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Article

Study on the Vibration-Damping Mechanism of a New Phononic Crystal Suspension Equipped on Underwater Gliders

by
Qindong Sun
1,2,†,
Yuhan Yang
1,†,
Pan Wu
2,
Ming Yang
1,2,
Tongshuai Sun
1,2,*,
Wendong Niu
1,2 and
Shaoqiong Yang
1,2
1
Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, School of Mechanical Engineering, Tianjin University, Tianjin 300350, China
2
Laoshan Laboratory, Qingdao 266237, China
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
J. Mar. Sci. Eng. 2024, 12(11), 2088; https://doi.org/10.3390/jmse12112088
Submission received: 1 September 2024 / Revised: 29 October 2024 / Accepted: 17 November 2024 / Published: 19 November 2024
(This article belongs to the Special Issue Marine Autonomous Vehicles: Design, Test and Operation)
Browse Figures
Versions Notes

Abstract

:
The vibration caused by the movement of internal actuating components within an acoustic underwater glider can interfere with onboard sensors. However, as a new vibration-damping material, phononic crystals can effectively reduce this impact. Using simulation and an underwater test, this work studied the vibration-damping mechanism of the phononic crystal suspension (PCS) designed by Tianjin University, China. The bandgaps and the modes of PCS were calculated first, which offered basic data for the following simulation. Then, the relationship between the modes and attenuation zones (AZs) were broadly considered to reveal the variation law of the AZs with the change in modes, both in the air and under water. Finally, an underwater test was carried out to verify the good vibration-damping effect of the PCS. The results show that the cutoff frequency of the AZs could be predicted by finding the relevant modes. The PCS showed a good vibration-damping effect from 170 Hz to 5000 Hz in the underwater test, with a maximum decrease of 6 dB at 2000 Hz. Finally, the damping of the PCS could suppress the overlap of modes that resulted from Bragg scattering. This work will also provide theoretical guidance for further study on the optimization of phononic crystal mechanisms for vibration damping.

