24th International Symposium on on Automation & Robotics in in Construction (ISARC 2007) Construction Automation Group I.I.T. Maras MR DAMPER OPTIMAL PLACEMENT FOR SEMI-ACTIVE CONTROL OF BUILDINGS USING AN EFFICIENT MULTI- OBJECTIVE BINARY GENETIC ALGORITHM N. M. Kwok Q. P. Ha an B. Samali Faculty of Engineering University of Technology Syney Broaway NSW 2007 Australia {ngai.kwok quang.ha bijan.samali}@eng.uts.eu.au ABSTRACT In orer to ensure the survival of builing structures uring earthquake perios inuce vibrations have to be mitigate. In this regar semi-active control of smart structures using magneto-rheological ampers is becoming an emerging technology. Improvements on vibration reuction are foreshaowe when the ampers are installe at critical locations on the builing structure. In this paper the placement of ampers is cast as a multi-objective optimization problem in the sense of minimum resultant vibration magnitues an with a minimum number of ampers. A binary-coe genetic algorithm is employe as the optimizer owing to its computational flexibility an high performance. Simulation results are inclue to illustrate the effectiveness of the propose approach on a high-rise builing moel subject to benchmark earthquake recors. KEYWORDS Builing Control Damper Placement Genetic Algorithm Multi-Objective Optimization. INTRODUCTION Casualties uring earthquake perios coul be very extensive. Particularly in urban areas loss of human lives an economic costs pai are mainly cause by collapsing of builing structures ue to inuce vibrations. Intuitively builings have to be built with sufficient resistances to vibrations but there is always a limit in the afforable capitals investe in their constructions. Control schemes aime at vibration reuctions [] hence are becoming attractive alternative approaches. Practically there are a number of evices available such as tune mass ampers [2]; however power sources may not be available uring earthquake perios. To this en ampers with controllable characteristics for example the magnetorheological (MR) ampers are wiely aopte in this problem omain [3]. Furthermore to reuce system complexity power consumption an to ensure satisfactory performances it is important to place the ampers in their critical locations in the builing. The importance of amper locations affecting the vibration characteristics of structures has been note more than a ecae ago [4] where a multiboy system was use as a stuy example. Vibration reuction in builing structures can be regare as the problem of maximizing the efficiency taking into account the energy issipate in ampers [5]. Obviously minimum vibration magnitues are crucial criteria for juging the effectiveness [6]. Therefore placing the ampers can be naturally tackle as an optimization process an was aresse in [7] where the use of steepest irection search might call for analytical system moels. Alternatively heuristic approaches [8] such as Tabu search or simulate annealing have been attempte. On the other han establishe techniques in the evolutionary computation area have been
362 recognise as attractive caniates in tackling the amper placement problem consiere. A genetic algorithm (GA) is use in [9] to obtain optimal placements of sensors/actuators on civil structures. The work reporte in [0] consiering the amper location problem also use the GA in conjunction with criteria erive from the system transfer function. The GA with a weighte-sum approach was propose in [] for optimal locations in a civil structure where ampers are installe. However assignment of proper important weights is generally ifficult. While evolutionary computation techniques are effective strategies to solve the amper placement problem optimization for multiple objectives has not been fully emphasize. In this irection the GA operating as a multi-objective optimizer has recently appeare in the literature [2] where the vibration magnitue an the number of ampers use are aopte as objective functions. The remainer of the paper is organize as follows. In Section 2 the semi-active control problem for smart structures with MR ampers is briefly presente. Section 3 reviews the use of binary-coe GA in the multi-objective optimization paraigm. The optimal placement of ampers is consiere in Section 4 an preliminary results are given in Section 5. In Section 6 a conclusion is rawn. 2. MR-DAMPER CIVIL STRUCTURES The civil structure consiere is a multi-storey builing consisting for simplicity of an aggregation of single egree-of-freeom bays. The builing is subject to simulate earthquake excitations e.g. from a shake-table facility an magneto-rheological (MR) ampers are to be installe to alleviate the vibrations inuce on the floors. 2. System Description Consier the n-storey builing structure shown in Fig. the equation of motion can be expresse as M & x + C + Kx = Γf + MΛ& g () where x is the floor isplacement vector M C K are the mass amping an stiffness matrices Γ Λ N. Kwok M. Kwok N. Q. Ha P. Q. Ha & Samali B. Samali B. are the istribution vectors for the amper force f an groun acceleration & g. An augmente system state equation is further formulate as z & = Az + Bu + E (2) T T T where z = [ x ] is the state vector = 0 I 0 A B = M K M C M (3) Γ are the system an gain matrices an earthquake inuce isturbance is given by E. The smart structure control u may be via the amper force f (t) or by applying irectly appropriate current i(t) to pairs of MR ampers set in a ifferential configuration as shown in Fig. to cancel out the offset forces. The control evelopment has been escribe in [3]. 2.2 Optimal Damper Placement A Rationale The response of the builing structure uner the effect of ranom earthquake excitation largely epens on the system matrix A an the control effort u. In fact installation of the MR ampers will affect the system matrix elements via its lumpe parameters on stiffness K an viscous amping C. Furthermore in irect control the influence of the current input woul also give way to the parametric epenence of the control gain matrix B whereby a zero entry is relate to the absence of a amper pair installe at the corresponing storey. The control goals require therefore not only robustness an feasibility of the system but also the satisfaction of seismic response performance in an economic manner. To achieve these goals a multi-objective optimization approach is propose as in the following section. 3. MULTI-OBJECTIVE GENETIC ALGORITHM In orer to achieve optimum for both the number of ampers use an for the suppression of quakeinuce vibrations the control problem is cast as an optimization process for which several approaches that can be applicable. Here for the sake of simplicity flexibility an goo performance a multi-objective binary-coe GA is use.
MR Damper Optimal Placement for Semi-Active Control of Builings Algorithm... Genetic Algorithm 363 Figure 2 Operations of Genetic Algorithm Figure Multi-Storey Builing Uner Control 3. Binary-Coe Genetic Algorithm Genetic algorithms are search routines inspire by the evolution of living species. In the course of evolution escenants inherit favourable genetic ingreients from their ascenants such that the survivability of the whole population increases over generations. When using the principle of survival of the fittest to optimization algorithms chromosomes of living organisms are coe in binary strings which represent potential solutions to the problem see Fig. 2. The quality of the solution to the problem is embee as a fitness value assigne to each chromosome. A near-optimal solution is obtaine by manipulating the population using genetic operators incluing selection crossover an mutation. At the first generation (iteration) binary strings are generate in ranom to cover the solution space. Their objective functions are evaluate an normalize to give the corresponing fitness. The chromosomes are selecte to the next generation accoring to their fitness. A pair of chromosomes is then ranomly picke an part of the string is exchange by the crossover operator. Before evaluating the fitness a ranom bit is flippe in the mutation process. The process is repeate until the number of generations reaches a pre-specifie count. 3.2 Multi-Objective Genetic Algorithm The problem of amper placement is concerne with obtaining minimum storey vibrations while using a minimum number of ampers. Without much a priori knowlege on control schemes our intention is to formulate the amper placement as a multi-objective optimization problem. The evaluation of the chromosome fitness has to be catere for this requirement. The objectives can be augmente into a single fitness by the use of importance weights. Although commonly-use in optimization there has not been a generic methoology for the choice of weights available in the relevant literature. In the context of evolutionary computation several approaches have been propose for this kin of optimization with genetic algorithms [2] incluing vector evaluate GA (VEGA) an fitness sharing (FSGA). In VEGA sub-populations are use for evaluating a single objective an then combine by the selection operator. On the other han FSGA moifies the fitness by sectoring the objective-space. The fitness of chromosomes in
364 close objective-space vicinities is reuce to promote searching for better solutions. In this work the principle of Pareto front is applie making use of the fact that it oes not nee to etermine the require sub-populations. In essence chromosomes at the frontier of the fitness lanscape are assigne with higher ranks. The ranking is efine as M m m Ri = N( fi < f j i ) M = 2 (4) m= where i is the chromosome inex m is the objective inex N(.) is the operator that gives the number of satisfactions for conition insie the bracket. 4. OPTIMAL DAMPER PLACEMENT The amping evices use are briefly presente in this section. Then a optimization proceure implemente with GA is evelope for placing them in the civil structure. 4. Damper Moel The MR amper use in this stuy is a LORD RD- 005-3 moel. The amper force generate is a function of the compression/extension isplacement an velocity escribe by a static hysteresis moel as given in [4]: f z i = c i + k = tanh( β i xi + αz + δsign( x i )) (5) to represent the non-linear force/isplacement relationship where z is the hysteretic variable an xi = xi + xi (6) is the inter-storey isplacement. The equation of motion can be written for an iniviual storey as mx && i + c = c or + kx = f + mx && i i i g i+ + c i k xi+ + k xi αz + mx && g mx && i + ( c c ) i + ( k k ) xi = c k x z + mx && i+ i+ α g (6) (7) which clearly shows the moification of the storey response when a amper is installe. N. Kwok M. Kwok N. Q. Ha P. Q. Ha & Samali B. Samali B. 4.2 GA Moifie Gain Matrix The gain matrix B given in (2) contains an upper n n zero matrix corresponing to the number of storeys. The lower part Γ is an ientity matrix also of n n scale inversely by the mass of the storey. If a amper is not installe the corresponing element of the matrix Γ becomes zero instea. This special matrix structure naturally poses the optimization problem in the omain of a binary-coe genetic algorithm (BCGA) optimization. Here the chromosome epicte in Fig. 2 is use to represent the iagonal elements in matrix Γ an will be manipulate through generations in the algorithm. 4.3 Optimization Problem The goal of reucing the root-mean-square vibration magnitues inuce on the builing an the use of a minimum number of ampers together constitutes to a multi-objective optimization (minimization) problem. It is state as follows: Minimize: Subject to: Storey RMS isplacement AND Number of ampers Supplie amper current i i. 0 max A chromosome in GA is encoe as a string of s an 0 s for example T C i = [ 0 L L] i = LP (8) where the length of the i-th chromosome correspons to the number of storeys n with entries corresponing to the amper-installe storey an P is the number of chromosomes to form a population. The fitness of one of the objectives the minimization of number of ampers use is i n f = C (9) k= i k where k is the inex for the gene insie a chromosome. The other objective fitness the rootmean-square (RMS) isplacement is calculate as ~ f r i = N τ t= ( x t 2 i ) f r i = n n j= ~ ( f r 2 j ) (0) here N is the number of isplacement time series t x i an τ is the maximum earthquake time consiere. Finally the overall Pareto-ranke
MR Damper Optimal Placement for Semi-Active Control of Builings Algorithm... Genetic Algorithm 365 fitness is given as in (4) where for the sake of simplicity M is set to 2 in this work. The final solution is given by the chromosome with highest rank in the terminating generation. maximum current (2A). Their corresponing effectiveness is illustrate as below. Figure 3 El-Centro Earthquake Recor 5. RESULTS Simulation stuies are conucte using the scale own (50%) benchmark El-Centro earthquake recor Fig. 3 for 20 secons. The builing moel is esigne such that the natural vibration moe frequencies are range from Hz to 23Hz for a 20- storey builing. The masses an stiffness are assume equal in every floor an a amping ratio of 2% also for every floor is assume. 5. Passive-off MR Dampers In this case ampers are supplie with zero currents (in passive-off moe) an the resultant RMS isplacements are epicte in Fig. 4. The otte line represents the isplacement when no ampers are installe an the builing structure is in the so-calle free vibration. The case with ampers installe at all the stores are given by the soli line where ots inicate the floor level. The effect of the use of a reuce number of ampers 8 amper pairs in this case is illustrate by the ark soli line with squares inicating the stores where ampers are employe. It is shown that the use of a smaller number of ampers is acceptable with the satisfactory resultant isplacement. 5.2 Passive-on MR Dampers Two cases are consiere when the ampers are supplie with a moerate current (A) an the Figure 4 RMS Displacements from Passive-off test (ash ot: no amper soli ot: max. no. of ampers use soli square: optimal place ampers) 5.2. Moerate supply current Figure 5 shows the RMS isplacements uner this test scenario. As illustrate an with the use of 9 amper pairs uner the effect of a moerate supply current the response is slightly better than the fullinstallation configuration with passive-off ampers. It is believe that ue to the moifie overall structural stiffness with the installation of ampers at critical points resonant characteristics are altere an vibrations have been reuce. On the other han full-installation may result in a too large rigiity where vibration energies have not been completely absorbe. 5.2.2 Maximum supply current Simulation results are illustrate in Fig. 6. It is note that the vibration of full-installe configuration is largest among the three cases. This coul be explaine that the structural rigiity is further increase an results in insufficient vibration absorption. However the isplacement is smaller than the free-vibration case. With the use of a reuce number of ampers 3 amper pairs use in this case the response is comparable to the previous case with a moerate supply current. It shoul be note that only a few ampers coul be use where larger currents supplie increases the capability of vibration
366 absorption. To this en simultaneous achievement of both optimization objectives is evient. Figure 5 RMS Displacements from Passive-on Test Moerate Supply Current Figure 6 RMS Displacements from Passive-on Test Maximum Supply Current 6. CONCLUSION This paper has aresse the problem of optimal placement of semi-active resources in the control of civil structures against earthquake excitations. The problem is tackle from a multi-objective optimization perspective where a binary-coe genetic algorithm is use as the optimizer to obtain near-optimum locations to install the MR ampers. The use of the BCGA together with the Paretofront approach is effective in this problem omain N. Kwok M. Kwok N. Q. Ha P. Q. Ha & Samali B. Samali B. in which the objectives for the use of resources an performance are simultaneously achieve. Satisfactory results using the El-Centro benchmark earthquake recors have verifie the propose approach. Future work will be irecte towars the incorporation of controllers on the amper supply currents together with an increase in the number of objectives in an integrate manner to aress optimality in the smart structure technology. 7. ACKNOWLEDGMENT This work is supporte by an Australian Research Council (ARC) Discovery Project Grant DP0559405 by the UTS Centre for Built Infrastructure Research an by the Centre of Excellence programme fune by the ARC an the New South Wales State Government. 8. REFERENCES [] Spencer B.F. Jr. & Sain M.K. (997) Controlling builings: a new frontier in feeback IEEE Control Sys. Magazine Vol. 7 No. 6 9-35. [2] Chen G. & Wu J. (200) Optimal placement of multiple tune mass ampers for seismic structures J. of Structural Engineering Vol. 27 No. 9 054-062. [3] Cho S.W. Jung H.J. & Lee I.W. (2005) Implementation of moal control for seismically excite structures using magnetorheological ampers J. of Engineering Mechanics Vol. 3 No. 2 77-84. [4] Gurgoze M. & Muller P.C. (992) Optimal positioning of ampers in multi-boy systems J. of Soun an Vibration Vo. 58 No. 3 57-530. [5] Wu B. Ou J.P. & Soong T.T. (997) Optimal placement of energy issipation evices for three-imensional structures Engineering Structures Vol. 9 No. 2 3-25. [6] Takewaki I. (2000) Optimal amper placement for critical excitation Probabilistic Engineering Mechanics Vol. 5 37-325. [7] Takewaki I. Yoshitomi S. Uetani K. & Tsuji M. (999) Non-monotonic optimal amper placement via steepest irection
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