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465

BUILDING HAZARD MITIGATION WITH PIEZOELECTRIC FRICTION DAMPERS G. D. Chen1 and C. Q. Chen1 1. Department of Civil Engineering, University of Missouri-Rolla, Rolla, MO 65409-0030, USA

ABSTRACT Semi-active piezoelectric friction dampers have been recently introduced to reduce the peak responses of buildings. They are regulated in real time with applied voltages according to a simple yet effective control algorithm that combines the viscous and nonlinear Reid damping mechanisms. This paper is aimed at further addressing some performance-related issues such as the stick and sliding features of a friction damper, optimum ratio of Reid and viscous damping in the control algorithm, performance comparison with Coulomb dampers, and optimal placement of friction dampers in a multi-story building. Numerical simulations of a single- and a 20- story building (statistical analysis under 10 earthquake ground motions) indicate that Coulomb dampers are effective when the external excitation on the building is known a prioror. Semi-active friction dampers, however, are effective in response reduction of a building subjected to excitations of various intensities. They have been applied to effectively mitigate the responses of buildings under both near-fault and far-field ground motions. The optimum ratio of control gain factors for the Reid and viscous damping is equal to 2/n times the excitation frequency. The application of piezoelectric actuators in civil engineering relies upon the distribution effect of multiple dampers. A sequential sub-optimal procedure for damper placement is developed for practical applications. Due to their adaptability to external disturbances, semi-active dampers can greatly enhance the multi-objective performance of buildings under multi-level excitations and will play an important role in performance-based engineering.

KEYWORDS: Piezoelectric actuators, active friction dampers, semi-active friction dampers, Coulomb dampers, seismic effectiveness, performance evaluation, near-fault effect, performance-based engineering

INTRODUCTION Friction dampers guarantee the dissipation of energy by friction and therefore do not cause instability of a structure being controlled. In this paper, piezoelectric actuators are used to modulate the clamping force of a fiction damper to make the damper adaptive to external disturbances. Such a device is referred to as a piezoelectric friction damper. Piezoelectric materials offer such unique features as effectiveness over wide frequency bands, high-speed actuation, low power consumption, simplicity,

466 reliability, and compactness as demanded in civil engineering applications (Housner et aL, 1994). However, these materials work under a small strain level and therefore are limited in their loading capacity. To generate a sufficient control force with piezoelectric materials in civil engineering applications must rely on the distribution effect of many actuators. Piezoelectric friction dampers have recently been used to mitigate the peak responses of elastic and inelastic buildings (Chen and Chen, 2000; 2002) under dynamic loading. A new control algorithm has been developed to command friction dampers. It has been demonstrated very effective in suppressing the harmonic responses of single-story structures and seismic responses of multi-story buildings. In this paper, some issues related to the control algorithm and optimal placement of multiple dampers in a building are addressed.

SEMI-ACTIVE PIEZOELECTRIC FRICTION DAMPERS A prototype damper and its schematic representation are shown in Figure 1. The schematic consists of two U-shaped bodies, one sliding against the other. The outer body is assumed to be rigid. The clamping force, N (t), acting on the sliding surfaces is controllable with four PZWT100 stack actuators. Each actuator consists of 24 discs with 0.02 in. thick each; they are connected mechanically in series and electrically in parallel. The clamping force on two friction surfaces is determined by

;V(0 = A U

+

±^M

(i)

h in which N is the pre-load on the stack actuators required for the generation of the passive friction force, E is the Young's Modulus of the PZWT material, A is the area of cross section of the stacks, h is the thickness of each layer, d33 is the piezoelectric strain coefficient and V(t) is the applied voltage on the stack actuators. When V{t) in Eqn.l is equal to zero, the clamping force is constant (=Npre) and it corresponds to a passive Coulomb friction damper. When Npre is negligible, the clamping force is proportional to the applied voltage V(t), which is fully adaptive to the feedback of the damper slippage and thus corresponds to an active friction damper. When both pre-load and voltage are applied, the damper is semi-active and requires less energy than an active damper to operate.

(a) Prototype

(b) Schematic representation

Figure 1: A piezoelectric friction damper Control Algorithm For a damper installed in the /th story of a building, a semi-active control strategy is proposed to regulate the damper with an applied voltage such that the following friction force is generated:

467 =

j

2M Npresgn[Xi(t)l

when e\Xi(t\ + g\x{t\ < Npre

in which e and g are positive gain factors, \xt(t \ and \xt(t\ are respectively the absolute values of the drift and drift rate of the ith building story, fi is the coefficient of friction and sgn[ ] represents the sign of the argument in the bracket. Note that the factor of 2 is used to account for two friction surfaces. The first expression represents the passive damper mechanism when structural responses are relatively small. When structural responses become significant, the active damper mechanism represented by the second expression is activated. Eqn. 2 indicates that, for relatively large structural responses, the active component of the proposed semi-active control strategy essentially combines both the viscous and the non-linear Reid damping mechanisms, which has been proved to be more effective than individual damping mechanisms (Chen and Chen, 2000). Optimum Gain Factor Ratio The active component of the proposed semi-active control strategy involves two gain factors, e and g, representing the weighting effect on the Reid and viscous damping mechanisms. Since the Reid damping force reaches its maximum when the viscous damping force is zero, there must exist an optimum gain ratio, e/g, which corresponds to the minimal responses of a structure under the same maximum control force. Consider the single-story frame structure with mass, natural frequency, and damping ratio respectively equal to m-2.92 kN-sec2/m, 600=3.47 Hz, and £=1.24% (Soong, 1990). The structure is controlled with an active friction damper (Npre=0). The proposed control algorithm is homogeneous to the first degree and is linearized, leading to an equivalent structure-damper system: mx\t) + m2£co0x(t) + m(olx{t) + Cex(t) = F0 sin(fttf) Ce=pg(l

+ 2e/nga))

in which Ce is the damping coefficient of the equivalent viscous damper for the active control force (Chen and Chen, 2000a). The active control force in the second expression of Eqn. 2 (i=l) can then be approximately represented by the stead state response from Eqn. 3 and its maximum value is flm&x=MgAjco2+(e/g)2

(4)

where A is the displacement amplitude of the structure described by Eqn. 3. After solving for jug from Eqn. 4 and plugging it into Ce of Eqn. 3, the equivalent damping coefficient Ce can be expressed as a function of fi,nax> A, and e/g, and A satisfies the following equation: [(1 / > ^ i

r

m (

A O T

Fa

S ^ i r

/ b

' ' ™ )+ ^ l ^ F0 Ji + b/ga,)*

Amcol ^LA^lelngwY F0 F02 l + (e/gco)2

,_„

in which ft is the ratio of excitation and natural frequencies. When f\nmx is constant, A can be solved from the above equation as a function of e/g as well as structural and excitation parameters. The displacement amplitude reaches its minimum value when e/g is approximately equal to 2co/7t. To obtain a real solution for A in Eqn. 5, the maximum control force must satisfy

468 Roles of Passive and Active Mechanisms in the Proposed Control Algorithm The same single-story frame structure is studied. When Coulomb damping is considered, the displacement and acceleration ratio of the controlled and uncontrolled structure are respectively shown in Figures 2(a, b). Their corresponding ratios when active damping is considered are presented in Figures 3(a, b), respectively. It can be observed from Figure 2 that, as the friction force increases or the excitation decreases, the displacement of the structure controlled with a Coulomb damper always decreases. However, its associated acceleration increases when j3 is smaller than 1.0 since the stick nature of the friction damper results in a sudden movement at the beginning of sliding stages. On the other hand, Figure 3 indicates that the reduction in both displacement and acceleration is nearly independent of the intensity of excitation due to the adaptability of an active damper.

0.0

0.0 0.2 0.4 0.6 0.8 1.0 Max friction force / excitation magnitude

0.2

0.4

0.6

0.8

1.0

Max friction force / excitation magnitude

(a) Reduction in displacement

(b) Reduction in acceleration

Figure 2: Performance of a Coulomb friction damper 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

2.0 • A A

^•••B

A

• 0=0-5

M.o • 8=1.5 I

0

• • • 1111

2

4 6 8 10 12 14 16 intensity of excitation

(a) Reduction in displacement

• • •

-m-+mm****

•

0=0-5 p=io

A-JH 1 ' 5 i

A

§ 0.4 (0

0.0 4 6 8 10 12 intensity of excitation

14

16

(b) Reduction in acceleration

Figure 3: Performance of an active damper with the proposed algorithm In building designs, both story drift and acceleration are important measures of building performance. Therefore, it would be beneficial to implement a semi-active control strategy for performance-based designs of buildings that are subjected to dynamic loads with uncertainties in magnitude and phase.