Graphical Abstract

1. Introduction

A buoyancy-driven underwater glider is a kind of low-power unmanned underwater vehicle that can carry various sensors to detect the underwater environment and collect significant marine data continuously. In terms of underwater sound recording, the hydrophones equipped on underwater gliders are still unable to meet the precision requirement of underwater observation missions. Although no propulsion system generates vibration noise, certain interference noises during the operation of underwater gliders may impact the signal acquisition of the hydrophone. For example, the motors used for the buoyancy system and pitch-regulating mechanism can result in significant mechanical noise, which may make the hydrophone misjudge or even lose the observation target. At the same time, the limited internal space for the vibration isolator greatly restricts the vibration-damping effect. The current vibration-damping methods for underwater gliders mainly involve passive noise control and active noise control. However, active noise control can increase the energy consumption of underwater gliders and interfere with the acoustic sensors. Therefore, passive noise control is the only option for acoustic underwater gliders.
At present, there are three main types of passive noise-control methods: acoustic coating, vibration-isolation mechanisms and metamaterials. Acoustic coating primarily eliminates noise through sound-absorbing materials or laser etching technology [1,2,3,4,5], while vibration-isolation mechanisms predominantly attenuate external vibrations through innovative mechanical designs [6,7,8,9]. However, neither acoustic coatings nor vibration isolators can effectively meet the requirement for a broad-bandgap damping effect at low or medium frequencies.
In contemporary studies on metamaterials designed for vibration damping, phononic crystals have gained widespread applications owing to their superior vibration-damping capability across the entire frequency bands. Yu et al. [10] designed a periodic elastic Helmholtz resonator for low-frequency noise mitigation and reported that it targeted low-frequency bands and achieved broad-spectrum attenuation within a compact cavity size, especially in the frequency band from 1 Hz to 378 Hz. Li et al. [11] studied the filtering properties of solid–liquid phononic crystals and obtained some low-frequency bandgaps from 50 Hz to 100 Hz by introducing a locally resonant envelope and considering the fluid–solid coupling effect. Li et al. [12] designed a controllable metamaterial damping mechanism and observed the creation of a novel low-frequency bandgap, resulting in significant vibration-draining effects within the band from 220 Hz to 290 Hz. The acceleration amplitude was notably reduced by a factor of approximately 1 × 104. Li et al. [13] designed a new structure with a wide bandgap in the low-frequency band from 100 Hz to 200 Hz using numerical simulation to improve the noise-damping effect of phononic crystals, which can broaden the total bandgap width by approximately 4.2 times. Jin et al. [14] introduced a layered honeycomb structure. By meticulously tuning the structural parameters and material properties, they effectively mitigated noise in high-frequency bands and achieved an exceptionally wide coupling bandgap that spanned from 600 Hz to 3000 Hz. Yin et al. [15] comprehensively reviewed the applications and development trends of mechanical metamaterials in the field of underwater acoustic control and vibration insulation. This review offered valuable insights for the applications of mechanical metamaterials in the field of underwater acoustics. Qiu et al. [16] innovatively introduced shape memory alloys to achieve the dynamic modulation of phononic crystal bandgaps. The results show that the shape memory alloy widened the bandgap by 103.9% in the XY mode and 437.5% in the Z mode when it transitioned from martensite to austenite. Sal-Anglada et al. [17] proposed a novel enhanced multiresonant layered acoustic metamaterial which improved the response of the device. The results show that the frequencies could be controlled through the structural features of the design and could be adjusted to enhance the effective attenuation bandwidth. Zuo et al. [18] designed a star-shaped metamaterial component applied to underwater vehicles as a vibration-damping mechanism. The results show that the radiation of the power cabin was significantly reduced at all frequencies, especially at 1500–4500 Hz. Zhang et al. [19] proposed a deliberate meta-structure hull design implemented with periodic phononic crystals for practical marine vehicle applications which achieves ultra-low transmission within the 50 Hz to 800 Hz.
Phononic crystals have now been widely studied in the field of noise damping, which primarily functions by creating specific bandgaps involving two mechanisms: Bragg scattering and local resonance. These two mechanisms have their own advantages and disadvantages, as summarized in Table 1.
Based on the above two mechanisms, most scholars explored the noise-damping mechanism from the perspectives of structure, material, mode, etc. Wang et al. [24] analyzed the bandgaps of the phononic crystal structure by modal-based analysis. They found that the characteristics of the bandgaps were determined by the structural parameters of the design, and the bandgap-generation mechanism should be modal resonance rather than local resonance. Ye et al. [25] proposed a phononic crystal structure with a gradient configuration, enabling the merging of adjacent bandgaps by utilizing the Bragg scattering mechanism. Amaral et al. [26] introduced a locally resonant metamaterial to gearbox housings for better vibration characteristics and lightweighting. The results show that the second LRM solution effectively suppressed vibration in the low-frequency range from 50 Hz to 1500 Hz caused by lightweighting. Xin et al. [27] applied Bloch’s theorem to design a snowflake-type phononic crystal structure, realizing 1.68-fold band broadening through structural dimension adjustments. Based on the mechanism of Bragg scattering, Cheng et al. [28] devised a cruciate ligament structure to establish an ultra-low bandgap from 274.3 Hz to 285.6 Hz and reported that the emergence of the low-frequency bandgap primarily stemmed from the resonance shift of the cross-shaped structure. Chu et al. [29] proposed a method to regulate the bandgaps of pentamode metamaterials under the local resonance of the phononic crystal. They obtained an extremely low-frequency (from 0.03 Hz to 0.13 Hz) band without changing the structural parameters. Li et al. [30] designed four vibrators with an acoustic metamaterial. They demonstrated that the increase in vibrator radius enhances the bandgap between each vibrator and the scatterer through locally resonant coupling, consequently widening the bandgap. Wang et al. [31] have developed a novel underwater acoustic model, consisting of two steel cylindrical elements and a central hollow cylinder encapsulated within a rubber matrix. This model is configured with either steel and air or water as the backing material, leveraging the principles of sound resonance absorption to achieve its design objectives. Wang et al. [32] introduced an innovative collaborative optimization framework for underwater sound absorption, integrating material properties, structural configurations, and functionalities into a cohesive design approach. By considering a comprehensive range of design variables, including variational material parameters and topological arrangements, the method is tailored for optimizing sound absorption within a specified frequency band. Consequently, it achieves remarkable performance, with an average sound absorption coefficient of 90% across the frequency range from 300 Hz to 10,000 Hz.
In summary, extensive research was carried out on the two mechanisms of phononic crystals, and the relevant methods for noise damping are summarized, as shown in Figure 1.
Previous studies of phononic crystals primarily treated the fluid as a unit of the periodic structure [33,34,35,36,37,38] rather than investigating the vibration-damping mechanism of the entire underwater phononic crystal. In this study, for the first time, phononic crystals were used to realize vibration-damping effects on the acoustic loads of underwater gliders. Based on the modal results, this study aimed to explore the reasons for the change in the attenuation zones (AZs) and the cutoff frequencies in air and under water. Furthermore, a new prediction method for the attenuation zones is summarized in detail from the perspective of modes. A theoretical analysis process is given for the optimization of phononic crystals in the vibration isolator applied on underwater vehicles. It can also provide theoretical guidance for further research on the optimization of phononic crystal mechanisms for vibration damping.
The rest of this paper is organized as follows. Section 2 discusses the bandgaps of single cells in phononic crystals and analyzes the modes of the phononic crystal suspension (PCS) and aluminum alloy suspension (AAS) in air and under water. Section 3 analyzes the changes in AZs in air and under water and reveals the relationship between the modes and intrinsic frequencies. Section 4 briefly introduces the underwater test and the impact of the damping of rubber on the vibration-damping effect of PCS. Finally, Section 5 presents the conclusions and future works.

2. Bandgap Study and Modal Analysis

2.1. The Structural Parameters

This study aimed to explore the vibration-damping mechanism of the PCS applied to the acoustic underwater glider. The structural parameters of the PCS and AAS are presented in Table 2. Thus, this study began with an exploration into the bandgaps of the PCS. Considering that the operational frequency band of the hydrophone applied on the acoustic underwater glider was concentrated from 50 Hz to 5000 Hz [39], the subsequent simulation set the frequency band range from 0 to 5000 Hz.

2.2. Phononic Crystal Bandgap Study

The PCS explored here consisted of infinite layers of aluminum alloy alternating with rubber layers. Each period was referred to as a single cell of a one-dimensional tubular infinite-period phononic crystal. According to the calculation of the transfer matrix, the bandgap of a single cell was analyzed using the commercial software Comsol Multiphysics. The structure and bandgap characteristics of this infinite-period phononic crystal were obtained, as shown in Figure 2.
Specifically, M in Figure 2 represents the lattice vector, which can be calculated using the following equation:
M = k × a π
where k is the wave vector and a is the lattice constant.
According to Figure 2, the frequency bands from 0 to 5000 Hz show two AZs: from 969 Hz to 2951 Hz and from 3180 Hz to 4681 Hz. Theoretically, the elastic waves that fall in these two AZs will be completely filtered. In practice, the dynamical properties of ideal infinite-period phononic crystals are different from those of finite-period phononic crystals. However, they share an equal vibration-damping effect [40,41]. Thus, the single-cell bandgap can be used to verify the calculations of the AZs.

