Optimal Design of MR Dampers Using NSGA-II Algorithm

1. Introduction

Controlling the response of structures against seismic excitations has always been one of the challenges of the structural engineering field. In recent years, new ideas and solutions have been proposed by scientists and researchers to reduce the structural response due to the effects of dynamic forces, the most important of which is the use of structural control systems. Tuned mass [ 1 ] and magneto-rheological dampers [ 2 ] are the most famous instruments to control and absorb earthquake loads. One important type of these systems is the semi-active control system.

In the semi-active control system, the required external energy is much smaller in size than in conventional active control systems, also while maintaining the reliability of passive control systems, it has advantages such as adjustable parametric characteristics of the active control system [ 3 ]. In these systems, the control action is achieved by the movement of the structure and regulated by an external energy source, and while determining the response of the structure at any time using a force control algorithm, the required amount of force can be changed.

Magnetorheological fluid dampers are modern control devices for structures having features such as small size, low energy consumption, adjustable force, high force capacity, fast response, and safe operation in the event of damage. These dampers are usually composed of a hydraulic cylinder containing micron-sized, magnetically polarized particles suspended inside an oily fluid [ 4 ]. In MR dampers, the viscosity of the fluid inside them can be changed. This change in viscosity occurs with an increase or decrease in the intensity of the magnetic field and changes the hardness and yield stress of these dampers. As a result, their energy absorption can change at any time. The damping force in an MR damper is a nonlinear function of voltage and velocity at both ends. The fluctuation in the damping ratio is obtained by changing the input voltage to the damper [ 5 ].

An important issue in the seismic control of structures is the position and number of control systems that must be determined and optimized for economic purposes [ 6 - 8 ]. Dyke and Spencer used the MR damper in 1996 to control a three-story structure [ 9 ]. The three-story model structure was subjected to the El Centro earthquake, and the MR dampers were set on two consecutive lower floors. The results showed that the acceleration and displacement of the structure decreased in the semi-active control. In 2001, Qu et al. Conducted a study on the use of ER/MR dampers to control the seismic responses of a structure containing a tower and a podium structure [ 10 ].

The ER/MR dampers were used to reduce the whipping effect on the tower structure and reduce the seismic responses of both the tower structure and the podium structure. In 2003, Zhou et al. used an adaptive fuzzy control strategy for protecting buildings against dynamic hazards, such as severe earthquakes and strong winds. The researchers concluded that the use of MR dampers reduced the response of linear and nonlinear multiple degrees of freedom systems [ 11 ]. In 2004, Renzi and Serino tested a four-story steel frame equipped with a bracing system including magnetorheological (MR) dampers operating in both passive and semi-active on a shaking table [ 12 ]. The structure was affected by two natural accelerograms. The results showed that under semi-active control, the structure's displacement was reduced by about 30 to 35% more than the passive one.

In 2006, Yan and Zhou controlled vibrations using a fuzzy controller and MR damper in a three-story structure [ 13 ]. In this study, a genetic optimization algorithm was used to control the vibrations of structures under the El Centro earthquake. The researchers concluded that in the case of using the semi-active control with the fuzzy optimized algorithm, displacement responses, and accelerations are effectively reduced. In 2012, Amini and Ghaderi used the improved ant colony algorithm to find the optimal location of three and four MR dampers in an eight-story and ten-story linear structure using only one shear force as the objective function [ 14 ]. In 2017, Askari et al. applied the genetic algorithm multi-objective optimization method to find the optimal location of a certain number of semi-active and active MR dampers in a twenty-story nonlinear structure [ 15 ]. They used LQG and clipped-optimal algorithms to regulate the input voltage of the MR damper. The results show that the MR dampers in optimal locations perform better in reducing inter-story drift. Instead, active dampers in the optimal positions work better at reducing the maximum acceleration and base shear.

2. NSGA-II algorithm

For solving optimization problems, there are a myriad of meta-heuristic algorithms and each one has its unique method of finding the optimum solution (e.g., ACO, ABC, PSO, GA, SBO, etc.) [ 16 - 30 ]. Also, there are plenty of studies related to the performance of these meta-heuristics on multi-objective optimizations [ 31 - 33 ].

In this study, the well-known evolutionary algorithm NSGA-II was used since it has shown its capabilities for solving various multi-objective optimization problems in recent years. It was first proposed by Deb and his colleagues in 2000 based on the genetic algorithm (GA) to enhance its capabilities in multi-objective optimization. Unlike the single objective optimization technique, non-dominated sorting algorithm II (NSGA-II) simultaneously optimizes each objective without being dominated by any other solution. Fig 1. demonstrates how the NSGA-II algorithm works. This algorithm has been applied for multi-objective optimization of steel buildings considering probabilistic performance-based design parameters [ 34 , 35 ].

3. Model of MR dampers

To develop control algorithms that take full advantage of the MR dampers, models must be defined that can accurately describe the inherent nonlinear behavior of dampers. To model the behavior of the MR damper, the Bouc-Wen behavioral model was used. In structural engineering, the Bouc-Wen model of hysteresis is one of the most used hysteretic models typically employed to describe non-linear hysteretic systems. It was introduced by Robert Bouc and extended by Yi-Kwei Wen, who demonstrated its versatility by producing a variety of hysteretic patterns. This model is able to capture, in analytical form, a range of hysteretic cycle shapes matching the behavior of a wide class of hysteretical systems. Due to its versatility and mathematical tractability, the Bouc-Wen model has gained popularity. Instead of modeling the MR damper itself, the force generated by it enters the model. The Bouc-Wen behavioral model for MR dampers consists of a Bouc-Wen element and a linear damper, which operate in parallel [ 36 ]. An overview of this behavioral model is shown in Figure 2.

Fig. 1. Flowchart of the NSGA-II algorithm.

Fig. 2. Bouc-Wen Behavioral Model for MR Dampers [37].

The force generated by the damper in this model is calculated through Equation (1):

$\begin{array}{cc}F=C_0\mathrm{x\; ̇}+\mathrm{az}& \mathrm{(1)}\end{array}$

$\begin{array}{cc}\mathrm{z\; ̇}=-\gamma \left|\mathrm{x\; ̇}\right|z|z{|}^{\mathrm{n-1}}-\beta \mathrm{x\; ̇}|z{|}^n+A_x\mathrm{x\; ̇}& \mathrm{(2)}\end{array}$

In Equations (1) and (2), F is the damping force, x is the damping displacement, z is the evolutionary variable, and the parameters n, γ, β, and A_m are fixed values that are determined experimentally on each damper. The parameters C₀ and α are determined using Equations (3) and (4):

$\alpha =\alpha \left(u\right)={\alpha }_a+{\alpha }_{b}u& \mathrm{(3)}$

$C_0=C_0\left(u\right)=C_{0_a}+C_{0_b}u& \mathrm{(4)}$

In these equations, the applied control voltage u and the parameters α_a, α_b, C_0a, C_0b are constant values. With nine fixed damper parameters, a Bouc-Wen behavioral model can be created.

In this study, an MR damper with a capacity of 1000 KN was used. The input voltage of the dampers can vary between zero and 10 volts. The optimal maximum voltage value for this model is 3.5 volts. The parameters for simulating the behavior of MR dampers are shown in Table 1.

One of the limitations of MR dampers is the time delay in applying the control force. The value of this delay, depending on the type of damper is about 0.02 to 0.1 seconds.

Equation (5) is used to model this slight delay in the system.

$u=-\eta (u-v)& \mathrm{(5)}$

In this equation ν is the command voltage and η is a constant value. To model the behavior of the MR damper, its behavioral equations are solved as recursive equations.

4. Optimization process

In this study, the positions of dampers are considered as design variables, which is a vector containing zero and one values. The vector is sent as an input to the Simulink model. All AISC design code requirements are considered constraints including stresses and displacements. After applying the earthquake force, the response of the damper is applied to the structure in each step of the earthquake. Consequently, the objective functions, which include either the maximum displacement and acceleration or the maximum inter-story displacement and acceleration, are sent to the algorithm as output. Thenceforth, the responses are sorted and queued after determining the crowding distance. Then, the first front of the answers is determined by using the crossover and mutation operators. In this study, the crossover rate is 0.7 and the mutation rate is 0.3.

Figure 3 shows the Simulink model of motion equation of the structure with the MR damper.

Fig. 3. Simulink model of structure’s motion equation with MR damper.

To investigate the structural response, the equations of motion of the structure in both controlled and uncontrolled states are created in Simulink. In the semi-active mode, a Mamdani fuzzy controller is used with two inputs and one output. The inputs are displacements and the relative speed of the floors, while the output is the damper voltage. To control the response of the structure, a two-objective optimization is used based on the NSGA-II algorithm. The NSGA-II algorithm is one of the best metaheuristics for solving multi-objective functions. The maximum displacement and acceleration of the highest floor of the structure are selected as the objective functions in the first problem and the maximum inter-story drift and acceleration of the highest floor are selected as the objective functions in the second problem. To satisfy these functions, the positions of MR dampers in the structure were optimized.

5. Experimental setup

5.1. Details of the frames

To evaluate and compare the performance of the control system, numerical studies have been performed on five, eight, and eleven-story steel frames. All of the structures are hypothetical schools located in Tehran. The design base acceleration is considered as 0.35, and the importance factor for important structures such as schools is considered as 1.2. Chevron braces are used in the middle span of the structures. The natural frequency of a five-story, eight-story and eleven-story structure is respectively 0.37 seconds, 0.68 seconds, and 0.92 seconds.

The frame specifications are presented in Table 2 and figures 4, 5, and 6.

Fig. 4. Five-story structure with Chevron bracing.

Fig. 5. Eight-story structure with Chevron bracing.

Fig. 6. Eleven-story structure with Chevron bracing.

5.2 Horizontal accelerogram of the earthquakes used

The analyses performed in this study are of linear time history analysis, and the earthquake records are applied horizontally to the structures. To perform time history analysis, seven far-field earthquake records are used, which are mentioned in Table 3. The Peak Ground Acceleration (PGA) is expressed in fractions of g (1 g= 9.81 m/s²).

Fig. 7. The response spectrum of earthquakes applied to the structure

Figure 7. shows the response spectrum of earthquakes applied to the structure (left figure) and compares the average spectrum and the code’s design spectrum of the five-story frame (right figure).

The values of the earthquake scale factor of all three frames are given in Table 4.

5.3. Behavior validation of MR dampers

To ensure that the behavior of the MR damper in the structural model is flawless, the damper is subjected to a cyclic deformation, and a diagram of the force generated by it for zero, five, and 10 volts is drawn. Figure 8 demonstrates the force diagram of dampers obtained from the present study which is similar to the graph attained by Ok et al. in 2007. Comparing the diagrams for all three voltages in the present study, it can be seen that the behavior of the MR damper is modeled close to reality.

Fig. 8. MR damper behavioral diagram in this study.

6. Experimental results

6.1. Five-story structure

According to the results obtained from the five-story frame, the optimal position for the MR damper is on the fifth floor. The stories selected for the three-damper mode are first, third, and fifth. The results for both modes are shown in Tables 5 and 6.

Figures 9 to 11 demonstrate the displacement, acceleration, and drift diagrams in both controlled and uncontrolled modes for the five-story frame.

Fig. 9. Maximum roof displacement response of the five-story frame for three modes of uncontrolled, controlled with one and, three dampers.

Looking at the details in Fig. 9, the uncontrolled structure has the maximum roof displacement of 8.7cm in the Loma Prieta earthquake, and the one-damper mode structure has 4.7cm, while the three-damper mode structure has 2.3cm which is more than 50% lower than the one-damper mode structure, and almost 75% lower than the uncontrolled structure.

Fig. 10. Maximum roof acceleration response of the five-story frame for three modes of uncontrolled, controlled with one, and three dampers.

In Fig. 10, the uncontrolled structure had the highest acceleration response of 27.5m/s² in the Loma Prieta earthquake, while the three-damper mode structure experienced only 10m/s² which is almost one-third of the uncontrolled structure.

Fig. 11. Maximum drift response of the five-story frame roof for three modes of uncontrolled, controlled with one, and three dampers.

Looking at Fig. 11, the results are the same as before, leaving the three-damper mode structure as the best performer among all with a drift response of 0.4cm which is 52% lower than the one-damper structure and 73% less than the uncontrolled structure.

As can be seen from the graphs above, the three-damper mode structure has achieved better results in all of the earthquakes in terms of roof displacement, roof acceleration, and drift.

6.2. Eight-story structure

For the eight-story frame, the optimal position for the MR damper was chosen as the seventh floor. For the three-damper mode, floors one, six, and eight were selected. The results are shown in Tables 7 and 8.

Figures 12 to 14 demonstrate the displacement, acceleration, and drift diagrams of the eight-story frame for three modes of uncontrolled, controlled with one and three dampers.

Fig. 12. Maximum roof displacement response of the eight-story frame for three modes of uncontrolled, controlled with one, and three dampers.

What stands out from Fig. 12 is that the three-damper mode structure outperforms the other two structures in all of the earthquakes.

Fig. 13. Maximum roof acceleration response of the eight-story frame for three modes of uncontrolled, controlled with one, and three dampers.

Regarding Fig. 13, the three-damper mode structure had better performance in all of the earthquakes except Landers in which the one-damper mode structure had a slightly lower acceleration response.

Fig. 14. Maximum drift response of the eight-story frame roof for three modes of uncontrolled, controlled with one, and three dampers.

If we look at Fig. 14, it can be seen that the three-damper mode structure had almost similar results to the one-damper mode structure in most of the earthquakes.

6.3. Eleven-story frame

For the eleven-story frame, the optimal position for the MR damper was chosen as the eleventh floor. For the three-damper mode, floors six, nine, and eleven were selected. The results are shown in Tables 9 and 10.

Figures 15 to 17 demonstrate the displacement, acceleration, and drift diagrams of the eleven-story frame for three modes of uncontrolled, controlled with one and three dampers.

Fig. 15. Maximum roof displacement of the eleven-story frame for three modes of uncontrolled, controlled with one and three dampers.

As regards Fig. 15 it is clear that the three-damper mode structure had the best result compared to the one-damper mode, and uncontrolled structure. An interesting point is that the uncontrolled structure had less roof displacement than the one-damper mode structure in the Kocaeli earthquake.

Fig. 16. Maximum roof acceleration of the eleven-story frame for three modes of uncontrolled, controlled with one and three dampers.

Looking at Fig. 16, the maximum roof acceleration belongs to the uncontrolled structure in the Loma Prieta earthquake with 16.5m/s², while the three-damper mode structure experienced 8m/s² in the same earthquake.

Fig. 17. Maximum drift of the eleven-story frame roof for three modes of uncontrolled, controlled with one and three dampers.

Regarding Fig. 17, the three-damper mode structure outperforms the other two structures in six out of seven earthquakes in terms of drift response. Figures 18 and 19 show the time history response of maximum displacement and acceleration for the eleven-story frame roof in both uncontrolled and controlled modes for the Imperial Valley earthquake with three dampers on floors six, nine, and eleven. It is clear that the controlled structure had much better results in terms of displacement and acceleration.

Fig. 18. Displacement time history record of the eleven-story frame in uncontrolled and controlled modes with three dampers.

Fig. 19. Acceleration time history of the eleven-story frame in uncontrolled and controlled modes with three dampers.

7. Conclusion

1- Using optimization, a suitable control level can be achieved with a smaller number of dampers. Normally, increasing the number of MR dampers improves the reduction of structural responses, but some unfavorable situations can lead to an increase in them.

2- According to the results obtained from the five-story frame, the optimal position for the MR damper is the fifth floor. The average reduction of responses in this case for maximum displacement, acceleration, and inter-story drift of the structure is 34.6%, 17.5%, and 27.4%, respectively. For the three dampers mode, the optimal positions are the first, third and fifth floors. The average reduction of responses in this case, is equal to 60.8%, 36.6%, and 46.8%, respectively.

3- For the eight-story frame, the optimal position for the MR damper is the seventh story. The average reduction of responses in this case for maximum displacement, acceleration, and inter-story drift of the structure is 34.3%, 11.5%, and 27.3%, respectively. For the three dampers mode, the optimal positions are floors one, six, and eight. The average reduction of responses, in this case, is 47.8%, 29%, and 34.6%, respectively.

4- For the eleven-story frame, the optimal position for the MR damper is the eleventh story. The average reduction of responses in this case for maximum displacement, acceleration, and inter-story drift of the structure is 15.8%, 2.2%, and 17.8%, respectively. For the three dampers mode, the optimal positions are floors six, nine, and eleven. The average reduction of maximal displacement, acceleration, and drift responses of the structure is 42.3%, 16%, and 32.7%, respectively.

5- The best position for dampers is often the upper floors of the structure. The optimal distribution of dampers is a function of structural and earthquake characteristics. As the results show, the MR damper has a better performance in reducing displacement responses and drift compared to reducing acceleration.

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