Research ArticleLoad Reconstruction Technique Using 119863-Optimal Design andMarkov Parameters

Deepak K Gupta1 and Anoop K Dhingra2

1Structural Engineering Konecranes Nuclear Equipment amp Services New Berlin WI 53151 USA2Department of Mechanical Engineering University of Wisconsin Milwaukee WI 53201 USA

Correspondence should be addressed to Deepak K Gupta deepakguptakonecranescom

Received 30 October 2014 Revised 20 January 2015 Accepted 27 January 2015

Academic Editor Mickael Lallart

Copyright copy 2015 D K Gupta and A K Dhingra This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper develops a technique for identifying dynamic loads acting on a structure based on impulse response of the structure alsoreferred to as the systemMarkov parameters and structure responsemeasured at optimally placed sensors on the structure InverseMarkov parameters are computed from the forward Markov parameters using a linear prediction algorithm and have the roles ofinput and output reversedThe applied loads are then reconstructed by convolving the inverse Markov parameters with the systemresponse to the loads measured at optimal locations on the structureThe structure essentially acts as its own load transducer It hasbeen noted that the computation of inverse Markov parameters like most other inverse problems is ill-conditioned which causestheir convolution with the measured response to become quite sensitive to errors in systemmodeling and response measurementsThe computation of inverse Markov parameters (and thereby the quality of load estimates) depends on the locations of sensorson the structure To ensure that the computation of inverse Markov parameters is well-conditioned a solution approach based onthe construction of 119863-optimal designs is presented to determine the optimal sensor locations such that precise load estimates areobtained

1 Introduction

It is often desired to have knowledge of loads acting on astructure early in design process so that effective analysisand design optimization ensuring the structural integrity ofthe structure may be performed Precise identification ofthe loads is imperative to having high level of confidencein numerical simulation such as finite element analysiswhich precludes the need for expensive and time consumingexperimental testing In some cases the loads acting on astructure may be measured by introducing load transducers(load cells) between the structure and the applied loadsThis method of load measurement has some disadvantagesIntroduction of load transducers may change the systemdynamic characteristics and the system may not behave inthe same fashion as it would if the load transducers were notpresent This leads to inaccuracies in the load estimation In

some applications the input locations may not be accessibleto insert a load transducer for themeasurement of loads beingtransmitted to the structure

The limitations posed by external load transducers maybe overcome by transforming the structure into a self-transducer where instead of measuring the loads directlyresponse of the structure to the unknown applied loads ismeasured Physically measurable quantities such as displace-ments accelerations and strains that depend on the loadscan be used as response Some kind of relationship betweenthe measured quantity and the loads to be estimated isthen utilized to estimate the applied loads The instrumentedstructure effectively becomes a load transducer This class ofproblems is known as the ldquoinverse problemrdquo

There are several challenges associated with solving theinverse problemThemajor challenge is that the inverse prob-lems tend to be highly ill-conditioned The difficulties posed

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 605695 9 pageshttpdxdoiorg1011552015605695

2 Shock and Vibration

by the inverse problems pertaining to load identification aredocumented by Stevens [1] Ewins [2] used displacementtransducers to identify input loads and found that very smallvariations (noise) in the response measurement cause largeerrors in the force estimation Load identification problemin frequency domain was studied by Desanghere and Snoeys[3] where they attributed the reason for ill-conditioning to afew dominant elements in the frequency response function(FRF) matrix Hillary and Ewins [4] and Okubo et al [5]investigated the influence of noise in the measured responseand noted that the inversion process is highly ill-conditioned

Another technique was proposed by Busby and Trujillo[6] where the load estimation problem is treated as mini-mization problem of error between the measured structuralresponse and the response predicted from model which isthen solved using dynamic programmingThis techniquewasextended by Hollandsworth and Busby [7] by applying it toactual experimental measurements Application of the tech-nique is limited since the amount of computation increasesdramatically with the model order

Genaro and Rade [8] suggested a load reconstructionmethod from acceleration response and its successive inte-gration Carne et al [9] developed a technique for loadidentification known as the Sum of Weighted AccelerationsTechnique (SWAT) An optimization problem for load recov-ery was formulated by Hashemi and Kargarnovin [10] wherethe objective function is the difference between analyticaland measured responses and the decision variables are thelocation and magnitude of the applied force Lu and Law [11]proposed a load estimation technique based on sensitivityof structural responses A gradient-based model updatingmethod based on dynamic response sensitivity is utilizedto identify the force and physical parameters Boukria et al[12] applied the FRF technique to identify the location andmagnitude of impact force on a circular plate

Lee [13] presented an impact load recovery technique onthick plates utilizing Greenrsquos function and acoustic waveformdue to impact load which are theoretically obtained based onplate theory Load identification method on a nonconserva-tive nonlinear system which has a nonlinear damping effectwas studied by Jang [14] Ahmari and Yang [15] proposed aninverse analysis technique based on mathematical forwardsolving model for a simply supported plate and ParticleSwarm Optimization (PSO) to identify impact location andimpact load time history Khoo et al [16] developed animpact force identification technique utilizing OperatingDeflection Shape (ODS) analysis FRF measurement andpseudoinverse method They concluded that the accuracy ofidentified loads depends on sensor locations and improveswith increased number of measured response locations Liuet al [17] proposed load reconstruction technique in timedomain based on shape function method of moving least-square fitting and regularization

The reason for ill-conditioning of the load predictionproblem in frequency domain was investigated by Starkeyand Merrill [18] where they attributed it to near-singularityof the FRF matrix Hansen and Starkey [19] furthered theinvestigation on the nature of ill-conditioning in modaldomainThey studied the effect of locations of accelerometers

on a steel beam on the condition number of the modalmatrix and concluded that the condition number of themodal matrix can be improved through proper placement ofaccelerometers on the beam and proper selection of modesincluded in the analysis

Kammer [20] presented a method that utilizes a set ofinverse system Markov parameters estimated from forwardsystem Markov parameters using a linear predictive schemeThis computation is ill-conditioned therefore a regulariza-tion technique is employed to stabilize the computationSteltzner and Kammer [21] suggested a technique for inputforce estimation using an Inverse Structural Filter (ISF) thatprocesses the structural response data and returns an estimateof the input forces In the aforementioned works the authorsmade no mention of any strategy to optimally place sensorson the structure that would result in minimizing the effectof ill-conditioning associated with the inversion A numberof these works [8 9 20] assume that the accelerometers arecollocated with the forces which may not always be feasible

A study on the effect of strain gage locations and angularorientations on the quality of load estimates was conductedby Masroor and Zachary [22] Through formulation of asensitivity parameter they argued that proper selection ofgage locations improves the condition of the inverse problemSince analysis based on all possible gage location combina-tions is not feasible they selected a few groups of gages basedon judgment for the analysis This approach may still resultin poor load estimates because the sensors may be placed ata location where they have relatively low sensitivity to theapplied loads

In view of the above discussion it can be concluded thatalthough a lot of researches have been dedicated towardsidentifying novel techniques for load estimation the idea ofdetermining optimal sensor locations so as to minimize theeffect ofmodeling andmeasurement errors has received littleattention It is recognized [16 19 22] that sensor location hassignificant influence on the overall quality of results To over-come the abovementioned shortcomings this work expandsupon that by Kammer [20] and Steltzner and Kammer [21]by proposing an approach for estimating time-varying loadsacting on a structure by measuring structure response atfinite number of optimum locations on the structure Theaccuracy of the load estimates is dependent on the locationsof the sensors The novel approach based on 119863-optimaldesign results in increased accuracy in the dynamic loadestimation Two examples dealing with numerical validationof the proposed approach are presented to illustrate theeffectiveness of the proposed technique

2 Theoretical Development

21 Markov Parameter Representation Any continuous sys-tem may be spatially discretized as an n degrees of freedom(DOF) system whereby its equilibrium equation of motioncan be written in matrix form as

[119872] (119905) + [119862] (119905) + [119870] 119909 (119905) = 119891 (119905) (1)For simple systems [119872] [119862] and [119870] may be obtained bywriting the system equations of motion for complex systems

Shock and Vibration 3

they can be generated from the finite element model of thestructure

Another way to characterize the input-output behaviorof a structural system is by writing the second-order (1) asa first-order equation or in continuous time-invariant state-space form as

(119905) = [119860119888] 119906 (119905) + [119861

119888] 119891 (119905) (2)

where [119860119888] is the system matrix and [119861

119888] is the input matrix

for continuous case given by

[119860119888] = [

[0] [119868]

minus [119872]minus1

[119870] minus [119872]minus1

[119862]

]

[119861119888] = [

[0]

[119872]minus1]

(3)

All physical phenomena fundamentally exist in con-tinuous time The experimentally measured response datahowever is available only at discrete time instants This callsfor the need to transform the continuous time-invariant state-space model into a discrete time-invariant state-space modelThe transformation takes the following form

119906119905119894+1

= [119860119889] 119906119905119894+ [119861119889] 119891119905119894 (4)

where 119905119894 is the subscript over the discretized time and [119860119889]

and [119861119889] are related to their continuous case counterparts as

[119860119889] = 119890[119860119888]Δ119905

[119861119889] = [119860

119888]minus1

([119860119889] minus [119868]) [119861

119888]

(5)

Equation (4) is the discrete linear time-invariant state-spacerepresentation of the structural system The correspondingsystem output is given by

119910119905119894= [119862119889] 119906119905119894+ [119863119889] 119891119905119894 (6)

Given that unit impulse load is applied to the system at time119905 = 0 that is 120575

0= 1 and assuming zero initial conditions the

impulse response at various time points is given by

[119867]0= [119862119889] 1199060+ [119863119889] 1205750= [119863119889]

1199061= [119860119889] 1199060+ [119861119889] 1205750= [119861119889]

[119867]1= [119862119889] 1199061+ [119863119889] 1205751= [119862119889] [119861119889]

1199062= [119860119889] 1199061+ [119861119889] 1205751= [119860119889] [119861119889]

[119867]2= [119862119889] 1199062+ [119863119889] 1205752= [119862119889] [119860119889] [119861119889]

1199063= [119860119889] 1199062+ [119861119889] 1205752= [119860119889]2

[119861119889]

[119867]3= [119862119889] 1199063+ [119863119889] 1205753= [119862119889] [119860119889]2

[119861119889]

[119867]119894= [119862119889] [119860119889]119894minus1

[119861119889]

(7)

Given the impulse response the response at any time isgiven by the convolution of the input force with the impulseresponse as

119910119905119894=

119905119894

sum

119894=0

[119867]119894119891119905119894minus119894 (8)

where the matrices [119867]119894are called forward Markov parame-

ters [20] and summarized as

[119867]0= [119863119889]

[119867]119894= [119862119889] [119860119889]119894minus1

[119861119889] 119894 = 1 2 3

(9)

The forward Markov parameters represent the responseof the discrete system to applied unit impulse and thuscontain the dynamic properties of the system They can beobtained either analytically from the discretized model ofthe structural system or experimentally by measuring theoutput of the system due to a known input computing thecorresponding frequency response function and then takingits inverse discrete Fourier transform Equation (8) is theMarkov parameter representation of a structural system

22 Problem Formulation In the inverse problem of interesthere the objective is to estimate the input forces 119891 giventhe forward Markov parameters and system responses

From (8) the system response at 119905119894 = 0 is given by

1199100= [119867]

01198910 (10)

Assuming the case where the number of sensors used is atleast equal to the number of loads the least-squares estimateof the loads at 119905119894 = 0 is given as

1198910= ([119867]

119879

0[119867]0)minus1

[119867]119879

01199100 (11)

Similarly at 119905119894 = 1 the system response is given by

1199101= [119867]

01198911+ [119867]

11198910

(12)

which can again be solved for the loads as

1198911= ([119867]

119879

0[119867]0)minus1

[119867]119879

0(1199101minus [119867]

11198910) (13)

By induction the loads at time ti can be verified to be

119891119905119894= ([119867]

119879

0[119867]0)minus1

[119867]119879

0(119910119905119894minus

119905119894

sum

119894=1

[119867]119905119894[119891]119905119894minus119894)

(14)

Next define a discrete inverse system similar to (4) and(6) where the roles of input forces 119891 and system response119910 are reversed as

119906119905119894+1

= [119860119889]119868119906119905119894+ [119861119889]119868119910119905119894

119891119905119894= [119862119889]119868119906119905119894+ [119863119889]119868119910119905119894

(15)

4 Shock and Vibration

where the inverse matrices [119860119889]119868 [119861119889]119868 [119862119889]119868 and [119863

119889]119868are

related to the corresponding forward matrices by

[119860119889]119868= [119860119889] minus [119861

119889] ([119863119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]

[119861119889]119868= [119861119889] ([119863119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]119868= minus ([119863

119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]

[119863119889]119868= ([119863

119889]119879

[119863119889])minus1

[119863119889]119879

(16)

Similar to (8) given the system response the input forces atany time are given by the convolution relation as

119891119905119894=

119905119894

sum

119894=0

[ℎ]119894119910119905119894minus119894 (17)

where the matrices [ℎ]119894are known as the inverse Markov

parameters [20] The inverse Markov parameters similar tothe forward Markov parameters contain the dynamic prop-erties of the inverse system On comparison of expansions of(14) and (17) it may be ascertained that the inverse Markovparameters [ℎ]

119894are related to the forwardMarkov parameters

[119867]119894by a linear predictive equation given by

[ℎ]0= ([119867]

119879

0[119867]0)minus1

[119867]119879

0

[ℎ]119896= minus [ℎ]

0

119896

sum

119894=1

[119867]119894[ℎ]119896minus119894

(18)

There exist cases where [119867]0is zero matrix In certain

other nonminimum phase structural systems [119867]0is rank

deficient and the least-squares inverse in (18) does not existThis renders the causal summation in (17) undefined To dealwith this limitation the system output (6) is stepped forwardin time as [21]

119910119905119894= [119862119889] 119906119905119894+ [119862119889] [119861119889] 119891119905119894 (19)

The inverse system associated with (19) is noncausal that isthe input force estimates at current time become a function ofthe structural response at future times In general if the first zMarkov parameters in (7) are zero that is [119863

119889] = [119862

119889][119861119889] =

[119862119889][119860119889][119861119889] = sdot sdot sdot = [119862

119889][119860119889]119911minus2

[119861119889] = [0] the noncausal

z-lead inverse system can be constructed as

[ℎ]119911= ([119867]

119879

119911[119867]119911)minus1

[119867]119879

119911

[ℎ]119896+119911

= minus [ℎ]119911

119896

sum

119894=1

[119867]119894+119911[ℎ]119896minus119894+119911

(20)

and the corresponding force estimates are given by

119891119905119894=

119905119894

sum

119894=0

[ℎ]119894+119911119910119905119894minus119894+119911

(21)

It must be noted in (21) that the input force estimates 119891 atcurrent time ti are dependent on the structural response 119910at future times 119905119894 minus 119894 + 119911

The determination of the inverse Markov parametersfrom the forward Markov parameters using (20) is com-putationally expensive however it must be noted that thecomputation of the inverse Markov parameters needs tobe performed only once for any particular system Oncethe inverse Markov parameters are computed the inputforces can be predicted from the response using (21) Thistechnique however suffers from a similar limitation as mostof the other inverse problemsmdashill-conditioning The inac-curacies in system modeling translate to errors in forwardMarkov parameters The computation of inverse Markovparameters from erroneous forwardMarkov parameters is ill-conditioned and the computed inverse Markov parametersbecome unbounded Thus the convolution of the inverseMarkov parameters [ℎ] with the noisy structural response119910 given by (21) is not always a converging sum Thesum becomes numerically unstable due to the intrinsic ill-conditioning of the inverse problem identified by (20)

Conditioning of the erroneous forward Markov param-eters depends on the locations of sensors on the structuretherefore the input loads 119891 can be estimated precisely frommeasured structural response 119910 by strategic placement ofsensors on the structure Since it is not feasible to placesensors at all the DOFs of a structure a methodology ispresented next to overcome the identified shortcoming

23 Candidate Set Optimal selection of sensor locationson the structure requires a candidate set to be defined Itcan be inferred from the load identification step in (21)that the quality of load estimates depends directly on howaccurately the inverse Markov parameters are computed Itcan further be assessed from (20) that the 119911th inverseMarkovparameter [ℎ]

119911is computed by the least-squares inversion

of the 119911th Markov parameter [119867]119911 All the other higher

order inverse Markov parameters depend on [ℎ]119911 Therefore

the accuracy in the computation of [ℎ]119911from [119867]

119911dictates

the accuracy of all the other higher order inverse Markovparameters Each row in [119867]

119911corresponds to a unique DOF

of the system output Typically there are a large number oflocations on a structure where sensors can potentially bemounted These locations may exclude inaccessible locationssuch as the regions of load application To form a candidateset for optimization the DOFs (rows) corresponding to theinaccessible locations for sensor placement are ignored from[119867]119911 the subset so obtained is called the candidate set [119867]

119911 csTo obtain ameaningful solution to (20) an optimal subset

[119867]119911 opt of the candidate set [119867]

119911 cs needs to be identifiedwhich in turn will lead to an increased accuracy in theestimation of 119891 As more sensors are used the additionalinformation helps to obtain a more precise estimate of 119891but practical and financial constraints place limitations onthe number of sensors that can be used If the number offorces to be estimated is 119899

119891 then in order to minimize the

error in 119891 estimates the number of sensors a must satisfythe criterion 119886 ge 119899

119891 Each sensor location referred to as

a candidate point provides a potential row for inclusion in[119867]119911 opt [119867]119911 opt needs to be such a subset of [119867]

119911 cs that

Shock and Vibration 5

provides the most precise estimates of 119891119905119894 An approach to

extract [119867]119911 opt from [119867]

119911 cs is described next

24 119863-Optimal Design If the errors in response measure-ments are statistically independent and the standard devia-tion of each of them is 120590 then the covariance matrix for 119891

119905119894

estimates is given by [22]

var (119891119905119894) = 1205902

([119867]119879

119911[119867]119911)minus1

(22)

The matrix ([119867]119879119911[119867]119911)minus1 is known as the sensitivity of

[119867]119911 For a given variance 1205902 in response measurements

minimization of the sensitivity of [119867]119911leads to an increased

precision in 119891119905119894

estimates The sensitivity of [119867]119911is a

function of the number and locations of the sensors on thestructure Therefore optimum selection of the number andlocations of the sensors on the structure so as to minimizethe sensitivity of [119867]

119911leads to the minimization of variation

in the 119891119905119894estimates

For a given number of sensors a the candidate set [119867]119911 cs

is searched to determine a sensor locations that providethe least variance in the load estimates One of the criteriafollowed involves the maximization of the determinant of[119867]119879

