#### Abstract

In order to mitigate the seismic response of a cable-stayed bridge, a new type damping device named asymmetric pounding tuned mass damper (APTMD) is developed in this paper on the basis of the traditional symmetric pounding tuned mass damper. The novel APTMD has three parameters to be determined: the left-side gap, the right-side gap, and the frequency ratio. A numerical model of the APTMD damping system is established with consideration of both the computational efficiency and accuracy to enable the parametric optimization of the damper. The numerical model is based on a simplified model of the cable-stayed bridge and a nonlinear pounding force model. The genetic algorithm is utilized for the optimization of the damper. Afterwards, the cable-stayed bridge is subjected to 20 recorded ground motions to evaluate the vibration control effectiveness of the APTMD. Four systems are considered: (1) without dampers; (2) with a TMD; (3) with a PTMD; and (4) with an APTMD. Time history analysis reveals the following: (1) those dampers can all effectively suppress the vibration of the bridge and (2) the vibration control effectiveness of the APTMD is slightly better than the TMD and the PTMD.

#### 1. Introduction

Cable-stayed bridges, typical long-span structures used in modern traffic and transportation systems, are featured by their large flexibility and low damping. During the past decades, structural damage induced by earthquakes has been extensively reported and investigated [1–4]. Therefore, it is of great necessity to reduce the seismic responses of a bridge and so as to improve its safety and reliability.

The seismic isolation technique is one of the most efficient vibration control techniques and its effectiveness has been verified by numerous numerical simulations and experimental investigations. The seismic isolation technique often employs flexible bearings to isolate the superstructure from its foundations and consequently interrupt the energy transfer between the superstructure and substructure. The most commonly adopted base isolators include lead rubber bearings (LRB) and friction pendulum systems (FPS), both exhibiting large vertical stiffness and low horizontal stiffness. Due to the large lateral flexibility, the isolating system often suffers from large displacements under earthquakes. Therefore, additional damping was introduced to the isolating systems to reduce the displacement of the isolators. Peng and Huang proposed to implant permanent magnets and elastomer fillings to the regular isolators. This new isolator was named the implant-magnetic bearing (IMB) and had been proved effective in mitigating the story accelerations and interstory drifts via both numerical studies [5] and experimental tests [6, 7]. Zheng [8] introduced the shape memory alloys (SMA) into the LRB system and proposed a superelastic-sliding lead rubber bearing (SSLRB) isolation system. Compared with the conventional LRBs, the SSLRB further reduce the residual displacement and the seismic demand of the piers. De Domenico et al. [9]combined the low-friction curved surface sliders (CSSs) with SMA gap dampers. The combined isolating system is featured by the negligible residual displacement under small earthquakes and reduced displacement demand under large earthquakes. Wang et al. [10] proposed a novel self-centering (SC) base isolating system by incorporating SMA U-shaped dampers (SMA-UD). The superelasticity behavior of the SMA-UDs provides sufficient energy dissipation and excellent SC force. Additionally, the hysteresis characteristics of the SMA-UD are nearly isotropic, indicating that these isolators can be effective in any horizontal direction. Another drawback of the FPS system is that the sliding friction coefficient is sensitive to the surrounding temperatures and may lose recentering capability at low temperature. Zheng et al. [11] proposed a shape memory alloys (SMA) based FPS system to address this issue. Analytical results demonstrate that the residual displacement of the regular FPB system can be effectively reduced without significantly increasing the base forces of the bridge piers. In a following parametric study [12], the influence of the SMA wire properties on the response of the bridge to be controlled was investigated and optimal design can be obtained. A case study further demonstrated that the SMA-based FPB system can reduce the residual displacement and peak displacement when subjected to near-fault ground motions. Gur et al. [13] proposed to incorporate the base isolators with thermally modulated SMA damper, of which the temperature is altered during loading-unloading cycles to improve the energy dissipation. It is demonstrated by a numerical simulation that the thermally modulated shape memory alloy friction pendulum (tmSMA-FP) showed improved control efficiency compared with the regular FPS and SMA-FPS.

The tuned mass damper (TMD) is another commonly adapted damping device for bridge structures. In concept, the TMD is a dynamic absorber, often installed at the place of the main structure where the largest motion occurs. When the TMD is tuned to the target frequency of the main structure, it can effectively absorb the kinetic energy of the primary structure and thus reduce its vibration [14]. However, the TMD often suffers from a limited energy dissipation capacity and a relatively narrow band of effective frequency. Therefore, a variety of damping elements are introduced to the TMD to improve vibration control efficiency and robustness, such as friction dampers [15, 16], eddy-current components [17–20], inerters [21–24], particle dampers [25, 26], and shape memory alloys [27–29]. Most of these damping elements are expensive in cost and difficult in maintenance.

Recently, a new passive damping device named pounding tuned mass damper (PTMD) was proposed for vibration control of flexible structures such as power transmission towers [30, 31], high-rise buildings [32], subsea pipelines [33, 34], and bridges [35–37]. The PTMD consists of a tuned mass whose stroke is restrained by a pair of delimiters covered with viscoelastic materials. When the tuned mass impacts on the viscoelastic delimiter, large amounts of energy can be dissipated via collision. Therefore, the PTMD has an additional energy consumption pattern compared with a regular TMD and thus has better vibration control effectiveness. Furthermore, this additional damping mechanism can also improve the robustness of the PTMD. Even if the PTMD is off-tuned from the target frequency by 15%, it still can effectively mitigate the vibration of the primary structure [33, 38].

In early studies of the PTMD [30], the mass block is located in the middle of the two delimiters, which implies that the gap of the left side is the same as the right side. Parametric studies revealed that the gap is the key parameter that influences the vibration control effectiveness [39, 40]. The optimal gap is sensitive to the input amplitude and mass. To avoid the design complexity, a single-sided pounding tuned mass damper (SSPTMD) is proposed [37, 41]. In the SSPTMD, the mass block is placed aside the delimiter, which means that the gap equals zero. The frequency of the SSPTMD is tuned to half of the value of the target frequency to achieve the optimal design. Numerical studies and experimental results also validated the effectiveness of the SSPTMD.

With the inspiration of the SSPTMD, a novel damper called asymmetric pounding tuned mass damper (APTMD) is proposed in this paper. Compared with the former symmetric PTMD, the two gaps of the APTMD can be set to different values. Additionally, the frequency ratio, which is defined as the frequency of the damper over that of the primary structure, is another parameter to be determined. In order to achieve the optimal vibration mitigation effectiveness for a realistic bridge, the genetic algorithm (GA) is employed, with the left-side gap, right-side gap, and frequency ratio defined as the optimal variables and the average value of dynamic responses under 20 ground motions taken as the objective function. Even though the finite element (FE) model of the cable-stayed bridge equipped with an APTMD can be established in ANSYS, it is time-consuming to perform a time history analysis of this numerical model. Therefore, a simplified model of the bridge is developed and utilized to calculate the fitness of each individual in the GA optimization process. Finally, a time history analysis is conducted on the FE model of the cable-stayed bridge damped by the optimized APTMD. A TMD and a PTMD with the same mass ratio are also attached to the bridge to compare their damping effect with the APTMD.

The remainder of this paper is arranged as follows: in Section 2, the damping mechanism of the APTMD will be further explained by comparison with the classical TMD and the previously proposed symmetric PTMD. Motion equations of the APTMD damping system will be established on the basis of a nonlinear pounding force model in MATLAB/Simulink environment. Afterwards, a simplified model of a real cable-stayed bridge is established in Section 3. Optimal design of the APTMD will be achieved using the genetic algorithm in Section 4. Then, in Section 5, the dynamic responses of the bridge equipped with optimal APTMD, former PTMD, and traditional TMD will be compared to evaluate the performance of the proposed APTMD. Finally, this paper concludes with a discussion of the overall results and suggestion for future studies.

#### 2. Development of APTMD Damping System

##### 2.1. Introduction of APTMD

In order to demonstrate the damping mechanism of the proposed APTMD, the schematics of the classical TMD, the previously proposed symmetric PTMD, and the APTMD, as well as the undamped system, are compared in Figure 1. In Figure 1(a), a mass block, , which is connected to the base by a spring, , and a dashpot, , represents the primary structure with being controlled. denotes the displacement of .

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In Figure 1(b), an auxiliary mass, , is connected to by a spring, , and a dashpot , composing a classical TMD. If optimally designed, the additional mass, , can generate an initial force toward the equilibrium position to damp the oscillation of the primary structure . However, the classical TMD suffers a deterioration in vibration control effectiveness when its parameters shift away from the optimal values. Additionally, the added mass often has an overlarge stroke, causing impacts between the added mass and the primary structure, which makes the TMD not applicable for structures with confined space. Therefore, the PTMDs are proposed to address these issues.

Figure 1(c) illustrates the previously proposed PTMD, which can be regarded as a restricted TMD. In the PTMD damping system, the motions of the additional mass *m*_{2} are restrained within a range of two delimiters (as illustrated by the two blue triangles in Figure 1(c)). The delimiters are covered with viscoelastic materials for two purposes: firstly, to dissipate the kinetic energy when collides with the delimiter and, secondly, to reduce the noise and excessive acceleration induced by the collision. In Figure 1(c), denotes the gap between the and the left delimiter, while is the gap of the right side. In the previous PTMD, equals . It is reported in much literature that there is an optimal gap for each given loading amplitude and mass ratio. For a severe dynamic excitation and relatively small mass ratio, a large gap shall be provided to allow for enough motion of the added mass, *m*_{2}. Otherwise, the energy dissipation during impact will be limited due to a small relative velocity between the added mass and the delimiter. However, if the gap is exceedingly large, the added mass will not impact on the two delimiters. In this case, the PTMD degrades to a spring-mass type dynamic absorber as Frahm invented in the 1900s.

Figure 1(d) shows the proposed APTMD. The major difference between the previous PTMD and the APTMD is that the and can be set to different values in the APTMD, which may further improve the vibration control effectiveness of the damper. Three key parameters of the APTMD are , , and the frequency ratio of the APTMD over that of the primary structure. These parameters can be optimized using the genetic algorithm in this paper.

##### 2.2. Nonlinear Pounding Force Model

Since the APTMD relies on the poundings between the added mass and the nonsymmetrically designed delimiters to dissipate energy, an accurate pounding force model is necessary to enable numerical study. In previous studies, the nonlinear pounding force model [42] based on the Hertz contact element and a nonlinear damping element has been proved to be relatively accurate and computationally efficient [30]. The mathematical expression of the pounding force is as follows:in which F denotes the nonlinear pounding force; and are the deformation and velocity of deformation, respectively; is the pounding stiffness; and *c* is the pounding damping, which can be determined bywhere and are the two colliding bodies and is the pounding damping ratio:where *e* is the restitution coefficient of the viscoelastic material. It can be obtained by dropping a sphere on the viscoelastic material, observing the initial height and rebound height :

##### 2.3. Governing Equation of APTMD Damping System

Based on the nonlinear pounding force model, the governing equations of a single degree of freedom (DOF) structure equipped with an APTMD arewhere , , and are the mass, damping, and stiffness of the primary structure, respectively; and and are the mass and stiffness of the APTMD. , , and denote the acceleration, velocity, and displacement of the primary structure, while , , and correspond to those quantities of the damper. In equation (5), is the recorded ground motion and is the pounding force computed by equation (1).

Here, the deformation, , and its velocity, , can be obtained by the relative displacement and velocity:where and are the gap of the left side and right side (as shown in Figure 2). In this equation, implies that collisions occur on the right-side delimiter, while implies that the collisions take place on the left-side delimiter. Otherwise, no collisions happen indicating = 0.

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Introducing the following mass, damping, and stiffness matrices,and acceleration, velocity, and displacement vector,and vectors indicating the location of the ground motion and pounding force.

Equation (5) can be simplified as

Introducing a state vector *Z* and the external force vector *U*, equation (10) can be expressed aswhere

Equation (11) can be solved using MATLAB/Simulink environment. Figure 3 demonstrates how equation (11) is established in Simulink. The Runge–Kutta method, which is already embedded in Simulink, is employed to solve this equation. Automatic solver selection and default settings of the software were applied.

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#### 3. Modelling of the Cable-Stayed Bridge

In order to demonstrate the vibration control effectiveness of the APTMD, a real cable-stayed bridge located in Dalian, Liaoning, China, is selected as the primary structure to be controlled. Figure 4 and 5 illustrate the configuration and appearance of the bridge. As shown in the figures, this bridge is a typical extradosed cable-stayed bridge, with a span of 140 m + 260 m + 140 m and a width of 25 m. The main girder is a continuous box girder with a variable cross section, supported by twin pylon and double-plane fan-type cables. The main pylon is 37.4 m high above the bridge deck. Concretes used for the main girder are C55 type concrete; those for the main pylon and piers are C50 and C40, respectively. The stay cables are made of *Ф*_{s} 15.2 type high yield steel strand. Properties of the concretes and cables are listed in Table 1 and 2.

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Every table must have a descriptive title and if numerical measurements are given, the units should be included in the column heading. Vertical rules should not be used (see Table 1). Tables should be cited consecutively in the text.

E denotes the elastic modulus of the concrete; *μ* is Poisson’s ratio; *G* is the shear modulus; *f*_{c} is the design value of axial compressive strength; and *f*_{t} is the design value of axial tensile strength.

##### 3.1. FE Model

A finite element model of the bridge is established in ANSYS. The FE model consists of 869 nodes and 639 elements (Figure 6). The main girder, pylons, and piers are modeled using BEAM188 element; the stay cables are modeled by the LINK10 element. Furthermore, the second phase loads, that is, the loads of pavements and other facilities, are modeled by MASS21 elements.

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In order to verify the FE model, a modal analysis was conducted using Block Lanczos method. The frequencies of the first 10 orders are listed in Table 3 and the first 3 modal shapes are presented in Figure 7. Compared with the technical report of the bridge, errors of the 1^{st} and 2^{nd} order frequencies are 1.3% and 2%, implying that the FE model can be used for time history analysis.

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##### 3.2. Simplified Model

Although the aforementioned nonlinear pounding force model can be simulated in ANSYS software using its APDL module, it is still time-consuming to conduct a time history analysis of the FE model of 5214 DOF. Consequently, a simplified model of the bridge is necessary to enable massive numerical studies and parametric optimization. In this paper, the bridge is represented by a concentrate mass connected to the ground with a spring and a damping element. The stiffness of the spring can be calculated using the FE model. The mass of the simplified model is kg and the stiffness of the spring is N/m. Figure 8 shows the displacement response of the simplified model and the FE model subjected to 2 recorded earthquakes. It can be observed that the dynamic responses of the two models are in satisfying agreement.

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##### 3.3. Selection of Earthquakes

According to the seismic design code of China [43], three earthquakes are adequate for time history analysis of an infrastructure. In order to fully consider the uncertainties of ground motions, a group of 20 seismic records (as listed in Table 4) are obtained from Pacific Earthquake Engineering Research Center (PEER). Peak accelerations are set to 139 gal, in accordance with the geological report and design manual of the bridge.

#### 4. Optimization of the APTMD with Genetic Algorithm

In order to evaluate the vibration control effectiveness of the proposed APTMD, the vibration reduction ratio is defined aswhere and are the average value of the relative displacement of the bridge deck without and with damper and is the vibration reduction ratio of the displacement.

There are three parameters to be determined to achieve the optimal design of the APTMD, that is, the left gap, , right gap, , and the frequency ratio, . The genetic algorithm is employed to obtain the optimum values of these parameters. The displacement reduction ratio is defined as the objective function or fitness function of the genetic algorithm. The three parameters , , and are the optimal variables. The range of and is from 0.005 m to 0.1 m at an interval of 0.005 m. The range of is from 0.5 to 1.5 at an interval of 0.05. Other parameters of the genetic algorithm, such as reproduction coefficient, mutation coefficient, crossover coefficient are set to the default value. The optimal individual of the final population yields the optimum APTMD: m, m, and .

An optimal PTMD is also designed to compare the vibration control effectiveness. The PTMD has two parameters to be designed, that is, the gap and the frequency ratio. This parameter is optimized by an enumeration algorithm. The value of the gap varies from 0.005 m to 0.1 m at an interval of 0.005 m. The range of is from 0.5 to 1.5 at an interval of 0.05. The optimized gap and frequency ratio are 0.095 m and 1, respectively.

Considering that the TMD, the PTMD, and the APTMD all belong to the dynamic absorber family, these dampers shall be installed at the place where the largest motion occurs to achieve the best vibration control effectiveness. As shown in Figure 7, the largest motion happens at the deck of the bridge. Therefore, the dampers shall be installed inside the hollowed girder, which is spacious to accommodate these dampers.

#### 5. Performance Evaluation of APTMD System

In order to validate the vibration control effectiveness of the APTMD, the seismic responses of the following systems are compared:(i)Undamped bridge(ii)Bridge equipped with TMD(iii)Bridge equipped with symmetric PTMD ( m, )(iv)Bridge equipped with APTMD ( m, m, )

For these damping systems, the mass ratio is all set to 5%, since increasement of mass ratio can slightly improve the vibration control effectiveness, after a certain value (normally 3%), as previous studies revealed [30, 39]. The damping ratio and frequency ratio of the TMD is determined using the following equations [14]:in which and are the frequency ratio and the damping ratio of the TMD. The pounding stiffness and restitution coefficient *e* are set to 17259 N/m^{1.5} and 0.2 based on experimental data of a previous study [30].

Figure 9–11 present the seismic response of the bridge under the Kobe earthquake. Due to the length limitation of the paper, the dynamic response corresponding to other ground motions is not plotted. The reduction ratios are listed in Table 5. It can be observed from these figures that all three dampers can effectively suppress the displacement, shear force, and bending moment of the bridge. However, the acceleration response is not satisfyingly reduced. The vibration reduction efficiency of the APTMD is slightly better than TMD and symmetric PTMD.

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Considering that the ground motions are with unignorable uncertainties, the bridges are subjected to 20 ground motions and the average vibration reduction ratios are listed in Table 5. This table also demonstrates the effectiveness of the TMD, symmetric PTMD, and APTMD. In terms of the maximum displacement, shear force, and bending moment, the vibration reduction ratios of these three dampers are very close. In terms of the root mean square (RMS) value of these responses, the APTMD is better than the TMD and PTMD.

#### 6. Conclusions

In this paper, a novel damping device, the asymmetric pounding tuned mass damper, is proposed to control the seismic-induced vibrations of a cable-stayed bridge. Compared with traditional symmetric PTMD, the APTMD has unsymmetrical gaps to be designed. A simplified model of the APTMD damping system with sufficient computation efficiency is established to enable optimization. GA is utilized in this paper to search for the optimal gaps and frequency ratio. Finally, time history analysis is conducted on the FE model of the bridge using ANSYS. Based on the numerical results, the following findings can be concluded:(1)The SDOF model can simulate the behavior of the cable-stayed bridge under earthquakes with satisfying accuracy and efficiency. Computation-intense tasks such as parametric study or optimization can be performed with this model.(2)The vibrations of the bridge are effectively mitigated by TMD, symmetric PTMD, and APTMD, indicating that impact damping can provide sufficient energy dissipation. Moreover, the impact damping is simple in design, installation, and maintenance, compared with other damping elements such as viscous dampers, SMA dampers, and eddy-current dampers. Therefore, the PTMD and APTMD are very promising for practical applications.(3)In this paper, the left-side gap, right-side gap, and frequency ratio were selected as the parameters to be designed. These parameters are optimized using GA. The vibration reduction ratio of the optimized APTMD is slightly better than the traditional symmetric PTMD.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was funded by National Natural Science Foundation of China (no. 51808092) and the United Navigation Foundation of Liaoning Province (No. 2020-HYLH-48). The financial supports are gratefully acknowledged.