Post on 06-Dec-2021
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 251935 7 pageshttpdxdoiorg1011552013251935
Research ArticleOptimal Control of Magnetorheological Fluid Dampers forSeismic Isolation of Structures
Ameen H El-Sinawi1 Mohammad H AlHamaydeh2 and Ali A Jhemi1
1 Department of Mechanical Engineering American University of Sharjah Sharjah 26666 United Arab Emirates2 Department of Civil Engineering American University of Sharjah Sharjah 26666 United Arab Emirates
Correspondence should be addressed to Ameen H El-Sinawi aelsinawiausedu
Received 27 February 2013 Accepted 21 April 2013
Academic Editor Chengjin Zhang
Copyright copy 2013 Ameen H El-Sinawi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper presents themodeling and control of amagnetorheological (MR) damper installed in Chevron configuration at the baseof a 20-story benchmark buildingThe building structural model is created using the commercial software package ETABSTheMRdamper model is derived from Bouc-Wen hysteresis model which provides the critical nonlinear dynamics that best represents theMR damper under a wide range of operating conditions System identification is used to derive a low-order nonlinear model thatbest mimics the nonlinear dynamics of the actual MR damper Dynamic behavior of this low-order model is tested and validatedover a range of inputs The damper model has proven its validity to a high degree of accuracy against the nonlinear model AKalman filter is designed to best estimate the state of the structure-damper system for feedback implementation purposes Usingthe estimated states an LQG-based compensator is designed to control the MR damper under earthquake loads To demonstratethe effectiveness of this control strategy four historical earthquakes are applied to the structure Controlled and uncontrolled flooraccelerations and displacements at key locations are compared Results of the optimally controlled model demonstrate superiorperformance in comparison to the uncontrolled model
1 Introduction
Protection of large structures against external disturbancessuch as earthquakes and wind has been a major concern toresearchers for decades Seismic isolation with and withoutsupplemental damping for energy dissipation has proven tobe very effective in protecting civil structures during seismicevents Most classical isolators are of the passive type suchas natural rubber bearings (NRBs) [1ndash3] These isolatorsare capable of providing adequate damping during low-to-moderate earthquakes With high-velocity pulses and highdisplacement demands many Near-Field (NF) situationsrequire impractical isolator bearing dimensions and designsIn such occasions utilizing high-damping rubber (HDR)bearings lead-rubber (LR) bearings or friction pendulumsystem (FPS) alone is not the best engineering solutionTypical FF response of base-isolated structures ismanageablecompared to the high demands of an NF event [4ndash8] The
combined isolation system of HDR or LR bearings withviscous dampers seems to work well in the NF regions wherethe ground shaking characteristics are capable of producingpulses with velocity of 05ndash15msec and durations of 1ndash3 secUnfortunately this combined system does not perform desir-ably in moderate or strong Far-Field (FF) events due to thesecondary forces produced by the dampers and their complexcoupling effects [9] Consequently supplemental dampingis needed to reduce the horizontal displacement demandsotherwise structural integrity could be jeopardized [10] Formoderate-to-severe earthquakes semiactive dampers suchas Magnetorheological (MR) dampers have proven to besignificantly more effective Jangid [11] investigated the opti-mum use of the FPS isolators for NF earthquake motion inmultistory buildings He evaluated the response of the systemto six records of NF earthquakes and derived the optimumfriction coefficient of the FPSThis was performed so that thetop floor acceleration and the total horizontal sliding distance
2 Mathematical Problems in Engineering
are minimized It was concluded that the optimum frictioncoefficient for FPS for NF earthquake motions is in the rangeof 005 to 015
When considering the challenge of limiting the totalmaximum displacement (119863TM) to practical limits especiallyin NF sites somtimes the designer relies on fluid viscousdampers (FVDs) The state of the practice involves carryingout preliminary calculations and analyses using typical HDRbearings These preliminary calculations could readily showwhether or not supplemental damping is required Oncesupplemental damping is deemed necessary many designerswould prefer utilizing the linear behavior of NRB isola-tors combined with the supplemental damping provided byFVDs the use of such a system often results in additionaluniformity in the induced superstructure story forces Thissystem has also been used in many projects in the USA [12ndash19] Recently AlHamaydeh et al [20] developed simplifieddesign equations for seismic isolation systems with dampersSeveral researchers [21ndash23] proposed a cost effective real-time hybrid simulation to evaluate different control strategiesfor advanced MR dampers Four semiactive control strate-gies based on the clipped-optimal controller were evaluatedexperimentally Force-tracking type controllers were foundto achieve excellent control performance while maintainingrelatively lowMRdamper forces Recently Zhao andZhu [24]introduced a stochastic optimal semiactive control law forcable-stayed bridges by solving the dynamical programmingequation produced by utilizing the Bingham model for anMR damper Since supplemental damping devices generallyprovide higher damping levels which are inversely propor-tional to their stiffness Hoslashgsberg [25] demonstrated thatMRdampers can be used to minimize dampers stiffness and evenhave equivalent negative stiffness Using linear equivalentmodels obtained by harmonic averaging improvement inresponse reduction is shown when compared to the corre-sponding case with optimal passive viscous dampers Mostrecently Assaleh et al [26] utilized group method of datahandling (GMDH) to model the MR damper behavior
This work presents a technique for seismic isolation of a20-story building adaptive control of anMRdamper installedin Chevron configuration between the base and first floorof a 20-story building The building structural model isderived from a benchmark structure model using ETABSTheMRdampermodel considered is derived fromWien-Bochysteresis model This model provides the critical nonlineardynamics that best represents the MR damper under a widerange of operating conditions System identification is usedto derive a low-order nonlinear model that best mimicsthe nonlinear dynamics of the actual MR damper Dynamicbehavior of this low order model is tested and validated overa wide range of inputs The damper model has proven itsvalidity to a high degree of accuracy against the nonlinearmodel A Kalman filter is designed to best estimate thestate of the structure-damper system for feedback imple-mentation purposes Using the estimated states an LQG-based compensator is designed to control the MR damperunder earthquake loads To demonstrate the effectiveness ofthis control strategy a wide range of historical earthquakesare applied to the structure and the MR damper is set
active Accelerations and drifts at all building floors arecomputed over the duration of each earthquake Results ofthe controlledmodel are compared to the uncontrolledmodeland the superior performance of the optimally controlledmodel is demonstrated
2 Structural Model
The proposed structure model was derived from a 20 storybenchmark building well studied in literature A full descrip-tion of the structure details is provided by others Spencer etal [27] and Ohtori et al [28] The building parameters werefed into ETABS and natural frequencies and mode shapeswere computed Figure 1 shows a sample mode of vibrationof the 20-story building under consideration
The structure is discretized into finite element modelforming an 119899-dimensional discrete spring-mass-damper sys-temwhose dynamics is described by the second-ordermatrixdifferential equation
119872 + 119862 + 119870119909 = 119906 (119905) (1)
where 119872 119862 and 119870 are the (20 times 20) square and symmetricmass stiffness and damping coefficient matrices respec-tively The variables 119909(119905) and 119906(119905) are the displacement andforce vectors respectively For systems with proportionaldamping the matrices 119872 119870 and 119862 can be diagonalized byemploying a proper normalized orthogonal transformationThis transformation yields
120578119894 (119905) + 2120589
119894120596119894120578119894 (119905) + 120596
119894120578119894 (119905) = 119881
119894119906 (119905) 119894 = 1 20 (2)
where 120578119894 120596119894 and 120589
119894represent the transformed coordinates
natural frequency and damping ratio of the structurersquos 119894thvibration mode respectively When the input is a pointforce (ie actuators) 119881
119894is the vector of the 119894th mode shape
evaluated at the force input locationFor flexible structures having point force(s) as the input(s)
and point displacement(s) as the measured output the state-space model of the flexible structures can be transformed asfollows
= [0 119868
minusΩ2
minus2120589Ω] 119911 + [
0
119881] 119906
119883119905= [119882 0] 119911 + 119863119906
(3)
where 119911(119905) = 120578(119905)
120578(119905) state vector 119873
119898 number of
modes 119873119906 number of inputs 119873
119910 number of outputs
120578(119905) = 1205781(119905) 1205782(119905) 120578
119873119898
(119905)119879 modal displacement
120578(119905) = 1205781(119905) 1205782(119905) 120578
119873119898
(119905)119879 modal velocity 119906(119905) =
1199061(119905) 1199062(119905) 119906
119873119906
(119905)119879 input vector 119903
119894 spatial coordi-
nates 119883119905(119905) = 119909(119903
1 119905) 119909(119903
2 119905) 119909(119903
119873119910
119905)119879 output vec-
tor Ω = diag1205961 1205962 120596
119873119898
natural frequency 120589 =
diag1205771 1205772 120577
119873119898
modal damping 120595119894119895 mode shape 119894 at
location 119895
Mathematical Problems in Engineering 3
1 1 1 1 1 1A B C D E F
119909
119911
Base
B minus 1
Ground
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
11th
12th
13th
14th
15th
16th
17th
18th
19th
20th
lowast lowast
Figure 1 Representation of building modes of vibration (5thhorizontal translational mode)
input matrix
119881 =[[
[
12059511
sdot sdot sdot 1205951119873119906
d
1205951198731198981 sdot sdot sdot 120595
119873119898119873119906
]]
]
(4)
output matrix
119882 =[[
[
12059511
sdot sdot sdot 1205951198731198981
d
1205951119873119910
sdot sdot sdot 120595119873119898119873119910
]]
]
(5)
The state-spacemodel of (3) can be expressed in the followingcompact form
= 119860119904 (120579) 119911 + 119861
119904 (120579) 119906 (6)
119883119905= 119862119904 (120579) 119911 + 119863
119904 (120579) 119906 (7)
where 119883119905is a vector of nodal displacement(s) at sensor(s)
location(s) and 119860119904 119861119904 119862119904 and 119863
119904matrices are functions
of the system (natural frequency damping ratio and modeshapes) (ie if we assume 120579 = 119891(120596
119894 120589119894 and 120595
119894)119894=1119899
)Information needed to construct matrices 119860
119904 119861119904 119862119904 and119863
119904
of (6) and (7) (iemode shapes andnatural frequencies)wereall obtained using ETABS
3 MR Damper Model
To make the simulation realistic the MR damper has tobe properly modeled The damper model must be accurateenough to capture the dynamic characteristics of the realdamper yet simple enough carry the computation in real timeon a low-power microprocessor To bridge the gap betweenthese two competing requirements system identification (SI)was used to derive an 8th-order nonlinear auto regression(Narx) model Spencer et al [27] have presented a model oftheMR damper based on the Bouc-WenHysteresis model Inthis model the force displacement force velocity and forceas a function of time were computed The damper modelequations are presented here for convenience
119865 = 1198621
119910 + 1198961(119909 minus 119909
119900) (8)
119910 =1
119888119900+ 1198881
[120572119911 + 119888119900 + 119896119900(119909 minus 119910)] (9)
= minus1205741003816100381610038161003816 minus 119910
1003816100381610038161003816 119911|119911|119899minus1
minus 120573 ( minus 119910) |119911|119899+ 119860 ( minus 119910) (10)
120572 = 120572119886+120572119887119906 (11)
1198881= 1198881119886+1198621119887119906 (12)
119888119900= 119888119900119886 +119862
119900119887119906 (13)
= minus120578 (119906 minus V) (14)
In this work numerical solutions of (8)ndash(14) have beenperformed and time history of all states was validated bychecking against previously published solutions by Spenceret al [27]
System identification (SI) relating input to output hasbeen performed to derive a simple nonlinearmodel DifferentSI techniques using the Matlab System Identification tool-box were employed yielding excellent matchingThe derivedmodels have been thoroughly tested and proved to matchthe nonlinear model behavior to a high degree of accuracyover a wide range of inputs Among the different methodsemployed the Narx method provided the best matchingResponses of various models obtained by the different SItechniques are shown in Figure 2
The polynomial model obtained from Narx is convertedto state-space format and compared to the nonlinear model
4 Mathematical Problems in Engineering
0
500
1000
Forc
e (N
)
01 02 03 04 05 06 07 08 09Time (s)
minus1500
minus1000
minus500
Force (sim)
ze measuredNarx3 fit 4097Narx8 fit 9305
Fit 9306Fit 8131
Figure 2 MR damper response versus different identified models
01 02 03 04 05 06 07 08
0
500
1000
1500
2000
Time (s)
MR
dam
per f
orce
(N)
Nonlinear versus linear response
Narx modelNonlinear model
minus1500
minus1000
minus500
Figure 3 Comparison between the state-spacemodel andnonlinearmodel
The comparison is shown in Figure 3 The state-space modelof the damper is expressed as
119909119891= 119860119891119909119891+ 119861119891119906119891
