A Review of Vibration Control Studies of Double-Layered Cylindrical Shells Under Transient Excitation in Water

Abstract

In recent years, with the wide application of underwater vehicles, the vibration and noise problems generated during their operation have attracted great attention from the academic community. Compared with the field of traditional mechanical noise, research on vibration control of the noise that is transiently excited underwater still has significant deficiencies in terms of its theoretical depth and systematicity. In this paper, we take transient noise control for underwater vehicles as the engineering entry point; systematically explain the vibration mechanisms and dynamic characteristics of underwater double-layered cylindrical shell structures; and discuss the vibration transmission paths and the development trends in the control technology in depth. This study mainly includes the following contents. Firstly, the vibration response mechanisms of underwater double-layered cylindrical shells are sorted through a bibliometric analysis, and the evolution laws for plate–shell structures and the vibration transmission paths for single–double-shell structures are summarized systematically; secondly, the multi-path vibration transmission characteristics of double-layered cylindrical shells are analyzed based on energy transfer theory, and the contribution to transient noise through different transmission paths is quantitatively evaluated; and thirdly, the vibration transmission characteristics of active control, passive control, and hybrid control are evaluated systematically in terms of the dimensions of the control mechanism. Then, the engineering applicability of active, passive, and hybrid control technologies is systematically reviewed. Finally, combined with the development of new intelligent materials and adaptive algorithms, the prospective outlook for vibration control technology for shell structures under transient excitation conditions is presented.

1. Introduction

As the core equipment in modern combat systems, underwater vehicles play a key role in underwater reconnaissance and tactical combat by virtue of their excellent stealth characteristics and flexible mobility. With iterative upgrading of related technologies, acoustic stealth performance has become an important indicator of the advanced nature of underwater equipment, and countries have increased their research investment in this field. For submarines, reductions in their launch noise are often a key part of transient noise control, which has been the focus of researchers attention. Reducing submarines’ launch noise is an important step in achieving the goal of “quiet” submarines [1].

Existing domestic and global underwater launchers experience challenges such as excessive launch noise, insufficient maneuvering accuracy, and difficulty in adjusting their launch energy, especially in terms of noise control, which has become a core factor limiting the stealth performance of underwater vehicles. The United States study “Naval Technology 2000–2035—Submarine Platform Technology” shows that “reducing weapons’ transient launch noise can greatly improve the survivability of submarines in combat”.

In order to improve the acoustic stealth performance of underwater vehicles, our research group estimated the distance between underwater launch noise and passive sonar by carrying out engineering tests in the early stage, and the results are shown in Figure 1. It can be seen that compared with conventional launchers, electromagnetic launchers’ noise has been substantially reduced. However, the noise of the new launcher is still greater than the navigation noise requirements for low-noise vehicles, and there is still significant room for further reductions in vibration and noise.

The electromagnetic transmitter is designed as a cylindrical body, which is externally mechanically coupled to a peripheral non-pressure-resistant shell by means of an elastic vibration isolator (made of rubber). Based on the geometric characteristics and the connection relationship, this combined system can be abstracted as a double-layered cylindrical shell model with co-axial characteristics (as shown in Figure 2). At present, vibration and noise reduction technologies for underwater cylindrical shells have shown diversified development trends, such as active control technology, passive control technology, active–passive hybrid control, and smart material technology, which involves a number of fields, such as fluid dynamics, structural dynamics, material science, and control theory. Despite a series of research results, there are still some unsolved technical problems and theoretical deficiencies. For example, underwater vibrations, transmission mechanisms, coupled vibrations, noise control efficiency, and damping mechanisms are still unclear, as is adapting to complex conditions in the actual marine environment.

Most of the existing research has focused on the application of a single technology or an analysis of specific working conditions and has lacked in-depth research into the interactions between different vibration and noise reduction methods and an evaluation of their comprehensive benefits. In addition, the design of new vibration and noise reduction programs applicable to the complex environment and the working conditions of ships is still in the exploratory stage, and systematic integration and innovative design are urgently needed to adapt to the complexity of engineering applications.

In view of this, this review synthesizes existing research to identify unresolved issues and required improvements in the field. Concurrently, by analyzing and comparing current technologies, it establishes a theoretical foundation and identifies innovations for subsequent research, thereby advancing vibration control studies for underwater cylindrical shell structures. This study encompasses the following aspects: Firstly, through bibliometric analysis, the vibration response mechanisms of underwater double-walled cylindrical shells are synthesized, systematically summarizing evolutionary patterns of vibration transmission paths in both single and double shell–plate structures. Secondly, based on energy transmission theory, the multi-path vibration transmission characteristics of double-walled cylindrical shells are comprehensively analyzed, with quantitative assessments of distinct transmission paths’ contributions to transient noise radiation. Thirdly, from control mechanism perspectives, a systematic review evaluates the engineering applicability of active, passive, and hybrid active–passive control techniques. Finally, integrating advancements in novel smart materials and adaptive algorithms, forward-looking prospects for vibration control technologies under transient excitation conditions are presented.

