study showed that damage detection using vibrationfrequencies is not very reliable

Alampalli and Fu [6] and Alampalli et al [7] conductedlaboratory and field studies on bridge structures to investigatethe feasibility of measuring bridge vibration for inspec-tion and evaluation These studies focused on sensitivityof measured modal parameters to damage Cross diagno-sis using multiple signatures involving natural frequenciesmode shapesmodal assurance criteria and coordinatemodalassurance criteria was shown to be necessary to detect thedamages Casas and Aparicio studied concrete bridge struc-tures and investigated dynamic response as an inspection toolto assess bearing conditions and girder cracking [8] Theirstudy showed the need to investigate more than one naturalfrequency and also to determine mode shapes in order thatthe damage could be successfully detected and located

The frequency-domain SI algorithms have been morewidely developed and applied as the amount ofmeasured datais reduced dramatically after the transform thus they can behandled easily Unfortunately the effects of local damages onthe natural frequencies and mode shapes of higher modesare greater than lower ones but they are usually difficult tomeasure from experiments In addition structural dampingproperties cannot be identified in frequency domain SI

The time-domain SImay be an attractive one to overcomethe drawbacks of the frequency-domain SI For time-domainSI the forced vibration responses of the structure are neededin the identification However in some cases it is eitherimpractical or impossible to use artificial inputs to excitethe civil engineering structures so natural excitation mustbe measured along with the structural responses to assessthe dynamic characteristics [9 10] In recent years someresearchers have investigated both the problem of loadidentification (moving load and impact load) and modalparameters identification under operational conditions [1112] In addition identification of the structural parametersapplying a moving load has been considered in many papersLaw et al [13] presented a novel moving force and prestressidentification method based on the finite element and thewavelet-based methods for a bridge-vehicle system Jianget al [14] identified the parameter of a vehicle moving onmultispan continuous bridges Zhu and Law [15] presented amethod for damage detection of a simply supported concretebridge structure in time domain using the interaction forcesfrom the moving vehicles as excitation Majumder andManohar [16] proposed a time-domain approach for damagedetection in beam structures using vibration data induced bya vehicle moving on a bridge deck

Sensitivity based methods allow a wide choice of phys-ically meaningful parameters and this advantage has led totheir widespread use in damage detection The most impor-tant difficulty in sensitivity based SI methods is calculationof sensitivity matrix Calculation of this massive matrix isrepeated in each iteration and it is so time-consuming andhas a significant effect on the efficiency ofmethodDespite thehigh importance of calculation method of sensitivity matrixand the optimization of its performance in SI procedure thereis no literature on this regard In this paper computationalmethods for sensitivity matrix are discussed and a novel

Starting values

Numerical modelParameters beam stiffness

ExperimentTest specimen

Improved parameters

Minimization cost function

Identified model parameters

Figure 1 General flowchart of a FEM updating

sensitivity based damage detection method in time-domainreferred to as ldquoAdjoint variable methodrdquo is developed Com-putational algorithm of proposedmethod is presented and itsperformance is comparedwith the conventionalmethods andit is shown that the numerical cost is considerably reduced byusing the concept of adjoint variable

The outline of the work is as follows inverse prob-lems along with model updating are briefly introduced inSection 2The basic theory of sensitivity analysis is addressedin Section 3 and the proposed algorithm will be presentedin Section 4 Numerical simulations along with comparisonstudies are presented in Section 5 with studies on the effectof different factors which may affect the accuracy of theproposed analysis in practice Conclusion will be drawn inthe last section

2 Finite Element Model Updating andInverse Problem

Since many algorithms of damage detection are based onthe difference between modified model before occurrenceof damage and after that problems such as parameteridentification and damage detection are closely related tomodel updating Discrepancy between twomodels is used fordetection and quantification of damage

A key step in model-based damage identification is theupdating of the finite element model of the structure in sucha way that the measured responses can be reproduced by theFE model A general flowchart of this operation is given inFigure 1The identification procedure presented in this paperis a sensitivity based model updating routine Sensitivitycoefficients are the derivatives of the system responses withrespect to the physical parameters or input excitation forceand are needed in the cost function of the flowchart ofFigure 1



[119872] 119911119905119905

+ [119862] 119911119905

+ [119870] 119911 = [119861] 119865 (1)


119905119905

119885119905



[119862] = 1198860[119872] + 119886

1[119870] (2)

where 1198860

and 1198861




119869 (120579) =1003817100381710038171003817119911119898 minus 119911(120572)

10038171003817100381710038172

= 120598119879

120598

120598 = 119911119898

minus 119911 (120572) (3)

Here 119911119898




[119876] 119877 = (4)


120597d1(tnt)

1205971205721

120597d1(tnt)

1205971205722

120597d2(tnt)

1205971205721

120597d2(tnt)

1205971205722

120597di(tnt)

1205971205721

120597di(tnt)

1205971205722

120597dn(tnt)

1205971205721

120597dn(tnt)

1205971205722

120597d1(tnt)

120597120572i

120597d1(tnt )

120597120572n

120597d2(tnt)

120597120572i

120597d2(tnt)

120597120572n

120597di(tnt)

120597120572i

120597di(tnt)

120597120572n

120597dn(tnt)

120597120572i

120597dn(tnt)

120597120572n

120597d1(t1)

1205971205721

120597d1(t1)

1205971205722

120597d2(t1)

1205971205721

120597d2(t1)

1205971205722

120597di(t1)

1205971205721

120597di(t1)

1205971205722

120597dn(t1)

1205971205721

120597dn(t1)

1205971205722

120597d1(t1)

120597120572i

120597d1(t1)

120597120572n

120597d2(t1)

120597120572i

120597d2(t1)

120597120572n

120597di(t1)

120597120572i

120597di(t1)

120597120572n

120597dn(t1)

120597120572i

120597dn(t1)

120597120572n

























Parameter

t

Dof


Let

120575119911 = minus [119876] 119877 = minus 119877cal (5)


120575119911 = [119878] 120575120572 (6)



120575119911 =

(1199051

)

(1199052

)

(119905119897

)

minus

119877cal (1199051)119877cal (1199052)

119877cal (119905119897)

(7)



120575120572 = [119878119879

119878]minus1

119878119879

120575119911 (8)

120572119895+1

= 120572119895

+ [119878119879

119895

119878119895

]minus1

119878119879

119895

( minus 119877cal) (9)





120575120572 = (119878119879

119878 + 120582119877

119868)minus1

119878119879

120575119911 (10)


119869 (120575120572 120582119877

) = 119878120575120572 minus 1205751199112

+ 1205821198771205751205722 (11)


119877


119877


119894


119894

EI119894


Δ119870119887119894

= (119870119887119894

minus 119887119894

) = (1 minus 120573119894

)119870119887119894

(12)

Sensitivity methods




Discrete approach



Continuum approach


Continuum-continuum

method


where 119870119887119894

and 119887119894


119887119894


119894

isin


119894


119894



119870119887

=

119873

sum119894=1

119860119879

119894

119887119894

119860119894

=

119873

sum119894=1

120573119894

119860119879

119894

119870119887119894

119860119894

(13)

where 119860119894









119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906)

Δ119906 (14)


119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906 minus Δ119906)

2Δ119906 (15)






119905

where 119911119905


119872(119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 = 119865 (119905 119887) (16)


119911 (0) = 1199110

119911119905

(0) = 1199110

119905

(17)


[119872] 120597119911119905119905

120597119887119894 + [119862]

120597119911119905

120597119887119894 + [119870]

120597119911

120597119887119894

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(18)

in which 120597119911120597119887119894

120597119911119905

120597119887119894

and 120597119911119905119905

120597119887119894


119884119905119905

=120597119911119905119905

120597119887119894 (19a)

119884119905

=120597119911119905

120597119887119894 (19b)

119884 =120597119911

120597119887119894 (19c)


[119872] 119884119905119905

+ [119862] 119884119905

+ [119870] 119884

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(20)





120595 = 119892 (119911 (119879) 119887) + int119879

0

119866 (119911 119887) 119889119905 (21)


Ω(119911 (119879) 119911119905

(119879) 119887) = 0 (22)


