research article biobjective optimization of vibration

Research ArticleBiobjective Optimization of Vibration Performance ofSteel-Spring Floating Slab Tracks by Four-Pole ParameterMethod Coupled with Ant Colony Optimization

Hao Jin1 Weining Liu2 Shunhua Zhou1 and Christoph Adam3

1Key Laboratory of Road and Traffic Engineering of Ministry of Education Tongji University Shanghai 201804 China2School of Civil Engineering Beijing Jiaotong University Beijing 100044 China3Department of Civil Engineering Sciences University of Innsbruck Technikerstraszlige 13 6020 Innsbruck Austria

Correspondence should be addressed to Hao Jin zhujijinhaogmailcom

Received 24 March 2015 Revised 10 July 2015 Accepted 13 July 2015

Academic Editor Domenico Mundo

Copyright copy 2015 Hao Jin et al This 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

Steel-spring floating slab tracks are one of the most effective methods to reduce vibrations from underground railways whichhas drawn more and more attention in scientific communities In this paper the steel-spring floating slab track located in TrackVibrationAbatement andControl Laboratorywasmodeledwith four-pole parametermethodThe influences of the fastener dampingratio the fastener stiffness the steel-spring damping ratio and the steel-spring stiffness were researched for the rail displacementand the foundation acceleration Results show that the rail displacement and the foundation acceleration will decrease with theincrease of the fastener stiffness or the steel-spring damping ratio However the rail displacement and the foundation accelerationhave the opposite variation tendency for the fastener damping ratio and the steel-spring stiffness In order to optimize the raildisplacement and the foundation acceleration affected by the fastener damping ratio and the steel-spring stiffness at the same timea multiobjective ant colony optimization (ACO) was employed Eventually Pareto optimal frontier of the rail displacement and thefoundation acceleration was derived Furthermore the desirable values of the fastener damping ratio and the steel-spring stiffnesscan be obtained according to the corresponding Pareto optimal solution set

1 Introduction

Vibrations generated by underground railways are one ofthe most serious engineering problems Waves induced bythe dynamic interaction between the train wheels and therails propagate from the surrounding soils to the foundationsof nearby buildings resulting in structural vibrations andreradiated noise One of themost effectivemethods to reducevibrations from underground railways is to use floating slabtracks that rest on the steel-springs In the last decadessteel-spring floating slab tracks are largely used especially inChina

Therefore many scientists apply themselves to improvevibration performance of steel-spring floating slab tracksZhai et al [1ndash3] developed a coupled dynamic computationmodel for metro vehicles along with a steel-spring float-ing slab track Using the developed model they analyzed

the influences of the thickness length and mass of floating-slab spring rate and its arrangement space running speedand so forth on the time and frequency domain character-istics of steel-spring fulcrum force Ding et al [4 5] alsoanalyzed the vibration parameters of the steel-spring floatingslab track using MIDASGTS Lei and Jiang [6] built themodel of a steel-spring floating slab track using FEM Thedensity the thickness and the length of the floating slab aswell as the isolator stiffness were researched based on modalanalysis and the harmonic response analysis Gu and Zhang[7] established a continuous 3D finite element model forsteel-spring floating slab track using ANSYS The dynamictransfer characteristics and the isolation effect of this kind oftrack structure were studied for different design parametersand actuation frequency Beside theoretical analysis some labtests and in-site tests were performed Liu et al [8] carried outlow-frequency vibration tests on the steel-spring floating slab

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 514612 8 pageshttpdxdoiorg1011552015514612

2 Mathematical Problems in Engineering

Figure 1 The steel-spring floating slab track in Track VibrationAbatement and Control Laboratory

track in Track Vibration Abatement and Control Laboratoryat Beijing Jiaotong University and studied the influence ofthe stiffness and spacing of steel-springs Li et al [9] testedvibration reduction ability of the steel-spring floating slabtrack of Beijing metro line 5 in China

The present contribution aims to optimize vibrationperformance of steel-spring floating slab tracks with four-pole parameter method coupled with ACO This paper isorganized as follows Section 2 describes how to model thesteel-spring floating slab track with four-pole parametermethod Then two optimization objectives are representedin Section 3 In Section 4 optimization variables are decidedA multiobjective ACO was introduced in Section 5 InSection 6 the floating slab track is optimized Conclusionsare in Section 7

2 Mathematical Model

There is one unit of steel-spring floating slab tracks (seeFigure 1) in Track Vibration Abatement and Control Labo-ratory which is the only underground laboratory for trackvibration research in Asia

Figure 2 shows plan and cross section of the steel-springfloating slab trackThe length 119897rail of each rail is 6000mmThedistance between the fasteners is 600mm and the distancebetween the steel-springs is 1200mm The length of thefloating slab 119897119887 is 6000mm the width 119889119887 = 3500mm andthe height ℎ119887 = 450mm

The steel-spring floating slab track was simplified (seeFigure 3)Then four-pole parameter method which is a sim-ple and effective method to solve the dynamic system [10]was employed to model it

According to Figure 3 the floating slab track system canbe divided into four subsystems the rail subsystem the fas-tener subsystem the floating slab subsystem and the steel-spring subsystemThe relations of the subsystems are demon-strated in Figure 4

Therefore the four-pole parameter matrix of the rail sub-system including only one rail is

S1(2)rail = [1 119895120596119898rail

0 1] (1)

The four-pole parametermatrix of the fastener subsystemconsisting of ten fasteners is

S1(2)119891

=[[

[

1 0119895120596

[10 times 119896119891 (1 + 2119895120585119891)]1]]

]

(2)

The four-pole parameter matrix of the floating slab sub-system is

S119887 = [1 119895120596119898119887

0 1] (3)

The four-pole parameter matrix of the steel-spring sub-system having five steel-springs is

S1(2)ss =[[

[

1 0119895120596

[5 times 119896ss (1 + 2119895120585ss)]1]]

]

(4)

where 119895 is the imaginary number 120596 is the angular frequency119898rail is the mass of one rail and119898119887 is the mass of the floatingslab The stiffness of a direction fixation fastener is 119896119891Nmand the damping ratio is 120585119891 The stiffness of a steel-spring is119896ss Nm and the damping ratio is 120585ss

The floating slab track is assumed to be located on aconcrete foundation which is a simply supported slab on fouredges The length of the foundation is 119887119888 the width is 119886119888 andthe height is 119888119888 119906 and V are the equivalent position where thefloating slab track forces The distance between 119906(V) and 119900 is1198871(1198872) Figure 5 gives the details about the foundation

As a consequence the mobility equation of the founda-tion is

1198811

119891

1198812

119891

= Y sdot 1198651

119891

1198652

119891

= [Y11 Y12

Y21 Y22] sdot

1198651

119891

1198652

119891

(5)

in which 1198651(2)119891(119891) is the response force of the foundation at

the position 119906(V) and 1198811(2)119891(119891) is the response velocity of the

foundation at the position 119906(V) Y is the mobility matrixwhich is presented as follows [11]

Y11 = (119895120596

119898119888

) sdot

infin

sum

119898=1

infin

sum

119899=11205932119898119899(1198861198882 minus 1198871 1198871198882)

[1205962119898119899sdot (1 + 119895120578119888) minus 1205962]

Y12 = (119895120596

119898119888

)

sdot

infin

sum

119898=1

infin

sum

119899=1120593119898119899 (1198861198882 minus 1198871 1198871198882) 120593119898119899 (1198861198882 + 1198872 1198871198882)

[1205962119898119899sdot (1 + 119895120578119888) minus 1205962]

Y21 = (119895120596

119898119888

)

sdot

infin

sum

119898=1

infin

sum

119899=1120593119898119899 (1198861198882 + 1198872 1198871198882) 120593119898119899 (1198861198882 minus 1198871 1198871198882)

[1205962119898119899sdot (1 + 119895120578119888) minus 1205962]

Y22 = (119895120596

119898119888

) sdot

infin

sum

119898=1

infin

sum

119899=11205932119898119899(1198861198882 + 1198872 1198871198882)

[1205962119898119899sdot (1 + 119895120578119888) minus 1205962]

(6)

Mathematical Problems in Engineering 3

Steel-spring

6000 6000 6000

6000

6000 6000 6000

3500

2200

(a) Plan

450

35002200

7175 7175

(b) Cross section

Figure 2 Plan and cross section of the steel-spring floating slab track (unit mm)

Steel-spring

Rail

Fastener

Floatingslab

F1rail

F2rail

Figure 3The simplifiedmodel of the steel-spring floating slab track

The rail subsystem

(1)

The rail subsystem

(2)

The fastener subsystem

(1)


(2)

The floating slab subsystem

The steel-spring subsystem (1)


Figure 4 The relations of the subsystems

u

x

y

z

ac

cc

bc

o

F1f

b1

b2F2f

bc2

bc2

Figure 5 The sizes of the foundation

where

(i) 120596119898119899 = 1205872(119898

2119886119888

2+ 119899

2119887119888

2)[119863(120588119888119888119888)]

12 is the res-onance frequency of the foundation the parameter119863 = 119864119888119888119888

3[12(1 minus 120592119888

2)]2 119898119888 is the mass of the

foundation and the angular frequency 120596 = 2120587119891

(ii) 120593119898119899 = 2 sin(119898120587119909119886119888) sin(119899120587119910119887119888) is the normalizedmode shape119898 = 1 2 infin 119899 = 1 2 infin

(iii) the density of the foundation 120588119888 = 2680 kgm3Youngrsquos modulus 119864119888 = 4 times 10

10 Pa the Poisson ratio120592119888 = 03 and the material loss factor 120578119888 = 0025

Suppose that 1198651119891= 1198652

119891 so we can get 1198811

1198911198651

119891= Y11 + Y12

and 11988121198911198652

119891= Y21 + Y22 from (5)

According to Figure 4 we can get the equation

1198651(2)rail

1198811(2)rail

= S

1198651(2)

119891

1198811(2)

119891

= SrailS119891S119887Sss

1198651(2)

119891

1198811(2)

119891

(7)


where the exciting force which is set in the middle of the rail(see Figure 3) 1198651(2)rail (119891) equals 1 at the frequency 119891

By (1)sim(5) and (7) the response force of the foundation atthe position 119906(V) can be written as

1198651(2)

119891(119891) = [1+ 119895120596119898rail sdot

119895120596

(10119896119891 (1 + 2119895120585119891))]

+[119895120596

5 (119896ss (1 + 2119895120585ss))+119872

1(2)]

sdot [119895120596119898119887(1+ 119895120596119898rail sdot119895120596

(10 times 119896119891 (1 + 2119895120585119891)))

+ 119895120596119898rail]

minus1

1198651(2)rail (119891)

(8)

in which the parameters1198721= 119881

1119891119865

1119891and1198722

= 1198812119891119865

2119891

The response velocity of the foundation at the position119906(V) is represented as follows

1198811(2)119891

(119891) = 1198721(2)sdot 119865

1(2)119891 (9)

The response velocity of the rail is

1198811(2)rail (119891) = S21 sdot 119865

1(2)119891

+ S22 sdot 1198811(2)119891 (10)

3 Optimization Objectives

The vibrations induced by the interaction between the wheelsand the rails are transferred from the rails to the foundationand then propagate to the foundations of the nearby buildingswhich will result in structure vibrations and reradiated noiseTherefore the foundation vibration reflects the structurevibration In order to reduce the structure vibration thefoundation vibration should be reduced firstly Consequentlythe foundation acceleration is set as the first optimizationobjective The objective function is

min 119886119891 = 20times log10(1198861015840

119891

1198860

) (11)

The large vibration displacement of the rail usually makesunsafety and instability of the railway operation Thereforethe vibration displacement of the rail is set as the second opti-mization objectiveThe objective function is given as follows

min119889rail = 20times log10 (1198891015840

rail1198890

) (12)

where

(i) the frequency band of interest for subway-inducedvibrations is 1 Hzsim80Hz

The damping ratio of the fastener000 002 004 006 008 010 012 014

Vibr

atio

n le

vel (

dB)

2437

2438

2439

2440

2441

2442

The acceleration of the foundation The displacement of the rail (+5962dB)

Figure 6 The displacement of the rail and the acceleration of thefoundation with 120585119891 when 119896119891 = 5 times 10

7Nm 120585ss = 0125 and 119896ss =5 times 10

6 Nm

(ii) 1198891015840rail = radicsum80119891=1 |119881

1(2)rail (119891)(119895120596)|

280 1198890 = 1 times 10minus12m

(iii) 1198861015840119891= radicsum

80119891=1 |119895120596 sdot 119881

1119891(119891) + 119895120596 sdot 1198812

119891(119891)|280 1198860 = 1 times

10minus6ms2

4 Optimization Variable Decision

According to the description of Sections 2 and 3 the vibrationdisplacement of the rail and the vibration acceleration of thefoundation will be influenced by the physical properties ofthe fasteners and the steel-springs that is the damping ratioof the fastener 120585119891 the stiffness of the fastener 119896119891 the dampingratio of the steel-spring 120585ss and the stiffness of the steel-spring119896ss In order to decide which ones should be set as optimiza-tion variables all physical parameters were analyzed firstly

Considering the application of the fastener and the steel-spring in practice vibration parameters of the floating slabtrack are supposed to be

(i) 0025 le 120585119891 le 025

(ii) 1 times 107 Nm le 119896119891 le 10 times 107Nm

(iii) 0025 le 120585ss le 025(iv) 1 times 106Nm le 119896ss le 10 times 10

6Nm

The calculation codes were built by the commercial soft-ware MATLAB Figures 6ndash9 show the calculation results

From Figures 7 and 8 the variation tendency of the dis-placement of the rail and the acceleration of the foundation isthe same for the stiffness of the fastener and the damping ratioof the steel-spring In other words the displacement of therail and the acceleration of the foundation will decrease withthe increase of the stiffness of the fastener or the dampingratio of the steel-spring So if we want to get the minimumdisplacement of the rail and the minimum acceleration of


The dynamic stiffness of the fastener0

Vibr

atio

n le

vel (

dB)

24

25

26

27

28

29

2e + 7 4e + 7 6e + 7 8e + 7 1e + 8


Figure 7 The displacement of the rail and the acceleration of thefoundation with 119896119891 when 120585119891 = 0125 120585ss = 0125 and 119896ss = 5 times

106 Nm

Vibr

atio

n le

vel (

dB)

The damping ratio of the steel-spring

16

18

20

22

24

26

28

000 002 004 006 008 010 012 014


Figure 8 The displacement of the rail and the acceleration of thefoundation with 120585ss when 120585119891 = 0125 119896119891 = 5 times 10

7 Nm and 119896ss =5 times 10

6Nm

the foundation we only need to make the stiffness of the fas-tener and the damping ratio of the steel-spring maximumFigure 6 shows that the vibration acceleration of the founda-tion will decrease and the rail displacement will increase withthe increase of the damping ratio of the fastener Figure 9 hasthe opposite regulation compared with Figure 6 In a wordthe variation tendency of the objectives (the displacement ofthe rail and the acceleration of the foundation) is in conflictwith the increase of the damping ratio of the fastener or thestiffness of the steel-spring

Therefore the damping ratio of the fastener and the stiff-ness of the steel-spring are set as optimization variables

Vibr

atio

n le

vel (

dB)

5

10

15

20

25

30

35

40

The dynamic stiffness of the steel-spring0 2e + 7 4e + 7 6e + 7 8e + 7 1e + 8



7 Nm and 120585ss =0125

5 Multiobjective ACO

With the development of computer technology swarm intel-ligence optimization algorithms that is Ant Colony Opti-mization (ACO) Particle Swarm Optimization (PSO) andso on have attracted more and more attention ACO is oneof the most successful swarm intelligence optimization algo-rithms It was proposed by Colorni et al [12] to solve theTraveling Salesman Problem (TSP) in 1991 named ant system(AS) which takes inspiration from the foraging behaviors ofthe Argentine ants In recent years ACO has solved manycombinatorial optimization problems of single-objective suc-cessfully [13 14] and is being extended to obtain the solutionsof the continuous problems [15] andmultiobjective problems[16 17]

In this section multiobjective FHACO was employed[18] The core thought of FHACO supposes that every antdeposits two kinds of pheromones that is the food phe-romone and the nest pheromone For this the process of theforaging-homing for ant colonies is described as follows

Each artificial ant randomly starts from the nest to thefood source following the food pheromone and depositingthe nest pheromone on the pathThen they comeback smell-ing the nest pheromone and laying the food pheromone onthe ground

51 Constructing the Initial Pareto Optimal Solution SetAccording to the number of the design variables the corre-sponding number of ant colonies was set Take two designvariables as an example

If the designing variables are 1199091 and 1199092 1198861 le 1199091 le 11988711198862 le 1199092 le 1198872 two groups of ants will be set And each grouphas Num ants which are distributed to the designing space ofthe corresponding variable uniformly


Therefore every ant of the first group has a value

1199091(119894) = 1198861 + (119894 minus 1) times(1198871 minus 1198861)

(Num minus 1) (13)

in which 119894 = 1 2 NumSimilarly each ant of the second group has a value given

as


(Num minus 1) (14)

where 119895 = 1 2 NumAn information exchange mechanism between two

groups of the ants was built upThen Num2 pairs of values ofdesign variables (1199091(119894) 1199092(119895)) are formed 119894 = 1 2 Num119895 = 1 2 Num

Substituting (1199091(119894) 1199092(119895)) to the objective functions theresults of the multiobjective functions are obtained thatis 1198911(1199091(119894) 1199092(119895)) and 1198912(1199091(119894) 1199092(119895)) 119894 = 1 2 Num119895 = 1 2 Num According to the constraint conditionsnonfeasible solutions are excluded Finally the initial Paretooptimal solution set is built by comparison of the dominationrelations of the feasible solutions

52 Updating the Initial Food Pheromone Given that the ini-tial food pheromone 120591food is the same it cannot guide ants tofind the optimal path Consequently wewill update the initialfood pheromone with the initial optimal paths obtained bythe initial Pareto optimal solution set as shown below

120591food larr997888 120591food +119876119888 (15)

where 120591food = 001 is the initial food pheromone of theinitial optimal paths and 119876119888 is the tuning factor Here we let119876119888 = 05

53 The Foraging Process Suppose that 119891(119909) is the originalobjective function and 119909 is the original design variable whoseminimum value is 119886 and the maximum value is 119887 Let 119891(119909) be119891(1199091015840) by simple mathematical transform in which 1199091015840 = (119909 minus

119886)(119887 minus 119886) 1199091015840 isin [0 1] Thus the process of 119891(119909) optimizationis the transformed process of 119891(1199091015840) optimization

Considering the searching space of 1199091015840 the function 119891(1199091015840)optimization process is simplified as an artificial ant makesa selection of ten decimal numbers whenever it takes a stepexcept the first floor and the last floor (see Figure 8) When|1199091015840minus 1| le 120576 we let 1199091015840 be equal to 1 in which 120576 is a

minimum value (1times1minus119889) According to the complexity of theoptimization problem the number of floors is set as 119889

Each artificial ant goes from the first floor (the nest)toward the last floor (the feeding source) 119879(119895 119896 minus 1) is thedecimal number when the ant 119895 is at the floor (119896 minus 1) The ant119895 selects the decimal number of the next floor according to

119879 (119895 119896) =

max 120591119896foodminus(119879(119895119896minus1)119879(119895119896)) 119902 lt 1198760

randperm 119902 ge 1198760

(16)

in which 119902 is a real random variable uniformly distributedin the internal [0 1] 1198760 is a tunable parameter controlling

90 8

0

0 91 8

0 91 8

0 91 8

0

The first floor (the nest)

The last floor(the feeding source)

1

Deposit

Follow

kth floor

(k minus 1)th floor

middot middot middot




120591kfoodminus(T(jkminus1)T(jk))

120591kminus1nestminus(T(jk)T(jkminus1))

Figure 10The foraging process to search the optimal results by ants

the influence of the pheromone randperm is a numberselected randomly from (0 1 9) and 120591119896foodminus(119879(119895119896minus1)119879(119895119896))is the intensity of the food pheromone laid on the pathbetween the number 119879(119895 119896 minus 1) and the number 119879(119895 119896)

When an ant finishes one step that is an ant arrives atthe floor 119896 from the floor (119896 minus 1) the strength of the nestpheromone 120591119896minus1nestminus(119879(119895119896)119879(119895119896minus1)) laid on the path between thenumber119879(119895 119896) and the number119879(119895 119896minus1) by the ant 119895 shouldbe updated as follows

120591119896minus1nestminus(119879(119895119896)119879(119895119896minus1)) larr997888 (1minus120588nest) times 120591

119896minus1nestminus(119879(119895119896)119879(119895119896minus1))

+120572

(17)

where 120572 is the modified coefficient of the intensity of the nestpheromone and 120588nest is the local evaporation factor of the nestpheromone (see Figure 10)

54 The Homing Process After all artificial ants arrive at thefood source each ant comes back smelling the nest phe-romone laid in the foraging processThemethod to select thedecimal number of the (119896 minus 1)th floor when the ant 119895 reachesat the floor 119896 is described as follows

119879 (119895 119896 minus 1) =

max 120591119896minus1nestminus(119879(119895119896)119879(119895119896minus1)) 119902 lt 1198760

randperm 119902 ge 1198760(18)

When the ant 119895 arrives at the position 119879(119895 119896 minus 1) ofthe floor (119896 minus 1) the intensity of the food pheromone120591119896

foodminus(119879(119895119896)119879(119895119896minus1)) laid on the path between the number119879(119895 119896) and the number 119879(119895 119896 minus 1) also should be updated

120591119896

foodminus(119879(119895119896)119879(119895119896minus1)) larr997888 (1minus120588food)

times 120591119896

foodminus(119879(119895119896)119879(119895119896minus1)) +120573(19)

in which 120573 is a constant to modify the intensity of the foodpheromone locally and 120588food is the local evaporation factor ofthe food pheromone


80 82 84 86 88 90 92 94 96 98 1005

10

15

20

25

30

35

a f(d

B)

drail (dB)

Figure 11 The Pareto optimal frontier of biobjective optimization

55 Updating the ParetoOptimal Solution Set After one itera-tion every ant gets its own solution of themultiobjective opti-mization problem In accordance with the constraint func-tions and the domination relations we can make the Paretooptimal solutions and the Pareto optimal frontier be updated

6 Biobjective Optimization

The parameters of the multiobjective ACO are set as followsthe floors 119889 are 3 the ant population 119899 is 20 the iterationnumber is 20 1198760 is 03 the local evaporation factor of thenest pheromone 120588nest is 05 the modified coefficient of thenest pheromone 120572 is 05 the local evaporation factor of thefood pheromone 120588food is 05 and the modified coefficient ofthe food pheromone 120573 is 05

Using the program of the mathematical model of thesteel-spring floating slab track coupled with the multiobjec-tive ACO we can obtain the Pareto optimal frontier (seeFigure 11) and the Pareto optimal solution set for the problemproposed in Section 4 In many engineering problems thedesigners only need to know the best optimum (lying on thePareto optimal frontier) in the preferred zone [19] In otherwords we can get the correspondent designing variables fromFigure 11 for the best objective values for the request of theengineering

For example if the planed 119889rail and 119886119891 are located in theblack box consequently we can obtain the correspondingvalues of the damping ratio of the fastener and the stiffnessof the steel-spring

7 Conclusions

In conclusion we modeled the floating slab track resting onthe steel-springs by the four-pole parameter method Withthis method the influence of four vibration parameters thatis the damping ratio of the fastener the stiffness of the fas-tener the damping ratio of the steel-spring and the stiffnessof the steel-spring for the displacement of the rail and theacceleration of the foundation was researched Results show

that the variation tendency of the displacement of the railand the acceleration of the foundation is the same for thestiffness of the fastener and the damping ratio of the steel-spring However they have the opposite variation tendencyfor the damping ratio of the fastener and the stiffness of thesteel-spring

In order to optimize the rail displacement and the foun-dation acceleration affected by the fastener damping ratio andthe steel-spring stiffness at the same time a multiobjectiveACO was employed Eventually Pareto optimal frontier ofthe rail displacement and the foundation acceleration wasderived Furthermore the desirable values of the fastenerdamping ratio and the steel-spring stiffness can be obtainedaccording to the corresponding Pareto optimal solution set

Conflict of Interests

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

Acknowledgment

This project was supported by National Natural ScienceFoundation of China 51478353

References

[1] W Zhai P Xu and K Wei ldquoAnalysis of vibration reductioncharacteristics and applicability of steel-spring floating-slabtrackrdquo Journal of Modern Transportation vol 19 no 4 pp 215ndash222 2011

[2] KWei S ZhouW Zhai J Xiao andM Zhou ldquoThe influencingfactors of natural frequency optimization for steel-spring float-ing slab track in themetro-building integrated structurerdquoChinaRailway Science vol 32 no 4 pp 8ndash13 2011

[3] KWei S ZhouW Zhai J Xiao andM Zhou ldquoApplicability ofsteel-spring floating-slab tracks of different natural frequenciesfor integrated metro-and-building structuresrdquo Tumu Gong-cheng XuebaoChina Civil Engineering Journal vol 44 no 11pp 134ndash142 2011

[4] D-Y Ding W-N Liu B-C Zhang and X-J Sun ldquoModalanalysis on the floating slab trackrdquo Journal of the China RailwaySociety vol 30 no 3 pp 61ndash64 2008

[5] D DingW Liu K Li and X Sun ldquoParametric study of the steelspring floating slab trackrdquo China Railway Science vol 32 no 1pp 30ndash47 2011

[6] X Lei and C Jiang ldquoAnalysis of vibration reduction effect ofsteel spring floating slab track with finite elementsrdquo Journal ofVibration and Control 2014

[7] A Gu and H Zhang ldquoAnalyses of vibration isolation effect indifferent frequency band for steel-spring floating slab trackrdquoNoise and Vibration Control vol 29 no 1 pp 39ndash42 2009

[8] W LiuDDing K Li andH Zhang ldquoExperimental study of thelow-frequency vibration characteristics of steel spring floatingslab trackrdquo Tumu Gongcheng XuebaoChina Civil EngineeringJournal vol 44 no 8 pp 118ndash125 2011

[9] K Li W Liu X Sun D Ding and Y Yuan ldquoIn-site test ofvibration attenuation of underground line of Beijing metro line5rdquo Journal of the ChinaRailway Society vol 33 no 4 pp 112ndash1182011


[10] H Jin and W-N Liu ldquoVibration reduction optimization ofladder track based on an ant colony algorithmrdquo Journal of Cen-tral South University (Science and Technology) vol 43 no 7 pp2751ndash2756 2012

[11] Z Cao Vibration Theory of Plates and Shells China RailwayPublishing House Beijing China 1983

[12] A ColorniMDorigo andVManiezzo ldquoDistributed optimiza-tion by ant coloniesrdquo in Proceedings of the 1st European Confer-ence on Artificial Life (ECAL rsquo91) Paris France December 1991

[13] C Blum ldquoAnt colony optimization introduction and recenttrendsrdquo Physics of Life Reviews vol 2 no 4 pp 353ndash373 2005

[14] B Chandra Mohan and R Baskaran ldquoA survey ant colonyoptimization based recent research and implementation onseveral engineering domainrdquo Expert Systems with Applicationsvol 39 no 4 pp 4618ndash4627 2012

[15] F A C Viana G I Kotinda D A Rade and V Steffen JrldquoTuning dynamic vibration absorbers by using ant colony opti-mizationrdquo Computers amp Structures vol 86 no 13-14 pp 1539ndash1549 2008

[16] M Chica O Cordon S Damas and J Bautista ldquoIncludingdifferent kinds of preferences in amulti-objective ant algorithmfor time and space assembly line balancing on different Nissanscenariosrdquo Expert Systems with Applications vol 38 no 1 pp709ndash720 2011

[17] B Yagmahan ldquoMixed-model assembly line balancing using amulti-objective ant colony optimization approachrdquo Expert Sys-tems with Applications vol 38 no 10 pp 12453ndash12461 2011

[18] H Jin andW Liu ldquoMulti-objective function optimization basedon ant colony algorithm with foraging-homing mechanismrdquoApplication Research of Computers vol 29 no 11 pp 4038ndash4040 2012

[19] S K Chaharsooghi and A H Meimand Kermani ldquoAn effectiveant colony optimization algorithm (ACO) for multi-objectiveresource allocation problem (MORAP)rdquo Applied Mathematicsand Computation vol 200 no 1 pp 167ndash177 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Figure 1 The steel-spring floating slab track in Track VibrationAbatement and Control Laboratory

track in Track Vibration Abatement and Control Laboratoryat Beijing Jiaotong University and studied the influence ofthe stiffness and spacing of steel-springs Li et al [9] testedvibration reduction ability of the steel-spring floating slabtrack of Beijing metro line 5 in China

The present contribution aims to optimize vibrationperformance of steel-spring floating slab tracks with four-pole parameter method coupled with ACO This paper isorganized as follows Section 2 describes how to model thesteel-spring floating slab track with four-pole parametermethod Then two optimization objectives are representedin Section 3 In Section 4 optimization variables are decidedA multiobjective ACO was introduced in Section 5 InSection 6 the floating slab track is optimized Conclusionsare in Section 7

2 Mathematical Model

There is one unit of steel-spring floating slab tracks (seeFigure 1) in Track Vibration Abatement and Control Labo-ratory which is the only underground laboratory for trackvibration research in Asia

Figure 2 shows plan and cross section of the steel-springfloating slab trackThe length 119897rail of each rail is 6000mmThedistance between the fasteners is 600mm and the distancebetween the steel-springs is 1200mm The length of thefloating slab 119897119887 is 6000mm the width 119889119887 = 3500mm andthe height ℎ119887 = 450mm

The steel-spring floating slab track was simplified (seeFigure 3)Then four-pole parameter method which is a sim-ple and effective method to solve the dynamic system [10]was employed to model it

According to Figure 3 the floating slab track system canbe divided into four subsystems the rail subsystem the fas-tener subsystem the floating slab subsystem and the steel-spring subsystemThe relations of the subsystems are demon-strated in Figure 4

Therefore the four-pole parameter matrix of the rail sub-system including only one rail is

S1(2)rail = [1 119895120596119898rail

0 1] (1)

The four-pole parametermatrix of the fastener subsystemconsisting of ten fasteners is

S1(2)119891

=[[

[

1 0119895120596

[10 times 119896119891 (1 + 2119895120585119891)]1]]

]

(2)

The four-pole parameter matrix of the floating slab sub-system is

S119887 = [1 119895120596119898119887

0 1] (3)

The four-pole parameter matrix of the steel-spring sub-system having five steel-springs is

S1(2)ss =[[

[

1 0119895120596

[5 times 119896ss (1 + 2119895120585ss)]1]]

]

(4)

where 119895 is the imaginary number 120596 is the angular frequency119898rail is the mass of one rail and119898119887 is the mass of the floatingslab The stiffness of a direction fixation fastener is 119896119891Nmand the damping ratio is 120585119891 The stiffness of a steel-spring is119896ss Nm and the damping ratio is 120585ss

The floating slab track is assumed to be located on aconcrete foundation which is a simply supported slab on fouredges The length of the foundation is 119887119888 the width is 119886119888 andthe height is 119888119888 119906 and V are the equivalent position where thefloating slab track forces The distance between 119906(V) and 119900 is1198871(1198872) Figure 5 gives the details about the foundation

As a consequence the mobility equation of the founda-tion is

1198811

119891

1198812

119891

= Y sdot 1198651

119891

1198652

119891

= [Y11 Y12

Y21 Y22] sdot

1198651

119891

1198652

119891

(5)

in which 1198651(2)119891(119891) is the response force of the foundation at

the position 119906(V) and 1198811(2)119891(119891) is the response velocity of the

foundation at the position 119906(V) Y is the mobility matrixwhich is presented as follows [11]

Y11 = (119895120596

119898119888

) sdot

infin

sum

119898=1

infin

sum

119899=11205932119898119899(1198861198882 minus 1198871 1198871198882)

[1205962119898119899sdot (1 + 119895120578119888) minus 1205962]

Y12 = (119895120596

119898119888

)

sdot

infin

sum

119898=1

infin

sum

119899=1120593119898119899 (1198861198882 minus 1198871 1198871198882) 120593119898119899 (1198861198882 + 1198872 1198871198882)

[1205962119898119899sdot (1 + 119895120578119888) minus 1205962]

Y21 = (119895120596

119898119888

)

sdot

infin

sum

119898=1

infin

sum

119899=1120593119898119899 (1198861198882 + 1198872 1198871198882) 120593119898119899 (1198861198882 minus 1198871 1198871198882)

[1205962119898119899sdot (1 + 119895120578119888) minus 1205962]

Y22 = (119895120596

119898119888

) sdot

infin

sum

119898=1

infin

sum

119899=11205932119898119899(1198861198882 + 1198872 1198871198882)

[1205962119898119899sdot (1 + 119895120578119888) minus 1205962]

(6)


Steel-spring

6000 6000 6000

6000

6000 6000 6000

3500

2200

(a) Plan

450

35002200

7175 7175

(b) Cross section


Steel-spring

Rail

Fastener

Floatingslab

F1rail

F2rail


The rail subsystem

(1)

The rail subsystem

(2)


(1)


(2)





u

x

y

z

ac

cc

bc

o

F1f

b1

b2F2f

bc2

bc2


where

(i) 120596119898119899 = 1205872(119898

2119886119888

2+ 119899

2119887119888

2)[119863(120588119888119888119888)]


3[12(1 minus 120592119888

2)]2 119898119888 is the mass of the





Suppose that 1198651119891= 1198652

119891 so we can get 1198811

1198911198651

119891= Y11 + Y12

and 11988121198911198652

119891= Y21 + Y22 from (5)


1198651(2)rail

1198811(2)rail

= S

1198651(2)

119891

1198811(2)

119891


1198651(2)

119891

1198811(2)

119891

(7)




1198651(2)

119891(119891) = [1+ 119895120596119898rail sdot

119895120596

(10119896119891 (1 + 2119895120585119891))]

+[119895120596

5 (119896ss (1 + 2119895120585ss))+119872

1(2)]

sdot [119895120596119898119887(1+ 119895120596119898rail sdot119895120596

(10 times 119896119891 (1 + 2119895120585119891)))

+ 119895120596119898rail]

minus1

1198651(2)rail (119891)

(8)


1119891119865

1119891and1198722

= 1198812119891119865

2119891


1198811(2)119891

(119891) = 1198721(2)sdot 119865

1(2)119891 (9)


1198811(2)rail (119891) = S21 sdot 119865

1(2)119891

+ S22 sdot 1198811(2)119891 (10)



min 119886119891 = 20times log10(1198861015840

119891

1198860

) (11)



rail1198890

) (12)

where



Vibr

atio

n le

vel (

dB)

2437

2438

2439

2440

2441

2442



7Nm 120585ss = 0125 and 119896ss =5 times 10

6 Nm

(ii) 1198891015840rail = radicsum80119891=1 |119881

1(2)rail (119891)(119895120596)|

280 1198890 = 1 times 10minus12m

(iii) 1198861015840119891= radicsum

80119891=1 |119895120596 sdot 119881

1119891(119891) + 119895120596 sdot 1198812

119891(119891)|280 1198860 = 1 times

10minus6ms2




(i) 0025 le 120585119891 le 025



6Nm





Vibr

atio

n le

vel (

dB)

24

25

26

27

28

29

2e + 7 4e + 7 6e + 7 8e + 7 1e + 8



106 Nm

Vibr

atio

n le

vel (

dB)


16

18

20

22

24

26

28

000 002 004 006 008 010 012 014




6Nm



Vibr

atio

n le

vel (

dB)

5

10

15

20

25

30

35

40




7 Nm and 120585ss =0125










(Num minus 1) (13)


as


(Num minus 1) (14)





120591food larr997888 120591food +119876119888 (15)







119879 (119895 119896) =


randperm 119902 ge 1198760

(16)


90 8

0

0 91 8

0 91 8

0 91 8

0



1

Deposit

Follow

kth floor

(k minus 1)th floor












+120572

(17)



119879 (119895 119896 minus 1) =


randperm 119902 ge 1198760(18)



120591119896


times 120591119896

foodminus(119879(119895119896)119879(119895119896minus1)) +120573(19)



80 82 84 86 88 90 92 94 96 98 1005

10

15

20

25

30

35

a f(d

B)

drail (dB)







7 Conclusions






Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Steel-spring

6000 6000 6000

6000

6000 6000 6000

3500

2200

(a) Plan

450

35002200

7175 7175

(b) Cross section


Steel-spring

Rail

Fastener

Floatingslab

F1rail

F2rail


The rail subsystem

(1)

The rail subsystem

(2)


(1)


(2)





u

x

y

z

ac

cc

bc

o

F1f

b1

b2F2f

bc2

bc2


where

(i) 120596119898119899 = 1205872(119898

2119886119888

2+ 119899

2119887119888

2)[119863(120588119888119888119888)]


3[12(1 minus 120592119888

2)]2 119898119888 is the mass of the





Suppose that 1198651119891= 1198652

119891 so we can get 1198811

1198911198651

119891= Y11 + Y12

and 11988121198911198652

119891= Y21 + Y22 from (5)


1198651(2)rail

1198811(2)rail

= S

1198651(2)

119891

1198811(2)

119891


1198651(2)

119891

1198811(2)

119891

(7)




1198651(2)

119891(119891) = [1+ 119895120596119898rail sdot

119895120596

(10119896119891 (1 + 2119895120585119891))]

+[119895120596

5 (119896ss (1 + 2119895120585ss))+119872

1(2)]

sdot [119895120596119898119887(1+ 119895120596119898rail sdot119895120596

(10 times 119896119891 (1 + 2119895120585119891)))

+ 119895120596119898rail]

minus1

1198651(2)rail (119891)

(8)


1119891119865

1119891and1198722

= 1198812119891119865

2119891


1198811(2)119891

(119891) = 1198721(2)sdot 119865

1(2)119891 (9)


1198811(2)rail (119891) = S21 sdot 119865

1(2)119891

+ S22 sdot 1198811(2)119891 (10)



min 119886119891 = 20times log10(1198861015840

119891

1198860

) (11)



rail1198890

) (12)

where



Vibr

atio

n le

vel (

dB)

2437

2438

2439

2440

2441

2442



7Nm 120585ss = 0125 and 119896ss =5 times 10

6 Nm

(ii) 1198891015840rail = radicsum80119891=1 |119881

1(2)rail (119891)(119895120596)|

280 1198890 = 1 times 10minus12m

(iii) 1198861015840119891= radicsum

80119891=1 |119895120596 sdot 119881

1119891(119891) + 119895120596 sdot 1198812

119891(119891)|280 1198860 = 1 times

10minus6ms2




(i) 0025 le 120585119891 le 025



6Nm





Vibr

atio

n le

vel (

dB)

24

25

26

27

28

29

2e + 7 4e + 7 6e + 7 8e + 7 1e + 8



106 Nm

Vibr

atio

n le

vel (

dB)


16

18

20

22

24

26

28

000 002 004 006 008 010 012 014




6Nm



Vibr

atio

n le

vel (

dB)

5

10

15

20

25

30

35

40




7 Nm and 120585ss =0125










(Num minus 1) (13)


as


(Num minus 1) (14)





120591food larr997888 120591food +119876119888 (15)







119879 (119895 119896) =


randperm 119902 ge 1198760

(16)


90 8

0

0 91 8

0 91 8

0 91 8

0



1

Deposit

Follow

kth floor

(k minus 1)th floor












+120572

(17)



119879 (119895 119896 minus 1) =


randperm 119902 ge 1198760(18)



120591119896


times 120591119896

foodminus(119879(119895119896)119879(119895119896minus1)) +120573(19)



80 82 84 86 88 90 92 94 96 98 1005

10

15

20

25

30

35

a f(d

B)

drail (dB)







7 Conclusions






Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198651(2)

119891(119891) = [1+ 119895120596119898rail sdot

119895120596

(10119896119891 (1 + 2119895120585119891))]

+[119895120596

5 (119896ss (1 + 2119895120585ss))+119872

1(2)]

sdot [119895120596119898119887(1+ 119895120596119898rail sdot119895120596

(10 times 119896119891 (1 + 2119895120585119891)))

+ 119895120596119898rail]

minus1

1198651(2)rail (119891)

(8)


1119891119865

1119891and1198722

= 1198812119891119865

2119891


1198811(2)119891

(119891) = 1198721(2)sdot 119865

1(2)119891 (9)


1198811(2)rail (119891) = S21 sdot 119865

1(2)119891

+ S22 sdot 1198811(2)119891 (10)



min 119886119891 = 20times log10(1198861015840

119891

1198860

) (11)



rail1198890

) (12)

where



Vibr

atio

n le

vel (

dB)

2437

2438

2439

2440

2441

2442



7Nm 120585ss = 0125 and 119896ss =5 times 10

6 Nm

(ii) 1198891015840rail = radicsum80119891=1 |119881

1(2)rail (119891)(119895120596)|

280 1198890 = 1 times 10minus12m

(iii) 1198861015840119891= radicsum

80119891=1 |119895120596 sdot 119881

1119891(119891) + 119895120596 sdot 1198812

119891(119891)|280 1198860 = 1 times

10minus6ms2




(i) 0025 le 120585119891 le 025



6Nm





Vibr

atio

n le

vel (

dB)

24

25

26

27

28

29

2e + 7 4e + 7 6e + 7 8e + 7 1e + 8



106 Nm

Vibr

atio

n le

vel (

dB)


16

18

20

22

24

26

28

000 002 004 006 008 010 012 014




6Nm



Vibr

atio

n le

vel (

dB)

5

10

15

20

25

30

35

40




7 Nm and 120585ss =0125










(Num minus 1) (13)


as


(Num minus 1) (14)





120591food larr997888 120591food +119876119888 (15)







119879 (119895 119896) =


randperm 119902 ge 1198760

(16)


90 8

0

0 91 8

0 91 8

0 91 8

0



1

Deposit

Follow

kth floor

(k minus 1)th floor












+120572

(17)



119879 (119895 119896 minus 1) =


randperm 119902 ge 1198760(18)



120591119896


times 120591119896

foodminus(119879(119895119896)119879(119895119896minus1)) +120573(19)



80 82 84 86 88 90 92 94 96 98 1005

10

15

20

25

30

35

a f(d

B)

drail (dB)







7 Conclusions






Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Vibr

atio

n le

vel (

dB)

24

25

26

27

28

29

2e + 7 4e + 7 6e + 7 8e + 7 1e + 8



106 Nm

Vibr

atio

n le

vel (

dB)


16

18

20

22

24

26

28

000 002 004 006 008 010 012 014




6Nm



Vibr

atio

n le

vel (

dB)

5

10

15

20

25

30

35

40




7 Nm and 120585ss =0125










(Num minus 1) (13)


as


(Num minus 1) (14)





120591food larr997888 120591food +119876119888 (15)







119879 (119895 119896) =


randperm 119902 ge 1198760

(16)


90 8

0

0 91 8

0 91 8

0 91 8

0



1

Deposit

Follow

kth floor

(k minus 1)th floor












+120572

(17)



119879 (119895 119896 minus 1) =


randperm 119902 ge 1198760(18)



120591119896


times 120591119896

foodminus(119879(119895119896)119879(119895119896minus1)) +120573(19)



80 82 84 86 88 90 92 94 96 98 1005

10

15

20

25

30

35

a f(d

B)

drail (dB)







7 Conclusions






Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(Num minus 1) (13)


as


(Num minus 1) (14)





120591food larr997888 120591food +119876119888 (15)







119879 (119895 119896) =


randperm 119902 ge 1198760

(16)


90 8

0

0 91 8

0 91 8

0 91 8

0



1

Deposit

Follow

kth floor

(k minus 1)th floor












+120572

(17)



119879 (119895 119896 minus 1) =


randperm 119902 ge 1198760(18)



120591119896


times 120591119896

foodminus(119879(119895119896)119879(119895119896minus1)) +120573(19)



80 82 84 86 88 90 92 94 96 98 1005

10

15

20

25

30

35

a f(d

B)

drail (dB)







7 Conclusions






Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










80 82 84 86 88 90 92 94 96 98 1005

10

15

20

25

30

35

a f(d

B)

drail (dB)







7 Conclusions






Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article biobjective optimization of vibration

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