Summarizing current global research, studies on the vibration characteristics of underwater cylindrical shells progress through three stages: basic theoretical frameworks and simplified models, emergence of numerical methods, and development of complex modeling systems. Theoretical models evolve toward increased precision, intelligent modeling, and multi-field coupling. This evolution is illustrated in Figure 3, with theoretical method comparisons detailed in

Table 1. Table summarizing vibration theory of double-walled cylindrical shells.

Theory Name	Mathematical Basis	Applicable Scenarios	Advantages	Limitations
Donnell–Mushtari Theory	Thin-shell approximation (neglects shear deformation)	Low-frequency vibration of thin-walled shells	Simple equations Accurate ring frequency prediction Mature analytical solutions	Neglects rotational inertia Unsuitable for stiffened structures
Flügge’s Precision Theory	Simplified 3D elasticity theory	Vibration of medium-thick shells	Includes shear/rotational inertia Higher accuracy than Donnell	Complex equations Difficult numerical solution
Point-Coupled Model	Discrete spring-mass system	Rib-connected double shells	Reveals modal coupling mechanisms Intuitive parameter influence analysis	Neglects shell continuity Cannot compute wave propagation
Equivalent Single-Layer Theory	Equivalent stiffness method	Overall stiffness assessment in preliminary design	Extremely fast computation Efficient conceptual design tool	Cannot capture inner/outer shell phase difference Errors up to 30%
Wave Propagation Theory	Traveling wave solution + boundary matching	Vibration propagation in infinitely long shells	Accurate vibration transmission loss prediction Reveals cut-off frequency phenomena	Difficult to handle finite-length boundary effects
Periodic Structure Theory	Floquet–Bloch theorem	Double shells with equally spaced ribs	Predicts bandgap characteristics Guides vibration suppression design	Fails for non-periodic structures Difficult to quantify damping effects
Fully Coupled FSI Theory	Potential flow theory + shell dynamics	Vibro-acoustic problems of underwater shells	Accurate added mass/radiation damping calculation Predicts acoustic radiation efficiency	High computational cost Difficult to simulate free surface effects

. This review addresses two key aspects: vibration characteristics research and vibration control technologies for cylindrical shells integrating active control, passive control, and active–passive hybrid control methodologies.

2. Vibration Characterization of Cylindrical Shells

As a common engineering construction form, cylindrical shell structures are widely used in underwater equipment and other applications. Studies on coupled vibration characteristics of cylindrical shells under steady-state conditions have been a major focus for scholars globally. Historically, research has predominantly focused on single-layer cylindrical shells; in recent years, however, scholarly attention has shifted to coupled vibration characteristics of double-walled submerged cylindrical shells.

2.1. Coupled Vibration Characteristics of Underwater Single-Layer Cylindrical Shells

In research on vibration characteristics of underwater single-layer cylindrical shells, scholars globally have conducted extensive studies on coupled vibrations of fluid-filled cylindrical shells, whose structural schematic is shown in Figure 4a. Zhang [2] employed the fluctuation method to analyze vibration characteristics under fluid-medium influence. Moeini [3] introduced the Ritz method to solve vibration problems in functionally gradient ring–rib cylindrical shells, achieving coupling analysis between ring–rib geometric parameters and gradient materials through parametric modeling, establishing a numerical model characterizing non-uniform rib size effects on structural dynamics. Based on Love’s theory, Dengfeng Yang [4] investigated free vibration characteristics in air/liquid media under simply supported boundary conditions. While most of the literature [5,6,7,8,9,10] considers only single-factor effects on vibration characteristics in aqueous media, Bin Liang [11] comprehensively examined liquid medium properties, cylindrical shell ring–rib parameters, shell geometry, boundary conditions, and other factors affecting coupled vibrations. Zhizhong Liu [12] established an acoustic–vibration coupling model for underwater cylindrical shells, demonstrating hydrostatic pressure’s positive correlation with coupling frequency.

Another underwater cylindrical shell structure (Figure 4b) contains an internal air medium. Chen Chen [13] applied non-destructive prediction methods to evaluate critical elastic instability loads in annular-ribbed cylindrical shells under hydrostatic pressure. Using transfer matrix methods, Lei Cao [14] investigated ring-parameter variation effects on vibration characteristics. Liu [15] analyzed hydrostatic pressure effects on input power flow in ring–ribbed cylindrical shells by incorporating pressure as additional stress in Flügge and Helmholtz fluctuation equations. Li and Zhang [16,17] constructed frequency-domain wave equation analytical models, revealing modal distribution patterns during free vibration. Xuebin Li [18] combined strain factors with energy methods to study ring–rib/cylindrical shell interactions and vibration characteristics. Bin Liang [19] independently investigated hydrostatic pressure, ring–rib parameters, and boundary condition effects on coupled vibration characteristics in aqueous media.

