Experimental Verification of Torsional Response Control of Asymmetric Buildings Using MR Dampers

Transcription

1 Submitted to the Earthquake Engineering & Structural Dynamics, June 22 Experimental Verification of orsional Response Control of Asymmetric Buildings Using MR Dampers Osamu Yoshida 1, Shirley J. Dyke 2, Luca M. Giacosa 3 and Kevin Z. ruman 4 SUMMARY his paper proposes a semiactive control system to reduce the coupled lateral and torsional motions in asymmetric buildings subjected to horizontal seismic excitations. netorheological (MR) dampers are applied as semiactive control devices and the control input determination is based on a clipped-optimal control algorithm which uses absolute acceleration feedback. he performance of this method is studied experimentally using a 2-story building model with an asymmetric stiffness distribution. An automated system identification methodology is implemented to develop a control-oriented model which has the natural frequencies observed in the experimental system. he parameters for the MR damper model are identified using experimental data to develop an integrated model of the structure and MR dampers. o demonstrate the performance of this control system on the experimental structure, a shake table is used to reproduce an El Centro 194 N-S earthquake as well as a random white noise excitation. he responses for the proposed control system are compared to those of passive control cases in which a constant voltage is applied to the MR damper. KEY WORDS: magnetorheological (MR) damper, semi-active, clipped-optimal, torsional response, irregular building 1. INRODUCION During the last three decades, various types of passive and active control systems have been developed and experimentally verified. A number of them have been implemented in full scale civil engineering structures. Recently, research efforts have been focused on semiactive control systems, because they combine the adaptability of active systems and reliabil- 1. Doctoral Candidate and Graduate Research Assistant, Department of Civil Engineering, Washington University, St. Louis, MO 6313, USA, yosam@cive.wustl.edu (On leave from Obayashi Corporation, echnical Research Institute, okyo , JAPAN) 2. Associate Professor, Department of Civil Engineering, Washington University, St. Louis, MO 6313, USA, sdyke@seas.wustl.edu 3. Undergrad. Research Asst, Department of Structural Engineering, Corso Duca degli Abruzzi, 24, Polytechnic of urin, 1129 urin, IALY, luca_mass@ciaoweb.it (Work performed while a visiting researcher at Wash Univ) 4. Professor and Chair, Department of Civil Engineering, Washington University, St. Louis, MO 6313, USA, ktrum@seas.wustl.edu 1

2 ity of passive systems. One semiactive device that appears to be particularly promising is the magnetorheological (MR) damper. he MR damper has attractive characteristics for civil applications, including low power requirements, high reliability, and inherent stability. For example, MR dampers were applied to a full scale analytical, nonlinear benchmark building demonstrating their performance for reducing the responses of the building under strong earthquakes [1]. One issue in structural control which has not been studied to a large extent is the control of coupled torsional-lateral behavior in irregular buildings. When a structure has an asymmetric distribution of mass or stiffness, torsional motions may be coupled with lateral responses. Kan and Chopra [2] studied the elastic earthquake responses of a torsionally-coupled single story building using response spectrum analysis. It was concluded that the maximum base shear in a torsionally-coupled system is smaller than in the corresponding uncoupled (symmetric) system, while the torque generally increases with the eccentricity between the center of resistance and the center of mass. It is also concluded that this effect depends strongly on the ratio of the natural frequencies of the torsional and lateral motions of the corresponding uncoupled system. Further study shows that this behavior causes concentrated deformations in some columns and amplifies accelerations at certain locations on the building [3]. As a practical example, a building structure, which had been moderately damaged during the Kocaeli Earthquake (urkey, August 17, 1999) and had been repaired only at one corner of the structure, was subjected to the Duzce Earthquake (urkey, November 12, 1999). As a result, the structure was severely damaged due to the torsional irregularity created by the partial strengthening of the structure [4]. One approach to minimize the detrimental effects of torsional motions is to implement structural control devices. Singh et al. [5] examined the use of tuned mass dampers (MD) to control torsional responses due to seismic excitations. In this study, four MDs are applied to control the torsional responses of a multi-story building. Optimal controller designs were obtained using a genetic algorithm. Chi et al. [6] studied the performance of passive, active and semiactive control of a base isolated, four story building with a setback on the third floor. he focus of this study was to develop a smart isolation system to reduce the motion of the structure. Gavin et al. [7] have also examined a base-isolated, L-shaped, eight story building, considering the effects of yield force, yield displacement, natural period and damping of the isolation devices. his building model is also the subject of a benchmark control problem for smart base isolation systems. his paper addresses the use of magnetorheological (MR) dampers to control the torsional response of asymmetric building structures subjected to seismic excitation. Control input determination is based on a clipped-optimal control algorithm which uses absolute acceleration feedback. he performance of this approach is studied experimentally in the Washington University Structural Control and Earthquake Engineering Lab using a 2-story building model with an asymmetric column distribution. he analytical model of the system is developed based on the structure, and this model is modified to have the natural frequencies observed in the experimental system. he parameters for the MR damper model are identified for the development of an integrated system model, which includes both the test structure and the MR dampers. 2

