Optimal Design of Liquid Dampers for Structural Vibration Control Based on GA and 𝐻_∞ Norm

Abstract

This paper focused on the optimal design of liquid dampers for the seismic response control of structures. The 𝐻_∞ norm of the transfer function from the ground motion to the structural response is selected as the optimal objective. The optimization procedure is carried out by using Genetic Algorithms (GAs) in order to reach an optimal solution. The proposed method has the advantages that it is unnecessary to solve the equation of motion for the control system and that the obtained optimal parameters of dampers are not dependent on the ground motion records. The influences of weighted functions on the optimization results are analyzed. The generality and effeteness of the proposed method are verified by the time history analysis of a 3-story structure subjected to earthquake records in different sites. The results show that the structural responses can be effectively reduced subjected to earthquake excitation at different sites.

1. Introduction

The installation of vibration absorbers on tall buildings or other flexible structures can be a successful method for reducing the effects of dynamic excitations, such as strong wind or earthquakes, which may exceed either serviceability or safety criteria. Tuned liquid column damper (TLCD) is an effective passive control device by the motion of liquid in a column container. A TLCD is a U-shaped tube of uniform rectangular or circle cross section, containing liquid. Vibration energy is transferred from the structure to the TLCD liquid through the motion of the rigid container exciting the TLCD liquid. And the vibration of a structure is suppressed by a TLCD through the gravitational restoring force acting on the displaced TLCD liquid and the energy is dissipated by the viscous interaction between the liquid and the rigid container, as well as liquid head loss due to orifices installed inside the TLCD container. The potential advantages of liquid vibration absorbers include low manufacturing and installation costs, the ability of the absorbers to be incorporated during the design stage of a structure or to be retrofitted to serve a remedial role, relatively low maintenance requirements, and the availability of the liquid to be used for emergency purposes or for the everyday function of the structure if fresh water is used [1, 2].

Analytical and experimental research works on this type of vibration reduction approach have been conducted, in which viscous interaction between a liquid and solid boundary has been investigated and used to control vibration [3, 4]. Their experiments, defining the relationship between the coefficient of liquid head loss (as well as its dependence on the orifice opening ratio) and the liquid damping, confirm the validity of their proposed equation of motion in describing liquid column relative motion under moderate excitation. A variation of TLCD, called a liquid column vibration absorber (LCVA), has also been investigated, which has different cross-sectional areas in its vertical and horizontal sections depending on performance requirements [5–8]. Yan and Li presented the adjustable frequency tuned liquid column damper by adding springs to the TLCD system, which modified the frequency of TLCD and expended its application ranges [9]. Gao et al. analyzed the characteristics of multiple liquid column dampers (MTLCD) [10]. It was found that the frequency of range and the coefficient of liquid head loss have significant effects on the performance of a MTLCD; increasing the number of TLCD can enhance the efficiency of MTLCD, but no further significant enhancement is observed when the number of TLCD is over five. It was also confirmed that the sensitivity of an optimized MTLCD to its central frequency ratio is not much less than that of an optimized single TLCD to its frequency ratio, and an optimized MTLCD is even more sensitive to the coefficient of head loss.

How to decide the parameters and location of dampers in structures to achieve the optimal response reduction is still an open problem currently. One of the commonly used methods for the optimization of the damper parameters is to set the objective function to be the stochastical response of structures based on the seismic frequency spectrum input and the optimal parameters can be achieved by Genetic Algorithms (GAs). Another commonly used method of optimization is to set the objective function to be the structural responses or their combination in time domain based on the actual earthquake records. The shortcoming for the above two methods is that the equation of motion of the structural system has to be solved in every optimization step, which is time consuming. Besides, the optimal parameters acquired by the above methods are related to specific earthquakes and may not be optimal in some earthquakes. Hence, it is necessary to find a more general and effective method to optimize the parameters of liquid dampers to reduce structural vibration. The active control theory has been used to optimize the parameters of passive control systems and a lot of results have been achieved [11–22]. Based on 𝐻_∞ norm and Genetic Algorithms (GAs), this paper proposed the optimization method for liquid dampers, which has the advantages that it is unnecessary to solve the equation of motion for the system and that the optimized parameters are not related to specific earthquake records.

2. Problem Descriptions

2.1. Equation of Motion for TLCD

The configuration of the tuned liquid column damper (TLCD) is shown in Figure 1. According to Lagrange theory, the equation of motion for TLCD on the structure subjected earthquake excitation is derived as ()̈1𝜌𝐴2𝐻+𝐵ℎ+2𝜌𝐴𝜉ℎ̇ℎ̇+2𝜌𝐴𝑔ℎ=−𝜌𝐴𝐵𝑥̈_𝑛+𝑥̈_𝑔, (1) where 𝜌 is the density of the liquid; ℎ is the relative movement of the liquid in TLCD; 𝐻 is the static height of the liquid in the container; 𝐴 is the cross-sectional area of TLCD; 𝑔 is the gravitational acceleration; 𝐵 is the length of the horizontal part of TLCD; 𝜉 is the coefficient of head loss, which can be controlled by varying the orifice area; 𝑥̈_𝑛 is the acceleration of the 𝑛th floor where TLCD is installed; 𝑥̈_𝑔 is the acceleration of ground motion.

Figure 1 

Configuration of TLCD.

Due to the nonlinear damping of the aforementioned equation, by the equivalently linearization of it (1) can be converted to 𝑚_𝑇ℎ̈+𝑐_𝑇ℎ̇+𝑘_𝑇ℎ=−𝛼_𝑥𝑚_𝑇𝑥̈_𝑛+𝑥̈_𝑔, (2) where 𝑚_𝑇=𝜌𝐴𝐿_𝑒 is the mass of the liquid in the TLCD, 𝐿_𝑒=2𝐻+𝐵 is the length of the liquid in the TLCD, 𝑐_𝑇=(𝜌𝐴𝜉|ℎ̇|)/2 is the equivalent damping of the TLCD, and 𝑘_𝑇=2𝜌𝐴𝑔 is the “stiffness” of the liquid in vibration. Hence, the circular frequency of the motion can be written as 𝜔_𝑇=𝑘_𝑇/𝑚_𝑇=2𝑔/𝐿_𝑒. 𝛼_𝑥 is the shape function of the liquid damper, written as 𝛼_𝑥=𝐵/𝐿_𝑒.

2.2. Mathematical Model of the Control System

For an 𝑛-story shear frame structure, the dynamics of the structural building without TLCD can be described as 𝑀𝑥̈+𝐶𝑥̇+𝐾𝑥=−𝑀𝐼𝑥̈_𝑔, (3) where 𝑀, 𝐶, and 𝐾 are the mass, damping, and stiffness matrices of the structural building with the dimension of 𝑛×𝑛, respectively; 𝐼 is an identity vector with the dimension of 𝑛×1; the relative displacement vector 𝑥 is defined as 𝑥=𝑥₁𝑥₂⋯𝑥_𝑛^𝑇, (4) where 𝑥_𝑖 is the relative displacement of the 𝑖th floor. So the equation of motion for the structural building instrumented with TLCD on the top floor as shown in Figure 2 can be formulated as 𝑀_𝑠𝑞̈+𝐶_𝑠𝑞̇+𝐾_𝑠𝑞=−𝑀_𝑠𝐼𝑥̈_𝑔, (5) where 𝑀_𝑠=𝑚₁0000⋱0000𝑚_𝑛+𝑚_𝑇𝛼𝑚_𝑇00𝛼𝑚_𝑇𝑚_𝑇,𝐾_𝑠=𝑘₁0000⋱0000𝑘_𝑛0000𝑘_𝑇,𝐶_𝑠=𝑐₁0000⋱0000𝑐_𝑛0000𝑐_𝑇,𝑞=𝑞₁⋯𝑞_𝑛𝑞_𝑛+1^𝑇=𝑥₁⋯𝑥_𝑛ℎ^𝑇. (6)

Figure 2 

The structure with TLCD.

The elements 𝑚_𝑖, 𝑐_𝑖, and 𝑘_𝑖 in matrices 𝑀_𝑠, 𝐶_𝑠, and 𝐾_𝑠 are the mass, damping, and stiffness of the 𝑖th floor. Equation (5) can be expressed by the state space form as 𝑧̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡)(𝑡),=𝐴𝑧+𝐵𝑤=𝐶𝑧+𝐷𝑤 (7) where 𝑧(𝑡) and 𝑦(𝑡) are, respectively, the state vector and output vector of the system and 𝑧(𝑡)=𝑞(𝑡)𝑞̇(𝑡),𝐴=0𝐼−𝑀_𝑠⁻¹𝐾_𝑠−𝑀_𝑠⁻¹𝐶_𝑠,0𝐵=(𝑡)=𝑢̈−𝐼,𝑤_𝑔(𝑡). (8)

The matrices 𝐶 and 𝐷 would be determined according to the optimization object.

2.3. 𝐻_∞ Norm

The transfer function of (7) can be expressed by 𝑇(𝑠)=𝐶(𝑠𝐼−𝐴)⁻¹𝐵+𝐷. (9)

The 𝐻_∞ norm is defined as ‖𝑇(𝑠)‖_∞=sup_𝑤𝜎_max𝑇𝑗𝑤. (10)

In another word, the 𝐻_∞ norm is the maximum singular value of frequency response of the system. In the frequency domain, the 𝐻_∞ norm can be derived by ‖𝑇(𝑠)‖_∞=sup_{‖‖𝑤(𝑡)2≠0}𝑦(𝑡)₂‖𝑤(𝑡)‖₂=sup_{‖𝑤(𝑡)‖2=1}𝑦(𝑡)₂, (11) where ‖𝑦(𝑡)‖₂ is the 𝐿₂ norm of the output 𝑦(𝑡) and can be determined by the following equation: 𝑦(𝑡)₂=^∞₀𝑦^𝑇(𝑡)𝑦(𝑡)𝑑𝑡^1/2. (12)

It can be found that ‖𝑦(𝑡)‖₂ is the energy measurement of the output 𝑦(𝑡). Similarly, the 𝐿₂ norm of 𝑤(𝑡) is the energy measurement of the seismic input, which can be determined by ‖𝑤(𝑡)‖₂=^∞₀𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡^1/2. (13)

If the 𝐻_∞ norm of the transfer function 𝑇(𝑠) achieves its minimum, the energy measurement of the output will meet its minimum and the control performance of the system will be in its best state based on 𝐻_∞ norm.

3. Selection of the Weighted Function

Theoretically, the optimum control based on 𝐻_∞ norm is to meet the minimum of 𝐻_∞ norm in the frequency domain. However, it is impractical to control the dynamic response of the system in the whole frequency domain due to the property limitation of dampers. Hence, the weighted functions are usually used to regulate the system’s performance in the active controller design or parameters optimization of passive dampers. The advantages of using weighted functions are obvious in controller design. First, some components of a vector signal are usually more important than others. Second, each component of the signal may not be measured in the same units. Also, we might be primarily interested in the dynamic response of a certain frequency range [23]. Therefore, some frequency-dependent weights must be chosen to obtain a high performance controller. Two weighted functions, 𝑊_𝑔(𝑠) and 𝑊₁(𝑠), will be used to facilitate the parameter optimization of liquid dampers, in which 𝑊_𝑔(𝑠) and 𝑊₁(𝑠) are used to weigh the seismic excitation and structural dynamics, respectively. However, there is no definite conclusion how the weighted functions will influence the optimization results and how to select appropriate weighted functions to get the optimal results with less computational efforts. This paper will analyze the influences of weighted functions on the optimal results of liquid dampers.

The weighted function 𝑊_𝑔 is used to reflect the frequency content of an earthquake. The most commonly used stochastic model of earthquakes is the Kanai-Tajimi spectrum, as shown in the following equation: 𝑆(𝜔)=𝑆₀𝜔⁴_𝑔+4𝜔²_𝑔𝜁²_𝑔𝜔²𝜔²−𝜔²_𝑔²+4𝜔²_𝑔𝜁²_𝑔𝜔². (14)

The weighted function 𝑊_𝑔 is chosen as the square root of the Kanai-Tajimi spectrum: 𝑊_𝑔(𝑠)=𝑆₀𝜔²_𝑔+2𝜁_𝑔𝜔_𝑔𝑠𝑠²+2𝜁_𝑔𝜔_𝑔𝑠+𝜔²_𝑔. (15)

The values of 𝜔_𝑔 and 𝜁_𝑔 can be determined according to the site types.

For the regulated response, we are only interested in the low-frequency response. Therefore, the weighted function 𝑊₁ can be selected as a low-pass filter. The influences of weighted functions on the optimization results will be discussed in the following section.

4. Genetic Algorithm

Genetic Algorithm (GA) is a search heuristic that mimics the process of natural evolution to generate useful solutions to optimization and search problems. In a genetic algorithm, a population of candidate solutions to an optimization problem is evolved toward better solutions. Each candidate solution has a set of properties which can be mutated and altered. The evolution usually starts from a population of randomly generated individuals and is an iterative process, with the population in each iteration called a generation. In each generation, the fitness of every individual in the population is evaluated; the fitness is usually the value of the objective function in the optimization problem being solved. The more fit individuals are stochastically selected from the current population, and each individual’s property is modified to form a new generation. The new generation of candidate solutions is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when either a maximum number of generations have been produced or a satisfactory fitness level has been reached for the population. A typical genetic algorithm requires a genetic representation of the solution domain and a fitness function to evaluate the solution domain. Once the genetic representation and the fitness function are defined, a GA proceeds to initialize a population of solutions and then to improve it through repetitive application of the mutation, crossover, inversion and selection operators. The steps of problem solution by genetic algorithm can be summarized as follows.(1)Determine variables and constraint conditions; in other words, determine the individuals’ form and the solution space.(2)Define the fitness function. The fitness function is defined as the 𝐻_∞ norm of the controlled system. (3)Find the chromosome coding and decoding method for the feasible solution.(4)Define the rules of transforming the value of objective function to individual fitness.(5)Define the genetic operators, which are the methods of selection, crossover, and mutation to generate the next generation.(6)Choose the parameters of the genetic algorithm, including the population size, iteration number, termination condition, selection probability, crossover probability, and mutation probability.(7)Run the genetic algorithm and acquire the individuals with maximum fitness values.

5. Numerical Analysis

5.1. Structural Parameters

A 3-story frame structure is used as a numerical example to verify the significance of weighted functions. The values of mass, stiffness and damping matrices are, respectively, as follows: 555𝑀=×10⁵Kg,𝐶=1.3506−0.52860−0.52861.3506−0.52860−0.52860.8220×10⁶N⋅s/m,𝐾=4−20−24−20−22×10⁸N/m. (16)

The natural frequencies of the structure without control are 1.58 Hz, 4.44 Hz, and 6.42 Hz, respectively. A TLCD is installed on the top floor to control the structural vibration.

With the consideration of structural safety, optimization objective is set to make the 𝐻_∞ norm of the transfer function of the top floor’s displacement minimum. Parameters of liquid dampers to be optimized include mass ratio 𝜇, frequency ratio 𝑓,and damping ratio 𝑑, of which their physical meanings and optimization ranges are presented as follows.(1)Mass ratio 𝜇 means the ratio of the mass of the liquid in the container to the one of the whole structure. In the paper, the ratio is supposed to be in the range from 0.1% to 3% according to the actual condition.(2)Frequency ratio 𝑓 is the ratio of the frequency of the damper to the first frequency of the structure. It is supposed to be in the range from 0 to 2.5.(3)Damping ratio 𝑑 denotes the ratio of the damper’s damping to its critical one. It is supposed to be in the range from 0% to 10%.

5.2. First-Order Weighted Function 𝑊₁(𝑠)

For building structures with multiple degrees of freedom, the first few modes will dominate the dynamic response. Generally, the high-order modes are usually ignored and only the first few modes are used for analysis. The low-pass filter can regulate the system response in the low-frequency range and can be used as the weighted function. The influence of the first-order weighted function 𝑊₁(𝑠) on the optimization result will be analyzed in this section. The first-order weighted function 𝑊₁(𝑠) can be written as 𝑊₁(𝑠)=𝑘𝑠/𝜔_𝑛,+1 (17) where 𝑘 means the system gain and 𝜔_𝑛 is the cut-off frequency.

5.2.1. The Influence of 𝜔_𝑛 on Optimization Results

The preliminary analysis shows that the frequency contents of the top floor’s displacement are below 40 rad/s. Given the value of 𝑘 to be 0.2 and 5, and 𝜔_𝑛 to be 10 rad/s, 30 rad/s, and 40 rad/s, then a series of frequency response curves of 𝑊₁(𝑠) can be obtained, as shown in Figures 3 and 4.

Figure 3 

Frequency response curve of 𝑊1(𝑘=0.2).

Figure 4 

Frequency response curve of 𝑊1(𝑘=5).

The parameters of TLCD with the above weighted functions are optimized by genetic algorithm and the results are shown in Table 1, in which 𝛾_∞ means the 𝐻_∞ norm of the system with optimized values. It can be seen from Table 1 that the values of 𝜔_𝑛 and 𝑘 have significant influences on the optimal mass ratio and frequency ratio of TLCD. The frequency response curves of the top floor’s displacement for the structure without control and with TLCD control in different optimal parameters are shown in Figures 5 and 6. It can be found from the figures that the top floor’s displacement of the structure can be suppressed obviously with TLCD control. For the controlled structure with optimized parameters in consideration of the weighted function, compared with the one without consideration of the weighted function, the peak response reduces the most when 𝜔_𝑛 is 10 rad/s.

Table 1 

The optimal parameters of TLCD.

Figure 5 

Frequency response curve of top floor’s displacement (𝑘=0.2).

Figure 6 

Frequency response curve of top floor’s displacement (𝑘=5).

5.2.2. The Influence of 𝑘 on Optimization Results

Given the value of 𝜔_𝑛 as 10 rad/s and 𝑘 as 0.2, 2, 5, and 10, respectively, the optimal parameters of liquid dampers can be obtained by GA, as shown in Table 2. The corresponding frequency response curves are shown in Figure 7.

Table 2 

The optimal parameters of liquid dampers.

Figure 7 

Frequency response curve of top floor’s displacement.

It can be seen from Table 2 that the variation of 𝑘 has significant influences on the optimal value of the mass ratio 𝜇 and little influence on the frequency ratio 𝑓 and damping ratio 𝑑. Figure 7 shows that the structural displacement can be effectively controlled with optimized TLCD parameters and the best control case is the one optimized with the first-order weighted function of 𝜔_𝑛=10 rad/s and 𝑘=2.

5.3. Second-Order-Weighted Function

The second-order weighted function can be written as 𝑊₂(𝑠)=𝑘𝑠/𝜔_𝑛²+2𝜁/𝜔_𝑛.𝑠+1 (18)

With fixed 𝑘=2 and 𝜔_𝑛=10 rad/s, the effect of structural response by different values of 𝜁 is analyzed in the following. Take 𝜁 as 0.1, 0.2, and 0.5, and then a series of frequency response curves of second-order weighted function can be obtained as shown in Figure 8.

Figure 8 

Frequency response curve of 𝑊2.

With the second-order weighted function, the parameters of the tuned liquid temper are optimized by genetic algorithm. The results are shown in Table 3, and the corresponding frequency response curves in are shown Figure 9. Table 3 shows that the optimized mass ratio 𝜇 decreases and better control effect can be achieved with the increasing of 𝜁. Figure 9 shows that the top floor’s response is suppressed effectively with optimized TLCD parameters. Different weighted functions result in the different optimized parameters and different dynamic response curves. The best control performance can be achieved with the 𝜁 of 0.1.

Table 3 

The optimal parameters of liquid dampers.

Figure 9 

Frequency response curve of top floor’s displacement.

To achieve the better vibration control effect, the use of the first-order and second-order weighted function is compared as follows. The mathematical formulation of the first-order weighted function 𝑊₁(𝑠) and second-order weighted function 𝑊₂(𝑠) is expressed by 𝑊₁(𝑠)=2/((𝑠/10)+1) and 𝑊₂(𝑠)=2/((𝑠²/100)+(0.2/10)𝑠+1). The frequency response curves of 𝑊₁(𝑠) and 𝑊₂(𝑠) are shown in Figure 10. The frequency response curves of top floor’s displacement and acceleration with optimized TLCD parameters weighted by 𝑊₁(𝑠) and 𝑊₂(𝑠) are shown in Figures 11 and 12. It can be seen from the figures that there is little difference for the control results with optimized TLCD parameters weighted by the first-order weighted function or the second-order one. Hence, the low-order weighted function should be selected to facilitate the controller design.

Figure 10 

Frequency response curves of 𝑊1(𝑠) and 𝑊2(𝑠).

Figure 11 

Frequency response curve of top floor’s displacement.

Figure 12 

Frequency response curve of top floor’s acceleration.

5.4. Time History Analysis

To verify the control performance of TLCD with optimized parameters, the seismic responses of the 3-story building are analyzed in the time domain. Four earthquake records, Kobe, El Centro, Northridge, and Hach, are used as excitation for different sites. The mass ratio, frequency, and damping ratio of TLCD are 0.013865, 0.94394, and 0.099712, as shown in Table 1. The time histories of top floor’s displacement and acceleration are shown in Figures 13, 14, 15, 16, 17, 18, 19, and 20. The detailed values of seismic responses and reduction ratios of the structure at different earthquake records are listed in Table 4. From these figures and the table, it can be concluded that the structural response can be effectively reduced subjected to earthquake excitation at different sites.

Table 4 

Reduction ratios of the top story of the structure at different earthquake records.

Figure 13 

Time history of top floor’s displacement (Kobe wave).

Figure 14 

Time history of top floor’s displacement (EL Centro wave).

Figure 15 

Time history of top floor’s displacement (Northridge wave).

Figure 16 

Time history of top floor’s displacement (Hach wave).

Figure 17 

Time history of top floor’s acceleration (Kobe wave).

Figure 18 

Time history of top floor’s acceleration (EL Centro wave).

Figure 19 

Time history of top floor’s acceleration (Northridge wave).

Figure 20 

Time history of top floor’s acceleration (Hach wave).

6. Conclusions

The paper proposed a more general and effective method to optimize the parameters of tuned liquid column dampers (TLCD) based on Genetics Algorithm (GA) and 𝐻_∞ norm. The weighted functions are necessary in the optimization process. For the weighted function of earthquake excitation, the square root of the Kanai-Tajimi spectrum is suggested. For the weighted function of output, the first-order function and second-order one are analyzed with different parameters. The results show that the weighted function of output has great influences on the optimized parameters. To facilitate the parameter optimization, the low-order weighted function of the output is suggested. The structure response with optimized liquid dampers subjected to actual earthquake records is analyzed in the time domain. The results show that TLCD with optimized parameters can effectively reduce the seismic responses at different site types, which verify the generality of the proposed optimization method.

Acknowledgment

The authors are grateful for the supports from the National Natural Science Foundation of China (Grant nos. 51261120375 and 51121005).