119911[119867]119911 |[119867]119879

119911[119867]119911| Designs that maximize |[119867]119879

119911[119867]119911|

are called 119863-optimal designs [23ndash25] where D denotesdeterminant In order to construct an a-point 119863-optimaldesign the a sensor locations that maximize |[119867]119879

119911[119867]119911|

must be selected from the candidate set [119867]119911 cs To select the

a-point 119863-optimal design sequential exchange algorithms[24] based on the principles of optimal augmentation andreduction of an existing design are used herein

The basic idea behind the sequential exchange algorithmis as follows Given the candidate set [119867]

119911 cs and the numberof sensors a the first step is to randomly select a distinctcandidate points from the candidate set to initialize [119867]

119911

Out of the remaining candidate set a candidate point is thenselected and the corresponding row is augmented to [119867]

119911

to form [119867]119911+

such that |[119867]119879119911 +[119867]119911+| is maximum Next

out of the 119886 + 1 rows in [119867]119911+ a row is deleted to arrive at

[119867]119911minus

such that |[119867]119879119911 minus[119867]119911minus| is maximum This process of

adding and deleting rows continues until there is no furtherimprovement in the value of |[119867]119879

119911[119867]119911| The final [119867]

119911so

obtained is the 119863-optimal design [119867]119911 opt whose row indices

provide the information on the optimum sensor locationsAs the candidate set gets bigger and bigger increasingly

large number of determinants and matrix inverses need to becalculated A very naıve way to compute the determinant isto obtain119872 = [119867]

119879

119911[119867]119911and then calculate the determinant

|119872| This approach of determinant calculation is computa-tionally very expensive An alternate formula for computingthe determinant |119872

+| = |[119867]

119879

119911 +[119867]119911+| from that of |119872|when

a row 119910119879 is augmented to [119867]119911is [25]

1003816100381610038161003816119872+1003816100381610038161003816 = |119872| (1 [+] 119910

119879

119872minus1

119910) (23)

where [+]denotes additionThe [+] is replaced by subtractionfor the case when a row 119910119879 is deleted from [119867]

119911+ In order to

k1

C1

m1

k2

C2

m2

k3

C3

middot middot middot

k14

C14

m14

k15

C15

m15

k16

C16

Figure 1 15-Dof spring-mass-damper system

be able to use (23) 119872minus1 can be maintained and updated asthe row 119910119879 is augmented to [119867]

119911by

1003816100381610038161003816119872+1003816100381610038161003816

minus1

= |119872|minus1

[minus]

(119872minus1

119910) (119872minus1

119910)119879

(1 [+] 119910119879119872minus1119910)

(24)

where [minus] denotes subtraction and is replaced by addition forthe case when a row 119910119879 is deleted from [119867]

119911+

Once [119867]119911 opt and in turn optimum sensor locations are

determined sensors are mounted at the identified optimumlocations and response is measured Equations (20) and (21)are then utilized to compute the input forces 119891 It is to benoted that in this approach it is not necessary that responsemeasurements be taken at locations where forces are applied

25 Error Quantification The percent root mean square(rms) error in the recovered loads with respect to the actualapplied loads can be calculated as

120576rms = (

radicsum(119891app minus 119891rec)2

radicsum1198912

app

times 100) (25)

The above equation can be used to quantify errors in the loadestimates

3 Example 1 15-DOFSpring-Mass-Damper System

The dynamic load estimation method from accelerationmeasurements is illustrated with the help of a numerical sim-ulation consisting of a 15 DOF spring-mass-damper systemshown in Figure 1 Masses119898

1and119898

15are connected to fixed

boundary A periodic forcing function is applied to mass1198987

The task is to determine the optimumaccelerometer locationsand reconstruct the input force based on the acceleration timeresponse at those locations

A total of 2 accelerometers were used to measure accel-erations of 2 masses The [119872] [119862] and [119870] matrices wereobtained by writing the equations of motion for the sys-tem Discrete time-invariant state-space form of the systemwas obtained using (3)ndash(6) Forward Markov parameterswere computed from the state-space model using (9) Allnumerical computations were performed in a MATLABprogramming environment

All DOFs except the DOF where the load was appliedwere selected to be the locations where accelerometers canpotentially bemounted that is theDOF corresponding to theapplied load did not form a part of the candidate set [119867]

119911 cs

6 Shock and Vibration

0 001 002 003 004 005 006 007 008 009 01minus1500

minus1000

minus500

0

500

1000

1500

Time

Load

AppliedRecovered

Figure 2 Applied and recovered loads at mass 7 with optimumaccelerometer placements

[119867]119911 cs was subjected to the 119863-optimal design algorithm

described in Section 24 to obtain [119867]119911 opt whereby the

optimum accelerometer locations were found to be at masses1198986and 119898

8 Having obtained the optimum accelerometer

locations the inverse Markov parameters were computedusing (20) In absence of any experimental data the acceler-ation time response at the optimum locations was obtainedby solving the ordinary differential equations numericallyUsing the acceleration data 119910 computed at the optimumaccelerometer locations the input force 119891 was recoveredusing (21) The applied and the recovered loads are plotted inFigure 2The rms error using (25)was calculated to be 07 Itcan be inferred from the plot that the applied load is recoveredaccurately

4 Example 2 Overhead Crane Girder

A more general numerical example is presented here wherea vertical dynamic load acting on an overhead crane girderthrough trolley wheels needs to be estimated For the sakeof illustration and simplicity the number of loads is assumedto be one and assumed to be applied at the girder mid-spanThe problem is to determine the optimum accelerometerlocations and reconstruct the input force based on theacceleration time response at those locations

A finite element model of the girder was developed inANSYS using BEAM188 elements The finite element modelof the girder along with the applied load and boundaryconditions is shown in Figure 3 The model consisted of 50beam elements and 49 unconstrained nodes with 6 degreesof freedom per node that is the total number of degrees offreedom of the model was 294

The [119872] and [119870] matrices were obtained using finiteelement method ANSYS provides data for [119872] and [119870]matrices in the Harwell-Boeing file format A routine waswritten in MATLAB to convert them into the matrix formatsuitable for current application All further calculations wereperformed in MATLAB Discrete time-invariant state-spaceform of the system was obtained using (3)ndash(6) Forward

Figure 3 Finite element model of girder with applied load andoptimum accelerometer locations

Markov parameters were computed from the state-spacemodel using (9) The number of accelerometers to be usedwas arbitrarily assumed to be equal to 3 All DOFs exceptthe DOF where the load was applied were selected to be thelocations where accelerometers can potentially be mountedthat is the DOF corresponding to the applied load didnot form a part of the candidate set [119867]

119911 cs [119867]119911 cs wassubjected to the 119863-optimal design algorithm described inSection 24 to obtain [119867]

119911 opt The optimum accelerometerlocations are also depicted in Figure 3 It is observed thatthe algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initiallybe excluded from the candidate set [119867]

119911 cs Furthermoreif the sensor positions seem to be too congested mutuallyadditional criteria may be instructed such as if a particularspot is chosen as a potential sensor location then certain areaaround that locationmay be excluded from the candidate set

Having obtained the optimum accelerometer locationsthe inverse Markov parameters were computed using (20)In absence of any experimental data the acceleration timeresponse at the optimum locations was obtained by solvingthe ordinary differential equations numerically Using theacceleration data 119910 computed at the optimum accelerom-eter locations the input force 119891 was recovered using (21)The applied and the recovered loads are plotted in Figure 4The rms error using (25) was calculated to be 01

In order to simulate a more realistic scenario whereaccelerations are measured experimentally the accelerationdata 119910 was corrupted with normally distributed randomerrors with zero mean and standard deviation of 10 ofits value The applied and recovered loads with errors inacceleration measurements are plotted in Figure 5 The rmserror using (25) was calculated to be 23

Next a different nature of force was considered where ahalf-sine wave loading was applied Both exact accelerationdata 119910 and acceleration data corrupted with normallydistributed random errors with zero mean and standarddeviation of 10 of its value were considered The appliedand recovered loads without and with errors in acceleration

Shock and Vibration 7

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 4 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 5 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

measurements are plotted in Figures 6 and 7 respectivelyThe rms error using (25) was calculated to be 13 forthe measurement involving acceleration error It can beinferred from the plots that the proposed approach is ableto successfully recover the applied load precisely even whenrealistic measurement errors are present in the structuralresponse

Based on the results for both examples it can be seenthat for discrete as well as continuous systems the proposedapproach is able to accurately estimate the loads acting on acomponent by measuring the acceleration response at a finitenumber of optimum locations

5 Conclusions

A computational technique is presented that allows forindirect measurement of dynamic loads acting on a structuresuch that the structure behaves as its own load transducer

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 6 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 7 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

This is achieved by placing finite number of sensors on thestructure The loads exciting a structure are estimated byconvolving the structural response with the inverse Markovparameters The inverse Markov parameters in turn arecomputed from the forward Markov parameters using alinear prediction algorithmThe forwardMarkov parametersrepresent the response of the system to applied unit impulseand thus contain the dynamic properties of the system Theycan be obtained analytically as well as experimentally

The computation of the inverse Markov parameters fromforward Markov parameters like all inverse problems suf-fers from ill-conditioning The computed inverse Markovparameters are not always bounded thus rendering theirconvolution with the structural response to diverge Theaccuracy of the inverse Markov parameters (and thereby thequality of input load estimates) depends on the locations ofsensors on the structure To improve the precision of loadestimates the technique of 119863-optimal design is utilized todetermine the optimum sensor locations It is observed that

8 Shock and Vibration

the algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initially beexcluded from the candidate set Furthermore if the sensorpositions seem to be too congested mutually additionalcriteriamay be instructed such as if a particular spot is chosenas a potential sensor location then certain area around thatlocation may be excluded from the candidate set The loadsrecovered based on accelerations measured from optimallyplaced accelerometers on the structure are observed to be inexcellent agreement with the applied loads

Nomenclature

[0] Zero matrix Derivative with respect to time

[119860119888] System matrix for continuous case

[119860119889] System matrix for discrete case

[119860119889]119868 Inverse system matrix for discrete case

119886 Number of sensors[119861119888] Input matrix for continuous case

[119861119889] Input matrix for discrete case

[119861119889]119868 Inverse input matrix for discrete case

[119862] Damping matrix[119862119889] Output matrix for discrete case

[119862119889]119868 Inverse output matrix for discrete case

[119863119889] Feedforward matrix for discrete case

[119863119889]119868 Inverse feedforward matrix for discrete

case119891(119905) Load vector119891app Applied load119891rec Reconstructed load[119867]119894 119894th forward Markov parameters matrix

[119867]119911 cs Candidate set

[119867]119911 opt Optimum subset of [119867]

119911 cs determined by119863-optimal design

[ℎ]119894 119894th inverse Markov parameters matrix

[119868] Identity matrix[119870] Stiffness matrix[119872] Mass matrix119899 Number of degrees of freedom of the

system119899119891 Number of applied forces

119905 Time119905119894 Discretized timeΔ119905 Time step119906 119906(119905) State variables vector119909(119905) Displacement vector119910 System output120575 Unit impulse load function120576rms Percent root mean square (rms) error

Conflict of Interests

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

References

[1] K K Stevens ldquoForce identification problemsmdashan overviewrdquo inProceedings of the SEM Conference on Experimental Mechanicspp 838ndash844 Houston Tex USA 1987

[2] D J Ewins Modal Testing Theory Practice and ApplicationResearch Studies Press 2000

[3] G Desanghere and R Snoeys ldquoIndirect identification of excita-tion forces bymodal coordinate transformationrdquo in Proceedingsof the 3rd International Modal Analysis Conference (IMAC rsquo85)pp 685ndash690 Orlando Fla USA 1985

[4] B Hillary and D J Ewins ldquoThe use of strain gauges inforce determination and frequency response functionmeasure-mentsrdquo in Proceedings of the 2nd International Modal AnalysisConference (IMAC rsquo84) pp 627ndash634 Orlando Fla USA 1984

[5] N Okubo S Tanabe and T Tatsuno ldquoIdentification of forcesgenerated by amachine under operating conditionrdquo in Proceed-ings of the 3rd International Modal Analysis Conference (IMACrsquo85) pp 920ndash927 Orlando Fla USA 1985

[6] H R Busby andDM Trujillo ldquoSolution of an inverse dynamicsproblem using an eigenvalue reduction techniquerdquo Computersand Structures vol 25 no 1 pp 109ndash117 1987

[7] P E Hollandsworth and H R Busby ldquoImpact force identifica-tion using the general inverse techniquerdquo International Journalof Impact Engineering vol 8 no 4 pp 315ndash322 1989

[8] GGenaro andDA Rade ldquoInput force identification in the timedomainrdquo in Proceedings of the 16th International Modal AnalysisConference (IMAC rsquo98) pp 124ndash129 Santa Barbara Calif USAFebruary 1998

[9] T G Carne R L Mayes and V I Bateman ldquoForce recon-struction using a sum of weighted accelerations techniquerdquo inProceedings of the 10th International Modal Analysis Conference(IMAC 92) pp 291ndash298 San Diego Calif USA 1992

[10] R Hashemi and M H Kargarnovin ldquoVibration base identifi-cation of impact force using genetic algorithmrdquo InternationalJournal of Mechanical Systems Science and Engineering vol 1no 4 pp 204ndash210 2007

[11] Z R Lu and S S Law ldquoIdentification of system parameters andinput force from output onlyrdquo Mechanical Systems and SignalProcessing vol 21 no 5 pp 2099ndash2111 2007

[12] Z Boukria P Perrotin and A Bennani ldquoExperimental impactforce location and identification using inverse problems appli-cation for a circular platerdquo International Journal of Mechanicsvol 5 no 1 pp 48ndash55 2011

[13] S K Lee ldquoIdentification of impact force in thick plates based onthe elastodynamics and time-frequency method (I)rdquo Journal ofMechanical Science and Technology vol 22 no 7 pp 1349ndash13582008

[14] T S Jang ldquoA novel method for the non-parametric identifica-tion of nonlinear restoring forces in nonlinear vibrations basedon response data a dissipative nonlinear dynamical systemrdquoShips and Offshore Structures vol 6 no 4 pp 257ndash263 2011

[15] S Ahmari and M Yang ldquoImpact location and load iden-tification through inverse analysis with bounded uncertainmeasurementsrdquo Smart Materials and Structures vol 22 no 8Article ID 085024 2013

[16] S Y Khoo Z Ismail K K Kong et al ldquoImpact force identifi-cation with pseudo-inverse method on a lightweight structurefor under-determined even-determined and over-determinedcasesrdquo International Journal of Impact Engineering vol 63 pp52ndash62 2014

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

by the inverse problems pertaining to load identification aredocumented by Stevens [1] Ewins [2] used displacementtransducers to identify input loads and found that very smallvariations (noise) in the response measurement cause largeerrors in the force estimation Load identification problemin frequency domain was studied by Desanghere and Snoeys[3] where they attributed the reason for ill-conditioning to afew dominant elements in the frequency response function(FRF) matrix Hillary and Ewins [4] and Okubo et al [5]investigated the influence of noise in the measured responseand noted that the inversion process is highly ill-conditioned

Another technique was proposed by Busby and Trujillo[6] where the load estimation problem is treated as mini-mization problem of error between the measured structuralresponse and the response predicted from model which isthen solved using dynamic programmingThis techniquewasextended by Hollandsworth and Busby [7] by applying it toactual experimental measurements Application of the tech-nique is limited since the amount of computation increasesdramatically with the model order

Genaro and Rade [8] suggested a load reconstructionmethod from acceleration response and its successive inte-gration Carne et al [9] developed a technique for loadidentification known as the Sum of Weighted AccelerationsTechnique (SWAT) An optimization problem for load recov-ery was formulated by Hashemi and Kargarnovin [10] wherethe objective function is the difference between analyticaland measured responses and the decision variables are thelocation and magnitude of the applied force Lu and Law [11]proposed a load estimation technique based on sensitivityof structural responses A gradient-based model updatingmethod based on dynamic response sensitivity is utilizedto identify the force and physical parameters Boukria et al[12] applied the FRF technique to identify the location andmagnitude of impact force on a circular plate

Lee [13] presented an impact load recovery technique onthick plates utilizing Greenrsquos function and acoustic waveformdue to impact load which are theoretically obtained based onplate theory Load identification method on a nonconserva-tive nonlinear system which has a nonlinear damping effectwas studied by Jang [14] Ahmari and Yang [15] proposed aninverse analysis technique based on mathematical forwardsolving model for a simply supported plate and ParticleSwarm Optimization (PSO) to identify impact location andimpact load time history Khoo et al [16] developed animpact force identification technique utilizing OperatingDeflection Shape (ODS) analysis FRF measurement andpseudoinverse method They concluded that the accuracy ofidentified loads depends on sensor locations and improveswith increased number of measured response locations Liuet al [17] proposed load reconstruction technique in timedomain based on shape function method of moving least-square fitting and regularization

The reason for ill-conditioning of the load predictionproblem in frequency domain was investigated by Starkeyand Merrill [18] where they attributed it to near-singularityof the FRF matrix Hansen and Starkey [19] furthered theinvestigation on the nature of ill-conditioning in modaldomainThey studied the effect of locations of accelerometers

on a steel beam on the condition number of the modalmatrix and concluded that the condition number of themodal matrix can be improved through proper placement ofaccelerometers on the beam and proper selection of modesincluded in the analysis

Kammer [20] presented a method that utilizes a set ofinverse system Markov parameters estimated from forwardsystem Markov parameters using a linear predictive schemeThis computation is ill-conditioned therefore a regulariza-tion technique is employed to stabilize the computationSteltzner and Kammer [21] suggested a technique for inputforce estimation using an Inverse Structural Filter (ISF) thatprocesses the structural response data and returns an estimateof the input forces In the aforementioned works the authorsmade no mention of any strategy to optimally place sensorson the structure that would result in minimizing the effectof ill-conditioning associated with the inversion A numberof these works [8 9 20] assume that the accelerometers arecollocated with the forces which may not always be feasible

A study on the effect of strain gage locations and angularorientations on the quality of load estimates was conductedby Masroor and Zachary [22] Through formulation of asensitivity parameter they argued that proper selection ofgage locations improves the condition of the inverse problemSince analysis based on all possible gage location combina-tions is not feasible they selected a few groups of gages basedon judgment for the analysis This approach may still resultin poor load estimates because the sensors may be placed ata location where they have relatively low sensitivity to theapplied loads

In view of the above discussion it can be concluded thatalthough a lot of researches have been dedicated towardsidentifying novel techniques for load estimation the idea ofdetermining optimal sensor locations so as to minimize theeffect ofmodeling andmeasurement errors has received littleattention It is recognized [16 19 22] that sensor location hassignificant influence on the overall quality of results To over-come the abovementioned shortcomings this work expandsupon that by Kammer [20] and Steltzner and Kammer [21]by proposing an approach for estimating time-varying loadsacting on a structure by measuring structure response atfinite number of optimum locations on the structure Theaccuracy of the load estimates is dependent on the locationsof the sensors The novel approach based on 119863-optimaldesign results in increased accuracy in the dynamic loadestimation Two examples dealing with numerical validationof the proposed approach are presented to illustrate theeffectiveness of the proposed technique

