Research ArticleOptimal Three-Dimensional Sensor Placement for Cable-StayedBridge Based on Dynamic Adjustment of Attenuation FactorGravitational Search Algorithm

Bo Gao Zhihui Bai and Yubo Song

School of Mechatronic Engineering Lanzhou Jiaotong University Lanzhou 730070 China

Correspondence should be addressed to Bo Gao gaobomaillzjtucn

Received 9 November 2020 Revised 7 February 2021 Accepted 22 February 2021 Published 31 March 2021

Academic Editor Yaobing Zhao

Copyright copy 2021 Bo Gao et alis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Structural health monitoring (SHM) is essential when detecting damage in large and complex structures in order to provide acomprehensive assessment of the structural health state Optimal sensor placement (OSP) is critical in the structural healthmonitoring system which aims to use a limited number of sensors to obtain high-quality structural health diagnosis dataHowever the current research mainly focuses on OSP for structures without considering the values contributed by differentmodes to the bridge structure In this article an optimal sensor placement method based on initial sensor layout using thedynamic adjustment of attenuation factor gravitational search algorithm (DGSA) is proposed e effective modal mass par-ticipation ratio is introduced to ensure the validity of the initial data of optimal sensor placement In view of the insufficientdevelopmental ability of the gravitational search algorithm the attenuation factor α adjusted dynamically aids the global search inthe early iteration and the local fine search in the late iteratione double coding method is used to apply the DGSA algorithm toOSP taking cable-stayed bridges as an example the feasibility of the algorithm is verified e results show that the improvedalgorithm has a good optimization ability and can accurately and efficiently determine the optimal placement of sensors

1 Introduction

With the progress being made in science and technologylarge buildings and bridge structures are becoming widelyused When they are subjected to harsh environments andextreme events such as strong winds and severe earth-quakes the functionality and safety of the large structuresbecome a vital issue and structural health monitoring(SHM) technology has been developed for damage detection[1] SHM uses nondestructive sensing technology to analyzethe system characteristics through the detected responseachieving the purpose of monitoring the structural healthstate or predicting the service life of the structure In theSHM system the main tasks of the sensor system includereal-time acquisition of the operating environment of thestructure the load on the structure and the dynamiccharacteristics of the structure [2 3] When the number ofsensors is large the monitoring cost will increase When

there are few sensors the measurement data will be in-complete and the measurement result will be unreasonableerefore adopting an optimal sensor placement scheme toobtain maximal structural information is critical in healthmonitoring

In the optimal sensor placement method the traditionaloptimization algorithm is incorporated with one of thefollowing the effective independence method [4ndash6] theGuyuan reduction method [7] the modal kinetic energymethod [8 9] and the information entropy method [10 11]e above mentioned traditional methods have difficulties inobtaining their global or near-global optimum In recentyears many optimal sensor placement methods based onintelligent algorithms have been proposed such as the ge-netic algorithm (GA) [12 13] the monkey algorithm (MA)[14] the harmony search (HS) algorithm [15] the sequencealgorithm (SA) [16] the particle swarm optimization (PSO)algorithm [17] and the differential evolution algorithm (DE)

HindawiShock and VibrationVolume 2021 Article ID 6664188 14 pageshttpsdoiorg10115520216664188

[18] For the objective function the modal assurance cri-terion (MAC) is widely used to quantify the collinearitybetween the modal shape vectors placed by the sensors estrategy involves minimizing the nondiagonal term of theMAC matrix to distinguish the modal shapes

Many researchers have attempted to improve the effi-ciency and performance of sensor optimization methodsZhou et al [19] proposed a one-dimensional binary codingmethod which was applied in the optimal arrangement ofcable-stayed bridge sensors Yi et al [12] improved thegenetic algorithm and used the double structure codingmethod to optimize sensor placement in high-rise buildingsYi et al [14] introduced the adaptive operator into themonkey algorithm increased the global search ability of thealgorithm and took the modal assurance criterion as theobjective function and the optimization is thus achievedthrough the experimental analysis Jin et al [15] combinedthe harmony search (HS) algorithm with the modal as-surance criterion and applied it to sensor placement on agantry crane e results show that the HS algorithm is apowerful search and optimization algorithm Yin et al [16]used the concept of Dijkstrarsquos edge relaxation operation togenerate the initial solution set from the sequence algorithmand improved this set via relaxation until the end of therelaxation operation Taking the truss structure as an ex-ample the effectiveness of the algorithm was verified Al-though the abovementioned swarm intelligence algorithmcan be applied to sensor layout optimization it has thedisadvantages of poor robustness insufficient developmentability and more control parameters

Although the above mentioned research has made sig-nificant achievements there are still three aspects that havenot been considered Firstly many sensor optimizationmethods do not consider the mode selection but select thefirst few modes based on experience which may not fullyreflect the information of the mechanism due to impropermode selection Secondly many researchers do not considerthe relationship between the number of sensors and theobjective function but give a set of sensor layout schemes fora given number of sensorsird there are many nodes to bemeasured in the optimal sensor layout Figuring out how touse OSP to solve high-dimensional problems with highaccuracy and efficiency is still a challenge erefore aninnovative method is needed to accurately and effectivelydetermine OSP in the structure

In view of these three defects a reworking of the dy-namic adjustment of attenuation factor gravitational searchalgorithm (DGSA) is presented in this papere cumulativeeffective modal participation ratio is the percentage of themodal effective mass to the total massis concept has beensuccessfully applied in seismic design and the optimalplacement of bridge sensors in order to determine thenumber of main modes [13 17 20] In this paper the ef-fective modal mass participation ratio is used to select themain modal of a bridge In order to improve the perfor-mance and search ability of the gravitational search algo-rithm (GSA) this paper improves the GSA by adjusting theattenuation factor α of the gravitational constantG tochange the step lengths of particles in different periods e

improved DGSA offers greater precision and a faster con-vergence speed e specific content of this paper is asfollows Section 2 introduces the improved GSA Section 3introduces the optimal sensor placement method based onthe DGSA and Section 4 takes a cable-stayed bridge as theobject reports on the example test and offers an analysis ofits results Section 5 contains the conclusion and future workarrangement

2 DGSA Algorithm

e gravitational search algorithm utilizes the gravitationalattraction between particles e particle with the largestmass in the given population occupies the optimal positionand the other particles move towards the particle with thehighest mass under the action of gravity thus the processof obtaining the global optimal solution of the problem iscarried out ere are M particles in the algorithm Takingone of the particle i as an example the particle i is subjectedto the gravitational attraction of the other particles andacceleration a and velocity v are generated under thiscombined force such that in the ensuing moments theparticle i approaches the global optimal solution egravitational constant runs through the entire searchprocess of the algorithm from the beginning to end echange process of the G value directly affects the searchefficiency and optimization precision of the algorithm evalue of the attenuation factor α in the GSA is constantwhich causes the particles to move with fixed step sizes inthe process of searching making it easy for the algorithm toreach the local extreme point In order to enhance theoptimization ability of the algorithm the DGSA proposedin this paper adaptively adjusts α At the beginning of thesearch the particle is far away from the optimal solutionand searches globally with a larger step size At a later stageof the search the particle is in closer proximity to theoptimal solution and employs smaller step sizes insearching which is conducive to the local search of thealgorithm and to effectively finding the global optimalsolution

21 Gravity Search Algorithm GSA [21] with its advantagesof easy implementation good operability and obviousoptimization effect has been widely used in productionpractice [22ndash24]

Assuming the population number is N the position ofparticle i in the Wdimensional space isXi [x1

i

xdi xW

i ] where (i 1 2 N) en the gravitationalattraction of particle i and particle j in the Wdimensionalspace at time t is defined as

Fdij(t) G(t)

Mi(t)lowastMj(t)

D + εx

di (t) minus x

dj (t)1113872 1113873 (1)

where Mi(t) is the gravitational mass related to particle iMj(t) is the gravitational mass related to particle j G(t)

is the gravitational constant at time t ε is a smallconstant and D is the Euclidian distance between twoagents i and j

2 Shock and Vibration

Formula for calculating G(t) is defined as

G(t) G0eminus αlowastIImax( ) (2)

where I is the number of the iterations and Imax refers to themaximum number of iterations

e resultant force Fdi (t) of particle i in d dimensional

space is defined as

Fdi (t) 1113944

jisinbestrandlowastF

dij(t) (3)

where best is a collection of particles that have gravitationaleffects on particle i

e acceleration adi (t) of particle i is defined as

adi (t)

Fdi (t)

Massi(t)

1113936jisinbestrandlowastFdij(t)

Massi(t) (4)

where Massi(t) is the inertia mass of particle i and itsformula is defined as

mi(t) fiti(t) minus fitw(t)

fitb(t) minus fitw(t) (5)

Massi(t) mi(t)

1113936nj1 mi(t)

(6)

where fitb(t) represents the best fitness value and fitw(t)

represents the worst fitness valuee formula for the velocity and position of the particle i

at the next moment is

vdi (t) randlowast v

di (t) + a

di (t)

xdi (t + 1) x

di (t) + v

di (t + 1)

⎧⎨

⎩ (7)

Note 1rand is a random variable which is uniformlydistributed among [0 1]

22 DynamicAdjustment of Attenuation FactorGravitationalSearch Algorithm e gravitational constant G(t) has asignificant influence over the convergence speed and opti-mization precision of the algorithme αin formula (2) hasthe effect of adjusting the convergence speed of the algo-rithm Aiming at the optimization efficiency and searchaccuracy of GSA this paper improves the GSA and proposesthe DGSA Considering that the value of α in the GSA is aconstant in the iterative process of the algorithm regardlessof whether the particle is far from or near to the globaloptimal solution the particle moves with a fixed step size Asa result when the particle is far away from the optimalsolution its step length is too small it moves too slowly andit takes too long Furthermore when the particle is close tothe optimal solution the step length is too large which maycause it to deviate from the optimal solution and fall into thelocal optimal solution erefore DGSA selects a smallerattenuation factor α in the early stage of the search and theparticles perform the global search with a larger step sizewhich is beneficial in reducing the search time of the al-gorithm and improving the overall optimization efficiency

e algorithm selects a larger attenuation factor α in the laterstage and the particles employ a smaller step size to com-plete the local search so as to avoid the algorithm fallingupon the local extreme point e expression of adaptiveattenuation factor α is defined as

α(t) βlowast eminus 1minus tImax| | (8)

where t represents the current number of iterations and β isthe initial parameter According to experience when β 1the optimization effect of the algorithm is the best eschematic diagram of the DGSA is shown in Figure 1

Particle M1 generates resultant force F1 and accelerationa1 under the action of universal gravitation of M2 M3 andM4 rough the adaptive mechanism of attenuationfactorα particle M1 approaches the global optimal solutionwith different steps

3 Optimal Sensor Placement Based onDGSA Method

In this section the dual-structure coding method is used tosolve the problem of DGSA being unable to realize optimalsensor placement and the flow of optimal sensor placementis shown in Figure 2 In the mode selection by calculatingthe effective modal mass participation ratio the mode orderwith a larger contribution value is selected

31 Modal Assurance Criterion e existing sensor opti-mization approaches such as the modal kinetic energymethod are highly dependent on the accuracy of the finiteelement mesh which will affect the reliability of the data It iseasy to lose the mode to be measured when using the Guyanreduction method which increases the difficulty of identi-fication e modal assurance criterion is one of the mostwidely used evaluation criteria for optimal sensor placementIt can enable the choosing of a larger space intersection angleand can retain the characteristics of the original model asmuch as possible

311 Mode Selection e selection of mode is the premiseto determine the number of sensors and then determine thelocation of sensor If too many modes are selected not only alot of calculation time and space will be consumed but also itis difficult to accurately define the correctness of the cal-culation results If too few modes are selected the reliabilityof the optimal sensor placement results will be too small torepresent the complete information of the bridge Atpresent many scholars have applied the effective modalmass participation factor to the selection of the modal orderand calculated it by using the ratio of the influence of thecurrent modal mass to the influence of all modal masses

e dynamic formula of n-DOF system is expressed as

Meurou + C _u + Ku minusMeeuroua(t) (9)

where M is the mass matrix C is the damping matrix K isthe stiffness matrix e is the direction matrix under theexcitation force euroua(t) is the acceleration produced by the

Shock and Vibration 3

F1

X0

Y

M3

Optimalsolution

M2

M4

F14

M1

a 1

Dynamic step

Adaptive adjustment

of attenuation factor

F12

F 13

M1

Step

Figure 1 Schematic diagram of DGSA

Start

Input modal matrix

Dual-structure coding

m sensorsNo

Attenuation factoradaptive adjustment

Update G forceand acceleration

Update speed and position

Update the optimal fitness valueand the optimal particle pb

f (pb) lt f (pg) pi = pb

Termination

Input parameters

Calculate N fitness values

Get the population size N

Individual optimal solution piglobal optimal solution pg

Yes

Yes

No

Yes

No

Update pg

End

Output pg

Figure 2 Optimal sensor placement based on DGSA

4 Shock and Vibration

excitation force u _u and eurou are modal displacement re-sponses velocity response and acceleration responserespectively

e relationship between the abovementioned responseand the mode shape matrix ϕ is as follows

u ϕ middot q _u ϕ middot _q eurou ϕ middot euroq (10)

Based on the orthogonality of M C and K we can getthe following formula from (9) and (10)

euroqi + 2ξiωi _qi + ω2i qi minus

ϕTi Me

ϕTi Mϕi

euroua(t) (11)

where qi is the modal coordinatesωi is the natural frequencyof the structure and ξi is the modal damping ratio

e participation factor is defined by the following

σi ϕT

i Me

ϕTi Mϕi

(12)

According to the principle of mass normalization ofmode shape matrix ϕi the following formula is obtained

ϕTi Mϕ 1 (13)

en formula (12) can be converted into the following

σi ϕTi Me (14)

e effective modal mass expression of the ith mode is

Mi σ2i

ϕTi Mϕi

(15)

erefore the total modal mass is

1113944

n

i1Mi 1113944

n

i1σ2i σTσ (16)

e expression of the ith order effective mode massparticipation ratio ri is

ri Mi

eT

Me (17)

For the first m modes (mlt n) the total modal massparticipation ratio R of the first m modes is

R 1113944m

i1ri (18)

According to the research when the value of R is morethan 90 it shows that the selected modal order containsenough structural information which can be used as themode shape of optimal sensor placement

312 Fitness Function e most important step in theproblem of optimal sensor placement is to establish themathematical model of the structure and construct the objectivefunction Actually due to the external factors the accuracy errorof the measuring instrument and the noise interference of the

test site the included angle of the modal vector is too small oreven 0 which leads to the loss of modal informationereforethe optimal sensor placement should select the points with largemodal angle and be easy to identify e modal assurancecriterion (MAC) is usually used to compare the correlation andindependence of experimental modes [25ndash27] so MAC can beused as the fitness function of optimal sensor placement toevaluate the independence of each mode

e expression of MAC is as follows

MACij ϕT

i middot ϕj1113872 11138732

ϕTi middot ϕi1113872 1113873 ϕT

j middot ϕj1113872 1113873 (19)

where ϕi and ϕj represent the ith and jth column vectors inmatrixϕ

In the above formula maximum off-diagonal elementranges from 0 to 1 e smaller the maximum off-diagonalelement means the larger the corresponding space angle andthe easier to distinguish the modal vectors e smaller thevalue of the off-diagonal element of the MAC matrix thebetter the calculated modal vector independence and theeasier it is to identify the mode shape erefore theminimization of the maximum off-diagonal element ofMAC matrix can be taken as the fitness function of optimalsensor placement of bridge and the expression is as follows

fit minf(x) (20)

where f(x) maxinej|MACij| e smaller the value of theobjective function the better the independence of the testedmodal vector and the better the optimal sensor placement

32 Method of Coding e binary coding method can beused to solve the optimal sensor placement problem 1indicates the sensor is placed and 0 indicates the sensor isnot placed but this codingmethod will lead to changes in thenumber of sensors Here dual-structure coding is adopted toovercome this probleme dual-structure codingmethod isshown in Table 1

e dual-structure coding method involves extra codeand variable code e extra code represents the positionvector of the individual and the variable code representsbinary vector 0 or 1 where 0 means no sensor is placed and1 means placement of sensor is dual coding methodenables the DGSA to solve the problem of optimal sensorplacement For example there are known to be nine pointsthat must be selected e dual-structure coding results areshown in Table 2 e sensor arrangement we were able toderive is s (3 4 6 8)

33 Optimal Sensor Placement Process Based on DGSA

Step 1 Select and import mode matrix e order choice ofthe mode matrix has an important influence on the sensorplacement results On the one hand the appropriate modematrix can represent the overall performance of the struc-ture and ensure the accuracy of the optimal sensor place-ment On the other hand the optimal sensor placement is

Shock and Vibration 5

different with different modes e first and foremost themode shape matrix of all nodes of the model is obtained bymode analysis method and the obtained mode shape matrixis taken as the input value and the degrees of freedomcorresponding to all nodes are used as the candidate pointsof the optimal sensor placement Secondly assuming thatthe number of candidate points is n and the number ofsensors ism the n candidate positions are numbered from 1to n successively

Step 2 Taking the ith particle in the population as an ex-ample (i 1 2 N in whichNis the number of pop-ulation size) the corresponding solution to i can beexpressed as xs(i)(xisi) (xi1 si1)(xi2 si2) (xin sin)e position vectors of xi are obtained by formula (21) andthe binary vectors of si are obtained by formula (22) wherexdown minus8 xup 8

xij rand times xup minus xdown1113872 1113873 + xdown (21)

where rand is a random number between 0 and 1

sij xij

1 + x

2ij

1113969

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(22)

In formula (22) the binary vectors calculated by differentposition components xij are different so a threshold δ needsto be set to satisfy the following

sij 1 if V xij1113872 1113873gt δ

0 else

⎧⎨

⎩ (23)

wherej isin 1 2 n δ 05 and the components of eachposition of xij are substituted into formula (23) If thefunction value is greater than 05 the additional code value is1 indicating that the sensor is placed at this candidate pointIf the function value is less than 05 the additional code valueis 0 indicating that the sensor is not placed at this candidatepoint It can be obtained by calculation that the value of xij istaken as [minus8 8] and the value of sij is 0leV(xij)le 09923 Itcan be approximated instead of [0 1]

Step 3 Repeat Steps 1 and 2 until N particles are generatedduring initialization the total number of sensors in si is notequal to the number of sensors arranged m In order toensure that all possible solutions in the population meet therequirements when we encounter this problem we should

reinitialize at this time and repeat Step 2 until the initialsolution meets the requirements of coding

Step 4 e N particles in the population are substituted intoformula (20) e best fitness value is labeled fitb(xsi) andthe worst fitness value is labeled fitw(xsi) e optimal ar-rangement corresponding to the fitness value is recorded aspi(xsi) pi(xsi) represents the individual optimal solution ofparticles in the population and it is the optimal solution ineach particle iteration pg(xsi) represents the global optimalsolution which is the optimal solution among all particles inthe population

Step 5 e inertia mass of each particle is calculated usingformulas (5) and (6) Adaptively change the attenuationfactor α according to formula (2) and update the gravita-tional constant G(t)

Step 6 Taking particle i as an example the correspondingposition component is xi xi1 xi2 middot middot middot xim1113864 1113865 where m is thenumber of sensors that need to be arranged and the Eu-clidean distance between the particle i and other particles iscalculated Using formulas (1) and (3) the resultant force ofparticle i under the action of other particles is obtainedUsing formula (4) to calculate the acceleration of particle isimilarly the gravitational and acceleration forces of otherparticles in the population are calculated

Step 7 In order to speed up the convergence of particlesimprove the optimization performance e mutation op-erator is introduced into the velocity formula of the particleto increase the guiding effect of the current optimal solutionand the global optimal solution on the particle and improvethe convergence speed of the algorithm e expression ofthe mutation operator is as in formula (8) In the process ofupdating the speed and position the calculated accelerationcomponent and velocity component may appear as non-integer erefore we round the speed formula and theposition formula e specific expression is as follows

η rand pdg(t) minus x

di (t)1113872 1113873 (24)

vdi (t) round rand times v

di (t) + a

di (t) + η1113872 1113873

xdi (t + 1) round x

di (t) + v

di (t + 1)1113872 1113873

⎧⎪⎨

⎪⎩(25)

where the round is a function to ensure that the updatedparticle position component is an integer e positionalcomponent of the particle is an integer randomly generatedfrom [minus8 8] In Step 7 the particle may exceed the range ofvalues in the position updateerefore this paper stipulatesthat when the position component is greater than 8 it takes8 when the position component is less than minus8 it takes minus8

Step 8 Substituting the mode displacement correspondingto the updated particle into formula (20) if the updatedfitness value is less than before the update the particleposition changes Otherwise the particle position remainsunchanged e optimal placement corresponding to the

Table 1 Dual-structure coding method

Extra code x(1) x(2) middot middot middot x(i) middot middot middot x(f )Variable code sx(1) sx(2) middot middot middot sx(i) middot middot middot sx(f )

Table 2 e result of dual-structure coding method

Extra code 5 3 6 1 4 7 2 8Variable code 0 1 1 0 1 0 0 1

6 Shock and Vibration

updated optimal fitness value is denoted as pb(xsi) If theupdated optimal fitness value fit(pb(xsi)) is smaller thanthe fitness value fit(pg(xsi)) corresponding to the globaloptimal value pg(xsi) of the previous generation thecurrent particle is used instead of the global optimal so-lution Otherwise the global optimal solution remainsunchanged

Step 9 Determine whether the algorithm reaches the presetaccuracy requirement or the maximum number of itera-tions If it is satisfied stop the iteration and output the globaloptimal solution which is the required sensor placement ifnot return to Step 6 e flow chart based on DGSA isshown in Figure 2

4 Model Validation

In this section the DGSA will be used in the optimal sensorplacement on the test object so as to verify the feasibility andeffectiveness of this algorithm

41 Modeling and Modal Analysis e example selected inthis paper is a single-tower cable-stayed bridge with a totallength of 264m Its main components include a mainbeam a bridge tower and a stay cable ANSYS 192 wasused to establish the finite element model of the bridge asshown in Figure 3 rough modal analysis the model wasdetermined to have 1588 nodes and 2422 elements eachnode incorporates three degrees of freedom corre-sponding to the modal information of the x y and zdirections Figures 4ndash6 show the effective modal masscorresponding to the first 50 modes of the cable-stayedbridge in the x y and z directions respectively

42ModeNumber Considering that the selection of mode isvery important in the optimal placement of sensors a matrixof the first 50 modes of the cable-stayed bridge is derivedthrough the modal analysis process e effective modalmass participation ratio in three directions is calculated andthe modal order with an R value greater than 90 is selectedas the mode to be used e effective mass corresponding tothe first 50 modes is shown e main modes are selected inthe x y and z directions and the correspondingMi ri and Rare determined

ree and four modes with high effective participationmass and twelve modes in total are selected in the x y and zdirections respectively and the corresponding values of Rand the other parameters are shown in the table It can beseen from Table 3 that the value of R for the 12 modes is09235 the value of R for the first 8 modes is 08335 and thevalue of R for the last 4 modes is 00900 e R values of the2nd 3rd 6th 9th 11th 16th 30th and 48th modes occupythe main part of the 12th mode in the table which can beused as the main mode to represent the information of thecable-stayed bridge erefore the 2nd 3rd 6th 9th 11th16th 30th and 48th modes are chosen as the initial modes ofthe optimal sensor placement

43 Analysis of the Sensor Placement Results

431 Number of Sensors Generally speaking the higher thenumber of sensors arranged the greater the amount ofobtainable information pertaining to structural arrange-ment and the better the reflection of the structurersquos healthcondition However in practical applications because of thehigh cost of sensors having a very high number of sensors isoften impossible so the selection of the number of sensors isvery important erefore this paper takes the number ofsensors as the independent variable and formula (20) as theobjective function Each group of variables is calculated 10times and the average value of the results of 10 operations isthe corresponding variable

It can be seen that when the optimal sensors quantity isbetween 1 and 11 the decline curve of the objective functionis steeper When the number of sensors is 11 the value of theobjective function is the lowest When the number of sensorschanges from 11 to 12 the value of the objective functionbegins to increase When the number is between 12 and 30the value of the objective function begins to decrease slowlyand tends to be stable Considering the cost of the sensorsm 11 is selected as the optimal number of sensors

432 Optimizing Performance Comparison Optimal sensorplacement is performed according to the DGSA proposed inthis paper with the following parameters population sizeN 100 maximum iteration times Imax 100 initial value ofgravitational constant G0 100 and attenuation factor β 40When the number of sensor m is 7 11 and 15 the algorithmrepeats 10 times to calculate the optimal solution the worstsolution and the average solution of the objective function

Figure 7 shows the curve of the objective functionchanging with the number of sensors when other parametersare unchanged It can be seen from Figures 8ndash10 that thefitness function values corresponding to different numbers ofsensor are different When the number of sensor m arrangedis 7 11 and 15 respectively the average values of the objectivefunction are 01866 00526 and 00489 e results show thatwith the increase of m the smaller the average value of theobjective function the easier to identify the modal vectorobtained by the search and the optimal value of the objectivefunction can reach 01815 00410 and 00455 in turn It showsthat within a certain value range as the number of sensorincreases the sensor placement results can be better

In order to further highlight the advantages of the DGSAthe results of the DGSA the GA algorithm and the PSO al-gorithm are compared when 11 sensors are arranged Figure 11shows the performance curve comparison for the three algo-rithms In the case of 100 iterations of three algorithms theconvergence speed and optimization accuracy of the DGSA arebetter than those of the GA and PSO Among them GA PSOandDGSA are different parameters and the optimal solution isobtained after many repetitions of the verification procedure

Figures 12ndash14 show a bar chart of the MAC matrices forthe three algorithms e maximum values of the off-di-agonal element of the MAC matrix obtained by the GAalgorithm and the PSO algorithm are 00741 and 00694

Shock and Vibration 7

1

XY

Z

Figure 3 Cable-stayed bridge model

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

11

Effec

tive m

odal

mas

s

times106

Figure 4 Effective modal mass in x direction

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 5 Effective modal mass in y direction

8 Shock and Vibration

0 5 10 15 20 25 30Sensor number

0

01

02

03

04

05

06

07

08

09

1

Best

fitne

ss

Figure 7 Curve of sensor number

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 6 Effective modal mass in z direction

Table 3 Calculation results based on response of cable-stayed bridge

Order Direction Naturalfrequency

Modal participationfactor

Effective modalmass

ri in correspondingdirection

R in threedirections

30 x 54742 32582 010616E+ 08 05661 017812 z 07627 30353 092131E+ 07 04477 033273 y 09631 28004 078423E+ 07 03869 046439 y 16349 25871 066931E+ 07 03302 0576648 z 10295 22577 050972E+ 07 02477 066216 z 12662 22155 049084E+ 07 02385 0744511 x 18105 minus14989 030582E+ 07 01631 0795816 x 26809 minus14989 022467E+ 07 01198 083357 y 13117 minus14938 022314E+ 07 01101 0870927 y 49842 13550 018360E+ 07 00906 0901734 x 65904 92595 085739E+ 06 00457 091618 z 15352 minus66225 043858E+ 06 00213 09235

Shock and Vibration 9

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

Formula for calculating G(t) is defined as

G(t) G0eminus αlowastIImax( ) (2)

where I is the number of the iterations and Imax refers to themaximum number of iterations

e resultant force Fdi (t) of particle i in d dimensional

space is defined as

Fdi (t) 1113944

jisinbestrandlowastF

dij(t) (3)

where best is a collection of particles that have gravitationaleffects on particle i

e acceleration adi (t) of particle i is defined as

adi (t)

Fdi (t)

Massi(t)

1113936jisinbestrandlowastFdij(t)

Massi(t) (4)

where Massi(t) is the inertia mass of particle i and itsformula is defined as

mi(t) fiti(t) minus fitw(t)

fitb(t) minus fitw(t) (5)

Massi(t) mi(t)

1113936nj1 mi(t)

(6)

where fitb(t) represents the best fitness value and fitw(t)

represents the worst fitness valuee formula for the velocity and position of the particle i

at the next moment is

vdi (t) randlowast v

di (t) + a

di (t)

xdi (t + 1) x

di (t) + v

di (t + 1)

⎧⎨

⎩ (7)

Note 1rand is a random variable which is uniformlydistributed among [0 1]

22 DynamicAdjustment of Attenuation FactorGravitationalSearch Algorithm e gravitational constant G(t) has asignificant influence over the convergence speed and opti-mization precision of the algorithme αin formula (2) hasthe effect of adjusting the convergence speed of the algo-rithm Aiming at the optimization efficiency and searchaccuracy of GSA this paper improves the GSA and proposesthe DGSA Considering that the value of α in the GSA is aconstant in the iterative process of the algorithm regardlessof whether the particle is far from or near to the globaloptimal solution the particle moves with a fixed step size Asa result when the particle is far away from the optimalsolution its step length is too small it moves too slowly andit takes too long Furthermore when the particle is close tothe optimal solution the step length is too large which maycause it to deviate from the optimal solution and fall into thelocal optimal solution erefore DGSA selects a smallerattenuation factor α in the early stage of the search and theparticles perform the global search with a larger step sizewhich is beneficial in reducing the search time of the al-gorithm and improving the overall optimization efficiency

e algorithm selects a larger attenuation factor α in the laterstage and the particles employ a smaller step size to com-plete the local search so as to avoid the algorithm fallingupon the local extreme point e expression of adaptiveattenuation factor α is defined as

α(t) βlowast eminus 1minus tImax| | (8)

where t represents the current number of iterations and β isthe initial parameter According to experience when β 1the optimization effect of the algorithm is the best eschematic diagram of the DGSA is shown in Figure 1

Particle M1 generates resultant force F1 and accelerationa1 under the action of universal gravitation of M2 M3 andM4 rough the adaptive mechanism of attenuationfactorα particle M1 approaches the global optimal solutionwith different steps

3 Optimal Sensor Placement Based onDGSA Method

In this section the dual-structure coding method is used tosolve the problem of DGSA being unable to realize optimalsensor placement and the flow of optimal sensor placementis shown in Figure 2 In the mode selection by calculatingthe effective modal mass participation ratio the mode orderwith a larger contribution value is selected

31 Modal Assurance Criterion e existing sensor opti-mization approaches such as the modal kinetic energymethod are highly dependent on the accuracy of the finiteelement mesh which will affect the reliability of the data It iseasy to lose the mode to be measured when using the Guyanreduction method which increases the difficulty of identi-fication e modal assurance criterion is one of the mostwidely used evaluation criteria for optimal sensor placementIt can enable the choosing of a larger space intersection angleand can retain the characteristics of the original model asmuch as possible

311 Mode Selection e selection of mode is the premiseto determine the number of sensors and then determine thelocation of sensor If too many modes are selected not only alot of calculation time and space will be consumed but also itis difficult to accurately define the correctness of the cal-culation results If too few modes are selected the reliabilityof the optimal sensor placement results will be too small torepresent the complete information of the bridge Atpresent many scholars have applied the effective modalmass participation factor to the selection of the modal orderand calculated it by using the ratio of the influence of thecurrent modal mass to the influence of all modal masses

e dynamic formula of n-DOF system is expressed as

Meurou + C _u + Ku minusMeeuroua(t) (9)

where M is the mass matrix C is the damping matrix K isthe stiffness matrix e is the direction matrix under theexcitation force euroua(t) is the acceleration produced by the

Shock and Vibration 3

F1

X0

Y

M3

Optimalsolution

M2

M4

F14

M1

a 1

Dynamic step

Adaptive adjustment

of attenuation factor

F12

F 13

M1

Step

Figure 1 Schematic diagram of DGSA

Start

Input modal matrix

Dual-structure coding

m sensorsNo

Attenuation factoradaptive adjustment

Update G forceand acceleration

Update speed and position

Update the optimal fitness valueand the optimal particle pb

f (pb) lt f (pg) pi = pb

Termination

Input parameters

Calculate N fitness values

Get the population size N

Individual optimal solution piglobal optimal solution pg

Yes

Yes

No

Yes

No

Update pg

End

Output pg

Figure 2 Optimal sensor placement based on DGSA

4 Shock and Vibration

excitation force u _u and eurou are modal displacement re-sponses velocity response and acceleration responserespectively

e relationship between the abovementioned responseand the mode shape matrix ϕ is as follows

u ϕ middot q _u ϕ middot _q eurou ϕ middot euroq (10)

Based on the orthogonality of M C and K we can getthe following formula from (9) and (10)

euroqi + 2ξiωi _qi + ω2i qi minus

ϕTi Me

ϕTi Mϕi

euroua(t) (11)

where qi is the modal coordinatesωi is the natural frequencyof the structure and ξi is the modal damping ratio

e participation factor is defined by the following

σi ϕT

i Me

ϕTi Mϕi

(12)

According to the principle of mass normalization ofmode shape matrix ϕi the following formula is obtained

ϕTi Mϕ 1 (13)

en formula (12) can be converted into the following

σi ϕTi Me (14)

e effective modal mass expression of the ith mode is

Mi σ2i

ϕTi Mϕi

(15)

erefore the total modal mass is

1113944

n

i1Mi 1113944

n

i1σ2i σTσ (16)

e expression of the ith order effective mode massparticipation ratio ri is

ri Mi

eT

Me (17)

For the first m modes (mlt n) the total modal massparticipation ratio R of the first m modes is

R 1113944m

i1ri (18)

According to the research when the value of R is morethan 90 it shows that the selected modal order containsenough structural information which can be used as themode shape of optimal sensor placement

312 Fitness Function e most important step in theproblem of optimal sensor placement is to establish themathematical model of the structure and construct the objectivefunction Actually due to the external factors the accuracy errorof the measuring instrument and the noise interference of the

test site the included angle of the modal vector is too small oreven 0 which leads to the loss of modal informationereforethe optimal sensor placement should select the points with largemodal angle and be easy to identify e modal assurancecriterion (MAC) is usually used to compare the correlation andindependence of experimental modes [25ndash27] so MAC can beused as the fitness function of optimal sensor placement toevaluate the independence of each mode

e expression of MAC is as follows

MACij ϕT

i middot ϕj1113872 11138732

ϕTi middot ϕi1113872 1113873 ϕT

j middot ϕj1113872 1113873 (19)

where ϕi and ϕj represent the ith and jth column vectors inmatrixϕ

In the above formula maximum off-diagonal elementranges from 0 to 1 e smaller the maximum off-diagonalelement means the larger the corresponding space angle andthe easier to distinguish the modal vectors e smaller thevalue of the off-diagonal element of the MAC matrix thebetter the calculated modal vector independence and theeasier it is to identify the mode shape erefore theminimization of the maximum off-diagonal element ofMAC matrix can be taken as the fitness function of optimalsensor placement of bridge and the expression is as follows

fit minf(x) (20)

where f(x) maxinej|MACij| e smaller the value of theobjective function the better the independence of the testedmodal vector and the better the optimal sensor placement

32 Method of Coding e binary coding method can beused to solve the optimal sensor placement problem 1indicates the sensor is placed and 0 indicates the sensor isnot placed but this codingmethod will lead to changes in thenumber of sensors Here dual-structure coding is adopted toovercome this probleme dual-structure codingmethod isshown in Table 1

e dual-structure coding method involves extra codeand variable code e extra code represents the positionvector of the individual and the variable code representsbinary vector 0 or 1 where 0 means no sensor is placed and1 means placement of sensor is dual coding methodenables the DGSA to solve the problem of optimal sensorplacement For example there are known to be nine pointsthat must be selected e dual-structure coding results areshown in Table 2 e sensor arrangement we were able toderive is s (3 4 6 8)

33 Optimal Sensor Placement Process Based on DGSA

Step 1 Select and import mode matrix e order choice ofthe mode matrix has an important influence on the sensorplacement results On the one hand the appropriate modematrix can represent the overall performance of the struc-ture and ensure the accuracy of the optimal sensor place-ment On the other hand the optimal sensor placement is

Shock and Vibration 5

different with different modes e first and foremost themode shape matrix of all nodes of the model is obtained bymode analysis method and the obtained mode shape matrixis taken as the input value and the degrees of freedomcorresponding to all nodes are used as the candidate pointsof the optimal sensor placement Secondly assuming thatthe number of candidate points is n and the number ofsensors ism the n candidate positions are numbered from 1to n successively

Step 2 Taking the ith particle in the population as an ex-ample (i 1 2 N in whichNis the number of pop-ulation size) the corresponding solution to i can beexpressed as xs(i)(xisi) (xi1 si1)(xi2 si2) (xin sin)e position vectors of xi are obtained by formula (21) andthe binary vectors of si are obtained by formula (22) wherexdown minus8 xup 8

xij rand times xup minus xdown1113872 1113873 + xdown (21)

where rand is a random number between 0 and 1

sij xij

1 + x

2ij

1113969

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(22)

In formula (22) the binary vectors calculated by differentposition components xij are different so a threshold δ needsto be set to satisfy the following

sij 1 if V xij1113872 1113873gt δ

0 else

⎧⎨

⎩ (23)

wherej isin 1 2 n δ 05 and the components of eachposition of xij are substituted into formula (23) If thefunction value is greater than 05 the additional code value is1 indicating that the sensor is placed at this candidate pointIf the function value is less than 05 the additional code valueis 0 indicating that the sensor is not placed at this candidatepoint It can be obtained by calculation that the value of xij istaken as [minus8 8] and the value of sij is 0leV(xij)le 09923 Itcan be approximated instead of [0 1]

Step 3 Repeat Steps 1 and 2 until N particles are generatedduring initialization the total number of sensors in si is notequal to the number of sensors arranged m In order toensure that all possible solutions in the population meet therequirements when we encounter this problem we should

reinitialize at this time and repeat Step 2 until the initialsolution meets the requirements of coding

Step 4 e N particles in the population are substituted intoformula (20) e best fitness value is labeled fitb(xsi) andthe worst fitness value is labeled fitw(xsi) e optimal ar-rangement corresponding to the fitness value is recorded aspi(xsi) pi(xsi) represents the individual optimal solution ofparticles in the population and it is the optimal solution ineach particle iteration pg(xsi) represents the global optimalsolution which is the optimal solution among all particles inthe population

Step 5 e inertia mass of each particle is calculated usingformulas (5) and (6) Adaptively change the attenuationfactor α according to formula (2) and update the gravita-tional constant G(t)

Step 6 Taking particle i as an example the correspondingposition component is xi xi1 xi2 middot middot middot xim1113864 1113865 where m is thenumber of sensors that need to be arranged and the Eu-clidean distance between the particle i and other particles iscalculated Using formulas (1) and (3) the resultant force ofparticle i under the action of other particles is obtainedUsing formula (4) to calculate the acceleration of particle isimilarly the gravitational and acceleration forces of otherparticles in the population are calculated

Step 7 In order to speed up the convergence of particlesimprove the optimization performance e mutation op-erator is introduced into the velocity formula of the particleto increase the guiding effect of the current optimal solutionand the global optimal solution on the particle and improvethe convergence speed of the algorithm e expression ofthe mutation operator is as in formula (8) In the process ofupdating the speed and position the calculated accelerationcomponent and velocity component may appear as non-integer erefore we round the speed formula and theposition formula e specific expression is as follows

η rand pdg(t) minus x

di (t)1113872 1113873 (24)

vdi (t) round rand times v

di (t) + a

di (t) + η1113872 1113873

xdi (t + 1) round x

di (t) + v

di (t + 1)1113872 1113873

⎧⎪⎨

⎪⎩(25)

where the round is a function to ensure that the updatedparticle position component is an integer e positionalcomponent of the particle is an integer randomly generatedfrom [minus8 8] In Step 7 the particle may exceed the range ofvalues in the position updateerefore this paper stipulatesthat when the position component is greater than 8 it takes8 when the position component is less than minus8 it takes minus8

Step 8 Substituting the mode displacement correspondingto the updated particle into formula (20) if the updatedfitness value is less than before the update the particleposition changes Otherwise the particle position remainsunchanged e optimal placement corresponding to the

Table 1 Dual-structure coding method

Extra code x(1) x(2) middot middot middot x(i) middot middot middot x(f )Variable code sx(1) sx(2) middot middot middot sx(i) middot middot middot sx(f )

Table 2 e result of dual-structure coding method

Extra code 5 3 6 1 4 7 2 8Variable code 0 1 1 0 1 0 0 1

6 Shock and Vibration

updated optimal fitness value is denoted as pb(xsi) If theupdated optimal fitness value fit(pb(xsi)) is smaller thanthe fitness value fit(pg(xsi)) corresponding to the globaloptimal value pg(xsi) of the previous generation thecurrent particle is used instead of the global optimal so-lution Otherwise the global optimal solution remainsunchanged

Step 9 Determine whether the algorithm reaches the presetaccuracy requirement or the maximum number of itera-tions If it is satisfied stop the iteration and output the globaloptimal solution which is the required sensor placement ifnot return to Step 6 e flow chart based on DGSA isshown in Figure 2

4 Model Validation

In this section the DGSA will be used in the optimal sensorplacement on the test object so as to verify the feasibility andeffectiveness of this algorithm

41 Modeling and Modal Analysis e example selected inthis paper is a single-tower cable-stayed bridge with a totallength of 264m Its main components include a mainbeam a bridge tower and a stay cable ANSYS 192 wasused to establish the finite element model of the bridge asshown in Figure 3 rough modal analysis the model wasdetermined to have 1588 nodes and 2422 elements eachnode incorporates three degrees of freedom corre-sponding to the modal information of the x y and zdirections Figures 4ndash6 show the effective modal masscorresponding to the first 50 modes of the cable-stayedbridge in the x y and z directions respectively

42ModeNumber Considering that the selection of mode isvery important in the optimal placement of sensors a matrixof the first 50 modes of the cable-stayed bridge is derivedthrough the modal analysis process e effective modalmass participation ratio in three directions is calculated andthe modal order with an R value greater than 90 is selectedas the mode to be used e effective mass corresponding tothe first 50 modes is shown e main modes are selected inthe x y and z directions and the correspondingMi ri and Rare determined

ree and four modes with high effective participationmass and twelve modes in total are selected in the x y and zdirections respectively and the corresponding values of Rand the other parameters are shown in the table It can beseen from Table 3 that the value of R for the 12 modes is09235 the value of R for the first 8 modes is 08335 and thevalue of R for the last 4 modes is 00900 e R values of the2nd 3rd 6th 9th 11th 16th 30th and 48th modes occupythe main part of the 12th mode in the table which can beused as the main mode to represent the information of thecable-stayed bridge erefore the 2nd 3rd 6th 9th 11th16th 30th and 48th modes are chosen as the initial modes ofthe optimal sensor placement

43 Analysis of the Sensor Placement Results

431 Number of Sensors Generally speaking the higher thenumber of sensors arranged the greater the amount ofobtainable information pertaining to structural arrange-ment and the better the reflection of the structurersquos healthcondition However in practical applications because of thehigh cost of sensors having a very high number of sensors isoften impossible so the selection of the number of sensors isvery important erefore this paper takes the number ofsensors as the independent variable and formula (20) as theobjective function Each group of variables is calculated 10times and the average value of the results of 10 operations isthe corresponding variable

It can be seen that when the optimal sensors quantity isbetween 1 and 11 the decline curve of the objective functionis steeper When the number of sensors is 11 the value of theobjective function is the lowest When the number of sensorschanges from 11 to 12 the value of the objective functionbegins to increase When the number is between 12 and 30the value of the objective function begins to decrease slowlyand tends to be stable Considering the cost of the sensorsm 11 is selected as the optimal number of sensors

432 Optimizing Performance Comparison Optimal sensorplacement is performed according to the DGSA proposed inthis paper with the following parameters population sizeN 100 maximum iteration times Imax 100 initial value ofgravitational constant G0 100 and attenuation factor β 40When the number of sensor m is 7 11 and 15 the algorithmrepeats 10 times to calculate the optimal solution the worstsolution and the average solution of the objective function

Figure 7 shows the curve of the objective functionchanging with the number of sensors when other parametersare unchanged It can be seen from Figures 8ndash10 that thefitness function values corresponding to different numbers ofsensor are different When the number of sensor m arrangedis 7 11 and 15 respectively the average values of the objectivefunction are 01866 00526 and 00489 e results show thatwith the increase of m the smaller the average value of theobjective function the easier to identify the modal vectorobtained by the search and the optimal value of the objectivefunction can reach 01815 00410 and 00455 in turn It showsthat within a certain value range as the number of sensorincreases the sensor placement results can be better

In order to further highlight the advantages of the DGSAthe results of the DGSA the GA algorithm and the PSO al-gorithm are compared when 11 sensors are arranged Figure 11shows the performance curve comparison for the three algo-rithms In the case of 100 iterations of three algorithms theconvergence speed and optimization accuracy of the DGSA arebetter than those of the GA and PSO Among them GA PSOandDGSA are different parameters and the optimal solution isobtained after many repetitions of the verification procedure

Figures 12ndash14 show a bar chart of the MAC matrices forthe three algorithms e maximum values of the off-di-agonal element of the MAC matrix obtained by the GAalgorithm and the PSO algorithm are 00741 and 00694

Shock and Vibration 7

1

XY

Z

Figure 3 Cable-stayed bridge model

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

11

Effec

tive m

odal

mas

s

times106

Figure 4 Effective modal mass in x direction

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 5 Effective modal mass in y direction

8 Shock and Vibration

0 5 10 15 20 25 30Sensor number

0

01

02

03

04

05

06

07

08

09

1

Best

fitne

ss

Figure 7 Curve of sensor number

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 6 Effective modal mass in z direction

Table 3 Calculation results based on response of cable-stayed bridge

Order Direction Naturalfrequency

Modal participationfactor

Effective modalmass

ri in correspondingdirection

R in threedirections

30 x 54742 32582 010616E+ 08 05661 017812 z 07627 30353 092131E+ 07 04477 033273 y 09631 28004 078423E+ 07 03869 046439 y 16349 25871 066931E+ 07 03302 0576648 z 10295 22577 050972E+ 07 02477 066216 z 12662 22155 049084E+ 07 02385 0744511 x 18105 minus14989 030582E+ 07 01631 0795816 x 26809 minus14989 022467E+ 07 01198 083357 y 13117 minus14938 022314E+ 07 01101 0870927 y 49842 13550 018360E+ 07 00906 0901734 x 65904 92595 085739E+ 06 00457 091618 z 15352 minus66225 043858E+ 06 00213 09235

Shock and Vibration 9

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

F1

X0

Y

M3

Optimalsolution

M2

M4

F14

M1

a 1

Dynamic step

Adaptive adjustment

of attenuation factor

F12

F 13

M1

Step

Figure 1 Schematic diagram of DGSA

Start

Input modal matrix

Dual-structure coding

m sensorsNo

Attenuation factoradaptive adjustment

Update G forceand acceleration

Update speed and position

Update the optimal fitness valueand the optimal particle pb

f (pb) lt f (pg) pi = pb

Termination

Input parameters

Calculate N fitness values

Get the population size N

Individual optimal solution piglobal optimal solution pg

Yes

Yes

No

Yes

No

Update pg

End

Output pg

Figure 2 Optimal sensor placement based on DGSA

4 Shock and Vibration

excitation force u _u and eurou are modal displacement re-sponses velocity response and acceleration responserespectively

e relationship between the abovementioned responseand the mode shape matrix ϕ is as follows

u ϕ middot q _u ϕ middot _q eurou ϕ middot euroq (10)

Based on the orthogonality of M C and K we can getthe following formula from (9) and (10)

euroqi + 2ξiωi _qi + ω2i qi minus

ϕTi Me

ϕTi Mϕi

euroua(t) (11)

where qi is the modal coordinatesωi is the natural frequencyof the structure and ξi is the modal damping ratio

e participation factor is defined by the following

σi ϕT

i Me

ϕTi Mϕi

(12)

According to the principle of mass normalization ofmode shape matrix ϕi the following formula is obtained

ϕTi Mϕ 1 (13)

en formula (12) can be converted into the following

σi ϕTi Me (14)

e effective modal mass expression of the ith mode is

Mi σ2i

ϕTi Mϕi

(15)

erefore the total modal mass is

1113944

n

i1Mi 1113944

n

i1σ2i σTσ (16)

e expression of the ith order effective mode massparticipation ratio ri is

ri Mi

eT

Me (17)

For the first m modes (mlt n) the total modal massparticipation ratio R of the first m modes is

R 1113944m

i1ri (18)

According to the research when the value of R is morethan 90 it shows that the selected modal order containsenough structural information which can be used as themode shape of optimal sensor placement

312 Fitness Function e most important step in theproblem of optimal sensor placement is to establish themathematical model of the structure and construct the objectivefunction Actually due to the external factors the accuracy errorof the measuring instrument and the noise interference of the

test site the included angle of the modal vector is too small oreven 0 which leads to the loss of modal informationereforethe optimal sensor placement should select the points with largemodal angle and be easy to identify e modal assurancecriterion (MAC) is usually used to compare the correlation andindependence of experimental modes [25ndash27] so MAC can beused as the fitness function of optimal sensor placement toevaluate the independence of each mode

e expression of MAC is as follows

MACij ϕT

i middot ϕj1113872 11138732

ϕTi middot ϕi1113872 1113873 ϕT

j middot ϕj1113872 1113873 (19)

where ϕi and ϕj represent the ith and jth column vectors inmatrixϕ

In the above formula maximum off-diagonal elementranges from 0 to 1 e smaller the maximum off-diagonalelement means the larger the corresponding space angle andthe easier to distinguish the modal vectors e smaller thevalue of the off-diagonal element of the MAC matrix thebetter the calculated modal vector independence and theeasier it is to identify the mode shape erefore theminimization of the maximum off-diagonal element ofMAC matrix can be taken as the fitness function of optimalsensor placement of bridge and the expression is as follows

fit minf(x) (20)

where f(x) maxinej|MACij| e smaller the value of theobjective function the better the independence of the testedmodal vector and the better the optimal sensor placement

32 Method of Coding e binary coding method can beused to solve the optimal sensor placement problem 1indicates the sensor is placed and 0 indicates the sensor isnot placed but this codingmethod will lead to changes in thenumber of sensors Here dual-structure coding is adopted toovercome this probleme dual-structure codingmethod isshown in Table 1

e dual-structure coding method involves extra codeand variable code e extra code represents the positionvector of the individual and the variable code representsbinary vector 0 or 1 where 0 means no sensor is placed and1 means placement of sensor is dual coding methodenables the DGSA to solve the problem of optimal sensorplacement For example there are known to be nine pointsthat must be selected e dual-structure coding results areshown in Table 2 e sensor arrangement we were able toderive is s (3 4 6 8)

33 Optimal Sensor Placement Process Based on DGSA

Step 1 Select and import mode matrix e order choice ofthe mode matrix has an important influence on the sensorplacement results On the one hand the appropriate modematrix can represent the overall performance of the struc-ture and ensure the accuracy of the optimal sensor place-ment On the other hand the optimal sensor placement is

Shock and Vibration 5

different with different modes e first and foremost themode shape matrix of all nodes of the model is obtained bymode analysis method and the obtained mode shape matrixis taken as the input value and the degrees of freedomcorresponding to all nodes are used as the candidate pointsof the optimal sensor placement Secondly assuming thatthe number of candidate points is n and the number ofsensors ism the n candidate positions are numbered from 1to n successively

Step 2 Taking the ith particle in the population as an ex-ample (i 1 2 N in whichNis the number of pop-ulation size) the corresponding solution to i can beexpressed as xs(i)(xisi) (xi1 si1)(xi2 si2) (xin sin)e position vectors of xi are obtained by formula (21) andthe binary vectors of si are obtained by formula (22) wherexdown minus8 xup 8

xij rand times xup minus xdown1113872 1113873 + xdown (21)

where rand is a random number between 0 and 1

sij xij

1 + x

2ij

1113969

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(22)

In formula (22) the binary vectors calculated by differentposition components xij are different so a threshold δ needsto be set to satisfy the following

sij 1 if V xij1113872 1113873gt δ

0 else

⎧⎨

⎩ (23)

wherej isin 1 2 n δ 05 and the components of eachposition of xij are substituted into formula (23) If thefunction value is greater than 05 the additional code value is1 indicating that the sensor is placed at this candidate pointIf the function value is less than 05 the additional code valueis 0 indicating that the sensor is not placed at this candidatepoint It can be obtained by calculation that the value of xij istaken as [minus8 8] and the value of sij is 0leV(xij)le 09923 Itcan be approximated instead of [0 1]

Step 3 Repeat Steps 1 and 2 until N particles are generatedduring initialization the total number of sensors in si is notequal to the number of sensors arranged m In order toensure that all possible solutions in the population meet therequirements when we encounter this problem we should

reinitialize at this time and repeat Step 2 until the initialsolution meets the requirements of coding

Step 4 e N particles in the population are substituted intoformula (20) e best fitness value is labeled fitb(xsi) andthe worst fitness value is labeled fitw(xsi) e optimal ar-rangement corresponding to the fitness value is recorded aspi(xsi) pi(xsi) represents the individual optimal solution ofparticles in the population and it is the optimal solution ineach particle iteration pg(xsi) represents the global optimalsolution which is the optimal solution among all particles inthe population

Step 5 e inertia mass of each particle is calculated usingformulas (5) and (6) Adaptively change the attenuationfactor α according to formula (2) and update the gravita-tional constant G(t)

Step 6 Taking particle i as an example the correspondingposition component is xi xi1 xi2 middot middot middot xim1113864 1113865 where m is thenumber of sensors that need to be arranged and the Eu-clidean distance between the particle i and other particles iscalculated Using formulas (1) and (3) the resultant force ofparticle i under the action of other particles is obtainedUsing formula (4) to calculate the acceleration of particle isimilarly the gravitational and acceleration forces of otherparticles in the population are calculated

Step 7 In order to speed up the convergence of particlesimprove the optimization performance e mutation op-erator is introduced into the velocity formula of the particleto increase the guiding effect of the current optimal solutionand the global optimal solution on the particle and improvethe convergence speed of the algorithm e expression ofthe mutation operator is as in formula (8) In the process ofupdating the speed and position the calculated accelerationcomponent and velocity component may appear as non-integer erefore we round the speed formula and theposition formula e specific expression is as follows

η rand pdg(t) minus x

di (t)1113872 1113873 (24)

vdi (t) round rand times v

di (t) + a

di (t) + η1113872 1113873

xdi (t + 1) round x

di (t) + v

di (t + 1)1113872 1113873

⎧⎪⎨

⎪⎩(25)

where the round is a function to ensure that the updatedparticle position component is an integer e positionalcomponent of the particle is an integer randomly generatedfrom [minus8 8] In Step 7 the particle may exceed the range ofvalues in the position updateerefore this paper stipulatesthat when the position component is greater than 8 it takes8 when the position component is less than minus8 it takes minus8

Step 8 Substituting the mode displacement correspondingto the updated particle into formula (20) if the updatedfitness value is less than before the update the particleposition changes Otherwise the particle position remainsunchanged e optimal placement corresponding to the

Table 1 Dual-structure coding method

Extra code x(1) x(2) middot middot middot x(i) middot middot middot x(f )Variable code sx(1) sx(2) middot middot middot sx(i) middot middot middot sx(f )

Table 2 e result of dual-structure coding method

Extra code 5 3 6 1 4 7 2 8Variable code 0 1 1 0 1 0 0 1

6 Shock and Vibration

updated optimal fitness value is denoted as pb(xsi) If theupdated optimal fitness value fit(pb(xsi)) is smaller thanthe fitness value fit(pg(xsi)) corresponding to the globaloptimal value pg(xsi) of the previous generation thecurrent particle is used instead of the global optimal so-lution Otherwise the global optimal solution remainsunchanged

Step 9 Determine whether the algorithm reaches the presetaccuracy requirement or the maximum number of itera-tions If it is satisfied stop the iteration and output the globaloptimal solution which is the required sensor placement ifnot return to Step 6 e flow chart based on DGSA isshown in Figure 2

4 Model Validation

In this section the DGSA will be used in the optimal sensorplacement on the test object so as to verify the feasibility andeffectiveness of this algorithm

41 Modeling and Modal Analysis e example selected inthis paper is a single-tower cable-stayed bridge with a totallength of 264m Its main components include a mainbeam a bridge tower and a stay cable ANSYS 192 wasused to establish the finite element model of the bridge asshown in Figure 3 rough modal analysis the model wasdetermined to have 1588 nodes and 2422 elements eachnode incorporates three degrees of freedom corre-sponding to the modal information of the x y and zdirections Figures 4ndash6 show the effective modal masscorresponding to the first 50 modes of the cable-stayedbridge in the x y and z directions respectively

42ModeNumber Considering that the selection of mode isvery important in the optimal placement of sensors a matrixof the first 50 modes of the cable-stayed bridge is derivedthrough the modal analysis process e effective modalmass participation ratio in three directions is calculated andthe modal order with an R value greater than 90 is selectedas the mode to be used e effective mass corresponding tothe first 50 modes is shown e main modes are selected inthe x y and z directions and the correspondingMi ri and Rare determined

ree and four modes with high effective participationmass and twelve modes in total are selected in the x y and zdirections respectively and the corresponding values of Rand the other parameters are shown in the table It can beseen from Table 3 that the value of R for the 12 modes is09235 the value of R for the first 8 modes is 08335 and thevalue of R for the last 4 modes is 00900 e R values of the2nd 3rd 6th 9th 11th 16th 30th and 48th modes occupythe main part of the 12th mode in the table which can beused as the main mode to represent the information of thecable-stayed bridge erefore the 2nd 3rd 6th 9th 11th16th 30th and 48th modes are chosen as the initial modes ofthe optimal sensor placement

43 Analysis of the Sensor Placement Results

431 Number of Sensors Generally speaking the higher thenumber of sensors arranged the greater the amount ofobtainable information pertaining to structural arrange-ment and the better the reflection of the structurersquos healthcondition However in practical applications because of thehigh cost of sensors having a very high number of sensors isoften impossible so the selection of the number of sensors isvery important erefore this paper takes the number ofsensors as the independent variable and formula (20) as theobjective function Each group of variables is calculated 10times and the average value of the results of 10 operations isthe corresponding variable

It can be seen that when the optimal sensors quantity isbetween 1 and 11 the decline curve of the objective functionis steeper When the number of sensors is 11 the value of theobjective function is the lowest When the number of sensorschanges from 11 to 12 the value of the objective functionbegins to increase When the number is between 12 and 30the value of the objective function begins to decrease slowlyand tends to be stable Considering the cost of the sensorsm 11 is selected as the optimal number of sensors

432 Optimizing Performance Comparison Optimal sensorplacement is performed according to the DGSA proposed inthis paper with the following parameters population sizeN 100 maximum iteration times Imax 100 initial value ofgravitational constant G0 100 and attenuation factor β 40When the number of sensor m is 7 11 and 15 the algorithmrepeats 10 times to calculate the optimal solution the worstsolution and the average solution of the objective function

Figure 7 shows the curve of the objective functionchanging with the number of sensors when other parametersare unchanged It can be seen from Figures 8ndash10 that thefitness function values corresponding to different numbers ofsensor are different When the number of sensor m arrangedis 7 11 and 15 respectively the average values of the objectivefunction are 01866 00526 and 00489 e results show thatwith the increase of m the smaller the average value of theobjective function the easier to identify the modal vectorobtained by the search and the optimal value of the objectivefunction can reach 01815 00410 and 00455 in turn It showsthat within a certain value range as the number of sensorincreases the sensor placement results can be better

In order to further highlight the advantages of the DGSAthe results of the DGSA the GA algorithm and the PSO al-gorithm are compared when 11 sensors are arranged Figure 11shows the performance curve comparison for the three algo-rithms In the case of 100 iterations of three algorithms theconvergence speed and optimization accuracy of the DGSA arebetter than those of the GA and PSO Among them GA PSOandDGSA are different parameters and the optimal solution isobtained after many repetitions of the verification procedure

Figures 12ndash14 show a bar chart of the MAC matrices forthe three algorithms e maximum values of the off-di-agonal element of the MAC matrix obtained by the GAalgorithm and the PSO algorithm are 00741 and 00694

Shock and Vibration 7

1

XY

Z

Figure 3 Cable-stayed bridge model

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

11

Effec

tive m

odal

mas

s

times106

Figure 4 Effective modal mass in x direction

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 5 Effective modal mass in y direction

8 Shock and Vibration

0 5 10 15 20 25 30Sensor number

0

01

02

03

04

05

06

07

08

09

1

Best

fitne

ss

Figure 7 Curve of sensor number

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 6 Effective modal mass in z direction

Table 3 Calculation results based on response of cable-stayed bridge

Order Direction Naturalfrequency

Modal participationfactor

Effective modalmass

ri in correspondingdirection

R in threedirections

30 x 54742 32582 010616E+ 08 05661 017812 z 07627 30353 092131E+ 07 04477 033273 y 09631 28004 078423E+ 07 03869 046439 y 16349 25871 066931E+ 07 03302 0576648 z 10295 22577 050972E+ 07 02477 066216 z 12662 22155 049084E+ 07 02385 0744511 x 18105 minus14989 030582E+ 07 01631 0795816 x 26809 minus14989 022467E+ 07 01198 083357 y 13117 minus14938 022314E+ 07 01101 0870927 y 49842 13550 018360E+ 07 00906 0901734 x 65904 92595 085739E+ 06 00457 091618 z 15352 minus66225 043858E+ 06 00213 09235

Shock and Vibration 9

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

excitation force u _u and eurou are modal displacement re-sponses velocity response and acceleration responserespectively

e relationship between the abovementioned responseand the mode shape matrix ϕ is as follows

u ϕ middot q _u ϕ middot _q eurou ϕ middot euroq (10)

Based on the orthogonality of M C and K we can getthe following formula from (9) and (10)

euroqi + 2ξiωi _qi + ω2i qi minus

ϕTi Me

ϕTi Mϕi

euroua(t) (11)

where qi is the modal coordinatesωi is the natural frequencyof the structure and ξi is the modal damping ratio

e participation factor is defined by the following

σi ϕT

i Me

ϕTi Mϕi

(12)

According to the principle of mass normalization ofmode shape matrix ϕi the following formula is obtained

ϕTi Mϕ 1 (13)

en formula (12) can be converted into the following

σi ϕTi Me (14)

e effective modal mass expression of the ith mode is

Mi σ2i

ϕTi Mϕi

(15)

erefore the total modal mass is

1113944

n

i1Mi 1113944

n

i1σ2i σTσ (16)

e expression of the ith order effective mode massparticipation ratio ri is

ri Mi

eT

Me (17)

For the first m modes (mlt n) the total modal massparticipation ratio R of the first m modes is

R 1113944m

i1ri (18)

According to the research when the value of R is morethan 90 it shows that the selected modal order containsenough structural information which can be used as themode shape of optimal sensor placement

312 Fitness Function e most important step in theproblem of optimal sensor placement is to establish themathematical model of the structure and construct the objectivefunction Actually due to the external factors the accuracy errorof the measuring instrument and the noise interference of the

test site the included angle of the modal vector is too small oreven 0 which leads to the loss of modal informationereforethe optimal sensor placement should select the points with largemodal angle and be easy to identify e modal assurancecriterion (MAC) is usually used to compare the correlation andindependence of experimental modes [25ndash27] so MAC can beused as the fitness function of optimal sensor placement toevaluate the independence of each mode

e expression of MAC is as follows

MACij ϕT

i middot ϕj1113872 11138732

ϕTi middot ϕi1113872 1113873 ϕT

j middot ϕj1113872 1113873 (19)

where ϕi and ϕj represent the ith and jth column vectors inmatrixϕ

In the above formula maximum off-diagonal elementranges from 0 to 1 e smaller the maximum off-diagonalelement means the larger the corresponding space angle andthe easier to distinguish the modal vectors e smaller thevalue of the off-diagonal element of the MAC matrix thebetter the calculated modal vector independence and theeasier it is to identify the mode shape erefore theminimization of the maximum off-diagonal element ofMAC matrix can be taken as the fitness function of optimalsensor placement of bridge and the expression is as follows

fit minf(x) (20)

where f(x) maxinej|MACij| e smaller the value of theobjective function the better the independence of the testedmodal vector and the better the optimal sensor placement

32 Method of Coding e binary coding method can beused to solve the optimal sensor placement problem 1indicates the sensor is placed and 0 indicates the sensor isnot placed but this codingmethod will lead to changes in thenumber of sensors Here dual-structure coding is adopted toovercome this probleme dual-structure codingmethod isshown in Table 1

e dual-structure coding method involves extra codeand variable code e extra code represents the positionvector of the individual and the variable code representsbinary vector 0 or 1 where 0 means no sensor is placed and1 means placement of sensor is dual coding methodenables the DGSA to solve the problem of optimal sensorplacement For example there are known to be nine pointsthat must be selected e dual-structure coding results areshown in Table 2 e sensor arrangement we were able toderive is s (3 4 6 8)

33 Optimal Sensor Placement Process Based on DGSA

Step 1 Select and import mode matrix e order choice ofthe mode matrix has an important influence on the sensorplacement results On the one hand the appropriate modematrix can represent the overall performance of the struc-ture and ensure the accuracy of the optimal sensor place-ment On the other hand the optimal sensor placement is

Shock and Vibration 5

different with different modes e first and foremost themode shape matrix of all nodes of the model is obtained bymode analysis method and the obtained mode shape matrixis taken as the input value and the degrees of freedomcorresponding to all nodes are used as the candidate pointsof the optimal sensor placement Secondly assuming thatthe number of candidate points is n and the number ofsensors ism the n candidate positions are numbered from 1to n successively

Step 2 Taking the ith particle in the population as an ex-ample (i 1 2 N in whichNis the number of pop-ulation size) the corresponding solution to i can beexpressed as xs(i)(xisi) (xi1 si1)(xi2 si2) (xin sin)e position vectors of xi are obtained by formula (21) andthe binary vectors of si are obtained by formula (22) wherexdown minus8 xup 8

xij rand times xup minus xdown1113872 1113873 + xdown (21)

where rand is a random number between 0 and 1

sij xij

1 + x

2ij

1113969

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(22)

In formula (22) the binary vectors calculated by differentposition components xij are different so a threshold δ needsto be set to satisfy the following

sij 1 if V xij1113872 1113873gt δ

0 else

⎧⎨

⎩ (23)

wherej isin 1 2 n δ 05 and the components of eachposition of xij are substituted into formula (23) If thefunction value is greater than 05 the additional code value is1 indicating that the sensor is placed at this candidate pointIf the function value is less than 05 the additional code valueis 0 indicating that the sensor is not placed at this candidatepoint It can be obtained by calculation that the value of xij istaken as [minus8 8] and the value of sij is 0leV(xij)le 09923 Itcan be approximated instead of [0 1]

Step 3 Repeat Steps 1 and 2 until N particles are generatedduring initialization the total number of sensors in si is notequal to the number of sensors arranged m In order toensure that all possible solutions in the population meet therequirements when we encounter this problem we should

reinitialize at this time and repeat Step 2 until the initialsolution meets the requirements of coding

Step 4 e N particles in the population are substituted intoformula (20) e best fitness value is labeled fitb(xsi) andthe worst fitness value is labeled fitw(xsi) e optimal ar-rangement corresponding to the fitness value is recorded aspi(xsi) pi(xsi) represents the individual optimal solution ofparticles in the population and it is the optimal solution ineach particle iteration pg(xsi) represents the global optimalsolution which is the optimal solution among all particles inthe population

Step 5 e inertia mass of each particle is calculated usingformulas (5) and (6) Adaptively change the attenuationfactor α according to formula (2) and update the gravita-tional constant G(t)

Step 6 Taking particle i as an example the correspondingposition component is xi xi1 xi2 middot middot middot xim1113864 1113865 where m is thenumber of sensors that need to be arranged and the Eu-clidean distance between the particle i and other particles iscalculated Using formulas (1) and (3) the resultant force ofparticle i under the action of other particles is obtainedUsing formula (4) to calculate the acceleration of particle isimilarly the gravitational and acceleration forces of otherparticles in the population are calculated

Step 7 In order to speed up the convergence of particlesimprove the optimization performance e mutation op-erator is introduced into the velocity formula of the particleto increase the guiding effect of the current optimal solutionand the global optimal solution on the particle and improvethe convergence speed of the algorithm e expression ofthe mutation operator is as in formula (8) In the process ofupdating the speed and position the calculated accelerationcomponent and velocity component may appear as non-integer erefore we round the speed formula and theposition formula e specific expression is as follows

η rand pdg(t) minus x

di (t)1113872 1113873 (24)

vdi (t) round rand times v

di (t) + a

di (t) + η1113872 1113873

xdi (t + 1) round x

di (t) + v

di (t + 1)1113872 1113873

⎧⎪⎨

⎪⎩(25)

where the round is a function to ensure that the updatedparticle position component is an integer e positionalcomponent of the particle is an integer randomly generatedfrom [minus8 8] In Step 7 the particle may exceed the range ofvalues in the position updateerefore this paper stipulatesthat when the position component is greater than 8 it takes8 when the position component is less than minus8 it takes minus8

Step 8 Substituting the mode displacement correspondingto the updated particle into formula (20) if the updatedfitness value is less than before the update the particleposition changes Otherwise the particle position remainsunchanged e optimal placement corresponding to the

Table 1 Dual-structure coding method

Extra code x(1) x(2) middot middot middot x(i) middot middot middot x(f )Variable code sx(1) sx(2) middot middot middot sx(i) middot middot middot sx(f )

Table 2 e result of dual-structure coding method

Extra code 5 3 6 1 4 7 2 8Variable code 0 1 1 0 1 0 0 1

6 Shock and Vibration

updated optimal fitness value is denoted as pb(xsi) If theupdated optimal fitness value fit(pb(xsi)) is smaller thanthe fitness value fit(pg(xsi)) corresponding to the globaloptimal value pg(xsi) of the previous generation thecurrent particle is used instead of the global optimal so-lution Otherwise the global optimal solution remainsunchanged

Step 9 Determine whether the algorithm reaches the presetaccuracy requirement or the maximum number of itera-tions If it is satisfied stop the iteration and output the globaloptimal solution which is the required sensor placement ifnot return to Step 6 e flow chart based on DGSA isshown in Figure 2

4 Model Validation

In this section the DGSA will be used in the optimal sensorplacement on the test object so as to verify the feasibility andeffectiveness of this algorithm

41 Modeling and Modal Analysis e example selected inthis paper is a single-tower cable-stayed bridge with a totallength of 264m Its main components include a mainbeam a bridge tower and a stay cable ANSYS 192 wasused to establish the finite element model of the bridge asshown in Figure 3 rough modal analysis the model wasdetermined to have 1588 nodes and 2422 elements eachnode incorporates three degrees of freedom corre-sponding to the modal information of the x y and zdirections Figures 4ndash6 show the effective modal masscorresponding to the first 50 modes of the cable-stayedbridge in the x y and z directions respectively

42ModeNumber Considering that the selection of mode isvery important in the optimal placement of sensors a matrixof the first 50 modes of the cable-stayed bridge is derivedthrough the modal analysis process e effective modalmass participation ratio in three directions is calculated andthe modal order with an R value greater than 90 is selectedas the mode to be used e effective mass corresponding tothe first 50 modes is shown e main modes are selected inthe x y and z directions and the correspondingMi ri and Rare determined

ree and four modes with high effective participationmass and twelve modes in total are selected in the x y and zdirections respectively and the corresponding values of Rand the other parameters are shown in the table It can beseen from Table 3 that the value of R for the 12 modes is09235 the value of R for the first 8 modes is 08335 and thevalue of R for the last 4 modes is 00900 e R values of the2nd 3rd 6th 9th 11th 16th 30th and 48th modes occupythe main part of the 12th mode in the table which can beused as the main mode to represent the information of thecable-stayed bridge erefore the 2nd 3rd 6th 9th 11th16th 30th and 48th modes are chosen as the initial modes ofthe optimal sensor placement

43 Analysis of the Sensor Placement Results

431 Number of Sensors Generally speaking the higher thenumber of sensors arranged the greater the amount ofobtainable information pertaining to structural arrange-ment and the better the reflection of the structurersquos healthcondition However in practical applications because of thehigh cost of sensors having a very high number of sensors isoften impossible so the selection of the number of sensors isvery important erefore this paper takes the number ofsensors as the independent variable and formula (20) as theobjective function Each group of variables is calculated 10times and the average value of the results of 10 operations isthe corresponding variable

It can be seen that when the optimal sensors quantity isbetween 1 and 11 the decline curve of the objective functionis steeper When the number of sensors is 11 the value of theobjective function is the lowest When the number of sensorschanges from 11 to 12 the value of the objective functionbegins to increase When the number is between 12 and 30the value of the objective function begins to decrease slowlyand tends to be stable Considering the cost of the sensorsm 11 is selected as the optimal number of sensors

432 Optimizing Performance Comparison Optimal sensorplacement is performed according to the DGSA proposed inthis paper with the following parameters population sizeN 100 maximum iteration times Imax 100 initial value ofgravitational constant G0 100 and attenuation factor β 40When the number of sensor m is 7 11 and 15 the algorithmrepeats 10 times to calculate the optimal solution the worstsolution and the average solution of the objective function

Figure 7 shows the curve of the objective functionchanging with the number of sensors when other parametersare unchanged It can be seen from Figures 8ndash10 that thefitness function values corresponding to different numbers ofsensor are different When the number of sensor m arrangedis 7 11 and 15 respectively the average values of the objectivefunction are 01866 00526 and 00489 e results show thatwith the increase of m the smaller the average value of theobjective function the easier to identify the modal vectorobtained by the search and the optimal value of the objectivefunction can reach 01815 00410 and 00455 in turn It showsthat within a certain value range as the number of sensorincreases the sensor placement results can be better

In order to further highlight the advantages of the DGSAthe results of the DGSA the GA algorithm and the PSO al-gorithm are compared when 11 sensors are arranged Figure 11shows the performance curve comparison for the three algo-rithms In the case of 100 iterations of three algorithms theconvergence speed and optimization accuracy of the DGSA arebetter than those of the GA and PSO Among them GA PSOandDGSA are different parameters and the optimal solution isobtained after many repetitions of the verification procedure

Figures 12ndash14 show a bar chart of the MAC matrices forthe three algorithms e maximum values of the off-di-agonal element of the MAC matrix obtained by the GAalgorithm and the PSO algorithm are 00741 and 00694

Shock and Vibration 7

1

XY

Z

Figure 3 Cable-stayed bridge model

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

11

Effec

tive m

odal

mas

s

times106

Figure 4 Effective modal mass in x direction

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 5 Effective modal mass in y direction

8 Shock and Vibration

0 5 10 15 20 25 30Sensor number

0

01

02

03

04

05

06

07

08

09

1

Best

fitne

ss

Figure 7 Curve of sensor number

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 6 Effective modal mass in z direction

Table 3 Calculation results based on response of cable-stayed bridge

Order Direction Naturalfrequency

Modal participationfactor

Effective modalmass

ri in correspondingdirection

R in threedirections

30 x 54742 32582 010616E+ 08 05661 017812 z 07627 30353 092131E+ 07 04477 033273 y 09631 28004 078423E+ 07 03869 046439 y 16349 25871 066931E+ 07 03302 0576648 z 10295 22577 050972E+ 07 02477 066216 z 12662 22155 049084E+ 07 02385 0744511 x 18105 minus14989 030582E+ 07 01631 0795816 x 26809 minus14989 022467E+ 07 01198 083357 y 13117 minus14938 022314E+ 07 01101 0870927 y 49842 13550 018360E+ 07 00906 0901734 x 65904 92595 085739E+ 06 00457 091618 z 15352 minus66225 043858E+ 06 00213 09235

Shock and Vibration 9

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

different with different modes e first and foremost themode shape matrix of all nodes of the model is obtained bymode analysis method and the obtained mode shape matrixis taken as the input value and the degrees of freedomcorresponding to all nodes are used as the candidate pointsof the optimal sensor placement Secondly assuming thatthe number of candidate points is n and the number ofsensors ism the n candidate positions are numbered from 1to n successively

Step 2 Taking the ith particle in the population as an ex-ample (i 1 2 N in whichNis the number of pop-ulation size) the corresponding solution to i can beexpressed as xs(i)(xisi) (xi1 si1)(xi2 si2) (xin sin)e position vectors of xi are obtained by formula (21) andthe binary vectors of si are obtained by formula (22) wherexdown minus8 xup 8

xij rand times xup minus xdown1113872 1113873 + xdown (21)

where rand is a random number between 0 and 1

sij xij

1 + x

2ij

1113969

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(22)

In formula (22) the binary vectors calculated by differentposition components xij are different so a threshold δ needsto be set to satisfy the following

sij 1 if V xij1113872 1113873gt δ

0 else

⎧⎨

⎩ (23)

wherej isin 1 2 n δ 05 and the components of eachposition of xij are substituted into formula (23) If thefunction value is greater than 05 the additional code value is1 indicating that the sensor is placed at this candidate pointIf the function value is less than 05 the additional code valueis 0 indicating that the sensor is not placed at this candidatepoint It can be obtained by calculation that the value of xij istaken as [minus8 8] and the value of sij is 0leV(xij)le 09923 Itcan be approximated instead of [0 1]

Step 3 Repeat Steps 1 and 2 until N particles are generatedduring initialization the total number of sensors in si is notequal to the number of sensors arranged m In order toensure that all possible solutions in the population meet therequirements when we encounter this problem we should

reinitialize at this time and repeat Step 2 until the initialsolution meets the requirements of coding

Step 4 e N particles in the population are substituted intoformula (20) e best fitness value is labeled fitb(xsi) andthe worst fitness value is labeled fitw(xsi) e optimal ar-rangement corresponding to the fitness value is recorded aspi(xsi) pi(xsi) represents the individual optimal solution ofparticles in the population and it is the optimal solution ineach particle iteration pg(xsi) represents the global optimalsolution which is the optimal solution among all particles inthe population

Step 5 e inertia mass of each particle is calculated usingformulas (5) and (6) Adaptively change the attenuationfactor α according to formula (2) and update the gravita-tional constant G(t)

Step 6 Taking particle i as an example the correspondingposition component is xi xi1 xi2 middot middot middot xim1113864 1113865 where m is thenumber of sensors that need to be arranged and the Eu-clidean distance between the particle i and other particles iscalculated Using formulas (1) and (3) the resultant force ofparticle i under the action of other particles is obtainedUsing formula (4) to calculate the acceleration of particle isimilarly the gravitational and acceleration forces of otherparticles in the population are calculated

Step 7 In order to speed up the convergence of particlesimprove the optimization performance e mutation op-erator is introduced into the velocity formula of the particleto increase the guiding effect of the current optimal solutionand the global optimal solution on the particle and improvethe convergence speed of the algorithm e expression ofthe mutation operator is as in formula (8) In the process ofupdating the speed and position the calculated accelerationcomponent and velocity component may appear as non-integer erefore we round the speed formula and theposition formula e specific expression is as follows

η rand pdg(t) minus x

di (t)1113872 1113873 (24)

vdi (t) round rand times v

di (t) + a

di (t) + η1113872 1113873

xdi (t + 1) round x

di (t) + v

di (t + 1)1113872 1113873

⎧⎪⎨

⎪⎩(25)

where the round is a function to ensure that the updatedparticle position component is an integer e positionalcomponent of the particle is an integer randomly generatedfrom [minus8 8] In Step 7 the particle may exceed the range ofvalues in the position updateerefore this paper stipulatesthat when the position component is greater than 8 it takes8 when the position component is less than minus8 it takes minus8

Step 8 Substituting the mode displacement correspondingto the updated particle into formula (20) if the updatedfitness value is less than before the update the particleposition changes Otherwise the particle position remainsunchanged e optimal placement corresponding to the

Table 1 Dual-structure coding method

Extra code x(1) x(2) middot middot middot x(i) middot middot middot x(f )Variable code sx(1) sx(2) middot middot middot sx(i) middot middot middot sx(f )

Table 2 e result of dual-structure coding method

Extra code 5 3 6 1 4 7 2 8Variable code 0 1 1 0 1 0 0 1

6 Shock and Vibration

updated optimal fitness value is denoted as pb(xsi) If theupdated optimal fitness value fit(pb(xsi)) is smaller thanthe fitness value fit(pg(xsi)) corresponding to the globaloptimal value pg(xsi) of the previous generation thecurrent particle is used instead of the global optimal so-lution Otherwise the global optimal solution remainsunchanged

Step 9 Determine whether the algorithm reaches the presetaccuracy requirement or the maximum number of itera-tions If it is satisfied stop the iteration and output the globaloptimal solution which is the required sensor placement ifnot return to Step 6 e flow chart based on DGSA isshown in Figure 2

4 Model Validation

In this section the DGSA will be used in the optimal sensorplacement on the test object so as to verify the feasibility andeffectiveness of this algorithm

41 Modeling and Modal Analysis e example selected inthis paper is a single-tower cable-stayed bridge with a totallength of 264m Its main components include a mainbeam a bridge tower and a stay cable ANSYS 192 wasused to establish the finite element model of the bridge asshown in Figure 3 rough modal analysis the model wasdetermined to have 1588 nodes and 2422 elements eachnode incorporates three degrees of freedom corre-sponding to the modal information of the x y and zdirections Figures 4ndash6 show the effective modal masscorresponding to the first 50 modes of the cable-stayedbridge in the x y and z directions respectively

42ModeNumber Considering that the selection of mode isvery important in the optimal placement of sensors a matrixof the first 50 modes of the cable-stayed bridge is derivedthrough the modal analysis process e effective modalmass participation ratio in three directions is calculated andthe modal order with an R value greater than 90 is selectedas the mode to be used e effective mass corresponding tothe first 50 modes is shown e main modes are selected inthe x y and z directions and the correspondingMi ri and Rare determined

ree and four modes with high effective participationmass and twelve modes in total are selected in the x y and zdirections respectively and the corresponding values of Rand the other parameters are shown in the table It can beseen from Table 3 that the value of R for the 12 modes is09235 the value of R for the first 8 modes is 08335 and thevalue of R for the last 4 modes is 00900 e R values of the2nd 3rd 6th 9th 11th 16th 30th and 48th modes occupythe main part of the 12th mode in the table which can beused as the main mode to represent the information of thecable-stayed bridge erefore the 2nd 3rd 6th 9th 11th16th 30th and 48th modes are chosen as the initial modes ofthe optimal sensor placement

43 Analysis of the Sensor Placement Results

431 Number of Sensors Generally speaking the higher thenumber of sensors arranged the greater the amount ofobtainable information pertaining to structural arrange-ment and the better the reflection of the structurersquos healthcondition However in practical applications because of thehigh cost of sensors having a very high number of sensors isoften impossible so the selection of the number of sensors isvery important erefore this paper takes the number ofsensors as the independent variable and formula (20) as theobjective function Each group of variables is calculated 10times and the average value of the results of 10 operations isthe corresponding variable

It can be seen that when the optimal sensors quantity isbetween 1 and 11 the decline curve of the objective functionis steeper When the number of sensors is 11 the value of theobjective function is the lowest When the number of sensorschanges from 11 to 12 the value of the objective functionbegins to increase When the number is between 12 and 30the value of the objective function begins to decrease slowlyand tends to be stable Considering the cost of the sensorsm 11 is selected as the optimal number of sensors

432 Optimizing Performance Comparison Optimal sensorplacement is performed according to the DGSA proposed inthis paper with the following parameters population sizeN 100 maximum iteration times Imax 100 initial value ofgravitational constant G0 100 and attenuation factor β 40When the number of sensor m is 7 11 and 15 the algorithmrepeats 10 times to calculate the optimal solution the worstsolution and the average solution of the objective function

Figure 7 shows the curve of the objective functionchanging with the number of sensors when other parametersare unchanged It can be seen from Figures 8ndash10 that thefitness function values corresponding to different numbers ofsensor are different When the number of sensor m arrangedis 7 11 and 15 respectively the average values of the objectivefunction are 01866 00526 and 00489 e results show thatwith the increase of m the smaller the average value of theobjective function the easier to identify the modal vectorobtained by the search and the optimal value of the objectivefunction can reach 01815 00410 and 00455 in turn It showsthat within a certain value range as the number of sensorincreases the sensor placement results can be better

In order to further highlight the advantages of the DGSAthe results of the DGSA the GA algorithm and the PSO al-gorithm are compared when 11 sensors are arranged Figure 11shows the performance curve comparison for the three algo-rithms In the case of 100 iterations of three algorithms theconvergence speed and optimization accuracy of the DGSA arebetter than those of the GA and PSO Among them GA PSOandDGSA are different parameters and the optimal solution isobtained after many repetitions of the verification procedure

Figures 12ndash14 show a bar chart of the MAC matrices forthe three algorithms e maximum values of the off-di-agonal element of the MAC matrix obtained by the GAalgorithm and the PSO algorithm are 00741 and 00694

Shock and Vibration 7

1

XY

Z

Figure 3 Cable-stayed bridge model

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

11

Effec

tive m

odal

mas

s

times106

Figure 4 Effective modal mass in x direction

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 5 Effective modal mass in y direction

8 Shock and Vibration

0 5 10 15 20 25 30Sensor number

0

01

02

03

04

05

06

07

08

09

1

Best

fitne

ss

Figure 7 Curve of sensor number

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 6 Effective modal mass in z direction

Table 3 Calculation results based on response of cable-stayed bridge

Order Direction Naturalfrequency

Modal participationfactor

Effective modalmass

ri in correspondingdirection

R in threedirections

30 x 54742 32582 010616E+ 08 05661 017812 z 07627 30353 092131E+ 07 04477 033273 y 09631 28004 078423E+ 07 03869 046439 y 16349 25871 066931E+ 07 03302 0576648 z 10295 22577 050972E+ 07 02477 066216 z 12662 22155 049084E+ 07 02385 0744511 x 18105 minus14989 030582E+ 07 01631 0795816 x 26809 minus14989 022467E+ 07 01198 083357 y 13117 minus14938 022314E+ 07 01101 0870927 y 49842 13550 018360E+ 07 00906 0901734 x 65904 92595 085739E+ 06 00457 091618 z 15352 minus66225 043858E+ 06 00213 09235

Shock and Vibration 9

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

updated optimal fitness value is denoted as pb(xsi) If theupdated optimal fitness value fit(pb(xsi)) is smaller thanthe fitness value fit(pg(xsi)) corresponding to the globaloptimal value pg(xsi) of the previous generation thecurrent particle is used instead of the global optimal so-lution Otherwise the global optimal solution remainsunchanged

Step 9 Determine whether the algorithm reaches the presetaccuracy requirement or the maximum number of itera-tions If it is satisfied stop the iteration and output the globaloptimal solution which is the required sensor placement ifnot return to Step 6 e flow chart based on DGSA isshown in Figure 2

4 Model Validation

In this section the DGSA will be used in the optimal sensorplacement on the test object so as to verify the feasibility andeffectiveness of this algorithm

41 Modeling and Modal Analysis e example selected inthis paper is a single-tower cable-stayed bridge with a totallength of 264m Its main components include a mainbeam a bridge tower and a stay cable ANSYS 192 wasused to establish the finite element model of the bridge asshown in Figure 3 rough modal analysis the model wasdetermined to have 1588 nodes and 2422 elements eachnode incorporates three degrees of freedom corre-sponding to the modal information of the x y and zdirections Figures 4ndash6 show the effective modal masscorresponding to the first 50 modes of the cable-stayedbridge in the x y and z directions respectively

42ModeNumber Considering that the selection of mode isvery important in the optimal placement of sensors a matrixof the first 50 modes of the cable-stayed bridge is derivedthrough the modal analysis process e effective modalmass participation ratio in three directions is calculated andthe modal order with an R value greater than 90 is selectedas the mode to be used e effective mass corresponding tothe first 50 modes is shown e main modes are selected inthe x y and z directions and the correspondingMi ri and Rare determined

ree and four modes with high effective participationmass and twelve modes in total are selected in the x y and zdirections respectively and the corresponding values of Rand the other parameters are shown in the table It can beseen from Table 3 that the value of R for the 12 modes is09235 the value of R for the first 8 modes is 08335 and thevalue of R for the last 4 modes is 00900 e R values of the2nd 3rd 6th 9th 11th 16th 30th and 48th modes occupythe main part of the 12th mode in the table which can beused as the main mode to represent the information of thecable-stayed bridge erefore the 2nd 3rd 6th 9th 11th16th 30th and 48th modes are chosen as the initial modes ofthe optimal sensor placement

43 Analysis of the Sensor Placement Results

431 Number of Sensors Generally speaking the higher thenumber of sensors arranged the greater the amount ofobtainable information pertaining to structural arrange-ment and the better the reflection of the structurersquos healthcondition However in practical applications because of thehigh cost of sensors having a very high number of sensors isoften impossible so the selection of the number of sensors isvery important erefore this paper takes the number ofsensors as the independent variable and formula (20) as theobjective function Each group of variables is calculated 10times and the average value of the results of 10 operations isthe corresponding variable

It can be seen that when the optimal sensors quantity isbetween 1 and 11 the decline curve of the objective functionis steeper When the number of sensors is 11 the value of theobjective function is the lowest When the number of sensorschanges from 11 to 12 the value of the objective functionbegins to increase When the number is between 12 and 30the value of the objective function begins to decrease slowlyand tends to be stable Considering the cost of the sensorsm 11 is selected as the optimal number of sensors

432 Optimizing Performance Comparison Optimal sensorplacement is performed according to the DGSA proposed inthis paper with the following parameters population sizeN 100 maximum iteration times Imax 100 initial value ofgravitational constant G0 100 and attenuation factor β 40When the number of sensor m is 7 11 and 15 the algorithmrepeats 10 times to calculate the optimal solution the worstsolution and the average solution of the objective function

Figure 7 shows the curve of the objective functionchanging with the number of sensors when other parametersare unchanged It can be seen from Figures 8ndash10 that thefitness function values corresponding to different numbers ofsensor are different When the number of sensor m arrangedis 7 11 and 15 respectively the average values of the objectivefunction are 01866 00526 and 00489 e results show thatwith the increase of m the smaller the average value of theobjective function the easier to identify the modal vectorobtained by the search and the optimal value of the objectivefunction can reach 01815 00410 and 00455 in turn It showsthat within a certain value range as the number of sensorincreases the sensor placement results can be better

In order to further highlight the advantages of the DGSAthe results of the DGSA the GA algorithm and the PSO al-gorithm are compared when 11 sensors are arranged Figure 11shows the performance curve comparison for the three algo-rithms In the case of 100 iterations of three algorithms theconvergence speed and optimization accuracy of the DGSA arebetter than those of the GA and PSO Among them GA PSOandDGSA are different parameters and the optimal solution isobtained after many repetitions of the verification procedure

Figures 12ndash14 show a bar chart of the MAC matrices forthe three algorithms e maximum values of the off-di-agonal element of the MAC matrix obtained by the GAalgorithm and the PSO algorithm are 00741 and 00694

Shock and Vibration 7

1

XY

Z

Figure 3 Cable-stayed bridge model

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

11

Effec

tive m

odal

mas

s

times106

Figure 4 Effective modal mass in x direction

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 5 Effective modal mass in y direction

8 Shock and Vibration

0 5 10 15 20 25 30Sensor number

0

01

02

03

04

05

06

07

08

09

1

Best

fitne

ss

Figure 7 Curve of sensor number

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 6 Effective modal mass in z direction

Table 3 Calculation results based on response of cable-stayed bridge

Order Direction Naturalfrequency

Modal participationfactor

Effective modalmass

ri in correspondingdirection

R in threedirections

30 x 54742 32582 010616E+ 08 05661 017812 z 07627 30353 092131E+ 07 04477 033273 y 09631 28004 078423E+ 07 03869 046439 y 16349 25871 066931E+ 07 03302 0576648 z 10295 22577 050972E+ 07 02477 066216 z 12662 22155 049084E+ 07 02385 0744511 x 18105 minus14989 030582E+ 07 01631 0795816 x 26809 minus14989 022467E+ 07 01198 083357 y 13117 minus14938 022314E+ 07 01101 0870927 y 49842 13550 018360E+ 07 00906 0901734 x 65904 92595 085739E+ 06 00457 091618 z 15352 minus66225 043858E+ 06 00213 09235

Shock and Vibration 9

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

1

XY

Z

Figure 3 Cable-stayed bridge model

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

11

Effec

tive m

odal

mas

s

times106

Figure 4 Effective modal mass in x direction

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 5 Effective modal mass in y direction

8 Shock and Vibration

0 5 10 15 20 25 30Sensor number

0

01

02

03

04

05

06

07

08

09

1

Best

fitne

ss

Figure 7 Curve of sensor number

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 6 Effective modal mass in z direction

Table 3 Calculation results based on response of cable-stayed bridge

Order Direction Naturalfrequency

Modal participationfactor

Effective modalmass

ri in correspondingdirection

R in threedirections

30 x 54742 32582 010616E+ 08 05661 017812 z 07627 30353 092131E+ 07 04477 033273 y 09631 28004 078423E+ 07 03869 046439 y 16349 25871 066931E+ 07 03302 0576648 z 10295 22577 050972E+ 07 02477 066216 z 12662 22155 049084E+ 07 02385 0744511 x 18105 minus14989 030582E+ 07 01631 0795816 x 26809 minus14989 022467E+ 07 01198 083357 y 13117 minus14938 022314E+ 07 01101 0870927 y 49842 13550 018360E+ 07 00906 0901734 x 65904 92595 085739E+ 06 00457 091618 z 15352 minus66225 043858E+ 06 00213 09235

Shock and Vibration 9

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

0 5 10 15 20 25 30Sensor number

0

01

02

03

04

05

06

07

08

09

1

Best

fitne

ss

Figure 7 Curve of sensor number

0 5 10 15 20 25 30 35 40 45 50Mode

0

1

2

3

4

5

6

7

8

9

10

Effec

tive m

odal

mas

s

times106

Figure 6 Effective modal mass in z direction

Table 3 Calculation results based on response of cable-stayed bridge

Order Direction Naturalfrequency

Modal participationfactor

Effective modalmass

ri in correspondingdirection

R in threedirections

30 x 54742 32582 010616E+ 08 05661 017812 z 07627 30353 092131E+ 07 04477 033273 y 09631 28004 078423E+ 07 03869 046439 y 16349 25871 066931E+ 07 03302 0576648 z 10295 22577 050972E+ 07 02477 066216 z 12662 22155 049084E+ 07 02385 0744511 x 18105 minus14989 030582E+ 07 01631 0795816 x 26809 minus14989 022467E+ 07 01198 083357 y 13117 minus14938 022314E+ 07 01101 0870927 y 49842 13550 018360E+ 07 00906 0901734 x 65904 92595 085739E+ 06 00457 091618 z 15352 minus66225 043858E+ 06 00213 09235

Shock and Vibration 9

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

Mode

Mode

0

02

04

1

06

MA

C

2

08

3

1

84 75 6

56 47 38 2

1

Figure 9 11-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 10 15-sensor iteration curve

ModeMode

0

02

04

1

06

MA

C

2

08

3

1

84 75 656 47 3

8 21

Figure 8 7-sensor iteration curve

10 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

respectively Although both values are less than 025 thereare still cases wherein it is not possible to distinguish in-dividual locations However the maximum value of the off-diagonal element of the DGSA is 00410 Considering thatthe off-diagonal elements of the MACmatrices in Figures 12and 14 are too small Figure 15 shows the maximum off-diagonal element of each modal column vector of the MACmatrices corresponding to the three algorithm optimizationschemesWe can see that the maximum nondiagonal elementin the MAC matrix of DGSA is obviously smaller than the

optimal sensor placement value of the GA algorithm and thePSO algorithm At the same time from the variation of thecurve we can see that the maximum nondiagonal elementvariation range for each order mode of the DGSA is smallerand the result obtained is more stable is also verifies thatthe modal vector corresponding to the DGSA offers greaterdiscrimination and a better optimization result

Figures 16ndash18 and Tables 4ndash6 show the optimal sensorplacement results and scheme of the three algorithms on thecable-stayed bridge

0 10 20 30 40 50 60 70 80 90 100Generation

015

02

025

03

035

04

045

05

055

06

Fitn

ess

Optimal solution

Average solution

Worst solution

Figure 12 MAC matrix of GA

0

005

01

015

02

025

03

035

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

GA

PSO

DGSA

Figure 11 Performance curves of three algorithms

0

005

01

015

02

025

03

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 13 MAC matrix of PSO

004

006

008

01

012

014

016

018

02

022

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100Generation

Optimal solution

Average solution

Worst solution

Figure 14 MAC matrix of DGSA

Shock and Vibration 11

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

1 2 3 4 5 6 7 8Mode

0

002

004

006

008

01

012

Max

imum

off-

diag

onal

elem

ent

GAPSODGSA

Figure 15 Maximum off-diagonal term at each mode in MAC matrix

y

xz

Figure 16 Optimal sensor placement of GA

Table 6 Optimal sensor locations of DGSA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 31 69 81 105 130 156 159 186 221 248 257Direction z z x z z x z z z z y

y

xz

Figure 17 Optimal sensor placement of PSO

y

xz

Figure 18 Optimal sensor placement of DGSA

Table 4 Optimal sensor locations of GA

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 27 27 53 88 109 156 158 186 220 235 240Direction x z z z z z y z x z z

Table 5 Optimal sensor locations of PSO

Sensor number 1 2 3 4 5 6 7 8 9 10 11Node 42 45 80 103 103 137 156 167 179 196 204Direction y z z x z z z x y z y

12 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

5 Conclusions and Future Work

is article presents an OSP method based on mode se-lection and an improved DGSA to solve the sensor place-ment optimization problem In order to improve theoptimization ability of the algorithm the DGSA proposed inthis paper dynamically adjusts the attenuation factor αAtthe beginning of the search the attenuation factor is smalland the global search step is large In the later stage of thesearch the attenuation factor is larger and the search stepsize of the particle is smaller which is conducive to thealgorithmrsquos local search e modal selection of the initialsensor layout is determined by the experience of the engi-neer In this paper through the calculation of the modalmass participation ratio the vibration mode which has agreat influence on the bridgersquos vibration response is de-termined so as to ensure the effectiveness of the selectedvibration mode matrix

e validity and effectiveness of the proposed sensorplacement method with DGSA are both demonstrated usingthe example of a cable-stayed bridge Compared with the GAalgorithm and the PSO algorithm the sensor placementmethod based on the DGSA has a greater optimizationefficiency and a higher convergence speed and the optimalplacement result is better In addition the concept of theDGSA can be applied not only in sensor placement problemsbut also in similar constrained optimization problems

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National KeyRampD Program of China (Grant no 2018YFB120602) and theNational Natural Science Foundation of China (Grant no61463028)

References

[1] J-F Lin Y-L Xu and S Zhan ldquoExperimental investigationon multi-objective multi-type sensor optimal placement forstructural damage detectionrdquo Structural Health Monitoringvol 18 no 3 pp 882ndash901 2019

[2] D C Kammer and L Yao ldquoEnhancement of on-orbit modalidentification of large space structures through sensorplacementrdquo Journal of Sound and Vibration vol 171 no 1pp 119ndash139 1994

[3] Y Fujino ldquoVibration control and monitoring of long-spanbridges-recent research developments and practice in JapanrdquoJournal of Constructional Steel Research vol 58 no 1pp 71ndash97 2002

[4] D C Kammer ldquoSensor placement for on-orbit modalidentification and correlation of large space structuresrdquo

Journal of Guidance Control and Dynamics vol 14 no 2pp 251ndash259 1991

[5] R Castro-Triguero S Murugan R Gallego andM I FriswellldquoRobustness of optimal sensor placement under parametricuncertaintyrdquo Mechanical Systems and Signal Processingvol 41 no 2 pp 268ndash287 2013

[6] T G Carne and C R Dohrmann ldquoA modal test designstrategy for modal correlationrdquo in Proceedings of the 13thInternational Modal Analysis Conference pp 927ndash933Nashville TN USA February 1995

[7] J E T Penny M I Friswell and S D Garvey ldquoAutomaticchoice of measurement locations for dynamic testingrdquo AIAAJournal vol 32 no 2 pp 407ndash414 1994

[8] G Heo M L Wang and D Satpathi ldquoOptimal transducerplacement for health monitoring of long span bridgerdquo SoilDynamics and Earthquake Engineering vol 16 no 7-8pp 495ndash502 1997

[9] N Debnath A Dutta and S K Deb ldquoPlacement of sensors inoperational modal analysis for truss bridgesrdquo MechanicalSystems and Signal Processing vol 31 pp 196ndash216 2012

[10] C Papadimitriou J L Beck and S-K Au ldquoEntropy-basedoptimal sensor location for structural model updatingrdquoJournal of Vibration and Control vol 6 no 5 pp 781ndash8002000

[11] H M Chow H F Lam T Yin and S K Au ldquoOptimal sensorconfiguration of a typical transmission tower for the purposeof structural model updatingrdquo Structural Control and HealthMonitoring vol 18 no 3 pp 305ndash320 2011

[12] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forhealth monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 12 pages 2011

[13] C He J Xing J Li and Q Yang ldquoA new optimal sensorplacement strategy based on modified modal assurance cri-terion and improved adaptive genetic algorithm for structuralhealth monitoringrdquo Mathematical Problems in Engineeringvol 2015 Article ID 626342 10 pages 2015

[14] T-H Yi H-N Li G Song and X-D Zhang ldquoOptimal sensorplacement for health monitoring of high-rise structure usingadaptive monkey algorithmrdquo Structural Control and HealthMonitoring vol 22 no 4 pp 667ndash681 2015

[15] H Jin J Xia and Y-Q Wang ldquoOptimal sensor placement forspace modal identification of crane structures based on animproved harmony search algorithmrdquo Journal of ZhejiangUniversity vol 16 no 6 pp 464ndash477 2015

[16] H Yin K Dong A Pan Z Peng Z Jiang and S Li ldquoOptimalsensor placement based on relaxation sequential algorithmrdquoNeurocomputing vol 344 pp 28ndash36 2019

[17] X Zhang J Li J Xing and P Wang ldquoOptimal sensorplacement for latticed shell structure based on an improvedparticle swarm optimization algorithmrdquo MathematicalProblems in Engineering vol 2014 Article ID 743904 2014

[18] W Deng ldquoDifferential evolution algorithm with wavelet basisfunction and optimal mutation strategy for complex optimiza-tion problemrdquo Applied Soft Computing Article ID 106724 2020

[19] G-D Zhou T-H Yi H Zhang and H-N Li ldquoA comparativestudy of genetic and firefly algorithms for sensor placement instructural health monitoringrdquo Shock and Vibration vol 2015Article ID 518692 10 pages 2015

[20] J Li X Zhang J Xing P Wang Q Yang and C HeldquoOptimal sensor placement for long-span cable-stayed bridgeusing a novel particle swarm optimization algorithmrdquo Journalof Civil Structural Health Monitoring vol 5 no 5 pp 677ndash685 2015

Shock and Vibration 13

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration

[21] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoA grav-itational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[22] T W Saucer and V Sih ldquoOptimizing nanophotonic cavitydesigns with the gravitational search algorithmrdquo Optics Ex-press vol 21 no 18 pp 20831ndash20836 2013

[23] P Niu C Liu P Li and G Li ldquoOptimized support vectorregression model by improved gravitational search algorithmfor flatness pattern recognitionrdquo Neural Computing andApplications vol 26 no 5 pp 1167ndash1177 2015

[24] F Chen J Zhou C Wang C Li and P Lu ldquoA modifiedgravitational search algorithm based on a non-dominatedsorting genetic approach for hydro-thermal-wind economicemission dispatchingrdquo Energy vol 121 pp 276ndash291 2017

[25] D C Kammer and M L Tinker ldquoOptimal placement oftriaxial accelerometers for modal vibration testsrdquo MechanicalSystems and Signal Processing vol 18 no 1 pp 29ndash41 2004

[26] L Zhang Y Xia J A Lozano-Galant and L Sun ldquoMass-stiffness combined perturbation method for mode shapemonitoring of bridge structuresrdquo Shock and Vibrationvol 2019 Article ID 7320196 14 pages 2019

[27] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimizationstrategiesrdquoDe Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

14 Shock and Vibration