1   Introduction

The emergence of underwater submersibles has greatly advanced humanity's exploration of the deep sea. To date, submersibles are widely used in education, scientific research, military, and tourism sectors with the introduction of composite materials. Compared with metal materials, composites offer clear advantages in terms of strength, vibration reduction, and corrosion resistance (Guo et al., 2023; Xie et al., 2023), which lead to their increasingly widespread application in submersibles. The industry has extensively examined the application of composites in submersibles and the vibration characteristics of these vehicles. In terms of experimentation, Wang et al. (2024a) conducted experimental research on the compressive dwell fatigue behavior of pressure hulls made of titanium alloy for deep-sea manned submersibles. This study provided guidance for the structural design of manned cabins in deep-depth transparent submersibles. Chen et al. (2009) experimentally investigated the impact of rubber coatings on the blast resistance performance of naval hulls. The results showed that the coating only slightly affected the low-frequency response but could mitigate the high-frequency response. With regard to numerical research, Wang et al. (2024b) analyzed the buckling behaviors under axial compression for cylindrical shells made of steel and composite materials. They also derived the theoretical formula of the linear elastic buckling for these shells. Feng et al. (2024) aimed to determine the damage accumulation and extension of composite laminates under secondary impact. For this purpose, they combined experimental and finite element simulation analyses to investigate the damage and load–time curve after secondary impact. McCoy and Sun (1997) employed finite element simulation techniques incorporating fluid–structure interaction effects. Moreover, they conducted a dynamic response analysis of laminated cylindrical shell structures constructed from graphite/epoxy resin subjected to underwater explosive loads. Their results highlighted the significant impact of fluid–structure interaction effects on stress distribution in composite structures. Liu et al. (2000) used finite element numerical analysis methods combined with a modified DAA method to study the dynamic response of hull structures made of glass fiber composites under underwater explosion loads. Their results were compared with experimental findings, which showed good agreement between finite element calculations and experimental results. Theoretically, Kandasamy et al. (2016) summarized various active, passive, and hybrid vibration mitigation methods, which widely employed the principles of adjusting stiffness and damping for vibration control. Shifting the natural frequency of the structure outside a certain bandwidth of the excitation frequency can prevent harmful energy superposition at resonance frequencies while increasing the damping of the structure can effectively reduce resonance peaks by dissipating vibrational energy. Yin et al. (2025) achieved effective control of the overall system vibration response by modifying the structural configuration, which resulted in a shift of peak frequencies in the structural vibration spectrum. Jiang et al. (2024) adjusted the inherent stiffness of the structure by adding mass locally, which successfully reduced the radiated noise by2–3dB. Li et al. (2016) performed modal and vibration noise experiments using an underwater vehicle model, which clarified the relationship between natural frequencies and the peak frequencies of the vibration noise spectrum. By modifying the structural configuration, they effectively controlled underwater acoustic radiation. Moon et al. (2010) used FEM to analyze the buckling failure of pressure cylindrical shells made of filament-wound composites for underwater vehicles. The results of their analysis, which was conducted using the self-developed finite element program ACOS, were compared with experimental results. This comparison revealed an error range of 2% to 23% between the analysis based on the program and the test results. Du and Cui (2014), and Du (2017) presented a finite element analysis (FEA) on spherical shells made of high-strength steel considering material and geometric nonlinearity, an elastic–plastic stress–strain relationship, and the initial deflection of the shell during the manufacturing process. The results were confirmed by model collapse tests. Moreover, Wang et al. (2019) proposed a relevant calculation and design method for spherical hulls made of ultrahigh-strength steel for full-ocean-depth manned cabins by using a direct method by FEA. The author also developed a sphere model and conducted a hydrostatic pressure experiment, which realized verification among different method results. Song et al. (2024) analyzed the natural frequency response of the FG bio-composite