2.3. Modal-Based Harmonic Response Analysis

The calculation of dry modes was realized through modal analysis with the commercial software Ansys Workbench. The commonly used computational methods for harmonic response analysis in ANSYS Workbench are the modal-based method and the complete method. The complete method uses a complete matrix, and the high-frequency response can be accurately calculated without performing modal analysis. Additionally, the complete method can accommodate a wider range of loads, but the computational cost is higher. The modal superposition method is based on modal analysis and uses the superposition of each mode to obtain the response at each frequency. Therefore, in general, the computational efficiency is higher. However, for the precise calculation of the high-frequency responses, the natural frequencies required by the modal superposition method must be 1.5 to 2 times the upper limit of the harmonic response analysis frequency, so the computational cost is also high for structures with a high modal density and significant modal overlap. Thus, this study utilized the modal superposition method for the harmonic response analysis.
For the modal-based harmonic response analysis, it was first necessary to obtain the modes of the PCS and the AAS, both in air and under water. In order to obtain enough modes within 0–5000 Hz, we selected 700 modes as the unified settings for simulation analysis. The designed module was meshed by importing the material and geometric characteristics of each module. The meshing of the two suspension structures is illustrated in Figure 3. In order to achieve a high speed of calculation on simulated cases and ensure the accuracy of the result, the optimal number of the mesh was selected, which is shown in Table 3. Table 3 compares the intrinsic frequency results of the first dry modes with different mesh numbers. It can be found that when the minimum size was 3 mm, the calculation cost was the lightest, and the difference from the results at 2 mm was less than 1%. Therefore, this size was selected as the optimal mesh setting.
The solid–liquid interactions needed to be considered for the underwater modal analysis of both models. Currently, the vibration characteristics of underwater structures are typically explored using the imaginary mass method and sound–solid coupling method [42,43,44]. In fact, the influence of structural damping and additional damping is larger in the subsequent solution of the harmonic response in the underwater condition. Thus, the acoustic–solid coupling method was selected to solve for the wet modes and underwater harmonic response of the suspensions.
Based on the finite element method, the vibration equation can be written as follows:
M { X ¨ } + C { X ˙ } + K X = { F }
where M is the mass matrix, C is the damping matrix, K is the stiffness matrix and F is the load vector. Considering the loading in the fluid domain, the vibration equation can be rewritten as follows:
K + i ω C + C * − ω 2 M + M ∗ X = F
where C * and M * are the additional damping and the additional mass in the fluid domain, respectively.
The expression for additional damping is
C * = G A R ( ω ) G T
where G is the transformation matrix, A is shape function matrix, R ( ω ) is the radiation resistance matrix and ω is the angular frequency.
The expression for additional mass is
M * = G A Y ( ω ) G T ω
where Y ( ω ) is the radiation resistance matrix.
It can be seen that both the additional mass and the additional damping are related to the shape and angular frequency of the structure, so the formation mechanism of the wet mode harmonious response of the suspension structure is complicated. Thus, by using Ansys, we could accurately and efficiently calculate the harmonic response in the fluid–structure coupling state.
Furthermore, the wet modes were solved by importing the modal information into the acoustic harmonic part for acoustic–solid coupling analysis. The dry and wet modes of the AAS and PCS and their intrinsic frequencies are shown in Table 4 and Table 5.
According to Table 4 and Table 5, the intrinsic frequencies of the wet modes for both the AAS and PCS were reduced compared with those of the dry modes. Most of the reductions did not exceed 40%, which aligns with the trends observed in the relevant literature [45]. Thus, the correctness of our modal results was verified. In brief, the AAS had a larger span between the adjacent modes due to the weak damping characteristics, while the PCS had a larger modal density and a slighter difference in the size of the adjacent intrinsic frequencies. Therefore, the first six dry and wet modes shared relatively similar rates of change in the intrinsic frequencies.

3. Determination of AZs and Modal Analysis

3.1. The AZ Calculation and Simulation Setup

After the dry and wet modes of both the AAS and PCS were obtained, the next step was to further investigate the variation law of their responses at different frequencies under the actions of specific excitations. First, the harmonic response analysis of the AAS and PCS in air was performed based on the modal state under the undamped condition. Force excitation was applied to the two surfaces connected by the left face of the suspension and the front fairing. The amplitude was 1 N, and its phase was 0. The sweep frequency ranged from 0 to 5000 Hz, and the displacement response amplitude was measured on the right face of the PCS and AAS. The analysis results are shown in Figure 4.
The displacement amplitude of the PCS demonstrated a better damping effect in the overall frequency than the AAS, and the displacement response curve exhibited notably lower magnitudes within the first AZ (969–2951 Hz) and second AZ (3180–4681 Hz). The comparison of the two suspensions showed that the PCS achieved a significant damping effect in both AZs.
Next, taking the dry modal results as a foundational basis, underwater harmonic response analysis was performed. Subsequently, an acoustic domain was defined to simulate an infinite water environment, which established an acoustic free-field space.
Within this setup, the interfaces of the acoustic and structural domains were designated as acoustic–solid coupling surfaces, whereas the connection between the suspension and the fairing was identified as the load application surface, where the excitation parameters remained consistent with those described earlier. In this arrangement, each suspension was categorized as the solid domain, and the water area was delineated as the acoustic domain, which ultimately culminated in the formulation of a comprehensive acoustic–solid coupling computational model (Figure 5). The vibration response results of the two suspensions under water were compared, and the results are shown in Figure 6. It can be seen that the response of the AAS was slightly shifted compared with that of air, while the PCS was significantly different in water and air. We supposed that the difference in the fluid–structure coupling effect was mainly due to the difference in the material and stiffness, which mainly changed the structures’ intrinsic frequencies of the modes. More detailed analysis is discussed in the following content.

3.2. Vibration Curve Analysis and Mechanism Study

3.2.1. Modal Characteristics at the Cutoff Frequency

The vibration-damping mechanisms that acted on the overall frequencies were investigated by analyzing the corresponding peaks of the modes. According to Figure 5 and Figure 6 in Section 3.1, the curve of the AAS’s displacement response was very smooth, and resonance existed only at 3372.3 Hz in the sixth-order dry mode, which resulted in a larger vibration response. However, the PCS had more resonance peaks due to the higher modal density of the harmonic response curve at approximately 993 Hz, 3084 Hz, 3321 Hz and 4971 Hz. Table 6 shows the fourth-order modal information of the excitation of these four resonance peaks.
According to this table, the four peaks of the PCS were caused by the 18th-, 162nd-, 218th- and 624th-order dry modes. The intrinsic frequencies of these four orders of dry mode were all coincident with the boundary frequencies of the two AZs given in Figure 4. The position of the resonance peak determined the onset frequency and boundary frequency of each AZ of the PCS, while the modal shapes determined the position of the resonance peak. Thus, the modal shapes were a fundamental factor in determining the boundary frequency of each AZ.
In addition, the dry and wet modes of the PCS also exhibited inconsistencies in the orders under similar vibration patterns. For example, the first and second orders of the axial vibrations were the sixth and thirteenth orders of the dry mode, respectively (Figure 7a,b), but the eighth and twelfth orders of the wet mode, respectively (Figure 7c,d).
Here, the first axial vibration patterns of both the dry and wet modes reflected the squeezing of all three rubber layers. In addition, the second-order patterns reflected the squeezing of the left rubber layer and the stretching of the right rubber layer. As a result, the axial vibration patterns of both the dry and wet modes were the same, but their modal orders exhibited opposite variation trends, which was consistent with the findings in Section 2.
After analyzing the changes in the peak values of the vibration curve, here, we further compare the displacement responses of the AAS and PCS in air and under water, as illustrated in Figure 8. Above the frequency of 955 Hz at b, the underwater vibration response of the PCS generally surpassed that in air but typically remained lower than that of the AAS. This indicates that the vibration-damping effect of the PCS under water could achieve a more stable vibration isolation effect, although it was slightly worse than that in air. In addition, the harmonic response curves of the AAS and the PCS showed high variation at the peak. The peak value (at a) of the underwater vibration response curve for the AAS was shifted toward the low frequency, which was caused by the water’s added mass effect on the structural modes, as well as the water’s additional damping effect on the resonance frequency.

3.2.2. Detailed Analysis of Band Migration

The peak at a in the harmonic response curve in air was caused by the resonance triggered by the sixth-order dry mode. According to Table 5, the sixth-order wet mode also caused a resonance peak at region a in the underwater harmonic response curve, and the vibration pattern of the sixth-order wet mode was similar to that of the sixth-order dry mode, while the intrinsic frequency was reduced. Therefore, the shapes of the two peaks at a were basically the same, with some slight difference, but that of the wet mode shifted to a lower frequency.
The half-power bandwidth Δ ω of the PCS can be expressed by
Δ ω = 2 ξ ω n
where ξ is the damping ratio and ω n is the resonance frequency.
Based on this equation, the half-power bandwidth will be increased when the additional damping introduced in the watershed increases the damping ratio. Considering the larger resonant frequency at a, it is normal for the bandwidth to show a significant change, which can explain the slight difference between the two peaks.
In the underwater vibration response curve (Figure 8), the resonance frequency at point a was 3190 Hz, while the inherent frequency of the sixth-order wet mode was 3212.6 Hz. Given Equation (7), it can be seen that the result of multiplying the intrinsic frequency ω 0 on the right side of Equation (7) decreases when the damping parameter ξ becomes larger. Thus, the resonance frequency ω p e a k on the left side of Equation (6) becomes smaller:
ω p e a k = ω 0 1 − 2 ξ 2
where ω p e a k denotes the resonance frequency, ω 0 denotes the intrinsic frequency and ξ denotes the damping ratio.
The difference of 22.6 Hz exceeded the error range introduced by the sweep interval (approximately five times the sweep interval), indicating that the effect of damping on the resonance frequency could not be ignored.
As shown in Figure 8, the peak frequencies of the PCS in air and under water were different at b and c. The peak frequencies at b were basically stable, while those at c were shifted to lower frequencies, which was also due to the damping of the intrinsic frequency caused by the additional mass effect of the fluid.
Table 7 shows the dry and wet mode information related to the peak at c. The resonance frequency at c was 3085 Hz, which corresponded to the 162nd-order dry mode intrinsic frequency of the PCS (3085.3 Hz). In addition, the peak resonance frequency at c of the underwater vibration response curve was 2670 Hz, and the intrinsic frequency of the wet mode close to this frequency was 2719.5 Hz (171st order). According to Table 7, the 162nd-order dry mode vibration pattern was characterized by the distortion of the three rubber layers in the axial direction, which contrasted with the 171st-order wet mode vibration pattern. Both patterns exhibited minimal alterations in the magnitudes of the modal displacements and the specific rubber layer deformation at the ends. However, the intrinsic frequency of the wet mode experienced a more pronounced decrease. As a result, it led to a large shift in the peak value at c toward low frequencies. Similar to the case of the AAS, the resonance frequency appeared to be smaller than the wet mode intrinsic frequency due to the additional damping effect of the water.
Similarly, the dry and wet modes associated with the peak resonance frequency at b could be obtained, and their modal information is given in Table 8. The two vibration modes were quite different. However, they both showed the deformation of the three rubber layers and the middle two aluminum alloy layers, and the intrinsic frequency was only reduced by a small magnitude. Therefore, the positions of the two resonance peaks at b were basically unchanged.
After the underwater harmonic response was obtained, we compared the exact vibration-damping effect of the AAS and PCS by using the tool of vibration transmittance, which can be calculated using
T = 20 lg U O U I
where T is the vibration transmission rate, Uo is the displacement amplitude at the output and UI is the displacement amplitude at the input.
The calculation results are given in Figure 9. The input response was the acceleration response of incident sound plane (the force application plane), and the output plane (transmission sound plane) was the other side of the AAS, as shown in Figure 5. Compared with Figure 8, some peaks of the curve disappeared in some frequency bands. From Equation (8), we can see that the closer the value between the input vibration and the response is, the closer to 0 the vibration transmissibility is. Therefore, when it comes to the reason for the disappearance of the vibration peak in Figure 9, it is due to the fact that the original vibration and the vibration response were almost the same, which also indicates that the AAS failed to present a better vibration-damping effect in this frequency band. This mainly depends on the bandgap characteristics and modal characteristics of the material itself. Furthermore, not every mode has its corresponding peak value; only the surface-localized modes can excite the peak value. Compared with air, the underwater onset frequency of the first AZ did not significantly change, but the cutoff frequency was reduced by approximately 278 Hz. Evidently, this boundary frequency coincided with the resonance frequency at b and c on the underwater vibration response curve of the PCS, which was consistent with the law obtained in air. According to the harmonic response analysis, the wet mode was the main reason for the peaks at b and c, and the cutoff frequency showed a large shift toward the low frequencies under the additional damping effect of water. Therefore, it was feasible to predict the cutoff frequency of the first AZ of the PCS using the wet mode. However, the underwater vibration transmittance of the PCS was generally less than 0 in the band from 2669 Hz to 5000 Hz, significantly lower than that of the AAS; therefore, it still exerted a significant effect on the vibration isolation. Notably, the curves of the PCS from 2669 Hz to 5000 Hz overlapped to a great extent, which is further discussed in the next section.
Two conclusions could be obtained based on the analysis in this section. First, the first AZ of the PCS under water was narrowed in the band from 1000 Hz to 5000 Hz, and the cutoff frequency shifted to a lower frequency. Second, the cutoff frequency of the first AZ could be predicted by the intrinsic frequency of the wet mode, and the modal vibration shapes of the resonance at the excitation boundary were more similar to those in air. However, the mode density of the second AZ was high, with a high level of mode overlapping, necessitating further research on the influencing factors of rubber damping.

4. Underwater Testing and Research on the Effect of Rubber Damping

4.1. Experimental Preparation

To examine the vibration-damping performance of the phononic crystals, an underwater test was designed for an acoustic underwater glider with the suspending acoustic payload at the front end, as illustrated in Figure 10b. Furthermore, the AAS was set as the experimental control group, whose specific structure is shown in Figure 10a. The black section at the front end represents the acoustic payload, while the yellow portion denotes the forebody fairing of the underwater glider. The connecting part between the acoustic load and the fairing was the suspension structure.
The underwater test was carried out in the anechoic pool laboratory of the National Deep Sea Base Center of China. The communication method of the underwater glider utilized a watertight cable for underwater communication. The baud rate of the wired communication was the same as that of radio, which is 19,200 Hz, and the communication interval was 1 s, so the accuracy of the excitation source control could be guaranteed. The excitation source was the rolling motor of the underwater glider. During the test, the rotating speed of the rolling motor remained at 400 rpm, and the motor moved from the left rudder position of the underwater glider to the right rudder position. Excitation source software was utilized for signal excitation of the acoustic underwater glider, and the frequencies of the axial channel signals were recorded in the signal-receiving interface. In order to acquire a smoother presentation of the data, the experimental spectrogram was deburred. The third-octave band was utilized to portray the signals acquired by the acoustic load of the AAS and PCS, both in air and under water, as shown in Figure 11.
Clearly, the received signal strength of the acoustic load with the PCS was generally lower than that with the AAS in most of the frequency bands from 0 to 5000 Hz, much closer to the ambient noise level. For example, the best vibration-damping effect in the air was achieved at 450 Hz, with a decrease of approximately 25 dB, indicating the significant vibration-damping effect of the PCS in the experiment. In the underwater environment, the PCS showed a good vibration-damping effect from 170 Hz to 5000 Hz, with a maximum decrease of 6 dB (at 2000 Hz). From 1300 Hz to 2000 Hz, the accepted signals of the axial channel were generally lower than the marine background noise of the Class 0 sea state (curve SS0).

4.2. Analysis of the Influence of Rubber Damping

Having noticed the different trends between the experimental results and the simulated results, we supposed that the damping of the rubber may have been the dominant factor. Thus, we continued to research the damping of the rubber, as is shown in Figure 12. As revealed by the underwater vibration-damping curve for the PCS, the axial reception was indeed weakened in the first AZ and the remaining frequency band, which was also consistent with the simulation results. The anastomotic region represents the region where the trend of the experimental data closely matched with the simulated data with a damping ratio of 0.11. The disturbed region represents the region where the trend of experimental data barely matched with the simulated data with a damping ratio of 0.11. However, the frequencies from 2669 Hz to 2950 Hz had a high peak value and a narrow interval of vibration transmittance of less than 0, which was not consistent with the simulation results. These results indicate that the vibration-damping effect in this frequency band should be worse than that of the first AZ, failing to demonstrate any significant vibration-damping ability. This was because the damping of the rubber was neglected in the simulation process.
Therefore, a damping ratio of 0.11 (matching the experimental condition) was added to the rubber structure to determine whether the vibration-damping effect in all the frequency ranges was caused by the Bragg scattering mechanism of the phononic crystal. Moreover, underwater harmonic response analysis was performed to calculate the vibration transmittance in the full frequency band. Figure 12 shows the simulation data of the vibration transmittance of the PCS with and without the existence of rubber damping.
When rubber damping was considered, the peak values in the band from 2669 Hz to 2950 Hz, which connected the first AZ with the remaining frequencies, were effectively suppressed. The consideration of rubber damping also broadened the AZs of the PCS. In addition, the vibration transmittance curve in the first AZ was very close to that without the rubber damping, with significant suppression only at the peaks. To conclude, the vibration-damping effect in the experimental measurements was mainly realized through the Bragg scattering mechanism. For the remaining frequencies, the vibration transmittance curves presented more peaks without considering the rubber damping. At high frequencies, the effect of rubber damping on the vibration transmission was higher than that of the Bragg scattering, which effectively inhibited the multi-peak interference. The peak density and noise level of each frequency segment could potentially be reduced through the adjustment of the rubber damping.

5. Conclusions and Future Work

To explore the primary mechanism behind the vibration-damping effect of a PCS applied acoustically under water, this study investigated the AZs and cutoff frequencies, both in air and under water. Having obtained the operational frequency bands of the hydrophone (50–5000 Hz), we set the simulated frequency bands from 0 to 5000 Hz. Then, our research began with the theoretical investigation of the bandgaps of phononic crystals and the vibration-damping mechanism of the PCS. Specifically, the dry and wet modes of both the AAS and PCS were calculated through model-based harmonic response analysis, and the relationship between the wet mode and dry mode was elucidated through simulation data analysis. The underwater vibration transmission characteristics of the AAS and PCS were also calculated using the acoustic–solid coupling method. The experiment verified the good vibration-damping effect of the PCS, both in air and under water. The difference between the simulation data and experimental data was also examined in this study. The main conclusions were as follows:
(1)
A comparison of the modes of the AAS and PCS showed that the modal vibration patterns shared the same trend, both in air and under water. Moreover, the frequency of the wet mode was smaller than that of the dry mode in cases of the same order. Although the wet mode had different mode orders compared with the similar vibration pattern of the dry mode, the intrinsic frequency of the wet mode was slightly lower than that of the dry mode.
(2)
The modes were a fundamental factor that determined the boundary frequencies of the AZs of the PCS. In the infinite water area, the cutoff frequency of the first AZ could be predicted by its intrinsic frequency in the wet mode. The prediction method is summarized as follows. Initially, the dry mode and vibration transmittance of the PCS should be calculated to obtain the dry modal shapes at the resonance peak of the cutoff frequency in each AZ. Then, it is necessary to find a more similar vibration pattern in the wet mode and predict the cutoff frequency of each AZ with the intrinsic frequency of the wet mode of this order. In summary, the prediction method proposed in this paper is based on known geometric structures and the modal prediction method to find the scale frequency at which the changes in the mode occur, thereby determining the damping frequency band. This is a closed-loop research approach, and the method proposed in this paper is innovative, guiding geometric optimization from the modal perspective. Thus, this method can be repeatable with different geometries.
(3)
Different from the AAS, the PCS still showed a stable vibration-damping effect from 260 Hz to 5000 Hz during the experiment performed in air, and the maximum damping effect could be up to 25 dB. For the underwater experiment, the PCS showed a significant damping effect within the target frequencies from 1000 Hz to 5000 Hz. The maximum damping effect could be up to 6 dB. The received signals of the axial channel of the acoustic load in the band from 1300 Hz to 5000 Hz were generally lower than that of the marine background noise in the sea state of Class 0.
(4)
The difference between the experimental results and the simulation results mainly lay in whether or not the damping of the rubber layers was considered. The vibration-damping effect of the PCS in the first AZ was mainly realized using the Bragg scattering mechanism of the phononic crystal. For the second AZ, the presence of damping had a significant influence on the vibration transmission of the PCS. As more peaks appeared in the second AZ, the contribution of the Bragg scattering mechanism was relatively lower compared with the influence of the rubber damping.
In the future, our work will focus on designing vibration-damping structures for specific frequency bands based on the acoustic loads carried by various unmanned underwater vehicles. The designed vibration-damping structure will be further optimized with the vibration-damping mechanism revealed in the current study. Additionally, the impact of the damping ratio on the vibration characteristics will be investigated in detail so that we can utilize “damping” as a tool to isolate vibration by suppressing excessive damping peaks. Then, the Bragg scattering principle or locally resonant principle can be used to set different solid–solid or solid–liquid combinations and design special pore structures to generate the required frequency bands. Finally, the noise-damping effect at the target frequencies can be successfully achieved.

Author Contributions

Q.S.: Conceptualization, Funding acquisition, Writing—Review & Editing. Y.Y.: Writing—Original Draft, Writing—Review & Editing, Methodology, Investigation, Validation. P.W.: Methodology, Investigation, Writing—Review & Editing. M.Y.: Funding acquisition, Resource. T.S.: Funding acquisition, Supervision, Resource. W.N.: Funding acquisition, Resource. S.Y.: Conceptualization, Resource. All authors have read and agreed to the published version of the manuscript.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and publication of this article: This work was jointly supported by the National Key R&D Program of China, the China Postdoctoral Science Foundation (2022TQ0233, 2022M722367), the Laoshan Laboratory Science and Technology Innovation Project (LSKJ202200200), National Key R&D Plan (Grant No. 2021YFC3100903) and Shandong Key R&D Plan (Grant No. 2021CXGC010708).

Institutional Review Board Statement

Informed consent was obtained from all subjects involved in the study.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Jmse 12 02088 g001
Figure 1. Classification and summary of the damping mechanism of phononic crystals.
Figure 1. Classification and summary of the damping mechanism of phononic crystals.
Jmse 12 02088 g001
Jmse 12 02088 g002
Figure 2. Structure and bandgap characteristics of a single cell of an infinite periodic phononic crystal. (a) Infinite-period phononic crystals and the structure of a single cell; (b) bandgap of an infinite-period phononic crystal.
Figure 2. Structure and bandgap characteristics of a single cell of an infinite periodic phononic crystal. (a) Infinite-period phononic crystals and the structure of a single cell; (b) bandgap of an infinite-period phononic crystal.
Jmse 12 02088 g002
Jmse 12 02088 g003
Figure 3. The mesh generation of the suspensions: (a) phononic crystal suspension (PCS); (b) aluminum alloy suspension (AAS).
Figure 3. The mesh generation of the suspensions: (a) phononic crystal suspension (PCS); (b) aluminum alloy suspension (AAS).
Jmse 12 02088 g003
Jmse 12 02088 g004
Figure 4. Harmonic response analysis results of AAS and PCS in the air.
Figure 4. Harmonic response analysis results of AAS and PCS in the air.
Jmse 12 02088 g004
Jmse 12 02088 g005
Figure 5. The computational model settings of both suspensions: (a) AAS; (b) PCS.
Figure 5. The computational model settings of both suspensions: (a) AAS; (b) PCS.
Jmse 12 02088 g005
Jmse 12 02088 g006
Figure 6. Comparison of harmonic analysis results in water between AAS and PCS.
Figure 6. Comparison of harmonic analysis results in water between AAS and PCS.
Jmse 12 02088 g006
Jmse 12 02088 g007
Figure 7. Comparison of first and second axial vibration shapes of PCS in dry and wet modes: (a) first order of dry axial mode; (b) second order of dry axial mode; (c) first order of wet axial mode; (d) second order of wet axial mode.
Figure 7. Comparison of first and second axial vibration shapes of PCS in dry and wet modes: (a) first order of dry axial mode; (b) second order of dry axial mode; (c) first order of wet axial mode; (d) second order of wet axial mode.
Jmse 12 02088 g007
Jmse 12 02088 g008
Figure 8. Vibration responses of two suspensions in air and under water. (a, b, and c represents the moving regions of the vibration peak in the water and the air of the AAS/ PCS respectively. Triangles, rhombus, and stars represent the vibration peak of the AAS/ PCS).
Figure 8. Vibration responses of two suspensions in air and under water. (a, b, and c represents the moving regions of the vibration peak in the water and the air of the AAS/ PCS respectively. Triangles, rhombus, and stars represent the vibration peak of the AAS/ PCS).
Jmse 12 02088 g008
Jmse 12 02088 g009
Figure 9. Underwater vibration transmittance of two suspensions.
Figure 9. Underwater vibration transmittance of two suspensions.
Jmse 12 02088 g009
Jmse 12 02088 g010
Figure 10. Physical design of two suspensions: (a) AAS; (b) PCS.
Figure 10. Physical design of two suspensions: (a) AAS; (b) PCS.
Jmse 12 02088 g010
Jmse 12 02088 g011
Figure 11. Signal characteristics of AAS and PCS under platform vibration excitation ((A, B) are enlarged images; 1:5 and 1:8 indicate the corresponding scales).
Figure 11. Signal characteristics of AAS and PCS under platform vibration excitation ((A, B) are enlarged images; 1:5 and 1:8 indicate the corresponding scales).
Jmse 12 02088 g011
Jmse 12 02088 g012
Figure 12. Simulation data of PCS’s vibration transmissibility with (yellow line) and without (blue line) rubber damping.
Figure 12. Simulation data of PCS’s vibration transmissibility with (yellow line) and without (blue line) rubber damping.
Jmse 12 02088 g012
Table 1. Advantages and disadvantages of the Bragg scattering mechanism and local resonance mechanism.
Table 1. Advantages and disadvantages of the Bragg scattering mechanism and local resonance mechanism.
MechanismAdvantagesDisadvantages
Bragg scattering🗹 High correlation with structural and material parameters.
🗹 Interaction with elastic waves based on periodic structures.
🗹 Production of a forbidden band effect [20].		⌧ Limited spacing between scatterers.
⌧ Challenges in practical applications.
⌧ Not suitable for disordered or non-periodic structures [21].
Local resonance🗹 No requirement for periodic structures.
🗹 Applicable for disordered or non-periodic structures.
🗹 The special structure of a single scatterer interacts with the incident wave [22].		⌧ Typically weak [23].
⌧ Enhancement may require a specific structural design or material selection.
⌧ The special structure necessitates precise design and manufacturing [22].
⌧ Requires desired bandgap characteristics in practical applications.
Table 2. Settings of structural parameters for the phononic crystal suspension (PCS) and aluminum alloy suspension (AAS).
Table 2. Settings of structural parameters for the phononic crystal suspension (PCS) and aluminum alloy suspension (AAS).
MaterialDensity
(kg/m3)		Young’s Modulus (MPa)Poisson’s RatioAxial Thickness (m)Inside Diameter (m)Outside Diameter (m)
Nitrile butadiene rubber1300120.470.0200.0400.060
Aluminum alloy277071,0000.330.0200.0400.060
Table 3. Verification of mesh independence.
Table 3. Verification of mesh independence.
Number of Cells
(Thousand)		Minimum Mesh Size (mm)Intrinsic Frequency of the First Dry Mode (Hz)Result Deviation
(%)
4486397.6-
8905400.57.3%
20283402.03.7%
65102402.70.17%
Table 4. Comparison of the dry and wet modes of the AAS with the same modal shapes.
Table 4. Comparison of the dry and wet modes of the AAS with the same modal shapes.
Modal ShapesModal OrdersIntrinsic Frequency (Hz)
Dry ModeWet ModeDry ModeWet ModeDry ModeWet Mode
Jmse 12 02088 i001Jmse 12 02088 i00211402.0276.5
Jmse 12 02088 i003Jmse 12 02088 i00422418.6280.8
Jmse 12 02088 i005Jmse 12 02088 i006432738.81920.1
Jmse 12 02088 i007Jmse 12 02088 i008542763.11966.6
Jmse 12 02088 i009Jmse 12 02088 i010352139.12096.5
Jmse 12 02088 i011Jmse 12 02088 i012663372.33212.6
Table 5. Comparison of the dry and wet modes of the PCS with the same modal shapes.
Table 5. Comparison of the dry and wet modes of the PCS with the same modal shapes.
Modal ShapesModal OrdersIntrinsic Frequency (Hz)
Dry ModeWet ModeDry ModeWet ModeDry ModeWet Mode
Jmse 12 02088 i013Jmse 12 02088 i0141126.417.7
Jmse 12 02088 i015Jmse 12 02088 i0162226.418.1
Jmse 12 02088 i017Jmse 12 02088 i0183376.875.0
Jmse 12 02088 i019Jmse 12 02088 i02044131.289.5
Jmse 12 02088 i021Jmse 12 02088 i02255131.889.9
Jmse 12 02088 i023Jmse 12 02088 i02486296.6219.2
Table 6. Four dry modes of PCS.
Table 6. Four dry modes of PCS.
Modal OrdersModal ShapesIntrinsic Frequency (Hz)
18Jmse 12 02088 i025993.19
162Jmse 12 02088 i0263083.4
218Jmse 12 02088 i0273321.0
624Jmse 12 02088 i0284971.1
Table 7. Dry and wet modes related to the peak of the displacement response curve at c.
Table 7. Dry and wet modes related to the peak of the displacement response curve at c.
Modal OrdersModal ShapesIntrinsic Frequency (Hz)
162 (dry)Jmse 12 02088 i0293085.3
171 (wet) Jmse 12 02088 i0302719.5
Table 8. Information on the dry and wet modes related to the peak at b.
Table 8. Information on the dry and wet modes related to the peak at b.
Modal OrdersModal ShapesIntrinsic Frequency (Hz)
18 (dry)Jmse 12 02088 i031913.2
19 (wet)Jmse 12 02088 i032904.83
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MDPI and ACS Style

Sun, Q.; Yang, Y.; Wu, P.; Yang, M.; Sun, T.; Niu, W.; Yang, S. Study on the Vibration-Damping Mechanism of a New Phononic Crystal Suspension Equipped on Underwater Gliders. J. Mar. Sci. Eng. 2024, 12, 2088. https://doi.org/10.3390/jmse12112088

AMA Style

Sun Q, Yang Y, Wu P, Yang M, Sun T, Niu W, Yang S. Study on the Vibration-Damping Mechanism of a New Phononic Crystal Suspension Equipped on Underwater Gliders. Journal of Marine Science and Engineering. 2024; 12(11):2088. https://doi.org/10.3390/jmse12112088

Chicago/Turabian Style

Sun, Qindong, Yuhan Yang, Pan Wu, Ming Yang, Tongshuai Sun, Wendong Niu, and Shaoqiong Yang. 2024. "Study on the Vibration-Damping Mechanism of a New Phononic Crystal Suspension Equipped on Underwater Gliders" Journal of Marine Science and Engineering 12, no. 11: 2088. https://doi.org/10.3390/jmse12112088

APA Style

Sun, Q., Yang, Y., Wu, P., Yang, M., Sun, T., Niu, W., & Yang, S. (2024). Study on the Vibration-Damping Mechanism of a New Phononic Crystal Suspension Equipped on Underwater Gliders. Journal of Marine Science and Engineering, 12(11), 2088. https://doi.org/10.3390/jmse12112088

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