Enhanced low-frequency vibration isolation via innovative double-resonator phononic crystals

Abstract

Ground vibrations induced by underground metro systems can adversely affect the surrounding environment, particularly in the context of low-frequency vibrations. Despite the introduction of various mitigation techniques, such as floating slab track (FST) over steel springs or phononic crystal vibration barriers, their efficacy in attenuating low-frequency vibrations is limited. Therefore, the main objective of this study is to develop a novel Double-Resonator Phononic Crystal (DRPC) vibration isolator. The Finite Element Method (FEM) is utilized to analyze the locally resonant bandgap characteristics, and the isolator’s shielding efficacy has been assessed through the transmission spectrum (TS). Furthermore, a detailed parametric study investigates the influence of Young modulus (MOE), density, geometric dimensions, and the damping (ƞ) of different components on the TS and bandgap properties of the DRPC. This study also investigates the impact of damping by analyzing the distribution of energy and velocity within the DRPC. The findings indicate that the DRPC vibration isolator substantially develops a low-frequency bandgap (36 Hz to 151 Hz) alongside a TS closely corresponding with the bandgap, thereby verifying the model’s accuracy. Moreover, among all rubber components, the damping in the middle rubber mainly governs the vibration attenuation efficiency and the bandgap properties of the isolator.

Introduction

Recently, traffic congestion on urban roads has plummeted due to the rapid development of the urban rail transit system. The underground metro system positively impacts the environment and passengers’ daily lives by improving accessibility, speed, energy efficiency, and cost-effectiveness. However, beyond these benefits, ground vibrations induced by the underground rail system negatively affect the surrounding environment^1,2,3,4,5. Subway lines in urban areas often pass through sensitive areas such as hospitals, schools, and commercial and residential places. Hence, there is a high demand to adopt measures that can sustainably lower the effect of these harmful vibrations^{6,7,8,9,10,11,12,13,14}. These vibrations mainly focus on low-frequency vibrations, especially those below 80 Hz. In commercial and residential buildings, such low-frequency vibrations can disrupt comfort¹⁵. Therefore, efforts to mitigate subway vibrations are primarily aimed at cutting down low-frequency vertical vibration^16,17,18.

Vibration control is predominantly achieved by utilizing soft elastic elements within track structures based on isolation principles found in mass-spring systems^19,20,21. Technologies such as rubber vibration isolation pads²², elastic sleepers²³, vibration-damping fasteners²⁴, and Floating Slab Tracks (FST)^25,26 have been widely implemented recently. Notably, FST stands out for its superior low-frequency vibration mitigation, reduced impact on rail corrugation, and prolonged durability²⁷. It is pertinent to mention that the Steel Spring Floating Slab Track (SSFST) has superior vibration attenuation capabilities across a broad spectrum of frequencies²⁸. Its operational mechanism involves the incorporation of a linear resonator characterized by a natural frequency substantially lower than the excitation frequency. It is positioned between the track slab and the underlying foundation²⁹. This configuration augments the system’s mass while diminishing its stiffness, enhancing its capacity for significant damping. Moreover, the SSFST demonstrates effective damping above its inherent frequency, albeit less effectively below the threshold above. As a result, a fundamental strategy in the design for vibration mitigation entails lowering the system’s natural frequency through an increase in the mass of the track slab, alongside a reduction in the stiffness of the supports beneath the slab³⁰. It is imperative to underscore that if the stiffness of these supports is excessively reduced or inaccurately calibrated concerning the load imposed by the train, the entire system may confront significant instability. Such instability would be reflected in the form of increased vibrations, which not only compromise the railway infrastructure’s structural integrity but also deteriorate passengers’ comfort and safety. Failures in steel spring vibration isolators due to conditions like water infiltration from heavy rainfall, fatigue, or resonance, further disrupt track uniformity and structural integrity, causing abrupt stiffness changes, uneven settlement, and increased maintenance³¹. Thus, the precise adjustment of support stiffness and advanced isolation techniques is necessary to maintain the delicate balance between vibration mitigation and system stability³².

Beyond the conventional steel spring isolator, Phononic crystal (PC) technology offers an innovative method for vibration isolation, utilizing two main bandgap mechanisms. Initially dominated by Bragg’s scattering due to structural periodicity, this approach encountered limitations in engineering applications. It is due to the frequency’s dependence on structural unit size³³. A novel vibration isolator generally designed to employ PC’s local resonance mechanism would match the static stiffness of standard steel springs. Simultaneously, it would reduce the vibration transfer from the track to the ballast, demonstrating significant isolation technology advancements³⁴. Using numerical simulations, Zhao et al.³⁵ developed and analyzed a PC meta barrier within a rail-train coupling model. They compared the vibration isolation performance of a PC isolator to that of a traditional steel spring isolator under train loading. It was demonstrated from their results that the PC isolator effectively reduced vibration, achieving a bandgap range of 63–140 Hz. In addition, Hao et al.³⁶ enhanced the vibration attenuation capabilities of PC under FST conditions by implementing an improved point support structure by local resonance type (IPSS by LRT). Through theoretical derivation and numerical simulation, they identified first and second-order bandgap ranges of 51–88 Hz and 131–166 Hz, respectively.

