Vibration Features of the Aft Shafting Subjected to Semi-Submerged Propeller Hydrodynamic Excitation

Abstract

To reduce the adverse effects of stern-shaft system vibration on ship performance, this work combined hydrodynamic excitations calculated for a semi-submerged propeller and established a multibody dynamics (MBDs) model of the stern shaft system that included a flexible shaft, propeller, and elastically damped support bearings. The MBDs model’s accuracy was verified through comparison between experimentally identified modal parameters and those computed by the model. It was found that the bearing stiffness and the hydrodynamic excitation frequency collectively determine the vibration amplitude and modal shape of the shaft system, based on an analysis of varied bearing stiffness and damping. Bearing displacement had a significant impact on shafting vibration. And the tie rod with a stiffness of 2.5 × 10⁷ N/m provided a noticeable vibration damping effect. The findings offered theoretical support for mitigating stern-shaft vibration in high-speed vessels subjected to hydrodynamic excitation from semi-submerged propellers.

1. Introduction

As the core component of a ship’s propulsion system, the stern shaft system undertakes the task of transmitting power from the main engine to the propeller, which has a direct impact on the vessel’s performance and safety [1]. Thanks to their excellent operational speed and mobility, high-speed craft find extensive applications in water transportation and offshore operations. Nevertheless, under high-speed sailing and semi-submerged conditions, the stern shaft system encounters a multitude of intricate working conditions. Specifically, the propeller–water interaction induces strong hydrodynamic excitation forces, resulting in intricate vibrations of both the stern shaft system and the hull [2]. Such vibrations not only impair the vessel’s maneuverability but also have the potential to hasten fatigue failure in the hull structure [3]. If the hydrodynamic excitation forces’ frequencies are close to the natural frequencies of the stern shaft system, resonance phenomena can induce intense structural responses, adversely affecting the design and operation of high-performance ships. Consequently, reducing the adverse impact of stern shaft vibrations on vessel performance remains a persistent and formidable challenge in the field.

Research on ship propulsion shaft vibrations spans several disciplinary fields, including structural science, statics, and rotor dynamics, and its inherent complexity renders it a significant challenge in system design [4,5,6,7,8,9,10,11]. In order to thoroughly comprehend these vibration phenomena and mitigate their effects, a substantial body of related research has been carried out by investigators. As an example, Lin et al. [12] introduced a three-degree-of-freedom modal coupling model to explore the mechanism for reducing vibration and noise caused by friction in marine stern tube bearings. This model highlights that friction coefficient and contact stiffness are key factors for system stability, and that randomly rough surfaces act as the main excitation source for friction-induced vibrations, whose peak frequency is governed by the system’s natural frequencies. In addition, vibration properties of propeller systems have been the subject of considerable research focus. Specifically addressing the problem of random vibrations induced by turbulence, Tong et al. [13] utilized a three-step approach to compute the propeller’s broadband forces and their effect on shaft system vibrations. A finite element model was developed along with a transfer frequency response function matrix to analyze the spectral features of the propeller’s hydrodynamic excitation forces and their impact on shaft vibration. This study offers a robust theoretical basis for comprehending the action mechanism of propeller excitation forces on marine shafting vibrations.

In recent years, researchers have proposed a variety of effective control methods for vibration suppression technology. Zhu et al. [14] introduced an active orthogonal support control technique that employs electromagnetic actuators to produce reaction forces, effectively mitigating the influence of underwater propulsion force variations on the ship’s hull. Likewise, Yang et al. [15] studied active magnetic bearings for the suppression of shaft system lateral vibrations. This approach markedly attenuates vibration transmission by minimizing friction at the shaft-stern bearing interface. Aboud et al. [16] conducted vibration modeling analysis using the Finite Element Method and MATLAB 2020, highlighting the effects of critical vibration-damping parameters in propeller systems, including diameter, stiffness, coupling, and mass unbalance. Studies show that the impact of these parameters on system stability is significant, and that proper vibration control can effectively improve vessel stability and safety.

Furthering this work, Su [17] introduced a Dynamic Vibration Absorber with Negative Stiffness (DVANS). This technique is effective in controlling longitudinal vibrations in propulsion shafting and exhibits enhanced vibration attenuation across a wide frequency band relative to conventional Dynamic Vibration Absorbers (DVAs). Xiao [18] integrated Timoshenko beam theory with the Finite Element Method to investigate the coupled bending, torsional, and axial vibration properties of shaft systems with variable cross-sections, offering theoretical insights for improving shafting stability. Additionally, Xie [19] presented an active control strategy that employs electromagnetic constraints to counteract static thrust and utilizes inertial electromagnetic actuators to dampen bearing foundation resonance. The results from the experiments and simulations indicate that this approach achieves notable performance in vibration damping, offering novel perspectives for the further optimization of propulsion shaft system design. Wang [20] employed a five-point smoothing algorithm to precisely measure torsional vibrations in a ship’s propulsion shaft and computed the maximum torque from instantaneous velocity data, thereby furnishing a dependable theoretical foundation for the safe operation of marine propulsion systems. In the realm of researching natural vibration characteristics of shaft systems, Li et al. [21] developed stochastic matrices for mass, stiffness, and damping uncertainties using a non-parametric approach. They analyzed the impact of these uncertainties on natural frequencies, stressing the importance of incorporating uncertainty factors in shaft vibration analysis. He [22] introduced an adaptive water-lubricated stern bearing.

