Experimental research on the multilayer compartmental particle damper and its application methods on long-period bridge structures

Abstract

Particle damping technology has attracted extensive research and engineering application interest in the field of vibration control due to its prominent advantages, including wide working frequency bands, ease of installation, longer durability and insensitivity to extreme temperatures. To introduce particle damping technology to long-period structure seismic control, a novel multilayer compartmental particle damper (MCPD) was proposed, and a 1/20 scale test model of a typical long-period self-anchored suspension bridge with a single tower was designed and fabricated. The model was subjected to a series of shaking table tests with and without the MCPD. The results showed that the seismic responses of the flexible or semi-flexible bridge towers of long-period bridges influence the seismic responses of the main beam. The MCPD can be conveniently installed on the main beam and bridge tower and can effectively reduce the longitudinal peak displacement and the root mean square acceleration of the main beam and tower. In addition, no particle accumulation was observed during the tests. A well-designed MCPD can achieve significant damping for long-period structures under seismic excitations of different intensities. These results indicate that the application of MCPDs for seismic control of single-tower self-anchored suspension bridges and other long-period structures is viable.

Keywords

energy dissipation devices / multilayer compartmental particle damper / self-anchored suspension bridges / shaking tables test / long-period structure / seismic control

Introduction

Energy dissipation devices can reduce the dynamic response of structures by dissipating the vibration energy caused by factors such as wind, earthquakes, and mechanical equipment. Additional energy dissipation devices are an important means of improving the seismic performance of engineering structures. Traditional energy dissipation devices for civil engineering structures mainly comprise viscous dampers [¹–³], metallic dampers [⁴–⁶], tuned mass dampers [⁷–⁹] and shock-absorbing bearings [¹⁰,¹¹]. In recent years, with developments and progress in energy dissipation structures, many energy dissipation technologies derived from the field of machinery and aircraft have been gradually introduced into the vibration control of civil engineering structures. Particle damping technology [¹²,¹³] is a passive control technology in which solid particles fill the interior or subsidiary cavity of the structure. When the controlled structure vibrates, the particles collide with, rub against, and impact each other or the cavity walls, which not only induces momentum exchange but also consumes the vibration power of the system, thus alleviating the structural vibration. Particle damping technology has been widely used for vibration control in the fields of aeronautical engineering and mechanical engineering. Skipor and Bain [¹⁴] conducted theoretical and experimental work leading to the application of an impact damper to the image-carrying cylinder of a web-fed printing press. Sims et al. [¹⁵] used particle dampers on a workpiece to mitigate chatter during milling. A new type of impact damper was invented by Olȩdzki [¹⁶], using for vibration control of long pipelines on a lightweight spacecraft. Duan et al. [¹⁷] introduced particle damping technology into the vibration control of helicopter rotor blades. Compared with traditional damping devices, particle dampers have some obvious advantages, including smaller changes to the primary structure, wider working frequency bands, less sensitivity to extreme temperature, longer durability and lower costs. Extensive basic research has focused on improving the application of particle damping technology to the vibration control of civil engineering structures. For example, the influence of particle damper cavity size on tuning, energy dissipation and equivalent damping characteristics was investigated by Liu et al. [¹⁸], and based on a large number of experimental results, a method of simulating the particle damper nonlinear characteristics with equivalent viscous damping was proposed. Hollkamp and Gordon [¹⁹] directly filled the holes of a single degree of freedom cantilever with damping particles and conducted a series of vibration mode tests. The influence laws of parameters such as the material, additional mass, filling ratio, filling position and the vibration amplitude of the structure on the vibration reduction performance of damping particles was studied, and it was found that the additional mass of particles, filling ratio, filling position and amplitude of the structure are the most important factors affecting the vibration reduction effect of damping particles. Lu et al. [²⁰–²²] conducted an experimental study of the damping effect of particle damping technology on the control of earthquake-induced structural vibration and proposed an approximate method for the numerical simulation of particle tuned mass dampers. Yan et al. [²³,²⁴] proposed a type of tuned particle damper (TPD), and a series of shaking table tests for the seismic control of a continuous viaduct showed that the TPD effectively reduced the acceleration response. Wang and Wu [²⁵] proposed a simulation method based on multiphase flow theory to evaluate particle damping characteristics. A numerical simulation of the frequency response functions of a plate with and without particle dampers under forced vibration was conducted, and the simulation results were compared with the experimental results. A new type of particle impact-damped dynamic absorber designed by Yang and Li [²⁶] was applied to the vibration control of a five-story frame structure and compared with a classic dynamic absorber of the same mass and stiffness. The results showed that the particle impact-damped dynamic absorber resulted in better damping of the frame structure subjected to random excitation than the classical dynamic absorber, with a wider working frequency. Egger et al. [²⁷] proposed a new type of distributed mass impact damper and applied it to the vibration control of a long, flexible, low-damping stay-cable from a real cable-stayed bridge. Full-scale experiments (free-decay tests) showed that the new device greatly increased the total damping ratios.

In summary, compared with traditional energy dissipation devices, well-designed particle dampers have good tuning and damping effects on the dynamic response of civil engineering structures. However, research on the application of particle damping technology to the vibration control of civil engineering structures has mostly employed single-component, medium-low-rise buildings or medium-small span bridges with high vibration frequency as the research objects. The damping effects of particle damping technology on long-period structures with low natural frequency, especially long-span, long-period bridges, have received less attention. In addition, traditional energy dissipation devices used in civil engineering structures have many inconveniences and limitations in vibration control applications on the high, flexible bridge towers (or piers) of long-period bridge structures such as suspension bridges, cable-stayed bridges, and rigid-frame bridges with high piers. However, the seismic responses of these bridge structures demand measures to reduce the dynamic responses of the high, flexible bridge towers. To investigate the vibration control and mechanism of particle dampers for long-period structures, a multilayer compartmental particle damper (MCPD) is proposed in this paper. A 1/20-scale test model of a typical long-period self-anchored suspension bridge with a single-tower was designed and fabricated, and the damping effects of the MCPD on the vibration control of the bridge model were examined under different excitation conditions induced by an earthquake-simulating shaking table system. The results provide a reference for the application of particle damping technology to the vibration control of self-anchored suspension bridges and other long-period structures.

Survey of the test for the bridge model

The prototype of the bridge model was an asymmetrical self-anchored suspension bridge with a single tower and double cables with a length of 370 m and a span arrangement of 35 m (first span) +135 m (second span) +165 m (third span) +35 m (fourth span). The prototype bridge main beam had a steel box girder structure with single-box double chambers and a calculated height of 3.0 m, whereas the bridge tower was a reinforced concrete structure with a single-box single chamber and a calculated height of 101.5 m. According to finite element modeling, the first vibration mode of the prototype bridge in the longitudinal direction was longitudinal drifting of the main beam, and the corresponding frequency was 0.215 Hz. The second vibration mode was longitudinal buckling of the bridge tower, and the corresponding frequency was 4.96 Hz. The bridge model was scaled to 1/20 of the prototype in geometric dimensions, considering the limitation of laboratory space and the actual capacity of the shaking tables. An elevation view of the bridge model is presented in Fig. 1(a). Furthermore, the design of the bridge model, the design and layout of the MCPD, the layout of the shaking tables and sensors, and the ground motion record selection are subsequently presented.

Design of the bridge model

The purpose of the experiment was to investigate the dynamic response of a self-anchored suspension bridge with a single-tower under seismic excitation in the horizontal direction with and without MCPDs. Therefore, a strength model with additional artificial mass ignoring the influence of gravity was adopted. Once the reduced scale of the bridge model and the materials were determined, the geometry, material and dynamic similarity coefficient of the bridge model was determined based on similarity theory [²⁸,²⁹], as shown in Table 1. The determination of the similarity coefficient S_ρe of equivalent mass density r_e needs to take into account its effect on the additional artificial mass and the time similarity coefficient S_T. If the similarity coefficient of the equivalent mass density is too low, the duration of the scaled ground motion will be too short, hindering the selection of the seismic wave and control of the precision of the tests. By contrast, if the similarity coefficient is too high, additional artificial mass will be needed, and the bridge model weight will exceed the actuation capacity of the shaking tables. Hence, considering both factors, the similarity coefficient of equivalent mass density S_ρe was set to 2.85. The additional artificial mass of the bridge model was calculated by Eq. (1):

(1)M_a{=M}_p{S}_L^{\mathrm{3}}{S}_{\rho e}-M_m,

where M_a is the additional artificial mass needed for the bridge model, M_p is the total mass of the prototype bridge, M_m is the total mass of the bridge model, and S_L is the reduced scale of the bridge model. The total additional artificial mass needed for the bridge model was 7.85 tons, according to Eq. (1) and Table 1. The additional artificial mass was arranged according to the principle of equal proportion and the mass distribution of the prototype bridge.

The bridge model mainly comprised the bridge tower, piers, main beam, cable system, bearings and other components. For the design of the complicated cross-sections, the equivalent stiffness, the same shape of the cross-section, and the similarity of the centroid position of the cross-section were based on the equivalent principle to avoid affecting the demand for the test. As with the prototype bridge, the longitudinal pot bearings are set between the main beam and the piers (or bridge tower). The details of the pot bearing are shown in Fig. 1(b). The force and cable curves of the main cable and boom were calculated and adjusted according to the similarity relation in Table 1. The details of the cable system are shown in Fig. 1(c). The bridge model was connected to the bends of the shaking tables with steel plate connectors, and the integral layout of the actual bridge model is shown in Fig. 1(d).

Design and layout of the MCPD

The main beam and bridge tower of self-anchored suspension bridges generally adopt a single box with a single chamber or a single box with a multi-chamber box cross-section. The main beam and bridge tower have sufficient, available cavity space for the application of particle damping technology vibration control without affecting the configuration of the bridge structures. According to conclusions [³⁰] from the observation of the movement patterns of particle groups under different vibration intensities if the particles in the cavity do not accumulate, the movement patterns are similar to the liquid in the cavity of a rectangular shallow TLD, and the energy dissipation mechanism is also similar. Therefore, a new type of MCPD is proposed in this paper, based on the damping control theory and design method of a rectangular shallow TLD [³¹], and its schematic diagram is shown in Fig. 2. L, D, and H are the length, width and total height of the damper cavity, and K_L, K_D and K_H are the connection stiffness between the MCPD and the structure.

In addition, studies [³²,³³] have shown that, the connected rigidity between particle dampers and the controlled structure has obvious influences on the particle damper’s vibration control effect. Thus, in the tests of this paper, the MCPD cavity was installed at the main beam and the bridge tower through the steel connectors that specially designed. According to the studies [²⁰,²⁴], the ratio R_f of the MCPD’ natural frequency to the natural frequencies of the controlled structure will be designed within a range of 0.9 to 1.1, and in this paper it was designed to be 1.0. Meanwhile, combined with the application parameters of TMD and TLD [³⁴], the total additional mass ratio R_m of the MCPD will be designed to be less than 5% to reduce the effect of MCPD on the model bridge. In this paper, the additional mass ratio of the MCPD was designed to be 2%, 3%, 4%, and 5% to study the influence law of the additional mass ratio. Then, the stiffness K of the steel connectors can be calculated through this formula f = {\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}}\sqrt{K/M} after the fundamental frequency f and weight M of the MCPD is determined.

If the bending rigidity of the MCPD cavity is sufficiently large, the MCPD can be regarded as multiple superposed monolayer compart-mental particle dampers along the height direction, and the particle pile height is the sum of the particle pile height in each layer. If the particles in the cavity do not accumulate or accumulate to a lesser height, the design parameters of the MCPD can be calculated by Eq. (2) [³⁴]

(2){\omega }_i={\left(g{\displaystyle \sum _{i\mathrm{=1}}^n{h}_i}\right)}^{\mathrm{1}/\mathrm{2}}{\displaystyle \frac{(\mathrm{2}i-\mathrm{1})\mathrm{\pi }}{l}}\left[\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\mathrm{6}}}{\left({\displaystyle \sum _{i\mathrm{=1}}^n{h}_i}\right)}^{\mathrm{2}}{\left({\displaystyle \frac{(\mathrm{2}i-\mathrm{1})\mathrm{\pi }}{l}}\right)}^{\mathrm{2}}\right] ,

where {\omega }_i is the vibration frequency of particles in the MCPD cavity, which is equal to the natural frequency of the MCPD and the fundamental frequency of the bridge model in the controlled direction, in this paper, g is the acceleration of gravity (g), h_i is the height of the particles in layer i(m), which is approximately equal to the particle’s radius r, and the length l can be gained based on Eq. (2), which is approximately equal to the gap l between the damping particle group and the cavity walls. The MCPD design steps are as follows:

(1) Determine the natural frequencies of the controlled structure. In this paper, the first mode along the longitudinal direction of the bridge model is obtained;

(2) Using the obtained natural frequency of the controlled structure and combined with the design methods of the Tuned Particle Damper [²⁰,²⁴], the ratio of the MCPD’s natural frequency to the natural frequencies of the controlled structure will be designed within a range of 0.9 to 1.1. And in this paper it was designed to be 1.0;

(3) Combined with application parameters of TMD and TLD [³²], the total additional mass ratio of MCPD will be designed to be less than 5% to reduce the effect of MCPD on the model bridge. In this paper, the additional mass ratio of the MCPD was designed to be 2%, 3%, 4%, and 5% to study the influence law of the additional mass ratio;

(4) According to the additional mass ratio and the natural frequency of MCPD, the horizontal and vertical stiffness of the MCPD (expressed as K_L, K_D, and K_H) will be determined. In this paper, the authors focus on the seismic control of the model’s first mode, the horizontal stiffness K_L corresponding to different additional masses was designed to be 7.07 kN/m, 10.62 kN/m, 14.14 kN/m, and 17.69 kN/m, respectively. And the horizontal stiffness K_D and the vertical stiffness K_H was designed 10 times as large as K_L;

(5) Considering the available space and the mass distribution of the controlled structure, the maximum usable geometry L, D, and H for the MCPD will be determined. In this paper, the MCPDs were installed to the first span, the fourth span, and the top of the bridge tower. And the usable space parameters of the MCPDs are shown in Table 2;

(6) According to the TLD’s shallow water vibration theory [³²], when the damping particles in the MCPD do not accumulate, the vibration law of the particles is similar to the shallow water in the TLD, and the vibration frequency of the particles in the cavity will approximately satisfy the Eq. (2), the gap l between the damping particles and the cavity walls will be obtained.

In the tests, the cavity of the MCPD was installed to the bridge model through the specially designed steel connections, which were installed between the bridge model and the cavity through welding. The dimensions of the connectors were designed to make the frequency ratio of the MCPD’s natural frequency to the controlled model fundamental frequency near to 1.0.

For the bridge model in this paper, the second and third span of the prototype bridge main beam are steel box girders, while the first and fourth span main beams are reinforced concrete box beams, which is the cable anchorage zone of self-anchored suspension bridge. Compared with the steel box girder, the self-weight of the reinforced concrete box beams is relatively larger, its self-weight accounts for 73.7% of the total weight of the whole main beam. Therefore, the effective mode-participation mass of the main beam first vibration mode (the longitudinal drift of the main beam) is mainly concentrated on the first and fourth span. Meanwhile, the dynamic responses at the ends of the main beam and the top of bridge tower are most significant under seismic excitation. Moreover, for the self-anchored suspension bridge, the dynamic response of the bridge tower has a non-negligible influence on the dynamic response of the main beam. Furthermore, considering the seismic control of the suspension bridge in this paper is mainly about the first mode, so the MCPDs were installed to the first, the fourth span and the top of bridge tower during the tests. The layout of the particle dampers is shown in Fig. 3.

Layout of the shaking table array and sensors

The tests were performed at the civil engineering structure test center of Beijing University of Technology (BJUT, Beijing, China) using a shaking table array system. The system consisted of eight independent shaking tables that could be used singly or in combination depending on the requirements of the experiment. Each shaking table had three horizontal actuators with three translational degrees of freedom (horizontal X, horizontal Y, and rotation). The performance parameters of the shaking table system are shown in Table 3. The layout of the shaking tables in the experiment was based on the span-arrangement of the bridge model, as shown in Fig. 4 and eight shaking tables and 20 actuators were used. Shaking Tables #1, #2, #7, and #8 correspond to the positions of piers #1, #2, #4, and #5, respectively, whereas the combined shaking table composed of Tables #3–€#6 corresponds to the position of pier #3 (bridge tower).

The experiment mainly measured the acceleration, displacement and deformation of the bridge model. The accelerometers, displacement meters and strain gauges were installed at the key positions of the main beam, bridge tower and piers. The layout and details of the installed sensors are given in Fig. 5. (1) Three 991B displacement sensors with a measuring range of ±200 mm and a frequency pass-band of 0.05 Hz–100 Hz, as shown in Fig. 5(a), were installed on the deck of the main beam and at the top of bridge tower to measure the main beam and bridge tower displacement. (2) Nine pull wire displacement meters with a measuring range of±100 mm, as shown in Fig. 5(b), were installed between the bottom and top plates of the pot bearing to measure the bearing displacement. (3) Nine strain sensors with a measuring range of 0.1–20000 me and a frequency pass-band of 0.01 Hz–50 Hz, as shown in Fig. 5(c), were installed at the bottom of the pier and bridge tower to measure their deformation. (4) Eleven 941B accelerometers with a measuring range of ±2g and a frequency pass-band of 0.05 Hz–100 Hz were installed on the deck of the main beam and the bridge tower to measure the acceleration, as shown in Fig. 5(d). Furthermore, the signal of each sensor was obtained using an Integrated Measurement and Control dynamic data-acquisition system.

Ground motion record selection and shaking table test cases

A self-anchored suspension bridge with a single-tower is a semi-flexible, long-period structure, and based on the site classification (class II) and seismic fortification intensity (7 degrees) [³⁵] of the prototype bridge, two actual ground motion acceleration time series were selected from the PEER (Pacific Earthquake Engineering Research) ground motion database: the EL-Centro wave and the ILA005 wave. In addition, according to the design response spectrum of the prototype bridge specified in the Chinese seismic code of highway bridges (JTG/T B02-01-2008, Ministry of Transport of the People’s Republic of China 2008), an artificial ground motion acceleration time series, Arti, was synthesized. The 90% energy durations of the three waves (that is, the time required for the energy of ground motion to increase from 5% to 95% of the total energy) are 24.5 s, 26.7 s, and 22.8 s, respectively. According to the code for the seismic design of buildings (GB 50011-2010, Ministry of Housing and Urban-Rural Development of the People’s Republic of China) [³⁶] and the similarity relations shown in Table 1, the peak accelerations of these three seismic waves were adjusted to three intensity levels: EA1= 0.035g, EA2= 0.1g, and EA3= 0.22g, respectively (corresponding to the design acceleration peak under a seismic fortification intensity of 7 degrees and exceeding probabilities of 63%, 10%, and 2%, respectively). The duration and amplitude of the ground motion were scaled and adjusted according to the similarity relations of time and acceleration shown in Table 1. The absolute durations of the scaled ground motions were 4.1 s, 7.4 s, and 8.0 s, respectively. The ground motion acceleration time series used in the tests and the corresponding Fourier amplitude spectra are given in Fig. 6 (EA2= 0.71g). As shown in Fig. 6, the spectral characteristics of the three ground motions used in the experiment are quite different: the energy of the EL-Centro wave is concentrated in the mid-high frequency band; the energy of the ILA005 wave is concentrated in the low-frequency band; and the energy of the artificial wave is distributed over a wider frequency band. These variations allowed the sensitivity of the MCPD to be assessed under seismic excitations with different spectral characteristics. Finally, after adjustment and scaling, the ground motions were imposed in the longitudinal direction of the bridge model, and a series of shaking table tests were performed sequentially as specified in Table 4 to investigate the damping effect of the MCPD. Factors influencing the shock-absorption effects of the MCPD were evaluated, including the additional mass ratio of particles, the spectrum characteristics and the intensity of excitation.

Comparison and analysis of the test results

The main objective of this paper was to investigate the damping effect of the MCPDs on the vibration control of long-period bridge structures. Because the bridge model exhibited long-period characteristics only in the longitudinal direction, ground motions were only imposed in the longitudinal direction in the test. The longitudinal displacement and acceleration of the main beam and the bridge tower with and without dampers were empirically analyzed.

Comparison of the dynamic characteristics of the bridge model with and without the MCPD

The vibration mode of the bridge model under white noise excitation along the longitudinal with the MCPDs was identified by the frequency domain decomposition (FDD) method [³⁷] with additional mass ratios of 0% (without MCPD), 2%, 3%, 4%, and 5%. The singular value decomposition (SVD) of the power spectra for the acceleration records from the main beam and bridge tower of the bridge model without the MPCDs is presented in Fig. 7. As shown in Fig. 7(a), the first 4 modes of the bridge model vibration were identified. In addition, as shown in Fig. 7(b), the vibration of the bridge model was mainly based on the first and second vibration modes, i.e., the longitudinal drift of the main beam and the longitudinal buckling of the bridge tower, with corresponding frequencies of 1.93 Hz and 4.96 Hz, respectively. The frequency and the corresponding modal damping ratio of the first vibration mode in the longitudinal direction of the main beam and bridge tower are given in Table 5. The fundamental frequencies of the longitudinal vibration for the main beam and the bridge tower decreased with increasing additional mass ratio of the particles, and the rate of decrease in amplitude slowed as the additional mass ratio of the particles increased. For additional mass ratios of the MCPDs of 2%, 3%, 4%, and 5%, the fundamental frequencies of the longitudinal vibration of the main beam decreased by 2.59%, 4.14%, 5.69%, and 7.77%, respectively, and the fundamental frequency of the longitudinal vibration for the bridge tower decreased by 2.82%, 4.05%, 5.48%, and 7.30%, respectively. These results demonstrate that the MCPDs have a significant tuning effect on the long-period self-anchored suspension bridge, mainly due to the characteristics of the dynamic response for long-period structures. Under seismic excitation, long-period structures often produce larger displacement, resulting in greater movement, collision, rubbing, and impact of the particles in the damper cavity. These motions of the particles in turn play a role in tuning the main beam of the bridge model.

As shown in Table 5, the damping ratio of the longitudinal vibration for the main beam and bridge tower without the MCPDs were relatively small, with values of only 1.72% and 1.85%, respectively. The equivalent damping ratio of the longitudinal vibration for the main beam and the bridge tower with MCPDs (corresponding to the first two vibration modes of the bridge model) were significantly improved. When the additional mass ratios of the particles were 2%, 3%, 4%, and 5%, the equivalent additional damping ratio of the MCPDs were 1.55%, 2.71%, 4.37%, and 4.52%, respectively, for the main beam and 1.61%, 2.54%, 4.11%, and 4.28%, respectively, for the bridge tower.

It should to be pointed out that the control effect of the MCPD includes tuning effects and energy dissipation effects. On the one hand, the MCPD play tuning effects by the change of the frequency ratio between the MCPD and the controlled structure (this is similar to the control mechanism of Tuned Mass Damper); On the other hand, the MCPD dissipates the vibration energy by the friction and collision between the particles, the particles and the cavity, and provide a certain equivalent damping effects. Meanwhile, the relative motion and collision between the particles and the cavity can also play tuning effects. The MCPD’s equivalent damping not only includes the tuning effect between the MCPD and the controlled structure, but also includes the tuning effect and the energy dissipation effects provided by the friction and collision between the particles, the particles and the cavity. Therefore, the equivalent damping ratio of the MCPD-bridge model system increases after the installation of MCPD. Referring to the existed research results about the Dual-layer Tuned Mass Damper [³³], the well-designed MCPD will have good robustness. And the parameter optimization method of MCPD’s will be conducted in further.

For long-period bridge structures, the increase in damping of the structure increase can effectively restrain the excessive displacement response, allow the dynamic response to attenuate more quickly under seismic excitation, and prevent damage to the bridge structure caused by excessive displacement, such as impact damage, bearing failure, and beam falling. These results indicate that the application of the MCPD is a viable method for vibration control of a self-anchored suspension bridge with a single tower and other long-period structures.

Comparison of the seismic responses of the bridge model with and without the MCPD

For the tests of the bridge model with and without the MCPD (with different particle additional mass ratios), the EL-Centro wave, ILA005 wave and artificial wave with peak accelerations of EA1= 0.25g, EA2= 0.71g, and EA3= 1.52g, respectively, were input along the longitudinal bridge, and the damping effect of the particle dampers with different additional mass ratios and excitation intensities on the test bridge model were investigated. During the experiments, no particle accumulation was observed in the cavity of the dampers under any test conditions. Compared with the traditional single-layer particle dampers, the MCPDs effectively solved the problem of reduced damping effects due to the accumulation of particles. Figures 8 and 9 show the time history of longitudinal displacement and acceleration of the bridge model main beam and bridge tower with and without the MCPD, respectively. The additional mass ratio was 4%, and the intensity of the wave was EA2.

According to Figs. 8 and 9, first, the MCPD exhibited positive damping under the excitation of all select seismic waves in the tests. No adverse effect of amplifying the structural response was observed, indicating that the particle damper exerted a damping effect over a wider working frequency band than the traditional tuned mass damper. Second, in the early stage of earthquake input, the effect of the MCPD on the dynamic response of the test bridge model was not obvious. Initially, the energy dissipation of the particle dampers due to collision, friction and impact between the particles and between the particles and the cavity is weak, and thus the tuning effect of the damper is also limited. Finally, the MCPD effectively reduced the dynamic response of the main beam and bridge tower, thereby demonstrating that the particle damper can be applied to control low-frequency vibration of a long-period bridge structure. These results are similar to the conclusions of a shaking table experimental study on a steel frame structure with particle-tuned mass dampers by Lu et al. [³⁸].

To measure the damping of the particle dampers more accurately, the root mean square (RMS) of the acceleration response was introduced as an index to reflect the strength of the structural acceleration response:

(3)A_{rms}\mathrm{=sqrt}\left({\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^n{\ddot{x}}^{\mathrm{2}}}\right).

Then, the decreasing ratio \betaand the average decreasing ratio \overline{\beta }of RMS acceleration can be defined as follows

(4)\beta =\left(A_{rms}^p-A_{rms}^d\right)/A_{rms}^p,

(5)\overline{\beta }={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}({\beta }_E+{\beta }_I+{\beta }_A),

Where \ddot{x} is the acceleration value of the bridge model at a certain time, A_{rms}^pand A_{rms}^d are the RMS values of the acceleration with and without the MCPD, respectively, and {\beta }_E, {\beta }_I and {\beta }_A are the decreasing ratio of the RMS acceleration response of the EL-Centro, IAL005 and artificial waves, respectively.

The peak decreasing ratio was again adopted as an index to evaluate the damping effect of the MCPDs on the displacement response of the bridge model. The peak displacement decreasing ratio R_d and the average decreasing ratio \overline{R} of the displacement response were calculated as follows:

(6)R_d=\left(D_M^p-D_M^q\right)/D_M^p,

(7)\overline{R}\hspace{0.17em}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}(R_E+R_I+R_A),

where D_M^pand D_M^qare the peaks of the displacement responses of the bridge model with and without MCPDs, respectively, and R_E, R_I and R_A are the peak displacement decreasing ratio of the EL-Centro, ILA005 and artificial waves, respectively.

Figures 10 and 11 show the peak displacement and RMS acceleration decreasing ratio of the MCPDs on the bridge model, respectively. The additional mass ratios were 2%, 3%, 4%, and 5%. The intensities of the waves were EA1= 0.25g, EA2= 0.71g, and EA3= 1.52g. As shown in Figs. 10 and 11, the MCPDs with different additional mass ratios had positive damping effects on the displacement and acceleration of the bridge model. For all select ground motions with different spectral characteristics, the MCPDs exhibited a similar damping control law:

(1) The damping effect of the MCPD first increases and then decreases with increasing additional particle mass. Taking the EA2 level seismic excitation as an example, when the additional mass ratio increased from 2% to 5%, the average decreasing ratio of the main beam displacement for the bridge model (data in parentheses are the decreasing ratio for the bridge tower) with the MCPDs were 16.6% (19.9%), 25.2% (23.7%), 36% (33.4%), and 34.1% (31.1%). The average decreasing ratio of the RMS acceleration for the main beam (bridge tower) were 19.6% (16.5%), 25.9% (21.8%), 29.1% (28.4%), and 25.7% (22.9%). The results show that the MCPD damping effect is the most significant when the additional mass ratio is 4%. In this case, the maximum decreasing ratio of the peak displacement for the main beam and bridge tower were 46.4% and 41.6%, respectively (as shown in Fig. 10), and the maximum decreasing ratio of the RMS acceleration for the main beam and the bridge tower were 42.7% and 38.3%, respectively (as shown in Fig. 11).

According to the test results, when the additional mass of the particles is relatively small (R_m= 2%), the distribution of particles in the cavity of the damper is sparse, and the effects of the collision, friction and impact of the particles with each other and with the cavity walls were not obvious. As a result, the damping effect was weaker. By contrast, when the additional mass of the particles is larger (R_m = 5%), the distribution of particles in the damper cavity is too dense, which limits the movement of the particles relative to each other and the cavity walls. As a result, the ability of the damper to dissipate energy is reduced. Thus, when the additional mass ratio of particles increased from 4% to 5%, the damping effect of the MCPDs did not increase but decreased slightly. In conclusion, there is an optimum additional mass ratio of particles according to the damping effect, economy and cost performance.

(2) The damping effect of the MCPD on the dynamic response of the bridge model increases significantly with increasing excitation intensity. Taking the additional mass ratio of 4% as an example, under the EA1, EA2 and EA3 seismic excitation levels, the average decreasing ratio of the main beam peak displacement for the bridge model (data in brackets below are the decreasing ratio for the bridge tower) with the MCPD were 17.5% (21.4%), 36% (33.4%), and 42.9% (39.3%), and the average decreasing ratio of the RMS acceleration responses of the main beam (bridge tower) were 19.2% (21.4%), 29.1% (28.4%), and 36.9% (35.2%). These results indicate that the damping effect of the MCPDs was worse under the EA1 seismic excitation level, and the decreasing ratio of the peak displacement and RMS acceleration of the main beam and bridge tower were less than 20%. However, due to the weak seismic response of the bridge model, the safety of the structure under seismic excitation was maintained. However, the damping effect of the MCPDs was markedly increased under the EA3 seismic excitation level, and the decreasing ratio of the peak displacement and RMS acceleration for the main beam and bridge tower were greater than 35%. Under severe seismic excitation, the damping effect of the MCPDs was acceptable and steady.

(3) In addition, as shown in Figs. 10 and 11, the damping effect of the MCPD is also influenced by ground motion and the dynamic characteristics of the structure itself. When the ILA005 wave was input into the bridge model, the damping effect of the MCPDs with the same R_m on the main beam was greater than that on the bridge tower. When the EL-Centro and artificial waves were input, the damping effect of the MCPDs with the same R_m was greater on the bridge tower than on the main beam. Overall, the damping effect of the MCPDs on the main beam was greater than that on the bridge tower, indicating that the MCPDs had greater damping control for the low-frequency dynamic responses of the long-period self-anchored suspension bridge than for the high-frequency dynamic responses. For the bridge model in this paper, the first vibration mode is the longitudinal drift of the main beam, and the corresponding fundamental frequency of the vibration is lower (w₁= 1.93 Hz). Under the excitation of the ILA005 wave with abundant low-frequency components (shown in Fig. 6(b)), the main beam has a larger displacement response (shown in Fig. 8(b)). The longitudinal buckling of the bridge tower is the second vibration mode of the bridge model, and the corresponding vibration frequency is higher (w₂ = 4.96 Hz). Under the excitation of the EL-Centro and artificial waves with abundant high-frequency components (shown in Figs. 6(a) and 6(c)), the bridge tower has a larger acceleration response (shown in Figs. 9(a) and 9(c)). The greater the dynamic response of the structures (or components) with the MCPD, the stronger the relative movement of particles with each other or the cavity walls and the more significant the energy dissipation. Thus, the shock energy of the structures with MCPDs can be dissipated to a greater degree, and better damping can be achieved.

In summary, long-period bridge structures generally contain one or several high, flexible or semi-flexible towers or piers. The dynamic responses of the main beam and bridge tower (or pier) of such bridges are significant, and there are some interactions. These features require new shock absorption measures for long-period bridges to control the dynamic responses of the main beam and bridge tower (or piers) simultaneously. The well-designed MCPD has a significant restraining effect on the first two modal low-frequency dynamic responses of the long-period self-anchored suspension bridge. The MCPD can effectively reduce the low-frequency dynamic responses of the bridge model, especially the peak of the displacement responses. The damping effect of the MCPDs significantly increases with the increased dynamic response of the controlled structures, which is of great significance and advantageous for restraining excessive dynamic responses under severe earthquakes. Therefore, particle damping technology can be applied to the seismic control of self-anchored suspension bridges and other long-period structures.

Compared with traditional energy dissipation devices, particle dampers have not shown significant advantages for vibration control of the bridge partial position [³⁹]. However, considering the characteristics of particle dampers described in the introduction, particle dampers can be conveniently installed on bridges in locations such as the internal cavities of the box girder and the box cross-section bridge tower. These installation points permit a greater degree of distributed damping while avoiding the disadvantage of local damage and destruction of the main structure due to the installation of traditional vibration absorbers. In addition, particle dampers can significantly reduce the maintenance costs of vibration absorbers in the life cycle of the structure. Furthermore, the MCPD also has good advantages for in-service bridges or building structures that need seismic strengthening.

Conclusions

To introduce particle damping technology into the seismic control of long-period engineering structures, a 1/20-scale self-anchored suspension bridge with a single tower was designed and fabricated. In addition, an MCPD suitable for the bridge model was designed and fabricated. A series of bridge model shaking table tests with and without the dampers was performed, and the following conclusions were obtained.

1) The dynamic responses of the main beam and bridge towers (or piers) of the long-period bridge structures with high, flexible or semi-flexible towers (or piers) are generally conspicuous. In addition, the seismic response of the bridge tower influences the seismic response of the main beam. The MCPD can be conveniently installed on the bridge tower to control the seismic responses of the main beam and bridge tower simultaneously.

2) No particle accumulation was observed during the test process. The MCPDs proposed in this paper can effectively solve the problem of reduced damping due to the accumulation of particles when the additional mass is relatively large, can effectively increase the damping of the bridge model, and have a significant tuning effect on the fundamental frequency of the longitudinal vibration.

3) The MCPD can greatly reduce the longitudinal peak displacement and RMS acceleration of the bridge model main beam and bridge tower. The maximum decreasing ratio of peak displacement for the main beam and the bridge tower were 46.4% and 41.6%, respectively. The maximum decreasing ratio of RMS acceleration for the main beam and bridge tower were 42.7% and 38.3%, respectively.

4) The low-frequency dynamic responses, especially the displacement response, of the long-period structures under severe seismic excitation can be well controlled by well-designed particle dampers. The particle damping technology can be applied to the seismic control of long-span bridges and other long-period structures, and the studies in this paper can provide a reference for the vibration control design of civil engineering structures.

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