Application of Active Vibration Control for Earthquake Protection of Multi-Structural Buildings

7/27/2019 Application of Active Vibration Control for Earthquake Protection of Multi-Structural Buildings

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I nternational Jour nal of Scientif ic Research in Knowledge (I JSRK), 1(11), pp. 502-513, 2013Available online at http://www.ijsrpub.com/ijsrk

ISSN: 2322-4541; 2013 IJSRPUB

http://dx.doi.org/10.12983/ijsrk-2013-p502-513

502

Full Length Research Paper

Application of Active Vibration Control for Earthquake Protection of Multi-Structural Buildings

Mohammad Gudarzi*, Hashem Zamanian

Department of Mechanical Engineering, Damavand Branch, Islamic Azad University, Damavand, Tehran, Iran

*Corresponding Author: [email protected]

Received 08 September 2013; Accepted 22 October 2013

Abstract. Investigating the effectiveness of LQG output-feedback active control for seismic alleviation of multi-structural

buildings is the main purpose of this study. A general model of a shear-frame multi-structure as a lumped-mass planar system

with movements in the path of the ground motion for the attached tree-building system is developed, first. Then an output-

feedback controller based on the Kalman filter and optimal control theories is designed for the main structure, to guarantee thestability of the closed-loop system to achieve an arranged level of disturbance attenuation. Consequently, for investigating the

effectiveness of the proposed method on the main and neighbor structures, some numerical simulations using historically

recorded ground accelerations, are considered and the results will be discussed.

Key words: Active control, seismic response, vibration attenuation, lumped-mass model

1. INTRODUCTION

More and more skyscrapers have been constructedbecause of the limitation of land, particularly in bigtowns in recent years. Nevertheless, at the same time,

strong earthquakes and wind occur frequently. Forexample, numerous structures were damaged orcollapsed during the 1994 Northridge, 1995 Kobe,1999 Duzce, 1999 Chi-Chi and the most recent 2008Winchuan earthquakes, resulted not only in significanteconomic losses but also large loss of lives. Asbuildings become higher, structural stability andstrength are challenged and cannot be guaranteed onlyby high quality materials. Therefore, the requirementof Structural Vibration Control (SVC) techniques forseismic-excited or wind-excited structurs becomesfurther important (Sandell et al., 1978; Housner et al.,

1997; Spencer and Sain, 1997; Alkhatib andGolnaraghi, 2003; Soong and Cimellaro, 2009).

Advanced means of SVC techniques can becategorized into the passive control systems andactive, semi active, or hybrid control schemes. Themain subcategories of passive control systems are thebase isolation and the passive energy dissipation

systems (Dolce et al., 2000; Ozbulut and Hurlebaus,2011; Chakraborty and Debbarma, 2011; Zhang andBalendra, 2013). Active, semi-active, and hybridcontrol schemes are a natural evolution of passivecontrol tools (Lee, 2012; Bitaraf et al., 2010; Park et

al., 2003). The devices of this category are part of anintegrated system, with real time processing

controllers and sensors, all installed to the structure.They act concurrently with the excitation to provide

improved structural behavior for progressed serviceand safety. The use of active, semi-active and thecombination of passive, active or semi-active systemsas a means to protect the structures against seismicexcitations has been attended considerably in the

recent years.During the last few years numerous control

algorithms and devices have been established,improved and studied by various groups ofresearchers in this field of research. To illustrate, Luet al. (2003) and Du and Zhang (2008) applied

classical and theories, Du and Zhang (2007)

and Zhang and Shi (2008) used and theories,Mahmoud et al. (2000) and Balandin and Kogan(2005) utilized optimal control, sliding mode controlwas used by Guclu (2006) and Manjunath andBandyopadhyay (2009), Bucz et al. (2008) developed

a new approach PID control, Madan (2005) andZapateiro et al. (2009) used neural networks, fuzzylogic was applied to structural systems by Pouezeunaliet al. (2007) and Shook et al. (2008), independentmodal space control was used by Park et al. (2004),to diminish building vibration under earthquake orwind loads. By the improvement of active structurevibration control methods, studies of buildingvibration control have already been converted fromtheories to practical applications, magnetorheological(MR) dampers, for instance, have been widely studiedand applied for vibration reduction by Zapateiro et al.(2009), in which a series of investigations run on areal-time hybrid testing have been done, and Goto etal. (2008) presented a new deterioration predictionmethod for conservation of rotating equipment.


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Moreover for practical aspects, ever more limitationsof real systems are taken into account, for exampleactuator saturation (Lim, 2008), uncertainties ofstructure models and controllers (Song et al., 2007;Wang et al., 2001), and controllers or sensors with

time delay (Li et al., 2008). In addition, Cole et al.(2008) presented a technique for reaching robustvibration control of flexible structures under contact

and establishes a robust cost bound for the closed-loop system; Du et al. (2008) and Karimi et al. (2009)

presented a mixed design scheme to control

tall buildings, while Du et al. (2008) considered thepole placement and a reduced-order model and Karimiet al. (2009) dealed with base-isolated models; Park etal. (2008) proposed a bilinear pole-shifting techniquewith control technique for dynamic response

attenuation of large structures. All the aforementionedworks reduce building vibration effectively.

Even though the most remarkable cases of SVC forseismic protection are mostly those including hugestructures as long-span bridges or tall skyscrapers, itshould also be emphasized that the existence of a widevariety of smaller structures for which a particularlevel of seismic protection might be requested such asemergency power plants, central police stations andhospitals. Besides, it is to be considered that in spiteof the medium or small size of individual structures,the whole structural system could be extremely

complex, including two or more adjacent buildingsand a group of attached substructures, which mayrequire different levels of seismic protection. Theseismic excitations on multi-structure systems canlead to undesirable interactions among the differentsubstructures such as inter-structure collisions thatmight cause severe structural damage. Additionally,the mechanical properties of the structural systemmight vary due to changes in masses of each part and

material characteristics. Accordingly, SVC designs formulti-structure systems should aim at two objectives:(i) decreasing the vibrational response of the mainsubstructure and (ii) avoiding unwanted inter-structureinteractions such as pounding events.

The main objective of this study is to demonstratethe positive features associated with the LQG output-feedback control for seismic alleviation of multi-structure buildings. For this purpose a general modelof a shear-frame multi-structure as a lumped-mass

planar system with movements in the path of theground motion for the attached tree-building system is

developed. An output-feedback controller based onthe Kalman filter theory and optimal control theory isdesigned for the main structure, to achieve anappropriate level of disturbance rejection. To illustrate

the performance of the proposed control method, anearthquake-excited multi-structure consisting of threebuildings with active controller system on the mainbuilding is used. For the numerical simulations,historically recorded ground accelerations, i.e., the ElCentro (1940) and Kobe (1995) earthquakes areconsidered as external disturbances. Finally, theresults of numerical simulations are discussed to

investigate the capability of the proposed method.

2. MATERIALS AND METHODS

2.1. Multi-structural building model

In this section, a simplified shear-frame structuremodeled as a lumped-mass planar system withdisplacements in the direction of the ground motionfor a three-building coupled system shown in Fig. 1 ispresented. The buildings motion can be described bythe following second-order model,

(1)

where is the mass matrix; and and are thetotal damping and the total stiffness matrices,respectively, including the damping and stiffnesscoefficients of the buildings, and also the damping

and stiffness coefficients of the linking systems. Thevector of story displacements with respect to theground is

(2)

where represents the displacement of the -th

story in the -th building corresponding to the time .

Regarding the actuation system, we assume that a setof interstory force actuators located between everytwo neighboring floors has been implemented in the

main building, as indicated in Fig. 1, where is

the actuation force exerted by actuator that is

implemented between stories and in the main

building at time . The vector of control forces, which

is produced by actuators, has the structure as

(3)


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Fig. 1: Schematic view of three-building system.

As shown in Fig. 1, the force delivered by theactuation device produces a pair of opposite forces

acting on floors and of the main building. Thisactuation scheme is modeled by means of the control

location matrix . Finally, is the disturbance input

matrix and is the ground acceleration. Note that

the explicit dependence on time of and has

been omitted in Fig. 1 to simplify the notation; thiswill also be done in the sequel when convenient. The

mass matrix in Eq. (1) has the block diagonal structureas,

(4)

where is an zero-matrix and is the

mass matrix of the -th building, and

(5)

The total damping matrix can be written in theform

(6)

where

(7)

is a block diagonal matrix corresponding to theinternal damping of the buildings with tridiagonal

blocks as


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(8)

and denotes the damping coefficient of the -th

story in the -th building. The damping matrix

corresponds to the linking system and can be writtenas a block matrix as follows

(9)

where

(10)

(11)

where is the damping coefficient of the -th

linking element between the first building and the -th

building, and is equal to the subscript in which

has the maximum value. Note that, it is

assumed the first building is the tallest one i.e.

.The total stiffness matrix can also be written in the

form , with

(12)

where matrices and can be obtained fromEqs. (8), (10) and (11), by replacing the damping

coefficients , and by the corresponding


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stiffness coefficients , and . The control

location matrix has the following structure

(13)

where denotes the control location matrix of

the -th building

(14)

Note that when there are not any actuators in the

neighborhood buildings, and are replaced by

and , respectively. Finally, the

disturbance input matrix can be written in the form

, where is a

column vector of dimension with all its

entries equal to 1.Now we take the state vector

(15)

and derive the following first-order state-space modelfrom the second-order model of the Eq. (1),

(16)

The state matrix in Eq. (16) can be written as

(17)

where denotes an zero-matrix and is

the identity matrix of order . The control anddisturbance input matrices have, respectively, thefollowing form

(18)

Regarding the output matrix, we will consider five

different cases. The vector of story displacementsrelative to the ground is

(19)

can be obtained using the output matrix (19).

(20)

The interstory drifts are the relative displacementsbetween consecutive floors of the same building, and

can be defined as

(21)

The vector of interstory drifts is as,

(22)

The vector of interstory drifts can be computed with the following output matrix.

(23)

where

(24)


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As a third case, we introduce the interbuildingapproaches to describe the approaching between the

stories placed at the th level in the adjacentbuildings

(25)

For considered three-building system the vector ofinterbuilding approaches will be as,

(26)

The vector of interbuilding approaches can beobtained using the output matrix (27).

(27)

Note that, as it was mentioned before, the middlebuilding is the tallest one.

It should be observed that positive values of theinterbuilding approaches (25) correspond to areduction in the distance between the correspondingstories. Clearly, for a given interbuilding separation,large values of the interbuilding approaches may

result in interbuilding collisions. To avoid thecomputational complexity associated with the

pounding impacts (Jankowski, 2005), the differentseismic response simulations presented in this paperare conducted under the assumption that theinterbuilding separation is large enough to avoidcollisions. In this case, the maximum values of the

interbuilding approaches can be understood as lowerbounds of safe interbuilding separation.

In the control of civil engineering structures,absolute acceleration measurements are readilyavailable (Tan and Agrawal, 2009). So, assuming that

accelerometers are employed and each of them islocated at each floor of the main building, the sensors

outputs vector will be as

(28)

And it can be obtained using the output matrices

(29)

where represents the th row of the matrix .It should be noted that the magnitude of the

disturbance input matrix must be added to the

calculated acceleration of each story.In this paper, our goal is to find an output-feedback

control system,

,(30)

Such that,1- The closed-loop system is asymptotically stable,

2- The objective functionis minimized.

2.2. Controller design

Now, following the standard Linear QuadraticRegulator (LQR) design procedure (Gawronski 2004),

for a given non-zero initial state , one should find

the control input (t) which puts the system (28) back

to the zero ( ) state in an optimal manner andattain the greatest possible reduction in the vibrationalresponse of the multi-structure system. This can beachieved by minimizing the deterministicperformance index (Ogata, 2009)

(31)

where and are suitably chosen constant weightingmatrices concerning the system response and control


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input, such that , ,and itshould be noted here that the linear quadratic regulator

with output weighting is equivalent to linear quadraticstate feedback regulator with weighting matrices

and obtained by solving the matrix equation (Lewiset al., 1995)

(32)

The first step in the solution of the LQG problemconsists of finding the optimal control to adeterministic linear quadratic regulator (LQR)

problem: namely, the above LQG problem without

and . The optimal solution to this problem can bewritten in terms of the simple state feedback law

(Ogata, 2009)

(33)

where is the estimated state, and in

which ( is the unique positive-semi

definite solution of the algebraic Riccati equation

. (34)

The next step is to find an optimal estimate of the

state so that is minimized, whereis the expectation operator.

The measurement noise, and the process noise,

are generally assumed to be uncorrelated zero-

mean Gaussian stochastic processes with constant

power spectral density matrices, and ,

respectively, whereand (Ogata,

2009). Also, the optimal state estimate is given by aKalman filter, which estimates the state of the system

in presence of noisy measurements, and isindependent of and . The Kalman filter has thestructure of an ordinary state estimator with

(35)

where is the optimal choice for

observer gain which minimizes the mean square error

and 0 is the uniquepositive-semi definite solution of the estimatoralgebraic Riccati equation (Ogata, 2009)

. (36)

Lastly, using the state-space realization, the statefeedback law (33), and the estimated state Eq. (35),after some straight forward manipulations, one canobtain the closed-loop system dynamic equations inthe form

(37)

where . This demonstrates that

the closed-loop poles are simply the combination ofthe poles of the deterministic LQR system (i.e., the

eigenvalues of ) and the poles of the Kalman

filter (i.e., the eigenvalues of ), which is

exactly as predicted by the Separation Theorem.

3. RESULTS AND DISCUSSIONS

To design the active controllers and to perform thenumerical simulation of the seismic responsecorresponding to the designed controller, a particular

three-building system formed by a main five-storybuilding adjacent to two three-story buildings hasbeen chosen. The nominal values of the systemparameters and their bounds of variation are used asthose are given in Table 1.

In the control design procedure ,

, and is considered.

A set of numerical simulations is conducted in

order to compare the vibrational response of the three-building structure with and without controller due to

two medium-sized and large-sized seismic excitations.For investigation of a medium-sized disturbance, the

full-scale North-South El Centro 1940 seismic recordobtained at the Imperial Valley Irrigation Districtsubstation in El Centro, CA, during the ImperialValley earthquake of May 18th, 1940, has been usedas a ground acceleration input. This is a medium-sized

seismic disturbance with an acceleration peak of about

(Ohtori et al., 2004) (see Fig. 2(a)). For

studying the effectiveness of proposed control strategywhen the three-building structure is excited by a largeseismic disturbance, we have considered the full scaleNorth-South component of the ground accelerationrecord obtained at the Kobe Japanese Meteorological

Agency station during the Hyogoken-Nanbuearthquake of January 17, 1995. This is a near-fieldrecord, corresponding to a close-to-epicentre station,that presents large acceleration peaks which areextremely destructive to tall structures (Huang and


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509

Chen, 2000; Ohtori et al., 2004). In order to give aclear idea of magnitude, the Kobe and El Centro

seismic records are presented together in Fig. 2 usingthe same acceleration range.

Table 1: The values of the multi-structure parameters.

Name Symbol Value

The mass of the -th story of the 1st building

The mass of the -th story of the 2nd

building

The mass of the -th story of the 3rd

building

The stiffness of the -th story of the 1st

building

The stiffness of the -th story of the 2nd

building

The stiffness of the -th story of the 3rd

building

The damping of the -th story of the 1st

building

The damping of the -th story of the 2nd

building

The damping of the -th story of the 3rd

building

The stiffness of the -th linking element between the 1st

building and the 2nd

building

The stiffness of the -th linking element between the 1st

building and the 3rd buildingThe damping of the -th linking element between the 1

st

building and the 2nd

building

The damping of the -th linking element between the 1st

building and the 3rd

building

Fig. 2: Seismic records: (a) Full scale NS El Centro 1940. (b) Full scale NS Kobe 1995.

As the first case, we consider the earthquakeresponse of the active and passive structures due toNorth-South El Centro seismic excitation. The

Maximum absolute story displacements relative to theground, maximum absolute interstory drifts,maximum interbuilding approaches, and maximum

absolute control forces are displayed in Fig. 3. Figures3(a) to 3(h) summarize the peak vibrational responsefor both passive and active structures. In addition, theideal active control forces with no saturation which

use the theoretical control action without

restriction are shown in Fig. 3(i).The graphics in Fig. 3 show clearly the appropriate

performance exhibited by the designed active controlwhere peak values for the absolute story

displacements relative to the ground, absoluteinterstory drifts and interbuilding approaches aredramatically smaller than those obtained from passivestructure. In addition, the vibrational response of the

adjacent buildings improved, although the controlsystem has only implemented in the main building.

As the second case, we consider the vibrational

response of the active and passive structures due toNorth-South component of the Kobe seismicexcitation. As the previous figure, the maximum

absolute story displacements relative to the ground,maximum absolute interstory drifts, maximuminterbuilding approaches, and maximum absolutecontrol forces are displayed in Fig. 4. Figures 4(a)through 4(h) summarize the peak vibrational response

for both passive and active structures using LQGcontroller. In addition, the ideal active control forceswith no saturation, those use the theoretical control

action without restriction, is shown in Fig. 4(i).

Figure 4 shows clearly the excellent performanceexhibited by the designed active control where peakvalues for the absolute story displacements relative tothe ground, absolute interstory drifts and interbuilding


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approaches are dramatically smaller than thoseobtained from passive structure, for both the mainbuilding and the adjacent ones.

By comparison of Figs. 3 and 4, one can see thatthe performance of the designed controller is more

appropriate in a large seismic disturbance. However,control forces in the case of large-sized seismicexcitation are dramatically larger than the medium-sized one. So, the capability of actuators for largeearthquakes must be considered.

Regarding the vibrational protection of the multi-structure system, the prime aspects associated with the

action of the designed output-feedback control system

consist of specific substructure protection, high-levelprotection for moderate and large externaldisturbances and high-level protection for moderateand large external disturbances. Specific substructureprotection implies that appropriate levels of

vibrational protection can be provided to individualsubstructures by implanting the output-feedbackcontrol systems in them. Additional protection againstinterstructure collisions is supplied by the actuators.For medium-sized and large-sized external

disturbances, high levels of vibrational attenuationand very small interstructure approaches can be

obtained using active devices.

Fig. 3: Peak vibrational responses and control forces for North-South El Centro seismic excitation.

4. CONCLUSION

An active structural vibration control strategy forseismic protection of general adjacent multi-structureshas been proposed. This strategy combines output-feedback control system implemented in the all floors

of the main building. Consequently, this strategy isconcerned with the LQG controller design considering

reasonable range of Gaussian noise and disturbance.From the time response analyses we can conclude thatthe applied controller attenuates vibration responses ofthe main and neighborhood buildings under medium-sized and large-sized seismic excitations. Therefore,this active strategy has the ability to maintain desired

performance of a complex multi-structural building.


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Fig. 4: Peak vibrational responses and control forces for North-South component of the Kobe seismic excitation.

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Mohammad Gudarzi was born in 1987. He received the B.S. degree in mechanical engineering from

Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran in 2009 and M.S. degree in

mechanical engineering from Sharif University of Technology, Tehran, Iran in 2011.

Now he is a lecturer at Islamic Azad University. His general research interests focus on smart structures and

new control strategies in vibration/noise reduction.

Hashem Zamanian was born in Iran in 1985. He received his B.Sc. degree in Mechanical Engineering in

2010 from the Amirkabir University of Technology and M.Sc. degree in Applied Mechanics in 2012 from

Sharif University of Technology.

Now he is a lecturer at Islamic Azad University. His main interest is Finite Element Analysis and

biomedical engineering.

Application of Active Vibration Control for Earthquake Protection of Multi-Structural Buildings

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