Structural Engineering
KSCE Journal of Civil Engineering Vol. 11, No. 2 / March 2007 pp. 121~126
Modified Energy Dissipation Algorithm for Seismic Structures Using Magnetorheological Damper By Kang Min Choi*, Heon-Jae Lee**, Sang-Won Cho***, and In Won Lee**** ···································································································································································································································
Abstract A simple modified energy dissipation algorithm is presented for seismic response reduction. The magnetorheological (MR) damper, an efficient semiactive device is controlled using a maximum energy dissipation algorithm (MEDA), as well as a newly proposed modified variable energy dissipation algorithm (VEDA) that supplies continuously varying command voltages. For MEDA, the command voltage is either zero or the maximum value. In some situations when the dominant frequencies of the system under control are low, large changes in the forces applied to the structure may result in high local acceleration values. The VEDA is proposed as a modification to the original MEDA to reduce this effect. Also the applicability of the VEDA-based semiactive control system using MR dampers is investigated through numerical simulation of a three-story shear building. Keywords: maximum energy dissipation algorithm, variable energy dissipation algorithm, semiactive control, MR damper, vibration control ···································································································································································································································
1. Introduction In the field of civil engineering, many control algorithms and devices have been proposed over the last few decades for the purpose of protecting structures and their contents from the damaging effects for environmental hazards such as earthquakes and wind loadings. Semiactive control has received a lot of attention recently because it offers great adaptability without a large power requirement. Previous research has demonstrated the effectiveness of semiactive systems in structural vibration control using viscous fluid dampers. Magnetorheological (MR) damper is one of the semiactive devices, it is reliable, fail-safe, and it is expected to be relatively inexpensive. The maximum energy dissipation algorithm (MEDA) for a semiactive control system represents one class of controller that employs Lyapunov’s direct approach to stability analysis in design of a feedback controller (Brogan, 1991). The approach requires the use of a Lyapunov function that must be a positive definite function of the states of the system. According to Lyapunov stability theory, if the rate of change of the Lyapunov function is negative semi-definite, the origin is stable in the sense of Lyapunov. Thus, in developing the control law based on Lyapunov stability theory, the goal is to choose control inputs for each device that result in making the rate of change of the Lyapunov function as negative as possible. Jansen and Dyke (2000) suggested MEDA as a variation of the decentralized bang-bang approach proposed by McClamroch and Gavin
(1995). It is noticeable that this control law requires only local measurements, which means that MEDA can be implemented without a design process. In the original MEDA, the command voltage takes on the value of either zero or the maximum value. In some situations when the dominant frequencies of the system under control are low, large changes in the forces applied to the structure may result in high local acceleration values. This behavior is dependent on the magnitude of the time lag in the generation of the control voltage. A modification to the original MEDA to reduce this effect is proposed in what follows.
2. Proposed Simple Semiactive Control Strategy Consider a seismically excited structure controlled with n MR dampers. The equation of motion can be written as Mx·· + Cx· + Kx = /f – M*x··g
(1)
where M, C, and K are the (nun) mass, damping, and stiffness matrices, respectively; x is the n-dimensional vector of the displacements of the floors of the structure relative to the ground; f is the vector of measured control forces generated by m control devices; x··g is ground acceleration; / is the matrix determined by the placement of control devices in the structure; * is the column vector of ones. This equation can be written in statespace form as
*Member, Post Doctoral Researcher, Dept. of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea (Corresponding Author, E-mail:
[email protected]) **Post Doctoral Researcher, Dept. of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea (Email:
[email protected]) ***Post Doctoral Researcher, Dept. of Civil and Environmental Engineering, The University of Western Ontario, London, Canada (E-mail: swcho815 @yahoo.co.kr) ****Member, Emeritus Professor, Dept. of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea (Email:
[email protected]) Vol. 11, No. 2 / March 2007
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z·· = Az + Bf + Ex··g
(2)
y = Cz + Df + Q
(3)
Table 1. Parameters for the Prototype MR Damper Model (Dyke et al., 1996c)
where z is a state vector; y is the vector of measured outputs; and v is a measurement noise vector. 2.1 Control Device An MR damper with capacity of 3,000N is considered as control devices. To accurately predict the behavior of controlled structure, an appropriate modeling of MR dampers is essential. Several types of control-oriented dynamic models have been investigated for modeling MR dampers. The modified BoucWen model (Spencer et al., 1997), which is numerically tractable and has been used extensively for modeling hysteretic systems, is considered for describing the behavior of the MR damper. The simple mechanical idealization of the MR damper depicted in Fig. 1 is based on a Bouc-Wen hysteresis model and is governed by the following seven simultaneous equations: f = c1 y· + k1 x – x0 1 y· = ------------------ ^ Dz + c0 x· + k0 x – y ` c0 + c1 z· = –J x· – y· z z n – 1 – E x· – y· z n + A x· – y· D = Da + Db u c1 = c1a + c1b u c0 = c0a + c0b u u· = –K u – Q
(4) (5) (6) (7) (8) (9) (10)
Parameter
Value
c0a(Ns/cm) c0b(Ns/cm) k0(N/cm) c1a(Ns/cm) c1b(Ns/cmV) k1(N/cm) x0 Da(N/cm) Db(N/cmV) J(cm-2) E(cm-2) A n K(s-1)
21.0 3.50 46.9 283 2.95 5.00 14.3 140 695 363 363 301 2 190
the decentralized bang-bang approach proposed by McClamroch and Gavin (1995). Lyapunov’s direct approach requires the use of a Lyapunov function, denoted V(x), which must be a positive definite function of the states of the system x. In the decentralized bang-bang approach, the Lyapunov function is chosen to represent total vibratory energy in the system. Jansen and Dyke (2000) instead consider a Lyapunov function that represents the relative vibratory energy in the structure as 1 1 V = --- x T Kx + --- x· T Mx· 2 2
(11)
Here, the accumulator stiffness is represented by k1, the viscous damping observed at large and low velocities by c0 and c1, respectively. k0 is present to control the stiffness at large velocities, and x0 is the initial displacement of spring k1 associated with the nominal damper force due to the accumulator. J, E and A are hysteresis parameters for the yield element. D is the evolutionary coefficient. A total of 14 model parameters are obtained to characterize the prototype MR damper using experimental data and a constrained nonlinear optimization algorithm. The resulting parameters are given in Table 1 (Dyke et al., 1996c).
According to Lyapunov stability theory, if the rate of change of the Lyapunov function V· x is negative semi-definite, the origin is stable in the sense of Lyapunov. Using Eq. (11), the rate of change of the Lyapunov function is then V· = x T Kx· + x· T M – Cx· – Kx – M*x· + /f (12)
2.2 Maximum Energy Dissipation Algorithm (MEDA) The following control algorithm is presented as a variation of
where /i is i th column of the / matrix; fi is i th column of the f matrix. Note that MEDA is very simple because only local measurements (i.e., the velocity and control force) are required to implement this control law. In Eq. (13), there is no design parameter to decide, which is an essential part in many other control laws. However, in the original MEDA, the command voltage takes on the value of either zero or the maximum value. In some situations when the dominant frequencies of the system under control are low, large changes in the forces applied to the structure may result in high local acceleration values. This behavior is partially dependent on the time lag in the generation of the control voltage.
Fig. 1. Simple Mechanical Model of the MR Damper (Spencer et al., 1997)
g
In this expression, the only way that an MR damper can directly affect V· is through the last term that contains the force vector f. To control this term and make V· as large and negative as possible, the following control law is obtained Qi = Vmax H – x· /i fi
(13)
2.3 Variable Energy Dissipation Algorithm (VEDA) In the modified version of the control algorithm that is proposed here, the control voltage can be any value between 0
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Modified Energy Dissipation Algorithm for Seismic Structures Using Magnetorheological Damper
3.1 Numerical Model To evaluate the performance of the proposed variable energy dissipation algorithm using the MR damper, the three-story shear building structure (Dyke et al., 1996c) is considered. The system matrices of a three-story shear building are Ms =
175 –50 0 N sec Cs = – 50 100 –50 --------------m 0 – 50 50
Fig. 2. Graphical Representation of the Variable Energy Dissipation Algorithm
and Vmax. The control voltage, denoted by Vi, is determined using a linear relationship between the maximum voltage and the maximum force of MR damper. This modified variable energy dissipation algorithm is graphically represented in Fig. 2 and can be given as Qi = Vi H –x· /i fi
(14)
3. Numerical Example Various approaches have been proposed for control of semiactive devices. In this study, three control algorithms are adopted: the clipped-optimal control algorithm, the maximum energy dissipation algorithm, and the proposed variable energy dissipation algorithm. A clipped-optimal controller was suggested and experimentally examined by Dyke et al. (1996a, b, c). The clipped-optimal control approach is to design a linear optimal controller that calculates desired control forces based on the measured structural responses and the measured control force applied to the structure. A force feedback loop is incorporated to induce the MR damper to generate approximately the desired optimal control force.
98.3 0 0 0 98.3 0 kg 0 0 98.3
12.0 –6.84 0 N Ks = – 6.84 13.7 – 6.84 ---m 0 –6.84 6.84
The MR damper is rigidly connected between the ground and the first floor of the structure. This system is a simple model of the scaled, three-story, test structure, that has been used in previous active control studies at the Structural Dynamics and Control/Earthquake Engineering Laboratory (SDC/EEL) at the University of Notre Dame (Dyke et al., 1996c). Because the MR damper is attached between the first floor and the ground, its displacement is assumed to be equal to the displacement of the first floor of the structure relative to the ground. The test structure is controlled under normal (100%), high (150%) and low (50%) excitation levels of the El Centro and Hachinohe earthquakes, and under normal (100%) and low (50%) excitation levels of the Northridge earthquake. Because the system under consideration is a scaled model, the earthquake must be reproduced at five times the recorded rate. 3.2 Control Performance Representative responses of the controlled MEDA and VEDA to the normal El Centro and Northridge earthquakes are shown in Figs. 3 and 4, respectively. As seen from the figures, compared to
Fig. 3. Comparison of Controlled and Uncontrolled Responses in the Third Floor for Normal El Centro Earthquake Vol. 11, No. 2 / March 2007
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Kang Min Choi, Heon-Jae Lee, Sang-Won Cho, and In Won Lee
Fig. 4. Comparison of Controlled and Uncontrolled Responses in the Third Floor for Normal Northridge Earthquake
the cases without an MR damper, the third floor drift and acceleration responses are reduced significantly. Figs. 5 and 6 show the command voltage sent to the MR damper due to the normal El Centro and Northridge earthquakes, respectively. The command voltage for original MEDA takes on the value of either zero or the maximum value (2.25 V). However, in these figures, modified version (VEDA) has any value between zero and the maximum value. The command voltage is determined using a linear relationship between the applied voltage and the maximum force of an MR damper. Table 2 shows the values for the maximum interstory drift and maximum absolute acceleration ratios normalized by uncontrolled cases under various earthquakes. To compare the performance of the controlled systems, the bar charts in Figs. 7 and 8 show the values for the maximum interstory drift and maximum absolute
acceleration ratios, respectively. Note that each controller is able to achieve a significant reduction in the interstory drift when compared with the uncontrolled system; the interstory drifts in MEDA and VEDA are reduced to 37~91% and 40~87% levels for the peak value and to 18~35% and 18~34% levels in the normed values for all earthquake excitations, respectively. The accelerations in MEDA and VEDA are reduced to 52~85% and 51~76% levels in the peak value and to 18~35% and 19~34% levels in the RMS values for all earthquake excitations, respectively. It is clear that most of structural responses are controlled well for MEDA and VEDA. However, note that the clipped-optimal controller fails to reduce the peak floor acceleration, giving a magnification factor of 102% for low El Centro earthquake; 143%, 150%, and 128% for low, normal, and high Hachinohe earthquake, respectively; and 102% for low
Fig. 5. Applied Voltage for Normal El Centro Earthquake
Fig. 6. Applied Voltage for Normal Northridge Earthquake 124
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Modified Energy Dissipation Algorithm for Seismic Structures Using Magnetorheological Damper Table 2. Comparison of Evaluation Criteria for the Three-story Building Clipped-optimal controller (CO) (Dyke et al., 1996c)
Maximum Energy Dissipation Algorithm (MEDA)
Variable Energy Dissipation Algorithm (VEDA)
Earthquake Intensity
El Centro 0.5/1.0/1.5
Hach. 0.5/1.0/1.5
North. 0.5/1.0
El Centro 0.5/1.0/1.5
Hach. 0.5/1.0/1.5
North. 0.5/1.0
El Centro 0.5/1.0/1.5
Hach. 0.5/1.0/1.5
North. 0.5/1.0
Peak drift ratio
0.34 0.50 0.47
0.51 0.67 0.71
0.73 0.67
0.50 0.54 0.49
0.37 0.37 0.37
0.91 0.79
0.47 0.43 0.42
0.40 0.48 0.53
0.86 0.87
Peak acceleration ratio
1.02 0.82 0.73
1.43 1.50 1.28
1.02 0.97
0.55 0.63 0.62
0.52 0.52 0.52
0.85 0.72
0.51 0.57 0.56
0.53 0.53 0.53
0.72 0.76
Norm drift ratio
0.24 0.24 0.24
0.18 0.30 0.28
0.35 0.49
0.25 0.23 0.24
0.18 0.19 0.18
0.35 0.33
0.25 0.25 0.24
0.18 0.20 0.20
0.32 0.34
Norm acceleration ratio
0.30 0.34 0.32
0.23 0.30 0.28
0.35 0.49
0.25 0.23 0.24
0.18 0.19 0.18
0.35 0.33
0.25 0.25 0.24
0.19 0.20 0.20
0.33 0.34
Fig. 7. Bar Chart Comparing the Peak Interstory Drift for Various Controllers
Fig. 8. Bar Chart Comparing the Peak Floor Acceleration for Various Controllers
Northridge earthquake, whereas MEDA and VEDA are successful. In comparing the MEDA and VEDA control systems, the resulting interstory drift ratios are similar; however, the VEDA is typically able to achieve a reduction in the peak floor accelerations over that of the MEDA. Therefore, if the control objective is to reduce accelerations, the proposed VEDA would offer better advantages over the original MEDA.
(VEDA) that supplies continuously varying command voltages. The MEDA controller leads to a command voltage that takes either zero or the maximum value. In some situations when the dominant frequencies of the system under control are low, large changes in the forces applied to the structure may result in high local acceleration values. This behavior is dependent on the time lag in the generation of the control voltage. A VEDA controller attempts a modification of the original MEDA controller to reduce this effect. Note that VEDA is very simple because only local measurements (i.e., the velocity and control force) are requires to implement this control law. There is no design parameter to determine, which gives it an advantage over many other control laws. Finally, we confirm that the semiactive VEDA control system
4. Conclusions A simple variable energy dissipation algorithm is presented for seismic response reduction with MR dampers. The MR damper, which is an efficient semiactive device, was controlled using an original maximum energy dissipation algorithm (MEDA), as well as a newly proposed variable energy dissipation algorithm Vol. 11, No. 2 / March 2007
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using an MR damper is efficient for a three-story shear building. If the control objective is to reduce accelerations, the VEDA is suitable and has advantages over the original MEDA.
Acknowledgement The writers gratefully acknowledge the partial support of this research by the Construction Core Technology Research and Development Project (Grant No. C105A1000021) supported by the Ministry of Construction and Transportation and the Smart Infra-Structure Technology Center (SISTeC) supported by the Korea Science Foundation and the Ministry of Science and Technology in Korea.
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Dyke, S. J., Spencer, B. F., Jr., Sain, M. K., and Carlson, J. D. (1996b). ‘‘Seismic response reduction using magnetorheological dampers,’’ Proc., IFAC World Congr., pp. 145-150. Dyke, S. J., Spencer, B. F., Jr., Sain, M. K., and Carlson, J. D. (1996c). ‘‘Modeling and control of magnetorheological dampers for seismic response reduction,” Smart Mat. And Struct., 5, pp. 565-575. Jansen, L. M. and Dyke, S. J. (2000). “Semiactive control strategies for MR dampers: Comparative study,” Journal of Engineering Mechanics, ASCE, Vol. 126, No. 8, pp. 795-803. McClamroch, N. H., and Gavin, H. P. (1995). ‘‘Closed loop structural control using electrorheological dampers,’’ Proc., Am. Control Conf., American Automatic Control Council, Washington, D.C., pp. 4173-4177. Spencer, Jr., B. F., Dyke, S. J., Sain, M. K., and Carlson, J. D. (1997). “Phenomenological model of a magnetorheological damper,” Journal of Engineering Mechanics, ASCE, Vol. 123, No. 3, pp. 230238. Yoshida, O., and Dyke, S. J. (2004). “Seismic control of a nonlinear benchmark building using smart dampers.” Journal of Engineering Mechanics: Special Issue on Structural Control Benchmark Problem, ASCE, Vol. 130, No. 4, pp. 386-392.
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(Received November 20, 2006/Accepted January 22, 2007)
KSCE Journal of Civil Engineering
