A New Algorithm of Semi-active Friction Dampers for Structures under Earthquakes

Qing Liu Chunxiang Li Department of Civil Engineering, Shanghai University, No.149 Yanchang Rd., Shanghai 200072,

P. R. China Abstract

A new algorithm of semi-active friction dampers

(SAFD) is proposed in this paper for controlling the seismic responses of structures, based on the linear quadratic regulator (LQR) control theory. In order to numerically verify the control effectiveness of the proposed algorithm, the time history analysis has been implemented to a seismically excited 3-storey frame structure with the friction dampers (FD). In the numerical simulations, five seismic waves, namely El Centro wave, Hachinohe wave, Kobe wave, Taft wave, and Shanghai artificial wave are taken into consideration, whose peak accelerations are scaled to 0.1gal and 0.3gal, respectively. For the purpose of comparison, the numerical results of the 3-storey frame structure with dampers designed by general semi-active damper control algorithm (referred herein to as the SADCA) are also taken into account. Numerical results demonstrate that the proposed semi-active control algorithm is better than the SADCA under the present five seismic waves. Especially, this new semi-active control algorithm will render higher level of the acceleration response reduction for higher floors of structures under earthquakes. Keywords: Earthquakes; Structures; Friction dampers; Semi-active control; Control algorithms 1. Introduction

Friction damper (referred hereafter to as the FD) dissipates the vibration energy of structures via friction. It may be manufactured as an axial member to replace an ordinary member, and its configuration is simple and compact. As an energy dissipating device (EDD), the FD can reduce all vibration modes, including seismic and wind-induced responses (Xu, Qu and Chen 2001). The passive friction dampers (PFD) have been used to improve earthquake-resistant performance of structures (Pall, Marsh and Fazio 1980; Aiken and Kelly 1990). But, passive measures reduce noise and vibration at low frequency domain (Paulin, Philippe and Patrice 2009). As a remedy, several researchers have made profound study in the semi-

active friction dampers (SAFD). Chen and Chen (2004) proposed a semi-active control algorithm. Ng and Xu (2002) made a comparison study on both the passive and semi-active FD control, which further pointed out the practical application value of the SAFD. The friction force of the FD should enhance as the interstory velocity of structures increase (Chen and Chen 2000). In view of this, this paper proposes a new SAFD control algorithm with variable stiffness and damping, based on the LQR optimal control theory. The seismic responses of a 3-storey frame structure with the semi-active FD devices are then analyzed. For the purpose of comparison, this study simultaneously calculates the seismic responses of the structure with dampers designed by general SADCA. Numerical simulations indicate that the new SAFD control algorithm is better than general SADCA, especially in controlling the top storey acceleration of structures. 2. Model of structures with the FD devices

Now, consider a 3-storey shear-type frame structure with the FD devices, as shown in Fig.1. Assuming that

kgMMM 5321 104 ×=== , mNKKK /106.1 7

321 ×=== . Likewise, the structural damping matrix may be obtained using the Rayleigh damping hypothesis. In each floor is one FD installed. Earthquake inputs in this study include the El Centro, Hachinohe, Kobe, Taft and Shanghai artificial waves.

Fig 1. Analytic model of the structure-FD system

2009 Sixth International Conference on Fuzzy Systems and Knowledge Discovery

978-0-7695-3735-1/09 $25.00 © 2009 IEEE

DOI 10.1109/FSKD.2009.454

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2009 Sixth International Conference on Fuzzy Systems and Knowledge Discovery

978-0-7695-3735-1/09 $25.00 © 2009 IEEE

DOI 10.1109/FSKD.2009.454

504

Establish the motion equations of the FD structure system as follows:

UBXMIKXXCXM sg +−=++ (1) Each floor installs one control device (FD), that is:

TuuuU ][ 321= (2)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−=

100110

011

sB (3)

Employ a state-space representation of the motion equations of the FD structure system subjected to a base acceleration

gX . Given then is the following matrix equation.

gXDBUAZZ ++= (4) where

⎥⎦

⎤⎢⎣

⎡−−

= −− CMKMI

A 11

0 ; ⎥⎦

⎤⎢⎣

⎡= −

sBMB 1

0 ; ⎥⎦

⎤⎢⎣

⎡−

= − MIMD 1

0

3. New SAFD control algorithm

The present SAFD control algorithm is realized

through the following two steps. Firstly, for the minimization of the structural control energy, a 3-

storey frame structure is controlled using the LQR control algorithm, so that the optimal control forces can be obtained. Secondly, the real-time control forces of friction dampers need to be regulated in terms of the predefined control law by using each floor’s optimal control force subjected to earthquakes. As a result, the real-time friction control force may be calculated. Subsequently, the SAFD is implemented to this structure.