plate within the framework of the newly developed refined higher-order shear deformation plate theory. In addition, the imperfection impact on frequency behavior was evaluated while considering three imperfection distribution patterns. Fathallah et al. (2015) optimized the layering design of different types of carbon fiber materials to minimize the buoyancy coefficient of elliptical pressure hulls made of composite materials for submersibles. Tang et al. (2019) and Li et al. (2020) investigated the acoustic radiation characteristics of composite shell structures based on composite shell theory. They studied the influence of elastic modulus and ply layup on structural radiation noise through numerical calculations and experimental methods. Moreover, a new semi-analytical method was proposed based on thin shell theory to analyze the free vibration characteristics of cylindrical shells with stepped thickness. This method is suitable for exploring the free vibration characteristics of cylindrical shells with stepped thicknesses under complex boundary conditions. It offers advantages such as good convergence, high computational accuracy, and simplified boundary condition simulation. Sun and Zou (2015) utilized a three-dimensional hydroelasticity method to analyze the vibration response characteristics during underwater navigation of a certain type of submersible. By solving the generalized coordinate responses of dry modes and the corresponding added mass of the surrounding water, they determined the correspondence between dry modal frequencies and wet resonant frequencies. Xu et al. (2011) used mechanical impedance theory to determine the mechanical impedance characteristics of pressure-resistant shells through FEA software modeling and simulation. Their analysis provided essential raw data and parameter inputs for the dynamic characteristic analysis of the entire system involving pipeline–elastic support–shell foundation. It also offered necessary theoretical guidance for system design and engineering installation. Chen and Dong (2023) strengthened external pressure vessels using the autofrettage approach to achieve an optimal balance between structural strength and weight. Zhang et al. (2022) designed a carbon fiber-reinforced polymer composite for full ocean depth applications, which not only reduces structural mass but also exhibits high hydrostatic load-bearing capacity. Lee et al. (2013) conducted an optimization study on the design loads of cylindrical shells made of composite sandwiches under hydrostatic external pressure. Structural buckling failure and material failure constraints were considered during the optimization calculations. Messager et al. (2022) used FEMs to perform layer optimization design on laminated cylindrical pressure hull structures made of composites for deep-sea submarines. This design aimed to improve the buckling resistance of the structure. Lopatin and Morozov (2015) employed the Galerkin method to perform a buckling analysis on cylindrical shells made of composite sandwiches with both ends clamped under uniform external pressure. The results were compared with those obtained from FEMs, which led to the derivation of an analytical formula for the critical load of the structure. Wu et al. (2023) modeled the submarine hull as a jointed conical-orthogonally stiffened cylindrical–spherical shell coupled with several circular plates. They also proposed an analytical method for the coupled shaft–submarine hull system. The hydrodynamic pressures acting on the rotor and the duct–stator system were considered single-point forces and multiple uniformly or non-uniformly spaced point forces acting on the shaft and the conical shell, respectively. The vibration displacements were analytically expanded with Fourier series in the circumferential direction. Ismail et al. (2023) investigated the buckling loading of imperfect steel spherical shells under external pressure, which presented a numerical calculation with a focus on the buckling behavior of imperfect shells subjected to external pressure. Bakshi and Chakravorty (2014) studied the first-ply failure of conoidal shells made of thin composites subjected to uniformly distributed load, which aimed to achieve high failure values. Walker and Smith (2003) described a technique for combining genetic algorithms with FEM to minimize the mass and deflection using a multi-objective approach. Ghandehari et al. (2024) examined the free vibration characteristics of coupled nested conical shells made of porous composite materials. Ghandehari et al. (2025) also assessed the vibration behavior of a novel shell structure, that is, nested dual conical shells, and a truncated conical shell under varying thermal conditions. Rezaiee‑Pajand and Masoodi (2019) investigated the buckling and post-buckling behavior of plates and shells with an efficient triangular shell element having six nodes.

