Insights into the Vibration Characteristics of Spatial Radial Gate Affected by Fluid–Structure Interaction

<jats:p>Radial gate, a spatial frame structure, is the key factor to control water discharge in dam structure and storm surge barriers. However, the fluid-induced vibration (FIV) problem always occurs owing to fluctuation loads exerted on the gate, threatening the safety of hydropower stations. In this work, two fluid–structure interaction (FSI) modal analysis methods—the coupled acoustics–structure method and the added-mass method—are provided. Further, a comprehensive investigation on the vibration characteristics of the spatial radial gate, considering spatial structural characteristics and the FSI effect, is conducted. The numerical results revealed that the feasibility of the proposed coupled acoustics–structure method in analyzing FSI modal analysis was demonstrated; moreover, a reasonable length of the fluid domain in front of the skinplate existed for efficient computation. Meanwhile, through the added-mass method, the rational added-mass discount factor of hydrodynamic loads obtained from the Westergaard formula was provided. The FSI effect induced whole-gate rotation vibration streamwise around trunnion pins, significantly reducing the gate’s fundamental vibration frequency. In addition, three typical dynamic-instability vibration patterns of radial gates were presented. These patterns were affected by spatial structural characteristics and FSI. It was demonstrated that the struts and skinplate coupled bending–torsional vibration would cause the radial gate frame structure to fail catastrophically. The proposed insights can provide guidelines of vibration characteristics analysis of the radial gate submerged in flow water in reservoir and storm surge barriers.</jats:p>

section_content: The radial gate, a spatial frame structure composed of a skinplate, horizontal girders, vertical girders, strut arms, and hydraulic actuating rods, is the main water-controlling structure in reservoir [1, 2] and storm surge barriers [3, 4] for regulating flood discharge and generating hydropower energy (as shown in Figure 1 ). Unfortunately, the FIV phenomenon, induced by fluctuation loads exerted on the gate, always occurs [5, 6]. Furthermore, serious vibration amplitude or even dynamic instability can be induced [6, 7], while fluctuation loads and gate dynamic characteristics satisfy certain coupling relationships, threatening the safety of the dam structure and people's lives downstream. Gate vibration characteristics play a major role in the gate FIV problem [8, 9]. Hence, it is essential to explore the internal insights into the vibration characteristics of the radial gate, the spatial frame structure submerged in flow water, during the water-discharging process. Island [3, 11]. 

In recent years, with the rapid development of hydropower construction and storm surge barriers, numerous radial gates have served for long periods, resulting in remarkable variation in structural characteristics [1]. In engineering practice, three methods-prototype observation, modeling experiments, and numerical simulation-are commonly adopted to analyze the vibration response of the gate structure [12] [13] [14] [15] [16] [17]. Subsequently, the gate's vibration safety can be determined according to the obtained vibration stress, displacement, and acceleration through the aforementioned methods. Moreover, whether the structural resonance occurs in the gate structure is usually determined according to the combination of the hydrodynamic load's frequency range and structural natural frequencies. Nonetheless, to date, the explanations for the mechanism of radial gate vibrations in different cases are quite controversial [2, 5, [18] [19] [20] [21]. It is still not clear about the relationship between gate vibration patterns and various fluctuation waters in front of the skinplate, as well as the fluid-structure coupling process. Hence, the underlying vibration failure patterns of spatial radial gates, considering the FSI process, still need to be further explored. 

During the water-discharging process, owing to the instability flow-fluctuation between complete separation and reattachment at the gate bottom, hydrodynamic fluctuation loads forcing on the skinplate induce the gate's vibration. In turn, the gate's motion changes the flow field of the water in front of the skinplate, generating additional dynamic water pressure [22] [23] [24] [25]. Furthermore, the separate flow, caused by shear-layer instability and motion-induced vortex shed at the edge of the gate bottom, also induces the gate's vibration. To date, several FIV mechanisms have been proposed, such as forced vibration, vortex-induced vibration, self-excited vibration, and parametric resonance [5]. Besides, the aforementioned additional dynamic water pressure can be transferred into added mass attached to the skinplate, which has been found to significantly affect the dynamic characteristics of gate structures based on remarkable reported works [26] [27] [28]. Therefore, the coupling effect between flow water and the gate structure should be considered when investigating the vibration characteristics of the radial gate. 

To date, two methods-the structural health monitoring method and the numerical simulation method-have been adopted for modal analysis of the gate's structure. Specifically, the former approach obtains gate vibration characteristics based on the input external excitation and the vibration response, using experimental modal analysis [29, 30] and operational modal analysis [31] [32] [33]. The latter method is usually adopted to conduct additional hydrodynamic pressure and gate modal analyses based on the finite element [5, 10]. (b) Fox Point Hurricane Barrier (Hurricane Barrier) in the City of Providence, Rhode Island [3, 11]. 

In recent years, with the rapid development of hydropower construction and storm surge barriers, numerous radial gates have served for long periods, resulting in remarkable variation in structural characteristics [1]. In engineering practice, three methods-prototype observation, modeling experiments, and numerical simulation-are commonly adopted to analyze the vibration response of the gate structure [12] [13] [14] [15] [16] [17]. Subsequently, the gate's vibration safety can be determined according to the obtained vibration stress, displacement, and acceleration through the aforementioned methods. Moreover, whether the structural resonance occurs in the gate structure is usually determined according to the combination of the hydrodynamic load's frequency range and structural natural frequencies. Nonetheless, to date, the explanations for the mechanism of radial gate vibrations in different cases are quite controversial [2, 5, [18] [19] [20] [21]. It is still not clear about the relationship between gate vibration patterns and various fluctuation waters in front of the skinplate, as well as the fluid-structure coupling process. Hence, the underlying vibration failure patterns of spatial radial gates, considering the FSI process, still need to be further explored. 