Regarding the influence of free liquid surfaces on the coupled vibrations of cylindrical shells, Peng Wang [20,21,22] established an acoustic–vibration coupling model for finite-length cylindrical shells based on the virtual source and mirror methods, accounting for free-liquid surface effects and hydrostatic pressure. Lingxiao Weng [23] constructed an infinite-length cylindrical shell acoustic–vibration coupling model using mirror methods and Graf’s addition theorem, deriving governing equations to examine complex acoustic boundary effects on shell vibrations and acoustic radiation. For cylindrical shells under arbitrary boundary conditions, Sun, Song, and Jin [24,25,26,27] analyzed vibration characteristics through Fourier series and energy methods, investigating the influence of boundary stiffness. Jinxiao Chen [28] employed Flugge theory combined with fluctuation methods and modified Fourier series to study natural frequency variations under elastic boundary constraints. Chen [29] examined cylindrical–cone coupled shell dynamics, quantitatively revealing boundary stiffness and ring–rib parameter modulation mechanisms’ influence on structural natural frequencies through three-dimensional vibration coupling analysis. Rui Zhang [30] established fluid–solid coupled vibration differential governing equations by integrating Donnell’s thin-shell theory with Helmholtz’s acoustic equations. When frequencies exceed critical thresholds, traditional thin-shell theory exhibits significant errors due to neglected shear deformation effects. Accordingly, an improved vibration prediction model based on equivalent plate theory was proposed, extending computational accuracy into high-frequency bands.

2.2. Coupled Vibration Characterization of Underwater Bilayer Plates and Shells

As an extended model of single-shell structures, double-layered cylindrical shells frequently serve as equivalent configurations for pressure-resistant hulls in mechanical analyses of submarines and other underwater equipment. This modeling approach is frequently adopted in Chinese academic research.

Takahashi [31] explored sound transfer characteristics in infinitely extended double-layer flat plates under point-connected conditions without accounting for fluid loading effects. Kyeong [32] further investigated the free vibration characteristics of submerged double-layered flat plates using Fourier series expansions for fluid velocity potentials and plate displacement functions, ensuring energy conservation. Lin [33] examined acoustic transfer characteristics in ribbed plate-connected double-layered infinite flat plates through Fourier transform methods, comparing contributions of ribbed plates and cavities to dual sound transfer pathways.

Cheng [34] developed a critical parameter determination method for vibration energy transfer in elastically connected double-layered plate systems with interlayer fluid. This approach reveals the underlying switching mechanism of energy transfer paths by quantifying the critical ratio of sandwich fluid equivalent stiffness to connection stiffness: when exceeding critical thresholds, the dominant energy transfer shifts between fluid coupling and structural coupling. Combining Euler–Bernoulli beam theory with modal analysis, Li Zeng [35] established a double-beam model containing entrained fluid and examined the entrained water’s effects on vibration characteristics. The specific model is shown in Figure 5. Extending this, Shitian Wen [36] developed a finite-large double-plate vibration transfer model incorporating elastic connections and interstitial fluid, investigating the effects of water layer thickness on vibration transfer, though computational accuracy requires improvement. Conversely, Zou Mingsong [37] constructed an acoustic–vibration model for water-filled double-layer spherical shells between outboards under harmonic force excitation, examining the inter-outboard water’s influence on underwater radiated noise. Xianzhong Wang [38] extended investigations into double-layered shell acoustic vibration characteristics, proposing a refined transfer matrix method under arbitrary boundary conditions for acoustic radiation calculations in underwater double-layered cylindrical shells, thereby enabling computation of radiated acoustic pressures.

Research on the coupled dynamics of double-layered cylindrical shells has yielded systematic results internationally. Yoshikawa [39] derived an infinite-length double-shell coupled dynamics model under point-source excitation using Fourier transforms, solving the inner shell displacement field via wavenumber-domain analytical methods. Wu and Zeng [40,41] pioneered an acoustic–structural coupling model for infinite-length ribbed double shells; their parametric analysis demonstrated that ribbed plates and inter-outboard fluid layers constitute dual-channel acoustic energy transfer mechanisms. For finite-length shells, Chen [42] established a coupling model based on Flügge shell theory, revealing that reduced double-shell spacing significantly enhances vibration coupling effects, with vibration energy levels exhibiting negative exponential correlation with spacing parameters. Integrating Flügge’s shell equations with Helmholtz’s acoustic radiation theory, Yao [43] quantified fluid–solid coupling frequency variations; water media dominate low-frequency energy transfer, while structural members modulate vibrations in high-frequency bands. Bai [44] systematically elucidated three vibration transmission pathways (structural conduction, fluid conduction, and hybrid conduction) between double-shell gangways, revealing solid ribbed plates’ advantages over fluid layers through transmitted power spectral density comparisons. Wang [45] investigated a two-dimensional acoustic–solid coupling analytical model, decoupling the differential influences of side-plate stiffness and water medium density on vibration transfer coefficients. These theoretical foundations support vibration prediction and control in complex shell systems.