The excellent damping properties of composite materials will not only greatly enhance the acoustic stealth performance of underwater submarines but also offer good impact protection for the equipment inside the submarine (Zhao et al., 2025; Ma et al., 2024). However, the composite material itself cannot adjust the balance of strength and stiffness well. Moreover, the production cost of high-performance composite materials, such as titanium alloy and carbon fiber, is dozens of times higher than that of steel products of equal quality. These factors have led to a lower maturity of composite materials in the submarine application. In response, this study takes the amidship section of the submersible as a sample and designs a laminated structure model with material properties between steel and composite materials. The manufacturing cost of this structure is only 3–5 times that of an equal-quality steel structure. A comparative model of steel with equal mass and shape is designed to compare the difference with the ordinary steel structure. Moreover, the modal and damping characteristics of the two models are analyzed and discussed through simulation and experiment. This study provides technical references for the design of submarines in the aspects of acoustic stealth, vibration reduction, and impact resistance protection.

2   Structural design

Laminated research models and steel comparison models with equal mass were designed to compare the structural characteristics of laminated and steel models. Based on literature research (Zhao, 2023; Xu, 2020; Yu et al., 2011; Zhang et al., 2011; Yang et al., 2011a; Yang et al., 2011b; Du et al., 2024; Cheng et al., 2024) and considering structural test conditions, the primary dimensions of the models were determined to be 0.6 m in diameter and 1 m in length. Numerical calculations indicated that reducing rib spacing beyond 50 mm did not significantly improve structural strength and stability but significantly increased overall model mass. Consequently, rib dimensions were finalized at 8 mm×32 mm, with a rib spacing of 50 mm and a model thickness of 5 mm, to achieve a balance between structural weight and strength.

During the design of the laminated model, the thickness of the steel structure portion of the shell was reduced to ensure equal mass with the steel comparison model after the carbon fiber layup. Numerical calculations showed that structural stability notably decreased when the steel shell thickness in the laminated model was below 3 mm. No significant improvement in strength was observed compared with the steel model at 4 mm thickness. Therefore, the final design of the laminated model included a 3 mm thickness for the steel shell portion and a 14 mm thickness for the external carbon fiber layup. The layup was applied using wet winding at orthogonal angles to the outer surface of the steel structure.

The ends of the model were designed flat to mitigate adverse effects from end effects during testing. The test assessed the midsection of the cylinder. Transition rings were installed at both ends of the shell to protect the central test section from external influences. Detailed model dimensions are provided in the following text.

2.1   Steel structure

Detailed structural parameters of the comparison model made of pure steel are shown in Table 1. The parameters are explained in Table 2.

The steel used for the model is Q235 steel, with a yield strength of 235 MPa. The ribs are made of rectangular steel, with a size of 8 mm×32 mm and a rib spacing of 50 mm. Material Density The total mass of the model is 471 kg. Detailed material properties can be found in Table 3.

The structural characteristics of the model are shown in Figure 1.

Figure  1  Schematic of steel cylinder structure

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2.2   Laminated structure

The laminated model involves laying a certain thickness of T700/M21 carbon fiber material on the external surface of the steel structure. The structural data for the laminated model are shown in Table 4, and the material data are shown in Table 5.

The schematic cross section of laminated structure and steel structure is shown in Figure 2. In the figure, t denotes the thickness of the structure.

Figure  2  Schematic cross section of laminated structure and steel structure

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The main body material of the model is Q235 steel with a yield strength of 235 MPa. Rectangular steel ribs with dimensions of 8 mm × 32 mm are used. In the midsection, a carbon fiber layup of 14 mm thickness is applied at orthogonal angles, with each layer being 0.2 mm thick and totaling 70 layers. The total mass of the laminated model is 479 kg. The mass error between the laminated model and the steel model is shown in Table 6.

Laminated models are constructed using the wet winding process. The wet winding process is a molding approach that uses a liquid resin system. In this process, continuous fiber bundle roving or fiber cloth tape is first soaked with glue. Then, it is directly wound onto the surface of the core mold or liner under the control of fiber tension by the tension controller. Finally, the material is cured. This processing method ensures the product has good air tightness. The fiber alignment parallelism is outstanding, and the resin glue also provides a level of protection against fiber abrasion.

2.3   Comparison

Compared with the steel model, the laminated model reduces the steel mass in the midsection of the cylinder by 40% by replacing it with carbon fiber material. Given that the structural intrinsic frequency is related to the stiffness K and mass M, the structural stiffness is determined by the elastic modulus E and moment of inertia I. As shown in Tables 1–5, the ratio of the elastic modulus of steel and carbon fiber is E_S/E_C = 3.82, and the ratio of the moment of inertia of the 2 mm steel ring and the 14 mm carbon fiber ring is I_S/I_C = 1/7.21. Therefore, E_SI_S = 0.53E_C I_C; that is, the laminated model stiffness should be higher than that of the steel model.

The structural mass of the two models is compared in Table 6.

Table 6 shows that the relative error in model mass is 1.7%, which is less than 2%. Considering that the laminated model has end fittings, the actual mass of the cylinder section should be slightly less than the data in Table 2. Given that the mass of the fittings is 2.4 kg, the actual mass error of the cylinder section is approximately 1.19%. Therefore, the two models can be assumed to have the same mass. Considering that the laminated model is stiffer than the steel model, the laminated model should have a higher intrinsic frequency than the steel model with equal mass.

The structural characteristics of the model are shown in Figures 3 and 4.

Figure  3  Schematic of laminated cylinder structure

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Figure  4  Physical models (Laminated model on the left, steel model on the right)

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In Figure 3, t_s denotes the thickness of the steel structure, while t_c denotes the thickness of the carbon fiber.

In this work, experimental and numerical analyses will be conducted to investigate the modal and damping characteristics.

2.4   Numerical modeling process

In this study, the numerical modeling and analysis is conducted using the ABAQUS numerical analysis software developed by Dassault Systèmes France. The effectiveness of this method for calculating the modal and underwater shock response of the model has been verified in the existing literature (Szturomski and Kiciński, 2017; Amirabadi and Ghazangian, 2018). The emphasis here is on the process of creating and analyzing the laminated model.

Referring to the modeling method in the references (Zhao, 2023; Zhang et al., 2021), the steel part of the laminated structure is modeled by "solid" elements with mesh type C3D8R, while the composite part is modeled by "shell" elements. These elements are created using the outer surface of the created solid part, with cell type "S4R", and peeled off from the solid part. The material properties of both parts are created in the "Property" module, which is shown in Figures 5 and 6.

Figure  5  "Create Composite Layup" in the property module

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Figure  6  Carbon fiber layup parameters of laminar numerical model

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Notably, the combination of the two is realized in the"Create Composite Layup" function in the "Property" module. In this function, "Region", "Material", "Thickness", and "Rotation Angle" can be set for each layer of the laminated structure. The module defaults to a "Tie" connection between two neighboring layers, which ignores the relative displacement of the layers under external forces. This negligence is a limitation of this method.

The modal analysis of the structure is conducted using the "frequency" function option in the "Linear Perturbation" option of the "Step" module. This module has a high level of application maturity, the description of which will be mentioned later.

The longitudinal wave velocity of steel is 5 500 m/s, and the analyzed frequency band of the model is predicted to be approximately 1 000 Hz at maximum. Thus, the wave-length should be not less than 5.5 m. This study will ensure that each wavelength contains at least 50 computational grids to ensure the accuracy of the calculation. Accordingly, the mesh size is set to 0.1 m.

3   Modal analysis

The experimental modal analysis aims to obtain modal shapes through experimental methods and determine the modal vectors of the response of the structural system. The process of experimental modal analysis generally involves capturing the time histories of excitation and response through experiments, applying digital signal processing techniques to obtain transfer functions or impulse response functions, and distinguishing modal parameters. For undamped and proportionally damped systems, the modal vectors representing the principal modes of the system are expressed using real numbers, which are referred to as real modal systems. By contrast, systems with structural damping or general viscous damping belong to complex modal systems. This study conducts a comparative analysis of the modal characteristics of the two models using experimental modal analysis methods.

3.1   Experimental principles

The motion differential equations for a multi-degree-of-freedom vibration system can be expressed as

In the equation, M represents the mass matrix, C repre sents the damping matrix, K represents the stiffness matrix, F(t) represents the load force vector, and x(t) represents the displacement vector.

In conducting Fourier transform on the physical model, we $\boldsymbol{x}(t)=\boldsymbol{X} \mathrm{e}^{\mathrm{j} \omega t}$ and $\boldsymbol{F}(t)=\boldsymbol{F} \mathrm{e}^{\mathrm{j} \omega t}$. When they are substi tuted into Equation (1), we obtain

The transfer function is defined as

We suppose φ_r is the r-th order modal shape (eigenvector). Then, the transfer function matrix can be expressed as

Here, m_r, k_r, and c_r represent the r-th order modal mass matrix, modal stiffness matrix, and modal damping matrix, respectively. Each element of H(ω) can be expressed as

Its physical meaning is the transfer function from excitation at point i to response at point j. For any row of the transfer matrix, we have

Evidently, each row of the transfer function matrix con tains all modal parameters, which correspond to the r-th order modal shape. Therefore, when vibration signals are collected at a fixed measurement point on the test structure and all measurement points are sequentially excited, one row of the transfer function matrix can be obtained. Accord ingly, all other modal analysis parameters can be derived. Similarly, when a specific point is excited and vibration responses at all points are measured, all modal parameters can be acquired.

3.2   Experimental conditions

The experiment employs transient excitation using a single-point hammer to conduct modal testing on the laminated model and the steel comparison model. Multiple vibration accelerometers are used to record vibration acceleration responses at typical structural assessment points. Modal shapes and natural frequency distributions of different structures are derived through modal testing and analysis using the LMS Testlab modal testing module. The principles and procedures of the experiment are illustrated in Figures 7 to 9.

Figure  7  Schematic of the modal testing experiment

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Figure  8  Measurement point in the experiment

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Figure  9  Excitation applied to the model during the experiment

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The structural boundary conditions should be free to avoid disturbance of structural vibration by external factors during the test. Therefore, the model should be kept in suspension by means of a lifting ring during the test. This way realizes the free boundary conditions of the model.

Computational modal analysis was conducted on the two models before the experiment to verify the rationality of the experimental scheme. The main methods for finite element modal analysis include the Block Lanczos method and the subspace method. The subspace method is appropriate for extracting fewer modes of large and medium-sized models and necessitates good element shapes when applied to solid and shell elements. For extracting a large number of modes from medium to large models, the Block method is more efficient due to its powerful functionality and is suitable for most modal problems. In models containing solid or shell elements, the Block method does not need to consider the initial truncation point and can handle rigid body modes effectively. However, it has higher memory space requirements. Given the regular shapes of the two models in the experiment, the subspace method was used for computational modal analysis.

In addition, considering that the load input frequency is not more than 20 000 Hz and the wave velocity of the stress wave in steel is 5 500 m/s, the minimum wavelength is not less than 0.28 m. In this work, the structural mesh is set to be 0.1 m, which indicates that the minimum number of structural units is included in one wavelength.

3.3   Equipment

In this experiment, the acceleration response of the model is collected using the acceleration sensor installed on the surface of the model. Moreover, a suitable test site must be provided to meet the requirements of lifting, lighting, loading, unloading, and other functions. A partial list of equipment is shown in Table 7.

The sampling frequency of the data acquisition instrument is set to 100 kHz, which can capture up to 10 000 Hz and can cover the first three orders of the intrinsic frequency of the model.

3.4   Experimental results

1) Verification of experimental scheme rationality.

The structural mode is determined by its inherent structural characteristics and is not influenced by the excitation force. To verify the rationality of the experimental scheme, excitations were applied to different points on the laminated structure, and the structural modal characteristics were measured accordingly.

The first three modal shapes and natural frequency distributions of the laminated structure under different excitation positions are shown in Table 8.

Table  8  Comparison of modal test results at different excitation positions

Incentive position 1	Incentive position 2	Intrinsic frequency offset (%)
		0.03
		0.03
		0.09
		0.07
		0.42

In the table, f represents the structural natural frequency; the subscripts 1, 2, and 3 indicate the order; "a" and "b" indicate two symmetric modes; "_Ⅰ"indicates incentive position 1; and "_Ⅱ" indicates incentive position 2. The modal test results show that the overall consistency of the modal vibration pattern is good before and after moving the excitation point. Differences exist in the amplitude of individual measurement points. The maximum shift of the intrinsic frequency is not more than 0.42%, which verifies the reasonableness of the test program.

2) Modal test results.

Based on the model tests, a FEM simulation solid calculation model with a scale of 1:1 was established. The size of the solid mesh satisfies the requirement of having at least 6 elements per wavelength. The structural mesh consists of 202 830 elements, and the actual structure and the simulation model are shown in Figure 10. Given that the magnitude and location of external excitation are independent of the natural frequency and modal shape, no additional excitation forces need to be applied in the FEM modal calculations.

Figure  10  Actual structure and FEM simulation model

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The test results of the laminated model using single-point excitation and multi-point pickup are compared with the modal calculation results of the finite element simulation model based on actual material properties and structural connection forms, as shown in Talbe 9. The modal shapes of the model are indicated by (m−n), where m is the axial half-wave number, and n is the circumferential half-wave number. In this way, a visual comparison of their modal shapes is provided.

Table  9  Modal comparison of the first three orders between simulation and experiment for the laminated model

Modal test results	Modal simulation results	Intrinsic frequency offset (%)
		0.62
		2.53
		3.39
		0.33
		5.24

The comparison of modal test results and simulation calculations shows that the maximum error of natural frequencies corresponding to the same modal shape obtained by the two modal analysis methods does not surpass 5.24%. This finding is similar to that obtained for common cylindrical structure modes. This similarity verifies the rationality and effectiveness of the experimental scheme.

The modal test results of the steel comparative model using single-point excitation and multi-point pickup are compared with the modal calculation results of the finite element simulation model, as shown in Table 10. The modal shapes of the structure are indicated by (m−n).

Table  10  Modal comparison of the first three orders between simulation and experiment for steel model

Steel model test results	Steel model simulation results	Intrinsic frequency offset (%)
		3.06
		3.23
		2.28
		2.88
		0.53
		3.46
		2.64

The comparison of modal test results and simulation calculations shows that the maximum error of natural frequencies corresponding to the same modal shape does not surpass 4.24%. The modal shapes also exhibit a superposition of axial and circumferential half-waves, which verifies the effectiveness of the modal analysis methods. The error between the experimental and simulated values is due to the modeling process and disturbances in the testing process.

The modal comparison between the laminated structure and the steel structure is shown in Table 11. The modal frequency shift of the laminated model reaches 40.81% for m = 0, n = 2 mode and 15.89% for m = 1, n = 2 mode. The formula for calculating natural frequency indicates that the overall stiffness of the steel model is slightly lower than that of the laminated model under equal mass conditions. The carbon fiber material laid on the outer surface causes the natural frequencies of the steel model to shift to higher frequencies. This conclusion is consistent with the theoretical derivation in Section 2.3.

In addition, the modal shape differences between the laminated model and the steel model are significant. For example, the laminated model exhibits a unique triangular mode in the circumferential direction, whereas the steel model exhibits two orthogonal elliptical modes in the same frequency range. Therefore, the modal shapes of cylindrical structures with the same dimensions but different material properties differ greatly.

Considering that the model used in this study is the submarine segment, its aspect ratio is relatively small. Accordingly, its modal characteristics are breathing modes with large intrinsic frequencies. For the complete submarine structure, its modal characteristics are reflected in the "whip-like" form. This form has a relatively low intrinsic frequency, which mainly reflects low-frequency characteristics. The load frequency that causes the overall response of the submarine is generally low. Only the intrinsic frequency that is close to the load frequency needs to be analyzed in the study of the total structural strength and vibration characteristics. When the intrinsic frequency is migrated to a higher place, the modal response provoked by the original excitation force will be reduced. As a result, the overall structure is more secure under the same external force conditions.

3) Transfer function test results.

1) Comparison of different measurement points.

For the laminated model, measurement points #1, #3, and #5 were taken along the longitudinal direction of the model. The transfer functions of the model in the range of 1–1200 Hz were measured, as shown in Figure 11.

