Vibration Control of Structures with Initially Accelerated Passive Tuned ...

Computer-Aided Civil and Infrastructure Engineering 25 (2010) 69–75

SHORT CONTRIBUTION

Vibration Control of Structures with Initially Accelerated Passive Tuned Mass Dampers under Near-Fault Earthquake Excitation Chi-Chang Lin∗ , Chi-Lun Chen & Jer-Fu Wang Department of Civil Engineering, National Chung Hsing University, Taichung, Taiwan

Abstract: This article deals with the applicability of passive tuned mass damper (PTMD) for reducing the dynamic responses of structures under near-fault earthquake excitations. Three types of pulse-like time functions are employed to simulate the near-fault ground motion. The vibration control effectiveness of PTMDs is extensively investigated through the comparisons of response spectrum and response time histories of a structure with and without PTMDs. To suppress structural peak responses, an initial velocity is applied to the PTMD to accelerate its motion. Numerical studies show that the more cycles of a pulse are contained in the ground motion, the more useful a PTMD becomes in reducing the peak structural responses. 1 INTRODUCTION Because of intensive analytical and experimental research, structural control has gained considerable acceptance, not only in the design of new structures and components but also in the retrofit of existing structures to enhance their reliability against winds, earthquakes, traffic loads, and human activities (Adeli and Saleh, 1997, 1998, 1999; Adeli and Kim, 2004; Jiang and Adeli, 2008a,b; Kim and Adeli, 2004, 2005a,b,c,d; Saleh and Adeli, 1994, 1997, 1998; Soong and Costantinou, 2002). Passive tuned mass damper (PTMD), first proposed by Frahm (Frahm, 1911) in 1909 for reducing the rocking motion of ships, is a device of great interest to those in civil engineering in recent decades. ∗ To

whom corrspondence should be addressed. E-mail: cclin3@ dragon.nchu.edu.tw.

 C 2009 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/j.1467-8667.2009.00607.x

Although previous studies on PTMDs are abundant (Satareh and Hanson, 1992a,b; Gu et al., 1994; Lin et al., 1994, 2001; Chang, 1999; Wang et al., 2003), the vibration control effectiveness of this device has been controversial, especially for seismic applications (Villaverde, 1994; Sadek et al., 1997). According to the study by Lin et al. (2001), the above inconsistency resulted mainly from using different dynamic characteristics of structure and external excitation, as well as different performance measures to determine the effectiveness of the PTMD. Theoretically, the damping effect of the PTMD depends on the fact that it delays the main structural response by a phase angle of 90◦ , such that the elastic force transmitted by the PTMD acts like a viscous force on the main structure. This condition does not occur unless the PTMD frequency is tuned to the natural frequency of the main structure and the excitation contains this frequency. A near-fault earthquake having the pulse-like characteristic may be a representative of transient earthquake excitations. Most of the measurements of nearfault earthquakes show that their velocity motions are similar to single impulse motions (Loh and Lee, 2000; Alavi and Krawinkler, 2004; Mavroeidis et al., 2004). Therefore, the performance of PTMDs in near-fault earthquake excitations becomes a subject worthy of further study. The objective of this article is to investigate the vibration control effectiveness of a PTMD for reducing the peak and root-mean-square responses of structures under near-fault earthquakes. Numerical studies are performed for a structure with and without a PTMD, subjected to simulated and real near-fault earthquakes. To

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(a)

0.6

(b)

0.6

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Accelerarion (g)

PGA=0.5g PGA = 0.50 g 0.4

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-0.4

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recorded Type-A Simulation

Velocity (m/s)

2 0.2 0

PGA=0.44g PGA = 0.44 g

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0

0

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-0.2 -2 -0.4

PGV=2.16m/sec PGV = 1.75 m/s

-2

PGV=2.75m/sec PGV = 2.75 m/s

PGV=0.36m/sec PGV = 0.36 m/sec m/s

-3 2

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0

0

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0.4 0.2 0 -0.2

Displacement PGD = 4.97 m =7.97m

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Fig. 1. Free-field measurements: (a) 1940 El Centro earthquake; (b) 1999 Chi-Chi earthquake in Taiwan at TCU068 station (EW); (c) 1999 Chi-Chi earthquake in Taiwan at TCU052 station (NS).

suppress the structural peak responses more effectively, an initial impulse is applied to the PTMD to accelerate its motion. The effect of the initial velocity, caused by the applied impulse, on the PTMD performance is discussed in the later sections.

respectively, to represent various types of near-fault ground acceleration, velocity, and displacement as 1. Type A vf sin ω f t (1a) x¨ g (t) = ω f 2

2 CHARACTERISTICS OF NEAR-FAULT GROUND MOTIONS Figure 1a illustrates the ground motions recorded during the 1940 El Centro earthquake, which is commonly used as the standard earthquake for seismic design of structures, whereas Figures 1b and 1c plot two nearfault ground motions recorded at the TCU068 and TCU052 stations near the epicenter of the 1999 ChiChi earthquake in Taiwan that led to a large fault rupture. The pulse-like feature of a near-fault ground motion can be clearly seen in the velocity–time history trace. Makris and Roussos (2000) developed three analytical expressions, named Type A, Type B, and Type C n ,

x˙ g (t) =

vf vf − cos ω f t 2 2

(1b)

xg (t) =

vf vf sin ω f t t− 2 2ω f

(1c)

2. Type B x¨ g (t) = ω f v f cos ω f t

(2a)

x˙ g (t) = v f sin ω f t

(2b)

xg (t) =

vf vf − cos ω f t ωf ωf

(2c)

where v f is the peak ground velocity, ω f = 2π /Tf , Tf is the period of pulse-like waves, and 0 ≤ t ≤ Tf .

Vibration control of structures with initially accelerated PTMDs

1.5

Relative Displacement (m)

0.30 Building A Type-A

0.25 0.23 (-8.0%)

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-0.866 (-17.5%)

Building B Type-C2

Relative Displacement (m)

Building B Type-A

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w/o PTMD with PTMD

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Fig. 2. Time histories of relative displacement of Buildings A and B with and without PTMD under Type A and Type C 2 impulsive ground motions.

3. Type C n x¨ g (t) = ω f v f cos(ω f t + φ)

(3a)

x˙ g (t) = v f sin(ω f t + φ) − v f sin φ

(3b)

vf xg (t) = − cos(ω f t + φ) ωf vf −v f t sin φ + cos φ ωf

3 PERFORMANCE OF CONVENTIONAL PTMD UNDER NEAR-FAULT EARTHQUAKES (3c)

where 0 ≤ t ≤ kn Tf ; kn = n + 1/2 − φ/π, n denotes the number of forward and backward cycles of the impulse, and φ denotes the phase angle that can be solved from Equation (4): cos [(2n + 1) π − φ] + [(2n + 1) π − 2φ] sin φ − cos φ = 0

the above-mentioned analytical expressions will be used to discuss the vibration control effectiveness of PTMDs for structures under near-fault ground motions.

(4)

Figure 1b shows that the Type A simulation agrees well with the measured ground acceleration, velocity, and displacement at the TCU068 station. In this study,

A single-degree-of-freedom (SDOF) building equipped with a PTMD was used to examine the PTMD vibration control effectiveness under near-fault earthquake excitations. The PTMD parameters used in this study were obtained on the basis of the optimal design procedure developed by the authors (Lin et al., 1994, 2001). When the ratio of the PTMD mass to the building mass, μ s , is equal to 2%, the optimal frequency ratio of the PTMD to the building, (rf ) opt , and the optimal damping ratio of the PTMD, (ξ ) opt , are 0.962 and 7%, respectively. Type A, Type B, Type C 1 , and Type C 2 near-fault ground motions with ω f /2π = 1 Hz and v f = 1 are chosen in this numerical study.

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Fig. 3. Normalized acceleration response spectrum with and without PTMD under various types of impulsive ground motions. 80

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Acceleration Reduction (%)

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Fig. 4. Acceleration reduction percentage against various PTMD initial velocities under various types of impulsive ground motions.

Vibration control of structures with initially accelerated PTMDs

1 Hz Structure under Type-A

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0.4 Hz Structure under TCU068

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Relative Displacement (m)

(a) 0.181 0.182 (-0.55%)

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1.497 0.119 (-25.3%)

v0 = 0

v0 = 0

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Relative Displacement (m)

(b) 0.235 0.159 (-12.2%)

1.497 0.908 (-39.3%)

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-1.0 m/s vv00==55m/sec

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-10m/sec m/s vv0 = 0 =-10

-2.0

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Time (s)

Fig. 5. Relative displacement of structures (1.0 and 0.4 Hz) without and with PTMD with and without initial velocity under Type A impulsive ground motion and the 1999 Chi-Chi earthquake in Taiwan at TCU068 station.

Two SDOF buildings with 2% damping ratio and different natural frequencies are chosen for analysis. Building A with ω p /2π = 1 Hz has the same frequency as those of the impulsive ground motions, whereas Building B with ω p /2π = 3 Hz, which is away from those of external excitations. Figure 2 illustrates the relative displacements of Building A and Building B subjected to Type A and Type C 2 impulsive ground motions, respectively. It is evident that the PTMD is more effective for Building A because a resonance condition leading to large PTMD stroke occurs, and less effective in reducing the first few local peaks of building vibration response. This is because the PTMD cannot immediately produce a complete hysteresis loop to dissipate the vibration energy. In other words, a PTMD for reducing the peak response of buildings excited by near-fault earthquake ground motions is less effective than that due to far-field earthquakes. Figure 3 illustrates the normalized response spectra for the four types of impulsive ground motions. Here, the peak values of the impulses are all scaled to unity.

It is seen that the peak response of the building with a PTMD is reduced as the impulse-to-structure period ratio, Tf /Tp , is close to unity. It also means that the PTMD is less effective in reducing the peak responses of structures that do not encounter resonance. From Figure 3, it is shown that the reduction of maximum response spectrum values for buildings under Type A and Type B impulsive ground motions are smaller than those under Type C 1 and Type C 2 motions, because Type C n reaches the peak response after few complete vibration cycles, which gives the PTMD more time to initiate its action.

4 EFFECT OF INITIAL VELOCITY ON PTMD To overcome the stated problem, a given initial PTMD velocity opposite to the direction of structural impulsive motion is considered to help the PTMD generate an initial force. Figure 4 shows the peak acceleration reduction of a structure (ω p /2π = 1 Hz and ξ p = 2%) with PTMDs (μ s = 2%, (rf ) opt = 0.962, and (ξ ) opt = 7%)

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2.0

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PTMD Stroke (m)

1.0

0.0

-0.701

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-0.825

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PTMD Initial Velocity 0 m/sec m/s m/s 5 m/sec

2

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PTMD Initial Velocity 0 m/sec m/s m/s -10 m/sec

-5.84

-10

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20

40

Another structure (ω p /2π = 0.4 Hz and ξ p = 2%) under the 1999 Chi-Chi earthquake, Taiwan, ground motion at TCU068 station (shown in Figure 1) is also investigated. With a PTMD having initial velocity of v 0 = −10 m/s, the peak displacement of the structure is significantly reduced, as shown in Figures 5c and 5d. However, Figure 6a shows that the stroke of PTMD could be larger than that of initially static PTMD. The inevitably large PTMD stroke makes it impractical to impose the PTMD with an optimum velocity. Therefore, the limitation of moving space for PTMD should be considered during the determination of initial velocity. The initial velocity v 0 of a PTMD can be generated by providing the PTMD with a force F in a short period, t. According to the impulse momentum theorem, the required impulse force F is equal to (v 0 /gt)ms g, where g is the acceleration of gravity. It reveals that F is large if t is short. This impulse force can be applied if a pulse-like ground motion over a certain threshold is monitored. Again, the optimum v 0 will correspond to impractically large F. Smaller v 0 needs to be selected in practice.

5 CONCLUSIONS 60

Time (s) Fig. 6. PTMD strokes without and with initial velocity for (a) 1-Hz structure under Type A impulse ground motion; (b) 0.4-Hz structure under the 1999 Chi-Chi earthquake in Taiwan at TCU068 station.

having various initial velocities. It is evident that there is an optimum PTMD initial velocity (about 20–30 m/s for v f = 1 m/s) for minimizing the structural acceleration response. Moreover, the reductions for Type A and Type B ground motions are more sensitive to the change of PTMD initial velocity than those for Type C n ground motion. In the latter case, the peak structural response does not occur in the first few vibration cycles with which the PTMD initial force is expected to work. The results also imply that a large PTMD initial velocity might even amplify the structural response (Type A case). Figures 5a and 5b plots the relative displacement– time history responses of the structure with the PTMD having an initial velocity of v 0 = 0 and 5 m/s, respectively, under Type A ground motion (ω f /2π = 1 Hz and v f = 1 m/s). It is clearly shown that the local peak structural response occurring at the beginning is reduced due to the PTMD initial velocity, and the peak value of the structural response is reduced from 0.55 to 12.2%.

The following conclusions can be drawn from this study: 1. PTMDs are more effective in vibration control of structures under impulse-like ground motion with more forward-and-backward cycles. 2. An appropriate PTMD initial velocity can be applied to effectively reduce the first few local peak responses of a structure under near-fault earthquake excitation. However, due to the limitation of PTMD’s stroke as well as the applied force, the initial velocity cannot be too large in practical applications. 3. This study primarily focuses on the PTMD control effectiveness for structures subjected to a transient loading, like a near-fault earthquake. Simple building and ground-motion models are used to avoid confusion on the results. The idea to give the PTMD an initial velocity is intuitive. Any more detailed study about the implementation of the initial velocity is not included and needs to be further examined.

ACKNOWLEDGMENTS This work was supported in part by the Ministry of Education and the National Science Council of the Republic

Vibration control of structures with initially accelerated PTMDs

of China in Taiwan under the ATU plan and grant NSC 90-2625-Z-005-007. These supports are greatly appreciated. The authors thank the reviewers for their constructive comments that improved the quality of this article.

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