Moreover, Gao et al.³⁷ developed a locally resonant PC isolator for rail systems, revealing a 49.7 to 104.9 Hz bandgap for vibration reduction through a numerical model. Comparative analysis demonstrated its superior ability over steel springs in mitigating low-frequency vibrations and protecting against environmental impacts. In addition, Li et al.³⁸ explored dual-sided composite tapered PC slabs, which identified a bandgap from 59 to 93Hz. Despite their unsuitability for railway loads, these designs introduce novel low-frequency vibration isolation concepts for rail infrastructure that emphasize the significance of PC in this context. In another study by Zhao et al.³⁹, they introduced an innovative vibration isolator for floating slab tracks by combining particle damping with bandgap methods to enhance vibration isolation between 50–150 Hz. It surpassed the conventional steel spring and PC vibration isolators by effectively reducing resonance peaks and bridge deck vertical accelerations. A PC vibration isolator, incorporating acoustic black hole (ABH) structures, demonstrates augmented vibration damping capabilities, which tend to outperform conventional isolators in mitigating train-induced vibrations⁴⁰. Wang et al.⁴¹ demonstrated that by employing local resonance mechanisms, the band gap for effective vibration and noise control in track structures is expanded. Their research indicated that attaching locally resonant units periodically formed a new PC structure. As a result, a broad and adjustable resonant band gap was created that significantly enhanced vibration and noise reduction in high-speed railway tracks. While most previous studies achieved low-frequency bandgaps above the 50 Hz range, the critical need to extend these bandgaps to lower frequencies still exists. This is because low-frequency longitudinal waves, which transmit substantial vertical vibrational energy and propagate vertically from the floating slab, can significantly harm humans and damage nearby structures. Consequently, it is imperative to counteract the detrimental effects of low-frequency vibrations more effectively. Notably, local resonance in PC meta barriers often leads to a narrower bandgap compared to Bragg scattering, thus presenting a substantial challenge in harmonizing both mechanisms. This limitation hinders the application of bandgap features in PC-based vibration isolators to enhance vibration isolation capabilities. It is primarily observed for frequencies below 50Hz, such as those generated by train loads. Therefore, exploring innovative methods that enable PC systems to achieve a wider bandgap at lower frequencies is crucial. Such advancements could provide effective solutions for mitigating the harmful impact of low-frequency vibrations without increasing system complexity or compromising engineering feasibility.

The current research incorporates the novel Double-Resonator Phononic Crystal (DRPC) vibration isolator produced for the streamlined decrease of low-frequency vibrations with a wider bandgap width. The Finite Element (FEM) technique is deployed to investigate the DRPC’s local resonance bandgap features and mode shapes. The preliminary analysis concentrates on inspecting the DRPC’s dispersion curves and assessing its vibration isolation performance. It is done by analyzing the transmission spectrum (TS) and identifying the bandgap characteristics. Afterward, the effects of Young modulus or Modulus of Elasticity (MOE), material density, geometric dimensions, and damping characteristics of the DRPC components are studied to observe their impact on vibration isolation capability. Lastly, the study investigates energy and velocity distribution processes within the DRPC by considering multiple damping conditions.

Mechanisms of traditional phononic crystals (PCs) and proposed double-resonator phononic crystals (DRPCs)

Phononic crystals (PCs) are manufactured materials crafted to regulate and direct the transmission of mechanical waves, including vibrations and noise. These configurations establish bandgaps, intervals of frequencies through which waves are unable to travel, effectively serving as vibration barriers. The single resonator PC, or the traditional PC, comprises connectors having a sandwiched internal rubber (IR), an external rubber (ER) layer, as well as a single steel resonator. The vibration isolation mechanism relies on the local resonant frequency of the steel resonator that interacts with the elastic properties of the rubber. Specifically, the starting frequency (f_start) of the bandgap is determined by the resonance of the steel resonator, while the connectors and IR are immovable. The cut-off frequency (f_cutoff) is determined by the antiphase resonance frequency of the steel resonator and the combination of connectors and IR³⁴. Additionally, increasing the weight of the resonator can decrease the f_start or may not affect it. However, it also lowers the f_cutoff, resulting in a narrower bandgap width, as reported in previous studies^37,42. Consequently, this design limitation inhibits the efficacy of traditional PCs in isolating low-frequency vibrations with a wider bandgap width.

The proposed DRPC enhances the traditional PC design by incorporating two steel resonators (SRs), as shown in Fig. 1. After the first resonator, an additional stiffer rubber layer is attached, followed by the second steel resonator. The DRPC’s mechanism leverages the coupling between the two resonators, which introduces further localized resonance modes alongside complex vibrational associations. The f_start of the bandgap in the DRPC is computed with the help of the resonance of the SRs. Note that the softer middle rubber (MR) layer allows for substantial distortion and energy absorption, thus vilifying the f_start. The antiphase resonance frequency of both the SRs having ER and the combination of connectors and IR governs the f_cutoff of a typical DRPC. However, the association between the SRs and the ER layer raises the f_cutoff by strengthening the coupling between the two resonators. This improved coupling creates hybridized vibrational modes, which result in a broader bandgap. Detailed numerical analyses are conducted in the following section to understand these mechanisms further and validate them.

Fig. 1

Simplified models of the DRPC vibration isolator.

DRPC meta barrier design

The DRPC vibration isolator comprises a periodic structure with five unit cells, as shown in Fig. 2a. Each unit cell in the DRPC includes two aluminium connectors with IR placed between them, a ring-shaped MR, and an ER, along with internal and external SRs. The connectors enable the connection of adjacent unit cells that form a giant supercell. The arrangement is designed so the ring-shaped MR, ER, and SRs generate a local resonance bandgap. At the same time, the aluminium connectors and the IR between them provide vertical support for the floating slab. Figure 2b displays the geometric dimensions of the unit cell, whereas Fig. 2c and d depict the top and side views of the DRPC’s unit cell, respectively. The characteristics of DRPC materials are presented in Table 1.

Fig. 2

Schematic diagram of the DRPC vibration isolator: (a) Cross-sectional view of the DRPC isolator; (b) Detailed geometrical parameters of the unit cell are as follows: a1 = 0.02 m; a2 = 0.008 m; a3 = 0.014 m; a4, a5, a6, a7 = 0.022 m; b1 = 0.0316 m; b2 = 0.0128 m; b3 = 0.0404 m; b4 = 0.032 m; b5 = 0.0484 m; (c and d) Top and side views of the unit cell, respectively.

Table 1 Characteristics of the DRPC material.

Analyzing bandgap characteristics and vibration isolation efficiency of DRPC meta barriers

Bandgap features of DRPC cell

The finite element analysis tool COMSOL Multiphysics was utilized to determine the energy band structure of DRPC, which provided a substantial benefit in managing the complexities related to geometrically complex and computationally intensive PC isolator models. Isolators were arranged vertically in a periodic pattern. The maximum size of the mesh elements was kept below 0.01 m. Based on Bloch’s theorem, one can obtain the calculation of the energy band structure by imposing Floquet’s periodic boundary conditions at both the top and bottom boundaries of the unit cell. The proposed meta barrier, characterized as a one-dimensional PC structure, necessitated that the wave vectors for band structure calculation and the irreducible Brillouin zone are one-dimensional. Consequently, the wave vector has been represented as a scalar and is denoted by k. Choosing the wave vector within the 1-d irreducible Brillouin zone, specifically in the range (-π/a, π/a), with “a” (equivalent to b5) being the lattice constant for the unit cell, yielded a sequence of associated angular frequency ω values. The discrete eigenvalue equation that details particle movement within the unit cell can be shown in Eq. (1).

$$(K - \omega^{2} M)U = 0$$

(1)

where U denotes the nodal displacement, while K and M represent the stiffness and mass matrices of the unit cell, respectively. The relationship between ω and k is expressed through the dispersion curve, also referred to as the energy band structure. Also, it is observed that the geometrical and material characteristics of the DRPC cell, as illustrated in Fig. 2 and detailed in Table 1, can be utilized. However, the analysis was conducted to obtain the energy band structure for zero damping. Figure 3 depicts the dispersion curves for the DRPC cell, derived through FEM analysis. Despite the propagation of waves (Fig. 3) occurring in the axial direction (i.e., vertically within the floating slab track system), it can be observed that not every mode represents a longitudinal wave. Additionally, shielding efforts predominantly concentrate on longitudinal waves, as these are the primary modes of vibration transmission into tunnels and soil via the vibration isolator. Hence, it is imperative to identify the mode shapes pertaining to longitudinal waves. Accordingly, the initial ten eigenmodes of the DRPC unit cell were analyzed to differentiate the relevant modes from the remaining, such that only the mode shapes related to longitudinal waves are shown in Fig. 4.

Fig. 3

First 10 energy band structures of the DRPC unit cell.

Fig. 4

Eigenmodes of the DRPC unit cell showing two longitudinal vibration modes with displacement vector fields.

The starting and cut-off bandgap frequencies of the DRPC meta barrier were governed by the vertical vibrational modes through the locally resonant bandgap mechanism. Specifically, the 4th mode, as shown in Fig. 4a, was identified as the starting bandgap frequency of the unit cell when the connector and IR remained stable. Notably, the other components underwent vertical vibrations. On the contrary, the 7th mode, depicted in Fig. 4b, revealed vertical vibrations of the connector and IR. In contrast, both SRs and the ER vibrated in the opposite direction to the connector and IR. Consequently, the starting and cut-off bandgap frequencies of the DRPC unit cell aligned with the 4th and 7th modes, respectively. The mode corresponding to the f_start possessed a wave vector of π/a, while the mode corresponding to the f_cutoff had a zero wave vector. The first bandgap frequency range of the unit cell relating to longitudinal waves ranged between 36 and 151 Hz, resulting in a bandgap width of 115 Hz (Fig. 3).

To further understand the low-frequency bandgap mechanism of the DRPC, the eigenmodes of longitudinal waves were examined thoroughly. As shown in Fig. 3, Point A on the energy band structure graph corresponds to the f_start of 36 Hz at the wave vector k = π/a. At this point, the resonators and the ER exhibit prominent translational vibrations along the z direction, as illustrated in Fig. 4a. This motion generates a resultant force in the z-direction, interacting with the traveling wave in the connector and IR. The coupling impact between these periodic forces and the traveling wave opens the bandgap. This phenomenon, known as local resonance, occurs when both resonators and the ER vibrate at or near their natural frequencies. At this frequency, the resonators and ER have significant back-and-forth motion, generating periodic forces that lead to destructive interference with the traveling waves in the connector. The double resonator and the ER are the desideratum in achieving the low-frequency bandgap. The presence of two resonators incorporates supplementary localized resonance modes, enhancing the overall vibrational interactions within the unit cell. Note that the softer MR layer allows for considerable distortion and energy absorption, lowering the start frequency of the bandgap.

For the f_cutoff of 151 Hz at the wave vector k = 0 (Point B, as given in Fig. 3), the connector and IR experience more significant vibrations while moving in the opposite direction to both the SRs and the ER (Fig. 4b). This opposite movement vilifies the coupling effect between the connector, the resonators, and the ER, causing the closure of the bandgap. Additionally, the stiffer ER layer extends the f_cutoff, broadening the bandgap. In addition, the interaction between the two resonators creates hybridized vibrational modes, further enhancing the bandgap’s width. This perplexing interplay of components ensures efficacious vibration isolation across a broader frequency spectrum (particularly at lower frequencies). Thus, the formation of the DRPC’s bandgap relies on the interaction between the connector, the SRs, and the ER in the z-direction. It is to say that the efficiency of this coupling interaction dictates the existence of the bandgap.

DRPC vibration isolation efficiency

After identifying the low-frequency bandgap and the inherent vibration modes of the DRPC, the vibration isolation characteristics of a DRPC composed of five unit cells were analyzed. The model deployed damping as an isotropic loss factor within a direct analysis approach for the calculations. A harmonic load, labeled as F_in, with a peak force of 17 kN, was applied to the DRPC’s upper surface (Fig. 5a), which led to the calculation of the amplitude, F_out, of the reaction force at the structure’s bottom fixed point. Moreover, the meshing criteria for the DRPC supercell resemble those described in “Bandgap features of DRPC cell” section, such as the meshing shown in Fig. 5b. Following this, the shielding performance of the DRPC meta barrier was evaluated across various harmonic loads using the TS, as depicted in Eq. (2). It is noted that when the longitudinal wave is effectively blocked, the TS value falls below zero.

$$TS \, = \, 20\lg \left| {\frac{{F_{out} }}{{F_{in} }}} \right|$$

(2)

Fig. 5

(a) Simplified diagram for TS evaluation of the DRPC supercell, and (b) Meshing of the supercell.

The results of the DRPC vibration isolator’s TS curve analysis (shown in Fig. 6) clearly show that the range of vibration attenuation matches the bandgap frequencies identified from the energy band structure (dispersion) curve. These observations, derived from infinite (unit cell) and finite (supercell) structures, underscore the precision and reliability of the numerical model employed. The results are almost similar, with a slight difference in f_cutoff due to the differences in boundary conditions. In the no-damping case, two significant peaks were observed at 34.8 Hz and 36.0 Hz, respectively. The corresponding vibration modes, along with the f_cutoff mode, identified at 144 Hz, are depicted in Fig. 7. As the incident elastic wave’s frequency approaches the threshold of the bandgap’s opening, it engages with the SRs within the isolator. This interaction excellently confines the elastic wave energy to the resonators, thereby avoiding its transmission through the connector to the isolator’s far end.

Fig. 6

TS of DRPC meta barrier with and without damping.

Fig. 7

Vertical vibration modes of the DRPC meta barrier without damping at (a) 34.8 Hz, (b) 36.0 Hz, and (c) 144 Hz.

At precisely 144 Hz, the scenario alters significantly; both SRs and the ER engage in inverse vibrations relative to the connector. This inverse vibrational dynamic facilitates the transfer of vibrational energy to the isolator’s opposite end via the connector, thus eliminating barriers to the energy’s transmission. Consequently, the vibration isolator’s f_cutoff can be determined by the anti-phase resonance frequency of its SRs and connectors. This defines the specific frequency at which the system is engineered to demonstrate vibrational modes that are out of phase. Moreover, introducing rubber damping, characterized as an isotropic loss factor, diminishes the DRPC meta barrier’s capacity to dampen vibrations within the attenuation zone. At the same time, there is a decrease in the TS peak before the opening of the bandgap (Fig. 6). Furthermore, the introduction of damping in the DRPC isolator leads to a slight broadening of the bandgap characteristics. The f_start and f_cutoff of the DRPC vibration isolator with damping were computed to be 34 Hz and 148 Hz, respectively. This suggests that while incorporating damping in the DRPC contributes to a broader bandgap, it also plummets the vibration isolation efficiency of the system. As a result, there is a compromise between bandgap broadening and vibration reduction efficiency in the damped DRPC isolator.

Parametric analysis of DRPC vibration isolator

In the preceding sections, a comprehensive analysis of the unit cell and supercell of the DRPC vibration isolator has been conducted to achieve a broader and lower frequency bandgap through local resonance phenomena. Based on this, the current section presents detailed parametric analyses of the DRPC to understand better how different geometric configurations and material characteristics influence the f_start and f_cutoff, as well as the bandgap width. Specifically, the impact of MOE and density of the IR, MR, and ER layers, along with the SRs, were analyzed. Furthermore, the influence of geometric parameters, especially the radius, was assessed exclusively for the SRs. These parametric analyses were performed by deploying a single-factor method, maintaining constant properties for other components (as described in “DRPC meta barrier design” section). It is essential to mention that these analyses were performed under no damping conditions.

Impact of rubber properties on DRPC bandgap

Figure 8a–c illustrates the effect of varying MOE_IR,MR,ER on the bandgap properties of the DRPC vibration isolator. The results indicate that increasing the MOE_IR,ER layers poses a negligible impact on the bandgap of the DRPC. However, increasing the MOE_MR layer increased both the f_start and the f_cutoff. Notably, the increase in the f_cutoff is more pronounced than that of the f_start, resulting in a broader bandgap when the MOE_MR is increased. This behavior is attributed to the increase in MOE_MR, which improves the shear stiffness of the rubber. Finally, the stiffer MR undergoes less elastic distortion and efficiently inhibits the motion of the SRs. As a result, the resonance scenario shifts to a more significant frequency, which leads to both the f_start and f_cutoff of the bandgap rising.

Fig. 8

Effect of material characteristics on the bandgap frequency of the DRPC vibration isolator: (a–c) Impact of the MOE of the IR, MR, and ER layers; (d–f) Impact of the density of the identical rubber layers.

Figure 8d–f depicts the impact of changing the density of the IR, MR, and ER layers on the bandgap characteristics of the DRPC vibration isolator. The results show that modifications in the density of the IR and MR layers substantially affect the bandgap characteristics of the isolator, primarily cutting down the f_cutoff. In contrast, the f_start shows only a slight variation. This reduction in the f_cutoff leads to a narrower bandgap as the density of the IR and MR layers rises. Whereas variations in the density of the ER minimally affect the f_cutoff of the DRPC vibration isolator. However, a little decrease in the f_start can be recorded at higher ER density, which leads to a broader bandgap. This behavior is attributed to the increased mass of the ER, which lessens the resonance frequency of the SRs and, hence, reduces the f_start of the DRPC vibration isolator.

Impact of steel resonator properties on DRPC bandgap

A parametric study was performed to explain the effect of both SRs on the bandgap characteristics of the DRPC vibration isolator. Figure 9a and b depict the impact of changing the MOE of the internal and external SRs on the bandgap characteristics of the isolator, respectively. It is revealed that the influence of MOE is substantial in the internal resonator in contrast to the external resonator. Specifically, Fig. 9a illustrates a notable rise in the f_cutoff at higher MOE values for the internal resonator, alongside a minimal increase in the f_start, which substantially widens the bandgap width. On the contrary, Fig. 9b shows the response of the external resonator, where the rise in the f_cutoff is less marked. At the same time, the f_start remains nearly unchanged, which leads to a marginal expansion in the bandgap width. This discrepancy is associated with the transmission pathway of vibrations and their relationships with the internal and external resonators. The internal resonator, centrally positioned, directly gains the vibrational energy, which leads to significant changes in its vibrational characteristics because of changes in stiffness and mass distribution. These alterations impact the resonance frequency of the steel resonator, substantially affecting the bandgap width. In contrast, the external resonator, located peripherally and receiving vibrations after passing through other components, has fewer resonance frequency changes. Therefore, it results in a lesser impact on the bandgap width. These observations suggest the critical role of resonator placement as well as the sequential relationship of components to tune the DRPC bandgap characteristics.

Fig. 9

Influence of inner and outer steel resonator characteristics on the bandgap frequency of the DRPC vibration isolator: (a–b) MOE; (c–d) Density; (e–f) Radius.

Figure 9c and d depict the effect of changing the density of SRs on the bandgap characteristics of the DRPC, with Fig. 9c concentrating on the inner steel resonator and Fig. 9d exploring the outer steel resonator. Both figures show that while the f_cutoff remains relatively stable across various densities, there is a minimal decrease in the f_start for both the internal and external resonators as density rises. Moreover, it can also be observed that the bandgap width is slightly increasing. This data shows that modifications in the density of the SRs have a negligible effect on the DRPC’s characteristics.

Figure 9e and f depict the effect of higher radius values of SRs on the bandgap characteristics of the DRPC, i.e., radius values change from 18 to 26 mm. With increasing radius, the f_start slightly decreased, the bandgap width slightly increased, and the f_cutoff remained relatively stable. Therefore, it is evident that extending the radius affects the resonators’ vibrational characteristics due to increased mass, which decreases the system’s natural frequency. This causes a reduction in the f_start and an expansion of the bandgap. The results reveal the sensitivity of the DRPC’s characteristics to modifications in its components’ physical dimensions, particularly the SRs’ radius.

Impact of individual rubber damping on DRPC TS and bandgaps

A complete parametric analysis of the DRPC vibration isolator has been performed in the analysis given above. The current section examines the effects of different damping values (ƞ) in individual rubber-damping systems within the model. It is done to analyze the influence of ƞ_IR, MR, ER on the performance of a DRPC meta barrier with no damping effects of aluminium connectors and SRs, mainly focusing on the TS and bandgap characteristics. Figure 10a illustrates the TS curve of the DRPC across various ƞ_IR values such that the ƞ_MR and ƞ_ER equal zero. The impact of ƞ_IR on the vibration attenuation capacity within the attenuation domain is minimal; nonetheless, it effectively reduces the resonance peak as the damping value increases.

Fig. 10

Effect of varying damping values in individual rubber damping systems on TS: (a) IR, (b) MR, and (c) ER.

In contrast, Fig. 10b shows that the effect of ƞ_MR on the DRPC meta barrier’s TS curve is significant when both ƞ_IR and ƞ_ER values are eliminated. Moreover, the capacity for vibration attenuation decreases near the local resonance peak as the damping value rises. This results from the inclusion of ƞ_MR, which dampens the vibrations of both the steel resonator and the ER. These observations are consistent with the findings of previous studies^37,39,40, thus affirming the accuracy and dependability of the model formulated here.

Furthermore, in the absence of ƞ_IR and ƞ_MR, ƞ_ER exhibited a minimal effect on suppressing the resonance peak. It occurs before the bandgap opens with a damping value of η = 0.1; however, it does lower the vibration attenuation capacity. In addition, it is noteworthy that elevating the ƞ_ER to 0.5 and 1.0 enhances the suppression of the resonance peak and markedly decreases the vibration attenuation capacity in the attenuation zone, as demonstrated in Fig. 10c.

From the above observations, it can be concluded that ƞ_IR decreases the resonance peaks with minimal impact on vibration attenuation. Secondly, the ƞ_MR has a significant influence, reducing overall vibration attenuation. Whereas ƞ_ER is initially less effective, it markedly enhances resonance suppression as damping increases, but with a reduced vibration attenuation.

Figure 11a illustrates the effects of various IR damping levels, with both ƞ_MR and ƞ_ER set to zero. As the ƞ_IR increases, the graph reveals a slight decrease in the f_start accompanied by a rise in the f_cutoff, leading to a minor expansion of the bandgap width. In contrast, Fig. 11b presents the results of varying ƞ_MR while maintaining the IR and ER damping at zero. The figure indicates that increasing ƞ_MR shifts both the f_start and f_cutoff upward, with a particularly sharp rise in the f_start for ƞ = 0.5 and 1.0. Consequently, this results in a noticeable decrease in the bandgap width. It is evident that ƞ_MR significantly influences the bandgap width and reduces the efficiency of the DRPC vibration isolator. An increase in ƞ_MR substantially lowers the motion of SRs by limiting energy transmission from the connector to the SRs and ER. As a result, the resonance frequency of the SRs shifts to a higher range, accompanied by a decline in the efficiency of the local resonance bandgap. Finally, Fig. 11c examines the effects of ƞ_ER, with ƞ_IR and ƞ_MR held at zero. The graph shows that as ƞ_ER increases, there is a slight decrease in the f_start and a corresponding increase in the f_cutoff, leading to a minor widening of the bandgap width.

Fig. 11

Impact of changing damping values in individual rubber damping system on the DRPC bandgap: (a) IR; (b) MR; and (c) ER.

Based on the above results of the analysis, the following conclusions are evident. Firstly, ƞ_IR facilitates the repression of resonance peaks with minimal impact on the overall vibration attenuation, thus rendering it an effective option to control particular frequencies without compromising the broader attenuation capacity. In contrast, ƞ_MR exhibits a distinct consequence, notably shifting the f_start upward and declining the bandgap width. This designates a noteworthy decrease in vibration attenuation as damping values surge. Lastly, in the case of ƞ_ER, although this one is initially less impactful, it converts to be progressively effective at suppressing resonance peaks with higher damping levels. However, this augmentation in resonance suppression comes at the cost of a lower vibration attenuation capacity. These results underscore the impact of each rubber layer damping on the dynamic performance of the DRPC vibration isolator. Thus, to optimize the efficacy of the DRPC vibration isolator, it is imperative to adjust damping values per the operational context’s unique stipulations.

Combined rubber damping effects on DRPC TS and bandgaps

This section presents the impact of changing damping values in various rubber layers within the combined rubber damping system on the TS and bandgap properties of the DRPC. The combined rubber damping system considers the damping impact across all rubber components during the analysis. For instance, Fig. 12a presents the TS of DRPC when only the ƞ_IR value is varied, with the ƞ_MR and ƞ_ER kept constant at 0.1. The results demonstrate that the combined damping system substantially decreases both the boosting and damping of vibrational energy compared to the individual rubber damping system (See Fig. 10a). This reduction is attributed to the introduction of ƞ_MR and ƞ_ER, which plummets the motion of both SRs. Furthermore, a minimal decrease in both the boosting and damping of vibration can be recorded at higher ƞ_IR.

Fig. 12

Impact of varying damping values in combined rubber damping system on TS: (a) IR; (b) MR; and (c) ER.

Figure 12b shows the effect of varying the ƞ_MR values while the ƞ_IR and ƞ_ER values are kept constant at 0.1. It can be revealed from the results that changes in the ƞ_MR lead to relatively higher distinct fluctuations in the transmission spectrum, especially before and after the f_start. Overall, an increase in the ƞ_MR considerably lowers the TS, emphasizing the critical role of the MR layer in determining the vibrational characteristics of the DRPC system. Furthermore, Fig. 12c focuses on the scenario where the ƞ_ER values are varied while keeping the ƞ_IR,MR constant at 0.1. The overall profound decrease in the TS_DRPC is evident, primarily due to the impact of the combined rubber damping system, with a particular emphasis on the role of ƞ_MR. Moreover, the transmission spectrum revealed that the variations above in ƞ_ER do not significantly affect the DRPC vibration energy, showing a lesser impact than ƞ_IR and ƞ_MR. These results also highlight that variations in ƞ_ER have a negligible influence on the overall transmission characteristics within the combined rubber damping system.

Figure 13 illustrates the influence of varying ƞ_IR,MR,ER components within the combined rubber damping system on the bandgap characteristics, mainly focusing on the f_start, f_cutoff, and bandgap width. In Fig. 13a, as the ƞ_IR increased from 0.1 to 1.0, while maintaining constant damping values for the ƞ_MR and ƞ_ER components at 0.1, a slight decrease in the f_start and an increase in the f_cutoff were observed. This resulted in slightly widening the bandgap width, suggesting that higher ƞ_IR led to a modest expansion of the frequency range in which vibrational isolation occurred. The impact of variations in ƞ_IR on the bandgap properties of the combined rubber damping system in the DRPC isolator is distinctly higher than that observed in the individual rubber damping system. This effect arises because, in the combined rubber damping system, ƞ_IR, ƞ_MR, and ƞ_ER collectively contribute to modifying the local resonance bandgap properties of the DRPC system. This influence is neglected in the individual rubber-damping system.

Fig. 13

Impact of varying damping values in combined rubber damping system on the DRPC bandgap: (a) Internal rubber; (b) Middle rubber; and (c) External rubber.

In Fig. 13b and c, the effects of varying middle and ƞ_ER were presented, with the other damping values held constant at η = 0.1. When the ƞ_MR was varied, both the f_start and f_cutoff shifted upwards, with the increase in f_start being more pronounced. This caused a minor reduction in the bandgap width. In contrast, the ƞ_ER induced only marginal changes in the f_start and f_cutoff, without significantly affecting the bandgap width. These results indicated that the combined effect of damping across the different rubber layers tended to induce slight shifts in the bandgap, primarily influencing the f_start, with ƞ_MR and ƞ_MR exerting the most significant impact.

In light of the current study’s results, it is evident that the combined rubber damping system largely influences the vibration transmission characteristics of the DRPC vibration isolator by efficiently dropping both resonance peaks and vibration attenuation in contrast to individual rubber damping systems. Incorporating of ƞ_MR substantially inhibits the motion of the resonators and ER components, leading to a prominent decrease in attenuation size. This finding points to a balance between minimizing resonance peaks and reducing vibration attenuation. Moreover, variations in damping values impact bandgap characteristics: increasing ƞ_IR results in a mild decrease in the f_start and an increase in the f_cutoff, gradually expanding the bandgap. Nevertheless, increasing the ƞ_MR shifts both f_start and f_cutoff upward, causing a minor bandgap narrowing. It is essential to state that the effect of ƞ_ER on both the TS and bandgap properties is minimal. Consequently, achieving optimal DRPC performance requires a prudent balance of ƞ_MR and ƞ_MR to improve vibration isolation while managing their impact on bandgap width and attenuation capacity.

Energy and velocity distribution mechanism of DRPC vibration isolator

This section evaluates each component within the DRPC vibration isolator, focusing on the mechanisms of energy distribution across different frequency ranges and the impact of changing ƞ_IR,MR,ER of the combined rubber damping system. In addition, it further analyzes the velocity amplitude field alongside the total energy distribution among the components of the DRPC vibration isolator at particular resonant frequencies. The elastic potential energy (E_PE) and kinetic energy (E_k) for the connectors, IR, MR, and ER layers, as well as the SRs, have been calculated using volume integrals, as detailed in Eqs. (3) and (4). Subsequently, the total energy (E_T) of these elements is determined based on Eq. (5).

$$E_{PE} = \iiint_{V} {W_{PE} dV}$$

(3)

$$E_{k} = \iiint_{V} {W_{k} dV}$$

(4)

$$E_{T} = E_{PE} + E_{k}$$

(5)

where W_PE and W_k represent the elastic potential and kinetic energy density.

Figure 14a displays the total energy (E_T) distribution among various components after reaching equilibrium under harmonic loading across a range of frequencies. The analysis included progressively increasing the ƞ_IR from 0.1 to 1.0, while the ƞ_MR and ƞ_ER were kept at 0.1. It is noted that all vibrational energy, defined here as the total energy comprising both elastic potential energy and kinetic energy induced by harmonic loading, is considered in the analysis. The vibrational energy within the SRs and the MR and ER layers remained limited at frequencies below 20 Hz. However, substantial vibrational energy was detected in the connector and IR at these lower frequencies, indicating the onset of vibration transmission outside the DRPC isolator. As the frequency value was higher, the vibrational energy in all components also progressively increased. At approximately 33 Hz, the vibrational energy peaked in the connector and IR. Beyond that, the vibrational energy in the connector and IR decreased while the energy in other components increased. At the same time, around 35 Hz, a noticeable peak in vibrational energy was recorded across all components except for the connector and IR, with the MR showing the maximum energy level, which marks the beginning of the bandgap in the DRPC vibration isolator. As the frequency crossed 35 Hz, the vibrational energy in the MR and ER, as well as in both resonators, began to decrease until it almost vanished, indicating the closure of the bandgap, despite the increased ƞ_IR, which led to a reduction in energy across all components. It is pertinent to mention that the order of increasing energy distribution remained as follows (excluding the connector and IR): MR > external steel resonator > internal steel resonator > ER.

Fig. 14

Energy distribution mechanism in a DRPC isolator with variable damping values: (a) IR, (b) MR, and (c) ER.

Figure 14b shows the significant impact of increased ƞ_MR on the vibrational energy across all components of the DRPC barrier. Under this condition, the ƞ_MR is increased, while the ƞ_IR,ER remains fixed at 0.1. This trend aligns with the results depicted in Fig. 14a, where a decreasing trend in vibrational energy was evident in all components except for the connector and IR, whereas the ƞ_MR increased. Despite the damping values ranging between η = 0.5 and η = 1.0, a noticeable reduction in energy intensity was observed in these components. However, in the f_start ranges, the vibrational energy in the connector and IR experienced negligible reduction with increased ƞ_MR. Moreover, the vibrational energy at the resonance peak was also observed to reduce with increased damping. Note that this trend did not result in a substantial overall decrease in vibration energy across the entire frequency spectrum for the connector and IR. However, it indicates a less effective vibration-damping capability of the meta barrier in the attenuation zone as the ƞ_MR increases. Thus, opting for lower ƞ_MR proves to be more productive for enhancing the DRPC meta barrier’s efficiency.

In addition, when the ƞ_ER is increased while the ƞ_MR is kept at 0.1, a slight decrease in vibrational energy across all components, excluding the connector and IR, can be recorded (See Fig. 14c). Despite these adjustments, the relative energy distribution among the components remains consistent with that shown in Fig. 14a and b, with the highest energy observed in the MR, followed by the external steel resonator, internal steel resonator, and ER. Although this energy pattern persists, the specific energy values vary in response to changes in the ƞ_ER. On the other hand, the connector and IR remain unchanged in vibrational energy as the ƞ_ER increases.

It can be concluded from the analysis that increasing the ƞ_IR reduces the overall vibrational energy across the components; however, it does not sufficiently diminish the energy concentration within the connector and IR, which implies that energy transmission may persist. The results suggest optimizing damping to effectively manage energy distribution and curtail transmission through the isolator. Along the same lines, while growing the ƞ_MR leads to a general decrease in vibrational energy across all components, the lessening is less noticeable in the connector and IR, demonstrating that higher ƞ_MR may be less viable in avoiding energy transmission. On the contrary, changes in the ƞ_ER result in only slight reductions in vibrational energy across most components, with minimal impact on the connector and IR. The consistently high energy levels in the connector and IR across various damping scenarios are significant. They highlight the critical need for tailored damping configurations to efficiently attenuate vibrational energy and minimize its transmission through the isolator.

Figure 15 demonstrates the velocity amplitude field of the DRPC vibration isolator under varying ƞ_{IR,MR, and ER} situations, offering a comprehensive analysis of the vibration energy distribution mechanism at the resonant frequency. Figure 15a depicts that the velocity amplitude systematically lowers as ƞ_IR increases, corroborating the observations in Fig. 14a, where an increase in ƞ_IR is attributed to a decrease in total energy. Notably, the resonator and ER layers constantly sustain a comparatively higher velocity amplitude across all levels of ƞ_IR. In contrast, the MR layer is categorized by the maximum energy concentration, as indicated in Fig. 14a. This phenomenon may be due to the increased elastic potential energy emanating from its deformation. Moreover, the velocity within the DRPC components, including the resonator and ER, progressively drops from top to bottom units. This pattern demonstrates that the units of the DRPC closer to the applied load, specifically the top units, have higher vibrational velocities compared to the units further away, such as the bottom ones. Furthermore, it is evident from the figure that the velocity in the connector and IR is significantly lower than that of other components of the DRPC isolator. This happens because the vibration energy is efficiently transmitted to the ER and both SRs through the MR (at the resonant frequency). It results in the connector and IR experiencing the lowest velocity amplitudes.

Fig. 15

Velocity amplitude fields of the DRPC vibration isolator at resonant peaks with varying rubber damping values: (a) IR damping, (b) MR damping, (c) ER damping.

The influence of ƞ_MR variation on velocity amplitude reduction is observed in the MR and ER layers and both resonators, as depicted in Fig. 15b. This effect arises because the ƞ_MR substantially impedes the transmission of vibrations from the connector to both the resonator and ER, which results in plummeting of the vibration amplitude. This observation aligns with the overall reduction in total energy, as illustrated in Fig. 14b. Consequently, there is a substantial reduction in both the TS and the bandgap width of the vibration isolator, as discussed in “Combined rubber damping effects on DRPC TS and bandgaps” section. In addition, Fig. 15c reveals variations in ƞ_ER that lead to a slight decrease in velocity amplitude. This declining pattern is consistent with the reduction in total energy attributable to increased ƞ_ER. Also, Fig. 15b and c demonstrate a similar decreasing trend in vibration velocity from the top units to the bottom units of the DRPC isolator, which can be observed in the case of both ƞ_MR and ƞ_ER.

It is essential to attain a more inclusive understanding of the contributions of each component within the DRPC vibration isolator and the impact of varying damping in the rubber elements. For this purpose, the total energy distribution at resonant frequency has been stated as a percentage and thoroughly analyzed. The investigation of this distribution explains how each factor contributes to the overall energy absorption at various damping levels (Fig. 16). It is demonstrated that increasing the ƞ_IR,ER minimally alters the order of energy distribution among the components of the vibration isolator. However, an increase in ƞ_MR leads to a noticeable shift in the energy distribution order between the DRPC components.

Fig. 16

Total energy distribution (%) in each component of the DRPC vibration isolator at resonant peaks for fluctuating damping values: (a1–a3) IR, (b1–b3) MR, and (c1–c3) ER.

As the ƞ_MR rises from 0.1 to 1.0, the energy absorption capacity of the connector and IR becomes significantly larger in contrast to other components, which results in negligible attenuation and high transmission of vibration through the isolator. These results further reveal that the ƞ_MR is immensely important to determine the energy distribution and overall efficacy of the DRPC vibration isolator, substantially affecting its ability to attenuate vibrations.

Conclusions

The novel DRPC vibration isolator, formulated using FEM, effectively attains a low-frequency bandgap by manipulating the local resonance mechanism inherent to DRPC. Both the dispersion curve (generated to assess various frequency bandgaps) and mode shapes analysis (performed to categorize the bandgap conforming to longitudinal waves) were crucial in understanding the isolator’s performance. In addition, detailed parametric studies of DRPC were conducted to examine the influence of different parameters on transmission spectrum (TS) and bandgap properties. Further investigation into the meta barrier’s damping characteristics was conducted by analyzing the energy and velocity distribution across various components of the DRPC vibration isolator. The key findings can be summarized as follows.

• 1. The DRPC meta barrier exhibited a bandgap within the low-frequency spectrum, with the f_start and f_cutoff identified at 36 Hz and 151 Hz, respectively. The TS of the DRPC vibration isolator aligns closely with the frequency range of the bandgap, resulting in a noticeable reduction in vibration within the attenuation domain.

• 2. The MOE_MR layer significantly affects the bandgap width. An increase in the MOE_MR raises both f_start and f_cutoff, broadening the bandgap, while changes in the IR and ER layers show no notable effects. Increasing the density of the IR and MR layers significantly lowers f_cutoff, resulting in a narrower bandgap.

• 3. Increasing the MOE of both resonators expands the bandgap width, with the internal resonator more sensitive to MOE variations. Higher density reduces both f_start and f_cutoff frequencies, resulting in a slight bandgap expansion. Additionally, increasing the dimensions of both resonators further widens the bandgap, with a significant decline in f_start.

• 4. In individual rubber damping systems, increasing MR damping (ƞ_MR) significantly reduces the vibration attenuation capacity of the DRPC isolator and narrows the bandgap width. Conversely, increasing IR damping (ƞ_IR) results in a slight decrease in TS, while the ER shows a moderate reduction in TS, with both rubbers displaying a minor rise in bandgap width. In combined rubber-damping systems, a marked decrease in attenuation capacity occurs across all rubber-damping layers, with ƞ_MR limiting the motion of both the resonator and the ER, impacting the TS of the DRPC more than that of the IR and ER. Minimizing ƞ_MR is recommended to improve overall efficiency, while increasing ƞ_IR,MR causes gradual broadening and narrowing of the bandgap, respectively; an increase in ƞ_ER does not significantly affect the bandgap.

• 5. The energy distribution analysis indicates that the MR exhibits more tremendous vibrational energy than the steel resonator and the ER, contributing to an expanded bandgap and improved vibration dissipation. A high ƞ_MR negatively affects vibration attenuation within the meta barrier, significantly reducing total energy in the middle and ER and both SRs. Simultaneously, the velocity distribution analysis reveals that the vibration velocity at resonance in the DRPC isolator decreases with increasing damping values, especially for ƞ_MR, supporting the energy distribution findings and highlighting the vibration attenuation mechanism.

The above observations reveal that the DRPC vibration isolator demonstrates superior efficiency in mitigating vibrations within lower frequency bandgaps, suggesting its potential as a suitable replacement for traditional isolators. The DRPC isolator can be effectively employed in real-world scenarios to reduce the transmission of vibrations generated from railways to nearby structures, particularly in low-frequency vibrations. Compared to traditional isolators, the DRPC isolator offers the potential for further modifications, allowing for the optimization of the vibration frequency bandgap, thereby enhancing its efficiency and usability in practical applications.

Future investigations would aim to conduct experiments on the DRPC isolator to solidify its practical application within the engineering domain. They would focus on its durability under prolonged train loading and evaluate its fatigue resistance. Furthermore, a rail-FST-tunnel coupled dynamic model would be developed to investigate the vibration attenuation capabilities of the DRPC under train loading conditions.