Figure  11  Comparison of transfer functions at different measurement points for laminated models

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In the figure, H (w) denotes the transfer function. The comparison of the transfer function line spectrum in Figure 11 shows that significant peaks appear at 365 600, and 966 Hz at measurement point #1 near the open end, measurement point #3 at the transition ring, and measurement point #5 at the center of the structure. They are consistent with the natural frequencies obtained from the modal analysis, which verifies the effectiveness of this analysis method. Conversely, from the overall amplitude of the transfer function line spectrum, the overall vibration response at measurement point #5 at the center of the structure is larger than that at measurement point #1 near the open end, which is related to the structural form. The steel thickness at the open end is 34 mm, while the thickness at the center of the structure is 5 mm, which results in higher stiffness at the open end and greater vibration response at the center under unit excitation.

Table 11 Comparison of modal frequencies of cylinders with different structures

2) Comparison of different materials.

As shown in Equation (6), the amplitude of the transfer function H (w) is related to the intrinsic frequency of the structure. When the frequency increases, H (w) decreases. The comparison of the transfer function between the steel model and the laminated model is shown in Figure 12 and the quantitative comparison of the two is shown in Table 12.

Figure  12  Comparison of transfer functions of laminated and steel models

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The comparison of the transfer function line spectrum of equal mass models with different material components in Figure 12 shows that the amplitudes of the transfer function for the steel model are generally higher than those of the laminated model. Therefore, the laminated structure made of steel and carbon fiber materials has better vibration-damping effects than the steel structure with equal mass. Meanwhile, the peak coordinates of the two curves correspond well to the natural frequencies obtained from modal analysis, which verifies the effectiveness of the analysis method. Therefore, replacing 40% of the steel thickness of the submarine shell with carbon fiber improves the structural damping characteristics by 9.6% while maintaining the same structural quality.

4   Damping characteristic analysis

The acceleration response of typical parts of the designed and processed laminated structure and the steel comparative structure with equal mass was tested using the hammering method. The loss factor values of the two structures were obtained using the transient decay method and compared.

4.1   Experimental principle

The internal loss factor is obtained through the transient decay method. The basic idea is to apply the Hilbert transform to the acceleration time-domain curve obtained from the experiment to obtain the envelope function of the structural response signal, which clearly reflects the relationship between signal amplitude and time decay. The slope of the envelope line is then calculated, which yields the damping ratio ξ. The internal loss factor η is obtained through the multiple relationship between the damping ratio and the internal loss factor. The specific method proceeds as follows:

The acceleration response signal of the structure after impact is obtained through experiments, and the Hilbert transform is applied to the signal:

Its inverse transform is

where a(t) is the acceleration response of the structure after impact, and the symbol * denotes the convolution.

The analytic signal of the real function is obtained:

The plural expression for u(t) is

where $a(t)=\sqrt{a^2(t)+\hat{a}^2(t)}$. For the response signal a(t) envelope, the instantaneous phase for the $\theta(t)=\arctan(\hat{a}(t) / a(t))$. Thus, any real function can be re-expressed as a(t) = A(t)cos(θ). The response signal envelope to take the logarithm of the attenuation curve can be obtained. The absolute value of the slope of the curve for the damping ratio and the product of the center frequency of ωξ, which is divided by the center frequency of the damping ratio of ξ, is the average value of the loss factor (Inner Loss Factors, ILF) in the frequency band. Twice the damping ratio is the average value of Inner Loss Factors (ILF) in the fre quency band, that is, η = 2ξ.

4.2   Test procedure

After the output of the test acquisition data, the usability of the data is first checked, and invalid data are eliminated. Then, the data are filtered, and the filtered data are processed according to the transient decay method to obtain the loss factor data. The sampling frequency was set to 100 kHz during the test. The flow of tests and the calculations are shown in Figures 13 and 14.

Figure  13  Data processing flow

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Figure  14  Attenuation curve of response signal

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4.3   Working condition settings

The experiment set up 10 measurement points, with 6 points on the middle section of the cylinder, 2 points on the transition section, and 2 points on the end faces. This test mainly focuses on #1–#6 test points, and #7–#10 points are set up to verify the reasonableness of the test.

The experiment set up 9 excitation points, with 3 excitation points on the middle section, 4 on the transition section, and 2 on the ends. A force hammer was used to input excitation force during the experiment. The layout of measurement and excitation points is shown in Figure 15.

