Research on Vibration Reduction Characteristics and Optimization of an Embedded Symmetric Distribution Multi-Level Acoustic Black Hole Floating Raft Isolation System

Abstract

The subject of ship structural dynamics has faced new technological obstacles due to scientific and technological advancements, and one of the main concerns in related sectors is how to effectively reduce the vibration levels of different ships. This article focuses on the application scenarios of ship floating raft isolation systems, establishing a wave propagation model for acoustic black hole (ABH) structures based on the idea of the ABH effect. Then, a transfer matrix model for serially connected ABH structures is derived, which serves as a basis for subsequent structural designs. Second, the finite element method is used to study the energy distribution and vibration characteristics of a symmetrically distributed periodic non-uniform multi-level ABH structure. Meanwhile, it examines its bandgap distribution under a one-dimensional periodic arrangement and then investigates the vibration properties of non-uniform multi-level ABH thin-plate constructions with different periods from the perspective of engineering applications. Moreover, parameter optimization studies of non-uniform multi-level ABH structures with finite periods are carried out with an emphasis on engineering applications. The first step is to use the design space to determine the range of values for the parameters that need to be optimized. The hyper Latin cubic sampling method is then employed to select samples, and the EI criterion and PSO optimization algorithm are applied to add new samples to improve the Kriging surrogate model’s accuracy. When the optimal structural parameters have been determined, they are applied to the raft rib plate to verify the isolation effect of the non-uniform multi-level ABH structure by analyzing the vibration level difference at specific raft positions before and after embedding it.

1. Introduction

Vibration and noise control technology is a crucial area of study in the shipbuilding industry, particularly in highly disguised equipment such as submarines and cruise ships. A large size, fast speed, and light weight are the goals of modern ships. As a result, the propulsion device’s power output is continuously rising—single-unit power can reach tens of thousands of kilowatts, and vibration levels can surpass 110 dB [1]. This has resulted in increasingly noticeable structural vibrations, which have a number of negative effects. The ship and the operating environment will be negatively impacted by the coupled propagation of these vibrations in the ship’s structure, which will shorten the ship’s service life and cause fatigue damage to the structure. It will also interfere with the operation of important instruments, affecting navigation performance and safety [2,3]. Increasing ships’ underwater radiation noise, particularly for submarines, has an impact on the acoustic stealth effect and decreases battlefield survivability [4]. Long-term exposure to vibration conditions can also negatively impact crew members’ mental and physical health, which lowers their productivity and standard of living. Furthermore, an increasing number of studies show that anthropogenic noise harms marine life [5]. Therefore, there is significant research and application value in figuring out how to separate the vibration produced by a ship’s main power equipment from the hull.

Two primary techniques for controlling structural vibration have been identified in several studies conducted by researchers. One is active control, which reduces vibrations by eliminating two kinds of inverse waves. However, its system design is complicated and requires other auxiliary equipment to supply energy. Subsequently, active control is rarely used in engineering due to its very low level of practicality and the high maintenance cost of the active control system [6]. For vibration control on large ships, another passive control technique is frequently employed. In 1989, Sakai et al. [7] proposed the tuned liquid column damper (TLCD), a passive vibration control device for motion suppression. Rijken, Bian, and Spillane et al. [8,9,10] used a shock absorber system that included water columns and dual-chamber air springs to suppress resonant motion. They also studied how the orifice ratio affected structural damping. Liu [11] suggested a topology-optimized structural vibration and noise reduction technique that is affordable, efficient at reducing noise, and appropriate for ship cabin acoustics. It provides a way to reduce vibration and optimize structural quality without the need for laborious tests. David and Geoffrey Kireitseu [12] developed an effective vibration damper based on carbon nanotube-reinforced composite materials. It has superior impact toughness and vibration-damping benefits across a broad temperature range, making it appropriate for structural environments that demand exceptional vibration reduction performance. However, these researchers’ suggested designs lack the benefit of broadband response properties. In addition, the “law of mass” has a major influence on the isolation effect—the greater the mass, the better the isolation effect [13,14]. The majority of ship power plants today use incorporated floating raft vibration isolation systems, which weigh hundreds of tons, significantly impact the overall design, and seriously affect the comprehensive achievement of overall indicators [15]. Further, local loads can cause resonance in a particular part of the structure, which intensifies the combined resonance under multi-frequency vibration. The combined resonance zone is a local load research area that requires special attention [16,17,18]. Symmetric structures reduce nonlinear coupling and preserve modal regularity, which effectively mitigates coupled resonances brought on by local loads. In conclusion, because of the tension–bending coupling, asymmetric damping plates have less stiffness than symmetric ones for the anti-vibration plate structure. This causes a wider range of unstable areas [19]. In conclusion, there is an urgent need to design a marine vibration and noise reduction device that can consider both sound and broadband isolation effects, with lightweight design characteristics.

An acoustic black hole (ABH) is a novel passive vibration reduction device that accumulates and dissipates elastic wave energy through a power square thickness gradient structure [20,21]. In ship constructions, when the thickness of the wedge-shaped structure reduces in a particular power function form, the wave velocity of elastic waves will likewise decrease. In certain circumstances, it can even be lowered to zero [22], which results in the zero reflection of elastic waves. However, in practical situations, the presence of truncation thickness necessitates the adoption of additional techniques, including overlaying damping materials at the structure’s edges to decrease the reflection coefficient. Thus, using the concepts of geometric acoustics, Krylov et al. [23] theoretically calculated the reflection coefficient of wedge-shaped structures under various damping layer thicknesses. The findings demonstrated that when the power exponent increases, the reflection coefficient of wedge-shaped structures falls. Bayod J et al. [24] successfully decreased the reflection of elastic waves at the structure’s edges by increasing the length of the ABH truncation section. Park S. et al. [25] suggested a spiral ABH structure that economizes the design area while simultaneously improving the absorption impact of structural damping on elastic waves. However, most previous studies have concentrated on a single basic ABH arrangement. Currently, the vibration and noise reduction capabilities of ABH structures are widely used in many industries, while their application in the shipping industry is rather limited [26]. However, there are drawbacks to using a single, basic form of ABH for ship raft systems in the context of ship acoustics applications. For instance, attaining efficient bending wave absorption while maintaining structural performance is challenging. Low local stiffness caused by simple shapes can quickly compromise the system’s overall stability [3]. Additionally, it will result in the concentration of stress and a decrease in structural strength and durability, which are detrimental to the ship raft systems’ capacity of functioning reliably over the long run. Even so, research on the dynamic characteristics of multi-layered periodic arrangement systems is still relatively limited. Tang et al. [27] embedded one-dimensional acoustic black hole structures into unit cells and, based on the local resonance effect of acoustic black hole structures, proved that a small number of periods can achieve a wide bandgap, which is beneficial for the application of acoustic black hole structures in low-frequency bands. Zhu et al. [28] embedded a two-dimensional acoustic black hole structure into a plate structure to form a square unit cell and studied its significant dispersion characteristics, demonstrating the significant effect of periodic acoustic black hole structures. Combining the high bandwidth of periodic structures and the energy dissipation principle of acoustic black hole structures, this study will propose a non-uniform multi-level acoustic black hole structure that differs from traditional acoustic black hole structures. Based on this, this study will establish a thin plate with periodic acoustic black hole structures to analyze and optimize its vibration characteristics.

Furthermore, the systematic application process of ABHs in engineering practice and the selection of ABH characteristics have received comparatively little research [26,29]. Multiple finite element calculations are necessary for parameter analysis and the optimization of finite-period non-uniform multi-level ABH structures. Analyzing parameterized scanning and optimal mesh partitioning techniques still takes an enormous amount of time and computing power [30]. In principle, a more attractive approach is to completely replace expensive computational models with fast alternative models. Merging data into a nested Kriging framework can greatly reduce the number of reference points required to construct inverse surrogate terms (for surrogate model definition). Compared with traditional modeling methods and primitive nested Kriging frameworks, this can significantly reduce the cost of setting up proxy models [31]. Moreover, the hyper Latin hypercube sampling (LHS) method can improve the distribution uniformity of samples in multidimensional parameter spaces, provide more representative samples for Kriging models, further enhance the fitting effect of the model on complex systems, such as finite-period non-uniform multi-level acoustic black hole structures, reduce computational costs, and improve the optimization efficiency [32]. Consequently, we created a Kriging surrogate model for non-uniform multi-level ABH structures and used the LHS and particle swarm optimization algorithms to optimize their structural parameters. Then, the optimized parameters were applied to the rib plates of the raft structure to explore its actual vibration isolation effect in practical engineering applications.

This research concentrates on the application scenarios of floating raft isolation devices on ships. First, the transfer matrix model of a series ABH structure is derived. Then, a symmetrically distributed periodic non-uniform multi-level ABH structure is proposed. Second, we investigated the vibration response, damping properties, and optimization techniques of periodic non-uniform multi-level ABH structures. Then, to confirm their efficacy, we ran related finite element simulations. Third, the technical applications and parameter optimization for non-uniform multi-level ABH systems with finite periods are completed. This offers a novel approach for enhancing the raft body’s vibration isolation capabilities in the floating valve isolation system.

2. Modeling

The professional symbols and their definitions involved in this article have been summarized in

Table 1. All nomenclature and acronyms in this paper.

L0: length of uniform part	L1/L2: length of black hole region	hb: maximum thickness	h0: cut off thickness
$w(x,t)$: lateral vibration displacement	E: Young’s modulus	$I\left(x\right)$: Moment of inertia of section	$A\left(x\right)$: section area
$\rho$: density	hi: thickness of each section	Ti: transfer matrix	${\mathit{u}}_i$: state vector
$W_i$: deflection	${\theta }_i$,: corresponding rotation angle	$M_i$: bending moment	$V_i$: shear force
J: transfer matrix at interface	$T_z$: Total transfer matrix	K: K-level acoustic black hole structure	rk: radius of class k
$h_k$: truncation thickness	$\epsilon$: scaling factor	$h_b$: height of plate class k	$h_0$: truncation thickness
m: power exponent	$La$: corresponding acceleration level	$a_0$: reference acceleration	l: lattice constant
$e_l/e_d$: the projection length of the center distance of non-uniform multi-stage acoustic black holes in the X or Y direction	$y\left(x\right)$: response function	$F(\beta ,x)$: regression model	$z\left(x\right)$: uncertain random error with mean equal to 0
${\omega }_t$: inertia weight factor	$t_{\mathrm{m}\mathrm{a}\mathrm{x}}$: maximum iteration number	P: number of particles	$c_1/c_2$: learning factor
$V_{\mathrm{m}\mathrm{a}\mathrm{x}}$: maximum velocities	$V_{\mathrm{m}\mathrm{i}\mathrm{n}}$: minimum velocities	F: incentive points	V: vibration level drop

.

2.1. Theoretical Model

In order to obtain the acoustic transfer matrix of an acoustic black hole, we need to establish a model of two non-ideal one-dimensional ABHs connected in series, as shown in Figure 1.

The length of the uniform part is $L_0$, and the lengths of the two ABH regions are $L_1$ and $L_2$, respectively. The maximum thickness and uniform part remain the same as $h_b$, and the truncated thickness is $h_0$. Therefore, the thickness distribution function is:

$$h(x)\left\{\begin{array}{c}{h}_b\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0\le x<L_0\hfill \\ h_b+(h_b-h_0){\left(1-{\displaystyle \frac{x-L_0}{L_1}}\right)}^m\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{L}_0\hspace{0.33em}\le x<L_0+L_1\hspace{0.33em}\hfill \\ h_b+(h_b-h_0){\left(1-{\displaystyle \frac{x-L_0-L_1}{L_2}}\right)}^m\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{L}_0+L_1\hspace{0.33em}\le x<L_0+L_1+L_2\hfill \end{array}\right.$$

(1)

Among them, $w(x,t)$ is the lateral vibration displacement, E is Young’s modulus, $I\left(x\right)$ is the moment of inertia of its cross-section, $A\left(x\right)$ is the cross-sectional area, $\rho$ is the density, and we set $w(x,t)=W\left(x\right)e^{i\omega t}$. The issues related to variable cross-section beams typically involve the dynamic response analysis, natural frequencies and modes, and the study of wave propagation along the length of the beam. To address these issues, the behavior of variable cross-section beams is typically modeled using the Euler Bernoulli equation [33]. The plate is 800 mm in length, 300 mm in breadth, and 5 mm in thickness. The typical ratio of thickness to length for two-dimensional structures applicable to beam theory is about 0.05 to 0.01 [34]. The geometric parameter is within the applicable range, supporting the rationality of applying the Euler Bernoulli beam theory. The wave control equation was obtained as follows:

$${\displaystyle \frac{{\partial }^2}{\partial x^2}}\left(EI(x){\displaystyle \frac{{\partial }^{2}w(x,t)}{\partial x^2}}\right)+\rho A(x){\displaystyle \frac{{\partial }^{2}w(x,t)}{\partial t^2}}=0$$

(2)

The wave equation can be solved within the L₀ length of the uniformly thick section. However, in the ABH region with variable thickness, this equation is difficult to solve analytically. We approximate the analysis in constant parameter form in each segment and divide the ABH structure into N segments, where each segment has a thickness of h_i and a length of $\mathsf{\Delta }x$. At this point, the vibration equation for segment I is:

$$D_i{\displaystyle \frac{{\partial }^4{W}_i}{\partial x^4}}+M_i{\displaystyle \frac{{\partial }^2{W}_i}{\partial t^2}}=0\hspace{0.33em},x_i\le x\le x_{i+1},i=\mathrm{1,2}I\dots ,N$$

(3)

We suppose $D\left(x\right)={\left[D_1,D_2,D_3\dots D_N\right]}^T$, $M\left(x\right)={\left[M_1,M_2,M_3\dots M_N\right]}^T$. Then, the general solution for a beam can be obtained as follows:

$$W(x)=W(x_i)f_1({\beta }_i)+{\displaystyle \frac{W^{\prime }(x_i)}{k_i}}{f}_2({\beta }_i)+{\displaystyle \frac{M(x_i)}{D_i{k}_i^2}}{f}_3({\beta }_i)+{\displaystyle \frac{Q(x_i)}{D_i{k}_i^3}}{f}_4({\beta }_i)$$

(4)

where:

$$k_i^4={\displaystyle \frac{M{\omega }_n^2}{D_i}}$$

(5)

$${\beta }_i=k_i(x-x_i)$$

(6)

$$M(x)=D_i{\displaystyle \frac{{\partial }^{2}W}{\partial x^2}}$$

(7)

$$Q(x)=D_i{\displaystyle \frac{{\partial }^{3}W}{\partial x^3}}$$

(8)

$$f_2({\beta }_i)={\displaystyle \frac{1}{2}}[\sin \mathrm{h}({\beta }_i)+\cos ({\beta }_i)]$$

$$f_1({\beta }_i)={\displaystyle \frac{1}{2}}[\cos \mathrm{h}({\beta }_i)+\cos ({\beta }_i)]$$

(9)

$$f_3({\beta }_i)={\displaystyle \frac{1}{2}}[\cos \mathrm{h}({\beta }_i)-\sin ({\beta }_i)]$$

$$f_4({\beta }_i)={\displaystyle \frac{1}{2}}[\sin \mathrm{h}({\beta }_i)-\sin ({\beta }_i)]$$

Then, we define the state vector:

$$u_i={\left[\begin{array}{cccc}W\left({\beta }_i\right),& W^{\prime }\left({\beta }_i\right),& M\left({\beta }_i\right),& Q\left({\beta }_i\right)\end{array}\right]}^T$$

(10)

The transfer matrix $T_i$ of the transfer matrix method can be constructed, and we can obtain that $u_{i+1}=T_i{u}_i$. By introducing boundary conditions $u_1$ and $u_{N+1}$, the overall transfer relationship obtained according to the chain rule is:

$$u_{N+1}=T_N{T}_{N-1}\cdots T_1{u}_1=T{u}_1$$

(11)

For the change in the $x=L_0+L_1$ cross-section at the connection between ABHs, we should consider the impedance variation. In a beam segment, the deflection $W_i$, corresponding to rotation angle ${\theta }_i$, bending moment $M_i$, and shear force $V_i$, can be obtained through wave solutions:

$$W_i(x)=A_i^{+}{e}^{j{k}_{i}x}+A_i^{-}{e}^{-j{k}_{i}x}$$

(12)

$${\theta }_i(x)={\displaystyle \frac{d{W}_i}{dx}}=j{k}_i(A_i^{+}{e}^{j{k}_{i}x}-A_i^{-}{e}^{-j{k}_{i}x})$$

(13)

$$M_i(x)=-E{I}_i\cdot {\displaystyle \frac{d^2{W}_i}{d{x}^2}}=-E{I}_i\cdot (-k_i^2)(A_i^{+}{e}^{j{k}_{i}x}+A_i^{-}{e}^{-j{k}_{i}x})=E{I}_i{k}_i^2{W}_i(x)$$

(14)

$$V_i(x)=-E{I}_i\cdot {\displaystyle \frac{d^3{W}_i}{d{x}^3}}=-E{I}_{i}j{k}_i^3(A_i^{+}{e}^{j{k}_{i}x}-A_i^{-}{e}^{-j{k}_{i}x})$$

(15)

The displacement should be continuous, so the transfer matrix J at the interface is:

$$J=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& {\displaystyle \frac{k_2}{k_1}}& 0& 0\\ 0& 0& {\displaystyle \frac{E{I}_2{k}_2^2}{E{I}_1{k}_1^2}}& 0\\ 0& 0& 0& {\displaystyle \frac{E{I}_2{k}_2^3}{E{I}_1{k}_1^3}}\end{array}\right]$$

(16)

The total transfer matrix of the serial ABH structure is as follows:

$$T_z=T_{L2}J{T}_{L1}$$

(17)

Depending on the input excitation and boundary conditions, the structure’s output can be obtained. The theory’s analysis shows that while the series connection method cannot completely remove reflection, it can create a stronger wave energy trap. Therefore, it improves the ABH structure’s energy aggregation and absorption while also producing a more consistent frequency response. The above research applies to the scenario of two ABHs in series. It can also be expanded to include numerous ABHs in series.

2.2. Structural Model

As shown in Figure 2, the non-uniform multi-level ABH structure proposed in this chapter is divided into four stages, starting from the center and extending outward. The subscript k represents the k-layer. This study assumes that the structural parameters of ABHs at the same level are equal. The radius, truncation thickness, scaling factor, and height of the k level is denoted as r_k, h_k, ${\epsilon }_k$, and h_b. We set the parameters as $r_k={\displaystyle \frac{90}{k+2}} \mathrm{m}\mathrm{m}, h_k=0.15\ast \left(5-k\right) \mathrm{m}\mathrm{m}$, ${\epsilon }_k=(h_b-h_k)/r_k^2$, and $h_b=2 \mathrm{m}\mathrm{m}$. Then, the values of the parameters are shown in

Table 2. ABH size parameters at all levels.

Level	rk (mm)	hk (mm)	${\mathit{\epsilon }}_{\mathit{k}}$
1	30	0.6	$7/4500$
2	22.5	0.45	$31/\mathrm{10,125}$
3	18	0.3	$17/3240$
4	15	0.15	$37/4500$

.

After rotating the cross-section of the ABH with the above parameters 180 degrees around the center and then performing mirror symmetry, the symmetrical non-uniform ABH structure shown in Figure 2 can be obtained. This structure uses the power law curve’s parameters to regulate the size of the two-dimensional multi-level ABH structure.

This structure controls the size of a two-dimensional multi-level ABH structure through the parameters of a power–law curve, with different structural parameters between each level. It adopts a multi-stage series connection method, consuming energy multiple times at the tip, which ensures that elastic waves can be gradually weakened during transmission [35] while also expanding the frequency band of action.

3. Analysis of Vibration Characteristics

3.1. Simulation Analysis

As shown in Figure 3 and Figure 4, this study uses COMSOL Multiphysics 6.3 software to conduct a harmonic response analysis on the non-uniform multi-level ABH model established in the previous section and explore its vibration characteristics.

We established a three-dimensional model of the acoustic black hole plate structure, as shown in Figure 3. The range of the acoustic black hole region is $x^2+y^2=r^2=100^2$, which means that the radius of the acoustic black hole is 100 mm, the plate thickness $h_b=5 \mathrm{mm}$, the truncation thickness $h_0=0.5 \mathrm{mm}$, the power exponent m = 2, the scaling factor $\epsilon =(h_b-h_0)/r^m$, the plate length is 800 mm, and the width is 300 mm.

A coordinate system is established with the bottom center of the acoustic black hole as the origin, the length direction as the x-axis, the width direction as the y-axis, and the thickness direction as the z-axis. The distribution range of the plate is $x\in \left[-\mathrm{400,400}\right] \mathrm{mm}$,$y\in \left[-\mathrm{150,150}\right] \mathrm{mm}$,$z\in \left[\mathrm{0,5}\right] \mathrm{mm}$, where the acoustic black hole structure is distributed within $(x^2+y^2{)}^{1/2}\le 100 \mathrm{mm}$. The research object of this simulation calculation is the air domain, using a free tetrahedral grid. When dividing the grid, it is necessary to ensure that the maximum cell size of the grid design is less than one-sixth of the wavelength corresponding to the highest calculated frequency point, and the minimum cell size is less than one-fifteenth of the wavelength corresponding to the lowest calculated frequency point. The specific inequality is shown in Equation (18). The approximate frequency range studied by this simulation model is 200~700 Hz; therefore, the size range of the grid cells is:

$$\left\{\begin{array}{c}\mathit{max}\le 343\left[\mathrm{m}/\mathrm{s}\right]/{f\left[\mathrm{H}\mathrm{z}\right]\left[\mathrm{m}/\mathrm{s}\right]\left[\mathrm{H}\mathrm{z}\right]}_{up-lim}\\ \mathit{min}\le 343\left[\mathrm{m}/\mathrm{s}\right]/{f\left[\mathrm{H}\mathrm{z}\right]\left[\mathrm{m}/\mathrm{s}\right]\left[\mathrm{H}\mathrm{z}\right]}_{low-lim}\end{array}\right.$$

(18)

The final configured unit size parameters are as follows: maximum unit size: 28; minimum unit size: 12; maximum unit growth rate: 1.35; curvature factor: 0.3; and narrow area resolution: 0.85.

We conducted a modal and harmonic response analysis on this structure in a free state, with a frequency range of $\left[10, 3000\right]$Hz and a step size of 5 Hz. A 1 N harmonic load was applied to a circle with a radius of 2 mm, 50 mm from the left edge of the flat plate. The point on the right edge of the thin plate was used as the response evaluation boundary, and its average vibration acceleration was taken as the evaluation index. The selected material was structural steel, and the specific material parameters are shown in

Table 3. Material parameters of the ABH thin plate.

Material	$\mathbf{Density} \mathit{\rho }$ (kg/m3)	Young’s Modulus e (GPa)	$\mathbf{Poisson}’\mathbf{s} \mathbf{Ratio} \mathit{v}$	$\mathbf{Isotropic} \mathbf{Loss} \mathbf{Factor} \mathit{\eta }$
structural steel	7850	210	0.3	0.04

.

We applied a harmonic load of 1 N at a point 50 mm from the left edge, extracted the effective vibration acceleration level L of each point on the right edge, and then took the average value as the evaluation index. For better observation, we converted this to the corresponding acceleration level $La=20\lg (a/a_0)$, with a reference acceleration value of $a_0=1\times 10^{-6}\hspace{0.33em}\mathrm{m}/{\mathrm{s}}^2$. The selected materials comprise structural steel, as shown in Table 3. According to the above theoretical model, the theoretical prediction results were compared with the finite element simulation, and the corresponding sound absorption coefficients were compared, as shown in Figure 5. It shows that the theoretical prediction values and the simulated sound absorption coefficient curves have the same trend of change in the frequency range of [1, 1800] Hz, with a high degree of agreement, confirming the accuracy and effectiveness of the above theoretical model.

We compared the non-uniform multi-level ABH with general ABH thin plates and ordinary thin plates of the same size. We calculated the frequency range in $\left[10, 2300\right]$ Hz with a step size of 5 Hz, as shown in Figure 6 and

Table 4. Comparison of the vibration responses of different structures.

Structure	L (dB)	Vibration Level Difference (dB)	Vibration Level Percentage
Ordinary thin plate	113.05	0	100%
General ABH	108.84	4.21	96.28%
Non-uniform and multi-level ABH	105.81	7.24	93.59%

.

Figure 6 shows that almost all resonance peaks are significantly reduced, and the entire curve is noticeably shifted downwards. The non-uniform multi-level acoustic black hole structure exhibits significant vibration reduction advantages in the [1900, 2315] Hz frequency band. The average vibration level in the [2000, 2150] Hz frequency band has decreased by more than 15 dB. At low frequencies below 350 Hz, the wavelength of bending waves is longer, making it difficult to confine their energy inside acoustic black holes. In the frequency band above 350 Hz, non-uniform multi-level acoustic black holes have a more significant vibration reduction effect compared to ordinary plates, with an average reduction of 6 dB. Taking the average acceleration vibration level of each frequency point within the research range, we found that the average vibration level of a general ABH structure decreased to 96.28% compared to a common thin plate, and that of non-uniform multi-level ABH structures decreased to 93.59%. From the vibration characteristic curve, we can see that compared with ordinary thin-plate structures, both general ABH structures and non-uniform multi-level ABH structures can significantly reduce the average acceleration at the evaluation boundary. Moreover, the non-uniform multi-level ABH structures are almost superior to the ordinary ABH structures over the entire frequency range, with a more significant effect.

The energy distribution in acoustic black hole thin-plate structures is also analyzed below. The following are energy distribution cloud maps of acoustic black hole thin-plate structures and ordinary thin-plate structures at several typical frequencies, measured in $\mathrm{J}/{\mathrm{m}}^3$.

Figure 7 shows that during the transmission process, vibrations will deflect and converge towards the central area of the acoustic black hole region. The acoustic black hole effect is not significant at 200 Hz, and the significant difference can only be seen at 350 Hz. Furthermore, it remains effective in the high-frequency range thereafter, verifying the deflection and convergence effect of the acoustic black hole on bending waves. It also indicates that in the analysis of acceleration vibration levels mentioned above, the decrease in the average acceleration vibration levels of acoustic black hole thin-plate structures compared to ordinary thin-plate structures is indeed a result of the acoustic black hole effect.

In addition, an analysis was conducted on the energy distribution in the non-uniform multi-level ABH thin-plate structure. The following energy distribution cloud maps were obtained for the non-uniform multi-level ABH thin-plate structure and the ordinary ABH thin-plate structure at 2000 Hz.

As can be seen from the energy distribution cloud map in Figure 8, the non-uniform multi-level ABH structure contains several rings of energy accumulation. At every step, elastic waves deflect and aggregate. The maximum value in its energy distribution cloud map is two orders of magnitude larger than that in the energy distribution of a typical ABH structure, further reflecting its convergence effect on energy. To ensure effective energy dissipation, only a small amount of damping layer material needs to be put in the core area.

3.2. Factors Affecting the Vibration Reduction Characteristics of ABH Structures

The vibration reduction effect of ABH structures is largely influenced by parameters such as the truncation thickness $h_0$, radius r, and power-law exponent m. This section will further analyze the influence of different $h_0$, r, and m values on the vibration characteristics of acoustic black hole thin plates in order to explore the specific impact of these parameters on the vibration reduction effect of acoustic black hole structures, as shown in Figure 9.

As shown in Figure 9, below 500 Hz, the vibration acceleration levels of the evaluation boundaries for the three truncated thickness thin plates are basically the same. This is because the vibration in this frequency band is mainly dominated by the material and structural parameters of the thin plate itself, and the acoustic black hole effect is not fully manifested. Therefore, it mainly targets the frequency band above 500 Hz. Figure 9a shows that when the frequency increases, the difference in the curve increases. The larger the $h_0$, the higher the vibration acceleration level. As the thickness increases, it weakens the deflection and aggregation effect of acoustic black holes on bending waves (affecting the reflection coefficient). However, in practical engineering, excessively reducing the cutting thickness will significantly increase the manufacturing difficulty and cost, so it needs to be controlled within a reasonable range. As shown in Figure 9c, with the increase of frequency, the difference in the curve increases, and the greater the $r$, the smaller the vibration acceleration level. Among them, the 125 mm diameter thin plate has a significant vibration reduction effect. Therefore, when material properties and structural dimensions allow, increasing the radius appropriately can improve the vibration reduction performance. Figure 9b shows that as the frequency increases, the difference in the curve increases. The larger the $m$, the smaller the vibration acceleration level. Theoretical analysis shows that a larger power exponent is beneficial for enhancing the aggregation effect of bending waves. However, in engineering, a large power exponent will rapidly reduce the thickness, leading to increased processing errors and uneven local stiffness. Considering the energy absorption effect and practical application stability, the power exponent value should be suitable.

Table 5. Average vibration acceleration levels at different $h_0$, r, and m values.

${\mathit{h}}_0$ (mm)	a (dB)	Vibration Level Difference (dB)	Vibration Level Percentage	$\mathit{r}$ (mm)	a (dB)	Vibration Level Difference (dB)	Vibration Level Percentage	$\mathit{m}$	a (dB)	Vibration Level Difference (dB)	Vibration Level Percentage
0.1	125.28	0	100%	75	126.88	0	100%	2	126.84	0	100%
0.5	126.84	1.56	101.25%	100	126.64	0.24	99.8%	2.5	125.98	0.86	99.32%
1	127.05	1.77	101.41%	125	123.54	3.34	97.3%	3	122.50	4.34	96.58%

evaluates the influence of $h_0$, $r$, and m on the vibration reduction performance of thin plates based on overall performance indicators. In theory, it would be better to reduce vibration when $h_0$ is small while $r$ and m are large within reasonable limits.

3.3. Periodic Non-Uniform Multi-Level Acoustic Modeling

Based on the non-uniform multi-level ABH structure proposed above, this section further extends it to the periodic arrangement structure shown in Figure 10.

According to the research and analysis of the distributed ABH structure by Zhao Nan et al. [29], the ABH structure contained in the primitive cell is extended to two different sizes of non-uniform multi-level ABH structures. Their specific size parameters are shown in

Table 6. ABH parameters at all levels.

Level	rk (mm)	hk (mm)	${\mathit{\epsilon }}_{\mathit{k}}$
1	16	0.6	$7/1280$
2	12	0.45	$31/2880$
3	9.6	0.3	$85/4608$
4	8	0.15	$37/1280$

.

According to Section 3.1, the thickness of the single-layer ABH structure is 2 mm, so the thickness of this three-dimensional primitive cell structure is 8 mm, with a length and width of 800 mm and 250 mm, respectively. When performing a periodic array in the x-direction, the lattice constant is l = 800 mm. Additionally, the $e_l=185 \mathrm{m}\mathrm{m}$ and $e_d=57 \mathrm{m}\mathrm{m}$ in Figure 11 represent the projection lengths of the center distances of the two types of non-uniform multi-level ABHs in the x- and y-directions, respectively.

This section utilizes the COMSOL Multiphysics simulation platform to conduct finite element analysis on the above-mentioned model and applies the Floquet periodic boundary condition:

$$u(x+l,y,z)=u(x,y,z)e^{-i{k}_{x}a}$$

(19)

The mesh division processing results of the COMSOL finite element analysis are shown in Figure 10.

The periodic symmetry of phononic crystals allows for the bandgap analysis to cover all the necessary information by just selecting the range of wave vector k within the first Brillouin zone. This makes the computation procedure simpler. Floquet boundary conditions are applied to the boundaries of the COMSOL Multiphysics simulation platform using finite element analysis:

$$u(x+l,y+l,z)=u(x,y,z)e^{-i{(k}_{x}a+k_{y}a)}$$

(20)

$$k_x=range\left(0,{\displaystyle \frac{\pi /m-0}{N-1}},{\displaystyle \frac{\pi }{m}}\right)$$

(21)

$$k_y=range\left(0,{\displaystyle \frac{\pi /n-0}{N-1}},{\displaystyle \frac{\pi }{n}}\right)$$

(22)

Among them, Range ($q_{1,}{q}_{2,}{q}_3$) is used to generate a numerical sequence, where $q_{1,}{q}_{2,}{q}_3$ represent the starting value, ending value, and step size, respectively. $a$ is the basis vector of the phononic lattice, $m$ represents the cell length, $n$ represents the cell width, and $N$ represents the number of discrete points. During the simulation analysis process, the band diagram of the phononic crystal structure is obtained by setting the scanning path of the wave vector k within the irreducible Brillouin zone.

Figure 12 shows that the periodic non-uniform multi-level ABH structure can produce several bandgaps within 2500 Hz, indicating that this structure can effectively block the propagation of elastic waves in multiple frequency bands and has good vibration isolation and vibration suppression capabilities.

3.4. Vibration Response Analysis of the Periodic Arrangement Number

The research on bandgaps is aimed at theoretically infinite periodic structures, but in practical engineering applications, the size of the structure is limited. Therefore, we chose a more suitable finite periodic structure based on the actual situation, and then analyzed the vibration characteristics of non-uniform ABH structures with finite periods.

Firstly, we established three finite periodic structures, as shown in Figure 13, whose overall dimensions remain consistent, with different numbers of periodic arrangements of non-uniform multi-level ABH units along the structural direction.

The unit cell size is the same as that of the infinite periodic structure mentioned above. The length of the plate is the length of the eight-period structure, and the lengths of the four-period and six-period structures are supplemented with ordinary flat plates. The material used is structural steel. A harmonic load of 1 N is applied at a point 50 mm away from the left edge to conduct the harmonic response analysis. The magnitudes of the effective acceleration a of each point on the right-side line are extracted, and then their average value is taken as the evaluation index. For the convenience of observation, we convert it into the corresponding acceleration level, and the reference acceleration is taken as $La=20\mathit{lg}(a/a_0)$, $a_0=1\times 10^{-6}\hspace{0.33em}\mathrm{m}/{\mathrm{s}}^2$. We calculated the frequency range in Hz with a step size of 5 Hz, as shown in Figure 14 and

Table 7. Average vibration acceleration level at different cycle numbers.

Cycles	Average Acceleration Vibration Level (dB)	Vibration Level Difference (dB)	Vibration Level
4 cycles	85.57	0	100%
6 cycles	81.42	4.15	95.15%
8 cycles	80.43	5.14	93.99%

.

It can be found from the vibration characteristic curve that the vibration acceleration at the response boundary decreases with the increase in the number of periods, and the vibration-damping performance of the structure is also improved accordingly. Moreover, the intervals where the structure obtains a large vibration attenuation are similar, and the vibration attenuation situation is in good agreement with the band-gap distribution results in Figure 14.

Compared with the four-period structure, the average vibration-damping effect of the six-period structure is improved by 4.15 dB, and that of the eight-period structure is improved by 5.14 dB. From the perspective of the absolute value and relative percentage of the attenuation of the vibration acceleration level, increasing the number of arrangement periods can enhance the suppression effect on vibration transmission, indicating that segmented acoustic black hole structures have better energy gathering effects compared to traditional acoustic black hole structures. However, the intensity of this enhancement will decrease with the increase in the number of periods and will not increase continuously with the increase in the number of periods. Therefore, if the design space allows, the vibration suppression effect of the structure can be improved by appropriately increasing the number of cycles.

4. Structural Optimization Methods

For non-uniform multi-level ABH structures with a certain number of cycles, there are other factors that affect their vibration characteristics. In this chapter, based on the actual layout requirements of raft structures, the layout parameters of four-period non-uniform multi-level ABHs are analyzed, and attempts are made to optimize them to improve the vibration reduction effect of finite-period non-uniform ABH structures.

4.1. Establishment of the Optimization Model

Referring to the model structure in Figure 13, to reduce the computational complexity, we selected a four-period structure and eliminated the complementary thin plates on both sides, as shown in Figure 15.

Figure 11 shows that the parameters that affect the structural effect of ABH include the lattice constant l and the projection length of the center distance in the x- and y-directions, $e_l$ and $e_d$. The objective of optimization is to minimize the average vibration acceleration level of the response boundary. Considering the practical application issues in engineering, the range of values is shown in

Table 8. Range of parameter values to be optimized.

Parameter	l (mm)	${\mathit{e}}_{\mathit{l}}$ (mm)	${\mathit{e}}_{\mathit{d}}$ (mm)
Range	[600, 800]	[150, 200]	[50, 65]

.

Considering the uniformity of sample distribution and the efficiency of computational resource utilization, the Latin hypercube sampling (LHS) method [30] is selected for sample point selection. The LHS method can evenly divide the three structural parameters into N intervals within their range of values and randomly select a value from each interval to generate an (N, 3) sample set. To balance the computational efficiency and predictive accuracy of the surrogate model, this paper uses a computer program to select five levels from each design variable, obtaining a total of 15 sets of sample points as the initial training set for constructing the surrogate model. Its distribution is shown in Figure 16, and the specific values of its data points are shown in

Table 9. Sample point data.

Sample Points	l (mm)	${\mathit{e}}_{\mathit{l}}$ (mm)	${\mathit{e}}_{\mathit{d}}$ (mm)
1	793.06	181.51	58.22
2	552.63	157.84	61.06
3	686.11	191.52	57.02
4	662.39	160.92	64.80
5	673.10	198.25	51.86
6	756.96	163.35	62.30
7	624.32	195.93	52.90
8	768.83	156.31	50.47
9	712.25	169.63	56.46
10	578.03	153.12	54.13
11	614.67	178.65	55.51
12	734.33	188.50	59.60
13	722.57	175.44	63.32
14	592.01	170.78	53.25
15	639.34	184.57	60.48

.

According to the parameters of the sample points, we parameterized the structural parameters in COMSOL and obtained the average vibration acceleration level of the response boundary under each parameter to obtain complete sample data.

4.2. EI-PSO Optimization Algorithm

When constructing a Kriging surrogate model for a finite-period non-uniform multi-level ABH structure, the Kriging surrogate model is first trained using the sample set in Section 4.1. However, when verifying its prediction results, we found that the model had significant errors. This result is mainly due to the uncertainty of the Kriging surrogate model in predicting unknown spaces. Therefore, this article uses particle swarm optimization algorithms to further iteratively optimize the Kriging surrogate model. Through the collaborative search and information sharing mechanism between individual particles, the risk of falling into local optima can be effectively avoided, and the global optimum can be approached more efficiently. The mathematical expression of the Kriging model is:

$$y\left(x\right)=F(\beta ,x)+z\left(x\right)$$

(23)

Among them, $y\left(x\right)$ is the response function and $F(\beta ,x)$ is the regression model. The updated expression for particle velocity and position is:

$$X^{t+1}=\left\{\begin{array}{c}{X}_{\max } X^t+V^{t+1}>X_{\max }\hfill \\ X^t+V^{t+1} X_{\min }\le X^t+V^{t+1}\le X_{\max }\hfill \\ X_{\min } X^t+V^{t+1}<X_{\min }\hfill \end{array}\right.$$

(24)

$$V^{t+1}=\omega V^t+c_1{r}_1(P_b^t-X^t)+c_2{r}_2(G_b^t-X^t)$$

(25)

$$V^{t+1}=\left\{\begin{array}{c}{V}_{\max } V^{t+1}>V_{\max }\hfill \\ V^{t+1} V_{\min }\le V^{t+1}\le V_{\max }\hfill \\ V_{\min } V^{t+1}<V_{\min }\hfill \end{array}\right.$$

(26)

Adopting a linear decreasing weight strategy, a larger inertia weight factor is selected at the beginning of the iteration to quickly explore the solution space. As the iteration progresses, the value of the inertia weight factor gradually decreases, so that the particle swarm can converge to the optimal solution in the later iteration process. During the t-th iteration, the inertia weight factor ${\omega }_t$ can be expressed as:

$${\omega }_t={\omega }_{max}-({\omega }_{max}-{\omega }_{min}){\displaystyle \frac{t}{t_{max}}}$$

(27)

Among them, $t_{max}$ is the maximum iteration number, and ${\omega }_{max}$ and ${\omega }_{min}$ represent the maximum and minimum values of the inertia weight factor, respectively. The relevant parameter values of the particle swarm optimization algorithm in this article are shown in

Table 10. Parameter values for the particle swarm optimization algorithm.

$\mathit{p}$	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{V}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$	${\mathit{V}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$	${\mathit{t}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$
50	2	2	1	−1	100

.

Among them, p is the number of particles$, c_1$ and $c_2$ are the learning factors, $V_{max}$ and $V_{min}$ are the maximum and minimum velocities, and $t_{max}$ is the maximum number of iterations. The difference between the actual average acceleration level a’ and the predicted value of the sample points added during each iteration of the Kriging surrogate model is shown in Figure 17.

After optimizing the parameters of the finite-period non-uniform multi-level ABH structure, the optimal parameters obtained are $l=643.2$ mm, $e_l=191.47$mm, and $e_d=52.28$mm. Then, mesh partitioning and finite element analyses are performed on the model in the COMSOL Multiphysics simulation platform. The excitation is a 1 N harmonic excitation at a distance of 50 mm from the left edge, and the response evaluation boundary is the rightmost boundary. Then, we obtain the vibration response curve of the structure at 10–1500 Hz. Taking the average of the acceleration levels at each frequency point, we can obtain the final evaluation index to compare and analyze it with the parameters before optimization. The results are shown in Figure 18 and

Table 11. Average vibration response before and after optimization.

	Average Acceleration Vibration Level (dB)	Vibration Level Difference (dB)	Vibration Level
Before optimization	100.61	0	100%
After optimization	96.25	4.36	95.6%

.

The vibration characteristic curve shows that the average acceleration vibration level of the optimized model response boundary has almost decreased throughout the entire analysis frequency band, especially in the frequency bands of [510, 1100] Hz and above 1350 Hz, indicating that the vibration reduction performance of the structure has been effectively improved. Overall, the optimized structure has improved the average vibration reduction effect within the analysis range by 4.36 dB compared to the pre-optimized structure. This improvement fully demonstrates the effectiveness of the optimization process in improving the vibration reduction effect of finite-period non-uniform multi-stage ABH structures.

4.3. Analysis of the Vibration Characteristics of the Raft System

We conducted a finite element analysis on the model shown in Figure 16 using the COMSOL Multiphysics simulation platform to investigate its vibration characteristics. Figure 19 shows that the structural composition mainly includes rigidly welded upper and lower partitions and intermediate ribs. The intermediate ribs are arranged in an orthogonal manner, and the upper and lower partitions are perforated to reduce the weight of the structure [36]. The plate’s thickness is 8 mm, and the rib height is d = 250 mm, as shown in Figure 11. In addition, each rib plate is distributed with a periodically arranged multi-level ABH structure, and its unit cell size parameters are optimized using the results in Section 4.2. The remaining specific dimensions and material parameters are shown in Figure 20 and

Table 12. Raft frame parameters.

Parameter	Value	Unit
Density	7850	kg/m3
Young’s modulus	210	GPa
Poisson’s ratio	0.3	-
Isotropic loss factor	0.04	-
Length	2798.56	mm
Width	1607.28	mm
Height	282	mm
Size of upper and lower plate openings	350 × 350	mm

.

Considering the constraints of actual installation conditions, simple support constraints are applied to the lower four sides of the model. To simulate the effects of simultaneous vibrations of multiple vibration sources, harmonic loads are applied at two points in the illustrated positions. These two points are symmetrical about the geometric center of the upper partition, with a distance of 595.64 mm between them, and the load magnitude is (2,2, −10) N. The evaluation point is located at the center point of the lower surface of the lower partition to calculate the effective vibration acceleration. The frequency range is taken as $\left[5, 1000\right]$ Hz, and the step size is taken as 5 Hz.

The vibration level drop method is widely used in engineering practice due to its simplicity and practicality [37]. The vibration level drop method is adopted to evaluate the isolation performance of floating raft systems. First, record two incentive points F₁ and F₂, extract their effective accelerations a₁ and a₂, and evaluate the effective acceleration of the point $a_0$. To intuitively judge the change in the acceleration magnitude of the evaluation point compared to the excitation point, we propose a new definition for the traditional formula of the vibration level drop: $V_1=20\mathit{lg}(a_0/a_1)$ for the vibration level drop of F₁, and $V_2=20\mathit{lg}(a_0/a_1)$ for the vibration level drop of F₂, with 0 as the boundary line. If it is greater than 0, it means that the vibration is amplified during the transmission process, and if it is less than 0, it means that the vibration is reduced when it is transmitted to the evaluation point. In Figure 18, it is shown that the lower the curve, the better the effect. A value greater than 0 indicates that the vibration is amplified during transmission, while a value less than 0 indicates that the vibration is attenuated when transmitted to the evaluation point. As shown in Figure 18, the lower the curve, the better the vibration isolation effect.

Similarly, to compare the vibration isolation effect of the non-uniform multi-level ABH raft structure, a conventional raft model with the same size was also established. The same excitation and constraints were applied, the vibration acceleration at the same evaluation points was extracted, and the vibration level drops $V_3$ and $V_4$ were obtained. The analysis results are shown in Figure 21 and

Table 13. Average vibration level drops.

Evaluation Points	Average Vibration Level Drop (dB)	Mean Value (dB)
V1	−3.427	−3.389
V2	−3.351
V3	−1.558	−1.517
V4	−1.476

.

Although there are slight differences in the vibration level reduction over several evaluation points of the same raft model, as shown in Figure 21, the general trend and difference are not statistically significant. This is because the chosen excitation locations have very similar vibration conditions because of their symmetrical arrangement, even if the raft system’s vibration mode will unavoidably provide varied vibration situations between different points. Furthermore, throughout the analyzed range, the raft structures have an isolation effect that is almost better than that of standard slab raft structures, particularly in the [700–800] Hz region, where the maximum difference can exceed 30 dB.

Overall, there is a 3.389 dB average vibration level difference between V₁ and V₂ in the analysis range, while there is only an average vibration level difference of 1.517 dB between V₃ and V₄ from Figure 16. The relative effect has improved by 123.4%, despite the fact that the absolute difference is just 1.872 dB. Further, the outcome of the chosen point should be the point with the largest vibration in the overall structural response since the selected response point is located at the center of the lower part of the raft frame, and our constraint on the raft frame is simply supported on its four sides at the bottom. It is clear that the raft body’s vibration isolation ability is enhanced by the addition of a non-uniform multi-level ABH structure.

In addition, the cavity introduced by the non-uniform multi-level ABH structure also reduces the overall volume of the raft. Calculations show that in this case, its volume is decreased by 3.109 × 10⁻³ m³. Given that the chosen material has a density of 7850 kg/m³, the construction’s total mass is decreased by 24.406 kg, which enhances the vibration isolation effect and makes it easier to lighten the structure.

5. Conclusions

This article focuses on the ship’s floating raft isolation system’s raft frame vibration issue, proposes a symmetrically distributed periodic non-uniform multi-level ABH structure, and assesses how well it suppresses structural vibration. Our primary findings and work content are as follows:

• A wave propagation model for ABH structures was developed based on the ABH effect principle, and a transfer matrix model for serially connected acoustic black hole structures has been developed. Additionally, a symmetrically distributed periodic non-uniform multi-level ABH structure was presented, which dissipates energy in multiple stages and has a better vibration reduction impact than traditional acoustic black hole structures of the same size. Its vibration reduction impact can be further enhanced by periodic arrangement, making it appropriate for various vibration isolation scenarios;
• We proposed a multi-level acoustic black hole structure optimization approach based on finite period arrangement. First, Latin hypercube sampling (LHS) is employed to obtain the vibration reduction effect under various parameter arrangement combinations. Then, we established a surrogate model that could predict the average vibration acceleration level. Finally, using the EI-PSO algorithm, we improved the surrogate model’s prediction accuracy and obtained the optimal structural parameter combination. This optimization model can be used to solve multi-parameter optimization problems in practical engineering applications;
• To improve the isolation effect of the floating raft vibration reduction system on ships, we applied a finite-period non-uniform multi-level acoustic black hole structure to the raft structure. This could provide a new idea for suppressing the vibration noise level during ship operations. The raft structure can also be used in the exterior of the vehicle’s internal power system or other power structures to achieve vibration reduction effects.

Prospects:

This paper is based on the practical needs of the raft structure in the floating raft vibration reduction system. Limited by computational cost and time, we only consider the vibration reduction performance of the proposed structure on flat plates but have not analyzed more complex cases, such as stiffened plates, composite plates, cylindrical surfaces, or the comparative analysis between the bandgap and frequency response function (FRF). Our future work can expand this to application scenarios with complex structures and further analyses of acoustic characteristics.