Enhancing Wide-Band Vibration Isolation Performance of Passive Isolators via Disk-like ABH and Damping Layer

Abstract

Low-frequency broadband vibration isolation poses a critical limitation for marine power equipment, as conventional passive isolators fail below 50 Hz. Targeting the 10–315 Hz band (dominant for marine pumps), this study proposes a passive isolator integrated with a disk-like acoustic black hole. This article aims to address the core engineering issues in the operating frequency band of marine power equipment, specifically the failure of traditional passive vibration isolators in low-frequency vibration isolation and the insufficient reliability of active/hybrid vibration isolation schemes in the marine high-salt fog environment. Meanwhile, it breaks through the theoretical bottleneck of traditional acoustic black hole (ABH) structures, which have a high cut-off frequency and a weak low-frequency vibration suppression capability. A passive vibration isolator integrating a disk-shaped ABH and a damping layer is proposed to achieve efficient low-frequency broadband vibration isolation. The modal participation factor was calculated via finite element modal superposition to identify the dominant low-frequency modes, and a high-fidelity dynamic model was established to analyze the key ABH parameters and damping layer configurations. A prototype validation was conducted on an ISG vertical centrifugal pump acceleration response. The results show that the isolator (L_ABH = 95 mm, h_uni = 10 mm, disk-shaped damping layer) achieves 8.87 dB and a higher vibration level drop of 17.52 dB in 10–315 Hz and 315 Hz–10 kHz, respectively, than non-ABH designs, with simulation–experiment errors of less than 5%. The ABH–dynamic vibration absorber synergistic mechanism overcomes the low-frequency limitation of conventional passive isolators, providing a reliable solution for marine power equipment vibration suppression.

1. Introduction

Vibrations, especially those in the low-frequency broadband range (typically below 300 Hz), pose a pervasive challenge across engineering fields—they compromise the precision and stability of mechanical systems [1,2], damage transportation infrastructure [3], and even threaten the safety of specialized equipment operating in harsh environments. For marine power systems, this problem is particularly acute: low-frequency vibrations (10–315 Hz, which is the dominant band for marine pumps) can accelerate equipment wear, degrade operational accuracy, and increase noise pollution, directly impacting ship navigation safety and crew comfort. Addressing low-frequency broadband vibration isolation has thus become a critical bottleneck in vibration control research, as conventional passive isolators often fail to suppress vibrations in the low-frequency range, and active/hybrid solutions struggle to adapt to the unique constraints of marine environments. An urgent need exists for tailored solutions that will bridge this gap.

Passive vibration isolators remain a cornerstone solution for marine power equipment isolation, valued for their technological maturity, cost-effectiveness, simple construction, and independence from external power supplies. These advantages are critical in marine environments, where active systems often face corrosion and power-supply challenges. To mitigate harmful vibrations, various approaches have been explored, including hybrid technologies [4,5], piezoelectric materials [6,7], mechanical metamaterials [8], quasi-zero stiffness (QZS) systems [9,10], and vibration energy harvesting structures [11]. While hybrid systems have demonstrated efficacy in enhancing isolation performance, their integration of active elements, such as piezoelectric air-springs [12,13], intelligent control algorithms [14], or magnetic suspension [15], introduces challenges related to stringent control requirements and poor reliability in salt-laden marine environments. QZS isolators offer potential for passive low-frequency isolation [16], yet their effectiveness is inherently constrained by the need to operate near the equilibrium point [17], limiting their performance in the 10–315 Hz band, which is dominated by marine pumps and engines. Furthermore, the practical implementation of QZS concepts—through magnetic springs [18], complex multi-mechanism designs [19], or bionic structures [20,21,22]—often introduces a structural complexity that hinders widespread engineering adoption. Consequently, existing technological paradigms struggle to meet the practical demands for an effective broadband low-frequency isolation in marine scenarios, necessitating the pursuit of innovative methodologies.

In recent years, acoustic black hole (ABH) structures—metamaterials engineered to manipulate vibrational energy and elastic waves—have garnered significant research interest. Building on this work, Krylov [23] pioneered research on wave propagation and damping effects in circular ABHs, providing theoretical insights into damping mechanisms [24]. The recent studies have focused on enhancing ABH performance at low frequencies, such as an integration with negative stiffness supports [25,26] and periodic arrays [27,28], but these designs either require complex structures that are incompatible with the compact marine equipment or lack broadband effectiveness [29]. Additionally, piezoelectric-integrated ABHs [29,30] face reliability issues in marine environments (e.g., piezoelectric degradation under humidity), creating a gap for passive low-frequency broadband solutions tailored to marine power systems.

The notable advances in ABH research include systematic analyses by Wang, who examined power flow and transmission in ABH beams [31] and coupled plate structures [32]. Early engineering applications—such as theoretical studies on vibration dissipation, damping, and noise reduction in ABH rectangular plates coupled with acoustic cavities [33]—further substantiate the practical potential of ABHs. Investigations into periodic ABH arrays [34,35] have highlighted the mechanical strength as a critical implementation factor. However, a persistent challenge for ABH structures is their limited efficacy in suppressing low-frequency vibrations, prompting significant research to focus on enhancing the low-frequency performance [36,37]. The strategies include integrating ABHs with heterogeneous periodic topologies or stiffness-graded structures [38,39] to boost the low-frequency attenuation, combining them with distributed mode absorbers [40] or local resonators [41,42] and augmenting ABH configurations with passive damping layers [43,44] or piezoelectric materials [45,46]. Notably, while piezoelectric materials present opportunities, their long-term reliability requires a deeper investigation compared to the well-established passive damping composites, making passive damping layers a more practical choice for low-frequency and broadband improvement.

Despite these advances, critical gaps remain for marine power equipment: QZS isolators exhibit a vibration level drop of only 35–40 dB in the 10–315 Hz band due to equilibrium point constraints; conventional ABH structures fail to suppress vibrations below 500 Hz, as their cutoff frequency is typically greater than 2000 Hz; and active/hybrid systems suffer from reliability issues in salt-laden marine environments. In response to the aforementioned research gaps and engineering pain points, this paper aims to address the problem of low-frequency broadband vibration isolation in the dominant frequency range of 10–315 Hz for power equipment such as marine vertical centrifugal pumps. By establishing a collaborative mechanism between the acoustic black hole (ABH) and dynamic vibration absorber (DVA), it can break through the low-frequency performance bottleneck of the traditional passive vibration isolators, develop a parameter design method for disk-shaped ABH vibration isolators, and verify the engineering applicability of the proposed scheme through simulations and prototype experiments. This provides a highly reliable, purely passive solution for the vibration suppression of equipment in marine environments.

2. Basic Theory of Additive ABH Disk-like Structure

The theoretical framework of the proposed passive isolator centers on the energy manipulation mechanism of disk-like acoustic black holes (ABHs), which are tailored to address the vibration isolation challenges of marine power equipment (10 Hz to 315 Hz). This section establishes the geometric–mechanical model of the disk-like ABH, quantifies the key performance-determining parameters, and reveals the underlying energy trapping mechanism.

2.1. Geometric and Mechanical Models of Disk-like ABH

The disk-like ABH is designed to be compact and compatible with marine equipment installation constraints, consisting of two functionally distinct regions (Figure 1): a central uniform cylinder (radius, r₁ and thickness, h_uni) for load bearing and the vibration input, and a radial variable-thickness region (L_ABH) (radial length from r₁ to r₂) for energy trapping. Theoretically, when a bending wave enters the ABH region, its wavelength shortens sharply, and its propagation speed decreases significantly. As a result, when the wave reaches the tip, it seems to stop moving, as if falling into a “black hole” from which it cannot escape or reflect.

The thickness of the variable-thickness region follows a power-law taper [33], which is a defining feature of ABH structures that enables wave velocity modulation. Mathematically, the thickness distribution h(r) is expressed as follows:

$$h(r)=\left\{\begin{array}{c}{h}_{uni},(0\le r\le r_1)\\ \epsilon {(r-r_2)}^m+h_0,(r_1\le r\le r_2)\end{array}\right.,$$

(1)

where h_ABH(r) is the thickness at radial position r, and m is the thickness gradient exponent (typically two for ideal ABH [33]). Due to the manufacturing constraints, a truncation thickness h₀(r) = 0.5 mm is retained at the outer edge of the variable-thickness region.

Based on Kirchhoff’s thin plate theory, the propagation of flexural waves (dominant in the structural vibration) in the disk-like ABH is governed by the following partial differential equation [46]:

$$\nabla \cdot \left(D(r)\nabla w(r,\theta ,t)\right)+\rho h(r){\displaystyle \frac{{\partial }^{2}w(r,\theta ,t)}{\partial t^2}}=0,$$

(2)

where w(r,θ,t) is the transverse displacement (z-direction, aligned with vertical vibration demand), D = Eh³/12(1 − ν²) is the bending stiffness of the ABH plate, and ρ = 7850 kg/m³, E = 206 GPa and ν = 0.3 are the density, Young’s modulus, and Poisson’s ratio of the structural steel, respectively.

For thick plate structures (with a thickness–span ratio of >1/5), the Mindlin theory takes into account the shear deformation and rotational inertia in the thickness direction of the plate, resulting in a higher solution accuracy. However, the derivation process and the complexity of the numerical calculations are significantly increased. In this study, the thickness–span ratio of the ABH structure is always less than 1/10, which falls into the category of typical thin plates. The influence of shear deformation on the propagation of the bending waves can be neglected. The Kirchhoff–Love theory can accurately describe the bending vibration characteristics of the structure while greatly simplifying the theoretical model and numerical calculation process. Therefore, this study adopts this theory for the dynamic modeling of the ABH structure.

A single harmonic analysis is the basis for analyzing the frequency domain characteristics of linear systems. The core objective of this study is to reveal the mechanisms by which the ABH structure regulates, concentrates, and dissipates the bending waves of different frequencies. A single harmonic analysis can accurately characterize the wave number, propagation characteristics, cut-off frequency, and energy concentration effect of the bending waves at any frequency, avoiding the interference of multi-frequency coupling on the core physical mechanisms. The ABH vibration structure in this study is a linear elastic system that satisfies the superposition principle. The total vibration response of the structure is the linear superposition of the harmonic responses of every single frequency, w(r,θ,t) = W(r,θ)exp(iωt), where ω = 2πf is the angular frequency. Equation (2) simplifies this to the Helmholtz equation:

$$\nabla \cdot \left(D(r)\nabla W\right)+\rho h(r){\omega }^{2}W=0.$$

(3)

Using the geometric acoustics approximation [35], the wavenumber k(r) in the variable-thickness region is derived by substituting h(r) into the expression for the uniform-plate wavenumber:

$$k(r)=k_0\cdot {\left({\displaystyle \frac{h_{\mathrm{uni}}}{h(r)}}\right)}^{1/2}=k_0\cdot {\left({\displaystyle \frac{r_1}{r}}\right)}^{m/2},$$

(4)

where D₀ = Eh_uni³/12(1 − ν2) is the uniform-region bending stiffness.

However, the cutoff frequency (f_c) is a crucial indicator for determining whether the ABH structure can absorb and reflect elastic waves. The f_c is the minimum frequency at which the ABH can effectively trap flexural waves, below which wave lengths are too large to interact with the ABH’s thickness gradient [47]. For the disk-like ABH, f_c is determined by the relationship between the ABH characteristic length (L_ABH) and the uniform height (h_uni):

$$f_c={\displaystyle \frac{h_{\mathrm{uni}}}{4L_{\mathrm{ABH}}}}\cdot \sqrt{{\displaystyle \frac{E}{24\rho (1-{\nu }^2)}}}.$$

(5)

where ρ denotes the material density. This is defined based on the wavelength–characteristic length relationship [48], where L_ABH = r₂ − r₁ represents the radial length of the variable-thickness region.

It can be clearly determined from the above formula that the cut-off frequency of the ABH structure is related to the length of the ABH region and the thickness of the uniform region. Therefore, in the process of parametric research on the ABH structure, the influence of the above two parameters on the ABH structure and the vibration isolation performance of the ABH vibration isolator should be analyzed.

2.2. Energy Transmission and Reflection Process of Disk-like ABH

The ABH’s energy manipulation capability—central to isolator performance—occurs in three sequential stages, quantified by wave propagation theory and validated by subsequent energy flow simulations. For the ABH structures with a truncation thickness of h₀, the energy flow in such structures always involves these two stages: wave convergence and energy accumulation and energy eddy formation caused by reflection.

For the wave convergence and energy accumulation stage, the flexural waves propagate from the uniform region into the variable-thickness region, where the wave velocity decreases with r. This velocity gradient causes the wavefronts to bend toward the ABH edge, concentrating energy into a small radial zone. The energy density (ε(r)) (kinetic and potential energy per unit volume) in this zone is derived by substituting h₀ and k(r) into the elastic wave energy equation [33]:

$$\mathcal{E}(r)={\displaystyle \frac{1}{2}}\rho {\omega }^2|W(r){|}^{2}h(r)+{\displaystyle \frac{1}{2}}D(r)|\nabla W(r){|}^2\propto {\displaystyle \frac{|W(r){|}^2}{r^2}}.$$

(6)

Furthermore, the truncation thickness h₀ prevents ideal wave absorption, causing partial wave reflection at the ABH edge. The reflection coefficient R₀ can be [46]:

$$R_0=\exp \left(-2{\displaystyle {\int }_{r_1}^{r_2}k\left(r\right)dr}\right)\left({\displaystyle \frac{k\left(r_2\right)-k\left(r_1\right)}{k\left(r_2\right)+k\left(r_1\right)}}\right).$$

(7)

The integral term, representing the cumulative phase shift, can be solved by substituting Equation (4), where m = 2:

$$\Phi ={\displaystyle {\int }_{r_1}^{r_2}k}(r)dr=2k_0{r}_1\left(1-\sqrt{{\displaystyle \frac{r_1}{r_2}}}\right).$$

(8)

Equation (8) shows the partial reflection ∣R₀∣ < 1 and arg(R₀) = −2Φ. The reflected waves interfere coherently with the incident waves, forming rotational energy eddies in the variable-thickness region. These eddies prolong the wave–structure interaction time, maximizing the energy exposure to damping materials.

During the processes of energy accumulation, transfer, and vortex formation, damping layers (DLs) can assist the ABH structure in achieving energy conversion more rapidly by relying on the hysteretic effect. A viscoelastic DL (for example, polyurethane materials, loss factor η = 0.35) can be integrated to convert trapped energy into heat. For the harmonic vibration, the unit volume damping capability D_cap can be given as:

$$D_{\mathit{cap}}=\pi \eta \epsilon \left(r\right).$$

(9)

Since D_cap reflects energy dissipation per cycle, the energy dissipation rate per unit volume D_diss is further correlated with excitation frequency f:

$$D_{diss}=\pi \eta \epsilon \left(r\right)f.$$

(10)

This rate directly translates to an improved structural damping performance. The system-level loss factor η_sys, a key indicator of overall vibration suppression capability, integrates the DL’s contribution with the base ABH structure. For the ABH-DL composite system, η_DL is calculated as the ratio of dissipated strain energy ΔU to the total stored elastic energy:

$$U_{\mathit{stored}}={\displaystyle \frac{\Delta U_{\mathit{ABH}}+\Delta U_{\mathit{DL}}}{U_{\mathit{stored},\mathit{ABH}}+U_{\mathit{stored},\mathit{DL}}}},$$

(11)

where ΔU_DL = η_{DLUstored, DL} (with η_DL = 0.35 for polyurethane), and ΔUABH is the inherent dissipation of the steel ABH (negligible, η_DL = 0.01). This confirms the DL as the dominant dissipation source.

3. Numerical Design of the ABH Isolator

From Figure 2, the additional ABH disk-like configuration primarily mitigates vibration energy through a connecting rod-constituted dynamic vibration absorber (DVA)-mimicking mechanism. The modal participation factor (MPF) is used to quantitatively characterize the participation weight of each natural mode of a structure in its overall dynamic response. Consequently, modal analysis was performed on the isolator to quantify each modal order’s contribution to the low-frequency structural vibration. Meanwhile, the relationship between the vibration isolation performance and the parameters of the disk-like ABH structure should be analyzed. Furthermore, the vibration energy transmission eddy current within the ABH disk-like structure was calculated and analyzed, serving as a critical criterion for validating the structural solution.

3.1. Modal Characteristics of the ABH Isolators

From Figure 2 the first twenty natural frequencies of the ABH disk-like structure (LABH = 95 mm, huni = 5 mm, h0 = 0.5 mm, and elastic stiffness kk = 6 × 10⁸ N/m) were calculated, with the results listed in

Table 1. The modal frequencies of the isolator in Figure 2.

Modal order	6th	7th	8th	9th	10th
Frequency/Hz	10.52	24.92	46.62	49.48	51.78
Modal order	11th	12th	13th	14th	15th
Frequency/Hz	149.97	181.78	205.98	232.62	256.71
Modal order	16th	17th	18th	19th	20th
Frequency/Hz	257.69	282.15	291.31	300.21	302.63

The first five natural frequencies are 1.81 Hz, 2.09 Hz, 3.33 Hz, 8.07 Hz, and 9.11 Hz, respectively. These five modal frequencies are in the ultra-low frequency range, under 10 Hz.

. The first five natural frequencies are 1.81 Hz, 2.09 Hz, 3.33 Hz, 8.07 Hz, and 9.11 Hz, respectively—these are ultra-low frequency modes with over 60% rotational deformation around the x/y axes. Per Section 2.1, in Kirchhoff’s thin plate theory, rotational modes contribute minimally to vertical vibration (the core demand for the marine equipment), so they are excluded from the subsequent analysis.

The physical significance of the MPF is the vibration contribution coefficient of a certain mode to a specific degree of freedom, and its value is calculated by the normalization of the inner product of the modal displacement vector and the excitation force vector. Therefore, the MPFs of the target orders within 10 Hz to 315 Hz can be evaluated easily based on the following model shown in Figure 3.

In this model, f(t) drives the isolator on (0 m, 0 m, and 0.191 m), where the amplitude is 1 N from 1 Hz to 500 Hz; kk = 6 × 10⁸ N/m is the elastic connection stiffness between the entire isolator system and the rigid ground.

Figure 4 presents the MPF contour map of the isolator in the 1 Hz to 500 Hz band, with distinct peaks marking the dominant modes: 24.92 Hz (7th mode), 49.48 Hz (9th mode), 51.78 Hz (10th mode), 149.97 Hz (11th mode), 300.21 Hz (19th mode), and 302.63 Hz (20th mode). These high MPF values indicate a strong resonance excitation, confirming the six modes as the primary contributors to the low-frequency vibration. Excluding the noise modes with the extraordinarily small MPFs, the remaining dominant modes are concentrated in the 10 Hz to 315 Hz band, aligning with the target frequency range.

To clarify the vibration characteristics of the dominant modes, the modal shapes of the six high-MPF orders were extracted (Figure 5), which all exhibited a DVA-like mechanism formed by the ABH disk and connecting rods.

For the seventh modal shape (24.3 Hz), the isolator base and shell deform slightly. The load-bearing end undergoes a 71% z-directional deformation (

Table 2. The proportion of deformation in the different DOF directions within the dominant mode shapes.

Modal Order	Frequency/Hz	z-Directional Translational Proportion	x, y-Directional Translational Proportion	z Axis-Wise Rotational Proportion
7th	24.92	71%	18%	11%
9th	49.48	82%	14%	4%
10th	51.78	76%	14%	10%
11th	149.97	79%	12%	9%
19th	300.21	78%	15%	10%
20th	302.63	81%	11%	8%

), directly responding to the vertical vibration demands. Meanwhile, for the 9th–10th modal shapes (50.2 Hz to 52.5 Hz), the ABH disk vibrates significantly. With the connecting rods synchronously vibrating (a swing angle smaller than 1° relative to the vertical axis) and the negligible load-bearing end motion, the resonance behavior enables effective energy trapping. The coupled vibration occurs in the 11th, 19th, and 20th modal shapes (148.9 Hz to 308.6 Hz). However, the z-directional motion accounts for 76–81% of the total deformation (Table 2), ensuring vertical vibration control. All six dominant modes function as DVA-like structures, laying the foundation for the low-frequency broadband vibration isolation.

All the modes with high MPFs can thus function like DVA structures, enabling the control and absorption of the vibration energy within the target frequency range. However, the extent to which the introduction of such a DVA-like structure can enhance the performance of the passive vibration isolators still requires further analysis and consideration.

3.2. Parameter Investigation and Analysis

After analyzing the modal characteristics of the vibration isolator in the low-frequency band, it is determined to exhibit a DVA-like structure between 10 Hz and 315 Hz. However, neither its broad frequency-band vibration isolation performance nor the influence of the ABH disk-like structural parameters has been confirmed or evaluated. Per Section 2.1’s cutoff frequency equation and DL theory, using the numerical model in Figure 6, the vibration isolation performance of the isolators with different structural parameters (such as L_ABH and h_uni and DLs) within 1 Hz to 10 kHz can be analyzed. Two measurement points are arranged on the isolators to calculate their vibration level drop (VLD).

The vibration acceleration responses can be measured at M1(−0.01 m, −0.01 m, and 0.191 m) and M2 (−0.175 m, −0.175 m, and 0.025 m), where the origin of the coordinate system is located at the center of the bottom surface of the isolator base. The excitation force (f(t)), located at the same point in Figure 4, with the amplitude 1 N. kk = 6 × 10⁸ N/m, is the elastic connection stiffness between the entire isolator system and the rigid ground, which is the same as the model in the above chapter.

The VLD is used as the performance metric:

$$VAL{R}_{M1,M2}=20\log (a_{M1,M2}/a_{ref}),$$

(12)

$$VLD=VAR{L}_{M1}-VAR{L}_{M2},$$

(13)

where a_M1 and a_M2 represent the vibration accelerations evaluated by the acceleration transducer, a_ref is the reference acceleration when calculating the vibration level, usually a_ref = 1 × 10⁻⁶ m/s². VARL_M1 and VARL_M2 represent the vibration acceleration response level of the M1 and M2 measurement points, respectively.

Initially, the improvement effect of the vibration isolation performance for the isolator system was evaluated upon introducing the initial ABH disk-like structure. The initial design is the vibration isolation system without the ABH disk-like structure, while Case 1 (L_ABH = 95 mm, h_uni = 5 mm) refers to the system that has an ABH initial structure.

Figure 7 illustrates that the ABH disk-like structure enhances the performance of the vibration isolation system.

As Case 1’s f_c = 2598 Hz, dividing the VLD spectrum into two segments, the vibration isolation performance can be discussed with the following two frequency ranges: below f_c (10–2598 Hz), where Case 1’s VLD is 2–4 dB higher than the initial design, which is attributed to DVA-like resonance, and above f_c (2598–10 kHz), where the VLD improves by 4–8 dB, resulting from the ABH’s flexural wave convergence and eddy formation. The initial design achieves 61.52 dB (10–315 Hz) and 39.79 dB (315 Hz–10 kHz), while Case 1 yields 65.78 dB and 45.11 dB (Appendix A,

Table A1. The comparison of the average VLD with and without the ABH disk-like structure.

Structural Characteristic	Average VLD/dB	Average VLD/dB
	10 Hz–315 Hz	10 Hz–315 Hz
Isolator without ABH disk-like structure (initial design)	61.52	61.52
Isolator with ABH disk-like structure (Case 1: LABH = 95 mm, huni = 5 mm)	65.78	65.78

), which is a 4.26 dB and 5.32 dB improvement, respectively. These results confirm the feasibility of ABH integration for the improvement of both the low- and high-frequency performance.

To further investigate the vibration energy confinement capability of ABHs, ABH disk-like structures with varying parameters were integrated into the isolation system, and their performance was evaluated. First, the influence of the different ABH region lengths on the vibration isolation performance was assessed. The vibration isolators incorporating the ABH structures, with L_ABH = 50 mm and L_ABH = 10 mm, were included as references, with their respective cutoff frequencies calculated as 9358 Hz and 233,973 Hz. Case 1, Case 2, and Case 3 refer to the isolation system with a L_ABH = 95 mm, L_ABH = 50 mm, and L_ABH = 10 mm ABH disk-like structure. The numerical analysis results of the vibration isolation performance for the above-mentioned three cases are shown in Figure 8.

From the Figure 8, the cutoff frequency increases are 9358 Hz (Case 2) and 23.39 kHz (Case 3). The cutoff frequency for Case 3 is well above 10 kHz, such that neither the VLD spectrum nor the one-third octave analysis captures its efficacy. As L_ABH decreases, fc increases sharply. Case 3’s fc far exceeds 10 kHz, so no ABH energy trapping occurs in the analyzed band. Its slight VLD improvement, from 39.79 dB to 40.04 dB, stems from inertial damping due to an increased structural mass, which is unrelated to the ABH wave manipulation. For the cutoff frequencies under 10 kHz (Case 1 and Case 2), the vibration isolation performance from their cutoff frequencies to 10 kHz is improved. However, Case 2’s f_c = 9358 Hz narrows the effective high-frequency band to 9358–10 kHz, with an average VL of 40.84 dB, which is 4.27 dB lower than Case 1. Case 1’s f_c = 2598 Hz covers 74.0% of the target high-frequency band, achieving the highest VLD with 45.11 dB. Furthermore, the one-third octave analysis further confirms that a larger L_ABH lowers f_c to match the target bandwidth, maximizing the energy dissipation (Appendix A,

Table A2. The comparison of the isolation systems’ average VLD with different L_ABH.

Structural Characteristic	Average VLD/dB	Average VLD/dB
	10 Hz–315 Hz	10 Hz–315 Hz
Isolator without ABH disk-like structure (initial design)	61.52	39.79
Isolator with ABH disk-like structure (Case 1: LABH = 95 mm, huni = 5 mm)	65.78	45.11
Isolator with ABH disk-like structure (Case 2: LABH = 50 mm, huni = 5 mm)	64.42	40.84
Isolator with ABH disk-like structure (Case 3: LABH = 10 mm, huni = 5 mm)	63.49	40.04

).

Subsequently, the uniform area height (h_uni) is selected for analysis. Due to the lower sensitivity to the cutoff frequency described in Equation (8), the ABH disk-like structure with h_uni = 10 mm has been analyzed as a reference case.

When h_uni = 10 mm, the f_c of Case 4 increases to 5185 Hz. The isolation performance depicted in Figure 9a is also enhanced. This indicates that a higher h_uni provides bending waves with a longer propagation path. During propagation, the waves continuously interact with the structure, undergoing physical phenomena such as reflection, refraction, and scattering, thereby enhancing the opportunities for vibrational energy dissipation. The one-third octave analysis results further validate this finding. Furthermore, Case 4’s average VLD is 6 dB higher than Case 1 due to an increased h_uni boosting modal mass participation in the high-MPF modes, which strengthen the DVA-like resonance. The one-third octave data (Appendix A,

Table A3. The comparison of the isolation systems’ average VLD with different h_uni.

Structural Characteristic	Average VLD/dB	Average VLD/dB
	10 Hz–315 Hz	10 Hz–315 Hz
Isolator without ABH disk-like structure (initial design)	61.52	39.79
Isolator with ABH disk-like structure (Case 1: LABH = 95 mm, huni = 5 mm)	65.78	45.11
Isolator with ABH disk-like structure (Case 4: LABH = 95 mm, huni = 10 mm)	68.29	48.19

) shows Case 4’s 10–315 Hz and 315 Hz–10 kHz VLD reach of 68.29 dB and 48.19 dB, confirming that h_uni = 10 mm balances the low-frequency resonance and high-frequency trapping.

The parametric analysis reveals that the larger values of L_ABH and h_uni enhance the energy dissipation above the cutoff frequency. The damping layers (DLs) enhance the energy dissipation via the hysteresis effects (Section 2.2). Thus, a detailed analysis was conducted to investigate the performance enhancement of the vibration isolation system enabled by the DL. Case 5 (ring-shaped DL, 40% coverage) and Case 6 (disk-shaped DL, 100% coverage) were compared to Case 1 (no DL). The isolators with DL structures are shown in Figure 10.

The results in Figure 11a demonstrate that integrating the DL significantly enhances the vibration isolation systems’ performance. From the 1–10 kHz VLD spectrum, both of the DL configurations improve the performance, with Case 6 outperforming Case 5—the disk-shaped DL’s full coverage maximizes the contact with the ABH region, accelerating the energy conversion to heat. Below f_c = 2598 Hz, Case 6 forms dense energy eddies by coupling with the DVA-like mechanism, boosting the 10 Hz to 315 Hz VLD by 7.14 dB (compared with Case 1). For the high frequency band, Case 6’s VLD has been improved to 57.31 dB in 315 Hz–10 kHz, which is 12.2 dB higher than Case 1. Meanwhile, Case 5 only reaches 46.78 dB in the high-frequency band. This difference stems from Case 6’s 92% energy dissipation efficiency. The one-third octave analysis (Appendix A,

Table A4. The comparison of the isolation systems’ average VLD with different h_uni.

Structural Characteristic	Average VLD/dB	Average VLD/dB
	10 Hz–315 Hz	10 Hz–315 Hz
Isolator without ABH disk-like structure (initial design)	61.52	39.79
Isolator with ABH disk-like structure (Case 1: LABH = 95 mm, huni = 5 mm)	65.78	45.11
Isolator with ABH disk-like structure (Case 5: LABH = 95 mm, huni = 10 mm, hDL = 2 mm ABH ring-like DL)	68.31	46.78
Isolator with ABH disk-like structure (Case 6: LABH = 95 mm, huni = 10 mm, hDL = 2 mm ABH disk-like DL)	72.52	57.31

) can also confirm that the disk-shaped DL is the configuration for the cross-band vibration suppression.

3.3. Energy Flow Analysis in the Designed Isolators

Previous studies have evaluated the vibration isolation capability of different isolators within the target frequency band, but the disk-like ABH structure’s exceptional ability to concentrate and dissipate vibrational energy lacks explicit demonstration. To address this, meticulous numerical computations were conducted to analyze the vibrational energy distribution and energy flow density within the disk-like ABH under various modal orders, aiming to establish quantitative criteria for a structural parameter design. The energy vortices in the ABH disk-like structures with varying configurations are depicted using vector arrows. Each unit-length vector represents an energy flux density of 1 × 10⁻⁴ W/m³ passing through the local region, whereas a higher density of vectors indicates more intricate energy flow patterns within that area. First, the vibration energy flow in the Case 1 ABH disk-like structure is evaluated as shown in Figure 12.

In the high-frequency band (after f_c, covering most of the 315 Hz–10 kHz effective bandwidth), the energy flow exhibits pronounced concentration: flexural waves converge radially outward from the uniform region, with wavenumber k(r) increasing per Equation (4) to reduce the wave velocity, driving the energy flow density from about 0.8 × 10⁻⁴ W/m² in the uniform region to about 6.1 × 10⁻⁴ W/m² in the ABH outer region. The partial reflection ∣R0∣ ≈ 0.3 forms dense energy eddies, which persist across 2598 Hz–10 kHz, resulting in a 45.11 dB VLD in the high-frequency segment of the effective bandwidth. Below f_c, the wave gradient effects weaken, but the ABH disk and the connecting rods form a DVA-like resonance structure (confirmed by MPF analysis), trapping the energy via mechanical resonance to yield faint eddies with a core density of 4.5 × 10⁻⁴ W/m², enabling basic sub-f_c manipulation.

Case 2 (L_ABH = 50 mm, f_c = 9358 Hz) and Case 3 (L_ABH = 10 mm, f_c = 233.97 kHz) reveal L_ABH’s critical role in the high-frequency effective bandwidth. The energy transmission vectors are shown in Figure 13.

Case 2’s f_c = 9358 Hz narrows the effective high-frequency band to 9358 Hz–10 kHz, with the energy flow density peaking at 3.8 × 10⁻⁴ W/m² (44% lower than Case 1) and eddies only forming in this limited range, reducing the effective bandwidth utilization to 68%. Case 3’s f_c far exceeds 10 kHz, so a negligible ABH-induced energy concentration occurs in the entire 315 Hz–10 kHz band; the energy flow density remains uniform (0.7 × 10⁻⁴ W/m² to 0.9 × 10⁻⁴ W/m²), with attenuation from inertial damping alone. By comparing Figure 12d–f and Figure 13, the results can further validate the conclusions drawn in Section 3.2. Specifically, the larger the value of L_ABH, the higher the energy density within the ABH structure and the more pronounced the improvement effect of the vibration isolation provided by the ABH.

The distribution characteristics of the energy transfer vectors within the ABH structure featuring a uniform thickness of 10 mm (Case 4) are analyzed, and the corresponding results are illustrated in the Figure 14.

In the high-frequency band (5185 Hz–10 kHz), the increased h_uni boosts the uniform-region bending stiffness D₀, raising the energy input to the ABH region (1.8 × 10⁻⁴ W/m² compared with Case 1’s 1.2 × 10⁻⁴ W/m²). The energy flow density peaks at 7.3 × 10⁻⁴ W/m², which is 20% higher than Case 1, with eddy density increasing 30% across the band, lifting the high-frequency VLD to 48.19 dB. Below f_c, the larger h_uni strengthens the ABH disk’s mass participation in the DVA-like resonance, driving the eddy density to 5.2 × 10⁻⁴ W/m² at 1000 Hz, balancing the high-frequency efficacy and the sub-f_c capture.

Furthermore, the energy transmission vectors in Case 5 and Case 6 were computed and evaluated to further demonstrate the enhancement effect of the damping layer on the vibration isolation performance of the vibration isolator, with the corresponding results compared and presented in Figure 15.

From the Figure 15, in the high-frequency band, Case 6’s full-coverage DL dissipates 92% of the trapped energy compared with Case 5, with the energy flow density that is downstream of the eddies dropping to 0.5 × 10⁻⁴ W/m², compared with Case 5’s 1.2 × 10⁻⁴ W/m², pushing the high-frequency VLD to 57.31 dB. The disk-shaped DL maximizes the contact with the ABH region, enhancing hysteretic dissipation (D_diss) by about 0.26 × 10⁻⁴ W/m² across the 315 Hz–10 kHz band. Below fc, Case 6’s DL induces pronounced eddies (absent in Case 5) by coupling with the DVA-like motion, with core density of 6.8 × 10⁻⁴ W/m², while Case 5’s edge-only DL fails to leverage the trapped energy, confirming that the full-coverage DL is key to cross-band efficacy.

In summary, high-frequency (post-f_c) energy manipulation hinges on matching the ABH parameters to the 315 Hz–10 kHz effective bandwidth: L_ABH = 95 mm covers most of the high frequencies, h_uni = 10 mm boosts the energy accumulation, and disk-shaped DL amplifies the dissipation, resulting in 57.31 dB VLD in the high-frequency segment. Below f_c, the DVA-like resonance of the ABH disk and connecting rods provides the foundational trapping mechanism, strengthened by h_uni’s enhanced modal participation and disk-shaped DL’s damping-induced eddy formation, overcoming traditional ABH sub-f_c ineffectiveness. This synergistic design enables a seamless energy manipulation across the entire effective bandwidth.

4. Experimental Verification and Results

To further validate the accuracy of the numerical results in Section 3, the design validity was verified by comparing numerical simulations with experimental data. Taking Case 6 as an example, the experimental validation commenced with a prototype fabrication. The acceleration sensors M1 and M2 were employed to measure the acceleration spectrum signals at the input and output ends of the ABH isolator, respectively. The VLD of the prototype was then calculated based on these measurements.

Consistent with the arrangement of the vibration acceleration response points in the numerical analysis that was mentioned earlier, in the experimental evaluation, all sensors are installed rigidly on the isolator’s structure, and their positions are the same:

M1 measuring point: This is arranged at the center of the top surface of the isolator bearing end (coaxial with the exciter output rod and impedance head), with coordinates (−0.01 m, −0.01 m, and 0.191 m), to measure the input vibration acceleration of the isolator.

M2 measuring point: This is arranged at the diagonal position of the isolator base flange surface, with coordinates (−0.175 m, −0.175 m, and 0.025 m), to measure the output vibration acceleration of the isolator (foundation end).

The measured acceleration signals from these transducers are normalized against the force signals that are obtained by the impedance head to derive the acceleration spectrum signals under unit force excitation. To further simulate the elastic constraints that vibration isolators typically encounter during use, this paper has provided the stiffness boundary conditions of the elastic constraints in the numerical calculations. In the experimental verification process, this paper uses a standard BE-type vibration isolator to simulate this condition. Finally, an experimental device schematic diagram as shown in Figure 16 and a test bench as shown in Figure 17 are formed.

4.1. Experimental Validation

First, a swept-frequency signal (1 Hz–10 kHz) was generated by a signal generator, amplified via a power amplifier, and input into a vibration exciter to induce vibrations in the isolator. The normalized vibration acceleration can be tested and compared with the numerical calculation.

As shown in Figure 18a,b, the maximum peak frequency error (2000–3000 Hz) is about 150 Hz (8%), which is attributed to the prototype’s truncation thickness of h₀ = 0.7 ± 0.1 mm (compared with the simulation model’s h₀ of 0.5 mm), which increases the actual cutoff frequency. In the 100 Hz to 500 Hz band, the experimental and simulated acceleration levels overlap perfectly. Three repeated experiments were conducted, with the VLD standard deviation less than 1.2 dB, confirming the reliability.

Table 3. The comparison of the average VLD between the simulation and experiment for Case 6.

Evaluation Method	Average VLD/dB
	10 Hz–315 Hz	315 Hz–10 kHz
Experiment	70.39	55.34
Simulation	72.52	57.31
Error	3.03%	3.56%

compares the average VLD of the simulation and experiment.

4.2. Feasibility Evaluation Under Marine Pump Load

To validate the engineering application effectiveness of the proposed ABH vibration isolator, the vibration characteristics of a marine vertical centrifugal pump foundation were selected as the excitation input. However, directly adopting the actual pump body for excitation is both difficult and hazardous: integrating a full-scale marine vertical centrifugal pump into the experimental setup requires complex auxiliary systems and substantial spatial resources, leading to great operational difficulty and poor experimental controllability. Moreover, such load data cannot be generated or output by a standard signal generator. To address this dual challenge of practical infeasibility, a visual programming software was employed to develop a simulation output program as shown in Figure 19, which enables the vertical centrifugal pump load to be input to the vibration exciter via a power amplifier.

The excitation is applied to the top of the isolator as an analog load, and its vibration response and vibration reduction characteristics are tested. The structural parameters of the vertical centrifugal pump and test operating conditions are listed in

Table 4. The performance parameters of the ISG vertical centrifugal pump.

Flow Rate/t/h	Speed Range/rpm	Rated Working Condition/rpm	Weight/kg	Number of Blades
260	600–1500	1500	417	6

.

When the pump operates under rated conditions, its shaft frequency, blade passing frequency, and double frequencies are 25 Hz, 50 Hz, 150 Hz, and 300 Hz, respectively. The vibration isolation performance at these characteristic frequencies deserves particular attention. As illustrated in the following Figure 20, the vibration acceleration response measured at the mounting foot of the water pump body was exerted on the input terminal of the ABH vibration isolator using the exciter.

At the pump’s shaft frequency, Figure 21b shows that the ABH isolator achieves a VLD of 42 dB, which is about a 12 dB improvement over the initial design. At the blade passing frequency of 150 Hz, the VLD reaches 58 dB, significantly reducing the vibration transmission to the foundation. Notably, even at the double shaft frequency and the blade passing frequency, the isolator maintains a VLD of 55 dB and 62 dB, ensuring a stable performance across the critical operational frequencies of power-generating equipment. These results not only validate the high consistency between the numerical model and experimental prototype but also confirm the isolator’s suitability for marine environments with multi-source low-frequency excitation.

5. Discussion

The passive isolator integrated with a disk-like ABH developed in this study achieves an average VLD of 70.39 dB in the 10–315 Hz band, which dominates the vibration of marine centrifugal pumps, and 55.34 dB in the 315 Hz–10 kHz band, with a deviation between the simulation and experiment of less than 5%. This work effectively addresses the critical challenge of the low-frequency broadband vibration isolation failure of conventional passive isolators in marine power equipment. To clarify the core advancements and scientific innovations of this work, a comparative analysis against the methodologies in the field of vibration isolation is presented herein.

Compared with the mainstream marine engineering isolators (rubber and QZS types), this isolator overcomes their inherent limitations. The rubber isolators lose efficiency below 50 Hz, while the QZS isolators have a constrained operating range, delivering only a 35–40 dB VLD in 10–315 Hz under the variable loads and posing stability risks due to nonlinear structures. The proposed isolator has no equilibrium constraints, maintains stable performance under variable loads, and achieves an 8.87 dB VLD improvement in 10–315 Hz vs. the non-ABH baseline, while retaining the merits of conventional passive isolators: simple structure, no external power, and high marine reliability.

This work fills the key gaps in ABH research and establishes a novel framework distinct from the traditional ABH-DVA combinations. The conventional disk-like ABHs have a cutoff frequency above 2000 Hz; split ABH-DVA designs only realize narrow-band single-frequency absorption, and the other ABH low-frequency expansion schemes suffer from bulkiness and instability. Its fundamental scientific novelty lies in three coupled innovations: an integrated ABH-DVA synergistic mechanism, achieving a deep coupling of energy focusing and a resonant absorption for 10–315 Hz broadband absorption; a MPF-based low-frequency mode matching method, expanding ABH’s effective band to 10 Hz by precise frequency matching; and a pure passive ABH engineering design method for harsh marine conditions, with a 17.52 dB VLD improvement in 315 Hz–10 kHz and outperforming the existing marine passive isolation schemes.

Compared with the active and hybrid isolation schemes, such as piezoelectric active control and maglev hybrid isolation, this work resolves the core reliability problem of active systems in marine environments. Although the active and hybrid schemes can achieve an excellent low-frequency isolation performance, they rely on external power supplies and complex sensing and control loops. Active components, such as piezoelectric wafers, are prone to degradation and failure in harsh marine environments, failing to meet the long-term operation requirements of marine equipment. The all-passive isolator that is proposed in this work has a reliability level comparable to that of the conventional rubber isolators, while achieving a low-frequency broadband isolation performance on par with the active schemes, giving it outstanding engineering applicability in marine engineering.

This work still has certain limitations: the cutoff frequency of the ABH structure is 2598 Hz, meaning the flexural wave modulation effect in the band below 2000 Hz still needs improvement. Future research will systematically analyze the thickness gradient exponent of the ABH structure in the range of 1.5 to 2.5, to further reduce the cutoff frequency to below 2000 Hz and to realize the efficient modulation of flexural waves in the full band from 10 Hz to 10 kHz.

6. Conclusions

Based on the practical engineering requirements, this study focuses on the passive vibration isolators for a vertical centrifugal pump. During the research process, mathematical modeling, simulation calculation, and experimental testing are integrated to conduct an in-depth analysis of the designed isolators, aiming to address the technical challenge that passive vibration isolators exhibit poor performance in isolating low-frequency broadband vibrations, with the key conclusions:

(1) The ABH-DVA synergistic mechanism enables targeted low-frequency vibration isolation: The disk-like ABH and connecting rods form a DVA-like structure, dominating the 10–315 Hz band (with z-directional deformation more than 75% in dominant modes). Compared with non-ABH isolators, the VLD is increased by 4.26 dB to 70.39 dB, breaking the low-frequency limitation of the conventional passive isolators.

(2) Parameter synergy ensures broadband efficacy: L_ABH = 95 mm regulates the cutoff frequency f_c = 2598 Hz, and h_uni = 10 mm enhances the modal mass participation for energy accumulation, and the disk-shaped damping layer maximizes hysteretic dissipation. The VLD reaches 55.34 dB in the 315 Hz–10 kHz band, meeting the vibration standards for marine equipment.

(3) Verification of engineering validity in marine scenarios: The isolator (L_ABH = 95 mm, h_uni = 10 mm, disk-shaped DL) exhibits a simulation–experiment error of less than 5%. For the analog load of an ISG centrifugal pump, the VLD achieves 42–62 dB at critical frequencies (25/50/150/300 Hz), confirming its applicability.

Future research will focus on a systematic improvement of the ABH structural parameters, particularly the gradient exponent within the range of 1.5 to 2.5, with the primary objective of reducing the characteristic cutoff frequency (f_c) below 2000 Hz.