By optimizing the bearing design, vibration transmission properties were markedly enhanced. Although substantial advancements have been achieved in understanding propeller system vibrations and their mitigation techniques, some research deficiencies persist. Previous studies predominantly rely on prescribed constant loads or simplified fixed-load assumptions, or alternatively consider random loads alone, resulting in limited fidelity to actual operating conditions. Therefore, incorporating realistic operational loads derived from propeller hydrodynamic characteristics into shafting vibration analysis is essential for a deeper understanding of propeller–shafting vibration behavior and for enhancing the engineering relevance of the results.

In the field of propeller hydrodynamics research, Tan [23] investigated the vibration behavior of a submerged propeller–shaft coupling system, noting that intense coupled vibrations around low-frequency modes can generate lateral excitation forces in inhomogeneous flows that propagate to the hull, resulting in significant vibration. Concurrently, Li [24] studied the hydroelastic dynamics of an underwater propeller-shaft system through a combined approach of a three-dimensional panel method and the Finite Element Method, underscoring the significance of elastic coupling effects in vibration control.

Notably, the effects of hydrodynamic excitation induced by partially submerged propellers on the vibration response of stern shafting systems remain inadequately explored. Research on semi-submerged propeller hydrodynamics [25,26] has mainly concentrated on evaluating performance indicators, including thrust coefficient, torque coefficient, and efficiency. Few studies have addressed the blade free surface coupled excitation, which constitutes a critical source of shaft vibration in propulsion systems. Consequently, this study introduces a vibration analysis methodology coupling multibody dynamics and propeller hydrodynamic computations. The objective is to examine how the stiffness of support bearings, their positional changes, and the support stiffness of tie rods in the stern shafting influence the shafting’s coupled vibrations. This approach seeks to mitigate resonance caused by hydrodynamic forces and improve the ship’s overall stability and safety.

2. Shaft System Vibration Analysis Model

2.1. Multibody Dynamics Modeling and Modal Validation

As shown in Figure 1a,b, the system comprises the primary drive shaft (Shaft 1), the secondary driven shaft (Shaft 2), a universal joint coupling the two shafts, four supporting bearings, the propeller, and the housing structure. The housing assembly attached to the Shaft1 is secured to the ship’s hull by several axially arranged bolts, maintaining a fixed position relative to the hull. Conversely, the housing associated with the Shaft2 is connected to the hull via telescoping tie rods, allowing the propeller’s water entry or exit condition to be modified as the rods extend or retract.

As shown in Figure 1a,b, a multibody dynamics model of the stern shafting system is developed, where Shaft 1 is rigidly coupled to the power source, Bearing1 and Bearing2 are positioned along the +Z direction, the right end of Shaft 1 is connected to the left end of Shaft 2 through a cross-type universal joint enabling torque transmission, and the propeller is attached to the right end of Shaft 2. Depicted in Figure 1c, the shafts are treated as flexible bodies in the computational model, based on beam theory [27]. The analysis accounts for bending deformation, torsional deformation, shear deformation, and the effects of rotational inertia. The support bearings are modeled with an elastic-damping constraint formulation, which supplies both elastic restoring forces and viscous damping forces in the radial (X, Y) and axial (Z) directions, while rigid constraints are neglected [28]. Both shafts are assumed to be viscously damped.

Assuming the system motion adheres to Newton’s second law, the overall dynamic equation of the system follows:

$$M\ddot{q} + C\dot{q }+ Kq = F_{b }+ F_p$$

(1)

M denotes the mass matrix, which characterizes the mass distribution of individual rigid bodies. C is the damping matrix, accounting for the energy dissipation within each body. K represents the stiffness matrix, describing the elastic interaction forces between rigid bodies. q is the system’s generalized coordinate vector. F_b signifies the bearing support force, while F_p represents the hydrodynamic force on the propeller surfaces.

Shaft 1 is 430 mm in length with a diameter of 32–38 mm, and Shaft 2 is 1221 mm in length with a diameter of 35–38 mm; both shafts are manufactured from 15 to 5 PH steel, exhibiting an elastic modulus in the range of 201–206 GPa and a Poisson’s ratio of 0.3.

The propeller is a six-bladed design, the blade diameter (D) is 640 mm, the hub diameter ratio is 0.17, and the chord length at 0.7R is 180 mm.

As illustrated in Figure 1d, propeller loads derived from hydrodynamic simulations are imposed, where the torque acting on the free surface generates excitation that is simplified in the model as a bending moment in the +Y direction and an axially distributed thrust accounting for free-surface-induced effects.

The natural frequencies of the shaft system were acquired through computation using the multibody dynamics model. A modal test rig for the shaft system was also established. The shaft was supported by air springs with low stiffness, which reduced the influence of support conditions on modal results, illustrated in Figure 2. Accelerometers were positioned in the axial and radial directions. Impact hammer excitation was employed, and the resulting multi-channel acceleration responses were recorded by an LMS SCADAS data acquisition system to perform modal analysis. The comparison of experiment and simulation results is shown in

Table 1. Comparison of natural frequencies.

Order	Experiment /Hz	Vibration Pattern	Type	Simulation /Hz	Modal Simulation	Error /%
1	275		bending	274		0.3
2	630		bending	647		2.6
3	1138		bending	1163		2.1
4	1724		bending	1724		0

, with a discrepancy within 2.6%, validating that the multibody dynamics model provides a relatively accurate prediction of the stern shafting’s vibrational response. The modal results indicate that the first four modes of shaft 2 are all bending modes, with each mode exhibiting distinct deformation characteristics.

2.2. Semi-Submerged Paddle Excitation

The main parameters of the semi-submerged propeller are listed in

Table 2. Key Parameters of semi-submerged propeller.

Parameters	Value
D/mm	640
Hub diameter ratio	0.17
Number of blades	6
Helical direction	Right

, where D denotes the propeller diameter. The immersion depth for the semi-submerged propeller is defined in Figure 3, with h being the distance from the blade tip to the free surface and the immersion ratio is defined as h/D.

This work calculates the open-water performance of a semi-submerged paddle with a 50% immersion ratio.

The semi-submerged propeller operates in close proximity to the free surface, resulting in a characteristic three-dimensional, unsteady, gas–liquid two-phase flow field. The numerical simulation approach adopted is grounded in viscous fluid mechanics theory. Within this framework, water and air are both modeled as incompressible fluids, and inter-phase energy exchange as well as surface tension effects are disregarded. The governing equations are the continuity equation and the Navier–Stokes equations, expressed as:

$${\displaystyle \frac{\mathsf{\partial }\rho }{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }\left(\rho u_i\right)}{\mathsf{\partial }{x}_i}}=0$$

(2)

$${\displaystyle \frac{\mathsf{\partial }\left(\rho u_i\right)}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }\left(\rho u_i{u}_j\right)}{\mathsf{\partial }{x}_j}}=-{\displaystyle \frac{\mathsf{\partial }P}{\mathsf{\partial }{x}_i}}+{\displaystyle \frac{\mathsf{\partial }{\tau }_{ij}}{\mathsf{\partial }{x}_j}}+\rho g_i+F_i$$

(3)

where P denotes hydrostatic pressure, ρ denotes the fluid density. u_i denotes the velocity component in the principal direction of the coordinate axis, g_i denotes the gravitational constant, τ_ij represents the shear stress tensor, and F_i stands for external body forces per unit volume. The τ_ij is defined as follows:

$${\tau }_{ij} =\left[\mu \left({\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{x}_j}}+{\displaystyle \frac{\mathsf{\partial }{u}_j}{\mathsf{\partial }{x}_i}}\right)\right]-{\displaystyle \frac{2}{3}}\mu {\displaystyle \frac{\mathsf{\partial }{u}_l}{\mathsf{\partial }{x}_l}}{\mathsf{\delta }}_{ij}$$

(4)

where $\mu$ represents the dynamic viscosity, and δ_ij denotes the Kronecker delta. Substituting the instantaneous velocity into the governing equations and combining Equations (2)–(4) yields the Reynolds-averaged momentum equation expressed as:

$${\displaystyle \frac{\mathsf{\partial }{(\rho u}_i)}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }{(\rho u}_i{u}_j)}{\mathsf{\partial }{x}_j}} = -{\displaystyle \frac{\mathsf{\partial }P}{\mathsf{\partial }{x}_i}}+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_j}}\left[\mu ({\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{x}_j}}+{\displaystyle \frac{\mathsf{\partial }{u}_j}{\mathsf{\partial }{x}_i}})-\left({\displaystyle \frac{2}{3}}\mu {\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{x}_i}}\right)\right]+\rho g_i-{\displaystyle \frac{\mathsf{\partial }\left(\overline{{{\rho u}_i}^{\prime }{u_j}^{\prime }}\right)}{\mathsf{\partial }{x}_j}}$$

(5)

The turbulence model is closed based on the Boussinesq hypothesis. The terms related to turbulence in Equation (5) are represented as follows:

$$\{\begin{array}{c}-\rho \overline{u_i^{\prime }{u}_j^{\prime }}=2{\mu }_t{S}_{ij}\hfill \\ S_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{x}_j}}+{\displaystyle \frac{\mathsf{\partial }{u}_j}{\mathsf{\partial }{x}_i}}\right)\hfill \end{array}$$

(6)

In this equation, μ_t and S_ij correspond to the turbulent eddy viscosity and the mean strain rate tensor, respectively. Accordingly, turbulence modeling under the Boussinesq hypothesis is reduced to the prediction of turbulent eddy viscosity using the auxiliary transport equations of the turbulence model. The SST k–ω turbulence model with enhanced wall treatment is employed, as it shows weak dependence on the wall Y+ value and provides reliable prediction of boundary-layer behavior and flow separation; the turbulence model equations are expressed as follows:

$${\displaystyle \frac{\mathsf{\partial }\left(\rho k\right)}{\mathsf{\partial }t}} +{\displaystyle \frac{\mathsf{\partial }\left(\rho u_{i}k\right)}{\mathsf{\partial }{x}_i}}=P_k-{\beta }^{\ast }\rho \omega k+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_i}}\left[\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{\mathsf{\partial }k}{\mathsf{\partial }{x}_i}}\right]$$

(7)

$${\displaystyle \frac{\mathsf{\partial }\left(\rho \omega \right)}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }\left(\rho u_i\omega \right)}{\mathsf{\partial }{x}_i}}={\displaystyle \frac{\lambda }{v_t}}{P}_k-{\beta }^{\ast }\rho {\omega }^2+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_i}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\mathsf{\partial }\omega }{\mathsf{\partial }{x}_i}}\right]+2\left(1-F_1\right){\displaystyle \frac{\rho {\sigma }_{\omega 2}}{\omega }}{\displaystyle \frac{\mathsf{\partial }K}{\mathsf{\partial }{x}_i}}{\displaystyle \frac{\mathsf{\partial }\omega }{\mathsf{\partial }{x}_i}}$$

(8)

where $k$ and $\omega$ are the turbulent kinetic energy and specific dissipation rate, respectively; ${\sigma }_k$ and ${\sigma }_{\omega }$ are the corresponding turbulent Prandtl numbers. The parameters ${\beta }^{\ast }$, $\lambda$, and ${\sigma }_{\omega 2}$ denote empirical constants of the turbulence model.

Under semi-submerged propeller conditions, the free surface separating water and air is captured using the Volume of Fluid (VOF) approach. In the VOF framework, two-phase flow is modeled by solving one common set of momentum equations throughout the computational domain. The interface between phases is tracked using a volume-fraction-based indicator function C.

When C = 0, the control volume contains no second phase; when C = 1, it is entirely filled with the second phase; intermediate values indicate that the control volume is partially occupied.

$$\rho ={\alpha }_1{\rho }_1+\left(1-{\alpha }_1\right){\rho }_2$$

(9)

$$\mu ={\alpha }_1{\mu }_1+\left(1-{\alpha }_1\right){\mu }_2$$

(10)

In the equation, ρ₁ and ρ₂ denote the densities of the primary and secondary phases, respectively, while μ₁ and μ₂ denote their viscosities respectively.

The semi-submerged propeller blades are discretized with a prismatic boundary layer mesh. The thickness of the first layer adjacent to the wall is determined by the following equation:

$$∆y =L·y^{+}{·Re}^{-{\displaystyle \frac{13}{14}}}·\sqrt{74}$$

(11)

where L is the characteristic length, which is taken as the propeller diameter D in the present computational case; Re is the Reynolds number; Δy is the distance from the first node to the wall. This study calculates the boundary layer parameters targeting a y⁺ value of 1. The boundary layer consists of 15 layers with a growth rate of 1.2.

Three mesh configurations were defined: 11.47 million, 14.90 million, and 22.54 million cells, respectively. Figure 4 displays the thrust pulsation curves for the three mesh configurations. The pulsation variations from the 14.9 million and 22.54 million mesh calculations are remarkably similar, whereas the 11.47 million mesh configuration exhibits significant divergence from the other two. To balance computational efficiency and accuracy, the 14.9 million mesh configuration was selected for subsequent calculations. The final mesh is illustrated in Figure 5.

The hydrodynamic simulation yielded the surface bending moment, thrust, and output torque, as depicted in Figure 6a–c. At steady-state conditions, the peak values were observed: a maximum surface bending moment of 458 N·m, a maximum thrust of 8642 N, and a maximum output torque of 1736 N·m. Spectral analysis of the hydrodynamic loads reveals, as shown in Figure 6d. Since the three spectra were similar, only the thrust spectrum is presented. The maximum excitation frequency of the hydrodynamic loads is 152.46 Hz, coinciding with the rotational frequency of the propeller.

3. Shaft System Vibration Analysis

3.1. Principal Factor Analysis of Shaft System Vibration

Numerical simulations were conducted at 1533 r/min using the developed MBDs model to investigate the global vibration behavior of the transmission shaft system. Figure 7a illustrates the vibration displacements of Shaft 1 in the X and Y directions. Figure 7b shows the vibration response spectrum of Shaft 1 in the Y direction, which corresponds to the hydrodynamic bending moment direction. Figure 7c shows the vibration displacements of Shaft 2 in the X and Y directions. Figure 7d presents the vibration response spectrum of Shaft 2 in the Y direction.

According to Ref. [4], bearing vibration increases with rotational speed, and the vibration amplitude in the horizontal direction is significantly larger than that in the vertical direction. In contrast, Figure 7a,c indicate that Shaft 1 and Shaft 2 exhibit comparable vibration amplitudes in both horizontal and vertical directions. This phenomenon arises because the hydrodynamic loads induced by the semi-submerged propeller act predominantly along the Y-axis, i.e., the vertical direction. As a result, a pronounced excitation effect is imposed on vertical vibrations. Accordingly, the vertical vibration response is used to evaluate and compare the vibration behavior of the shafting system in order to assess the effects of hydrodynamic excitation.

From the results, Shaft 1 demonstrates comparatively steady vibration with low amplitudes, while Shaft 2 exhibits more severe oscillatory behavior. The vibration response of Shaft 1 is primarily governed by a fundamental frequency of 24.46 Hz, nearly coincident with the first-order rotating frequency. Additional higher-order peaks are observed at 122.3 Hz and 171.2 Hz, which are close to the 5th and 7th rotational harmonics. These features suggest that Shaft 1 is primarily excited by rotational frequency components. Hydrodynamic bending moment excitation induced by the semi-submerged propeller is predominantly transferred to Shaft 2. Shaft 2 exhibits a frequency peak at 24.49 Hz, corresponding approximately to the first-order rotating frequency. Frequency components at 146.7 Hz and 293.9 Hz are associated with propeller-induced excitation. In high-speed shafting systems, increased rotational speeds can result in higher bearing loads, which, in turn, have a more significant impact on the bearings, potentially triggering vibrations [4]. The higher-order component at 440.9 Hz is identified as the constrained modal frequency with bearing support. The combined effect of multiple frequency excitations leads to amplified displacement fluctuations. These results demonstrate that Shaft 2 is subjected to both fundamental and harmonic excitations, resulting in more complex dynamic behavior. Both the long and short shafts have first-order natural frequencies near the first-order rotating frequency, suggesting a risk of resonance. In addition, the higher-order frequency components of Shaft 2 approach the hydrodynamic excitation frequencies, leading to an increased likelihood of resonance.

Torsional vibrations in a long shaft rotor system are measured by the relative torsional angle between adjacent moments of inertia. In agreement with Ref. [29], the torsional vibration of the stern shafting system exhibits clear coupling characteristics. Nevertheless, when subjected to hydrodynamic excitation, Shaft 1 and Shaft 2 each exhibit only one prominent torsional response peak. This behavior is attributed to the fact that, at the current operating condition, higher-order fixed frequencies are well separated from the primary hydrodynamic excitation frequency, leading to negligible torsional response amplitudes. As a result, the frequency spectrum exhibits only one pronounced peak. This mechanism further accounts for the relatively high frequency of the dominant torsional vibration peak. As illustrated in Figure 8a, the torsional vibration velocity of Shafts 1 and 2 in the angular domain are presented, and Figure 8b shows their frequency spectra, indicating that the driven shaft displays a more significant torsional vibration response. This phenomenon can be explained by the fact that the driven shaft is considerably longer than the driving shaft, leading to lower torsional and bending stiffness and thereby enhancing its susceptibility to self-excited angular vibration.

Thus, the elementary modal analysis reveals that the primary vibration excitation mechanism in a stern shafting system configured with a semi-submerged propeller stems from the coupled effects of two factors. Firstly, hydrodynamic surface excitation serves as the principal external driving source, supplying sustained vibrational input to the shafting. Secondly, the inherent structural properties of the driven shaft confer an increased susceptibility to torsional vibrations, leading to ultimately greater vibration amplitude and severity relative to the drive shaft.

To further analyze the vibration characteristics of the shafting system, the vibration responses of the support bearings (Bearing1, Bearing2, Bearing3, and Bearing4) for the drive shaft Shaft1 and the driven shaft Shaft2 were extracted. Figure 9a–d illustrate the vibration displacement along the X and Y axes for Bearing1, Bearing2, Bearing3, and Bearing4, respectively. The vibration condition for each bearing is monitored at three nodes. Node 2 serves as the central node, with Nodes 1 and 3 positioned symmetrically at a distance of 10 mm from Node 2.

As illustrated in Figure 9a, the proximity of Bearing1 to the power source means it is directly subjected to the rotational imbalance excitation, leading to vibrations dominated by high-frequency, small-amplitude oscillations. With bearing Nodes 1 and 3 positioned symmetrically, the vibration phase and amplitude differences between them and the central Node 2 are slight, influenced by the shaft system’s bending mode. This demonstrates a uniform vibration mode when the bearing provides adequate rigid support, and suggests the hydrodynamic surface excitation has a limited effect on Bearing1’s vibration.

Figure 9b shows that Bearing2, being adjacent to the universal joint, displays a vibration pattern where the amplitude first decays and then fluctuates in response to the excitation from the joint. This behavior arises because the variable-velocity transmission of the universal joint introduces additional alternating loads, leading to initially steady vibration that later becomes governed by joint excitation. Furthermore, the joint’s flexible connection dampens the transmission of power-source excitation, resulting in a lower vibration amplitude for Bearing2 compared to Bearing1.

Illustrated in Figure 9c, Bearing3, also proximate to the universal joint, experiences relatively large vibration amplitudes characterized by a pronounced low-frequency component. The bearing is subjected to additional bending moments from the torsional vibration transmitted through the universal joint and shaft misalignment excitation, resulting in primarily low-frequency vibrations. The driven shaft’s relatively low initial stiffness causes a more pronounced amplification of the universal joint excitation. This leads to significant disparities in vibration among Nodes 1, 3, and the central Node 2, reflecting a non-uniform vibration mode. Bearing3 demonstrates the greatest vibration amplitude across the whole shaft system.

Figure 9d indicates that Bearing4, situated close to the propeller, is directly excited by the propeller’s hydrodynamic pulsations. The flow reaction at the propeller end also contributes to a primarily high-frequency vibration signature. Furthermore, influenced by the vibration mode characteristic of the shaft end, Nodes 1, 3, and the central Node 2 vibrate with high synchronization, showing a highly concentrated vibration pattern at the terminus.

Therefore, since Shafts 1 and 2 have nearly identical diameters and the length of Shaft 2 is approximately three times that of Shaft 1, the driving shaft (Shaft 1) exhibits higher stiffness and stronger attenuation capability for excitation from the power source; however, the flexible connection of the universal joint allows the transmitted excitation to be insufficiently attenuated in the low-stiffness secondary shaft, and because Shaft 2 is closer to the propeller excitation, the corresponding harmonic vibration amplitudes are amplified through dynamic response, ultimately resulting in a significant increase in secondary shaft vibration and the maximum vibration amplitude occurring at Bearing3.

3.2. The Effect of Variations in Bearing Stiffness and Damping

Analysis of the shaft system’s vibration modes indicates that hydrodynamic excitation is a crucial factor affecting its vibrational response. Consequently, in order to study the impact of changes in the stern shaft bearing properties on the global vibration response under hydrodynamic excitation, six distinct combinations of bearing stiffness and damping were formulated, presented in

Table 3. Bearing stiffness and damping parameters.

Operating Conditions	Stiffness (N/m)	Damping (N·s/m)
1	3 × 107	5 × 106
2	3 × 107	1 × 107
3	3 × 108	5 × 106
4	3 × 108	1 × 107
5	3 × 109	5 × 106
6	3 × 109	1 × 107

. Among these, the bearing stiffness increases at a rate of 10 times.

Using the stern shafting multibody dynamics model, dynamic solutions were obtained for each of the six cases. The vibration response was analyzed at a node close to the hydrodynamic excitation source. Figure 10 presents the corresponding Y-axis vibration spectrum, with Figure 10a–c corresponding to condition 1 to condition 6, respectively.

As illustrated in Figure 10a–c, the vibration response in all cases is governed by a dominant frequency of 146.7 Hz, corresponding approximately to the first-order hydrodynamic excitation, together with secondary components at 24.45 Hz, 293.9 Hz, and 440.9 Hz. For the same stiffness level, an increase in the damping coefficient from 5 × 10⁶ N·s/m to 1 × 10⁷ N·s/m leads to a noticeable reduction in modal frequencies. When the damping coefficient is held constant, the vibration amplitude exhibits only mild variations.

As a multi-degree-of-freedom rotating mechanical system, the natural frequencies of the stern shaft system conform to the characteristic equation of multi-body dynamics:

$$[K-{\omega }^{2}M]\Phi = 0$$

(12)

ω is the natural angular frequency, and Φ is the mode shape vector. An increase in bearing stiffness effectively strengthens the support terms in the stiffness matrix K, leading to enhanced deformation resistance of the system and consequently lower response amplitudes under identical excitation. Higher bearing damping improves the dissipation of vibrational energy, thereby reducing the vibration response amplitude. Variations in bearing stiffness and damping modify only the response intensity of each modal order, whereas the mode shapes Φ remain unchanged, resulting in amplitude changes in the spectra but nearly identical vibration mode characteristics.

At 440.9 Hz, the vibration response shows evident nonlinear characteristics with increasing stiffness under the damping condition of 5 × 10⁶ N·s/m. The observed nonlinearity is attributed to the lower damping ratio at 5 × 10⁶ N·s/m compared with 1 × 10⁷ N·s/m; inherent nonlinear effects stemming from bearing support characteristics and structural coupling are magnified when the damping ratio is low.

Consequently, the combination of increased bearing stiffness and enhanced damping provides an effective approach for mitigating vibrations at nodes in proximity to hydrodynamic excitation sources.

3.3. The Effect of Bearing Position Changes

The vibration of the shaft system is influenced not only by bearing stiffness and damping but also significantly by the frequency content of the excitation forces. Acting as the support elements for the shafting, bearings can, under hydrodynamic surface excitation, allow excitation forces at particular frequencies to become close to or match the system’s natural frequencies, thereby inducing resonance. Optimization of bearing placement can aid in decoupling the excitation frequencies from the natural frequencies, thus mitigating the potential for resonance. Based on the dimensions of the Shaft 2 and the structural limitations of the stern tube, the Bearing3 can be adjusted within a range of −25 mm to 25 mm, whereas the Bearing4 has a maximum adjustment range of −30 mm to 30 mm. In order to examine the effects of varying bearing locations on the shaft system’s vibration, six distinct bearing configuration modifications were devised in this research:

• Keeping the separation between Bearing3 and Bearing4 constant, both bearings are moved 20 mm in the -Z direction. The resulting vibration acceleration is presented in Figure 11a;
• With the spacing between Bearing3 and Bearing4 fixed, both bearings are displaced 20 mm in the +Z direction. The corresponding vibration acceleration is depicted in Figure 11b;
• To increase the bearing spacing, Bearing3’s position is fixed while Bearing4 is moved 20 mm in the -Z direction. The vibration acceleration for this configuration is illustrated in Figure 11c;
• The bearing spacing is increased by maintaining Bearing4’s position relative to the excitation source and shifting Bearing3 by 20 mm in the +Z direction. Figure 11d shows the vibration acceleration outcome;
• The bearing spacing is minimized by concurrently moving Bearing3 20 mm in the +Z direction and Bearing4 20 mm in the -Z direction. The vibration acceleration profile for this minimum-spacing case is given in Figure 11e;
• To achieve the maximum bearing spacing, Bearing3 is moved 20 mm in the -Z direction while Bearing4 is moved 20 mm in the +Z direction. The vibration acceleration for this maximum-spacing configuration is shown in Figure 11f.

Figure 11a,b indicate that when both bearings move in the same direction with fixed spacing, the support span and stiffness distribution of Shaft 2 are preserved, resulting in only minor variations in dominant frequency amplitudes, approximately at the level of 4 × 10⁻⁷ m. In contrast, displacement of the bearings toward the propeller (+Z direction) leads to a pronounced reduction in the peak amplitudes associated with the first- and second-order hydrodynamic excitation frequencies. These results confirm that bearing displacement toward the propeller effectively mitigates hydrodynamically induced vibration. A comparison between Figure 11c,d shows that relocating a single bearing toward the propeller end effectively attenuates the first-order rotational vibration of the shafting.

Figure 11e,f illustrate the minimum and maximum spacing cases, revealing that the smallest spacing results in the most severe vibration due to concentrated support constraints. In this case, Shaft 2 exhibits increased vibration freedom, which enhances energy accumulation in the dominant modes and consequently intensifies the vibration response.

The vibration response of the shafting system is determined by the degree of matching between the excitation frequency and the system’s natural frequencies. Bearing positions influence the distribution of natural frequencies by altering the distribution of support stiffness within the shafting. When the bearing spacing increases, the support constraints become more dispersed, leading to a more uniform stiffness distribution. This causes the shafting’s natural frequencies to deviate from the water-surface excitation frequency, avoiding resonance and thus reducing vibration amplitude.

Therefore, comparison reveals that the original bearing spacing provides superior support dispersion. Moving both bearings simultaneously towards the propeller direction proves beneficial for shaft system vibration damping.

3.4. The Influence of External Support

The previous analysis examined the impact of bearings on stern shaft system vibrations through variations in bearing stiffness/damping and positional adjustments, establishing trends for bearing modifications that can mitigate semi-submerged propeller excitation. External supports constitute a vital part of the stern shafting system; therefore, studying the influence of their property changes on the hydrodynamic excitation is also a key approach to enhancing the overall vibrational performance of the stern shafting. Therefore, this study designed three tie rods with low, medium, and high stiffness levels, specifically 2.5 × 10⁷ N/m, 2.5 × 10⁸ N/m, and 2.5 × 10⁹ N/m, respectively. Under these three different support stiffness conditions, the vibration responses of Bearing1, Bearing2, and Bearing3 showed little variation, whereas the vibration response of Bearing4 exhibited significant differences. As shown in Figure 12, it presents the variation in vibration displacement of Bearing4 in the X and Y directions.

Located at the end of the stern shafting system and in the direct action zone of the semi-submerged propeller’s hydrodynamic excitation, the vibration response of Bearing4 exhibits distinct stratified characteristics as the tie rod stiffness changes. Figure 12a shows the vibration response of Bearing4 under low-stiffness support. The vibration amplitude is relatively minimal, with a gentle fluctuation trend. The differences in vibration phase and amplitude among the bearing nodes are relatively significant, indicating strong independence between nodes.

Figure 12b illustrates the vibration response of Bearing4 under medium-stiffness support. Here, the vibration amplitude increases significantly, with more pronounced high-frequency fluctuations. The vibrational synchrony among the bearing nodes is markedly enhanced, and the phase and amplitude differences between nodes diminish.

Figure 12c presents the vibration response of Bearing4 under high-stiffness support. The vibration amplitude of Bearing4 increases further, with more dense and energy-concentrated fluctuations. The degree of vibrational coupling among the bearing nodes is greatly enhanced, and differences between nodes virtually disappear. Therefore, the vibration characteristics of Bearing4 are governed jointly by the tie rod support stiffness, the transmission pattern of the hydrodynamic excitation, and the vibrational mode constraints at the shafting end. A compliant support acts to buffer and dissipate the propeller’s hydrodynamic pulsations. The vibrational energy is attenuated via the support’s elastic deformation, leading to lower amplitudes.

Furthermore, the weak constraint from low stiffness allows greater freedom for the bearing nodes (1, 2, 3) to vibrate, resulting in pronounced independence and observable differences among them. As the support stiffness nears the inherent stiffness of the shaft system, a resonant coupling tendency emerges. This dramatically increases the efficiency of hydrodynamic excitation transmission, causing a substantial rise in vibration amplitude. The enhanced constraint from the support on the shaft’s vibration modes improves the coordinated motion of all nodes, boosting their synchronization. With a high-stiffness support, the rigid constraint severely limits the vibrational freedom at the shaft end. Hydrodynamic pulsations struggle to dissipate through the support, and the rigid boundary condition actually amplifies the vibration amplitude. Additionally, the shaft’s vibration modes become intensely localized at the terminal region, and the coupling between the motions of the bearing nodes is greatly intensified, producing denser oscillations and minimal inter-node variation.

4. Conclusions

A robust framework coupling multibody dynamics (MBDs) with propeller hydrodynamic analysis was developed in this research. It effectively characterizes the coupled vibrational behavior of the stern shafting subjected to semi-submerged propeller excitation and precisely simulates the system’s dynamic response to variations in bearing stiffness, bearing location, and support rod stiffness. The findings of this study are of considerable importance for vibration mitigation and reliability improvement in high-speed vessel stern shafting systems. The principal findings are summarized below.

• Shaft 1 exhibits a dominant fundamental frequency of 24.44 Hz, approaching the first-order transition frequency. The frequency peaks on Shaft 2 include 24.45 Hz near the first-order transition frequency, alongside 146.7 Hz and 293.9 Hz approaching propeller excitation frequencies. The superposition of multiple excitation frequencies induces more pronounced displacement oscillations, rendering Shaft 2 at higher resonance risk.
• The combination of increased bearing stiffness and enhanced damping provides an effective approach for mitigating vibrations at nodes in proximity to hydrodynamic excitation sources.
• Maintaining the spacing between Bearing3 and Bearing4 while moving them towards the propeller direction is beneficial for shaft system vibration damping. Thus, vibration response analysis can be employed in the design process of the stern shafting system to optimize both the casing and shaft designs, improving the overall reliability of the system.
• The support stiffness of the tie rod exerts a nonlinear influence on the vibration of the Bearing4. A compliant stiffness (2.5 × 10⁷ N/m) provides significant vibration reduction by buffering and dissipating vibrational energy through elastic deformation.
• The investigation of vibration characteristics of the stern shafting under hydrodynamic excitation provides important vibration-reduction references for the development of high-performance fast craft. All investigations in this study were conducted under the operational condition of 1533 r/min, where the stern shaft system exhibits relatively stable performance. Future work could explore the vibration response characteristics of the stern shafting under different rotational speeds to investigate the coupled effects of speed on hydrodynamics and shafting vibrations.