119884119891= 119862119891119909119891+119863119891119906119891
(15)
where 119909119891
is the state vector of the MR damper state-space model 119906
119891is the input and 119884
119891is the output (ie
damper force) The quadruple (119860119891 119861119891 119862119891 119863119891) represents
the dynamic input output and direct input matrices of theMR damper model The following figure shows the resultsobtained from the Narx model and the nonlinear modelExcellent matching of dynamic behavior is demonstrated inFigure 3
The state-space model obtained from Narx polynomial isutilized in the control scheme of the multidegree of freedomstructure found in the preceding The controller design ispresented next
4 Controller Design
Assuming that the control effort will be utilized to isolatethe passive structure from ground excitation as shown inFigure 4 A schematic of the control process implemented onthe structure is shown in Figure 5
The (KAFB) dynamic model is formed from the com-bined dynamics of the MR damper transmitted force repre-sented by (15) and the dynamics of the structure representedby (6) and (7)
The equivalent continuous state-spacemodel of the trans-mitted forcewhere acceleration is the input to the forcemodelis
119891= 119860119891119909119891+ 119861119891119884119884
119865 = 119862119891119909119891+ 119863119891119884119884
(16)
where (119884119884(119905)) is the base acceleration and 119909119891is a vector of the
transmitted force states 119860119891 119861119891 119862119891 and 119863
119891are dynamics
input output and direct input matrices of the transmittedforce block of Figure 5 respectively The term 119865 denotes theforce transmitted to the structure through its elastic base andis a function of time
Using the above analysis we can express the continuousstate-space model of the structure described by (7) and (8) as
119891= 119860119904119911 + 119861119904119865
119883119905= 119862119904119911 + 119863
119904119865
(17)
where that the term 119906(119905) of (6) and (7) has been replacedby 119865 in (17) to indicate that the input to the structure is theforce transmitted to it through its elastic base and includesthe seismic excitation and the damper control force
A state-space model of the beam-base system (the trans-mitted force represented by (16) and the structure representedby (17)) can now be constructed by augmenting the two partstogether such that
119860119886= [
119860119891
0
119861119904119862119891
119860119904
]
119861119886= [
119861119891
119861119904119863119891
]
119862119886= lfloor119863119904119862119891 119862
119904rfloor
119863119886= lfloor119863119891119863119904rfloor
(18)
where 119860119886 119861119886 119862119886 and 119863
119886represent the state-space matrices
of the augmented base-beam system in which the first twostates belong to the transmitted force part and the remainingstates belong to the structure mounted on the base
Mathematical Problems in Engineering 5
11990911199092
Figure 4 Passive structure
Structure
Nonlinear MRdamper model
Gain
MR damper force
Structure model
Controller
1199091 1199092
119870119886
119870119888
sum
sum
Figure 5 Active structure
Matrices 119860119886 119861119886 and 119862
119886are used for designing the
(KAFB) matrix of gains (119870119886) such that
119870119886= 119878∘119862119879
119886119877minus1 (19)
The column vector 119870119886in this case is a (2 + 2119899) times 1 column
vector and the first two rows are the Kalman gains of thestates of the transmitted force and the remaining 2119899 gainsare those of the states of the structure 119878
∘is the steady-state
solution of the following filter algebraic Riccati equation
119878 = 119860119886119878 + 119878119860
119879minus 119878119862119879
119886119877minus1119862119886119878 + 119861119886119876119861119879
119886 (20)
Matrices 119877 and 119876 are positive definite and positive semidef-inite matrices respectively [29 30] Proper choice of 119877 and119876 is important because both matrices are heavily involved
0 5 10 15 20 25 30
0
5El Centro
0 5 10 15 20 25 30
0010203
Time (s)
Time (s)
minus5
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus02
minus01
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
Figure 6 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)El Centro earthquake input
in the construction of the cost function In this work thevalue of 119876119877 ranges from 1 to 100 This ratio is limitedby the maximum force that can be provided by the MRdamper and keep the control force within its saturationlimit If the 119877 is very high sensitivity to measurementnoise will be accentuated and the controller performancemight be degraded After many iterations and driven bythe objective of minimizing accelerations and displacementswhile maintaining damper force with allowable limits thefinal 119877 and 119876matrices were selected
For a specific value of 119877 and 119876 Kalman matrix of gains(119870119886) of (19) is
119870119886= [
[119870119891]
[119870119904]
] =
[[[[[[[
[
[1198701
1198702
]
[[
[
1198703
1198702119899+2
]]
]
]]]]]]]
]
(21)
Equation (21) shows that 119870119886is partitioned into two parts
namely 119870119891which corrects the estimates of 119909
119891in (16) and
119870119904which corrects the estimates beam states (ie119883
119905of (17))
In general the structure of the Kalman estimator takeson a particularly simple structure that closely resembles theoriginal dynamic system [29 30] The complete vibrationisolation scheme proposed by this study is shown in Figure 5119870119888in Figure 5 is the linear quadratic regulator (LQR) gain
obtained with a similar procedure used to obtain119870119886
It is well known that the Kalman estimator is subject toall deterministic inputs that the plant is subject to including
6 Mathematical Problems in Engineering
Time (s)0 5 10 15 20 25 30
0123
Hachinohe
0 5 10 15 20 25 30
00102
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus03
minus02
minus01
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
minus1
minus2
Figure 7 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Hachinohe earthquake
0 5 10 15 20 25 30
0
10
20 Kobe
0 5 10 15 20 25 30
0
05
1
Time (s)
Time (s)
minus20
minus10
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
Figure 8 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Kobe earthquake
the estimated damper control force shown in Figure 5 Thisis why the realization of the structure inside the controllerin Figure 5 is subject to the estimated transmitted damperforce twice These two forces have the same magnitudeand like the two forces acting on the structure (plant) theyare opposite in sign nullifying the net force seen by the
Time (s)0 5 10 15 20 25 30
05
10Northridge
0 5 10 15 20 25 30
0
05
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
minus15
minus10
minus5
Figure 9 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Northridge earthquake
realization of the structure inside the controller Thereforethe Kalman estimate of the acceleration of any point onthe beam is identically zero which eliminates the need forrealizing (including) the structure inside the controller whichsubsequently yields a second-order control scheme regardlessof the order of the plant model This lowers the complexityof the controller and therefore significantly reduces thecomputational time Simulated results of the 20th floor arepresented in Figures 6 7 8 and 9 Each figure has two plotsone for acceleration and the other for displacement Each plotshows the uncontrolled and the controlled response
5 Conclusions
To demonstrate the advanced performance of a semiactiveMR damper used in the protection of a 20-story structurefrom earthquake damage a control strategy based on alinear quadratic Gaussian regulator is proposed First a linearmodel of the structure is derived using ETABS SecondSystem identification is used to derive a linear low-ordermodel of the MR damper from the Bouc-Wen nonlinearmodel Third a Kalman filter is designed to best estimatethe states of the system for feedback implementation pur-poses Finally an LQG controller is designed to minimizedynamic loads and structural damage Extensive simulationis performed to test and validate the effectiveness of theMR damper control strategy Four historical earthquakes areapplied to the structure and theMRdamper is set active Aftersuccessful simulation accelerations and drifts at all structurefloors are computed Proposed optimal control ofMRdamper
Mathematical Problems in Engineering 7
effectiveness is demonstrated by significantly reducing theaccelerations and displacements at all floors of the structure
Acknowledgment
The authors acknowledge the support of the AmericanUniversity of Sharjah
References
[1] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation John Wiley amp Sons Chichester UK1993
[2] J M Kelly Earthquake-Resistant Design With Rubber SpringerLondon UK 2nd edition 1997
[3] F Naeim and J M Kelly Design of Seismic Isolated StructuresJohn Wiley amp Sons New York NY USA 1999
[4] R S Jangid and J M Kelly ldquoBase isolation for near-faultmotionsrdquoEarthquake Engineering and Structural Dynamics vol30 no 5 pp 691ndash707 2001
[5] A Rodriguez-Marek Near fault seismic site response [PhDthesis] Civil Engineering University of California BerkeleyCalif USA 2000
[6] G A MacRae D V Morrow and C W Roeder ldquoNear-faultground motion effects on simple structuresrdquo ASCE Journal ofStructural Engineering vol 127 no 9 pp 996ndash1004 2001
[7] A K Chopra and C Chintanapakdee ldquoComparing responseof SDF systems to near-fault and far-fault earthquake motionsin the context of spectral regionsrdquo Earthquake Engineering andStructural Dynamics vol 30 no 12 pp 1769ndash1789 2001
[8] N Makris and C J Black ldquoDimensional analysis of bilinearoscillators under pulse-type excitationsrdquo ASCE Journal of Engi-neering Mechanics vol 130 no 9 pp 1019ndash1031 2004
[9] C P Providakis ldquoEffect of LRB isolators and supplementalviscous dampers on seismic isolated buildings under near-faultexcitationsrdquo Engineering Structures vol 30 no 5 pp 1187ndash11982008
[10] C P Providakis ldquoEffect of supplemental damping on LRB andFPS seismic isolators under near-fault ground motionsrdquo SoilDynamics and Earthquake Engineering vol 29 no 1 pp 80ndash902009
[11] R S Jangid ldquoOptimum friction pendulum system for near-faultmotionsrdquoEngineering Structures vol 27 no 3 pp 349ndash3592005
[12] S M Hussain J W Asher and R D Ewing ldquoSeismic baseisolation design for the San Bernardino CountyMedical Centerreplacement projectrdquo in Proceedings of the Symposium onStructural Engineering in Natural Hazards Mitigation vol 1 pp760ndash765 Irvine Calif USA April 1993
[13] S Hussain ldquoPerformance of base isolated buildings in thenorthridge earthquakerdquo in Seismic Base Isolation State of thePractice Seminar Structural Engineers Association of SouthernCalifornia 1994
[14] S Hussain and E Retamal ldquoA hybrid seismic isolation systemmdashisolators with supplemental viscous dampersrdquo in Proceedings ofthe 1st World Conference on Structural Control vol 3 pp FA2-53ndashFA2-62 International Association for Structural ControlLos Angeles Calif USA August 1994
[15] M Al Satari and J Abdalla ldquoOptimization of a base-isolationsystem consisting of natural rubber bearings and fluid viscous
dampersrdquo in Proceedings of the 11thWorld Conference on SeismicIsolation Energy Dissipation and Active Vibration Control ofStructures Guangzhou China 2009
[16] S Hussain and M Al Satari ldquoInnovative design of a seismicisolation supplemental viscous damping systems of an essentialservices facility in a near-fault regionrdquo in Proceedings of the 14thWorld Conference on Earthquake Engineering Beijing China2008
[17] S Hussain and M Al Satari ldquoDesign of a seismic isolationsystem with supplemental viscous damping for a near-faultessential services facilityrdquo in Proceedings of the 76th StructuralEngineers Association of California Annual Convention SquawCreek Calif USA 2007
[18] S Hussain andM Al Satari ldquoViscous-damped seismic isolationsystem for a near-fault essential services facilityrdquo in Proceedingsof the 10th World Conference on Seismic Isolation EnergyDissipation and Active Vibration Control of Structures IstanbulTurkey 2007
[19] M AlHamaydeh and S Hussain ldquoInnovative design of aseismically-isolated building with supplemental dampingrdquo inProceedings of 14th European Conference on Earthquake Engi-neering (ECEE rsquo10) Ohrid Republic of Macedonia 2010
[20] M AlHamaydeh S Barakat and F Abed ldquoMultiple regressionmodeling of natural rubber seismic-isolation systems withsupplemental viscous damping for near-field ground motionrdquoJournal of Civil Engineering and Management In press
[21] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo ASCE Journal of Engineering Mechanics vol128 no 10 pp 1088ndash1099 2002
[22] A Friedman J Zhang Y Cha et al ldquoAccommodating MRdamper dynamics for control of large scale structures systemsrdquoin Proceddings of the 5thWorld Conference on Structural Controland Monitoring 2010
[23] H J Jung B F Spencer Jr and I W Lee ldquoControl ofseismically excited cable-stayed bridge employing magnetorhe-ological fluid dampersrdquo Journal of Structural Engineering vol129 no 7 pp 873ndash883 2003
[24] M Zhao andWQ Zhu ldquoStochastic optimal semi-active controlof stay cables by using magneto-rheological damperrdquo Journal ofVibration and Control vol 17 no 13 pp 1921ndash1929 2011
[25] J Hoslashgsberg ldquoThe role of negative stiffness in semi-activecontrol of magneto-rheological dampersrdquo Structural Controland Health Monitoring vol 18 no 3 pp 289ndash304 2011
[26] K Assaleh T Shanableh and Y Kheil ldquoGroup method of datahandling for modeling magnetorheological dampersrdquo Intelli-gent Control and Automation vol 4 no 1 pp 70ndash79 2013
[27] B F Spencer S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997
[28] Y Ohtori R E Christenson B F Spencer Jr and S J DykeldquoBenchmark control problems for seismically excited nonlinearbuildingsrdquo ASCE Journal of Engineering Mechanics vol 130 no4 pp 366ndash385 2004
[29] AH El-Sinawi ldquoActive vibration isolation of a flexible structuremounted on a vibrating elastic baserdquo Journal of Sound andVibration vol 271 no 1-2 pp 323ndash337 2004
[30] A El-Sinawi and A R Kashani ldquoActive isolation using aKalman estimator-based controllerrdquo Journal of Vibration andControl vol 7 no 8 pp 1163ndash1173 2001
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
are minimized It was concluded that the optimum frictioncoefficient for FPS for NF earthquake motions is in the rangeof 005 to 015
When considering the challenge of limiting the totalmaximum displacement (119863TM) to practical limits especiallyin NF sites somtimes the designer relies on fluid viscousdampers (FVDs) The state of the practice involves carryingout preliminary calculations and analyses using typical HDRbearings These preliminary calculations could readily showwhether or not supplemental damping is required Oncesupplemental damping is deemed necessary many designerswould prefer utilizing the linear behavior of NRB isola-tors combined with the supplemental damping provided byFVDs the use of such a system often results in additionaluniformity in the induced superstructure story forces Thissystem has also been used in many projects in the USA [12ndash19] Recently AlHamaydeh et al [20] developed simplifieddesign equations for seismic isolation systems with dampersSeveral researchers [21ndash23] proposed a cost effective real-time hybrid simulation to evaluate different control strategiesfor advanced MR dampers Four semiactive control strate-gies based on the clipped-optimal controller were evaluatedexperimentally Force-tracking type controllers were foundto achieve excellent control performance while maintainingrelatively lowMRdamper forces Recently Zhao andZhu [24]introduced a stochastic optimal semiactive control law forcable-stayed bridges by solving the dynamical programmingequation produced by utilizing the Bingham model for anMR damper Since supplemental damping devices generallyprovide higher damping levels which are inversely propor-tional to their stiffness Hoslashgsberg [25] demonstrated thatMRdampers can be used to minimize dampers stiffness and evenhave equivalent negative stiffness Using linear equivalentmodels obtained by harmonic averaging improvement inresponse reduction is shown when compared to the corre-sponding case with optimal passive viscous dampers Mostrecently Assaleh et al [26] utilized group method of datahandling (GMDH) to model the MR damper behavior
This work presents a technique for seismic isolation of a20-story building adaptive control of anMRdamper installedin Chevron configuration between the base and first floorof a 20-story building The building structural model isderived from a benchmark structure model using ETABSTheMRdampermodel considered is derived fromWien-Bochysteresis model This model provides the critical nonlineardynamics that best represents the MR damper under a widerange of operating conditions System identification is usedto derive a low-order nonlinear model that best mimicsthe nonlinear dynamics of the actual MR damper Dynamicbehavior of this low order model is tested and validated overa wide range of inputs The damper model has proven itsvalidity to a high degree of accuracy against the nonlinearmodel A Kalman filter is designed to best estimate thestate of the structure-damper system for feedback imple-mentation purposes Using the estimated states an LQG-based compensator is designed to control the MR damperunder earthquake loads To demonstrate the effectiveness ofthis control strategy a wide range of historical earthquakesare applied to the structure and the MR damper is set
active Accelerations and drifts at all building floors arecomputed over the duration of each earthquake Results ofthe controlledmodel are compared to the uncontrolledmodeland the superior performance of the optimally controlledmodel is demonstrated
2 Structural Model
The proposed structure model was derived from a 20 storybenchmark building well studied in literature A full descrip-tion of the structure details is provided by others Spencer etal [27] and Ohtori et al [28] The building parameters werefed into ETABS and natural frequencies and mode shapeswere computed Figure 1 shows a sample mode of vibrationof the 20-story building under consideration
The structure is discretized into finite element modelforming an 119899-dimensional discrete spring-mass-damper sys-temwhose dynamics is described by the second-ordermatrixdifferential equation
119872 + 119862 + 119870119909 = 119906 (119905) (1)
where 119872 119862 and 119870 are the (20 times 20) square and symmetricmass stiffness and damping coefficient matrices respec-tively The variables 119909(119905) and 119906(119905) are the displacement andforce vectors respectively For systems with proportionaldamping the matrices 119872 119870 and 119862 can be diagonalized byemploying a proper normalized orthogonal transformationThis transformation yields
120578119894 (119905) + 2120589
119894120596119894120578119894 (119905) + 120596
119894120578119894 (119905) = 119881
119894119906 (119905) 119894 = 1 20 (2)
where 120578119894 120596119894 and 120589
119894represent the transformed coordinates
natural frequency and damping ratio of the structurersquos 119894thvibration mode respectively When the input is a pointforce (ie actuators) 119881
119894is the vector of the 119894th mode shape
evaluated at the force input locationFor flexible structures having point force(s) as the input(s)
and point displacement(s) as the measured output the state-space model of the flexible structures can be transformed asfollows
= [0 119868
minusΩ2
minus2120589Ω] 119911 + [
0
119881] 119906
119883119905= [119882 0] 119911 + 119863119906
(3)
where 119911(119905) = 120578(119905)
120578(119905) state vector 119873
119898 number of
modes 119873119906 number of inputs 119873
119910 number of outputs
120578(119905) = 1205781(119905) 1205782(119905) 120578
119873119898
(119905)119879 modal displacement
120578(119905) = 1205781(119905) 1205782(119905) 120578
119873119898
(119905)119879 modal velocity 119906(119905) =
1199061(119905) 1199062(119905) 119906
119873119906
(119905)119879 input vector 119903
119894 spatial coordi-
nates 119883119905(119905) = 119909(119903
1 119905) 119909(119903
2 119905) 119909(119903
119873119910
119905)119879 output vec-
tor Ω = diag1205961 1205962 120596
119873119898
natural frequency 120589 =
diag1205771 1205772 120577
119873119898
modal damping 120595119894119895 mode shape 119894 at
location 119895
Mathematical Problems in Engineering 3
1 1 1 1 1 1A B C D E F
119909
119911
Base
B minus 1
Ground
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
11th
12th
13th
14th
15th
16th
17th
18th
19th
20th
lowast lowast
Figure 1 Representation of building modes of vibration (5thhorizontal translational mode)
input matrix
119881 =[[
[
12059511
sdot sdot sdot 1205951119873119906
d
1205951198731198981 sdot sdot sdot 120595
119873119898119873119906
]]
]
(4)
output matrix
119882 =[[
[
12059511
sdot sdot sdot 1205951198731198981
d
1205951119873119910
sdot sdot sdot 120595119873119898119873119910
]]
]
(5)
The state-spacemodel of (3) can be expressed in the followingcompact form
= 119860119904 (120579) 119911 + 119861
119904 (120579) 119906 (6)
119883119905= 119862119904 (120579) 119911 + 119863
119904 (120579) 119906 (7)
where 119883119905is a vector of nodal displacement(s) at sensor(s)
location(s) and 119860119904 119861119904 119862119904 and 119863
119904matrices are functions
of the system (natural frequency damping ratio and modeshapes) (ie if we assume 120579 = 119891(120596
119894 120589119894 and 120595
119894)119894=1119899
)Information needed to construct matrices 119860
119904 119861119904 119862119904 and119863
119904
of (6) and (7) (iemode shapes andnatural frequencies)wereall obtained using ETABS
3 MR Damper Model
To make the simulation realistic the MR damper has tobe properly modeled The damper model must be accurateenough to capture the dynamic characteristics of the realdamper yet simple enough carry the computation in real timeon a low-power microprocessor To bridge the gap betweenthese two competing requirements system identification (SI)was used to derive an 8th-order nonlinear auto regression(Narx) model Spencer et al [27] have presented a model oftheMR damper based on the Bouc-WenHysteresis model Inthis model the force displacement force velocity and forceas a function of time were computed The damper modelequations are presented here for convenience
119865 = 1198621
119910 + 1198961(119909 minus 119909
119900) (8)
119910 =1
119888119900+ 1198881
[120572119911 + 119888119900 + 119896119900(119909 minus 119910)] (9)
= minus1205741003816100381610038161003816 minus 119910
1003816100381610038161003816 119911|119911|119899minus1
minus 120573 ( minus 119910) |119911|119899+ 119860 ( minus 119910) (10)
120572 = 120572119886+120572119887119906 (11)
1198881= 1198881119886+1198621119887119906 (12)
119888119900= 119888119900119886 +119862
119900119887119906 (13)
= minus120578 (119906 minus V) (14)
In this work numerical solutions of (8)ndash(14) have beenperformed and time history of all states was validated bychecking against previously published solutions by Spenceret al [27]
System identification (SI) relating input to output hasbeen performed to derive a simple nonlinearmodel DifferentSI techniques using the Matlab System Identification tool-box were employed yielding excellent matchingThe derivedmodels have been thoroughly tested and proved to matchthe nonlinear model behavior to a high degree of accuracyover a wide range of inputs Among the different methodsemployed the Narx method provided the best matchingResponses of various models obtained by the different SItechniques are shown in Figure 2
The polynomial model obtained from Narx is convertedto state-space format and compared to the nonlinear model
4 Mathematical Problems in Engineering
0
500
1000
Forc
e (N
)
01 02 03 04 05 06 07 08 09Time (s)
minus1500
minus1000
minus500
Force (sim)
ze measuredNarx3 fit 4097Narx8 fit 9305
Fit 9306Fit 8131
Figure 2 MR damper response versus different identified models
01 02 03 04 05 06 07 08
0
500
1000
1500
2000
Time (s)
MR
dam
per f
orce
(N)
Nonlinear versus linear response
Narx modelNonlinear model
minus1500
minus1000
minus500
Figure 3 Comparison between the state-spacemodel andnonlinearmodel
The comparison is shown in Figure 3 The state-space modelof the damper is expressed as
119909119891= 119860119891119909119891+ 119861119891119906119891
119884119891= 119862119891119909119891+119863119891119906119891
(15)
where 119909119891
is the state vector of the MR damper state-space model 119906
119891is the input and 119884
119891is the output (ie
damper force) The quadruple (119860119891 119861119891 119862119891 119863119891) represents
the dynamic input output and direct input matrices of theMR damper model The following figure shows the resultsobtained from the Narx model and the nonlinear modelExcellent matching of dynamic behavior is demonstrated inFigure 3
The state-space model obtained from Narx polynomial isutilized in the control scheme of the multidegree of freedomstructure found in the preceding The controller design ispresented next
4 Controller Design
Assuming that the control effort will be utilized to isolatethe passive structure from ground excitation as shown inFigure 4 A schematic of the control process implemented onthe structure is shown in Figure 5
The (KAFB) dynamic model is formed from the com-bined dynamics of the MR damper transmitted force repre-sented by (15) and the dynamics of the structure representedby (6) and (7)
The equivalent continuous state-spacemodel of the trans-mitted forcewhere acceleration is the input to the forcemodelis
119891= 119860119891119909119891+ 119861119891119884119884
119865 = 119862119891119909119891+ 119863119891119884119884
(16)
where (119884119884(119905)) is the base acceleration and 119909119891is a vector of the
transmitted force states 119860119891 119861119891 119862119891 and 119863
119891are dynamics
input output and direct input matrices of the transmittedforce block of Figure 5 respectively The term 119865 denotes theforce transmitted to the structure through its elastic base andis a function of time
Using the above analysis we can express the continuousstate-space model of the structure described by (7) and (8) as
119891= 119860119904119911 + 119861119904119865
119883119905= 119862119904119911 + 119863
119904119865
(17)
where that the term 119906(119905) of (6) and (7) has been replacedby 119865 in (17) to indicate that the input to the structure is theforce transmitted to it through its elastic base and includesthe seismic excitation and the damper control force
A state-space model of the beam-base system (the trans-mitted force represented by (16) and the structure representedby (17)) can now be constructed by augmenting the two partstogether such that
119860119886= [
119860119891
0
119861119904119862119891
119860119904
]
119861119886= [
119861119891
119861119904119863119891
]
119862119886= lfloor119863119904119862119891 119862
119904rfloor
119863119886= lfloor119863119891119863119904rfloor
(18)
where 119860119886 119861119886 119862119886 and 119863
119886represent the state-space matrices
of the augmented base-beam system in which the first twostates belong to the transmitted force part and the remainingstates belong to the structure mounted on the base
Mathematical Problems in Engineering 5
11990911199092
Figure 4 Passive structure
Structure
Nonlinear MRdamper model
Gain
MR damper force
Structure model
Controller
1199091 1199092
119870119886
119870119888
sum
sum
Figure 5 Active structure
Matrices 119860119886 119861119886 and 119862
119886are used for designing the
(KAFB) matrix of gains (119870119886) such that
119870119886= 119878∘119862119879
119886119877minus1 (19)
The column vector 119870119886in this case is a (2 + 2119899) times 1 column
vector and the first two rows are the Kalman gains of thestates of the transmitted force and the remaining 2119899 gainsare those of the states of the structure 119878
∘is the steady-state
solution of the following filter algebraic Riccati equation
119878 = 119860119886119878 + 119878119860
119879minus 119878119862119879
119886119877minus1119862119886119878 + 119861119886119876119861119879
119886 (20)
Matrices 119877 and 119876 are positive definite and positive semidef-inite matrices respectively [29 30] Proper choice of 119877 and119876 is important because both matrices are heavily involved
0 5 10 15 20 25 30
0
5El Centro
0 5 10 15 20 25 30
0010203
Time (s)
Time (s)
minus5
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus02
minus01
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
Figure 6 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)El Centro earthquake input
in the construction of the cost function In this work thevalue of 119876119877 ranges from 1 to 100 This ratio is limitedby the maximum force that can be provided by the MRdamper and keep the control force within its saturationlimit If the 119877 is very high sensitivity to measurementnoise will be accentuated and the controller performancemight be degraded After many iterations and driven bythe objective of minimizing accelerations and displacementswhile maintaining damper force with allowable limits thefinal 119877 and 119876matrices were selected
For a specific value of 119877 and 119876 Kalman matrix of gains(119870119886) of (19) is
119870119886= [
[119870119891]
[119870119904]
] =
[[[[[[[
[
[1198701
1198702
]
[[
[
1198703
1198702119899+2
]]
]
]]]]]]]
]
(21)
Equation (21) shows that 119870119886is partitioned into two parts
namely 119870119891which corrects the estimates of 119909
119891in (16) and
119870119904which corrects the estimates beam states (ie119883
119905of (17))
In general the structure of the Kalman estimator takeson a particularly simple structure that closely resembles theoriginal dynamic system [29 30] The complete vibrationisolation scheme proposed by this study is shown in Figure 5119870119888in Figure 5 is the linear quadratic regulator (LQR) gain
obtained with a similar procedure used to obtain119870119886
It is well known that the Kalman estimator is subject toall deterministic inputs that the plant is subject to including
6 Mathematical Problems in Engineering
Time (s)0 5 10 15 20 25 30
0123
Hachinohe
0 5 10 15 20 25 30
00102
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus03
minus02
minus01
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
minus1
minus2
Figure 7 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Hachinohe earthquake
0 5 10 15 20 25 30
0
10
20 Kobe
0 5 10 15 20 25 30
0
05
1
Time (s)
Time (s)
minus20
minus10
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
Figure 8 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Kobe earthquake
the estimated damper control force shown in Figure 5 Thisis why the realization of the structure inside the controllerin Figure 5 is subject to the estimated transmitted damperforce twice These two forces have the same magnitudeand like the two forces acting on the structure (plant) theyare opposite in sign nullifying the net force seen by the
Time (s)0 5 10 15 20 25 30
05
10Northridge
0 5 10 15 20 25 30
0
05
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
minus15
minus10
minus5
Figure 9 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Northridge earthquake
realization of the structure inside the controller Thereforethe Kalman estimate of the acceleration of any point onthe beam is identically zero which eliminates the need forrealizing (including) the structure inside the controller whichsubsequently yields a second-order control scheme regardlessof the order of the plant model This lowers the complexityof the controller and therefore significantly reduces thecomputational time Simulated results of the 20th floor arepresented in Figures 6 7 8 and 9 Each figure has two plotsone for acceleration and the other for displacement Each plotshows the uncontrolled and the controlled response
5 Conclusions
To demonstrate the advanced performance of a semiactiveMR damper used in the protection of a 20-story structurefrom earthquake damage a control strategy based on alinear quadratic Gaussian regulator is proposed First a linearmodel of the structure is derived using ETABS SecondSystem identification is used to derive a linear low-ordermodel of the MR damper from the Bouc-Wen nonlinearmodel Third a Kalman filter is designed to best estimatethe states of the system for feedback implementation pur-poses Finally an LQG controller is designed to minimizedynamic loads and structural damage Extensive simulationis performed to test and validate the effectiveness of theMR damper control strategy Four historical earthquakes areapplied to the structure and theMRdamper is set active Aftersuccessful simulation accelerations and drifts at all structurefloors are computed Proposed optimal control ofMRdamper
Mathematical Problems in Engineering 7
effectiveness is demonstrated by significantly reducing theaccelerations and displacements at all floors of the structure
Acknowledgment
The authors acknowledge the support of the AmericanUniversity of Sharjah
References
[1] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation John Wiley amp Sons Chichester UK1993
[2] J M Kelly Earthquake-Resistant Design With Rubber SpringerLondon UK 2nd edition 1997
[3] F Naeim and J M Kelly Design of Seismic Isolated StructuresJohn Wiley amp Sons New York NY USA 1999
[4] R S Jangid and J M Kelly ldquoBase isolation for near-faultmotionsrdquoEarthquake Engineering and Structural Dynamics vol30 no 5 pp 691ndash707 2001
[5] A Rodriguez-Marek Near fault seismic site response [PhDthesis] Civil Engineering University of California BerkeleyCalif USA 2000
[6] G A MacRae D V Morrow and C W Roeder ldquoNear-faultground motion effects on simple structuresrdquo ASCE Journal ofStructural Engineering vol 127 no 9 pp 996ndash1004 2001
[7] A K Chopra and C Chintanapakdee ldquoComparing responseof SDF systems to near-fault and far-fault earthquake motionsin the context of spectral regionsrdquo Earthquake Engineering andStructural Dynamics vol 30 no 12 pp 1769ndash1789 2001
[8] N Makris and C J Black ldquoDimensional analysis of bilinearoscillators under pulse-type excitationsrdquo ASCE Journal of Engi-neering Mechanics vol 130 no 9 pp 1019ndash1031 2004
[9] C P Providakis ldquoEffect of LRB isolators and supplementalviscous dampers on seismic isolated buildings under near-faultexcitationsrdquo Engineering Structures vol 30 no 5 pp 1187ndash11982008
[10] C P Providakis ldquoEffect of supplemental damping on LRB andFPS seismic isolators under near-fault ground motionsrdquo SoilDynamics and Earthquake Engineering vol 29 no 1 pp 80ndash902009
[11] R S Jangid ldquoOptimum friction pendulum system for near-faultmotionsrdquoEngineering Structures vol 27 no 3 pp 349ndash3592005
[12] S M Hussain J W Asher and R D Ewing ldquoSeismic baseisolation design for the San Bernardino CountyMedical Centerreplacement projectrdquo in Proceedings of the Symposium onStructural Engineering in Natural Hazards Mitigation vol 1 pp760ndash765 Irvine Calif USA April 1993
[13] S Hussain ldquoPerformance of base isolated buildings in thenorthridge earthquakerdquo in Seismic Base Isolation State of thePractice Seminar Structural Engineers Association of SouthernCalifornia 1994
[14] S Hussain and E Retamal ldquoA hybrid seismic isolation systemmdashisolators with supplemental viscous dampersrdquo in Proceedings ofthe 1st World Conference on Structural Control vol 3 pp FA2-53ndashFA2-62 International Association for Structural ControlLos Angeles Calif USA August 1994
[15] M Al Satari and J Abdalla ldquoOptimization of a base-isolationsystem consisting of natural rubber bearings and fluid viscous
dampersrdquo in Proceedings of the 11thWorld Conference on SeismicIsolation Energy Dissipation and Active Vibration Control ofStructures Guangzhou China 2009
[16] S Hussain and M Al Satari ldquoInnovative design of a seismicisolation supplemental viscous damping systems of an essentialservices facility in a near-fault regionrdquo in Proceedings of the 14thWorld Conference on Earthquake Engineering Beijing China2008
[17] S Hussain and M Al Satari ldquoDesign of a seismic isolationsystem with supplemental viscous damping for a near-faultessential services facilityrdquo in Proceedings of the 76th StructuralEngineers Association of California Annual Convention SquawCreek Calif USA 2007
[18] S Hussain andM Al Satari ldquoViscous-damped seismic isolationsystem for a near-fault essential services facilityrdquo in Proceedingsof the 10th World Conference on Seismic Isolation EnergyDissipation and Active Vibration Control of Structures IstanbulTurkey 2007
[19] M AlHamaydeh and S Hussain ldquoInnovative design of aseismically-isolated building with supplemental dampingrdquo inProceedings of 14th European Conference on Earthquake Engi-neering (ECEE rsquo10) Ohrid Republic of Macedonia 2010
[20] M AlHamaydeh S Barakat and F Abed ldquoMultiple regressionmodeling of natural rubber seismic-isolation systems withsupplemental viscous damping for near-field ground motionrdquoJournal of Civil Engineering and Management In press
[21] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo ASCE Journal of Engineering Mechanics vol128 no 10 pp 1088ndash1099 2002
[22] A Friedman J Zhang Y Cha et al ldquoAccommodating MRdamper dynamics for control of large scale structures systemsrdquoin Proceddings of the 5thWorld Conference on Structural Controland Monitoring 2010
[23] H J Jung B F Spencer Jr and I W Lee ldquoControl ofseismically excited cable-stayed bridge employing magnetorhe-ological fluid dampersrdquo Journal of Structural Engineering vol129 no 7 pp 873ndash883 2003
[24] M Zhao andWQ Zhu ldquoStochastic optimal semi-active controlof stay cables by using magneto-rheological damperrdquo Journal ofVibration and Control vol 17 no 13 pp 1921ndash1929 2011
[25] J Hoslashgsberg ldquoThe role of negative stiffness in semi-activecontrol of magneto-rheological dampersrdquo Structural Controland Health Monitoring vol 18 no 3 pp 289ndash304 2011
[26] K Assaleh T Shanableh and Y Kheil ldquoGroup method of datahandling for modeling magnetorheological dampersrdquo Intelli-gent Control and Automation vol 4 no 1 pp 70ndash79 2013
[27] B F Spencer S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997
[28] Y Ohtori R E Christenson B F Spencer Jr and S J DykeldquoBenchmark control problems for seismically excited nonlinearbuildingsrdquo ASCE Journal of Engineering Mechanics vol 130 no4 pp 366ndash385 2004
[29] AH El-Sinawi ldquoActive vibration isolation of a flexible structuremounted on a vibrating elastic baserdquo Journal of Sound andVibration vol 271 no 1-2 pp 323ndash337 2004
[30] A El-Sinawi and A R Kashani ldquoActive isolation using aKalman estimator-based controllerrdquo Journal of Vibration andControl vol 7 no 8 pp 1163ndash1173 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
1 1 1 1 1 1A B C D E F
119909
119911
Base
B minus 1
Ground
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
11th
12th
13th
14th
15th
16th
17th
18th
19th
20th
lowast lowast
Figure 1 Representation of building modes of vibration (5thhorizontal translational mode)
input matrix
119881 =[[
[
12059511
sdot sdot sdot 1205951119873119906
d
1205951198731198981 sdot sdot sdot 120595
119873119898119873119906
]]
]
(4)
output matrix
119882 =[[
[
12059511
sdot sdot sdot 1205951198731198981
d
1205951119873119910
sdot sdot sdot 120595119873119898119873119910
]]
]
(5)
The state-spacemodel of (3) can be expressed in the followingcompact form
= 119860119904 (120579) 119911 + 119861
119904 (120579) 119906 (6)
119883119905= 119862119904 (120579) 119911 + 119863
119904 (120579) 119906 (7)
where 119883119905is a vector of nodal displacement(s) at sensor(s)
location(s) and 119860119904 119861119904 119862119904 and 119863
119904matrices are functions
of the system (natural frequency damping ratio and modeshapes) (ie if we assume 120579 = 119891(120596
119894 120589119894 and 120595
119894)119894=1119899
)Information needed to construct matrices 119860
119904 119861119904 119862119904 and119863
119904
of (6) and (7) (iemode shapes andnatural frequencies)wereall obtained using ETABS
3 MR Damper Model
To make the simulation realistic the MR damper has tobe properly modeled The damper model must be accurateenough to capture the dynamic characteristics of the realdamper yet simple enough carry the computation in real timeon a low-power microprocessor To bridge the gap betweenthese two competing requirements system identification (SI)was used to derive an 8th-order nonlinear auto regression(Narx) model Spencer et al [27] have presented a model oftheMR damper based on the Bouc-WenHysteresis model Inthis model the force displacement force velocity and forceas a function of time were computed The damper modelequations are presented here for convenience
119865 = 1198621
119910 + 1198961(119909 minus 119909
119900) (8)
119910 =1
119888119900+ 1198881
[120572119911 + 119888119900 + 119896119900(119909 minus 119910)] (9)
= minus1205741003816100381610038161003816 minus 119910
1003816100381610038161003816 119911|119911|119899minus1
minus 120573 ( minus 119910) |119911|119899+ 119860 ( minus 119910) (10)
120572 = 120572119886+120572119887119906 (11)
1198881= 1198881119886+1198621119887119906 (12)
119888119900= 119888119900119886 +119862
119900119887119906 (13)
= minus120578 (119906 minus V) (14)
In this work numerical solutions of (8)ndash(14) have beenperformed and time history of all states was validated bychecking against previously published solutions by Spenceret al [27]
System identification (SI) relating input to output hasbeen performed to derive a simple nonlinearmodel DifferentSI techniques using the Matlab System Identification tool-box were employed yielding excellent matchingThe derivedmodels have been thoroughly tested and proved to matchthe nonlinear model behavior to a high degree of accuracyover a wide range of inputs Among the different methodsemployed the Narx method provided the best matchingResponses of various models obtained by the different SItechniques are shown in Figure 2
The polynomial model obtained from Narx is convertedto state-space format and compared to the nonlinear model
4 Mathematical Problems in Engineering
0
500
1000
Forc
e (N
)
01 02 03 04 05 06 07 08 09Time (s)
minus1500
minus1000
minus500
Force (sim)
ze measuredNarx3 fit 4097Narx8 fit 9305
Fit 9306Fit 8131
Figure 2 MR damper response versus different identified models
01 02 03 04 05 06 07 08
0
500
1000
1500
2000
Time (s)
MR
dam
per f
orce
(N)
Nonlinear versus linear response
Narx modelNonlinear model
minus1500
minus1000
minus500
Figure 3 Comparison between the state-spacemodel andnonlinearmodel
The comparison is shown in Figure 3 The state-space modelof the damper is expressed as
119909119891= 119860119891119909119891+ 119861119891119906119891
119884119891= 119862119891119909119891+119863119891119906119891
(15)
where 119909119891
is the state vector of the MR damper state-space model 119906
119891is the input and 119884
119891is the output (ie
damper force) The quadruple (119860119891 119861119891 119862119891 119863119891) represents
the dynamic input output and direct input matrices of theMR damper model The following figure shows the resultsobtained from the Narx model and the nonlinear modelExcellent matching of dynamic behavior is demonstrated inFigure 3
The state-space model obtained from Narx polynomial isutilized in the control scheme of the multidegree of freedomstructure found in the preceding The controller design ispresented next
4 Controller Design
Assuming that the control effort will be utilized to isolatethe passive structure from ground excitation as shown inFigure 4 A schematic of the control process implemented onthe structure is shown in Figure 5
The (KAFB) dynamic model is formed from the com-bined dynamics of the MR damper transmitted force repre-sented by (15) and the dynamics of the structure representedby (6) and (7)
The equivalent continuous state-spacemodel of the trans-mitted forcewhere acceleration is the input to the forcemodelis
119891= 119860119891119909119891+ 119861119891119884119884
119865 = 119862119891119909119891+ 119863119891119884119884
(16)
where (119884119884(119905)) is the base acceleration and 119909119891is a vector of the
transmitted force states 119860119891 119861119891 119862119891 and 119863
119891are dynamics
input output and direct input matrices of the transmittedforce block of Figure 5 respectively The term 119865 denotes theforce transmitted to the structure through its elastic base andis a function of time
Using the above analysis we can express the continuousstate-space model of the structure described by (7) and (8) as
119891= 119860119904119911 + 119861119904119865
119883119905= 119862119904119911 + 119863
119904119865
(17)
where that the term 119906(119905) of (6) and (7) has been replacedby 119865 in (17) to indicate that the input to the structure is theforce transmitted to it through its elastic base and includesthe seismic excitation and the damper control force
A state-space model of the beam-base system (the trans-mitted force represented by (16) and the structure representedby (17)) can now be constructed by augmenting the two partstogether such that
119860119886= [
119860119891
0
119861119904119862119891
119860119904
]
119861119886= [
119861119891
119861119904119863119891
]
119862119886= lfloor119863119904119862119891 119862
119904rfloor
119863119886= lfloor119863119891119863119904rfloor
(18)
where 119860119886 119861119886 119862119886 and 119863
119886represent the state-space matrices
of the augmented base-beam system in which the first twostates belong to the transmitted force part and the remainingstates belong to the structure mounted on the base
Mathematical Problems in Engineering 5
11990911199092
Figure 4 Passive structure
Structure
Nonlinear MRdamper model
Gain
MR damper force
Structure model
Controller
1199091 1199092
119870119886
119870119888
sum
sum
Figure 5 Active structure
Matrices 119860119886 119861119886 and 119862
119886are used for designing the
(KAFB) matrix of gains (119870119886) such that
119870119886= 119878∘119862119879
119886119877minus1 (19)
The column vector 119870119886in this case is a (2 + 2119899) times 1 column
vector and the first two rows are the Kalman gains of thestates of the transmitted force and the remaining 2119899 gainsare those of the states of the structure 119878
∘is the steady-state
solution of the following filter algebraic Riccati equation
119878 = 119860119886119878 + 119878119860
119879minus 119878119862119879
119886119877minus1119862119886119878 + 119861119886119876119861119879
119886 (20)
Matrices 119877 and 119876 are positive definite and positive semidef-inite matrices respectively [29 30] Proper choice of 119877 and119876 is important because both matrices are heavily involved
0 5 10 15 20 25 30
0
5El Centro
0 5 10 15 20 25 30
0010203
Time (s)
Time (s)
minus5
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus02
minus01
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
Figure 6 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)El Centro earthquake input
in the construction of the cost function In this work thevalue of 119876119877 ranges from 1 to 100 This ratio is limitedby the maximum force that can be provided by the MRdamper and keep the control force within its saturationlimit If the 119877 is very high sensitivity to measurementnoise will be accentuated and the controller performancemight be degraded After many iterations and driven bythe objective of minimizing accelerations and displacementswhile maintaining damper force with allowable limits thefinal 119877 and 119876matrices were selected
For a specific value of 119877 and 119876 Kalman matrix of gains(119870119886) of (19) is
119870119886= [
[119870119891]
[119870119904]
] =
[[[[[[[
[
[1198701
1198702
]
[[
[
1198703
1198702119899+2
]]
]
]]]]]]]
]
(21)
Equation (21) shows that 119870119886is partitioned into two parts
namely 119870119891which corrects the estimates of 119909
119891in (16) and
119870119904which corrects the estimates beam states (ie119883
119905of (17))
In general the structure of the Kalman estimator takeson a particularly simple structure that closely resembles theoriginal dynamic system [29 30] The complete vibrationisolation scheme proposed by this study is shown in Figure 5119870119888in Figure 5 is the linear quadratic regulator (LQR) gain
obtained with a similar procedure used to obtain119870119886
It is well known that the Kalman estimator is subject toall deterministic inputs that the plant is subject to including
6 Mathematical Problems in Engineering
Time (s)0 5 10 15 20 25 30
0123
Hachinohe
0 5 10 15 20 25 30
00102
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus03
minus02
minus01
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
minus1
minus2
Figure 7 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Hachinohe earthquake
0 5 10 15 20 25 30
0
10
20 Kobe
0 5 10 15 20 25 30
0
05
1
Time (s)
Time (s)
minus20
minus10
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
Figure 8 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Kobe earthquake
the estimated damper control force shown in Figure 5 Thisis why the realization of the structure inside the controllerin Figure 5 is subject to the estimated transmitted damperforce twice These two forces have the same magnitudeand like the two forces acting on the structure (plant) theyare opposite in sign nullifying the net force seen by the
Time (s)0 5 10 15 20 25 30
05
10Northridge
0 5 10 15 20 25 30
0
05
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
minus15
minus10
minus5
Figure 9 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Northridge earthquake
realization of the structure inside the controller Thereforethe Kalman estimate of the acceleration of any point onthe beam is identically zero which eliminates the need forrealizing (including) the structure inside the controller whichsubsequently yields a second-order control scheme regardlessof the order of the plant model This lowers the complexityof the controller and therefore significantly reduces thecomputational time Simulated results of the 20th floor arepresented in Figures 6 7 8 and 9 Each figure has two plotsone for acceleration and the other for displacement Each plotshows the uncontrolled and the controlled response
5 Conclusions
To demonstrate the advanced performance of a semiactiveMR damper used in the protection of a 20-story structurefrom earthquake damage a control strategy based on alinear quadratic Gaussian regulator is proposed First a linearmodel of the structure is derived using ETABS SecondSystem identification is used to derive a linear low-ordermodel of the MR damper from the Bouc-Wen nonlinearmodel Third a Kalman filter is designed to best estimatethe states of the system for feedback implementation pur-poses Finally an LQG controller is designed to minimizedynamic loads and structural damage Extensive simulationis performed to test and validate the effectiveness of theMR damper control strategy Four historical earthquakes areapplied to the structure and theMRdamper is set active Aftersuccessful simulation accelerations and drifts at all structurefloors are computed Proposed optimal control ofMRdamper
Mathematical Problems in Engineering 7
effectiveness is demonstrated by significantly reducing theaccelerations and displacements at all floors of the structure
Acknowledgment
The authors acknowledge the support of the AmericanUniversity of Sharjah
References
[1] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation John Wiley amp Sons Chichester UK1993
[2] J M Kelly Earthquake-Resistant Design With Rubber SpringerLondon UK 2nd edition 1997
[3] F Naeim and J M Kelly Design of Seismic Isolated StructuresJohn Wiley amp Sons New York NY USA 1999
[4] R S Jangid and J M Kelly ldquoBase isolation for near-faultmotionsrdquoEarthquake Engineering and Structural Dynamics vol30 no 5 pp 691ndash707 2001
[5] A Rodriguez-Marek Near fault seismic site response [PhDthesis] Civil Engineering University of California BerkeleyCalif USA 2000
[6] G A MacRae D V Morrow and C W Roeder ldquoNear-faultground motion effects on simple structuresrdquo ASCE Journal ofStructural Engineering vol 127 no 9 pp 996ndash1004 2001
[7] A K Chopra and C Chintanapakdee ldquoComparing responseof SDF systems to near-fault and far-fault earthquake motionsin the context of spectral regionsrdquo Earthquake Engineering andStructural Dynamics vol 30 no 12 pp 1769ndash1789 2001
[8] N Makris and C J Black ldquoDimensional analysis of bilinearoscillators under pulse-type excitationsrdquo ASCE Journal of Engi-neering Mechanics vol 130 no 9 pp 1019ndash1031 2004
[9] C P Providakis ldquoEffect of LRB isolators and supplementalviscous dampers on seismic isolated buildings under near-faultexcitationsrdquo Engineering Structures vol 30 no 5 pp 1187ndash11982008
[10] C P Providakis ldquoEffect of supplemental damping on LRB andFPS seismic isolators under near-fault ground motionsrdquo SoilDynamics and Earthquake Engineering vol 29 no 1 pp 80ndash902009
[11] R S Jangid ldquoOptimum friction pendulum system for near-faultmotionsrdquoEngineering Structures vol 27 no 3 pp 349ndash3592005
[12] S M Hussain J W Asher and R D Ewing ldquoSeismic baseisolation design for the San Bernardino CountyMedical Centerreplacement projectrdquo in Proceedings of the Symposium onStructural Engineering in Natural Hazards Mitigation vol 1 pp760ndash765 Irvine Calif USA April 1993
[13] S Hussain ldquoPerformance of base isolated buildings in thenorthridge earthquakerdquo in Seismic Base Isolation State of thePractice Seminar Structural Engineers Association of SouthernCalifornia 1994
[14] S Hussain and E Retamal ldquoA hybrid seismic isolation systemmdashisolators with supplemental viscous dampersrdquo in Proceedings ofthe 1st World Conference on Structural Control vol 3 pp FA2-53ndashFA2-62 International Association for Structural ControlLos Angeles Calif USA August 1994
[15] M Al Satari and J Abdalla ldquoOptimization of a base-isolationsystem consisting of natural rubber bearings and fluid viscous
dampersrdquo in Proceedings of the 11thWorld Conference on SeismicIsolation Energy Dissipation and Active Vibration Control ofStructures Guangzhou China 2009
[16] S Hussain and M Al Satari ldquoInnovative design of a seismicisolation supplemental viscous damping systems of an essentialservices facility in a near-fault regionrdquo in Proceedings of the 14thWorld Conference on Earthquake Engineering Beijing China2008
[17] S Hussain and M Al Satari ldquoDesign of a seismic isolationsystem with supplemental viscous damping for a near-faultessential services facilityrdquo in Proceedings of the 76th StructuralEngineers Association of California Annual Convention SquawCreek Calif USA 2007
[18] S Hussain andM Al Satari ldquoViscous-damped seismic isolationsystem for a near-fault essential services facilityrdquo in Proceedingsof the 10th World Conference on Seismic Isolation EnergyDissipation and Active Vibration Control of Structures IstanbulTurkey 2007
[19] M AlHamaydeh and S Hussain ldquoInnovative design of aseismically-isolated building with supplemental dampingrdquo inProceedings of 14th European Conference on Earthquake Engi-neering (ECEE rsquo10) Ohrid Republic of Macedonia 2010
[20] M AlHamaydeh S Barakat and F Abed ldquoMultiple regressionmodeling of natural rubber seismic-isolation systems withsupplemental viscous damping for near-field ground motionrdquoJournal of Civil Engineering and Management In press
[21] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo ASCE Journal of Engineering Mechanics vol128 no 10 pp 1088ndash1099 2002
[22] A Friedman J Zhang Y Cha et al ldquoAccommodating MRdamper dynamics for control of large scale structures systemsrdquoin Proceddings of the 5thWorld Conference on Structural Controland Monitoring 2010
[23] H J Jung B F Spencer Jr and I W Lee ldquoControl ofseismically excited cable-stayed bridge employing magnetorhe-ological fluid dampersrdquo Journal of Structural Engineering vol129 no 7 pp 873ndash883 2003
[24] M Zhao andWQ Zhu ldquoStochastic optimal semi-active controlof stay cables by using magneto-rheological damperrdquo Journal ofVibration and Control vol 17 no 13 pp 1921ndash1929 2011
[25] J Hoslashgsberg ldquoThe role of negative stiffness in semi-activecontrol of magneto-rheological dampersrdquo Structural Controland Health Monitoring vol 18 no 3 pp 289ndash304 2011
[26] K Assaleh T Shanableh and Y Kheil ldquoGroup method of datahandling for modeling magnetorheological dampersrdquo Intelli-gent Control and Automation vol 4 no 1 pp 70ndash79 2013
[27] B F Spencer S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997
[28] Y Ohtori R E Christenson B F Spencer Jr and S J DykeldquoBenchmark control problems for seismically excited nonlinearbuildingsrdquo ASCE Journal of Engineering Mechanics vol 130 no4 pp 366ndash385 2004
[29] AH El-Sinawi ldquoActive vibration isolation of a flexible structuremounted on a vibrating elastic baserdquo Journal of Sound andVibration vol 271 no 1-2 pp 323ndash337 2004
[30] A El-Sinawi and A R Kashani ldquoActive isolation using aKalman estimator-based controllerrdquo Journal of Vibration andControl vol 7 no 8 pp 1163ndash1173 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
0
500
1000
Forc
e (N
)
01 02 03 04 05 06 07 08 09Time (s)
minus1500
minus1000
minus500
Force (sim)
ze measuredNarx3 fit 4097Narx8 fit 9305
Fit 9306Fit 8131
Figure 2 MR damper response versus different identified models
01 02 03 04 05 06 07 08
0
500
1000
1500
2000
Time (s)
MR
dam
per f
orce
(N)
Nonlinear versus linear response
Narx modelNonlinear model
minus1500
minus1000
minus500
Figure 3 Comparison between the state-spacemodel andnonlinearmodel
The comparison is shown in Figure 3 The state-space modelof the damper is expressed as
119909119891= 119860119891119909119891+ 119861119891119906119891
119884119891= 119862119891119909119891+119863119891119906119891
(15)
where 119909119891
is the state vector of the MR damper state-space model 119906
119891is the input and 119884
119891is the output (ie
damper force) The quadruple (119860119891 119861119891 119862119891 119863119891) represents
the dynamic input output and direct input matrices of theMR damper model The following figure shows the resultsobtained from the Narx model and the nonlinear modelExcellent matching of dynamic behavior is demonstrated inFigure 3
The state-space model obtained from Narx polynomial isutilized in the control scheme of the multidegree of freedomstructure found in the preceding The controller design ispresented next
4 Controller Design
Assuming that the control effort will be utilized to isolatethe passive structure from ground excitation as shown inFigure 4 A schematic of the control process implemented onthe structure is shown in Figure 5
The (KAFB) dynamic model is formed from the com-bined dynamics of the MR damper transmitted force repre-sented by (15) and the dynamics of the structure representedby (6) and (7)
The equivalent continuous state-spacemodel of the trans-mitted forcewhere acceleration is the input to the forcemodelis
119891= 119860119891119909119891+ 119861119891119884119884
119865 = 119862119891119909119891+ 119863119891119884119884
(16)
where (119884119884(119905)) is the base acceleration and 119909119891is a vector of the
transmitted force states 119860119891 119861119891 119862119891 and 119863
119891are dynamics
input output and direct input matrices of the transmittedforce block of Figure 5 respectively The term 119865 denotes theforce transmitted to the structure through its elastic base andis a function of time
Using the above analysis we can express the continuousstate-space model of the structure described by (7) and (8) as
119891= 119860119904119911 + 119861119904119865
119883119905= 119862119904119911 + 119863
119904119865
(17)
where that the term 119906(119905) of (6) and (7) has been replacedby 119865 in (17) to indicate that the input to the structure is theforce transmitted to it through its elastic base and includesthe seismic excitation and the damper control force
A state-space model of the beam-base system (the trans-mitted force represented by (16) and the structure representedby (17)) can now be constructed by augmenting the two partstogether such that
119860119886= [
119860119891
0
119861119904119862119891
119860119904
]
119861119886= [
119861119891
119861119904119863119891
]
119862119886= lfloor119863119904119862119891 119862
119904rfloor
119863119886= lfloor119863119891119863119904rfloor
(18)
where 119860119886 119861119886 119862119886 and 119863
119886represent the state-space matrices
of the augmented base-beam system in which the first twostates belong to the transmitted force part and the remainingstates belong to the structure mounted on the base
Mathematical Problems in Engineering 5
11990911199092
Figure 4 Passive structure
Structure
Nonlinear MRdamper model
Gain
MR damper force
Structure model
Controller
1199091 1199092
119870119886
119870119888
sum
sum
Figure 5 Active structure
Matrices 119860119886 119861119886 and 119862
119886are used for designing the
(KAFB) matrix of gains (119870119886) such that
119870119886= 119878∘119862119879
119886119877minus1 (19)
The column vector 119870119886in this case is a (2 + 2119899) times 1 column
vector and the first two rows are the Kalman gains of thestates of the transmitted force and the remaining 2119899 gainsare those of the states of the structure 119878
∘is the steady-state
solution of the following filter algebraic Riccati equation
119878 = 119860119886119878 + 119878119860
119879minus 119878119862119879
119886119877minus1119862119886119878 + 119861119886119876119861119879
119886 (20)
Matrices 119877 and 119876 are positive definite and positive semidef-inite matrices respectively [29 30] Proper choice of 119877 and119876 is important because both matrices are heavily involved
0 5 10 15 20 25 30
0
5El Centro
0 5 10 15 20 25 30
0010203
Time (s)
Time (s)
minus5
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus02
minus01
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
Figure 6 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)El Centro earthquake input
in the construction of the cost function In this work thevalue of 119876119877 ranges from 1 to 100 This ratio is limitedby the maximum force that can be provided by the MRdamper and keep the control force within its saturationlimit If the 119877 is very high sensitivity to measurementnoise will be accentuated and the controller performancemight be degraded After many iterations and driven bythe objective of minimizing accelerations and displacementswhile maintaining damper force with allowable limits thefinal 119877 and 119876matrices were selected
For a specific value of 119877 and 119876 Kalman matrix of gains(119870119886) of (19) is
119870119886= [
[119870119891]
[119870119904]
] =
[[[[[[[
[
[1198701
1198702
]
[[
[
1198703
1198702119899+2
]]
]
]]]]]]]
]
(21)
Equation (21) shows that 119870119886is partitioned into two parts
namely 119870119891which corrects the estimates of 119909
119891in (16) and
119870119904which corrects the estimates beam states (ie119883
119905of (17))
In general the structure of the Kalman estimator takeson a particularly simple structure that closely resembles theoriginal dynamic system [29 30] The complete vibrationisolation scheme proposed by this study is shown in Figure 5119870119888in Figure 5 is the linear quadratic regulator (LQR) gain
obtained with a similar procedure used to obtain119870119886
It is well known that the Kalman estimator is subject toall deterministic inputs that the plant is subject to including
6 Mathematical Problems in Engineering
Time (s)0 5 10 15 20 25 30
0123
Hachinohe
0 5 10 15 20 25 30
00102
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus03
minus02
minus01
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
minus1
minus2
Figure 7 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Hachinohe earthquake
0 5 10 15 20 25 30
0
10
20 Kobe
0 5 10 15 20 25 30
0
05
1
Time (s)
Time (s)
minus20
minus10
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
Figure 8 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Kobe earthquake
the estimated damper control force shown in Figure 5 Thisis why the realization of the structure inside the controllerin Figure 5 is subject to the estimated transmitted damperforce twice These two forces have the same magnitudeand like the two forces acting on the structure (plant) theyare opposite in sign nullifying the net force seen by the
Time (s)0 5 10 15 20 25 30
05
10Northridge
0 5 10 15 20 25 30
0
05
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
minus15
minus10
minus5
Figure 9 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Northridge earthquake
realization of the structure inside the controller Thereforethe Kalman estimate of the acceleration of any point onthe beam is identically zero which eliminates the need forrealizing (including) the structure inside the controller whichsubsequently yields a second-order control scheme regardlessof the order of the plant model This lowers the complexityof the controller and therefore significantly reduces thecomputational time Simulated results of the 20th floor arepresented in Figures 6 7 8 and 9 Each figure has two plotsone for acceleration and the other for displacement Each plotshows the uncontrolled and the controlled response
5 Conclusions
To demonstrate the advanced performance of a semiactiveMR damper used in the protection of a 20-story structurefrom earthquake damage a control strategy based on alinear quadratic Gaussian regulator is proposed First a linearmodel of the structure is derived using ETABS SecondSystem identification is used to derive a linear low-ordermodel of the MR damper from the Bouc-Wen nonlinearmodel Third a Kalman filter is designed to best estimatethe states of the system for feedback implementation pur-poses Finally an LQG controller is designed to minimizedynamic loads and structural damage Extensive simulationis performed to test and validate the effectiveness of theMR damper control strategy Four historical earthquakes areapplied to the structure and theMRdamper is set active Aftersuccessful simulation accelerations and drifts at all structurefloors are computed Proposed optimal control ofMRdamper
Mathematical Problems in Engineering 7
effectiveness is demonstrated by significantly reducing theaccelerations and displacements at all floors of the structure
Acknowledgment
The authors acknowledge the support of the AmericanUniversity of Sharjah
References
[1] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation John Wiley amp Sons Chichester UK1993
[2] J M Kelly Earthquake-Resistant Design With Rubber SpringerLondon UK 2nd edition 1997
[3] F Naeim and J M Kelly Design of Seismic Isolated StructuresJohn Wiley amp Sons New York NY USA 1999
[4] R S Jangid and J M Kelly ldquoBase isolation for near-faultmotionsrdquoEarthquake Engineering and Structural Dynamics vol30 no 5 pp 691ndash707 2001
[5] A Rodriguez-Marek Near fault seismic site response [PhDthesis] Civil Engineering University of California BerkeleyCalif USA 2000
[6] G A MacRae D V Morrow and C W Roeder ldquoNear-faultground motion effects on simple structuresrdquo ASCE Journal ofStructural Engineering vol 127 no 9 pp 996ndash1004 2001
[7] A K Chopra and C Chintanapakdee ldquoComparing responseof SDF systems to near-fault and far-fault earthquake motionsin the context of spectral regionsrdquo Earthquake Engineering andStructural Dynamics vol 30 no 12 pp 1769ndash1789 2001
[8] N Makris and C J Black ldquoDimensional analysis of bilinearoscillators under pulse-type excitationsrdquo ASCE Journal of Engi-neering Mechanics vol 130 no 9 pp 1019ndash1031 2004
[9] C P Providakis ldquoEffect of LRB isolators and supplementalviscous dampers on seismic isolated buildings under near-faultexcitationsrdquo Engineering Structures vol 30 no 5 pp 1187ndash11982008
[10] C P Providakis ldquoEffect of supplemental damping on LRB andFPS seismic isolators under near-fault ground motionsrdquo SoilDynamics and Earthquake Engineering vol 29 no 1 pp 80ndash902009
[11] R S Jangid ldquoOptimum friction pendulum system for near-faultmotionsrdquoEngineering Structures vol 27 no 3 pp 349ndash3592005
[12] S M Hussain J W Asher and R D Ewing ldquoSeismic baseisolation design for the San Bernardino CountyMedical Centerreplacement projectrdquo in Proceedings of the Symposium onStructural Engineering in Natural Hazards Mitigation vol 1 pp760ndash765 Irvine Calif USA April 1993
[13] S Hussain ldquoPerformance of base isolated buildings in thenorthridge earthquakerdquo in Seismic Base Isolation State of thePractice Seminar Structural Engineers Association of SouthernCalifornia 1994
[14] S Hussain and E Retamal ldquoA hybrid seismic isolation systemmdashisolators with supplemental viscous dampersrdquo in Proceedings ofthe 1st World Conference on Structural Control vol 3 pp FA2-53ndashFA2-62 International Association for Structural ControlLos Angeles Calif USA August 1994
[15] M Al Satari and J Abdalla ldquoOptimization of a base-isolationsystem consisting of natural rubber bearings and fluid viscous
dampersrdquo in Proceedings of the 11thWorld Conference on SeismicIsolation Energy Dissipation and Active Vibration Control ofStructures Guangzhou China 2009
[16] S Hussain and M Al Satari ldquoInnovative design of a seismicisolation supplemental viscous damping systems of an essentialservices facility in a near-fault regionrdquo in Proceedings of the 14thWorld Conference on Earthquake Engineering Beijing China2008
[17] S Hussain and M Al Satari ldquoDesign of a seismic isolationsystem with supplemental viscous damping for a near-faultessential services facilityrdquo in Proceedings of the 76th StructuralEngineers Association of California Annual Convention SquawCreek Calif USA 2007
[18] S Hussain andM Al Satari ldquoViscous-damped seismic isolationsystem for a near-fault essential services facilityrdquo in Proceedingsof the 10th World Conference on Seismic Isolation EnergyDissipation and Active Vibration Control of Structures IstanbulTurkey 2007
[19] M AlHamaydeh and S Hussain ldquoInnovative design of aseismically-isolated building with supplemental dampingrdquo inProceedings of 14th European Conference on Earthquake Engi-neering (ECEE rsquo10) Ohrid Republic of Macedonia 2010
[20] M AlHamaydeh S Barakat and F Abed ldquoMultiple regressionmodeling of natural rubber seismic-isolation systems withsupplemental viscous damping for near-field ground motionrdquoJournal of Civil Engineering and Management In press
[21] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo ASCE Journal of Engineering Mechanics vol128 no 10 pp 1088ndash1099 2002
[22] A Friedman J Zhang Y Cha et al ldquoAccommodating MRdamper dynamics for control of large scale structures systemsrdquoin Proceddings of the 5thWorld Conference on Structural Controland Monitoring 2010
[23] H J Jung B F Spencer Jr and I W Lee ldquoControl ofseismically excited cable-stayed bridge employing magnetorhe-ological fluid dampersrdquo Journal of Structural Engineering vol129 no 7 pp 873ndash883 2003
[24] M Zhao andWQ Zhu ldquoStochastic optimal semi-active controlof stay cables by using magneto-rheological damperrdquo Journal ofVibration and Control vol 17 no 13 pp 1921ndash1929 2011
[25] J Hoslashgsberg ldquoThe role of negative stiffness in semi-activecontrol of magneto-rheological dampersrdquo Structural Controland Health Monitoring vol 18 no 3 pp 289ndash304 2011
[26] K Assaleh T Shanableh and Y Kheil ldquoGroup method of datahandling for modeling magnetorheological dampersrdquo Intelli-gent Control and Automation vol 4 no 1 pp 70ndash79 2013
[27] B F Spencer S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997
[28] Y Ohtori R E Christenson B F Spencer Jr and S J DykeldquoBenchmark control problems for seismically excited nonlinearbuildingsrdquo ASCE Journal of Engineering Mechanics vol 130 no4 pp 366ndash385 2004
[29] AH El-Sinawi ldquoActive vibration isolation of a flexible structuremounted on a vibrating elastic baserdquo Journal of Sound andVibration vol 271 no 1-2 pp 323ndash337 2004
[30] A El-Sinawi and A R Kashani ldquoActive isolation using aKalman estimator-based controllerrdquo Journal of Vibration andControl vol 7 no 8 pp 1163ndash1173 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
11990911199092
Figure 4 Passive structure
Structure
Nonlinear MRdamper model
Gain
MR damper force
Structure model
Controller
1199091 1199092
119870119886
119870119888
sum
sum
Figure 5 Active structure
Matrices 119860119886 119861119886 and 119862
119886are used for designing the
(KAFB) matrix of gains (119870119886) such that
119870119886= 119878∘119862119879
119886119877minus1 (19)
The column vector 119870119886in this case is a (2 + 2119899) times 1 column
vector and the first two rows are the Kalman gains of thestates of the transmitted force and the remaining 2119899 gainsare those of the states of the structure 119878
∘is the steady-state
solution of the following filter algebraic Riccati equation
119878 = 119860119886119878 + 119878119860
119879minus 119878119862119879
119886119877minus1119862119886119878 + 119861119886119876119861119879
119886 (20)
Matrices 119877 and 119876 are positive definite and positive semidef-inite matrices respectively [29 30] Proper choice of 119877 and119876 is important because both matrices are heavily involved
0 5 10 15 20 25 30
0
5El Centro
0 5 10 15 20 25 30
0010203
Time (s)
Time (s)
minus5
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus02
minus01
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
Figure 6 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)El Centro earthquake input
in the construction of the cost function In this work thevalue of 119876119877 ranges from 1 to 100 This ratio is limitedby the maximum force that can be provided by the MRdamper and keep the control force within its saturationlimit If the 119877 is very high sensitivity to measurementnoise will be accentuated and the controller performancemight be degraded After many iterations and driven bythe objective of minimizing accelerations and displacementswhile maintaining damper force with allowable limits thefinal 119877 and 119876matrices were selected
For a specific value of 119877 and 119876 Kalman matrix of gains(119870119886) of (19) is
119870119886= [
[119870119891]
[119870119904]
] =
[[[[[[[
[
[1198701
1198702
]
[[
[
1198703
1198702119899+2
]]
]
]]]]]]]
]
(21)
Equation (21) shows that 119870119886is partitioned into two parts
namely 119870119891which corrects the estimates of 119909
119891in (16) and
119870119904which corrects the estimates beam states (ie119883
119905of (17))
In general the structure of the Kalman estimator takeson a particularly simple structure that closely resembles theoriginal dynamic system [29 30] The complete vibrationisolation scheme proposed by this study is shown in Figure 5119870119888in Figure 5 is the linear quadratic regulator (LQR) gain
obtained with a similar procedure used to obtain119870119886
It is well known that the Kalman estimator is subject toall deterministic inputs that the plant is subject to including
6 Mathematical Problems in Engineering
Time (s)0 5 10 15 20 25 30
0123
Hachinohe
0 5 10 15 20 25 30
00102
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus03
minus02
minus01
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
minus1
minus2
Figure 7 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Hachinohe earthquake
0 5 10 15 20 25 30
0
10
20 Kobe
0 5 10 15 20 25 30
0
05
1
Time (s)
Time (s)
minus20
minus10
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
Figure 8 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Kobe earthquake
the estimated damper control force shown in Figure 5 Thisis why the realization of the structure inside the controllerin Figure 5 is subject to the estimated transmitted damperforce twice These two forces have the same magnitudeand like the two forces acting on the structure (plant) theyare opposite in sign nullifying the net force seen by the
Time (s)0 5 10 15 20 25 30
05
10Northridge
0 5 10 15 20 25 30
0
05
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
minus15
minus10
minus5
Figure 9 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Northridge earthquake
realization of the structure inside the controller Thereforethe Kalman estimate of the acceleration of any point onthe beam is identically zero which eliminates the need forrealizing (including) the structure inside the controller whichsubsequently yields a second-order control scheme regardlessof the order of the plant model This lowers the complexityof the controller and therefore significantly reduces thecomputational time Simulated results of the 20th floor arepresented in Figures 6 7 8 and 9 Each figure has two plotsone for acceleration and the other for displacement Each plotshows the uncontrolled and the controlled response
5 Conclusions
To demonstrate the advanced performance of a semiactiveMR damper used in the protection of a 20-story structurefrom earthquake damage a control strategy based on alinear quadratic Gaussian regulator is proposed First a linearmodel of the structure is derived using ETABS SecondSystem identification is used to derive a linear low-ordermodel of the MR damper from the Bouc-Wen nonlinearmodel Third a Kalman filter is designed to best estimatethe states of the system for feedback implementation pur-poses Finally an LQG controller is designed to minimizedynamic loads and structural damage Extensive simulationis performed to test and validate the effectiveness of theMR damper control strategy Four historical earthquakes areapplied to the structure and theMRdamper is set active Aftersuccessful simulation accelerations and drifts at all structurefloors are computed Proposed optimal control ofMRdamper
Mathematical Problems in Engineering 7
effectiveness is demonstrated by significantly reducing theaccelerations and displacements at all floors of the structure
Acknowledgment
The authors acknowledge the support of the AmericanUniversity of Sharjah
References
[1] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation John Wiley amp Sons Chichester UK1993
[2] J M Kelly Earthquake-Resistant Design With Rubber SpringerLondon UK 2nd edition 1997
[3] F Naeim and J M Kelly Design of Seismic Isolated StructuresJohn Wiley amp Sons New York NY USA 1999
[4] R S Jangid and J M Kelly ldquoBase isolation for near-faultmotionsrdquoEarthquake Engineering and Structural Dynamics vol30 no 5 pp 691ndash707 2001
[5] A Rodriguez-Marek Near fault seismic site response [PhDthesis] Civil Engineering University of California BerkeleyCalif USA 2000
[6] G A MacRae D V Morrow and C W Roeder ldquoNear-faultground motion effects on simple structuresrdquo ASCE Journal ofStructural Engineering vol 127 no 9 pp 996ndash1004 2001
[7] A K Chopra and C Chintanapakdee ldquoComparing responseof SDF systems to near-fault and far-fault earthquake motionsin the context of spectral regionsrdquo Earthquake Engineering andStructural Dynamics vol 30 no 12 pp 1769ndash1789 2001
[8] N Makris and C J Black ldquoDimensional analysis of bilinearoscillators under pulse-type excitationsrdquo ASCE Journal of Engi-neering Mechanics vol 130 no 9 pp 1019ndash1031 2004
[9] C P Providakis ldquoEffect of LRB isolators and supplementalviscous dampers on seismic isolated buildings under near-faultexcitationsrdquo Engineering Structures vol 30 no 5 pp 1187ndash11982008
[10] C P Providakis ldquoEffect of supplemental damping on LRB andFPS seismic isolators under near-fault ground motionsrdquo SoilDynamics and Earthquake Engineering vol 29 no 1 pp 80ndash902009
[11] R S Jangid ldquoOptimum friction pendulum system for near-faultmotionsrdquoEngineering Structures vol 27 no 3 pp 349ndash3592005
[12] S M Hussain J W Asher and R D Ewing ldquoSeismic baseisolation design for the San Bernardino CountyMedical Centerreplacement projectrdquo in Proceedings of the Symposium onStructural Engineering in Natural Hazards Mitigation vol 1 pp760ndash765 Irvine Calif USA April 1993
[13] S Hussain ldquoPerformance of base isolated buildings in thenorthridge earthquakerdquo in Seismic Base Isolation State of thePractice Seminar Structural Engineers Association of SouthernCalifornia 1994
[14] S Hussain and E Retamal ldquoA hybrid seismic isolation systemmdashisolators with supplemental viscous dampersrdquo in Proceedings ofthe 1st World Conference on Structural Control vol 3 pp FA2-53ndashFA2-62 International Association for Structural ControlLos Angeles Calif USA August 1994
[15] M Al Satari and J Abdalla ldquoOptimization of a base-isolationsystem consisting of natural rubber bearings and fluid viscous
dampersrdquo in Proceedings of the 11thWorld Conference on SeismicIsolation Energy Dissipation and Active Vibration Control ofStructures Guangzhou China 2009
[16] S Hussain and M Al Satari ldquoInnovative design of a seismicisolation supplemental viscous damping systems of an essentialservices facility in a near-fault regionrdquo in Proceedings of the 14thWorld Conference on Earthquake Engineering Beijing China2008
[17] S Hussain and M Al Satari ldquoDesign of a seismic isolationsystem with supplemental viscous damping for a near-faultessential services facilityrdquo in Proceedings of the 76th StructuralEngineers Association of California Annual Convention SquawCreek Calif USA 2007
[18] S Hussain andM Al Satari ldquoViscous-damped seismic isolationsystem for a near-fault essential services facilityrdquo in Proceedingsof the 10th World Conference on Seismic Isolation EnergyDissipation and Active Vibration Control of Structures IstanbulTurkey 2007
[19] M AlHamaydeh and S Hussain ldquoInnovative design of aseismically-isolated building with supplemental dampingrdquo inProceedings of 14th European Conference on Earthquake Engi-neering (ECEE rsquo10) Ohrid Republic of Macedonia 2010
[20] M AlHamaydeh S Barakat and F Abed ldquoMultiple regressionmodeling of natural rubber seismic-isolation systems withsupplemental viscous damping for near-field ground motionrdquoJournal of Civil Engineering and Management In press
[21] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo ASCE Journal of Engineering Mechanics vol128 no 10 pp 1088ndash1099 2002
[22] A Friedman J Zhang Y Cha et al ldquoAccommodating MRdamper dynamics for control of large scale structures systemsrdquoin Proceddings of the 5thWorld Conference on Structural Controland Monitoring 2010
[23] H J Jung B F Spencer Jr and I W Lee ldquoControl ofseismically excited cable-stayed bridge employing magnetorhe-ological fluid dampersrdquo Journal of Structural Engineering vol129 no 7 pp 873ndash883 2003
[24] M Zhao andWQ Zhu ldquoStochastic optimal semi-active controlof stay cables by using magneto-rheological damperrdquo Journal ofVibration and Control vol 17 no 13 pp 1921ndash1929 2011
[25] J Hoslashgsberg ldquoThe role of negative stiffness in semi-activecontrol of magneto-rheological dampersrdquo Structural Controland Health Monitoring vol 18 no 3 pp 289ndash304 2011
[26] K Assaleh T Shanableh and Y Kheil ldquoGroup method of datahandling for modeling magnetorheological dampersrdquo Intelli-gent Control and Automation vol 4 no 1 pp 70ndash79 2013
[27] B F Spencer S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997
[28] Y Ohtori R E Christenson B F Spencer Jr and S J DykeldquoBenchmark control problems for seismically excited nonlinearbuildingsrdquo ASCE Journal of Engineering Mechanics vol 130 no4 pp 366ndash385 2004
[29] AH El-Sinawi ldquoActive vibration isolation of a flexible structuremounted on a vibrating elastic baserdquo Journal of Sound andVibration vol 271 no 1-2 pp 323ndash337 2004
[30] A El-Sinawi and A R Kashani ldquoActive isolation using aKalman estimator-based controllerrdquo Journal of Vibration andControl vol 7 no 8 pp 1163ndash1173 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Time (s)0 5 10 15 20 25 30
0123
Hachinohe
0 5 10 15 20 25 30
00102
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus03
minus02
minus01
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
minus1
minus2
Figure 7 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Hachinohe earthquake
0 5 10 15 20 25 30
0
10
20 Kobe
0 5 10 15 20 25 30
0
05
1
Time (s)
Time (s)
minus20
minus10
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)
Uncontrolled displacementControlled displacement
Figure 8 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Kobe earthquake
the estimated damper control force shown in Figure 5 Thisis why the realization of the structure inside the controllerin Figure 5 is subject to the estimated transmitted damperforce twice These two forces have the same magnitudeand like the two forces acting on the structure (plant) theyare opposite in sign nullifying the net force seen by the
Time (s)0 5 10 15 20 25 30
05
10Northridge
0 5 10 15 20 25 30
0
05
Time (s)
Uncontrolled accelerationControlled acceleration
Acce
lera
tion
(ms2)
minus1
minus05
Disp
lace
men
t (m
)Uncontrolled displacementControlled displacement
minus15
minus10
minus5
Figure 9 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Northridge earthquake
realization of the structure inside the controller Thereforethe Kalman estimate of the acceleration of any point onthe beam is identically zero which eliminates the need forrealizing (including) the structure inside the controller whichsubsequently yields a second-order control scheme regardlessof the order of the plant model This lowers the complexityof the controller and therefore significantly reduces thecomputational time Simulated results of the 20th floor arepresented in Figures 6 7 8 and 9 Each figure has two plotsone for acceleration and the other for displacement Each plotshows the uncontrolled and the controlled response
5 Conclusions
To demonstrate the advanced performance of a semiactiveMR damper used in the protection of a 20-story structurefrom earthquake damage a control strategy based on alinear quadratic Gaussian regulator is proposed First a linearmodel of the structure is derived using ETABS SecondSystem identification is used to derive a linear low-ordermodel of the MR damper from the Bouc-Wen nonlinearmodel Third a Kalman filter is designed to best estimatethe states of the system for feedback implementation pur-poses Finally an LQG controller is designed to minimizedynamic loads and structural damage Extensive simulationis performed to test and validate the effectiveness of theMR damper control strategy Four historical earthquakes areapplied to the structure and theMRdamper is set active Aftersuccessful simulation accelerations and drifts at all structurefloors are computed Proposed optimal control ofMRdamper
Mathematical Problems in Engineering 7
effectiveness is demonstrated by significantly reducing theaccelerations and displacements at all floors of the structure
Acknowledgment
The authors acknowledge the support of the AmericanUniversity of Sharjah
References
[1] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation John Wiley amp Sons Chichester UK1993
[2] J M Kelly Earthquake-Resistant Design With Rubber SpringerLondon UK 2nd edition 1997
[3] F Naeim and J M Kelly Design of Seismic Isolated StructuresJohn Wiley amp Sons New York NY USA 1999
[4] R S Jangid and J M Kelly ldquoBase isolation for near-faultmotionsrdquoEarthquake Engineering and Structural Dynamics vol30 no 5 pp 691ndash707 2001
[5] A Rodriguez-Marek Near fault seismic site response [PhDthesis] Civil Engineering University of California BerkeleyCalif USA 2000
[6] G A MacRae D V Morrow and C W Roeder ldquoNear-faultground motion effects on simple structuresrdquo ASCE Journal ofStructural Engineering vol 127 no 9 pp 996ndash1004 2001
[7] A K Chopra and C Chintanapakdee ldquoComparing responseof SDF systems to near-fault and far-fault earthquake motionsin the context of spectral regionsrdquo Earthquake Engineering andStructural Dynamics vol 30 no 12 pp 1769ndash1789 2001
[8] N Makris and C J Black ldquoDimensional analysis of bilinearoscillators under pulse-type excitationsrdquo ASCE Journal of Engi-neering Mechanics vol 130 no 9 pp 1019ndash1031 2004
[9] C P Providakis ldquoEffect of LRB isolators and supplementalviscous dampers on seismic isolated buildings under near-faultexcitationsrdquo Engineering Structures vol 30 no 5 pp 1187ndash11982008
[10] C P Providakis ldquoEffect of supplemental damping on LRB andFPS seismic isolators under near-fault ground motionsrdquo SoilDynamics and Earthquake Engineering vol 29 no 1 pp 80ndash902009
[11] R S Jangid ldquoOptimum friction pendulum system for near-faultmotionsrdquoEngineering Structures vol 27 no 3 pp 349ndash3592005
[12] S M Hussain J W Asher and R D Ewing ldquoSeismic baseisolation design for the San Bernardino CountyMedical Centerreplacement projectrdquo in Proceedings of the Symposium onStructural Engineering in Natural Hazards Mitigation vol 1 pp760ndash765 Irvine Calif USA April 1993
[13] S Hussain ldquoPerformance of base isolated buildings in thenorthridge earthquakerdquo in Seismic Base Isolation State of thePractice Seminar Structural Engineers Association of SouthernCalifornia 1994
[14] S Hussain and E Retamal ldquoA hybrid seismic isolation systemmdashisolators with supplemental viscous dampersrdquo in Proceedings ofthe 1st World Conference on Structural Control vol 3 pp FA2-53ndashFA2-62 International Association for Structural ControlLos Angeles Calif USA August 1994
[15] M Al Satari and J Abdalla ldquoOptimization of a base-isolationsystem consisting of natural rubber bearings and fluid viscous
dampersrdquo in Proceedings of the 11thWorld Conference on SeismicIsolation Energy Dissipation and Active Vibration Control ofStructures Guangzhou China 2009
[16] S Hussain and M Al Satari ldquoInnovative design of a seismicisolation supplemental viscous damping systems of an essentialservices facility in a near-fault regionrdquo in Proceedings of the 14thWorld Conference on Earthquake Engineering Beijing China2008
[17] S Hussain and M Al Satari ldquoDesign of a seismic isolationsystem with supplemental viscous damping for a near-faultessential services facilityrdquo in Proceedings of the 76th StructuralEngineers Association of California Annual Convention SquawCreek Calif USA 2007
[18] S Hussain andM Al Satari ldquoViscous-damped seismic isolationsystem for a near-fault essential services facilityrdquo in Proceedingsof the 10th World Conference on Seismic Isolation EnergyDissipation and Active Vibration Control of Structures IstanbulTurkey 2007
[19] M AlHamaydeh and S Hussain ldquoInnovative design of aseismically-isolated building with supplemental dampingrdquo inProceedings of 14th European Conference on Earthquake Engi-neering (ECEE rsquo10) Ohrid Republic of Macedonia 2010
[20] M AlHamaydeh S Barakat and F Abed ldquoMultiple regressionmodeling of natural rubber seismic-isolation systems withsupplemental viscous damping for near-field ground motionrdquoJournal of Civil Engineering and Management In press
[21] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo ASCE Journal of Engineering Mechanics vol128 no 10 pp 1088ndash1099 2002
[22] A Friedman J Zhang Y Cha et al ldquoAccommodating MRdamper dynamics for control of large scale structures systemsrdquoin Proceddings of the 5thWorld Conference on Structural Controland Monitoring 2010
[23] H J Jung B F Spencer Jr and I W Lee ldquoControl ofseismically excited cable-stayed bridge employing magnetorhe-ological fluid dampersrdquo Journal of Structural Engineering vol129 no 7 pp 873ndash883 2003
[24] M Zhao andWQ Zhu ldquoStochastic optimal semi-active controlof stay cables by using magneto-rheological damperrdquo Journal ofVibration and Control vol 17 no 13 pp 1921ndash1929 2011
[25] J Hoslashgsberg ldquoThe role of negative stiffness in semi-activecontrol of magneto-rheological dampersrdquo Structural Controland Health Monitoring vol 18 no 3 pp 289ndash304 2011
[26] K Assaleh T Shanableh and Y Kheil ldquoGroup method of datahandling for modeling magnetorheological dampersrdquo Intelli-gent Control and Automation vol 4 no 1 pp 70ndash79 2013
[27] B F Spencer S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997
[28] Y Ohtori R E Christenson B F Spencer Jr and S J DykeldquoBenchmark control problems for seismically excited nonlinearbuildingsrdquo ASCE Journal of Engineering Mechanics vol 130 no4 pp 366ndash385 2004
[29] AH El-Sinawi ldquoActive vibration isolation of a flexible structuremounted on a vibrating elastic baserdquo Journal of Sound andVibration vol 271 no 1-2 pp 323ndash337 2004
[30] A El-Sinawi and A R Kashani ldquoActive isolation using aKalman estimator-based controllerrdquo Journal of Vibration andControl vol 7 no 8 pp 1163ndash1173 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
effectiveness is demonstrated by significantly reducing theaccelerations and displacements at all floors of the structure
Acknowledgment
The authors acknowledge the support of the AmericanUniversity of Sharjah
References
[1] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation John Wiley amp Sons Chichester UK1993
[2] J M Kelly Earthquake-Resistant Design With Rubber SpringerLondon UK 2nd edition 1997
[3] F Naeim and J M Kelly Design of Seismic Isolated StructuresJohn Wiley amp Sons New York NY USA 1999
[4] R S Jangid and J M Kelly ldquoBase isolation for near-faultmotionsrdquoEarthquake Engineering and Structural Dynamics vol30 no 5 pp 691ndash707 2001
[5] A Rodriguez-Marek Near fault seismic site response [PhDthesis] Civil Engineering University of California BerkeleyCalif USA 2000
[6] G A MacRae D V Morrow and C W Roeder ldquoNear-faultground motion effects on simple structuresrdquo ASCE Journal ofStructural Engineering vol 127 no 9 pp 996ndash1004 2001
[7] A K Chopra and C Chintanapakdee ldquoComparing responseof SDF systems to near-fault and far-fault earthquake motionsin the context of spectral regionsrdquo Earthquake Engineering andStructural Dynamics vol 30 no 12 pp 1769ndash1789 2001
[8] N Makris and C J Black ldquoDimensional analysis of bilinearoscillators under pulse-type excitationsrdquo ASCE Journal of Engi-neering Mechanics vol 130 no 9 pp 1019ndash1031 2004
[9] C P Providakis ldquoEffect of LRB isolators and supplementalviscous dampers on seismic isolated buildings under near-faultexcitationsrdquo Engineering Structures vol 30 no 5 pp 1187ndash11982008
[10] C P Providakis ldquoEffect of supplemental damping on LRB andFPS seismic isolators under near-fault ground motionsrdquo SoilDynamics and Earthquake Engineering vol 29 no 1 pp 80ndash902009
[11] R S Jangid ldquoOptimum friction pendulum system for near-faultmotionsrdquoEngineering Structures vol 27 no 3 pp 349ndash3592005
[12] S M Hussain J W Asher and R D Ewing ldquoSeismic baseisolation design for the San Bernardino CountyMedical Centerreplacement projectrdquo in Proceedings of the Symposium onStructural Engineering in Natural Hazards Mitigation vol 1 pp760ndash765 Irvine Calif USA April 1993
[13] S Hussain ldquoPerformance of base isolated buildings in thenorthridge earthquakerdquo in Seismic Base Isolation State of thePractice Seminar Structural Engineers Association of SouthernCalifornia 1994
[14] S Hussain and E Retamal ldquoA hybrid seismic isolation systemmdashisolators with supplemental viscous dampersrdquo in Proceedings ofthe 1st World Conference on Structural Control vol 3 pp FA2-53ndashFA2-62 International Association for Structural ControlLos Angeles Calif USA August 1994
[15] M Al Satari and J Abdalla ldquoOptimization of a base-isolationsystem consisting of natural rubber bearings and fluid viscous
dampersrdquo in Proceedings of the 11thWorld Conference on SeismicIsolation Energy Dissipation and Active Vibration Control ofStructures Guangzhou China 2009
[16] S Hussain and M Al Satari ldquoInnovative design of a seismicisolation supplemental viscous damping systems of an essentialservices facility in a near-fault regionrdquo in Proceedings of the 14thWorld Conference on Earthquake Engineering Beijing China2008
[17] S Hussain and M Al Satari ldquoDesign of a seismic isolationsystem with supplemental viscous damping for a near-faultessential services facilityrdquo in Proceedings of the 76th StructuralEngineers Association of California Annual Convention SquawCreek Calif USA 2007
[18] S Hussain andM Al Satari ldquoViscous-damped seismic isolationsystem for a near-fault essential services facilityrdquo in Proceedingsof the 10th World Conference on Seismic Isolation EnergyDissipation and Active Vibration Control of Structures IstanbulTurkey 2007
[19] M AlHamaydeh and S Hussain ldquoInnovative design of aseismically-isolated building with supplemental dampingrdquo inProceedings of 14th European Conference on Earthquake Engi-neering (ECEE rsquo10) Ohrid Republic of Macedonia 2010
[20] M AlHamaydeh S Barakat and F Abed ldquoMultiple regressionmodeling of natural rubber seismic-isolation systems withsupplemental viscous damping for near-field ground motionrdquoJournal of Civil Engineering and Management In press
[21] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo ASCE Journal of Engineering Mechanics vol128 no 10 pp 1088ndash1099 2002
[22] A Friedman J Zhang Y Cha et al ldquoAccommodating MRdamper dynamics for control of large scale structures systemsrdquoin Proceddings of the 5thWorld Conference on Structural Controland Monitoring 2010
[23] H J Jung B F Spencer Jr and I W Lee ldquoControl ofseismically excited cable-stayed bridge employing magnetorhe-ological fluid dampersrdquo Journal of Structural Engineering vol129 no 7 pp 873ndash883 2003
[24] M Zhao andWQ Zhu ldquoStochastic optimal semi-active controlof stay cables by using magneto-rheological damperrdquo Journal ofVibration and Control vol 17 no 13 pp 1921ndash1929 2011
[25] J Hoslashgsberg ldquoThe role of negative stiffness in semi-activecontrol of magneto-rheological dampersrdquo Structural Controland Health Monitoring vol 18 no 3 pp 289ndash304 2011
[26] K Assaleh T Shanableh and Y Kheil ldquoGroup method of datahandling for modeling magnetorheological dampersrdquo Intelli-gent Control and Automation vol 4 no 1 pp 70ndash79 2013
[27] B F Spencer S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997
[28] Y Ohtori R E Christenson B F Spencer Jr and S J DykeldquoBenchmark control problems for seismically excited nonlinearbuildingsrdquo ASCE Journal of Engineering Mechanics vol 130 no4 pp 366ndash385 2004
[29] AH El-Sinawi ldquoActive vibration isolation of a flexible structuremounted on a vibrating elastic baserdquo Journal of Sound andVibration vol 271 no 1-2 pp 323ndash337 2004
[30] A El-Sinawi and A R Kashani ldquoActive isolation using aKalman estimator-based controllerrdquo Journal of Vibration andControl vol 7 no 8 pp 1163ndash1173 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of