Ω119905

=120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119911119905119905

(119879) = 0 (23)



119887120591

= 119887 + 120591120575119887 (24)



120591


1205951015840

=120597119892

120597119887120575119887 +

120597119892

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] + 119866 (119911 (119879) 119887) 1198791015840

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905

(25)

where

1199111015840

= 1199111015840

(119887 120575119887) equiv119889

119889120591119911 (119905 119887 + 120591120575119887)|

120591=0

=119889

119889119887[119911 (119905 119887)] 120575119887

1198791015840

= 1198791015840

(119887 120575119887) equiv119889

119889120591119879 (119887 + 120591120575119887)|

120591=0

=119889119879

119889119887120575119887

(26)



120597Ω

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] +120597Ω

120597119911119905

[1199111015840

119905

(119879) + 119911119905119905

(119879) 1198791015840

]

+120597Ω

120597119887120575119887 = 0

(27)


Ω119905

1198791015840

= [120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119905

119911119905119905

(119879)]1198791015840

= minus (120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887)

(28)


= 0 then

1198791015840

= minus1

Ω119905

(120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887) (29)


1205951015840

= [120597119892

120597119911minus (

120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887))1

Ω119905

120597Ω

120597119911] 1199111015840

(119879)

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119911119905

1199111015840

119905

(119879)

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905 +

120597119892

120597119887120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

(30)


1015840

119905





int119879

0

120582119879

[119872 (119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 minus 119865 (119905 119887)] 119889119905 = 0

(31)


120591


int119879

0

[120582119879

119872(119887) 1199111015840

119905119905

+ 120582119879

119862 (119887) 1199111015840

119905

+ 120582119879

119870 (119887) 1199111015840

minus120597119877

120597119887120575119887] 119889119905 = 0

(32)

where

119877 = 119879

119865 (119905 119887) minus 119879

119872(119887) 119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887) (33)




120582119879

119872(119887) 1199111015840

119905

(119879) minus 120582119879

119905

(119879)119872 (119887) 1199111015840

(119879) + 120582119879

119862 (119887) 1199111015840

(119879)

+ int119879

0

[120582119879

119905119905

119872(119887) minus 120582119879

119905

119862 (119887) + 120582119879

119870 (119887)] 1199111015840

minus120597119877

120597119887120575119887 119889119905

= 0

(34)


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 in (30) and


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 can be


119872(119887) 120582 (119879) = minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911119905

(35)

119872(119887) 120582119905

(119879) = 119862119879

(119887) 120582 (119879) minus120597119892119879

120597119911

+ [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911

(36)

119872(119887) 120582119905119905

minus 119862119879

(119887) 120582119905

+ 119870 (119887) 120582 =120597119866119879

120597119911 0 le 119905 le 119879

(37)



119905

(119879)120597Ω120597119911 120597Ω120597119911

119905

and Ω119905




1205951015840

=120597119892

120597119887120575119887 + int

119879

0

[120597119866

120597119887+

120597119877

120597119887] 119889119905120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

equiv120597120595

120597119887120575119887

(38)



119889120595

119889119887

=120597119892

120597119887(119911 (119879) 119887)

+ int119879

0

[120597119866

120597119887(119911 119887) +

120597119877

120597119887(120582 (119905) 119911 (119905) 119911

119905

(119905) 119911119905119905

(119905) 119887)] 119889119905

minus1

Ω119905

[120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]120597Ω

120597119887

(39)









4 Proposed Method


119898120582119905119905

minus 119888120582119905

+ 119896120582 = 0


120582119879

(119905) = 119890120585120596(119905minus119879)

(1198601

sin (120596119863

119905) + 1198611

cos (120596119863

119905))

1198601

= (120582119905

(119879)

120596119863

minus120585

radic1 minus 1205852120582 (119879)) cos (120596

119863

119879)

+ 120582 (119879) sin (120596119863

119879)

1198611

=120582 (119879)

cos (120596119863

119905)minus 1198601

tan (120596119863

119879)

(40)


in which

120585 =119888

2119898120596=

119888

119888crlt 1 120596

119863

= 120596radic1 minus 1205852 (41)


120582 (119879) = 0 (42)

120582119905

(119879) = 119872minus1

(119887) times (minus120597119892119879

120597119911) (43)


1198601

=120582119905

(119879)

120596119863

cos (120596119863

119879)

1198611

= minus120582119905

(119879)

120596119863

sin (120596119863

119879)

(44)

Note that 120597119892120597119911 like 1198601

and 1198611


separately So

120582119879

(119905) = 119890120585120596(119905minus119879)

(120582119905

(119879)

120596119863

cos (120596119863

119879) sin (120596119863

119905)

minus120582119905

(119879)

120596119863

sin (120596119863

119879) cos (120596119863

119905))

= 119875119879

119891 (119905) + 119876119879

119892 (119905)

(45)

in which

119875119879

= 119890minus120585120596119879

120582119905

(119879)

120596119863

cos (120596119863

119879)

119891 (119905) = 119890120585120596119905 sin (120596

119863

119905)

119876119879

= minus119890minus120585120596119879

120582119905

(119879)

120596119863

sin (120596119863

119879)

119892 (119905) = 119890120585120596119905 cos (120596

119863

119905)

(46)


119889120595

119889119887= int119879

0

120597119877

120597119887119889119905 (47)

In this equation

119877 = 119879

119865 (119905) minus 119879

119872119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887)

119862 = 1198860

119870 (119887) + 1198861

119872(48)


120597119877

120597119887= minus120582119879119886

0

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887 (49)


119889120595

119889119887(119879) = int

119879

0

(minus1205821198791198860

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887) 119889119905 (50)


120582 = [120601] 119884 (51)


119884 (119879) = 119872minus1

[120601]119879

[119898] 120582 (119879) (52)

119884119905

(119879) = 119872minus1

[120601]119879

[119898] 120582119905

(119879) (53)



[119872] 119884119905119905

minus [119862] 119884119905

+ [119870] 119884 = 0 (54)


119872119894

119884119905119905119894

minus 119862119894

119884119905119894 + 119870119894

119884119894

= 0 (55)

119889120595

119889119887(119879) = minus int

119879

0

⟨119884⟩ times [120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

+ ⟨119884⟩ times [120601]119879

times [120597119896

120597119887] times 119911 119889119905

(56)

Consider

[120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

= 119911119911119905

[120601]119879

times [120597119896

120597119887] times 119911 = 119911119911

(57)


119889120595

119889119887(119879) = minusint

119879

0

⟨119884⟩ times 119911119911119905

+ ⟨119884⟩ times 119911119911 119889119905 (58)


119884 = 119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905) (59)


119889120595

119889119887(119879)

= minusint119879

0

(119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911119905

+ (119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911 119889119905

(60)



119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)


119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)


119889120595


(63)


119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)


120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)


119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)



(119879) from (43)

Step 2 Calculate 120596 120596119863



and 119911119911 andconsider 119895 = 1


119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)


Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899


= 119879119899

and 119879119899

= 119879119899





Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[119872] 119911119905119905

+ [119862] 119911119905

+ [119870] 119911 = [119861] 119865 (1)


119905119905

119885119905



[119862] = 1198860[119872] + 119886

1[119870] (2)

where 1198860

and 1198861




119869 (120579) =1003817100381710038171003817119911119898 minus 119911(120572)

10038171003817100381710038172

= 120598119879

120598

120598 = 119911119898

minus 119911 (120572) (3)

Here 119911119898




[119876] 119877 = (4)


120597d1(tnt)

1205971205721

120597d1(tnt)

1205971205722

120597d2(tnt)

1205971205721

120597d2(tnt)

1205971205722

120597di(tnt)

1205971205721

120597di(tnt)

1205971205722

120597dn(tnt)

1205971205721

120597dn(tnt)

1205971205722

120597d1(tnt)

120597120572i

120597d1(tnt )

120597120572n

120597d2(tnt)

120597120572i

120597d2(tnt)

120597120572n

120597di(tnt)

120597120572i

120597di(tnt)

120597120572n

120597dn(tnt)

120597120572i

120597dn(tnt)

120597120572n

120597d1(t1)

1205971205721

120597d1(t1)

1205971205722

120597d2(t1)

1205971205721

120597d2(t1)

1205971205722

120597di(t1)

1205971205721

120597di(t1)

1205971205722

120597dn(t1)

1205971205721

120597dn(t1)

1205971205722

120597d1(t1)

120597120572i

120597d1(t1)

120597120572n

120597d2(t1)

120597120572i

120597d2(t1)

120597120572n

120597di(t1)

120597120572i

120597di(t1)

120597120572n

120597dn(t1)

120597120572i

120597dn(t1)

120597120572n

























Parameter

t

Dof


Let

120575119911 = minus [119876] 119877 = minus 119877cal (5)


120575119911 = [119878] 120575120572 (6)



120575119911 =

(1199051

)

(1199052

)

(119905119897

)

minus

119877cal (1199051)119877cal (1199052)

119877cal (119905119897)

(7)



120575120572 = [119878119879

119878]minus1

119878119879

120575119911 (8)

120572119895+1

= 120572119895

+ [119878119879

119895

119878119895

]minus1

119878119879

119895

( minus 119877cal) (9)





120575120572 = (119878119879

119878 + 120582119877

119868)minus1

119878119879

120575119911 (10)


119869 (120575120572 120582119877

) = 119878120575120572 minus 1205751199112

+ 1205821198771205751205722 (11)


119877


119877


119894


119894

EI119894


Δ119870119887119894

= (119870119887119894

minus 119887119894

) = (1 minus 120573119894

)119870119887119894

(12)

Sensitivity methods




Discrete approach



Continuum approach


Continuum-continuum

method


where 119870119887119894

and 119887119894


119887119894


119894

isin


119894


119894



119870119887

=

119873

sum119894=1

119860119879

119894

119887119894

119860119894

=

119873

sum119894=1

120573119894

119860119879

119894

119870119887119894

119860119894

(13)

where 119860119894









119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906)

Δ119906 (14)


119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906 minus Δ119906)

2Δ119906 (15)






119905

where 119911119905


119872(119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 = 119865 (119905 119887) (16)


119911 (0) = 1199110

119911119905

(0) = 1199110

119905

(17)


[119872] 120597119911119905119905

120597119887119894 + [119862]

120597119911119905

120597119887119894 + [119870]

120597119911

120597119887119894

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(18)

in which 120597119911120597119887119894

120597119911119905

120597119887119894

and 120597119911119905119905

120597119887119894


119884119905119905

=120597119911119905119905

120597119887119894 (19a)

119884119905

=120597119911119905

120597119887119894 (19b)

119884 =120597119911

120597119887119894 (19c)


[119872] 119884119905119905

+ [119862] 119884119905

+ [119870] 119884

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(20)





120595 = 119892 (119911 (119879) 119887) + int119879

0

119866 (119911 119887) 119889119905 (21)


Ω(119911 (119879) 119911119905

(119879) 119887) = 0 (22)


Ω119905

=120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119911119905119905

(119879) = 0 (23)



119887120591

= 119887 + 120591120575119887 (24)



120591


1205951015840

=120597119892

120597119887120575119887 +

120597119892

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] + 119866 (119911 (119879) 119887) 1198791015840

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905

(25)

where

1199111015840

= 1199111015840

(119887 120575119887) equiv119889

119889120591119911 (119905 119887 + 120591120575119887)|

120591=0

=119889

119889119887[119911 (119905 119887)] 120575119887

1198791015840

= 1198791015840

(119887 120575119887) equiv119889

119889120591119879 (119887 + 120591120575119887)|

120591=0

=119889119879

119889119887120575119887

(26)



120597Ω

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] +120597Ω

120597119911119905

[1199111015840

119905

(119879) + 119911119905119905

(119879) 1198791015840

]

+120597Ω

120597119887120575119887 = 0

(27)


Ω119905

1198791015840

= [120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119905

119911119905119905

(119879)]1198791015840

= minus (120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887)

(28)


= 0 then

1198791015840

= minus1

Ω119905

(120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887) (29)


1205951015840

= [120597119892

120597119911minus (

120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887))1

Ω119905

120597Ω

120597119911] 1199111015840

(119879)

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119911119905

1199111015840

119905

(119879)

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905 +

120597119892

120597119887120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

(30)


1015840

119905





int119879

0

120582119879

[119872 (119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 minus 119865 (119905 119887)] 119889119905 = 0

(31)


120591


int119879

0

[120582119879

119872(119887) 1199111015840

119905119905

+ 120582119879

119862 (119887) 1199111015840

119905

+ 120582119879

119870 (119887) 1199111015840

minus120597119877

120597119887120575119887] 119889119905 = 0

(32)

where

119877 = 119879

119865 (119905 119887) minus 119879

119872(119887) 119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887) (33)




120582119879

119872(119887) 1199111015840

119905

(119879) minus 120582119879

119905

(119879)119872 (119887) 1199111015840

(119879) + 120582119879

119862 (119887) 1199111015840

(119879)

+ int119879

0

[120582119879

119905119905

119872(119887) minus 120582119879

119905

119862 (119887) + 120582119879

119870 (119887)] 1199111015840

minus120597119877

120597119887120575119887 119889119905

= 0

(34)


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 in (30) and


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 can be


119872(119887) 120582 (119879) = minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911119905

(35)

119872(119887) 120582119905

(119879) = 119862119879

(119887) 120582 (119879) minus120597119892119879

120597119911

+ [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911

(36)

119872(119887) 120582119905119905

minus 119862119879

(119887) 120582119905

+ 119870 (119887) 120582 =120597119866119879

120597119911 0 le 119905 le 119879

(37)



119905

(119879)120597Ω120597119911 120597Ω120597119911

119905

and Ω119905




1205951015840

=120597119892

120597119887120575119887 + int

119879

0

[120597119866

120597119887+

120597119877

120597119887] 119889119905120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

equiv120597120595

120597119887120575119887

(38)



119889120595

119889119887

=120597119892

120597119887(119911 (119879) 119887)

+ int119879

0

[120597119866

120597119887(119911 119887) +

120597119877

120597119887(120582 (119905) 119911 (119905) 119911

119905

(119905) 119911119905119905

(119905) 119887)] 119889119905

minus1

Ω119905

[120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]120597Ω

120597119887

(39)









4 Proposed Method


119898120582119905119905

minus 119888120582119905

+ 119896120582 = 0


120582119879

(119905) = 119890120585120596(119905minus119879)

(1198601

sin (120596119863

119905) + 1198611

cos (120596119863

119905))

1198601

= (120582119905

(119879)

120596119863

minus120585

radic1 minus 1205852120582 (119879)) cos (120596

119863

119879)

+ 120582 (119879) sin (120596119863

119879)

1198611

=120582 (119879)

cos (120596119863

119905)minus 1198601

tan (120596119863

119879)

(40)


in which

120585 =119888

2119898120596=

119888

119888crlt 1 120596

119863

= 120596radic1 minus 1205852 (41)


120582 (119879) = 0 (42)

120582119905

(119879) = 119872minus1

(119887) times (minus120597119892119879

120597119911) (43)


1198601

=120582119905

(119879)

120596119863

cos (120596119863

119879)

1198611

= minus120582119905

(119879)

120596119863

sin (120596119863

119879)

(44)

Note that 120597119892120597119911 like 1198601

and 1198611


separately So

120582119879

(119905) = 119890120585120596(119905minus119879)

(120582119905

(119879)

120596119863

cos (120596119863

119879) sin (120596119863

119905)

minus120582119905

(119879)

120596119863

sin (120596119863

119879) cos (120596119863

119905))

= 119875119879

119891 (119905) + 119876119879

119892 (119905)

(45)

in which

119875119879

= 119890minus120585120596119879

120582119905

(119879)

120596119863

cos (120596119863

119879)

119891 (119905) = 119890120585120596119905 sin (120596

119863

119905)

119876119879

= minus119890minus120585120596119879

120582119905

(119879)

120596119863

sin (120596119863

119879)

119892 (119905) = 119890120585120596119905 cos (120596

119863

119905)

(46)


119889120595

119889119887= int119879

0

120597119877

120597119887119889119905 (47)

In this equation

119877 = 119879

119865 (119905) minus 119879

119872119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887)

119862 = 1198860

119870 (119887) + 1198861

119872(48)


120597119877

120597119887= minus120582119879119886

0

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887 (49)


119889120595

119889119887(119879) = int

119879

0

(minus1205821198791198860

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887) 119889119905 (50)


120582 = [120601] 119884 (51)


119884 (119879) = 119872minus1

[120601]119879

[119898] 120582 (119879) (52)

119884119905

(119879) = 119872minus1

[120601]119879

[119898] 120582119905

(119879) (53)



[119872] 119884119905119905

minus [119862] 119884119905

+ [119870] 119884 = 0 (54)


119872119894

119884119905119905119894

minus 119862119894

119884119905119894 + 119870119894

119884119894

= 0 (55)

119889120595

119889119887(119879) = minus int

119879

0

⟨119884⟩ times [120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

+ ⟨119884⟩ times [120601]119879

times [120597119896

120597119887] times 119911 119889119905

(56)

Consider

[120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

= 119911119911119905

[120601]119879

times [120597119896

120597119887] times 119911 = 119911119911

(57)


119889120595

119889119887(119879) = minusint

119879

0

⟨119884⟩ times 119911119911119905

+ ⟨119884⟩ times 119911119911 119889119905 (58)


119884 = 119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905) (59)


119889120595

119889119887(119879)

= minusint119879

0

(119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911119905

+ (119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911 119889119905

(60)



119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)


119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)


119889120595


(63)


119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)


120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)


119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)



(119879) from (43)

Step 2 Calculate 120596 120596119863



and 119911119911 andconsider 119895 = 1


119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)


Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899


= 119879119899

and 119879119899

= 119879119899





Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120575120572 = [119878119879

119878]minus1

119878119879

120575119911 (8)

120572119895+1

= 120572119895

+ [119878119879

119895

119878119895

]minus1

119878119879

119895

( minus 119877cal) (9)





120575120572 = (119878119879

119878 + 120582119877

119868)minus1

119878119879

120575119911 (10)


119869 (120575120572 120582119877

) = 119878120575120572 minus 1205751199112

+ 1205821198771205751205722 (11)


119877


119877


119894


119894

EI119894


Δ119870119887119894

= (119870119887119894

minus 119887119894

) = (1 minus 120573119894

)119870119887119894

(12)

Sensitivity methods




Discrete approach



Continuum approach


Continuum-continuum

method


where 119870119887119894

and 119887119894


119887119894


119894

isin


119894


119894



119870119887

=

119873

sum119894=1

119860119879

119894

119887119894

119860119894

=

119873

sum119894=1

120573119894

119860119879

119894

119870119887119894

119860119894

(13)

where 119860119894









119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906)

Δ119906 (14)


119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906 minus Δ119906)

2Δ119906 (15)






119905

where 119911119905


119872(119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 = 119865 (119905 119887) (16)


119911 (0) = 1199110

119911119905

(0) = 1199110

119905

(17)


[119872] 120597119911119905119905

120597119887119894 + [119862]

120597119911119905

120597119887119894 + [119870]

120597119911

120597119887119894

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(18)

in which 120597119911120597119887119894

120597119911119905

120597119887119894

and 120597119911119905119905

120597119887119894


119884119905119905

=120597119911119905119905

120597119887119894 (19a)

119884119905

=120597119911119905

120597119887119894 (19b)

119884 =120597119911

120597119887119894 (19c)


[119872] 119884119905119905

+ [119862] 119884119905

+ [119870] 119884

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(20)





120595 = 119892 (119911 (119879) 119887) + int119879

0

119866 (119911 119887) 119889119905 (21)


Ω(119911 (119879) 119911119905

(119879) 119887) = 0 (22)


Ω119905

=120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119911119905119905

(119879) = 0 (23)



119887120591

= 119887 + 120591120575119887 (24)



120591


1205951015840

=120597119892

120597119887120575119887 +

120597119892

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] + 119866 (119911 (119879) 119887) 1198791015840

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905

(25)

where

1199111015840

= 1199111015840

(119887 120575119887) equiv119889

119889120591119911 (119905 119887 + 120591120575119887)|

120591=0

=119889

119889119887[119911 (119905 119887)] 120575119887

1198791015840

= 1198791015840

(119887 120575119887) equiv119889

119889120591119879 (119887 + 120591120575119887)|

120591=0

=119889119879

119889119887120575119887

(26)



120597Ω

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] +120597Ω

120597119911119905

[1199111015840

119905

(119879) + 119911119905119905

(119879) 1198791015840

]

+120597Ω

120597119887120575119887 = 0

(27)


Ω119905

1198791015840

= [120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119905

119911119905119905

(119879)]1198791015840

= minus (120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887)

(28)


= 0 then

1198791015840

= minus1

Ω119905

(120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887) (29)


1205951015840

= [120597119892

120597119911minus (

120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887))1

Ω119905

120597Ω

120597119911] 1199111015840

(119879)

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119911119905

1199111015840

119905

(119879)

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905 +

120597119892

120597119887120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

(30)


1015840

119905





int119879

0

120582119879

[119872 (119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 minus 119865 (119905 119887)] 119889119905 = 0

(31)


120591


int119879

0

[120582119879

119872(119887) 1199111015840

119905119905

+ 120582119879

119862 (119887) 1199111015840

119905

+ 120582119879

119870 (119887) 1199111015840

minus120597119877

120597119887120575119887] 119889119905 = 0

(32)

where

119877 = 119879

119865 (119905 119887) minus 119879

119872(119887) 119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887) (33)




120582119879

119872(119887) 1199111015840

119905

(119879) minus 120582119879

119905

(119879)119872 (119887) 1199111015840

(119879) + 120582119879

119862 (119887) 1199111015840

(119879)

+ int119879

0

[120582119879

119905119905

119872(119887) minus 120582119879

119905

119862 (119887) + 120582119879

119870 (119887)] 1199111015840

minus120597119877

120597119887120575119887 119889119905

= 0

(34)


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 in (30) and


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 can be


119872(119887) 120582 (119879) = minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911119905

(35)

119872(119887) 120582119905

(119879) = 119862119879

(119887) 120582 (119879) minus120597119892119879

120597119911

+ [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911

(36)

119872(119887) 120582119905119905

minus 119862119879

(119887) 120582119905

+ 119870 (119887) 120582 =120597119866119879

120597119911 0 le 119905 le 119879

(37)



119905

(119879)120597Ω120597119911 120597Ω120597119911

119905

and Ω119905




1205951015840

=120597119892

120597119887120575119887 + int

119879

0

[120597119866

120597119887+

120597119877

120597119887] 119889119905120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

equiv120597120595

120597119887120575119887

(38)



119889120595

119889119887

=120597119892

120597119887(119911 (119879) 119887)

+ int119879

0

[120597119866

120597119887(119911 119887) +

120597119877

120597119887(120582 (119905) 119911 (119905) 119911

119905

(119905) 119911119905119905

(119905) 119887)] 119889119905

minus1

Ω119905

[120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]120597Ω

120597119887

(39)









4 Proposed Method


119898120582119905119905

minus 119888120582119905

+ 119896120582 = 0


120582119879

(119905) = 119890120585120596(119905minus119879)

(1198601

sin (120596119863

119905) + 1198611

cos (120596119863

119905))

1198601

= (120582119905

(119879)

120596119863

minus120585

radic1 minus 1205852120582 (119879)) cos (120596

119863

119879)

+ 120582 (119879) sin (120596119863

119879)

1198611

=120582 (119879)

cos (120596119863

119905)minus 1198601

tan (120596119863

119879)

(40)


in which

120585 =119888

2119898120596=

119888

119888crlt 1 120596

119863

= 120596radic1 minus 1205852 (41)


120582 (119879) = 0 (42)

120582119905

(119879) = 119872minus1

(119887) times (minus120597119892119879

120597119911) (43)


1198601

=120582119905

(119879)

120596119863

cos (120596119863

119879)

1198611

= minus120582119905

(119879)

120596119863

sin (120596119863

119879)

(44)

Note that 120597119892120597119911 like 1198601

and 1198611


separately So

120582119879

(119905) = 119890120585120596(119905minus119879)

(120582119905

(119879)

120596119863

cos (120596119863

119879) sin (120596119863

119905)

minus120582119905

(119879)

120596119863

sin (120596119863

119879) cos (120596119863

119905))

= 119875119879

119891 (119905) + 119876119879

119892 (119905)

(45)

in which

119875119879

= 119890minus120585120596119879

120582119905

(119879)

120596119863

cos (120596119863

119879)

119891 (119905) = 119890120585120596119905 sin (120596

119863

119905)

119876119879

= minus119890minus120585120596119879

120582119905

(119879)

120596119863

sin (120596119863

119879)

119892 (119905) = 119890120585120596119905 cos (120596

119863

119905)

(46)


119889120595

119889119887= int119879

0

120597119877

120597119887119889119905 (47)

In this equation

119877 = 119879

119865 (119905) minus 119879

119872119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887)

119862 = 1198860

119870 (119887) + 1198861

119872(48)


120597119877

120597119887= minus120582119879119886

0

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887 (49)


119889120595

119889119887(119879) = int

119879

0

(minus1205821198791198860

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887) 119889119905 (50)


120582 = [120601] 119884 (51)


119884 (119879) = 119872minus1

[120601]119879

[119898] 120582 (119879) (52)

119884119905

(119879) = 119872minus1

[120601]119879

[119898] 120582119905

(119879) (53)



[119872] 119884119905119905

minus [119862] 119884119905

+ [119870] 119884 = 0 (54)


119872119894

119884119905119905119894

minus 119862119894

119884119905119894 + 119870119894

119884119894

= 0 (55)

119889120595

119889119887(119879) = minus int

119879

0

⟨119884⟩ times [120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

+ ⟨119884⟩ times [120601]119879

times [120597119896

120597119887] times 119911 119889119905

(56)

Consider

[120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

= 119911119911119905

[120601]119879

times [120597119896

120597119887] times 119911 = 119911119911

(57)


119889120595

119889119887(119879) = minusint

119879

0

⟨119884⟩ times 119911119911119905

+ ⟨119884⟩ times 119911119911 119889119905 (58)


119884 = 119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905) (59)


119889120595

119889119887(119879)

= minusint119879

0

(119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911119905

+ (119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911 119889119905

(60)



119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)


119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)


119889120595


(63)


119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)


120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)


119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)



(119879) from (43)

Step 2 Calculate 120596 120596119863



and 119911119911 andconsider 119895 = 1


119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)


Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899


= 119879119899

and 119879119899

= 119879119899





Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906)

Δ119906 (14)


119889120595

119889119906asymp

120595 (119906 + Δ119906) minus 120595 (119906 minus Δ119906)

2Δ119906 (15)






119905

where 119911119905


119872(119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 = 119865 (119905 119887) (16)


119911 (0) = 1199110

119911119905

(0) = 1199110

119905

(17)


[119872] 120597119911119905119905

120597119887119894 + [119862]

120597119911119905

120597119887119894 + [119870]

120597119911

120597119887119894

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(18)

in which 120597119911120597119887119894

120597119911119905

120597119887119894

and 120597119911119905119905

120597119887119894


119884119905119905

=120597119911119905119905

120597119887119894 (19a)

119884119905

=120597119911119905

120597119887119894 (19b)

119884 =120597119911

120597119887119894 (19c)


[119872] 119884119905119905

+ [119862] 119884119905

+ [119870] 119884

= minus120597 [119870]

120597119887119894119911 minus 120572

2

120597 [119870]

120597119887119894119911119905

(20)





120595 = 119892 (119911 (119879) 119887) + int119879

0

119866 (119911 119887) 119889119905 (21)


Ω(119911 (119879) 119911119905

(119879) 119887) = 0 (22)


Ω119905

=120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119911119905119905

(119879) = 0 (23)



119887120591

= 119887 + 120591120575119887 (24)



120591


1205951015840

=120597119892

120597119887120575119887 +

120597119892

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] + 119866 (119911 (119879) 119887) 1198791015840

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905

(25)

where

1199111015840

= 1199111015840

(119887 120575119887) equiv119889

119889120591119911 (119905 119887 + 120591120575119887)|

120591=0

=119889

119889119887[119911 (119905 119887)] 120575119887

1198791015840

= 1198791015840

(119887 120575119887) equiv119889

119889120591119879 (119887 + 120591120575119887)|

120591=0

=119889119879

119889119887120575119887

(26)



120597Ω

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] +120597Ω

120597119911119905

[1199111015840

119905

(119879) + 119911119905119905

(119879) 1198791015840

]

+120597Ω

120597119887120575119887 = 0

(27)


Ω119905

1198791015840

= [120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119905

119911119905119905

(119879)]1198791015840

= minus (120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887)

(28)


= 0 then

1198791015840

= minus1

Ω119905

(120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887) (29)


1205951015840

= [120597119892

120597119911minus (

120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887))1

Ω119905

120597Ω

120597119911] 1199111015840

(119879)

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119911119905

1199111015840

119905

(119879)

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905 +

120597119892

120597119887120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

(30)


1015840

119905





int119879

0

120582119879

[119872 (119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 minus 119865 (119905 119887)] 119889119905 = 0

(31)


120591


int119879

0

[120582119879

119872(119887) 1199111015840

119905119905

+ 120582119879

119862 (119887) 1199111015840

119905

+ 120582119879

119870 (119887) 1199111015840

minus120597119877

120597119887120575119887] 119889119905 = 0

(32)

where

119877 = 119879

119865 (119905 119887) minus 119879

119872(119887) 119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887) (33)




120582119879

119872(119887) 1199111015840

119905

(119879) minus 120582119879

119905

(119879)119872 (119887) 1199111015840

(119879) + 120582119879

119862 (119887) 1199111015840

(119879)

+ int119879

0

[120582119879

119905119905

119872(119887) minus 120582119879

119905

119862 (119887) + 120582119879

119870 (119887)] 1199111015840

minus120597119877

120597119887120575119887 119889119905

= 0

(34)


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 in (30) and


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 can be


119872(119887) 120582 (119879) = minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911119905

(35)

119872(119887) 120582119905

(119879) = 119862119879

(119887) 120582 (119879) minus120597119892119879

120597119911

+ [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911

(36)

119872(119887) 120582119905119905

minus 119862119879

(119887) 120582119905

+ 119870 (119887) 120582 =120597119866119879

120597119911 0 le 119905 le 119879

(37)



119905

(119879)120597Ω120597119911 120597Ω120597119911

119905

and Ω119905




1205951015840

=120597119892

120597119887120575119887 + int

119879

0

[120597119866

120597119887+

120597119877

120597119887] 119889119905120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

equiv120597120595

120597119887120575119887

(38)



119889120595

119889119887

=120597119892

120597119887(119911 (119879) 119887)

+ int119879

0

[120597119866

120597119887(119911 119887) +

120597119877

120597119887(120582 (119905) 119911 (119905) 119911

119905

(119905) 119911119905119905

(119905) 119887)] 119889119905

minus1

Ω119905

[120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]120597Ω

120597119887

(39)









4 Proposed Method


119898120582119905119905

minus 119888120582119905

+ 119896120582 = 0


120582119879

(119905) = 119890120585120596(119905minus119879)

(1198601

sin (120596119863

119905) + 1198611

cos (120596119863

119905))

1198601

= (120582119905

(119879)

120596119863

minus120585

radic1 minus 1205852120582 (119879)) cos (120596

119863

119879)

+ 120582 (119879) sin (120596119863

119879)

1198611

=120582 (119879)

cos (120596119863

119905)minus 1198601

tan (120596119863

119879)

(40)


in which

120585 =119888

2119898120596=

119888

119888crlt 1 120596

119863

= 120596radic1 minus 1205852 (41)


120582 (119879) = 0 (42)

120582119905

(119879) = 119872minus1

(119887) times (minus120597119892119879

120597119911) (43)


1198601

=120582119905

(119879)

120596119863

cos (120596119863

119879)

1198611

= minus120582119905

(119879)

120596119863

sin (120596119863

119879)

(44)

Note that 120597119892120597119911 like 1198601

and 1198611


separately So

120582119879

(119905) = 119890120585120596(119905minus119879)

(120582119905

(119879)

120596119863

cos (120596119863

119879) sin (120596119863

119905)

minus120582119905

(119879)

120596119863

sin (120596119863

119879) cos (120596119863

119905))

= 119875119879

119891 (119905) + 119876119879

119892 (119905)

(45)

in which

119875119879

= 119890minus120585120596119879

120582119905

(119879)

120596119863

cos (120596119863

119879)

119891 (119905) = 119890120585120596119905 sin (120596

119863

119905)

119876119879

= minus119890minus120585120596119879

120582119905

(119879)

120596119863

sin (120596119863

119879)

119892 (119905) = 119890120585120596119905 cos (120596

119863

119905)

(46)


119889120595

119889119887= int119879

0

120597119877

120597119887119889119905 (47)

In this equation

119877 = 119879

119865 (119905) minus 119879

119872119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887)

119862 = 1198860

119870 (119887) + 1198861

119872(48)


120597119877

120597119887= minus120582119879119886

0

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887 (49)


119889120595

119889119887(119879) = int

119879

0

(minus1205821198791198860

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887) 119889119905 (50)


120582 = [120601] 119884 (51)


119884 (119879) = 119872minus1

[120601]119879

[119898] 120582 (119879) (52)

119884119905

(119879) = 119872minus1

[120601]119879

[119898] 120582119905

(119879) (53)



[119872] 119884119905119905

minus [119862] 119884119905

+ [119870] 119884 = 0 (54)


119872119894

119884119905119905119894

minus 119862119894

119884119905119894 + 119870119894

119884119894

= 0 (55)

119889120595

119889119887(119879) = minus int

119879

0

⟨119884⟩ times [120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

+ ⟨119884⟩ times [120601]119879

times [120597119896

120597119887] times 119911 119889119905

(56)

Consider

[120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

= 119911119911119905

[120601]119879

times [120597119896

120597119887] times 119911 = 119911119911

(57)


119889120595

119889119887(119879) = minusint

119879

0

⟨119884⟩ times 119911119911119905

+ ⟨119884⟩ times 119911119911 119889119905 (58)


119884 = 119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905) (59)


119889120595

119889119887(119879)

= minusint119879

0

(119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911119905

+ (119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911 119889119905

(60)



119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)


119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)


119889120595


(63)


119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)


120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)


119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)



(119879) from (43)

Step 2 Calculate 120596 120596119863



and 119911119911 andconsider 119895 = 1


119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)


Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899


= 119879119899

and 119879119899

= 119879119899





Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120591


1205951015840

=120597119892

120597119887120575119887 +

120597119892

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] + 119866 (119911 (119879) 119887) 1198791015840

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905

(25)

where

1199111015840

= 1199111015840

(119887 120575119887) equiv119889

119889120591119911 (119905 119887 + 120591120575119887)|

120591=0

=119889

119889119887[119911 (119905 119887)] 120575119887

1198791015840

= 1198791015840

(119887 120575119887) equiv119889

119889120591119879 (119887 + 120591120575119887)|

120591=0

=119889119879

119889119887120575119887

(26)



120597Ω

120597119911[1199111015840

(119879) + 119911119905

(119879) 1198791015840

] +120597Ω

120597119911119905

[1199111015840

119905

(119879) + 119911119905119905

(119879) 1198791015840

]

+120597Ω

120597119887120575119887 = 0

(27)


Ω119905

1198791015840

= [120597Ω

120597119911119911119905

(119879) +120597Ω

120597119911119905

119911119905119905

(119879)]1198791015840

= minus (120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887)

(28)


= 0 then

1198791015840

= minus1

Ω119905

(120597Ω

1205971199111199111015840

(119879) +120597Ω

120597119911119905

1199111015840

119905

(119879) +120597Ω

120597119887120575119887) (29)


1205951015840

= [120597119892

120597119911minus (

120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887))1

Ω119905

120597Ω

120597119911] 1199111015840

(119879)

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119911119905

1199111015840

119905

(119879)

+ int119879

0

[120597119866

1205971199111199111015840

+120597119866

120597119887120575119887] 119889119905 +

120597119892

120597119887120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

(30)


1015840

119905





int119879

0

120582119879

[119872 (119887) 119911119905119905

+ 119862 (119887) 119911119905

+ 119870 (119887) 119911 minus 119865 (119905 119887)] 119889119905 = 0

(31)


120591


int119879

0

[120582119879

119872(119887) 1199111015840

119905119905

+ 120582119879

119862 (119887) 1199111015840

119905

+ 120582119879

119870 (119887) 1199111015840

minus120597119877

120597119887120575119887] 119889119905 = 0

(32)

where

119877 = 119879

119865 (119905 119887) minus 119879

119872(119887) 119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887) (33)




120582119879

119872(119887) 1199111015840

119905

(119879) minus 120582119879

119905

(119879)119872 (119887) 1199111015840

(119879) + 120582119879

119862 (119887) 1199111015840

(119879)

+ int119879

0

[120582119879

119905119905

119872(119887) minus 120582119879

119905

119862 (119887) + 120582119879

119870 (119887)] 1199111015840

minus120597119877

120597119887120575119887 119889119905

= 0

(34)


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 in (30) and


1015840

(119879) 1199111015840119905

(119879) and 1199111015840 can be


119872(119887) 120582 (119879) = minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911119905

(35)

119872(119887) 120582119905

(119879) = 119862119879

(119887) 120582 (119879) minus120597119892119879

120597119911

+ [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω119879

120597119911

(36)

119872(119887) 120582119905119905

minus 119862119879

(119887) 120582119905

+ 119870 (119887) 120582 =120597119866119879

120597119911 0 le 119905 le 119879

(37)



119905

(119879)120597Ω120597119911 120597Ω120597119911

119905

and Ω119905




1205951015840

=120597119892

120597119887120575119887 + int

119879

0

[120597119866

120597119887+

120597119877

120597119887] 119889119905120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

equiv120597120595

120597119887120575119887

(38)



119889120595

119889119887

=120597119892

120597119887(119911 (119879) 119887)

+ int119879

0

[120597119866

120597119887(119911 119887) +

120597119877

120597119887(120582 (119905) 119911 (119905) 119911

119905

(119905) 119911119905119905

(119905) 119887)] 119889119905

minus1

Ω119905

[120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]120597Ω

120597119887

(39)









4 Proposed Method


119898120582119905119905

minus 119888120582119905

+ 119896120582 = 0


120582119879

(119905) = 119890120585120596(119905minus119879)

(1198601

sin (120596119863

119905) + 1198611

cos (120596119863

119905))

1198601

= (120582119905

(119879)

120596119863

minus120585

radic1 minus 1205852120582 (119879)) cos (120596

119863

119879)

+ 120582 (119879) sin (120596119863

119879)

1198611

=120582 (119879)

cos (120596119863

119905)minus 1198601

tan (120596119863

119879)

(40)


in which

120585 =119888

2119898120596=

119888

119888crlt 1 120596

119863

= 120596radic1 minus 1205852 (41)


120582 (119879) = 0 (42)

120582119905

(119879) = 119872minus1

(119887) times (minus120597119892119879

120597119911) (43)


1198601

=120582119905

(119879)

120596119863

cos (120596119863

119879)

1198611

= minus120582119905

(119879)

120596119863

sin (120596119863

119879)

(44)

Note that 120597119892120597119911 like 1198601

and 1198611


separately So

120582119879

(119905) = 119890120585120596(119905minus119879)

(120582119905

(119879)

120596119863

cos (120596119863

119879) sin (120596119863

119905)

minus120582119905

(119879)

120596119863

sin (120596119863

119879) cos (120596119863

119905))

= 119875119879

119891 (119905) + 119876119879

119892 (119905)

(45)

in which

119875119879

= 119890minus120585120596119879

120582119905

(119879)

120596119863

cos (120596119863

119879)

119891 (119905) = 119890120585120596119905 sin (120596

119863

119905)

119876119879

= minus119890minus120585120596119879

120582119905

(119879)

120596119863

sin (120596119863

119879)

119892 (119905) = 119890120585120596119905 cos (120596

119863

119905)

(46)


119889120595

119889119887= int119879

0

120597119877

120597119887119889119905 (47)

In this equation

119877 = 119879

119865 (119905) minus 119879

119872119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887)

119862 = 1198860

119870 (119887) + 1198861

119872(48)


120597119877

120597119887= minus120582119879119886

0

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887 (49)


119889120595

119889119887(119879) = int

119879

0

(minus1205821198791198860

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887) 119889119905 (50)


120582 = [120601] 119884 (51)


119884 (119879) = 119872minus1

[120601]119879

[119898] 120582 (119879) (52)

119884119905

(119879) = 119872minus1

[120601]119879

[119898] 120582119905

(119879) (53)



[119872] 119884119905119905

minus [119862] 119884119905

+ [119870] 119884 = 0 (54)


119872119894

119884119905119905119894

minus 119862119894

119884119905119894 + 119870119894

119884119894

= 0 (55)

119889120595

119889119887(119879) = minus int

119879

0

⟨119884⟩ times [120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

+ ⟨119884⟩ times [120601]119879

times [120597119896

120597119887] times 119911 119889119905

(56)

Consider

[120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

= 119911119911119905

[120601]119879

times [120597119896

120597119887] times 119911 = 119911119911

(57)


119889120595

119889119887(119879) = minusint

119879

0

⟨119884⟩ times 119911119911119905

+ ⟨119884⟩ times 119911119911 119889119905 (58)


119884 = 119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905) (59)


119889120595

119889119887(119879)

= minusint119879

0

(119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911119905

+ (119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911 119889119905

(60)



119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)


119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)


119889120595


(63)


119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)


120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)


119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)



(119879) from (43)

Step 2 Calculate 120596 120596119863



and 119911119911 andconsider 119895 = 1


119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)


Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899


= 119879119899

and 119879119899

= 119879119899





Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119905

(119879)120597Ω120597119911 120597Ω120597119911

119905

and Ω119905




1205951015840

=120597119892

120597119887120575119887 + int

119879

0

[120597119866

120597119887+

120597119877

120597119887] 119889119905120575119887

minus [120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]1

Ω119905

120597Ω

120597119887120575119887

equiv120597120595

120597119887120575119887

(38)



119889120595

119889119887

=120597119892

120597119887(119911 (119879) 119887)

+ int119879

0

[120597119866

120597119887(119911 119887) +

120597119877

120597119887(120582 (119905) 119911 (119905) 119911

119905

(119905) 119911119905119905

(119905) 119887)] 119889119905

minus1

Ω119905

[120597119892

120597119911119911119905

(119879) + 119866 (119911 (119879) 119887)]120597Ω

120597119887

(39)









4 Proposed Method


119898120582119905119905

minus 119888120582119905

+ 119896120582 = 0


120582119879

(119905) = 119890120585120596(119905minus119879)

(1198601

sin (120596119863

119905) + 1198611

cos (120596119863

119905))

1198601

= (120582119905

(119879)

120596119863

minus120585

radic1 minus 1205852120582 (119879)) cos (120596

119863

119879)

+ 120582 (119879) sin (120596119863

119879)

1198611

=120582 (119879)

cos (120596119863

119905)minus 1198601

tan (120596119863

119879)

(40)


in which

120585 =119888

2119898120596=

119888

119888crlt 1 120596

119863

= 120596radic1 minus 1205852 (41)


120582 (119879) = 0 (42)

120582119905

(119879) = 119872minus1

(119887) times (minus120597119892119879

120597119911) (43)


1198601

=120582119905

(119879)

120596119863

cos (120596119863

119879)

1198611

= minus120582119905

(119879)

120596119863

sin (120596119863

119879)

(44)

Note that 120597119892120597119911 like 1198601

and 1198611


separately So

120582119879

(119905) = 119890120585120596(119905minus119879)

(120582119905

(119879)

120596119863

cos (120596119863

119879) sin (120596119863

119905)

minus120582119905

(119879)

120596119863

sin (120596119863

119879) cos (120596119863

119905))

= 119875119879

119891 (119905) + 119876119879

119892 (119905)

(45)

in which

119875119879

= 119890minus120585120596119879

120582119905

(119879)

120596119863

cos (120596119863

119879)

119891 (119905) = 119890120585120596119905 sin (120596

119863

119905)

119876119879

= minus119890minus120585120596119879

120582119905

(119879)

120596119863

sin (120596119863

119879)

119892 (119905) = 119890120585120596119905 cos (120596

119863

119905)

(46)


119889120595

119889119887= int119879

0

120597119877

120597119887119889119905 (47)

In this equation

119877 = 119879

119865 (119905) minus 119879

119872119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887)

119862 = 1198860

119870 (119887) + 1198861

119872(48)


120597119877

120597119887= minus120582119879119886

0

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887 (49)


119889120595

119889119887(119879) = int

119879

0

(minus1205821198791198860

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887) 119889119905 (50)


120582 = [120601] 119884 (51)


119884 (119879) = 119872minus1

[120601]119879

[119898] 120582 (119879) (52)

119884119905

(119879) = 119872minus1

[120601]119879

[119898] 120582119905

(119879) (53)



[119872] 119884119905119905

minus [119862] 119884119905

+ [119870] 119884 = 0 (54)


119872119894

119884119905119905119894

minus 119862119894

119884119905119894 + 119870119894

119884119894

= 0 (55)

119889120595

119889119887(119879) = minus int

119879

0

⟨119884⟩ times [120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

+ ⟨119884⟩ times [120601]119879

times [120597119896

120597119887] times 119911 119889119905

(56)

Consider

[120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

= 119911119911119905

[120601]119879

times [120597119896

120597119887] times 119911 = 119911119911

(57)


119889120595

119889119887(119879) = minusint

119879

0

⟨119884⟩ times 119911119911119905

+ ⟨119884⟩ times 119911119911 119889119905 (58)


119884 = 119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905) (59)


119889120595

119889119887(119879)

= minusint119879

0

(119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911119905

+ (119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911 119889119905

(60)



119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)


119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)


119889120595


(63)


119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)


120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)


119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)



(119879) from (43)

Step 2 Calculate 120596 120596119863



and 119911119911 andconsider 119895 = 1


119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)


Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899


= 119879119899

and 119879119899

= 119879119899





Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










in which

120585 =119888

2119898120596=

119888

119888crlt 1 120596

119863

= 120596radic1 minus 1205852 (41)


120582 (119879) = 0 (42)

120582119905

(119879) = 119872minus1

(119887) times (minus120597119892119879

120597119911) (43)


1198601

=120582119905

(119879)

120596119863

cos (120596119863

119879)

1198611

= minus120582119905

(119879)

120596119863

sin (120596119863

119879)

(44)

Note that 120597119892120597119911 like 1198601

and 1198611


separately So

120582119879

(119905) = 119890120585120596(119905minus119879)

(120582119905

(119879)

120596119863

cos (120596119863

119879) sin (120596119863

119905)

minus120582119905

(119879)

120596119863

sin (120596119863

119879) cos (120596119863

119905))

= 119875119879

119891 (119905) + 119876119879

119892 (119905)

(45)

in which

119875119879

= 119890minus120585120596119879

120582119905

(119879)

120596119863

cos (120596119863

119879)

119891 (119905) = 119890120585120596119905 sin (120596

119863

119905)

119876119879

= minus119890minus120585120596119879

120582119905

(119879)

120596119863

sin (120596119863

119879)

119892 (119905) = 119890120585120596119905 cos (120596

119863

119905)

(46)


119889120595

119889119887= int119879

0

120597119877

120597119887119889119905 (47)

In this equation

119877 = 119879

119865 (119905) minus 119879

119872119905119905

minus 119879

119862 (119887) 119905

minus 119879

119870 (119887)

119862 = 1198860

119870 (119887) + 1198861

119872(48)


120597119877

120597119887= minus120582119879119886

0

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887 (49)


119889120595

119889119887(119879) = int

119879

0

(minus1205821198791198860

120597119870

120597119887119911119905

minus 120582119879120597119870

120597119887) 119889119905 (50)


120582 = [120601] 119884 (51)


119884 (119879) = 119872minus1

[120601]119879

[119898] 120582 (119879) (52)

119884119905

(119879) = 119872minus1

[120601]119879

[119898] 120582119905

(119879) (53)



[119872] 119884119905119905

minus [119862] 119884119905

+ [119870] 119884 = 0 (54)


119872119894

119884119905119905119894

minus 119862119894

119884119905119894 + 119870119894

119884119894

= 0 (55)

119889120595

119889119887(119879) = minus int

119879

0

⟨119884⟩ times [120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

+ ⟨119884⟩ times [120601]119879

times [120597119896

120597119887] times 119911 119889119905

(56)

Consider

[120601]119879

times 1198860

[120597119896

120597119887] times 119911

119905

= 119911119911119905

[120601]119879

times [120597119896

120597119887] times 119911 = 119911119911

(57)


119889120595

119889119887(119879) = minusint

119879

0

⟨119884⟩ times 119911119911119905

+ ⟨119884⟩ times 119911119911 119889119905 (58)


119884 = 119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905) (59)


119889120595

119889119887(119879)

= minusint119879

0

(119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911119905

+ (119875 (119879) sdot 119891 (119905) + 119876 (119879) sdot 119892 (119905))119879

times 119911119911 119889119905

(60)



119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)


119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)


119889120595


(63)


119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)


120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)


119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)



(119879) from (43)

Step 2 Calculate 120596 120596119863



and 119911119911 andconsider 119895 = 1


119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)


Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899


= 119879119899

and 119879119899

= 119879119899





Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119889120595

119889119887(119879)

= minusint119879

0

⟨119875 (119879)⟩ times (119891 (119905) sdot 119911119911119905

+ 119891 (119905) sdot 119911119911119905

)

+ ⟨119876 (119879)⟩ times (119892 (119905) sdot 119911119911119905

+ 119892 (119905) sdot 119911119911119905

) 119889119905

(61)


119860 = int119879

0

119891 (119905) sdot 119911119911119905

119889119905

119861 = int119879

0

119892 (119905) sdot 119911119911119905

119889119905

119862 = int119879

0

119891 (119905) sdot 119911119911 119889119905

119863 = int119879

0

119892 (119905) sdot 119911119911 119889119905

(62)


119889120595


(63)


119860119879+Δ119879

= int119879+Δ119879

0

119891 (119905) sdot 119911119911119905

119889119905

= int119879

0

119891 (119905) sdot 119911119911119905

119889119905 + int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

(64)

119860119879+Δ119879

= 119860119879

+ 120575119860

120575119860 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911119905

119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

(65)


120575119861 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911119905

119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911

119905

(119879 +Δ119879

2)

120575119862 = int119879+Δ119879

119879

119891 (119905) sdot 119911119911 119889119905

cong 119891(119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

120575119863 = int119879+Δ119879

119879

119892 (119905) sdot 119911119911 119889119905

cong 119892 (119879 +Δ119879

2) sdot 119911119911 (119879 +

Δ119879

2)

(66)


119889120595

119889119887(119879 + Δ119879) = minus ⟨119875 (119879 + Δ119879)⟩ times (119860

119879+Δ119879

+ 119862119879+Δ119879

)

minus ⟨119876 (119879 + Δ119879)⟩ times (119861119879+Δ119879

+ 119863119879+Δ119879

)

(67)



(119879) from (43)

Step 2 Calculate 120596 120596119863



and 119911119911 andconsider 119895 = 1


119905

(119879) from Step 1 and 119884119905

(119879) from (53) and 119879119899

= Δ119905

and 119879119900

= 0

Step 5 Consider 119860 = 119861 = 119862 = 119863 = 0


= 1198790

+ (Δ1199052) and calculate 119875(119879119899

) minus

119876(119879119899

) minus 119891(119879119898

) minus 119892(119879119898

) from (45)


Step 8 Calculate 119889120595119889119887(119879119899

) from (67)

Step 9 If 119879119899


= 119879119899

and 119879119899

= 119879119899





Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Sensors



10000mm10000mm 10000mm

P

VZ

X1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30




1003817100381710038171003817119864119894+1 minus 119864119894

10038171003817100381710038171003817100381710038171003817119864119894

1003817100381710038171003817times 100 le Tol1


10038171003817100381710038171003817100381710038171003817Response119894

1003817100381710038171003817times 100 le Tol2

(68)




5 Numerical Results



RPE =


1003817100381710038171003817119864True1003817100381710038171003817

times 100 (69)






0

and 1198861






120572V =119881

119881cr (70)









M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














M1-1 00465 00461 00457 00454 0045 00743 00416 00471 0088M1-2 03135 0317 03165 03157 0315 02937 0291 02967 00038M1-3 00273 00268 00265 00262 00259 00281 00007 00007 00007M1-4 0052 00525 00516 00522 00531 00382 00576 00346 00155M1-5 00411 00395 00408 00367 00403 006 00542 00207 00091M1-6 00502 00546 00485 00471 00431 0046 00422 0041 00007








0 5 10 15 20 25 30 350

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35Element number

Erro

r (

)

0 5 10 15 20 25 30 3505

101520

Dam

age i

ndex

Element number


minus004

minus002

0

002

004

times105




119905119905119899119901


119885119905119905119899119901

= 119885119905119905119899119891

+ RMS (119885119905119905119899119891





M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












M1-1 00354 00346 00342 00338 00337 00003 00214 00107 00079M1-2 00496 00338 00493 00585 00575 00294 0024 00396 00214M1-3 00008 00005 00005 00005 00005 00007 00007 00007 00006M1-4 00271 00247 00222 00077 00071 00161 00006 00287 00007M1-5 00051 00047 00028 00035 00031 01971 00171 0001 00134M1-6 00526 00237 00156 00009 00008 0065 0001 00008 00007

Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

40

45

(a)

Loops

Noi

se

2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

12

14

16

18

2

5

10

15

20

25

30

35

DDM method

(b)


where RMS(119885119905119905119899119891









REP =STDDMSTADM

(72)










02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














02468101214

0103

0507

09



Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6

003

05





0

and 1198861


minus5










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Sensors

Element number


Node number

P V

Moving vehicle

7000mm

3000mmXZY

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

43

44

45

46

42

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

41

32

33

34

35

36

37

38

39

40




M2-1 00003 00003 00003 00003 00004 00004 00003 00006 00006M2-2 00005 00006 00005 00004 00003 00004 00004 00005 00006M2-3 00004 00004 00006 00003 00006 00005 00003 00005 00003M2-4 00006 00006 00004 00005 00005 00005 00004 00002 00004M2-5 00005 00006 00006 00004 00004 00003 00005 00004 00003M2-6 00004 00004 00003 00005 00004 00004 00006 00003 00004



M2-1 00002 00002 00001 00009 00011 00034 00014 00007 00007M2-2 00006 00008 00004 00011 0001 00014 00015 00012 00007M2-3 00005 00007 00097 0001 00011 00013 00018 0001 00007M2-4 00003 00003 00007 00013 00007 0001 00012 00008 00011M2-5 0001 0001 00008 00009 0001 0001 00014 00007 00012M2-6 00007 00007 00007 00009 00011 00011 00011 00011 00011




6 Conclusion





0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0 5 10 15 20 25 30 35 40 45 500

051

152

25

Element number

Mod

ule o

f ela

stici

ty

0 5 10 15 20 25 30 35 40 45 50Element number

Erro

r (

)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Dam

age i

ndex

Element number


minus1

minus05

0

05

1times10

minus3

times105









Nomenclature


119911 119911119905

119911119905119905


119865 = 1198651

(119905) 1198652

(119905) 119865119873119865


119870119887119894

and 119887119894


Δ119870119887119894



1198860

and 1198861


119911119898





parameters


Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Loops

Noi

se

ADM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

10

20

30

40

50

60

(a)

Loops

Noi

se

DDM method

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

5

10

15

20

25

30

(b)


005115225335

0102

0304

0506

0708

09

Scen

ario

1Sc

enar

io 2

Scen

ario

3Sc

enar

io 4

Scen

ario

5Sc

enar

io 6




120582119877








and 119885119905119905119899119901





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









Research Article Damage Detection of Bridges Using...

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