Dynamic response analysis method of floating breakwater anchor chain system

Abstract

The invention discloses a dynamic response analysis method of a floating breakwater anchor chain system. The dynamic response analysis method comprises the following steps: determining parameters of a floating box and an anchor chain system in the floating breakwater system; a sea condition environment where the system is located is given; motion of the anchor chain unit is described, and a global position vector of any point P on the axis of the anchor chain unit is given; constructing an expression of an anchor chain unit kinetic energy and mass matrix; calculating strain energy and elastic force of the anchor chain unit; calculating the environmental load borne by the floating breakwater anchor chain system; assembling a system mass matrix and a constraint equation of the system; establishing a dynamic equation set of the system, and correcting to obtain a first-order differential equation set; performing simulation to extract displacement, speed and acceleration data of the top end of the anchor chain rope, and substituting the data as external constraint conditions of the top end of the anchor chain into the dynamic equation set; and performing dynamic solution on the first-order differential equation set. According to the invention, the accuracy of dynamic response calculation is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of dynamic response analysis, and particularly to a method for analyzing the dynamic response of a floating breakwater anchor chain system. Background Art

[0002] Floating breakwaters are widely used to protect ports, wharves and waterways due to their low cost, easy disassembly, little environmental impact and non-destruction of the marine ecosystem. Waves are the key factors affecting the performance of floating breakwaters. Although existing studies have extensively explored the hydrodynamic characteristics of floating bodies under regular and irregular waves, there is still a lack of systematic analysis of the forces and motion characteristics of the anchor chains of floating bodies. In particular, the specific influence of the motion of the floating body, which is the main source of the dynamic load of the anchor chain, still needs to be further studied. In terms of modeling, the lumped mass method is often used for anchor chain analysis, but it has limitations in describing the continuity and complex deformation of the anchor chain. In contrast, the absolute nodal coordinate formulation (ANCF) describes the rotation and strain of elements through position and gradient vectors, and can more accurately handle nonlinear and large deformation problems. However, existing ANCF-based studies mainly focus on the anchor chain itself, and insufficient consideration is given to the modeling of the floating body and the influence of its motion on the dynamic response of the anchor chain. For example, by simplifying the floating body or applying a motion function to the top node of the anchor chain, the complexity under actual working conditions cannot be truly reflected.

[0003] Therefore, it is urgent to solve the above problems. Summary of the Invention

[0004] Object of the Invention: The object of the present invention is to provide a method for analyzing the dynamic response of a floating breakwater anchor chain system, which improves the accuracy of dynamic response calculation.

[0005] Technical Solution: To achieve the above object, the present invention discloses a method for analyzing the dynamic response of a floating breakwater anchor chain system, including the following steps:

[0006] (1) Determine the parameters of the floating box and the anchor chain system. The parameters of the floating box include the geometric dimensions, material properties and mass distribution of the floating box. The center of the floating box is located at the origin of coordinates, and the bottom surface of the floating box is parallel to the X-O-Y plane. The anchor chain system includes four anchor chains arranged symmetrically about the center. One end of each anchor chain is connected to the floating box, and the other end is fixed to the seabed. Each single anchor chain is divided into several elements;

[0007] (2) Give the sea condition environment. The sea condition environment parameters include the water depth H, the seawater density ρ s , the wave type, wave direction, wave height η, amplitude η x , the wave number k in the X direction x , the frequency ω x , the sea current velocity components V x , V z and the sea current acceleration components where the wave direction is along the positive X-axis;

[0008] (3) Describe the motion of the anchor chain element. First, based on the absolute nodal coordinate formulation (ANCF), give the initial configuration and the deformed configuration of the anchor chain element, as well as the global position vector of any point P on the axis of the anchor chain element;

[0009] (4) Construct the expressions for the kinetic energy and mass matrix of the anchor chain element;

[0010] (5) Calculate the strain energy and elastic force of the anchor chain element through the length, stiffness, damping of the anchor chain element and the position of the mooring point;

[0011] (6) Calculate the environmental loads on the floating breakwater anchor chain system. The environmental loads on each anchor chain include the wet weight of the anchor chain, the seabed soil force, the buoyancy of the floating box and the current force;

[0012] (7) Assemble the mass matrices of each element on each anchor chain to obtain the system mass matrix, and according to the constraints between the anchor chains, assemble to obtain the system constraint equations;

[0013] (8) Based on the Lagrangian method, for the constrained anchor chain system, establish the dynamic equations of the system and correct them to obtain the first-order differential equations;

[0014] (9) First, simulate the floating breakwater anchor chain system through OrcaFlex software, and extract the displacement, velocity and acceleration data of the top of the anchor chain; then use the displacement, velocity and acceleration data of the top of the anchor chain as external constraint conditions and substitute them into the dynamic equations;

[0015] (10) Solve the dynamics of the first-order differential equations to obtain the absolute nodal coordinates of the system, and calculate the kinetic energy and strain energy of the anchor chain system through the absolute nodal coordinates of the system.

[0016] Optionally, in step (3), the initial configuration means that the anchor chain element initially appears as a straight line, and its axis extends in a certain direction, defining the original positions of each point in the unloaded state. The extension direction can be along the X-axis; the deformed configuration means that the axis of the anchor chain element bends or stretches due to external forces, forming a new curved shape, reflecting the new positions of each point relative to the initial position in the loaded state;

[0017] For an anchor chain element with an element length of L, its absolute nodal coordinate q is composed of the positions q1 and q2 of the two ends of the element, and its absolute nodal coordinate q is:

[0018]

[0019] where r T (x = 0) and rT (x = L) represent the position vectors of the two endpoints of the element, and represent the rotation vectors of the two endpoints of the element respectively;

[0020] For any point on the anchor chain element, the global position coordinate r(x, t) is expressed by the cubic Hermite interpolation of the absolute nodal coordinates of the element:

[0021] r(x, t) = S(x)q(t)

[0022] where q(t) is the absolute nodal coordinate of any point on the anchor chain element, and S(x) is the shape function of any point on the anchor chain element. The shape function S(x) is expressed as cubic Hermite interpolation, and the specific form is:

[0023]

[0024] where x is the local coordinate of the element, ranging from 0 to 1, representing the position of the point within the element; The global velocity and acceleration vectors of point P are functions of time t of the absolute nodal coordinates of the element. Therefore, the global velocity and acceleration vector are expressed by differentiating the position vector with respect to time:

[0025]

[0026] where: and represent the velocity vector and acceleration vector of the absolute nodal coordinates of point P; The first derivative r x (x, t) and the second derivative r xx (x, t) of the global position vector of point P with respect to the local coordinate of the element are expressed as:

[0027] r x (x, t) = S x (x)q(t)

[0028] r xx (x, t) = S xx (x)q(t)

[0029] where: S x (x) and S xx (x) represent the first and second derivatives of the shape function with respect to the local coordinate respectively.

[0030] Optionally, for the anchor chain element with length L, single density ρ, and cross-sectional area A in step (4), the expression of the kinetic energy T of the anchor chain element is:

[0031]

[0032] Among them, represents the velocity vector of the unit, represents the first derivative of the absolute nodal coordinates of the unit with respect to t, S is the shape function of the unit, and M i is the element mass matrix. Through the derivation of the kinetic energy T expression, the anchor chain element mass matrix is obtained as:

[0033]

[0034] In the formula and mean that the element mass matrix is divided into four 6×6 matrices.

[0035] Optionally, in step (5), it is assumed that the shear and torsion effects of the anchor chain element are ignored, and the strain energy only includes the energy caused by axial deformation and bending deformation; according to the Euler - Bernoulli beam theory, the strain energy U of the anchor chain element is expressed as:

[0036]

[0037] Among them, E is the Young's modulus of the anchor chain, J is the cross - sectional moment of inertia of the beam, ε and κ are the axial strain and curvature respectively, and the specific calculations of the axial strain and curvature are as follows:

[0038]

[0039] Among them, r x and r xx represent the first and second derivatives of the position vector of the mooring element with respect to the local coordinates of the element respectively;

[0040] By taking the partial derivatives of the strain energy U with respect to the element nodal coordinates, the generalized elastic force of the element is obtained:

[0041]

[0042] Among them, Q e1 and Q e2 are the elastic forces generated by the axial tension and bending deformation of the element respectively, and K1 and K2 are their corresponding stiffness matrices;

[0043] Q e1 is expressed as:

[0044]

[0045] Q e2 is expressed as:

[0046]

[0047] Optionally, the wet weight of the anchor chain in step (6) includes its own gravity load f g and its own buoyancy load f b , and the gravity and buoyancy loads acting on the infinitesimal element dx of the anchor chain arc length are respectively expressed as:

[0048] f g = [0 0 -ρAg] T

[0049] f b = [0 0 ρ s Ag] T

[0050] where ρ is the density of the anchor chain material, g is the acceleration due to gravity, and A is the cross-sectional area.

[0051] Optionally, the seabed soil force in step (6) approximately describes the normal support force of the seabed on the anchor chain using a linear spring model. For any point P on the anchor chain unit in contact with the seabed, the normal support force it receives is the seabed soil force f z , and the seabed soil force f z is expressed as:

[0052] f z = k·Δz (Δz≥0)

[0053] f z = 0 (Δz<0)

[0054] Δz = r P ·e Z -Z bed

[0055]

[0056] where k refers to the equivalent spring stiffness, d is the anchor chain diameter, Z bed is the seabed depth coordinate, and e Z is the unit direction vector in the Z direction.

[0057] Optionally, the buoyancy force of the floating box on the top node of the anchor chain in step (6) is represented by Q f as:

[0058] Q f = ρgV s -Mg

[0059] where M is the mass of the floating box, and V s is the volume of the floating box underwater. Therefore, V s can be expressed as:

[0060]

[0061] The buoyancy force on the pontoon is:

[0062] Q f = ρgV s - Mg

[0063] Where: ρ s is the seawater density, g is the acceleration due to gravity, S buoy is the cross-sectional area of the pontoon, and z is the depth of the pontoon below the water surface;

[0064] The movement of the anchor chain unit in water is affected by hydrodynamic forces. The non-linearity of the hydrodynamic forces is described by the Morison equation. The non-linear resistance in the Morison equation includes tangential force and normal force. By setting the tangential unit vector of the anchor chain unit, the tangential component and normal component of the hydrodynamic force are calculated. The tangential component f Dt can be expressed by the following formula:

[0065]

[0066] The normal component f Dn can be expressed by the following formula:

[0067]

[0068] Where, C d is the drag coefficient, f t and f n are loading functions, d is the diameter of the anchor chain, is the relative velocity vector, U is the incoming flow velocity, V Rt and V Rn are the tangential and normal components of the relative velocity;

[0069] The drag coefficient C d is closely related to the Reynolds number Re, and its calculation formula is:

[0070]

[0071] Where, μ is the dynamic viscosity coefficient of water, The tangential component and normal component are vectorially summed to obtain the ocean current force f D .

[0072] Optionally, in step (7), for an anchor chain with N units, each anchor chain unit corresponds to a mass matrix M i ; then the mass matrix of a single anchor chain is obtained by assembling the mass matrices of all units. The specific method is:

[0073]

[0074] For an anchor chain system, which generally contains several anchor chains, its system mass matrix is as follows:

[0075]

[0076] Optionally, the constraints between the anchor chains in step (7) are mainly divided into displacement constraints and rotational constraints. Select a connection point i of an anchor chain unit in the anchor chain system and a connection point j on another anchor chain unit. If the absolute nodal coordinates q of the connection point i i are:

[0077]

[0078] where r i and r j represent the position vectors of the connection points i and j respectively, and r x,i and r x,j represent the rotational vectors of the connection points i and j respectively;

[0079] Then the constraint equation C(q, t) between the connection points i and j can be expressed as:

[0080]

[0081] Assembling the constraint equations of each anchor chain into a matrix, the constraint equation of the system is obtained as:

[0082] C(q s , t) = 0.

[0083] Optionally, according to the principle of virtual work in step (8), the virtual work δW done by the anchor chain system s is:

[0084]

[0085] where q s and represent the absolute nodal coordinates and acceleration vectors of the anchor chain system, Q s is the generalized force of the anchor chain system, which includes the generalized elastic force of the system the buoyancy Q of the floating box f and the generalized external force of the system. The generalized external force of the system includes the seabed soil force of the system and the sea current force of the system Q s can then be expressed as:

[0086]

[0087] For the constrained anchor chain system, its dynamic equations are expressed as:

[0088]

[0089] Among them, is the Jacobian matrix of the system constraint equation C(q s , t), and λ is the Lagrange multiplier vector;

[0090] By solving it using the Baumgarte method, a differential-algebraic equation system can be obtained:

[0091]

[0092] In the formula: Q d is the second-order partial derivative of the constraint equation with respect to the independent variable. The first-order and second-order partial derivatives of the system constraint equation C(q, t) with respect to time are respectively:

[0093]

[0094] Based on the first-order and second-order partial derivatives, it can be obtained:

[0095]

[0096] A set of ordinary differential equation systems is obtained:

[0097]

[0098] By introducing a feedback system to the constraint equation, the final constraint equation is obtained:

[0099]

[0100] Among them, α and β are stability coefficients. The empirical values of α and β are taken between 5 and 50. When α = β, the stable response is fast, and a first-order differential equation system can be obtained:

[0101]

[0102] Beneficial effects: Compared with the prior art, the present invention has the following remarkable advantages: The method of the present invention can accurately predict the dynamic behavior of the floating breakwater system under the actions of waves, wind, etc., especially the influence on the anchor chain rope under dynamic conditions such as inclination and rotation; The absolute nodal coordinate method is adopted to describe the motion of the anchor chain unit. The anchor chain unit is divided into multiple finite elements, and the global position vector of any point is described by using the global shape function and the absolute nodal coordinate vector. The geometric deformation and dynamic response of each unit can be obtained by numerical solution; The present invention adopts a mass matrix and kinetic energy calculation method, which can accurately describe the mass distribution and energy transfer of the anchor chain under dynamic loads. The mass matrix of the anchor chain unit is obtained through numerical integration, so as to calculate its dynamic response more accurately; In the present invention, the potential energy and generalized force of the unit are obtained under the condition of neglecting the torsional deformation. The strain energy of the anchor chain unit is represented by the axial strain and the bending curvature. The elastic force of the anchor chain unit is caused by the unit bending and axial deformation, and can be calculated by taking the partial derivative of the strain energy with respect to the generalized coordinate; In the present invention, the generalized external forces acting on the unit mainly consider the self-gravity of the anchor chain, buoyancy, concentrated force of the buoy, the acting force of the seabed soil, and the Morison force of the wave current, etc.; Since the size of the anchor chain is small and it is in the ocean fluid, the fluid resistance it receives cannot be ignored. The present invention uses the Morison equation to describe the hydrodynamic force received by the anchor chain, considering the influence of the flow velocity and the water flow direction on the anchor chain, thus improving the calculation method of the force received by the anchor chain; The present invention considers the interaction between the anchor chain and the soil force. One is the traditional spring model, which is applicable to the situation where the anchor chain only contacts the seabed. By effectively modeling the soil force, the interaction between the soil and the anchor chain can be simulated more accurately, and the actual working conditions can be better reflected; The present invention establishes the dynamic equation of the anchor chain system by assembling the stiffness matrix, mass matrix and external loads of each unit. Using these equations, the displacement vector of the anchor chain in the ocean environment can be effectively solved, and then its dynamic response can be analyzed. The establishment of these equations can support the analysis of the anchor chain under complex working conditions, including dynamic factors such as waves, sea currents, and buoy actions; The present invention improves the accuracy of the dynamic response calculation by analyzing the dynamic response of the floating breakwater anchor chain system in detail, providing a reliable theoretical basis for the design and safety assessment of the floating breakwater. Brief Description of the Drawings

[0103] Figure 1 is a schematic flow chart of the present invention;

[0104] Figure 2 is a mooring and anchoring schematic diagram of the single-box floating breakwater structure provided for the implementation of the present invention;

[0105] Figure 3 is a three-dimensional mooring unit model diagram provided for the implementation of the present invention;

[0106] Figure 4Schematic diagram of the interaction between the anchor chain rope and the seabed soil provided for the implementation of the present invention;

[0107] Figure 5 Displacements of the floating ball in the X, Y, and Z directions provided for the implementation of the present invention;

[0108] Figure 6 Vertical displacement diagram of the free end of the anchor chain rope provided for the implementation of the present invention;

[0109] Figure 7 Free-fall structure diagram of the anchor chain rope making a flexible pendulum provided for the implementation of the present invention;

[0110] Figure 8 Tension diagram of the top end of the anchor chain rope and the anchor point provided for the implementation of the present invention;

[0111] Figure 9 Flow chart for dynamic calculation of the present invention. Detailed implementation manners

[0112] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0113] As Figure 1 shown, a method for analyzing the dynamic response of an anchor chain system of a floating breakwater according to the present invention includes the following steps:

[0114] (1) As Figure 2 shown, determine the parameters of the floating box and the anchor chain system in the single-box floating breakwater anchor chain system. The parameters of the floating box include the geometric dimensions, material properties, and mass distribution of the floating box. The center of the floating box is located at the origin of coordinates, and the bottom surface of the floating box is parallel to the X-O-Y plane. The anchor chain system includes four symmetrically arranged anchor chains. One end of each anchor chain is connected to the floating box, and the other end is fixed to the seabed. The length of a single anchor chain is set to 44 meters and is divided into 22 anchor chain units;

[0115] (2) Give the sea condition environment where the single-box floating breakwater anchor chain system is located. The sea condition environment parameters include water depth H, seawater density ρ s , wave type, wave direction, wave height η, amplitude η x , wave number k in the X direction x , frequency ω x , sea current velocity components V x , V z and sea current acceleration components where the wave direction is along the positive X-axis; Table 1 gives the specific parameters of the sea condition;

[0116] Table 1

[0117]

[0118] (3) Describe the motion of the anchor chain unit. First, based on the absolute nodal coordinate formulation (ANCF), give the initial configuration and the deformed configuration of the anchor chain unit, as well as the global position vector r of any point P on the axis of the anchor chain unit, as Figure 3 shown;

[0119] The initial configuration means that the anchor chain unit initially appears as a straight line, and its axis extends in a certain direction, defining the original positions of all points in the unloaded state. The extension direction can be along the X-axis;

[0120] The deformed configuration means that the axis of the anchor chain unit bends or stretches due to external forces, forming a new curved shape, which reflects the new positions of all points relative to their initial positions under the loaded state;

[0121] For an anchor chain unit with a unit length of L, where L = 2 m, its absolute nodal coordinates q are composed of the positions q1 and q2 of the two endpoints of the unit. Its absolute nodal coordinates q are:

[0122]

[0123] where r T (x = 0) and r T (x = L) represent the position vectors of the two endpoints of the unit respectively, and represent the rotation vectors of the two endpoints of the unit respectively;

[0124] For the global position coordinates r(x, t) of any point on the anchor chain unit, they are represented by cubic Hermite interpolation of the absolute nodal coordinates of the unit:

[0125] r(x, t) = S(x)q(t)

[0126] where q(t) are the absolute nodal coordinates of any point on the anchor chain unit, and S(x) is the shape function of any point on the anchor chain unit. The shape function S(x) is expressed as cubic Hermite interpolation, and its specific form is:

[0127]

[0128] where x is the local coordinate of the unit, ranging from 0 to 1, representing the position of the point within the unit; The global velocity and acceleration vectors of point P are functions of time t of the absolute nodal coordinates of the unit. Therefore, the global velocity and acceleration vector of point P are represented by taking the time derivative of the position vector:

[0129]

[0130] where: and The velocity vector and acceleration vector representing the absolute nodal coordinates of point P;

[0131] The first derivative r of the position vector of point P with respect to the local coordinates of the element x (x, t) and the second derivative r xx (x, t) are expressed as:

[0132] r x (x, t) = S x (x)q(t)

[0133] r xx (x, t) = S xx (x)q(t)

[0134] Where: S x (x) and S xx (x) represent the first and second derivatives of the shape function with respect to the local coordinates, respectively;

[0135] (4) Construct the kinetic energy and mass matrix of the anchor chain element,

[0136] For an anchor chain element with length L, unit density ρ, and cross-sectional area A, the expression for the kinetic energy T of the anchor chain element is:

[0137]

[0138] Where, represents the velocity vector of the element, represents the first derivative of the absolute nodal coordinates of the element with respect to t, S is the shape function of the element, and M i is the element mass matrix. By deriving the expression for the kinetic energy T, the mass matrix of the anchor chain element can be obtained as:

[0139]

[0140] In the formula and refer to dividing the element mass matrix into four 6×6 matrices;

[0141] (5) Calculate the strain energy and elastic force of the anchor chain element through the length, stiffness, damping, and mooring point position of the anchor chain element;

[0142] Assume that the shear and torsion effects of the anchor chain element are ignored, and the strain energy only includes the energy caused by axial deformation and bending deformation; according to Euler-Bernoulli beam theory, the strain energy U of the anchor chain element is expressed as:

[0143]

[0144] Among them, E is the Young's modulus of the anchor chain, J is the cross-sectional moment of inertia of the beam, ε and κ are the axial strain and curvature respectively, and the specific calculations of the axial strain and curvature are as follows:

[0145]

[0146] Among them, r x and r xx respectively represent the derivatives of the position vector of the mooring unit with respect to the local coordinates of the unit;

[0147] By taking the partial derivatives of the strain energy U with respect to the nodal coordinates of the element, the generalized elastic force of the element is obtained:

[0148]

[0149] Among them, Q e1 and Q e2 are the elastic forces generated by the axial tension and bending deformation of the element respectively, and K1 and K2 are their corresponding stiffness matrices;

[0150] The elastic force Q e1 due to axial deformation is expressed as:

[0151]

[0152] The axial stiffness matrix Q e2 is expressed as:

[0153]

[0154] (6) Calculate the environmental loads on the floating breakwater anchor chain system. The environmental loads include the wet weight of the anchor chain, the seabed soil force, the buoyancy of the floating box, and the ocean current force.

[0155] (6.1) Wet weight of the anchor chain:

[0156] The wet weight of the anchor chain includes its own gravity load f g and its own buoyancy load f b . The gravity and buoyancy loads acting on the infinitesimal arc length dx of the anchor chain are respectively expressed as:

[0157] f g =[0 0 -ρAg] T

[0158] f b =[0 0 ρ s Ag] T

[0159] Among them, ρ is the density of the anchor chain material, g is the acceleration due to gravity, and A is the cross-sectional area;

[0160] (6.2) Seabed soil force:

[0161] The linear spring model is adopted to approximately describe the normal supporting force of the seabed on the anchor chain, as Figure 4 shown; for any point P on the anchor chain unit in contact with the seabed, the normal supporting force f z is expressed as:

[0162] f z = k·Δz (Δz≥0)

[0163] f z = 0 (Δz<0)

[0164] Δz = r P ·e Z -Z bed

[0165]

[0166] where k refers to the equivalent spring stiffness, d is the diameter of the anchor chain, Z bed is the seabed depth coordinate, and e Z is the unit direction vector in the Z direction;

[0167] (6.3) Buoyancy of the pontoon:

[0168] Ignoring the added mass effect and the hydrodynamic resistance acting on the pontoon, the force on the top node of the anchor chain from the pontoon is denoted by Q f and expressed as:

[0169] Q f = ρgV s - Mg

[0170] where M is the mass of the pontoon, and V s is the volume of the pontoon underwater, which can be calculated according to its specific position. The basic parameters of the pontoon are shown in Table 2, so V s can be expressed as:

[0171]

[0172] The buoyancy force on the pontoon is:

[0173] Q f = ρgV s - Mg

[0174] In the formula: ρ s is the density of seawater, g is the acceleration due to gravity, S buoy is the cross-sectional area of the pontoon, and z is the depth of the pontoon below the water surface;

[0175] Table 2

[0176]

[0177] (6.4) Ocean current force f D :

[0178] The movement of the anchor chain unit in water is affected by hydrodynamic forces. The non-linearity of the hydrodynamic forces is described by the Morison equation. The non-linear resistance in the Morison equation includes tangential force and normal force. By setting the tangential unit vector of the anchor chain unit, the tangential component and normal component of the hydrodynamic force are calculated. The tangential component f Dt can be expressed by the following formula:

[0179]

[0180] The normal component f Dn can be expressed by the following formula:

[0181]

[0182] where C d is the drag force coefficient, f t and f n are loading functions, d is the diameter of the anchor chain, is the relative velocity vector, U is the incoming flow velocity, V Rt and V Rn are the tangential and normal components of the relative velocity;

[0183] The drag force coefficient C d is closely related to the Reynolds number Re, and its calculation formula is:

[0184]

[0185] where μ is the dynamic viscosity coefficient of water,

[0186] The tangential component and normal component are vectorially summed to obtain the ocean current force f of the anchor chain unit D .

[0187] (7) Assemble the mass matrix of each unit on each anchor chain to obtain the system mass matrix, and according to the constraints between the anchor chains, assemble to obtain the constraint equations of the system;

[0188] (7.1) System mass matrix:

[0189] For an anchor chain unit with N units, each anchor chain unit corresponds to a mass matrix M i ; then the mass matrix of a single anchor chain is obtained by assembling the mass matrices of all units. The specific method is:

[0190]

[0191] For an anchor chain system, which generally contains several anchor chains, its system mass matrix is as follows:

[0192]

[0193] (7.2) Constraint equations of the system:

[0194] The constraints between the anchor chains are mainly divided into displacement constraints and rotational constraints. Select a connection point i on one anchor chain and a connection point j on another anchor chain in the anchor chain system. If the absolute nodal coordinate q of the connection point i i is:

[0195]

[0196] where r i and r j represent the position vectors of the connection point i and the connection point j respectively, and r x,i and r x,j represent the rotational vectors of the connection point i and the connection point j respectively;

[0197] Then, the constraint equation C(q, t) between the connection points i and j can be expressed as:

[0198]

[0199] Assembling the constraint equations of each anchor chain into a matrix, the constraint equation of the system is obtained as:

[0200] C(q s , t) = 0

[0201] (8) System dynamics equation:

[0202] According to the principle of virtual work, the virtual work δW done by the anchor chain system s is:

[0203]

[0204] where q s and represent the absolute nodal coordinates and acceleration vectors of the anchor chain system, Q s is the generalized force of the anchor chain system, which includes the generalized elastic force of the system the buoyancy Q of the pontoon f and the generalized external force of the system. The generalized external force of the system includes the seabed soil force and the ocean current force Q s Then it can be expressed as:

[0205]

[0206] For a constrained anchor chain system, its dynamic equations are expressed as:

[0207]

[0208] where is the Jacobian matrix of the system constraint equation C(q s ,t), and λ is the Lagrange multiplier vector;

[0209] By solving it using the Baumgarte method, a differential-algebraic equation system of index-1 can be obtained:

[0210]

[0211] In the formula: Q d is the second-order partial derivative of the constraint equation with respect to the independent variable. The first-order and second-order partial derivatives of the system constraint equation C(q,t) with respect to time are respectively:

[0212]

[0213] According to the first-order and second-order partial derivatives, we can get:

[0214]

[0215] A set of ordinary differential equation systems is obtained:

[0216]

[0217] By introducing a feedback system to the constraint equation, the final constraint equation is obtained:

[0218]

[0219] where α and β are stability coefficients. The empirical values of α and β are taken between 5 and 50. When α = β, the stable response is relatively fast, and a first-order differential equation system can be obtained:

[0220]

[0221] (9) Under the action of waves, the floating caisson often exhibits dynamic behaviors such as tilting or rotation during movement, which will lead to differences in the movement and stretching of the tops of each mooring chain. Therefore, it is difficult to describe the movement of the tops of each mooring chain with accurate motion equations. If modeling the floating caisson, the flexible-rigid coupling problem needs to be solved, and the advantage of the absolute nodal coordinate method for modeling is mainly reflected in describing the behavior of flexible bodies under complex loadings. To solve the above problems, the OrcaFlex-MATLAB combined numerical model proposed in the present invention first simulates the single-caisson floating breakwater system through the OrcaFlex software to extract the displacement, velocity, and acceleration data of the top of the mooring chain rope. Then, the displacement, velocity, and acceleration data of the top of the mooring chain rope are used as the external constraint conditions at the top of the mooring chain in the ANCF model, and the external constraint conditions are substituted into the dynamic equations. Among them, the constraint equation C(q s ,t) of the top of the mooring chain system can be expressed as:

[0222]

[0223] where: r t ,r x,t are the position vector and rotation vector of the mooring chain vertex respectively;

[0224] The present invention can accurately predict the dynamic behavior of the floating breakwater system under the action of waves, wind force, etc., especially the influence on the mooring chain rope under dynamic conditions such as tilting and rotation.

[0225] Finally, the present invention writes its program through Matlab and uses the ode113 solver to perform dynamic solution of the first-order differential equations to obtain the absolute nodal coordinates q s , and the kinetic energy and strain energy of the mooring chain system can be calculated through q s .

[0226] As Figure 9 shown, the dynamic solution process includes: First, initialization: determine the initial coordinates, initial velocity, and mass matrix of the mooring chain; Second, force solution: solve the generalized force matrix according to the position coordinates, velocity, and other parameters of the mooring chain at the current step size; Third, equation construction: give the acceleration of the mooring chain according to Newton's second law; Fourth, iterative solution: solve the differential equation at the current step size to obtain the position / velocity vector at the next step size.

[0227] As Figure 5 shown, for the verification of the accuracy of the modeling method proposed in the present invention, Figure 5 is the comparison of the displacement results in the X, Y, and Z directions of the top of the mooring chain fixed floating ball model based on the ANCF method provided in the present invention with the results of the NM method in the literature. It is found from the figure that the two results are consistent, verifying the accuracy of the program.

[0228] AsFigure 6 and Figure 7 As shown in Figure 7 , by comparing the program developed in the present invention with the existing flexible pendulum model based on ANCF, the results show that the two are highly consistent, verifying that the method of the present invention is also efficient.

[0229] As Figure 8 shown in Figure 8 , the tensions at the top and bottom of the anchor chain 1 obtained by the present invention based on the ANCF method are consistent with the anchor chain tension results obtained by the commercial software OrcaFlex, verifying the feasibility of the OrcaFlex-MATLAB combined numerical model method proposed in the present invention.

Claims

1. A method for analyzing the dynamic response of a floating breakwater anchor chain system, characterized in that The steps are as follows: (1) Determine the parameters of the pontoon and the anchor chain system. The pontoon parameters include the geometric dimensions, material properties, and mass distribution of the pontoon. The center of the pontoon is located at the origin of coordinates, and its bottom surface is parallel to the X-O-Y plane. The anchor chain system includes four anchor chains arranged symmetrically about the center. One end of each anchor chain is connected to the pontoon, and the other end is fixed to the seabed. Each single anchor chain is divided into several units. (2) Specify the sea condition environment. The sea condition environment parameters include water depth, seawater density, wave type, wave direction, wave height, amplitude, wave number in the X direction, frequency, sea current velocity components, and sea current acceleration components in the sea condition. The wave direction is along the positive X axis. (3) Describe the motion of the anchor chain unit. Based on the absolute nodal coordinate method, give the initial configuration and the deformed configuration of the anchor chain unit, as well as the global position vector of any point P on the axis of the anchor chain unit. (4) Construct the expressions for the kinetic energy and mass matrix of the anchor chain unit. (5) Calculate the strain energy and elastic force of the anchor chain unit through the length, stiffness, damping, and mooring point position of the anchor chain unit. (6) Calculate the environmental loads on the floating breakwater anchor chain system. The environmental loads on each anchor chain include the wet weight of the anchor chain, seabed soil force, buoyancy of the pontoon, and sea current force. (7) Assemble the mass matrix of each unit on each anchor chain to obtain the system mass matrix, and assemble according to the constraints between the anchor chains to obtain the constraint equations of the system. (8) Based on the Lagrangian method, for the constrained anchor chain system, establish the dynamic equations of the system and correct them to obtain the first-order differential equations. (9) Simulate the floating breakwater anchor chain system through OrcaFlex software, and extract the displacement, velocity, and acceleration data at the top of the anchor chain. Then, take the displacement, velocity, and acceleration data at the top of the anchor chain as external constraint conditions and substitute them into the dynamic equations. (10) Perform dynamic solution on the first-order differential equations to obtain the absolute nodal coordinates of the system, and calculate the kinetic energy and strain energy of the anchor chain system through the absolute nodal coordinates of the system.

2. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 1, characterized in that In step (3), the initial configuration means that the anchor chain unit initially appears as a straight line, and its axis extends in a certain direction, defining the original positions of each point in the unloaded state. The extension direction can be along the X axis. The deformed configuration means that the axis of the anchor chain unit bends or stretches due to external forces, forming a new curve shape, reflecting the new positions of each point relative to the initial positions in the loaded state. For an anchor chain unit with a unit length of L, its absolute nodal coordinate q is composed of the positions q1 and q2 of the two endpoints of the unit, and its absolute nodal coordinate q is: where r T (x = 0) and r T (x = L) represent the position vectors of the two end points of the element respectively, and represent the rotation vectors of the two end points of the element respectively; For the global position coordinate r(x,t) of any point on the anchor chain unit, it is represented by cubic Hermite interpolation of the unit absolute nodal coordinates: r(x,t) = S(x)q(t) where q(t) is the absolute nodal coordinate of any point on the anchor chain unit, and S(x) is the shape function of any point on the anchor chain unit. The shape function S(x) is expressed as cubic Hermite interpolation, and its specific form is: where \(x\) is the local coordinate of the element, ranging from 0 to 1, representing the point position within the element; the global velocity and acceleration vectors of point \(P\) are functions of the absolute nodal coordinates of the element with respect to time \(t\), so the global velocity of point \(P\) and acceleration vectors are expressed by differentiating the position vector with respect to time: Wherein: and represent the velocity vector and acceleration vector of the absolute nodal coordinate of point P; the first derivative r x (x, t) and the second derivative r xx (x, t) are expressed as: r x ψ(x,t) = S x (x)q(t) r xx (x, t) = S xx (x)q(t) Where: S x (x) and S xx (x) represent the first-order and second-order derivatives of the shape function with respect to the local coordinates, respectively.

3. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 2, characterized in that In step (4), for an anchor chain unit with length L, single density ρ, and cross-sectional area A, the expression for the kinetic energy T of the anchor chain unit is: Among them, represents the velocity vector of the element, represents the first derivative of the absolute nodal coordinates of the element with respect to t, S is the shape function of the element, M i is the element mass matrix. By deriving the expression of kinetic energy T, the mass matrix of the anchor chain element is obtained as follows: where M 11 i , M 12 i , M 21 i and M 22 i means dividing the element mass matrix into four 6×6 matrices.

4. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 3, characterized in that In step (5), the strain energy only includes the energy caused by axial deformation and bending deformation. The strain energy U of the anchor chain unit is expressed as: where E is the Young's modulus of the anchor chain, J is the moment of inertia of the beam cross-section, ε and κ are the axial strain and curvature in the axis direction respectively. The specific calculations of the axial strain and curvature in the axis direction are as follows: where r x and r xx represent the first and second order derivatives of the position vector of the mooring unit with respect to the local coordinates of the unit, respectively; By taking the partial derivatives of the strain energy U with respect to the unit node coordinates, the generalized elastic force of the unit is obtained: Among them, Q e1 and Q e2 are the elastic forces generated by the axial tension and bending deformation of the unit respectively, and K1 and K2 are their corresponding stiffness matrices; Q e1 Expressed as: Q e2 Expressed as:

5. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 4, characterized in that In the step (6), the wet weight of the anchor chain includes its own gravity load f g and its own buoyancy load f b , and the gravity and buoyancy loads acting on the infinitesimal element dx of the anchor chain arc length are respectively expressed as: f g = [0 0 -ρAg] T f b = [0 0 ρ s Ag] T where ρ is the density of the anchor chain material, g is the acceleration due to gravity, and A is the cross-sectional area.

6. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 5, characterized in that In step (6), the seabed soil force is approximately described by a linear spring model to represent the normal support force of the seabed on the anchor chain. For any point P on the anchor chain unit in contact with the seabed, the normal support force it receives is the seabed soil force f z , the seabed soil force f z is expressed as: f z = k·Δz (Δz ≥ 0) f z = 0 (Δz < 0) Δz = r P ·e Z -Z bed where k refers to the equivalent stiffness of the spring, d is the diameter of the anchor chain, Z bed is the seabed depth coordinate, and e Z is the unit direction vector in the Z direction.

7. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 6, characterized in that The buoyancy of the floating box on the top node of the anchor chain in step (6) is denoted by Q f It is expressed as: Q f = ρgV s - Mg Among them, M is the mass of the floating box, and V s is the volume of the floating box underwater. Therefore, V s can be expressed as: The buoyancy force on the floating box is: Q f = ρgV s - Mg Where: ρ s is the seawater density, g is the acceleration due to gravity, S buoy is the cross-sectional area of the pontoon, and z is the depth of the pontoon below the water surface; The movement of the anchor chain unit in water is affected by hydrodynamic forces. The non-linearity of the hydrodynamic forces is described by the Morison equation. The non-linear resistance in the Morison equation includes tangential force and normal force. By setting the tangential unit vector of the anchor chain unit, the tangential component and normal component of the hydrodynamic force are calculated. The tangential component f Dt can be expressed by the following formula: Normal component f Dn It can be expressed by the following formula: Among them, C d is the drag force coefficient, f t and f n are the loading functions, d is the diameter of the anchor chain, is the relative velocity vector, U is the incoming flow velocity, V Rt and V Rn are the tangential and normal components of the relative velocity; Drag force coefficient C d Is closely related to the Reynolds number Re, and its calculation formula is: where μ is the dynamic viscosity coefficient of water, The tangential component and the normal component are vectorially summed to obtain the ocean current force f of the anchor chain element D .

8. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 7, characterized in that In step (7), for the anchor chain with N units, each anchor chain unit corresponds to a mass matrix M i ; the mass matrix of a single anchor chain is obtained by assembling the mass matrices of all units. The specific method is as follows: For an anchor chain system, generally containing several anchor chains, its system mass matrix is:

9. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 8, characterized in that, The constraints between the anchor chains in step (7) are mainly divided into displacement constraints and rotational constraints. Select the connection point i of an anchor chain unit in the anchor chain system and the connection point j on another anchor chain unit. If the absolute nodal coordinate q of the connection point i i is as follows: where r i and r j represent the position vectors connecting joint point i and joint point j, respectively, and r x,i and r x,j represent the rotation vectors of joint point i and joint point j, respectively; Then the constraint equation C(q,t) between connection points i and j can be expressed as: Assembling the constraint equations of each anchor chain into a matrix, the system constraint equation is obtained as: C(q s , t) = 0.

10. The dynamic response analysis method of a floating breakwater anchor chain system according to claim 9, characterized in that, In step (8), according to the principle of virtual work, the virtual work δW done by the anchor chain system s is as follows: where q s and represent the absolute nodal coordinates and acceleration vectors of the anchor chain system, and Q s is the generalized force of the anchor chain system, which includes the generalized elastic force Q e s , the buoyancy Q f of the pontoon, and the generalized external force of the system. The generalized external force of the system includes the seabed soil force f z s and the ocean current force f D s ; Q s can be expressed as: Q s = Q e s + Q f + f z s + f D s For the constrained anchor chain system, its dynamic equations are expressed as: Among them, is the Jacobian matrix of the system constraint equation C(q s , t), and λ is the Lagrange multiplier vector; By solving it using the Baumgarte method, the differential-algebraic equations can be obtained: Where: Q d is the second-order partial derivative of the constraint equation with respect to the independent variable, and the first- and second-order partial derivatives of its system constraint equation C(q, t) with respect to time are respectively: According to the first-order and second-order partial derivatives, it can be obtained: A set of ordinary differential equations is obtained: Introducing a feedback system to the constraint equation, the final constraint equation is obtained: Among them, α and β are stability coefficients, and the empirical values of α and β are taken between 5 and 50. When α = β, the stable response is fast, and a first-order differential equation system can be obtained:

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