3 o demonstrate the performance of this control system on the experimental structure, a shake table is used to reproduce an El Centro 194 N-S earthquake as well as a random white noise excitation. he responses for the proposed control system are compared to those of passive control cases where a constant voltage is applied to the MR damper. 2. MOIVAION o examine the effect of asymmetry on the dynamic behavior of a typical structure, a parametric study is performed using a mathematical model of a one-story building with an asymmetric stiffness distribution along one axis. he model is subjected to a uniaxial lateral disturbance, exciting both lateral and torsional motions. he equation of motion of this structure can be written as follows [2] x rθ + ω x 2 e y e y ω r x ω r x ω θ x rθ = 1 x g (1) where x is the relative displacement of mass center to ground, θ is the rotation about the vertical axis, x g is the ground acceleration along x-direction, e y is the static eccentricity (distance from center of mass along y-axis to center of resistance), r is the radius of gyration of the deck about a vertical axis through the center of mass. he frequency parameters, ω x, and ω θ, may be interpreted as the natural frequencies of the system if it were not coupled (i.e., e y = ). he variation of the responses for various eccentricity ratios and frequency ratios is examined herein. he structural responses considered in this study include normalized measures of the base shear and base torque (previously considered in [2]), as well as the maximum acceleration and column drift. he rms response values for a stationary, white noise excitation are determined for comparison, and results are obtained by solving the associated Lyapunov equation [8]. he maximum responses for various eccentricity values, e y r, are shown in Fig. 1 as a function of the uncoupled frequency ratio, ω θ ω x. he base shear and the base torque are normalized using the base shear of the uncoupled system,, given by V xo V x = V x r r = V x rv x. (2) From these results, it is clear that base torque increases and base shear decreases with an increasing eccentricity, and this effect is most pronounced when the translational and torsional natural frequencies of the uncoupled system are equal. hese results agree with the results provided by Kan and Chopra [2]. Note that the results in [2] are determined based on re- 3

4 sponse spectrum analysis, whereas the results obtained in this study are determined as rms responses due to a stationary, white noise excitation. o examine the acceleration and column drift responses, the parameters of the one story building model are selected to be similar to those of an experimental structure available in the lab. he mass of the floor is 23.3 kg (51.12 lb). he stiffness is supplied by a permanent column at each corner, plus additional columns that are employed to vary the eccentricity of the system. Every column, permanent and variable column, in this numerical study has the stiffness of a circular steel rod with a.64cm (1/4in) outer diameter. he following two practical cases are considered (see Fig. 2 for schematic). Vx r ey/r= ωθ/ωx CASE2 CASE1 1. ey/r= CASE2 CASE1 ωθ/ωx Figure 1. Normalized Base Shear and orque. Case1: he floor mass is supported by four identical columns (one per corner), and two extra columns whose locations move from the center to one edge of the mass along the y-axis. Case2: he floor mass is supported by four identical columns (one per corner), plus one column at the center to have the same uncoupled torsional and translational frequencies, and two extra columns whose locations move from the center to one edge of the mass along the y-axis. y y x x CASE1 CASE2 Figure 2. Schematic of Cases Studied. he base shear and torque responses for these two cases are shown in Fig. 1. he maximum acceleration and maximum column deformation responses, which are normalized by the acceleration and column deformation of the corresponding uncoupled system, are shown in Fig. 3 as a function of the location of the extra columns. From these figures, it is clear that asymmetry of the building results in an increase in torsional response and a decrease in translational response, which concentrates the deformation at some columns and amplifies the maximum acceleration of the floors. In the example, the maximum column deformation and maximum floor acceleration are 4 6% and 3 4% larger than those of uncoupled building, respectively. 4

5 According to FEMA recommendations [9], an irregular structure is defined as one in which the ratio e y d is greater than.1, where d is a building dimension perpendicular to the direction of the seismic excitation. he set of parameters that defines the boundary for an irregular structure is identified in Fig. 3 for Case 1. For Case 2, when the extra columns are located at the edge of the structure, the ratio e y d is only.78, well within the range of a regular structure, although the responses are significantly larger than those of the irregular building of Case 1. his is the reason that a core-type building does not perform well in strong earthquakes, although it is not classified as an irregular building. Acceleration Column Deformation Center Extra Column Location Side Case 1: Irregular Boundary Center Extra Column Location Side Figure 3. Normalized Responses for an Asymmetric Building. CASE2 CASE1 CASE2 CASE1 3. SEMIACIVE CONROL SYSEM USING MR DAMPER 3.1 Model of MR damper he semiactive control device used in this q study is the MR damper. he MR dampers have demonstrated a great deal of promise Bouc-Wen for civil engineering applications in both analytical and experimental studies. Adequate modeling of the control devices is essential for the adequate prediction of the behavior of the controlled system. he simple mechanical model shown in Fig. 4 was developed c f and shown to accurately predict the behavior of a prototype shear-mode MR Figure 4. Mechanical Model of the MR Damper. damper over a wide range of inputs in a set of experiments [1-13]. he equations governing the force f produced by this device model are f = c q + αz z = γ q zz n 1 βq z n + Aq (3) where q is the displacement of the device, and z is the evolutionary variable that accounts for the history dependence of the response. By adjusting the parameters of the model γ, β, n, and A, one can control the linearity in the unloading and the smoothness of the transition 5

6 from the preyield to the postyield region. he functional dependence of the device parameters on the command input u is modeled as α = α( u) = α a + α b u c = c ( u) = c a + c b u. (4) In addition, the current driver circuit of the MR damper introduces dynamics into the system. hese dynamics are typically considered to be a first order time lag in the response of the device to changes in the command input. hese dynamics are accounted for with the first order filter on the control input given by u = η( u v) (5) where v is the command voltage applied to the control circuit. 3.2 Semiactive Control Algorithm Dyke et al. [14-15] proposed a clipped-optimal control strategy based on acceleration feedback for controlling a single MR damper. Dyke and Spencer [16] extended the control algorithm for multiple MR devices, and Yi et al. [12-13] experimentally verified its performance. Jansen and Dyke [11] also compared the performance of this algorithm to that of several other approaches, and for civil engineering applications this algorithm was found to be among the best performing controllers. his control algorithm was selected as a control algorithm for this study and is summarized herein. In the clipped-optimal controller, the approach is to append n force feedback loops to induce each MR damper to produce approximately a desired control force. Because the force generated in the ith MR v i = damper f i is dependent on the local responses of the structural system, the f i desired optimal control force f ci cannot v i = always be produced by the MR damper. v i = Only the control voltage v i can be directly controlled to increase or decrease the force produced by the device. hus, a force feedback v i = V max loop is incorporated to induce the MR Figure 5. Graphical Representation of Clipped- damper to generate approximately the Optimal Control Algorithm. desired optimal control force. o induce the MR damper to generate approximately the desired optimal control force, the command signal is selected as follows. When the ith MR damper is providing the desired optimal force (i.e., f i = f ci ), the voltage applied to the damper should remain at the present level. If the magnitude of the force produced by the damper is smaller than the magnitude of the desired optimal force and the two forces have the same sign, the voltage applied to the current driver is increased to the maximum level so f ci v i = v i = V max 6

7 as to increase the force produced by the damper to match the desired control force. Otherwise, the commanded voltage is set to zero. he algorithm for selecting the command signal for the ith MR damper is represented in Fig. 5 and can be stated as v i = V max H( { f ci f i }f i ) (6) where V max is the maximum voltage to the current driver, and ( ) is the Heaviside step function. In this study, H 2 /LQG (Linear Quadratic Gaussian) strategies are employed as nominal controller to have desired control forces, because of the stochastic nature of earthquake ground motions and because of their successful application in other civil engineering structural control applications [12 17]. Note that a modified version of this algorithm was developed by Yoshida and Dyke [1] but the modified version was found to perform well for structures with lower natural frequencies, and was not very effective for this structure. 4. EXPERIMENAL SUDY 4.1 Experimental Setup H An experimental model has been designed and constructed for this study. he model is a 2-story frame building with an asymmetric column distribution (see Figs. 6 and Fig. 7). he mass of each story is simulated by two steel plates, cm (12 2 3/8 in) and cm ( /2 in), and has a weight of 23.3 kg (51.12 lb). his mass is supported by a total of six columns, one at each corner plus two additional columns along one side of the building to create an asymmetry. hese columns are threaded steel rods with an outer diameter of.64 cm (1/4 in) and a length of 3.48 cm (12 in). wo MR dampers are installed between the first floor and the ground and equally spaced from the center of mass. Four accelerometers (two on each floor) are installed as shown in Fig. 7. Control actions are computed using a DSP-based, real-time controller manufactured by dspace, Inc. MR dampers Figure 6. Photo of Experiment. 4.2 Identification of Experimental Structure he first step in the experiment is to obtain a model of the structural system that is appropriate for control design purposes. Herein we implement an automated approach that was developed specifically for control-oriented structural modeling [18]. he method is based on the Eigensystem Realization Algorithm (ERA) [19] and integrates the results of this automated system identification technique with an analytical model of the structural system. 7

8 Force ransducer Accelerometer MR damper Accelerometer 17.46cm (6 7/8 in.) 17.46cm (6 7/8 in.) 17.15cm (6 3/4 in.) 17.15cm (6 3/4 in.) MR damper 3.48cm (12in.) 3.48cm (12in.) 2.22cm (7/8in.) 2.22cm (7/8in.) 13.49cm (5 5/16in.) 13.49cm (5 5/16 in.) 3.48cm (12in.) 5.8cm (2in.) A block diagram of the system to be identified is shown in Fig. 8. he three inputs to the system include the ground acceleration x g and the two control force inputs, f 1 and f 2 at the weak and strong sides of the structure where the MR devices will be placed. Four outputs to the system include the accelerations of weak and strong side on 1FL, x 11 and x 12, and the accelerations of weak and strong side Figure 7. Schematic View of est Structure. 2-Story Asymmetric Building on 2FL, x 21 and x 22. o obtain a realization of the structure which has the frequencies observed in the experimental system, an analytical model of the system is developed based on the structure, and this model is modified [13,18]. he parameters used for this analytical model are based on the physical dimensions of the members and the materials. Fixed connections are assumed at column-beam joints. he lumped parameter model takes the form x g f 1 f 2 x 11 x 12 x 21 x 22 Figure 8. Block Diagram of System to be Identified. M x s + C s x + K s x = M s Γx g + Λf (7) 8

9 2nd Floor st Floor st Mode 2.68 Hz 2nd Mode 4.56 Hz 3rd Mode 7.4 Hz 4th Mode Figure 9. Mode Shapes of the est Structure. where x = x 1 x 2 θ 1 θ 2, x 1 and x 2 are the relative displacements of the mass center of the 1st floor and 2nd floor, respectively, θ1 and θ 2 are the rotation about vertical axis of the 1st floor and 2nd floor, respectively, x g is a one-dimensional ground acceleration, f is the vector of control forces, Γ is a [ 1 1 ], and M s m 1 2k x k x 2k xθ k xθ 1 1 m 2 k =, K x k x k x k xθ s =, Λ = (8) I 1 2k xθ k xθ 2k θ k θ l 1 l 2 I 2 k xθ k xθ k θ k θ where l 1 = cm and l 2 =17.15 cm are the coordinates of the control forces input locations. From the experimental structure, m =.272 N/(cm/sec 2 ), =.241 N/(cm/sec 2 1 m 2 ), I =75.95 N cm/(rad/sec 2 ), =62.55 N cm/(rad/sec 2 1 I 2 ), k x =212. N/cm, k xθ = N, and k θ = N cm. he frequencies of this lumped mass model are f n =[ ] (Hz). Here the ERA [19] was applied to experimental data to determine the natural frequencies and the damping ratios of the experimental model. he identified natural frequencies and damping ratios are f e =[ ] (Hz), and h e =[ ] (%), respectively. Also, the identified mode shapes are shown in Fig. 9. o obtain a realization of the structure which has the frequencies observed in the experimental system, the analytical model is modified [13]. In this approach, the modal matrix, 1 Φ = [ φ 1 φ 2 φ n ], is used where φ i are the eigenvectors of M s Ks. he new stiffness matrix is computed using 9

10 K r = M s Φdiag( [ 2πf e ] 2 )Φ (9) yielding the modified stiffness matrix, K r. Note that this approach results in a model of the system which maintains the mode shapes of the analytical model, but has the frequencies of the experimental system. In addition, the damping matrix C s is determined to have the modal damping ratios which are identified by ERA method as follows C s = M s Φdiag( 2h e [ 2πf e ])Φ. (1) hese updated stiffness matrix and damping matrix are used to form the state space equations for this system as z = Az + Bf + Ex g y = Cz + Df (11) where z= [ x ] is the state vector, and x A = I,,. (12) 1 1 B = E = 1 M s Kr M s Cs M s Λ Γ Here, the vector of outputs is taken as y= [ ], where y m = x 11 x 12 x 21 x 22 is the vector of acceleration measurements, and y d x is the vector of relative velocities d1 x d2 = across the MR dampers. hus the matrices for the output equation C and D have the following form: y m yd C = 1 P m M Kr s 1 P m M Cs s, D = 1 P m M s Λ, (13) P d where P m 1 l m1 1 l 1 l = m2, P 1 d =, (14) 1 l 1 l 2 m1 1 l m2 and l m1 = cm and l m2 =17.46 cm are the coordinates of the acceleration measurements. 1

11 Figure 1 provides a representative comparison of the identified model and the experimentally obtained data. he transfer functions of the model appear to match the experimental data well in general. Because the mode shapes from the analytical model are used, small errors in the zeros of the transfer functions may occur, although this is not expected to be problematic for semiactive control systems. Further examination of this control-oriented system identification method is being performed for more realistic structures that may not behave as shear buildings. 4.3 Identification of Applied MR Damper. 1 2 Input:Xga - Output:X11a Input:Xga - Output:X12a (a) ransfer Functions from Ground Acceleration he MR devices employed in this experiment are shear mode MR dampers, shown schematically in Fig. 11. he experimental 1 Input:F1 - Output:X21a devices were obtained from the Lord Corporation < he device consists of two steel parallel plates. 1 Input:F1 - Output:X22a he dimensions of the device are cm ( in). 1 1 he magnetic field produced in the device 1 1 Frequency (Hz) is generated by an electromagnet consisting (b) ransfer Functions from Force 1 of a coil at one end of the device. Forc- Experimental Data Input:F2 - Output:X11a 1 es are generated when the moving plate, Identified Model coated with a thin foam saturated with MR fluid, slides between the two parallel 1 Input:F2 - Output:X12a plates. 1 2 Power is supplied to the device by a 1 1 regulated voltage power supply driving a Input:F2 - Output:X21a DC to pulse-width modulator (PWM). As 1 2 a PWM unit, RD-32 Rheonetic Device 1 1 Input:F2 - Output:X22a 1 Controller (Load Corporation) is used in 1 1 this study. his PWM unit supplies regulated current to the MR damper at a frequency of 3kHz. he maximum output Frequency (Hz) current is 2 amps with input voltage of 5 V. Although the relationship between input voltage and output current is linear, there (c) ransfer Functions from Force 2 Figure 1. ransfer Functions of est Structure. exists a small dead zone in the input voltage, as described in the following section Input:Xga - Output:X21a Input:Xga - Output:X22a Frequency (Hz) 1 4 Input:F1 - Output:X11a Input:F1 - Output:X12a 1 4 Experimental Data Identified Model Experimental Data Identified Model 11

12 he model parameters for this shear mode MR damper had been already identified in previous experiments [12-13]. However, it is necessary to update some parameters of the MR damper model because the MR damper is at a different operating point when it is employed in this study. Additionally, a new PMW unit is used in this study and the relationship between input command voltage and force generated by MR damper will change. Direction of Motion Front View Figure 11. Schematic Diagram of a Shear Mode MR Damper. he first step in developing a new set of model parameters is to obtain the characteristics of the MR damper itself. he load Force ransducer frame shown in Fig. 12 was used to obtain MRdamper this data. he MR damper is cycled using sinusoidal displacements of.318 cm (1/8 in.) and.635 cm (1/4 in.) at 3Hz, while various command voltage levels were applied. Representative results for a sinusoidal displacement of.635 cm (1/4 in.) are shown in Actuator Fig. 13, including force-displacement and force-velocity loops. he maximum force generated by this Figure 12. Photo of Load Frame est. MR damper was found to be about 2 25 N depending on relative velocity across the MR damper with saturation voltage of 2 V. So, in this study, the range of voltage applied to the MR dampers was set to be 2 V, and the dynamic range, defined as the ratio of the maximum force with maximum control input of 2 V to the maximum force with minimum V, is approximately 4. Also it was found that the input voltage range.8 V is determined to be a dead zone, assuming the linear relationship using the data of 1 V and 2 V. his dead zone is taken into account in the program implementing the digital controller within the dspace environment on the computer. o identify the new set of model parameters, a series of tests was conducted to measure the response of the system with MR dampers in the test structure due to sinusoidal excitation with the first natural frequency of the test structure and random white noise excitation, while applying voltage of V, 1V, and 2V to MR damper. he identified new set of parameters are as follows: α a = 13.8 N/cm, α b = 62.1 N/(cm V), c a =.454 N sec/cm, c b =.195 N sec/(cm V), n = 1, A = 12, γ = 3 cm -1, β = 3 cm -1, and η = 8 sec -1. Coil Side View MR Fluid Saturated Foam 12

13 3 2 1 V 1.V 1.5V 2.V 3.V 4.V V 1.V 1.5V 2.V 3.V 4. Force (N) -1 Force (N) Displacement (cm) Velocity (cm/s) (a) Force-Displacement Hysteresis Loop Figure 13. Characteristics of Applied MR damper. (b) Force-Velocity Hysteresis Loop X11a (cm/s2) X21a (cm/s2) Experiment Simulation X11a (cm/s2) X21a (cm/s2) 4 Experiment Simulation F1 (N) ime (sec) ime (sec) (a) Command Voltage of V (b) Command Voltage of 2V Figure 14. Responses of Integrated System Model (Sinusoidal Excitation at 2.68Hz). F1 (N) he responses of the integrated system model, which is a model of the test structure combined with MR damper, are shown in Fig. 14. he results are shown for the acceleration outputs of the weak side of the building on each floor and the force applied at the weak side of the building by the MR damper with sinusoidal input excitation at the first natural frequency of the structure (2.68 Hz) and the command voltage to MR dampers of V and 2 V. he identified integrated system model is adequate to represent the experimental system. 13

14 4.4 Design of Nominal Controller for Semiactive Control Algorithm In this study a clipped-optimal control is chosen as a semiactive control algorithm where /LQG controller is employed as a nominal controller. he feedback measurements in- H 2 cluded the four accelerations on the structure y m = x 11 x 12 x 21 x 22, as well as measure- ments of the forces provided by the MR damper = f 1 f 2. H 2 In the design of the /LQG controller, the ground acceleration input, x g, is taken to be a stationary white noise, and an infinite horizon performance index is chosen as f m J = 1 τ lim -- E y τ { r Qyr + f Rf} dt τ (15) where Q and R are weighting matrices for the vectors of regulated responses y r = and of control forces f=, respectively. For design purposes, the measurement noise vector, v, is assumed to contain identically distributed, statistical- x 11 x 12 x 21 x 22 f 1 f 2 ly independent Gaussian white noise processes, with S x gx g S vi v i = γ g = 25. he nominal controller is represented as ẑ = ( A LC)ẑ + Ly m + ( B LD)f m (16) f c = Kẑ (17) where L is the gain matrix for state estimator and K is the gain matrix for linear quadratic regulator. For more information on the determination of these gain matrices, see [13] or [17]. As described in the previous chapter, the control force determined using this algorithm is compared to the measured control force, and, using Eq. (6), the appropriate control voltage is applied to the control devices. o design the nominal controller, parametric studies were performed where the weighting matrix Q for the regulated responses are selected as Q = q 1 diag 11q 2 1, while the weighting matrix R for control forces remains unit values as R = diag( 11). o find the optimal values of q 1 and q 2, the maximum and rms acceleration responses are calculated due to both broadband white noise ( 2 Hz) and El Centro earthquake (194NS) ground excitations. he El Centro earthquake is scaled by.4 in time to cause resonance at the first natural frequency of the test structure. Because the integrated system with MR dampers is nonlinear and the responses depend highly on input excitation level, three differ- 14

15 Amax/Amax ELCentro.15 ELCentro.3 ELCentro Amax/Amax ELCentro.15 ELCentro.3 ELCentro Arms/Arms Arms/Arms Fmax (N) 2 1 Fmax (N) Weighting Parameter q Weighting Parameter q2 (a) For Weighting Parameter q1 (q2=1) (b) For Weighting Parameter q2 (q1=.1) ent input levels are considered for each input. Maximum accelerations of 1, 2, and 3 (cm/s 2 ) are chosen for the white noise excitation, and El Centro earthquake is scaled in magnitude by.15,.3, and.45. he optimal values of q 1 and q 2 are determined as follows. As a first step, q 2 is set as q 2 = 1, parametric studies are performed for various values of q 1. he results for the scaled El Centro earthquake are shown in Fig. 15(a). Similar results are also obtained for the random white noise excitations. From these results, the optimal value is chosen as q 1 =.1. As a next step, parametric studies are performed for various values of q 2. he results for the scaled El Centro earthquake are shown in Fig. 15(b) and similar results are also obtained for the random white noise excitations. hus, the optimal value is chosen as q 2 = Experimental Results Figure 15. Parametric Study for Weighting Parameter q1 and q2 (Scaled EL Centro Earthquake). o demonstrate the performance of this control system on the experimental structure, the shaking table was used to reproduce an El Centro 194 N-S earthquake. o test this experimental structure, the original earthquake was scaled by.45,.3, and.15 in magnitude and by.4 in time. A second set of tests was also performed in which a random white noise (-2 Hz) was used as the input ground excitation. wo input levels were considered for a random white noise excitation with maximum acceleration of 25 and 122 (cm/s 2 ). hree cases were studied for each input, including semiactive control with the clipped-optimal controller, passive-off where a constant V was applied to the MR dampers, and passive-on where a constant 2 V was applied to the MR dampers. Note that an optimal constant voltage might be determined for the controller for a given excitation amplitude, but the optimal voltage level is likely to be different for different input amplitudes. 15

16 X21a (cm/s2) X21a (cm/s2) X21a (cm/s2) 5 Experiment Simulation ime (sec.) (a) Acceleration Responses Passive-Off Passive-On Clipped-Optimal Figure 16. ypical Responses Due to Scaled El Centro Earthquake (.45 in magnitude). F1 (N) F1 (N) F1 (N) (b) MR damper Forces Experiment Simulation Passive-Off Passive-On Clipped-Optimal ime (sec.) ABLE 1. Maximum and rms Responses Due to Scaled El Centro Earthquake. 5.1 Scaled El Centro Earthquake Results Acceleration (cm/s 2 ) Control Force (N) Maximum rms Maximum rms Large Amplitude Scaled El Centro Earthquake (45%) Passive-Off Passive-On 343 (.72) 7.4 (.89) a Clipped-Optimal 293 (.61) 46.3 (.59) Medium Amplitude Scaled El Centro Earthquake (3%) Passive-Off Passive-On 245 (.92) 54.3 (1.18) Clipped-Optimal 191 (.72) 3.5 (.66) Small Amplitude Scaled El Centro Earthquake (15%) Passive-Off Passive-On 157 (1.28) 42. (2.2) Clipped-Optimal 93.4 (.76) 17.8 (.93) a. Parenthesis indicate percent of passive-off results. Fig. 16 shows typical responses of the test structure due to the.45 scaled El Centro Earthquake, including the acceleration response of the weak side on 2FL, which is the maximum among all of the acceleration outputs, and the control force of the MR damper attached to the weak side, which is also the maximum among all of the control forces. In these figures, 16

17 the experimental results are compared with the simulation results for each case. Good agreement is found between the experimental and the simulation results in general. he experimental responses are provided in able 1. In the case of the large amplitude earthquake, the passive-on control reduces the maximum acceleration of the structure by 28% of the passive-off case, while reducing the rms acceleration responses by only 11%. his occurs because the maximum control voltage is applied to the MR dampers at all times in the passive-on tests and the control forces are quite large even after the main event of the earthquake. hus, the first floor is rigid and the second floor can move freely. However, in the case of the clipped-optimal control, the maximum and rms responses can each be reduced to 6% of the passive-off results while using smaller control forces than the passiveon controller. In the case of the medium and small amplitude earthquakes, good control performance cannot be achieved using the passive-on controller and some responses are larger than the passive-off controller. Here the clipped-optimal controller can reduce the maximum accelerations by 28% and 24%, and can reduce the maximum rms accelerations by 34% and 7% as compared to the passive-off results. 5.2 Random White Noise Excitation Results Fig. 17 shows the experimental results of the power spectral densities due to random white noise excitations which have a flat power spectral density up to 2 Hz. he excitation signal here was developed by inverting the transfer function of the shake table and amplifying the power of the lower frequency signals appropriately to achieve a band-limited white noise. he power spectral densities of the weak side accelerations on each floor are shown for the large and small amplitude excitations. he figures include results for the clipped-optimal controller as well as the passive-on and passive-off cases. With the passive-on controller, the response peaks on the first floor are reduced, while another resonant peak is produced in the second floor responses around 4Hz. his result clearly demonstrates that when the passiveon controller is applied, the first floor become rigid and the second floor moves freely. For small amplitude excitations, the magnitude of this resonant peak becomes larger than that of the first mode peak for the passive-off controller. It is also noted that the passive-on controller makes the responses in the higher frequency range larger, especially the responses at the first floor. However, the clipped-optimal controller can reduce the resonant peaks effectively without exciting other modes. his is especially true for the small amplitude excitation in which, even though the passive-off control works well, the clipped-optimal controller achieves higher performance reduction in the system. he experimental responses due to random white noise excitations are provided in able 2. With the large amplitude excitation, the passive-on control strategy reduces the maximum and rms acceleration of the structure by 19% and 9%, respectively. However, with the clipped-optimal controller, the maximum and rms responses can each be reduced to 35 37% of the passive-off case. In the case of the smaller amplitude, the clipped-optimal control can reduce the maximum and rms responses by 2% and 11% compared to the passive-off case, while the passive-on control increases these responses to 144% and 17% of the passive-off case. 17

18 Pxx Passive-Off Passive-On Clipped-Optimal Pxx Passive-Off Passive-On Clipped-Optimal Pxx Frequency (Hz) Pxx Frequency (Hz) (a) Large Amplitude White Noise Excitation (b) Small Amplitude White Noise Excitation (max=25cm/s 2 ) (max=122cm/s 2 ) Figure 17. Power Spectral Densities Due to Random White Noise Excitation. 6. CONCLUSION ABLE 2. Maximum and rms Responses Due to Random White Noise. Acceleration (cm/s 2 ) Control Force (N) Maximum rms Maximum rms Large Amplitude Random White Noise (max=25cm/s 2 rms=54cm/s 2 ) Passive-Off Passive-On 347 (.81) 86.4 (.91) a Clipped-Optimal 27 (.63) 62.4 (.65) Small Amplitude Random White Noise (max=122cm/s 2 rms=26cm/s 2 ) Passive-Off Passive-On 219 (1.44) 6.8 (1.7) Clipped-Optimal 122 (.8) 31.6 (.89) a. Parenthesis indicate percent of passive-off results. When a structure has an asymmetric distribution of mass or stiffness, torsional motions, coupled with lateral responses, may be excited by lateral seismic loads. his behavior results in concentrated deformations in some of the columns of the structure, and amplifies accelerations at certain floors of the building. o reduce the response of asymmetric buildings, a semiactive controller using MR dampers was proposed. A clipped-optimal controller was 18

19 used as a nonlinear, clipped-optimal, control algorithm where /LQG controller is employed as a nominal linear controller. Experimental studies were conducted using a 2-story, asymmetric building model with four degrees of freedom. o obtain a control-oriented model of this experimental structure, the analytical model of the system was developed based on the structural parameters, and this model was modified to have the frequencies observed in the experimental system. he parameters for the MR damper model were identified for the integrated system model, which is a model of the test structure combined with the MR dampers. he obtained integrated system model was found to represent the experimental system well. he optimal nominal controller was designed through a series of parametric studies. High performance controllers were designed by placing a higher weighting on the acceleration responses of the weak side on the 2nd floor. he experimental results demonstrate that the performance of a semiactive controller using MR dampers is significantly better than passive control system where constant voltages are applied to the MR dampers. When the large constant voltage is applied to the MR damper, the first floor becomes rigid and the second floor can move freely with its own natural frequency, resulting in an increase in the maximum response, especially for small ground excitations. However, when the proposed semiactive controller is applied, all responses of the building can be effectively reduced. For further information, see: < H 2 ACKNOWLEDGMENS his research is partially supported by National Science Foundation Grant No. CMS (Dr. Chi Liu, Program Director). his support is gratefully acknowledged. he authors would also like to thank Lord Corporation for supplying the prototype MR damper used in this study. Additional support for the lead author from Obayashi Corporation, and for the third author from urin Polytechnic, is also acknowledged. REFERENCES 1. Yoshida O, Dyke SJ. Seismic Control of a Nonlinear Benchmark Building Using Smart Dampers. Journal of Engineering Mechanics ASCE 22 (in press). 2. Kan CL, Chopra AK. Effects of orsional Coupling on Earthquake Forces in Buildings. Journal of the Structural Division ASCE 1977; 13: Yoshida O, Dyke SJ, Giacosa LM, ruman KZ. orsional Response Control of Asymmetric Buildings Using Smart Dampers. Proc. of 15th ASCE Engineering Mechanics Conference, New York, NY, June ezcan SS, Alhan C. Parametric Analysis of Irregular Structures under Seismic Loading According to the New urkish Earthquake Code. Engineering Structures, 21; 23(6): Singh MP, Singh S, Moreschi LM. uned Mass Dampers for Response Control of or- 19

20 sional Buildings. Earthquake Engineering and Structural Dynamics, 22; 31: Chi Y, Sain MK, Pham KD, Spencer Jr. BF. Structural Control Paradigms for an Asymmetric Buildings. Proc. of 8th ASCE Special Conference on Probabilistic Mechanics and Structural Reliability, PMC Gavin H, Alhan C. Control of orsionally Asymmetric Structures. Proc. of the American Control Conference, Anchorage, AK, May Soong, Grigoriu M. Random Vibration of Mechanical and Structural Systems. Prentice Hall: Englewood Cliffs, New Jersey, FEMA Edition NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings - Part2 Commentary, FEMA Dyke SJ, Yi F, Carlson JD. Application of netorheological Dampers to Seismically Excited Structures. Proc. of the International. Modal Analysis Conference, Kissimmee, FL, Feb Jansen LM, Dyke SJ. Semiactive Control Strategies for MR Dampers: Comparison Study. Journal of Engineering Mechanics ASCE 2; 126(8): Yi F, Dyke SJ, Caicedo JM, Carlson JD. Seismic Response Control Using Smart Dampers. Proc. of 1999 American Control Conference, San Diego, CA, June ; Yi F, Dyke SJ, Caicedo JM, Carlson JD. Experimental Verification of Multi-Input Seismic Control Strategies for Smart Dampers. Journal of Engineering Mechanics ASCE 21; 127(11): Dyke SJ, Spencer Jr. BF, Sain MK, Carlson JD. Modeling and Control of netorheological Dampers for Seismic Response Reduction. Smart Materials and Structures 1996; 5: Dyke SJ, Spencer Jr. BF, Sain MK, Carlson JD. Seismic Response Reduction Using netorheological Dampers. Proc. of IFAC World Congress, San Francisco, CA, Jun. 3 Jul Dyke SJ, Spencer Jr. BF. Seismic Response Control Using Multiple MR Dampers. Proc. of the 2nd Intl. Workshop on Structural Control, Hong Kong, 1996; Dyke SJ, Spencer Jr. BF, Sain MK, Carlson JD. An Experimental Study of MR Dampers for Seismic Protection. Smart Materials and Structures, 1998; 7: Giacosa LM, Yoshida O, Dyke SJ. Control-oriented System Identification Using ERA. Proc. of the hird World Conf. on Structural Control, Como, Italy, 22 (in press). 19. Juang JN, Pappa RS. An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction. Journal of Guidance Control and Dynamics, 1985; 8: