A method for analyzing dynamic response of a floating bridge in transit considering multiple-bridge-section and multiple-cabin water ingress

Abstract

This invention discloses a method for analyzing the dynamic response of a multi-section, multi-compartment, damaged floating bridge under load. The method dynamically updates the liquid level in each compartment, the mass of water entering the compartment, its center of mass, and the change in moment of inertia of the bridge sections caused by water ingress, based on the size and location of the breach. After determining the position and inclination angle of each bridge section in a stable equilibrium state in still water, the method calculates the generalized added mass coefficient, radiation damping coefficient, and wave excitation force of the floating bridge system under water ingress conditions based on potential flow theory. Finally, it uses the multi-body time-domain hydrodynamic analysis equations to analyze the dynamic response of the damaged floating bridge under load. This invention constructs a theoretical framework for the dynamic response analysis of a damaged floating bridge in still water using multi-body dynamics methods, constructs a multi-section, multi-compartment water ingress model using the modified Bernoulli equation, and then uses potential flow theory to conduct time-domain hydrodynamic analysis, achieving efficient dynamic response prediction and improving calculation accuracy and efficiency.

Description

Technical Field

[0001] This invention belongs to the field of floating bridge technology, and in particular relates to a dynamic response analysis method for a fully loaded floating bridge under the condition of multi-section and multi-compartment damage and water ingress. Background Technology

[0002] Floating bridges, as transportation engineering facilities, play an important role in emergency rescue and the construction of remote islands and reefs. Floating bridges are typically composed of multiple bridge sections connected longitudinally by a connecting mechanism.

[0003] In practical use, floating bridges may experience damage and water ingress into multiple sections or compartments within sections due to corrosion, collisions, or explosions. This can lead to changes in the bridge section's attitude and the stress on the connectors, resulting in reduced passage capacity. Therefore, it is necessary to assess the operational safety of floating bridges after damage. Currently, the hydrodynamic or hydroelastic response analysis techniques for intact floating bridges in wave environments are relatively mature. However, research on the dynamic response of floating bridges under water ingress after damage is limited, especially studies that simultaneously consider the effects of wave environments and water ingress into multiple sections and compartments. Relevant research mainly focuses on the dynamic response analysis of multi-floating body systems with liquid tank swaying. The research methods for the dynamic response of multi-floating body systems with liquid tanks mainly include:

[0004] (1) Time-domain hydrodynamic analysis method based on potential flow theory: The potential flow theory is used to consider the flow field inside and outside the bridge section of the floating bridge and the hydrodynamic interference between the bridge sections. This method can only carry out hydrodynamic analysis based on a given water inflow, and cannot determine the water inflow and floating bridge morphology after damage to multiple bridge sections and multiple compartments, and it is difficult to consider the coupling effect between the floating body and the moving load;

[0005] (2) Fully Coupled Simulation Method Based on CFD (Computational Fluid Dynamics): While using computational fluid dynamics to simulate the water inflow process for fluid-structure interaction studies offers high accuracy, the computational cost is enormous, especially when dealing with nonlinear connections and moving loads, requiring in-depth code development and integration. Therefore, this method is difficult to apply for efficient prediction and multi-condition analysis of large-scale floating bridge systems;

[0006] (3) Model test method based on similarity theory: Based on the dominant force, the similarity criterion is determined, and a scaled-down model test is carried out to obtain the actual dynamic response data of the floating bridge after the hull is damaged, and to verify the numerical model. However, this method also has disadvantages such as scale effect, high cost, long cycle, and difficulty in comprehensively and without interference measuring certain local physical quantities.

[0007] In summary, existing methods have significant shortcomings in determining the water inflow into multi-section pontoon bridges and the motion response after a pontoon bridge is breached. There is a lack of a method for analyzing the dynamic response of a pontoon bridge under wave-induced water inflow and with both computational efficiency and high accuracy. Summary of the Invention

[0008] The purpose of this invention is to provide a dynamic response analysis method for a floating bridge under continuous load that takes into account the damage and water ingress of multiple bridge sections and multiple compartments. The method constructs a theoretical framework for the dynamic response analysis of a floating bridge with damaged compartments in still water using multibody dynamics, and constructs a multi-bridge section and multiple compartment water ingress model by combining the modified Bernoulli equation. Then, the method uses potential flow theory to carry out time-domain hydrodynamic analysis to achieve efficient dynamic response prediction.

[0009] To achieve the objective of this invention, a method for analyzing the dynamic response of a continuously loaded floating bridge considering water ingress due to damage to multiple bridge sections and compartments is provided, comprising the following steps:

[0010] Step 1: Based on the parameters of the damaged and flooded floating bridge system, establish a dynamic analysis model of the multi-section and multi-compartment damaged and flooded floating bridge in still water, and determine the water ingress mass of each compartment and the water ingress mass, position, tilt angle, and changes in the rotational inertia of the bridge sections caused by water ingress when the floating bridge system reaches a stable state after water ingress.

[0011] Step 2: Based on the water ingress mass of each compartment and the water ingress mass, position, tilt angle, and change in moment of inertia of each bridge section caused by water ingress, establish a numerical model based on potential flow theory to determine the additional hydrodynamic coefficient and wave excitation force of each bridge section.

[0012] Step 3: Using the time-domain hydrodynamic analysis equations for a multi-section, multi-compartment, flooded floating bridge subjected to water ingress, the changes in bridge section mass and moment of inertia caused by water ingress, the wave loads on each bridge section, the water ingress loads, the hydrostatic restoring force loads, the moving loads, and the mooring loads, the dynamic response results of the flooded floating bridge are obtained.

[0013] The parameters of the breached and flooded floating bridge system include: the dimensions, draft, center of mass position, mass, moment of inertia, additional mass, additional moment of inertia, initial velocity and initial angular velocity of each bridge section, the number and location of the breached bridge sections, the number, location and dimensions of the breached compartments, and the location and dimensions of the breach.

[0014] Step 1 includes the following steps:

[0015] Step 1-1: Based on the breach size and the difference in liquid level inside and outside the compartment, determine the water ingress mass of each compartment by integrating the flow rate over time;

[0016] Steps 1-2: Determine the water ingress mass of the damaged bridge section based on the water ingress mass of each compartment, and determine the centroid coordinates of the water ingress mass in the compartment by combining the location of the damaged compartment, the size of the compartment, the water ingress mass, the shape of the water ingress volume, the bridge section inclination angle and the bridge section draft;

[0017] Steps 1-3: Based on the mass of water entering the compartment and the coordinates of its center of mass, determine the forces and torques caused by the water ingress, as well as the change in the moment of inertia of the bridge section caused by the damage of a single compartment;

[0018] Steps 1-4: Determine the hydrostatic restoring force and moment, as well as the mooring force and moment, based on the bridge section inclination angle and draft.

[0019] Steps 1-5: Considering the combined effects of the inflow load, still water restoring force load, and mooring load on the floating bridge, establish the dynamic analysis equations for a multi-section, multi-compartment floating bridge that has suffered damage and inflow in still water.

[0020] In step 1-1, the calculation is performed by integrating the flow rate over time. Time of the first The bridge section The water ingress mass of each damaged compartment is shown in the following formula:

[0021] ;

[0022] in, The flow coefficient is related to bridge joint motion, tank sloshing, breach geometry, and compartment layout. The density of water; An operator for determining the sign of the function value within parentheses; and The first The bridge section The height of the external and internal liquid levels in each damaged compartment; It is the acceleration due to gravity; This represents the area of ​​the breach.

[0023] The steps 1-2 determine the amount of water entering each compartment based on the water quality of each compartment. The water ingress quality of the damaged bridge section in each compartment: .

[0024] Steps 1-3 are specifically as follows:

[0025] The weight increase after water enters the bridge section is The resulting heeling moment and pitching moment are respectively and ;

[0026] in, and These are the x and y coordinates, respectively, in a coordinate system where the center of mass of the water ingress into the compartment is located at the center of the float on the surface of the intact bridge water-stop line at the origin.

[0027] No. The first bridge section The change in the moment of inertia matrix of a single bridge segment caused by water ingress into a damaged compartment can be calculated using the mass of the water entering the compartment. The coordinates of the centroid and its center of mass in the connected coordinate system with the origin located at the center of the waterline of the intact bridge section are calculated as follows:

[0028] ;

[0029] in, , and These are the abscissa, ordinate, and vertical coordinates of the combined coordinate system in which the center of mass of the water ingress in the compartment is located at the center of the float on the surface of the intact bridge water-stop line, respectively.

[0030] For those The first compartment to be flooded The change in the moment of inertia of each bridge section caused by water ingress is obtained by the superposition of the effects of each damaged compartment. .

[0031] The dynamic analysis equations for the multi-section, multi-compartment floating bridge damaged and flooded in still water in steps 1-5 are shown in the following formulas:

[0032] ;

[0033] in, , q represents the system's degree of freedom number; and These are the generalized velocities corresponding to the p-th and q-th degrees of freedom of the system, respectively, where the generalized velocities of the system are the relative rotational and translational velocities between the bridge sections; and The first The m-th component of the partial velocity array of the bridge section relative to the l-th and p-th degrees of freedom of the system; and The first The m-th component of the angular velocity array of the bridge section relative to the l-th and p-th degrees of freedom of the system; For the first The r-th component of the deflection angular velocity array of the p-th degree of freedom of the bridge section relative to the system; For the first The nth component of the angular velocity array of the bridge section relative to the qth degree of freedom of the system; and The first The m-th component of the first-order derivative array of the deflection velocity and deflection angular velocity of the bridge section relative to the p-th degree of freedom of the system; and The first The mass and moment of inertia matrix of the nth bridge segment Line number Column elements; and The first Elements in the matrix of the water inflow mass of each bridge section and the resulting change in the moment of inertia of the bridge section; and The first The additional mass and additional inertia matrix elements of each bridge section; Let the generalized rate be the value corresponding to the p-th degree of freedom of the system. The first derivative; and They represent the first The increased weight of each bridge section after water ingress and the resulting torque component; and The first The hydrostatic restoring force and moment components of each bridge section; and The first The mooring force and mooring moment components of each bridge section; These are the elements of the permutation matrix.

[0034] Step 2 specifically involves: based on the water ingress mass of each compartment and the water ingress mass, position, and tilt angle of each bridge section determined in Step 1, solving the boundary values ​​of the Laplace equation for different wave frequencies based on potential flow theory, and obtaining the additional hydrodynamic coefficients and wave excitation forces experienced by each bridge section in the multi-bridge-section, multi-compartment damaged and water-ingressed floating bridge at different wave frequencies. The additional hydrodynamic coefficients include the generalized additional mass coefficient and the radiation damping coefficient.

[0035] The generalized added mass coefficient and radiation damping coefficient are obtained by having a bridge section move with unit amplitude at wave frequency in each degree of freedom, determining the generated flow field potential, integrating the fluid force of the motion mode on all degrees of freedom, and combining the relevant results of each bridge section to obtain the generalized added mass coefficient and radiation damping coefficient at the wave frequency.

[0036] The wave excitation force is obtained by fixing all bridge sections, determining the velocity potential of the superimposed incident and reflected waves at the specified wave frequency, and integrating the results to obtain the wave excitation force at the specified wave frequency exerted by the wave on each bridge section. .

[0037] Step 3 specifically involves:

[0038] Step 3-1: Using the Kramer-Kronig relation, determine the generalized additional mass coefficient matrix of each degree of freedom of each bridge section within the damaged pontoon bridge as the wave frequency approaches infinity, based on the matrix composed of the additional hydrodynamic coefficients. ;

[0039] Specifically, it is shown in the following formula.

[0040] ;

[0041] in, When the wave frequency is The generalized additional mass matrix for each degree of freedom of each bridge section within the damaged pontoon bridge, wherein the generalized additional mass matrix includes the additional mass and the additional moment of inertia matrix. 0- A certain frequency in For all frequencies, and The generalized additional mass coefficient and radiation damping coefficient and The matrix formed;

[0042] Step 3-2: Radiation damping coefficient matrix for different wave frequencies Integrate to determine the velocity delay function matrix. ;

[0043] Step 3-3: Calculate the vertical force of all moving loads acting on the bridge section at time t. lateral moment and longitudinal moment ;

[0044] in, The number of moving loads acting on the k-th bridge section; For the action on the k-th bridge segment The mass of a moving load; and The first The lateral and longitudinal eccentric displacements of the position of the moving load relative to the center of gravity of the intact bridge section;

[0045] Steps 3-4: Consider the combined effects of wave excitation load, moving load, still water restoring load, mooring load, etc., and substitute the water ingress mass of the bridge section determined in Step 1 and the resulting change in the moment of inertia of the bridge section into the time-domain hydrodynamic analysis equation of the multi-bridge section and multi-compartment damaged water-filled floating bridge to determine the motion response of each bridge section of the floating bridge system and the connector load at time t.

[0046] The specific time-domain hydrodynamic analysis equations for the multi-section, multi-compartment damaged, water-intake floating bridge in steps 3-4 are shown below:

[0047] ;

[0048] in, The generalized mass matrix represents the state of the multi-bridge joints in good condition. The change in the generalized mass matrix caused by water ingress into the bridge section; The damping matrix excluding the effects of linear radiation damping; The total stiffness matrix of hydrostatic restoring force includes mooring constraints and considers the effects of liquid tanks; The array of wave excitation forces acting on the multi-bridge segment at time t; The array of moving loads experienced by the multi-bridge section at time t; , and These represent the acceleration, velocity, and displacement response arrays of the multi-bridge section, respectively. The velocity delay function matrix is ​​defined as follows: the generalized mass includes mass and moment of inertia; the generalized additional mass includes additional mass and additional moment of inertia.

[0049] Compared with the prior art, the significant progress of the present invention is that the present invention can consider the dynamic response results of the bridge sections and connector loads when a multi-section, multi-compartment damaged floating bridge is subjected to water inrush in waves. The calculation accuracy is high and the calculation efficiency is higher than the dynamic response analysis method of damaged floating bridges based on computational fluid dynamics. At the same time, it overcomes the disadvantages of model test methods, such as high cost, long implementation cycle and difficulty in conducting multi-parameter influence studies.

[0050] To more clearly illustrate the functional characteristics and structural parameters of the present invention, further explanation is provided below in conjunction with the accompanying drawings and specific embodiments. Attached Figure Description

[0051] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0052] Figure 1 This is a flowchart illustrating the overall process of analyzing the dynamic response of a multi-section, multi-compartment, damaged, flooded floating bridge according to an embodiment of the present invention.

[0053] Figure 2 This is a flowchart of the dynamic response analysis method for a multi-section, multi-compartment floating bridge damaged and flooded in still water according to an embodiment of the present invention;

[0054] Figure 3 This is a top view schematic diagram of the multi-section, multi-compartment floating bridge structure that has been damaged and flooded according to the present invention;

[0055] Figure 4 This is a side view schematic diagram of the multi-section, multi-compartment floating bridge structure that has been damaged and flooded according to the present invention;

[0056] Figure 5 This is a top view of a vehicle on one of the bridge sections of the multi-section, multi-compartment damaged and flooded floating bridge of the present invention;

[0057] Figure 6This is a diagram showing the vertical displacement of the middle damaged bridge section of the floating bridge obtained by using the dynamic response analysis method of the multi-section, multi-compartment damaged and flooded floating bridge of this application when a vehicle is traveling on a floating bridge with two consecutive damaged bridge sections and flooded under the action of regular waves of different wave frequencies in an example of this invention.

[0058] Figure 7 This is a diagram showing the vertical displacement of the middle damaged bridge section of the floating bridge obtained by using the dynamic response analysis method for multi-section, multi-compartment, flooded floating bridges of this application when a vehicle is traveling on a floating bridge with a different number of damaged bridge sections under the action of regular waves in an example of this invention.

[0059] The attached diagram is labeled as follows: 1-compartment, 2-bridge section, 3-connector, 4-cable, 5-anchor, 6-vehicle. Detailed Implementation

[0060] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0061] This invention provides a dynamic response analysis method for a continuously loaded floating bridge that takes into account water ingress due to damage to multiple bridge sections and compartments, combined with... Figure 1 and Figure 2 This includes the following steps:

[0062] Step 1: Based on the parameters of the damaged and flooded floating bridge system, establish a dynamic analysis model of the multi-section and multi-compartment damaged and flooded floating bridge in still water, and determine the water ingress mass of each compartment and the water ingress mass, position, tilt angle, and changes in the rotational inertia of the bridge sections caused by water ingress when the floating bridge system reaches a stable state after water ingress.

[0063] Step 2: Based on the water ingress mass of each compartment and the water ingress mass, position, tilt angle, and change in moment of inertia of each bridge section caused by water ingress, establish a numerical model based on potential flow theory to solve the multi-buoy radiation problem and diffraction problem, and determine the additional hydrodynamic coefficient and wave excitation force of each bridge section.

[0064] Step 3: Substitute the changes in bridge section mass and moment of inertia caused by water ingress, the wave load on each bridge section, the water ingress load, the hydrostatic restoring force load, the moving load, and the mooring load into the time-domain hydrodynamic analysis equation for a multi-section, multi-compartment damaged, water-ingressed floating bridge in waves, and obtain the dynamic response results of the damaged, water-ingressed floating bridge at time t.

[0065] The parameters of the breached and flooded floating bridge system include: the dimensions, draft, center of mass position, mass, moment of inertia, additional mass, additional moment of inertia, initial velocity and initial angular velocity of each bridge section, the number and location of the breached bridge sections, the number, location and dimensions of the breached compartments, and the location and dimensions of the breach.

[0066] Step 1 includes the following steps:

[0067] Step 1-1: Based on the breach size and the difference in liquid level inside and outside the compartment, determine the water ingress mass of each compartment by integrating the flow rate over time;

[0068] Steps 1-2: Determine the water ingress mass of the damaged bridge section based on the water ingress mass of each compartment, and determine the centroid coordinates of the water ingress mass in the compartment at time t by combining the location of the damaged compartment, the size of the compartment, the water ingress mass, the shape of the water ingress volume, the bridge section inclination angle and the bridge section draft.

[0069] Steps 1-3: Based on the mass of water entering the compartment and the coordinates of its center of mass, determine the forces and torques caused by the water ingress, as well as the change in the moment of inertia of the bridge section caused by the damage of a single compartment;

[0070] Steps 1-4: Determine the hydrostatic restoring force and moment, as well as the mooring force and moment, based on the bridge section inclination angle and draft.

[0071] Steps 1-5: Considering the combined effects of the inflow load, still water restoring force load, and mooring load on the floating bridge, establish the dynamic analysis equations for a multi-section, multi-compartment floating bridge that has suffered damage and inflow in still water.

[0072] In step 1-1, the calculation is performed by integrating the flow rate over time. Time of the first The bridge section The water ingress mass of each damaged compartment is shown in the following formula:

[0073] ;

[0074] in, The flow coefficient is related to bridge joint motion, tank sloshing, breach geometry, and compartment layout. The density of water; An operator for determining the sign of the function value within parentheses; and The first The bridge section The height of the external and internal liquid levels in each damaged compartment; It is the acceleration due to gravity; This represents the area of ​​the breach.

[0075] The steps 1-2 determine the amount of water entering each compartment based on the water quality of each compartment. The water ingress quality of the damaged bridge section in each compartment: .

[0076] Steps 1-3 are specifically as follows:

[0077] The weight increase after water enters the bridge section is The resulting heeling moment and pitching moment are respectively and ;

[0078] in, and These are the x and y coordinates, respectively, in a coordinate system where the center of mass of the water ingress into the compartment is located at the center of the float on the surface of the intact bridge water-stop line at the origin.

[0079] No. The first bridge section The change in the moment of inertia matrix of a single bridge segment caused by water ingress into a damaged compartment can be calculated using the mass of the water entering the compartment. The coordinates of the centroid and its center of mass in the connected coordinate system with the origin located at the center of the waterline of the intact bridge section are calculated as follows:

[0080] ;

[0081] in, , and These are the abscissa, ordinate, and vertical coordinates of the combined coordinate system in which the center of mass of the water ingress in the compartment is located at the center of the float on the surface of the intact bridge water-stop line, respectively.

[0082] For those The first compartment to be flooded The change in the moment of inertia of each bridge section caused by water ingress is obtained by the superposition of the effects of each damaged compartment. .

[0083] The dynamic analysis equations for the multi-section, multi-compartment floating bridge damaged and flooded in still water in steps 1-5 are shown in the following formulas:

[0084] ;

[0085] in, , q represents the system's degree of freedom number; and These are the generalized velocities corresponding to the p-th and q-th degrees of freedom of the system, respectively, where the generalized velocities of the system are the relative rotational and translational velocities between the bridge sections; and The first The m-th component of the partial velocity array of the bridge section relative to the l-th and p-th degrees of freedom of the system; and The first The m-th component of the angular velocity array of the bridge section relative to the l-th and p-th degrees of freedom of the system; For the first The r-th component of the deflection angular velocity array of the p-th degree of freedom of the bridge section relative to the system; For the first The nth component of the angular velocity array of the bridge section relative to the qth degree of freedom of the system; and The first The m-th component of the first-order derivative array of the deflection velocity and deflection angular velocity of the bridge section relative to the p-th degree of freedom of the system; and The first The mass and moment of inertia matrix of the nth bridge segment Line number Column elements; and The first Elements in the matrix of the water inflow mass of each bridge section and the resulting change in the moment of inertia of the bridge section; and The first The additional mass and additional inertia matrix elements of each bridge section; Let the generalized rate be the value corresponding to the p-th degree of freedom of the system. The first derivative; and They represent the first The increased weight of each bridge section after water ingress and the resulting torque component; and The first The hydrostatic restoring force and moment components of each bridge section; and The first The mooring force and mooring moment components of each bridge section; These are the elements of the permutation matrix.

[0086] The dynamic analysis equations for a multi-section, multi-compartment floating bridge undergoing water ingress in still water are first-order ordinary differential equations concerning the generalized velocity. A fourth-order Runge-Kutta method is used for time-domain numerical solution to determine the generalized velocity of the system at time t. This allows for the determination of the position and tilt angle of each bridge section at time t, and the updating of information such as the internal and external liquid levels. Calculations are made regarding the mass of water entering the compartment at any given time, the change in rotational inertia caused by the water ingress, and other factors to determine... The dynamic response of the time-step system. As time progresses, the multi-section, multi-compartment floating bridge system eventually reaches a stable equilibrium state in still water. The inclination angle, position, and change in moment of inertia of each section are obtained when the multi-section, multi-compartment floating bridge reaches a stable equilibrium state in still water.

[0087] Step 2 specifically involves: based on the water inflow mass of each compartment and the water inflow mass, location, and tilt angle of each bridge section determined in Step 1, solving the boundary values ​​of the Laplace equation for different wave frequencies using potential flow theory, thereby obtaining the additional hydrodynamic coefficients and wave excitation forces experienced by each bridge section of the multi-section, multi-compartment damaged floating bridge at different wave frequencies. The additional hydrodynamic coefficients include the generalized additional mass coefficient. and radiation damping coefficient ;

[0088] Wherein, the generalized additional mass coefficient and radiation damping coefficient The solution can be considered as a radiation problem, by letting a certain bridge section be distributed at a certain frequency in each degree of freedom. By performing unit amplitude motion, the resulting flow field potential is calculated, and then integrated to obtain the fluid forces exerted by this motion mode on all degrees of freedom. Finally, by combining the relevant results from each bridge section, the wave frequency is obtained. Generalized additional mass coefficient and radiation damping coefficient ;

[0089] The solution to the wave excitation force can be considered as a diffraction problem. By fixing all bridge sections, the wave frequency can be calculated. The velocity potential of the superimposed incident and reflected waves, when integrated, yields the force exerted by the wave on each bridge segment, which is the wave frequency. Wave excitation force .

[0090] Step 3 specifically involves:

[0091] Step 3-1: The additional hydrodynamic coefficient and The matrix formed and Using the Kramer-Kronig relation, the generalized additional mass coefficient matrix for each degree of freedom of each bridge section within the breached pontoon bridge is determined as the wave frequency approaches infinity. ;

[0092] Specifically, it is shown in the following formula.

[0093] ;

[0094] in, When the wave frequency is The generalized added mass (including added mass and added moment of inertia) matrix of each degree of freedom of each bridge section within the damaged pontoon bridge. 0- A certain frequency in For all frequencies;

[0095] Step 3-2: Radiation damping coefficient matrix for different wave frequencies Integrate to determine the velocity delay function matrix. ;

[0096] Step 3-3: Calculate the vertical force of all moving loads acting on the bridge section at time t. lateral moment and longitudinal moment ;

[0097] in, The number of moving loads acting on the k-th bridge section; For the action on the k-th bridge segment The mass of a moving load; and The first The lateral and longitudinal eccentric displacements of the position of the moving load relative to the center of gravity of the intact bridge section;

[0098] Steps 3-4: Consider the combined effects of wave excitation load, moving load, still water restoring load, mooring load, etc., and substitute the water ingress mass of the bridge section determined in Step 1 and the resulting change in the moment of inertia of the bridge section into the time-domain hydrodynamic analysis equation of the multi-bridge section and multi-compartment damaged water-filled floating bridge to determine the motion response of each bridge section of the floating bridge system and the connector load at time t.

[0099] The specific time-domain hydrodynamic analysis equations for the multi-section, multi-compartment damaged, water-intake floating bridge in steps 3-4 are shown below:

[0100] ;

[0101] in, The generalized mass matrix represents the state of the multi-bridge joints in good condition. The change in the generalized mass matrix caused by water ingress into the bridge section; The damping matrix excluding the effects of linear radiation damping; The total stiffness matrix of hydrostatic restoring force includes mooring constraints and considers the effects of liquid tanks; The array of wave excitation forces acting on the multi-bridge segment at time t; The array of moving loads experienced by the multi-bridge section at time t; , and These represent the acceleration, velocity, and displacement response arrays of the multi-bridge section, respectively. The velocity delay function matrix is ​​defined as follows: the generalized mass includes mass and moment of inertia; the generalized additional mass includes additional mass and additional moment of inertia.

[0102] Example

[0103] This invention provides a method for analyzing the dynamic response of a continuously loaded floating bridge that has suffered water ingress due to damage to multiple bridge sections and compartments. Figure 1 As shown, the method includes the following steps:

[0104] Step 1: Based on the structural parameters of the floating bridge and the location and size of the damaged compartments and breaches, construct a dynamic analysis model of the multi-section, multi-compartment floating bridge damaged and flooded in still water to conduct hydrodynamic analysis and determine the equilibrium state of the multi-section, multi-compartment floating bridge after the breach.

[0105] The structural parameters of a floating bridge include: the number of bridge sections, the parameters of each section, the type of connectors between sections, and the parameters of the mooring system. For example... Figures 3-4 As shown, a single bridge section 2 comprises several compartments 1 arranged laterally and longitudinally. The floating bridge consists of several bridge sections 2, which are sequentially connected along the length to form the main structure of the shore-connecting floating bridge. The parameters of each bridge section 2 include the number and size of the compartments 1, the length, width, height, mass, moment of inertia, and center of mass position of the bridge section 2. The bridge sections are connected by connectors 3. The mooring system of the floating bridge mainly includes anchors 5 for positioning and cables 4 for connecting the bridge sections 2 to the anchors 5.

[0106] When establishing a dynamic analysis model for a multi-section, multi-compartment floating bridge in still water, the connector 3 between the bridge sections is established using hinge constraints. The model is solved as follows: Figure 2 As shown, firstly, based on the breach size and the difference in liquid level inside and outside the compartment, the flow rate is calculated by integrating over time. Time of the first The bridge section The water ingress quality of each damaged compartment:

[0107] ;

[0108] in, The flow coefficient is related to bridge joint motion, tank sloshing, breach geometry, and compartment layout. The density of water; An operator for determining the sign of the function value within parentheses; and The first The bridge section The height of the external and internal liquid levels in each damaged compartment; This is the acceleration due to gravity.

[0109] Secondly, the quality of water entering each compartment was used to determine the presence of... The quality of water ingress into each compartment and the damaged bridge section Based on the location, size, mass, volume and shape of the damaged compartment, bridge section inclination angle, and bridge section draft, the time was determined. The center of mass coordinates of the water ingress mass within the compartment at any given moment. Damaged bridge section. When floating upright, the center of mass of the incoming liquid is located at the origin in a coordinate system connected to the center of float of the intact bridge section. The coordinates in the middle are:

[0110] ;

[0111] in, For the first The first bridge section The longitudinal coordinate of the center of each flooded compartment; The initial draft for a complete bridge section; The density of water; and These are the length and width of the i-th water intake compartment in the k-th bridge section, respectively.

[0112] When the damaged bridge section is trimmed, ① when the trim angle is... The The water quality of the compartments in each water intake bridge section At this time, the side projection of the liquid inside the compartment is a triangle. The coordinates of the internal liquid's center of mass in the connected coordinate system with the center of gravity of the intact bridge section as the origin are:

[0113] ;

[0114] in, The contact length between the water entering the i-th water intake chamber in the k-th bridge section and the bottom of the chamber.

[0115] ② When the water quality of the compartments in the water intake bridge section At that time, the side projection of the liquid inside the compartment is a right trapezoid, and the coordinates of the center of mass of the ingress water in the connected coordinate system with the center of the float as the origin are:

[0116] ;

[0117] Next, based on the mass of water entering the compartment and the coordinates of its center of mass, the forces and moments caused by the water ingress, as well as the change in the moment of inertia of the bridge section due to the failure of a single compartment, were determined. The weight increase of the bridge section after water ingress is... The resulting heeling moment and pitching moment are respectively and ;No. The first bridge section The change in the moment of inertia matrix of a single bridge segment caused by water ingress into a damaged compartment can be calculated using the mass of the water entering the compartment. Calculation of the coordinates of its center of mass in a coordinate system where the origin is located at the center of the floating surface of the intact bridge section:

[0118] ;

[0119] For those The first compartment to be flooded The change in the moment of inertia of each bridge section caused by water ingress is obtained by the superposition of the effects of each damaged compartment. .

[0120] At the same time, the hydrostatic restoring force and moment, as well as the mooring force and moment, are determined based on the bridge section inclination angle and draft.

[0121] Then, considering the combined effects of the inflow load, still water restoring force load, and mooring load on the floating bridge, the dynamic analysis equations for a multi-section, multi-compartment floating bridge subjected to inflow and damage in still water are established:

[0122] ;

[0123] in, , q represents the system's degree of freedom number; and These are the generalized velocities corresponding to the p-th and q-th degrees of freedom of the system, respectively, where the generalized velocities of the system are the relative rotational and translational velocities between the bridge sections; and The first The m-th component of the partial velocity array of the bridge section relative to the l-th and p-th degrees of freedom of the system; and The first The m-th component of the angular velocity array of the bridge section relative to the l-th and p-th degrees of freedom of the system; For the first The r-th component of the deflection angular velocity array of the p-th degree of freedom of the bridge section relative to the system; For the first The nth component of the angular velocity array of the bridge section relative to the qth degree of freedom of the system; and The first The m-th component of the first-order derivative array of the deflection velocity and deflection angular velocity of the bridge section relative to the p-th degree of freedom of the system; and The first The water inflow mass of each bridge section and the resulting changes in the elements of the bridge section's moment of inertia matrix; and The first The additional mass and additional moment of inertia matrix elements of each bridge section; Let the generalized rate be the value corresponding to the p-th degree of freedom of the system. The first derivative; and They represent the first The increased weight of each bridge section after water ingress and the resulting torque component; and The first The hydrostatic restoring force and moment components of each bridge section; and The first The mooring force and mooring moment components of each bridge section; These are the elements of the permutation matrix;

[0124] At time t, the partial velocity array elements of each bridge segment and its derivative Angular velocity array elements and its derivative ,quality Matrix of inertia Inlet water quality Changes in moment of inertia caused by water ingress Additional mass and additional rotational inertia matrix Force caused by water ingress and torque Still water restoring force and torque Mooring force And torque All quantities are known. The dynamic analysis equations for a multi-section, multi-compartment floating bridge undergoing flooding in still water are first-order ordinary differential equations concerning the generalized velocity. A fourth-order Runge-Kutta method is used for time-domain numerical solution to determine the generalized velocity of the system at time t. This allows for the determination of the position and tilt angle of each bridge section at time t, and updates to information such as the internal and external liquid levels. Calculations are made regarding the mass of water entering the compartment at any given time, the change in rotational inertia caused by the water ingress, and other factors to determine... The dynamic response of the time-tracking system. As time progresses, the multi-section, multi-compartment floating bridge system eventually reaches a stable equilibrium state in still water after damage and water ingress.

[0125] Assuming the external liquid level of the bridge section in still water is zero, the internal liquid level of the damaged compartment needs to be determined comprehensively based on factors such as compartment size, location, incoming water quality, bridge section attitude, and draft. Damaged bridge section When it is floating, its first The internal liquid level of a damaged compartment can be determined by... Time of the first The draft of each bridge section , No. The first bridge section The length of each flooded compartment Hekuan The calculation yielded: .

[0126] Damaged bridge section During trimming, the liquid level inside the damaged compartment will continuously change due to the ingress rate and volume. ① When the trim angle is... The The water quality of the compartments in each water intake bridge section At this time, the side projection of the liquid inside the compartment is triangular, and the contact length between the incoming water and the bottom of the compartment is... ,when ( When the distance from the bottom orifice to the bottom corner of the liquid surface is given, and the liquid level inside the tank does not reach the orifice, the height of the internal liquid level relative to the external liquid level should be calculated using the vertical coordinate at the orifice. At this point, the liquid level inside the tank has already submerged the orifice. Therefore, the actual internal liquid level relative to the external liquid level should be used as the reference point, i.e.:

[0127] ;

[0128] In the formula, For the first The vertical distance from the center of each bridge section to its bottom.

[0129] ② When the water quality of the compartments in the water intake bridge section At that time, the side projection of the liquid inside the compartment is a right trapezoid, and the liquid level inside the compartment is:

[0130] .

[0131] Step 2: Based on the bridge section structural parameters and the information such as the inclination angle, position, and influent mass of each bridge section determined in Step 1, establish the geometric model of the bridge section and divide it into compartments and allocate the influent mass within each compartment. After completing the wet surface mesh generation, sequentially set each bridge section in the system to operate at wave frequency in each degree of freedom direction. Perform simple harmonic motion with unit amplitude. For each unit motion, solve for the radiation potential by combining the boundary conditions of the object surface, the free surface, the bottom boundary conditions, and the far-field radiation conditions. Laplace equation We obtain the radiation potential corresponding to each degree of freedom. Then, we use the linearized Bernoulli equation... Converting radiation potential into dynamic pressure and dynamic pressure Integrating across the wetted surfaces of all bridge sections yields the generalized force acting on each degree of freedom in each bridge section. Extracting the coefficients of acceleration and velocity from these generalized forces provides the generalized added mass coefficients for each degree of freedom in each bridge section. and radiation damping coefficient In the formula, For the first Spatial radiation potential generated by unit amplitude motion in one degree of freedom direction; For the first The generalized normal vector corresponding to each degree of freedom.

[0132] Based on the known incident potential Solving the reflection potential of a multi-bridge joint system with boundary conditions The dynamic pressure generated by the incident potential and the reflected potential Integrating on the wetted surfaces of all bridge sections yields the array of wave excitation forces acting on each degree of freedom of each bridge section. For different wave frequencies Repeat the above solution process after meshing to obtain the generalized additional mass coefficient, radiation damping coefficient, and wave excitation force database.

[0133] Step 3, apply the additional hydrodynamic coefficient determined in Step 2. and The matrix formed and Substituting into the Kramer-Kronig relation Determine the generalized additional mass coefficient matrix for each degree of freedom of each bridge section within the breached pontoon bridge when the wave frequency approaches infinity. Then, the radiation damping coefficient matrix at different frequencies... Integrate to determine the velocity delay function matrix. Next, calculate the vertical forces of all moving loads acting on the bridge section at time t. lateral moment and longitudinal moment .

[0134] in, The number of moving loads acting on the k-th bridge section; For the action on the k-th bridge segment The mass of a moving load; and The first The lateral and longitudinal eccentric displacements of the moving load relative to the center of gravity of the intact bridge section, such as Figure 5 As shown.

[0135] The specific calculation methods for still water restoring force and mooring force can be found in existing methods, and will not be elaborated upon here. Considering the combined effects of wave excitation force, moving load, still water restoring force, and mooring force, the water ingress mass of the bridge section determined in step 1 and the resulting change in the moment of inertia of the bridge section are substituted into the time-domain hydrodynamic analysis equations for a multi-section, multi-compartment damaged, water-filled floating bridge. ;

[0136] in, This is the generalized mass (including mass and moment of inertia) matrix for a multi-bridge joint in good condition. The change in the generalized mass matrix (including mass and moment of inertia) caused by water ingress into the bridge section; This is the generalized additional mass (including additional mass and additional moment of inertia) matrix for each degree of freedom of each bridge section within the damaged pontoon bridge when the wave frequency approaches infinity. The damping matrix excluding the effects of linear radiation damping; The total stiffness matrix of hydrostatic restoring force includes mooring constraints and considers the effects of liquid tanks; The array of wave excitation forces acting on the multi-bridge segment at time t; The array of moving loads experienced by the multi-bridge section at time t; , and These represent the acceleration, velocity, and displacement response arrays of the multi-bridge section, respectively. This is the velocity delay function matrix.

[0137] Solving the time-domain hydrodynamic analysis equations for the multi-section, multi-compartment floating bridge in waves yields the hydrodynamic response analysis results of the damaged floating bridge at time t under the combined action of moving loads and wave loads. This includes the floating bridge's motion response and connector loads.

[0138] The dynamic response analysis method for multi-section, multi-compartment damaged floating bridges under water ingress of the present invention can take into account the dynamic response results of the bridge sections and connector loads when the multi-section, multi-compartment damaged floating bridges under water ingress are under load in waves. By comprehensively applying multibody dynamics methods and potential flow theory, the calculation efficiency of dynamic response analysis of under-load damaged floating bridges is improved.

[0139] In this embodiment, the vertical displacement results of the damaged bridge sections of the floating bridge obtained when vehicle 6 travels on two consecutive damaged and flooded floating bridge sections under the action of three different frequency regular waves are as follows: Figure 6 As shown in the figure. The horizontal axis represents time, and the vertical axis represents the vertical displacement of the damaged bridge section. The purple solid line, red dashed line, and blue dotted line represent the results for periods of 5s, 6s, and 7s, respectively. Furthermore, the vertical displacement results of the middle damaged bridge section of the floating bridge obtained when vehicle 6 travels on the three different numbers of damaged bridge sections under the action of regular waves are shown in the figure. Figure 7As shown in the figure. The horizontal axis represents time, and the vertical axis represents the vertical displacement of the damaged bridge section. The purple solid line, red dashed line, and blue dotted line represent the results of single bridge section failure, consecutive double bridge section failure, and consecutive triple bridge section failure, respectively.

[0140] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0141] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for analyzing the dynamic response of a continuously loaded floating bridge considering water ingress due to damage to multiple bridge sections and compartments, characterized in that, Includes the following steps: Step 1: Based on the parameters of the damaged and flooded floating bridge system, establish a dynamic analysis model of the multi-section and multi-compartment damaged and flooded floating bridge in still water, and determine the water ingress mass of each compartment and the water ingress mass, position, tilt angle, and changes in the rotational inertia of the bridge sections caused by water ingress when the floating bridge system reaches a stable state after water ingress. Step 2: Based on the water ingress mass of each compartment and the water ingress mass, position, tilt angle, and change in moment of inertia of each bridge section caused by water ingress, establish a numerical model based on potential flow theory to determine the additional hydrodynamic coefficient and wave excitation force of each bridge section. Step 3: The changes in bridge section mass and moment of inertia caused by water ingress, the wave loads on each bridge section, the water ingress loads, the hydrostatic restoring force loads, the moving loads, and the mooring loads are analyzed using the time-domain hydrodynamic analysis equations for a multi-section, multi-compartment, damaged, water-ingress floating bridge in waves. The dynamic response results of the damaged, water-ingress floating bridge are obtained. Step 2 specifically involves: based on the water ingress mass of each compartment and the water ingress mass, position, and tilt angle of each bridge section determined in Step 1, solving the boundary values ​​of the Laplace equation for different wave frequencies based on potential flow theory, and obtaining the additional hydrodynamic coefficients and wave excitation forces experienced by each bridge section in the multi-bridge-section, multi-compartment damaged and water-ingressed floating bridge at different wave frequencies. The additional hydrodynamic coefficients include the generalized additional mass coefficient and the radiation damping coefficient. The generalized added mass coefficient and radiation damping coefficient are obtained by having a bridge section move with unit amplitude at wave frequency in each degree of freedom, determining the generated flow field potential, integrating the fluid force of the motion mode on all degrees of freedom, and combining the relevant results of each bridge section to obtain the generalized added mass coefficient and radiation damping coefficient at the wave frequency. The wave excitation force is obtained by fixing all bridge sections, determining the velocity potential of the superimposed incident and reflected waves at the specified wave frequency, and integrating the force exerted by the wave on each bridge section as the wave excitation force at the specified wave frequency. Step 3 specifically involves: Step 3-1: Using the Kramer-Kronig relation, determine the generalized additional mass coefficient matrix for each degree of freedom of each bridge section within the breached pontoon bridge when the wave frequency approaches infinity, by constructing the matrix composed of the additional hydrodynamic coefficients. ; Specifically, it is shown in the following formula. ； in, When the wave frequency is The generalized additional mass matrix for each degree of freedom of each bridge section within the damaged pontoon bridge, wherein the generalized additional mass matrix includes the additional mass and the additional moment of inertia matrix. 0- A certain frequency in For all frequencies, and The generalized additional mass coefficient and radiation damping coefficient and The matrix formed; Step 3-2: Radiation damping coefficient matrix for different wave frequencies Integrate to determine the velocity delay function matrix. ; Step 3-3: Calculate the vertical force of all moving loads acting on the bridge section at time t. lateral moment and longitudinal moment ; in, The number of moving loads acting on the k-th bridge section; For the action on the k-th bridge segment The mass of a moving load; and The first The lateral and longitudinal eccentric displacements of the position of the moving load relative to the center of gravity of the intact bridge section; Steps 3-4: Based on the combined effects of wave excitation load, moving load, still water restoring load, and mooring load, and substituting the water ingress mass of the bridge section determined in Step 1 and the resulting change in the moment of inertia of the bridge section into the time-domain hydrodynamic analysis equation of the multi-bridge section, multi-compartment damaged water-filled floating bridge, determine the motion response of each bridge section of the floating bridge system and the connector load at time t.

2. The method for analyzing the dynamic response of a floating bridge under load, considering water ingress due to damage to multiple bridge sections and compartments, as described in claim 1, is characterized in that... The parameters of the breached and flooded floating bridge system include: the dimensions, draft, center of mass position, mass, moment of inertia, additional mass, additional moment of inertia, initial velocity and initial angular velocity of each bridge section, the number and location of the breached bridge sections, the number, location and dimensions of the breached compartments, and the location and dimensions of the breach.

3. The method for analyzing the dynamic response of a continuously loaded floating bridge considering water ingress due to damage to multiple bridge sections and compartments, as described in claim 1, is characterized in that... Step 1 includes the following steps: Step 1-1: Based on the breach size and the difference in liquid level inside and outside the compartment, determine the water ingress mass of each compartment by integrating the flow rate over time; Steps 1-2: Determine the water ingress mass of the damaged bridge section based on the water ingress mass of each compartment, and determine the centroid coordinates of the water ingress mass in the compartment by combining the location of the damaged compartment, the size of the compartment, the water ingress mass, the shape of the water ingress volume, the bridge section inclination angle and the bridge section draft; Steps 1-3: Based on the mass of water entering the compartment and the coordinates of its center of mass, determine the forces and torques caused by the water ingress, as well as the change in the moment of inertia of the bridge section caused by the damage of a single compartment; Steps 1-4: Determine the hydrostatic restoring force and moment, as well as the mooring force and moment, based on the bridge section inclination angle and draft. Steps 1-5: Considering the combined effects of the inflow load, still water restoring force load, and mooring load on the floating bridge, establish the dynamic analysis equations for a multi-section, multi-compartment floating bridge that has suffered damage and inflow in still water.

4. The dynamic response analysis method for a continuously loaded floating bridge considering water ingress due to damage to multiple bridge sections and compartments, as described in claim 3, is characterized in that... In step 1-1, the calculation is performed by integrating the flow rate over time. Time of the first The bridge section The water ingress mass of each damaged compartment is shown in the following formula: ； in, The flow coefficient is related to bridge joint motion, tank sloshing, breach geometry, and compartment layout. The density of water; An operator for determining the sign of the function value within parentheses; and The first The bridge section The height of the external and internal liquid levels in each damaged compartment; It is the acceleration due to gravity; This represents the area of ​​the breach.

5. The dynamic response analysis method for a continuously loaded floating bridge considering water ingress due to damage to multiple bridge sections and compartments, as described in claim 4, is characterized in that... The steps 1-2 determine the amount of water entering each compartment based on the water quality of each compartment. The water ingress quality of the damaged bridge section in each compartment: .

6. The method for analyzing the dynamic response of a continuously loaded floating bridge considering water ingress due to damage to multiple bridge sections and compartments, as described in claim 5, is characterized in that... Steps 1-3 are specifically as follows: The weight increase after water enters the bridge section is The resulting heeling moment and pitching moment are respectively and ; in, and These are the x and y coordinates, respectively, in a coordinate system where the center of mass of the water ingress into the compartment is located at the center of the float on the surface of the intact bridge water-stop line at the origin. No. The first bridge section The change in the moment of inertia matrix of a single bridge segment caused by water ingress into a damaged compartment can be calculated using the mass of the water entering the compartment. The coordinates of the centroid and its center of mass in the connected coordinate system with the origin located at the center of the waterline of the intact bridge section are calculated as follows: ； in, , and These are the abscissa, ordinate, and vertical coordinates of the combined coordinate system in which the center of mass of the water ingress in the compartment is located at the center of the float on the surface of the intact bridge water-stop line, respectively. For those The first compartment to be flooded The change in the moment of inertia of each bridge section caused by water ingress is obtained by the superposition of the effects of each damaged compartment. .

7. The dynamic response analysis method for a continuously loaded floating bridge considering water ingress due to damage to multiple bridge sections and compartments, as described in claim 6, is characterized in that... The dynamic analysis equations for the multi-section, multi-compartment floating bridge damaged and flooded in still water in steps 1-5 are shown in the following formulas: ； in, , q represents the system's degree of freedom number; and These are the generalized velocities corresponding to the p-th and q-th degrees of freedom of the system, respectively, where the generalized velocities of the system are the relative rotational and translational velocities between the bridge sections; and The first The m-th component of the partial velocity array of the bridge section relative to the l-th and p-th degrees of freedom of the system; and The first The m-th component of the angular velocity array of the bridge section relative to the l-th and p-th degrees of freedom of the system; For the first The r-th component of the deflection angular velocity array of the p-th degree of freedom of the bridge section relative to the system; For the first The nth component of the angular velocity array of the bridge section relative to the qth degree of freedom of the system; and The first The m-th component of the first-order derivative array of the deflection velocity and deflection angular velocity of the bridge section relative to the p-th degree of freedom of the system; and The first The mass and moment of inertia matrix of the nth bridge segment Line number Column elements; and The first Elements in the matrix of the water inflow mass of each bridge section and the resulting change in the moment of inertia of the bridge section; and The first The additional mass and additional inertia matrix elements of each bridge section; Let the generalized rate be the value corresponding to the p-th degree of freedom of the system. The first derivative; and They represent the first The increased weight of each bridge section after water ingress and the resulting torque component; and The first The hydrostatic restoring force and moment components of each bridge section; and The first The mooring force and mooring moment components of each bridge section; These are the elements of the permutation matrix.

8. The method for analyzing the dynamic response of a continuously loaded floating bridge considering water ingress due to damage to multiple bridge sections and compartments, as described in claim 1, is characterized in that... The specific time-domain hydrodynamic analysis equations for the multi-section, multi-compartment damaged, water-intake floating bridge in steps 3-4 are shown below: ； in, The generalized mass matrix represents the state of the multi-bridge joints in good condition. The change in the generalized mass matrix caused by water ingress into the bridge section; The generalized additional mass matrix of each degree of freedom of each bridge section in the breached pontoon bridge when the wave frequency approaches infinity. The damping matrix excluding the effects of linear radiation damping; The total stiffness matrix of hydrostatic restoring force includes mooring constraints and considers the effects of liquid tanks; The array of wave excitation forces acting on the multi-bridge segment at time t; The array of moving loads experienced by the multi-bridge section at time t; , and These represent the acceleration, velocity, and displacement response arrays of the multi-bridge section, respectively. The velocity delay function matrix is ​​defined as follows: the generalized mass includes mass and moment of inertia; the generalized additional mass includes additional mass and additional moment of inertia.

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