Figure  15  Test conditions and measurement points

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4.4   Experimental results

The loss factors of the steel and laminated cylindrical middle sections obtained at each measurement point using the transient decay method are shown in Figure 16.

Figure  16  Structural loss factor measurement point average

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In Figure 16, Place_iindicates the loss factor of the i-th test point. The average of each measurement point is shown by the dark red curve in the figure. As shown in Figure 16, the ILF values obtained at each measurement point exhibit a consistent trend. Similar amplitudes are observed in the high-frequency region, and slight fluctuations are found in the mid-low frequency region. Nevertheless, the fluctuations are insignificant.

Given that the structural loss factor is an inherent property, the amplitude is determined by material properties and geometric attributes and is independent of external loads. Therefore, the loss factors under each condition were averaged, and the results are shown in Figure 17. In the figure, P_i denotes the structural loss factor measured at the ith operating condition, and P_average denotes the average of the loss factors measured at each operating condition.

Figure  17  Average values at various operating conditions

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The final average ILF values obtained for the steel-carbon laminated structure and the equal mass steel comparison model are shown in Table 13.

In Table 13, ILF_S denotes the ILF values for steel structures, and ILF_L denotes the ILF values for laminated structures. The table shows that the loss factor values of the laminated cylindrical shell structure are approximately 1.77–4.56 times those of the steel structure.

The data in the table are shown through bar and line charts, as shown in Figure 18, to more intuitively compare the ILF distribution of the two structures in each frequency band.

Figure  18  Comparison of the mean square values of ILF

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The abovementioned information suggests that, on the one hand, the loss factors of steel and laminated structures exhibit the characteristic of "large at low frequencies, small at high frequencies". The overall amplitudes range from 10⁻⁴‒10⁻², and the ILF values at each measurement point insignificantly differ. On the other hand, the loss factor of the laminated structure increases by approximately 1.77–4.56 times that of the steel structure, and the maximum loss factor improves to 1.77 times.

The ratio was plotted, as shown in Figure 19, to explore the variation in the loss factor ratio between the two models with the frequency band.

Figure  19  Variation in the ratio of the loss factor of the laminated model to that of the steel model with frequency

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In Figure 19, the "ratio" value is the ratio of the loss factor of the laminated model to that of the steel model in the corresponding frequency band. The figure shows that the ratio of the loss factors of the two models peaks at 355, 562, and 1 000 Hz. This result indicates that the loss factor increases the fastest at the three frequencies after changing the structural material properties. Comparing the conclusions of the previous modal analysis suggests that the three frequencies correspond to the first three natural frequencies measured in the laminated model experiment. Therefore, the loss factor of the model increases the fastest at the shifted natural frequencies after the structural material properties are changed.

5   Conclusions

This study aims to improve the vibration and noise characteristics of submarines, balance the properties of the construction materials in terms of damping, strength, and stiffness, and control the manufacturing costs. Thus, in this study, a model of a laminated cylindrical shell made of steel and carbon fiber and a steel comparison model with equal mass and shape are designed based on the tectonics of the submersible segment. The modal and damping characteristics of the two models are compared and analyzed through experimentations and simulations. The following conclusions are obtained:

1) The relative error of the modal between experiment and computation does not surpass 6%, which verifies the rationality of the modal analysis method.

2) The modal frequency shift of the laminated model reaches 40.81% in the m = 0, n = 2 mode and 15.89% in the m = 1, n = 2 mode, which provides a large modal avoidance space for the submarine design. This condition avoids intense resonance and effectively suppresses sound radiation caused by resonance.

3) The amplitudes of the transfer function for the steel model are generally higher than those of the laminated model. Up to a 9.6% reduction in the amplitude of the transfer function is observed, which means that the response of the laminated model provoked by the same excitation force will be lower than that of the steel model. Therefore, the vibration characteristics of the submarine have been improved.

4) Compared with that for the steel structure, the loss factor for the laminated structure of the same mass increases by approximately 1.77– 4.56 times. The peak growth rate of the loss factor appears near the natural frequencies, which indicates that the laminated structures have better acoustic stealth capabilities.