1. Introduction
In the past, there have been significant advancements in the fields of transportation engineering safety and civil engineering safety [
1,
2,
3,
4]. There is significant concern about the seismic safety of structures built in earthquake-prone regions [
5,
6]. To mitigate the negative impacts of seismic activities on human life and productivity, numerous technologies concerning vibration control have been extensively developed. Vibration control techniques play a significant role in reducing displacement and force response when subjected to seismic [
7], ice [
8], wind [
9], and other types of loads. Recent trends in civil engineering have led to a proliferation of studies focusing on the damping effects of spring-mass-damper systems [
10,
11,
12,
13,
14,
15,
16]. The approach of [
17], namely the tuned mass damper (TMD), first proposed by Frahm, may offer a viable solution. Fujino Y. et al. conducted a study on the TMD system [
18,
19] and used a combination of vertical plates and TMDs to effectively control the vibration of the Trans Tokyo Bay Crossing Bridge [
20]. After conducting research on the use of tuned mass dampers (TMDs) in a five-story reinforced concrete building, Mazzon et al. concluded that the middle-rise structure with TMDs would not need repair intervention after the earthquake [
21]. There has been keen interest among researchers in TMD-based derivative devices, with the multiple tuned mass dampers (MTMD) emerging as a prominent example. Significant advancements have been achieved in studying the impact of structural frequency change on MTMD, as seen in the studies conducted by academics [
22,
23,
24,
25]. There is very extensive literature on the topic of the Inerter, which is a mechanical device with two terminals [
26,
27,
28,
29,
30,
31,
32,
33]. This device serves the functions of inertia amplification and damping enhancement by transforming the linear motion of its moving terminal into the rotational motion of a flywheel, thereby generating significant inertia [
34,
35]. According to Tang et al.’s research [
36], the tuned mass damper with Inerter (TMDI) can dissipate structural kinetic energy more efficiently during seismic events. The optimally designed Tuned Inerter Damper (TID) developed by Shen et al. effectively suppresses structural displacement responses across a wide range of natural periods [
37]. Furthermore, Li et al. introduced the dual-layer multiple tuned mass damper (DMTMD), which offers improved damping efficiency. However, one drawback of the DMTMD is that its secondary mass experiences an excessively long stroke [
38].
An important constraint on all the work discussed in this area is the excessively long stroke of the secondary mass present in both TMDs and TMD-derived devices. This limitation substantially restricts their practical application in engineering, particularly in terms of displacement and force along the longitudinal direction of the bridge. The approaches of [
39] may offer a viable solution to tackle this issue. This innovative device comprises a transmission mechanism (specifically, a rack-and-pinion system), a spring-connected secondary mass, and a damping component. The transmission device amplifies the relative displacement between the damper and the primary structure, thereby augmenting its energy absorption and vibration mitigation efficacy while simplifying the technical complexities. Nonetheless, there are two things that need to be improved in AOD.
Improvement needed 1: Installing the secondary structure of AOD in actual bridge applications, which consists of secondary mass, secondary damping, and secondary spring, is challenging. Additionally, the amplification coefficient of the secondary damping and secondary spring can be increased.
Improvement needed 2: As a generalized single-degree-of-freedom (SDOF) system, the AOD exhibits a remarkable sensitivity to the equivalent natural frequency of the system, especially in the near-resonance region [
40].
Based on the fundamental structure of the Accelerated Oscillator Damper, this study employs two primary strategies: firstly, providing sufficient space to install the secondary mass, spring, and damper by connecting the secondary system to the primary structure (main girder), and simultaneously strengthening the property of the original AOD system; and secondly, introducing nonlinear devices to mitigate substantial vibration responses within the near-resonance range. The work presented here provides one of the new investigations into how to reduce the response of bridges to earthquake action. Furthermore, this study improves our understanding of the methodology involved in ice- and wind-induced vibration, such as Yellow River.
The organization of this paper is as follows: In
Section 2, a strengthened AOD design is introduced by connecting the secondary spring and damper to the primary structure. Additionally, the motion equations for the strengthened AOD with linear springs (SAOD-LS) are derived. The conceptual design and mechanical characteristics analysis of the nonlinear spring device (NSD) is proposed in
Section 3.
Section 4 describes the performance of both SAOD-LS and the strengthened AOD equipped with the NSD (SAOD-NSD), where comparisons with the traditional AOD are also given. In
Section 5, we apply the proposed SAOD-LS and SAOD-NSD to an actual bridge project. Conclusions are drawn in
Section 6.
The research methods employed, and the data analysis procedures utilized in each section, are outlined below. In
Section 2, the Lagrange Dynamic method is used for mathematical derivation, and the finite element (FE) analysis is employed to validate the motion equations of the SAOD-LS system. We utilize the response comparison approach to analyze the shock absorption data of SAOD-LS. In
Section 3, the mechanical properties of the NSD are quantitatively derived and validated using the FE method in
Section 3. In
Section 4, we gain the response of various cases using the FE method. The comparison of the response amplitude shows that the SAOD-NSD has significant damping performance in the near resonance region.
Section 5 of the study involves modeling an actual bridge project using the FE approach. The response data obtained are then processed using programming. The results demonstrate the outstanding practicality and robustness of the the SAOD-LS and SAOD-NSD methods.
4. Comparison of Shock Absorption Performance between SAOD-NSD and SAOD-LS
Figure 5 presents the mechanical model diagram of the SAOD-NSD, which is created by replacing the NSD with the constant value
k2 of SAOD-LS. The only distinction between SAOD-NSD and SAOD-LS would be the secondary spring
k2. In the SAOD-LS,
k2 is an ideal spring with a certain stiffness coefficient, but in the SAOD-NSD,
k2 refers to the NSD presented in
Section 3.
To provide a clearer understanding of the exceptional damping abilities of the SAOD-NSD, we have selected Example 3 for illustration in this section. The parameters for the SAOD-LS and external excitation are set based on the values listed in
Table 2.
By differentiating Equation (20) with respect to
z, we can derive a function that describes the NSD’s equivalent stiffness, which corresponds to a specific
z. This enables us to ascertain the equivalent stiffness for a particular set of geometric parameters (specifically,
l1,
l2, and
l3) and spring constants (
kA and
kB). Referring to Example 3,
Table 3 outlines the NSD parameters as mentioned. The NSD equivalent stiffness is calculated to be 1389.4 kN/m, which coincides with the SAOD-LS
k2 in
Table 3.
Using
Table 2 and Equation (7), it can be determined that the natural frequency of SAOD-LS (
fSAOD-LS) is 4.53 Hz. In Example 3, the value of
fload falls within the range of 0.95
fSAOD-LS to 1.05
fSAOD-LS, which is classified as the near-resonance region according to reference [
45].
The value
x1max is defined as the peak value of the
m1’s time-history displacement response to external excitation.
Figure 6 compares the
x1max curve between the SAOD-NSD and the SAOD-LS under 50 s harmonic loads at different frequencies (i.e.,
fload). In this instance,
x1max is calculated by the FE method. During the FE modeling process, the NSD is integrated into a spring element that exhibits nonlinear behavior depicted in Equation (20). The other modeling methods remain consistent with those used in Example 1.
Figure 6 clearly illustrates that at the exact resonance point, the
x1max of the SAOD-NSD is 22 mm, notably less than the 162 mm observed in the SAOD-LS. Throughout the entire near-resonance region, the
x1max of the SAOD-NSD remains below 40% of that of SAOD-LS. This indicates that the NSD exhibits exceptional damping effectiveness in the near-resonance region.
Similar to many SDOF dampers, time-history analysis reveals a distinct resonance phenomenon when the parameters of the SAOD-LS and external stimulation remain constant. In addition, despite the SAOD-NSD demonstrating resonance-like characteristics, its resonance peak can be significantly reduced. This study utilizes Example 4, which involves three distinct cases [
40].
Case 4 is the classic SDOF mass-spring-damper system, as depicted in Figure 1.6.1a in [
46]. In Case 4, the mass (
m1) is set at 29,485 kg, the spring constant (
k1) at 3800 kN/m, and the damping coefficient (
c1) at 3740 Ns/m. Please refer to
Figure 7 for schematic sketches depicting Case 5 and Case 6. The main structural forms of Case 5 and Case 6 are exactly the same, except for the fact that
k2 and
k3 in Case 5 are LSs (i.e., linear springs), whereas in Case 6, they are NSDs (i.e., the nonlinear spring devices).
In Case 5, the values of
k2 and
k3 are set at 1389.4 kN/m. As for Case 6, the NSD parameters
k2 and
k3 are consistent with the values specified in
Table 3. Additionally, the external excitation in Example 4 is represented by the formula
38sin(2π·1.81t). For further details on the other parameters of both Case 5 and Case 6, please refer to
Table 4.
The natural frequencies of Case 4 and Case 5 are both 1.81 Hz, thereby satisfying resonance. The main objective of Case 6’s structure design is to minimize the impact of the force change of NSD during both the tension and compression stages. The organization of Case 5 facilitates the comparison of Case 6. The NSD’s equivalent stiffness in Case 6 is identical to that of Case 5, making both cases comparable.
Figure 8a displays the time-history displacements of
m1 in Case 4, Case 5, and Case 6, specifically for the first 12 s. The FE technique is used to calculate the results. We use NSD for a nonlinear-spring element. The following modeling steps are identical to those in Example 1.
Figure 8a demonstrates that in Case 5, the SAOD-LS exhibits a notable damping effect at resonance, resulting in a reduction in the displacement peak at resonance. In comparison, the peak resonance of the SAOD-NSD in Case 6 appears lower. Meanwhile, the time-history response indicates that the SAOD-NSD exhibits a resonance-like phenomenon, but with a significant reduction in its response peak.
Subsequently, aiming to facilitate a more in-depth investigation of the damping mechanism of the NSD in comparison to LS, the amplitude spectrum is employed for analyzing both the frequency components and the amplitude of the response. The
m1 displacement responses of Case 4, Case 5, and Case 6 during the first 300 s are subjected to Fourier transformation. We analyze the amplitude spectra of the filtered data and present the results in
Figure 8b.
Upon examination of
Figure 8b, it is evident that the frequency of 1.81 Hz corresponds to both the natural frequency of the structure and the frequency of external excitation in Case 4 and Case 5. Notably, the peak value at Point B (Case 5) is considerably lower than that at Point A (Case 4), corroborating the results obtained through time-history analysis.
Given that the equivalent stiffness of the NSD in Case 6 is equal to the stiffness of the LS in Case 5, it can be inferred that the frequency at Point D in Case 6 is also 1.81 Hz. Additionally, the NSD contributes to Point C, which displays a greater amplitude than Point D. When comparing Case 5 and Case 6, it becomes evident that the NSD not only consumes low-frequency energy but also demonstrates remarkable energy dissipation capacity.
To gain a deeper understanding of how amplitude spectra evolve over time, the evolutionary power spectral density (EPSD) method has been adopted. Since both Case 4 and Case 5 in Example 4 are SDOF structures, their EPSD results exhibit substantial similarity. Therefore, there is no need to analyze them separately in this context. Consequently, we have generated a graph illustrating the EPSD outcome specifically for Case 6 over the initial 420 s, as depicted in
Figure 9.
By investigating
Figure 8b and
Figure 9 together, it becomes evident that the highest energy consumption at Point C, caused by NSD in
Figure 8b, is more effective than that at Point D at the beginning. However, the peak energy consumption at Point C decreases over time. Nonetheless, in practical seismic problems, the displacement peak typically occurs during the initial several minutes of vibration, known as the transient term. Therefore, the practical application of the NSD to vibration concerns remains valuable.
In summary, the SAOD-LS demonstrates a significant capacity to reduce the peak amplitude of the response. As for the SAOD-NSD, although the NSD cannot fully eliminate the resonance effect, it still manages to notably diminish the maximum amplitude of the response.
5. Damping Performance of SAOD-LS and SAOD-NSD in an Actual Bridge Example
To illustrate the profound effect of the SAOD-LS and SAOD-NSD in mitigating vibrations of real bridge structures, this section employs a six-span continuous beam bridge from the Puwan 14th Bridge located in the Puwan New Area of Dalian as Example 5. This bridge is a twin-deck concrete continuous beam bridge that plays a vital role in Puwan New Area’s transportation network.
As shown in
Figure 10, Example 5 has a span arrangement of 70 m + 120 m + 120 m + 120 m + 70 m, summing up to a total length of 500 m. The piles, piers, and main girder were constructed using concrete grades C30, C40, and C50, respectively. The seismic fortification intensity for Dalian stands at 8 degrees, with a corresponding design seismic acceleration peak value of 0.2 g [
47,
48,
49].
The El Centro wave, scaled to a magnitude of 2.1 times, was selected as the external excitation for our analysis. We conducted a comparative study on the displacement of the main girder in three different cases: the original bridge (labeled as Case 7), the bridge equipped with two identical SAOD-LS devices on both abutments (Case 8), and the bridge equipped with two identical SAOD-NSD devices on both abutments (Case 9). Regarding the SAOD-LS and SAOD-NSD, it can be considered that the main beam of the bridge is the primary structure, while the transmission device can be positioned on the abutment. Additionally, the secondary structure can be located within the bridge’s box-girder. This arrangement effectively addresses the challenge of installing the secondary structure of the AOD. The specific parameters chosen for the SAOD-NSD devices in Case 9 on one side of the abutment are presented in
Table 5.
Apart from the
k2 of the SAOD-LS in Case 8, which is precisely set to 100 kN/m, all the other parameters for the SAOD-LS in Case 8 are identical to those in Case 9 (i.e.,
m2 = 180 t,
c2 = 0 Ns/m and
r2 = 5). We compute the displacement and force results in
Section 5 using the FE method. The general FE modeling method (i.e., the finite element analysis) for beam bridges models the Puwan 14th Bridge part. The modeling methods for the SAOD-LS and SAOD-NSD are similar to Example 4, with the distinction that the main girder of the bridge is treated as the primary structure, including
m1, c
1, and
k1.
Figure 11 illustrates a comparison of the main girder’s time-history displacement response during the first 10 s for Case 7, Case 8, and Case 9.
After researching
Figure 11, it is observed that within the first 10 s, the maximum value for
x1-Case 7 reaches 17.93 cm, while
x1-Case 8 is 13.55 cm (reflecting a seismic reduction rate of 24.43%), and
x1-Case 9 is 13.15 cm (showing a seismic reduction rate of 26.66%). This suggests that both SAOD-LS and SAOD-NSD exhibit exceptional shock absorption capabilities for the actual bridge.
Note: seismic reduction rate = (maximum structural response without control − maximum structural response with control)/maximum structural response without control [
50].
A vibration reduction system with remarkable robustness implies that, under different types of seismic waves, the system demonstrates superior damping performance. Therefore, in the following study, we selected three near-field and three far-field seismic waves [
51,
52,
53,
54], listed in
Table 6.
We calculate the seismic reduction rate for the main girder displacement and the bending moment at the bottom of the middle pier equipped with the SAOD-LS and SAOD-NSD, respectively.
Figure 12 illustrates that the SAOD-LS and SAOD-NSD exhibit effective damping properties against various types of seismic waves, indicating their outstanding robustness as damping devices.
6. Concluding Remarks
In order to solve the problem of the secondary system arrangement and improve the damping efficiency of the AOD with equivalent values of parameters (i.e., Improvement needed 1 mentioned in Introduction), this paper introduces the SAOD-LS, in which the secondary spring and damper are connected to the primary structure. We derive the SAOD-LS system’s motion equation from the Lagrange Dynamic and verify it using the FE approach. The SAOD-LS demonstrates a significant capacity to reduce the peak amplitude of the response. Considering that in practical applications,
r can reach about 10, raising
r will further improve the damping performance [
40]. As a result, the SAOD-LS offers a wide range of useful applications. SAOD-LS’s rapid response to providing the damping effect is a significant mechanical characteristic that is strongly tied to its unique nature as a SDOF damper. Future anti-ice collisions on piers may utilize this mechanical characteristic.
Additionally, an innovative nonlinear spring device (NSD) was proposed, wherein connecting rods and linear springs are arranged in a diamond pattern. Its constitutive relationship can be expressed as a piecewise function, which is checked by the FE analysis. In summary, the device exhibits the following mechanical characteristics: when the relative displacement between the NSD’s two terminals is small, the force (F) is also small; however, if the relative displacement between the NSD’s two terminals is large, the device will generate a considerably large force (F). The device lays the groundwork for future applications in fields such as civil engineering and machinery. For instance, in the context of controlling displacement and force along the longitudinal direction of the bridge, it enables the controlled release of small displacements caused by temperature changes and regular vehicle loads. However, in the event of abnormal excitations, such as earthquakes, the device exerts significant force to prevent excessive displacement of the main girder.
Also, in order to realize the Improvement needed 2, mentioned in Introduction, the SAOD-NSD, which is an SDOF system made by combining the NSD and SAOD-LS, has a much better damping effect in the near-resonance area than the SAOD-LS. At exact resonance, the SAOD-NSD exhibits a resonance-like phenomenon, but the peak value of the response is much smaller. When the characteristic frequency of an earthquake closely matches the natural frequency of the structure, the utilization of NSD for shock absorption becomes highly significant. It is noteworthy that while Case 3 in Example 4 fully utilizes the nonlinear stage of the NSD, it also stays within the maximal applicability range of the NSD. In future research, on the one hand, it is crucial to ensure a sufficient safety margin for NSDs to prevent damage from excessive force. On the other hand, it is critical that we make great efforts to ensure that NSD’s strong nonlinear stage plays an active role in the task. Otherwise, the NSD cannot fully utilize its distinctive features.
Last but not least, this research applies the SAOD-LS and SAOD-NSD to an actual bridge project, demonstrating that both devices effectively absorb shocks in practical applications. By analyzing the response to various seismic waves, it is evident that both devices exhibit strong robustness.
It is noteworthy that the SAOD-LS and SAOD-NSD also have certain restrictions on their applications. With the SAOD-LS as an example, engineers ought to calculate the values of M, C, and K in the Equation (7) for practical use while designing the device. Practical engineering generally establishes m1, k1, and c1, so the focus should solely be on determining the values of , , and . The approach presented in this study attempts to determine the values of m2, k2, and c2 under the condition of a constant r. Determining the values of m2, k2, and c2 first, and then figuring out r’s value, is particularly easily accessible for engineering practice. In subsequent investigation, it appears that this issue can be resolved by adjusting the values of rm2, rk2, and rc2, (i.e., different r for m2, k2, and c2).
This study has also laid the groundwork for future research. Simultaneously, it is hoped that this study will lead to new insights into bridge earthquake resistance.