During the water-discharging process, owing to the instability flow-fluctuation between complete separation and reattachment at the gate bottom, hydrodynamic fluctuation loads forcing on the skinplate induce the gate's vibration. In turn, the gate's motion changes the flow field of the water in front of the skinplate, generating additional dynamic water pressure [22] [23] [24] [25]. Furthermore, the separate flow, caused by shear-layer instability and motion-induced vortex shed at the edge of the gate bottom, also induces the gate's vibration. To date, several FIV mechanisms have been proposed, such as forced vibration, vortex-induced vibration, self-excited vibration, and parametric resonance [5]. Besides, the aforementioned additional dynamic water pressure can be transferred into added mass attached to the skinplate, which has been found to significantly affect the dynamic characteristics of gate structures based on remarkable reported works [26] [27] [28]. Therefore, the coupling effect between flow water and the gate structure should be considered when investigating the vibration characteristics of the radial gate. 

To date, two methods-the structural health monitoring method and the numerical simulation method-have been adopted for modal analysis of the gate's structure. Specifically, the former approach obtains gate vibration characteristics based on the input external excitation and the vibration response, using experimental modal analysis [29, 30] and operational modal analysis [31] [32] [33]. The latter method is usually adopted to conduct additional hydrodynamic pressure and gate modal analyses based on the finite element method, finite volume method, and particle method. In this study, in order to explore the FSI effect on the vibration characteristics of the radial gate, two numerical approaches, namely, the added-mass method [34, 35] and the coupled fluid-structure method [2, 8, [36] [37] [38] [39], are considered. In the added-mass method, according to the specifications referenced in the gate design stage [40, 41], the Westergaard formula [42, 43] is adopted to calculate the hydrodynamic pressure. The added mass can be yielded from this formula. However, the discount factor of added mass is still not clear, and the rational value of added mass attached to the radial gate skinplate during the water-discharging process is still puzzling [41], which need to be studied in depth. 

Furthermore, vibration patterns and the corresponding frequencies are the key factors of concern in structural dynamic characteristics analysis. In the past few decades, many works on modal analysis of hydraulic gates in air or water have been conducted [2, 8, [36] [37] [38] [39] [44] [45] [46] [47]. Specifically, Li et al. [8] investigated the structural self-vibration characteristics of the radial gate in partial gate opening using analytical sub-space iteration and the unsymmetric method. Lee et al. [36] established a three-dimensional finite element model to analyze the structural vibration responses and added a supplementary plate in the gate structure for mitigating the FIV. Lian et al. [2] analyzed the cause of hydraulic gate vibration from the perspective of structural dynamics and found two vibration patterns of the surface outlet gate in certain cases. Zhao et al. [37] explored the self-vibration characteristics of a large-scale radial gate under fluctuating pressure using the added-mass method, while the dynamic response of the gate structure was obtained through the random vibration method. Sun et al. [38] carried out a numerical simulation of the radial gate based on building information modeling under different opening degrees. Wang et al. [39] conducted vibration analysis of a large-scale radial gate considering the elasticity of the water stop structure. In addition, Xu et al. [5, 9, 21, 48] carried out a series of studies on the dynamic instability mechanism of the radial gate, finding that the frame's flexural-torsional vibration deformation can significantly enlarge the size and intensity of dynamic instability regions. Hence, the vibration characteristics analysis is critical for vibration safety evaluation of the spatial radial gate. However, to date, the explanations of vibration patterns of the radial gate in different cases are still diverse, and a consensus has not yet been reached, especially regarding the vibration patterns that easily cause whole-gate destruction. 

In this work, to address the aforementioned research gap, we carry out a systematic investigation on the vibration characteristics of the radial gate affected by FSI and spatial structural characteristics during the water-discharging process to reveal the typical dynamic-instability vibration patterns of the spatial radial gate. Specifically, two FSI modal analysis methods-the coupled acoustics-structure method and the added-mass method-for exploring the vibration characteristics of the radial gate submerged in water are proposed. Meanwhile, the effect of the fluid domain volume in front of the gate skinplate on the vibration characteristics of the radial gate is demonstrated, the rational value of the discount factor for the added-mass method is provided, and the applicability conditions of the above two methods are discussed, respectively. Subsequently, the effect of FSI on the vibration patterns and frequencies of the spatial radial gate is investigated. The influence of the structural spatial characteristics on the radial gate's vibration patterns is investigated as well. Furthermore, the typical dynamic-instability vibration patterns of the radial gate, affected by FSI and structural spatial characteristics, are provided.
section_name: Introduction

section_name: Materials and Methods

section_content: FSI occurs during the water-discharging process, leading to remarkable vibration problems in the gate structure. Specifically, the unstable flow fluctuation between complete separation and reattachment at the gate skinplate induces the gate structure's vibration motion. Meanwhile, the structural movement reacts on the aforementioned flow water, generating additional hydrodynamic pressure exerted on the gate structure. Hence, additional hydrodynamic loads would affect the structural dynamic characteristics of the whole gate, and it is necessary to uncover this influence. 

Nevertheless, the gate's water-discharging dynamic system is complicated, and the closed-form solutions generally do not exist using traditionally analytical methods. Particularly, in engineering practice, a discretized numerical model of the gate's water-discharging dynamic system could be established, taking into account the added mass generated by flow water. The differential equation of motion of the gate's water-discharging dynamic system is presented as Equation (1). Notably, owing to the slight effect of damping on the structural dynamic characteristics, the damping term caused by added hydrodynamic pressure is not considered in the governing equation:

where Q is the displacement vector of the spatial radial gate, M is the global mass matrix of the gate structure, M w is the added-mass matrix induced by water sloshing, M + M w is the global mass matrix of the coupled water-structure model, K is the global stiffness matrix, and F(t) is the external dynamic loads, excluding added hydrodynamic pressure induced by structural motion. Subsequently, when considering the additional hydrodynamic pressure caused by structural motion, the free vibration equation of the gate structure is taken as Equation ( 2 ). Further, the vibration frequencies of the discrete gate model and corresponding vibration patterns of the spatial radial gate can be calculated through Equation (3). Here, the addedmass matrix, M w, denotes the effect of additional hydrodynamic pressure, caused by the fluid-structure interaction, on the gate structure:

where w i is the i-th-order natural frequency, and φ i is the corresponding i-th vibration mode.
section_name: Mathematical Model of Free Vibration in Water

section_content: In this study, two methods-the added-mass method and the coupled acousticsstructure method-were adopted to calculate and analyze the vibration characteristics of the spatial radial gate. Furthermore, the efficiency and accuracy of these methods for analyzing the FSI effect were demonstrated by comparing the obtained numerical results.
section_name: Methods in Modal Analysis Considering the FSI Effect

section_content: In order to explore the coupling effects of additional hydrodynamic pressure on the gate structure's vibration, the added-mass method, proposed in the current specification [41] reference for gate design, was adopted in this study. Specifically, the detailed analysis process is described below. 