While earlier studies focused on infinite domains, subsequent research extended to finite basins. Based on two-dimensional Flügge and Love theories, Liao [46] analyzed vibration characteristics in semi-infinite and right-angle domains. For submerged cylindrical shells under complex conditions, Yu [47] constructed a multi-physics coupling model incorporating fluid–structure interactions and hydrostatic pressure using Flügge theory and elastic wave propagation theory. Through characteristic frequency spectrum analysis and transient response decoupling, this model reveals energy conversion mechanisms between free vibration mode distributions and forced vibration frequency-domain responses. Shao [48] examined the free vibrations of orthotropic anisotropic cylindrical shells under hydrodynamic loading. Han [49] proposed governing motion equations using Flügge shell theory, developing a semi-analytical method for vibration responses in elastic cylindrical shells with internal liquid sloshing. Yang [50] studied cylindrical shell-acoustic cavity coupling effects, identifying peak vibroacoustic radiation factors and demonstrating radiation reduction through structural stiffness adjustments, acoustic cavity redistribution, and damping layers. Liu [51] analyzed the effects of flow field and internal excitation on cylindrical shell vibration noise, revealing flow-excited noise dominance at 20 Hz–100 Hz and radiated noise dominance at 160 Hz–200 Hz.

3. Research on Vibration Control Technology of Cylindrical Shells

Traditional vibration reduction and isolation methods primarily include vibration dissipation, vibration isolation, and vibration absorption, as shown in Figure 6. Vibration dissipation is achieved by controlling vibration sources, typically through low-noise equipment, while vibration isolation and vibration absorption focus on controlling vibration transmission paths—approaches widely applied in ship machinery systems. Vibration isolation aims to impede vibration transmission between sources and controlled objects, commonly implemented in shipboard floating raft isolation systems; vibration absorption employs elastic subsystems attached to target objects to offset source vibrations. In modern engineering, passive control technologies (encompassing discrete components like dynamic absorbers and dampers, and continuous media such as viscoelastic composite damping layers) have become standardized solutions. However, they still face dual challenges of mass-space constraints and modal coupling effects in low-frequency broadband spectral domains, leading to limited vibration energy dissipation efficiency [52].

3.1. Overview of Passive Control of Cylindrical Shell Vibration Research

Passive vibration control employs mechanical impedance matching principles to achieve energy isolation. The core technology configures elastic coupling units between vibration sources and hull transmission paths, utilizing inherent parameters of isolation systems (mass distribution, stiffness matrices, damping coefficients) to establish mechanical impedance that effectively suppresses vibration energy transfer from mechanical equipment and piping to hull structures. This approach’s advantage lies in its independent energy dissipation characteristics; broadband vibration attenuation is achievable without external energy inputs or active control circuits.

However, electromagnetic launch systems generate significant low-frequency vibration noise where traditional mechanical passive control faces limitations. Linear vibration isolators exhibit suboptimal low-frequency performance due to effective isolation frequency constraints and may even amplify vibration transmission within system resonance bands. To address complex vibration modes and multi-source low-frequency excitations induced by transient noise in electromagnetic launch systems, researchers have proposed novel materials and structures to expand vibration control bandwidths and enhance isolation performance [53].

3.1.1. Constrained Damping Layer Passive Vibration Isolation

In practical engineering applications, cylindrical shell vibration control primarily employs constrained damping treatments. Constrained damped structures consist of a base matrix, damping layer, and constrained layer. Research on damping methods originated in the 1950s, when Kerwin [54] utilized complex stiffness methods to study beam structures with constrained damping layers. Subsequently, Mead et al. [55] further investigated beam vibrations with constrained damping layers using sixth-order partial differential equations. Thereafter, constrained damped structures attracted significant research attention, yielding numerous formulations for damped beams and plates [56,57,58]. Pan [59] pioneered constrained damped cylindrical shell studies. The published literature confirms that polymeric viscoelastic damping materials consume substantial vibrational energy. The modal loss factor serves as a key metric for structural damping performance. Building on this foundation, scholars globally have conducted extensive research and published relevant studies [60,61,62,63].

Current research methodologies for constrained damped structures include analytical, semi-analytical, finite element, and experimental approaches. Chen [64,65] explored damped shell vibration responses through hypothetical modeling. Cao [66] applied Sander’s theory and improved wave propagation methods to investigate natural frequencies and loss factors in constrained damped cylindrical shells. Xiang [67] adopted a transfer matrix-based semi-analytical method to analyze free vibration characteristics. Zheng [68] extended transfer matrix research to examine multilayer damping effects. Mokhtari [69] employed Lagrange’s equations and Rayleigh–Ritz methods to study vibration characteristics while exploring control parameter effects on frequencies and loss factors. Permoona [70] investigated constrained damped conical shells using Rayleigh–Ritz methods, describing viscoelastic layer properties via Zener fractional models. Chen [71] analyzed the vibration characteristics of double-layer cylindrical shells with surface-applied damping materials (with the computational model shown in Figure 7), demonstrating that increased damping thickness elevates loss factors and enhances vibration/noise reduction. Zhang [72] experimentally validated vibration damping characteristics in viscoelastic composite-damped double-layer cylindrical shells.

Recently, finite element methods have become efficient tools for solving complex engineering structures. Within this framework, Wang [73] developed governing equations for constrained damped cylindrical shells. Ramesh [74] proposed a novel three-node cell (six DOFs per node) to establish governing equations. Mohammadi [75] introduced a semi-analytical finite element method based on linear/nonlinear displacement models for solving vibration characteristics. Biswall [76] developed finite element methods for multilayered spherical shells with damping material layups, analyzing geometrical parameters’ effects on damping characteristics.

Since viscoelastic damping materials are polymeric, their dynamic properties exhibit temperature- and frequency-dependent behavior. Accounting for these factors in numerical simulations is often challenging, so viscoelastic material models are typically validated experimentally. Masti [77] combined finite element and experimental methods to investigate the vibration characteristics of free-boundary cylindrical shells under partial damping treatments. Kumar [78] performed experimental studies on optimal constrained damping layer designs for curved plates. Etienne [79] conducted modal tests on CFM56 aero-engine compressor drums treated with SMACTANE50 damping material. Song [80,81,82] applied constrained damping layers to single-layer cylindrical shells (fixed at one end, free at the other), testing modal shapes and natural frequencies to validate theoretical models for arbitrarily constrained damped cylindrical shells. Han [83] measured the vibroacoustic characteristics of pallet-connected damped double-layered cylindrical shells, testing vibration acceleration and sound pressure while comparing the control effects of different damping layouts. The results indicate that pallets significantly influence inner-shell vibration transmission, with optimal control achieved when damping materials cover the inner shell.

In engineering practice, non-ideal boundaries necessitate elastic boundary simulations. Qu [84] introduced Chebyshev orthogonal polynomials to enhance regional decomposition methods for analyzing constrained damped cylindrical shell vibrations. Jin [27,85,86] examined composite cylindrical shell vibrations under elastic boundaries using standard Fourier series functions. Wu [87] modeled aero-engine magazines as cylindrical shells, establishing theoretical models for arbitrarily constrained damped cylindrical shells in air media by incorporating artificial springs and energy methods, then investigating dynamic characteristics under elastic supports.

3.1.2. New Vibration Isolation Materials

New passive vibration isolation materials—including acoustic metamaterials (e.g., phononic crystals), localized resonance-type vibration isolators, and quasi-zero stiffness isolators—effectively reduce volume and mass in electromagnetic launch isolation systems while enhancing vibration damping and noise reduction. Recent research has extensively explored wave modulation mechanisms in mechanical metamaterials and low-frequency broadband vibration damping design, encompassing bandgap broadening and metamaterial unit configuration. Yin [52] demonstrated that integrating mechanical metamaterials in structural designs enhances structure–low-frequency wave coupling, improving wave energy control and dissipation. Addressing marine water pump vibration control, Li [88] developed an iterative floating raft design method using topology-optimized GFRP ply (z glass fiber-reinforced resin matrix composite) platens. The novel composite floating raft achieves 3 dB improvement in vibration transmission loss (10–500 Hz) with 18% structural lightweighting. Jin [89] reconfigured bow sonar array base damping via polyurethane composite layering, reducing structural acoustic radiation intensity by 8 dB experimentally.

Teng et al. [90] proposed solid–liquid hybrid materials for isolators, demonstrating performance correlation with solid unit count to enhance design accuracy, as shown in Figure 8. Exact integration methods revealed that hybrid isolators with 120 solid units achieved 15 dB vibration reduction at 20 Hz while maintaining serviceability. Liu et al. [91] integrated negative stiffness beams in parallel to isolators, preserving load capacity and volume while boosting low-frequency isolation. Hao [92] designed a high-load-capacity quasi-zero-stiffness oscillator with dynamic stiffness, deriving harmonic-force transmittance rate equations. Li et al. [93] established a high-dimensional mathematical model for quasi-zero-stiffness floating raft isolation systems, exploring rafts’ physical parameters’ effects on ship machinery vibration/noise reduction.

The design of acoustic metamaterials has evolved from single bandgap regulation to multi-physics field coupling. Chenhui Ren [94] designed double-negative metamaterial cylindrical shells with both negative stiffness effects and negative Poisson’s ratio characteristics through cytosolic topology optimization technology. In the field of marine acoustic metamaterials’ application, Lifu Xia [95,96] reconstructed vibration energy transfer paths by constructing double-shell coupling systems with negative-Poisson-ratio metamaterial rib plates. When rib plate width decreases, the number of primitive layers increases, and radiated acoustic power decreases. Jinyu Luo [97,98] investigated elastic wave transfer mechanisms in cylindrical shell mechanical metamaterials, proposed periodic arrangements of multilevel resonators to broaden bandgap ranges, and designed cylindrical vibration isolators with effective low-frequency vibration damping (below 100 Hz). Hongsai Huang [99] proposed locally resonant phononic crystal structures with spring oscillators arranged circumferentially within cylindrical shells, forming low-frequency bandgaps to suppress vibrations. Xinmei Yang [50] analyzed sound absorption characteristics using JCA equivalent fluid theory and multilayer medium transfer matrix methods; the results showed that sound absorption coefficients relate to porous material properties and excitation frequencies, suggesting static flow resistance and material layouts should be rationally designed to improve absorption performance. Siwei Chen [100] constructed bandgap optimization algorithms for acoustic metamaterials and predicts intrinsic models through finite element simulations and machine learning, providing new ideas for metamaterial structural optimization. These innovations significantly enhanced passive control energy dissipation in the 20-–500 Hz frequency band, establishing new paradigms for broadband stealth design in submarines and similar equipment.

Wan et al. [101] proposed a new locally resonant shell structure with gradient design strategies to improve broadband vibration isolation performance while considering frequency spacing, damping, and array period effects on vibration characteristics. An et al. [102] combined Bloch’s theorem with coordinate transformation methods to design cylindrical shell metamaterial structures (LCSMs) with broadband vibration isolation and high load-bearing capacity, maintaining isolation performance while improving load capacity. Zuo et al. [103] utilized time-domain finite element methods coupled with boundary element methods to predict the vibration and acoustic radiation characteristics of navigators under transient shock loading, designing star-shaped metamaterials for broadband transient vibration and noise control. The structure achieved maximum sound pressure attenuation of 75 dB within 1500–4500 Hz, reducing overall sound pressure levels by 15.78 dB (air) and 19.21 dB (water).

3.1.3. New Vibration Isolation Structures

In the design of ship vibration isolation systems, the development of new vibration isolation materials must be closely integrated with the actual working conditions of electromagnetic transmitters and meet the dual requirements of pressure resistance and vibration isolation at the same time. Hualiang Zhang et al. [104] revealed the quantitative relationship between the vibration isolation performance of floating rafts and the mass, stiffness, and damping parameters of the base through a systematic study; they also established design guidelines of floating rafts based on multi-parameter optimization. The core of passive control technology is to seek the best balance of mass, stiffness, and damping to maximize the vibration isolation effect under the premise of ensuring structural load-carrying capacity and installation convenience.

Jin et al. [105] designed lightweight cylindrical honeycomb sandwich structures for low-frequency isolation based on Love’s shell theory, fabricating cylindrical-shell honeycomb sandwich metamaterials. Ke et al. [106] periodically attached spring–mass resonators to cylindrical shells, proposing gradient-configured multi-resonant cylindrical shells that extend bandwidth by 170.7% and reduce total mass by 41.5%, achieving low-frequency broadband vibration control. Tang et al. [107] investigated sandwich thin-plate acoustic metamaterials (SPAMs), using machine learning to optimize sound transmission loss (STL) at 100–315 Hz; results showed average STL increases of 11.94 dB in this range.

As ships’ primary passive isolation form, floating raft structures undergo continuous optimization. Machens [108] demonstrated that truss-type floating rafts’ assembly joints effectively inhibit mechanical vibration transmission to hulls. Zhang et al. [109] innovatively integrated dynamic vibration absorbers within raft truss structures to enhance isolation. Xu [110] established a cabin raft system frequency response matrix model considering attached equipment coupling effects, proposing a chiral raft design method with directional vibration energy guidance based on periodic structures’ bandgap regulation.

Recently, acoustic black hole (ABH) technology has emerged in ship vibration control. Mironov [111] first proposed ABH principles, envisioning an ideal cone where bending waves never reach the wedge tip, thus trapping vibration energy. Applying elastic damping materials at ABH tips effectively suppresses machinery-to-hull vibration noise [112]. Deng et al. [113,114] calculated cylindrical hull modal parameters using the Gaussian expansion method (GEM), establishing Gaussian expansion component modal synthesis (GECMS) to describe system modal states; studies confirm ABH’s significant damping effect suppresses cylindrical hull vibration and acoustic power levels. Wang [115] established a finite element model of annular ribbed cylindrical shells with helical ABHs, calculating natural frequencies and vibration responses; results showed that mean square velocity levels decreased significantly—first-order resonance peaks reduced by >6 dB, other resonance peaks by >13 dB, and shell velocity levels by >8 dB. Using finite element methods, Wei [116] investigated ABH-embedded cylindrical shells, finding that vibration attenuation increases with inner diameter and that ABH oscillator mass/stiffness parameters significantly affect vibration characteristics. For shell vibration control, Liu [117] introduced ABH effects into cylindrical shell–circular plate coupled systems, constructing ABH cylindrical shell energy equations based on Love’s shell theory and analyzing excitation position effects on coupled system vibration responses via Rayleigh–Ritz variational principle discretization (Figure 9).

Quasi-zero stiffness vibration isolation technology achieves breakthroughs in low-frequency vibration control through nonlinear stiffness regulation. Its quasi-static stiffness—defined as the ratio of force to deformation—characterizes structural load-bearing capacity, while dynamic stiffness primarily accounts for small vibrations near equilibrium positions, determining system frequency response characteristics. Unlike traditional linear isolators with equal dynamic and static stiffness, quasi-zero stiffness systems reduce dynamic stiffness at equilibrium positions while maintaining equivalent static load capacity, enabling near-zero natural frequencies.Schenk et al. [118] explored stable zero-stiffness modes via pre-stressing tension topology optimization. Carrella et al. [119] established a triple-spring negative-stiffness model, deriving expressions for relationships between geometric–physical parameters and stiffness. For engineering applications, Wang et al. [120] developed a segmented nonlinear model showing isolation performance approximating traditional quasi-zero stiffness systems under small excitation amplitudes and linear isolation systems under large amplitudes. Changqi Cai [121] proposed a flexible constant-force structure–local resonance composite structure; by adjusting vibrator mass, stiffness, and shell parameters to study bandgap effects, it formed low-frequency local resonance bandgaps, enabling vibration isolation.

3.2. Overview of Active Vibration Control Research on Cylindrical Shells

Active control technology, as a key vibration control method for electromagnetic transmitters, plays a vital role in reducing low-frequency noise. Its core principle involves real-time reception of the controlled system’s interference and response signals, applying counter-phase vibrations through actuators to achieve mutual vibrational offset. In recent years, scholars globally have focused on optimizing active control actuator designs and conducting in-depth studies of control strategies.

3.2.1. Vibration Isolation Actuator

The core unit of ship active vibration control systems consists of sensors and actuators. Displacement and acceleration sensors detect and transmit vibration characteristic signals in real time, while intelligent actuators generate inverse driving force fields through reverse-phase control algorithms. Current actuator technologies encompass pneumatic servo mechanisms, hydraulic actuators, electromagnetic exciters, and smart material actuators, forming advanced clusters including piezoelectric ceramic stacked actuators (PZTs), Terfenol-D magnetostrictive actuators, and magnetorheological dampers (MRDs).

In ship active vibration control, researchers globally have conducted in-depth optimization studies on electromagnetic actuator structures. Gardonio et al. [122] proposed a semi-active vibration absorber tuning strategy that effectively reduces ship structural resonance responses in low-frequency bands. Wang et al. [123] constructed an actuator position optimization system enabling automatic mounting position optimization. Hu et al. [124] overcame traditional single-parameter optimization limitations by proposing a multivariate synergistic optimization model for tilt angles and actuator sizes, significantly improving damping effects. Recent trends indicate actuator research is evolving toward deep multidisciplinary integration, requiring systematic innovation through intelligent control algorithms, novel material properties, and structural dynamics theories.

Regarding smart actuation applications, Zhang et al. [125] studied functionally graded graphene–piezoelectric composite plates, reducing vibration acceleration levels via positive feedback control. He et al. [126] designed a TBS hybrid vibration isolation device integrating PZT actuator arrays with rubber isolation layers, modeling it using substructure dynamics approaches and experimentally verifying further harmonic reduction beyond broadband isolation. Sun et al. [127] established a piezoelectric shell LQR control model based on Hamiltonian principles and Lagrangian equations, optimizing actuator layouts to enhance modal control efficiency. Niasar et al. [128] derived system ordinary differential equations using actuation efficiency indices from Love’s shell theory, demonstrating that power–form piezoelectric characteristic adjustments improve vibration isolation effects by modifying actuator positions and performance metrics.

In actuator position optimization research, Geng et al. [129] developed a genetic algorithm platform for cylindrical shells achieving optimal actuator group topological configurations with improved computational efficiency compared with traditional methods.

3.2.2. Active Control Strategy

The choice of active control strategy decisively impacts actuators’ ability to effectively cancel input vibrations. Recently, researchers have focused on overcoming the limitations of traditional methods through in-depth improvements to mechanical system active controls. Soni et al. [130] proposed ternary and quaternary control strategies, comparing them with PD/PID controls; the results demonstrated significant vibration transmission suppression. Marinangeli et al. [131] developed a fractional-order feedback compensator to inhibit closed-loop effect overflow caused by high-order transfer function instability. Yuan et al. [132] introduced acceleration feedback mechanisms to positive position control, with multimodal experiments showing enhanced mid-high frequency vibration suppression. Yang [133] combined neural networks with active control, optimizing low-frequency parameters via time series prediction to improve system adaptability against time-varying disturbances.

For nonlinear control challenges, Kamaruzaman et al. [134] proposed multipoint linearized approximation methods achieving local optimal control while preserving system nonlinearity. Wang et al. [135] developed adaptive algorithms by dynamically adjusting control gains to maintain stable isolation under strong disturbances. Cao [136] improved variable step-size algorithms to autonomously adapt parameters to signal non-stationarity, enhancing tracking accuracy in random vibration environments. Zhang [137] adopted adaptive filtering techniques to eliminate sound field time-delay effects, significantly improving active noise-control phase synchronization accuracy.

In structural vibration control, Zhang [138] achieved precise suppression of low-frequency line spectral vibrations in ribbed cylindrical shells using a semi-analytical fine-area decomposition spectral element method with modal decomposition and energy optimization. For ribbed cylindrical shells, He [139] designed an active–passive support vibration isolation system integrating piezoelectric actuation and fluctuation propagation theories to simultaneously attenuate structural vibrations and acoustic radiation energy across broad frequency bands, as shown in Figure 10. This research advances ship vibration control toward intelligent high-precision solutions through methodological integration and theoretical innovation. These algorithmic innovations drive active control systems toward lightweight design (40% volume reduction) and high real-time performance (bandwidth >1 kHz), laying the foundation for precise vibration suppression in complex sea conditions.

3.3. Overview of Active–Passive Hybrid Control of Vibration in Cylindrical Shells

Hybrid active–passive vibration control for cylindrical shells aims to construct mechanical vibration reduction systems capable of effectively attenuating vibrations across all frequency bands. Existing studies indicate that while passive vibration isolators perform well in mid- to high-frequency bands, their low-frequency isolation effectiveness remains unsatisfactory. This limitation stems from passive isolators’ operational principles restricting performance below the system’s intrinsic frequency, potentially exacerbating vibrations at resonance. Consequently, scholars are exploring strategies combining active and passive vibration isolation to achieve superior damping effects over broader frequency ranges.

For active–passive composite vibration isolation technology, Wu [140] proposed a design combining active control systems with double-layered passive isolators for marine diesel engine vibration characteristics, as shown in Figure 11, verifying system effectiveness through multi-case vibration spectrum analysis. Li et al. [141] studied non-contact magnetic levitation coupling mechanisms, integrating magnetic levitation active control with air spring passive isolation while utilizing magnetic field flexibility to achieve precise micro-amplitude vibration suppression. Niu et al. [142] synergized floating raft passive control with machinery active control, using energy flow transfer efficiency as an evaluation metric to realize broadband vibration control. Regarding system parameter optimization, Kamaruzaman et al. [134] resolved cross-vibration coupling interference in six-degrees-of-freedom magnetic levitation systems via decentralized PD control based on active–passive stiffness matching. Ma et al. [143] designed a parallel actuator–hydraulic passive isolation device incorporating adaptive algorithms for dynamic adjustment, significantly enhancing low-frequency noise suppression in ship power equipment. Hasheminejad et al. [144] constructed piezoelectric actuator/current-variable fluid core layer composites, implementing hierarchical vibration energy dissipation through hybrid active–semi-active control strategies. Notably, compact active–passive integrated designs represent an emerging trend. Wang [145] developed a feed-forward adaptive hybrid control model balancing effectiveness and system volume through optimized actuator layouts, offering novel solutions for traditional integrated systems’ spatial constraints. These innovations demonstrate that material–structure–control tri-dimensional optimization progressively resolves low-frequency control versus compactness conflicts in hybrid systems.

4. Conclusions

In recent years, significant progress has been made in the vibration transfer mechanisms and vibration control technologies of double-layered cylindrical shells in aqueous media, and future development trends and research prospects show a diversified direction.

• With the continuous progress of material science, the development of new composite materials, functional gradient materials, and smart materials provides a wide range of possibilities for vibration control technology. Research on emerging vibration control technologies such as quasi-zero stiffness vibration isolators, acoustic black holes, and phononic crystals has significantly improved the performance of passive control in vibration damping and isolation performance. Currently, the main research objectives in passive control are combining the research results of new materials with practical application contexts to reduce frequency range and expand bandwidth.
• With in-depth research into algorithms, especially the development of algorithms based on artificial intelligence, machine learning and neural networks, it is possible to adjust the vibration control strategy in real time to adapt to the complex and changeable underwater environment so as to improve the vibration reduction and isolation effect of active control. Currently, improving the calculation and feedback speed of the active control system while reducing the size of the system has become the focus of research.
• As an important trend in adapting to a wide bandwidth and underwater complex environment, active–passive hybrid control technology combines the advantages of active control and passive control in the low-frequency and high-frequency bands, which can significantly improve the overall vibration damping effect. Vibration control is carried out by utilizing the unique sound-absorbing and damping characteristics or intelligent regulation capability of new materials and combining them with active control technology to effectively suppress low-frequency noise.