In recent years, there has been a tendency to control the seismic behavior of structures with vibrational absorption tools such as seismic base isolators, viscose dampers, friction dampers, and pendulum dampers. A Tuned Mass Damper (TMD) has a mass, a spring, and a damper that is added to the main structure and vibrates with the structure. The TMD’s frequency is tuned in resonance with the frequency of the main structure, so a large amount of the structural vibrating energy is transferred to the TMD. Frahm [ 1 ] reported the first study on the TMD to control the vibrations of the ship’s lounge, which invented a device for damping resonant vibrations in 1911. This device was effective only when the frequency of the TMD was close to the excitation frequency. The main weakness in this device’s use is that the TMD’s inherent damping is ignored. Ormondroyd [ 2 ] updated this old type by adding a viscose damper to the regulated TMD and acquired good results. Den Hartog [ 3 ] developed several techniques that could be only used for single-degree-of-freedom systems to optimize damper parameters, including frequency and damping ratios of TMD. In a different method presented by Warburton and Ayorinde [ 4 ], if a system’s natural frequencies are separated, an SDOF can be used to design the parameters of TMD. Warburton [ 5 ] presented a newly regulated damper that estimated the effects of harmonic loading and random white noise in an SDOF. Villaverde et al. [ 6 ] found that the optimal performance of TMDs is achieved when the damping ratio of the two primary modes is equal. Other methods have been suggested to improve the TMD performance [ 7 - 9 ].
In recent decades, metaheuristic algorithms have been used to solve and optimize engineering problems. These algorithms are particularly effective for problems in which the definitive solution is complex or unavailable. An example of these algorithms is the genetic algorithm [ 10 , 11 ], the particle swarm algorithm [ 12 ], the harmony search algorithm [ 13 ], and the charged system search algorithm [ 14 ]. Recently, these developed metaheuristic algorithms are used to solve many engineering problems. For example, Babaei et al. [ 15 ] employed NSGA-II algorithm to optimize MR semi-active control systems. Ghiasi et al. [ 16 ] utilized the invasive weed algorithm to optimize the dimensions of the Koyna weight concrete dam in India, intending to achieve optimal concrete consumption.
The use of metaheuristic algorithms to optimize the parameters of TMD was initially expressed by Hadi and Arfiadi [ 17 ]. Lee et al. [ 18 ] presented a numerical method that ultimately minimized system responses in the frequency domain. In another study, harmony search algorithm was used as an optimization method by Bekdas et al. [ 19 ]. Their study estimated the optimal parameters of TMD by assuming the structure under harmonic loading. Araz et al. [ 20 ] proposed a methodology for the optimization of double TMDs of installed on the top floor of a building under seismic loads. Chowdhury et al. [ 21 ] used H2 and H∞ optimization techniques for optimized system parameters of the negative stiffness inertial amplifier tuned mass dampers (NSIA-TMD). Khatibinia et al. [ 22 ] found the optimum design of a TMD for a 10-story inelastic steel moment-resisting frame (SMRF) using the passive congregation particle swarm and grey wolf optimization techniques. Domizio et al. [ 23 ] evaluated the performance of the three optimal TMD configurations on three SDOF structures with different fundamental periods subjected to a set of far-field records and a set of near-fault ground motions with strong velocity pulses.
This research investigates the optimization of passive mass damper parameters (TMD) to reduce dynamic responses of a multi-degree-of-freedom structure with this type of damper under seismic loading. The particle swarm optimization (PSO) algorithm, whale optimization algorithm (WOA), and the combination of these two algorithms (Hybrid PSO-WOA or HPW) have been selected as efficient and effective optimization algorithms for this purpose.
2. Equation of motion
Consider an n-floor shear frame with a mass damper mounted on its roof. The equations of motion of the structural system can be written as Eq. (1).
That is, , and x, respectively, the vectors of acceleration, velocity, and displacement of the system are multi-degree-of-freedom. Γ is also the impact vector of earth acceleration, a column vector, unit, and co-order with the number of degrees of system freedom. The structures’ mass, stiffness, and damping matrices are defined according to Eqs. (2), (3), and (4) .
mi, ki and ci are the mass, stiffness, and damping coefficients of floor i (i=1,2,...,n), md, kd, and cd, respectively, TMD’s mass, damping, and stiffness. There are two methods for determining the system response: the time integration method and using state-space equations. Equations of motion can be converted into a set of state-space equations according to Eqs. (5) and (6):
The parameters are determined as equations Eqs. (7) to (12):
 and [I] are zero and identity matrices, respectively. When values shown in the above equations are defined, iterative solutions can be used to determine the system response with different loading types, such as seismic excitation.
3. Particle swarm optimization (PSO) algorithm
The particle swarm optimization (PSO) algorithm is a metaheuristic algorithm proposed by Russell Ebrahatt and James Kennedy in 1995. This algorithm works so that a group of particles is randomly distributed in the search space, and their responses are determined. Then, the current position information, the best position in which the particle is located (Pbest), and the best position discovered in the whole particle (Gbest) are recorded. This data is used to determine and modify the new position and velocity of particles (Fig. 1). This step is repeated several times to get the best answer. In each step, the algorithm updates each particle’s new velocity and position according to Eqs. (13) and (14) after finding the two Pbest and Gbest values from the previous step. This procedure will continue until the termination conditions (time limit, maximum number of repetitions, and error limits) [ 12 ].
That vi (t) and xi (t) respectively is the velocity and position of the i-particles in the t repetition. r1 and r2 random numbers are between zero and one. c1 and c2 constants are called PSO algorithm acceleration coefficients, and ω is the weighted coefficient of inertia, which increasing iterations reduces its value from one to zero. The particle optimization algorithm flowchart is presented in Fig. 2.
4. Whale optimization algorithm (WOA)
This algorithm was proposed by Mirjalili and Lewis [ 25 ]. The most exciting thing about whales is their specific hunting method. This exploratory behavior is known as the bubble-net feeding method. Exploration and hunting are accomplished by creating index bubbles along a circle or paths. This feeding behavior is performed by placing specific bubbles in spiral shapes, according to Fig. 3.
Whales can identify and surround the hunting site. The WOA algorithm assumes that the best candidate solution at the moment is to hunt the target or be close to the desired state. After the best search agent is identified, other search agents try to update their location relative to the best search agent. The whale swims around the prey along a contractile circle and simultaneously in a spiral-shaped path. To model this behavior, at the same time, it is assumed that the whale with a 50 percent probability chooses between the mechanism of shrinking the siege or the spiral model to update the position of the whales during optimization (where the random numerical p is between 0 and 1) if p<0.5, the mechanism for shrinking the siege is used according to the following equations.
Where and are determined based on the following equations:
a decrease linearly from 2 to 0 during repetitions, and r is a random vector at a distance of 0 to 1 (Fig. 4). If the value || is larger than one, the Eqs. (19) and (20) will replace with Eqs. (15) and (16):
If p≥0.5, the spiral position update method (Fig. 5) is used according to the following equations:
5. Proposed the Hybrid PSO-WOA (HPW) algorithm
In this section, a hybrid algorithm based on the combination of the PSO algorithm and the WOA algorithm is presented. In the PSO algorithm, the position of particles is updated based on either the best solution found for the particle and the best solution found for all particles. On the other hand, in the WOA algorithm, only the best solution found for all whales is used to update whales’ positions. These two different features together may result in a more accurate method that is more efficient than either of the two methods alone. The basis of the proposed method is that after performing PSO operators on the particles and updating the best solution of all particles, all particles are considered as whales for applying the WOA method. Then, all the solutions should be updated according to the WOA algorithm. The best solution for each particle and the best solution between all particles are updated again. At the current stage, it is checked that whether the convergence conditions are satisfied or not. In the case that convergence is not occurred, the current solutions are given as particles to the PSO algorithm, and the mentioned process is repeated again. This process is continued until the converge condition is satisfied. The schematic flowchart for the proposed method is shown in Fig. 6.