First, the hydrodynamic pressure exerted on the radial gate's skinplate was calculated using the classical Westergaard formula [42, 43]. This was derived by Westergaard in 1933 for analyzing the infinite reservoir hydrodynamic pressure on a straight rigid dam, with the vertical upstream face of the Hoover Dam during an earthquake, as shown in Equation (4). Here, Westergaard's hydrodynamic pressure equation was induced by hypothesizing that the dam body is a rigid body without any vibration deformation. It should be mentioned that, for calculating the hydrodynamic pressure exerted on the radial gate's skinplate, the inclination degree of the skinplate facing water and the vibration deformation of the gate structure should be considered, which are the main differences compared to these hypotheses used in Westergaard's hydrodynamic pressure equation:

where P h is the seismic-induced hydrodynamic pressure at water depth h, and a g is seismic acceleration, which was taken as the unit value for calculating the hydrodynamic pressure. 

Here, ρ w is the water density, H is the water depth from the surface to the reservoir bottom, and h is the calculated water depth from the water surface to the gate bottom, which vary in different gate openings. Subsequently, based on the Westergaard formula, through Equation (4) we converted the calculated hydrodynamic pressure equivalently into an additional added mass at each node attached on the gate skinplate. In addition, the nodal added mass was calculated based on the nodal bearing pressure and forcing area, where the seismic acceleration was adopted as the unit value, as shown in Equation (5). In particular, owing to the aforementioned differences between the straight rigid dam in the original Westergaard formula and the spatial radial gate submerged in water, namely, the curved gate skinplate and vibration deformation of the radial gate, the induced additional dynamic pressure would be weakened [41, 49]. Hence, the added-mass discount factor of the seismic action effect was proposed to consider the effect of the above differences in Equation ( 5 ), with the value varying from 0. 0~1. 0:

where m hi is the added mass of the i-th node in the gate skinplate, A i denotes the effectiveness area of the element corresponding to the i-th node in the gate skinplate, and ξ is the added-mass discount factor of the seismic action effect, a dimensionless parameter with a value ranging from 0. 0 to 1. 0. Furthermore, the aforementioned added mass, m hi, was added into the gate's waterdischarging numerical dynamic system and, subsequently, we assembled the added mass into the added-mass matrix, M w. It is worth stressing that the added-mass discount factor, ξ, should be considered in Equation ( 5 ), owing to the lacking mutual coupling effect caused by the gate's deformation and the curved surface of the radial gate. Finally, the vibration characteristics of the radial gate submerged in flow water during the fluid-discharging process can be calculated using Equations (1)-(3). 

Accordingly, Figure 2 shows an intuitive schematic of the above added-mass method of the radial gate. The added-mass method is a quasi-static, simplified method, with which it is difficult to accurately consider the mutual coupling effect between the fluid and the gate structure; hence, the effect of the added-mass discount factor, ξ, of seismic action needs to be considered. However, for the radial gate submerged in water, the rational value of coefficient ξ remains unclear in the current specification [41], and it needs to be further explored.
section_name: Added-Mass Method

section_content: In order to accurately analyze the effect of water flow on the vibration characteristics of the gate structure, it was necessary to establish the coupled acoustics-structure model, including water in front of the gate skinplate. The detailed acoustics-structure model of the spatial radial gate is depicted in Figure 3. 

Specifically, the analysis process of this coupled acoustics-structure method was as follows: (1) First, we established the spatial radial gate structure, containing strut arms, a trunnion pin, hydraulic actuating rods, girders, and the skinplate. (2) Subsequently, we established the fluid domain based on the water volume in front of the gate skinplate and the meshing fluid elements. It should be noted that the common nodes should be required in the intersection between the fluid and the gate structure. (3) Further, the equations of motion of the fluid and the gate structure were confirmed, respectively. (4) Finally, the fluid-structure coupling interface at the intersection between the fluid domain and the gate structure, labeled red in Figure 3, were established, through which the nodal data were transferred. Accordingly, the coupling calculation between the fluid and the gate structure was realized.
section_name: Coupled Acoustics-Structure Method

section_content: In order to accurately analyze the effect of water flow on the vibration characteristics of the gate structure, it was necessary to establish the coupled acoustics-structure model, including water in front of the gate skinplate. The detailed acoustics-structure model of the spatial radial gate is depicted in Figure 3. 

Specifically, the analysis process of this coupled acoustics-structure method was as follows: (1) First, we established the spatial radial gate structure, containing strut arms, a trunnion pin, hydraulic actuating rods, girders, and the skinplate. (2) Subsequently, we established the fluid domain based on the water volume in front of the gate skinplate and the meshing fluid elements. It should be noted that the common nodes should be required in the intersection between the fluid and the gate structure. (3) Further, the equations of motion of the fluid and the gate structure were confirmed, respectively. (4) Finally, the fluid-structure coupling interface at the intersection between the fluid domain and the gate structure, labeled red in Figure 3, were established, through which the nodal data were transferred. Accordingly, the coupling calculation between the fluid and the gate structure was realized. 

In engineering practice, the length of the fluid domain in front of the gate skinplate is infinite; in this study, this length was taken as L. Moreover, the influence of the fluid domain length on the dynamic characteristics of the spatial gate structure, considering FSI, was further analyzed.
section_name: Coupled Acoustics-Structure Method

section_content: In this study, a spatial radial gate structure in engineering practice was considered. The geometric and material parameters adopted for the gate are provided in Table 1. The commercial finite element analysis software ANSYS 2022 was used in the FSI modal analysis; specifically, the ANSYS Parametric Design Language (APDL) was adopted in the added-mass method and ANSYS Workbench was used in the coupled acoustics-structure method. MATLAB software R2023a was employed for calculations and data processing. 

Based on ANSYS software, the finite element model of the skinplate, strut arms, and girders in the radial gate was established using the SHELL181 element, a four-node structural shell element, with six degrees of freedom at each node, translations in the x, y, and z directions, and rotations about the x, y, and z axes. Meanwhile, the hydraulic actuating rod and trunnion pin in the radial gate were established using the SOLID185 element, a three-dimensional, eight-node structural solid element, with three degrees of freedom at each node and translations in the nodal x, y, and z directions. Moreover, the fluid domain, as shown in Figure 3, was meshed by the FLUID30 element, a three-dimensional, eightnode acoustic fluid element used for modeling the interface in the fluid-structure interac- In engineering practice, the length of the fluid domain in front of the gate skinplate is infinite; in this study, this length was taken as L. Moreover, the influence of the fluid domain length on the dynamic characteristics of the spatial gate structure, considering FSI, was further analyzed.
section_name: Spatial Radial Gate Model

section_content: In this study, a spatial radial gate structure in engineering practice was considered. The geometric and material parameters adopted for the gate are provided in Table 1. The commercial finite element analysis software ANSYS 2022 was used in the FSI modal analysis; specifically, the ANSYS Parametric Design Language (APDL) was adopted in the added-mass method and ANSYS Workbench was used in the coupled acoustics-structure method. MATLAB software R2023a was employed for calculations and data processing. Based on ANSYS software, the finite element model of the skinplate, strut arms, and girders in the radial gate was established using the SHELL181 element, a four-node structural shell element, with six degrees of freedom at each node, translations in the x, y, and z directions, and rotations about the x, y, and z axes. Meanwhile, the hydraulic actuating rod and trunnion pin in the radial gate were established using the SOLID185 element, a three-dimensional, eight-node structural solid element, with three degrees of freedom at each node and translations in the nodal x, y, and z directions. Moreover, the fluid domain, as shown in Figure 3, was meshed by the FLUID30 element, a three-dimensional, eight-node acoustic fluid element used for modeling the interface in the fluid-structure interaction problem. In the numerical simulation analysis, for ensuring the accuracy of this FSI calculation, a mesh refinement study was carried out. In the study, the length of per FLUID30 element in the fluid domain was taken as 200 mm; when the length of the fluid domain L = 30 m, the fluid domain was discretized into nearly 3. 0 million FLUID elements. For the gate structure, the length of SOLID elements in the gate was taken to be 200 mm as well, and the length of SHELL181 elements in the gate structure was taken as 50 mm. The total number of radial gate structural elements, including SHELL and SOLID elements, was close to 1. 5 million. 

In addition, the discrete model of the spatial radial gate in the space coordinate system with x, y, and z directions is shown in Figure 3. The x direction was along the streamwise direction in water discharge, the y direction was the connection between two trunnion pins, and the y direction was vertically upward and opposite to gravity. During the waterdischarging process, the trunnion pin, hydraulic actuating road, and lateral stop water position at both sides of the skinplate were constrained. Specifically, the trunnion pin only had a rotational degree around the y axis, and all displacement degrees of freedom were restricted. Besides, for the lateral stop water position at both sides of the skinplate, the displacement degrees of freedom in the y direction were constrained. The ends of the hydraulic actuating rods, connected in the dam structure, were restricted in both displacement and rotation degrees of freedom. These constraints are presented visually in Figure 3.
section_name: Spatial Radial Gate Model

section_content: The vibration patterns and frequencies of the gate structure are the critical issues for analyzing the dynamic stability of the radial gate's water-discharging process. Compared to the limitations in the hydro-elastic modeling experiment method, such as the experimental period's long duration, high cost, and unavoidable operating errors, the individual numerical simulation of the radial gate submerged in flow water could efficiently obtain the vibration characteristics of the gate structure in various operating cases. Hence, the dynamic characteristics of the spatial radial gate, considering FSI, were thoroughly investigated using the two proposed FSI modal analysis methods. 

For the sake of organization, this section is presented as three subsections. First, the fundamental vibration patterns and frequencies of the spatial radial gate, using the coupled acoustics-structure method and added-mass method, are studied. Furthermore, the rational value of the added-mass discount factor, ξ, is presented. Second, the influence of FSI on the dynamic characteristics of the spatial radial gate is explored through comparing high-order vibration frequencies and modes of the gate structure in water and air, respectively. Third, the effects of the spatial structural characteristics on the vibration patterns of the radial gate are investigated. In addition, three typical dynamic-instability vibration patterns of radial gates are presented.
section_name: Results and Discussions

section_content: To verify the efficiency and accuracy of the two proposed FSI modal analysis methods, the dynamic characteristics of the spatial radial gate in the fluid domain with different lengths, L, were studied using the coupled acoustics-structure method. Furthermore, the vibration patterns and frequencies of the spatial radial gate were studied using the addedmass method with different discount factors, ξ. In addition, the rational discount factor, ξ, was provided by comparing the analysis results from the coupled acoustics-structure method and the added-mass method.
section_name: Validation of FSI Modal Analysis Methods

section_content: Figure 4 depicts the fundamental vibration pattern and corresponding frequency of the spatial radial gate, with the length of the fluid domain, L, varying from 6 m to 36 m and the length ratio of the fluid domain, L/h, varying from 0. 5 to 3. 0. It can be seen that the fundamental vibration frequency of the spatial radial gate decreased from 7. 3 Hz to 5. 66 Hz when increasing the fluid length ratio, L/h, from 0. 5 to 1. 0. On the contrary, when increasing L/h from 1. 0 to 3. 0, the fundamental vibration frequency initially increased, from 5. 66 Hz to 6. 25 Hz, until it reached a smooth horizontal trend. There was no difference in the fundamental frequency when changing L/h from 2. 5 m to 3. 0; hence, the rational fluid length ratio L/h = 2. 5. from 5. 66 Hz to 6. 25 Hz, until it reached a smooth horizontal trend. There was no difference in the fundamental frequency when changing L/h from 2. 5 m to 3. 0; hence, the rational fluid length ratio L/h = 2. 5. 

Additionally, Table 2 shows the comparison of high-order vibration frequencies of the spatial radial gate, with different length ratios, L/h, of the fluid domain from 0. 5 to 3. 0. It was found that the difference in fundamental frequencies was small (see Table 2 for ∆). When increasing the length of the fluid domain in front of the gate skinplate, the higherorder vibration frequencies gradually converged to the same trend. Hence, a rational value of the fluid domain length existed in analyzing the dynamic characteristics of the spatial radial gate using the coupled acoustics-structure method. Further, from the presented fundamental vibration patterns of the spatial radial gate in Figure 4, it was seen that the fundamental vibration pattern was bending deformation of the hydraulic actuating rods, with the length ratio of the fluid domain L/h = 0. 5, which is the local vibration of components in the radial gate. On the contrary, when increasing the length of the fluid domain in front of the gate skinplate, the whole-gate vibration de- Additionally, Table 2 shows the comparison of high-order vibration frequencies of the spatial radial gate, with different length ratios, L/h, of the fluid domain from 0. 5 to 3. 0. It was found that the difference in fundamental frequencies was small (see Table 2 for ∆). When increasing the length of the fluid domain in front of the gate skinplate, the higher-order vibration frequencies gradually converged to the same trend. Hence, a rational value of the fluid domain length existed in analyzing the dynamic characteristics of the spatial radial gate using the coupled acoustics-structure method. Further, from the presented fundamental vibration patterns of the spatial radial gate in Figure 4, it was seen that the fundamental vibration pattern was bending deformation of the hydraulic actuating rods, with the length ratio of the fluid domain L/h = 0. 5, which is the local vibration of components in the radial gate. On the contrary, when increasing the length of the fluid domain in front of the gate skinplate, the whole-gate vibration deformation occurred in the fundamental vibration pattern, with the whole-gate rotational vibration around the trunnion pin in the downstream direction. Hence, the fundamental vibration pattern of the radial gate was changed from local vibration to whole-gate vibration when considering the fluid domain ahead of the gate skinplate in dynamic characteristics analysis of the radial gate. The corresponding vibration frequency decreased as well. Notably, when the length ratio of the fluid domain of L/h increased from 1. 5 to 3. 0, the vibration frequencies for each order remained unchanged (see Table 2 ), indicating that the rational fluid domain length existed here. In order to improve the calculation efficiency in the coupled acoustics-structure modal analysis, the rational value of the fluid domain's length ratio, L/h, can be taken as 2. 5.
section_name: Coupled Acoustics-Structure Method

section_content: The added mass attached in the gate skinplate, from the added-mass method, can be yielded from the Westergaard formula [42]. Accordingly, the accurate added mass can be altered by changing the value of the discount factor, ξ, which in turn affects the vibration characteristics of the gate structure. However, the rational value still remains unclear in the current specification [41]. In this subsection, by using the added-mass method with different discount factors, ξ, the vibration characteristics of the radial gate are investigated. 

Figure 5 depicts the natural fundamental vibration frequency and corresponding vibration pattern of the radial gate, with the discount factor, ξ, varying from 0. 0~1. 0 (as shown in Table 1 ). It can be observed that as the discount factor, ξ, increased, the fundamental vibration frequency gradually decreased from 7. 03 Hz to 3. 12 Hz. Besides, from the corresponding vibration mode, it was seen that when the added mass attached in the skinplate was smaller, the first-order vibration pattern could be described as the bending deformation of the hydraulic actuating rods, which is a kind of local structural vibration. Subsequently, with the decreasing discount factor, ξ, the hydrodynamic added mass in the gate structure increased. The fundamental vibration mode of the radial gate was the rotational vibration of the whole gate around the trunnion pin, and the skinplate showed bending deformation along the streamwise direction as well, which is a whole-gate vibration deformation. Additionally, Table 3 provides high-order vibration frequencies of the radial gate, with the added-mass discount factor, ξ, ranging from 0. 0 to 1. 0. It can be seen that each high-order vibration frequency also decreased as the discount factor, ξ, increased. Specifically, when ξ increased from 0. 1 to 1. 0, the second-order vibration frequency decreased from 9. 32 Hz to 5. 73 Hz, the third-order vibration frequency reduced from 9. 8 Hz to 6. 75 Hz, and the fourth-order vibration frequency reduced from 15. 38 Hz to 6. 94 Hz, which is also depicted in Figure 6 intuitively. Furthermore, from the variation of these high-order vibration patterns and frequencies with ξ taken as 0. 1, 0. 5, 0. 75, and 1. 0, respectively, it was observed that when ξ was adopted as 0. 1, the high-order vibration mode was mainly the bending deformation of the strut arms. Conversely, when increasing the added mass, the high-order vibration patterns changed, as bending deformation occurred in the streamwise direction of the skinplate (for instance, ξ = 1. 0). Essentially, the value of the discount factor, ξ, determined the magnitude of added mass applied in the numerical model of the gate structure, and by increasing the value of ξ, the attached added mass increased as well. Indeed, owing to the large added mass attached in the skinplate, it would induce streamwise vibration deformation of the skinplate. Clearly, it was seen that the value of the added-mass discount factor, ξ, determined the calculated vibration characteristics of the gate structure; however, it should be pointed out that according to the current specification, the rational value of ξ for the gate structure remains unclear. 

Furthermore, we explored the reasonable value of the added-mass discount factor, ξ, through comparison with the accurate result yielded by the coupled acoustics-structure method demonstrated in Section 3. 1. 1. The fundamental vibration frequency of the radial gate submerged in water, in this case, was 6. 25 Hz (presented in Figure 4 ). As shown in Figure 5, comparing the fundamental vibration mode and frequency obtained from the Additionally, Table 3 provides high-order vibration frequencies of the radial gate, with the added-mass discount factor, ξ, ranging from 0. 0 to 1. 0. It can be seen that each highorder vibration frequency also decreased as the discount factor, ξ, increased. Specifically, when ξ increased from 0. 1 to 1. 0, the second-order vibration frequency decreased from 9. 32 Hz to 5. 73 Hz, the third-order vibration frequency reduced from 9. 8 Hz to 6. 75 Hz, and the fourth-order vibration frequency reduced from 15. 38 Hz to 6. 94 Hz, which is also depicted in Figure 6 intuitively. Furthermore, from the variation of these high-order vibration patterns and frequencies with ξ taken as 0. 1, 0. 5, 0. 75, and 1. 0, respectively, it was observed that when ξ was adopted as 0. 1, the high-order vibration mode was mainly the bending deformation of the strut arms. Conversely, when increasing the added mass, the high-order vibration patterns changed, as bending deformation occurred in the streamwise direction of the skinplate (for instance, ξ = 1. 0). Essentially, the value of the discount factor, ξ, determined the magnitude of added mass applied in the numerical model of the gate structure, and by increasing the value of ξ, the attached added mass increased as well. Indeed, owing to the large added mass attached in the skinplate, it would induce streamwise vibration deformation of the skinplate. Clearly, it was seen that the value of the added-mass discount factor, ξ, determined the calculated vibration characteristics of the gate structure; however, it should be pointed out that according to the current specification, the rational value of ξ for the gate structure remains unclear. 

Furthermore, we explored the reasonable value of the added-mass discount factor, ξ, through comparison with the accurate result yielded by the coupled acoustics-structure method demonstrated in Section 3. 1. 1. The fundamental vibration frequency of the radial gate submerged in water, in this case, was 6. 25 Hz (presented in Figure 4 ). As shown in Figure 5, comparing the fundamental vibration mode and frequency obtained from the coupled acoustics-structure method and those from the added-mass method, evidently, when the discount factor, ξ, was equal to 0. 15, a good match for vibration modes and frequencies between the two methods was observed. Therefore, the added-mass discount factor, ξ, in the specification for yielding the added mass of the radial gate can be taken as 0. 15, in this case. coupled acoustics-structure method and those from the added-mass method, evidently, when the discount factor, ξ, was equal to 0. 15, a good match for vibration modes and frequencies between the two methods was observed. Therefore, the added-mass discount factor, ξ, in the specification for yielding the added mass of the radial gate can be taken as 0. 15, in this case.
section_name: Added-Mass Method

section_content: Subsequently, the influence of FSI on the vibration patterns and frequencies of the spatial radial gate was analyzed by comparing the dynamic characteristics of the gate structure in air and water. Moreover, for the gate in air, the pre-stress modal analysis was conducted, considering the structural deformation caused by hydrostatic pressure in front of the skinplate. The coupled acoustics-structure method was adopted to analyze the radial gate submerged in water.
section_name: Influences of FSI on the Vibration Patterns and Frequencies of the Spatial Radial Gate

section_content: Subsequently, the influence of FSI on the vibration patterns and frequencies of the spatial radial gate was analyzed by comparing the dynamic characteristics of the gate structure in air and water. Moreover, for the gate in air, the pre-stress modal analysis was conducted, considering the structural deformation caused by hydrostatic pressure in front of the skinplate. The coupled acoustics-structure method was adopted to analyze the radial gate submerged in water. 

Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. 

Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. 

Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. 

Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. 

J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 13 of 21

Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. 

J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 13 of 21

Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. 

J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 13 of 21

Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. 

Table 4 shows the comparison of the first-five order vibration frequencies and vibration modes of the radial gate in air and water. Specifically, it can be seen that the fundamental vibration frequency of the gate structure was 9. 15 Hz, and the corresponding vibration mode presented bending deformation of the strut arms and hydraulic actuating rods. Conversely, for this gate submerged in water, the first-order natural frequency was 6. 25 Hz, an almost 31. 67% reduction compared with the gate in air (see ∆ in Table 4 ). The vibration pattern presented as rotational vibration of the whole gate around the trunnion pins, with skinplate bending deformation streamwise, which differed significantly from the mode of the gate in air. Furthermore, for the second-and third-order modes, the vibration patterns of the gate in air showed strut arms' bending deformation orthogonal to streamwise, corresponding to natural frequencies of 9. 36 Hz and 10. 18 Hz, respectively. Similarly, the strut arms' bending vibration, along the direction of the line connected to the trunnion pins, also occurred in the second-and third-order modes of the radial gate submerged in water, corresponding to vibration frequencies of 9. 27 Hz and 9. 57 Hz, respectively. It can be seen that the differences in these natural frequencies of the gate in air and water, performing strut arms' bending vibration, were small (see ∆ in Table 4 ). Hence, fluid-structure coupling had a light effect on strut arms' bending vibration orthogonal to streamwise when the radial gate discharged water. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise. In addition, the fourth-order vibration of the gate in air performed strut arms' torsional vibration deformation, and the corresponding natural frequency was 16. 35 Hz. In contrast, the gate structure in water performed counterclockwise rotational vibration around the trunnion pins, with a vibration frequency of 28. 59 Hz. On this basis, it can be concluded that for the radial gate discharging water in partial gate opening, the FSI effect could induce rotational vibration of the whole-gate structure, reducing the natural vibration frequency of the gate structure, in turn. 

Therefore, FSI affects the vibration characteristics of the radial gate, but owing to many components in the spatial radial gate, the magnitude of this effect depends on the vibration patterns of the gate structure. During the gate's water-discharging process, when the radial gate displayed strut arms' bending vibration, the fluid-structure coupling effect was slight. However, by inducing hydrodynamic pressure in front of the skinplate, the FSI could cause whole-gate rotational vibration with skinplate bending deformation streamwise.
section_name: Influences of FSI on the Vibration Patterns and Frequencies of the Spatial Radial Gate

section_content: The radial gate is a spatial frame structure, composed of a skinplate, horizontal girders, vertical girders, hydraulic actuating rods, and strut arms. Owing to the spatial structural characteristics, the vibration patterns of the radial gate are complicated. Subsequently, the influences of the spatial structural characteristics on the vibration patterns and frequencies of the spatial radial gate are systematically investigated in this section. In addition, the main vibration modes inducing whole-gate dynamic instability are proposed.
section_name: Influences of Spatial Structural Characteristics on Vibration Patterns and Frequencies of the Radial Gate

section_content: Figure 7 depicts the rotational vibration pattern of the whole radial gate around the trunnion pins. Specifically, it can be seen that the first-and fourth-order modes showed this rotational vibration pattern. The former presented as rotational vibration downstream of the whole-gate structure around the trunnion pins, with a frequency of 6. 25 Hz. Conversely, the latter pattern was described as anticlockwise rotational vibration streamwise of the whole gate, with a frequency of 11. 67 Hz. The maximum magnitude of vibration deformation occurred at the bottom of the gate structure. 

and frequencies of the spatial radial gate are systematically investigated in this section. In addition, the main vibration modes inducing whole-gate dynamic instability are proposed.
section_name: Rotational Vibration of Whole Spatial Radial Gate

section_content: Figure 7 depicts the rotational vibration pattern of the whole radial gate around the trunnion pins. Specifically, it can be seen that the first-and fourth-order modes showed this rotational vibration pattern. The former presented as rotational vibration downstream of the whole-gate structure around the trunnion pins, with a frequency of 6. 25 Hz. Conversely, the latter pattern was described as anticlockwise rotational vibration streamwise of the whole gate, with a frequency of 11. 67 Hz. The maximum magnitude of vibration deformation occurred at the bottom of the gate structure. 

Notably, according to experimental and numerical works reported in the authoritative literature [5, 50], the pulsation frequency of the fluctuation load exerted on the skinplate ranges from 0. 0 to 20. 0 Hz. Here, the vibration frequencies of the rotational vibration pattern of the whole gate, presented in Figure 6, were close to this fluctuation frequency range. This would lead to prominent resonance or parametric resonance phenomena of the gate structure [9, 21], even causing dynamic-instability destruction of the radial gate structure. Notably, according to experimental and numerical works reported in the authoritative literature [5, 50], the pulsation frequency of the fluctuation load exerted on the skinplate ranges from 0. 0 to 20. 0 Hz. Here, the vibration frequencies of the rotational vibration pattern of the whole gate, presented in Figure 6, were close to this fluctuation frequency range. This would lead to prominent resonance or parametric resonance phenomena of the gate structure [9, 21], even causing dynamic-instability destruction of the radial gate structure.
section_name: Rotational Vibration of Whole Spatial Radial Gate

section_content: Figure 8 shows the bending vibration pattern of the spatial radial gate, with double strut arms bending orthogonal to the water-flow direction. The bending vibration deformation of strut arms is the most likely vibration type of the radial gate owing to the influence of the spatial structural characteristics. The strut arms have also become the components that are most likely to be damaged by vibrations. In engineering practice, numerous dynamic failure destructions of radial gates are caused by the vibration instability of the strut arms [5]. From Figure 8, it can be observed that the bending vibration deformation of the double strut arms occurred in the second-and third-order vibration modes. Specifically, the second vibration pattern presented as bending deformation of two strut arms along the same direction, and the corresponding vibration frequency was 9. 27 Hz. The third vibration mode was also bending deformation of two strut arms, with a vibration frequency of 9. 57 Hz. On the contrary, the two strut arms' bending vibration was along the opposite direction. Figure 8 shows the bending vibration pattern of the spatial radial gate, with double strut arms bending orthogonal to the water-flow direction. The bending vibration deformation of strut arms is the most likely vibration type of the radial gate owing to the influence of the spatial structural characteristics. The strut arms have also become the components that are most likely to be damaged by vibrations. In engineering practice, numerous dynamic failure destructions of radial gates are caused by the vibration instability of the strut arms [5]. From Figure 8, it can be observed that the bending vibration deformation of the double strut arms occurred in the second-and third-order vibration modes. Specifically, the second vibration pattern presented as bending deformation of two strut arms along the same direction, and the corresponding vibration frequency was 9. 27 Hz. The third vibration mode was also bending deformation of two strut arms, with a vibration frequency of 9. 57 Hz. On the contrary, the two strut arms' bending vibration was along the opposite direction. In summary, the maximum magnitude of bending deformation in the two vibration modes occurred in the connection rod of the strut arms near the trunnion pins. Hence, this position, with the rod connected to upper and lower struts in a plane, should be the vibration monitoring location of the strut arms. According to the monitoring vibration magnitude and acceleration in this location, the vibration safety of the radial gate can be determined. Additionally, based on the above analysis, it can be seen that the two vibration In summary, the maximum magnitude of bending deformation in the two vibration modes occurred in the connection rod of the strut arms near the trunnion pins. Hence, this position, with the rod connected to upper and lower struts in a plane, should be the vibration monitoring location of the strut arms. According to the monitoring vibration magnitude and acceleration in this location, the vibration safety of the radial gate can be determined. Additionally, based on the above analysis, it can be seen that the two vibration frequencies of the gate structure were 9. 27 Hz and 9. 57 Hz, respectively, which are both in the range of the pulsation frequency of the fluctuation load exerted on the skinplate. The resonance phenomenon of the strut arms may occur, resulting in structural failure destruction of the radial gate.
section_name: Bending Vibration of Strut Arms

section_content: Figure 9 presents the torsional vibration pattern of this spatial frame structure of the radial gate. It can be seen that this vibration mode can be described as bending and torsional vibration deformation of the strut arms and skinplate in the streamwise direction, which is the typical vibration deformation of the whole-gate structure induced by the spatial structural characteristics of the radial gate. The connection between the strut arms and horizontal girders forms a spatial frame structure. Notably, the bending-torsion coupling vibration of this frame structure would significantly induce the whole gate's vibration. It is also the typical vibration pattern most likely to cause vibration failure of the radial gate. In summary, under the effects of the spatial structural characteristics, the three vib tion patterns mentioned above would occur in the spatial radial gate, namely, whole-ga rotational vibration around trunnion pins, strut arms' bending vibration orthogonal streamwise, and coupled bending-torsional vibration of the spatial frame structure. T maximum magnitude of vibration deformation in the fundamental vibration pattern o curred in the bottom position of the gate skinplate, while in the last two vibration patter the largest vibration magnitude of the radial gate occurred in the strut arms. In practi engineering, the bottom of the gate skinplate and strut arms always showed significa vibration deformation. Consequently, owing to the FSI effect, the rotational vibrati streamwise of the radial gate occurred. Further, affected by the spatial frame character tics, the bending-torsion coupling vibration pattern of the whole structure occurred in t radial gate. 

Nonetheless, in the current specification for designing the radial gate, the plane d sign principle and strength limit state principle are the main guidelines [40, 41]. Hence, t spatial frame characteristics of the radial gate usually cannot be considered, and the ca strophic damage of the radial gate induced by structural bending-torsion coupling vib tion is often ignored. Therefore, under the effect of the FSI and spatial frame character Specifically, the fifth-and sixth-order vibration modes of the radial gate both presented this typical vibration deformation. The fifth-order vibration pattern was described as bending-torsion coupling vibration deformation of the frame composite of the skinplate, girders, and strut arms, with a vibration frequency of 16. 11 Hz; meanwhile, torsional deformation occurred in strut arms along the water flow direction. In the sixth-order vibration mode, only the double strut arms presented torsional vibration deformation along the same direction, and the corresponding vibration frequency was 18. 22 Hz. 

Accordingly, the bending-torsion coupling vibration deformation in all vibration patterns of the gate structure was most likely to significantly cause structural damage to the radial gate. Compared to the sluice gate, owing to the effect of the spatial structural characteristics, the bending-torsion coupling vibration problem of the frame structure in the radial gate is critical. Besides, the above vibration frequencies in the fifth-and sixth-order modes were both in the range of the pulsation frequency of the fluctuation load. In engineering practice, this vibration pattern would still occur in the radial gate structure during the water-discharging process. Furthermore, according to the reported research [45], it has been found that in parametric resonance, the dynamic instability regions caused by bending-torsion vibration deformation would significantly enlarge the size and intensity of the dynamic instability regions. Hence, this bending-torsion coupling vibration of the spatial frame structure would cause disastrous damage in the radial gate. In the structural design stage, this vibration pattern should be avoided through structural design and reinforcement. 

In summary, under the effects of the spatial structural characteristics, the three vibration patterns mentioned above would occur in the spatial radial gate, namely, whole-gate rotational vibration around trunnion pins, strut arms' bending vibration orthogonal to streamwise, and coupled bending-torsional vibration of the spatial frame structure. The maximum magnitude of vibration deformation in the fundamental vibration pattern occurred in the bottom position of the gate skinplate, while in the last two vibration patterns, the largest vibration magnitude of the radial gate occurred in the strut arms. In practical engineering, the bottom of the gate skinplate and strut arms always showed significant vibration deformation. Consequently, owing to the FSI effect, the rotational vibration streamwise of the radial gate occurred. Further, affected by the spatial frame characteristics, the bending-torsion coupling vibration pattern of the whole structure occurred in the radial gate. 

Nonetheless, in the current specification for designing the radial gate, the plane design principle and strength limit state principle are the main guidelines [40, 41]. Hence, the spatial frame characteristics of the radial gate usually cannot be considered, and the catastrophic damage of the radial gate induced by structural bending-torsion coupling vibration is often ignored. Therefore, under the effect of the FSI and spatial frame characteristics, an accurate solution of gate vibration characteristics is of significant value for engineering applications.
section_name: Bending-Torsion Coupling Vibration of the Whole Spatial Radial Gate

section_content: In this work, the overall vibration patterns and frequencies of the spatial frame structure of the radial gate, affected by FSI during water-discharging, were systematically investigated. A comparison of two FSI modal analysis approaches-the added-mass method and the coupled acoustics-structure method-was conducted. Accordingly, a thorough investigation about the effects of the FSI and spatial frame characteristics on the vibration characteristics of the spatial radial gate was carried out. Three typical vibration patterns of the radial gate causing structural dynamic instability were proposed. The conclusions of this study are summarized as follows:

(1) The efficiency and accuracy of the coupled acoustics-structure method and addedmass method in analyzing the FSI modal analysis were demonstrated. In the coupled acoustics-structure method, the rational length of the fluid domain in front of the gate skinplate satisfied the length ratio L/h = 2. 5. Limitations for using the added-mass method to analyze FSI modal analysis of the radial gate exist. The value of the addedmass discount factor determined the calculated vibration characteristics of the gate structure, so the reasonable added-mass discount factor in the specification can be taken as 0. 15 empirically. However, the added-mass discount factor is still a puzzling problem in theoretical solutions. In future research, the theoretical solution of the added-mass discount factor in the added-mass method should be explored. (2) Fluid-structure coupling caused by fluctuation in front of the skinplate can induce the whole-gate rotational vibration streamwise of the radial gate, including rotational vibration downstream and anticlockwise rotational vibration streamwise of the whole-gate structure. Meanwhile, the FSI effect significantly reduced the gate's fundamental vibration frequency and high-order vibration frequencies, with a reduction of amplitude by 31%, which could not be found in the modal analysis of the gate in air. 

(3) Under the influence of spatial frame characteristics, three typical vibration patterns of the whole radial gate structure causing structural dynamic instability were proposed, namely, whole-gate rotational vibration around trunnion pins, strut arms' bending vibration orthogonal to streamwise, and coupled bending-torsional vibration of the spatial frame structure. Bending-torsional coupling vibration would induce catastrophic structural damages. Additionally, based on the plane design principle and strength limit state principle in the current specification for designing the radial gate, the spatial frame characteristics of the radial gate usually cannot be considered when analyzing the gate's dynamic characteristics. The catastrophic damage in the radial gate induced by structural bending-torsion coupling vibration is always ignored.
section_name: Conclusions

section_content: Funding: This work has been supported by funding from the Fundamental Research Funds for the Central Universities ( 2023MS073 ), the Scientific Project of China Energy Science and Technology Research Institute ( CD2023Y01 ), the National Natural Science Foundation of China (Grant Number 52209032 ), and the Special Research Fund of Nanjing Hydraulic Research Institute ( Y123001 ). These supports are gratefully acknowledged. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable.

section_content: Data Availability Statement: Data are contained within the article.

section_name: Conflicts of Interest:

section_content: Authors Feng Liu, Min Liu and Ruiji Yi were employed by the company China Energy Science and Technology Research Institute Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
section_name: Conflicts of Interest: