Research Article Optimal Design of Liquid Dampers...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 191279 10 pageshttpdxdoiorg1011552013191279

Research ArticleOptimal Design of Liquid Dampers for Structural VibrationControl Based on GA and119867

infinNorm

Linsheng Huo Wenhe Shen Hongnan Li and Yaowen Zhang

State Key Laboratory of Coastal and Offshore Engineering Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Linsheng Huo lshuodluteducn

Received 4 July 2013 Accepted 7 October 2013

Academic Editor Gang Li

Copyright copy 2013 Linsheng Huo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper focused on the optimal design of liquid dampers for the seismic response control of structures The 119867infin

norm ofthe transfer function from the ground motion to the structural response is selected as the optimal objective The optimizationprocedure is carried out by using Genetic Algorithms (GAs) in order to reach an optimal solution The proposed method has theadvantages that it is unnecessary to solve the equation of motion for the control system and that the obtained optimal parametersof dampers are not dependent on the ground motion records The influences of weighted functions on the optimization resultsare analyzed The generality and effeteness of the proposed method are verified by the time history analysis of a 3-story structuresubjected to earthquake records in different sitesThe results show that the structural responses can be effectively reduced subjectedto earthquake excitation at different sites

1 Introduction

The installation of vibration absorbers on tall buildings orother flexible structures can be a successful method forreducing the effects of dynamic excitations such as strongwind or earthquakes which may exceed either serviceabilityor safety criteria Tuned liquid column damper (TLCD) isan effective passive control device by the motion of liquidin a column container A TLCD is a U-shaped tube ofuniform rectangular or circle cross section containing liquidVibration energy is transferred from the structure to theTLCD liquid through the motion of the rigid containerexciting the TLCD liquid And the vibration of a structureis suppressed by a TLCD through the gravitational restoringforce acting on the displaced TLCD liquid and the energy isdissipated by the viscous interaction between the liquid andthe rigid container as well as liquid head loss due to orificesinstalled inside theTLCDcontainerThepotential advantagesof liquid vibration absorbers include low manufacturingand installation costs the ability of the absorbers to beincorporated during the design stage of a structure or to beretrofitted to serve a remedial role relatively lowmaintenancerequirements and the availability of the liquid to be used

for emergency purposes or for the everyday function of thestructure if fresh water is used [1 2]

Analytical and experimental research works on this typeof vibration reduction approach have been conducted inwhich viscous interaction between a liquid and solid bound-ary has been investigated and used to control vibration [34] Their experiments defining the relationship between thecoefficient of liquid head loss (as well as its dependence on theorifice opening ratio) and the liquid damping confirm thevalidity of their proposed equation of motion in describingliquid column relative motion under moderate excitation Avariation of TLCD called a liquid column vibration absorber(LCVA) has also been investigated which has differentcross-sectional areas in its vertical and horizontal sectionsdepending on performance requirements [5ndash8] Yan andLi presented the adjustable frequency tuned liquid columndamper by adding springs to the TLCD system which mod-ified the frequency of TLCD and expended its applicationranges [9] Gao et al analyzed the characteristics of multipleliquid column dampers (MTLCD) [10] It was found that thefrequency of range and the coefficient of liquid head losshave significant effects on the performance of a MTLCDincreasing the number of TLCD can enhance the efficiency of

2 Mathematical Problems in Engineering

Orifice

B

h

H

Figure 1 Configuration of TLCD

MTLCD but no further significant enhancement is observedwhen the number of TLCD is over five It was also confirmedthat the sensitivity of an optimized MTLCD to its centralfrequency ratio is not much less than that of an optimizedsingle TLCD to its frequency ratio and an optimizedMTLCDis even more sensitive to the coefficient of head loss

How to decide the parameters and location of dampersin structures to achieve the optimal response reduction isstill an open problem currently One of the commonly usedmethods for the optimization of the damper parameters isto set the objective function to be the stochastical responseof structures based on the seismic frequency spectrum inputand the optimal parameters can be achieved by GeneticAlgorithms (GAs) Another commonly used method of opti-mization is to set the objective function to be the structuralresponses or their combination in time domain based on theactual earthquake records The shortcoming for the abovetwo methods is that the equation of motion of the structuralsystem has to be solved in every optimization step which istime consuming Besides the optimal parameters acquiredby the above methods are related to specific earthquakesand may not be optimal in some earthquakes Hence itis necessary to find a more general and effective methodto optimize the parameters of liquid dampers to reducestructural vibration The active control theory has been usedto optimize the parameters of passive control systems anda lot of results have been achieved [11ndash22] Based on 119867

infin

norm and Genetic Algorithms (GAs) this paper proposedthe optimization method for liquid dampers which has theadvantages that it is unnecessary to solve the equation ofmotion for the system and that the optimized parameters arenot related to specific earthquake records

2 Problem Descriptions

21 Equation of Motion for TLCD The configuration of thetuned liquid column damper (TLCD) is shown in Figure 1According to Lagrange theory the equation of motion forTLCD on the structure subjected earthquake excitation isderived as

120588119860 (2119867 + 119861)ℎ +

1

2

120588119860120585

10038161003816100381610038161003816

ℎ

10038161003816100381610038161003816

ℎ + 2120588119860119892ℎ = minus120588119860119861 (

119899+ 119892)

(1)

where 120588 is the density of the liquid ℎ is the relativemovementof the liquid in TLCD 119867 is the static height of the liquid inthe container 119860 is the cross-sectional area of TLCD 119892 is thegravitational acceleration 119861 is the length of the horizontalpart of TLCD 120585 is the coefficient of head loss which can becontrolled by varying the orifice area

119899is the acceleration of

the 119899th floor where TLCD is installed 119892is the acceleration

of ground motionDue to the nonlinear damping of the aforementioned

equation by the equivalently linearization of it (1) can beconverted to

119898119879

ℎ + 119888119879

ℎ + 119896119879ℎ = minus120572

119909119898119879(119899+ 119892) (2)

where119898119879= 120588119860119871

119890is the mass of the liquid in the TLCD 119871

119890=

2119867+119861 is the length of the liquid in the TLCD 119888119879= (120588119860120585|

ℎ|)2

is the equivalent damping of the TLCD and 119896119879

= 2120588119860119892 isthe ldquostiffnessrdquo of the liquid in vibration Hence the circularfrequency of the motion can be written as 120596

119879= radic119896119879119898119879=

radic2119892119871119890120572119909is the shape function of the liquid damperwritten

as 120572119909= 119861119871

119890

22MathematicalModel of the Control System For an 119899-storyshear frame structure the dynamics of the structural buildingwithout TLCD can be described as

119872 + 119862 + 119870119909 = minus119872119868119892 (3)

where 119872 119862 and 119870 are the mass damping and stiffnessmatrices of the structural buildingwith the dimension of 119899times119899respectively 119868 is an identity vectorwith the dimension of 119899times1the relative displacement vector 119909 is defined as

119909 = [1199091

1199092

sdot sdot sdot 119909119899]

119879

(4)

where 119909119894is the relative displacement of the 119894th floor So the

equation of motion for the structural building instrumentedwith TLCD on the top floor as shown in Figure 2 can beformulated as

119872119904119902 + 119862119904119902 + 119870119904119902 = minus119872

119904119868119892 (5)

where

119872119904=

[

[

[

[

1198981

0 0 0

0 d 0 0

0 0 119898119899+ 119898119879

120572119898119879

0 0 120572119898119879

119898119879

]

]

]

]

119870119904=

[

[

[

[

1198961

0 0 0

0 d 0 0

0 0 119896119899

0

0 0 0 119896119879

]

]

]

]

119862119904=

[

[

[

[

1198881

0 0 0

0 d 0 0

0 0 119888119899

0

0 0 0 119888119879

]

]

]

]

119902 = (1199021

sdot sdot sdot 119902119899

119902119899+1

)

119879

= (1199091


ℎ)

119879

(6)

The elements119898119894 119888119894 and 119896

119894in matrices119872

119904 119862119904 and119870

119904are

the mass damping and stiffness of the 119894th floor Equation (5)can be expressed by the state space form as

(119905) = 119860119911 (119905) + 119861119908 (119905)

119910 (119905) = 119862119911 (119905) + 119863119908 (119905)

(7)

Mathematical Problems in Engineering 3

Figure 2 The structure with TLCD

where 119911(119905) and 119910(119905) are respectively the state vector andoutput vector of the system and

119911 (119905) = [

119902 (119905)

119902 (119905)] 119860 = [

0 119868

minus119872minus1

119904119870119904minus119872minus1

119904119862119904

]

119861 = [

0

minus119868] 119908 (119905) =

119892(119905)

(8)

The matrices 119862 and119863 would be determined according tothe optimization object

23119867infinNorm The transfer function of (7) can be expressed

by

119879 (119904) = 119862(119904119868 minus 119860)minus1

119861 + 119863 (9)

The119867infin

norm is defined as

119879 (119904)infin

= sup119908

120590max [119879 (119895119908)] (10)

In another word the119867infinnorm is the maximum singular

value of frequency response of the system In the frequencydomain the119867

infinnorm can be derived by

119879 (119904)infin

= sup119908(119905)2 = 0

1003817100381710038171003817119910 (119905)

10038171003817100381710038172

119908 (119905)2

= sup119908(119905)2=1

[1003817100381710038171003817119910 (119905)

10038171003817100381710038172] (11)

where 119910(119905)2is the 119871

2norm of the output 119910(119905) and can be

determined by the following equation

1003817100381710038171003817119910 (119905)

10038171003817100381710038172

= (int

infin

0

119910119879

(119905) 119910 (119905) 119889119905)

12

(12)

It can be found that 119910(119905)2is the energy measurement of

the output 119910(119905) Similarly the 1198712norm of 119908(119905) is the energy

measurement of the seismic input which can be determinedby

119908 (119905)2= (int

infin

0

119908119879

(119905) 119908 (119905) 119889119905)

12

(13)

If the119867infin

norm of the transfer function 119879(119904) achieves itsminimum the energy measurement of the output will meetits minimum and the control performance of the system willbe in its best state based on119867

infinnorm

3 Selection of the Weighted Function

Theoretically the optimum control based on 119867infin

norm is tomeet the minimum of 119867

infinnorm in the frequency domain

However it is impractical to control the dynamic response ofthe system in thewhole frequency domain due to the propertylimitation of dampers Hence the weighted functions areusually used to regulate the systemrsquos performance in theactive controller design or parameters optimization of passivedampers The advantages of using weighted functions areobvious in controller design First some components of avector signal are usuallymore important than others Secondeach component of the signal may not be measured in thesame units Also we might be primarily interested in thedynamic response of a certain frequency range [23] There-fore some frequency-dependent weights must be chosen toobtain a high performance controller Two weighted func-tions119882

119892(119904) and119882

1(119904) will be used to facilitate the parameter

optimization of liquid dampers in which119882119892(119904) and119882

1(119904) are

used to weigh the seismic excitation and structural dynamicsrespectively However there is no definite conclusion howthe weighted functions will influence the optimization resultsand how to select appropriate weighted functions to get theoptimal results with less computational efforts This paperwill analyze the influences of weighted functions on theoptimal results of liquid dampers

The weighted function119882119892is used to reflect the frequency

content of an earthquakeThe most commonly used stochas-tic model of earthquakes is the Kanai-Tajimi spectrum asshown in the following equation

119878 (120596) = 1198780

[

[

1205964

119892+ 41205962

1198921205772

1198921205962

(1205962minus 1205962

119892)

2

+ 41205962

1198921205772

1198921205962

]

]

(14)

The weighted function119882119892is chosen as the square root of

the Kanai-Tajimi spectrum

119882119892(119904) =

radic1198780(1205962

119892+ 2120577119892120596119892119904)

1199042+ 2120577119892120596119892119904 + 1205962

119892

(15)

The values of 120596119892and 120577119892can be determined according to

the site typesFor the regulated response we are only interested in the

low-frequency responseTherefore theweighted function1198821

can be selected as a low-pass filterThe influences of weightedfunctions on the optimization results will be discussed in thefollowing section

4 Genetic Algorithm

Genetic Algorithm (GA) is a search heuristic that mimics theprocess of natural evolution to generate useful solutions tooptimization and search problems In a genetic algorithm


a population of candidate solutions to an optimizationproblem is evolved toward better solutions Each candidatesolution has a set of properties which can be mutated andaltered The evolution usually starts from a population ofrandomly generated individuals and is an iterative processwith the population in each iteration called a generationIn each generation the fitness of every individual in thepopulation is evaluated the fitness is usually the value ofthe objective function in the optimization problem beingsolved The more fit individuals are stochastically selectedfrom the current population and each individualrsquos propertyis modified to form a new generation The new generation ofcandidate solutions is then used in the next iteration of thealgorithm Commonly the algorithm terminates when eithera maximum number of generations have been produced or asatisfactory fitness level has been reached for the populationA typical genetic algorithm requires a genetic representationof the solution domain and a fitness function to evaluatethe solution domain Once the genetic representation andthe fitness function are defined a GA proceeds to initializea population of solutions and then to improve it throughrepetitive application of the mutation crossover inversionand selection operators The steps of problem solution bygenetic algorithm can be summarized as follows

(1) Determine variables and constraint conditions inother words determine the individualsrsquo form and thesolution space

(2) Define the fitness function The fitness function isdefined as the119867

infinnorm of the controlled system

(3) Find the chromosome coding and decoding methodfor the feasible solution

(4) Define the rules of transforming the value of objectivefunction to individual fitness

(5) Define the genetic operators which are the methodsof selection crossover and mutation to generate thenext generation

(6) Choose the parameters of the genetic algorithmincluding the population size iteration number ter-mination condition selection probability crossoverprobability and mutation probability

(7) Run the genetic algorithm and acquire the individualswith maximum fitness values

5 Numerical Analysis

51 Structural Parameters A 3-story frame structure is usedas a numerical example to verify the significance of weighted

functionsThe values of mass stiffness and dampingmatricesare respectively as follows

119872 =[

[

5

5

5

]

]

times 105 Kg

119862 =[

[

13506 minus05286 0

minus05286 13506 minus05286

0 minus05286 08220

]

]

times 106N sdot sm

119870 =[

[

4 minus2 0

minus2 4 minus2

0 minus2 2

]

]

times 108Nm

(16)

The natural frequencies of the structure without controlare 158Hz 444Hz and 642Hz respectively A TLCD isinstalled on the top floor to control the structural vibration

With the consideration of structural safety optimizationobjective is set to make the119867

infinnorm of the transfer function

of the top floorrsquos displacement minimum Parameters of liq-uid dampers to be optimized include mass ratio 120583 frequencyratio119891and damping ratio119889 of which their physicalmeaningsand optimization ranges are presented as follows

(1) Mass ratio 120583means the ratio of the mass of the liquidin the container to the one of the whole structure Inthe paper the ratio is supposed to be in the range from01 to 3 according to the actual condition

(2) Frequency ratio 119891 is the ratio of the frequency of thedamper to the first frequency of the structure It issupposed to be in the range from 0 to 25

(3) Damping ratio 119889 denotes the ratio of the damperrsquosdamping to its critical one It is supposed to be in therange from 0 to 10

52 First-OrderWeighted Function1198821(119904) For building struc-

tures with multiple degrees of freedom the first few modeswill dominate the dynamic response Generally the high-order modes are usually ignored and only the first fewmodesare used for analysis The low-pass filter can regulate thesystem response in the low-frequency range and can be usedas the weighted function The influence of the first-orderweighted function 119882

1(119904) on the optimization result will be

analyzed in this section The first-order weighted function1198821(119904) can be written as

1198821(119904) =

119896

119904120596119899+ 1

(17)

where 119896 means the system gain and 120596119899is the cut-off

frequency

521 The Influence of 120596119899on Optimization Results The pre-

liminary analysis shows that the frequency contents of the topfloorrsquos displacement are below 40 rads Given the value of 119896to be 02 and 5 and 120596

119899to be 10 rads 30 rads and 40 rads

then a series of frequency response curves of 1198821(119904) can be

obtained as shown in Figures 3 and 4


Frequency (rads)

Mag

nitu

de (d

B)

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10

Figure 3 Frequency response curve of1198821(119896 = 02)

0

5

10

15

Frequency (rads)

Mag

nitu

de (d

B)

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10

minus25

minus20

minus15

minus10

minus5


The parameters of TLCD with the above weighted func-tions are optimized by genetic algorithm and the results areshown in Table 1 in which 120574

infinmeans the 119867

infinnorm of the

system with optimized values It can be seen from Table 1that the values of 120596

119899and 119896 have significant influences on the

optimal mass ratio and frequency ratio of TLCD The fre-quency response curves of the top floorrsquos displacement for thestructure without control and with TLCD control in differentoptimal parameters are shown in Figures 5 and 6 It can be

Table 1 The optimal parameters of TLCD

119896 120596119899

120583 119891 119889 120574infin

0210 0018016 090971 0099155 0007488330 0016761 093214 0099347 0009785440 0013865 094394 0099712 0010142

510 0023371 088797 0098486 01862930 0015675 093541 0096908 02482840 0018858 092501 0099791 024220

Table 2 The optimal parameters of liquid dampers

120596119899

119896 120583 119891 119889 120574infin

10 02 0018016 090971 0099155 0055610 2 0027219 087352 009968 0055410 5 001647 09038 0099394 0055610 10 0019613 090336 009991 00500

found from the figures that the top floorrsquos displacement of thestructure can be suppressed obviously with TLCD controlFor the controlled structure with optimized parameters inconsideration of the weighted function compared with theone without consideration of the weighted function the peakresponse reduces the most when 120596

119899is 10 rads

522 The Influence of 119896 on Optimization Results Given thevalue of 120596

119899as 10 rads and 119896 as 02 2 5 and 10 respectively

the optimal parameters of liquid dampers can be obtainedby GA as shown in Table 2 The corresponding frequencyresponse curves are shown in Figure 7

It can be seen from Table 2 that the variation of 119896 hassignificant influences on the optimal value of the mass ratio120583 and little influence on the frequency ratio 119891 and dampingratio 119889 Figure 7 shows that the structural displacement canbe effectively controlled with optimized TLCD parametersand the best control case is the one optimized with the first-order weighted function of 120596

119899= 10 rads and 119896 = 2

53 Second-Order-Weighted Function The second-orderweighted function can be written as

1198822(119904) =

119896

(119904120596119899)2

+ 2 (120577120596119899) 119904 + 1

(18)

With fixed 119896 = 2 and120596119899= 10 rads the effect of structural

response by different values of 120577 is analyzed in the followingTake 120577 as 01 02 and 05 and then a series of frequencyresponse curves of second-order weighted function can beobtained as shown in Figure 8

With the second-order weighted function the param-eters of the tuned liquid temper are optimized by geneticalgorithm The results are shown in Table 3 and the corre-sponding frequency response curves in are shown Figure 9Table 3 shows that the optimized mass ratio 120583 decreasesand better control effect can be achieved with the increas-ing of 120577 Figure 9 shows that the top floorrsquos response issuppressed effectively with optimized TLCD parameters


Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control

Figure 5 Frequency response curve of top floorrsquos displacement (119896 =

02)

Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


5)


120577 120583 119891 119889 120574infin

01 0028854 092344 0099003 02533902 0027841 089932 0099837 01939605 0011828 092507 0098664 010967

Different weighted functions result in the different optimizedparameters and different dynamic response curves The bestcontrol performance can be achieved with the 120577 of 01

Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

k = 02

k = 2

k = 5

k = 10

101

Wo control

Figure 7 Frequency response curve of top floorrsquos displacement

0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

120577 = 05

120577 = 02

120577 = 01

100 101 102

Figure 8 Frequency response curve of1198822

To achieve the better vibration control effect the use ofthe first-order and second-order weighted function is com-pared as follows The mathematical formulation of the first-order weighted function 119882

1(119904) and second-order weighted

function 1198822(119904) is expressed by 119882

1(119904) = 2((11990410) + 1)

and 1198822(119904) = 2((119904

2

100) + (0210)119904 + 1) The frequencyresponse curves of 119882

1(119904) and 119882

2(119904) are shown in Figure 10

The frequency response curves of top floorrsquos displacementand acceleration with optimized TLCD parameters weightedby 119882

1(119904) and 119882

2(119904) are shown in Figures 11 and 12


Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

120577 = 01 120577 = 02

120577 = 05

101

Wo control


0

20

40

Frequency (rads)

Mag

nitu

de (d

B)

minus80

minus60

minus40

minus20

100 101 102 10310minus1

W2(s)

W1(s)

Figure 10 Frequency response curves of1198821(119904) and119882

2(119904)

It can be seen from the figures that there is little differencefor the control results with optimized TLCD parametersweighted by the first-order weighted function or the second-order one Hence the low-order weighted function should beselected to facilitate the controller design

54 TimeHistory Analysis To verify the control performanceof TLCD with optimized parameters the seismic responsesof the 3-story building are analyzed in the time domainFour earthquake records Kobe El Centro Northridge andHach are used as excitation for different sitesThemass ratio

Frequency (rads)

Mag

nitu

de (d

B)

101

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)


Frequency (rads)

Mag

nitu

de (d

B)

101

minus20

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)

Figure 12 Frequency response curve of top floorrsquos acceleration

frequency and damping ratio of TLCD are 0013865 094394and 0099712 as shown in Table 1 The time histories of topfloorrsquos displacement and acceleration are shown in Figures 1314 15 16 17 18 19 and 20 The detailed values of seismicresponses and reduction ratios of the structure at differentearthquake records are listed in Table 4 From these figuresand the table it can be concluded that the structural responsecan be effectively reduced subjected to earthquake excitationat different sites


0 5 10 15 20

0

1

2

Time (s)

Disp

lace

men

t (cm

)

Control

minus1

minus2

minus3

Wo control

Figure 13 Time history of top floorrsquos displacement (Kobe wave)

0 5 10 15 20

0

05

1

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control

Figure 14 Time history of top floorrsquos displacement (EL Centrowave)

6 Conclusions

The paper proposed a more general and effective methodto optimize the parameters of tuned liquid column dampers(TLCD) based on Genetics Algorithm (GA) and 119867

infinnorm

The weighted functions are necessary in the optimizationprocess For the weighted function of earthquake excitationthe square root of the Kanai-Tajimi spectrum is suggestedFor the weighted function of output the first-order function

05

0 5 10 15 20

0

1

15

2

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control

Figure 15 Time history of top floorrsquos displacement (Northridgewave)

0 5 10 15 20Time (s)

Disp

lace

men

t (cm

)

minus04

minus02

0

02

04

06

ControlWo control

Figure 16 Time history of top floorrsquos displacement (Hach wave)

and second-order one are analyzed with different parametersThe results show that the weighted function of output hasgreat influences on the optimized parameters To facilitatethe parameter optimization the low-order weighted functionof the output is suggested The structure response withoptimized liquid dampers subjected to actual earthquakerecords is analyzed in the time domain The results showthat TLCD with optimized parameters can effectively reduce


Table 4 Reduction ratios of the top story of the structure at different earthquake records

Earthquakerecords

Maximum displacements Maximum accelerationsWithout control

(cm) TLCD control (cm) Reduction ratio()

Without control(g) TLCD control (g) Reduction ratio

()Kobe 209 175 1627 223 187 1614El Centro 098 078 2041 102 085 1667Northridge 173 152 1214 183 160 1257Hach 048 037 2292 048 039 1875

0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control

Figure 17 Time history of top floorrsquos acceleration (Kobe wave)

Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control

Figure 18 Time history of top floorrsquos acceleration (ELCentrowave)

Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control

Figure 19 Time history of top floorrsquos acceleration (Northridgewave)

Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control

Figure 20 Time history of top floorrsquos acceleration (Hach wave)


the seismic responses at different site types which verify thegenerality of the proposed optimization method

Acknowledgment

The authors are grateful for the supports from the NationalNatural Science Foundation ofChina (Grant nos 51261120375and 51121005)

References

[1] P A Hitchcock K C S Kwok R D Watkins and BSamali ldquoCharacteristics of liquid column vibration absorbers(LCVA)mdashIrdquo Engineering Structures vol 19 no 2 pp 126ndash1341997

[2] P A Hitchcock K C S Kwok R D Watkins and BSamali ldquoCharacteristics of liquid column vibration absorbers(LCVA)mdashIIrdquo Engineering Structures vol 19 no 2 pp 135ndash1441997

[3] W L Qu Z Y Li and G Q Li ldquoExperimental study on U typewater tank for tall buildings and high-rise structuresrdquo ChinaJournal of Building Structures vol 14 no 5 pp 37ndash43 1993

[4] F Sakai S Takaeda and T Tamake ldquoTuned liquid columndampermdashnew type device for suppression of building vibra-tionsrdquo in Proceedings of the International Conference on EuropersquosHigh-Rise Building pp 926ndash931 Nanjing China 1989

[5] H Gao K C S Kwok and B Samali ldquoOptimization of tunedliquid column dampersrdquo Engineering Structures vol 19 no 6pp 476ndash486 1997

[6] S Yan H N Li and G Lin ldquoStudies on control parameters ofadjustable tuned liquid column damperrdquo Earthquake Engineer-ing and Engineering Vibration vol 18 no 4 pp 96ndash101 1998

[7] C C Chang and C T Hsu ldquoControl performance of liquidcolumn vibration absorbersrdquo Engineering Structures vol 20 no7 pp 580ndash586 1998

[8] C C Chang ldquoMass dampers and their optimal designs forbuilding vibration controlrdquo Engineering Structures vol 21 no5 pp 454ndash463 1999

[9] S Yan andHN Li ldquoPerformance of vibration control forU typewater tank with variable cross sectionrdquo Earthquake Engineeringand Engineering Vibration vol 19 no 1 pp 197ndash201 1999

[10] H Gao K S C Kwok and B Samali ldquoCharacteristics of mul-tiple tuned liquid column dampers in suppressing structuralvibrationrdquo Engineering Structures vol 21 no 4 pp 316ndash3311999

[11] N Gluck A M Reinhorn J Gluck and R Levy ldquoDesignof supplemental dampers for control of structuresrdquo Journal ofStructural Engineering vol 122 no 12 pp 1394ndash1399 1996

[12] C H Loh P Y Lin and N H Chung ldquoDesign of dampersfor structures based on optimal control theoryrdquo EarthquakeEngineering amp Structural Dynamics vol 29 no 9 pp 1307ndash13232000

[13] A K Agrawal and J N Yang ldquoDesign of passive energydissipation systems based on LQR control methodsrdquo Journal ofIntelligent Material Systems and Structures vol 10 no 12 pp933ndash944 2000


[15] J N Yang S Lin J-H Kim andA K Agrawal ldquoOptimal designof passive energy dissipation systems based on Hinfin and H2performancesrdquo Earthquake Engineering amp Structural Dynamicsvol 31 no 4 pp 921ndash936 2002

[16] G Song P Qiao V Sethi and A Prasad ldquoActive vibrationcontrol of a smart pultruded fiber-reinforced polymer I-beamrdquoSmart Materials and Structures vol 13 no 4 pp 819ndash827 2004

[17] V Sethi M A Franchek and G Song ldquoActive multimodalvibration suppression of a flexible structure with piezoceramicsensor and actuator by using loop shapingrdquo Journal of Vibrationand Control vol 17 no 13 pp 1994ndash2006 2011

[18] J L Meyer W B Harringtont B N Agrawal and G SongldquoVibration suppression of a spacecraft flexible appendage usingsmart materialrdquo Smart Materials and Structures vol 7 no 1 pp95ndash104 1998

[19] V Sethi andG Song ldquoMultimodal vibration control of a flexiblestructure using piezoceramic sensor and actuatorrdquo Journal ofIntelligentMaterial Systems and Structures vol 19 no 5 pp 573ndash582 2008

[20] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexible beamrdquoSmart Materials and Structures vol 17 no 3 Article ID 0350072008

[21] L Y Li G Song and J P Ou ldquoAdaptive fuzzy sliding modebased active vibration control of a smart beam with massuncertaintyrdquo Structural Control ampHealthMonitoring vol 18 no1 pp 40ndash52 2011

[22] HGu andG Song ldquoActive vibration suppression of a compositeI-beam using fuzzy positive position controlrdquo Smart Materialsand Structures vol 14 no 4 pp 540ndash547 2005

[23] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall Upper Saddle River NJ USA 1996

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

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

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


Orifice

B

h

H

Figure 1 Configuration of TLCD

MTLCD but no further significant enhancement is observedwhen the number of TLCD is over five It was also confirmedthat the sensitivity of an optimized MTLCD to its centralfrequency ratio is not much less than that of an optimizedsingle TLCD to its frequency ratio and an optimizedMTLCDis even more sensitive to the coefficient of head loss

How to decide the parameters and location of dampersin structures to achieve the optimal response reduction isstill an open problem currently One of the commonly usedmethods for the optimization of the damper parameters isto set the objective function to be the stochastical responseof structures based on the seismic frequency spectrum inputand the optimal parameters can be achieved by GeneticAlgorithms (GAs) Another commonly used method of opti-mization is to set the objective function to be the structuralresponses or their combination in time domain based on theactual earthquake records The shortcoming for the abovetwo methods is that the equation of motion of the structuralsystem has to be solved in every optimization step which istime consuming Besides the optimal parameters acquiredby the above methods are related to specific earthquakesand may not be optimal in some earthquakes Hence itis necessary to find a more general and effective methodto optimize the parameters of liquid dampers to reducestructural vibration The active control theory has been usedto optimize the parameters of passive control systems anda lot of results have been achieved [11ndash22] Based on 119867

infin

norm and Genetic Algorithms (GAs) this paper proposedthe optimization method for liquid dampers which has theadvantages that it is unnecessary to solve the equation ofmotion for the system and that the optimized parameters arenot related to specific earthquake records

2 Problem Descriptions

21 Equation of Motion for TLCD The configuration of thetuned liquid column damper (TLCD) is shown in Figure 1According to Lagrange theory the equation of motion forTLCD on the structure subjected earthquake excitation isderived as

120588119860 (2119867 + 119861)ℎ +

1

2

120588119860120585

10038161003816100381610038161003816

ℎ

10038161003816100381610038161003816

ℎ + 2120588119860119892ℎ = minus120588119860119861 (

119899+ 119892)

(1)

where 120588 is the density of the liquid ℎ is the relativemovementof the liquid in TLCD 119867 is the static height of the liquid inthe container 119860 is the cross-sectional area of TLCD 119892 is thegravitational acceleration 119861 is the length of the horizontalpart of TLCD 120585 is the coefficient of head loss which can becontrolled by varying the orifice area

119899is the acceleration of

the 119899th floor where TLCD is installed 119892is the acceleration

of ground motionDue to the nonlinear damping of the aforementioned

equation by the equivalently linearization of it (1) can beconverted to

119898119879

ℎ + 119888119879

ℎ + 119896119879ℎ = minus120572

119909119898119879(119899+ 119892) (2)

where119898119879= 120588119860119871

119890is the mass of the liquid in the TLCD 119871

119890=

2119867+119861 is the length of the liquid in the TLCD 119888119879= (120588119860120585|

ℎ|)2

is the equivalent damping of the TLCD and 119896119879

= 2120588119860119892 isthe ldquostiffnessrdquo of the liquid in vibration Hence the circularfrequency of the motion can be written as 120596

119879= radic119896119879119898119879=

radic2119892119871119890120572119909is the shape function of the liquid damperwritten

as 120572119909= 119861119871

119890

22MathematicalModel of the Control System For an 119899-storyshear frame structure the dynamics of the structural buildingwithout TLCD can be described as

119872 + 119862 + 119870119909 = minus119872119868119892 (3)

where 119872 119862 and 119870 are the mass damping and stiffnessmatrices of the structural buildingwith the dimension of 119899times119899respectively 119868 is an identity vectorwith the dimension of 119899times1the relative displacement vector 119909 is defined as

119909 = [1199091

1199092

sdot sdot sdot 119909119899]

119879

(4)

where 119909119894is the relative displacement of the 119894th floor So the

equation of motion for the structural building instrumentedwith TLCD on the top floor as shown in Figure 2 can beformulated as

119872119904119902 + 119862119904119902 + 119870119904119902 = minus119872

119904119868119892 (5)

where

119872119904=

[

[

[

[

1198981

0 0 0

0 d 0 0

0 0 119898119899+ 119898119879

120572119898119879

0 0 120572119898119879

119898119879

]

]

]

]

119870119904=

[

[

[

[

1198961

0 0 0

0 d 0 0

0 0 119896119899

0

0 0 0 119896119879

]

]

]

]

119862119904=

[

[

[

[

1198881

0 0 0

0 d 0 0

0 0 119888119899

0

0 0 0 119888119879

]

]

]

]

119902 = (1199021


119902119899+1

)

119879

= (1199091


ℎ)

119879

(6)

The elements119898119894 119888119894 and 119896

119894in matrices119872

119904 119862119904 and119870

119904are

the mass damping and stiffness of the 119894th floor Equation (5)can be expressed by the state space form as

(119905) = 119860119911 (119905) + 119861119908 (119905)

119910 (119905) = 119862119911 (119905) + 119863119908 (119905)

(7)




119911 (119905) = [

119902 (119905)

119902 (119905)] 119860 = [

0 119868

minus119872minus1

119904119870119904minus119872minus1

119904119862119904

]

119861 = [

0

minus119868] 119908 (119905) =

119892(119905)

(8)



by

119879 (119904) = 119862(119904119868 minus 119860)minus1

119861 + 119863 (9)

The119867infin

norm is defined as

119879 (119904)infin

= sup119908

120590max [119879 (119895119908)] (10)




119879 (119904)infin

= sup119908(119905)2 = 0

1003817100381710038171003817119910 (119905)

10038171003817100381710038172

119908 (119905)2

= sup119908(119905)2=1

[1003817100381710038171003817119910 (119905)

10038171003817100381710038172] (11)

where 119910(119905)2is the 119871



1003817100381710038171003817119910 (119905)

10038171003817100381710038172

= (int

infin

0

119910119879

(119905) 119910 (119905) 119889119905)

12

(12)




119908 (119905)2= (int

infin

0

119908119879

(119905) 119908 (119905) 119889119905)

12

(13)

If the119867infin


infinnorm






119892(119904) and119882



1(119904) are




119878 (120596) = 1198780

[

[

1205964

119892+ 41205962

1198921205772

1198921205962

(1205962minus 1205962

119892)

2

+ 41205962

1198921205772

1198921205962

]

]

(14)



119882119892(119904) =

radic1198780(1205962

119892+ 2120577119892120596119892119904)

1199042+ 2120577119892120596119892119904 + 1205962

119892

(15)





4 Genetic Algorithm















119872 =[

[

5

5

5

]

]

times 105 Kg

119862 =[

[

13506 minus05286 0

minus05286 13506 minus05286

0 minus05286 08220

]

]

times 106N sdot sm

119870 =[

[

4 minus2 0

minus2 4 minus2

0 minus2 2

]

]

times 108Nm

(16)












1198821(119904) =

119896

119904120596119899+ 1

(17)


frequency







Frequency (rads)

Mag

nitu

de (d

B)

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10


0

5

10

15

Frequency (rads)

Mag

nitu

de (d

B)

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10

minus25

minus20

minus15

minus10

minus5




infinnorm of the





119896 120596119899

120583 119891 119889 120574infin

0210 0018016 090971 0099155 0007488330 0016761 093214 0099347 0009785440 0013865 094394 0099712 0010142

510 0023371 088797 0098486 01862930 0015675 093541 0096908 02482840 0018858 092501 0099791 024220


120596119899

119896 120583 119891 119889 120574infin

10 02 0018016 090971 0099155 0055610 2 0027219 087352 009968 0055410 5 001647 09038 0099394 0055610 10 0019613 090336 009991 00500


119899is 10 rads





119899= 10 rads and 119896 = 2


1198822(119904) =

119896

(119904120596119899)2

+ 2 (120577120596119899) 119904 + 1

(18)





Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


02)

Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


5)


120577 120583 119891 119889 120574infin

01 0028854 092344 0099003 02533902 0027841 089932 0099837 01939605 0011828 092507 0098664 010967


Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

k = 02

k = 2

k = 5

k = 10

101

Wo control


0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

120577 = 05

120577 = 02

120577 = 01

100 101 102





1(119904) = 2((11990410) + 1)

and 1198822(119904) = 2((119904

2


1(119904) and 119882



1(119904) and 119882



Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

120577 = 01 120577 = 02

120577 = 05

101

Wo control


0

20

40

Frequency (rads)

Mag

nitu

de (d

B)

minus80

minus60

minus40

minus20

100 101 102 10310minus1

W2(s)

W1(s)


2(119904)



Frequency (rads)

Mag

nitu

de (d

B)

101

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)


Frequency (rads)

Mag

nitu

de (d

B)

101

minus20

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)




0 5 10 15 20

0

1

2

Time (s)

Disp

lace

men

t (cm

)

Control

minus1

minus2

minus3

Wo control


0 5 10 15 20

0

05

1

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


6 Conclusions


infinnorm


05

0 5 10 15 20

0

1

15

2

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


0 5 10 15 20Time (s)

Disp

lace

men

t (cm

)

minus04

minus02

0

02

04

06

ControlWo control





Earthquakerecords





0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control




Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119911 (119905) = [

119902 (119905)

119902 (119905)] 119860 = [

0 119868

minus119872minus1

119904119870119904minus119872minus1

119904119862119904

]

119861 = [

0

minus119868] 119908 (119905) =

119892(119905)

(8)



by

119879 (119904) = 119862(119904119868 minus 119860)minus1

119861 + 119863 (9)

The119867infin

norm is defined as

119879 (119904)infin

= sup119908

120590max [119879 (119895119908)] (10)




119879 (119904)infin

= sup119908(119905)2 = 0

1003817100381710038171003817119910 (119905)

10038171003817100381710038172

119908 (119905)2

= sup119908(119905)2=1

[1003817100381710038171003817119910 (119905)

10038171003817100381710038172] (11)

where 119910(119905)2is the 119871



1003817100381710038171003817119910 (119905)

10038171003817100381710038172

= (int

infin

0

119910119879

(119905) 119910 (119905) 119889119905)

12

(12)




119908 (119905)2= (int

infin

0

119908119879

(119905) 119908 (119905) 119889119905)

12

(13)

If the119867infin


infinnorm






119892(119904) and119882



1(119904) are




119878 (120596) = 1198780

[

[

1205964

119892+ 41205962

1198921205772

1198921205962

(1205962minus 1205962

119892)

2

+ 41205962

1198921205772

1198921205962

]

]

(14)



119882119892(119904) =

radic1198780(1205962

119892+ 2120577119892120596119892119904)

1199042+ 2120577119892120596119892119904 + 1205962

119892

(15)





4 Genetic Algorithm















119872 =[

[

5

5

5

]

]

times 105 Kg

119862 =[

[

13506 minus05286 0

minus05286 13506 minus05286

0 minus05286 08220

]

]

times 106N sdot sm

119870 =[

[

4 minus2 0

minus2 4 minus2

0 minus2 2

]

]

times 108Nm

(16)












1198821(119904) =

119896

119904120596119899+ 1

(17)


frequency







Frequency (rads)

Mag

nitu

de (d

B)

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10


0

5

10

15

Frequency (rads)

Mag

nitu

de (d

B)

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10

minus25

minus20

minus15

minus10

minus5




infinnorm of the





119896 120596119899

120583 119891 119889 120574infin

0210 0018016 090971 0099155 0007488330 0016761 093214 0099347 0009785440 0013865 094394 0099712 0010142

510 0023371 088797 0098486 01862930 0015675 093541 0096908 02482840 0018858 092501 0099791 024220


120596119899

119896 120583 119891 119889 120574infin

10 02 0018016 090971 0099155 0055610 2 0027219 087352 009968 0055410 5 001647 09038 0099394 0055610 10 0019613 090336 009991 00500


119899is 10 rads





119899= 10 rads and 119896 = 2


1198822(119904) =

119896

(119904120596119899)2

+ 2 (120577120596119899) 119904 + 1

(18)





Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


02)

Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


5)


120577 120583 119891 119889 120574infin

01 0028854 092344 0099003 02533902 0027841 089932 0099837 01939605 0011828 092507 0098664 010967


Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

k = 02

k = 2

k = 5

k = 10

101

Wo control


0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

120577 = 05

120577 = 02

120577 = 01

100 101 102





1(119904) = 2((11990410) + 1)

and 1198822(119904) = 2((119904

2


1(119904) and 119882



1(119904) and 119882



Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

120577 = 01 120577 = 02

120577 = 05

101

Wo control


0

20

40

Frequency (rads)

Mag

nitu

de (d

B)

minus80

minus60

minus40

minus20

100 101 102 10310minus1

W2(s)

W1(s)


2(119904)



Frequency (rads)

Mag

nitu

de (d

B)

101

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)


Frequency (rads)

Mag

nitu

de (d

B)

101

minus20

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)




0 5 10 15 20

0

1

2

Time (s)

Disp

lace

men

t (cm

)

Control

minus1

minus2

minus3

Wo control


0 5 10 15 20

0

05

1

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


6 Conclusions


infinnorm


05

0 5 10 15 20

0

1

15

2

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


0 5 10 15 20Time (s)

Disp

lace

men

t (cm

)

minus04

minus02

0

02

04

06

ControlWo control





Earthquakerecords





0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control




Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















119872 =[

[

5

5

5

]

]

times 105 Kg

119862 =[

[

13506 minus05286 0

minus05286 13506 minus05286

0 minus05286 08220

]

]

times 106N sdot sm

119870 =[

[

4 minus2 0

minus2 4 minus2

0 minus2 2

]

]

times 108Nm

(16)












1198821(119904) =

119896

119904120596119899+ 1

(17)


frequency







Frequency (rads)

Mag

nitu

de (d

B)

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10


0

5

10

15

Frequency (rads)

Mag

nitu

de (d

B)

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10

minus25

minus20

minus15

minus10

minus5




infinnorm of the





119896 120596119899

120583 119891 119889 120574infin

0210 0018016 090971 0099155 0007488330 0016761 093214 0099347 0009785440 0013865 094394 0099712 0010142

510 0023371 088797 0098486 01862930 0015675 093541 0096908 02482840 0018858 092501 0099791 024220


120596119899

119896 120583 119891 119889 120574infin

10 02 0018016 090971 0099155 0055610 2 0027219 087352 009968 0055410 5 001647 09038 0099394 0055610 10 0019613 090336 009991 00500


119899is 10 rads





119899= 10 rads and 119896 = 2


1198822(119904) =

119896

(119904120596119899)2

+ 2 (120577120596119899) 119904 + 1

(18)





Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


02)

Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


5)


120577 120583 119891 119889 120574infin

01 0028854 092344 0099003 02533902 0027841 089932 0099837 01939605 0011828 092507 0098664 010967


Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

k = 02

k = 2

k = 5

k = 10

101

Wo control


0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

120577 = 05

120577 = 02

120577 = 01

100 101 102





1(119904) = 2((11990410) + 1)

and 1198822(119904) = 2((119904

2


1(119904) and 119882



1(119904) and 119882



Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

120577 = 01 120577 = 02

120577 = 05

101

Wo control


0

20

40

Frequency (rads)

Mag

nitu

de (d

B)

minus80

minus60

minus40

minus20

100 101 102 10310minus1

W2(s)

W1(s)


2(119904)



Frequency (rads)

Mag

nitu

de (d

B)

101

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)


Frequency (rads)

Mag

nitu

de (d

B)

101

minus20

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)




0 5 10 15 20

0

1

2

Time (s)

Disp

lace

men

t (cm

)

Control

minus1

minus2

minus3

Wo control


0 5 10 15 20

0

05

1

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


6 Conclusions


infinnorm


05

0 5 10 15 20

0

1

15

2

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


0 5 10 15 20Time (s)

Disp

lace

men

t (cm

)

minus04

minus02

0

02

04

06

ControlWo control





Earthquakerecords





0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control




Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Frequency (rads)

Mag

nitu

de (d

B)

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10


0

5

10

15

Frequency (rads)

Mag

nitu

de (d

B)

10minus1 100 101 102 103

120596n = 40

120596n = 30

120596n = 10

minus25

minus20

minus15

minus10

minus5




infinnorm of the





119896 120596119899

120583 119891 119889 120574infin

0210 0018016 090971 0099155 0007488330 0016761 093214 0099347 0009785440 0013865 094394 0099712 0010142

510 0023371 088797 0098486 01862930 0015675 093541 0096908 02482840 0018858 092501 0099791 024220


120596119899

119896 120583 119891 119889 120574infin

10 02 0018016 090971 0099155 0055610 2 0027219 087352 009968 0055410 5 001647 09038 0099394 0055610 10 0019613 090336 009991 00500


119899is 10 rads





119899= 10 rads and 119896 = 2


1198822(119904) =

119896

(119904120596119899)2

+ 2 (120577120596119899) 119904 + 1

(18)





Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


02)

Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


5)


120577 120583 119891 119889 120574infin

01 0028854 092344 0099003 02533902 0027841 089932 0099837 01939605 0011828 092507 0098664 010967


Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

k = 02

k = 2

k = 5

k = 10

101

Wo control


0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

120577 = 05

120577 = 02

120577 = 01

100 101 102





1(119904) = 2((11990410) + 1)

and 1198822(119904) = 2((119904

2


1(119904) and 119882



1(119904) and 119882



Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

120577 = 01 120577 = 02

120577 = 05

101

Wo control


0

20

40

Frequency (rads)

Mag

nitu

de (d

B)

minus80

minus60

minus40

minus20

100 101 102 10310minus1

W2(s)

W1(s)


2(119904)



Frequency (rads)

Mag

nitu

de (d

B)

101

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)


Frequency (rads)

Mag

nitu

de (d

B)

101

minus20

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)




0 5 10 15 20

0

1

2

Time (s)

Disp

lace

men

t (cm

)

Control

minus1

minus2

minus3

Wo control


0 5 10 15 20

0

05

1

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


6 Conclusions


infinnorm


05

0 5 10 15 20

0

1

15

2

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


0 5 10 15 20Time (s)

Disp

lace

men

t (cm

)

minus04

minus02

0

02

04

06

ControlWo control





Earthquakerecords





0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control




Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


02)

Frequency (rads)

Mag

nitu

de (d

B)

101

minus15

minus20

minus25

minus30

minus35

minus40

120596n = 10

120596n = 30

120596n = 40

Wo control


5)


120577 120583 119891 119889 120574infin

01 0028854 092344 0099003 02533902 0027841 089932 0099837 01939605 0011828 092507 0098664 010967


Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

k = 02

k = 2

k = 5

k = 10

101

Wo control


0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

120577 = 05

120577 = 02

120577 = 01

100 101 102





1(119904) = 2((11990410) + 1)

and 1198822(119904) = 2((119904

2


1(119904) and 119882



1(119904) and 119882



Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

120577 = 01 120577 = 02

120577 = 05

101

Wo control


0

20

40

Frequency (rads)

Mag

nitu

de (d

B)

minus80

minus60

minus40

minus20

100 101 102 10310minus1

W2(s)

W1(s)


2(119904)



Frequency (rads)

Mag

nitu

de (d

B)

101

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)


Frequency (rads)

Mag

nitu

de (d

B)

101

minus20

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)




0 5 10 15 20

0

1

2

Time (s)

Disp

lace

men

t (cm

)

Control

minus1

minus2

minus3

Wo control


0 5 10 15 20

0

05

1

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


6 Conclusions


infinnorm


05

0 5 10 15 20

0

1

15

2

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


0 5 10 15 20Time (s)

Disp

lace

men

t (cm

)

minus04

minus02

0

02

04

06

ControlWo control





Earthquakerecords





0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control




Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Frequency (rads)

Mag

nitu

de (d

B)

minus15

minus20

minus25

minus30

minus35

minus40

120577 = 01 120577 = 02

120577 = 05

101

Wo control


0

20

40

Frequency (rads)

Mag

nitu

de (d

B)

minus80

minus60

minus40

minus20

100 101 102 10310minus1

W2(s)

W1(s)


2(119904)



Frequency (rads)

Mag

nitu

de (d

B)

101

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)


Frequency (rads)

Mag

nitu

de (d

B)

101

minus20

minus25

minus30

minus35

minus40

minus45

W1(s)

W2(s)




0 5 10 15 20

0

1

2

Time (s)

Disp

lace

men

t (cm

)

Control

minus1

minus2

minus3

Wo control


0 5 10 15 20

0

05

1

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


6 Conclusions


infinnorm


05

0 5 10 15 20

0

1

15

2

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


0 5 10 15 20Time (s)

Disp

lace

men

t (cm

)

minus04

minus02

0

02

04

06

ControlWo control





Earthquakerecords





0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control




Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20

0

1

2

Time (s)

Disp

lace

men

t (cm

)

Control

minus1

minus2

minus3

Wo control


0 5 10 15 20

0

05

1

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


6 Conclusions


infinnorm


05

0 5 10 15 20

0

1

15

2

Time (s)

Disp

lace

men

t (cm

)

minus1

Control

minus05

Wo control


0 5 10 15 20Time (s)

Disp

lace

men

t (cm

)

minus04

minus02

0

02

04

06

ControlWo control





Earthquakerecords





0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control




Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Earthquakerecords





0 5 10 15 20

0

1

2

3

Time (s)

Acce

lera

tion

(g)

minus2

minus1

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

0

1

minus1

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

05

15

0

1

minus1

minus2

minus05

minus15

ControlWo control


Acce

lera

tion

(g)

0 5 10 15 20Time (s)

minus04

minus02

minus06

0

02

04

ControlWo control




Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Optimal Design of Liquid Dampers...

Documents

Transcript of Research Article Optimal Design of Liquid Dampers...