6. The optimum parameters of passive tuned mass damper
In this research, the system’s response is obtained using the transient integration methods in OpenSees and controlled by the state-space equation. In this section, both methods are first explained, and then the responses of two numerical examples are extracted using them and compared with the results of previous studies.
6.1. The state-space equations
In this study, TMD’s optimal stiffness and damping parameters are calculated by considering a constant mass for TMD. Then, the TMD’s optimal mass is obtained according to the other two parameters. The optimization process minimizes the maximum floor drift relative to the ground when the structure is excited under an earthquake. This procedure can be done by adding a transfer function between changing the upstairs location and the ground acceleration to the objective function. Generally, the transfer function is defined as a criterion for evaluating the number of input components transferred to the system. By taking the Laplace conversion (with zero initial conditions) from the space system, we have the state defined in Eqs.(23) and (24):
By solving these equations:
The transfer function was defined as an output Laplace transformation adjusted to the Laplace transform of the input function (external forces).
Where, T.F is the transfer function between the input and output of the system, which can be used for displacement or acceleration, it should be noted that, as can be seen in Eq. (26), the transfer function is independent of the type of output component and is considered to be an inherent feature of the system. Considering the force entered into the first floor as input and displacement of the first floor of the MDOF as an output (in the transmission function in both controlled and uncontrolled) and displacement ratio (in both controlled and uncontrolled) as a factor for controlling the behavior of the structure, the objective function can be defined as follows:
6.2. The transient integration method in OpenSees
The structures are initially modeled in the OpenSees software to extract the systems’ responses using the transient integration method. For this purpose, a ten-story shear building with one span is considered. In this building, the height of columns and span width were assumed to be 3 meters. In the first example, a 10-story frame with similar properties on each floor, and in the second example, a 10-story shear frame with different properties on each floor are investigated. Elastic beam-column element is also used to model the samples. The beam of the floors is assumed to be rigid, and using the stiffness presented in previous studies and the relation K=24EIc/Lc3, the columns’ characteristics are entered into the software [ 26 ]. Uniaxial materials viscous and element two node link are used to consider damping in floors. The schematic of the structure used is depicted in Fig. 7. In the transient integration method, the ratio of maximum displacements of the top floor of the frame with and without TMD is minimized for the optimization objective.
6.3. Numerical examples
6.3.1. Example 1
This example investigates a ten-story frame with a TMD attached to the top floor and under seismic loading (El Centro earthquake). The characteristics of this building are shown in Table 1. TMD mass is taken at 108 tons. TMD stiffness and damping values are defined as optimization algorithm variables. The lower and upper bounds of stiffness are 0 and 5000 kN/m. Also, the lower and upper bounds of the damping coefficients are 0 and 1000 kN.s/m.
The results of the genetic algorithm (GA) [ 17 ], the method presented by Lee et al. [ 18 ], the search of the charged system (CSS) [ 26 ], and the three methods presented in this study are demonstrated in Table 2. According to this table, the WOA and HPW algorithms have predicted the lowest value among the optimal values for cd and kd. The optimal value for cd and kd in the PSO algorithm is 173.86 and 122.91% higher than the values obtained from the WOA algorithm, respectively. The optimal values of WOA and HPW algorithms are also close to each other.