OPTIMAL PLACEMENT OF DAMPERS IN MULTI-STORY BUILDINGS The 20-story steel frame structure and 10 earthquake ground motions studied by Ohtori et al. (2000) are used to demonstrate the distribution and seismic effectiveness of piezoelectric friction dampers. For practical applications, a heuristic approach is taken to develop a simple and effective procedure for damper placement. The optimum gain factor ratio, e/g, is determined with the average dominant frequency (C0u= 4.73 rad/sec) of the uncontrolled displacement responses under the 10 earthquake

469 inputs. The gain factor g is taken to be 1.9 xlO5 N-sec/m to ensure that the dampers can generate the maximum friction force nearly equal to their capacity (93.1 kN) without saturation under the earthquake excitations. The pre-load Npre is selected to provide a friction force equal to 10% of the damper capacity. This algorithm is implemented in MATLAB (Chen and Chen, 2002). The following sequential procedure is used for the sub-optimal placement of dampers in a building: Step 1. Select an optimization index such as peak story drift or peak floor acceleration of the building. Step 2. Place one or several dampers on the structure toward the reduction of the optimization index. Specifically, place one damper on each permissible location of the building and evaluate the performance index. If the index is reduced with the installation of the damper at various locations, the damper is placed at the location corresponding to the maximum reduction. If adding one damper at a time cannot further reduce the performance index, two or more dampers should be placed on the structure simultaneously. Step 3. Repeat Step 2 until all the dampers have been placed on the structure. The peak story drift and the peak floor acceleration are used as potential optimization indices to facilitate the placement of dampers. To determine which index is more effective in reducing the overall building responses, both peak story drift and peak acceleration of the building under six earthquake ground motions are presented in Figures 4(a, b) and 5(a, b) as the number of dampers on the structure increases. It can be observed from Figure 4 that the use of peak floor acceleration as an optimization index results in comparable reduction in both acceleration and drift of the structure except for the Kobe earthquake. On the other hand, if the peak drift ratio is used as an index as indicated in Figure 5, it is very difficult to reduce the peak floor acceleration simultaneously. Sometimes it even increases the acceleration as illustrated in Figure 5(b). Therefore, it is recommended that the peak floor acceleration be used as the optimization index for optimal damper placement.

20

40 60 No. of dampers

40 60 No. of dampers

(a) Peak drift ratio

80

(b) Peak floor acceleration

Figure 4: Effectiveness of various numbers of dampers with acceleration-based damper profiles

40 60 No. of dampers

(a) Peak drift ratio

80

40 60 No. of dampers

80

(b) Peak floor acceleration

Figure 5: Effectiveness of various numbers of dampers with drift-based damper profiles

470

Figures 4 and 5 indicate the comparable reductions in both story drift and floor acceleration under the Northridge earthquake of different modification factors of 62% and 100%. However, under the Kobe earthquake, dampers can suppress significantly more story drift under the high excitation (100%) than that under the low excitation (58.3%). This result implies the occurrence of substantial plastic deformation at several locations under the 100%Kobe earthquake. Considering various earthquakes of the same intensity, e.g. 150%E1 Centro vs. 62%Northridge earthquake, it is observed from Figures 4 and 5 that both story drift and floor acceleration can be reduced more effectively under the El Centro earthquake than those under the Northridge earthquake. Those under the 150%Hachinohe are mitigated slightly less effectively than under the 58.3%Kobe earthquake. In general, a structure subjected to farfield ground motions, such as the El Centro earthquake, can be more easily controlled with dampers than that under near-fault ground motions such as the Northridge and Kobe earthquakes. However, it is not difficult to control the excessive inelastic story drift of a structure under strong near-fault earthquakes such as the 100%Kobe. To select the optimal distribution of the 80 dampers over the building height for overall performance under various earthquakes, six damper profiles were determined using the proposed sequential procedure with the peak floor acceleration as the optimization index. Each profile corresponds to one of the six earthquake inputs used in Figures 4 and 5. These profiles were then employed to calculate the peak accelerations at all floors of the building subjected to the 10 earthquake excitations (Ohtori et al.y 2000). The maximum value of the peak accelerations at all floors was normalized with that of the uncontrolled structure under each of the 10 earthquake excitations. The statistical characteristics of the 10 samples for peak acceleration ratio (ra) such as the maximum value [max(ra)], average (r fl ), and standard deviation [ s(ra) ] are shown in Table 1. It can be observed from the table that Profile8 has the lowest maximum and average accelerations. The standard deviation associated with this profile is also very small, indicating the consistent damper performance under the 10 earthquake ground motions. Thus, Profile8, determined under the Northridge earthquake, achieves the best overall performance in reducing the peak floor acceleration of the structure. The 80 dampers in this case are distributed along the building height from top to bottom as: [26, 1, 3, 0, 7, 0, 0, 1, 7, 0, 7, 5, 4, 2, 1, 0, 6, 1, 0, 9]. TABLE 1 PEAK ACCELERATION RATIOS WITH 10% CAPACITY FROM THE PASSIVE COMPONENT Damper Profile Profile3 (150%E1 Centro EQ) Profile6 (150%Hachinohe EQ) Profile8a (62%Northridge EQ) Profile8 (100%Northridge EQ) ProfilelOa (58.3%Kobe EQ) ProfilelO (100%Kobe EQ)

max(ra) 1.111 0.985 1.085 0.989 1.004 1.067

a

*(0

0.882 0.891 0.882 0.878 0.885 0.882

0.122 0.059 0.110 0.081 0.085 0.104

r

PERFORMANCE OF PASSIVE, SEMI-ACTIVE, AND ACTIVE FRICTION DAMPERS To understand the pre-load effect on the damper performance, the statistical characteristics similar to those in Table 1 under the Northridge earthquake are given in Table 2 for various Npre. It can be clearly seen from the table that all statistical characteristics of the peak floor acceleration increases as the preload increases. This means less effectiveness of dampers in reducing the structural acceleration when the percentage of the passive friction force increases in the proposed semi-active strategy. Similarly, more pre-load on the passive component of the semi-active damper results in the increasing maximum and average peak drift ratio. On the other hand, Table 2 indicates that the required maximum active

471 control force (normalized by the total building weight, 1.09x10s kN) significantly drops as the pre-load increases. In conclusion, the introduction of more passive friction force in the semi-active friction dampers significantly degrades the performance of the dampers but reduces the active control force. To maintain the effectiveness of dampers with reasonable power demand, a pre-load corresponding to 10% of the damper capacity was used in the previous section. TABLE 2 PERFORMANCE COMPARISON BETWEEN ACTIVE AND SEMI-ACTIVE DAMPERS Damper Profile Profile8b (Npre=>0% capacity) (active) Profile8 (Npre=>10% capacity) (semi-active) Profile8d (Npre=>30% capacity) (semi-active) Profile8f (Npre=>50% capacity) (semi-active)

Response Quantity Story drift ratio Acceleration ratio Control force (xl0~4) Story drift ratio Acceleration ratio Control force (xlO"4) Story drift ratio Acceleration ratio Control force (xl0~4) Story drift ratio Acceleration ratio Control force (xl0~4)

max(r) 0.957 0.944 8.186 0.962 0.980 7.470 0.976 1.428 5.339 0.982 2.255 4.164

r 0.877 0.872 4.228 0.875 0.878 3.362 0.860 0.976 1.799 0.854 1.211 0.847

s{f) 0.718 0.044 2.099 0.078 0.081 2.174 0.095 0.191 1.827 0.100 0.481 1.422

To compare the performance between Coulomb dampers and semi-active friction dampers, Table 3 lists the statistical characteristics of the peak story drift and peak floor acceleration ratios associated with the optimal damper placement, Profile8, determined under the Northridge earthquake. For passive dampers, different levels of friction forces have been used in numerical simulations. It is seen from Table 3 that the maximum values of both story-drift and floor acceleration ratios can be reduced more by semi-active dampers. As the friction force of Coulomb dampers increases, both the maximum and average drifts decreases and their standard deviation increases due to stick effects. All the statistical quantities of the floor acceleration substantially increase with the friction force of Coulomb dampers, indicating degrading performance. These results indicate that semi-active dampers outperform Coulomb dampers for the multi-level earthquake design of buildings. TABLE 3 PERFORMANCE COMPARISON BETWEEN SEMI-ACTIVE AND PASSIVE DAMPERS Control Strategy Semi-active (NDre=>10% capacity) Passive (Npre=>10% capacity) Passive (Npre=>50% capacity) Passive (NDre=»100% capacity)

Response Quantity Story drift ratio Acceleration ratio Story drift ratio Acceleration ratio Story drift ratio Acceleration ratio Story drift ratio Acceleration ratio

max(r) 0.962 0.980 1.012 1.036 0.986 2.249 0.964 4.224

r 0.875 0.878 0.932 0.938 0.833 1.173 0.778 1.850

sir) 0.078 0.081 0.078 0.074 0.101 0.431 0.109 0.994

The optimum ratio of control gain factors, e/g, was derived from a single-story building subjected to harmonic loading. To see how sensitive the dampers' performance is to the change of the gain factor ratio, Table 4 shows the statistical results of peak story-drift and peak acceleration ratios when ±10%

472

uncertainties in the determination of the gain factor ratio is introduced. Clearly, it is observed that the changes in maximum and average story-drift ratios or acceleration ratios are all within 1%. The optimum ratio of two gain factors derived in this study is thus valid for the practical design of dampers. TABLE 4 PERFORMANCE SENSITIVITY TO OPTIMUM GAIN FACTOR RATIO Excitation Frequency 0.9(0u l.OODu l.lCOu

Response Quantity Story drift ratio Acceleration ratio Story drift ratio Acceleration ratio Story drift ratio Acceleration ratio

max(r) 0.965 0.978 0.962 0.980 0.959 0.980

r 0.880 0.880 0.875 0.878 0.870 0.874

s(r) 0.076 0.081 0.078 0.081 0.082 0.079

CONCLUDING REMARKS Based on the extensive numerical simulations in this study, it is concluded that Coulomb dampers can be designed to effectively reduce the dynamic responses of a building when the intensity of an external disturbance exerted on the building is known during the design period. Semi-active friction dampers, however, are effective in response reduction of a building subjected to excitations of various intensities. The proposed algorithm is effective in mitigating the inelastic responses of buildings under near-fault ground motions. The optimum ratio of control gain factors for the Reid and viscous damping is equal to 2/n times the excitation frequency. The application of piezoelectric actuators in civil engineering relies upon the distribution effect of multiple dampers. The proposed sequential procedure for damper placement is very efficient.

ACKNOWLEDGEMENTS This study was sponsored by the U.S. National Science Foundation under Award No. CMS-9733123 with Drs. S. C. Liu and P. Chang as Program Directors. The results, opinions and conclusions expressed in this paper are solely those of the authors and do not necessarily represent those of the sponsor.

REFERENCES Chen, G. D. and Chen, C. C. (2000). Behavior of Piezoelectric Friction Dampers under Dynamic Loading. Proc. SPIE Symposium on Smart Structures and Materials: Smart Systems for Bridges, Structures, and Highways, Newport Beach, CA, Vol. 3988, 54-63. Chen, G. D. and Chen, C. C. (2002). Semi-Active Control of the 20-Story Benchmark Building with Piezoelectric Friction Dampers. Journal of Engineering Mechanics, ASCE (accepted for publication). Housner, G. W., Soong T. T. and Masri, S. F. (1994). Second Generation of Active Structural Control in Civil Engineering. Proc. 1st World Conference on Structural Control, Los Angeles, CA, Panel: 3-18. Ohtori, Y., Christenson, R. E. and Spencer, Jr., B. F. (2000), "Benchmark control problems for seismically excited nonlinear buildings", http://www.nd.edu/~quake/nlbench.html. Soong T.T. (1990). Active Structural Control: Theory & Practice, Longman & Scientific Technical, New York.