2 Theoretical Development

21 Markov Parameter Representation Any continuous sys-tem may be spatially discretized as an n degrees of freedom(DOF) system whereby its equilibrium equation of motioncan be written in matrix form as

[119872] (119905) + [119862] (119905) + [119870] 119909 (119905) = 119891 (119905) (1)For simple systems [119872] [119862] and [119870] may be obtained bywriting the system equations of motion for complex systems

Shock and Vibration 3

they can be generated from the finite element model of thestructure

Another way to characterize the input-output behaviorof a structural system is by writing the second-order (1) asa first-order equation or in continuous time-invariant state-space form as

(119905) = [119860119888] 119906 (119905) + [119861

119888] 119891 (119905) (2)

where [119860119888] is the system matrix and [119861

119888] is the input matrix

for continuous case given by

[119860119888] = [

[0] [119868]

minus [119872]minus1

[119870] minus [119872]minus1

[119862]

]

[119861119888] = [

[0]

[119872]minus1]

(3)

All physical phenomena fundamentally exist in con-tinuous time The experimentally measured response datahowever is available only at discrete time instants This callsfor the need to transform the continuous time-invariant state-space model into a discrete time-invariant state-space modelThe transformation takes the following form

119906119905119894+1

= [119860119889] 119906119905119894+ [119861119889] 119891119905119894 (4)

where 119905119894 is the subscript over the discretized time and [119860119889]

and [119861119889] are related to their continuous case counterparts as

[119860119889] = 119890[119860119888]Δ119905

[119861119889] = [119860

119888]minus1

([119860119889] minus [119868]) [119861

119888]

(5)

Equation (4) is the discrete linear time-invariant state-spacerepresentation of the structural system The correspondingsystem output is given by

119910119905119894= [119862119889] 119906119905119894+ [119863119889] 119891119905119894 (6)

Given that unit impulse load is applied to the system at time119905 = 0 that is 120575

0= 1 and assuming zero initial conditions the

impulse response at various time points is given by

[119867]0= [119862119889] 1199060+ [119863119889] 1205750= [119863119889]

1199061= [119860119889] 1199060+ [119861119889] 1205750= [119861119889]

[119867]1= [119862119889] 1199061+ [119863119889] 1205751= [119862119889] [119861119889]

1199062= [119860119889] 1199061+ [119861119889] 1205751= [119860119889] [119861119889]

[119867]2= [119862119889] 1199062+ [119863119889] 1205752= [119862119889] [119860119889] [119861119889]

1199063= [119860119889] 1199062+ [119861119889] 1205752= [119860119889]2

[119861119889]

[119867]3= [119862119889] 1199063+ [119863119889] 1205753= [119862119889] [119860119889]2

[119861119889]

[119867]119894= [119862119889] [119860119889]119894minus1

[119861119889]

(7)

Given the impulse response the response at any time isgiven by the convolution of the input force with the impulseresponse as

119910119905119894=

119905119894

sum

119894=0

[119867]119894119891119905119894minus119894 (8)

where the matrices [119867]119894are called forward Markov parame-

ters [20] and summarized as

[119867]0= [119863119889]

[119867]119894= [119862119889] [119860119889]119894minus1

[119861119889] 119894 = 1 2 3

(9)

The forward Markov parameters represent the responseof the discrete system to applied unit impulse and thuscontain the dynamic properties of the system They can beobtained either analytically from the discretized model ofthe structural system or experimentally by measuring theoutput of the system due to a known input computing thecorresponding frequency response function and then takingits inverse discrete Fourier transform Equation (8) is theMarkov parameter representation of a structural system

22 Problem Formulation In the inverse problem of interesthere the objective is to estimate the input forces 119891 giventhe forward Markov parameters and system responses

From (8) the system response at 119905119894 = 0 is given by

1199100= [119867]

01198910 (10)

Assuming the case where the number of sensors used is atleast equal to the number of loads the least-squares estimateof the loads at 119905119894 = 0 is given as

1198910= ([119867]

119879

0[119867]0)minus1

[119867]119879

01199100 (11)

Similarly at 119905119894 = 1 the system response is given by

1199101= [119867]

01198911+ [119867]

11198910

(12)

which can again be solved for the loads as

1198911= ([119867]

119879

0[119867]0)minus1

[119867]119879

0(1199101minus [119867]

11198910) (13)

By induction the loads at time ti can be verified to be

119891119905119894= ([119867]

119879

0[119867]0)minus1

[119867]119879

0(119910119905119894minus

119905119894

sum

119894=1

[119867]119905119894[119891]119905119894minus119894)

(14)

Next define a discrete inverse system similar to (4) and(6) where the roles of input forces 119891 and system response119910 are reversed as

119906119905119894+1

= [119860119889]119868119906119905119894+ [119861119889]119868119910119905119894

119891119905119894= [119862119889]119868119906119905119894+ [119863119889]119868119910119905119894

(15)

4 Shock and Vibration

where the inverse matrices [119860119889]119868 [119861119889]119868 [119862119889]119868 and [119863

119889]119868are

related to the corresponding forward matrices by

[119860119889]119868= [119860119889] minus [119861

119889] ([119863119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]

[119861119889]119868= [119861119889] ([119863119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]119868= minus ([119863

119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]

[119863119889]119868= ([119863

119889]119879

[119863119889])minus1

[119863119889]119879

(16)

Similar to (8) given the system response the input forces atany time are given by the convolution relation as

119891119905119894=

119905119894

sum

119894=0

[ℎ]119894119910119905119894minus119894 (17)

where the matrices [ℎ]119894are known as the inverse Markov

parameters [20] The inverse Markov parameters similar tothe forward Markov parameters contain the dynamic prop-erties of the inverse system On comparison of expansions of(14) and (17) it may be ascertained that the inverse Markovparameters [ℎ]

119894are related to the forwardMarkov parameters

[119867]119894by a linear predictive equation given by

[ℎ]0= ([119867]

119879

0[119867]0)minus1

[119867]119879

0

[ℎ]119896= minus [ℎ]

0

119896

sum

119894=1

[119867]119894[ℎ]119896minus119894

(18)

There exist cases where [119867]0is zero matrix In certain

other nonminimum phase structural systems [119867]0is rank

deficient and the least-squares inverse in (18) does not existThis renders the causal summation in (17) undefined To dealwith this limitation the system output (6) is stepped forwardin time as [21]

119910119905119894= [119862119889] 119906119905119894+ [119862119889] [119861119889] 119891119905119894 (19)

The inverse system associated with (19) is noncausal that isthe input force estimates at current time become a function ofthe structural response at future times In general if the first zMarkov parameters in (7) are zero that is [119863

119889] = [119862

119889][119861119889] =

[119862119889][119860119889][119861119889] = sdot sdot sdot = [119862

119889][119860119889]119911minus2

[119861119889] = [0] the noncausal

z-lead inverse system can be constructed as

[ℎ]119911= ([119867]

119879

119911[119867]119911)minus1

[119867]119879

119911

[ℎ]119896+119911

= minus [ℎ]119911

119896

sum

119894=1

[119867]119894+119911[ℎ]119896minus119894+119911

(20)

and the corresponding force estimates are given by

119891119905119894=

119905119894

sum

119894=0

[ℎ]119894+119911119910119905119894minus119894+119911

(21)

It must be noted in (21) that the input force estimates 119891 atcurrent time ti are dependent on the structural response 119910at future times 119905119894 minus 119894 + 119911

The determination of the inverse Markov parametersfrom the forward Markov parameters using (20) is com-putationally expensive however it must be noted that thecomputation of the inverse Markov parameters needs tobe performed only once for any particular system Oncethe inverse Markov parameters are computed the inputforces can be predicted from the response using (21) Thistechnique however suffers from a similar limitation as mostof the other inverse problemsmdashill-conditioning The inac-curacies in system modeling translate to errors in forwardMarkov parameters The computation of inverse Markovparameters from erroneous forwardMarkov parameters is ill-conditioned and the computed inverse Markov parametersbecome unbounded Thus the convolution of the inverseMarkov parameters [ℎ] with the noisy structural response119910 given by (21) is not always a converging sum Thesum becomes numerically unstable due to the intrinsic ill-conditioning of the inverse problem identified by (20)

Conditioning of the erroneous forward Markov param-eters depends on the locations of sensors on the structuretherefore the input loads 119891 can be estimated precisely frommeasured structural response 119910 by strategic placement ofsensors on the structure Since it is not feasible to placesensors at all the DOFs of a structure a methodology ispresented next to overcome the identified shortcoming

23 Candidate Set Optimal selection of sensor locationson the structure requires a candidate set to be defined Itcan be inferred from the load identification step in (21)that the quality of load estimates depends directly on howaccurately the inverse Markov parameters are computed Itcan further be assessed from (20) that the 119911th inverseMarkovparameter [ℎ]

119911is computed by the least-squares inversion

of the 119911th Markov parameter [119867]119911 All the other higher

order inverse Markov parameters depend on [ℎ]119911 Therefore

the accuracy in the computation of [ℎ]119911from [119867]

119911dictates

the accuracy of all the other higher order inverse Markovparameters Each row in [119867]

119911corresponds to a unique DOF

of the system output Typically there are a large number oflocations on a structure where sensors can potentially bemounted These locations may exclude inaccessible locationssuch as the regions of load application To form a candidateset for optimization the DOFs (rows) corresponding to theinaccessible locations for sensor placement are ignored from[119867]119911 the subset so obtained is called the candidate set [119867]

119911 csTo obtain ameaningful solution to (20) an optimal subset

[119867]119911 opt of the candidate set [119867]

119911 cs needs to be identifiedwhich in turn will lead to an increased accuracy in theestimation of 119891 As more sensors are used the additionalinformation helps to obtain a more precise estimate of 119891but practical and financial constraints place limitations onthe number of sensors that can be used If the number offorces to be estimated is 119899

119891 then in order to minimize the

error in 119891 estimates the number of sensors a must satisfythe criterion 119886 ge 119899

119891 Each sensor location referred to as

a candidate point provides a potential row for inclusion in[119867]119911 opt [119867]119911 opt needs to be such a subset of [119867]

119911 cs that

Shock and Vibration 5

provides the most precise estimates of 119891119905119894 An approach to

extract [119867]119911 opt from [119867]

119911 cs is described next

24 119863-Optimal Design If the errors in response measure-ments are statistically independent and the standard devia-tion of each of them is 120590 then the covariance matrix for 119891

119905119894

estimates is given by [22]

var (119891119905119894) = 1205902

([119867]119879

119911[119867]119911)minus1

(22)

The matrix ([119867]119879119911[119867]119911)minus1 is known as the sensitivity of

[119867]119911 For a given variance 1205902 in response measurements

minimization of the sensitivity of [119867]119911leads to an increased

precision in 119891119905119894

estimates The sensitivity of [119867]119911is a

function of the number and locations of the sensors on thestructure Therefore optimum selection of the number andlocations of the sensors on the structure so as to minimizethe sensitivity of [119867]

119911leads to the minimization of variation

in the 119891119905119894estimates

For a given number of sensors a the candidate set [119867]119911 cs

is searched to determine a sensor locations that providethe least variance in the load estimates One of the criteriafollowed involves the maximization of the determinant of[119867]119879

119911[119867]119911 |[119867]119879

119911[119867]119911| Designs that maximize |[119867]119879

119911[119867]119911|

are called 119863-optimal designs [23ndash25] where D denotesdeterminant In order to construct an a-point 119863-optimaldesign the a sensor locations that maximize |[119867]119879

119911[119867]119911|

must be selected from the candidate set [119867]119911 cs To select the

a-point 119863-optimal design sequential exchange algorithms[24] based on the principles of optimal augmentation andreduction of an existing design are used herein

The basic idea behind the sequential exchange algorithmis as follows Given the candidate set [119867]

119911 cs and the numberof sensors a the first step is to randomly select a distinctcandidate points from the candidate set to initialize [119867]

119911

Out of the remaining candidate set a candidate point is thenselected and the corresponding row is augmented to [119867]

119911

to form [119867]119911+

such that |[119867]119879119911 +[119867]119911+| is maximum Next

out of the 119886 + 1 rows in [119867]119911+ a row is deleted to arrive at

[119867]119911minus

such that |[119867]119879119911 minus[119867]119911minus| is maximum This process of

adding and deleting rows continues until there is no furtherimprovement in the value of |[119867]119879

119911[119867]119911| The final [119867]

119911so

obtained is the 119863-optimal design [119867]119911 opt whose row indices

provide the information on the optimum sensor locationsAs the candidate set gets bigger and bigger increasingly

large number of determinants and matrix inverses need to becalculated A very naıve way to compute the determinant isto obtain119872 = [119867]

119879

119911[119867]119911and then calculate the determinant

|119872| This approach of determinant calculation is computa-tionally very expensive An alternate formula for computingthe determinant |119872

+| = |[119867]

119879

119911 +[119867]119911+| from that of |119872|when

a row 119910119879 is augmented to [119867]119911is [25]

1003816100381610038161003816119872+1003816100381610038161003816 = |119872| (1 [+] 119910

119879

119872minus1

119910) (23)

where [+]denotes additionThe [+] is replaced by subtractionfor the case when a row 119910119879 is deleted from [119867]

119911+ In order to

k1

C1

m1

k2

C2

m2

k3

C3

middot middot middot

k14

C14

m14

k15

C15

m15

k16

C16

Figure 1 15-Dof spring-mass-damper system

be able to use (23) 119872minus1 can be maintained and updated asthe row 119910119879 is augmented to [119867]

119911by

1003816100381610038161003816119872+1003816100381610038161003816

minus1

= |119872|minus1

[minus]

(119872minus1

119910) (119872minus1

119910)119879

(1 [+] 119910119879119872minus1119910)

(24)

where [minus] denotes subtraction and is replaced by addition forthe case when a row 119910119879 is deleted from [119867]

119911+

Once [119867]119911 opt and in turn optimum sensor locations are

determined sensors are mounted at the identified optimumlocations and response is measured Equations (20) and (21)are then utilized to compute the input forces 119891 It is to benoted that in this approach it is not necessary that responsemeasurements be taken at locations where forces are applied

25 Error Quantification The percent root mean square(rms) error in the recovered loads with respect to the actualapplied loads can be calculated as

120576rms = (

radicsum(119891app minus 119891rec)2

radicsum1198912

app

times 100) (25)

The above equation can be used to quantify errors in the loadestimates

3 Example 1 15-DOFSpring-Mass-Damper System

The dynamic load estimation method from accelerationmeasurements is illustrated with the help of a numerical sim-ulation consisting of a 15 DOF spring-mass-damper systemshown in Figure 1 Masses119898

1and119898

15are connected to fixed

boundary A periodic forcing function is applied to mass1198987

The task is to determine the optimumaccelerometer locationsand reconstruct the input force based on the acceleration timeresponse at those locations

A total of 2 accelerometers were used to measure accel-erations of 2 masses The [119872] [119862] and [119870] matrices wereobtained by writing the equations of motion for the sys-tem Discrete time-invariant state-space form of the systemwas obtained using (3)ndash(6) Forward Markov parameterswere computed from the state-space model using (9) Allnumerical computations were performed in a MATLABprogramming environment

All DOFs except the DOF where the load was appliedwere selected to be the locations where accelerometers canpotentially bemounted that is theDOF corresponding to theapplied load did not form a part of the candidate set [119867]

119911 cs

6 Shock and Vibration

0 001 002 003 004 005 006 007 008 009 01minus1500

minus1000

minus500

0

500

1000

1500

Time

Load

AppliedRecovered

Figure 2 Applied and recovered loads at mass 7 with optimumaccelerometer placements

[119867]119911 cs was subjected to the 119863-optimal design algorithm

described in Section 24 to obtain [119867]119911 opt whereby the

optimum accelerometer locations were found to be at masses1198986and 119898

8 Having obtained the optimum accelerometer

locations the inverse Markov parameters were computedusing (20) In absence of any experimental data the acceler-ation time response at the optimum locations was obtainedby solving the ordinary differential equations numericallyUsing the acceleration data 119910 computed at the optimumaccelerometer locations the input force 119891 was recoveredusing (21) The applied and the recovered loads are plotted inFigure 2The rms error using (25)was calculated to be 07 Itcan be inferred from the plot that the applied load is recoveredaccurately

4 Example 2 Overhead Crane Girder

A more general numerical example is presented here wherea vertical dynamic load acting on an overhead crane girderthrough trolley wheels needs to be estimated For the sakeof illustration and simplicity the number of loads is assumedto be one and assumed to be applied at the girder mid-spanThe problem is to determine the optimum accelerometerlocations and reconstruct the input force based on theacceleration time response at those locations

A finite element model of the girder was developed inANSYS using BEAM188 elements The finite element modelof the girder along with the applied load and boundaryconditions is shown in Figure 3 The model consisted of 50beam elements and 49 unconstrained nodes with 6 degreesof freedom per node that is the total number of degrees offreedom of the model was 294

The [119872] and [119870] matrices were obtained using finiteelement method ANSYS provides data for [119872] and [119870]matrices in the Harwell-Boeing file format A routine waswritten in MATLAB to convert them into the matrix formatsuitable for current application All further calculations wereperformed in MATLAB Discrete time-invariant state-spaceform of the system was obtained using (3)ndash(6) Forward

Figure 3 Finite element model of girder with applied load andoptimum accelerometer locations

Markov parameters were computed from the state-spacemodel using (9) The number of accelerometers to be usedwas arbitrarily assumed to be equal to 3 All DOFs exceptthe DOF where the load was applied were selected to be thelocations where accelerometers can potentially be mountedthat is the DOF corresponding to the applied load didnot form a part of the candidate set [119867]

119911 cs [119867]119911 cs wassubjected to the 119863-optimal design algorithm described inSection 24 to obtain [119867]

119911 opt The optimum accelerometerlocations are also depicted in Figure 3 It is observed thatthe algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initiallybe excluded from the candidate set [119867]

119911 cs Furthermoreif the sensor positions seem to be too congested mutuallyadditional criteria may be instructed such as if a particularspot is chosen as a potential sensor location then certain areaaround that locationmay be excluded from the candidate set

Having obtained the optimum accelerometer locationsthe inverse Markov parameters were computed using (20)In absence of any experimental data the acceleration timeresponse at the optimum locations was obtained by solvingthe ordinary differential equations numerically Using theacceleration data 119910 computed at the optimum accelerom-eter locations the input force 119891 was recovered using (21)The applied and the recovered loads are plotted in Figure 4The rms error using (25) was calculated to be 01

In order to simulate a more realistic scenario whereaccelerations are measured experimentally the accelerationdata 119910 was corrupted with normally distributed randomerrors with zero mean and standard deviation of 10 ofits value The applied and recovered loads with errors inacceleration measurements are plotted in Figure 5 The rmserror using (25) was calculated to be 23

Next a different nature of force was considered where ahalf-sine wave loading was applied Both exact accelerationdata 119910 and acceleration data corrupted with normallydistributed random errors with zero mean and standarddeviation of 10 of its value were considered The appliedand recovered loads without and with errors in acceleration

Shock and Vibration 7

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 4 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 5 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

measurements are plotted in Figures 6 and 7 respectivelyThe rms error using (25) was calculated to be 13 forthe measurement involving acceleration error It can beinferred from the plots that the proposed approach is ableto successfully recover the applied load precisely even whenrealistic measurement errors are present in the structuralresponse

Based on the results for both examples it can be seenthat for discrete as well as continuous systems the proposedapproach is able to accurately estimate the loads acting on acomponent by measuring the acceleration response at a finitenumber of optimum locations

5 Conclusions

A computational technique is presented that allows forindirect measurement of dynamic loads acting on a structuresuch that the structure behaves as its own load transducer

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 6 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 7 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

This is achieved by placing finite number of sensors on thestructure The loads exciting a structure are estimated byconvolving the structural response with the inverse Markovparameters The inverse Markov parameters in turn arecomputed from the forward Markov parameters using alinear prediction algorithmThe forwardMarkov parametersrepresent the response of the system to applied unit impulseand thus contain the dynamic properties of the system Theycan be obtained analytically as well as experimentally

The computation of the inverse Markov parameters fromforward Markov parameters like all inverse problems suf-fers from ill-conditioning The computed inverse Markovparameters are not always bounded thus rendering theirconvolution with the structural response to diverge Theaccuracy of the inverse Markov parameters (and thereby thequality of input load estimates) depends on the locations ofsensors on the structure To improve the precision of loadestimates the technique of 119863-optimal design is utilized todetermine the optimum sensor locations It is observed that

8 Shock and Vibration

the algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initially beexcluded from the candidate set Furthermore if the sensorpositions seem to be too congested mutually additionalcriteriamay be instructed such as if a particular spot is chosenas a potential sensor location then certain area around thatlocation may be excluded from the candidate set The loadsrecovered based on accelerations measured from optimallyplaced accelerometers on the structure are observed to be inexcellent agreement with the applied loads

Nomenclature

[0] Zero matrix Derivative with respect to time

[119860119888] System matrix for continuous case

[119860119889] System matrix for discrete case

[119860119889]119868 Inverse system matrix for discrete case

119886 Number of sensors[119861119888] Input matrix for continuous case

[119861119889] Input matrix for discrete case

[119861119889]119868 Inverse input matrix for discrete case

[119862] Damping matrix[119862119889] Output matrix for discrete case

[119862119889]119868 Inverse output matrix for discrete case

[119863119889] Feedforward matrix for discrete case

[119863119889]119868 Inverse feedforward matrix for discrete

case119891(119905) Load vector119891app Applied load119891rec Reconstructed load[119867]119894 119894th forward Markov parameters matrix

[119867]119911 cs Candidate set

[119867]119911 opt Optimum subset of [119867]

119911 cs determined by119863-optimal design

[ℎ]119894 119894th inverse Markov parameters matrix

[119868] Identity matrix[119870] Stiffness matrix[119872] Mass matrix119899 Number of degrees of freedom of the

system119899119891 Number of applied forces

119905 Time119905119894 Discretized timeΔ119905 Time step119906 119906(119905) State variables vector119909(119905) Displacement vector119910 System output120575 Unit impulse load function120576rms Percent root mean square (rms) error

Conflict of Interests

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

References

[1] K K Stevens ldquoForce identification problemsmdashan overviewrdquo inProceedings of the SEM Conference on Experimental Mechanicspp 838ndash844 Houston Tex USA 1987

[2] D J Ewins Modal Testing Theory Practice and ApplicationResearch Studies Press 2000

[3] G Desanghere and R Snoeys ldquoIndirect identification of excita-tion forces bymodal coordinate transformationrdquo in Proceedingsof the 3rd International Modal Analysis Conference (IMAC rsquo85)pp 685ndash690 Orlando Fla USA 1985

[4] B Hillary and D J Ewins ldquoThe use of strain gauges inforce determination and frequency response functionmeasure-mentsrdquo in Proceedings of the 2nd International Modal AnalysisConference (IMAC rsquo84) pp 627ndash634 Orlando Fla USA 1984

[5] N Okubo S Tanabe and T Tatsuno ldquoIdentification of forcesgenerated by amachine under operating conditionrdquo in Proceed-ings of the 3rd International Modal Analysis Conference (IMACrsquo85) pp 920ndash927 Orlando Fla USA 1985

[6] H R Busby andDM Trujillo ldquoSolution of an inverse dynamicsproblem using an eigenvalue reduction techniquerdquo Computersand Structures vol 25 no 1 pp 109ndash117 1987

[7] P E Hollandsworth and H R Busby ldquoImpact force identifica-tion using the general inverse techniquerdquo International Journalof Impact Engineering vol 8 no 4 pp 315ndash322 1989

[8] GGenaro andDA Rade ldquoInput force identification in the timedomainrdquo in Proceedings of the 16th International Modal AnalysisConference (IMAC rsquo98) pp 124ndash129 Santa Barbara Calif USAFebruary 1998

[9] T G Carne R L Mayes and V I Bateman ldquoForce recon-struction using a sum of weighted accelerations techniquerdquo inProceedings of the 10th International Modal Analysis Conference(IMAC 92) pp 291ndash298 San Diego Calif USA 1992

[10] R Hashemi and M H Kargarnovin ldquoVibration base identifi-cation of impact force using genetic algorithmrdquo InternationalJournal of Mechanical Systems Science and Engineering vol 1no 4 pp 204ndash210 2007

[11] Z R Lu and S S Law ldquoIdentification of system parameters andinput force from output onlyrdquo Mechanical Systems and SignalProcessing vol 21 no 5 pp 2099ndash2111 2007

[12] Z Boukria P Perrotin and A Bennani ldquoExperimental impactforce location and identification using inverse problems appli-cation for a circular platerdquo International Journal of Mechanicsvol 5 no 1 pp 48ndash55 2011

[13] S K Lee ldquoIdentification of impact force in thick plates based onthe elastodynamics and time-frequency method (I)rdquo Journal ofMechanical Science and Technology vol 22 no 7 pp 1349ndash13582008

[14] T S Jang ldquoA novel method for the non-parametric identifica-tion of nonlinear restoring forces in nonlinear vibrations basedon response data a dissipative nonlinear dynamical systemrdquoShips and Offshore Structures vol 6 no 4 pp 257ndash263 2011

[15] S Ahmari and M Yang ldquoImpact location and load iden-tification through inverse analysis with bounded uncertainmeasurementsrdquo Smart Materials and Structures vol 22 no 8Article ID 085024 2013

[16] S Y Khoo Z Ismail K K Kong et al ldquoImpact force identifi-cation with pseudo-inverse method on a lightweight structurefor under-determined even-determined and over-determinedcasesrdquo International Journal of Impact Engineering vol 63 pp52ndash62 2014

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

they can be generated from the finite element model of thestructure

Another way to characterize the input-output behaviorof a structural system is by writing the second-order (1) asa first-order equation or in continuous time-invariant state-space form as

(119905) = [119860119888] 119906 (119905) + [119861

119888] 119891 (119905) (2)

where [119860119888] is the system matrix and [119861

119888] is the input matrix

for continuous case given by

[119860119888] = [

[0] [119868]

minus [119872]minus1

[119870] minus [119872]minus1

[119862]

]

[119861119888] = [

[0]

[119872]minus1]

(3)

All physical phenomena fundamentally exist in con-tinuous time The experimentally measured response datahowever is available only at discrete time instants This callsfor the need to transform the continuous time-invariant state-space model into a discrete time-invariant state-space modelThe transformation takes the following form

119906119905119894+1

= [119860119889] 119906119905119894+ [119861119889] 119891119905119894 (4)

where 119905119894 is the subscript over the discretized time and [119860119889]

and [119861119889] are related to their continuous case counterparts as

[119860119889] = 119890[119860119888]Δ119905

[119861119889] = [119860

119888]minus1

([119860119889] minus [119868]) [119861

119888]

(5)

Equation (4) is the discrete linear time-invariant state-spacerepresentation of the structural system The correspondingsystem output is given by

119910119905119894= [119862119889] 119906119905119894+ [119863119889] 119891119905119894 (6)

Given that unit impulse load is applied to the system at time119905 = 0 that is 120575

0= 1 and assuming zero initial conditions the

impulse response at various time points is given by

[119867]0= [119862119889] 1199060+ [119863119889] 1205750= [119863119889]

1199061= [119860119889] 1199060+ [119861119889] 1205750= [119861119889]

[119867]1= [119862119889] 1199061+ [119863119889] 1205751= [119862119889] [119861119889]

1199062= [119860119889] 1199061+ [119861119889] 1205751= [119860119889] [119861119889]

[119867]2= [119862119889] 1199062+ [119863119889] 1205752= [119862119889] [119860119889] [119861119889]

1199063= [119860119889] 1199062+ [119861119889] 1205752= [119860119889]2

[119861119889]

[119867]3= [119862119889] 1199063+ [119863119889] 1205753= [119862119889] [119860119889]2

[119861119889]

[119867]119894= [119862119889] [119860119889]119894minus1

[119861119889]

(7)

Given the impulse response the response at any time isgiven by the convolution of the input force with the impulseresponse as

119910119905119894=

119905119894

sum

119894=0

[119867]119894119891119905119894minus119894 (8)

where the matrices [119867]119894are called forward Markov parame-

ters [20] and summarized as

[119867]0= [119863119889]

[119867]119894= [119862119889] [119860119889]119894minus1

[119861119889] 119894 = 1 2 3

(9)

The forward Markov parameters represent the responseof the discrete system to applied unit impulse and thuscontain the dynamic properties of the system They can beobtained either analytically from the discretized model ofthe structural system or experimentally by measuring theoutput of the system due to a known input computing thecorresponding frequency response function and then takingits inverse discrete Fourier transform Equation (8) is theMarkov parameter representation of a structural system

22 Problem Formulation In the inverse problem of interesthere the objective is to estimate the input forces 119891 giventhe forward Markov parameters and system responses

From (8) the system response at 119905119894 = 0 is given by

1199100= [119867]

01198910 (10)

Assuming the case where the number of sensors used is atleast equal to the number of loads the least-squares estimateof the loads at 119905119894 = 0 is given as

1198910= ([119867]

119879

0[119867]0)minus1

[119867]119879

01199100 (11)

Similarly at 119905119894 = 1 the system response is given by

1199101= [119867]

01198911+ [119867]

11198910

(12)

which can again be solved for the loads as

1198911= ([119867]

119879

0[119867]0)minus1

[119867]119879

0(1199101minus [119867]

11198910) (13)

By induction the loads at time ti can be verified to be

119891119905119894= ([119867]

119879

0[119867]0)minus1

[119867]119879

0(119910119905119894minus

119905119894

sum

119894=1

[119867]119905119894[119891]119905119894minus119894)

(14)

Next define a discrete inverse system similar to (4) and(6) where the roles of input forces 119891 and system response119910 are reversed as

119906119905119894+1

= [119860119889]119868119906119905119894+ [119861119889]119868119910119905119894

119891119905119894= [119862119889]119868119906119905119894+ [119863119889]119868119910119905119894

(15)

4 Shock and Vibration

where the inverse matrices [119860119889]119868 [119861119889]119868 [119862119889]119868 and [119863

119889]119868are

related to the corresponding forward matrices by

[119860119889]119868= [119860119889] minus [119861

119889] ([119863119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]

[119861119889]119868= [119861119889] ([119863119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]119868= minus ([119863

119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]

[119863119889]119868= ([119863

119889]119879

[119863119889])minus1

[119863119889]119879

(16)

Similar to (8) given the system response the input forces atany time are given by the convolution relation as

119891119905119894=

119905119894

sum

119894=0

[ℎ]119894119910119905119894minus119894 (17)

where the matrices [ℎ]119894are known as the inverse Markov

parameters [20] The inverse Markov parameters similar tothe forward Markov parameters contain the dynamic prop-erties of the inverse system On comparison of expansions of(14) and (17) it may be ascertained that the inverse Markovparameters [ℎ]

119894are related to the forwardMarkov parameters

[119867]119894by a linear predictive equation given by

[ℎ]0= ([119867]

119879

0[119867]0)minus1

[119867]119879

0

[ℎ]119896= minus [ℎ]

0

119896

sum

119894=1

[119867]119894[ℎ]119896minus119894

(18)

There exist cases where [119867]0is zero matrix In certain

other nonminimum phase structural systems [119867]0is rank

deficient and the least-squares inverse in (18) does not existThis renders the causal summation in (17) undefined To dealwith this limitation the system output (6) is stepped forwardin time as [21]

119910119905119894= [119862119889] 119906119905119894+ [119862119889] [119861119889] 119891119905119894 (19)

The inverse system associated with (19) is noncausal that isthe input force estimates at current time become a function ofthe structural response at future times In general if the first zMarkov parameters in (7) are zero that is [119863

119889] = [119862

119889][119861119889] =

[119862119889][119860119889][119861119889] = sdot sdot sdot = [119862

119889][119860119889]119911minus2

[119861119889] = [0] the noncausal

z-lead inverse system can be constructed as

[ℎ]119911= ([119867]

119879

119911[119867]119911)minus1

[119867]119879

119911

[ℎ]119896+119911

= minus [ℎ]119911

119896

sum

119894=1

[119867]119894+119911[ℎ]119896minus119894+119911

(20)

and the corresponding force estimates are given by

119891119905119894=

119905119894

sum

119894=0

[ℎ]119894+119911119910119905119894minus119894+119911

(21)

It must be noted in (21) that the input force estimates 119891 atcurrent time ti are dependent on the structural response 119910at future times 119905119894 minus 119894 + 119911

The determination of the inverse Markov parametersfrom the forward Markov parameters using (20) is com-putationally expensive however it must be noted that thecomputation of the inverse Markov parameters needs tobe performed only once for any particular system Oncethe inverse Markov parameters are computed the inputforces can be predicted from the response using (21) Thistechnique however suffers from a similar limitation as mostof the other inverse problemsmdashill-conditioning The inac-curacies in system modeling translate to errors in forwardMarkov parameters The computation of inverse Markovparameters from erroneous forwardMarkov parameters is ill-conditioned and the computed inverse Markov parametersbecome unbounded Thus the convolution of the inverseMarkov parameters [ℎ] with the noisy structural response119910 given by (21) is not always a converging sum Thesum becomes numerically unstable due to the intrinsic ill-conditioning of the inverse problem identified by (20)

Conditioning of the erroneous forward Markov param-eters depends on the locations of sensors on the structuretherefore the input loads 119891 can be estimated precisely frommeasured structural response 119910 by strategic placement ofsensors on the structure Since it is not feasible to placesensors at all the DOFs of a structure a methodology ispresented next to overcome the identified shortcoming

23 Candidate Set Optimal selection of sensor locationson the structure requires a candidate set to be defined Itcan be inferred from the load identification step in (21)that the quality of load estimates depends directly on howaccurately the inverse Markov parameters are computed Itcan further be assessed from (20) that the 119911th inverseMarkovparameter [ℎ]

119911is computed by the least-squares inversion

of the 119911th Markov parameter [119867]119911 All the other higher

order inverse Markov parameters depend on [ℎ]119911 Therefore

the accuracy in the computation of [ℎ]119911from [119867]

119911dictates

the accuracy of all the other higher order inverse Markovparameters Each row in [119867]

119911corresponds to a unique DOF

of the system output Typically there are a large number oflocations on a structure where sensors can potentially bemounted These locations may exclude inaccessible locationssuch as the regions of load application To form a candidateset for optimization the DOFs (rows) corresponding to theinaccessible locations for sensor placement are ignored from[119867]119911 the subset so obtained is called the candidate set [119867]

119911 csTo obtain ameaningful solution to (20) an optimal subset

[119867]119911 opt of the candidate set [119867]

119911 cs needs to be identifiedwhich in turn will lead to an increased accuracy in theestimation of 119891 As more sensors are used the additionalinformation helps to obtain a more precise estimate of 119891but practical and financial constraints place limitations onthe number of sensors that can be used If the number offorces to be estimated is 119899

119891 then in order to minimize the

error in 119891 estimates the number of sensors a must satisfythe criterion 119886 ge 119899

119891 Each sensor location referred to as

a candidate point provides a potential row for inclusion in[119867]119911 opt [119867]119911 opt needs to be such a subset of [119867]

119911 cs that

Shock and Vibration 5

provides the most precise estimates of 119891119905119894 An approach to

extract [119867]119911 opt from [119867]

119911 cs is described next

24 119863-Optimal Design If the errors in response measure-ments are statistically independent and the standard devia-tion of each of them is 120590 then the covariance matrix for 119891

119905119894

estimates is given by [22]

var (119891119905119894) = 1205902

([119867]119879

119911[119867]119911)minus1

(22)

The matrix ([119867]119879119911[119867]119911)minus1 is known as the sensitivity of

[119867]119911 For a given variance 1205902 in response measurements

minimization of the sensitivity of [119867]119911leads to an increased

precision in 119891119905119894

estimates The sensitivity of [119867]119911is a

function of the number and locations of the sensors on thestructure Therefore optimum selection of the number andlocations of the sensors on the structure so as to minimizethe sensitivity of [119867]

119911leads to the minimization of variation

in the 119891119905119894estimates

For a given number of sensors a the candidate set [119867]119911 cs

is searched to determine a sensor locations that providethe least variance in the load estimates One of the criteriafollowed involves the maximization of the determinant of[119867]119879

119911[119867]119911 |[119867]119879

119911[119867]119911| Designs that maximize |[119867]119879

119911[119867]119911|

are called 119863-optimal designs [23ndash25] where D denotesdeterminant In order to construct an a-point 119863-optimaldesign the a sensor locations that maximize |[119867]119879

119911[119867]119911|

must be selected from the candidate set [119867]119911 cs To select the

a-point 119863-optimal design sequential exchange algorithms[24] based on the principles of optimal augmentation andreduction of an existing design are used herein

The basic idea behind the sequential exchange algorithmis as follows Given the candidate set [119867]

119911 cs and the numberof sensors a the first step is to randomly select a distinctcandidate points from the candidate set to initialize [119867]

119911

Out of the remaining candidate set a candidate point is thenselected and the corresponding row is augmented to [119867]

119911

to form [119867]119911+

such that |[119867]119879119911 +[119867]119911+| is maximum Next

out of the 119886 + 1 rows in [119867]119911+ a row is deleted to arrive at

[119867]119911minus

such that |[119867]119879119911 minus[119867]119911minus| is maximum This process of

adding and deleting rows continues until there is no furtherimprovement in the value of |[119867]119879

119911[119867]119911| The final [119867]

119911so

obtained is the 119863-optimal design [119867]119911 opt whose row indices

provide the information on the optimum sensor locationsAs the candidate set gets bigger and bigger increasingly

large number of determinants and matrix inverses need to becalculated A very naıve way to compute the determinant isto obtain119872 = [119867]

119879

119911[119867]119911and then calculate the determinant

|119872| This approach of determinant calculation is computa-tionally very expensive An alternate formula for computingthe determinant |119872

+| = |[119867]

119879

119911 +[119867]119911+| from that of |119872|when

a row 119910119879 is augmented to [119867]119911is [25]

1003816100381610038161003816119872+1003816100381610038161003816 = |119872| (1 [+] 119910

119879

119872minus1

119910) (23)

where [+]denotes additionThe [+] is replaced by subtractionfor the case when a row 119910119879 is deleted from [119867]

119911+ In order to

k1

C1

m1

k2

C2

m2

k3

C3

middot middot middot

k14

C14

m14

k15

C15

m15

k16

C16

Figure 1 15-Dof spring-mass-damper system

be able to use (23) 119872minus1 can be maintained and updated asthe row 119910119879 is augmented to [119867]

119911by

1003816100381610038161003816119872+1003816100381610038161003816

minus1

= |119872|minus1

[minus]

(119872minus1

119910) (119872minus1

119910)119879

(1 [+] 119910119879119872minus1119910)

(24)

where [minus] denotes subtraction and is replaced by addition forthe case when a row 119910119879 is deleted from [119867]

119911+

Once [119867]119911 opt and in turn optimum sensor locations are

determined sensors are mounted at the identified optimumlocations and response is measured Equations (20) and (21)are then utilized to compute the input forces 119891 It is to benoted that in this approach it is not necessary that responsemeasurements be taken at locations where forces are applied

25 Error Quantification The percent root mean square(rms) error in the recovered loads with respect to the actualapplied loads can be calculated as

120576rms = (

radicsum(119891app minus 119891rec)2

radicsum1198912

app

times 100) (25)

The above equation can be used to quantify errors in the loadestimates

3 Example 1 15-DOFSpring-Mass-Damper System

The dynamic load estimation method from accelerationmeasurements is illustrated with the help of a numerical sim-ulation consisting of a 15 DOF spring-mass-damper systemshown in Figure 1 Masses119898

1and119898

15are connected to fixed

boundary A periodic forcing function is applied to mass1198987

The task is to determine the optimumaccelerometer locationsand reconstruct the input force based on the acceleration timeresponse at those locations

A total of 2 accelerometers were used to measure accel-erations of 2 masses The [119872] [119862] and [119870] matrices wereobtained by writing the equations of motion for the sys-tem Discrete time-invariant state-space form of the systemwas obtained using (3)ndash(6) Forward Markov parameterswere computed from the state-space model using (9) Allnumerical computations were performed in a MATLABprogramming environment

All DOFs except the DOF where the load was appliedwere selected to be the locations where accelerometers canpotentially bemounted that is theDOF corresponding to theapplied load did not form a part of the candidate set [119867]

119911 cs

6 Shock and Vibration

0 001 002 003 004 005 006 007 008 009 01minus1500

minus1000

minus500

0

500

1000

1500

Time

Load

AppliedRecovered

Figure 2 Applied and recovered loads at mass 7 with optimumaccelerometer placements

[119867]119911 cs was subjected to the 119863-optimal design algorithm

described in Section 24 to obtain [119867]119911 opt whereby the

optimum accelerometer locations were found to be at masses1198986and 119898

8 Having obtained the optimum accelerometer

locations the inverse Markov parameters were computedusing (20) In absence of any experimental data the acceler-ation time response at the optimum locations was obtainedby solving the ordinary differential equations numericallyUsing the acceleration data 119910 computed at the optimumaccelerometer locations the input force 119891 was recoveredusing (21) The applied and the recovered loads are plotted inFigure 2The rms error using (25)was calculated to be 07 Itcan be inferred from the plot that the applied load is recoveredaccurately

4 Example 2 Overhead Crane Girder

A more general numerical example is presented here wherea vertical dynamic load acting on an overhead crane girderthrough trolley wheels needs to be estimated For the sakeof illustration and simplicity the number of loads is assumedto be one and assumed to be applied at the girder mid-spanThe problem is to determine the optimum accelerometerlocations and reconstruct the input force based on theacceleration time response at those locations

A finite element model of the girder was developed inANSYS using BEAM188 elements The finite element modelof the girder along with the applied load and boundaryconditions is shown in Figure 3 The model consisted of 50beam elements and 49 unconstrained nodes with 6 degreesof freedom per node that is the total number of degrees offreedom of the model was 294

The [119872] and [119870] matrices were obtained using finiteelement method ANSYS provides data for [119872] and [119870]matrices in the Harwell-Boeing file format A routine waswritten in MATLAB to convert them into the matrix formatsuitable for current application All further calculations wereperformed in MATLAB Discrete time-invariant state-spaceform of the system was obtained using (3)ndash(6) Forward

Figure 3 Finite element model of girder with applied load andoptimum accelerometer locations

Markov parameters were computed from the state-spacemodel using (9) The number of accelerometers to be usedwas arbitrarily assumed to be equal to 3 All DOFs exceptthe DOF where the load was applied were selected to be thelocations where accelerometers can potentially be mountedthat is the DOF corresponding to the applied load didnot form a part of the candidate set [119867]

119911 cs [119867]119911 cs wassubjected to the 119863-optimal design algorithm described inSection 24 to obtain [119867]

119911 opt The optimum accelerometerlocations are also depicted in Figure 3 It is observed thatthe algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initiallybe excluded from the candidate set [119867]

119911 cs Furthermoreif the sensor positions seem to be too congested mutuallyadditional criteria may be instructed such as if a particularspot is chosen as a potential sensor location then certain areaaround that locationmay be excluded from the candidate set

Having obtained the optimum accelerometer locationsthe inverse Markov parameters were computed using (20)In absence of any experimental data the acceleration timeresponse at the optimum locations was obtained by solvingthe ordinary differential equations numerically Using theacceleration data 119910 computed at the optimum accelerom-eter locations the input force 119891 was recovered using (21)The applied and the recovered loads are plotted in Figure 4The rms error using (25) was calculated to be 01

In order to simulate a more realistic scenario whereaccelerations are measured experimentally the accelerationdata 119910 was corrupted with normally distributed randomerrors with zero mean and standard deviation of 10 ofits value The applied and recovered loads with errors inacceleration measurements are plotted in Figure 5 The rmserror using (25) was calculated to be 23

Next a different nature of force was considered where ahalf-sine wave loading was applied Both exact accelerationdata 119910 and acceleration data corrupted with normallydistributed random errors with zero mean and standarddeviation of 10 of its value were considered The appliedand recovered loads without and with errors in acceleration

Shock and Vibration 7

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 4 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 5 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

measurements are plotted in Figures 6 and 7 respectivelyThe rms error using (25) was calculated to be 13 forthe measurement involving acceleration error It can beinferred from the plots that the proposed approach is ableto successfully recover the applied load precisely even whenrealistic measurement errors are present in the structuralresponse

Based on the results for both examples it can be seenthat for discrete as well as continuous systems the proposedapproach is able to accurately estimate the loads acting on acomponent by measuring the acceleration response at a finitenumber of optimum locations

5 Conclusions

A computational technique is presented that allows forindirect measurement of dynamic loads acting on a structuresuch that the structure behaves as its own load transducer

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 6 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 7 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

This is achieved by placing finite number of sensors on thestructure The loads exciting a structure are estimated byconvolving the structural response with the inverse Markovparameters The inverse Markov parameters in turn arecomputed from the forward Markov parameters using alinear prediction algorithmThe forwardMarkov parametersrepresent the response of the system to applied unit impulseand thus contain the dynamic properties of the system Theycan be obtained analytically as well as experimentally

The computation of the inverse Markov parameters fromforward Markov parameters like all inverse problems suf-fers from ill-conditioning The computed inverse Markovparameters are not always bounded thus rendering theirconvolution with the structural response to diverge Theaccuracy of the inverse Markov parameters (and thereby thequality of input load estimates) depends on the locations ofsensors on the structure To improve the precision of loadestimates the technique of 119863-optimal design is utilized todetermine the optimum sensor locations It is observed that

8 Shock and Vibration

the algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initially beexcluded from the candidate set Furthermore if the sensorpositions seem to be too congested mutually additionalcriteriamay be instructed such as if a particular spot is chosenas a potential sensor location then certain area around thatlocation may be excluded from the candidate set The loadsrecovered based on accelerations measured from optimallyplaced accelerometers on the structure are observed to be inexcellent agreement with the applied loads

Nomenclature

[0] Zero matrix Derivative with respect to time

[119860119888] System matrix for continuous case

[119860119889] System matrix for discrete case

[119860119889]119868 Inverse system matrix for discrete case

119886 Number of sensors[119861119888] Input matrix for continuous case

[119861119889] Input matrix for discrete case

[119861119889]119868 Inverse input matrix for discrete case

[119862] Damping matrix[119862119889] Output matrix for discrete case

[119862119889]119868 Inverse output matrix for discrete case

[119863119889] Feedforward matrix for discrete case

[119863119889]119868 Inverse feedforward matrix for discrete

case119891(119905) Load vector119891app Applied load119891rec Reconstructed load[119867]119894 119894th forward Markov parameters matrix

[119867]119911 cs Candidate set

[119867]119911 opt Optimum subset of [119867]

119911 cs determined by119863-optimal design

[ℎ]119894 119894th inverse Markov parameters matrix

[119868] Identity matrix[119870] Stiffness matrix[119872] Mass matrix119899 Number of degrees of freedom of the

system119899119891 Number of applied forces

119905 Time119905119894 Discretized timeΔ119905 Time step119906 119906(119905) State variables vector119909(119905) Displacement vector119910 System output120575 Unit impulse load function120576rms Percent root mean square (rms) error

Conflict of Interests

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

References

[1] K K Stevens ldquoForce identification problemsmdashan overviewrdquo inProceedings of the SEM Conference on Experimental Mechanicspp 838ndash844 Houston Tex USA 1987

[2] D J Ewins Modal Testing Theory Practice and ApplicationResearch Studies Press 2000

[3] G Desanghere and R Snoeys ldquoIndirect identification of excita-tion forces bymodal coordinate transformationrdquo in Proceedingsof the 3rd International Modal Analysis Conference (IMAC rsquo85)pp 685ndash690 Orlando Fla USA 1985

[4] B Hillary and D J Ewins ldquoThe use of strain gauges inforce determination and frequency response functionmeasure-mentsrdquo in Proceedings of the 2nd International Modal AnalysisConference (IMAC rsquo84) pp 627ndash634 Orlando Fla USA 1984

[5] N Okubo S Tanabe and T Tatsuno ldquoIdentification of forcesgenerated by amachine under operating conditionrdquo in Proceed-ings of the 3rd International Modal Analysis Conference (IMACrsquo85) pp 920ndash927 Orlando Fla USA 1985

[6] H R Busby andDM Trujillo ldquoSolution of an inverse dynamicsproblem using an eigenvalue reduction techniquerdquo Computersand Structures vol 25 no 1 pp 109ndash117 1987

[7] P E Hollandsworth and H R Busby ldquoImpact force identifica-tion using the general inverse techniquerdquo International Journalof Impact Engineering vol 8 no 4 pp 315ndash322 1989

[8] GGenaro andDA Rade ldquoInput force identification in the timedomainrdquo in Proceedings of the 16th International Modal AnalysisConference (IMAC rsquo98) pp 124ndash129 Santa Barbara Calif USAFebruary 1998

[9] T G Carne R L Mayes and V I Bateman ldquoForce recon-struction using a sum of weighted accelerations techniquerdquo inProceedings of the 10th International Modal Analysis Conference(IMAC 92) pp 291ndash298 San Diego Calif USA 1992

[10] R Hashemi and M H Kargarnovin ldquoVibration base identifi-cation of impact force using genetic algorithmrdquo InternationalJournal of Mechanical Systems Science and Engineering vol 1no 4 pp 204ndash210 2007

[11] Z R Lu and S S Law ldquoIdentification of system parameters andinput force from output onlyrdquo Mechanical Systems and SignalProcessing vol 21 no 5 pp 2099ndash2111 2007

[12] Z Boukria P Perrotin and A Bennani ldquoExperimental impactforce location and identification using inverse problems appli-cation for a circular platerdquo International Journal of Mechanicsvol 5 no 1 pp 48ndash55 2011

[13] S K Lee ldquoIdentification of impact force in thick plates based onthe elastodynamics and time-frequency method (I)rdquo Journal ofMechanical Science and Technology vol 22 no 7 pp 1349ndash13582008

[14] T S Jang ldquoA novel method for the non-parametric identifica-tion of nonlinear restoring forces in nonlinear vibrations basedon response data a dissipative nonlinear dynamical systemrdquoShips and Offshore Structures vol 6 no 4 pp 257ndash263 2011

[15] S Ahmari and M Yang ldquoImpact location and load iden-tification through inverse analysis with bounded uncertainmeasurementsrdquo Smart Materials and Structures vol 22 no 8Article ID 085024 2013

[16] S Y Khoo Z Ismail K K Kong et al ldquoImpact force identifi-cation with pseudo-inverse method on a lightweight structurefor under-determined even-determined and over-determinedcasesrdquo International Journal of Impact Engineering vol 63 pp52ndash62 2014

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

where the inverse matrices [119860119889]119868 [119861119889]119868 [119862119889]119868 and [119863

119889]119868are

related to the corresponding forward matrices by

[119860119889]119868= [119860119889] minus [119861

119889] ([119863119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]

[119861119889]119868= [119861119889] ([119863119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]119868= minus ([119863

119889]119879

[119863119889])minus1

[119863119889]119879

[119862119889]

[119863119889]119868= ([119863

119889]119879

[119863119889])minus1

[119863119889]119879

(16)

Similar to (8) given the system response the input forces atany time are given by the convolution relation as

119891119905119894=

119905119894

sum

119894=0

[ℎ]119894119910119905119894minus119894 (17)

where the matrices [ℎ]119894are known as the inverse Markov

parameters [20] The inverse Markov parameters similar tothe forward Markov parameters contain the dynamic prop-erties of the inverse system On comparison of expansions of(14) and (17) it may be ascertained that the inverse Markovparameters [ℎ]

119894are related to the forwardMarkov parameters

[119867]119894by a linear predictive equation given by

[ℎ]0= ([119867]

119879

0[119867]0)minus1

[119867]119879

0

[ℎ]119896= minus [ℎ]

0

119896

sum

119894=1

[119867]119894[ℎ]119896minus119894

(18)

There exist cases where [119867]0is zero matrix In certain

other nonminimum phase structural systems [119867]0is rank

deficient and the least-squares inverse in (18) does not existThis renders the causal summation in (17) undefined To dealwith this limitation the system output (6) is stepped forwardin time as [21]

119910119905119894= [119862119889] 119906119905119894+ [119862119889] [119861119889] 119891119905119894 (19)

The inverse system associated with (19) is noncausal that isthe input force estimates at current time become a function ofthe structural response at future times In general if the first zMarkov parameters in (7) are zero that is [119863

119889] = [119862

119889][119861119889] =

[119862119889][119860119889][119861119889] = sdot sdot sdot = [119862

119889][119860119889]119911minus2

[119861119889] = [0] the noncausal

z-lead inverse system can be constructed as

[ℎ]119911= ([119867]

119879

119911[119867]119911)minus1

[119867]119879

119911

[ℎ]119896+119911

= minus [ℎ]119911

119896

sum

119894=1

[119867]119894+119911[ℎ]119896minus119894+119911

(20)

and the corresponding force estimates are given by

119891119905119894=

119905119894

sum

119894=0

[ℎ]119894+119911119910119905119894minus119894+119911

(21)

It must be noted in (21) that the input force estimates 119891 atcurrent time ti are dependent on the structural response 119910at future times 119905119894 minus 119894 + 119911

The determination of the inverse Markov parametersfrom the forward Markov parameters using (20) is com-putationally expensive however it must be noted that thecomputation of the inverse Markov parameters needs tobe performed only once for any particular system Oncethe inverse Markov parameters are computed the inputforces can be predicted from the response using (21) Thistechnique however suffers from a similar limitation as mostof the other inverse problemsmdashill-conditioning The inac-curacies in system modeling translate to errors in forwardMarkov parameters The computation of inverse Markovparameters from erroneous forwardMarkov parameters is ill-conditioned and the computed inverse Markov parametersbecome unbounded Thus the convolution of the inverseMarkov parameters [ℎ] with the noisy structural response119910 given by (21) is not always a converging sum Thesum becomes numerically unstable due to the intrinsic ill-conditioning of the inverse problem identified by (20)

Conditioning of the erroneous forward Markov param-eters depends on the locations of sensors on the structuretherefore the input loads 119891 can be estimated precisely frommeasured structural response 119910 by strategic placement ofsensors on the structure Since it is not feasible to placesensors at all the DOFs of a structure a methodology ispresented next to overcome the identified shortcoming

23 Candidate Set Optimal selection of sensor locationson the structure requires a candidate set to be defined Itcan be inferred from the load identification step in (21)that the quality of load estimates depends directly on howaccurately the inverse Markov parameters are computed Itcan further be assessed from (20) that the 119911th inverseMarkovparameter [ℎ]

119911is computed by the least-squares inversion

of the 119911th Markov parameter [119867]119911 All the other higher

order inverse Markov parameters depend on [ℎ]119911 Therefore

the accuracy in the computation of [ℎ]119911from [119867]

119911dictates

the accuracy of all the other higher order inverse Markovparameters Each row in [119867]

119911corresponds to a unique DOF

of the system output Typically there are a large number oflocations on a structure where sensors can potentially bemounted These locations may exclude inaccessible locationssuch as the regions of load application To form a candidateset for optimization the DOFs (rows) corresponding to theinaccessible locations for sensor placement are ignored from[119867]119911 the subset so obtained is called the candidate set [119867]

119911 csTo obtain ameaningful solution to (20) an optimal subset

[119867]119911 opt of the candidate set [119867]

119911 cs needs to be identifiedwhich in turn will lead to an increased accuracy in theestimation of 119891 As more sensors are used the additionalinformation helps to obtain a more precise estimate of 119891but practical and financial constraints place limitations onthe number of sensors that can be used If the number offorces to be estimated is 119899

119891 then in order to minimize the

error in 119891 estimates the number of sensors a must satisfythe criterion 119886 ge 119899

119891 Each sensor location referred to as

a candidate point provides a potential row for inclusion in[119867]119911 opt [119867]119911 opt needs to be such a subset of [119867]

119911 cs that

Shock and Vibration 5

provides the most precise estimates of 119891119905119894 An approach to

extract [119867]119911 opt from [119867]

119911 cs is described next

24 119863-Optimal Design If the errors in response measure-ments are statistically independent and the standard devia-tion of each of them is 120590 then the covariance matrix for 119891

119905119894

estimates is given by [22]

var (119891119905119894) = 1205902

([119867]119879

119911[119867]119911)minus1

(22)

The matrix ([119867]119879119911[119867]119911)minus1 is known as the sensitivity of

[119867]119911 For a given variance 1205902 in response measurements

minimization of the sensitivity of [119867]119911leads to an increased

precision in 119891119905119894

estimates The sensitivity of [119867]119911is a

function of the number and locations of the sensors on thestructure Therefore optimum selection of the number andlocations of the sensors on the structure so as to minimizethe sensitivity of [119867]

119911leads to the minimization of variation

in the 119891119905119894estimates

For a given number of sensors a the candidate set [119867]119911 cs

is searched to determine a sensor locations that providethe least variance in the load estimates One of the criteriafollowed involves the maximization of the determinant of[119867]119879

119911[119867]119911 |[119867]119879

119911[119867]119911| Designs that maximize |[119867]119879

119911[119867]119911|

are called 119863-optimal designs [23ndash25] where D denotesdeterminant In order to construct an a-point 119863-optimaldesign the a sensor locations that maximize |[119867]119879

119911[119867]119911|

must be selected from the candidate set [119867]119911 cs To select the

a-point 119863-optimal design sequential exchange algorithms[24] based on the principles of optimal augmentation andreduction of an existing design are used herein

The basic idea behind the sequential exchange algorithmis as follows Given the candidate set [119867]

119911 cs and the numberof sensors a the first step is to randomly select a distinctcandidate points from the candidate set to initialize [119867]

119911

Out of the remaining candidate set a candidate point is thenselected and the corresponding row is augmented to [119867]

119911

to form [119867]119911+

such that |[119867]119879119911 +[119867]119911+| is maximum Next

out of the 119886 + 1 rows in [119867]119911+ a row is deleted to arrive at

[119867]119911minus

such that |[119867]119879119911 minus[119867]119911minus| is maximum This process of

adding and deleting rows continues until there is no furtherimprovement in the value of |[119867]119879

119911[119867]119911| The final [119867]

119911so

obtained is the 119863-optimal design [119867]119911 opt whose row indices

provide the information on the optimum sensor locationsAs the candidate set gets bigger and bigger increasingly

large number of determinants and matrix inverses need to becalculated A very naıve way to compute the determinant isto obtain119872 = [119867]

119879

119911[119867]119911and then calculate the determinant

|119872| This approach of determinant calculation is computa-tionally very expensive An alternate formula for computingthe determinant |119872

+| = |[119867]

119879

119911 +[119867]119911+| from that of |119872|when

a row 119910119879 is augmented to [119867]119911is [25]

1003816100381610038161003816119872+1003816100381610038161003816 = |119872| (1 [+] 119910

119879

119872minus1

119910) (23)

where [+]denotes additionThe [+] is replaced by subtractionfor the case when a row 119910119879 is deleted from [119867]

119911+ In order to

k1

C1

m1

k2

C2

m2

k3

C3

middot middot middot

k14

C14

m14

k15

C15

m15

k16

C16

Figure 1 15-Dof spring-mass-damper system

be able to use (23) 119872minus1 can be maintained and updated asthe row 119910119879 is augmented to [119867]

119911by

1003816100381610038161003816119872+1003816100381610038161003816

minus1

= |119872|minus1

[minus]

(119872minus1

119910) (119872minus1

119910)119879

(1 [+] 119910119879119872minus1119910)

(24)

where [minus] denotes subtraction and is replaced by addition forthe case when a row 119910119879 is deleted from [119867]

119911+

Once [119867]119911 opt and in turn optimum sensor locations are

determined sensors are mounted at the identified optimumlocations and response is measured Equations (20) and (21)are then utilized to compute the input forces 119891 It is to benoted that in this approach it is not necessary that responsemeasurements be taken at locations where forces are applied

25 Error Quantification The percent root mean square(rms) error in the recovered loads with respect to the actualapplied loads can be calculated as

120576rms = (

radicsum(119891app minus 119891rec)2

radicsum1198912

app

times 100) (25)

The above equation can be used to quantify errors in the loadestimates

3 Example 1 15-DOFSpring-Mass-Damper System

The dynamic load estimation method from accelerationmeasurements is illustrated with the help of a numerical sim-ulation consisting of a 15 DOF spring-mass-damper systemshown in Figure 1 Masses119898

1and119898

15are connected to fixed

boundary A periodic forcing function is applied to mass1198987

The task is to determine the optimumaccelerometer locationsand reconstruct the input force based on the acceleration timeresponse at those locations

A total of 2 accelerometers were used to measure accel-erations of 2 masses The [119872] [119862] and [119870] matrices wereobtained by writing the equations of motion for the sys-tem Discrete time-invariant state-space form of the systemwas obtained using (3)ndash(6) Forward Markov parameterswere computed from the state-space model using (9) Allnumerical computations were performed in a MATLABprogramming environment

All DOFs except the DOF where the load was appliedwere selected to be the locations where accelerometers canpotentially bemounted that is theDOF corresponding to theapplied load did not form a part of the candidate set [119867]

119911 cs

6 Shock and Vibration

0 001 002 003 004 005 006 007 008 009 01minus1500

minus1000

minus500

0

500

1000

1500

Time

Load

AppliedRecovered

Figure 2 Applied and recovered loads at mass 7 with optimumaccelerometer placements

[119867]119911 cs was subjected to the 119863-optimal design algorithm

described in Section 24 to obtain [119867]119911 opt whereby the

optimum accelerometer locations were found to be at masses1198986and 119898

8 Having obtained the optimum accelerometer

locations the inverse Markov parameters were computedusing (20) In absence of any experimental data the acceler-ation time response at the optimum locations was obtainedby solving the ordinary differential equations numericallyUsing the acceleration data 119910 computed at the optimumaccelerometer locations the input force 119891 was recoveredusing (21) The applied and the recovered loads are plotted inFigure 2The rms error using (25)was calculated to be 07 Itcan be inferred from the plot that the applied load is recoveredaccurately

4 Example 2 Overhead Crane Girder

A more general numerical example is presented here wherea vertical dynamic load acting on an overhead crane girderthrough trolley wheels needs to be estimated For the sakeof illustration and simplicity the number of loads is assumedto be one and assumed to be applied at the girder mid-spanThe problem is to determine the optimum accelerometerlocations and reconstruct the input force based on theacceleration time response at those locations

A finite element model of the girder was developed inANSYS using BEAM188 elements The finite element modelof the girder along with the applied load and boundaryconditions is shown in Figure 3 The model consisted of 50beam elements and 49 unconstrained nodes with 6 degreesof freedom per node that is the total number of degrees offreedom of the model was 294

The [119872] and [119870] matrices were obtained using finiteelement method ANSYS provides data for [119872] and [119870]matrices in the Harwell-Boeing file format A routine waswritten in MATLAB to convert them into the matrix formatsuitable for current application All further calculations wereperformed in MATLAB Discrete time-invariant state-spaceform of the system was obtained using (3)ndash(6) Forward

Figure 3 Finite element model of girder with applied load andoptimum accelerometer locations

Markov parameters were computed from the state-spacemodel using (9) The number of accelerometers to be usedwas arbitrarily assumed to be equal to 3 All DOFs exceptthe DOF where the load was applied were selected to be thelocations where accelerometers can potentially be mountedthat is the DOF corresponding to the applied load didnot form a part of the candidate set [119867]

119911 cs [119867]119911 cs wassubjected to the 119863-optimal design algorithm described inSection 24 to obtain [119867]

119911 opt The optimum accelerometerlocations are also depicted in Figure 3 It is observed thatthe algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initiallybe excluded from the candidate set [119867]

119911 cs Furthermoreif the sensor positions seem to be too congested mutuallyadditional criteria may be instructed such as if a particularspot is chosen as a potential sensor location then certain areaaround that locationmay be excluded from the candidate set

Having obtained the optimum accelerometer locationsthe inverse Markov parameters were computed using (20)In absence of any experimental data the acceleration timeresponse at the optimum locations was obtained by solvingthe ordinary differential equations numerically Using theacceleration data 119910 computed at the optimum accelerom-eter locations the input force 119891 was recovered using (21)The applied and the recovered loads are plotted in Figure 4The rms error using (25) was calculated to be 01

In order to simulate a more realistic scenario whereaccelerations are measured experimentally the accelerationdata 119910 was corrupted with normally distributed randomerrors with zero mean and standard deviation of 10 ofits value The applied and recovered loads with errors inacceleration measurements are plotted in Figure 5 The rmserror using (25) was calculated to be 23

Next a different nature of force was considered where ahalf-sine wave loading was applied Both exact accelerationdata 119910 and acceleration data corrupted with normallydistributed random errors with zero mean and standarddeviation of 10 of its value were considered The appliedand recovered loads without and with errors in acceleration

Shock and Vibration 7

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 4 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 5 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

measurements are plotted in Figures 6 and 7 respectivelyThe rms error using (25) was calculated to be 13 forthe measurement involving acceleration error It can beinferred from the plots that the proposed approach is ableto successfully recover the applied load precisely even whenrealistic measurement errors are present in the structuralresponse

Based on the results for both examples it can be seenthat for discrete as well as continuous systems the proposedapproach is able to accurately estimate the loads acting on acomponent by measuring the acceleration response at a finitenumber of optimum locations

5 Conclusions

A computational technique is presented that allows forindirect measurement of dynamic loads acting on a structuresuch that the structure behaves as its own load transducer

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 6 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 7 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

This is achieved by placing finite number of sensors on thestructure The loads exciting a structure are estimated byconvolving the structural response with the inverse Markovparameters The inverse Markov parameters in turn arecomputed from the forward Markov parameters using alinear prediction algorithmThe forwardMarkov parametersrepresent the response of the system to applied unit impulseand thus contain the dynamic properties of the system Theycan be obtained analytically as well as experimentally

The computation of the inverse Markov parameters fromforward Markov parameters like all inverse problems suf-fers from ill-conditioning The computed inverse Markovparameters are not always bounded thus rendering theirconvolution with the structural response to diverge Theaccuracy of the inverse Markov parameters (and thereby thequality of input load estimates) depends on the locations ofsensors on the structure To improve the precision of loadestimates the technique of 119863-optimal design is utilized todetermine the optimum sensor locations It is observed that

8 Shock and Vibration

the algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initially beexcluded from the candidate set Furthermore if the sensorpositions seem to be too congested mutually additionalcriteriamay be instructed such as if a particular spot is chosenas a potential sensor location then certain area around thatlocation may be excluded from the candidate set The loadsrecovered based on accelerations measured from optimallyplaced accelerometers on the structure are observed to be inexcellent agreement with the applied loads

Nomenclature

[0] Zero matrix Derivative with respect to time

[119860119888] System matrix for continuous case

[119860119889] System matrix for discrete case

[119860119889]119868 Inverse system matrix for discrete case

119886 Number of sensors[119861119888] Input matrix for continuous case

[119861119889] Input matrix for discrete case

[119861119889]119868 Inverse input matrix for discrete case

[119862] Damping matrix[119862119889] Output matrix for discrete case

[119862119889]119868 Inverse output matrix for discrete case

[119863119889] Feedforward matrix for discrete case

[119863119889]119868 Inverse feedforward matrix for discrete

case119891(119905) Load vector119891app Applied load119891rec Reconstructed load[119867]119894 119894th forward Markov parameters matrix

[119867]119911 cs Candidate set

[119867]119911 opt Optimum subset of [119867]

119911 cs determined by119863-optimal design

[ℎ]119894 119894th inverse Markov parameters matrix

[119868] Identity matrix[119870] Stiffness matrix[119872] Mass matrix119899 Number of degrees of freedom of the

system119899119891 Number of applied forces

119905 Time119905119894 Discretized timeΔ119905 Time step119906 119906(119905) State variables vector119909(119905) Displacement vector119910 System output120575 Unit impulse load function120576rms Percent root mean square (rms) error

Conflict of Interests

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

References

[1] K K Stevens ldquoForce identification problemsmdashan overviewrdquo inProceedings of the SEM Conference on Experimental Mechanicspp 838ndash844 Houston Tex USA 1987

[2] D J Ewins Modal Testing Theory Practice and ApplicationResearch Studies Press 2000

[3] G Desanghere and R Snoeys ldquoIndirect identification of excita-tion forces bymodal coordinate transformationrdquo in Proceedingsof the 3rd International Modal Analysis Conference (IMAC rsquo85)pp 685ndash690 Orlando Fla USA 1985

[4] B Hillary and D J Ewins ldquoThe use of strain gauges inforce determination and frequency response functionmeasure-mentsrdquo in Proceedings of the 2nd International Modal AnalysisConference (IMAC rsquo84) pp 627ndash634 Orlando Fla USA 1984

[5] N Okubo S Tanabe and T Tatsuno ldquoIdentification of forcesgenerated by amachine under operating conditionrdquo in Proceed-ings of the 3rd International Modal Analysis Conference (IMACrsquo85) pp 920ndash927 Orlando Fla USA 1985

[6] H R Busby andDM Trujillo ldquoSolution of an inverse dynamicsproblem using an eigenvalue reduction techniquerdquo Computersand Structures vol 25 no 1 pp 109ndash117 1987

[7] P E Hollandsworth and H R Busby ldquoImpact force identifica-tion using the general inverse techniquerdquo International Journalof Impact Engineering vol 8 no 4 pp 315ndash322 1989

[8] GGenaro andDA Rade ldquoInput force identification in the timedomainrdquo in Proceedings of the 16th International Modal AnalysisConference (IMAC rsquo98) pp 124ndash129 Santa Barbara Calif USAFebruary 1998

[9] T G Carne R L Mayes and V I Bateman ldquoForce recon-struction using a sum of weighted accelerations techniquerdquo inProceedings of the 10th International Modal Analysis Conference(IMAC 92) pp 291ndash298 San Diego Calif USA 1992

[10] R Hashemi and M H Kargarnovin ldquoVibration base identifi-cation of impact force using genetic algorithmrdquo InternationalJournal of Mechanical Systems Science and Engineering vol 1no 4 pp 204ndash210 2007

[11] Z R Lu and S S Law ldquoIdentification of system parameters andinput force from output onlyrdquo Mechanical Systems and SignalProcessing vol 21 no 5 pp 2099ndash2111 2007

[12] Z Boukria P Perrotin and A Bennani ldquoExperimental impactforce location and identification using inverse problems appli-cation for a circular platerdquo International Journal of Mechanicsvol 5 no 1 pp 48ndash55 2011

[13] S K Lee ldquoIdentification of impact force in thick plates based onthe elastodynamics and time-frequency method (I)rdquo Journal ofMechanical Science and Technology vol 22 no 7 pp 1349ndash13582008

[14] T S Jang ldquoA novel method for the non-parametric identifica-tion of nonlinear restoring forces in nonlinear vibrations basedon response data a dissipative nonlinear dynamical systemrdquoShips and Offshore Structures vol 6 no 4 pp 257ndash263 2011

[15] S Ahmari and M Yang ldquoImpact location and load iden-tification through inverse analysis with bounded uncertainmeasurementsrdquo Smart Materials and Structures vol 22 no 8Article ID 085024 2013

[16] S Y Khoo Z Ismail K K Kong et al ldquoImpact force identifi-cation with pseudo-inverse method on a lightweight structurefor under-determined even-determined and over-determinedcasesrdquo International Journal of Impact Engineering vol 63 pp52ndash62 2014

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

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Navigation and Observation

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

provides the most precise estimates of 119891119905119894 An approach to

extract [119867]119911 opt from [119867]

119911 cs is described next

24 119863-Optimal Design If the errors in response measure-ments are statistically independent and the standard devia-tion of each of them is 120590 then the covariance matrix for 119891

119905119894

estimates is given by [22]

var (119891119905119894) = 1205902

([119867]119879

119911[119867]119911)minus1

(22)

The matrix ([119867]119879119911[119867]119911)minus1 is known as the sensitivity of

[119867]119911 For a given variance 1205902 in response measurements

minimization of the sensitivity of [119867]119911leads to an increased

precision in 119891119905119894

estimates The sensitivity of [119867]119911is a

function of the number and locations of the sensors on thestructure Therefore optimum selection of the number andlocations of the sensors on the structure so as to minimizethe sensitivity of [119867]

119911leads to the minimization of variation

in the 119891119905119894estimates

For a given number of sensors a the candidate set [119867]119911 cs

is searched to determine a sensor locations that providethe least variance in the load estimates One of the criteriafollowed involves the maximization of the determinant of[119867]119879

119911[119867]119911 |[119867]119879

119911[119867]119911| Designs that maximize |[119867]119879

119911[119867]119911|

are called 119863-optimal designs [23ndash25] where D denotesdeterminant In order to construct an a-point 119863-optimaldesign the a sensor locations that maximize |[119867]119879

119911[119867]119911|

must be selected from the candidate set [119867]119911 cs To select the

a-point 119863-optimal design sequential exchange algorithms[24] based on the principles of optimal augmentation andreduction of an existing design are used herein

The basic idea behind the sequential exchange algorithmis as follows Given the candidate set [119867]

119911 cs and the numberof sensors a the first step is to randomly select a distinctcandidate points from the candidate set to initialize [119867]

119911

Out of the remaining candidate set a candidate point is thenselected and the corresponding row is augmented to [119867]

119911

to form [119867]119911+

such that |[119867]119879119911 +[119867]119911+| is maximum Next

out of the 119886 + 1 rows in [119867]119911+ a row is deleted to arrive at

[119867]119911minus

such that |[119867]119879119911 minus[119867]119911minus| is maximum This process of

adding and deleting rows continues until there is no furtherimprovement in the value of |[119867]119879

119911[119867]119911| The final [119867]

119911so

obtained is the 119863-optimal design [119867]119911 opt whose row indices

provide the information on the optimum sensor locationsAs the candidate set gets bigger and bigger increasingly

large number of determinants and matrix inverses need to becalculated A very naıve way to compute the determinant isto obtain119872 = [119867]

119879

119911[119867]119911and then calculate the determinant

|119872| This approach of determinant calculation is computa-tionally very expensive An alternate formula for computingthe determinant |119872

+| = |[119867]

119879

119911 +[119867]119911+| from that of |119872|when

a row 119910119879 is augmented to [119867]119911is [25]

1003816100381610038161003816119872+1003816100381610038161003816 = |119872| (1 [+] 119910

119879

119872minus1

119910) (23)

where [+]denotes additionThe [+] is replaced by subtractionfor the case when a row 119910119879 is deleted from [119867]

119911+ In order to

k1

C1

m1

k2

C2

m2

k3

C3

middot middot middot

k14

C14

m14

k15

C15

m15

k16

C16

Figure 1 15-Dof spring-mass-damper system

be able to use (23) 119872minus1 can be maintained and updated asthe row 119910119879 is augmented to [119867]

119911by

1003816100381610038161003816119872+1003816100381610038161003816

minus1

= |119872|minus1

[minus]

(119872minus1

119910) (119872minus1

119910)119879

(1 [+] 119910119879119872minus1119910)

(24)

where [minus] denotes subtraction and is replaced by addition forthe case when a row 119910119879 is deleted from [119867]

119911+

Once [119867]119911 opt and in turn optimum sensor locations are

determined sensors are mounted at the identified optimumlocations and response is measured Equations (20) and (21)are then utilized to compute the input forces 119891 It is to benoted that in this approach it is not necessary that responsemeasurements be taken at locations where forces are applied

25 Error Quantification The percent root mean square(rms) error in the recovered loads with respect to the actualapplied loads can be calculated as

120576rms = (

radicsum(119891app minus 119891rec)2

radicsum1198912

app

times 100) (25)

The above equation can be used to quantify errors in the loadestimates

3 Example 1 15-DOFSpring-Mass-Damper System

The dynamic load estimation method from accelerationmeasurements is illustrated with the help of a numerical sim-ulation consisting of a 15 DOF spring-mass-damper systemshown in Figure 1 Masses119898

1and119898

15are connected to fixed

boundary A periodic forcing function is applied to mass1198987

The task is to determine the optimumaccelerometer locationsand reconstruct the input force based on the acceleration timeresponse at those locations

A total of 2 accelerometers were used to measure accel-erations of 2 masses The [119872] [119862] and [119870] matrices wereobtained by writing the equations of motion for the sys-tem Discrete time-invariant state-space form of the systemwas obtained using (3)ndash(6) Forward Markov parameterswere computed from the state-space model using (9) Allnumerical computations were performed in a MATLABprogramming environment

All DOFs except the DOF where the load was appliedwere selected to be the locations where accelerometers canpotentially bemounted that is theDOF corresponding to theapplied load did not form a part of the candidate set [119867]

119911 cs

6 Shock and Vibration

0 001 002 003 004 005 006 007 008 009 01minus1500

minus1000

minus500

0

500

1000

1500

Time

Load

AppliedRecovered

Figure 2 Applied and recovered loads at mass 7 with optimumaccelerometer placements

[119867]119911 cs was subjected to the 119863-optimal design algorithm

described in Section 24 to obtain [119867]119911 opt whereby the

optimum accelerometer locations were found to be at masses1198986and 119898

8 Having obtained the optimum accelerometer

locations the inverse Markov parameters were computedusing (20) In absence of any experimental data the acceler-ation time response at the optimum locations was obtainedby solving the ordinary differential equations numericallyUsing the acceleration data 119910 computed at the optimumaccelerometer locations the input force 119891 was recoveredusing (21) The applied and the recovered loads are plotted inFigure 2The rms error using (25)was calculated to be 07 Itcan be inferred from the plot that the applied load is recoveredaccurately

4 Example 2 Overhead Crane Girder

A more general numerical example is presented here wherea vertical dynamic load acting on an overhead crane girderthrough trolley wheels needs to be estimated For the sakeof illustration and simplicity the number of loads is assumedto be one and assumed to be applied at the girder mid-spanThe problem is to determine the optimum accelerometerlocations and reconstruct the input force based on theacceleration time response at those locations

A finite element model of the girder was developed inANSYS using BEAM188 elements The finite element modelof the girder along with the applied load and boundaryconditions is shown in Figure 3 The model consisted of 50beam elements and 49 unconstrained nodes with 6 degreesof freedom per node that is the total number of degrees offreedom of the model was 294

The [119872] and [119870] matrices were obtained using finiteelement method ANSYS provides data for [119872] and [119870]matrices in the Harwell-Boeing file format A routine waswritten in MATLAB to convert them into the matrix formatsuitable for current application All further calculations wereperformed in MATLAB Discrete time-invariant state-spaceform of the system was obtained using (3)ndash(6) Forward

Figure 3 Finite element model of girder with applied load andoptimum accelerometer locations

Markov parameters were computed from the state-spacemodel using (9) The number of accelerometers to be usedwas arbitrarily assumed to be equal to 3 All DOFs exceptthe DOF where the load was applied were selected to be thelocations where accelerometers can potentially be mountedthat is the DOF corresponding to the applied load didnot form a part of the candidate set [119867]

119911 cs [119867]119911 cs wassubjected to the 119863-optimal design algorithm described inSection 24 to obtain [119867]

119911 opt The optimum accelerometerlocations are also depicted in Figure 3 It is observed thatthe algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initiallybe excluded from the candidate set [119867]

119911 cs Furthermoreif the sensor positions seem to be too congested mutuallyadditional criteria may be instructed such as if a particularspot is chosen as a potential sensor location then certain areaaround that locationmay be excluded from the candidate set

Having obtained the optimum accelerometer locationsthe inverse Markov parameters were computed using (20)In absence of any experimental data the acceleration timeresponse at the optimum locations was obtained by solvingthe ordinary differential equations numerically Using theacceleration data 119910 computed at the optimum accelerom-eter locations the input force 119891 was recovered using (21)The applied and the recovered loads are plotted in Figure 4The rms error using (25) was calculated to be 01

In order to simulate a more realistic scenario whereaccelerations are measured experimentally the accelerationdata 119910 was corrupted with normally distributed randomerrors with zero mean and standard deviation of 10 ofits value The applied and recovered loads with errors inacceleration measurements are plotted in Figure 5 The rmserror using (25) was calculated to be 23

Next a different nature of force was considered where ahalf-sine wave loading was applied Both exact accelerationdata 119910 and acceleration data corrupted with normallydistributed random errors with zero mean and standarddeviation of 10 of its value were considered The appliedand recovered loads without and with errors in acceleration

Shock and Vibration 7

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 4 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 5 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

measurements are plotted in Figures 6 and 7 respectivelyThe rms error using (25) was calculated to be 13 forthe measurement involving acceleration error It can beinferred from the plots that the proposed approach is ableto successfully recover the applied load precisely even whenrealistic measurement errors are present in the structuralresponse

Based on the results for both examples it can be seenthat for discrete as well as continuous systems the proposedapproach is able to accurately estimate the loads acting on acomponent by measuring the acceleration response at a finitenumber of optimum locations

5 Conclusions

A computational technique is presented that allows forindirect measurement of dynamic loads acting on a structuresuch that the structure behaves as its own load transducer

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 6 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 7 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

This is achieved by placing finite number of sensors on thestructure The loads exciting a structure are estimated byconvolving the structural response with the inverse Markovparameters The inverse Markov parameters in turn arecomputed from the forward Markov parameters using alinear prediction algorithmThe forwardMarkov parametersrepresent the response of the system to applied unit impulseand thus contain the dynamic properties of the system Theycan be obtained analytically as well as experimentally

The computation of the inverse Markov parameters fromforward Markov parameters like all inverse problems suf-fers from ill-conditioning The computed inverse Markovparameters are not always bounded thus rendering theirconvolution with the structural response to diverge Theaccuracy of the inverse Markov parameters (and thereby thequality of input load estimates) depends on the locations ofsensors on the structure To improve the precision of loadestimates the technique of 119863-optimal design is utilized todetermine the optimum sensor locations It is observed that

8 Shock and Vibration

the algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initially beexcluded from the candidate set Furthermore if the sensorpositions seem to be too congested mutually additionalcriteriamay be instructed such as if a particular spot is chosenas a potential sensor location then certain area around thatlocation may be excluded from the candidate set The loadsrecovered based on accelerations measured from optimallyplaced accelerometers on the structure are observed to be inexcellent agreement with the applied loads

Nomenclature

[0] Zero matrix Derivative with respect to time

[119860119888] System matrix for continuous case

[119860119889] System matrix for discrete case

[119860119889]119868 Inverse system matrix for discrete case

119886 Number of sensors[119861119888] Input matrix for continuous case

[119861119889] Input matrix for discrete case

[119861119889]119868 Inverse input matrix for discrete case

[119862] Damping matrix[119862119889] Output matrix for discrete case

[119862119889]119868 Inverse output matrix for discrete case

[119863119889] Feedforward matrix for discrete case

[119863119889]119868 Inverse feedforward matrix for discrete

case119891(119905) Load vector119891app Applied load119891rec Reconstructed load[119867]119894 119894th forward Markov parameters matrix

[119867]119911 cs Candidate set

[119867]119911 opt Optimum subset of [119867]

119911 cs determined by119863-optimal design

[ℎ]119894 119894th inverse Markov parameters matrix

[119868] Identity matrix[119870] Stiffness matrix[119872] Mass matrix119899 Number of degrees of freedom of the

system119899119891 Number of applied forces

119905 Time119905119894 Discretized timeΔ119905 Time step119906 119906(119905) State variables vector119909(119905) Displacement vector119910 System output120575 Unit impulse load function120576rms Percent root mean square (rms) error

Conflict of Interests

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

References

[1] K K Stevens ldquoForce identification problemsmdashan overviewrdquo inProceedings of the SEM Conference on Experimental Mechanicspp 838ndash844 Houston Tex USA 1987

[2] D J Ewins Modal Testing Theory Practice and ApplicationResearch Studies Press 2000

[3] G Desanghere and R Snoeys ldquoIndirect identification of excita-tion forces bymodal coordinate transformationrdquo in Proceedingsof the 3rd International Modal Analysis Conference (IMAC rsquo85)pp 685ndash690 Orlando Fla USA 1985

[4] B Hillary and D J Ewins ldquoThe use of strain gauges inforce determination and frequency response functionmeasure-mentsrdquo in Proceedings of the 2nd International Modal AnalysisConference (IMAC rsquo84) pp 627ndash634 Orlando Fla USA 1984

[5] N Okubo S Tanabe and T Tatsuno ldquoIdentification of forcesgenerated by amachine under operating conditionrdquo in Proceed-ings of the 3rd International Modal Analysis Conference (IMACrsquo85) pp 920ndash927 Orlando Fla USA 1985

[6] H R Busby andDM Trujillo ldquoSolution of an inverse dynamicsproblem using an eigenvalue reduction techniquerdquo Computersand Structures vol 25 no 1 pp 109ndash117 1987

[7] P E Hollandsworth and H R Busby ldquoImpact force identifica-tion using the general inverse techniquerdquo International Journalof Impact Engineering vol 8 no 4 pp 315ndash322 1989

[8] GGenaro andDA Rade ldquoInput force identification in the timedomainrdquo in Proceedings of the 16th International Modal AnalysisConference (IMAC rsquo98) pp 124ndash129 Santa Barbara Calif USAFebruary 1998

[9] T G Carne R L Mayes and V I Bateman ldquoForce recon-struction using a sum of weighted accelerations techniquerdquo inProceedings of the 10th International Modal Analysis Conference(IMAC 92) pp 291ndash298 San Diego Calif USA 1992

[10] R Hashemi and M H Kargarnovin ldquoVibration base identifi-cation of impact force using genetic algorithmrdquo InternationalJournal of Mechanical Systems Science and Engineering vol 1no 4 pp 204ndash210 2007

[11] Z R Lu and S S Law ldquoIdentification of system parameters andinput force from output onlyrdquo Mechanical Systems and SignalProcessing vol 21 no 5 pp 2099ndash2111 2007

[12] Z Boukria P Perrotin and A Bennani ldquoExperimental impactforce location and identification using inverse problems appli-cation for a circular platerdquo International Journal of Mechanicsvol 5 no 1 pp 48ndash55 2011

[13] S K Lee ldquoIdentification of impact force in thick plates based onthe elastodynamics and time-frequency method (I)rdquo Journal ofMechanical Science and Technology vol 22 no 7 pp 1349ndash13582008

[14] T S Jang ldquoA novel method for the non-parametric identifica-tion of nonlinear restoring forces in nonlinear vibrations basedon response data a dissipative nonlinear dynamical systemrdquoShips and Offshore Structures vol 6 no 4 pp 257ndash263 2011

[15] S Ahmari and M Yang ldquoImpact location and load iden-tification through inverse analysis with bounded uncertainmeasurementsrdquo Smart Materials and Structures vol 22 no 8Article ID 085024 2013

[16] S Y Khoo Z Ismail K K Kong et al ldquoImpact force identifi-cation with pseudo-inverse method on a lightweight structurefor under-determined even-determined and over-determinedcasesrdquo International Journal of Impact Engineering vol 63 pp52ndash62 2014

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

0 001 002 003 004 005 006 007 008 009 01minus1500

minus1000

minus500

0

500

1000

1500

Time

Load

AppliedRecovered

Figure 2 Applied and recovered loads at mass 7 with optimumaccelerometer placements

[119867]119911 cs was subjected to the 119863-optimal design algorithm

described in Section 24 to obtain [119867]119911 opt whereby the

optimum accelerometer locations were found to be at masses1198986and 119898

8 Having obtained the optimum accelerometer

locations the inverse Markov parameters were computedusing (20) In absence of any experimental data the acceler-ation time response at the optimum locations was obtainedby solving the ordinary differential equations numericallyUsing the acceleration data 119910 computed at the optimumaccelerometer locations the input force 119891 was recoveredusing (21) The applied and the recovered loads are plotted inFigure 2The rms error using (25)was calculated to be 07 Itcan be inferred from the plot that the applied load is recoveredaccurately

4 Example 2 Overhead Crane Girder

A more general numerical example is presented here wherea vertical dynamic load acting on an overhead crane girderthrough trolley wheels needs to be estimated For the sakeof illustration and simplicity the number of loads is assumedto be one and assumed to be applied at the girder mid-spanThe problem is to determine the optimum accelerometerlocations and reconstruct the input force based on theacceleration time response at those locations

A finite element model of the girder was developed inANSYS using BEAM188 elements The finite element modelof the girder along with the applied load and boundaryconditions is shown in Figure 3 The model consisted of 50beam elements and 49 unconstrained nodes with 6 degreesof freedom per node that is the total number of degrees offreedom of the model was 294

The [119872] and [119870] matrices were obtained using finiteelement method ANSYS provides data for [119872] and [119870]matrices in the Harwell-Boeing file format A routine waswritten in MATLAB to convert them into the matrix formatsuitable for current application All further calculations wereperformed in MATLAB Discrete time-invariant state-spaceform of the system was obtained using (3)ndash(6) Forward

Figure 3 Finite element model of girder with applied load andoptimum accelerometer locations

Markov parameters were computed from the state-spacemodel using (9) The number of accelerometers to be usedwas arbitrarily assumed to be equal to 3 All DOFs exceptthe DOF where the load was applied were selected to be thelocations where accelerometers can potentially be mountedthat is the DOF corresponding to the applied load didnot form a part of the candidate set [119867]

119911 cs [119867]119911 cs wassubjected to the 119863-optimal design algorithm described inSection 24 to obtain [119867]

119911 opt The optimum accelerometerlocations are also depicted in Figure 3 It is observed thatthe algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initiallybe excluded from the candidate set [119867]

119911 cs Furthermoreif the sensor positions seem to be too congested mutuallyadditional criteria may be instructed such as if a particularspot is chosen as a potential sensor location then certain areaaround that locationmay be excluded from the candidate set

Having obtained the optimum accelerometer locationsthe inverse Markov parameters were computed using (20)In absence of any experimental data the acceleration timeresponse at the optimum locations was obtained by solvingthe ordinary differential equations numerically Using theacceleration data 119910 computed at the optimum accelerom-eter locations the input force 119891 was recovered using (21)The applied and the recovered loads are plotted in Figure 4The rms error using (25) was calculated to be 01

In order to simulate a more realistic scenario whereaccelerations are measured experimentally the accelerationdata 119910 was corrupted with normally distributed randomerrors with zero mean and standard deviation of 10 ofits value The applied and recovered loads with errors inacceleration measurements are plotted in Figure 5 The rmserror using (25) was calculated to be 23

Next a different nature of force was considered where ahalf-sine wave loading was applied Both exact accelerationdata 119910 and acceleration data corrupted with normallydistributed random errors with zero mean and standarddeviation of 10 of its value were considered The appliedand recovered loads without and with errors in acceleration

Shock and Vibration 7

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 4 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 5 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

measurements are plotted in Figures 6 and 7 respectivelyThe rms error using (25) was calculated to be 13 forthe measurement involving acceleration error It can beinferred from the plots that the proposed approach is ableto successfully recover the applied load precisely even whenrealistic measurement errors are present in the structuralresponse

Based on the results for both examples it can be seenthat for discrete as well as continuous systems the proposedapproach is able to accurately estimate the loads acting on acomponent by measuring the acceleration response at a finitenumber of optimum locations

5 Conclusions

A computational technique is presented that allows forindirect measurement of dynamic loads acting on a structuresuch that the structure behaves as its own load transducer

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 6 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 7 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

This is achieved by placing finite number of sensors on thestructure The loads exciting a structure are estimated byconvolving the structural response with the inverse Markovparameters The inverse Markov parameters in turn arecomputed from the forward Markov parameters using alinear prediction algorithmThe forwardMarkov parametersrepresent the response of the system to applied unit impulseand thus contain the dynamic properties of the system Theycan be obtained analytically as well as experimentally

The computation of the inverse Markov parameters fromforward Markov parameters like all inverse problems suf-fers from ill-conditioning The computed inverse Markovparameters are not always bounded thus rendering theirconvolution with the structural response to diverge Theaccuracy of the inverse Markov parameters (and thereby thequality of input load estimates) depends on the locations ofsensors on the structure To improve the precision of loadestimates the technique of 119863-optimal design is utilized todetermine the optimum sensor locations It is observed that

8 Shock and Vibration

the algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initially beexcluded from the candidate set Furthermore if the sensorpositions seem to be too congested mutually additionalcriteriamay be instructed such as if a particular spot is chosenas a potential sensor location then certain area around thatlocation may be excluded from the candidate set The loadsrecovered based on accelerations measured from optimallyplaced accelerometers on the structure are observed to be inexcellent agreement with the applied loads

Nomenclature

[0] Zero matrix Derivative with respect to time

[119860119888] System matrix for continuous case

[119860119889] System matrix for discrete case

[119860119889]119868 Inverse system matrix for discrete case

119886 Number of sensors[119861119888] Input matrix for continuous case

[119861119889] Input matrix for discrete case

[119861119889]119868 Inverse input matrix for discrete case

[119862] Damping matrix[119862119889] Output matrix for discrete case

[119862119889]119868 Inverse output matrix for discrete case

[119863119889] Feedforward matrix for discrete case

[119863119889]119868 Inverse feedforward matrix for discrete

case119891(119905) Load vector119891app Applied load119891rec Reconstructed load[119867]119894 119894th forward Markov parameters matrix

[119867]119911 cs Candidate set

[119867]119911 opt Optimum subset of [119867]

119911 cs determined by119863-optimal design

[ℎ]119894 119894th inverse Markov parameters matrix

[119868] Identity matrix[119870] Stiffness matrix[119872] Mass matrix119899 Number of degrees of freedom of the

system119899119891 Number of applied forces

119905 Time119905119894 Discretized timeΔ119905 Time step119906 119906(119905) State variables vector119909(119905) Displacement vector119910 System output120575 Unit impulse load function120576rms Percent root mean square (rms) error

Conflict of Interests

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

References

[1] K K Stevens ldquoForce identification problemsmdashan overviewrdquo inProceedings of the SEM Conference on Experimental Mechanicspp 838ndash844 Houston Tex USA 1987

[2] D J Ewins Modal Testing Theory Practice and ApplicationResearch Studies Press 2000

[3] G Desanghere and R Snoeys ldquoIndirect identification of excita-tion forces bymodal coordinate transformationrdquo in Proceedingsof the 3rd International Modal Analysis Conference (IMAC rsquo85)pp 685ndash690 Orlando Fla USA 1985

[4] B Hillary and D J Ewins ldquoThe use of strain gauges inforce determination and frequency response functionmeasure-mentsrdquo in Proceedings of the 2nd International Modal AnalysisConference (IMAC rsquo84) pp 627ndash634 Orlando Fla USA 1984

[5] N Okubo S Tanabe and T Tatsuno ldquoIdentification of forcesgenerated by amachine under operating conditionrdquo in Proceed-ings of the 3rd International Modal Analysis Conference (IMACrsquo85) pp 920ndash927 Orlando Fla USA 1985

[6] H R Busby andDM Trujillo ldquoSolution of an inverse dynamicsproblem using an eigenvalue reduction techniquerdquo Computersand Structures vol 25 no 1 pp 109ndash117 1987

[7] P E Hollandsworth and H R Busby ldquoImpact force identifica-tion using the general inverse techniquerdquo International Journalof Impact Engineering vol 8 no 4 pp 315ndash322 1989

[8] GGenaro andDA Rade ldquoInput force identification in the timedomainrdquo in Proceedings of the 16th International Modal AnalysisConference (IMAC rsquo98) pp 124ndash129 Santa Barbara Calif USAFebruary 1998

[9] T G Carne R L Mayes and V I Bateman ldquoForce recon-struction using a sum of weighted accelerations techniquerdquo inProceedings of the 10th International Modal Analysis Conference(IMAC 92) pp 291ndash298 San Diego Calif USA 1992

[10] R Hashemi and M H Kargarnovin ldquoVibration base identifi-cation of impact force using genetic algorithmrdquo InternationalJournal of Mechanical Systems Science and Engineering vol 1no 4 pp 204ndash210 2007

[11] Z R Lu and S S Law ldquoIdentification of system parameters andinput force from output onlyrdquo Mechanical Systems and SignalProcessing vol 21 no 5 pp 2099ndash2111 2007

[12] Z Boukria P Perrotin and A Bennani ldquoExperimental impactforce location and identification using inverse problems appli-cation for a circular platerdquo International Journal of Mechanicsvol 5 no 1 pp 48ndash55 2011

[13] S K Lee ldquoIdentification of impact force in thick plates based onthe elastodynamics and time-frequency method (I)rdquo Journal ofMechanical Science and Technology vol 22 no 7 pp 1349ndash13582008

[14] T S Jang ldquoA novel method for the non-parametric identifica-tion of nonlinear restoring forces in nonlinear vibrations basedon response data a dissipative nonlinear dynamical systemrdquoShips and Offshore Structures vol 6 no 4 pp 257ndash263 2011

[15] S Ahmari and M Yang ldquoImpact location and load iden-tification through inverse analysis with bounded uncertainmeasurementsrdquo Smart Materials and Structures vol 22 no 8Article ID 085024 2013

[16] S Y Khoo Z Ismail K K Kong et al ldquoImpact force identifi-cation with pseudo-inverse method on a lightweight structurefor under-determined even-determined and over-determinedcasesrdquo International Journal of Impact Engineering vol 63 pp52ndash62 2014

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 4 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

15

Time

Load

AppliedRecovered

times104

Figure 5 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

measurements are plotted in Figures 6 and 7 respectivelyThe rms error using (25) was calculated to be 13 forthe measurement involving acceleration error It can beinferred from the plots that the proposed approach is ableto successfully recover the applied load precisely even whenrealistic measurement errors are present in the structuralresponse

Based on the results for both examples it can be seenthat for discrete as well as continuous systems the proposedapproach is able to accurately estimate the loads acting on acomponent by measuring the acceleration response at a finitenumber of optimum locations

5 Conclusions

A computational technique is presented that allows forindirect measurement of dynamic loads acting on a structuresuch that the structure behaves as its own load transducer

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 6 Applied and recovered loads using optimum accelerome-ter placements no acceleration errors

0 002 004 006 008 01 012 014 0160

500100015002000250030003500400045005000

Time

Load

AppliedRecovered

Figure 7 Applied and recovered loads using optimum accelerome-ter placements with acceleration errors

This is achieved by placing finite number of sensors on thestructure The loads exciting a structure are estimated byconvolving the structural response with the inverse Markovparameters The inverse Markov parameters in turn arecomputed from the forward Markov parameters using alinear prediction algorithmThe forwardMarkov parametersrepresent the response of the system to applied unit impulseand thus contain the dynamic properties of the system Theycan be obtained analytically as well as experimentally

The computation of the inverse Markov parameters fromforward Markov parameters like all inverse problems suf-fers from ill-conditioning The computed inverse Markovparameters are not always bounded thus rendering theirconvolution with the structural response to diverge Theaccuracy of the inverse Markov parameters (and thereby thequality of input load estimates) depends on the locations ofsensors on the structure To improve the precision of loadestimates the technique of 119863-optimal design is utilized todetermine the optimum sensor locations It is observed that

8 Shock and Vibration

the algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initially beexcluded from the candidate set Furthermore if the sensorpositions seem to be too congested mutually additionalcriteriamay be instructed such as if a particular spot is chosenas a potential sensor location then certain area around thatlocation may be excluded from the candidate set The loadsrecovered based on accelerations measured from optimallyplaced accelerometers on the structure are observed to be inexcellent agreement with the applied loads

Nomenclature

[0] Zero matrix Derivative with respect to time

[119860119888] System matrix for continuous case

[119860119889] System matrix for discrete case

[119860119889]119868 Inverse system matrix for discrete case

119886 Number of sensors[119861119888] Input matrix for continuous case

[119861119889] Input matrix for discrete case

[119861119889]119868 Inverse input matrix for discrete case

[119862] Damping matrix[119862119889] Output matrix for discrete case

[119862119889]119868 Inverse output matrix for discrete case

[119863119889] Feedforward matrix for discrete case

[119863119889]119868 Inverse feedforward matrix for discrete

case119891(119905) Load vector119891app Applied load119891rec Reconstructed load[119867]119894 119894th forward Markov parameters matrix

[119867]119911 cs Candidate set

[119867]119911 opt Optimum subset of [119867]

119911 cs determined by119863-optimal design

[ℎ]119894 119894th inverse Markov parameters matrix

[119868] Identity matrix[119870] Stiffness matrix[119872] Mass matrix119899 Number of degrees of freedom of the

system119899119891 Number of applied forces

119905 Time119905119894 Discretized timeΔ119905 Time step119906 119906(119905) State variables vector119909(119905) Displacement vector119910 System output120575 Unit impulse load function120576rms Percent root mean square (rms) error

Conflict of Interests

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

References

[1] K K Stevens ldquoForce identification problemsmdashan overviewrdquo inProceedings of the SEM Conference on Experimental Mechanicspp 838ndash844 Houston Tex USA 1987

[2] D J Ewins Modal Testing Theory Practice and ApplicationResearch Studies Press 2000

[3] G Desanghere and R Snoeys ldquoIndirect identification of excita-tion forces bymodal coordinate transformationrdquo in Proceedingsof the 3rd International Modal Analysis Conference (IMAC rsquo85)pp 685ndash690 Orlando Fla USA 1985

[4] B Hillary and D J Ewins ldquoThe use of strain gauges inforce determination and frequency response functionmeasure-mentsrdquo in Proceedings of the 2nd International Modal AnalysisConference (IMAC rsquo84) pp 627ndash634 Orlando Fla USA 1984

[5] N Okubo S Tanabe and T Tatsuno ldquoIdentification of forcesgenerated by amachine under operating conditionrdquo in Proceed-ings of the 3rd International Modal Analysis Conference (IMACrsquo85) pp 920ndash927 Orlando Fla USA 1985

[6] H R Busby andDM Trujillo ldquoSolution of an inverse dynamicsproblem using an eigenvalue reduction techniquerdquo Computersand Structures vol 25 no 1 pp 109ndash117 1987

[7] P E Hollandsworth and H R Busby ldquoImpact force identifica-tion using the general inverse techniquerdquo International Journalof Impact Engineering vol 8 no 4 pp 315ndash322 1989

[8] GGenaro andDA Rade ldquoInput force identification in the timedomainrdquo in Proceedings of the 16th International Modal AnalysisConference (IMAC rsquo98) pp 124ndash129 Santa Barbara Calif USAFebruary 1998

[9] T G Carne R L Mayes and V I Bateman ldquoForce recon-struction using a sum of weighted accelerations techniquerdquo inProceedings of the 10th International Modal Analysis Conference(IMAC 92) pp 291ndash298 San Diego Calif USA 1992

[10] R Hashemi and M H Kargarnovin ldquoVibration base identifi-cation of impact force using genetic algorithmrdquo InternationalJournal of Mechanical Systems Science and Engineering vol 1no 4 pp 204ndash210 2007

[11] Z R Lu and S S Law ldquoIdentification of system parameters andinput force from output onlyrdquo Mechanical Systems and SignalProcessing vol 21 no 5 pp 2099ndash2111 2007

[12] Z Boukria P Perrotin and A Bennani ldquoExperimental impactforce location and identification using inverse problems appli-cation for a circular platerdquo International Journal of Mechanicsvol 5 no 1 pp 48ndash55 2011

[13] S K Lee ldquoIdentification of impact force in thick plates based onthe elastodynamics and time-frequency method (I)rdquo Journal ofMechanical Science and Technology vol 22 no 7 pp 1349ndash13582008

[14] T S Jang ldquoA novel method for the non-parametric identifica-tion of nonlinear restoring forces in nonlinear vibrations basedon response data a dissipative nonlinear dynamical systemrdquoShips and Offshore Structures vol 6 no 4 pp 257ndash263 2011

[15] S Ahmari and M Yang ldquoImpact location and load iden-tification through inverse analysis with bounded uncertainmeasurementsrdquo Smart Materials and Structures vol 22 no 8Article ID 085024 2013

[16] S Y Khoo Z Ismail K K Kong et al ldquoImpact force identifi-cation with pseudo-inverse method on a lightweight structurefor under-determined even-determined and over-determinedcasesrdquo International Journal of Impact Engineering vol 63 pp52ndash62 2014

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

the algorithm predicts the optimum sensor locations to be asclose to the loads as possible This should not be consideredas a limitation to the application of the proposed techniquesince if certain locations around the force application pointsare not available for sensor placement they can initially beexcluded from the candidate set Furthermore if the sensorpositions seem to be too congested mutually additionalcriteriamay be instructed such as if a particular spot is chosenas a potential sensor location then certain area around thatlocation may be excluded from the candidate set The loadsrecovered based on accelerations measured from optimallyplaced accelerometers on the structure are observed to be inexcellent agreement with the applied loads

Nomenclature

[0] Zero matrix Derivative with respect to time

[119860119888] System matrix for continuous case

[119860119889] System matrix for discrete case

[119860119889]119868 Inverse system matrix for discrete case

119886 Number of sensors[119861119888] Input matrix for continuous case

[119861119889] Input matrix for discrete case

[119861119889]119868 Inverse input matrix for discrete case

[119862] Damping matrix[119862119889] Output matrix for discrete case

[119862119889]119868 Inverse output matrix for discrete case

[119863119889] Feedforward matrix for discrete case

[119863119889]119868 Inverse feedforward matrix for discrete

case119891(119905) Load vector119891app Applied load119891rec Reconstructed load[119867]119894 119894th forward Markov parameters matrix

[119867]119911 cs Candidate set

[119867]119911 opt Optimum subset of [119867]

119911 cs determined by119863-optimal design

[ℎ]119894 119894th inverse Markov parameters matrix

[119868] Identity matrix[119870] Stiffness matrix[119872] Mass matrix119899 Number of degrees of freedom of the

system119899119891 Number of applied forces

119905 Time119905119894 Discretized timeΔ119905 Time step119906 119906(119905) State variables vector119909(119905) Displacement vector119910 System output120575 Unit impulse load function120576rms Percent root mean square (rms) error

Conflict of Interests

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

References

[1] K K Stevens ldquoForce identification problemsmdashan overviewrdquo inProceedings of the SEM Conference on Experimental Mechanicspp 838ndash844 Houston Tex USA 1987

[2] D J Ewins Modal Testing Theory Practice and ApplicationResearch Studies Press 2000

[3] G Desanghere and R Snoeys ldquoIndirect identification of excita-tion forces bymodal coordinate transformationrdquo in Proceedingsof the 3rd International Modal Analysis Conference (IMAC rsquo85)pp 685ndash690 Orlando Fla USA 1985

[4] B Hillary and D J Ewins ldquoThe use of strain gauges inforce determination and frequency response functionmeasure-mentsrdquo in Proceedings of the 2nd International Modal AnalysisConference (IMAC rsquo84) pp 627ndash634 Orlando Fla USA 1984

[5] N Okubo S Tanabe and T Tatsuno ldquoIdentification of forcesgenerated by amachine under operating conditionrdquo in Proceed-ings of the 3rd International Modal Analysis Conference (IMACrsquo85) pp 920ndash927 Orlando Fla USA 1985

[6] H R Busby andDM Trujillo ldquoSolution of an inverse dynamicsproblem using an eigenvalue reduction techniquerdquo Computersand Structures vol 25 no 1 pp 109ndash117 1987

[7] P E Hollandsworth and H R Busby ldquoImpact force identifica-tion using the general inverse techniquerdquo International Journalof Impact Engineering vol 8 no 4 pp 315ndash322 1989

[8] GGenaro andDA Rade ldquoInput force identification in the timedomainrdquo in Proceedings of the 16th International Modal AnalysisConference (IMAC rsquo98) pp 124ndash129 Santa Barbara Calif USAFebruary 1998

[9] T G Carne R L Mayes and V I Bateman ldquoForce recon-struction using a sum of weighted accelerations techniquerdquo inProceedings of the 10th International Modal Analysis Conference(IMAC 92) pp 291ndash298 San Diego Calif USA 1992

[10] R Hashemi and M H Kargarnovin ldquoVibration base identifi-cation of impact force using genetic algorithmrdquo InternationalJournal of Mechanical Systems Science and Engineering vol 1no 4 pp 204ndash210 2007

[11] Z R Lu and S S Law ldquoIdentification of system parameters andinput force from output onlyrdquo Mechanical Systems and SignalProcessing vol 21 no 5 pp 2099ndash2111 2007

[12] Z Boukria P Perrotin and A Bennani ldquoExperimental impactforce location and identification using inverse problems appli-cation for a circular platerdquo International Journal of Mechanicsvol 5 no 1 pp 48ndash55 2011

[13] S K Lee ldquoIdentification of impact force in thick plates based onthe elastodynamics and time-frequency method (I)rdquo Journal ofMechanical Science and Technology vol 22 no 7 pp 1349ndash13582008

[14] T S Jang ldquoA novel method for the non-parametric identifica-tion of nonlinear restoring forces in nonlinear vibrations basedon response data a dissipative nonlinear dynamical systemrdquoShips and Offshore Structures vol 6 no 4 pp 257ndash263 2011

[15] S Ahmari and M Yang ldquoImpact location and load iden-tification through inverse analysis with bounded uncertainmeasurementsrdquo Smart Materials and Structures vol 22 no 8Article ID 085024 2013

[16] S Y Khoo Z Ismail K K Kong et al ldquoImpact force identifi-cation with pseudo-inverse method on a lightweight structurefor under-determined even-determined and over-determinedcasesrdquo International Journal of Impact Engineering vol 63 pp52ndash62 2014

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

[17] J Liu X Sun X Han C Jiang and D Yu ldquoA novel computa-tional inverse technique for load identification using the shapefunction method of moving least square fittingrdquo Computers ampStructures vol 144 pp 127ndash137 2014

[18] JM Starkey andG LMerrill ldquoOn the Ill-conditioned nature ofindirect force measurement techniquesrdquo International Journalof Analytical and Experimental Modal Analysis vol 4 no 3 pp103ndash108 2011

[19] M Hansen and J M Starkey ldquoOn predicting and improving thecondition of modal-model-based indirect force measurementalgorithmsrdquo in Proceedings of the 8th International ModalAnalysis Conference (IMAC rsquo90) pp 115ndash120 Kissimmee FlaUSA 1990

[20] D C Kammer ldquoInput force reconstruction using a time domaintechniquerdquo Transactions of the ASMEmdashJournal of Vibration andAcoustics vol 120 no 4 pp 868ndash874 1998

[21] A D Steltzner and D C Kammer ldquoInput force estimationusing an inverse structural filterrdquo in Proceedings of the 17thInternational Modal Analysis Conference (IMAC rsquo99) pp 954ndash960 Kissimmee Fla USA February 1999

[22] S A Masroor and L W Zachary ldquoDesigning an all-purposeforce transducerrdquo Experimental Mechanics vol 31 no 1 pp 33ndash35 1991

[23] T J Mitchell ldquoAn algorithm for the construction of lsquoD-optimalrsquoexperimental designsrdquoTechnometrics vol 16 no 2 pp 203ndash2101974

[24] Z Galil and J Kiefer ldquoTime- and space-saving computer meth-ods related to Mitchellrsquos DETMAX for finding D-optimumdesignsrdquo Technometrics vol 22 no 3 pp 301ndash313 1980

[25] M E Johnson and C J Nachtsheim ldquoSome guidelines forconstructing exact d-optimal designs on convex design spacesrdquoTechnometrics vol 25 no 3 pp 271ndash277 1983

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of