In order to fulfill the design requirement of structural control under earthquakes, some methods, such as adding stiffness or damping of structures, can be adopted. However, using either variable stiffness control or variable damping control may lead to amplifying the structural acceleration (Yamada and Kobori 1995; Yang, Wu and Li 1996; Loh and Ma 1996).

Consequently, a new semi-active friction damper (SAFD) control algorithm, namely the predefined control law, is proposed in this paper. In order to conduct semi-active control on the structures under earthquakes, the control forces of the SAFD will be calculated using a new algorithm, which can be represented as follows:

[ ] [ ][ ]

[ ]

max max

max

min

2 ( ) 2 ( ) ( ) ( ) 0 and ( ) / ( )( ) 2 ( ) / ( ) ( ) 2 ( ) ( ) ( ) 0 and ( ) / ( )

( ) ( ) 02 ( )

id i id i i i i i id

i i i i id i i i i i id

i iid i

C x t K x t u t x t u t x t CU t u t x t x t K x t u t x t u t x t C

u t x tC x t

μ μμ μ

μ

⎧ + < >⎪

= ⎡ ⎤ + < <⎨ ⎣ ⎦⎪ ≥⎩

(5)

max max max/id i iC u x= (6)

sCC idid /maxmin = (7)

maxmax / iiid xuK = (8)

in which maxidC and minidC = maximum and minimum damping coefficients of the FD in the ith floor; s means adjustable diameter of the FD damping coefficient (in this study s = 8), which can be set according to the control requirements; idK = stiffness coefficient of the FD in the ith floor; )(txi and )(txi = interstory velocity and drift of the ith floor at the instant time t ; )(tui = obtained optimal control force of the ith floor at the instant time t , based on the LQR optimal control algorithm; each FD has two sliding friction surfaces, and its friction coefficient is μ (in the present paper μ = 0.8). 4. Numerical verification and comparison

In order to numerically verify the advance of this new SAFD control algorithm, it is programmed by using the MATLAB software package to a 3-storey shear-type frame structure.

For the purpose of comparisons, the numerical results of general SADCA (see Appendix A) are also taken into account. It is hypothesized that each floor installs one control device. Five earthquake waves, namely the El Centro wave, Hachinohe wave, Kobe wave, Taft wave, and Shanghai artificial wave are taken into consideration, whose peak ground accelerations (PGA) are scaled to 0.1gal. Figure 2 presents the time history of the top storey seismic responses of the structure-control system under these five earthquake waves (Peak ground acceleration, PGA = 0.1gal).

505505

0 5 10 15-0.05

0

0.05

Displ

acem

ent (

m)

Uncontrolgeneral SADCANew Algorithm

0 5 10 15-0.5

0

0.5

Vel

ocity

(m

/s)

0 5 10 15-2

0

2

Time (s)

Acc

(m

/s2 )

0 5 10 15-0.05

0

0.05

Displ

acem

ent (

m)

0 5 10 15-0.5

0

0.5

Vel

ocity

(m

/s)

0 5 10 15-2

0

2

Time (s)

Acc

(m

/s2)

Uncontrolgeneral SADCANew Algorithm

(a) Under El Centro wave with PGA=0.1gal (b) Under Hachinohe wave with PGA=0.1gal

5 10 15 20-0.05

0

0.05

Displ

acem

ent (

m)

5 10 15 20-0.5

0

0.5

Vel

ocity

(m

/s)

5 10 15 20-5

0

5

Time (s)

Acc

(m

/s2 )

Uncontrolgeneral SADCANew Algorithm

0 5 10 15 20 25-0.05

0

0.05

Displ

acem

ent (

m)

Uncontrolgeneral SADCANew Algorithm

0 5 10 15 20 25-0.5

0

0.5

Vel

ocity

(m

/s)

0 5 10 15 20 25-2

0

2

Time (s)

Acc

(m

/s2 )

(c) Under Kobe wave with PGA=0.1gal (d) Under Taft wave with PGA=0.1gal

5 10 15 20-0.05

0

0.05

Displ

acem

ent (

m)

5 10 15 20-0.5

0

0.5

Vel

ocity

(m

/s)

5 10 15 20-2

0

2

Time (s)

Acc

(m

/s2)

Uncontrolgeneral SADCANew Algorithm

(e) Under Shanghai artificial wave with PGA=0.1gal

Fig 2. Time history of the top storey seismic responses of the SAFD structure system

Time-history of the top storey seismic responses of the structure-control system are shown in Figure 2(a)-(e) under the El Centro wave (0.1gal), Hachinohe wave (0.1gal), Kobe wave (0.1gal), Taft wave (0.1gal), and Shanghai artificial wave (0.1gal), respectively. The displacement, velocity, and acceleration responses of top storey are all decreased significantly by resorting to the control devices. Likewise, the control effectiveness of the new semi-active control algorithm is better than general SADCA, when the structure is subjected to the El Centro wave (0.1gal), Hachinohe wave (0.1gal) and Taft wave (0.1gal), as shown in Figure 2(a), 2(b), and 2(d). It is seen from Figure 2(a) that the displacement and velocity responses of structural top storey are

decreased remarkably by resorting to the control devices, when subjected to Kobe wave. However, unfortunately, the acceleration response of structural top storey using general SADCA is beyond the uncontrolled structure acceleration at some instant time points. But the structural acceleration response can be effectively controlled when using the new semi-active control algorithm. From figure 2(e), when the structure suffers from Shanghai artificial wave, the displacement and velocity responses of structural top storey are also decreased significantly with resorting to the control devices. Same, the acceleration response of structural top storey using general SADCA is beyond the acceleration response of the uncontrolled structure at

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some instant time points. The structural acceleration response can be effectively controlled when employing the new semi-active control algorithm. Likewise, this new semi-active control algorithm will render higher level of the acceleration response reduction for higher floors of structures under earthquakes. A comparison in details is given in Table 1.

Furthermore, the present study also calculates the seismic responses of the structure-control system under these earthquakes but with their peak ground accelerations (PGA) scaled to 0.3gal. The numerical results, as listed in Table 2, indicate that the new SAFD control algorithm is same undertaking an excellent e f f e c t i v e n e s s i n a l l e v i a t i n g t h e s e i s m i c responses of structures.

Table 1. Control efficiencies of the semi-active control devices for the structure

(PGA=0.1gal)

Control Algorithm Earthquake waves

Control efficiency for the top storey displacement

Control efficiency for the top storey

velocity

Control efficiency for the top storey

absolute acceleration

Uncontrolled

El Centro --- --- --- Hachinohe --- --- --- Kobe --- --- --- Shanghai artificial --- --- --- Taft --- --- ---

Employing general SADCA

El Centro 54.7% 45.8% 30.6% Hachinohe 41.7% 38.5% 14.6% Kobe 40.1% 33.1% -3.3% Shanghai artificial 47.0% 35.8% -1.1% Taft 60.8% 54.9% 26.6%

Employing the new semi-

active control algorithm

El Centro 60.0% 52.4% 49.1% Hachinohe 45.9% 46.8% 24.7% Kobe 71.5% 65.3% 29.6% Shanghai artificial 55.8% 43.6% 24.0% Taft 77.2% 65.3% 30.4%

Table 2. Control efficiencies of the semi-active control devices for the structure

(PGA=0.3gal)

Control Algorithm Earthquake waves

Control efficiency for the top storey displacement

Control efficiency for the top storey

velocity

Control efficiency for the top storey

absolute acceleration

Uncontrolled

El Centro --- --- --- Hachinohe --- --- --- Kobe --- --- --- Shanghai artificial --- --- --- Taft --- --- ---

Employing general SADCA

El Centro 54.5% 45.8% 30.6% Hachinohe 41.6% 38.5% 14.6% Kobe 40.4% 33.1% -3.3% Shanghai artificial 46.8% 35.8% -1.1% Taft 60.8% 54.9% 26.6%

Employing the new semi-

active control algorithm

El Centro 59.7% 52.4% 49.0% Hachinohe 45.5% 46.9% 24.7% Kobe 71.4% 65.2% 29.5% Shanghai artificial 55.7% 43.6% 17.9% Taft 77.1% 65.3% 30.4%

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5. Conclusions

A new algorithm of semi-active friction dampers (SAFD) is proposed in the present paper to mitigate the earthquake-induced vibration, based on the linear quadratic regulator (LQR) control theory. The performance of the proposed control scheme has been numerically investigated through implementation of a 3-storey frame structure. The obtained results using a set of earthquake records have been displayed and compared with those employing general SADCA. It has demonstrated that the proposed semi-active control algorithm is quite competitive as compared with general SADCA. Likewise, the numerical simulations show that general SADCA has a poorly controlled undertaking in reducing the structural acceleration responses, and even it sometimes may amplify the structure acceleration responses. However, as a remedy, the new semi-active control algorithm has an advantage in providing more effective structural acceleration attenuation and moreover, the simulation results have verified that it will render higher level of the acceleration response reduction for higher floors of structures under earthquakes. Evidently, the proposed SAFD control algorithm possesses useful reference values in the mitigation of earthquake-or wind-induced vibration of multi-storey, high-rise, even super high-rise buildings. Appendix A: General semi-active damper control algorithm (referred herein to as general SADCA) (Hrovat and Barak 1983; Ou 2003)

In order to verify the superiority of this new algorithm, general SADCA is also taken into consideration in this study. Essentially, general SADCA is a semi-active variable damping control strategy. Its semi-active control force also is based on the LQR optimal control theory; this control algorithm can be represented as follows:

)()()( txtCtU iidi = (9)

max max

max

min

( ) ( ) 0 and ( ) / ( )( ) ( ) / ( ) ( ) ( ) 0 and ( ) / ( )

( ) ( ) 0

id i i i i id

id i i i i i i id

i iid

C u t x t u t x t CC t u t x t u t x t u t x t C

u t x tC

⎧ < >⎪

= ⎡ ⎤ < <⎨⎣ ⎦⎪ ≥⎩

(10)

in which maxidC and minidC = maximum and minimum damping coefficients of the control device in the ith floor; iu = obtained optimal control force of the ith

floor on the instant time t , by resorting to the LQR optimal control algorithm. Acknowledgements

The authors would like to acknowledge the financial contributions received from the Innovation Project of Shanghai Board of Education (Contract/grant number: 09-YZ-35). References [1] Y.L. Xu, W.L. Qu, Z.H. Chen, “Control of wind-

excited truss tower using semi-active friction damper”, Journal of Structural Engineering, ASCE 2001, 127(8) pp. 861-868.

[2] A.S. Pall, C. Marsh, P. Fazio, “Friction joints for seismic control of large panel structures”, Journal of Prestressed Concrete Institute, 1980, 25(6) pp. 38-61.

[3] I.D. Aiken, J.M. Kelly, “Earthquake simulator testing and analytical studies of two energy-absorbing systems for multi-storey structures”, Rep. No.UCB/EERC-90/03, University of California, Berkeley, 1990.

[4] B.M. Paulin, M. Philippe, M. Patrice, “Nonlinear phase shift control of semi-active friction devices for optimal energy dissipation”, Journal of Sound and Vibration, 2009, 320 pp. 16-18.

[5] G.D. Chen, C.C. Chen, “Semi-active control of the 20-storey benchmark building with piezoelectric friction dampers”, Journal of Engineering Mechanics, ASCE 2004, 130(4) pp. 393-400.

[6] C.L. Ng, Y.L. Xu, “Seismic response control of multi-storey buildings using semi-active friction dampers”, Advance in Building Technology, 2002, 2 pp. 993-1000.

[7] G.D. Chen, C.C. Chen, “Behavior of piezoelectric friction dampers under dynamic loading”, Proceedings of SPIE, 2000, 3988 pp. 54–63.

[8] K. Yamada, T. Kobori, “Control algorithm for estimating future response of active variable stiffness structure”, Earthquake Engineering and Structure Dynamics, 1995, 24 pp. 1085-1099.

[9] J.N. Yang, J.C. Wu, Z. Li, “Control of seismic-excited buildings using active variable stiffness systems”, Engineering Structures, 1996, 18(8) pp. 589-596.

[10] C.H. Loh, M.J. Ma, “Control of seismically excited building structure using variable damper systems”, Engineering Structures, 1996, 18(4) pp. 279-287.

[11] Hrovat D, Barak P, Robins M, “Semi-active versus passive or active tuned mass dampers for structural control”, Journal of Engineering Mechanics, ASCE

1983, 109(3): 691-705. [12] J.P. Ou, “Structural vibration control-Active, Semi-

active and Intelligent control [in Chinese]”, Science Press, Beijing, 2003.

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