|Story||Mass (ton)||Stiffness (kN/m)||Damping (kN.s/m)|
|TMD parameters||Optimum values|
|Ref. ||Ref. ||Ref. ||PSO||WOA||HPW|
The maximum absolute displacement of each floor relative to the ground (in both controlled and uncontrolled) is summarized in Table 3. The percentage of displacement reduction is shown in Table 4. To better understand the results of Table 4, these results are presented again in Fig. 8. The results show that by using a TMD attached to the top floor in a ten-story shear frame, the maximum absolute displacement is reduced. The amount of reduction predicted for absolute displacement by PSO, WOA, and HPW algorithms are equal to 37.21, 38.54%, and 38.85, respectively. The changes in the objective function versus the number of iterations for the first example come in three optimization algorithms in Fig. 9(a) to 9(c). As can be observed, the PSO, WOA, and HPW algorithms have converged to the optimum solution before 40, 90, and 70 iterations, respectively.
|Story||Without TMD||With TMD|
|Ref. ||Ref. ||Ref. ||PSO||WOA||HPW|
|Story||Ref. ||Ref. ||Ref. ||PSO||WOA||HPW|
The time history displacement of the structure from the first to the tenth floor is depicted in Fig. 10(a) to 10(j). When the optimum values of TMD stiffness and damping coefficient are determined, the system performance can now be improved by changing the mass of TMD, which was initially selected at 108 tons. For this purpose, the system response should be investigated by applying changes in the range of 90 to 116 tons (with the same desired stiffness values and damping coefficient as previously determined). The results of these changes for the PSO method are shown in Table 5. According to Table 5, increasing the TMD mass from 90 to 100 will lead to a decrease in the maximum displacement. For example, the PSO algorithm has predicted that the mass increase from 90 to 100 tons and the maximum displacement on the top floor has decreased by 8.8%, but increasing the mass from 100 to 116 tons has led to an increase in the maximum displacement.
|Story||TMD mass (ton)|
The effect of ground motion (GM) record change on the three algorithms’ performance was also evaluated in this example. For this purpose, six GM records from the far-field records provided in FEMA P695 methodology [ 27 ] were selected according to Table 6. These GM records were scaled based on the first-mode period (Sa(T1) scaling method), considering the El Centro earthquake as the target spectral acceleration.
|ID No.||Event name||Year||Magnitude||Station||Site class (NEHRP)||PGAmax (g)||PGVmax (cm/s)|
|5||Landers||1992||7.3||Yermo Fire Station||D||0.24||52|
The maximum absolute displacement of each floor relative to the ground (in both controlled and uncontrolled) under these three GM records is summarized in Table 7. Also, the percentage of displacement reduction is shown in Table 8. For clearer comprehension of the results, a bar chart was drawn according to the percentage of displacement reduction provided for each record in Fig.11(a). Additionally, the mean response distribution for the percentage of displacement reduction due to record-to-record (RTR) variability per three algorithms is shown in Fig. 11(b) using lognormal probability distribution functions (PDFs).
|Duzce, Turkey||Without TMD||0.06490||0.12757||0.18661||0.24087||0.28937||0.33128||0.36597||0.39271||0.41089||0.42008|
|Hector Mine||Without TMD||0.03643||0.07159||0.10486||0.13563||0.16330||0.18740||0.20746||0.22296||0.23351||0.23886|
|Kobe, Japan||Without TMD||0.19068||0.37658||0.55568||0.72297||0.87439||1.00629||1.11551||1.19956||1.25660||1.28541|
|Manjil, Iran||Without TMD||0.02973||0.05810||0.08391||0.10647||0.12599||0.14258||0.15598||0.16543||0.17165||0.17492|
Figure 11(a) shows that using the HPW algorithm in seismic excitation under Northridge, Duzce, Hector Mine, and Landers earthquakes has caused a greater reduction in the displacement of floors than the other two algorithms. As a result, this algorithm has provided more optimum parameters for the passive-tuned mass damper. Figure 11(b) shows that due to the nearly equal extracted responses’ standard deviation of all three algorithms, the distribution of the responses is likewise equal, and eventually, RTR variability is nearly the same per three algorithms.
6.3.2. Example 2
This example investigates another 10-story frame with a TMD attached to the top floor and undergoing seismic loading (El Centro earthquake). The properties of this building are shown in Table 9. Like the first example, TMD stiffness and damping values are defined as variables of the optimization algorithm. The lower and upper bounds of stiffness are 0 and 500 kN/m, respectively, and the lower and upper bounds of the damping coefficient are 0 and 150 kN.s/m, respectively. The optimal TMD values estimated in this research and previous research are shown in Table 10. The optimal value of cd and kd obtained from the PSO algorithm is 404.76 and 194.78% higher than the optimal value obtained from the WOA algorithm. Like the previous example, the WOA and HPW algorithms have predicted the lowest optimal value of cd and kd, and the optimal values of WOA and HPW algorithms are close to each other.
|Story||Mass (ton)||Stiffness (kN/m)||Damping (kN.s/m)|
|TMD parameters||Optimum values|
|Ref. ||Ref. ||Ref. ||PSO||WOA||HPW|
The maximum absolute displacement of each floor relative to the ground (in both controlled and uncontrolled) and the percentage of displacement reduction are shown in Tables 11 and 12, also changes in the objective function versus the number of iterations for the second example in three optimization algorithms are depicted in Fig. 12(a) to 12(c). Also, the time history of floor displacement of the structure from the first floor to the tenth floor is shown in Fig. 13(a) to 13(j). The HPW algorithm has predicted a greater reduction in displacement than the other two algorithms (PSO algorithm, 26.91%, WOA algorithm, 26.88%, and HPW algorithm, 28.09). According to Fig. 12, the PSO, WOA, and HPW algorithms have converged to the optimum solution before 20, 50, and 40 iterations, respectively.
|Story||Without TMD||With TMD|
|Ref. ||Ref. ||Ref. ||Ref. ||Ref. ||PSO||WOA||HPW|
|Story||Ref. ||Ref. ||Ref. ||Ref. ||Ref. ||PSO||WOA||HPW|
The primary purpose of this study was to estimate the optimum parameters of a passive mass damper to reduce the system’s response under earthquake loading. The emphasis on the inactive type of TMD damper is because this type of TMD does not delay responding to the system reaction and usually requires lower costs than other types. The particle swarm optimization (PSO) algorithm, whale optimization algorithm (WOA), and the combination of these two algorithms (Hybrid PSO-WOA or HPW) have been used to determine the optimum stiffness and damping of TMD. The system’s response was obtained using the transient integration methods in OpenSees and controlled by the state-space equation. In the first example, a 10-story shear frame with similar properties on each floor, and in the second example, a 10-story shear frame with different properties on each floor under the seismic loading of the El Centro earthquake is presented to demonstrate the effectiveness of the proposed method. The optimal value predicted for cd and kd from the PSO algorithm in example 1 is 1.74 and 1.23 times the optimal value predicted by the WOA algorithm, respectively. Also, the optimal value predicted for cd and kd from the PSO algorithm in example 2 is 4.05 and 1.98 times the optimal value predicted by the WOA algorithm, respectively. In both examples, the WOA and HPW algorithms have predicted the lowest optimal value of cd and kd, and the optimal values of WOA and HPW algorithms are near each other, and the HPW algorithm has predicted a near each other reduction in displacement than the other two algorithms. Moreover, The effect of ground motion (GM) record change on the three algorithms’ performance by considering six far-field GM records provided in the FEMA P695 methodology was assessed in the first example. Like the earlier outcomes, the HPW algorithm has provided more optimum parameters for the passive-tuned mass damper, and RTR variability is approximately the same per three algorithms. The practical advantage of using these algorithms, among other metaheuristic optimization algorithms, is that these algorithms have the most uncomplicated relationships and the slightest need for external parameters. The proposed methods have good performance and are recommended as three approximate and rapid methods for the optimal design of these dampers.
The authors gratefully acknowledge the useful comments of anonymous reviewers on the draft version of this paper.
This research received no external funding.
Conflicts of interest
The authors declare no conflict of interest.
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